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1600	https://ssmr.org/patients/patient-resources/patient-education-forum/is-there-a-paternal-(sperm)-contribution-to-embryo.aspx	Patient Education Forum \| Male Infertility \| SSMR.org \| Society for the Study of Male Reproduction (SSMR) HomeMember Login)SearchContact About Mission Statement Board of Directors Past Presidents Bylaws Contact Patients What Is Male Infertility? 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Causes of Male Infertility Diagnosing Male Infertility Treatment for Male Infertility Patient Support Resources Patient Education Forum Fertility & Cancer Is There a Paternal (Sperm) Contribution to Embryo Development? _Taylor P. Kohn Baylor College of Medicine_ _Alexander W. Pastuszak, MD, PhD Assistant Professor Division of Male Reproductive Medicine and Surgery Scott Department of Urology Baylor College of Medicine_ Human embryo development begins when the paternal sperm fuses with the maternal egg, forming a zygote. Both the sperm and the egg contribute DNA, or genetic material, to the newly formed zygote, and this genetic material is packaged as chromosomes. Humans have two copies of 22 unique chromosomes in each cell – one copy from the egg and one from the sperm, as well as a combination of X and Y chromosomes that determine sex – for a total of 46 chromosomes (Table 1). The chromosomal DNA is expressed only beginning on the third day of the embryo’s development.1 While it is well known that both the sperm and egg supply genetic material to the developing embryo, there are other contributions that the sperm makes to the embryo that are important for its development.. In addition to contributing half of the genetic material to the embryo, the sperm also contributes the centrosome, a subcellular organelle that helps the embryo grow..2 Another important sperm contribution to the zygote is the “oocyte activation factor,” which stimulates the zygote to complete its first cellular division and become an embryo. Finally, the sperm also contributes messenger RNA (mRNA - another form of genetic material), which contains instructions for protein production and can be used immediately to make proteins that are important for early development.. mRNA’s exact role in early embryo formation is not completely clear, but it appears to have an effect early in embryo development.1 Development of an embryo may be affected at any point. Abnormal sperm contributions resulting in either “early” or “late” sperm effects on embryo development have been identified. “Early” paternal effects are primarily defined by the absence of fertilization or poor zygote morphology (how the zygote looks under a microscope), and correlate with an absent or dysfunctional oocyte activation factor or a dysfunctional centrosome. These “early” sperm effects are evident during in vitro fertilization (IVF) or intracytoplasmic sperm injection (ICSI) cycles when oocytes fail to fertilize or initially develop.1 In contrast, “late” paternal effects are manifested by poor embryo progression or development resulting in failure of the embryo to implant into the uterus. “Late” sperm effects correlate with abnormalities in sperm DNA, such as increased fragmentation (broken pieces) of sperm DNA, an indicator of low genetic quality, or an abnormal number of chromosomes coming from the male partner.1 While specific genes that may be affected by DNA breaks are unknown, studies examining fragmented or broken sperm DNA have shown that high levels of fragmentation are associated with decreased fertilization and pregnancy outcomes naturally or with intrauterine insemination (IUI), and assisted reproductive techniques such as IVF and ICSI.3 Additionally, the presence of too many or too few chromosomes in sperm cells, or additions or deletions in the sperm DNA, can hinder embryo development and induce embryo loss.1 Recent studies have also shown a higher rate of early embryo failure when levels of certain paternal mRNAs are decreased.4 While there is clearly a contribution of the sperm to fertilization and embryo development, the exact nature and extent of this contribution remains to be determined. In couples with recurrent miscarriage or IVF cycle failures, the reason for embryo loss is often debated and a cause is searched for. While the timing of the embryo loss may correlate with “early” or “late” effects that may be caused by sperm, it is often difficult to definitively attribute the cause of failure to either sperm or egg. The timing of embryo loss, however, can help guide clinicians in subsequent testing, particularly if a male factor has not been ruled out. In the future, definitive identification of sperm or egg factors resulting in fertility problems will become easier, paving the way for a more complete understanding of how the sperm contributes to this process.5 Table 1 – Paternal Sperm Contributions to Embryo Development \| Paternal Sperm Contributions \| Purpose \| --- \| \| 23 Chromosomes \| (DNA) Genetic Material \| \| Centrosome \| Cellular Division \| \| Oocyte-Activation Factor \| Maturation of the Oocyte \| \| mRNA \| Post-Fertilization Development \| REFERENCES Barroso G, Valdespin C, Vega E, et al: Developmental sperm contributions: fertilization and beyond. Fertility and Sterility 2009; 92: 835–848. Schatten G: The centrosome and its mode of inheritance: the reduction of the centrosome during gametogenesis and its restoration during fertilization. Dev. Biol. 1994; 165: 299–335. Kumar M, Kumar K, Jain S, et al: Novel insights into the genetic and epigenetic paternal contribution to the human embryo. Clinics (Sao Paulo) 2013; 68: 5–14. Simon A and Laufer N: Assessment and treatment of repeated implantation failure (RIF). J Assist Reprod Genet 2012; 29: 1227–1239. Bhattacharya SM: Association of various sperm parameters with unexplained repeated early pregnancy loss—which is most important? Int Urol Nephrol 2007; 40: 391–395. Be sure to check back later for our next installment of the Patient Education Forum. Find a Male Infertility Specialist An SSMR member physician is just a click away. Find a male infertility specialist » Patient Education Forum Check out our collection of articles to help patients stay informed about male infertility. Visit the forum » © 2011 – 2025 Society for the Study of Male Reproduction, Inc. All rights reserved. (847) 517-7225 (847) 517-7229 info@ssmr.org Privacy PolicyFacebookInstagramX.comYoutube Website Designed and Hosted by WJ Weiser & Associates, Inc.
1601	https://www.reddit.com/r/learnmath/comments/lejfo6/help_me_understand_this/	Help me understand this. : r/learnmath Skip to main contentHelp me understand this. : r/learnmath Open menu Open navigationGo to Reddit Home r/learnmath A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to learnmath r/learnmath r/learnmath Post all of your math-learning resources here. Questions, no matter how basic, will be answered (to the best ability of the online subscribers). 403K Members Online •5 yr. ago Defiant_Items Help me understand this. So, i was hoping somebody could clear some confusion for me. I'm slowly working through a math book, and i stumbled upon this problem. The problem is as following; Px -Q = Rx -S , find x. Apparently the answer is x = Q -S / P -R. The pages don't give any further details on how i should have solved this or anything, and i don't know how to look for to understand the how and why. Thank you in advance. Read more Share Related Answers Section Related Answers Effective strategies for mastering algebra Tips for improving mental math skills Exploring real-world uses of number theory Comparing different methods of integration Best practices for preparing for math exams New to Reddit? Create your account and connect with a world of communities. Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community 0 0 Top Posts Reddit reReddit: Top posts of February 7, 2021 Reddit reReddit: Top posts of February 2021 Reddit reReddit: Top posts of 2021 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
1602	https://iovs.arvojournals.org/article.aspx?articleid=2122987	Differential Recovery of Retinal Function after Mitochondrial Inhibition by Methanol Intoxication \| IOVS \| ARVO Journals iovs Issues Topics For Authors About Editorial Board Journals Home MANAGE ALERTS Forgot password? Create an Account To View More... Purchase this article with an account. or Subscribe Now Advanced Search All Journals All Journals IOVS JOV TVST Issues Topics For Authors About Editorial Board March 2001 Volume 42, Issue 3 ‹ Issue › Jump To... Methods Results Discussion Free Retinal Cell Biology\| March 2001 Differential Recovery of Retinal Function after Mitochondrial Inhibition by Methanol Intoxication Marina T. Seme; Phyllis Summerfelt; Jay Neitz; Janis T. Eells; Michele M. Henry Author Affiliations Marina T. Seme From the Department of Pharmacology and Toxicology and the Phyllis Summerfelt Department of Cell Biology, Neurobiology, and Anatomy, Medical College of Wisconsin, Milwaukee. Jay Neitz Department of Cell Biology, Neurobiology, and Anatomy, Medical College of Wisconsin, Milwaukee. Janis T. Eells From the Department of Pharmacology and Toxicology and the Michele M. Henry From the Department of Pharmacology and Toxicology and the Investigative Ophthalmology & Visual Science March 2001, Vol.42, 834-841. doi: Views Full Article Figures PDF Share email facebook twitter linkedin digg tumblr Tools AlertsUser Alerts You are adding an alert for: Differential Recovery of Retinal Function after Mitochondrial Inhibition by Methanol Intoxication You will receive an email whenever this article is corrected, updated, or cited in the literature. You can manage this and all other alerts in My Account The alert will be sent to: Confirm × ###### This feature is available to authenticated users only. Sign In or Create an Account× Get Citation Citation Marina T. Seme, Phyllis Summerfelt, Jay Neitz, Janis T. Eells, Michele M. Henry; Differential Recovery of Retinal Function after Mitochondrial Inhibition by Methanol Intoxication. Invest. Ophthalmol. Vis. Sci. 2001;42(3):834-841. Download citation file: Ris (Zotero) EndNote BibTex Medlars ProCite RefWorks Reference Manager © ARVO (1962-2015); The Authors (2016-present) × Get Permissions Abstract purpose. The authors’ laboratory has previously documented formate-induced retinal toxicity in a rodent model of methanol intoxication. These studies determined functional, bioenergetic, and structural recovery of the retina after methanol intoxication. methods. Rats were intoxicated with methanol, and retinal function was assessed by electroretinography 72 hours after the initial dose of methanol and after a 72-hour recovery period. Retinal energy metabolites, glutathione (GSH) concentrations, and histology were determined at the same time points. results. Both rod-dominated and UV-cone–mediated electroretinogram responses were profoundly attenuated in methanol-intoxicated rats. In rats allowed to recover from methanol intoxication, there was significant, although incomplete, recovery of rod-dominated retinal function. However, there was no demonstrable improvement in UV-cone–mediated responses. Retinal adenosine triphosphate (ATP), adenosine diphosphate (ADP), and GSH concentrations were significantly reduced after intoxication. Although retinal energy metabolites returned to control values after the recovery period, retinal GSH remained significantly depleted. Histopathologic changes were apparent in the photoreceptors after methanol intoxication, with evidence of inner segment swelling and mitochondrial disruption. In animals allowed to recover from methanol intoxication, there was no evidence of histopathology at the light microscopic level; however, ultrastructural studies revealed subtle photoreceptor mitochondrial alterations. conclusions. These findings support the hypothesis that formate inhibits retinal mitochondrial function and increases oxidative stress. They also provide evidence for a differential sensitivity of photoreceptors to the cytotoxic actions of formic acid, with a partial recovery of rod-dominated responses and no recovery of UV-cone–mediated responses. Methanol has been recognized as a human visual neurotoxin for more than a century, and the clinical features of acute human methanol toxicity have been extensively documented.123456 Toxic exposure to methanol typically results in an initial transient central nervous system depression, followed by an asymptomatic latent period lasting 12 to 24 hours. This latent period is then followed by the development of formic acidemia, uncompensated metabolic acidosis, visual toxicity, coma, and in extreme cases, death. Visual disturbances generally develop between 18 and 48 hours after methanol ingestion and range from mild photophobia and misty or blurred vision to markedly reduced visual acuity and complete blindness. Susceptibility among persons to the acute effects of methanol is highly variable, and the minimum lethal dose is considered to be between 300 mg/kg and 1 g/kg.456 The minimum dose causing permanent visual defects is unknown, although blindness has been reported after ingestion of as little as 4 ml of methanol.78 Methanol toxicity is primarily attributable to its metabolite, formic acid. Formic acid is the toxic metabolite responsible for the metabolic acidosis and visual toxicity observed in human methanol poisoning.569 Formate has been hypothesized to produce retinal and optic nerve toxicity by disrupting mitochondrial energy production.1011 In vitro studies have shown that formate inhibits the activity of cytochrome oxidase, the terminal electron acceptor of the mitochondrial electron transport chain involved in adenosine triphosphate (ATP) synthesis.1213 Inhibition occurs subsequent to the binding of formic acid to the ferric heme iron in cytochrome oxidase, with inhibition constants between 5 and 30 mM.1213 Permanent visual damage in methanol-intoxicated humans5 and nonhuman primates1014 has been associated with prolonged exposures (usually longer than 24 hours) to blood formate concentrations in excess of 7 mM. However, very little information is available on the potential for recovery of retinal function after toxic exposure to methanol-derived formate. Our laboratory has developed a rodent model of methanol intoxication in which formate oxidation is selectively inhibited by treatment with nitrous oxide (N 2 O). Subanesthetic concentrations of nitrous oxide inactivate the enzyme methionine synthetase, reducing the production of tetrahydrofolate, a necessary cofactor for formate oxidation.141516171819 This allows formate to accumulate to toxic concentrations after methanol administration.141516171819 In methanol-intoxicated rats, formic acidemia, metabolic acidosis, and visual toxicity develop, analogous to the toxicity seen in methanol intoxicated humans. Previous studies in our laboratory have established this rodent model of methanol-induced visual toxicity and have documented abnormalities in the flash-evoked visual potential and electroretinogram (ERG).16171819 The clinical features of methanol intoxication are remarkably similar to those of Leber’s hereditary optic neuropathy, nutritional amblyopia, and the recent Cuban epidemic of optic neuropathy. In each case, there is evidence that a common pathophysiological mechanism involving mitochondrial dysfunction contributes to the retinal and optic nerve dysfunction characteristic of the disease.2021 We hypothesize that the retinal pathophysiology of methanol intoxication is a consequence of formate-induced mitochondrial dysfunction. In this study, we examined the effect of methanol intoxication on retinal function and retinal energy metabolism and assessed the potential for recovery of retinal function after intoxication. Our findings indicate that formate accumulation after methanol intoxication inhibited retinal energy metabolism, increased oxidative stress in the retina, and profoundly attenuated retinal function. These studies also provide evidence for a complete recovery of retinal energy metabolites and a partial recovery of retinal glutathione (GSH) and retinal function in animals allowed to recover for 72 hours from methanol intoxication. Furthermore, our results are indicative of a differential sensitivity of photoreceptors to the cytotoxic actions of formic acid with a partial recovery of rod-dominated responses and no recovery of UV-cone–mediated responses. Methods Materials Methanol (high-performance liquid chromatography [HPLC] grade) was obtained from Sigma (St. Louis, MO). Thiobutabarbital (Inactin) was purchased from Research Biochemicals (Natick, MA), atropine sulfate from AmVet (Fort Collins, CO), hydroxypropyl methylcellulose (2.5%) drops from Iolab (Claremont, CA), and atropine sulfate ophthalmic solution drops from Phoenix (St. Joseph, MO). All other chemicals were reagent grade or better. Animals Adult (250–350 g) male Long–Evans rats (Harlan Sprague–Dawley, Madison, WI) were used throughout the experiments. Animals were supplied food and water ad libitum and maintained on a 12-hour light–dark schedule in a temperature- and humidity-controlled environment. All animal experiments were performed in accordance with the ARVO Statement for the Use of Animals in Ophthalmic and Vision Research. Methanol Intoxication Protocol Rats were placed in a thermostatically controlled plexiglas chamber (22 × 55 × 22 cm; maintained at 22–23°C) and exposed to a mixture of N 2 O/O 2 (1:1; flow rate, 2 l/min) for 4 hours before the administration of methanol or saline. N 2 O/O 2 exposure was continued for 72 hours after the initial dose of methanol. Two treatment protocols were used. In the first protocol (intoxication), rats were intoxicated with methanol (25% wt/vol in saline, 4 g/kg, intraperitoneally, followed by 2-g/kg supplemental doses at 24 and 48 hours) in the presence of N 2 O/O 2 for 72 hours. At 72 hours, ERG analysis was performed, animals were killed, and retinal tissue was prepared for biochemical and histologic analysis. In the second protocol (recovery), rats were intoxicated with methanol (same dosage regimen as above) for 72 hours in the presence of N 2 O/O 2 and allowed to recover from methanol intoxication for an additional 72 hours in the absence of N 2 O/O 2. At 144 hours, ERG analysis was performed, animals were killed, and retinal tissue was prepared for biochemical and histologic analysis. Controls for these experiments included groups of rats treated with saline and exposed to N 2 O/O 2 (nitrous oxide-control), rats treated with methanol, but not exposed to N 2 O/O 2 (methanol-control), and untreated rats (untreated-control). Formate concentrations were determined from tail vein blood samples by fluorometric analysis as previously described.1718 ERG Procedures and Analyses ERG experiments were performed as previously described.18 The light-stimulation apparatus consisted of a three-beam optical system.22 All three beams were derived from tungsten-halide lamps (50 W, 12 V), and beam intensity was controlled by using neutral-density step filters. Each of the beams contained a high-speed, computer-driven shutter (Uniblitz; Vincent Associates, Rochester, NY). One beam had a wavelength computer controlled by a tunable band-pass filter (Varispec; Cambridge Research Instruments, Wilmington, DE; half-energy pass-band, 7 nm). The second beam was used with a short-pass UV filter (half pass, 380 nm) in experiments designed to isolate UV-cones. The third beam was used as a chromatic adapting light to suppress responses from rods and M-cones in the UV-cone isolation experiments. For this purpose, a glass long-wavelength pass filter (half pass, 590 nm) was used. The three beams were optically superimposed and focused on the lens to illuminate a 70° patch of retina in Maxwellian view. Light calibrations were made with a silicon photodiode (PIN 10 DF; United Detector Technology, Hawthorne, CA). ERG recordings were differentially amplified and computer averaged. The amplified signal was processed through a two-stage active narrow band-pass filter, the half voltage of which was 0.2 times the center frequency. To ensure that any transients in the response that occur at the onset of the stimulus pulses were not included in the average, the initiation of signal averaging was delayed by a preset number of stimulus cycles (typically a minimum of 20). The resultant ERG is a noise-free, single-cycle, sinusoidal waveform. The averaged responses were measured (peak-to-trough amplitude) from a calibrated digital oscilloscope display.22 Before ERG analysis, ophthalmoscopic examination confirmed that all eyes were free of lenticular opacities or other gross anomalies. Rats were anesthetized with thiobutabarbital sodium (100 mg/kg, intraperitoneally), positioned in a stereotaxic apparatus (David Kopf, Tujunga, CA), and placed on a heating pad to maintain core body temperature at 37°C. Atropine sulfate (0.05 mg/kg, subcutaneously) was administered to inhibit respiratory tract secretions. The pupil of the eye to be tested was dilated by topical application of 1% atropine sulfate. Methylcellulose was topically applied as a lubricant and to enhance electrical conduction. A circular, silver wire recording electrode was positioned on the cornea, a reference electrode was placed above the eye, and a ground electrode was placed on the tongue. Recordings were obtained under ambient light conditions from cool white fluorescent room lights approximately 100 candelas[ cd]/m 2 at the rat’s eye. Flickering stimuli (light–dark ratio, 0.25:0.75) were presented. Responses to 60 successive flashes were averaged for each stimulus condition. At each test wavelength, a minimum of four stimulus intensities, spaced at intervals of 0.3 log units, were presented. The stimulus intensity yielding a 5-μV criterion response was determined by extrapolating between the two intensity points that bracketed the 5-μV response for each animal. All sensitivity measures were made in triplicate. After ERG analysis, anesthetized rats were killed by decapitation, and retinal tissue was prepared for histologic and biochemical analysis. One retina from each animal was prepared for histology, and the other retina was prepared for analysis of retinal energy metabolites and GSH concentrations. Two experimental protocols were used to evaluate retinal function. (1) The 15-Hz/510-nm ERG response: ERGs were recorded in response to a 15-Hz flickering light at a wavelength of 510 nm over a 3-log-unit range of light intensity. For these studies the unattenuated stimulus (log relative retinal illumination [LRRI], 0) had an irradiance of 25μ W distributed over the 70° patch of illuminated retina. This can be calculated to produce retinal illumination equivalent to approximately 10 4 scotopic trolands (scot td). These recording conditions disadvantage rods; however, because at least 97% of rat photoreceptors are rods and ERGs are recorded at luminance intensities ranging from 10 1 to 10 4 scot td, it is likely that the responses to the 15-Hz/510-nm light are drawn from both rods and medium-wavelength cones (M-cones).232425 (2) 25-Hz/UV ERG response: Cone responses were elicited by a 25-Hz flickering UV light (380-nm cutoff) in the presence of an intense chromatic adapting light (445 μW), which eliminated responses mediated by rods and M-cones.26 Recording conditions were the same as those used by Jacobs et al.26 (except that the intensity of the chromatic adapting light was lower in our studies). In Jacobs et al.,26 complete spectral sensitivity functions were measured in the rat, and it was demonstrated that UV-cone responses are separated from rod and M-cone responses. The 25-Hz/UV ERG responses were recorded over a 1-log-unit range of light intensity. For these studies, the unattenuated stimulus (LRRI, 0) had an irradiance of 12.5μ W distributed over the 70° patch of illuminated retina. By equating the effectiveness of this light to the 510-nm stimulus, we estimate that the unattenuated light produced the equivalent of 10 2.5 scot td in the rat eye. Determination of Retinal Energy Metabolites Retinas were rapidly dissected and frozen in liquid nitrogen. Frozen retinas were extracted in 2.5% trichloroacetic acid (TCA), the suspension centrifuged, and the supernatant neutralized with 1.0 M Tris base. Protein concentrations in the pellet were determined using a modification of the dye-binding method of Bradford.27 The neutralized supernatant was assayed for ATP, adenosine diphosphate (ADP), and adenosine monophosphate (AMP) by the high-performance liquid chromatography (HPLC) method of Bernocchi et al.,28 using a system (1090L; Hewlett Packard, Palo Alto, CA) with a diode array detector and a 3-μm reversed-phase column (15 cm × 4.6 mm; Supelcosil LC-18; Supelco, Bellefonte, PA). Peak identities were confirmed by comparison of the peak retention time and peak spectral characteristics of samples with those of known standards. Quantitative measurements were made on the basis of the injection of standard solutions in known concentrations. Metabolite concentrations are expressed per milligram of protein. Determination of Retinal GSH Concentrations Retinas were rapidly dissected and frozen in liquid nitrogen. Frozen retinas were extracted in 2.5% TCA, the suspension centrifuged, and the supernatant assayed for GSH. Protein concentrations in the pellet were determined using a modification of the dye-binding method of Bradford.27 GSH was assayed by the fluorometric method of Morkrasch and Teschke,29 using o-phthalaldehyde. Fluorescence was measured at an excitation wavelength of 345 nm and an emission wavelength of 425 nm. GSH concentrations are expressed per milligram of protein. Histopathologic Analysis Retinal tissue was prepared for histology as previously described.1819 Briefly, eyes were enucleated, hemisected, and immersed in fixative (2.67% glutaraldehyde in 0.1 M phosphate buffer at 4°C [pH 7.3]) for 72 hours, then transferred to 4% sucrose. The anterior segment and vitreous were removed and full-thickness pieces of eye wall were dissected from the posterior pole, including the optic nerve. Tissues were postfixed in phosphate-buffered 2% osmium tetroxide (OsO 4), dehydrated in a graded ethanol series, and embedded in epoxy resin. Thick sections (1 μm) for light microscopy were stained with toluidine blue; thin sections for electron microscopy were stained with uranyl acetate-lead citrate.1819 Statistical Analysis All values are expressed as means ± SEM. A one-way analysis of variance (ANOVA) with Bonferroni’s test was used to determine significant differences among groups for blood formate concentrations, energy metabolites, and GSH concentrations. For ERG studies, a one-way ANOVA with repeated measures was performed, followed by Scheffé’s F procedure. In all cases, the minimum level of significance was taken as P< 0.05. Results Accumulation of Formic Acid In the present studies, methanol was administered to N 2 O/O 2-exposed rats at an initial dose of 4 g/kg, followed by supplemental doses of 2 g/kg at 24 and 48 hours. This treatment protocol has been shown to produce a state of prolonged formic acidemia with formate concentrations between 7 and 10 mM for 40 hours in methanol-intoxicated rats, resulting in visual dysfunction.16171819 Moreover, similar concentrations of blood formate over similar periods have been shown to produce ocular toxicity experimentally in monkeys and have been associated with visual toxicity in human methanol intoxication.56 As shown in Figure 1 , blood formate concentrations increased linearly in both the intoxicated and recovery groups of animals during the initial 72 hours of intoxication. In the intoxicated group, formate concentrations increased from endogenous concentrations of 0.6 ± 0.3 (mean ± SEM) to 7.8 ± 0.1 mM by 72 hours. In animals allowed to recover after intoxication, there was a similar increase in blood formate concentrations during the initial 72-hour intoxication period from endogenous concentrations of 0.5 + 0.3 to 7.0 ± 0.4 mM. Formate concentrations did not differ between the intoxicated and recovery groups at any sampling point during the 72-hour intoxication period. After the 72-hour recovery period (in the absence of N 2 O/O 2 and methanol), blood formate declined to endogenous concentrations in the recovery group of animals. 15-Hz/510-nm ERG Response We have previously reported that a 15-Hz/510-nm light produced a robust and reproducible ERG response in our rodent model.18 As stated in the Methods section, these recording conditions disadvantage rods; however, because at least 97% of rat photoreceptors are rods and ERGs are recorded at luminance intensities ranging from 10 1 to 10 4 scot td, it is likely that the responses to the 15-Hz/510-nm light were drawn from both rods and M-cones.23242530 Moreover, it is likely that ERG responses recorded at the lower luminance intensities (<10 3 scot td) reflect responses with a robust rod component, whereas, those responses recorded at the higher luminance intensities (>10 3 scot td) may be dominated by the activity of M-cones. We base this interpretation on flicker photometry studies in the gerbil, which show a shift from rods to cones at 10 3 scot td.23 The effects of methanol intoxication and recovery on the 15-Hz/510-nm ERG responses are shown in Figure 2 . In the control group, 15-Hz/510-nm ERG amplitude increased linearly over the 3-log-unit range of retinal illumination intensities, achieving a maximal amplitude of 57.1 ± 3.1 μV at maximal retinal illumination (0 LRRI, equivalent to 10 4 scot td). A consistent 5-μV criterion threshold response was obtained in control animals at −2.7 ± 0.1 LRRI. We observed a decrease in retinal responsiveness and attenuation of maximal ERG amplitude in both the intoxicated and the recovery groups in comparison with untreated control animals. In intoxicated animals, ERG responses were at or below the 5-μV threshold response over the entire range of luminance intensities, indicative of a severe deficit in retinal function. In animals allowed to recover from intoxication for 72 hours, the 5-μV threshold response was not different from the control group; however, the ERG intensity response curve was significantly attenuated at all light intensities of more than −2.1 LRRI, and the maximal response to light stimulation was reduced to 33.9 ± 5.4 μV. These data are indicative of a partial recovery of this component of retinal function. 25-Hz/UV ERG Response The function of UV-sensitive cones was examined by recording the retinal response to a 25-Hz flickering UV light (380-nm cutoff) in the presence of an intense chromatic adapting light. These conditions have been shown to isolate the UV-cone response in the rat retina.26 The effects of methanol intoxication and recovery on 25-Hz/UV ERG responses are shown in Figure 3 . In the control group, 25-Hz/UV ERG amplitude increased from a minimal value of 1.9 ± 0.4 μV to a maximal value of 19.6 ± 1.2μ V over the log unit range of retinal illumination used in these studies. A consistent 5-μV threshold criterion response was obtained in control animals at −0.4 ± 0.1 LRRI. In intoxicated animals, the 25-Hz/UV ERG responses over the entire range of luminance intensities were at or below the 5-μV threshold response. These findings are indicative of a severe deficit of cone-mediated retinal function similar to the deficit observed in the 15-Hz/510-nm response. In contrast to the partial recovery observed in the 15-Hz/510-nm ERG response, the 25-Hz/UV ERG response remained fully attenuated in animals allowed to recover from methanol intoxication, indicating that UV-cone–mediated function does not recover after methanol intoxication. Retinal Energy Metabolites Methanol intoxication has been hypothesized to disrupt retinal mitochondrial energy production, secondary to formate induced inhibition of mitochondrial cytochrome oxidase activity.61011 To assess the effect of formate on retinal energy metabolism in vivo, we measured concentrations of energy metabolites (ATP, ADP, and AMP) in the retinas of control, intoxicated, and recovery groups of rats (Fig. 4) . In untreated control animals, concentrations of ATP, ADP, and AMP were 5.1 ± 0.6 nanomoles/mg protein, 12.5 ± 1.2 nanomoles/mg protein, and 23.7 ± 3.4 nanomoles/mg protein, respectively. After intoxication, there was a significant decrease in ATP (1.2 ± 0.2 nanomoles/mg protein) and ADP (6.2 ± 0.5 nanomoles/mg protein), and a corresponding increase in AMP (37.2 ± 4.0 nanomoles/mg protein). In animals allowed to recover from intoxication, retinal energy metabolite concentrations were not significantly different from energy metabolite concentrations measured in the retinas of control animals, consistent with a restoration of mitochondrial bioenergetics. Retinal GSH Concentrations GSH plays a central role in the antioxidant defenses of the cell.3132333435 Inhibition of mitochondrial function has been shown to increase oxidative stress and deplete tissue GSH.3132333435 As an index of formate-induced oxidative stress after methanol intoxication and recovery, we measured GSH concentrations in the retinas of control, intoxicated, and recovery groups of rats (Fig. 5) . In the control group, retinal GSH concentrations were 29.4 ± 2.0 nanomoles/mg protein. After intoxication, retinal GSH concentrations were significantly reduced to 12.7 ± 3.4 nanomoles/mg protein. After recovery, GSH concentrations were 22.3 ± 2.0 nanomoles/mg—-significantly higher than the intoxicated group, but also significantly lower than control, which suggests that oxidative stress may play an important role in methanol intoxication. Retinal Histology and Photoreceptor Ultrastructure The effects of formate after methanol intoxication on retinal histology and ultrastructure were assessed by light and electron microscopy after intoxication and recovery. Figure 6 illustrates outer retinal morphology in representative control (Fig. 6A) , intoxicated (Fig. 6B) , and recovery (Fig. 6C) retinas. Retinas prepared from control animals had ordered photoreceptor inner segments with no evidence of vacuolization or swelling. The outer nuclear layer of the control retina was compact, with round and well-defined nuclei. In contrast, retinas prepared from rats intoxicated for 72 hours showed evidence of retinal edema, swelling of photoreceptor inner segments, and morphologic changes in photoreceptor nuclei. Retinal edema was evidenced by the spacing between the photoreceptor inner segments and the spacing of the nuclei in the outer nuclear layer. Photoreceptor inner segments were profoundly swollen and enlarged. Changes in the appearance of photoreceptor nuclei were also apparent in intoxicated rats. Nuclei appeared somewhat enlarged with irregularly stained chromatin. The chromatin staining pattern in photoreceptor nuclei in intoxicated animals ranged from tightly compact to dispersed and fragmented. In animals allowed to recover from methanol intoxication, retinal morphology was similar to control, with the only histologic alteration being an increased spacing between nuclei in the outer nuclear layer. The retinas of control, intoxicated, and recovery rats were also examined by electron microscopy. We focused our ultrastructural observations on the mitochondria of the inner segments of the photoreceptors, because formate is known to act as a mitochondrial toxin, and the inner segments of the photoreceptors contain the highest density of mitochondria in the retina. Figure 7 illustrates the inner segment region in representative control (Fig. 7A) , intoxicated (Fig. 7B) , and recovery (Fig. 7C) retinas. Mitochondria in the photoreceptor inner segments of control animals exhibited normal morphology with well-defined cristae. The most obvious ultrastructural change observed in the outer retina of methanol-intoxicated rats was the swelling and disruption of photoreceptor mitochondria. In rats intoxicated for 72 hours, numerous photoreceptor mitochondria were profoundly swollen, with severely disrupted cristae. In intoxicated animals, there was swelling and disruption of photoreceptor mitochondria to various degrees within the retina and within the individual cells. Some mitochondria were swollen and contained expanded cristae, and other mitochondria were disrupted and showed no evidence of cristae. Photoreceptor mitochondria in animals allowed to recover from methanol intoxication showed less evidence of disruption than photoreceptor mitochondria in intoxicated animals. In recovery animals, photoreceptor mitochondrial morphology ranged from normal with well-defined cristae to rounded with expanded cristae. Discussion We report the nature of the functional, biochemical, and structural changes produced in the retina after methanol intoxication and recovery. Several important findings are reported in this study. We provide in vivo evidence of a significant alteration in retinal energy metabolism, which supports previous in vitro studies showing that formate is a mitochondrial toxin.1213363738 These data also indicate that formate-induced mitochondrial dysfunction produced GSH depletion and increased oxidative stress. In addition, we describe the nature of the recovery of retinal function after methanol intoxication. We provide evidence for partial recovery of retinal function in rod-dominated pathways, but no recovery of UV-cone–mediated responses. These findings are indicative of a differential sensitivity of photoreceptors to the cytotoxic actions of formic acid. Formic acid has been hypothesized to produce ocular toxicity by a disruption of mitochondrial energy production in the retina and optic nerve.10113940 In vitro studies in our laboratory and by other investigators have shown that formate inhibits cytochrome oxidase, the terminal electron acceptor of the mitochondrial electron transport chain involved in ATP synthesis.12133738 Inhibition occurs subsequent to the binding of formic acid with the ferric heme iron of cytochrome oxidase, and the apparent inhibition constant is between 5 to 30 mM.1213 Blood formate concentrations in methanol-intoxicated rats in the present study fall within this range, as do blood formate concentrations in methanol-poisoned humans and monkeys.5910 Moreover, retinal and vitreous humor formate concentrations closely parallel blood formate concentrations.1617 Additional in vitro studies in isolated mitochondria and cultured neuronal cells have shown that formate inhibits mitochondrial ATP synthesis and decreases cellular ATP content.3638 The present studies provide evidence that formate inhibits mitochondrial energy metabolism in vivo. These studies document formate-induced depletion of retinal ATP and ADP and a corresponding increase in retinal AMP after methanol intoxication. After recovery, energy metabolites returned to control concentrations, providing evidence of bioenergetic recovery. These findings strongly support the hypothesis that formate inhibits retinal mitochondrial energy metabolism and oxidative phosphorylation in methanol intoxication and are consistent with the documented actions of formate in isolated mitochondria.121338 The vertebrate retina has several features that render it vulnerable to damage from reactive oxygen species, including abundant mitochondria and a high percentage of polyunsaturated fatty acids in photoreceptor membranes that are susceptible to lipid peroxidation.3435 Because of its constant exposure to irradiation and high metabolic activity, the retina has a great need for antioxidant protection.3435 Reduced GSH is one of the most abundant intracellular thiols in the central nervous system and acts as a major cellular antioxidant by supporting GSH peroxidase-dependent reduction of hydrogen peroxide and organic peroxides.313233 GSH is normally present in high concentration in the retina and has been shown to play a key role in antioxidant defenses in the retina.3435 Studies have shown that retinal GSH may be depleted during periods of oxidative stress.3435 In the present studies, we observed a significant reduction in retinal GSH concentrations after methanol intoxication. Moreover, in contrast to our findings with energy metabolites, GSH concentrations did not return to control concentrations after recovery. We hypothesize that the observed depletion of retinal GSH is a consequence of formate-induced mitochondrial inhibition. Depletion of GSH by formate could result directly from formate-induced peroxidative stress, because inhibition of mitochondrial electron transport has been shown to profoundly increase the production of reactive oxygen species including superoxide and hydrogen peroxide.313233 Alternatively, GSH depletion could occur as a consequence of formate-induced ATP depletion, because GSH synthesis is ATP dependent.313233 In support of the latter mechanism, studies in cultured hepatocytes have shown that inhibition of cellular energy metabolism and ATP synthesis by mitochondrial poisons lead to a rapid decline in GSH content that precedes cell death by several hours.31 Additional studies are under way to determine the mechanism of formate-induced GSH depletion. Because of the critical involvement of GSH in cellular defense mechanisms, depletion of intracellular GSH under conditions of mitochondrial impairment may augment the susceptibility of the retina to oxidative stress. Thus, formate-induced mitochondrial inhibition may not only increase the production of reactive oxygen species, but may also predispose the retina to increased oxidative stress through a perturbation of GSH status. A similar mechanism of tissue injury has been proposed to lead to neuronal degeneration in Parkinson’s disease.31 The morphologic changes in the present study are also consistent with formate-induced inhibition of photoreceptor energy metabolism, GSH depletion, and increased oxidative stress. The most profound ultrastructural alterations observed in methanol-intoxicated rats were mitochondrial swelling and disruption in the inner segments of the photoreceptor cells. Similar mitochondrial changes have been associated with the production of GSH deficiency after inhibition of GSH synthesis with buthionine sulfoximine.32 GSH plays a major role in the maintenance of mitochondrial function.313233 Reduction in cytosolic and mitochondrial GSH has been shown to increase mitochondrial susceptibility to oxidative stress, disrupt mitochondrial structure and function, and promote cytotoxicity.3233 In addition, mitochondrial disruption has also been reported in the retinas of patients with mitochondrial diseases that inhibit electron transport414243 and in certain forms of light-induced retinal degeneration in which inactivation of cytochrome oxidase is postulated to play a role in the disease.444546 The present studies confirm and extend our previous investigations that showed that rod- and cone-mediated ERG responses were profoundly attenuated in rats intoxicated with methanol for 72 hours.14 We also observed a differential recovery of retinal function after methanol intoxication. In rats allowed to recover for 72 hours from methanol intoxication, there was a significant, although incomplete recovery of the 15-Hz/510-nm ERG response. In contrast, there was no evidence of recovery of UV-cone–mediated function. The rat retina contains three types of photoreceptors, rods, M-cones, and UV-cones.242526 Rods comprise approximately 97% of rat photoreceptors.242526 Although our recording conditions can clearly discriminate UV-cone function, the 15-Hz/510-nm ERG measurements cannot discriminate between rod and M-cone function. In the absence of a definitive assessment of M-cone function, it is unknown whether the observed sensitivity to the cytotoxic actions of formate is specific for UV-cones or reflects a general property of cones. However, it is clear that UV-cone responses are more severely affected after methanol intoxication than rod and M-cone responses. Previous studies have shown that rod-dominated retinal responses are affected earlier in the course of intoxication and at lower formate concentrations than the UV-cone–mediated responses. Taken together, these data suggest that although UV-cones may be resistant to the initial cytotoxic actions of formate,18 once poisoned, they do not recover, or their recovery is delayed. These findings have important implications. In cases of human methanol intoxication, the most common outcome is loss of central, but not peripheral, vision. This has been attributed to the loss of central fibers in the optic nerve.1011 However, the present findings raise the possibility that the loss of central vision may also involve the loss of cone function, because the density of cones is greatest in the central retina. One potential explanation for the initial resistance and subsequent vulnerability of UV-cone photoreceptors to the toxic actions of formate may be the greater numbers of mitochondria present in cones in comparison with rods.47 We have previously postulated that cones have a greater metabolic reserve, allowing them to continue to function for a longer period in the presence of a metabolic toxin, to explain the delayed attenuation of UV-cone function relative to rod-dominated function.18 It is also likely that prolonged metabolic inhibition in cells containing many mitochondria could generate excessive amounts of reactive oxygen species, overwhelming antioxidant protection systems and resulting in cell death.48 Of importance, because decrements in retinal energy production and oxidative stress have been postulated to be involved in the pathogenesis of numerous retinal diseases including age-related macular degeneration and diabetic retinopathy, studies of formate-induced retinal dysfunction may provide valuable insight into the pathogenesis of other acquired and genetic retinal disorders. Supported in part by Grant ES06648 from the National Institute of Environmental Health Sciences and by Grants EY01931 and EY11396 from the National Eye Institute, National Institutes of Health. Submitted for publication April 17, 2000; revised October 2, 2000; accepted October 26, 2000. Commercial relationships policy: N. Corresponding author: Janis T. Eells, Department of Pharmacology and Toxicology, Medical College of Wisconsin, 8701 Watertown Plank Road, Milwaukee, WI 53226. jeells@mcw.edu F igure 1. View OriginalDownload Slide Effect of methanol intoxication and recovery on blood formate concentrations. Rats were exposed to a mixture of N 2 O/O 2 (1:1) 4 hours before the administration of methanol and for 72 hours after the initial dose of methanol. Methanol was administered at a dose of 4 g/kg at the zero time point, followed by 2 g/kg at 24 and 48 hours. In the recovery group, the N 2 O/O 2 exposure was discontinued at the 72-hour time point, and animals were allowed to recover for an additional 72 hours. Blood formate concentrations were determined before methanol administration and at 24-hour intervals after methanol administration for 72 hours. In the recovery animals, blood formate concentrations were also determined 72 hours after the N 2 O/O 2 exposure was discontinued at 144 hours after the initial dose of methanol. Shown are the mean values ± SEM from five to six rats in each experimental group. Formate concentrations did not differ between the intoxicated and recovery group over the 72-hour intoxication period (ANOVA with Bonferroni’s test; P< 0.05). Blood formate concentrations measured in N 2 O-control and methanol-control rats were not significantly different from blood formate concentrations measured in untreated-control rats (data not shown). F igure 1. Effect of methanol intoxication and recovery on blood formate concentrations. Rats were exposed to a mixture of N 2 O/O 2 (1:1) 4 hours before the administration of methanol and for 72 hours after the initial dose of methanol. Methanol was administered at a dose of 4 g/kg at the zero time point, followed by 2 g/kg at 24 and 48 hours. In the recovery group, the N 2 O/O 2 exposure was discontinued at the 72-hour time point, and animals were allowed to recover for an additional 72 hours. Blood formate concentrations were determined before methanol administration and at 24-hour intervals after methanol administration for 72 hours. In the recovery animals, blood formate concentrations were also determined 72 hours after the N 2 O/O 2 exposure was discontinued at 144 hours after the initial dose of methanol. Shown are the mean values ± SEM from five to six rats in each experimental group. Formate concentrations did not differ between the intoxicated and recovery group over the 72-hour intoxication period (ANOVA with Bonferroni’s test; P< 0.05). Blood formate concentrations measured in N 2 O-control and methanol-control rats were not significantly different from blood formate concentrations measured in untreated-control rats (data not shown). View OriginalDownload Slide F igure 2. View OriginalDownload Slide Effect of methanol intoxication and recovery on 15-Hz/510-nm ERG responses. Flicker ERG analyses were performed at 72 hours after the initial dose of methanol and 144 hours after the initial dose of methanol, after a 72-hour recovery. Shown are the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats; †significant differences from values obtained in intoxicated rats (repeated measures ANOVA with Scheffé’s F test; P< 0.05). 15-Hz/510-nm luminance–response relationships measured in N 2 O-control and methanol-control rats were not significantly different from 15-Hz/510-nm luminance–response relationships measured in untreated-control rats (data not shown). F igure 2. Effect of methanol intoxication and recovery on 15-Hz/510-nm ERG responses. Flicker ERG analyses were performed at 72 hours after the initial dose of methanol and 144 hours after the initial dose of methanol, after a 72-hour recovery. Shown are the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats; †significant differences from values obtained in intoxicated rats (repeated measures ANOVA with Scheffé’s F test; P< 0.05). 15-Hz/510-nm luminance–response relationships measured in N 2 O-control and methanol-control rats were not significantly different from 15-Hz/510-nm luminance–response relationships measured in untreated-control rats (data not shown). View OriginalDownload Slide F igure 3. View OriginalDownload Slide Effect of methanol intoxication and recovery on 25-Hz/UV ERG responses. ERG analyses were performed at 72 and 144 hours after the initial dose of methanol, after a 72-hour recovery. UV-cone–sensitive ERGs were recorded from the same animals in which 15-Hz/510-nm ERG responses were recorded. Shown are mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats; †significant differences from values obtained in intoxicated rats (repeated measures ANOVA with Scheffé’s F test; P< 0.05). UV-cone–mediated luminance–response relationships measured in N 2 O-control and methanol-control rats were not significantly different from UV-cone–mediated luminance–response relationships measured in untreated-control rats (data not shown). F igure 3. Effect of methanol intoxication and recovery on 25-Hz/UV ERG responses. ERG analyses were performed at 72 and 144 hours after the initial dose of methanol, after a 72-hour recovery. UV-cone–sensitive ERGs were recorded from the same animals in which 15-Hz/510-nm ERG responses were recorded. Shown are mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats; †significant differences from values obtained in intoxicated rats (repeated measures ANOVA with Scheffé’s F test; P< 0.05). UV-cone–mediated luminance–response relationships measured in N 2 O-control and methanol-control rats were not significantly different from UV-cone–mediated luminance–response relationships measured in untreated-control rats (data not shown). View OriginalDownload Slide F igure 4. View OriginalDownload Slide Effect of methanol intoxication and recovery on retinal energy metabolites. ATP, ADP, and AMP concentrations were measured in retinas of rats 72 and 144 hours after the initial dose of methanol, after a 72-hour recovery. Data are expressed as the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats (ANOVA with Bonferroni’s test; P< 0.05). Retinal energy metabolite concentrations measured in N 2 O-control and methanol-control rats were not significantly different from energy metabolite concentrations measured in untreated-control rats (data not shown). F igure 4. Effect of methanol intoxication and recovery on retinal energy metabolites. ATP, ADP, and AMP concentrations were measured in retinas of rats 72 and 144 hours after the initial dose of methanol, after a 72-hour recovery. Data are expressed as the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats (ANOVA with Bonferroni’s test; P< 0.05). Retinal energy metabolite concentrations measured in N 2 O-control and methanol-control rats were not significantly different from energy metabolite concentrations measured in untreated-control rats (data not shown). View OriginalDownload Slide F igure 5. View OriginalDownload Slide Effect of methanol intoxication and recovery on retinal GSH concentrations. GSH concentrations were measured in retinas of rats 72 and 144 hours, after a 72-hour recovery. Data are expressed as the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats;† significant differences from values obtained in intoxicated animals (ANOVA with Bonferroni’s test; P< 0.05). GSH concentrations measured in N 2 O-control and methanol-control rats were not significantly different from GSH concentrations measured in untreated-control rats (data not shown). F igure 5. Effect of methanol intoxication and recovery on retinal GSH concentrations. GSH concentrations were measured in retinas of rats 72 and 144 hours, after a 72-hour recovery. Data are expressed as the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats;† significant differences from values obtained in intoxicated animals (ANOVA with Bonferroni’s test; P< 0.05). GSH concentrations measured in N 2 O-control and methanol-control rats were not significantly different from GSH concentrations measured in untreated-control rats (data not shown). View OriginalDownload Slide F igure 6. View OriginalDownload Slide Effect of methanol intoxication and recovery on retinal histology. Outer retinal morphology in representative untreated-control (A), intoxicated (B), and recovery (C) rats. Sections were taken from the posterior pole of the retina within two disc diameters of the optic nerve in any direction. (A) rpe, retinal pigment epithelium; os, photoreceptor outer segments; is, photoreceptor inner segments; onl, outer nuclear layer; opl, outer plexiform layer; ipl, inner plexiform layer. (B) Arrowheads: Enlargement and swelling of the photoreceptor inner segments; short arrows: fragmented appearance of photoreceptor nuclei; long arrow: retinal edema, as evidenced by increased spacing between the nuclei in the outer nuclear layer. (C) Arrows: Increased spacing between the nuclei in the outer nuclear layer. No histopathologic changes were apparent at the light microscopic level in the N 2 O-control or methanol-control groups (data not shown). Toluidine blue; magnification, ×450. F igure 6. Effect of methanol intoxication and recovery on retinal histology. Outer retinal morphology in representative untreated-control (A), intoxicated (B), and recovery (C) rats. Sections were taken from the posterior pole of the retina within two disc diameters of the optic nerve in any direction. (A) rpe, retinal pigment epithelium; os, photoreceptor outer segments; is, photoreceptor inner segments; onl, outer nuclear layer; opl, outer plexiform layer; ipl, inner plexiform layer. (B) Arrowheads: Enlargement and swelling of the photoreceptor inner segments; short arrows: fragmented appearance of photoreceptor nuclei; long arrow: retinal edema, as evidenced by increased spacing between the nuclei in the outer nuclear layer. (C) Arrows: Increased spacing between the nuclei in the outer nuclear layer. No histopathologic changes were apparent at the light microscopic level in the N 2 O-control or methanol-control groups (data not shown). Toluidine blue; magnification, ×450. View OriginalDownload Slide F igure 7. View OriginalDownload Slide Effect of methanol intoxication and recovery on photoreceptor ultrastructure. Electron micrographs of the rod inner segment region in representative control (A), intoxicated (B), and recovery (C) rats. Abnormal mitochondrial morphology was present in photoreceptor inner segments. Arrowheads: Swollen mitochondria with expanded cristae; arrows: completely disrupted mitochondria with no evidence of cristae. Photoreceptor mitochondria from N 2 O-control or methanol-control rats exhibited normal morphology with well-defined cristae (data not shown). Magnification, ×5000 F igure 7. Effect of methanol intoxication and recovery on photoreceptor ultrastructure. Electron micrographs of the rod inner segment region in representative control (A), intoxicated (B), and recovery (C) rats. Abnormal mitochondrial morphology was present in photoreceptor inner segments. Arrowheads: Swollen mitochondria with expanded cristae; arrows: completely disrupted mitochondria with no evidence of cristae. Photoreceptor mitochondria from N 2 O-control or methanol-control rats exhibited normal morphology with well-defined cristae (data not shown). Magnification, ×5000 View OriginalDownload Slide The authors thank Anna Fekete for excellent technical assistance. 1 Wood CA, Buller F. Poisoning by wood alcohol: cases of death and blindness from Colombian spirits and other methylated preparations. JAMA . 1904;43:972–977. 2 Benton CD, Calhoun FP. The ocular effects of methyl alcohol poisoning: report of a catastrophe involving three hundred and twenty persons. Trans Am Acad Ophthalmol Otolaryngol . 1952;56:875–883.[PubMed] 3 Roe O. 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Brain Res Brain Res Rev . 1999;29:1–25.[CrossRef][PubMed] F igure 1. View OriginalDownload Slide Effect of methanol intoxication and recovery on blood formate concentrations. Rats were exposed to a mixture of N 2 O/O 2 (1:1) 4 hours before the administration of methanol and for 72 hours after the initial dose of methanol. Methanol was administered at a dose of 4 g/kg at the zero time point, followed by 2 g/kg at 24 and 48 hours. In the recovery group, the N 2 O/O 2 exposure was discontinued at the 72-hour time point, and animals were allowed to recover for an additional 72 hours. Blood formate concentrations were determined before methanol administration and at 24-hour intervals after methanol administration for 72 hours. In the recovery animals, blood formate concentrations were also determined 72 hours after the N 2 O/O 2 exposure was discontinued at 144 hours after the initial dose of methanol. Shown are the mean values ± SEM from five to six rats in each experimental group. Formate concentrations did not differ between the intoxicated and recovery group over the 72-hour intoxication period (ANOVA with Bonferroni’s test; P< 0.05). Blood formate concentrations measured in N 2 O-control and methanol-control rats were not significantly different from blood formate concentrations measured in untreated-control rats (data not shown). F igure 1. Effect of methanol intoxication and recovery on blood formate concentrations. Rats were exposed to a mixture of N 2 O/O 2 (1:1) 4 hours before the administration of methanol and for 72 hours after the initial dose of methanol. Methanol was administered at a dose of 4 g/kg at the zero time point, followed by 2 g/kg at 24 and 48 hours. In the recovery group, the N 2 O/O 2 exposure was discontinued at the 72-hour time point, and animals were allowed to recover for an additional 72 hours. Blood formate concentrations were determined before methanol administration and at 24-hour intervals after methanol administration for 72 hours. In the recovery animals, blood formate concentrations were also determined 72 hours after the N 2 O/O 2 exposure was discontinued at 144 hours after the initial dose of methanol. Shown are the mean values ± SEM from five to six rats in each experimental group. Formate concentrations did not differ between the intoxicated and recovery group over the 72-hour intoxication period (ANOVA with Bonferroni’s test; P< 0.05). Blood formate concentrations measured in N 2 O-control and methanol-control rats were not significantly different from blood formate concentrations measured in untreated-control rats (data not shown). View OriginalDownload Slide F igure 2. View OriginalDownload Slide Effect of methanol intoxication and recovery on 15-Hz/510-nm ERG responses. Flicker ERG analyses were performed at 72 hours after the initial dose of methanol and 144 hours after the initial dose of methanol, after a 72-hour recovery. Shown are the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats; †significant differences from values obtained in intoxicated rats (repeated measures ANOVA with Scheffé’s F test; P< 0.05). 15-Hz/510-nm luminance–response relationships measured in N 2 O-control and methanol-control rats were not significantly different from 15-Hz/510-nm luminance–response relationships measured in untreated-control rats (data not shown). F igure 2. Effect of methanol intoxication and recovery on 15-Hz/510-nm ERG responses. Flicker ERG analyses were performed at 72 hours after the initial dose of methanol and 144 hours after the initial dose of methanol, after a 72-hour recovery. Shown are the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats; †significant differences from values obtained in intoxicated rats (repeated measures ANOVA with Scheffé’s F test; P< 0.05). 15-Hz/510-nm luminance–response relationships measured in N 2 O-control and methanol-control rats were not significantly different from 15-Hz/510-nm luminance–response relationships measured in untreated-control rats (data not shown). View OriginalDownload Slide F igure 3. View OriginalDownload Slide Effect of methanol intoxication and recovery on 25-Hz/UV ERG responses. ERG analyses were performed at 72 and 144 hours after the initial dose of methanol, after a 72-hour recovery. UV-cone–sensitive ERGs were recorded from the same animals in which 15-Hz/510-nm ERG responses were recorded. Shown are mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats; †significant differences from values obtained in intoxicated rats (repeated measures ANOVA with Scheffé’s F test; P< 0.05). UV-cone–mediated luminance–response relationships measured in N 2 O-control and methanol-control rats were not significantly different from UV-cone–mediated luminance–response relationships measured in untreated-control rats (data not shown). F igure 3. Effect of methanol intoxication and recovery on 25-Hz/UV ERG responses. ERG analyses were performed at 72 and 144 hours after the initial dose of methanol, after a 72-hour recovery. UV-cone–sensitive ERGs were recorded from the same animals in which 15-Hz/510-nm ERG responses were recorded. Shown are mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats; †significant differences from values obtained in intoxicated rats (repeated measures ANOVA with Scheffé’s F test; P< 0.05). UV-cone–mediated luminance–response relationships measured in N 2 O-control and methanol-control rats were not significantly different from UV-cone–mediated luminance–response relationships measured in untreated-control rats (data not shown). View OriginalDownload Slide F igure 4. View OriginalDownload Slide Effect of methanol intoxication and recovery on retinal energy metabolites. ATP, ADP, and AMP concentrations were measured in retinas of rats 72 and 144 hours after the initial dose of methanol, after a 72-hour recovery. Data are expressed as the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats (ANOVA with Bonferroni’s test; P< 0.05). Retinal energy metabolite concentrations measured in N 2 O-control and methanol-control rats were not significantly different from energy metabolite concentrations measured in untreated-control rats (data not shown). F igure 4. Effect of methanol intoxication and recovery on retinal energy metabolites. ATP, ADP, and AMP concentrations were measured in retinas of rats 72 and 144 hours after the initial dose of methanol, after a 72-hour recovery. Data are expressed as the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats (ANOVA with Bonferroni’s test; P< 0.05). Retinal energy metabolite concentrations measured in N 2 O-control and methanol-control rats were not significantly different from energy metabolite concentrations measured in untreated-control rats (data not shown). View OriginalDownload Slide F igure 5. View OriginalDownload Slide Effect of methanol intoxication and recovery on retinal GSH concentrations. GSH concentrations were measured in retinas of rats 72 and 144 hours, after a 72-hour recovery. Data are expressed as the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats;† significant differences from values obtained in intoxicated animals (ANOVA with Bonferroni’s test; P< 0.05). GSH concentrations measured in N 2 O-control and methanol-control rats were not significantly different from GSH concentrations measured in untreated-control rats (data not shown). F igure 5. Effect of methanol intoxication and recovery on retinal GSH concentrations. GSH concentrations were measured in retinas of rats 72 and 144 hours, after a 72-hour recovery. Data are expressed as the mean values ± SEM from five to six rats in each treatment group. Significant differences from values obtained in control rats;† significant differences from values obtained in intoxicated animals (ANOVA with Bonferroni’s test; P< 0.05). GSH concentrations measured in N 2 O-control and methanol-control rats were not significantly different from GSH concentrations measured in untreated-control rats (data not shown). View OriginalDownload Slide F igure 6. View OriginalDownload Slide Effect of methanol intoxication and recovery on retinal histology. Outer retinal morphology in representative untreated-control (A), intoxicated (B), and recovery (C) rats. Sections were taken from the posterior pole of the retina within two disc diameters of the optic nerve in any direction. (A) rpe, retinal pigment epithelium; os, photoreceptor outer segments; is, photoreceptor inner segments; onl, outer nuclear layer; opl, outer plexiform layer; ipl, inner plexiform layer. (B) Arrowheads: Enlargement and swelling of the photoreceptor inner segments; short arrows: fragmented appearance of photoreceptor nuclei; long arrow: retinal edema, as evidenced by increased spacing between the nuclei in the outer nuclear layer. (C) Arrows: Increased spacing between the nuclei in the outer nuclear layer. No histopathologic changes were apparent at the light microscopic level in the N 2 O-control or methanol-control groups (data not shown). Toluidine blue; magnification, ×450. F igure 6. Effect of methanol intoxication and recovery on retinal histology. Outer retinal morphology in representative untreated-control (A), intoxicated (B), and recovery (C) rats. Sections were taken from the posterior pole of the retina within two disc diameters of the optic nerve in any direction. (A) rpe, retinal pigment epithelium; os, photoreceptor outer segments; is, photoreceptor inner segments; onl, outer nuclear layer; opl, outer plexiform layer; ipl, inner plexiform layer. (B) Arrowheads: Enlargement and swelling of the photoreceptor inner segments; short arrows: fragmented appearance of photoreceptor nuclei; long arrow: retinal edema, as evidenced by increased spacing between the nuclei in the outer nuclear layer. (C) Arrows: Increased spacing between the nuclei in the outer nuclear layer. No histopathologic changes were apparent at the light microscopic level in the N 2 O-control or methanol-control groups (data not shown). Toluidine blue; magnification, ×450. View OriginalDownload Slide F igure 7. View OriginalDownload Slide Effect of methanol intoxication and recovery on photoreceptor ultrastructure. Electron micrographs of the rod inner segment region in representative control (A), intoxicated (B), and recovery (C) rats. Abnormal mitochondrial morphology was present in photoreceptor inner segments. Arrowheads: Swollen mitochondria with expanded cristae; arrows: completely disrupted mitochondria with no evidence of cristae. Photoreceptor mitochondria from N 2 O-control or methanol-control rats exhibited normal morphology with well-defined cristae (data not shown). Magnification, ×5000 F igure 7. Effect of methanol intoxication and recovery on photoreceptor ultrastructure. Electron micrographs of the rod inner segment region in representative control (A), intoxicated (B), and recovery (C) rats. Abnormal mitochondrial morphology was present in photoreceptor inner segments. Arrowheads: Swollen mitochondria with expanded cristae; arrows: completely disrupted mitochondria with no evidence of cristae. Photoreceptor mitochondria from N 2 O-control or methanol-control rats exhibited normal morphology with well-defined cristae (data not shown). Magnification, ×5000 View OriginalDownload Slide Copyright 2001 The Association for Research in Vision and Ophthalmology, Inc. 5,149 Views 45Web of Science View Metrics × Related Articles P23H Rhodopsin Transgenic Rat: Correlation of Retinal Function with Histopathology Hyperoxia Promotes Electroretinogram Recovery after Retinal Artery Occlusion in Cats Retinal Function Loss after Monocarboxylate Transport Inhibition n-3 Fatty Acid Deficiency Alters Recovery of the Rod Photoresponse in Rhesus Monkeys Electroretinogram recovery in the rabbit after repetitive short-term ischemia in light and dark. 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1603	https://www.combinatorics.org/ojs/index.php/eljc/article/download/v22i4p17/pdf/	Rank and crank analogs for some colored partitions Roberta R. Zhou ∗ School of Mathematics and Statistics Northeastern University at Qinhuangdao Hebei 066004, P. R. China zhourui@neuq.edu.cn Wenlong Zhang † School of Mathematical Sciences Dalian University of Technology Dalian 116024, P. R. China wenlongzhang@dlut.edu.cn Submitted: Apr 27, 2015; Accepted: Oct 10, 2015; Published: Oct 30, 2015 Mathematics Subject Classifications: Primary 11P83 and Secondary 05A17, 05A19, 11F03, 11P81 Abstract We establish some rank and crank analogs for partitions into distinct colors and give combinatorial interpretations for colored partitions such as partitions defined by Toh, Zhang and Wang congruences modulo 5, 7. Keywords: Partition congruences; rank analogs; Jacobi’s triple product identity; Winquist’s product identity 1 Introduction and Motivation Let p(n) be the number of unrestricted partitions of n, where n is nonnegative integer. In 1921, Ramanujan discovered the following congruences p(5 n + 4) ≡ 0 (mod 5) p(7 n + 5) ≡ 0 (mod 7) . There exist many proofs in mathematical literature, for example [6, 7, 19]. In 1944, F. J. Dyson defined the rank of a partition as the largest part minus the number of parts. Let N (m, n ) denote the number of partitions of n with rank m and let ∗ The first author’s research was partially supported by the Fundamental Research Funds for the Central Universities (N142303009), the Natural Science Foundation of Hebei Province (A2015501066) and NSFC(11501090). † The second author’s research was partially supported by the Fundamental Research Funds for the Central Universities. the electronic journal of combinatorics 22(4) (2015), #P4.17 1N (m, t, n ) denote the number of partitions of n with rank congruent to m modulo t. In 1953, A. O. L. Atkin and H. P. F. Swinnerton-Dyer proved N (0 , 5, 5n + 4) = N (1 , 5, 5n + 4) = · · · = N (4 , 5, 5n + 4) = p(5 n + 4) 5and N (0 , 7, 7n + 5) = N (1 , 7, 7n + 5) = · · · = N (6 , 7, 7n + 5) = p(7 n + 5) 7 . Following from the fact that the operation of conjugation reverses the sign of the rank, the trivial consequences are N (m, n ) = N (−m, n ) and N (m, t, n ) = N (t − m, t, n ). Hammond and Lewis defined birank and explained that the residue of the birank mod 5 divided 2-colored partitions of n into 5 equal classes provided n ≡ 2, 3 or 4 (mod 5). F. G. Garvan found two other analogs the Dyson-birank and the 5-core-birank. In 2010, Chan introduced the cubic partition a(n) as the number of 2-color parti-tions of n with colors red and blue subjecting to the restriction that the color blue appears only in even parts, and obtained the following congruence a(3 n + 2) ≡ 0 (mod 3) . Another proof has been given by B. Kim . He defined a crank analog M ′(m, N, n ) for a(n) and proved that M ′(0 , 3, 3n + 2) ≡ M ′(1 , 3, 3n + 2) ≡ M ′(2 , 3, 3n + 2) (mod 3) , for all nonnegative integers n, where M ′(m, N, n ) is the number of partition of n with crank congruent to m modulo N . Later, B. Kim gave two partition statistics which explained the partition congruences about cubic partition pairs b(n). Here, b(n) is the number of 4-color partitions of n with colors red, yellow, orange, and blue subjecting to the restriction that the colors orange and blue appear only in even parts. About further research of arithmetic properties of cubic partitions, overcubic partitions and other colored partitions, some interesting results can be found in [18, 21, 23, 24]. The first author of the present paper generalized Hammond-Lewis birank and gave combinatorial interpretations for some colored partitions. The paper is organized as follows. In Section 2, we introduce necessary notation and some preliminary results. In Section 3, we aim to provide two partition statistics for two colored partition congruences modulo 5. We establish six rank or crank analogs for six colored partition with modulus 7 and give combinatorial interpretations in Section 4. 2 Preliminary results For the two indeterminates q and z with \|q\| < 1, the q-shifted factorial of infinite order is defined by (z; q)∞ = ∞ ∏ n=0 (1 − zq n) the electronic journal of combinatorics 22(4) (2015), #P4.17 2 where the multi-parameter expression for the former will be abbreviated as [α, β, · · · , γ ; q]∞ = ( α; q)∞(β; q)∞ · · · (γ; q)∞. The main purpose of this paper is to define rank and crank analogs for partition into colors and prove colored partition congruences applying the method of , which uses roots of unity. Jacobi triple product identity, the modified Jacobi triple product identity and Winquist product identity are given as follows: • Jacobi triple product identity [1, 4, 5, 13, 15]: +∞ ∑ n=−∞ (−1) n q(n 2 ) xn = [ q, x, q/x ; q]∞ . (1) • Modified Jacobi triple product identity : [q, zq, q/z ; q]∞ = ∑ n>0 (−1) n(zn + zn−1 + · · · + z−n)q(n+1 2 ). (2) • Winquist product identity [9, 22]: (q; q)4 ∞ [x, q/x ; q]2 ∞ [x2, q/x 2; q] ∞ = +∞ ∑ i,j =−∞ (−1) i+j q3(i 2 )+3 (j 2 )+j (1 − 3i + 3 j){x3i+3 j − x4−3i−3j }. (3) By replacing q by q2 in (3), splitting into two bilateral sums on right hand side of the resulting equation, and replacing j → j − 1 in the first double sum, and i → i + 1 in the second double sum, the resulting formula can be transformed as • Modified Winquist product identity (q2; q2)4 ∞ [x, q 2/x ; q2]2 ∞ [x2, q 2/x 2; q2] ∞ = +∞ ∑ i,j =−∞ (−1) i+j q6(i 2 )+6 (j 2 )+3 i−j+1 (2 + 3 i − 3j){(x/q )3i+3 j−3 − (x/q )1−3i−3j }. (4) Dividing both sides by 1 + x in (3) and applying L’Hˆ opital’s rule for the limit x → − 1, we have (q; q)2 ∞ (q2; q2)4 ∞ = +∞ ∑ i,j =−∞ q3(i 2 )+3 (j 2 )+j (1 − 3i + 3 j)(2 − 3i − 3j)4 . (5) Divide both sides by 1 − q2/x 2 in (4) and utilize L’Hˆ opital’s rule for the limit x → q to obtain (q; q)4 ∞ (q2; q2)2 ∞ = +∞ ∑ i,j =−∞ (−1) i+j q6(i 2 )+6 (j 2 )+3 i−j+1 (2 + 3 i − 3j)(3 i + 3 j − 2) . (6) the electronic journal of combinatorics 22(4) (2015), #P4.17 3After Andrews and Garvan , for a partition λ, we define #( λ) is the number of parts in λ and σ(λ) is the sum of the parts of λ with the convention #( λ) = σ(λ) = 0 for the empty partition λ. Let P be the set of all ordinary partitions, DO be the set of all partitions into distinct odd parts. For a given partition λ, the crank c(λ) of a partition is defined as c(λ) := { `(λ), if r = 0; ω(λ) − r, if r > 1, where r is the number of 1’s in λ, ω(λ) is the number of parts in λ that are strictly larger than r and `(λ) is the largest part in λ. By extending the set of partitions P to a new set P∗ by adding two additional copies of the partition 1, say 1 ∗ and 1 ∗∗ , B. Kim [16, 17] obtains (q; q)∞ [zq, z −1q; q]∞ = ∑ λ∈P ∗ wt (λ)zc∗(λ)qσ∗(λ), where wt (λ), c∗(λ), and σ∗(λ) are defined as follows. Denote the weight wt (λ) for λ ∈ P ∗ by wt (λ) := { 1, if λ ∈ P , λ = 1 ∗, or λ = 1 ∗∗ ; −1, if λ = 1 , and denote the extended crank c∗(λ) by c∗(λ) :=  c(λ), if λ ∈ P ;0, if λ = 1; 1, if λ = 1 ∗; −1, if λ = 1 ∗∗ . Finally, denote the extended sum parts function σ∗(λ) in the following way: σ∗(λ) := { σ(λ), if λ ∈ P ;1, otherwise . 3 Rank analogs for colored partitions congruences modulo 5 In this section, we establish two statistics that divide the relevant partitions into equinu-merous classes and present the combinatorial interpretation for colored partition congru-ences modulo 5. We denote C1232 = {(λ1, λ 2, 3λ3, 3λ4) \| λ1, λ 2, λ 3, λ 4 ∈ P} . For λ ∈ C1232 , we define the sum of parts s1232 (λ), and rank analog r1232 (λ) by s1232 (λ) = σ(λ1) + σ(λ2) + 3 σ(λ3) + 3 σ(λ4) r1232 (λ) = #( λ1) − #( λ2) + #( λ3) − #( λ4). the electronic journal of combinatorics 22(4) (2015), #P4.17 4 The number of 4-colored partitions of n if s1232 (λ) = n having r1232 (λ) = m will be written as NC1232 (m, n ), and NC1232 (m, t, n ) is the number of such 4-colored partitions of n having rank analog r1232 (λ) ≡ m (mod t). Now, summing over all 4-colored partitions λ ∈ C1232 gives NC1232 (m, n ) = ∑ λ∈C1232,s 1232(λ)= n, r1232(λ)= m 1. Since r1232 (λ1, λ 2, λ 3, λ 4) = −r1232 (λ2, λ 1, λ 4, λ 3), hence NC1232 (m, n ) = NC1232 (−m, n ) and NC1232 (m, t, n ) = NC1232 (t − m, t, n ). Then we have ∑ m∈Z ∞ ∑ n=0 NC1232 (m, n )zmqn = 1(zq ; q)∞(z−1q; q)∞(zq 3; q3)∞(z−1q3; q3)∞ . (7) By putting z = 1 in the identity (7), we find ∞ ∑ m=−∞ NC1232 (m, n ) = c(n), where c(n) is defined by ∑∞ n=0 c(n)qn = 1(q;q)2 ∞(q3;q3)2 ∞ . Theorem 1. For n > 0, NC1232 (0 , 5, 5n + 2) = NC1232 (1 , 5, 5n + 2) = NC1232 (2 , 5, 5n + 2) = c(5 n + 2) 5 . It can also prove the identity in Zhang and Wang : c(5 n + 2) ≡ 0 (mod 5). Proof. Suppose ζ is primitive 5 th root of unity. By setting z = ζ in (7), we write ∑ m∈Z ∞ ∑ n=0 NC1232 (m, n )ζmqn = 1[ζq, ζ −1q; q]∞ [ζq 3, ζ −1q3; q3]∞ = 1(q5; q5)∞(q15 ; q15 )∞ × [q, ζ 2q, ζ −2q; q] ∞ [q3, ζ 2q3, ζ −2q3; q3] ∞ . Using modified Jacobi triple product identity (2), the last two infinite products have the following series representation ∞ ∑ i,j =0 (−1) i+j q(i+1 2 )+3 (j+1 2 ){ζ2i + ζ2i−2 + · · · + ζ−2i}{ ζ2j + ζ2j−2 + · · · + ζ−2j }. the electronic journal of combinatorics 22(4) (2015), #P4.17 5Observe the congruence relation (i + 1 2 ) +3 (j + 1 2 ) +3 ≡ 8 {(i + 1 2 ) +3 (j + 1 2 ) +3 } ≡ (2 i+1) 2+3(2 j+1) 2 ≡ 0 (mod 5) , which can be reached only if i ≡ 2 (mod 5) and j ≡ 2 (mod 5) since the corresponding residues modulo 5 read respectively as (2 i + 1) 2 ≡ 0, 1, 4 (mod 5) , and 3(2 j + 1) 2 ≡ 0, 2, 3 (mod 5) . When i ≡ 2 (mod 5) and j ≡ 2 (mod 5), we have {ζ2i + ζ2i−2 + · · · + ζ−2i}{ ζ2j + ζ2j−2 + · · · + ζ−2j } = 0. We see that in the q-expansion on the right side of the last equation the coefficient of qn is zero when n ≡ 2 (mod 5). The proof of Theorem 1 has been finished. Let C2232 = {(2 λ1, 2λ2, 3λ3, 3λ4) \| λ1, λ 2, λ 3, λ 4 ∈ P} . For λ ∈ C2232 , we define the sum of parts s2232 (λ), and rank analog r2232 (λ) by s2232 (λ) = 2σ(λ1) + 2 σ(λ2) + 3 σ(λ3) + 3 σ(λ4) r2232 (λ) = #( λ1) − #( λ2) + #( λ3) − #( λ4). Define NC2232 (m, n ) as the number of 4-colored partitions of n if s2232 (λ) = n having r2232 (λ) = m, and NC2232 (m, t, n ) as the number of such 4-colored partitions of n having rank analog r2232 (λ) ≡ m (mod t). Now, summing over all 4-colored partitions λ ∈ C2232 gives NC2232 (m, n ) = ∑ λ∈C2232,s 2232(λ)= n, r2232(λ)= m 1. Since r2232 (λ1, λ 2, λ 3, λ 4) = −r2232 (λ2, λ 1, λ 4, λ 3), hence NC2232 (m, n ) = NC2232 (−m, n ) and NC2232 (m, t, n ) = NC2232 (t − m, t, n ). Then we have ∑ m∈Z ∞ ∑ n=0 NC2232 (m, n )zmqn = 1(zq 2; q2)∞(z−1q2; q2)∞(zq 3; q3)∞(z−1q3; q3)∞ . (8) By putting z = 1 in the identity (8), we find ∞ ∑ m=−∞ NC2232 (m, n ) = ρ(n), where ρ(n) is defined by ∑∞ n=0 ρ(n)qn = 1(q2;q2)2 ∞(q3;q3)2 ∞ . the electronic journal of combinatorics 22(4) (2015), #P4.17 6Theorem 2. For n > 0, NC2232 (0 , 5, 5n + k) = NC2232 (1 , 5, 5n + k) = NC2232 (2 , 5, 5n + k) = ρ(5 n + k)5 ; k = 1 , 4. It can also prove the identity in Zhang and Wang : ρ(5 n + 1) ≡ 0 (mod 5) and ρ(5 n + 4) ≡ 0 (mod 5). Proof. The proof of Theorem 2 is similar to Theorem 1. Replacing z by ζ in (8), we get ∑ m∈Z ∞ ∑ n=0 NC2232 (m, n )ζmqn = 1[ζq 2, ζ −1q2; q2]∞ [ζq 3, ζ −1q3; q3]∞ = 1(q10 ; q10 )∞(q15 ; q15 )∞ × [q2, ζ 2q2, ζ −2q2; q2] ∞ [q3, ζ 2q3, ζ −2q3; q3] ∞ . Applying modified Jacobi triple product identity (2), we transform the last two infinite products as follows ∞ ∑ i,j =0 (−1) i+j q2(i+1 2 )+3 (j+1 2 ){ζ2i + ζ2i−2 + · · · + ζ−2i}{ ζ2j + ζ2j−2 + · · · + ζ−2j }. It is not hard to check that the residues of q-exponent in the formal power series just displayed 2 (i+1 2 ) + 3 (j+1 2 ) modulo 5 are given by the following table: j\i 0 1 2 3 40 0 2 1 2 01 3 0 4 0 32 4 1 0 1 43 3 0 4 0 34 0 2 1 2 0When i ≡ 2 (mod 5) or j ≡ 2 (mod 5), we have {ζ2i + ζ2i−2 + · · · + ζ−2i}{ ζ2j + ζ2j−2 + · · · + ζ−2j } = 0. We observe that in the q-expansion on the right side of the last equation the coefficient of qn is zero when n ≡ 1 (mod 5) and n ≡ 4 (mod 5). The proof of Theorem 2 has been completed. 4 Rank and crank analogs for colored partitions congruences modulo 7 In this section, we define statistics that divide the relevant partitions into equinumerous classes and provide the combinatorial interpretation according to for colored partitions congruences modulo 7 given in Toh , Zhang and Wang . If we denote C132−2 = {(2 λ1, 2λ2, λ 3, λ 4, λ 5) \| λ1, λ 2 ∈ P , λ 3, λ 4, λ 5 ∈ P ∗}, the electronic journal of combinatorics 22(4) (2015), #P4.17 7then we can call them as partitions into 5-colors. For the set of the colored partitions, we define the sum of parts s132−2 (λ), a weight wt 132−2 (λ) and a crank analog c132−2 (λ) by s132−2 (λ) = 2σ(λ1) + 2 σ(λ2) + σ∗(λ3) + σ∗(λ4) + σ∗(λ5) wt 132−2 (λ) = (−1) #( λ1)+#( λ2)wt (λ3)wt (λ4)wt (λ5) c132−2 (λ) = c∗(λ3) + 2 c∗(λ4) + 3 c∗(λ5), where the definitions of P∗, σ∗(λ), wt (λ) and c∗(λ) are presented in section 2. Let MC132−2 (m, n ) denote the number of 5-colored partitions of n if s132−2 (λ) = n (counted ac-cording to the weight wt 132−2 (λ)) with analog of crank c132−2 (λ) = m, and MC132−2 (m, t, n )denote the number of 5-colored partitions of n with analog of crank c132−2 (λ) congruent to m (mod t), so that MC132−2 (m, n ) = ∑ λ∈C132−2,s 132−2(λ)= n, c132−2(λ)= m wt 132−2 (λ). Then we have ∑ m∈Z ∞ ∑ n=0 MC132−2 (m, n )zmqn = (q2; q2)2 ∞ (q; q)3 ∞ [zq, z −1q, z 2q, z −2q, z 3q, z −3q; q]∞ . (9) By putting z = 1 in the identity (9), we find ∞ ∑ m=−∞ MC132−2 (m, n ) = Q(po,p )(n), where Q(po,p )(n) is defined by ∑∞ n=0 Q(po,p )(n)qn := (−q;q)∞ (q;q2)∞(q;q)∞ = (q2;q2)2 ∞ (q;q)3 ∞ .Suppose $ is primitive seventh root of unity. By letting z = $ in (9), we have ∑ m∈Z ∞ ∑ n=0 MC132−2 (m, n )$mqn = (q2; q2)2 ∞ (q; q)3 ∞ [$q, $ −1q, $ 2q, $ −2q, $ 3q, $ −3q; q]∞ = (q2; q2)2 ∞ (q; q)4 ∞ (q7; q7)∞ . Utilizing product identity (6), we compute the numerator of the right hand side of last identity as follows +∞ ∑ i,j =−∞ (−1) i+j q6(i 2 )+6 (j 2 )+3 i−j+1 (2 + 3 i − 3j)(3 i + 3 j − 2) . (10) We illustrate that the residues of q-exponent in the formal power series just displayed 6(i 2 ) + 6 (j 2 ) + 3 i − j + 1 modulo 7 are given by the following table: the electronic journal of combinatorics 22(4) (2015), #P4.17 8j\i 0 1 2 3 4 5 60 1 4 6 0 0 6 41 0 3 5 6 6 5 32 5 1 3 4 4 3 13 2 5 0 1 1 0 54 5 1 3 4 4 3 15 0 3 5 6 6 5 36 1 4 6 0 0 6 4The power of q is congruent to 2 modulo 7 only when i ≡7 0 and j ≡7 3. Since the coefficient of qn on the right side of the last identity is a multiple of 7 when n ≡ 2(mod 7), and 1 + $ + $2 + $3 + $4 + $5 + $6 is a minimal polynomial in Z[$], we must have the result following as Theorem 3. For n > 0 and 0 6 i < j 6 6, we have MC132−2 (i, 7, 7n + 2) ≡ MC132−2 (j, 7, 7n + 2) (mod 7) . It can also prove the identity in Toh : Q(po,p )(7 n + 2) ≡ 0 (mod 7). Next we define C14442−7 = {(λ1, λ 2, λ 3, λ 4, 2λ5, 2λ6, 2λ7, 2λ8, 2λ9) \|λ1, λ 2, λ 3, λ 4 ∈ DO ,λ5, λ 6 ∈ P , λ 7, λ 8, λ 9 ∈ P ∗}. For λ = ( λ1, λ 2, λ 3, λ 4, 2λ5, 2λ6, 2λ7, 2λ8, 2λ9), we denote the sum of parts s14442−7 (λ), a weight wt 14442−7 (λ) and a crank analog c14442−7 (λ) by s14442−7 (λ) = σ(λ1) + σ(λ2) + σ(λ3) + σ(λ4)+ 2 σ(λ5) + 2 σ(λ6) + 2 σ∗(λ7) + 2 σ∗(λ8) + 2 σ∗(λ9) wt 14442−7 (λ) =( −1) #( λ5)+#( λ6)wt (λ7)wt (λ8)wt (λ9) c14442−7 (λ) = c∗(λ7) + 2 c∗(λ8) + 3 c∗(λ9). Finally define MC14442−7 (m, n ) as the number of 9-colored partitions of n if s14442−7 (λ) = n with crank analog c14442−7 (λ) = m counted according to the weight wt 14442−7 (λ) as follows:, MC14442−7 (m, n ) = ∑ λ∈C14442−7,s 14442−7(λ)= n, c14442−7(λ)= m wt 14442−7 (λ). Let MC14442−7 (m, t, n ) denote the number of 9-colored partitions of n with crank analog c14442−7 (λ) congruent to m (mod t). Then we have ∑ m∈Z ∞ ∑ n=0 MC14442−7 (m, n )zmqn = (−q; q2)4 ∞ (q2; q2)5 ∞ [zq 2, z −1q2, z 2q2, z −2q2, z 3q2, z −3q2; q2]∞ . (11) the electronic journal of combinatorics 22(4) (2015), #P4.17 9By replacing z by 1 in the identity (11), we discover ∞ ∑ m=−∞ MC14442−7 (m, n ) = γ(n), where γ(n) is defined by ∑∞ n=0 γ(n)qn = (−q;q2)4 ∞ (q2;q2)∞ . (see ). Theorem 4. For n > 0, MC14442−7 (0 , 7, 7n + 2) ≡ MC14442−7 (1 , 7, 7n + 2) ≡ · · · ≡ MC14442−7 (6 , 7, 7n + 2) (mod 7) . It can also prove the identity γ(7 n + 2) ≡ 0 (mod 7). Proof. Put z = $ in (11) and apply product identity (6) substituting q → − q to obtain ∑ m∈Z ∞ ∑ n=0 MC14442−7 (m, n )$mqn = (−q; q2)4 ∞ (q2; q2)5 ∞ [$q 2, $ −1q2, $ 2q2, $ −2q2, $ 3q2, $ −3q2; q2]∞ = (−q; q2)4 ∞ (q2; q2)6 ∞ (q14 ; q14 )∞ = −1(q14 ; q14 )∞ +∞ ∑ i,j =−∞ q6(i 2 )+6 (j 2 )+3 i−j+1 (2 + 3 i − 3j)(3 i + 3 j − 2) . We discover that the double sum of the last identity is similar as (10). Then we can use the same congruence relations. Since the coefficient of qn on the right side of the last identity is a multiple of 7 when n ≡ 2 (mod 7), and 1 + $ + $2 + $3 + $4 + $5 + $6 is a minimal polynomial in Z[$], we deduce the theorem. If we denote C122 = {(λ1, λ 2, 2λ3, 2λ4, 2λ5) \| λ1, λ 2, λ 3, λ 4, λ 5 ∈ P} . It can be said as partitions into 5-colors. For λ = ( λ1, λ 2, 2λ3, 2λ4, 2λ5) ∈ C122, we define the sum of parts s122(λ), a weight w122(λ) and a rank analog r122(λ) by s122(λ) = σ(λ1) + σ(λ2) + 2 σ(λ3) + 2 σ(λ4) + 2 σ(λ5) w122(λ) = (−1) #( λ5) r122(λ) = #( λ1) − #( λ2) + 3#( λ3) − 3#( λ4). Let NC122 (m, n ) denote the number of 5-colored partitions of n if s122(λ) = n (counted according to the weight w122(λ)) with rank analog r122(λ) = m, and NC122 (m, t, n ) denote the number of 5-colored partitions of n with rank analog r122(λ) congruent to m (mod t), hence NC122 (m, n ) = ∑ λ∈C122,s (λ)= n, r122(λ)= m w122(λ). By considering the transformation that interchanges λ1 and λ2, λ3 and λ4, we get NC122 (m, n ) = NC122 (−m, n ), NC122 (m, t, n ) = NC122 (t − m, t, n ). the electronic journal of combinatorics 22(4) (2015), #P4.17 10 Then we have ∑ m∈Z ∞ ∑ n=0 NC122 (m, n )zmqn = (q2; q2)∞ (zq ; q)∞(z−1q; q)∞(z3q2; q2)∞(z−3q2; q2)∞ . (12) By putting z = 1 in the identity (12), we check ∞ ∑ m=−∞ NC122 (m, n ) = α(n), where α(n) is defined by ∑∞ n=0 α(n)qn = 1(q;q)2 ∞(q2;q2)∞ . (see ). Suppose $ is primitive 7 th root of unity. Substituting z = $ into (12), we have ∑ m∈Z ∞ ∑ n=0 NC122 (m, n )$mqn = (q2; q2)∞ [$q, q/$ ; q]∞ [$3q2, q 2/$ 3; q2]∞ = [q, $ 2q, q/$ 2; q]∞ [q2, $ 3q, q/$ 3; q2]∞ (q7; q7)∞ = ∑ i>0 (−1) i($2i + $2i−2 + · · · + $−2i)q(i+1 2 ) ∑∞ j=−∞ (−1) j $3j q2(j 2 )+j (q7; q7)∞ . The last line depends only on modified Jacobi identity (2) and classical Jacobi identity (1). If and only if i ≡7 3, we have $2i + $2i−2 + · · · + $−2i = 0. Obviously (i + 1 2 ) ≡7  0, i ≡7 0, 6; 1, i ≡7 1, 5; 3, i ≡7 2, 4; 6, i ≡7 3; and 2 (j 2 ) j ≡7  0, j ≡7 0; 1, j ≡7 1, 6; 4, j ≡7 2, 5; 2, m ≡7 3, 4. (13) The power of q is congruent to 6 modulo 7 only when (i+1 2 ) ≡7 6 and 2 (j 2 ) + j ≡7 0 in which case i ≡7 3 and j ≡7 0 and the coefficient of q7n+6 in the last identity is zero. Since 1 + $ + $2 + $3 + $4 + $5 + $6 is a minimal polynomial in Z[$], our main result is as follows. Theorem 5. For n > 0, NC122 (0 , 7, 7n + 6) = NC122 (1 , 7, 7n + 6) = NC126 (2 , 7, 7n + 6) = · · · = NC126 (6 , 7, 7n + 6) =α(7 n + 6) 7 . It can also prove the identity α(7 n + 6) ≡ 0 (mod 7). Denote C15452−7 = {(λ1, λ 2, λ 3, λ 4, λ 5, 2λ6, 2λ7, 2λ8) \| λ1, λ 2, λ 3, λ 4, λ 5 ∈ DO , λ 6, λ 7, λ 8 ∈ P ∗}. the electronic journal of combinatorics 22(4) (2015), #P4.17 11 We call the elements of C15452−7 8-colored partitions. For λ ∈ C15452−7 , we define the sum of parts s15452−7 (λ), a weight wt 15452−7 (λ) and a crank analog c15452−7 (λ) by s15452−7 (λ) = σ(λ1) + σ(λ2) + σ(λ3) + σ(λ4) + σ(λ5) + 2 σ∗(λ6) + 2 σ∗(λ7) + 2 σ∗(λ8) wt 15452−7 (λ) = wt (λ6)wt (λ7)wt (λ8) c15452−7 (λ) = c∗(λ6) + 2 c∗(λ7) + 3 c∗(λ8). Let MC15452−7 (m, n ) denote the number of 8-colored partitions of n if s15452−7 (λ) = n (counted according to the weight wt 15452−7 (λ)) with crank analog c15452−7 (λ) = m, and MC15452−7 (m, t, n ) denote the number of 8-colored partitions of n with crank analog c15452−7 (λ) ≡ t (mod m), so that MC15452−7 (m, n ) = ∑ λ∈C15452−7,s 15452−7(λ)= n, c15452−7(λ)= m wt 15452−7 (λ). Then the generating function is ∑ m∈Z ∞ ∑ n=0 MC15452−7 (m, n )zmqn = (−q; q2)5 ∞ (q2; q2)3 ∞ [zq 2, z −1q2, z 2q2, z −2q2, z 3q2, z −3q2; q2]∞ . (14) By putting z = 1 in the identity (14), we discover ∞ ∑ m=−∞ MC15452−7 (m, n ) = ν(n), where ν(n) is defined by ∑∞ n=0 ν(n)qn = (−q,q 2)5 ∞ (q2;q2)3 ∞ . Theorem 6. For n > 0, MC15452−7 (0 , 7, 7n + 6) ≡ MC15452−7 (1 , 7, 7n + 6) ≡ · · · ≡ MC15452−7 (6 , 7, 7n + 6) (mod 7) . It can also prove the identity ν(7 n + 6) ≡ 0 (mod 7). Proof. By replacing z by $ in (14), we write ∑ m∈Z ∞ ∑ n=0 NC13412−3 (m, n )$mqn = (−q; q2)5 ∞ (q2; q2)3 ∞ [$q 2, $ −1q2, $ 2q2, $ −2q2, $ 3q2, $ −3q2; q2]∞ = (−q; q2)5 ∞ (q2; q2)4 ∞ (q14 ; q14 )∞ . Consider (q; q)3 ∞ [q2, q, q ; q2] ∞ = ∞ ∑ i=0 ∞ ∑ j=−∞ (−1) i+j (2 i + 1) q(i+1 2 )+j2 , the electronic journal of combinatorics 22(4) (2015), #P4.17 12 which can be deduced by Jacobi identity (1) and (2). Replacing q by −q in the last identity, we have the following series representation (−q; q2)5 ∞ (q2; q2)4 ∞ = ∞ ∑ i=0 ∞ ∑ j=−∞ (−1) (i 2 )(2 i + 1) q(i+1 2 )+j2 . If and only if i ≡7 3, we have 2 i + 1 ≡7 0. We see that in the q-expansion on the right side of the last equation the coefficient of qn is a multiple of 7 when n ≡ 6 (mod 7) referring to (13). The proof of Theorem 6 has been finished. Let C1523 = {(λ1, λ 2, λ 3, λ 4, λ 5, 2λ6, 2λ7, 2λ8) \| λ1, λ 2, λ 3, λ 4 ∈ P , λ 5, λ 6, λ 7, λ 8 ∈ P ∗}. We can say them as partitions into 8-colors. For λ ∈ C1523 , we denote the sum of parts s1523 (λ), a weight wt 1523 (λ) and a crank analog c1523 (λ) by s1523 (λ) = σ(λ1) + σ(λ2) + σ(λ3) + σ(λ4) + σ∗(λ5) + 2 σ∗(λ6) + 2 σ∗(λ7) + 2 σ∗(λ8) wt 1523 (λ) = wt (λ5)wt (λ6)wt (λ7)wt (λ8) c1523 (λ) = #( λ1) − #( λ2) + 2#( λ3) − 2#( λ4) + 3 c∗(λ5) + c∗(λ6) + 2 c∗(λ7) + 3 c∗(λ8). The number of 8-colored partitions of n if s1523 (λ) = n with crank analog c1523 (λ) = m counted according to the weight wt 1523 (λ) is denoted by MC1523 (m, n ), so that MC1523 (m, n ) = ∑ λ∈C1523,s 1523(λ)= n, c1523(λ)= m wt 1523 (λ). The number of 8-colored partitions of n with crank analog c1523 (λ) congruent to m (mod t)is denoted by MC1523 (m, t, n ). The following generating function for MC1523 (m, n ) is ∑ m∈Z ∞ ∑ n=0 MC1523 (m, n )zmqn = (q; q)∞(q2; q2)3 ∞ [zq, z −1q, z 2q, z −2q, z 3q, z −3q; q]∞ [zq 2, z −1q2, z 2q2, z −2q2, z 3q2, z −3q2; q2]∞ . (15) By setting z = 1 in the identity (15), we find ∞ ∑ m=−∞ MC1523 (m, n ) = μ(n), where μ(n) is defined by ∑∞ n=0 μ(n)qn = 1(q;q)5 ∞(q2;q2)3 ∞ . Theorem 7. For n > 0, MC1523 (0 , 7, 7n + 6) ≡ MC1523 (1 , 7, 7n + 6) ≡ · · · ≡ MC1523 (6 , 7, 7n + 6) (mod 7) . the electronic journal of combinatorics 22(4) (2015), #P4.17 13 It can also prove the identity μ(7 n + 6) ≡ 0 (mod 7). Proof. By letting z = $ in (15), we get ∑ m∈Z ∞ ∑ n=0 MC1523 (m, n )$mqn = (q; q)∞(q2; q2)3 ∞ [$q, $ −1q, $ 2q, $ −2q, $ 3q, $ −3q; q]∞ [$q 2, $ −1q2, $ 2q2, $ −2q2, $ 3q2, $ −3q2; q2]∞ = (q; q)2 ∞ (q2; q2)4 ∞ (q7; q7)∞(q14 ; q14 )∞ . Investigating product identity (5), splitting the bilateral sum with respect to i into two unilateral sums, the numerator infinite products on the last line of the last formula have the following series expression (q; q)2 ∞ (q2; q2)4 ∞ = ∞ ∑ i=0 +∞ ∑ j=−∞ q3(i+1 2 )+3 (j 2 )+j (1 + 3 i + 3 j)(2 + 3 i − 3j)2 . (16) We check that the residues of q-exponent in the formal power series displayed 3 (i+1 2 ) +3(j 2 ) + j modulo 7 are presented by the following table: j\i 0 1 2 3 4 5 60 0 3 2 4 2 3 01 1 4 3 5 3 4 12 5 1 0 2 0 1 53 5 1 0 2 0 1 54 1 4 3 5 3 4 15 0 3 2 4 2 3 06 2 5 4 6 4 5 2If and only if 3 (i+1 2 ) + 3 (j 2 ) + j ≡ 6 (mod 7), we have i ≡7 3 and j ≡7 6. Since the coefficient of qn on the right side of the last identity is a multiple of 7 when n ≡ 6(mod 7), and 1 + $ + $2 + $3 + $4 + $5 + $6 is a minimal polynomial in Z[$], we finish the proof of Theorem 7. Consider C12422−3 = {(λ1, λ 2, 2λ3, 2λ4, 2λ5, 2λ6, 2λ7) \| λ1, λ 2 ∈ DO , λ 3, λ 4 ∈ P , λ 5, λ 6, λ 7 ∈ P ∗}. We call them as partitions into 7-colors. For λ ∈ C12422−3 , we define the sum of parts s12422−3 (λ), a weight wt 12422−3 (λ) and a crank analog c12422−3 (λ) by s12422−3 (λ) = σ(λ1) + σ(λ2) + 2 σ(λ3) + 2 σ(λ4) + 2 σ∗(λ5) + 2 σ∗(λ6) + 2 σ∗(λ7) wt 12422−3 (λ) = (−1) #( λ3)+#( λ4)wt (λ5)wt (λ6)wt (λ7) c12422−3 (λ) = c∗(λ5) + 2 c∗(λ6) + 3 c∗(λ7). the electronic journal of combinatorics 22(4) (2015), #P4.17 14 Let MC12422−3 (m, n ) denote the number of 7-colored partitions of n if s12422−3 (λ) = n (counted according to the weight wt 12422−3 (λ)) with crank analog c12422−3 (λ) = m, so that MC12422−3 (m, n ) = ∑ λ∈C12422−3,s 12422−3(λ)= n, c12422−3(λ)= m wt 12422−3 (λ). The number of 7-colored partitions of n with crank analog c12422−3 (λ) ≡ m (mod t) is de-noted by MC12422−3 (m, t, n ). Then the two variable generating function for MC12422−3 (m, n )is ∑ m∈Z ∞ ∑ n=0 MC12422−3 (m, n )zmqn = (−q; q2)2 ∞ (q2; q2)5 ∞ [zq 2, z −1q2, z 2q2, z −2q2, z 3q2, z −3q2; q2]∞ . (17) If we simply put z = 1 in the identity (17), and read off the coefficients of like powers of q, we find ∞∑ m=−∞ MC12422−3 (m, n ) = β(n), where β(n) is defined by ∑∞ n=0 β(n)qn = (−q;q2)2 ∞ (q2;q2)∞ .Putting z = $ in (17) gives ∑ m∈Z ∞ ∑ n=0 MC12422−3 (m, n )$mqn = (−q; q2)2 ∞ (q2; q2)5 ∞ [$q 2, $ −1q2, $ 2q2, $ −2q2, $ 3q2, $ −3q2; q2]∞ = (−q; q2)2 ∞ (q2; q2)6 ∞ (q14 ; q14 )∞ . Substituting q → − q into identity (16), the numerator infinite products have the following series expression (−q; q2)2 ∞ (q2; q2)6 ∞ = ∞ ∑ i=0 +∞ ∑ j=−∞ (−1) (i+1 2 )+(j+1 2 )q3(i+1 2 )+3 (j 2 )+j (1 + 3 i + 3 j)(2 + 3 i − 3j)2 . It is easy to find that the power of q is congruent to 6 modulo 7 if and only if i ≡7 3and j ≡7 6 considering the congruence relations in the proof of Theorem 7. Since the coefficient of qn on the right side of the last identity is a multiple of 7 when n ≡7 6, and 1 + $ + $2 + $3 + $4 + $5 + $6 is a minimal polynomial in Z[$], our main result is as follows: Theorem 8. For n > 0 and 0 6 i < j 6 6, we obtain MC12422−3 (i, 7, 7n + 6) ≡ MC12422−3 (j, 7, 7n + 6) (mod 7) . It can also prove the identity β(7 n + 6) ≡ 0 (mod 7). the electronic journal of combinatorics 22(4) (2015), #P4.17 15 References G. E. Andrews. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. CBMS Regional Conference Series in Mathematics, No. 66; Amer. Math. Society , 1986. G. E. Andrews and F. G. Garvan. Dyson’s crank of a partition. Bull. Amer. Math. Soc. (N.S.) , 18:167–171, 1988. A. O. L. Atkin and H. P. F. Swinnerton-Dyer. Some properties of partitions. Proc. London Math. Soc. , 4(3): 84–106, 1954. W. N. Bailey. Generalized Hypergeometric Series. Cambridge University Press, Cambridge , 1935. B. C. Berndt. Ramanujan’s Notebooks (Part III) . Springer-Verlag, New York ,pp. 510+xiv, 1991. B. C. Berndt. Number Theory in the Spirit of Ramanujan. American Mathematical Society, Providence , 2004. B. C. Berndt and K. Ono. Ramanujan’s unpublished manuscript on the partition and tau functions. The Andrews Festschrift (Ed. D. Foata and G.-N. Han), Springer-Verlag, Berlin , 39–110, 2001. H. C. Chan. Ramanujan’s cubic continued fraction and a generalization of his “most beautiful identity”. Int. J. Number Theory , 6:673–680, 2010. W. Chu and L. Di Claudio. Classical Partition Identities and Basic Hypergeometric Series. Universit` a degli Studi di Lecce, Edizioni del Grifo , 2004. F. J. Dyson. Some guesses in the theory of partitions. Eureka (Cambridge) , 8:10–15, 1944. F. G. Garvan. New combinatorial interpretations of Ramanujan’s partition congru-ences mod 5, 7, and 11. Trans. Amer. Math. Soc. , 305:47–77, 1988. F. G. Garvan. Biranks for partitions into 2 colors. Ramanujan rediscovered , 87–111, Ramanujan Math. Soc. Lect. Notes Ser. , 14, Ramanujan Math. Soc. , Mysore, 2010. G. Gasper and M. Rahman. Basic Hypergeometric Series (2nd ed.) . Cambridge University Press, Cambridge , 2004. P. Hammond and R. Lewis. Congruences in ordered pairs of partitions. Int. J. Math. Math. Sci. , 47:2509–2512, 2004. G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. The Clarendon Press, Oxford University Press, New York , 1979. B. Kim. A crank analog on a certain kind of partition function arising from the cubic continued fraction. Acta. Arith. , 148:1–19, 2011. B. Kim. Partition statistics for cubic partition pairs. Electron. J. Combin. , 18:#P128, 2011. B. L. S. Lin. Arithmetic properties of overcubic partition pairs. Electron. J. Combin. ,21(3):#P3.35, 2014. the electronic journal of combinatorics 22(4) (2015), #P4.17 16 S. Ramanujan. Some properties of p(n), the number of partitions of n. Proc. Cam-bridge Philos. Soc. , 19:207–210, 1919. S. Ramanujan. Congruence properties of partitions. Math. Zeit. , 9:147–153, 1921. P. C. Toh. Ramanujan type identities and congruences for partition pairs. Discrete Mathematics , 312:1244–1250, 2012. L. Winquist. An elementary proof of p(11 m + 6) ≡ 0( mod 11). J. Combin. Theory ,6:56–59, 1969. W. Zhang and C. Wang. Ramanujan-type congruences for several partition functions. Int. J. Number Theory , 10(3):641–652, 2014. H. Zhao and Z. Zhong. Ramanujan type congruences for a partition function. Elec-tron. J. Combin. , 18:#P58, 2011. R. R. Zhou. Multiranks for partitions into Multi-Colors. Electron. J. Combin. ,19:#P8, 2012.
1604	https://www.geteasysolution.com/10a+b=3ab	10a+b=3ab - solution Toggle navigationGetEasySolution.com Math Solvers Equations solver - equations involving one unknown Quadratic equations solver Percentage Calculator - Step by step Derivative calculator - step by step Graphs of functions Factorization Greatest Common Factor Least Common Multiple System of equations - step by step solver Fractions calculator - step by step Theory in mathematics Roman numerals conversion Tip calculator Numbers as decimals, fractions, percentages More or less than - questions Math Theory Numbers and activities 4th grade help Math Games and Apps Version en español 10a+b=3ab Simple and best practice solution for 10a+b=3ab equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so don`t hesitate to use it as a solution of your homework. If it's not what You are looking for type in the equation solver your own equation and let us solve it. Equation:SOLVE Discover more Mathematics mathematics Math math Derivative calculator Solution for 10a+b=3ab equation: Simplifying 10a + b = 3ab Solving 10a + b = 3ab Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3ab' to each side of the equation. 10a + -3ab + b = 3ab + -3ab Combine like terms: 3ab + -3ab = 0 10a + -3ab + b = 0 Add '-1b' to each side of the equation. 10a + -3ab + b + -1b = 0 + -1b Combine like terms: b + -1b = 0 10a + -3ab + 0 = 0 + -1b 10a + -3ab = 0 + -1b Remove the zero: 10a + -3ab = -1b Combine like terms: -1b + b = 0 10a + -3ab + b = 0 The solution to this equation could not be determined. Discover more math Mathematics Math Derivative calculator mathematics Derivative Calculator See similar equations: \| 2(4p+6)-4(p+1)= \|\| (5x/8)4 \|\| mn=2m-5n \|\| f(x)=2x^2-12x+19 \|\| 3-11x=36 \|\| 5-3(b+2)=2b+7-(b-2) \|\| 6x+1y=-29y \|\| 9t+2=20 \|\| 22+2m=37+6+m \|\| -9x+y=-6 \|\| 5y^2+15y= \|\| 529n=215 \|\| 3(7-2x)=5(4-2x) \|\| 2+14x=9x-8 \|\| -3=13-2x \|\| 4z^2-6z-4=0 \|\| 15/7=5w \|\| 7x+151=5x-89 \|\| 9(6-2v)= \|\| f(x)=ln^2x-4 \|\| a^2+11a-26=0 \|\| 5c-4=55 \|\| -4x-3-8x+1=-5x+9 \|\| 2x+3y=745 \|\| -14=-4u+6(u-3) \|\| 10+5(z-10)=4z+1 \|\| 1/2-23=41 \|\| k^2=27 \|\| 20y+y=26 \|\| 5x-3(x-2)=x+6 \|\| -4(x-3)=-4x+12 \|\| 5x+2(6x-3)=7x-2 \| Equations solver categories Equations solver - equations involving one unknown Quadratic equations solver Percentage Calculator - Step by step Derivative calculator - step by step Graphs of functions Factorization Greatest Common Factor Least Common Multiple System of equations - step by step solver Fractions calculator - step by step Theory in mathematics Roman numerals conversion Tip calculator Numbers as decimals, fractions, percentages More or less than - questions How to solve complicated linear equation Discover more mathematics math Mathematics Math Derivative calculator Derivative Calculator Copyright © 2011-2023 Get Easy Solution 2x-2=8x-3=53x+2=182x+10=126x-2=143x=124x-2=129x-3=612+x=5x+8=13all equations
1605	https://brilliant.org/wiki/sat-counter-examples/	Counter-Examples Sign up with Facebook or Sign up manually Already have an account? Log in here. Tatiana Georgieva, Calvin Lin, and Jimin Khim contributed Some questions ask you to find a counter-example to a given statement. This means that you must find an example which renders the conclusion of the statement false. If you must select a counter-example among multiple choices, often you can use the trial and error approach to determine which of those choices leads to a contradiction. Other questions are more open-ended and require you to think more creatively. Common values that lead to contradictions are -1, 0, 1, and 2, but each problem has unique givens and restrictions, and you must keep those in mind. Also, there isn't a counter-example to a true statement. If you find yourself testing value after value to no avail, you should consider proving the statement true. Finding a counter-example to each answer choice may be the fastest way to solve the problem. Remember, one counter-example to a statement is enough to disprove it. Which of the following numbers is a counter-example to the following claim? If p is an odd prime, then p+2 is also a prime. (A) 3 (B) 5 (C) 7 (D) 9 (E) 11 Correct Answer: C Solution: Because we are looking for a counter-example, we must find an odd prime number p, such that p+2 is NOT a prime. We analyze each of the choices. (A) 3 is a prime, and 3+2=5 is a prime. This is not a counter-example. (B) 5 is a prime, and 5+2=7 is a prime. This is not a counter-example. (C) 7 is a prime, and 7+2=9 is NOT a prime. This is a counter-example. (D) 9 is a not a prime. This is not a counter-example. (E) 11 is a prime, and 11+2=13 is a prime. This is not a counter-example. Hence, the answer is (C). Incorrect Choices: (A), (B), (D), and (E) The solution explains how to eliminate these choices. If you got this problem wrong, you should review Prime Numbers. A B C D E If p is a prime number, which of the following must be true? (\begin{array}{r r l} &\text{I.} & p\ \text{is odd.}\ &\text{II.} & p\ \text{is not divisible by}\ 6.\ &\text{III.} & p\ \text{is not divisible by}\ 7.\ \end{array}) (A) I only (B) II only (C) I and II only (D) I and III only (E) II and III only The correct answer is: B Which of the following numbers is a counter-example to the following claim? If n is an integer, then n2+1 is a prime. (A) 1 (B) 2 (C) 3 (D) 4 (E) 32 Correct Answer: C Solution: Because we are looking for a counter-example, we must find an integer n such that n2+1 is NOT a prime. We analyze each of the choices. (A) 1 is an integer, and 12+1=2, which is a prime. This is not a counter-example. (B) 2 is an integer, and 22+1=5, which is a prime. This is not a counter-example. (C) 3 is an integer, and 32+1=10=2×5, which is NOT a prime. This is a counter-example. (D) 4 is an integer, and 42+1=17, which is a prime. This is not a counter-example. (E) 32 is not an integer. This is not a counter-example. Thus, the answer is (C). Incorrect Choices: (A), (B), (D), and (E) The solution explains how to eliminate these choices. If you got this problem wrong, you should review Prime Numbers. Which of the following cannot be the perimeter of a triangle whose side lengths are all integers? (A) 2 (B) 3 (C) 5 (D) 7 (E) 11 Correct Answer: A Solution: Let's consider each choice. (A) If the side lengths are positive integers, then they are at least 1, and hence the perimeter cannot be 2. Thus, this choice is not possible. (B) The triangle with side lengths 1−1−1 has a perimeter of 3. (C) The triangle with side lengths 2−2−1 has a perimeter of 5. (D) The triangle with side lengths 3−3−1 has a perimeter of 7. (E) The triangle with side lengths 5−5−1 has a perimeter of 11. Hence, the answer is (A). Incorrect Choices: (B), (C), (D), and (E) The solution explains how to eliminate these choices. If you got this problem wrong, you should review Triangles. Cite as: Counter-Examples. Brilliant.org. Retrieved 10:58, September 28, 2025, from
1606	https://books-library.website/files/download-pdf-ebooks.org-1519394376Bw5X0.pdf	ii Part II Clinical and Laboratory Alan B. Carr, DMD, MS Professor Department of Dental Specialties Mayo Clinic Rochester, Minnesota David T. Brown, DDS, MS Chair Department of Restorative Dentistry Indiana University School of Dentistry Indianapolis, Indiana 3251 Riverport Lane St. Louis, Missouri 63043 McCRACKEN’S REMOVABLE PARTIAL PROSTHODONTICS, TWELFTH EDITION ISBN: 978-0-323-06990-8 Copyright © 2011, 2005, 2000, 1995, 1989, 1985, 1981, 1977, 1973, 1969, 1964, 1960 by Mosby, Inc., an affiliate of Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies, and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier. com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. With respect to any drug or pharmaceutical products identified, readers are advised to check the most current information provided (i) on procedures featured or (ii) by the manufacturer of each product to be administered, to verify the recommended dose or formula, the method and duration of administration, and contraindications. It is the responsibility of practitioners, relying on their own experience and knowledge of their patients, to make diagnoses, to determine dosages and the best treatment for each individual patient, and to take all appropriate safety precautions. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Control Number 2010928146 Vice President and Publishing Director: Linda Duncan Executive Editor: John Dolan Senior Developmental Editor: Courtney Sprehe Publishing Services Manager: Catherine Jackson Senior Project Manager: Karen M. Rehwinkel Design Direction: Maggie Reid Printed in Canada Last digit is the print number: 9 8 7 6 5 4 3 2 1 Working together to grow libraries in developing countries www.elsevier.com \| www.bookaid.org \| www.sabre.org v Chapter v Preface With this new edition, we once again recognize the oppor-tunity to provide useful and updated instruction in the field of removable partial prosthodontics as a significant respon-sibility. Removable partial dentures (RPDs) continue to be an option for tooth replacements worldwide. As dental implant use continues to grow in both application and scope as an option for stabilizing replacement teeth, RPDs remain the prosthesis of choice for many patients and clinical situ-ations, making the need for a continued sound educational foundation necessary. Though dentists will always be challenged as to whether they should try new techniques, devices, and materials in practice, appropriate care of the partially edentulous patient continues to require careful attention to basic prosthodontic principles. At issue with RPD care is always how well a new approach augments provision of the required support, stabil-ity, and retention—foundational principles for all tooth replacement prostheses. In the Preface of the last edition, functional stability was highlighted as a critical consideration. As a therapeutic goal this remains to be the case as the evidence that brought the functional stability concern to light has not been countered. What has emerged as a means to efficiently address the issue of instability under function with RPDs is an increased application of implants for stabilizing RPDs in patients who are able to take advantage of them. The authors consider selective use of dental implants to address meaningful con-cerns identified in an individual patient related to support and stability to be laudable goals of treatment using RPDs. This edition of McCracken’s Removable Partial Prosth-odontics has attempted to embed implant considerations into the diagnostic and treatment considerations basic to RPD care. Implant use in RPDs is also addressed in Chapter 25, which reinforces implant application specific to limita-tions found in clinical examination. Where support is found to be a major limitation, or when functional instability is likely to be related to chronic mucosal pain, these are appli-cations in which treatment goals are positively impacted through selective location of dental implants. When accom-plished with a consideration of future patient needs based on an individual risk assessment, maintenance of a func-tional occlusion over time becomes a proactive, patient care planning, and education process, not a reactive response. It is obvious that continued application of implants with RPDs will necessarily lead to attempts at modification of prosthesis designs. The rationale of such modifications must balance the material requirements for durability, patient perception advantages to reduced bulk, and the biological tolerance of all oral tissues involved. This edition continues to use modifications incorporated in recent past editions that were responses to evolving student learning needs. To provide a perspective for under-standing the impact of removable partial denture prosth-odontics, a review of tooth loss and its sequelae, functional restoration with prostheses, and prosthesis outcomes is pro-vided at the outset to define a context for the student. Addi-tionally, new art and color clinical photographs have been provided to completely update material that supplements the text. We continued the strategy begun with the last edition of making a content distinction to facilitate both the first-time learner and the more experienced clinician. The convention used was to separate the advanced level material by shaded boxes. This was accomplished with the full under-standing that such a distinction is not always clear, and that some readers will object to how certain material was classi-fied. Our hope is that it will help clarify the material for the majority of readers. Alan B. Carr David T. Brown vi NEW TO THIS EDITION • Additional information on the use of implants • Chapter 4: New section on impact of implants on movements of partial denture • Chapter 6: New section on implants as a rest • Chapter 7: New section on implants as direct retainers • Chapter 10: New section on implant considerations in design • Chapter 25: A new chapter, Considerations for the use of dental implants with removable partial dentures, presents basic considerations when selecting implants to improve prosthesis performance through increasing functional stability • Revamped art program. The line art for the book has been completely redrawn in full color to better show techniques and anatomic detail. In addition, new full-color photos have been added where appropriate. KEY FEATURES • Content that is considered beyond the basic level is set within a shaded box. • A wide selection of relevant references is presented at the back of the text in Appendix B for quick and easy access. • Various philosophies and techniques are presented throughout, facilitating the selection and incorporation of the applicable techniques on a case-by-case basis. • Chapters presented in three logically-sequenced sections: • General Concepts/Treatment Planning • Clinical and Laboratory • Maintenance About the Book vii Chapter vii We would like to express our gratitude to many who contributed to this text in a variety of ways. Contributions to the text were provided by Dr. Tom Salinas, who assisted with the Implant chapter, and Dr. Vanchit John, who provided input regarding periodontal therapy in mouth preparation. We also would like to acknowledge the following contribu-tors to the clinical images: Drs. Ned van Roekel, James Taylor, Miguel Alfaro, and Carl Andres. We also acknowledge the helpful work of a dedicated group of laboratory techni-cians who contributed to the updates of many laboratory procedure images: Mr. Joe Bly, Mr. Albert Evans, and Mr. Rick Lee. The clerical assistance of Mrs. Melanie Budihas and Mrs. Barbara Jarjoura is also acknowledged and greatly appreciated. Alan B. Carr David T. Brown Acknowledgments viii Dr. Alan B. Carr, Department of Dental Specialties at Mayo Clinic, is a consultant in the Division of Prosthodontics and a professor of dentistry at the Mayo Clinic College of Medi-cine. Dr. Carr received his prosthodontics training at Mayo. Following his training he was an assistant professor at Marquette University and then became a full professor at The Ohio State University, where his clinical duties included Director of Maxillofacial Prosthetics at the James Cancer Hospital. He returned to Mayo in 2000. Dr. Carr is board certified by the American Board of Prosthodontics. He served in the Air Force and has degrees from the University of Southern Mississippi, University of Mississippi, and Mayo Graduate School of Medicine. He also is a member of numerous professional organizations, including the Ameri-can Academy of Maxillofacial Prosthetics, the American College of Prosthodontists, and the American Dental Asso-ciation. He has made dozens of international and national presentations. His clinical practice focuses on combined prosthodontics and reconstruction of patients with disabling oral conditions. His research interests include oral and craniofacial endosseous implants, tobacco cessation, and the impact of oral health on general health, especially for the elderly and patients with chronic illness. Dr. David T. Brown, Indiana University School of Dentistry, is the Chair of the Restorative Dentistry Department and Professor of Prosthodontics. Dr. Brown is a Summa Cum Laude graduate of The Ohio State University College of Den-tistry, and received his prosthodontic training at the Mayo Clinic/Mayo Graduate School of Medicine. He has been a full-time faculty member at Indiana University since 1986, teaching in both the predoctoral curriculum and the post-graduate prosthodontic program. Dr. Brown is board certi-fied by the American Board of Prosthodontics. He has been a reviewer for a number of professional journals, and is a member of several dental and prosthodontic organizations. He currently serves as a member of the Executive Council for the Academy of Prosthodontics. Dr. Brown maintains a part-time practice limited to prosthodontics. About the Authors CHAPTER 1 Partially Edentulous Epidemiology, Physiology, and Terminology Chapter Outline Tooth Loss and Age Consequences of Tooth Loss Anatomic Physiologic Functional Restoration With Prostheses Mastication Food reduction Current Removable Partial Denture Use Need for Removable Partial Dentures This textbook focuses on what the clinician should know about partially edentulous patients to appropriately provide comfortable and useful tooth replacements in the form of removable partial dentures. Removable partial dentures are a component of prosthodontics, which denotes the branch of dentistry pertaining to the restoration and maintenance of oral function, comfort, appearance, and health of the patient by the restoration of natural teeth and/or the replace-ment of missing teeth and craniofacial tissues with artificial substitutes. Current practice in the management of partial tooth loss involves consideration of various types of prostheses (Figure 1-1). Each type of prosthesis requires the use of various remaining teeth, implants, and/or tissues, and consequently demands appropriate application of knowledge and critical thinking to ensure the best possible outcome given patient needs and desires. Although more than one prosthesis may serve the needs of a patient, any prosthesis should be pro-vided as part of overall management that meets the basic objectives of prosthodontic treatment, which include (1) the elimination of oral disease to the greatest extent possible; (2) the preservation of the health and relationships of the teeth and the health of oral and paraoral structures, which will enhance the removable partial denture design; and (3) the restoration of oral functions that are comfortable, are esthet-ically pleasing, and do not interfere with the patient’s speech. It is critically important to emphasize that the preservation of health requires proper maintenance of removable partial dentures. To provide a perspective for understanding the impact of removable partial denture prosthodontics, a review of tooth loss and its sequelae, functional restoration with prostheses, and prosthesis use and outcomes is in order. Familiarity with accepted prosthodontic terminology related to removable partial dentures is necessary. Figures 1-2 and 1-3 provide prosthesis terms related to mandibular and maxillary frameworks, and Appendix A provides a 2 3 Chapter 1 Partially Edentulous Epidemiology, Physiology, and Terminology If one accepts that tooth loss and age are linked, how will this affect current and future dental practice? Replacement of missing teeth is a common patient need, and patients will demand it well into their elderly years. Current population estimates show that 13% of the U.S. population is 65 years of age or older. By the year 2030, this percentage is expected to double, with a significant increase also expected world-wide. These individuals are expected to be in better health, and health care strategies for this group should focus on maintenance of active and productive lives. Oral health care is expected to be a highly sought after and significant com-ponent of overall health care. Tooth loss patterns associated with age are also evolving. The proportion of edentulous adults has been reported to be decreasing, although this varies widely by state. However, it has been reported that the absolute number of edentulous patients who need care is actually increasing. More pertinent to this text, estimates suggest that the need for restoration of partially edentulous conditions will also be increasing. An explanation for this is presented in an argument that 62% of Americans of the “baby boomer” generation and younger review of selected prosthodontic terms. Additional termi-nology can be reviewed in The Glossary of Prosthodontic Terms1 and a glossary of accepted terms in all disciplines of dentistry, such as Mosby’s Dental Dictionary, second edition.2 Tooth Loss and Age It should come as no surprise that tooth loss and age are linked. A specific tooth loss relationship has been docu-mented with increasing age because some teeth are retained longer than others. It has been suggested that in general an interarch difference in tooth loss occurs, with maxillary teeth demonstrating loss before mandibular teeth. An intraarch difference has also been suggested, with posterior teeth lost before anterior teeth. These observations are likely related to respective caries susceptibilities, which have been reported (Table 1-1). Frequently the last remaining teeth in the mouth are the mandibular anterior teeth, especially the mandibular canines, and it is a common finding to see an edentulous maxilla opposing mandibular anterior teeth. A C D B Figure 1-1 A, Fixed partial dentures that restore missing anterior (#10) and posterior (#5, #13) teeth. Teeth bordering edentulous spaces are used as abutments. B, Clasp-type removable partial denture restoring missing posterior teeth. Teeth adjacent to edentulous spaces serve as abutments. C, Tooth-supported removable partial denture restoring missing anterior and posterior teeth. Teeth bound-ing edentulous spaces provide support, retention, and stability for restoration. D, Mandibular bilateral distal extension removable partial denture restoring missing premolars and molars. Support, retention, and stability are shared by abutment teeth and residual ridges. 4 Part I General Concepts/Treatment Planning D B A F B C B D B C C B D E A B C D E C B D Figure 1-2 Mandibular framework designed for a partially edentulous arch with a Kennedy Classification II, modification 1 (see Chapter 3). Various component parts of the framework are labeled for identification. Subsequent chapters will describe their function, fabrication, and use. A, Major connector. B, Rests. C, Direct retainer. D, Minor connector. E, Guide plane. F, Indirect retainer. D F A F B D D B E C C B B A B C B E D Figure 1-3 Maxillary framework designed for a partially edentulous arch with a Kennedy Classification I (see Chapter 3). As in Figure 1-2, component parts are labeled for identification. A, Major connector. B, Rests. C, Direct retainer. D, Minor connector. E, Guide plane. F, Indirect retainer. 5 Chapter 1 Partially Edentulous Epidemiology, Physiology, and Terminology Physiologic What are we replacing when we consider managing missing teeth? We are replacing both the physical anatomic tools for mastication and the oral capacity for neuromuscular functions to manipulate food. Chewing studies have shown that the oral sensory feedback that guides movement of the mandible in chewing comes from a variety of sources. The most sensitive input, which means the input that provides the most refined and precisely controlled movement, comes from periodontal mechanoreceptors (PMRs), with addi-tional input coming from the gingiva, mucosa, periosteum/ bone, and temporomandibular joint (TMJ) complex. Chewing as a learned behavior has a basic pattern of movement that is generated from within the central nervous system. In typical function, this patterned movement is moderated on the basis of food and task needs by oral sensory input from various sources. With loss of the finely tuned contribution from tooth PMRs, the resulting periph-eral receptor influence is less precise in muscular guidance, producing more variable masticatory function, and the type of prosthesis selected to replace missing teeth may poten-tially contribute to functional impediments. The esthetic impact of tooth loss can be highly significant and may be more of a concern to a patient than loss of func-tion. It is generally perceived that in today’s society, loss of visible teeth, especially in the anterior region of the mouth, carries with it a significant social stigma. With loss of teeth and diminishing residual ridge, facial features can change as the result of altered lip support and/or reduced facial height caused by a reduction in occlusal vertical dimension. Restor-ing facial esthetics in a manner that maintains an appropri-ate appearance can be a challenge and is a major factor in restoration and maintenance decisions made for various prosthetic treatments. Functional Restoration with Prostheses Individuals with a full complement of teeth report some variation in their levels of masticatory function. The loss of teeth may lead patients to seek care for functional reasons if they notice diminished function to a level that is unaccept-able to them. The level at which a patient finds function to be unacceptable varies among individuals. This variability increases with accelerating tooth loss. These variables may be confusing to clinicians, who may perceive that they have provided prostheses of equal quality to different patients with the same tooth loss patterns, and yet have received dif-ferent patient reports of success. An understanding of these variations among individuals with a full complement of teeth and those with prostheses can help clinicians formulate realistic treatment goals that can be communicated to the patient. A review of oral func-tion, especially mastication, may help interested clinicians better understand issues related to the impact of removable partial denture function. have benefited from fluoridated water. The result of such exposure has been a decrease in caries-associated tooth loss. In addition, current estimates suggest that patients are keeping more teeth longer, demonstrated by the fact that 71.5% of 65- to 74-year-old individuals are partially eden-tulous (mean number of retained teeth = 18.9). It has been suggested that partially edentulous conditions are more common in the maxillary arch, and that the most commonly missing teeth are first and second molars. Consequences of Tooth Loss Anatomic With the loss of teeth, the residual ridge no longer benefits from the functional stimulus it once experienced. Because of this, a loss of ridge volume—both height and width—can be expected. This finding is not predictable for all individuals with tooth loss, as the change in anatomy has been reported to be variable across patient groups. In general, bone loss is greater in the mandible than in the maxilla and more pro-nounced posteriorly than anteriorly, and it produces a broader mandibular arch while constricting the maxillary arch. These anatomic changes can present challenges to fab-rication of prostheses, including implant-supported pros-theses and removable partial dentures. Associated with this loss of bone is an accompanying alteration in the oral mucosa. The attached gingiva of the alveolar bone can be replaced with less keratinized oral mucosa, which is more readily traumatized. Table 1-1 Caries Risk Assessment High Risk Lower 6 and 7 Mandibular 1st and 2nd molars Upper 6 and 7 Maxillary 1st and 2nd molars Lower 5 Upper 1, 2, 4, 5 Mandibular 2nd premolar Maxillary central, Lateral incisors Maxillary 1st and 2nd premolars Low Risk Upper 3 and lower 4 Maxillary canine, Mandibular 1st premolar Lower 1, 2, 3 Mandibular central, Lateral incisors, Canines If tooth loss parallels caries activity, caries risk may be a proxy for tooth loss. Data from Klein H, Palmer CE: Studies on Dental Caries: XII. Comparison of the caries susceptibility of the various morphological types of permanent teeth. J Dent Res 20:203-216, 1941. 6 Part I General Concepts/Treatment Planning Mastication Although functionally considered as a separate act, mas-tication as part of the feeding continuum precedes swal-lowing and is not an end in itself. The interaction of the two distinct but coordinated aspects of feeding suggests that some judgment of mastication termination or com-pleteness precedes the initiation of swallowing. Although the mastication-swallowing sequence is obvious, the interaction of the two functions is not widely understood and may be important to prosthesis use when removable partial dentures are considered. Mastication involves two discrete but well- synchronized activities: subdivision of food by applied force, and selective manipulation by the tongue and cheeks to sort out coarse particles and bring them to the occlusal surfaces of teeth for further breakdown. The initial subdivision or comminution phase involves the processes of selection, which refers to the chance that a particle is placed between the teeth in position to be broken, and breakage, which is the degree of fragmenta-tion of a particle once selected. The size, shape, and texture of food particles provide the sensory input that influences the configuration and area of each chewing stroke. Larger particles are selectively reduced in size more rapidly than fine particles in efficient mastication. The process of mastication is therefore greatly influenced by factors that affect physical ability to reduce food and to monitor the reduction process by neurosensory means. Food Reduction Teeth or prostheses serve the role of reducing food to a point ready for swallowing. An index of food reduction is described as masticatory efficiency, or the ability to reduce food to a certain size in a given time frame. A strong correlation has been shown between masticatory efficiency and the number of occluding teeth in dentate individuals, which would suggest variability of particle selection related to contacting teeth. Performance mea-sures reveal a great deal of functional variability in patients with similar numbers of contacting teeth, and even greater variability is seen within populations with greater loss of teeth (increasing degrees of edentulousness). Because occlusal contact area is highly correlated with masticatory performance, the loss of molar teeth would be expected to have a greater impact on measures of performance in that the molar has a larger occlusal contact area. This effect has been demonstrated in indi-viduals with missing molars who reveal a greater number of chewing strokes required and a greater mean particle size before swallowing. The point at which an individual is prepared to swallow the food bolus is another measure of performance and is described as the swallowing thresh-old. Superior masticatory ability that is highly correlated with occlusal contact area also achieves greater food reduction at the swallowing threshold. Conversely, a diminished ability to chew is reflected in larger particles at the swallowing threshold. These objective measures, which show a benefit to molar contact in dentate individuals, are in conflict with some subjective measures from patients who express no perceived functional problems associated with having only premolar occlusion. This shortened dental arch concept has highlighted that patient perceptions of func-tional compromise, as well as benefit, should be consid-ered when it is decided whether to replace missing molars. When the loss of posterior teeth results in an unstable tooth position, such as distal or labial migration, tooth replacement should be carefully considered; this is a sepa-rate situation from the shortened dental arch concept. It has been reported that prosthetic replacement of teeth provides function that is often less than that seen in the complete, natural dentition state. Functional mea-sures are closest to the natural state when replacements are fixed partial dentures rigidly supported by teeth or implants, intermediate in function when replacements are removable and supported by teeth, lower in function when replacements are removable and supported by teeth and edentulous ridges, and lowest in function when replacements are removable and supported by edentu-lous ridges alone. Objective and subjective measures of a patient’s oral function often are not in agreement. It has been shown that subjective measures of masticatory ability are often overrated compared with objective functional tests, and that for complete denture wearers, the subjective criteria may be more appropriate in monitoring perceived out-comes. Some literature reports that removable partial dentures can be described by patients as adding very little benefit over no prostheses. However, these findings may be related to a number of factors, including lack of main-tenance of occluding tooth relationships, limitations of this form of dental prosthesis for patient populations that may be unreliable in maintaining follow-up visits, and intrinsic variation in patient response to prostheses. Food reduction is also influenced by the ability to monitor the process required to determine the point at which swallowing is initiated. As was mentioned earlier, the size, shape, and texture of food are monitored during mastication to allow modification in mandibular move-ment for efficient food reduction. This has been demon-strated in dentate individuals given food particles of varying size and concentration suspended in yogurt, who revealed that increased concentrations and particle size required more time to prepare for swallowing (i.e., greater swallowing threshold). These findings suggest that the oral mucosa has a critical role in detecting characteristics necessary for efficient mastication. The influence of the removable partial denture on the ability of the mucosa to perform this role in mastication is not known. 7 Chapter 1 Partially Edentulous Epidemiology, Physiology, and Terminology fied as common flaws associated with poor removable partial dentures. These characteristics are directly related to the functional stability of prostheses. Need for Removable Partial Dentures What does all this information mean to us today? It means a number of things that are important to consider. The need for partially edentulous management will be increasing. Patient use of removable partial dentures has been high in the past and is expected to continue in the future. Some patients who are given the choice between a prosthesis entirely supported by implants or a removable partial denture are not able to pursue implant care. This contributes to higher use of removable partial dentures. Finally, these findings suggest that we should strive to understand how to maximize the opportunity for providing and maintaining stable prostheses because this is the most frequently deficient aspect of removable partial denture service. Consequently, throughout this text the basic prin-ciples of diagnosis, mouth preparation, prosthesis design, fabrication, placement, and maintenance will be reinforced to improve the reader’s understanding of care of removable partial denture prostheses. References 1. The glossary of prosthodontic terms, ed 8. From The Journal of Prosthetic Dentistry 94:10-81, 2005. 2. Mosby’s dental dictionary, ed 2, St Louis, 2008, Mosby/Elsevier. Current Removable Partial Denture Use Given an understanding of the relationship between tooth loss and age, the consequences of tooth loss, and our ability to restore function with removable partial prostheses, what do we know about current prosthesis use for these condi-tions, and what are some common clinical outcomes? A recent study estimated 21.4% prosthesis use among indi-viduals aged 15 to 74. In the 55- to 64-year-old group, 22.2% were found to wear a removable partial denture. This age group has the highest use of removable partial dentures among those reviewed. It has been suggested that the use of removable partial dentures among individuals aged 55 years or older is even greater. Analysis of this study provides some useful information for consideration. Partially edentulous individuals not wearing a prosthesis were 6 times more likely to have missing mandibular teeth (19.4%) than missing maxillary teeth (2.2%). This might suggest greater difficulty in the use of a mandibular prosthesis. The distribution of prostheses used in this large patient group is shown in Table 1-2. The pros-theses in this large study were evaluated on the basis of five technical quality characteristics: integrity, excessive wear of posterior denture teeth, the presence of temporary reline material or tissue conditioner or adhesive, stability, and retention. As seen in Table 1-3, lack of stability was the most common characteristic noted. In the maxilla, lack of stability was 7 times more prevalent than lack of retention. In the mandible, lack of stability was 1.8 times more prevalent than lack of retention. In another study, rest form, denture base extension, stress distribution, and framework fit were identi-Table 1-2 Distribution of Prostheses Removable partial dentures RPD/RPD 9.0% RPD/–15.3%, –/RPD 4.5% Complete dentures CU/CL 3.8% CU/–20.7% Combination CU/RPD 11.5% RPD/CL 0.3% CL, Complete lower denture; CU, complete upper denture; RPD, removable partial denture. Natural teeth denoted with dash (–). Table 1-3 Technical Quality Concerns for Removable Partial Dentures Lack Stability Lack Integrity Lack Retention Reline/Adhesive Excessive Wear Maxillary RPD 43.9% 24.3% 6.2% 3.9% 21.6% Mandibular RPD 38.2% 13.2% 21.2% 21.6% 7.1% RPD, Removable partial denture. 8 CHAPTER 2 Considerations for Managing Partial Tooth Loss Tooth Replacements From the Patient Perspective Chapter Outline Points of View Tooth replacements from the patient’s perspective Shared decision making Tooth-Supported Prostheses Tooth and Tissue–Supported Prostheses Six Phases of Partial Denture Service Education of patient Diagnosis, treatment planning, design, treatment sequencing, and mouth preparation Support for distal extension denture bases Establishment and verification of occlusal relations and tooth arrangements Initial placement procedures Periodic recall Reasons for Failure of Clasp-Retained Partial Dentures Points of View Do we treat or do we manage tooth loss? Is the distinction important as we attempt to help our patients decide which type of prosthesis to choose? For patients who want to know what to expect now and in the future, it is helpful to make this distinction, as it helps them realize that the decision has implications for future needs that may be different between prostheses. Tooth Replacements From the Patient’s Perspective Tooth loss is a permanent condition in that the natural order has been disrupted, and in this sense it is much like a chronic medical condition. Like hypertension and diabetes, two medical conditions that are not reversible and that require medical management to monitor care to ensure appropriate response over time, tooth replacement prostheses must be managed to ensure appropriate response over time. The term management suggests a focus on meeting needs that may change over time. These needs may be expected or unexpected. Expected outcomes are those that accompany the common clinical course for a type of prosthesis that is related to the tooth-tissue response. This biological toll response is heavily influenced by the type of prosthesis chosen. In addition, various needs due to prosthesis degra-dation and related to expected time-to-retreatment concerns of life expectancy are seen. Unexpected needs are those that might involve factors related to our control of manipula-tions (such as tissue damage or abuse, material design flaws, or prosthesis design) or to those out of our control (such as parafunction or accidental trauma). With this in mind, it is helpful to consider how we approach educating our patients about management of 9 Chapter 2 Considerations for Managing Partial Tooth Loss decision. Ultimately, it is our role to help patients consider important differences between different prosthesis types. What then defines important differences? Multiple out-comes combine to describe the overall impact of prosthetic care for all patients. These include technical outcomes, physical outcomes, esthetic outcomes, various maintenance needs, initial and future costs, and even physiologic out-comes that suggest to what extent prostheses “feel” like teeth. When tooth replacement prostheses are considered from a patient’s perspective, it can be seen that the desire is to replace teeth that serve functional and social roles in every-day life. In considering how well various types of prostheses may meet patients’ specific needs, it is helpful to note what features of the original dentition—the gold standard, in this instance—we strive to duplicate in the replacement. Although it is common to find that existing oral conditions do not easily allow complete restoration to the state of a fully dentate patient, considering the respective strengths and weaknesses of the prosthodontic options (compared with this “gold standard”) helps in identification of realistic expectations. In this text, the focus will be on a type of replacement prosthesis for patients with some, but not all, missing teeth. The replacement prosthesis ideally should provide function and a level of comfort as equivalent as possible to normal dentition. In achieving this, stability while chewing is a primary focus of attention, and we should strive to deter-mine what is required to ensure it. If the prosthesis will be visible during casual speaking, smiling, and/or laughing, it is obvious that the replacement should look as natural as the surrounding environment. In summary, tooth replacement prostheses should provide a combination of several features of natural teeth: socially acceptable in appearance, comfort-able and stable in function, and maintainable throughout their serviceable lifetime at a reasonable cost. Tooth-Supported Prostheses For partially edentulous patients, available prosthetic options include natural tooth–supported fixed partial dentures, removable partial dentures, and implant-supported fixed partial dentures. How well these options restore and main-tain the features of natural teeth mentioned previously depends to a large extent on the numbers and locations of the missing teeth. The major categories of partial tooth loss (see Chapter 3) are those (1) with teeth both anterior and posterior to the space (a tooth-supported space), and (2) with teeth either anterior or posterior to the space (a tooth- and tissue-supported space). All prosthetic options listed are available for the tooth-bound space (although they are not necessarily indicated for every clinical situation), but only removable partial dentures and implant-supported prosthe-ses are available for the distal extension (recognizing limited application of cantilevers). Removable partial dentures can be designed in various ways to allow use of abutment teeth and supporting tissue missing teeth. Most often, a typical sequence is used to discuss tooth replacement options with patients: dental implant–supported prostheses, fixed prostheses, and, finally, removable partial dentures. When removable partial den-tures are suggested, they are seldom described in the detail in which fixed or implant prostheses are described, as they generally are considered less like teeth and not as desirable a replacement. The desirability of a prosthesis is important to consider, and because removable partial dentures (RPDs) are less like teeth than other replacements, it is important to recognize what this suggests from the patient’s perspective. Patients’ experiences have involved natural teeth, and their expectations of replacements would best be described within this context. The order with which we provide replacement prosthesis options for consideration is likely developed on the basis of numerous factors, including the following: we may believe we know what’s best for patients, our practice style may not include removable options, we may not have had good experience with removable prosthe-ses and this lessens our confidence in their use, or RPDs do not match our practice resources. Although these are important factors, the reason to include RPDs in the discussion is related to identifying whether such a prosthesis is viable, and, if so, whether it is the best option for the patient. We discover this only by interacting with our patients regarding their expectations and understanding their capacity to benefit from options of management that have trade-offs unique to each type of prosthesis. Shared Decision Making When patients are given information regarding their oral health status, which includes disease and functional deficits, as well as the means to address both, what do they need to hear? To achieve a state of oral health, they need to recognize behavioral issues related to plaque control, so that once active disease is controlled, they have an understanding that best ensures future health. For tooth replacement decisions, complex trade-offs in care choice are often required. The “shared decision making” approach addresses the need to fully inform patients about risks and benefits of care, and ensures that the patient’s values and preferences play a prominent role in the ultimate decision. It is recognized that patients vary in their desire to par-ticipate in such decisions, thus our active inquiry is required to engage them in discussion. This becomes especially important when elective care, which involves potentially high-burden, costly options with highly variable mainte-nance requirements, is considered. When patients wish to participate, it is our responsibility to provide them with specific and sufficient information that they can use to decide between treatment options. Specific information ideally comes from our own practice outcomes, in that such information provides effectiveness information and is provider specific. Sufficient information describes exactly what aspects of care are important to the overall 10 Part I General Concepts/Treatment Planning tissue support in the tooth-tissue removable partial denture predictably changes over time, to adequately manage partial tooth loss with a removable prosthesis, we must carefully monitor our patients to maintain support and ensure maximum prosthetic function. The clasp-retained partial denture, with extracoronal direct retainers, is used significantly more frequently than the precision attachment partial denture (Figure 2-1). It is capable of providing physiologically sound treatment for most patients who need partial denture restorations. Although the clasp-retained partial denture has disadvan-tages, its advantages of lower cost and shorter fabrication time ensure that it will continue to be widely used. Following are some possible disadvantages of a clasp-retained partial denture: 1. Strain on the abutment teeth often is caused by improper tooth preparation or clasp design, and/or loss of tissue support under distal extension partial denture bases. 2. Clasps can be unesthetic, particularly when they are placed on visible tooth surfaces without consideration of esthetic impact. 3. Caries may develop beneath clasp components, especially if the patient fails to keep the prosthesis and the abut-ments clean. Despite these disadvantages, the use of removable pros-theses may be preferred whenever tooth-bounded edentu-lous spaces are too large to be restored safely with fixed prostheses, or when cross-arch stabilization and wider dis-tribution of forces to supporting teeth and tissues are desir-able. Fixed partial dentures, however, should always be considered and used when indicated. The removable partial denture retained by internal attachments eliminates some of the disadvantages of clasps, but it has other disadvantages, one of which is higher cost, which makes it more difficult to obtain for a large percentage of patients who need partial dentures. However, when align-ment of the abutment teeth is favorable and periodontal health and bone support are adequate, when the clinical crown is of sufficient length and the pulp morphology can accommodate the required tooth preparation, and when the economic status of the patient permits, an internal attach-ment prosthesis provides an unquestionable advantage for esthetic reasons. When this situation exists, carefully weigh-ing of tooth attachment versus implant attachment options is required (see RPDs and Implants, Chapter 25). In most instances, if the extracoronal clasp-retained partial denture is designed properly, the only advantage of the internal attachment denture is esthetic, because abut-ment protection and stabilizing components should be used with both internal and external retainers. However, eco-nomics permitting, esthetics alone may justify the use of internal attachment retainers, especially when a crown is indicated for non-RPD reasons. Injudicious use of internal attachments can lead to exces-sive torsional load on the abutments supporting distal exten-sion removable partial dentures, especially in the mandible. for stability, support, and retention of the prosthesis. In terms of tooth-bound spaces, the removable partial denture is like a fixed partial denture because natural teeth alone provide direct resistance to functional forces. Because natural teeth support the prosthesis, it should not move under these functional forces. In this condition, the interface between, or relationship of, the removable partial denture framework and the abutment teeth should be designed to take advantage of tooth support—similar to the relationship between a fixed partial denture retainer and a prepared tooth. This means that it should provide positive vertical support (rest preparations) and a restrictive angle of dislodg-ment (opposing guide planes). Put another way, when the removable partial denture is selected for a tooth-bound situ-ation, stability under functional load should be as well con-trolled as a fixed partial denture when appropriate tooth preparation is provided. Because removable partial denture clasps do not completely encircle the tooth, as a fixed partial denture retainer does, they must be designed to engage more than half the circumference to allow the prosthesis to main-tain position under the influence of horizontal chewing loads. It should be obvious that careful planning and execu-tion of the necessary natural tooth contour modifications are required to ensure movement control and functional stabil-ity for removable partial dentures supported by teeth. Simi-larities between the prosthesis-tooth interface for fixed partial dentures and for removable partial dentures are high-lighted to emphasize the modification principles required to ensure stability for movement control in removable partial dentures. Over time, natural tooth support can be main-tained as with the fixed partial denture. Chapter 14 helps to explain how this is accomplished when natural tooth modi-fications or surveyed crowns are produced. Tooth and Tissue–Supported Prostheses For removable partial dentures that do not have the benefit of natural tooth support at each end of the replacement teeth (extension base removable partial dentures), it is necessary that the residual ridge be used to assist in the functional stability of the prosthesis. When a removable partial denture is selected for a tooth-tissue–supported arch, the prosthesis must be designed to allow functional movement of the base to the extent expected by the residual ridge mucosa. This mucosa movement is variable, but for healthy residual ridge (masticatory) mucosa, movement from 1 to 3 mm can be expected. Consequently, unlike with the tooth-bound space, tooth modification for the tooth-tissue–supported prosthe-sis must be designed with the dual goal of framework tooth contact to allow appropriate functional stability from the tooth, but with allowance for the anticipated vertical and/or horizontal movement of the extension base. This introduces the concept of anticipated movement with a prosthesis and the requirement that we have a role in designing prostheses to appropriately control movement. Additionally, because 11 Chapter 2 Considerations for Managing Partial Tooth Loss removable partial denture can move under function (because it is not cemented to teeth like a fixed partial denture). We should take steps to prescribe the necessary prosthetic fit to teeth (and tissue) to control movement as much as possible. This entails providing appropriate natural tooth mouth preparations, ensuring an accurate frame fit at tooth and tissue, providing a simultaneous contacting relationship between natural and prosthetic opposing teeth, and provid-ing and maintaining optimum support from the soft tissue and teeth. As we will review in Chapter 4, control of the anticipated movement of your prosthesis is addressed by assigning the appropriate component part of the prosthesis to contact/ engage the tooth or tissue in a manner that allows movement and removal of the prosthesis. Are there movements that we should control that are more important than others? Although we recognize the need to resist movement away The use of hinges or other types of stress breakers is discour-aged in these situations. It is not that they are ineffective, but that they are frequently misused. As an example, in the man-dibular arch, a stress-broken distal extension partial denture does not provide for cross-arch stabilization and frequently subjects the edentulous ridge to excessive trauma from hori-zontal and torquing forces. Therefore a rigid design is pre-ferred, and some type of extracoronal clasp retainer is still the most logical and the most frequently used. It seems likely that its use will continue until a more widely acceptable retainer is devised. As was reviewed in Chapter 1, the most commonly cited problem associated with removable partial dentures is insta-bility. Healthy natural teeth should not move when used; therefore we should strive to provide and maintain as stable a prosthesis as possible given the means available. How do we ensure functional stability? By understanding that a A B C D Figure 2-1 A, Maxillary and mandibular clasp-retained removable partial dentures. All clasps are extracoronal retainers (clasps) on abutments. B, Prostheses from (A) shown intraorally in occlusion. C, Maxillary prosthesis using intracoronal retainers and full palatal coverage. The male portions of the attachments are shown at the mesial position of the artificial teeth and will fit into intracoronal rests. D, Internal attachment prosthesis in the patient’s mouth. Note the precise fit of male and female portions of the attachments. 12 Part I General Concepts/Treatment Planning educational procedure is especially important when the treatment plan and prognosis are discussed with the patient. Limitations imposed on the success of treatment through failure of the patient to accept responsibility must be explained before definitive treatment is undertaken. A patient usually will not retain all the information presented in the oral educational instructions. For this reason, patients should be given written suggestions to reinforce the oral presentations. Diagnosis, Treatment Planning, Design, Treatment Sequencing, and Mouth Preparation Treatment planning and design begin with thorough medical and dental histories. The complete oral examination must include both clinical and radiographic interpretation of (1) caries, (2) the condition of existing restorations, (3) peri-odontal conditions, (4) responses of teeth (especially abut-ment teeth) and residual ridges to previous stress, and (5) the vitality of remaining teeth. In addition, evaluation of the occlusal plane, the arch form, and the occlusal relations of the remaining teeth must be meticulously accomplished by clinical visual evaluation and diagnostic mounting. After a complete diagnostic examination has been accomplished and a removable partial denture has been selected as the treatment of choice, a treatment plan is sequenced and a partial denture design is developed in accordance with avail-able support. The dental cast surveyor (Figure 2-2) is an absolute neces-sity in any dental office in which patients are being treated with removable partial dentures. The surveyor is instrumen-tal in diagnosing and guiding the appropriate tooth prepara-tion and verifying that the mouth preparation has been performed correctly. There is no more reason to justify its omission from a dentist’s armamentarium than there is to ignore the need for roentgenographic equipment, the mouth mirror and explorer, or the periodontal probe used for diag-nostic purposes. Several moderately priced surveyors that adequately accomplish the diagnostic procedures necessary for design-ing the partial denture are available. In many dental offices, this most important phase of dental diagnosis is delegated to the commercial dental laboratory because this invaluable diagnostic tool is absent, or because the dentist feels inexpe-rienced or is apathetic. This situation places the technician in the role of diagnostician. Any clinical treatment based on the diagnosis of the technician remains the responsibility of the dentist. This makes no more sense than relying on the technician to interpret radiographs and to render a diagnosis. After treatment planning, a predetermined sequence of mouth preparations can be performed with a definite goal in mind. It is mandatory that the treatment plan be reviewed to ensure that the mouth preparation necessary to accom-modate the removable partial denture design has been prop-erly sequenced. Mouth preparations, in the appropriate sequence, should be oriented toward the goal of providing from the teeth and tissue to keep prostheses from falling out of mouths, the most damaging forces are those resulting from functional closure during chewing (and in some patients, parafunction). Consequently, control of combined vertical (tissue-ward) and horizontal movement is most critical and places a premium on tooth modifications (rest and stabilizing component preparations) and verification of adequate fit of the frame to the teeth. Six Phases of Partial Denture Service Partial denture service may be logically divided into six phases. The first phase is related to patient education. The second phase includes diagnosis, treatment planning, design of the partial denture framework, treatment sequencing, and execution of mouth preparations. The third phase is the provision of adequate support for the distal extension denture base. The fourth phase is establishment and verifica-tion of harmonious occlusal relationships and tooth rela-tionships with opposing and remaining natural teeth. The fifth phase involves initial placement procedures, including adjustments to the contours and bearing surfaces of denture bases, adjustments to ensure occlusal harmony, and a review of instructions given the patient to optimally maintain oral structures and provided restorations. The sixth and final phase of partial denture service consists of follow-up services by the dentist through recall appointments for periodic evaluation of the responses of oral tissue to restorations and of the acceptance of restorations by the patient. The follow-ing is an overview of these phases. The context of each phase is discussed in greater detail in the respective chapters of this book. Education of Patient The term patient education is described in Mosby’s Dental Dictionary as “the process of informing a patient about a health matter to secure informed consent, patient coopera-tion, and a high level of patient compliance.” The dentist and the patient share responsibility for the ultimate success of a removable partial denture. It is folly to assume that a patient will have an understanding of the benefits of a removable partial denture unless he or she is so informed. It is also unlikely that the patient will have the knowledge to avoid misuse of the restoration or will be able to provide the required oral care and maintenance proce-dures to ensure the success of the partial denture unless he or she is adequately advised. The finest biologically oriented removable partial denture is often doomed to limited success if the patient fails to exercise proper oral hygiene habits or ignores recall appoint-ments. Preservation of the oral structures, one of the primary objectives of prosthodontic treatment, will be compromised without the patient’s cooperation in oral hygiene and regular maintenance visits. Patient education should begin at the initial contact with the patient and should continue throughout treatment. This 13 Chapter 2 Considerations for Managing Partial Tooth Loss adequate support, stability, retention, and a harmonious occlusion for the partial denture. Placing a crown or restor-ing a tooth out of sequence may result in the need to restore teeth that were not planned for restoration, or it may neces-sitate remaking a restoration or even seriously jeopardizing the success of the removable partial denture. Through the aid of diagnostic casts on which the tentative design of the partial denture has been outlined and the mouth prepara-tions have been indicated in colored pencil, occlusal adjust-ments, abutment restorations, and abutment modifications can be accomplished. Support for Distal Extension Denture Bases The third of the six phases in the treatment of a patient with a partial denture involves obtaining adequate support for distal extension bases. Therefore it does not apply to tooth-supported removable partial dentures. With the latter, support comes entirely from the abutment teeth through the use of rests. For the distal extension partial denture, however, a base made to fit the anatomic ridge form does not provide ade-quate support under occlusal loading (Figure 2-3). Neither does it provide for maximum border extension nor accurate border detail. Therefore some type of corrected impression is necessary. This may be accomplished by several means, any of which satisfy the requirements for support of any distal extension partial denture base. Foremost is the requirement that certain soft tissue in the primary supporting area should be recorded or related under some loading, so that the base may be made to fit the form of the ridge when under function. This provides support and ensures maintenance of that support for the longest possible time. This requirement makes the distal extension partial denture unique in that support from the tissue underlying the distal extension base must be made as equal to and com-patible with the tooth support as possible. A complete denture is entirely tissue supported, and the entire denture can move toward the tissue under function. In contrast, any movement of a partial denture base is inevi-tably a rotational movement that, if toward the tissue, may result in undesirable torquing forces to the abutment teeth and loss of planned occlusal contacts. Therefore every effort must be made to provide the best possible support for the distal extension base to minimize these forces. Usually no single impression technique can adequately record the anatomic form of the teeth and adjacent struc-tures and at the same time record the supporting form of the mandibular edentulous ridge. A method should be used that can record these tissues in their supporting form or in a supporting relationship to the rest of the denture (see Figure 2-3). This may be accomplished by one of several methods that will be discussed in Chapter 16. Establishment and Verification of Occlusal Relations and Tooth Arrangements Whether the partial denture is tooth supported or has one or more distal extension bases, the recording and verifica-tion of occlusal relationships and tooth arrangement are important steps in the construction of a partial denture. For the tooth-supported partial denture, ridge form is of less significance than it is for the tooth- and tissue-supported prosthesis because the ridge is not called on to support the prosthesis. For the distal extension base, however, jaw rela-tion records should be made only after the best possible support is obtained for the denture base. This necessitates the making of a base or bases that will provide the same support as the finished denture. Therefore the final jaw rela-tions should not be recorded until after the denture frame-work has been returned to the dentist, the fit of the framework to the abutment teeth and opposing occlusion has been veri-fied and corrected, and a corrected impression has been made. Then a new resin base or a corrected base must be used to record jaw relations. Occlusal records for a removable partial denture may be made by the various methods described in Chapter 17. Initial Placement Procedures The fifth phase of treatment occurs when the patient is given possession of the removable prosthesis. Inevitably it seems that minute changes in the planned occlusal relation-ships occur during processing of the dentures. Not only must occlusal harmony be ensured before the patient is given possession of the dentures, but the processed bases Figure 2-2 Dental cast surveyor facilitates the design of a removable partial denture. It is an instrument by which parallel-ism or lack of parallelism of abutment teeth and other oral struc-tures, on a stone cast, can be determined (magnified view shows parallel guide plane surface). Use of the surveyor is discussed in later chapters. 14 Part I General Concepts/Treatment Planning A B C Figure 2-3 A, Occlusal view of a cast from a preliminary impression, which produced an anatomic ridge form (left), and an altered cast of the same ridge showing a functional or supportive form (right). The altered cast impression selectively placed pressure on the buccal shelf region, which is the primary stress-bearing area of the mandibular posterior residual ridge. B, Buccal view of anatomic ridge form. C, Buccal view of functional or supportive ridge form. Note that the supportive form of the ridge clearly delineates the extent of coverage available for a denture base and is most different from the anatomic form when the mucosa is easily displaced. must be reasonably perfected to fit the basal seats. It must be ascertained that the patient understands the sugges-tions and recommendations given by the dentist for care of the dentures and oral structures and understands about expectations (based on the “Shared Decision Making” discussion) in the adjustment phases and the use of restora-tions. These facets of treatment are discussed in detail in Chapter 20. Periodic Recall Initial placement and adjustment of the prosthesis certainly is not the end of treatment for the partially edentulous patient. Periodic reevaluation of the patient is critical for early recognition of changes in oral structures to allow steps to be taken to maintain oral health. These examinations must monitor the condition of the oral tissue, the response to tooth restorations, the prosthesis, the patient’s accep-tance, and the patient’s commitment to maintain oral hygiene. Although a 6-month recall period is adequate for most patients, more frequent evaluation may be required for some. Chapter 20 contains some suggestions concerning this sixth phase of treatment. Reasons for Failure of Clasp-Retained Partial Dentures Experience with the clasp-retained partial denture made by the methods outlined has proved its merit and justifies its continued use. The occasional objection to the visibil-ity of retentive clasps can be minimized through the use of wrought-wire clasp arms. Few contraindications for use of a properly designed clasp-retained partial denture are known. Practically all objections to this type of denture can be eliminated by pointing to deficiencies in mouth preparation, denture design and fabrication, and patient education. These include the following: Diagnosis and treatment planning 1. Inadequate diagnosis 2. Failure to use a surveyor or to use a surveyor prop-erly during treatment planning Mouth preparation procedures 1. Failure to properly sequence mouth preparation procedures 2. Inadequate mouth preparations, usually resulting from insufficient planning of the design of the 15 Chapter 2 Considerations for Managing Partial Tooth Loss partial denture or failure to determine that mouth preparations have been properly accomplished 3. Failure to return supporting tissue to optimum health before impression procedures are performed 4. Inadequate impressions of hard and soft tissue Design of the framework 1. Failure to use properly located and sized rests 2. Flexible or incorrectly located major and minor connectors 3. Incorrect use of clasp designs 4. Use of cast clasps that have too little flexibility, are too broad in tooth coverage, and have too little consideration for esthetics Laboratory procedures 1. Problems in master cast preparation a. Inaccurate impression b. Poor cast-forming procedures c. Incompatible impression materials and gypsum products 2. Failure to provide the technician with a specific design and necessary information to enable the technician to execute the design 3. Failure of the technician to follow the design and written instructions Support for denture bases 1. Inadequate coverage of basal seat tissue 2. Failure to record basal seat tissue in a supporting form Occlusion 1. Failure to develop a harmonious occlusion 2. Failure to use compatible materials for opposing occlusal surfaces Patient-dentist relationship 1. Failure of the dentist to provide adequate dental health care information, including details on care and use of the prosthesis 2. Failure of the dentist to provide recall opportuni-ties on a periodic basis 3. Failure of the patient to exercise a dental health care regimen and respond to recall A removable partial denture designed and fabricated so that it avoids the errors and deficiencies listed is one that proves the clasp-type partial denture can be made functional, esthetically pleasing, and long lasting without damage to supporting structures. The proof of the merit of this type of restoration lies in the knowledge that (1) it permits treatment for the largest number of patients at a reasonable cost; (2) it provides restorations that are comfortable and efficient over a long period of time, with adequate support and maintenance of occlusal contact relations; (3) it can provide for healthy abutments, free of caries and periodontal disease; (4) it can provide for the continued health of restored, healthy tissue of the basal seats; and (5) it makes possible a partial denture service that is definitive and not merely an interim treatment. Removable partial dentures thus made will contribute to a concept of prosthetic dentistry that has as its goal the promotion of oral health, the restoration of partially edentulous mouths, and elimination of the ultimate need for complete dentures. 16 CHAPTER 3 Classification of Partially Edentulous Arches Chapter Outline Requirements of an Acceptable Method of Classification Kennedy Classification Applegate’s Rules for Applying the Kennedy Classification Even though recent reports have shown a consistent decline in the prevalence of tooth loss during the past few decades, significant variation in tooth loss distribution remains. It would be most helpful to consider which combinations of tooth loss are most common and to classify these for the purpose of assisting our management of partially edentulous patients. Several classifications of partially edentulous arches have been proposed and are in use. This variety has led to some confusion and disagreement concerning which classi-fication best describes all possible configurations and should be adopted. The most familiar classifications are those originally pro-posed by Kennedy, Cummer, and Bailyn. Beckett, Godfrey, Swenson, Friedman, Wilson, Skinner, Applegate, Avant, Miller, and others have also proposed classifications. It is evident that an attempt should be made to combine the best features of all classifications so that a universal classification can be adopted. A classification that is based on diagnostic criteria has been proposed recently for partial edentulism.1 The purpose of this system of classification is to facilitate treatment deci-sions on the basis of treatment complexity. Complexity is determined from four broad diagnostic categories that include location and extent of the edentulous areas, condi-tion of the abutments, occlusal characteristics and require-ments, and residual ridge characteristics. The advantage of this classification system over those in standard use has yet to be documented. Today, the Kennedy method is probably the most widely accepted classification of partially edentulous arches. In an attempt to simplify the problem and encourage more uni-versal use of a classification, and in the interest of adequate communication, the Kennedy classification will be used in this textbook. The student can refer to the “Selected Reading Resources” section for information relative to other classifications. 17 Chapter 3 Classification of Partially Edentulous Arches Kennedy Classification The Kennedy method of classification was originally pro-posed by Dr. Edward Kennedy in 1925. It attempts to classify the partially edentulous arch in a manner that suggests certain principles of design for a given situation (Figure 3-1). Kennedy divided all partially edentulous arches into four basic classes. Edentulous areas other than those that deter-mine the basic classes were designated as modification spaces (Figure 3-2). The following is the Kennedy classification: Class I Bilateral edentulous areas located posterior to the natural teeth Class II A unilateral edentulous area located posterior to the remaining natural teeth Class III A unilateral edentulous area with natural teeth remaining both anterior and posterior to it Class IV A single, but bilateral (crossing the midline), edentulous area located anterior to the remaining natural teeth Although classifications are actually descriptive of the partially edentulous arches, the removable partial denture that restores a particular class of arch is described as a denture of that class. For example, we speak of a Class III or Class I removable partial denture. It is simpler to say “a Class II partial denture” than it is to say “a partial denture restor-ing a Class II partially edentulous arch.” Requirements of an Acceptable Method of Classification The classification of a partially edentulous arch should satisfy the following requirements: 1. It should permit immediate visualization of the type of partially edentulous arch that is being considered. 2. It should permit immediate differentiation between the tooth-supported and the tooth- and tissue-supported removable partial denture. 3. It should be universally acceptable. Figure 3-1 Representative examples of partially edentulous arches classified by the Kennedy method. A B C D E F G H Figure 3-2 Kennedy classification with examples of modifications. A, Class I maxillary arch. B, Class II mandibular arch. C, Class III mandibular arch. D, Class IV maxillary arch. E, Class II, modification 1 mandibular arch. F, Class II, modification 1 maxillary arch. G, Class II, modification 2 mandibular arch. H, Class III, modification 2 maxillary arch. 19 Chapter 3 Classification of Partially Edentulous Arches A B C D E F G H I Figure 3-3 Nine partially edentulous arch configurations. Identify each. Answers can be found at the end of the chapter, after the reference. Box 3-1 RULES GOVERNING APPLICATION OF THE KENNEDY METHOD Rule 1 Classification should follow rather than precede any extractions of teeth that might alter the original classification. Rule 2 If a third molar is missing and is not to be replaced, it is not considered in the classification. Rule 3 If a third molar is present and is to be used as an abutment, it is considered in the classification. Rule 4 If a second molar is missing and is not to be replaced, it is not considered in the classification (e.g., if the opposing second molar is likewise missing and is not to be replaced). Rule 5 The most posterior edentulous area (or areas) always deter-mines the classification. Rule 6 Edentulous areas other than those that determine the classifi-cation are referred to as modifications and are designated by their number. Rule 7 The extent of the modification is not considered, only the number of additional edentulous areas. Rule 8 No modification areas can be included in Class IV arches. (Other edentulous areas that lie posterior to the single bilateral areas crossing the midline would instead determine the clas-sification; see Rule 5.) 20 Part I General Concepts/Treatment Planning One of the principal advantages of the Kennedy method is that it permits immediate visualization of the partially edentulous arch and allows easy distinction between tooth-supported versus tooth- and tissue-supported prostheses. Those schooled in its use and in the principles of partial denture design can readily relate the arch configuration design to be used in the basic partial denture. This method permits a logical approach to the problems of design. It makes possible the application of sound principles of partial denture design and is therefore a logical method of classifica-tion. However, a classification system should not be used to stereotype or limit the concepts of design. Applegate’s Rules for Applying the Kennedy Classification The Kennedy classification would be difficult to apply in every situation without certain rules for application. Apple-gate provided eight rules that govern application of the Kennedy method (Box 3-1). Although some confusion may occur initially as to why Class I should refer to two edentulous areas and Class II should refer to one, the principles of design make this distinction logical. Kennedy placed the Class II unilateral distal extension type between the Class I bilateral distal extension type and the Class III tooth-supported classifica-tion because the Class II partial denture must embody fea-tures of both, especially when tooth-supported modifications are present. Because it has a tissue-supported extension base, the denture must be designed similarly to a Class I partial denture. Often, however, a tooth-supported, or Class III, component is present elsewhere in the arch. Thus the Class II partial denture rightly falls between the Class I and the Class III, because it embodies design features common to both. In keeping with the principle that design is based on classification, the application of such principles of design is simplified by retaining the original classification of Kennedy. Figure 3-3 presents a chance to assess your skills. Review the figure and classify the partially edentulous arches illus-trated. The answers are provided below. Reference 1. McGarry TJ, Nimmo A, Skiba JF, et al: Classification system for partial edentulism, J Prosthodont 11(3):181-193, 2002. Answer to Figure 3-3: A. CL IV B. CL II Mod 2 C. CL I Mod 1 D. CL III Mod 3 E. CL III Mod 1 F. CL III Mod 1 G. CL IV H. CL II I. CL III Mod 5 21 CHAPTER 4 Biomechanics of Removable Partial Dentures Chapter Outline Biomechanics and Design Solutions Biomechanical Considerations Possible Movements of Partial Dentures Impact of Implants on Movements of Partial Dentures As was stated in Chapter 1, the goal is to provide useful, functional removable partial denture prostheses by striving to understand how to maximize every opportunity for pro-viding and maintaining a stable prosthesis. Because remov-able partial dentures are not rigidly attached to teeth, the control of potential movement under functional load is criti-cal to providing the best chance for stability and patient accommodation. The consequence of prosthesis movement under load is an application of stress to the teeth and tissue that are contacting the prosthesis. It is important that the stress not exceed the level of physiologic tolerance, which is a range of mechanical stimulus that a system can resist without disruption or traumatic consequences. In the termi-nology of engineering mechanics, the prosthesis induces stress in the tissue equal to the force applied across the area of contact with the teeth and/or tissue. This same stress acts to produce strain in the supporting tissue, which results in load displacement in the teeth and tissue. The understanding of how these mechanical phenomena act within a biological environment that is unique to each patient can be discussed in terms of biomechanics. In the design of removable partial dentures, with a focus on the goal of providing and main-taining stable prostheses, consideration of basic biomechan-ical principles associated with the unique features of each mouth is essential. Oral hygiene and appropriate prosthesis maintenance procedures are required for continued benefit of optimum biomechanical principles. Biomechanics and Design Solutions Removable partial dentures by design are intended to be placed into and removed from the mouth. Because of this, they cannot be rigidly connected to the teeth or tissue. This makes them subject to movement in response to functional loads, such as those created by mastication. It is important for clinicians who provide removable partial denture service 22 Part I General Concepts/Treatment Planning to understand the possible movements in response to func-tion and to be able to logically design the component parts of the removable partial denture to help control these move-ments. Just how this is accomplished in a logical manner may not be clear to a clinician who is new to this exercise. One method of helping to organize design thought is to consider it as an exercise in creating a design solution. Designing a removable partial denture can be considered similar to the classic, multifaceted design problem in con-ventional engineering, which is characterized by being open ended and ill structured. Open ended means that problems typically have more than one solution, and ill structured means that solutions are not the result of standard mathe-matical formulas used in some structured manner. The design process, which is a series of steps that lead toward a solution of the problem, includes identifying a need, defin-ing the problem, setting design objectives, searching for background information and data, developing a design rationale, devising and evaluating alternative solutions, and providing the solution (i.e., decision making and communi-cation of solutions) (Box 4-1). The rationale for design should logically develop from analysis of the unique oral condition of each mouth under consideration. However, it is possible that alternative design “solutions” could be applied, and it is the evaluation of perceived merits of these various designs that seems most confusing to clinicians. The following biomechanical considerations provide a background related to principles of the potential movement associated with removable partial dentures, and the subse-quent chapters covering the various component parts describe how these components are applied in designs to control the resultant movements of prostheses. Box 4-1 DESIGN PROCESS FOR REMOVABLE PARTIAL DENTURES Need Tooth replacement Definition of problem Provision of stable removable prosthesis Objectives Limited functional movement within tooth-tissue tolerance Background information Forces of occlusion, tissue “load-displacement” character and potential for movement, biomechanical principles applied to specific features of this unique mouth, removable partial denture component parts assigned to control movement Choice of a solution (among alternatives) for application Based on prior experience, principles and concepts learned from school and textbooks, and applicable clinical research Biomechanical Considerations As Maxfield states, “Common observation clearly indicates that the ability of living things to tolerate force is largely dependent upon the magnitude or intensity of the force.” The supporting structures for removable partial dentures (abutment teeth and residual ridges) are living things that are subjected to forces. Whether the supporting structures are capable of resisting the applied forces depends on (1) what typical forces require resistance, (2) what duration and intensity these forces have, (3) what capacity the teeth and/ or mucosae have to resist these forces, (4) how material use and application influence this teeth-tissue resistance, and (5) whether resistance changes over time. Consideration of the forces inherent in the oral cavity is critical. This includes the direction, duration, frequency, and magnitude of the force. In the final analysis, it is bone that provides the support for a removable prosthesis (i.e., the alveolar bone by way of the periodontal ligament and the residual ridge bone through its soft tissue covering). If potentially destructive forces can be minimized, then the physiologic tolerances of the supporting structures are not exceeded and pathologic change does not occur. The forces that occur with removable prosthesis function can be widely distributed and directed, and their effect mini-mized by appropriate design of the removable partial denture. An appropriate design includes the selection and location of components in conjunction with a harmonious occlusion. Unquestionably the design of removable partial dentures necessitates mechanical and biological considerations. Most dentists are capable of applying simple mechanical principles to the design of a removable partial denture. For example, the lid of a paint can is more easily removed with a screwdriver than with a half dollar. The longer the handle, the less effort (force) it takes. This is a simple application of the mechanics of leverage. By the same token, a lever system represented by a distal extension removable partial denture could magnify the applied force of occlusion to the terminal abutments, which would be undesirable. Tylman states, “Great caution and reserve are essential whenever an attempt is made to interpret biological phe-nomena entirely by mathematical computation.” However, an understanding of simple machines applied to the design of removable partial dentures helps to accomplish the objec-tive of preservation of oral structures. Without such under-standing, a removable partial denture can be inadvertently designed as a destructive machine. Machines may be classified into two general categories: simple and complex. Complex machines are combinations of many simple machines. The six simple machines are lever, wedge, screw, wheel and axle, pulley, and inclined plane (Figure 4-1). Of the simple machines, the lever, the wedge, and the inclined plane should be avoided in the design of removable partial dentures. 23 Chapter 4 Biomechanics of Removable Partial Dentures Lever Wedge Inclined plane Screw Pulley Wheel and axle F Figure 4-1 The six simple machines include lever, wedge, inclined plane, screw, pulley, and wheel and axle. The fulcrum, wedge, and inclined plane are matters of concern in removable partial denture designs because of the potential for harm if they are not appropriately controlled. F, Fulcrum. In its simplest form, a lever is a rigid bar supported some-where along its length. It may rest on the support or may be supported from above. The support point of the lever is called the fulcrum, and the lever can move around the fulcrum (Figure 4-2; see Figure 6-6). The rotational movement of an extension base type of removable partial denture, when a force is placed on the denture base, is illustrated in Figure 4-3. It will rotate in relation to the three cranial planes because of differences in the support characteristics of the abutment teeth and the soft tissue covering the residual ridge. Even though the actual movement of the denture may be small, a lever force may be imposed on abutment teeth. This is especially detrimental when prosthesis maintenance is neglected. Three types of levers are used: first, second, and third class (see Figure 4-2). The potential of a lever system to magnify a force is illus-trated in Figure 4-4. A cantilever is a beam supported at one end that can act as a first-class lever (Figure 4-5). A cantilever design should be avoided (Figure 4-6). Examples of other lever designs and suggestions for alternative designs to avoid or minimize their destructive potential are illustrated in Figures 4-7 and 4-8. The most efficient means of addressing the potential effects of a lever is to provide a rigid element at the unsup-ported end to disallow movement. This is the most beneficial use of dental implants in conjunction with removable partial dentures (RPDs) and should be considered when support capacity for a distal extension is considered significantly poor. A tooth is apparently better able to tolerate vertically directed forces than nonvertical, torquing, or horizontal forces. This characteristic is observed clinically, and it seems rational that more periodontal fibers are activated to resist the application of vertical forces to teeth than are activated to resist the application of nonvertical forces (Figure 4-9). Again, a distal extension removable partial denture rotates when forces are applied to the artificial teeth attached to the extension base. Because it can be assumed that this rotation must create predominantly nonvertical forces, the location of stabilizing and retentive components in relation to the horizontal axis of rotation of the abutment becomes extremely important. An abutment tooth will better tolerate nonvertical forces if these forces are applied as near as pos-sible to the horizontal axis of rotation of the abutment (Figure 4-10). The axial surface contours of abutment teeth must be altered to locate components of clasp assemblies more favorably in relation to the abutment’s horizontal axis (Figure 4-11). 24 Part I General Concepts/Treatment Planning E (Gravity) E (Sticky foods) E F F F R R R A B C Figure 4-2 A-C, The three classes of levers. Classification is based on location of the fulcrum (F), resistance (R), and direc-tion of effort (force) (E). In dental terms, E can represent the force of occlusion or gravity; F can be a tooth surface such as an occlusal rest; and R is the resistance provided by a direct retainer or a guide plane surface. Possible Movements of Partial Dentures If it is presumed that direct retainers are functioning to minimize vertical displacement, rotational movement will occur about some axis as the distal extension base or bases move toward, away, or horizontally across the underlying tissue. Unfortunately, these possible movements do not occur singularly or independently but tend to be dynamic, and all occur at the same time. The greatest movement pos-sible is found in the tooth-tissue–supported prosthesis because of reliance on the distal extension supporting tissue to share the functional loads with the teeth. Movement of a distal extension base toward the ridge tissue will be propor-tionate to the quality of that tissue, the accuracy and extent of the denture base, and the applied total functional load. A review of prosthesis rotational movement that is possible around various axes in the mouth provides some under-standing of how component parts of removable partial dentures should be prescribed to control prosthesis movement. One movement is rotation about an axis through the most posterior abutments. This axis may pass through occlusal rests or any other rigid portion of a direct retainer assembly located occlusally or incisally to the height of contour of the primary abutments (see Figures 4-6 and 4-7). This axis, known as the fulcrum line, is the center of rotation as the distal extension base moves toward the supporting tissue when an occlusal load is applied. The axis of rotation may shift toward more anteriorly placed components, occlu-sal or incisal to the height of contour of the abutment, as the base moves away from the supporting tissue when vertical dislodging forces act on the partial denture. These dislodging forces result from the vertical pull of food between opposing tooth surfaces, the effects of moving border tissue, and the forces of gravity against a maxillary partial denture. If it is presumed that the direct retainers are functional and that the supportive anterior components remain seated, rota-tion—rather than total displacement—should occur. Verti-cal tissue-ward movement of the denture base is resisted by the tissue of the residual ridge in proportion to the support-ing quality of that tissue, the accuracy of the fit of the denture base, and the total amount of occlusal load applied. Move-ment of the base in the opposite direction is resisted by the action of the retentive clasp arms on terminal abutments and the action of stabilizing minor connectors in conjunction with seated, vertical support elements of the framework anterior to the terminal abutments acting as indirect retain-ers. Indirect retainers should be placed as far as possible from the distal extension base, affording the best possible leverage against lifting of the distal extension base. A second movement is rotation about a longitudinal axis as the distal extension base moves in a rotary direction about the residual ridge (see Figure 4-3). This movement is resisted primarily by the rigidity of the major and minor connectors and their ability to resist torque. If the connectors are not rigid, or if a stress-breaker exists between the distal extension base and the major connector, this rotation about a longitu-dinal axis applies undue stress to the sides of the supporting ridge or causes horizontal shifting of the denture base. A third movement is rotation about an imaginary vertical axis located near the center of the dental arch (see Figure 4-4). This movement occurs under function because diago-nal and horizontal occlusal forces are brought to bear on the partial denture. It is resisted by stabilizing components, such as reciprocal clasp arms and minor connectors that are in contact with vertical tooth surfaces. Such stabilizing compo-nents are essential to any partial denture design, regardless of the manner of support and the type of direct retention employed. Stabilizing components on one side of the arch 25 Chapter 4 Biomechanics of Removable Partial Dentures Sagittal A A B B C C Frontal Horizontal Figure 4-3 Distal extension removable partial dentures will rotate when force is directed on the denture base. Differences in dis-placeability of the periodontal ligament of the supporting abutment teeth and soft tissue covering the residual ridge permit this rotation. It would seem that rotation of the prosthesis occurs in a combination of directions rather than in a unidirectional way. The three pos-sible movements of distal extension partial dentures are (A) rotation around a fulcrum line passing through the most posterior abut-ments when the denture base moves vertically toward or away from the supporting residual ridges; (B) rotation around a longitudinal axis formed by the crest of the residual ridge; and (C) rotation around a vertical axis located near the center of the arch. 26 Part I General Concepts/Treatment Planning E R2 R1 Figure 4-4 The length of a lever from fulcrum (F) (see Figure 4-7) to resistance (R) is called the resistance arm. That portion of a lever from the fulcrum to the point of application of force (E) is called the effort arm. Whenever the effort arm is longer than the resistance arm, mechanical advantage favors the effort arm, proportionately to the difference in length of the two arms. In other words, when the effort arm is twice the length of the resis-tance arm, a 25-lb weight on the effort arm will balance a 50-lb weight at the end of the resistance arm. The opposite is also true and helps illustrate cross-arch stabilization. When the resistance arm is lengthened (cross-arch clasp assembly placed on a second molar (R2) versus a second premolar [R1]), the effort arm is more efficiently counteracted. E E F F R R Figure 4-5 A cantilever can be described as a rigid beam supported only at one end. When force is directed against the unsupported end of the beam (as in this rest placed on a canti-levered pontic), the cantilever can act as a first-class lever. The mechanical advantage in this illustration favors the effort arm. Occlusal load Figure 4-6 Design often seen for a distal extension remov-able partial denture. A cast circumferential direct retainer engages the mesiobuccal undercut and is supported by the disto-occlusal rest. If it is rigidly attached to the abutment tooth, this could be considered a cantilever design, and detrimental first-class lever force may be imparted to the abutment if tissue support under the extension base allows excessive vertical movement toward the residual ridge. act to stabilize the partial denture against horizontal forces applied from the opposite side. It is obvious that rigid con-nectors must be used to make this effect possible. Horizontal forces always will exist to some degree because of lateral stresses that occur during mastication, bruxism, clenching, and other patient habits. These forces are accen-tuated by failure to consider the orientation of the occlusal plane, the influence of malpositioned teeth in the arch, and the effects of abnormal jaw relationships. Fabricating an occlusion that is in harmony with the opposing dentition and that is free of lateral interference during eccentric jaw movements may minimize the magnitude of lateral stress. The amount of horizontal movement occurring in the partial denture therefore depends on the magnitude of the lateral forces that are applied and on the effectiveness of the stabi-lizing components. In a tooth-supported partial denture, movement of the base toward the edentulous ridge is prevented primarily by the rests on the abutment teeth and to some degree by any 27 Chapter 4 Biomechanics of Removable Partial Dentures Occlusal load F R F Figure 4-7 As is shown in Figure 4-6, the potential for first-class lever action can also exist in Class II, modification 1 designs for removable partial denture frameworks. If a cast circumferen-tial direct retainer with a mesiobuccal undercut on the right first premolar were used, force placed on the denture base could impart upward and posteriorly moving force on the premolar, resulting in loss of contact between premolar and canine. Tissue support from the extension base area is most important to mini-mize the lever action of the clasp. The retainer design could help accommodate more of an anteriorly directed force during rota-tion of the denture base in an attempt to maintain tooth contact. Other alternatives to the first premolar design of the direct retainer would include a tapered wrought-wire retentive arm that uses mesiobuccal undercut, or that just has a buccal stabilizing arm above the height of contour. Occlusal load Occlusal load A B F F Figure 4-8 Mesial rest concept for distal extension removable partial dentures. With recognition that clasp movement occurs with functional displacement of the distal extension base, the primary aim of a mesial rest is to alter the fulcrum position and resultant clasp movement, disallowing harmful engagement of the abutment tooth. A, Bar type of retainer, minor connector contacting the guiding plane on the distal surface of the premolar, and mesio-occlusal rest used to reduce cantilever or first-class lever force when and if the denture rotates toward the residual ridge. B, Tapered wrought-wire retentive arm, minor connector contacting guiding plane on the distal surface of the premolar, and the mesio-occlusal rest. This design is applicable when the distobuccal undercut cannot be found or created, or when the tissue undercut contraindicates placement of a bar-type retentive arm. This design would be kinder to the periodontal ligament than would a cast, half-round retentive arm. Again, tissue support of the extension base is a key factor in reducing the lever action of the clasp arm. Note: Depending on the amount of contact of the minor connector proximal plate with the guiding plane, the fulcrum point will change. rigid portion of the framework located occlusal to the height of contour. Movement away from the edentulous ridge is prevented by the action of direct retainers on the abutments that are situated at each end of each edentulous space and by the rigid, minor connector stabilizing components. Therefore the first of the three possible movements can be controlled in the tooth-supported denture. The second pos-sible movement, which occurs along a longitudinal axis, is prevented by the rigid components of the direct retainers on the abutment teeth and by the ability of the major connector to resist torque. This movement is much less in the tooth-supported denture because of the presence of posterior abut-ments. The third possible movement occurs in all partial dentures. Therefore stabilizing components against horizon-tal movement must be incorporated into any partial denture design. For prostheses capable of movement in three planes, occlusal rests should provide occlusal support only to resist tissue-ward movement. All other movements of the partial denture should be resisted by components other than occlu-sal rests. Entrance of the occlusal rest into a stabilizing func-tion would result in direct transfer of torque to the abutment tooth. Because movements around three different axes are possible in a distal extension partial denture, an occlusal rest for such a partial denture should not have steep vertical walls or locking dovetails. This rest design is characterized by lack of free movement, which could cause horizontal and torqu-ing forces to be applied intracoronally to the abutment tooth. In the tooth-supported denture, the only movements of any significance are horizontal, and these may be resisted by the stabilizing effects of components placed on the axial surfaces of the abutments. Therefore in the tooth-supported denture, the use of intracoronal rests is permissible. In these instances, the rests provide not only occlusal support but also notable horizontal stabilization. In contrast, all Class I and Class II partial dentures, which have one or more distal extension bases, are not totally tooth 28 Part I General Concepts/Treatment Planning Lingual Buccal Figure 4-11 The abutment has been contoured (see shaded region) to allow rather favorable locations of retentive and recip-rocal stabilizing components (mirror view). Impact of Implants on Movements of Partial Dentures Similar to the process of considering how an individual tooth is best used in RPD design to control prosthesis move-ment, use of an implant should be directed toward the most beneficial movement control. Although possible roles for implant use include all three desired principles demon-strated by prostheses—support, stability, and retention— the major functional demand is imposed by chewing, and therefore the greatest benefit of implant use involves resist-ing instability by improving support. In this context, because implants are being considered to augment RPD design and not provide support for a fixed prosthesis, other benefits for the patient include reduced cost and less surgical morbidity. Because cost is typically a major consideration when one is choosing among prosthe-ses, and because use of an RPD provides a distinct cost advantage, selection of the most advantageous position of the implant(s) based on design goals is a primary consider-ation. Minimizing rotation about an axis in a Kennedy Class I or II arch, or any long modification span, is important to consider. Figure 4-10 Clasps placed closer to the occlusal/incisal surface have a greater likelihood of imparting tipping forces to the abutments. Figure 4-9 More periodontal fibers are activated to resist forces directed vertically on the tooth than are activated to resist horizontally (off-vertical) directed force. The horizontal axis of rotation is located somewhere in the root of the tooth. supported. Neither are they completely retained by bound-ing abutments. Any extensive Class III or Class IV partial denture that does not have adequate abutment support falls into the same category. These latter dentures may derive some support from the edentulous ridge and therefore may have composite support from both teeth and ridge tissue. 29 CHAPTER 5 Major and Minor Connectors Chapter Outline Role of Major Connectors in Control of Prosthesis Movement Location Mandibular major connectors Maxillary major connectors Minor Connectors Functions Form and location Tissue stops Finishing Lines Reaction of Tissue to Metallic Coverage Major Connectors in Review Components of a typical removable partial denture are illus-trated in Figure 5-1. 1. Major connectors 2. Minor connectors 3. Rests 4. Direct retainers 5. Stabilizing or reciprocal components (as parts of a clasp assembly) 6. Indirect retainers (if the prosthesis has distal extension bases) 7. One or more bases, each supporting one to several replacement teeth (see Figure 5-1) When a prosthesis that can be removed from the mouth is used, the prosthesis must extend to both sides of the arch. This enables transfer of functional forces of occlusion from the denture base to all supporting teeth and tissues within an arch for optimum stability. It is through this cross-arch tooth contact, which occurs at some distance from the func-tional force, that optimum resistance can be achieved. This is most effectively accomplished when a rigid major connec-tor joins the portion of the prosthesis receiving the function to selected regions throughout the arch. The chief functions of a major connector include unification of the major parts of the prosthesis, distribution of the applied force throughout the arch to selected teeth and tissue, and minimization of torque to the teeth. A properly designed rigid major connector effectively distrib-utes forces throughout the arch and acts to reduce the load to any one area while effectively controlling prosthesis movement. The principle of leverage is connected with this compo-nent part. A rigid major connector will limit movement possibilities by acting as a counteracting lever. This phenom-enon is referred to as cross-arch stability. Cross-arch stability becomes more important in situations associated with high potential for greater prosthesis movement (e.g., distal extensions). In this chapter, major and minor connectors are consid-ered separately as to their function, location, and design 30 Part I General Concepts/Treatment Planning criteria. Other components are presented in designated chapters. Role of Major Connectors in Control of Prosthesis Movement A major connector is the component of the partial denture that connects the parts of the prosthesis located on one side of the arch with those on the opposite side. It is that unit of the partial denture to which all other parts are directly or indirectly attached. This component also provides cross-arch stability to help resist displacement by functional stresses. The major connector may be compared with the frame of an automobile or with the foundation of a building. It is through the major connector that other components of the partial denture become unified and effective. If the major connector is flexible, the ineffectiveness of connected com-ponents jeopardizes the supporting oral structures and can be a detriment to the comfort of the patient. Failure of the major connector to provide rigidity may be manifest by traumatic damage to periodontal support of the abutment teeth, injury to residual ridges, or impingement of underly-ing tissue. It is the dentist’s responsibility to ensure that appropriate design and fabrication of the major connector are accomplished. Location Major connectors should be designed and located with the following guidelines in mind: 1. Major connectors should be free of movable tissue. 2. Impingement of gingival tissue should be avoided. Figure 5-1 A, Framework for mandibular removable partial denture with the following components: 1, lingual bar major connector; 2a, minor connector by which the resin denture base will be attached; 2b, minor connector, proximal plate, which is part of clasp assembly; 2c, minor connector used to connect rests to major connectors; 3, occlusal rests; 4, direct retainer arm, which is part of the total clasp assembly; 5, stabilizing or reciprocal components of clasp assembly (includes minor connectors); and 6, an indirect retainer consisting of a minor connector and an occlusal rest. B, Maxillary removable partial denture with resin denture bases supporting artificial posterior teeth. Bases are attached to metal framework by ladderlike minor connectors similar to those seen in 2a. C, Mandibular bilateral distal extension removable partial denture with resin denture bases supporting artificial posterior teeth. A B C 31 Chapter 5 Major and Minor Connectors 3. Bony and soft tissue prominences should be avoided during placement and removal. 4. Relief should be provided beneath a major connector to prevent its settling into areas of possible interference, such as inoperable tori or elevated median palatal sutures. 5. Major connectors should be located and/or relieved to prevent impingement of tissue that occurs because the distal extension denture rotates in function. Appropriate relief beneath the major connector avoids the need for its adjustment after tissue damage has occurred. In addition to being time consuming, grinding to provide relief from impingement may seriously weaken the major connector, which can result in flexibility or possibly fracture. Major connectors should be carefully designed for proper shape, thickness, and location. Alteration of these dimen-sions by grinding can only be detrimental. Relief is covered at the end of this chapter and is expanded in Chapter 11. Margins of major connectors adjacent to gingival tissue should be located far enough from the tissue to avoid any possible impingement. To accomplish this, it is recom-mended that the superior border of a lingual bar connector be located a minimum of 4 mm below the gingival margin(s) (Figure 5-2). At the inferior border of the lingual bar con-nector, the limiting factor is the height of the moving tissue in the floor of the mouth. Because the connector must have sufficient width and bulk to provide rigidity, a linguoplate is commonly used when space is insufficient for a lingual bar. In the maxillary arch, because no moving tissue is present in the palate as in the floor of the mouth, the borders of the major connector may be placed well away from gingival tissue. Structurally, the tissue covering the palate is well suited for placement of the connector because of the pres-ence of firm submucosal connective tissue and an adequate, deep blood supply. However, when soft tissue covering the midline of the palate is less displaceable than the tissue cov-ering the residual ridge, varying amounts of relief under the connectors must be provided to avoid impingement of tissue. The amount of relief required is directly proportional to the difference in displaceability of the tissue covering the midline of the palate and the tissue covering the residual ridges. The gingival tissue, on the other hand, must have an unrestricted superficial blood supply to remain healthy. To Figure 5-2 A, Lingual bar major connector should be located at least 4 mm inferior to gingival margins and farther if possible. The vertical height of a finished lingual bar should be at least 4 mm for strength and rigidity. If less than 8 mm exists between gingival margins and the movable floor of the mouth, a linguoplate (B), a sublingual bar (C), or a continuous bar (D) is preferred as a major connector. Relief is provided for soft tissue under all portions of the mandibular major connector and at any location where the frame-work crosses the marginal gingiva. The inferior border of mandibular major connectors should be gently rounded after being cast to eliminate a sharp edge. Half-pear-shaped lingual bar pattern Linguoplate pattern 4 mm Metric 4 mm Rounded after being cast in metal Rounded after being cast in metal Continuous (cingulum) bar 4 mm Sublingual bar B A D C 32 Part I General Concepts/Treatment Planning accomplish this, it is recommended that the borders of the palatal connector be placed a minimum of 6 mm away from and parallel to the gingival margins. Minor connectors that must cross gingival tissue should do so abruptly, joining the major connector at nearly a right angle (Figure 5-3). In this way, maximum freedom is ensured for gingival tissue. Except for a palatal torus or a prominent median palatal suture area, palatal connectors ordinarily require no relief. Intimate contact between the connector and the supporting tissue adds much to the support, stability, and retention of the denture. Except for gingival areas, intimacy of contact elsewhere in the palate is not detrimental to the health of the tissue if rests are provided on abutment teeth to prevent tissue-ward movement. An anterior palatal strap or the anterior border of a palatal plate also should be located as far as possible poste-riorly to avoid interference with the tongue in the area of the rugae. It should be uniformly thin and its anterior border should be located to follow the contours between the crests of the rugae. The anterior borders of such palatal major con-nectors therefore will be irregular in outline as they follow the contours between the rugae. The tongue may then pass from one ruga prominence to another without encountering the border of the connector. When the connector border must cross a ruga crest, this should be done abruptly, while avoiding the crest as much as possible. The posterior limita-tion of a maxillary major connector should be just anterior to the vibrating line. A useful rule applied to major connec-tors and throughout partial denture design is to try to avoid adding any part of the denture framework to an already convex surface. Characteristics of major connectors that contribute to the maintenance of health of the oral environment and the well-being of the patient may be listed as shown in Box 5-1. Mandibular Major Connectors The six types of mandibular major connectors include the following: 1. Lingual bar (Figure 5-4, A) 2. Linguoplate (Figure 5-4, B) 3. Sublingual bar (Figure 5-4, C) 4. Lingual bar with cingulum bar (continuous bar) (Figure 5-4, D) 5. Cingulum bar (continuous bar) (Figure 5-4, E) 6. Labial bar (Figure 5-4, F) The lingual bar and the linguoplate are by far the most common major connectors used in mandibular removable partial dentures. Lingual Bar The basic form of a mandibular major connector is a half-pear shape, located above moving tissue but as far below the gingival tissue as possible. It is usually made of reinforced, 6-gauge, half-pear–shaped wax or a similar plastic pattern (Figure 5-5). The major connector must be contoured so that it does not present sharp margins to the tongue and cause irritation or annoyance by an angular form. The superior border of a lingual bar connector should be tapered toward the gingival tissue superiorly, with its greatest bulk at the inferior border, resulting in a contour that has a half-pear shape. Lingual bar patterns, both wax and plastic, are made in this conventional shape. However, the inferior border of the lingual bar should be slightly rounded when the framework is polished. A rounded border will not impinge on the lingual tissue when the denture bases rotate inferiorly under occlusal loads. Frequently, additional bulk is necessary to provide rigidity, particularly when the bar is long or when a less rigid alloy is used. This is accomplished by lining the ready-made form Figure 5-3 Palatal major connector should be located at least 6 mm away from gingival margins and parallel to their mean curvature. All adjoining minor connectors should cross gingival tissues abruptly and should join major connectors at nearly a right angle. 6 mm 6 mm Box 5-1 CHARACTERISTICS OF MAJOR CONNECTORS CONTRIBUTING TO HEALTH AND WELL-BEING 1. Are made from an alloy compatible with oral tissue 2. Are rigid and provide cross-arch stability through the principle of broad distribution of stress 3. Do not interfere with and are not irritating to the tongue 4. Do not substantially alter the natural contour of the lingual surface of the mandibular alveolar ridge or of the palatal vault 5. Do not impinge on oral tissue when the restoration is placed, is removed, or rotates in function 6. Cover no more tissue than is absolutely necessary 7. Do not contribute to retention or trapping of food particles 8. Have support from other elements of the framework to minimize rotation tendencies in function 9. Contribute to the support of the prosthesis 33 Chapter 5 Major and Minor Connectors Figure 5-4 Mandibular major connectors. A, Lingual bar. B, Linguoplate. C, Sublingual bar. D, Lingual bar with continuous bar (cingulum bar). E, Cingulum bar. F, Labial bar. A B C D E F Figure 5-5 Sagittal section showing half-pear shape of lingual bar. A taper of the superior border of the bar to the soft tissues above will minimize interference with the tongue and will be more acceptable to the patient than would a dissimilar contour. Tissue relief is necessary to protect the soft tissue of the floor of the mouth. Relief underneath with a sheet of 24-gauge casting wax rather than altering the original half-pear shape. The inferior border of a lingual mandibular major con-nector must be located so that it does not impinge on the tissue in the floor of the mouth because it changes elevations during the normal activities of mastication, swallowing, speaking, licking the lips, and so forth. Yet at the same time, it seems logical to locate the inferior border of these connec-tors as far inferiorly as possible to avoid interference with the resting tongue and trapping of food substances when they are introduced into the mouth. In addition, the more inferiorly a lingual bar can be located, the farther the supe-rior border of the bar can be placed from the lingual gingival crevices of adjacent teeth, thereby avoiding impingement on the gingival tissue. 34 Part I General Concepts/Treatment Planning At least two clinically acceptable methods may be used to determine the relative height of the floor of the mouth and locate the inferior border of a lingual mandibular major connector. The first method is to measure the height of the floor of the mouth in relation to the lingual gingival margins of adjacent teeth with a periodontal probe (Figure 5-6). When these measurements are taken, the tip of the patient’s tongue should just lightly touch the vermilion border of the upper lip. Recording of these measurements permits their transfer to both diagnostic and master casts, thus ensuring a rather advantageous location of the inferior border of the major connector. The second method is to use an individu-alized impression tray for which lingual borders are 3 mm short of the elevated floor of the mouth, and then to use an impression material that will permit the impression to be accurately molded as the patient licks the lips. The inferior border of the planned major connector can then be located at the height of the lingual sulcus of the cast resulting from such an impression. Of the two methods, we have found measuring the height of the floor of the mouth to be less variable and more clinically acceptable. Linguoplate If the rectangular space is bounded by the lingual bar, the anterior tooth contacts, and the cingula, and the bordering minor connectors are filled in, a linguoplate results (Figure 5-7). A linguoplate should be made as thin as is technically feasible and should be contoured to follow the contours of the teeth and the embrasures (Figure 5-8). The patient should be aware of as little added bulk and as few altered contours as possible. The upper border should follow the natural curvature of the supracingular surfaces of the teeth and should not be located above the middle third of the lingual surface, except to cover interproximal spaces to the contact points. The half-pear shape of a lingual bar should still form the inferior border that provides the greatest bulk and rigidity. All gingival crevices and deep embrasures must Figure 5-6 A, Height of floor of the mouth (tongue elevated) in relation to lingual gingival sulci measured with a periodontal probe. B, Recorded measurements are transferred to a diagnostic cast and then to a master cast after mouth preparations are completed. The line connecting marks indicates the location of the inferior border of the major connector. If periodontal surgery is performed, the line on the cast can be related to incisal edges of teeth and the measurements recorded for subsequent use. C, Impression made with functional movement of the tongue to demonstrate maximum shortening of the floor of the mouth. This allows visualization of the anatomic feature that establishes the inferior extent of a major connector. If a stock tray causes impingement on this functional posi-tion, an individualized or custom tray may be used for the same purpose. A D B C 35 Chapter 5 Major and Minor Connectors Figure 5-7 View of mandibular Class I design with contoured linguoplate. Linguoplate is made as thin as possible and should follow the lingual contours of the teeth contacted. Doing so will often result in a scalloped superior margin. In this example, the straight superior margin can be bulky at the cingulum region, causing tongue discomfort. Figure 5-8 Apron of linguoplate (tissue side) is closely adapted to the teeth extending into nonundercut interproximal embrasures, resulting in a scalloped form. When well adapted, this form will benefit from some anterior teeth acting together to help resist horizontal rotational tendencies of the prosthesis, especially if the posterior ridge form does not resist such movement. Figure 5-9 If a linguoplate major connector was indicated for this patient with overlapped anterior teeth, judicious recontour-ing of the lingual proximal surfaces of right lateral, right central, and left lateral incisors would eliminate excessive undercuts and permit closer adaptation of the lingual apron of the major connector. be blocked out parallel to the path of placement to avoid gingival irritation and any wedging effect between the teeth. In many instances, judicious recontouring of the lingual proximal surfaces of overlapped anterior teeth permits closer adaptation of the linguoplate major connector, eliminating otherwise deep interproximal embrasures to be covered (Figure 5-9). The linguoplate does not in itself serve as an indirect retainer. When indirect retention is required, definite rests must be provided for this purpose. Both the linguoplate and the cingulum bar ideally should have a terminal rest at each end, regardless of the need for indirect retention. However, when indirect retainers are necessary, these rests may also serve as terminal rests for the linguoplate or continuous bar. Because no component part of a removable partial denture should be added arbitrarily, each component should be added to serve a definite purpose. Indications for the use of a linguoplate may be listed as follows: 1. When the lingual frenum is high or the space available for a lingual bar is limited. In either instance, the superior border of a lingual bar would have to be placed too close to the gingival tissue. Irritation could be avoided only by generous relief, which might be annoying to the tongue and create an undesirable food trap. When a clinical mea-surement from the free gingival margins to the slightly elevated floor of the mouth is less than 8 mm, a linguo-plate is indicated in lieu of a lingual bar. The use of a linguoplate permits the inferior border to be placed more superiorly without tongue and gingival irritation and without compromise of rigidity. 2. In Class I situations in which the residual ridges have undergone excessive vertical resorption. Flat residual ridges offer little resistance to the horizontal rotational tendencies of a denture. The bracing effect provided by the remaining teeth must be depended upon to resist such rotation. A correctly designed linguoplate will engage the remaining teeth to help resist horizontal rotations. 3. For stabilizing periodontally weakened teeth, splinting with a linguoplate can be of some value when used with definite rests on sound adjacent teeth. As was described previously, a cingulum bar may be used to accomplish the same purpose because it actually represents the supe-rior border of a linguoplate without the gingival apron. The cingulum bar accomplishes stabilization along with the other advantages of a linguoplate. However, it is fre-quently more objectionable to the patient’s tongue and is certainly more of a food trap than is the contoured apron of a linguoplate. 36 Part I General Concepts/Treatment Planning 4. When the future replacement of one or more incisor teeth will be facilitated by the addition of retention loops to an existing linguoplate. Mandibular incisors that are periodontally weak may thus be retained, with provisions for possible loss and future additions. The same reasons for use of a linguoplate anteriorly apply to its use elsewhere in the mandibular arch. If a lingual bar alone is to be used anteriorly, there is no reason to add an apron elsewhere. However, when auxiliary splinting is used for stabilization of the remaining teeth or for horizontal stabilization of the prosthesis, or for both, small rectangular spaces sometimes remain. Tissue response to such small spaces is better when they are bridged with an apron than when they are left open. Generally, the apron is used to avoid gingival irritation or entrapment of food debris or to cover generously relieved areas that would be irritating to the tongue (Figure 5-10). Sometimes a dentist is faced with a clinical situation wherein a linguoplate is indicated as the major connector of choice even though the anterior teeth are quite spaced and the patient strenuously objects to metal showing through the spaces. The linguoplate can then be constructed so that the metal will not appreciably show through the spaced anterior teeth (Figure 5-11). The rigidity of the major connector is not greatly altered. However, such a design may be as much of a food trap as the continuous bar type of major connector. Design of Mandibular Major Connectors The following systematic approach to the design of a man-dibular lingual bar and linguoplate major connectors can be readily used with diagnostic casts after the diagnostic data are considered and related to the basic principles of major connector design: Figure 5-10 Sagittal section through the linguoplate demon-strating a basic half-pear–shaped inferior border with the metallic apron extending superiorly. Extension of linguoplate to the height of contour on the premolar was accomplished to enclose a rather large triangular interproximal space inferior to the contact point between the canine and premolar. Such spaces may often be bridged to eliminate obvious food traps. Relief is provided for soft tissue under all portions of the mandibular major connector and at any location where the framework crosses the marginal gingiva. Relief Figure 5-11 Interrupted linguoplate in the presence of inter-proximal spaces. Step 1: Outline the basal seat areas on the diagnostic cast (Figure 5-12, A) Step 2: Outline the inferior border of the major connector (Figure 5-12, B) Step 3: Outline the superior border of the major connector (Figure 5-12, C) Step 4: Connect the basal seat area to the inferior and supe-rior borders of the major connector, and add minor con-nectors to retain the acrylic resin denture base material (Figure 5-12, D) Sublingual Bar A modification of the lingual bar that has been demon-strated to be useful when the height of the floor of the mouth does not allow placement of the superior border of the bar at least 4 mm below the free gingival margin is the sublingual bar. The bar shape remains essentially the same as that of a lingual bar, but placement is inferior and posterior to the usual placement of a lingual bar, lying over and parallel to the anterior floor of the mouth. It is generally accepted that a sublingual bar can be used in lieu of a lingual plate if the lingual frenum does not interfere, or in the presence of an anterior lingual under-cut that would require considerable blockout for a conventional lingual bar. Contraindications include interfering lingual tori, high attachment of a lingual frenum, and interference with elevation of the floor of the mouth during functional movements. Cingulum Bar (Continuous Bar) When a linguoplate is the major connector of choice, but the axial alignment of the anterior teeth is such that excessive blockout of interproximal undercuts must be made, a cingulum bar may be considered. A cingulum bar located on or slightly above the cingula of the anterior teeth may be added to the lingual bar or can be used independently (Figure 5-13). In addition, when wide dia-stemata exist between the lower anterior teeth, a continu-37 Chapter 5 Major and Minor Connectors Figure 5-12 Sequence of design considerations for a mandibular major connector. A, Diagnostic cast with basal seat areas outlined. B, Inferior border of the major connector is outlined. Location of the inferior border was determined as suggested in Figure 5-6 and extends to the mesial of the mandibular right molar. C, Superior border of the major connector is outlined. Limited space for the lingual bar requires use of the linguoplate major connector. Linguoplate requires that rest seats be used on canines and the first premolar for positive support. D, Rest seat areas on the posterior teeth are outlined, and minor connectors for retention of resin denture bases are sketched. A B D C Figure 5-13 A, Lingual bar and cingulum bar (continuous bar) major connector. Upper portion of this major connector is located on the cingula of anterior teeth. The requirement of positive support by rest seats, at least as far anteriorly as the canines, is critical. Note that the superior border of the lingual bar portion is often placed objectionably close to the gingival margins if sufficient bulk for rigidity is to be obtained. This type of major connector easily traps food and is often more objectionable to patients than a linguoplate. B, Cingulum bar (continuous bar) major connector. Although this design may reduce the possibility of food entrapment, it may not provide adequate rigidity. A B ous bar retainer may be more esthetically acceptable than a linguoplate. Labial Bar Fortunately, in only a few situations does extreme lingual inclination of the remaining lower premolar and incisor teeth prevent the use of a lingual bar major connector. With the use of conservative mouth preparations in the form of recontouring and blockout, a lingual major con-nector can almost always be used. Lingually inclined teeth sometimes may have to be reshaped by means of crowns. Although the use of a labial major connector may be 38 Part I General Concepts/Treatment Planning necessary in rare instances, this should be avoided by resorting to necessary mouth preparations rather than by accepting a condition that is otherwise correctable (Figure 5-14). The same applies to the use of a labial bar when a mandibular torus interferes with placement of a lingual bar. Unless surgery is definitely contraindicated, interfer-ing mandibular tori should be removed so that the use of a labial bar connector may be avoided. A modification to the linguoplate is the hinged con-tinuous labial bar. This concept is incorporated in the Swing-Lock design, which consists of a labial or buccal bar that is connected to the major connector by a hinge at one end and a latch at the other end (Figure 5-15). Support is provided by multiple rests on the remaining natural teeth. Stabilization and reciprocation are pro-vided by a linguoplate that contacts the remaining teeth and are supplemented by the labial bar with its retentive struts. Retention is provided by a bar type of retentive clasp with arms projecting from the labial or buccal bar and contacting the infrabulge areas on the labial surfaces of the teeth. Use of the Swing-Lock concept would seem primarily indicated when the following conditions are present: 1. Missing key abutments. When all remaining teeth are used for retention and stability, the absence of a key abutment (such as a canine) may not present as serious a treatment problem with this concept as with more conventional designs (Figure 5-16). 2. Unfavorable tooth contours. When existing tooth contours (uncorrectable by recontouring with appro-priate restorations) or excessive labial inclinations of anterior teeth prevent conventional clasp designs, the basic principles of removable partial design may be better implemented with the Swing-Lock concept. Swing-Lock Inc., Milford, Texas. Figure 5-14 A, Lingual inclination of patient’s canines and premolars precludes use of the lingual bar. B, Labial bar major connector was used in treatment. Retention was obtained on terminal abutments. Support and stabilization were gained by using rests, minor connectors arising from the labial bar, and well-fitting denture bases. A B Figure 5-15 The hinge for this continuous labial bar connec-tor is located buccal and distal to the remaining dentition (area of tooth #21). The latching mechanism is opposite to the hinge, adjacent to tooth #28. In this location, it will be housed within the buccal flange of the denture. Figure 5-16 Absence of the mandibular canine requires that all remaining anterior teeth be used for stabilization and reten-tion of the replacement restoration. The Swing-Lock concept can be used to ensure that all remaining teeth share in stabilization and retention of the prosthesis. 39 Chapter 5 Major and Minor Connectors the health of the surrounding tissue is usually impaired. Similarly, interproximal projections of the major connector that rest on the gingival third of the tooth and on gingival tissues that are structurally unable to render support may be traumatized. To prevent these sequelae, one should support the major connector with definite rests on the teeth, provide adequate gingival relief, and/or locate the connector far enough away from the gingival margin to avoid any possible restriction of blood supply and entrapment of food debris. All gingival crossings should be abrupt and at right angles to the major connector. Creating a sharp, angular form on any portion of a palatal connector should be avoided, and all borders should be tapered toward the tissue. Single Palatal Strap Bilateral tooth-supported prostheses, even those with short edentulous spaces, are effectively connected with a single, broad palatal strap connector, particularly when the edentu-lous areas are located posteriorly (Figure 5-18). Such a con-nector can be made rigid without objectionable bulk and interference with the tongue, provided the cast framework material is distributed in three planes. Suitable rigidity, without excessive bulk, may be obtained for a single palatal strap by the laboratory technician by casting a 22-gauge matte plastic pattern. For reasons of torque and leverage, a single palatal strap major connector should not be used to connect anterior replacements with distal extension bases. To be rigid enough to resist torque and to provide adequate vertical support and horizontal stabilization, a single palatal strap would have to be objectionably bulky. When placed anteriorly, this bulk would become even more objectionable to the patient because it could interfere with speech. Combination Anterior and Posterior Palatal Strap–type Connector Structurally, this is a rigid palatal major connector. The ante-rior and posterior palatal strap combination may be used in almost any maxillary partial denture design (Figure 5-19). A posterior palatal strap should be flat and a minimum of 8 mm wide. Posterior palatal connectors should be located as far posterior as possible to avoid interference with the tongue but anterior to the line of flexure formed by the junc-tion of the hard and soft palates. The only condition that prevents their use is an inoperable maxillary torus that extends posterior to the soft palate. In this situation, a broad, U-shaped major connector may be used, as described else-where in this chapter. The strength of this major connector design lies in the fact that the anterior and posterior components are joined together by longitudinal connectors on either side, which form a square or rectangular frame. Each component braces the others against possible torque and flexure. Flexure is practically nonexistent in such a design. The anterior connector may be extended anteriorly to support anterior tooth replacements. In this form, a 3. Unfavorable soft tissue contours. Extensive soft tissue undercuts may prevent proper location of component parts of a conventional removable partial denture or an overdenture. The hinged continuous labial bar concept may provide an adjunctive modality to accom-modate such unfavorable soft tissue contours. 4. Teeth with questionable prognoses. The possibility of losing a key abutment tooth with a guarded prognosis seriously affects the stability and retention of a con-ventional prosthesis. Because all remaining teeth func-tion as abutments in the Swing-Lock denture, it seems that the loss of a tooth would not compromise reten-tion and stability to such a degree. The hinged labial bar type of restoration can be used satisfactorily in certain clinically compromised situations. As is true with any type of removable restoration, good oral hygiene, maintenance, regular recall, and close atten-tion to details of design are paramount to successful implementation of this treatment concept. Obvious contraindications to the use of this hinged labial bar concept are apparent. The most obvious is poor oral hygiene or lack of motivation for plaque control by the patient. Other contraindications include the presence of a shallow buccal or labial vestibule or a high frenal attachment. Any of these factors would prevent the proper placement of components of the Swing-Lock partial denture. Maxillary Major Connectors Six basic types of maxillary major connectors are considered: 1. Single palatal strap (Figure 5-17, A) 2. Combination anterior and posterior palatal strap–type connector (Figure 5-17, B) 3. Palatal plate-type connector (Figure 5-17C) 4. U-shaped palatal connector (Figure 5-17, D) 5. Single palatal bar (Figure 5-17, E) 6. Anterior-posterior palatal bars (Figure 5-17, F) Whenever it is necessary for the palatal connector to make contact with the teeth for reasons of support, definite tooth support must be provided. This is best accomplished by establishing definite rest seats on the predetermined abut-ment teeth. These should be located far enough above the gingival attachment to provide for bridging of the gingival crevice with blockout. At the same time, they should be low enough on the tooth to avoid unfavorable leverage and low enough on the maxillary incisors and canine teeth to avoid incisal interference of the opposing dentition. Major connector components resting on unprepared inclined tooth surfaces can lead to slippage of the denture or to orthodontic movement of the tooth, or to both. In either situation, settling into gingival tissue is inevitable. In the absence of the required vertical support provided by rests, 40 Part I General Concepts/Treatment Planning Figure 5-17 Maxillary major connectors: A, Single palatal strap. B, Anterior-posterior palatal strap. C, Palatal plate. D, U-shaped. E, Single palatal bar. F, Anterior-posterior palatal bars. A B D C E F U-shaped connector is made rigid by the horizontal strap that has been added posteriorly. If a maxillary torus exists, it may be encircled by this type of major strap-type connec-tor without reduced rigidity. The combination anterior-posterior connector design may be used with any Kennedy class of partially edentulous arch. It is used most frequently in Classes II and IV, whereas the single wide palatal strap is used more frequently in Class III situations. The palatal plate–type or complete coverage connector, described in this chapter, is used most frequently in Class I situations for reasons to be explained subsequently. All maxillary major connectors should cross the midline at a right angle rather than on a diagonal. It has been suggested that the tongue will accept symmetrically placed compo-nents far more readily than those placed without regard for symmetry. Palatal Plate–type Connector For lack of better terminology, the words palatal plate are used to designate any thin, broad, contoured palatal cover-age used as a maxillary major connector and covering one half or more of the hard palate (Figure 5-20). Anatomic replica palatal castings have uniform thickness and strength by reason of their corrugated contours. Through the use of 41 Chapter 5 Major and Minor Connectors Figure 5-18 A, This single palatal strap–type major connector is better suited for the restoration of short-span tooth-supported bilateral edentulous areas. It may also be used in tooth-sup-ported unilateral edentulous situations with provision for cross-arch attachment by extracoronal retainers or internal attachments. Width of the palatal strap should be confined within the boundar-ies of supporting rests. B, Sagittal section. Midportion of the major connector demonstrates slight elevation to provide rigid-ity. Such thickness of the major connector does not appreciably alter palatal contours. A B Figure 5-19 Anterior-posterior palatal strap–type major con-nector. The anterior component is a flat strap located as far posteriorly as possible to avoid rugae coverage and tongue inter-ference. The anterior border of this strap should be located just posterior to a rugae crest or in the valley between two crests. The posterior strap is thin, a minimum of 8 mm wide, and located as far posteriorly as possible, yet entirely on the hard palate. It should be located at right angles to midline rather than diagonally. Figure 5-20 Palatal major connector covering two thirds of the palate. The anterior border follows valleys between rugae and does not extend anteriorly to indirect retainers on the first pre-molars. The posterior border is located at the junction of the hard and soft palates but does not extend onto the soft palate. In the bilateral distal extension situation illustrated, indirect retainers are a must to aid in resisting horizontal rotation of the restora-tion. Note that provisions have been made for a butt-type joint joining the denture bases and framework as the denture base on each side passes through the pterygomaxillary notch. electrolytic polishing, uniformity of thickness can be main-tained, and the anatomic contours of the palate will be faith-fully reproduced in the finished denture. The anatomic replica palatal major connector has several potential advantages: 1. It permits the making of a uniformly thin metal plate that reproduces faithfully the anatomic contours of the patient’s own palate. Its uniform thinness and the thermal conductivity of the metal are designed to make the palatal plate more readily acceptable to the tongue and underly-ing tissue. 2. The corrugation in the anatomic replica adds strength to the casting; thus a thinner casting with adequate rigidity can be made. 3. Surface irregularities are intentional rather than acciden-tal; therefore electrolytic polishing is all that is needed. The original uniform thickness of the plastic pattern is thus maintained. 4. By virtue of intimate contact, interfacial surface tension between metal and tissue provides the prosthesis with greater retention. Retention must be adequate to resist 42 Part I General Concepts/Treatment Planning the pull of sticky foods, the action of moving border tissue against the denture, the forces of gravity, and the more violent forces of coughing and sneezing. These are all resisted to some extent by the retention of the base itself, which is proportional to the total area of denture base contact with supporting tissue. The required amount of both direct and indirect retention will depend on the amount of retention provided by the denture base. The palatal plate may be used in any one of three ways. It may be used as a plate of varying width that covers the area between two or more edentulous areas, as a complete or partial cast plate that extends posterior to the junction of the hard and soft palates (Figures 5-21 and 5-22), or in the form of an anterior palatal connector with a provision for extending an acrylic resin denture base in a posterior direc-tion (Figure 5-23). The palatal plate should be located anterior to the poste-rior palatal seal area. The maxillary complete denture’s typical posterior palatal seal is not necessary with a maxillary partial denture’s palatal plate because of the accuracy and stability of the cast metal. When the last remaining abutment tooth on either side of a Class I arch is the canine or first premolar tooth, com-plete palatal coverage is strongly advised, especially when the residual ridges have undergone excessive vertical resorption. This may be accomplished in one of two ways. One method is to use a complete cast plate that extends to the junction of the hard and soft palates (see Figure 5-22). The other method is to use a cast major connector anteriorly, with retention posteriorly, for the attachment of an acrylic-resin denture base that extends posterior to the anatomic land-marks previously described (see Figure 5-23). Figure 5-21 Palatal plate major connector for a Class I, modi-fication 1 removable partial denture. The posterior border lies on the immovable hard palate and crosses the midline at a right angle. Total contact provides excellent support. Figure 5-22 Complete coverage palatal major connector. The posterior border terminates at the junction of the hard and soft palates. The anterior portion, in the form of the palatal linguo-plate, is supported by positive lingual rest seats on canines. The location of finishing lines is most important in this type of major connector. Anteroposteriorly, they should be parallel to a line along the center of the ridge crest and located just lingual to an imaginary line contacting the lingual surfaces of missing natural teeth. Alteration of the natural palatal contour should be antici-pated with its attendant detrimental effects on speech if these contours are not followed. Despite increased costs, the advantages of a cast palate make it preferable to an acrylic-resin palate. However, the latter method may be used satisfactorily when relining is anticipated or cost is a factor. The complete palatal plate is not a connector that has received universal use. It has, however, become accepted as a satisfactory palatal connector for many maxillary partial dentures. In all circumstances, the portion contacting the teeth must have positive support from adequate rest seats. The dentist should be familiar with its use and, at the same time, with its limitations, so that it may be used intelligently and to fullest advantage. Design of Maxillary Major Connectors In 1953, Blatterfein described a systematic approach to designing maxillary major connectors. His method involves five basic steps and is certainly applicable to most maxillary removable partial denture situations. When using a diagnos-tic cast and knowledge of the relative displaceability of the palatal tissue, including that covering the median palatal raphe, he recommends the following basic steps: Step 1: Outline of primary bearing areas. The primary bearing areas are those that will be covered by the denture base(s) (Figure 5-24, A and B). Step 2: Outline of nonbearing areas. The nonbearing areas are the lingual gingival tissue within 5 to 6 mm of the remaining teeth, hard areas of the medial palatal raphe (including tori), and palatal tissue posterior to the vibrat-ing line (Figure 5-24, C). 43 Chapter 5 Major and Minor Connectors Figure 5-23 A, Maxillary major connector in the form of a palatal linguoplate with provisions for attaching the full-coverage resin denture base. B, Completed removable partial denture with resin base. The palatal linguoplate is supported by rests occupying lingual rest seats prepared in cast restorations on canines. This type of removable partial denture is particularly applicable when (1) residual ridges have undergone extreme vertical resorption, and (2) terminal abutments have suffered some bone loss and splinting cannot be accomplished. A B Figure 5-24 A, Diagnostic cast of partially edentulous maxillary arch. B, The palatal extent of the denture base areas are located 2 mm from the palatal surface of the posterior teeth. C, Nonbearing areas outlined in black, which include lingual soft tissue within 5 to 6 mm of teeth, an unyielding median palatal raphe area, and the soft palate. The space bounded by bearing and nonbearing area outlines is available for placement of the major connector. D, The major connector selected will be rigid and noninterfering with the tongue and will cover a minimum of the palate. A B C D 44 Part I General Concepts/Treatment Planning Step 3: Outline of connector areas. Steps 1 and 2, when completed, provide an outline or designate areas that are available to place components of major connectors (see Figure 5-24, C). Step 4: Selection of connector type. Selection of the type of connector(s) is based on four factors: mouth comfort, rigidity, location of denture bases, and indirect retention. Connectors should be of minimum bulk and should be positioned so that interference with the tongue during speech and mastication is not encountered. Connectors must have a maximum of rigidity to distribute stress bilaterally. The double-strap type of major connector provides the maximum rigidity without bulk and total tissue coverage. In many instances the choice of a strap type of major connector is limited by the location of the edentulous ridge areas. When edentulous areas are located anteriorly, the use of only a posterior strap is not recommended. By the same token, when only posterior edentulous areas are present, the use of only an anterior strap is not recommended. The need for indirect reten-tion influences the outline of the major connector. Provi-sion must be made in the major connector so that indirect retainers may be attached. Step 5: Unification. After selection of the type of major con-nector based on considerations in Step 4, the denture base areas and connectors are joined (Figure 5-24, D). Indications for the use of complete palatal coverage have been previously discussed in this chapter. Although many variations in palatal major connectors have been noted, a thorough comprehension of all factors that influence their design will lead to the best design for each patient. U-shaped Palatal Connector From both the patient’s standpoint and a mechanical standpoint, the U-shaped palatal connector is the least desirable of maxillary major connectors. It should never be used arbitrarily. When a large inoperable palatal torus exists, and occasionally when several anterior teeth are to be replaced, the U-shaped palatal connector may have to be used (Figure 5-25). In most instances, however, other designs will serve more effectively. The following are the principal objections to use of the U-shaped connector: 1. Its lack of rigidity (compared with other designs) can allow lateral flexure under occlusal forces, which may induce torque or direct lateral force to abutment teeth. 2. The design fails to provide good support characteris-tics and may permit impingement of underlying tissue when subjected to occlusal loading. 3. Bulk to enhance rigidity results in increased thickness in areas that are a hindrance to the tongue. Many maxillary partial dentures have failed for no other reason than the flexibility of a U-shaped major con-nector (Figure 5-26). To be rigid, the U-shaped palatal Figure 5-25 U-shaped palatal connector is probably the least rigid type of maxillary major connector and should be used only when a large inoperable palatal torus prevents the use of palatal coverage or combination anterior-posterior palatal strap–type designed framework. Figure 5-26 Removable partial denture design that uses an objectionable U-shaped palatal major connector. Such a connec-tor lacks necessary rigidity, places bulk where it is most objec-tionable to the patient, and impinges on gingival tissue lingual to remaining teeth. connector must have bulk where the tongue needs freedom the most—the rugae area. Without sufficient bulk, the U-shaped design leads to increased flexibility and movement at the open ends. In distal extension partial dentures, when tooth support posterior to the edentulous area is nonexistent, movement is particularly noticeable and is traumatic to the residual ridge. No matter how well the extension base is supported or how harmonious the occlusion, without a rigid major connec-tor the residual ridge suffers. The wider the coverage of a U-shaped major connec-tor, the more it resembles a palatal plate–type connector with its several advantages. But when used as a narrow 45 Chapter 5 Major and Minor Connectors U design, the necessary rigidity is usually lacking. A U-shaped connector may be made more rigid with mul-tiple tooth support through definite rests. A common error in the design of a U-shaped connector, however, is its proximity to, or actual contact with, gingival tissue. The principle that the borders of major connectors should be supported by rests in prepared rest seats or should be located well away from gingival tissue has been stated previously. Most U-shaped connectors fail to do either, with resulting gingival irritation and periodontal damage to the tissue adjacent to remaining teeth. Single Palatal Bar To differentiate between a palatal bar and a palatal strap, a palatal connector component less than 8 mm in width is referred to as a bar in this textbook. The single palatal bar is perhaps the most widely used and yet the least logical of all palatal major connectors. It is difficult to say whether the bar or the U-shaped palatal connector is the more objectionable of palatal connectors. For a single palatal bar to have the necessary rigidity for cross-arch distribution of stress, it must have concen-trated bulk, which, unfortunately, is all too often ignored. For a single palatal bar to be effective, it must be rigid enough to provide support and cross-arch stabilization and must be centrally located between the halves of the denture. Mechanically, this practice may be sound enough. However, from the standpoint of patient com fort and alteration of palatal contours, it is highly objectionable. A partial denture made with a single palatal bar is often too thin and flexible or too bulky and objectionable to the patient’s tongue. The decision to use a single palatal bar instead of a strap should be based on the size of the denture-bearing areas that are connected and on whether a single connector located between them would be rigid without objectionable bulk. Combination Anterior and Posterior Palatal Bar–type Connectors Structurally, this combination of major connectors exhibits many of the same disadvantages as the single palatal bar (Figure 5-27). To be sufficiently rigid and to provide needed support and stability, these connectors could be too bulky and could interfere with tongue function. Beading of the Maxillary Cast Beading is a term used to denote the scribing of a shallow groove on the maxillary master cast outlining the palatal major connector exclusive of rugae areas (Figure 5-28). The purposes of beading are as follows: 1. To transfer the major connector design to the invest-ment cast (Figure 5-29, A and B) 2. To provide a visible finishing line for the casting (Figure 5-30) 3. To ensure intimate tissue contact of the major connec-tor with selected palatal tissue Beading is readily accomplished by using an appropri-ate instrument, such as a cleoid carver. Care must be exercised to create a groove not in excess of 0.5 mm in width or depth (Figure 5-31). Figure 5-27 Combination anterior-posterior palatal bar. To be sufficiently rigid to provide required support and stability, these major connectors must be excessively bulky. Because of its bulk and location, the anterior bar often interferes with the tongue. Figure 5-28 Framework design on master cast before prepa-ration for duplication in refractory investment. A shallow groove (0.5 mm) has been scribed on the outline of anterior and poste-rior borders of the major connector. The anterior outline follows the valleys of rugae. Beading is readily accomplished with a cleoid carver. A slightly rounded groove is preferred to a V-shaped groove. 46 Part I General Concepts/Treatment Planning Figure 5-29 A, Refractory cast. Note the definitive outline of the major connector indicated by beading lines that were transferred in duplicating the master cast. B, The wax pattern for the major connector is accurately executed by following the beading lines. The major connector is confined to previously scribed beading. A B Figure 5-30 Cast and framework showing metal margin pro-duced by the 0.5-mm beading line scribed on the cast. Such a margin is easily finished in the lab and provides intimate tissue contact, preventing food from easily dislodging the prosthesis. Care should be exercised in adapting such a beaded margin to noncompressible tissue, such as the median palatal raphe. Minor Connectors Minor connectors are those components that serve as the connecting link between the major connector or the base of a removable partial denture and the other components of the prosthesis, such as the clasp assembly, indirect retainers, occlusal rests, or cingulum rests. In many instances, a minor connector may be continuous with some other part of the denture. For example, an occlusal rest at one end of a lin-guoplate is actually the terminus of a minor connector, even though that minor connector is continuous with the linguo-plate. Similarly the portion of a partial denture framework that supports the clasp and the occlusal rest is a minor con-nector, which joins the major connector with the clasp proper. Those portions of a removable partial denture framework that retain the denture bases are also minor connectors. Functions In addition to joining denture parts, the minor connector serves two other purposes. 1. Transfers functional stress to the abutment teeth. This is a prosthesis-to-abutment function of the minor connec-tor. Occlusal forces applied to the artificial teeth are transmitted through the base to the underlying ridge tissue if that base is primarily tissue supported. Occlusal forces applied to the artificial teeth are also transferred to abutment teeth through occlusal rests. The minor con-nectors arising from a rigid major connector make pos-sible this transfer of functional stress throughout the dental arch. 2. Transfers the effects of the retainers, rests, and stabilizing components throughout the prosthesis. This is an abut-ment-to-prosthesis function of the minor connector. Thus forces applied on one portion of the denture may be resisted by other components placed elsewhere in the arch for that purpose. A stabilizing component on one side of the arch may be placed to resist horizontal forces that originate on the opposite side. This is possible only because of the transferring effect of the minor connector, which supports that stabilizing component, and the rigid-ity of the major connector. 47 Chapter 5 Major and Minor Connectors Form and Location Like the major connector, the minor connector must have sufficient bulk to be rigid; otherwise the transfer of func-tional stresses to the supporting teeth and tissue will not be effective. At the same time, the bulk of the minor connector should not be objectionable. A minor connector that contacts the axial surface of an abutment should not be located on a convex surface. Instead it should be located in an embrasure (Figure 5-32), where it will be least noticeable to the tongue. It should conform to the interdental embrasure, passing vertically from the major connector so that the gingival crossing is abrupt and covers as little of the gingival tissue as possible. It should be thickest toward the lingual surface, tapering toward the contact area (Figure 5-33). The deepest part of the interdental embrasure should have been blocked out to avoid interference during place-ment and removal, and to avoid any wedging effect on the contacted teeth. Figure 5-31 A, Tissue side of casting. Note slightly elevated ridges outlining anterior, posterior, and midpalatal opening regions of this anterior-posterior palatal strap major connector. B, Casting finished to a demarcated outline and seated on the master cast showing intimate adaptation. A B Figure 5-32 In an embrasure space, the minor connector is tapered to the tooth to avoid bulk and to accommodate the tongue. Figure 5-33 The minor connector that contacts the guiding plane is part of a clasp assembly. It can be separate from the other parts, or, as in this case, it can be connected to the lingual stabilizing portion of the clasp assembly. The proximal plate minor connector contact is about one-half the distance between tips of adjacent buccal and lingual cusps of the abutment tooth, and it extends gingivally, contacting an area of the abutment from the marginal ridge to two-thirds the length of the enamel crown. Viewed from above, it is triangular, the apex of the triangle being located buccally and the base of the triangle being located lingually. Less interference with the arrangement of the adjacent artificial tooth is encountered with minor connectors so shaped. 48 Part I General Concepts/Treatment Planning A modification of the conventional removable partial denture minor connector has been proposed. This applica-tion was suggested to be limited to the maxillary arch, with the minor connector located in the center of the lingual surface of the maxillary abutment tooth. It is suggested that this modification reduces the amount of gingival tissue coverage, provides enhanced guidance for the partial denture during insertion and removal, and increases stabilization against horizontal and rotational forces. However, because of its location, such a design varia-tion could encroach on the tongue space and create a greater potential space for food entrapment. The proposed variation should be used with careful application. When a minor connector contacts tooth surfaces on either side of the embrasure in which it lies, it should be tapered to the teeth. This avoids sharp angles, which could hinder tongue movement, and eliminates spaces that could trap food (Figure 5-34). It is a minor connector that contacts the guiding plane surfaces of the abutment teeth, whether as a connected part of a direct retainer assembly or as a separate entity (see Figure 5-33). Here the minor connector must be wide enough that the guiding plane can be used to fullest advan-tage. When it gives rise to a clasp arm, the connector should be tapered to the tooth below the origin of the clasp. If no clasp arm is formed (as when a bar clasp arm originates elsewhere), the connector should be tapered to a knife-edge the full length of its buccal aspect. Figure 5-34 Finishing line at the junction of the ladderlike minor connector and the major connector blends smoothly into the minor connector contacting the distal guiding plane on the second premolar. The framework is feathered toward tissue ante-rior to the finishing line to avoid as much bulk in this area as possible without compromising the strength of the butt-type joint. When an artificial tooth will be placed against a proximal minor connector, the minor connector’s greatest bulk should be toward the lingual aspect of the abutment tooth. This way sufficient bulk is ensured with the least interference with placement of the artificial tooth. Ideally the artificial tooth should contact the abutment tooth with only a thin layer of metal intervening buccally. Lin-gually the bulk of the minor connector should lie in the interdental embrasure—the same as between two natural teeth. As was stated previously, those portions of a denture framework by which acrylic-resin denture bases are attached are minor connectors. This type of minor con-nector should be so designed that it will be completely embedded within the denture base. The junctions of these mandibular minor connectors with the major connectors should be strong butt-type joints but without appreciable bulk (see Figure 5-34). Angles formed at the junctions of the connectors should not be greater than 90 degrees, thus ensuring the most advantageous and the strongest mechanical connection between the acrylic-resin denture base and the major connector. An open latticework or ladder type of design is prefer-able and is conveniently made by using preformed 12-gauge half-round and 18-gauge round wax strips. The minor connector for the mandibular distal extension base should extend posteriorly about two-thirds the length of the edentulous ridge and should have elements on both lingual and buccal surfaces. Such an arrangement not only will add strength to the denture base but may mini-mize distortion of the cured base from its inherent strains caused by processing. The minor connector must be planned with care so that it will not interfere with the arrangement of artificial teeth (Figure 5-35). A means to attach acrylic-resin individualized trays to the mandibular framework when a corrected impression is planned must be arranged when the framework pattern is being developed. Three nailhead minor connectors fab-ricated as part of the denture base minor connector serve this purpose well. Unless some similar arrangement is made, the resin trays may become detached or loosened during impression-making procedures. Minor connec-tors for maxillary distal extension denture bases should extend the entire length of the residual ridge and should be of a ladderlike and loop design (Figure 5-36). Tissue Stops Tissue stops are integral parts of minor connectors designed for retention of acrylic-resin bases. They provide stability to the framework during the stages of transfer and processing. They are particularly useful in preventing distortion of the framework during acrylic-resin process-ing procedures. Tissue stops can engage buccal and lingual slopes of the residual ridge for stability (Figure 5-37). Altered cast impression procedures often necessitate that tissue stops be augmented subsequent to the devel-opment of the altered cast. This can be readily accom-plished with the addition of autopolymerizing acrylic resin (Figure 5-38). Another integral part of the minor connector designed to retain the acrylic-resin denture base is similar to a tissue stop but serves a different purpose. It is located distal to the terminal abutment and is a continuation of 49 Chapter 5 Major and Minor Connectors Figure 5-35 The minor connector for attaching the resin denture base should be designed so that denture tooth place-ment is not compromised. The minor connector design should not include a main lattice strut at the ridge crest or in a desired tooth location. the minor connector contacting the guiding plane. Its purpose is to establish a definitive finishing index tissue stop for the acrylic-resin base after processing (Figure 5-39). Finishing Lines The finishing line junction with the major connector should take the form of an angle not greater than 90 degrees, therefore being somewhat undercut (Figure 5-40). Of course the medial extent of the minor connec-tor depends on the lateral extent of the major palatal connector. Too little attention is given to this finishing line location in many instances. If the finishing line is located too far medially, the natural contour of the palate will be altered by the thickness of the junction and the acrylic resin supporting the artificial teeth (Figure 5-41). If, on the other hand, the finishing line is located too far buccally, it will be most difficult to create a natural contour of the acrylic resin on the lingual surface of the artificial teeth. The location of the finishing line at the junction of the major and minor connectors should be based on restoration of the natural palatal shape, with consideration given to the location of the replacement teeth. Equal consideration must be given to the junction of minor connectors and bar-type direct retainer arms. These junctions are 90-degree butt-type joints and should follow the guidelines for base contour and clasp length. Figure 5-36 Extension of the finishing line to the area of the pterygomaxillary notch provides a butt-type joint for attachment of the border portion of the resin base through the pterygomaxil-lary notch (arrows). must be covered or crossed by elements of the partial denture. Lack of adequate hygiene measures can result in tissue reactions caused by an accumulation of food debris and bacteria. Coverage of oral tissue with partial dentures that are not kept clean irritates those tissues because of an accu-mulation of irritating factors. This has led to misinterpreta-tion of the effects of tissue coverage by prosthetic restorations. An additional hygiene concern is related to the problem of maintaining cleanliness of the tissue surface of the prosthesis. The first two causes of untoward tissue reaction can be accentuated the longer a prosthesis is worn. It is apparent that mucous membranes cannot tolerate this constant Reaction of Tissue to Metallic Coverage The reaction of tissue to coverage by the metallic compo-nents of a removable partial denture has been the subject of significant controversy, particularly in regions of marginal gingiva and broad areas of tissue contact. These tissue reac-tions can result from pressure caused by lack of support, lack of adequate hygiene measures, and prolonged contact through continual use of a prosthesis. Pressure occurs at regions where relief over gingival crossings and other areas of contact with tissue that are incapable of supporting the prosthesis is inadequate. Impingement will likewise occur if the denture settles because of loss of tooth and/or tissue support. This may be caused by failure of the rest areas resulting from improper design, caries involvement, fracture of the rest itself, or intrusion of abutment teeth under occlusal loading. It is important to maintain adequate relief and support from both teeth and tissue. Settling of the denture caused by loss of tissue support may also produce pressure elsewhere in the arch, such as beneath major connectors. Settling of a pros-thesis must be prevented or corrected if it has occurred. Excessive pressure must be avoided whenever oral tissue 50 Part I General Concepts/Treatment Planning Figure 5-37 A, Arrow points to location of the tissue stop. B, Master cast partially prepared for duplication in refractory investment. Posterior to relief wax, at the distal of the residual ridge (arrow), a tissue stop will be waxed. C, Wax tissue stop placed distal to relief (arrow). After casting, this will result in tissue stop contact of the framework. D, Tissue stop seen from labial position. E, Framework on cast shows tissue contact posterior to the minor connector with planned relief. Arrow points to the created tissue stop. A B C D E contact with a prosthesis without resultant inflammation and breakdown of the epithelial barrier. Some patients become so accustomed to wearing a removable restoration that they neglect to remove it often enough to give the tissue any respite from constant contact. This is frequently true when anterior teeth are replaced by the partial denture and the individual does not allow the prosthesis to be out of the mouth at any time except in the privacy of the bathroom during toothbrushing. Living tissues should not be covered all the time, or changes in those tissues will occur. Partial dentures should be removed for several hours each day so that the effects of tissue contact can subside and the tissue can return to a normal state. Clinical experience with the use of linguoplates and com-plete metallic palatal coverage has shown conclusively that when factors of pressure, cleanliness, and time are con-trolled, tissue coverage is not in itself detrimental to the health of oral tissue. 51 Chapter 5 Major and Minor Connectors Figure 5-38 A, Lower half of the flask in which the distal extension denture was invested. Note that the terminal portion of the minor connector (original tissue stop) is elevated from the residual ridge. The framework was developed on cast, with the residual ridge recorded in its anatomic form. The residual ridge was later recorded in its functional form by a corrected impression, thus the elevated tissue stop. B, Autopolymerizing resin is painted on between the tissue stop and the ridge to maintain the position of the minor con-nector during packing and processing procedures for a resin denture base. A B Figure 5-39 Finishing index tissue stop. A, Designed to facilitate finishing of the denture base resin at the region of the terminal abutment. Note the space at the anterior region of wax relief. Framework will be waxed to fill this space and provide positive tissue contact. B, Refractory cast shows space distal to the abutment. C, Wax pattern filling space for future tissue index contact. D, Framework index tissue stop anterior to relief beneath the minor connector of the distal extension base and posterior to the primary abutment. A B D C 52 Part I General Concepts/Treatment Planning Figure 5-40 Frontal sections through lingual finishing lines of palatal major connectors. The right image is through the full cast metal base major connector; the left image is through the resin denture base. In both situations, the location of finishing lines minimizes the bulk of resin attaching the artificial teeth. Palatal contours are restored, enhancing speech and contributing to a natural feeling for the patient. Figure 5-41 Junction of the major connector and the minor connector at palatal finishing lines should be located 2 mm medial from an imaginary line that would contact the lingual surfaces of missing posterior teeth. The finish line on the right is too far toward midline of the palate. The natural contours of the palate will be altered. Correct Incorrect Major Connectors in Review Mandibular Lingual Bar Indications for Use: The lingual bar should be used for man-dibular removable partial dentures when sufficient space exists between the slightly elevated alveolar lingual sulcus and the lingual gingival tissue. Characteristics and Location: (1) Half-pear shaped with bulkiest portion inferiorly located. (2) Superior border tapered to soft tissue. (3) Superior border located at least 4 mm inferior to gingival margins and farther if possible. (4) Inferior border located at the ascertained height of the alveolar lingual sulcus when the patient’s tongue is slightly elevated. Blockout and Relief of Master Cast: (1) All tissue undercuts parallel to the path of placement. (2) An additional thick-ness of 32-gauge sheet wax when the lingual surface of the alveolar ridge is undercut or parallel to the path of placement (see Figures 11-23 and 11-24). (3) No relief is necessary when the lingual surface of the alveolar ridge slopes inferiorly and posteriorly. (4) One thickness of baseplate wax over basal seat areas (to elevate minor con-nectors for attaching acrylic-resin denture bases). Waxing Specifications: (1) Six-gauge, half-pear–shaped wax form reinforced by 22- to 24-gauge sheet wax or similar plastic pattern adapted to the design width. (2) Long bar requires more bulk than short bar; however, cross-sec-tional shape is unchanged. Finishing Lines: Butt-type joint(s) with minor connector(s) for retention of denture base(s). Mandibular Linguoplate Indications for Use: (1) When the alveolar lingual sulcus so closely approximates the lingual gingival crevices that adequate width for a rigid lingual bar does not exist. (2) In those instances in which the residual ridges in Class I arch have undergone such vertical resorption that they will offer only minimal resistance to horizontal rotations of the denture through its bases. (3) For using periodon-tally weakened teeth in group function to furnish support to the prosthesis and to help resist horizontal (off-verti-cal) rotation of the distal extension type of denture. (4) When the future replacement of one or more incisor teeth will be facilitated by the addition of retention loops to an existing linguoplate. Characteristics and Location: (1) Half-pear shaped with bulkiest portion inferiorly located. (2) Thin metal apron extending superiorly to contact cingula of anterior teeth and lingual surfaces of involved posterior teeth at their height of contour. (3) Apron extended interproximally to 53 Chapter 5 Major and Minor Connectors the height of contact points (i.e., closing interproximal spaces). (4) Scalloped contour of apron as dictated by interproximal blockout. (5) Superior border finished to continuous plane with contacted teeth. (6) Inferior border at the ascertained height of the alveolar lingual sulcus when the patient’s tongue is slightly elevated. Blockout and Relief of Master Cast: (1) All involved under-cuts of contacted teeth parallel to the path of placement. (2) All involved gingival crevices. (3) Lingual surfaces of alveolar ridge and basal seat areas the same as for a lingual bar. Waxing Specifications: (1) Inferior border: 6-gauge, half-pear–shaped wax form reinforced with 24-gauge sheet wax or similar plastic pattern. (2) Apron: 24-gauge sheet wax. Finishing Lines: Butt-type joint(s) with minor connector(s) for retention of denture base(s). Mandibular Sublingual Bar Indications for Use: The sublingual bar should be used for mandibular removable partial dentures when the height of the floor of the mouth in relation to the free gingival margins will be less than 6 mm. It also may be indicated whenever it is desirable to keep the free gingival margins of the remaining anterior teeth exposed and depth of the floor of the mouth is inadequate to place a lingual bar. Contraindications for Use: Remaining natural anterior teeth severely tilted toward the lingual. Characteristics and Location: The sublingual bar is essen-tially the same half-pear shape as a lingual bar, except that the bulkiest portion is located to the lingual and the tapered portion is toward the labial. The superior border of the bar should be at least 3 mm from the free gingival margin of the teeth. The inferior border is located at the height of the alveolar lingual sulcus when the patient’s tongue is slightly elevated. This necessitates a functional impression of the lingual vestibule to accurately register the height of the vestibule. Blockout and Relief of Master Cast: (1) All tissue undercuts parallel to path of placement. (2) An additional thickness of 32-gauge sheet wax when the lingual surface of the alveolar ridge is undercut or parallel to the path of place-ment. (3) One thickness of baseplate wax over basal seat areas (to elevate minor connectors for attaching acrylic-resin denture bases). Waxing Specifications: (1) Six-gauge, half-pear–shaped wax form reinforced by 22- to 24-gauge sheet wax or similar plastic pattern adapted to design width. (2) Long bar bulkier than short bar; however, cross-sectional shape unchanged. Finishing Lines: Butt-type joint(s) with minor connector(s) for retention of denture base(s). Mandibular Lingual Bar With Continuous Bar (Cingulum Bar) Indications for Use: (1) When a linguoplate is otherwise indicated but the axial alignment of anterior teeth is such that excessive blockout of interproximal undercuts would be required. (2) When wide diastemata exist between mandibular anterior teeth and a linguoplate would objec-tionably display metal in a frontal view. Characteristics and Location: (1) Conventionally shaped and located same as lingual bar major connector compo-nent when possible. (2) Thin, narrow (3 mm) metal strap located on cingula of anterior teeth, scalloped to follow interproximal embrasures with inferior and superior borders tapered to tooth surfaces. (3) Originates bilater-ally from incisal, lingual, or occlusal rests of adjacent principal abutments. Blockout and Relief of Master Cast: (1) Lingual surfaces of alveolar ridge and basal seat areas same as for lingual bar. (2) No relief for continuous bar except blockout of inter-proximal spaces parallel to path of placement. Waxing Specifications: (1) Lingual bar major connector component waxed and shaped same as lingual bar. (2) Continuous bar pattern formed by adapting two strips (3 mm wide) of 28-gauge sheet wax, one at a time, over the cingula and into interproximal embrasures. Finishing Lines: Butt-type joint(s) with minor connector(s) for retention of denture base(s). Mandibular Continuous Bar (Cingulum Bar) Indications for Use: When a lingual plate or sublingual bar is otherwise indicated but the axial alignment of the ante-rior teeth is such that excessive blockout of interproximal undercuts would be required. Contraindications for Use: (1) Anterior teeth severely tilted to the lingual. (2) When wide diastemata that exist between the mandibular anterior teeth and the cingulum bar would objectionably display metal in a frontal view. Characteristics and Location: (1) Thin, narrow (3 mm) metal strap located on cingula of anterior teeth, scalloped to follow interproximal embrasures with inferior and superior borders tapered to tooth surfaces. (2) Originates bilaterally from incisal, lingual, or occlusal rests of adja-cent principal abutments. Blockout and Relief of Master Cast: No relief for cingulum bar except blockout of interproximal spaces parallel to the path of placement. Waxing Specifications: Cingulum bar pattern formed by adapting two strips (3 mm wide) of 28-gauge sheet wax, one at a time, over the cingula and into interproximal embrasures. Finishing Lines: Butt-type joint(s) with minor connector(s) for retention of denture base(s). 54 Part I General Concepts/Treatment Planning Mandibular Labial Bar Indications for Use: (1) When lingual inclinations of remain-ing mandibular premolar and incisor teeth cannot be corrected, preventing placement of a conventional lingual bar connector. (2) When severe lingual tori cannot be removed and prevent the use of a lingual bar or lingual plate major connector. (3) When severe and abrupt lingual tissue undercuts make it impractical to use a lingual bar or a lingual plate major connector. Characteristics and Location: (1) Half-pear shaped with bulkiest portion inferiorly located on the labial and buccal aspects of the mandible. (2) Superior border tapered to soft tissue. (3) Superior border located at least 4 mm inferior to labial and buccal gingival margins and farther if possible. (4) Inferior border located in the labial-buccal vestibule at the juncture of attached (immo-bile) and unattached (mobile) mucosae. Blockout and Relief of Master Cast: (1) All tissue undercuts parallel to path of placement, plus an additional thickness of 32-gauge sheet wax when the labial surface is undercut or parallel to the path of placement. (2) No relief neces-sary when the labial surface of the alveolar ridge slopes inferiorly to the labial or buccal. (3) Basal seat areas same as for lingual bar major connector. Waxing Specifications: (1) Six-gauge, half-pear–shaped wax form reinforced with 22- to 24-gauge sheet wax or similar plastic pattern. (2) Long bar necessitates more bulk than short bar; however, cross-sectional shape unchanged. (3) Minor connectors joined with occlusal or other superior components by a labial or buccal approach. (4) Minor connectors for base attachment joined by a labial or buccal approach. Finishing Lines: Butt-type joint(s) with minor connector(s) for retention of denture base(s). Single Palatal Strap–type Major Connector Indications for Use: Bilateral edentulous spaces of short span in a tooth-supported restoration. Characteristics and Location: (1) Anatomic replica form. (2) Anterior border follows the valleys between rugae as nearly as possible at right angles to median suture line. (3) Posterior border at right angle to median suture line. (4) Strap should be 8 mm wide or approximately as wide as the combined width of a maxillary premolar and first molar. (5) Confined within an area bounded by the four principal rests. Blockout and Relief of Master Cast: (1) Usually none required except slight relief of elevated medial palatal raphe or any exostosis crossed by the connector. (2) One thickness of baseplate wax over basal seat areas (to elevate minor connectors for attaching acrylic-resin denture bases). Beading: (See Figures 5-37 to 5-40.) Waxing Specifications: Anatomic replica pattern equivalent to 22- to 24-gauge wax, depending on arch width. Finishing Lines: (1) Undercut and slightly elevated. (2) No farther than 2 mm medial from an imaginary line con-tacting lingual surfaces of principal abutments and teeth to be replaced. (3) Follow curvature of arch. Single Broad Palatal Major Connector Indications for Use: (1) Class I partially edentulous arches with residual ridges that have undergone little vertical resorption and will lend excellent support. (2) V- or U-shaped palates. (3) Strong abutments (single or made so by splinting). (4) More teeth in arch than six remain-ing anterior teeth. (5) Direct retention not a problem. (6) No interfering tori. Characteristics and Location: (1) Anatomic replica form. (2) Anterior border following valleys of rugae as near right angle to median suture line as possible and not extending anterior to occlusal rests or indirect retainers. (3) Posterior border located at junction of hard and soft palate but not extended onto soft palate; at right angle to the median suture line; extended to pterygomaxillary notches. Blockout and Relief of Master Cast: (1) Usually none required except relief of elevated median palatal raphe or any small exostoses covered by the connector. (2) One thickness of baseplate wax over basal seat areas (to elevate minor connectors for attaching acrylic-resin denture bases). Beading: (See Figures 5-37 to 5-40.) Waxing Specifications: Anatomic replica pattern equivalent to 24-gauge sheet wax thickness. Finishing Lines: (1) Provision for butt-type joint at ptery-gomaxillary notches. (2) Undercut and slightly elevated. (3) No farther than 2 mm medial from an imaginary line contacting the lingual surfaces of the missing natural teeth. (4) Following curvature of arch. Anterior-Posterior Strap–type Major Connector Indications for Use: (1) Class I and II arches in which excel-lent abutment and residual ridge support exists, and direct retention can be made adequate without the need for indirect retention. (2) Long edentulous spans in Class II, modification 1 arches. (3) Class IV arches in which anterior teeth must be replaced with a removable partial denture. (4) Inoperable palatal tori that do not extend posteriorly to the junction of the hard and soft palates. 55 Chapter 5 Major and Minor Connectors Characteristics and Location: (1) Parallelogram shaped and open in center portion. (2) Relatively broad (8 to 10 mm) anterior and posterior palatal straps. (3) Lateral palatal straps (7 to 9 mm) narrow and parallel to curve of arch; minimum of 6 mm from gingival crevices of remaining teeth. (4) Anterior palatal strap: anterior border not placed farther anteriorly than anterior rests and never closer than 6 mm to lingual gingival crevices; follows the valleys of the rugae at right angles to the median palatal suture. Posterior border, if in rugae area, follows valleys of rugae at right angles to the median palatal suture. (5) Posterior palatal connector: posterior border located at junction of hard and soft palates and at right angles to median palatal suture and extended to hamular notch area(s) on distal extension side(s). (6) Anatomic replica or matte surface. Blockout and Relief of Master Cast: (1) Usually none required except slight relief of elevated median palatal raphe where anterior or posterior straps cross the palate. (2) One thickness of baseplate wax over basal seat areas (to elevate minor connectors for attaching acrylic-resin denture bases). Beading: (See Figures 5-37 to 5-40.) Waxing Specifications: (1) Anatomic replica patterns or matte surface forms of 22-gauge thickness. (2) Posterior palatal component: a strap of 22-gauge thickness, 8 to 10 mm wide (a half-oval form of approximately 6-gauge thickness and width) may also be used. Finishing Lines: Same as for single broad palatal major connector. Complete Palatal Coverage Major Connector Indications for Use: (1) In most situations in which only some or all anterior teeth remain. (2) Class II arch with a large posterior modification space and some missing anterior teeth. (3) Class I arch with one to four premolars and some or all anterior teeth remaining, when abutment support is poor and cannot otherwise be enhanced; resid-ual ridges have undergone extreme vertical resorption; direct retention is difficult to obtain. (4) In the absence of a pedunculated torus. Characteristics and Location: (1) Anatomic replica form for full palatal metal casting supported anteriorly by positive rest seats. (2) Palatal linguoplate supported anteriorly and designed for attachment of acrylic-resin extension posteriorly. (3) Contacts all or almost all of the teeth remaining in the arch. (4) Posterior border: terminates at the junction of the hard and soft palates; extended to hamular notch area(s) on distal extension side(s); at a right angle to median suture line. Blockout and Relief of Master Cast: (1) Usually none required except relief of elevated median palatal raphe or any small palatal exostosis. (2) One thickness of baseplate wax over basal seat areas (to elevate minor connectors for attaching acrylic-resin denture bases). Beading: (See Figures 5-37 to 5-40.) Waxing Specifications: (1) Anatomic replica pattern equiva-lent to 22- to 24-gauge sheet wax thickness. (2) Acrylic-resin extension from linguoplate the same as for a complete denture. Finishing Lines: As illustrated here and previously discussed. U-Shaped Palatal Major Connector This connector should be used only in those situations in which inoperable tori extend to the posterior limit of the hard palate. The U-shaped palatal major connector is the least favor-able design of all palatal major connectors because it lacks the rigidity of other types of connectors. When it must be used, indirect retainers must support any portion of the connector that extends anteriorly from the principal occlusal rests. Anterior border areas of this type of connector must be kept at least 6 mm away from adjacent teeth. If for any reason the anterior border must contact the remaining teeth, the connector must again be supported by rests placed in properly prepared rest seats. It should never be supported even temporarily by inclined lingual surfaces of anterior teeth. Waxing specifications, finishing lines, and so forth, are the same as for full palatal castings or other previously dis-cussed similar major connectors. 56 CHAPTER 6 Rests and Rest Seats Chapter Outline Role of Rests in Control of Prosthesis Movement Form of the Occlusal Rest and Rest Seat Extended Occlusal Rest Interproximal Occlusal Rest Seats Internal Occlusal Rests Support for Rests Lingual Rests on Canines and Incisor Teeth Incisal Rests and Rest Seats Implants as a Rest The capacity for teeth to resist functional forces and remain stable over time is provided through their sophisticated support mechanisms. Studies have shown that displacement and recovery following loading are far better for natural teeth than for oral mucosa. Consequently, appropriate use of the teeth to help resist functional forces in removable prostheses is a critical strategy to control prosthesis move-ment and achieve functional stability. Role of Rests in Control of Prosthesis Movement Appropriate use of teeth requires consideration as to how best to engage teeth for the supportive qualities they provide. Because the most effective resistance can be provided if the tooth is stressed along its long axis, the prosthesis framework should engage the tooth in a manner that encourages axial loading. The various forms of rests have as a main goal a form that allows for axial loading. It is important to realize that this goal can be achieved only through some form of tooth modification. Vertical support must be provided for a removable partial denture. Any component of a partial denture on a tooth surface that provides vertical support is called a rest (Figure 6-1). Rests should always be located on properly prepared tooth surfaces. The prepared surface of an abutment to receive the rest is called the rest seat. Rests are designated by the surface of the tooth prepared to receive them (occlusal rest, lingual rest, and incisal rest). The topography of any rest should restore the topography of the tooth that existed before the rest seat was prepared. The primary purpose of the rest is to provide vertical support for the partial denture. In doing so, it also does the following: 1. Maintains components in their planned positions 2. Maintains established occlusal relationships by prevent-ing settling of the denture 3. Prevents impingement of soft tissue 4. Directs and distributes occlusal loads to abutment teeth 57 Chapter 6 Rests and Rest Seats When rests prevent movement of the denture in an apical direction, the position of the retentive portion of the clasp arms can be maintained in designated relation to the tooth undercuts. Although passive when it is in its terminal posi-tion, the retentive portion of the clasp arm should remain in contact with the tooth, ready to resist a vertical dislodging force. Then, when a dislodging force is applied, the clasp arm should immediately become actively engaged to resist verti-cal displacement. If settling of the denture results in clasp arms that stand away from the tooth, some vertical displace-ment is possible before the retainer can become functional. The rest serves to prevent such settling and thereby helps to maintain the vertical stability of the partial denture. The use of an implant as a rest can also be considered. In this application, the implant eliminates compression of supporting soft tissues, controls vertical movement of the denture base, eliminates or alters fulcrum lines, and serves to increase support and stability of the prosthesis. Thus rests serve to support the position of a partial denture and to resist movement toward the tissue. They serve to transmit vertical forces to the abutment teeth and to direct those forces along the long axes of the teeth. In this respect, tooth-supported removable partial denture rests function in a manner similar to fixed abutment retainers. It is obvious that for this degree of stability to exist, the rests must be rigid and must receive positive support from the abutment teeth, which means that under occlusal loading, the rest and the tooth remain in stable contact and no inde-pendent movement or slippage occurs. In a removable partial denture that has one or more distal extension bases, the denture becomes increasingly tissue supported as the distance from the abutment increases. Closer to the abutment, however, more of the occlusal load is transmitted to the abutment tooth by means of the rest. The load is thereby distributed between the abutment and the supporting residual ridge tissue. A B C D Figure 6-1 A, Occlusal rest seats have been prepared on molar and premolar teeth to provide vertical support for the removable partial denture. B, Framework for tooth-supported removable partial denture. Rests on patient’s right provide vertical support to the replacement dentition; rests on patient’s left provide cross-arch support and stabilization. C, Tooth support for this prosthesis is pro-vided by rests that occupy definite, prepared, rest seats on the canine and occlusal surfaces of selected posterior teeth. D, Kennedy Class III, modification 1 maxillary arch with rest seats prepared on the lingual surfaces of the canine and lateral incisor, and on the occlusal surfaces of the premolar and molar. 58 Part I General Concepts/Treatment Planning axis of the abutment tooth. This also permits slippage of the prosthesis away from the abutment, which can result in orthodontic-like forces being applied to an inclined plane on the abutment, with possible tooth movement (Figure 6-6). When an existing occlusal rest preparation is inclined apically toward the reduced marginal ridge and cannot be modified or deepened because of fear of perforation of the enamel or restoration, then a secondary occlusal rest must be employed to prevent slippage of the primary rest and Less than 90 Figure 6-4 The occlusal rest should be spoon shaped and slightly inclined apically from the marginal ridge. The rest should restore the occlusal morphology of the tooth that existed before preparation of the rest seat. Less than 90 Figure 6-5 The floor of the occlusal rest seat should be inclined apically from the lowered marginal ridge. Any angle less than 90 degrees is acceptable as long as the preparation of the proximal surface and lowering and rounding of the marginal ridge precede completion of the rest seat itself. Deepest part of rest seat Figure 6-2 The deepest part of an occlusal rest preparation should be inside the lowered marginal ridge at X. The marginal ridge is lowered to provide bulk and to accommodate the origin of the occlusal rest with the least occlusal interference. Figure 6-3 Occlusal rest seat preparation on molar. The preparation is rounded, and the triangular concavity has smooth margins on the occlusal surface and a lowered, rounded mar-ginal ridge. Form of the Occlusal Rest and Rest Seat 1. The outline form of an occlusal rest seat should be a rounded triangular shape with the apex toward the center of the occlusal surface (Figure 6-2). 2. It should be as long as it is wide, and the base of the tri-angular shape (at the marginal ridge) should be at least 2.5 mm for both molars and premolars. Rest seats of smaller dimensions do not provide an adequate bulk of metal for rests, especially if the rest is contoured to restore the occlusal morphology of the abutment tooth. 3. The marginal ridge of the abutment tooth at the site of the rest seat must be lowered to permit a sufficient bulk of metal for strength and rigidity of the rest and the minor connector. This means that a reduction of the marginal ridge of approximately 1.5 mm is usually necessary. 4. The floor of the occlusal rest seat should be apical to the marginal ridge and the occlusal surface and should be concave, or spoon shaped (Figure 6-3). Caution should be exercised in preparing a rest seat to avoid creating sharp edges or line angles in the preparation. 5. The angle formed by the occlusal rest and the vertical minor connector from which it originates should be less than 90 degrees (Figures 6-4 and 6-5). Only in this way can the occlusal forces be directed along the long axis of the abutment tooth. An angle greater than 90 degrees fails to transmit occlusal forces along the supporting vertical 59 Chapter 6 Rests and Rest Seats effect of the occlusal rest, which will cause the application of leverages to the abutment tooth. Extended Occlusal Rest In Kennedy Class II, modification 1, and Kennedy Class III situations in which the most posterior abutment is a mesially tipped molar, an extended occlusal rest should be designed and prepared to minimize further tipping of the abutment and to ensure that the forces are directed down the long axis of the abutment. This rest should extend more than one-half the mesiodistal width of the tooth, should be approximately one-third the buccolingual width of the tooth, and should allow for a minimum of 1-mm thickness of the metal; the preparation should be rounded with no undercuts or sharp angles (Figure 6-8). In situations in which the abutment is severely tilted, the extended occlusal rest may take the form of an onlay to restore the occlusal plane (Figure 6-9). The tooth prepara-tion for this type of extended rest must include removing or restoring pits, fissures, and grooves; placing a 1- to 2-mm bevel on the buccal and lingual occlusal surfaces to allow the extended rest (onlay) to provide stabilization; allowing the rest to restore the contour and occlusion of the natural tooth; and ensuring that the rest directs the forces down the long axis of the tooth. Tooth preparation must also include a 1- to 2-mm guiding plane on the mesial surface of the abutment. Interproximal Occlusal Rest Seats The design of a direct retainer assembly may require the use of interproximal occlusal rests (Figure 6-10). These rest seats are prepared as individual occlusal rest seats, with the excep-A D C F B Figure 6-6 The result of force applied to an inclined plane when the floor of the occlusal rest preparation inclines apically toward the marginal ridge of the abutment tooth. F, Occlusal force applied to the abutment tooth. AB, Relationship of the occlusal rest to the abutment tooth when the angle is greater than 90 degrees. ABC, Removable partial denture framework. ABD, Abutment tooth. Figure 6-7 Diagnostic cast evaluation of mesially tipped molar abutment. Anterior tilt of molar precludes preparation of acceptable rest seat on the mesio-occlusal surface. The patient could not afford a crown to improve axial alignment or orthodon-tic treatment to upright the molar. Occlusal rests will be used on mesio-occlusal and disto-occlusal surfaces to support restora-tion and direct forces over the greatest root mass of the abut-ment. The proposed ring clasp design is outlined. orthodontic movement of the abutment tooth (Figure 6-7). Such a rest should pass over the lowered marginal ridge on the side of the tooth opposite the primary rest and should, if possible, be inclined slightly apically from the marginal ridge. However, two opposing occlusal rests on diverging tooth inclines will function to prevent unfavorable forces if all related connectors are sufficiently rigid. In any tooth-tissue–supported partial denture, the relation of the occlusal rest to the abutment should be that of a shallow ball-and-socket joint, to prevent a possible transfer of horizontal stresses to the abutment tooth. The occlusal rest should provide only occlusal support. Stabilization against horizon-tal movement of the prosthesis must be provided by other components of the partial denture rather than by any locking Figure 6-8 Cast shows extended occlusal rest on the man-dibular first molar, designed to ensure maximum bracing from the tooth. If placed on a mesially inclined molar next to a modi-fication space (as in Figure 6-7), the extended rest would ensure that the forces are directed down the long axis of the abutment, and therefore the disto-occlusal rest would not have been needed. 60 Part I General Concepts/Treatment Planning When such rest seats are prepared, care must be exercised to avoid reducing or eliminating contact points of abutment teeth. However, sufficient tooth structure must be removed to allow adequate bulk of the component for strength and to permit the component to be so shaped that occlusion will not be altered. Therefore analysis of mounted diagnostic casts is mandatory for assessment of interocclusal contact areas in which rests are to be placed. Sufficient space must be present or created to avoid interference with the place-ment of rests (Figure 6-12). tion that the preparations must be extended farther lingually than is ordinarily accomplished (Figure 6-11). Adjacent rests, rather than a single rest, are used to prevent interproxi-mal wedging by the framework. In addition, the joined rests are designed to shunt food away from contact points. Figure 6-10 Drawing on cast shows the desired design of a direct retainer assembly on mandibular premolar and molar abutments that incorporates interproximal occlusal rests. The direct retainers on the distobuccal undercut of the molar and on the mesiobuccal undercut of the premolar are extended from the joined occlusal rests, which occupy specifically prepared adjoin-ing rest seats. Buccal Lingual Figure 6-11 Rest seat preparations on the premolar and molar fulfill requirements of properly prepared rest seats. Prepa-rations are extended lingually to provide strength (through bulk) without overly filling the interproximal space with a minor con-nector. This type of preparation is challenging for natural tooth modification, and care must be exercised to avoid violation of contact points—yet the marginal ridge of each abutment should be sufficiently lowered (1.5 mm). Figure 6-12 View of mounted casts with the framework fully seated illustrates that interocclusal space was made available by a properly prepared rest seat. Figure 6-9 The intaglio surface of an onlay occlusal rest restoring contour and occlusion for this maxillary molar. 61 Chapter 6 Rests and Rest Seats Rests placed on sound enamel are not conducive to caries in a mouth with a low-caries index, provided that good oral hygiene is maintained. Proximal tooth surfaces are much more vulnerable to caries attack than are the occlusal sur-faces supporting an occlusal rest. The decision to use abut-ment coverage is usually based on needed mouth preparation, determined from the survey of diagnostic casts, to accom-modate modifications of abutment teeth necessary to fabri-cate a removable partial denture (see Chapter 11). When precarious fissures are found in occlusal rest areas in teeth that are otherwise sound, they may be removed and appro-priately restored without resorting to more extensive abut-ment protection. Although it cannot be denied that the best protection from caries for an abutment tooth is full cover-age, it is imperative that such crowns be contoured properly to provide support, stabilization, and retention for the partial denture. When the decision of whether to use unprotected enamel surfaces for rests is made, the future vulnerability of each tooth must be considered because it is not easy to fabricate full crowns to accommodate rests and clasp arms after the partial denture has been made. In many instances, sound enamel may be used safely for the support of occlusal rests. In such situations, the patient should be advised that future susceptibility to caries is not predictable, and that much depends on oral hygiene and possible future changes in caries susceptibility. Although the decision to use unpro-tected abutments logically should be left up to the dentist, economic factors may influence the final decision. The patient should be informed of the risks involved and of his or her responsibility for maintaining good oral hygiene and for returning periodically for observation. Rest seat preparations should be made in sound enamel. In most instances, preparation of proximal tooth surfaces is necessary to provide proximal guiding planes and to elimi-nate undesirable undercuts that rigid parts of the framework must pass over during placement and removal. The prepara-tion of occlusal rest seats always must follow proximal prep-aration, never precede it. Only after the alteration of proximal The lingual interproximal area requires only minor prep-aration. Creation of a vertical groove must be avoided to prevent a torquing effect on the abutments by the minor connector. Internal Occlusal Rests A partial denture that is totally tooth supported by means of cast retainers on all abutment teeth may use intracoronal rests for both occlusal support and horizontal stabilization (Figure 6-13). An intracoronal rest is not a retainer and should not be confused with an attachment. Occlusal support is derived from the floor of the rest seat. Horizontal stabilization is derived from the near-vertical walls of this type of rest seat. The form of the rest should be parallel to the path of place-ment, slightly tapered occlusally, and slightly dovetailed to prevent dislodgment proximally. The main advantages of the internal rest are that it facili-tates the elimination of a visible clasp arm buccally and permits the location of the rest seat in a more favorable posi-tion in relation to the tipping axis (horizontal) of the abut-ment. Retention is provided by a lingual clasp arm, either cast or of wrought wire, lying in a natural or prepared infra-bulge area on the abutment tooth. Internal rests are carved in wax or spark eroded in abut-ment castings. Ready-made plastic rest patterns are readily available and can be waxed into crown or partial-veneer patterns, invested, and cast after having been positioned par-allel to the path of placement with a dental cast surveyor. Future developments and techniques promise more wide-spread use of the internal occlusal rest but only for tooth-supported partial dentures. Support For Rests Rests may be placed on sound enamel or on any restoration material that has been proven scientifically to resist fracture and distortion when subjected to applied forces. A B Figure 6-13 Maxillary tooth-supported removable partial denture utilizing internal occlusal rests. A, Wax pattern developed utilizing internal rests on canine, premolar, and molars. B, Maxillary framework on cast with the internal rests fitted within surveyed crowns. 62 Part I General Concepts/Treatment Planning the outline form of the occlusal rest seat sometimes is irrepa-rably altered. The possibility that an existing restoration may be perfo-rated in the process of preparation of an ideal occlusal rest seat is always present. Although some compromise is per-missible, the effectiveness of the occlusal rest seat should not be jeopardized for fear of perforating an existing restoration. The rest seat may be widened to compensate for shallowness, but the floor of the rest seat should still be slightly inclined apically from the marginal ridge. When this is not possible, a secondary occlusal rest should be used on the opposite side of the tooth to prevent slipping of the primary rest. When perforation does occur, it may be repaired, but occasionally the making of a new restoration is unavoidable. In such a situation, the original preparation should be modi-fied to accommodate the occlusal rest, thereby avoiding the risk of perforating the completed restoration or fabricating a restoration with an inadequate rest seat. Occlusal rest seat location in new restorations should be known when the tooth is prepared so that sufficient clear-ance may be provided for the rest seat within the prepara-tion. The final step in the preparation of the tooth should be to make sure such clearance exists and, if not, to make a depression to accommodate the depth of the rest (Figure 6-15). Occlusal rest seats in crowns and inlays generally are made somewhat larger and deeper than those in enamel. Those made in abutment crowns for tooth-supported den-tures may be made slightly deeper than those in abutments that support a distal extension base; thus they approach the effectiveness of boxlike internal rests. Internal rest seats also should be created first in wax, either with suitable burs in a handpiece holder or by waxing around a lubricated mandrel, which is held in the surveyor. In either situation, the rest preparation must be finished on the casting with burs in a handpiece holder or with a preci-tooth surfaces is completed may the location of the occlusal rest seat in relation to the marginal ridge be determined. When proximal preparation follows occlusal rest seat prepa-ration, the inevitable consequence is that the marginal ridge is too low and too sharp, with the center of the floor of the rest seat too close to the marginal ridge. Therefore it is often impossible to correct the rest preparation without making it too deep, which causes irreparable damage to the tooth. Occlusal rest seats in sound enamel may be prepared with burs and polishing points that leave the enamel surface as smooth as the original enamel (Figure 6-14). The larger round bur is used first to lower the marginal ridge and to establish the outline form of the rest seat. The resulting occlusal rest seat is then complete except that the floor is not sufficiently concave. A slightly smaller round bur then is used to deepen the floor of the occlusal rest seat. At the same time, it forms the desired spoon shape inside the lowered marginal ridge. The preparation is smoothed by a polishing point of suitable size and shape. When a small enamel defect is encountered in the prepa-ration of an occlusal rest seat, it is usually best to ignore it until the rest preparation has been completed. Then, with small burs, the remaining defect should be prepared to receive a small restoration. This may be finished flush with the floor of the rest preparation that was previously established. A fluoride gel should be applied to abutment teeth fol-lowing enamel recontouring. If the master cast will be fab-ricated from an irreversible hydrocolloid impression, application of the gel should be delayed until after impres-sions are made because some fluoride gels and irreversible hydrocolloids may be incompatible. Occlusal rest seat preparations in existing restorations are treated the same way as those in sound enamel. Any proxi-mal preparations must be done first, for if the occlusal rest seat is placed first and then the proximal surface is prepared, A B Figure 6-14 Recontouring of axial surfaces and rest seat preparations in enamel may be readily accomplished with selected use of accessories. A, The two pear-shaped, multifluted burs (two burs on the left) can be used for cingulum rests and rounding marginal ridges; the longer straight, multifluted enamelplasty bur (middle bur) is ideal for height of contour adjustments and guide plane prepa-ration; round multifluted or carbide burs (three burs at right of middle) are used for occlusal rest preparation; and the inverted cone (far right bur) can be used for cingulum rests as well. B, Various abrasive rubber polishing points are necessary to ensure a smooth surface finish following any enamelplasty procedure. Following the manufacturer’s recommended sequence of abrasives should return the surface to smoothness comparable with the original condition. Diamonds are not recommended for this type of tooth reduction. 63 Chapter 6 Rests and Rest Seats may be the only abutment available for occlusal support of the denture. Also an anterior tooth occasionally must be used to support an indirect retainer or an auxiliary rest. A canine is much preferred over an incisor for this purpose. When a canine is not present, multiple rests that are spread over several incisor teeth are preferable to the use of a single incisor. Root form, root length, inclination of the tooth, and ratio of the length of the clinical crown to the alveolar support must be considered when the site and form of rests placed on incisors are determined. A lingual rest is preferable to an incisal rest because it is placed closer to the horizontal axis of rotation (tipping axis) of the abutment and therefore will have less of a tendency to tip the tooth. In addition, lingual rests are more esthetically acceptable than are incisal rests. If an anterior tooth is sound and the lingual slope is gradual rather than perpendicular, a lingual rest sometimes may be placed in an enamel seat at the cingulum or just incisally to the cingulum (Figure 6-16). This type of lingual rest is usually confined to maxillary canines that have a gradual lingual incline and a prominent cingulum. In a few instances, such a rest also may be placed on maxillary central incisors. The lingual slope of the mandibular canine is usually too steep for an adequate lingual rest seat to be placed in the enamel, and some other provision for rest support must be made. Lingual rest seat preparations in enamel are rarely satisfactory on mandibular anterior teeth because of lack of thickness of the enamel in which a seat of adequate form to be truly supportive is prepared. The preparation of an anterior tooth to receive a lingual rest may be accomplished in one of two ways: 1. A slightly rounded V is prepared on the lingual surface at the junction of the gingival and the middle one third of the tooth. The apex of the V is directed incisally. This preparation may be started by using an inverted, cone-shaped diamond stone and progressing to smaller, tapered stones with round ends to complete the preparation. All sion drill press. Plastic and metal shoes that fit over a mandrel are also available for this purpose. Thus a smooth casting is ensured, and the need for finishing the inside of the internal rest with burs is eliminated. Sufficient clearance must be provided in the preparation of the abutment to accommo-date the depth of the internal rest. Lingual Rests on Canines and Incisor Teeth Analysis of mounted diagnostic casts is mandatory in assess-ment of incisal and lingual contact areas where rests are to be placed. Sufficient space must be present or created to avoid interference with placement of rests. Although the preferred site for an external rest is the occlusal surface of a molar or a premolar, an anterior tooth Figure 6-15 Preparation on mandibular premolar for sur-veyed crown incorporates space for mesio-occlusal rest seat. Adequate occlusal reduction was accomplished to accommodate the depth of the rest seat in the abutment crown. The modifica-tion for the rest seat is performed following the standard crown preparation. Figure 6-16 Three views of lingual rest seat prepared in enamel of the maxillary canine. The rest seat, from the lingual aspect, assumes the form of a broad inverted V, maintaining the natural contour sometimes seen in a maxillary canine cingulum. An inverted V notch form is self-centering for the rest and at the same time directs forces rather favorably in an apical direction. From the incisal view, it will be noted that the rest seat preparation is broadest at the most lingual aspect of the canine. As the preparation approaches the proximal surface of the tooth, it is less broad than at any other area. The proximal view demonstrates correct taper of the floor of the rest seat. It also should be noted that the borders of the rest seat are slightly rounded to avoid line angles in its preparation. The mesiodistal length of the preparation should be a minimum of 2.5 to 3 mm, labiolingual width about 2 mm, and incisal-apical depth a minimum of 1.5 mm. It is a risky preparation and should not be attempted on lower anterior teeth. 64 Part I General Concepts/Treatment Planning restoration. The three-quarter crown may be used if the labial surface of the tooth is sound, and if the retentive con-tours are satisfactory. However, if the labial surface presents inadequate or excessive contours for placement of a reten-tive clasp arm, or if gingival decalcification or caries is present, a veneered complete coverage restoration should be used. In some instances, ball types of rests may be used in pre-pared seats. Such rest seats may be cautiously prepared in tooth surfaces with overly sufficient enamel thickness or may be prepared in restorations placed in teeth in which enamel thickness is inadequate. Conservative restorations (e.g., silver amalgam, pin inlays, composite resin) in anterior teeth may be better suited for ball types of rest seats than are the less conservative inverted V types of rest seats. Some scientific evidence demonstrates that individually cast chromium-cobalt alloy rest seat forms (attached to lingual surfaces of anterior teeth by the use of composite resin cements with acid-etched tooth preparation), lami-nates, and composite resins have been successfully used as conservative approaches to forming rest seats on teeth with unacceptable lingual contours (Figure 6-19). Sapphire ceramic orthodontic brackets have also been bonded to the lingual surfaces of mandibular canines and shaped as rest seats. These have advantages over the metal acid-etched retained rests in that a laboratory step is avoided and increased bond strengths are achieved. The major disadvan-tage of using orthodontic brackets is that removal of the rest seat would necessitate that they be ground off with the potential of heat generation and possible pulpal damage. Incisal Rests and Rest Seats Incisal rests are placed at the incisal angles of anterior teeth and on prepared rest seats. Although this is the least desir-line angles must be eliminated, and the rest seat must be prepared within the enamel and must be highly polished. Shaped, abrasive rubber polishing points, followed by flour of pumice, produce an adequately smooth and pol-ished rest seat. A predetermined path of placement for the denture must be kept in mind when the rest seat is prepared. The lingual rest seat must not be prepared as though it was going to be approached from a direction perpendicular to the lingual slope. The floor of the rest seat should be toward the cingulum rather than the axial wall. Care must be taken not to create an enamel under-cut, which interferes with placement of the denture (see Figure 6-16). 2. The most satisfactory lingual rest from the standpoint of support is one that is placed on a prepared rest seat in a cast restoration (Figure 6-17). This is done most effec-tively by planning and executing a rest seat in the wax pattern rather than by attempting to cut a rest in a cast restoration in the mouth. The contour of the framework may then restore the lingual form of the tooth. When the cingulum in the wax pattern is accentuated, the floor of the rest seat is readily carved to be the most apical portion of the preparation. A saddlelike shape, which pro-vides a positive rest seat located favorably in relation to the long axis of the tooth, is formed. The framework of the denture is made to fill out the continuity of the lingual surface so that the tongue contacts a smooth surface without the patient being conscious of bulk or irregularities. The lingual rest may be placed on the lingual surface of a cast veneer crown (Figure 6-18), a three-quarter crown, an inlay, a laminate veneer, a composite restoration, or an etched metal restoration. The latter displays less metal than the three-quarter crown, especially on the mandibular canine, where the lingual rest that was placed on a cast res-toration is frequently used, and it is a more conservative Figure 6-17 Rest seat preparation can be exaggerated for better support when it is prepared in cast restoration. Figure 6-18 Positive vertical support for a prosthesis is fur-nished by rest seats prepared in splinted metal ceramic crowns. The cingulum rest seats are optimally placed as near the hori-zontal axis of rotation as possible to minimize tipping forces. 65 Chapter 6 Rests and Rest Seats In the event that full incisal rests are considered, the patient should be thoroughly informed regarding their loca-tion, form, and esthetic impact. It is, of course, essential that both the master cast and the casting be accurate if rests are to seat properly. The incisal rest should be overcontoured slightly to allow for labial and incisal finishing to the adjoining enamel, in much the same manner as a three-quarter crown or inlay margin is finished to enamel. In this way, minimal display of metal is possible without jeopardizing the effectiveness of the rest. Care taken in selecting the type of rest seat to be used, in preparing it, and in fabricating the framework casting does much to ensure the success of any type of rest. The topog-raphy of any rest should be such that it restores the topog-raphy of the tooth that exists before the rest seat is prepared. able placement of a rest seat for reasons previously men-tioned, it may be used successfully for selected patients when the abutment is sound and when a cast restoration is not otherwise indicated. Therefore incisal rests generally are placed on enamel (Figure 6-20). Incisal rests are used pre-dominantly as auxiliary rests or as indirect retainers. Although the incisal rest may be used on a canine abut-ment in either arch, it is more applicable to the mandibular canine. This type of rest provides definite support with rela-tively little loss of tooth structure and little display of metal. Esthetically it is preferable to the three-quarter crown. The same criteria may be applied when one is deciding whether to use unprotected enamel for an occlusal rest on a molar or premolar. An incisal rest is more likely than a lingual rest to lead to some orthodontic movement of the tooth because of unfavorable leverage factors. An incisal rest seat is prepared in the form of a rounded notch at the incisal angle of a canine or on the incisal edge of an incisor, with the deepest portion of the preparation apical to the incisal edge (Figure 6-21). The notch should be beveled both labially and lingually, and the lingual enamel should be partly shaped to accommodate the rigid minor connector connecting the rest to the framework. An incisal rest seat should be approximately 2.5 mm wide and 1.5 mm deep so that the rest will be strong without having to exceed the natural contour of the incisal edge (Figure 6-22). In the absence of other suitable placements for incisal rests and rest seats, incisal rests on multiple mandibular incisors may be considered. Use of such rests may be justi-fied by the following factors: 1. They may take advantage of natural incisal faceting. 2. Tooth morphology does not permit other designs. 3. Such rests can restore defective or abraded tooth anatomy. 4. Incisal rests provide stabilization. 5. Full incisal rests may restore or provide anterior guidance. Figure 6-19 Chromium-cobalt lingual rest seat on mandibu-lar canine retained by resin cement with acid-etched tooth preparation. Figure 6-20 A, Incisal rest seat placed in the mesial incisal edge of the lower canine. Note that the contact point is not involved in preparation of the rest seat. B, Mesial incisal rests on the canines will furnish excellent vertical support and indirect retention for this prosthesis upon completion. The incisal rest on tooth #27 will also provide a third point of reference when frame orientation is established during maintenance reline procedures. A B 66 Part I General Concepts/Treatment Planning 2.5 mm 1.5 mm Figure 6-22 Dimensions given in the illustration for an incisal rest seat preparation will provide adequate strength of the frame-work at the junction of the rest and the minor connector. Rest seats of smaller dimension have proved unsatisfactory regard-less of the metal alloy from which the framework is made. Figure 6-21 Three views of an incisal rest seat preparation on the mandibular canine adjacent to a modification space. The labial view demonstrates inclination of the floor of the rest seat, which allows forces to be directed along the long axis of the tooth as nearly as possible. Note that the floor of the rest seat has been extended slightly onto the labial aspect of the tooth. As can be seen from a proximal view, the proximal edge of the rest seat is rounded rather than straight. The lingual view shows that all borders of the rest seat are rounded to avoid sharp line angles. It is especially important to avoid a line angle at the junction of the axial wall of the prepa-ration and the floor of the rest seat. The rest that occupies such a preparation should be able to move slightly in a lateral direction to avoid torquing of the abutment tooth. Implants As a Rest Implants can also be considered to serve as a rest when one takes advantage of the vertical stiffness characteristic they possess. In this application, they can serve to resist tissue-ward movement alone and may be considered useful for retentive needs as well. When a dual role is considered, selec-tion of the retentive element will require consideration of how much stress from vertical movement resistance (the rest function) will affect the retentive function of the attachment mechanism. When used as a rest, implants can serve to efficiently resist vertical movement and provide positive support. This use allows a low profile connection (i.e., close to the ridge), imparting less torque to the implant. Location can be pre-scribed for maximum advantage, and the implants can effec-tively alter rotation around a fulcrum line—eliminating it if applied to an unsupported end of a denture base, or reduc-ing the effects by decreasing the effective lever arm. 67 CHAPTER 7 Direct Retainers Chapter Outline Direct Retainer’s Role in Control of Prosthesis Movement Basic Principles of Clasp Design Reciprocal arm functions Types of Direct Retainers Criteria for Selecting a Given Clasp Design Types of Clasp Assemblies Clasps designed to accommodate functional movement Clasps designed without movement accommodation Analysis of Tooth Contours for Retentive Clasps Amount of Retention Size of and distance into the angle of cervical convergence Length of clasp arm Diameter of clasp arm Cross-sectional form of the clasp arm Material used for the clasp arm Relative uniformity of retention Stabilizing-reciprocal cast clasp arm Implants as Direct Retainers Other Types of Retainers Lingual retention in conjunction with internal rests Internal Attachments Direct Retainer’s Role in Control of Prosthesis Movement Retention of a removable prosthesis is a unique concern when compared with other prostheses. When one is dealing with a crown or fixed partial denture, the combined use of preparation geometry (i.e., resistance and retention form) and a luting agent can fix the prosthesis to the tooth in a manner that resists all forces to which the teeth are sub-jected. As was mentioned in Chapter 4, the direction of forces can be toward, across, or away from the tissue. In general, the forces acting to move prostheses toward and across the supporting teeth and/or tissue are the greatest in intensity. This is because most often they are forces of occlusion. Forces acting to displace the prosthesis from the tissue can consist of gravity acting against a maxillary prosthesis, the action of adherent foods acting to displace the prosthesis on opening of the mouth in chewing, or functional forces acting across a fulcrum to unseat the prosthesis. The first two of these forces are seldom at the magnitude of functional forces, and the latter is minimized through the use of ade-quate support. The component part applied to resist this movement away from the teeth and/or tissue provides reten-tion for the prosthesis and is called the direct retainer. A direct retainer is any unit of a removable dental prosthesis that engages an abutment tooth or implant to resist displace-ment of the prosthesis away from basal seat tissue. The direct retainer’s ability to resist this movement is greatly influenced by the stability and support of the prosthesis provided by major and minor connectors, rests, and tissue bases. This relationship of the supportive and retentive components highlights the relative importance of these component parts. Although the forces working against a removable partial denture to move it away from the tissue generally are not as great as the functional forces causing stress toward the tissue, the removable partial denture must have retention appropri-ate to resist reasonable dislodging forces. Too often reten-tion concerns are given greater importance than is 68 Part I General Concepts/Treatment Planning Buccal Lingual Buccal Lingual A B Figure 7-1 A, Line drawn through the illustration represents 180 degrees of greatest circumference of abutment from the occlusal rest. Unless portions of the lingual reciprocal arm and the retentive buccal arm are extended beyond the line, the clasp would not accomplish its intended purpose. If respective arms of the retainer were not extended beyond the line, the abutment tooth could be forced away from the retainer through torquing action of the clasp, or the removable partial denture could move away from the abutment. B, Bar-type clasp assembly engagement of more than 180 degrees of circumference of the abutment is realized by the minor connector for the occlusal rest, the minor connector contacting the guiding plane on the distal proximal surface, and the retentive bar arm. A B Figure 7-2 A, Flexing action of the retentive clasp arm initi-ates medially directed pressure on the abutment teeth as its retentive tip springs over the height of contour. B, Reciprocation to medially directed pressure is counteracted by rigid lingually placed clasp arms contacting the abutments simultaneously with the buccal arms, or by rigid stabilizing components of the frame-work contacting the lingual guiding planes when the buccal arms begin to flex. appropriate, especially if such a focus detracts from more serious consideration of the resistance of typical functional forces. Sufficient retention is provided by two means. Primary retention for the removable partial denture is accomplished mechanically by placing retaining elements (direct retainers) on the abutment teeth. Secondary retention is provided by the intimate relationship of the minor connector contact with the guiding planes and denture bases, and of the major connector (maxillary) with underlying tissue. The latter is similar to the retention of a complete denture. It is propor-tionate to the accuracy of the impression registration, the accuracy of the fit of the denture bases, and the total involved area of contact. Retention can also be provided through engagement of an attachment mechanism on a dental implant. Basic Principles of Clasp Design The clasp assembly serves a similar function for a removable partial denture that a retainer crown serves for a fixed partial denture. Both must encircle the prepared tooth in a manner that prevents movement of the tooth separate from the retainer. To borrow from a fixed prosthodontic term, limit-ing the freedom of displacement refers to the effect of one cylindrical surface (the framework encircling the tooth) on another cylindrical surface (the tooth). It implies that the curve that defines the framework is properly shaped if it prevents movement at right angles to the tooth axis. This basic principle of clasp design offers a two-way benefit. First, it ensures the stability of the tooth position because of the restraint from encirclement, and second, it ensures stability of the clasp assembly because of the controlled position of the clasp in three dimensions. Therefore the basic principle of clasp design, referred to as the principle of encirclement, means that more than 180 degrees in the greatest circumference of the tooth, passing from diverging axial surfaces to converging axial surfaces, must be engaged by the clasp assembly (Figure 7-1). The engagement can occur in the form of continuous contact, such as in a circumferential clasp, or discontinuous contact, such as in the use of a bar clasp. Both provide tooth contact in at least three areas encircling the tooth: the occlusal rest area, the retentive clasp terminal area, and the reciprocal clasp terminal area. In addition to encirclement, other basic principles of clasp design are as follows: 1. The occlusal rest must be designed to prevent movement of the clasp arms toward the cervical. 2. Each retentive terminal should be opposed by a reciprocal component capable of resisting any transient pressures exerted by the retentive arm during placement and removal. Stabilizing and reciprocal components must be rigidly connected bilaterally (cross-arch) to realize recip-rocation of the retentive elements (Figure 7-2). 69 Chapter 7 Direct Retainers caused by a reduction in the effort arm as described in Chapter 4. Reciprocal Arm Functions As was mentioned earlier, reciprocal arms are intended to resist tooth movement in response to deformation of the retainer arm as it engages a tooth height of contour. The opposing clasp arm reciprocates the effect of this deforma-tion as it prevents tooth movement. For this to occur, the reciprocal arm must be in contact during the time of retainer arm deformation. Unless the abutment tooth has been spe-cifically contoured, the reciprocal clasp arm will not come into contact with the tooth until the denture is fully seated and the retentive clasp arm has again become passive. When this happens, a momentary tipping force is applied to the abutment teeth during each placement and removal. This may not be a damaging force—because it is transient—so long as the force does not exceed the normal elasticity of the periodontal attachments. True reciprocation during place-ment and removal is possible only through the use of crown surfaces made parallel to the path of placement. The use of cast restorations permits the parallel surfaces to be contacted by the reciprocal arm in such a manner that true reciproca-tion is made possible. This is discussed in Chapter 14. Reciprocal arms can have additional functions as well. The reciprocal clasp arm should be located so that the denture is stabilized against horizontal movement. Stabiliza-tion is possible only through the use of rigid clasp arms, rigid minor connectors, and a rigid major connector. Horizontal forces applied on one side of the dental arch are resisted by the stabilizing components on the opposite side, providing cross-arch stability. Obviously the greater the number of such components, within reason, the greater will be the dis-tribution of horizontal stresses. b a a b a b b a a b Figure 7-3 Retentive clasps should be bilaterally opposed. This means bilateral buccal or bilateral lingual undercuts should be used, as shown on this Class III, modification 2 arch, in which the retention may be (a) bilaterally buccal or (b) bilaterally lingual. 3. Clasp retainers on abutment teeth adjacent to distal extension bases should be designed so that they will avoid direct transmission of tipping and rotational forces to the abutment. In effect, they must act as stress-breakers, either by their design or by their construction. This is accomplished through proper location of the retentive terminal relative to the rest, or by the use of a more flex-ible clasp arm in relation to the anticipated rotation of the denture under functional forces. 4. Unless guiding planes will positively control the path of removal and will stabilize abutments against rotational movement, retentive clasps should be bilaterally opposed (i.e., buccal retention on one side of the arch should be opposed by buccal retention on the other, or lingual on one side opposed by lingual on the other). In Class II situ-ations, the third abutment may have buccal or lingual retention. In Class III situations, retention may occur bilaterally or may be diametrically opposed (Figure 7-3). 5. The path of escapement for each retentive clasp terminal must be other than parallel to the path of removal for the prosthesis to require clasp engagement with the resistance to deformation that is retention. 6. The amount of retention should always be the minimum necessary to resist reasonable dislodging forces. 7. Reciprocal elements of the clasp assembly should be located at the junction of the gingival and middle thirds of the crowns of abutment teeth. The terminal end of the retentive arm is optimally placed in the gingival third of the crown (Figures 7-4 through 7-6). These locations permit better resistance to horizontal and torquing forces Occlusal third Stabilizing Retentive Middle third Survey line Gingival third Figure 7-4 Simple mechanical laws demonstrate that the nearer stabilizing-reciprocal and retentive elements of direct retainer assemblies are located horizontal to the axis of rotation of the abutment, the less likely it is that physiologic tolerance of the periodontal ligament will be exceeded. The horizontal axis of rotation of the abutment tooth is located somewhere in its root. 70 Part I General Concepts/Treatment Planning Types of Direct Retainers Mechanical retention of removable partial dentures is accomplished by means of direct retainers of one type or another. Retention is accomplished by using frictional means, by engaging a depression in the abutment tooth, or by engaging a tooth undercut lying cervically to its height of contour. Two basic types of direct retainers are available: the intracoronal retainer and the extracoronal retainer. The extracoronal (clasp-type) retainer is the most commonly used retainer for removable partial dentures. The reciprocal clasp arm also may act to a minor degree as an indirect retainer. This is true only when it rests on a suprabulge surface of an abutment tooth lying anterior to the fulcrum line (see Figure 8-8). Lifting of a distal extension base away from the tissue is thus resisted by a rigid arm, which is not easily displaced cervically. The effectiveness of such an indirect retainer is limited by its proximity to the fulcrum line, which gives it a relatively poor leverage advan-tage, and by the fact that slippage along tooth inclines is always possible. The latter may be prevented by the use of a ledge on a cast restoration; however, enamel surfaces are not ordinarily so prepared. Lingual Occlusal Buccal Support Stabilization Retention Occlusal third Middle third Gingival third Support Stabilization Retention Occlusal third Middle third Gingival third A B C Figure 7-5 A bar-type clasp on the mandibular premolar. A, Support is provided by the occlusal rest. B, Stabilization is provided by the occlusal rest and the mesial and distal minor connectors. C, Retention is provided by the buccal I-bar. Reciprocation is obtained through the location of the minor connectors. Engagement of more than 180 degrees of circumference of the abutment is accomplished by proper location of components contacting the axial surfaces. (The minor connector supports the occlusal rest, the proximal plate minor connector, and the buccal I-bar.) 71 Chapter 7 Direct Retainers In situations where support requirements are adequately met by available teeth and/or oral tissues, dental implants can be used for retention and provide the advantage of elimi-nation of a visible clasp. Criteria for Selecting a Given Clasp Design When a particular clasp design is selected for a given situa-tion, its function and limitations must be carefully evaluated. Extracoronal direct retainers, as part of the clasp assembly, should be considered as components of a removable partial denture framework. They should be designed and located to perform the specific functions of support, stabilization, reciprocation, and retention. It does not matter whether the direct retainer-clasp assembly components are physically attached to each other, or originate from major and minor connectors of the framework (see Figures 1-2 and 1-3, B-E). If attention is directed to the separate function of each com-ponent of the direct retainer-clasp assembly, then selection of a direct retainer is simplified. Although some rather complex designs are used for clasp arms, they all may be classified into one of two basic catego-ries. One is the circumferential clasp arm, which approaches the retentive undercut from an occlusal direction. The other is the bar clasp arm, which approaches the retentive under-cut from a cervical direction. A clasp assembly may comprise various retentive arms (i.e., a cast circumferential, a bar clasp arm, or a wrought-wire retainer), depending on the specific requirement for retainer construction, given the necessary adjustability, clasp approach position, and survey line location. A clasp assembly should consist of four component parts. First, one or more minor connectors should be present, from which the clasp components originate. Second, a principal rest should be designed to direct stress along the long axis of the tooth. Third, a retentive arm should engage a tooth undercut. For most clasps, the retentive region is only at its terminus. Fourth, a nonretentive arm (or other component) should be present on the opposite side of the tooth for sta-bilization and reciprocation against horizontal movement of the prosthesis (rigidity of this clasp arm is essential to its purpose). No confusion should exist between the choice of clasp arm and the purpose for which it is used. Either type of cast clasp arm (bar or circumferential) may be made tapered and retentive, or nontapered (rigid) and nonretentive. The choice depends on whether it is used for retention, stabiliza-tion, or reciprocation. An occlusal rest, such as in the RPI (rest, proximal plate, and I-bar component parts of the clasp assembly) concept, may be used rather than a reciprocal clasp arm to satisfy the need for encirclement, provided it is located in such a way that it can accomplish the same purpose (Figure 7-10; see also Figure 7-9). The addition of a lingual apron to a cast reciprocal clasp arm alters neither its primary The intracoronal retainer may be cast or may be attached totally within the restored natural contours of an abutment tooth. It is typically composed of a prefabricated machined key and keyway with opposing vertical parallel walls, which serve to limit movement and resist removal of the partial denture through frictional resistance (Figure 7-7). The intra-coronal retainer is usually regarded as an internal, or preci-sion, attachment. The principle of the internal attachment was first formulated by Dr. Herman E.S. Chayes in 1906. The extracoronal retainer uses mechanical resistance to displacement through components placed on or attached to the external surfaces of an abutment tooth. The extracoronal retainer is available in three principal forms. The clasp-type retainer (Figures 7-8 and 7-9), the form used most com-monly, retains through a flexible clasp arm. This arm engages an external surface of an abutment tooth in an area cervical to the greatest convexity of the tooth, or it engages a depres-sion prepared to receive the terminal tip of the arm. The other forms both involve manufactured attachments and include interlocking components or the use of a spring-loaded device that engages a tooth contour to resist occlusal displacement. Another type is a manufactured attachment, which uses flexible clips or rings that engage a rigid compo-nent that is cast or attached to the external surface of an abutment crown. Support Stabilization Retention Occlusal third Buccal Lingual Middle third Gingival third Support Stabilization Retention Occlusal third Middle third Gingival third Figure 7-6 Circumferential clasp on the mandibular premo-lar. Support is provided by the occlusal rest; stabilization is provided by the occlusal rest, proximal minor connector, lingual clasp arm, and rigid portion of the buccal retentive clasp arm occlusal to the height of contour; retention is realized by the retentive terminal of the buccal clasp arm; reciprocation is pro-vided by the nonflexible lingual clasp arm. Assembly engages more than 180 degrees of the abutment tooth’s circumference. 72 Part I General Concepts/Treatment Planning Some clasp assemblies are designed to accommodate prosthesis functional movement (as mentioned in the basic principles above), and others do not incorporate such design features. Although it has been demonstrated by Kapur and others that adverse outcomes are not always associated with the use of rigid clasp assemblies in distal extension classifica-tions, the following clasp assemblies will be described as clasps designed to accommodate distal extension functional movement and clasps designed without movement accom-modation. The clinician should not interpret these catego-ries as mutually exclusive because most any clasp assembly can be used to retain a well-supported and maintained prosthesis. purpose nor the need for proper location to accomplish that purpose. Types of Clasp Assemblies A wide variety of clasp assemblies are available for clinicians to use. This variety exists largely because of the imagination of clinicians and technicians who provided prostheses when tooth modification was not or could not be provided. To simplify clasp design and to improve the functional predict-ability of prostheses, today’s clinician must realize the need for tooth modification. A B C Figure 7-7 A, The intracoronal retainer consists of a key and keyway with extremely small tolerance. Keyways are contained within abutment crowns, and (B) keys are attached to the removable partial denture framework. C, Frictional resistance to removal and place-ment and limitation of movement serve to retain and stabilize the prosthesis. 73 Chapter 7 Direct Retainers Clasps Designed to Accommodate Functional Movement RPI, RPA, and Bar Clasp Clasp assemblies that accommodate functional prosthesis movement are designed to address the concern of a Class I lever. The concern is that the distal extension acts as a long “effort arm” across the distal rest “fulcrum” to cause the clasp tip “resistance arm” to engage the tooth undercut. This results in harmful tipping or torquing of the tooth, which is greater with stiff clasps and increased denture base move-ment. Two strategies may be adopted to change the fulcrum location and subsequently the “resistance arm” engaging effect (mesial rest concept clasp assemblies), or to minimize the effect of the lever through the use of a flexible arm (wrought-wire retentive arm). Mesial rest concept clasps have been proposed to accom-plish movement accommodation by changing the fulcrum location. This concept includes the RPI and RPA [rest, prox-imal plate, Akers] clasps. The RPI is a current concept of bar clasp design that refers to the rest, proximal plate, and I-bar component parts of the clasp assembly. Basically, this clasp assembly consists of a mesio-occlusal rest with the minor connector placed into the mesiolingual embrasure, but not contacting the adjacent tooth (Figure 7-11, A). A distal guiding plane, extending from the marginal ridge to the junction of the middle and gingival thirds of the abutment Support C A B Stabilization Retention Support Stabilization Retention Lingual Buccal Figure 7-8 Extracoronal circumferential direct retainer. Assembly consists of (A) the buccal retentive arm; (B) the rigid lingual stabilizing (reciprocal) arm; and (C) the supporting occlu-sal rest. The terminal portion of the retentive arm is flexible and engages measured undercut. Assembly remains passive until activated by placement or removal of the restoration, or when subjected to masticatory forces that tend to dislodge the denture base. Support Stabilization Retention Support Stabilization HOC UC Retention Buccal Lingual B C D D B A Figure 7-9 Extracoronal bar-type direct retainer. Assembly consists of (A) the buccal retentive arm engaging the measured undercut (with slight occlusal extension for stabilization, see insert where HOC is height of contour and UC is undercut); (B) the stabilizing (reciprocal) elements, with the proximal plate minor connector on the distal; (C) the lingually placed mesial minor connector for the occlusal rest, which also serves as a stabilizing (reciprocal) component; and (D) the mesially placed supporting occlusal rest. Assembly remains passive until activated. 74 Part I General Concepts/Treatment Planning should extend the entire length of the proximal tooth surface, with physiologic tissue relief eliminating impingement of the free gingival margin (Figure 7-12). A second approach sug-gests that the guiding plane and the corresponding proximal plate minor connector should extend from the marginal ridge to the junction of the middle and gingival thirds of the proximal tooth surface (Figure 7-13). Both approaches rec-ommend that the retaining clasp arm should be located in the gingival third of the buccal or labial surface of the abut-ment in a 0.01-inch undercut. Placement of the retaining clasp arm generally occurs in an undercut located at the greatest mesiodistal prominence of the tooth or adjacent to the extension base area (Figure 7-14, A and B). The third approach favors a proximal plate minor connector that con-tacts approximately 1 mm of the gingival portion of the guiding plane (Figure 7-15, A) and a retentive clasp arm located in a 0.01-inch undercut in the gingival third of the tooth at the greatest prominence or toward the mesial away from the edentulous area (Figure 7-14, C). If the abutment teeth demonstrate contraindications for a bar-type clasp (i.e., exaggerated buccal or lingual tilts, severe tissue under-cut, or a shallow buccal vestibule) and the desirable undercut is located in the gingival third of the tooth away from the extension base area, a modification should be considered for the RPI system (the RPA clasp) (Figure 7-15, B). Application of each approach is predicated on the distribution of load to be applied to the tooth and edentulous ridge. The bar clasp, which gave rise to the RPI, is discussed here because of this association. It may not be configured to allow functional movement, but it can be. The term bar clasp is generally preferred over the less descriptive term Roach clasp arm. Reduced to its simplest term, the bar clasp arm arises from the denture framework or a metal base and approaches the retentive undercut from a gingival direction (see Figure 7-11). The bar clasp arm has been classified by the shape of the retentive terminal. Thus it has been identified as a T, modified T, I, or Y. The form the terminal takes is of little significance as long as it is mechanically and functionally effective, covers as little tooth surface as possible, and dis-plays as little metal as possible. In most situations, the bar clasp arm can be used with tooth-supported partial dentures, with tooth-supported modification areas, or when an undercut that can be logi-cally approached with a bar clasp arm lies on the side of an abutment tooth adjacent to a distal extension base (Figure 7-16). If a tissue undercut prevents the use of a bar clasp arm, a mesially originating ring clasp, a cast, or a wrought-wire clasp or reverse-action clasp may be used. Preparation of adjacent abutments (natural teeth) to receive any type of interproximal direct retainer, traversing from lingual to buccal surfaces, is most difficult to adequately accomplish. Inevitably the relative size of the occlusal table is increased, contributing to undesirable and additional functional loading. Specific indications for use of a bar clasp arm include (1) when a small degree of undercut (0.01 inch) exists in tooth, is prepared to receive a proximal plate (Figure 7-11, B). The buccolingual width of the guiding plane is deter-mined by the proximal contour of the tooth (Figure 7-11, A and C). The proximal plate, in conjunction with the minor connector supporting the rest, provides the stabilizing and reciprocal aspects of the clasp assembly. The I-bar should be located in the gingival third of the buccal or labial surface of the abutment in a 0.01-inch undercut (Figure 7-11, D). The whole arm of the I-bar should be tapered to its terminus, with no more than 2 mm of its tip contacting the abutment. The retentive tip contacts the tooth from the undercut to the height of contour (Figure 7-11, E). This area of contact, along with the rest and proximal plate contact, provides stabilization through encirclement (see Figure 7-11, C). The horizontal portion of the approach arm must be located at least 4 mm from the gingival margin and even farther if possible. Three basic approaches to the application of the RPI system may be used. The location of the rest, the design of the minor connector (proximal plate) as it relates to the guiding plane, and the location of the retentive arm are factors that influence how this clasp system functions. Varia-tions in these factors provide the basis for differences among these approaches. All advocate the use of a rest located mesi-ally on the primary abutment tooth adjacent to the extension base area. One approach recommends that the guiding plane and the corresponding proximal plate minor connector Figure 7-10 The auxiliary occlusal rest (mirror view) may be used rather than the reciprocal clasp arm without violating any principle of clasp design. Its greatest disadvantages are that the second rest seat must be prepared, and that enclosed tissue space at the gingival margin can result in a food trap. The auxil-iary occlusal rest is also sometimes used to prevent slippage when the principal occlusal rest seat cannot be inclined apically from the marginal ridge. Minor connectors used to close the interproximal space most often require rests on adjacent teeth to avoid a wedging effect when force is placed on the denture. 75 Chapter 7 Direct Retainers undercut exists, either of which must be bridged by excessive blockout. When severe tooth and tissue undercuts exist, a bar clasp arm usually is an annoyance to the tongue and cheek and may traps food debris. Other limiting factors in the selection of a bar clasp assembly include a shallow vesti-bule or an excessive buccal or lingual tilt of the abutment tooth (Figure 7-18). Some common errors in the design of bar-type clasps are illustrated in Figure 7-19. the cervical third of the abutment tooth, which may be approached from a gingival direction; (2) on abutment teeth for tooth-supported partial dentures or tooth-supported modification areas (Figure 7-17); (3) in distal extension base situations; and (4) in situations in which esthetic consider-ations must be accommodated and a cast clasp is indicated. Thus use of the bar clasp arm is contraindicated when a deep cervical undercut exists or when a severe tooth and/or tissue A B C D E Support Lingual Stabilization Retention Support Stabilization Buccal Retention Occlusal third Middle third Gingival third Figure 7-11 Bar-type clasp assembly. A, Occlusal view. Component parts (proximal plate minor connector, rest with minor connec-tor, and retentive arm) tripod the abutment to prevent its migration. B, The proximal plate minor connector extends just far enough lingually so that it combines with the mesial minor connector to prevent lingual migration of the abutment. C, On narrow or tapered abutments (mandibular first premolars), the proximal plate should be designed to be as narrow as possible but still sufficiently wide to prevent lingual migration. D, I-bar retainer located at greatest prominence of tooth in the gingival third. E, Mesial view of I-bar illus-trating the retentive tip relationship to the undercut and a region superior to the height of contour, which serves a stabilization function in encirclement. 76 Part I General Concepts/Treatment Planning Combination Clasp Another strategy that may be used to reduce the effect of the Class I lever in distal extension situations includes a flexible component in the “resistance arm”; this strategy is employed in the combination clasp. The combination clasp consists of The bar clasp arm is not a particularly flexible clasp arm because of the effects of its half-round form and its several planes of origin. Although the cast circumferential clasp arm can be made more flexible than the bar clasp arm, the combination clasp is preferred for use on terminal abut-ments when torque and tipping are possible, because of engagement of an undercut away from the distal extension base. Situations often exist, however, in which a bar clasp arm may be used to advantage without jeopardizing a ter-minal abutment. A bar clasp arm swinging distally into the undercut may be a logical choice, because movement of the clasp on the abutment as the distal extension base moves tissue-ward is minimized by the distal location of the clasp terminal. Minimum tissue relief Occlusal force GP PP Figure 7-12 Bar clasp assembly in which the guiding plane (GP) and the corresponding proximal plate (PP) extend the entire length of the proximal tooth surface. Physiologic relief is required to prevent impingement of the gingival tissues during function. Extending the proximal plate to contact a greater surface area of the guide plane directs functional forces in a horizontal direction, thus the teeth are loaded to a greater extent than the edentulous ridge. Occlusal force GP PP Figure 7-13 Bar clasp assembly in which the guiding plane (GP) and the corresponding proximal plate (PP) extend from the marginal ridge to the junction of the middle and gingival thirds of the proximal tooth surface. This decrease (compared with Figure 7-23) in amount of surface area contact of the proximal plate on the guiding plane more evenly distributes functional force between the tooth and the edentulous ridge. C A B Figure 7-14 Occlusal view of the RPI (rest, proximal plate, and I-bar component parts) bar clasp assembly. Placement of the I-bar in a 0.01-inch undercut (A) on the distobuccal surface; (B) at the greatest mesiodistal prominence; and (C) on the mesio-buccal surface. Some advantages attributed to the infrabulge clasp are (1) its interproximal location, which may be used to esthetic advantage; (2) increased retention without tipping action on the abutment; and (3) less chance of accidental distortion resulting from its proximity to the denture border. The wearer should be meticulous in the care of a denture so made, not only for reasons of oral hygiene but also to prevent cariogenic debris from being held against tooth surfaces. The T and Y clasp arms are the most frequently misused. It is unlikely that the full area of a T or Y termi-nal is ever necessary for adequate clasp retention. Although the larger area of contact would provide greater frictional resistance, this is not true clasp retention, and only that portion engaging an undercut area should be considered retentive. Only one terminal of such a clasp arm should be placed in an undercut area (Figure 7-20). The remainder of the clasp arm may be superfluous unless it is needed as part of the clasp assembly to encircle the abutment tooth by more than 180 degrees at its great-est circumference. If the bar clasp arm is made to be flexible for retentive purposes, any portion of the clasp above the height of contour will provide only limited stabilization, because it is also part of the flexible arm. Therefore in many instances, this suprabulge portion of a T or Y clasp arm may be removed, and the retentive terminal of the bar clasp should be designed to be biologi-cally and mechanically sound rather than to conform to any alphabetical configuration. 77 Chapter 7 Direct Retainers Occlusal force A B GP GP PP PP Figure 7-15 A, Bar clasp assembly in which the proximal plate (PP) contacts approximately 1 mm of the gingival portion of the guiding plane (GP). During function, the proximal plate and the I-bar clasp arm are designed to move in a mesiogingival direction, disengaging the tooth. Lack of sustained contact between the proximal plate and the guiding plane distributes increased functional force to the edentulous ridge. Asterisk () indicates center of rotation. B, Modification of the RPI (rest, proximal plate, and I-bar com-ponent parts) system (rest, proximal plate, Akers or RPA clasp) is indicated when a bar-type clasp is contraindicated and a desirable undercut is located in the gingival third of the tooth away from the extension base area. Figure 7-16 Bar clasp arm properly used on the terminal abutment. The combination of rest, proximal plate, and bar clasp contacting the abutment tooth provides more than 180 degrees encirclement. A uniform taper to the bar ensures proper flexibility and internal stress distribution. Taper can originate from the minor connector junction or at a finishing line indicating the anterior extent of the denture base. Encirclement does not require that retentive tip modification (in the form of a T) be provided, and such a modification adds little to the clasp assembly. Figure 7-17 A bar retainer is used on the anterior abutment of the modification space, and its terminus engages the disto-buccal undercut. The denture is designed to rotate around the terminal abutments when force is directed toward the basal seat on the left. Such rotation would impart force on the right premo-lar directed superiorly and anteriorly. However, this direction of force is resisted in great part by mesial contact with the canine. A direct retainer on the right premolar engaging the mesiobuccal undercut would tend to force the tooth superiorly and posteriorly. 78 Part I General Concepts/Treatment Planning retainer is contraindicated. It may be used for its adjustabil-ity when precise retentive requirements are unpredictable and later adjustment to increase or decrease retention may be necessary. A third justification for its use is its esthetic advantage over cast clasps. Wrought in structure, it may be used in smaller diameters than a cast clasp, with less danger of fracture. Because it is round, light is reflected in such a manner that the display of metal is less noticeable than with the broader surfaces of a cast clasp. The most common use of the combination clasp is on an abutment tooth adjacent to a distal extension base where a wrought-wire retentive clasp arm and a cast reciprocal clasp arm (Figure 7-21). Although the latter may be seen in the form of a bar clasp arm, it is usually a circumferential arm. The retentive arm is almost always circumferential, but it also may be used in the manner of a bar, originating gin-givally from the denture base. Advantages of the combination clasp include its flexibility and adjustability and the appearance of the wrought-wire retentive arm. It is used when maximum flexibility is desir-able, such as on an abutment tooth adjacent to a distal exten-sion base or on a weak abutment when a bar-type direct A B C Figure 7-18 Contraindications for selection of bar-type clasps. A, Severe buccal or lingual tilts of abutment teeth. B, Severe tissue undercuts. C, Shallow buccal or labial vestibules. A B Occlusal third Middle third Gingival third C D E F Occlusal third Middle third Gingival third Figure 7-19 Common errors and recommended corrections in the design of bar-type clasp assemblies. A, Survey line unsuitable for the bar clasp (too high). B, Retentive portion of the bar clasp arm improperly contoured to resist dislodging force in the occlusal direction. C, Retentive tip not located in the gingival third of the abutment. D, Contour of abutment correctly altered to receive bar clasp. E and F, Correct position of bar clasp assembly. 79 Chapter 7 Direct Retainers it is bent by hand, it may be less accurately adapted to the tooth and therefore may provide less stabilization in the suprabulge portion; and (4) it may distort with function and not engage the tooth. The disadvantages of the wrought-wire clasp are offset by its several advantages, which include (1) its flexibility; (2) its adjustability; (3) its esthetic advantage over other retentive circumferential clasp arms; (4) coverage of a minimum of tooth surface as a result of its line contact with the tooth, rather than having the surface contact a cast clasp arm; and (5) a less likely occurrence of fatigue failure in service with the tapered wrought-wire retentive arm versus the cast, half-round retentive arm. The disadvantages listed previously should not prevent its use regardless of the type of alloy that is used for the cast framework. Technical problems are minimized by selecting only a mesial undercut exists on the abutment or where a large tissue undercut contraindicates a bar-type retainer (Figure 7-22). When a distal undercut exists that may be approached with a properly designed bar clasp arm or with a ring clasp (despite its several disadvantages), a cast clasp can be located so that it will not cause abutment tipping as the distal extension base moves tissue-ward. When the undercut is on the side of the abutment away from the exten-sion base, the tapered wrought-wire retentive arm offers greater flexibility than does the cast clasp arm and therefore better dissipates functional stresses. For this reason, the combination clasp is preferred (see Figure 7-22, D). The combination clasp has several disadvantages: (1) it involves extra steps in fabrication, particularly when high-fusing chromium alloys are used; (2) it may be distorted by careless handling on the part of the patient; (3) because Support Stabilization Retention Buccal Occlusal third Middle third Gingival third Figure 7-20 Only one terminal of the retentive arm engages the undercut in the gingival third of the abutment. The suprabulge portion of the retentive clasp arm provides only limited stabilization and may be eliminated. Path of placement 18-gauge round wrought wire B A Height of contour Figure 7-21 A, A combination clasp consists of a cast reciprocal arm and a tapered, round wrought-wire retentive clasp arm. The latter is cast to, or soldered to, a cast framework. This design is recommended for the anterior abutment of the posterior modification space in a Class II partially edentulous arch, where only a mesiobuccal undercut exists, to minimize the effects of a first-class lever system. B, In addition to the advantages of flexibility, adjustability, and appearance, a wrought-wire retentive arm makes only line contact with the abutment tooth, rather than broader contact with the cast clasp. 80 Part I General Concepts/Treatment Planning The patient may be taught to avoid distortion of the wrought wire by the explanation that to remove a partial denture, the fingernail should always be applied to its point of origin, where it is held rigid by the casting, rather than to the flexible terminal end. Often, lingual retention may be used rather than buccal retention, especially on a mandibu-lar abutment, so that the patient never touches the wrought-wire arm during removal of the denture. Instead, removal may be accomplished by lifting against the cast reciprocal arm located on the buccal side of the tooth. This may negate the esthetic advantage of the wrought-wire clasp arm, and esthetics should be given preference when the choice must be made between buccal and lingual retention. In most situ-ations, however, retention must be used where it is possible, and the clasp designed accordingly. Clasps Designed Without Movement Accommodation Circumferential Clasp Although a thorough knowledge of the principles of clasp design should lead to a logical application of those princi-ples, it is better when some of the more common clasp designs are considered individually. The circumferential clasp will be considered first as an all-cast clasp. The circumferential clasp is usually the most logical clasp to use with all tooth-supported partial dentures because of its retentive and stabilizing ability (Figure 7-23). Only when the retentive undercut may be approached better with a bar clasp arm or when esthetics will be enhanced should the latter be used. The circumferential clasp arm has the follow-ing disadvantages: 1. More tooth surface is covered than with a bar clasp arm because of its occlusal origin. 2. On some tooth surfaces, particularly the buccal surfaces of mandibular teeth and the lingual surfaces of maxillary teeth, its occlusal approach may increase the width of the occlusal surface of the tooth. 3. In the mandibular arch, more metal may be displayed than with the bar clasp arm. A B C D E F Figure 7-22 Five types of extracoronal direct retainer assem-blies that may be used on abutments adjacent to the distal extension base to avoid or minimize the effects of the cantilever design. Arrows indicate the general direction of movement of retentive tips of retainer arms when the denture base rotates toward and away from the edentulous ridge. A, Distobuccal undercut engaged by one-half T-type bar clasp. The portion of the clasp arm on and above the height of contour might afford some stabilization against horizontal rotation of the denture base. B, I-bar placed in undercut at the middle (anteroposteri-orly) of the buccal surface. This retainer contacts the tooth only at its tip. Note that the guiding plane on the distal aspect of the abutment is contacted by the metal of the denture framework, and that a mesial rest is used. C, Interproximal ring clasp engag-ing the distobuccal undercut. Bar-type retainer cannot be used because of tissue undercuts inferior to the buccal surface of the abutment. D, Round, uniformly tapered 18-gauge wrought-wire circumferential retainer arm engaging the mesiobuccal undercut. A wrought-wire arm, instead of a cast arm, should be used in this situation because of the ability of wrought wire to flex omni-directionally. A cast half-round retainer arm would not flex edge-wise, which could result in excessive stress on the tooth when rotation of the denture base occurs. E, A hairpin clasp may be used when the undercut lies cervical to the origin of the retainer arm. Both hairpin and interproximal ring clasps may be used to engage the distobuccal undercut on the terminal abutment of the distal extension denture. However, the distobuccal undercut on the terminal abutment should be engaged by a bar-type clasp in the absence of a large buccal tissue undercut cervical to the terminal abutment. Hairpin and interproximal ring clasps are the least desirable of the clasping situations illustrated here. F, Lingual view shows the use of double occlusal rests, connected to the lingual bar by the minor connector in illustrated designs. This design eliminates the need for a lingual clasp arm, places the fulcrum line anteriorly to make better use of the residual ridge for support, and provides stabilization against horizontal rota-tion of the denture base. Figure 7-23 Cast circumferential retentive clasp arms prop-erly designed. They originate on or occlusal to the height of contour, which they then cross in their terminal third, and engage retentive undercuts progressively as their taper decreases and their flexibility increases. the best wrought wire for this purpose, then casting to it or soldering it to the cast framework. Selection of wrought wire, attachment of it to the framework, and subsequent labora-tory procedures to maintain its optimum physical properties are presented in Chapter 12. 81 Chapter 7 Direct Retainers Figure 7-24 Cast circumferential retentive clasp arm. 4. As with all cast clasps, its half-round form prevents adjustment to increase or decrease retention. Adjust-ments in the retention afforded by a cast clasp arm should be made by moving a clasp terminal cervically into the angle of cervical convergence or occlusally into a lesser area of undercut. Tightening a clasp against the tooth or loosening it away from the tooth increases or decreases frictional resistance and does not affect the retentive potential of the clasp. True adjustment is, therefore, impossible with most cast clasps. Despite its disadvantages, the cast circumferential clasp arm may be used effectively, and many of these disadvan-tages may be minimized by mouth preparation. Adequate mouth preparation will permit its point of origin to be placed far enough below the occlusal surface to avoid poor esthetics and increased tooth dimension (see Figure 7-23). Although some of the disadvantages listed imply that the bar-type clasp may be preferable, the circumferential clasp is actually superior to a bar clasp arm that is improperly used or poorly designed. Experience has shown that faulty application and design too often negate the possible advantages of the bar clasp arm, whereas the circumferential clasp arm is not as easily misused. The basic form of the circumferential clasp is a buccal and lingual arm originating from a common body (Figure 7-24). This clasp is used improperly when two retentive clasp arms originate from the body and occlusal rest areas and approach bilateral retentive areas on the side of the tooth away from the point of origin. The correct form of this clasp has only one retentive clasp arm, opposed by a nonretentive recipro-cal arm on the opposite side. A common error is to use this clasp improperly by making both clasp terminals retentive. This not only is unnecessary but also disregards the need for reciprocation and bilateral stabilization. Other common errors in the design of circumferential clasps are illustrated in Figure 7-25. A C E F G B D Figure 7-25 Some improper applications of circumferential clasp design and their recommended corrections. A, Tooth with undesirable height of contour in its occlusal third. B, Unsuitable contour and location of retentive clasp arm on an unmodified abutment. C, More favorable height of contour achieved by modi-fication of the abutment. D, Retentive clasp arm properly designed and located on a modified abutment. E, Unsuitable contour and location of the retentive arm in relation to the height of contour (straight arm configuration provides poor approach to the retentive area and is less resistant to dislodging force). F, Terminal portion of the retentive clasp arm located too close to the gingival margin. G, Clasp arm that is properly designed and located. Ring Clasp. The circumferential type of clasp may be used in several forms. It appears as though many of these forms of the basic circumferential clasp design were devel-oped to accommodate situations in which corrected tooth modifications could not be or were not accomplished by the dentist. One is the ring clasp, which encircles nearly all of a tooth from its point of origin (Figure 7-26). It is used when a proximal undercut cannot be approached by other means. For example, when a mesiolingual undercut on a lower molar abutment cannot be approached directly because of its proximity to the occlusal rest area and cannot be approached with a bar clasp arm because of lingual inclina-tion of the tooth, the ring clasp encircling the tooth allows the undercut to be approached from the distal aspect of the tooth. The clasp should never be used as an unsupported ring (Figure 7-27) because if it is free to open and close as a ring, it cannot provide reciprocation or stabilization. Instead the ring-type clasp should always be used with a supporting strut on the nonretentive side, with or without an auxiliary occlu-sal rest on the opposite marginal ridge. The advantage of an auxiliary rest is that further movement of a mesially inclined tooth is prevented by the presence of a distal rest. In any event, the supporting strut should be regarded as a minor 82 Part I General Concepts/Treatment Planning connector from which the flexible retentive arm originates. Reciprocation then comes from the rigid portion of the clasp lying between the supporting strut and the principal occlusal rest. The ring-type clasp should be used on protected abut-ments, whenever possible, because it covers such a large area of tooth surface. Esthetics need not be considered on such a posteriorly located tooth. A ring-type clasp may be used in reverse on an abutment located anterior to a tooth-bounded edentulous space (Figure 7-28). Although potentially an effective clasp, this clasp covers an excessive amount of tooth surface and can be esthetically objectionable. The only justification for its use is when a distobuccal or distolingual undercut cannot be approached directly from the occlusal rest area and/or tissue undercuts prevent its approach from a gingival direction with a bar clasp arm. Embrasure Clasp. In the fabrication of an unmodified Class II or Class III partial denture, no edentulous spaces are available on the opposite side of the arch to aid in clasping. Mechanically, this is a disadvantage. However, when the A B Figure 7-26 Ring clasp(s) encircling nearly all of the tooth from its point of origin. A, Clasp originates on the mesiobuccal surface and encircles the tooth to engage the mesiolingual undercut. B, Clasp originates on the mesiolingual surface and encircles the tooth to engage the mesiobuccal undercut. In either example, a supporting strut is used on the nonretentive side (drawn both as direct view of near side of tooth and as mirror view of opposite side). Figure 7-27 Improperly designed ring clasp lacking neces-sary support. Such a clasp lacks any reciprocating or stabilizing action because the entire circumference of the clasp is free to open and close. A supporting strut should always be added on the nonretentive side of the abutment tooth, which then becomes, in effect, a minor connector from which a tapered and flexible retentive clasp arm originates. 83 Chapter 7 Direct Retainers Figure 7-28 Ring clasp may be used in reverse on the abut-ment located anterior to the tooth-bound edentulous space. Figure 7-29 Embrasure clasps extend through occlusal embrasures to engage facial undercuts when no modification spaces are present. teeth are sound and retentive areas are available, or when multiple restorations are justified, clasping can be accom-plished by means of an embrasure clasp (Figure 7-29). Sufficient space must be provided between the abutment teeth in their occlusal third to make room for the common body of the embrasure clasp (Figure 7-30), yet the contact area should not be eliminated entirely. Historically, this clasp assembly demonstrates a high percentage of fracture caused by inadequate tooth preparation in the contact area. Because vulnerable areas of the teeth are involved, abutment protection with inlays or crowns is recommended. The deci-sion to use unprotected abutments must be made at the time of oral examination and should be based on the patient’s age, Figure 7-30 Embrasure and hairpin circumferential retentive clasp arms. The terminus of each engages a suitable retentive undercut. Use of a hairpin-type clasp on the second molar is made necessary by the fact that the only available undercut lies directly below the point of origin of the clasp arm. caries index, and oral hygiene, as well as on whether existing tooth contours are favorable or can be made favorable by tooth modification. Preparation of adjacent, contacting, uncrowned abutments to receive any type of embrasure clasp of adequate interproximal bulk is difficult, especially when it is opposed by natural teeth. The embrasure clasp always should be used with double occlusal rests, even when definite proximal shoulders can be established (Figure 7-31). This is done to avoid interproxi-mal wedging by the prosthesis, which could cause separation of the abutment teeth, resulting in food impaction and clasp displacement. In addition to providing support, occlusal rests serve to shunt food away from contact areas. For this reason, occlusal rests should always be used whenever food impaction is possible. Embrasure clasps should have two retentive clasp arms and two reciprocal clasp arms that are bilaterally or diago-nally opposed. An auxiliary occlusal rest or a bar clasp arm can be substituted for a circumferential reciprocal arm, as long as definite reciprocation and stabilization result. A lin-gually placed retentive bar clasp arm may be substituted if a rigid circumferential clasp arm is placed on the buccal surface for reciprocation, provided lingual retention is used on the opposite side of the arch. Other less commonly used modifications of the cast circumferential clasp that are of historical interest are the multiple clasp, the half-and-half clasp, and the reverse-action clasp. Back-action Clasp. The back-action clasp is a modi-fication of the ring clasp; it has all of the same disadvan-tages and no apparent advantages (Figure 7-32). Its use is difficult to justify. The undercut can usually be approached just as well by using a conventional circum-ferential clasp, with less tooth coverage and less display 84 Part I General Concepts/Treatment Planning A B Figure 7-31 A, Example of the use of an embrasure clasp for a Class II partially edentulous arch. Embrasure clasp on the two left molar abutments was used in the absence of posterior modi-fication space. B, Occlusal and proximal surfaces of adjacent molar and premolar prepared for embrasure clasp. Note that rest seat preparations are extended both buccally and lingually to accommodate the retentive and reciprocal clasp arms. Adequate preparation confined to the enamel can rarely be accomplished for such a clasp, especially when clasped teeth are opposed by natural teeth. Figure 7-32 Back-action circumferential clasp used on pre-molar abutment anterior to the edentulous space. Figure 7-33 Multiple clasp is actually two opposing circum-ferential clasps joined at the terminal end of two reciprocal arms (mirror view). of metal. With the circumferential clasp, the proximal tooth surface can be used as a guiding plane, as it should be, and the occlusal rest can have the rigid support it requires. An occlusal rest always should be attached to some rigid minor connector and should never be sup-ported by a clasp arm alone. If the occlusal rest is part of a flexible assembly, it cannot function adequately as an occlusal rest. Multiple Clasp. The multiple clasp simply consists of two opposing circumferential clasps joined at the termi-nal end of the two reciprocal arms (Figure 7-33). It is used when additional retention and stabilization are needed, usually on tooth-supported partial dentures. It may be used for multiple clasping in instances in which the partial denture replaces an entire half of the dental arch. It may be used rather than an embrasure clasp when the only available retentive areas are adjacent to each other. Its disadvantage is that two embrasure approaches are necessary rather than a single common embrasure for both clasps. Half-and-half Clasp. The half-and-half clasp consists of a circumferential retentive arm arising from one direc-tion and a reciprocal arm arising from another (Figure 7-34). The second arm must arise from a second minor connector, and this arm is used with or without an aux-iliary occlusal rest. Reciprocation arising from a second minor connector usually can be accomplished with a short bar or with an auxiliary occlusal rest, thereby avoid-ing so much tooth coverage. There is little justification for the use of the half-and-half clasp in bilateral extension 85 Chapter 7 Direct Retainers Figure 7-34 Half-and-half clasp consists of one circumferen-tial retentive arm arising from the distal aspect and a second circumferential arm arising from the mesial aspect on the oppo-site side, with or without a secondary occlusal rest. Broken line illustrates a nonretentive reciprocal clasp arm used without a secondary occlusal rest (mirror view). Figure 7-35 A reverse-action, or hairpin, clasp arm may be used on abutments of tooth-supported dentures when the proxi-mal undercut lies below the point of origin of the clasp (mirror view). It may be esthetically objectionable and covers consider-able tooth surface. It should be used only when a bar-type reten-tive arm is contraindicated because of a tissue undercut, a tilted tooth, or a shallow vestibule. base partial dentures. Its design was originally intended to provide dual retention, a principle that should be applied only to unilateral partial denture design. Reverse-action Clasp. The reverse-action, or hairpin, clasp arm is designed to allow a proximal undercut to be engaged from an occlusal approach (Figure 7-35). Other methods of accomplishing the same result involve use of a ring clasp originating on the opposite side of the tooth or a bar clasp arm originating from a gingival direction. However, when a proximal undercut must be used on a posterior abutment, and when tissue undercuts, tilted teeth, or high tissue attachments prevent the use of a bar clasp arm, the reverse-action clasp may be used success-fully. Although the ring clasp may be preferable, lingual undercuts may prevent the placement of a supporting strut without tongue interference. In this limited situa-tion, the hairpin clasp arm serves adequately, despite its several disadvantages. The clasp covers considerable tooth surface and may trap debris; its occlusal origin may increase the functional load on the tooth, and its flexibil-ity is limited. Esthetics usually need not be considered when the clasp is used on a posterior abutment, but the hairpin clasp arm does have the additional disadvantage of displaying too much metal for use on an anterior abutment. When properly designed, the reverse-action clasp should make a hairpin turn to engage an undercut below the point of origin (see Figure 7-35). The upper part of the arm of this clasp should be considered a minor con-nector, giving rise to the tapered lower part of the arm. Therefore only the lower part of the arm should be flex-ible. With the retentive portion beginning beyond the turn, only the lower part of the arm should flex over the height of contour to engage a retentive undercut. The bend that connects the upper and lower parts of the arm should be rounded to prevent stress accumulation and fracture of the arm at the bend. The clasp should be designed and fabricated with this in mind. These are the various types of cast circumferential clasps. As was mentioned previously, they may be used in combination with bar clasp arms as long as differentia-tion is made between retention and reciprocation by their form and location. Circumferential and bar clasp arms may be made flexible (retentive) or rigid (reciprocal) in any combination, as long as each retentive clasp arm is opposed by a rigid reciprocal component. Use of many of the less desirable clasp forms can be avoided by changing the crown forms of the abutments through tooth modification within the enamel or with restorations. When abutment coverage is fabricated, tooth contours should be established that will permit the use of the most desirable clasp forms, rather than a form that makes it necessary to use a less desirable clasp design. This is best accomplished by first altering the crown contour of abutment teeth not designated for restoration to meet the requirements of guiding planes and survey line location. This is followed by the prescribed crown preparations. Before tooth reduction is performed for the prescribed crown preparations, the requirements of guiding planes and survey line location should be met. 86 Part I General Concepts/Treatment Planning Analysis of Tooth Contours for Retentive Clasps Although the extracoronal, or clasp-type, direct retainer is used more often than the internal attachment, it is com-monly misused. It is hoped that a better understanding of the principles of clasp design will lead to more intelligent use of this retainer. To best gain this understanding, it is vitally important for the clinician to consider how tooth contour and removable partial denture components must interact (be related) to allow stable prosthetic function. Just as unal-tered natural teeth are not appropriately contoured to receive fixed partial dentures without preparation, the teeth that are engaged by a removable partial denture must be contoured to support, stabilize, and retain the functioning prosthesis. Analysis and accomplishment of tooth modifica-tion when it is required for optimum stability and retention are necessary for the success of the prosthesis. Critical areas of an abutment that provide for retention and stabilization (reciprocation) can be identified only with the use of a dental cast surveyor (Table 7-1). To enhance the understanding of direct retainers, an introduction of the dental cast surveyor is appropriate at this time. Surveying will be covered in detail in Chapter 11. The cast surveyor (Figure 7-36) is a simple instrument that is essential for planning partial denture treatment. Its main working parts are the vertical arm and the adjustable table, which hold the cast in a fixed relation to the vertical arm. This relationship of the vertical arm to the cast repre-sents the path of placement that the partial denture will ultimately take when inserted or removed from the mouth (Figure 7-37). The adjustable table may be tilted in relation to the verti-cal arm of the surveyor until a path can be found that best satisfies all involved factors (Figure 7-38). A cast in a hori-Table 7-1 Function and Position of Clasp Assembly Parts Component Part Function Location Rest Support Occlusal, lingual, incisal Minor connector Stabilization Proximal surfaces extending from a prepared marginal ridge to the junction of the middle and gingival one third of the abutment crown Clasp arms Stabilization (reciprocation) Middle one third of crown Retention Gingival one third of crown in measured undercut Figure 7-36 Essential parts of a dental surveyor (Ney Paral-lelometer, Dentsply Ceramco, Burlington, NJ), showing the verti-cal spindle in relation to an adjustable table. Path of placement A B Height of contour x x Figure 7-37 Angle of cervical convergence on two teeth pre-senting dissimilar contours. A greater angle of cervical conver-gence on tooth A necessitates placement of a clasp terminus, X, nearer the height of contour than when a lesser angle exists, as in B. It is apparent that uniform clasp retention depends on depth (amount) of the tooth undercut rather than on distance below the height of the contour at which the clasp terminus is placed. zontal relationship to the vertical arm represents a vertical path of placement; a cast in a tilted relationship represents a path of placement toward the side of the cast that is on an upward slant. The vertical arm, when brought in contact with a tooth surface, identifies the location on the clinical crown where the greatest convexity exists. This line, called the height of contour (specific to the surveyor-defined path), 87 Chapter 7 Direct Retainers triangle is at the point of contact of the surveyor blade with the tooth, and the base is the area of the cast representing the gingival tissue (see Figure 11-18). The apical angle is called the angle of cervical convergence (see Figure 7-37). This angle may be measured as described in Chapter 11, or it may be estimated by observing the triangle of light visible between the tooth and the surveyor blade. For this reason, a wide surveyor blade rather than a small cylindrical tool is used so it is easier to see the triangle of light. The importance of this angle lies in its relationship to the amount of retention. is the boundary between (1) an occlusal or incisal region of the tooth that is freely accessible to a prosthesis and (2) a gingival region of the tooth that can be accessed only when a portion of the prosthesis elastically deforms and recovers to contact the tooth. This surveyor-defined path and the subsequent tooth height of contour will indicate the areas available for retention and those available for support and stabilization, as well as the existence of tooth and other tissue interference with the path of placement. When the surveyor blade contacts a tooth on the cast at its greatest convexity, a triangle is formed. The apex of the A B C Figure 7-38 Relationship of height of contour, suprabulge, and infrabulge. A, When an egg is placed with its long axis parallel to the surveying tool, the height of contour is found at its greatest circumference, here designated by the arrow. In this example, the second line is diagonal to the line outlining the height of contour and is either above the height of contour (right side of egg), referred to as the suprabulge region, or below the height of contour (left side of egg), referred to as the infrabulge region. B, If the long axis of the egg is reoriented so that the previous diagonal line is now at the greatest circumference, the original “height of contour line” no longer is at the greatest circumference. The line segment at A is in the suprabulge region, and the line segment B is in the infrabulge region. Chang-ing the orientation alters the relationship of surfaces relative to the greatest circumference and consequently alters suprabulge and infrabulge locations. C, Just as the height of contour changes as orientation changes for the egg, when a tooth orientation changes, the height of contour is altered. For this molar, the line H was produced with a horizontal orientation. When the tooth was inclined buccally, the height of contour moved to B, relative to the horizontal location. Alternatively, when the tooth was inclined lingually, the height of contour moves to L, relative to the horizontal location. 88 Part I General Concepts/Treatment Planning (regardless of its cross-sectional form), clasp cross-sectional form or shape (whether it is round, half-round, or some other form), and the material used in making the clasp. The retention characteristics of gold alloy, chrome alloy, tita-nium, or titanium alloy depend on whether it is in cast or wrought form. Size of and Distance Into the Angle of Cervical Convergence To be retentive, a tooth must have an angle of convergence cervical to the height of contour. When it is surveyed, any single tooth will have a height of contour or an area of great-est convexity. Areas of cervical convergence may not exist when the tooth is viewed in relation to a given path of place-ment. Also, certain areas of cervical convergence may not be usable for the placement of retentive clasps because of their proximity to gingival tissue. This is best illustrated by mounting a spherical object, such as an egg, on the adjustable table of a dental surveyor (see Figure 7-38). The egg now represents the cast of a dental arch or, more correctly, one tooth of a dental arch. The egg first is placed perpendicular to the base of the surveyor and is surveyed so the height of contour can be determined. The vertical arm of the surveyor represents the path of placement that a denture would take, and conversely its path of removal. With a carbon marker, a circumferential line has been drawn on an egg at its greatest circumference, as shown by the arrow in Figure 7-38, A. This line, which Kennedy called the height of contour, is its greatest convexity. Cummer spoke of it as the guideline because it is used as a guide in the place-ment of retentive and nonretentive clasps. To this, DeVan added the terms suprabulge, denoting the surfaces sloping superiorly, and infrabulge, denoting the surfaces sloping inferiorly. Any areas cervical to the height of contour may be used for the placement of retentive clasp components, whereas areas occlusal to the height of contour may be used for the placement of nonretentive, stabilizing, or reciprocating components. Obviously, only flexible components may be placed gingivally to the height of contour because rigid ele-ments would not flex over the height of contour or contact the tooth in the undercut area. With the original height of contour marked on the egg, the egg is now tilted from the perpendicular to an angular relation with the base of the surveyor (see Figure 7-38, B). Its relation to the vertical arm of the surveyor has now been changed, just as a change in the position of a dental cast would bring about a different relationship with the surveyor. The vertical arm of the surveyor still represents the path of placement. However, its relation to the egg is totally different. Again, the carbon marker is used to delineate the height of contour or the greatest convexity. It will be seen that areas that were formerly infrabulge are now suprabulge, and vice versa. A retentive clasp arm placed below the height of contour in the original position may now be excessively Amount of Retention Clasp retention is based on resistance to deformation of the metal. For a clasp to be retentive, it must be placed in an undercut area of the tooth, where it is forced to deform upon application of a vertical dislodging force (Figure 7-39). It is this resistance to deformation along an appropriately selected path that generates retention (Figure 7-40). Such resistance to deformation is dependent on several factors and is also proportionate to the flexibility of the clasp arm. The interaction of a number of factors under the control of the clinician combines to produce retention. These include both tooth (planned and executed by the clinician) and prosthesis (to be planned by the dentist and executed by the laboratory technician) factors. Tooth factors include the size of the angle of cervical convergence (depth of undercut) and how far the clasp ter-minal is placed into the angle of cervical convergence. Pros-thesis factors include the flexibility of the clasp arm. Clasp flexibility is the product of clasp length (measured from its point of origin to its terminal end), clasp relative diameter Lifting force Lifting force Figure 7-39 Retention is provided primarily by the flexible portion of the clasp assembly. Retentive terminals are ideally located in measured undercuts in the gingival third of abutment crowns. When force acts to dislodge the restoration in an occlu-sal direction, the retentive arm is forced to deform as it passes from the undercut location over the height of contour. The amount of retention provided by the clasp arm is determined by its length, diameter, taper, cross-sectional form, contour, type of alloy, and location and depth of undercut engaged. 89 Chapter 7 Direct Retainers should be conducted for a different path of placement. The cast is merely tilted in relation to the vertical arm until the most suitable path is found. The most suitable path of place-ment is generally considered to be the path of placement that will require the least amount of mouth preparation neces-sary to place the components of the partial denture in their ideal position on the tooth surfaces and in relation to the soft tissue. Then mouth preparations are planned with a definite path of placement in mind. It is important to remember that tooth surfaces can be recontoured through selective grinding or the placement of restorations (mouth preparations) to achieve a more suit-able path of placement. The path of placement also must take into consideration the presence of tissue undercuts that retentive or totally nonretentive, whereas a nonretentive sta-bilizing or reciprocal arm that is located above the height of contour in the first position may now be located in an area of undercut. Figure 7-35 C illustrates this principle com-pared with a tooth example showing that changes in tilt can significantly alter heights of contour. The location and depth of a tooth undercut available for retention are therefore relative only to the path of placement and removal of the partial denture. At the same time, non-retentive areas on which rigid components of the clasp may be placed exist only for a given path of placement (see Figure 7-39). If conditions are found that are not favorable for the particular path of placement under consideration, a study A B C D Caramel candy Figure 7-40 A, Retentive areas are not sufficient to resist reasonable dislodging forces when a cast is surveyed at its most advanta-geous position (occlusal plane parallel to the surveyor table), even though guide planes could be established with minor tooth modifica-tion. B, Tilting cast creates functionally ineffective tooth contours, which are present only in relation to the surveying rod and do not exist when compared with the most advantageous position (position in which restoration will be subject to dislodging forces in an occlusal direction). C-D, Clasps designed at tilt are ineffective without the development of corresponding guide planes to resist displace-ment when the restoration is subject to dislodging forces in the occlusal direction. 90 Part I General Concepts/Treatment Planning One tenth or less of clasp length T T T T ½ T ½ T ½ T ½ T Figure 7-41 The retentive cast clasp arm should be tapered uniformly from its point of attachment at the clasp body to its tip. Dimensions at the tip are about half those at the point of attachment. The clasp arm so tapered is approximately twice as flexible as one without any taper. T, Clasp thickness. (Courtesy The Argen Corporation, New York, NY.) Figure 7-42 Length of the tapered cast retentive clasp arm is measured along the center portion of the arm until it joins the clasp body (circumferential) or becomes part of the denture base or is embedded in the base (bar-type clasp). would interfere with the placement of major connectors, the location of vertical minor connectors, the origin of bar clasp arms, and the denture bases. When the theory of clasp retention is applied to the abut-ment teeth in a dental arch during surveying of the dental cast, each tooth may be considered individually and in rela-tion to the other abutment teeth as far as the designs of retentive and stabilizing (reciprocating) components are concerned. This is necessary because the relationship of each tooth to the entire arch and to the design of the whole pros-thesis has been considered previously when teeth were selected or modified to achieve the most suitable path of placement. Once this relationship of the cast to the surveyor has been established, the height of contour on each abut-ment tooth becomes fixed, and the clasp design for each must be considered separately. A positive path of placement and removal is made pos-sible by the contact of rigid parts of the denture framework with parallel tooth surfaces, which act as guiding planes. Because guiding planes control the path of placement and removal, they can provide additional retention for the partial denture by limiting the possibilities that exist for its dislodg-ment. The more vertical are the walls (guiding planes) that are prepared parallel to the path of insertion, the fewer the possibilities that exist for dislodgment. If some degree of parallelism does not exist during placement and removal, trauma to the teeth and supporting structures and strain on the denture parts are inevitable. This ultimately results in damage to the teeth and their periodontal support or to the denture itself, or both. Therefore without guiding planes, clasp retention will be detrimental or practically nonexistent. If clasp retention is frictional only because of an active rela-tionship of the clasp to the teeth, then orthodontic move-ment or damage to periodontal tissue, or both, will result. Instead a clasp should bear a passive relationship to the teeth, except when a dislodging force is applied. In addition to the degree of the angle of cervical convergence and the distance a clasp is placed into the angle of cervical conver-gence, the amount of retention generated by a clasp depends on its flexibility. This is a function of clasp length, diameter, taper, cross-sectional form, and material. The amount of retention also depends on the flexibility of a clasp arm. This is a function of clasp length, diameter, cross-sectional form, and material. Length of Clasp Arm The longer the clasp arm is, the more flexible it will be, all other factors being equal. The length of a circumferential clasp arm is measured from the point at which a uniform taper begins. Tooth modification providing increased length to a suprabulge retentive clasp by allowing the retentive tip to approach the undercut from a gingival direction opti-mizes clasp retention (see Figure 7-8). The retentive circum-ferential clasp arm should be tapered uniformly from its point of origin through the full length of the clasp arm (Figure 7-41). The length of a bar clasp arm also is measured from the point at which a uniform taper begins. Generally the taper of a bar clasp arm should begin at its point of origin from a metal base, or at the point at which it emerges from a resin base (Figure 7-42). Although a bar clasp arm usually will be longer than a circumferential clasp arm, its flexibility will be less because its half-round form lies in several planes; this prevents its flexibility from being proportionate to its total length. Tables 7-2 and 7-3 give an approximate depth of undercut that may be used for the cast gold and chromium-cobalt retentive clasp arms of circumferential and bar-type clasps. Based on a proportional limit of 60,000 psi and on the assumption that the clasp arm is properly tapered, the clasp arm should be able to flex repeatedly within the limits stated without hardening or rupturing because of fatigue. It has been estimated that alternate stress applications of the fatigue type are placed on a retainer arm during mastication and other force-producing functions about 300,000 times a year. 91 Chapter 7 Direct Retainers tional movement of the distal extension base. It must have universal flexibility to avoid transmission of tipping stresses to the abutment tooth, or it must be capable of disengaging the undercut when vertical forces directed against the denture are toward the residual ridge. A round clasp is the only circumferential clasp form that may be safely used to engage a tooth undercut on the side of an abutment tooth away from the distal extension base. The location of the undercut is perhaps the single most important factor in selection of a clasp for use with distal extension partial dentures. Material Used for the Clasp Arm Although all cast alloys used in partial denture construction possess flexibility, their flexibility is proportionate to their bulk. If this were not true, other components of the partial denture could not have the necessary rigidity. A disadvan-tage of a cast gold partial denture is that its bulk must be increased to obtain needed rigidity at the expense of added weight and increased cost. It cannot be denied that greater rigidity with less bulk is possible through the use of chro-mium-cobalt alloys. Although cast gold alloys may have greater resiliency than do cast chromium-cobalt alloys, the fact remains that the structural nature of the cast clasp does not approach the flexibility and adjustable nature of the wrought-wire clasp. Because it was formed by being drawn into a wire, the wrought-wire clasp arm has toughness exceeding that of a cast clasp arm. The tensile strength of a wrought structure is at least 25% greater than that of the cast alloy from which it was made. It may therefore be used in smaller diameters to provide greater flexibility without fatigue and ultimate fracture. Relative Uniformity of Retention Now that the factors inherent to a determination of the amount of retention from individual clasps have been reviewed, it is important to consider the coordination of relative retention between various clasps in a single prosthesis. The size of the angle of convergence will determine how far a given clasp arm should be placed into that angle. By disregarding—for the time being—variations in clasp flexi-bility, the relative uniformity of retention will depend on the location of the retentive part of the clasp arm, not in relation to the height of contour, but in relation to the angle of cervi-cal convergence. The retention on all principal abutments should be as equal as possible. Although esthetic placement of clasp arms is desirable, it may not be possible to place all clasp arms in the same occlusocervical relationship because of variations in tooth contours. However, retentive surfaces may be made similar by altering tooth contours or by using cast restora-tions with similar contours. Retentive clasp arms must be located so that they lie in the same approximate degree of undercut on each abutment Table 7-2 Permissible Flexibilities of Retentive Cast Circumferential and Bar-Type Clasp Arms of Type IV Gold Alloys CIRCUMFERENTIAL BAR-TYPE Length, inches Flexibility, inches Length, inches Flexibility, inches 0 to 0.3 0.01 0 to 0.7 0.01 0.3 to 0.6 0.02 0.7 to 0.9 0.02 0.6 to 0.8 0.03 0.9 to 1.0 0.03 Based on the approximate dimensions of Jelenko preformed plastic pat-terns (JF Jelenko, New York, NY). Table 7-3 Permissible Flexibilities of Retentive Cast Circumferential and Bar-Type Clasp Arms for Chromium-Cobalt Alloys CIRCUMFERENTIAL BAR-TYPE Length, inches Flexibility, inches Length, inches Flexibility, inches 0 to 0.3 0.004 0 to 0.7 0.004 0.3 to 0.6 0.008 0.7 to 0.9 0.008 0.6 to 0.8 0.012 0.9 to 1.0 0.012 Based on the approximate dimensions of Jelenko preformed plastic pat-terns (JF Jelenko, New York, NY). Diameter of Clasp Arm The greater the average diameter of a clasp arm is, the less flexible it will be, all other factors being equal. If its taper is absolutely uniform, the average diameter will be at a point midway between its origin and its terminal end. If its taper is not uniform, a point of flexure—and therefore a point of weakness—will exist; this then will be the determining factor in its flexibility, regardless of the average diameter of its entire length. Cross-sectional Form of the Clasp Arm Flexibility may exist in any form, but it is limited to one direction in the case of the half-round form. The only uni-versally flexible form is the round form, which is practically impossible to obtain by casting and polishing. Because most cast clasps are essentially half-round in form, they may flex away from the tooth, but edgewise flexing (and edgewise adjustment) is limited. For this reason, cast retentive clasp arms are more acceptable in tooth- supported partial dentures in which they are called on to flex only during placement and removal of the prosthesis. A retentive clasp arm on an abutment adjacent to a distal extension base not only must flex during placement and removal but also must be capable of flexing during func-92 Part I General Concepts/Treatment Planning tage of eliminating a visible clasp. The unique aspect of implant use with removable partial dentures (RPDs) is that their location generally can be prescribed, meaning that the clinician selects the best location. The location within the modification space must first consider anatomic character-istics of bone availability. It would not be a great advantage to a patient if extensive augmentation procedures were required to allow implant placement in conjunction with an RPD. If anatomic needs are met, placement of an implant within a modification space to the advantage of retentive needs requires consideration of anterior, mid, or distal placement. Because retainers utilizing teeth have always been restricted to tooth locations at either end of a span, the mid-span location typically is not considered. However, the application of implants allows consideration of where the most beneficial location of retention can be provided, which forces consideration of where the most efficient resistance to movement away from the denture base can be provided. Placement at either extreme of the denture base may allow greater movement than placement at a mid-point, and should be taken into consideration. tooth. In Figure 7-37, this is seen at point X on both teeth— A and B—despite variation in the distance below the height of contour. Should both clasp arms be placed equidistant below the height of contour, the higher location on tooth B would have too little retention, whereas the lower location on tooth A would be too retentive. Measurement of the degree of undercut by mechanical means with the use of a surveyor is important. However, undercut identification is only one factor that is important to consider when one is providing appropriate retention for a removable partial denture. Stabilizing-Reciprocal Cast Clasp Arm When the direct retainer comes into contact with the tooth, the framework must be stabilized against horizontal move-ment for the required clasp deformation to occur. This sta-bilization is derived from cross-arch framework contacts or from a stabilizing or reciprocal clasp in the same clasp assembly. To provide true reciprocation, the reciprocal clasp must be in contact during the entire period of retentive clasp deformation. This is best provided with lingual-palatal, guide-plane surfaces. A stabilizing (reciprocal) clasp arm should be rigid. Therefore it is shaped somewhat differently than is the cast retentive clasp arm, which must be flexible. Its average diam-eter must be greater than the average diameter of the oppos-ing retentive arm, to increase desired rigidity. A cast retentive arm is tapered in two dimensions, as illustrated in Figure 7-41, whereas a reciprocal arm should be tapered in one dimension only, as shown in Figure 7-43. Achieving such a form for the arm requires freehand waxing of patterns. Implants as Direct Retainers As was stated earlier, in situations where support require-ments are adequately met by available oral tissues, dental implants can be used for retention and provide the advan-One tenth or less of clasp length T T ½ T ½ T T ½ T Figure 7-43 The reciprocal arm of the direct retainer assem-bly should be rigid. An arm tapered both lengthwise and width-wise is more flexible than an arm of the same dimensions tapered only lengthwise. T, Clasp thickness. Other Types of Retainers Lingual Retention in Conjunction With Internal Rests The internal rest is covered in Chapter 6. It is emphasized that the internal rest is not used as a retainer, but that its near-vertical walls provide for reciprocation against a lin-gually placed retentive clasp arm. For this reason, visible clasp arms may be eliminated, thus avoiding one of the principal objections to the extracoronal retainer. Such a retentive clasp arm, terminating in an existing or prepared infrabulge area on the abutment tooth, may be of any acceptable design. It is usually a circumferential arm arising from the body of the denture framework at the rest area. It should be wrought because the advantages of adjustability and flexibility make the wrought clasp arm preferable. It may be cast with gold or a low-fusing, chromium-cobalt alloy, or it may be assembled by being soldered to one of the higher-fusing, chromium-cobalt alloys. In any event, future adjustment or repair is facilitated. The use of lingual extracoronal retention avoids much of the cost of the internal attachment, yet disposes of a visible clasp arm when esthetics must be considered. Often it is employed with a tooth-supported partial denture only on the anterior abutments, and when esthet-ics is not a consideration, the posterior abutments are clasped in the conventional manner. One of the dentist’s prime considerations in clasp selection is the control of stress transferred to the abut-ment teeth when the patient exerts an occluding force on 93 Chapter 7 Direct Retainers Occlusal force Minor connector in contact with tooth Occlusal force No contact A B Figure 7-44 A, Minor connector supporting the distal rest does not contact the prepared guiding plane, resulting in uncon-trolled stress to the abutment tooth. B, A minor connector con-tacts the prepared guiding plane and directs stresses around the arch through the proximal contacts. the artificial teeth. The location and design of rests, the clasp arms, and the position of minor connectors as they relate to guiding planes are key factors in controlling transfer of stress to abutments. Errors in the design of a clasp assembly can result in uncontrolled stress to abut-ment teeth and their supporting tissues. Some common errors and their corrections are illustrated in Figures 7-44 and 7-45. No contact Occlusal force Possible wedge No contact Occlusal force Occlusal force gp A B C D Figure 7-45 A, Clasp assembly designed so that the vertical occlusal force results in movement of the proximal plate cervically and out of contact with the guiding plane, as illustrated in (B). This lack of contact may contribute to a possible wedging effect. C, Extending contact of the proximal plate on the prepared guiding plane or, as in (D), eliminating space between the artificial tooth and the guiding plane (gp) will help direct stresses around the arch through proximal contacts. The choice of clasp design should be based on biologi-cal and mechanical principles. The dentist responsible for the treatment being rendered must be able to justify the clasp design used for each abutment tooth in keeping with these principles. Internal Attachments As was mentioned earlier in the chapter, the principle of the internal attachment was first formulated by Dr. Herman E.S. Chayes in 1906. One such attachment man-ufactured commercially still carries his name. Although it may be fabricated by the dental technician as a cast dovetail fitting into a counterpart receptacle in the abut-ment crown, the alloys used in manufactured attach-ments and the precision with which they are constructed make the ready-made attachment preferable to any of this type that can be fabricated in the dental laboratory. Much credit is due the manufacturers of metals used in dentistry for continued improvements in the design of internal attachments. The internal attachment has two major advantages over the extracoronal attachment: elimination of visible retentive and support components, and better vertical support through a rest seat located more favorably in relation to the horizontal axis of the abutment tooth. For these reasons, the internal attachment may be preferable in selected situations. It provides horizontal stabilization similar to that of an internal rest. However, additional extracoronal stabilization is usually desirable. It has been 94 Part I General Concepts/Treatment Planning claimed that stimulation to the underlying tissue is greater when internal attachments are used because of intermit-tent vertical massage. This is probably no more than is possible with extracoronal retainers of similar construction. Some of the disadvantages of internal attachments include the following: (1) they require prepared abut-ments and castings; (2) they require somewhat compli-cated clinical and laboratory procedures; (3) they eventually wear, with progressive loss of frictional resis-tance to denture removal; (4) they are difficult to repair and replace; (5) they are effective in proportion to their length and are therefore least effective on short teeth; (6) they are difficult to place completely within the circum-ference of an abutment tooth because of the size of the pulp; and (7) they are considered more costly. Because the principle of the internal attachment does not permit horizontal movement, all horizontal, tipping, and rotational movements of the prosthesis are transmit-ted directly to the abutment tooth. The internal attach-ment therefore should not be used in conjunction with extensive tissue-supported distal extension denture bases unless some form of stress-breaker is used between the movable base and the rigid attachment. Although stress-breakers may be used, they do have some disadvantages— which are discussed later—and their use adds to the cost of the partial denture. Numerous other types of retainers for partial dentures have been devised that cannot be classified as primarily of the intracoronal or extracoronal type. Neither can they be classified as relying primarily on frictional resistance or placement of an element in an undercut to prevent displacement of the denture. However, all of these use some type of locking device, located intracoronally or extracoronally, for providing retention without visible clasp retention. Although the motivation behind the development of other types of retainers has usually been a desire to eliminate visible clasp retainers, the desire to minimize torque and tipping stresses on the abutment teeth has also been given consideration. All of the retainers that are discussed herein have merit, and much credit is due to those who have devel-oped specific devices and techniques for the retention of partial dentures. The use of patented retaining devices and other techniques falls in the same limited category as the internal attachment prosthesis and is for economic and technical reasons available to only a small percentage of those patients who need partial denture service. Internal attachments of the locking or dovetail type unquestionably have many advantages over the clasp-type denture in tooth-supported situations. However, it is questionable whether the locking type of internal attachment is indicated for distal extension removable partial dentures, with or without stress-breakers and with or without splinted abutments, because of inherent excessive leverages most often associated with these attachments. The nonlocking type of internal attachment, in con-junction with sound prosthodontic principles, can be advantageously used in many instances in Class I and Class II partially edentulous situations. However, unless the cross-arch axis of rotation is common to the bilater-ally placed attachments, torque on the abutments may be experienced (Figure 7-46). Excellent textbooks devoted to the use of manufactured intracoronal and extracoronal retainer systems are available. For this reason, this text concerns itself primarily with the extracoronal type of direct retainer assembly (clasps). Numerous well-designed internal attachments are available in the dental market that may be used in situations requiring special retention. Descriptive literature and technique manuals are available from the manufacturers. Other conservative treatment of partially edentulous arches with removable partial dentures can be accom-A B Figure 7-46 A, Axes of rotation, although parallel, are not common because one axis is located anterior to the other axis. B, When one nonlocking internal attachment is elevated farther from the residual ridge than its cross-arch counterpart, the axes of rotation do not fall on a common line; thus some torquing of abutments should be anticipated. However, in many instances, the effect produced by this situation will not exceed physiologic tolerance of the supporting structures of the abutments—all other torquing factors being equal. 95 Chapter 7 Direct Retainers Figure 7-47 Classification II, modification 1 maxillary remov-able partial denture with three internal attachments. Gold frame-work with soldered attachments that gain retention through activation of the gingival region of the male component (see inset picture). Figure 7-48 Same prosthesis as Figure 7-47, showing the attachment position relative to the framework palatal and distal minor connector components. Figure 7-49 Similar photo as in Figure 7-48, showing the attachment at the anterior modification space. Careful arrange-ment of the components is required to maximize the esthetic advantage of attachment use in such a region. Figure 7-50 Prosthesis with three internal attachments shown from the tissue side, depicting the common path for each. Notice the broad ridge coverage and the use of an anterior-posterior palatal strap major connector to help support the pros-thesis, minimizing stress to the attachments. plished in a variety of ways. Treatment is still contingent on the location and condition of the remaining teeth and the contour and quality of the residual ridges. Basic prin-ciples and concepts of design relative to support and sta-bility must be respected even though a variety of retaining devices can be incorporated. Examples of some of these retaining devices are illustrated in Figures 7-47 through 7-50. 96 CHAPTER 8 Indirect Retainers Chapter Outline Role of Indirect Retainers in Control of Prosthesis Movement Factors Influencing Effectiveness of Indirect Retainers Auxiliary Functions of Indirect Retainers Forms of Indirect Retainers Auxiliary occlusal rest Canine rests Canine extensions from occlusal rests Cingulum bars (continuous bars) and linguoplates Modification areas Rugae support Role of Indirect Retainers in Control of Prosthesis Movement As was described in Chapter 4, partial denture movement can exist in three planes. Tooth-supported partial dentures effectively use teeth to control movement away from the tissues. Tooth-tissue–supported partial dentures do not have this capability because one end of the prosthesis is free to move away from the tissue. This may occur because of the effects of gravity in the maxillary arch or adhesive foods in either arch. Attention to the details of design and location of partial denture component parts in control of functional movement is the strategy used in partial denture design. When the distal extension denture base is dislodged from its basal seat, it tends to rotate around the fulcrum lines. Theoretically, this movement away from the tissues can be resisted by activation of the direct retainer, the stabilizing components of the clasp assembly, and the rigid components of the partial denture framework, which are located on defi-nite rests on the opposite side of the fulcrum line away from the distal extension base. These components are referred to as indirect retainers (Figures 8-1 and 8-2). Indirect retainer components should be placed as far as possible from the distal extension base, which provides the best leverage advantage against dislodgment (Figure 8-3). For the sake of clarity in discussion of the location and functions of indirect retainers, fulcrum lines should be con-sidered the axis about which the denture will rotate when the bases move away from the residual ridge. An indirect retainer consists of one or more rests and the supporting minor connectors (Figures 8-4 and 8-5). The proximal plates, adjacent to the edentulous areas, also provide indirect retention. Although it is customary to iden-tify the entire assembly as the indirect retainer, it should be remembered that the rest is actually the indirect retainer united to the major connector by a minor connector. This is noted to avoid interpretation of any contact with tooth inclines as part of the indirect retainer. An indirect retainer should be placed as far from the distal extension base as 97 Chapter 8 Indirect Retainers E E R R F F Figure 8-1 Mandibular distal extension removable partial denture showing the distal extension base being lifted from the ridge and the clasp assembly being activated and engaged, with the indirect retainer providing stabilization against dislodgment. A C E G B D F H Figure 8-2 Fulcrum lines found in various types of partially edentulous arches, around which the denture may rotate when bases are subjected to forces directed toward or away from the residual ridge. Arrows indicate the most advantageous position of indirect retainer(s). A-B, In a Class I arch, the fulcrum line passes through the most posterior abutments, provided some rigid component of the framework is occlusal to the abutment’s heights of contour. C, In a Class II arch, the fulcrum line is diago-nal, passing through the abutment on the distal extension side and the most posterior abutment on the opposite side. D, If the abutment tooth anterior to the modification space lies far enough removed from the fulcrum line, it may be used effectively for support of the indirect retainer. E-F, In a Class IV arch, the fulcrum line passes through two abutments adjacent to the single edentulous space. G, In a Class III arch with a posterior tooth on the right side, which has a poor prognosis and eventu-ally will be lost, the fulcrum line is considered the same as though posterior tooth were not present. Thus its future loss may not necessitate altering the original design of the removable partial denture framework. H, In a Class III arch with nonsup-porting anterior teeth, the adjacent edentulous area is consid-ered to be the tissue-supported end, with a diagonal fulcrum line passing through the two principal abutments, as in a Class II arch. 98 Part I General Concepts/Treatment Planning F Fulcrum F F Fulcrum Force Fulcrum Force Direct retainer DR DR DR Fulcrum Force Indirect retainer IR IR IR DR DR DR A C D B Figure 8-3 Indirect retainer principle. A, Beams are supported at various points. B, A lifting force will displace the entire beam in the absence of retainers. C, With direct retainers (dr) at the fulcrum, the lifting force will depress one end of the beam and elevate the other end. D, With both direct and indirect retainers (ir) functioning, the lifting force will not displace beam. The farther the indirect retainer is from the fulcrum, the more efficiently it should control movement. Figure 8-4 Planning the location for an indirect retainer for a Class II modification 2 removable partial denture. The greatest distance from the axis of rotation around most distal rests (fulcrum line) would fall on #22. The decision to use an incisal rest or cingulum rest will depend on the patient’s concern for the esthetic impact of an incisal rest versus having a crown (for the cingulum rest). Figure 8-5 Example of indirect retention used in conjunction with a palatal plate–type major connector. Indirect retainers are proximal plates on second premolars and occlusal rests located on first premolars. A secondary function of auxiliary occlusal rest assemblies is to prevent settling of the anterior portion of the major connector and to provide stabilization against horizontal rotation. 99 Chapter 8 Indirect Retainers possible in a prepared rest seat on a tooth capable of sup-porting its function. Although the most effective location of an indirect retainer is commonly in the vicinity of an incisor tooth, that tooth may not be strong enough to support an indirect retainer and may have steep inclines that cannot be favorably altered to support a rest. In such a situation, the nearest canine tooth or the mesio-occlusal surface of the first pre-molar may be the best location for the indirect retention, despite the fact that it is not as far removed from the fulcrum line. Whenever possible, two indirect retainers closer to the fulcrum line are used to compensate for the compromise in distance. Factors Influencing Effectiveness of Indirect Retainers The following factors influence the effectiveness of an indi-rect retainer: 1. The principal occlusal rests on the primary abutment teeth must be reasonably held in their seats by the reten-tive arms of the direct retainers. If rests are held in their seats, rotation about an axis should occur, which subse-quently would activate the indirect retainers. If total dis-placement of the rests occurs, no rotation about the fulcrum would occur, and the indirect retainers would not be activated. 2. Distance from the fulcrum line. The following three areas must be considered: a. Length of the distal extension base b. Location of the fulcrum line c. How far beyond the fulcrum line the indirect retainer is placed 3. Rigidity of the connectors supporting the indirect retainer. All connectors must be rigid if the indirect retainer is to function as intended. 4. Effectiveness of the supporting tooth surface. The indi-rect retainer must be placed on a definite rest seat on which slippage or tooth movement will not occur. Tooth inclines and weak teeth should never be used to support indirect retainers. Auxiliary Functions of Indirect Retainers In addition to effectively activating the direct retainer to prevent movement of a distal extension base away from the tissues, an indirect retainer may serve the following auxiliary functions: 1. It tends to reduce anteroposterior tilting leverages on the principal abutments. This is particularly important when an isolated tooth is being used as an abutment—a situa-tion that should be avoided whenever possible. Ordinar-ily, proximal contact with the adjacent tooth prevents such tilting of an abutment as the base lifts away from the tissues. 2. Contact of its minor connector with axial tooth surfaces aids in stabilization against horizontal movement of the denture. Such tooth surfaces, when made parallel to the path of placement, may also act as auxiliary guiding planes. 3. Anterior teeth supporting indirect retainers are stabilized against lingual movement. 4. It may act as an auxiliary rest to support a portion of the major connector, facilitating stress distribution. For example, a lingual bar may be supported against settling into the tissues by the indirect retainer acting as an aux-iliary rest. One must be able to differentiate between an auxiliary rest placed for support for a major connector, one placed for indirect retention, and one serving a dual purpose. Some auxiliary rests are added solely to provide rest support to a segment of the denture and should not be confused with indirect retention. 5. It may provide the first visual indications for the need to reline an extension base partial denture. Deficiencies in basal seat support are manifested by the dislodgment of indirect retainers from their prepared rest seats when the denture base is depressed and rotation occurs around the fulcrum. These auxiliary functions derived from indirect retainers are important to consider, especially given the reported con-troversy as to the effectiveness of indirect retainers. Forms of Indirect Retainers The indirect retainer may take any one of several forms. All are effective in proportion to their support and distance from the fulcrum line. Auxiliary Occlusal Rest The most commonly used indirect retainer is an auxiliary occlusal rest located on an occlusal surface and as far away from the distal extension base as possible. In a mandibular Class I arch, this location is usually on the mesial marginal ridge of the first premolar on each side of the arch (see Figure 8-4). The ideal position for the indirect retainer per-pendicular to the fulcrum line would be in the vicinity of the central incisors, which are too weak and have lingual sur-faces that are too perpendicular to support a rest. Bilateral rests on the first premolars are quite effective, even though they are located closer to the axis of rotation. The same principle applies to any maxillary Class I partial denture when indirect retainers are used. Bilateral rests on the mesial marginal ridge of the first premolars generally are used in preference to rests on incisor teeth (see Figure 8-5). Not only are they effective without jeopardizing the weaker single-rooted teeth, but interference with the tongue is far less when the minor connector can be placed in the embra-sure between canine and premolar rather than anterior to the canine teeth. Indirect retainers for Class II partial dentures are usually placed on the marginal ridge of the first premolar tooth on 100 Part I General Concepts/Treatment Planning Figure 8-6 Mandibular Class II design showing a favorable location for the indirect retainer on the mesio-occlusal of the first premolar #28. This location is at 90 degrees to the fulcrum line between primary rests—DO of #20 and DO of #31—and pro-vides efficient resistance to a denture base lift based on the longest distance to resistant rest support and because the occlu-sal rest is perpendicular to the load. Figure 8-7 Mandibular Class I design using canine exten-sions from occlusal rests as indirect retainers. The canine exten-sion must be placed on prepared rest seats, so that resistance will be directed as nearly as possible along the long axis of the canine abutment. the opposite side of the arch from the distal extension base (Figure 8-6). Bilateral rests are seldom indicated except when an auxiliary occlusal rest is needed for support of the major connector, or when the prognosis for the distal abut-ment is poor and provision is being considered for later conversion to a Class I partial denture. Canine Rests When the mesial marginal ridge of the first premolar is too close to the fulcrum line, or when the teeth are overlapped so that the fulcrum line is not accessible, a rest on the adja-cent canine tooth may be used. Such a rest may be made more effective by placing the minor connector in the embra-sure anterior to the canine, either curving back onto a pre-pared lingual rest seat or extending to a mesioincisal rest. The same types of canine rests as those previously outlined— lingual or incisal rests—may be used (see Chapter 6). Canine Extensions From Occlusal Rests Occasionally, a finger extension from a premolar rest is placed on the prepared lingual slope of the adjacent canine tooth (Figure 8-7). Such an extension is used to effect indirect retention by increasing the distance of a resisting element from the fulcrum line. This method is particularly applicable when a first premolar must serve as a primary abutment. The distance anterior to the fulcrum line is only the distance between the mesio-occlusal rest and the anterior terminal of the finger exten-sion. In this instance, although the extension rests on a prepared surface, it is used in conjunction with a terminal rest on the mesial marginal ridge of the premolar tooth. Even when they are not used as indirect retainers, canine extensions, continuous bar retainers, and linguoplates should never be used without terminal rests because of the resultant forces effective when they are placed on inclined planes alone. Cingulum Bars (Continuous Bars) and Linguoplates Technically, cingulum bars (continuous bars) and lin-guoplates are not indirect retainers because they rest on unprepared lingual inclines of anterior teeth. The indirect retainers are actually the terminal rests at either end that occur in the form of auxiliary occlusal rests or canine rests (see Chapter 5). In Class I and Class II partial dentures, a cingulum bar or linguoplate may extend the effectiveness of the indirect retainer if it is used with a terminal rest at each end. In tooth-supported partial dentures, a cingulum bar or lin-guoplate is placed for other reasons but always with ter-minal rests (see Chapter 5). In Class I and Class II partial dentures especially, a continuous bar retainer or the superior border of the linguoplate should never be placed above the middle third of the teeth, so that orthodontic movement during the rotation of a distal extension denture is avoided. This guideline is not so important when the six anterior teeth are in nearly a straight line, but when the arch is narrow and tapering, a cingulum bar or linguoplate on the ante-rior teeth extends well beyond the terminal rests, and orthodontic movement of those teeth is more likely. Although these are intended primarily to stabilize weak anterior teeth, they may have the opposite effect if not used with discretion. 101 Chapter 8 Indirect Retainers Figure 8-8 Class II, modification 1 removable partial denture framework. The fulcrum line, when the denture base is displaced toward the residual ridge, runs from the left second premolar to the right second molar. When forces tend to displace the denture away from its basal seat, the supportive element (distal occlusal rest) of the direct retainer assembly on the right first premolar serves as an indirect retainer. Figure 8-9 Class II maxillary removable partial denture frame-work design. The fulcrum line runs from the patient’s right canine to left second molar. Forces that tend to unseat the denture from its basal seat will be resisted by activation of reten-tive elements on canine and molar, with the use of supportive elements on the left first premolar as an indirect retainer. Modification Areas Occasionally, the occlusal rest on a secondary abutment in a Class II partial denture may serve as an indirect retainer. This use will depend on how far from the fulcrum line the secondary abutment is located. The primary abutments in a Class II, modification 1 partial denture are the abutment adjacent to the distal extension base and the most distal abutment on the tooth-supported side. The fulcrum line is a diagonal axis between the two terminal abutments (Figure 8-8). The anterior abutment on the tooth-supported side is a secondary abutment, which serves to support and retain one end of the tooth-supported segment and adds hori-zontal stabilization to the denture. If the modification space were not present, as in an unmodified Class II arch, auxiliary occlusal rests and stabilizing components in the same position would still be essential to the design of the denture (Figure 8-9). However, the presence of a modi-fication space conveniently provides an abutment tooth for support, stabilization, and retention. If the occlusal rest on the secondary abutment lies far enough from the fulcrum line, it may serve adequately as an indirect retainer. Its dual function then is tooth support for one end of the modification area and support for an indirect retainer. The most typical example is a distal occlusal rest on a first premolar when a second premolar and the first molar are missing and the second molar serves as one of the primary abutments. The longest perpendicular to the fulcrum line falls in the vicinity of the first premolar, making the location of the indirect retainer nearly ideal. On the other hand, if only one tooth, such as a first molar, is missing on the modification side, the occlu-sal rest on the second premolar abutment is too close to the fulcrum line to be effective. In such a situation, an auxiliary occlusal rest on the mesial marginal ridge of the first premolar is needed, both for indirect retention and for support for an otherwise unsupported major connector. Support for a modification area extending anteriorly to a canine abutment is obtained by any one of the accepted canine rest forms, as previously outlined in Chapter 6. In this situation, the canine tooth provides nearly ideal indirect retention and support for the major connector as well. Rugae Support Some clinicians consider coverage of the rugae area of the maxillary arch as a means of indirect retention because the rugae area is firm and usually well situated to provide indirect retention for a Class I removable partial denture. Although it is true that broad coverage over the rugae area can conceivably provide some support, the facts remain that tissue support is less effective than positive tooth support, and that rugae coverage is undesirable if it can be avoided. The use of rugae support for indirect retention is usually part of a palatal horseshoe design. Because posterior retention is usually inadequate in this situation, the requirements for indirect retention are probably greater than can be satisfied by this type of tissue support alone. In the mandibular arch, retention from the distal extension base alone is usually inadequate to prevent lifting of the base away from the tissues. In the maxillary 102 Part I General Concepts/Treatment Planning arch, where only anterior teeth remain, full palatal cover-age is usually necessary. In fact, with any Class I maxillary removable partial denture that extends distally from the first premolar teeth, except when a maxillary torus pre-vents its use, palatal coverage may be used to advantage. Although complete coverage may be seen in the form of a resin base, the added retention and reduced bulk of a cast metal palate make the latter preferable (see Chapter 5). However, in the absence of full palatal coverage, an indirect retainer should be used with other designs of major palatal connectors for the Class I removable partial denture. 103 CHAPTER 9 Denture Base Considerations Chapter Outline Functions of Denture Bases in Control of Prosthesis Movement Tooth-supported partial denture base Distal extension partial denture base Methods of Attaching Denture Bases Ideal Denture Base Material Advantages of Metal Bases Accuracy and permanence of form Comparative tissue response Thermal conductivity Weight and bulk Methods of Attaching Artificial Teeth Porcelain or acrylic-resin artificial teeth attached with acrylic-resin Porcelain or resin tube teeth and facings cemented directly to metal bases Resin teeth processed directly to metal bases Metal teeth Chemical bonding Need for Relining Stress-Breakers (Stress Equalizers) Functions of Denture Bases in Control of Prosthesis Movement The denture base supports the artificial teeth and conse-quently receives the functional forces from occlusion and transfers functional forces to supporting oral structures (Figure 9-1). This function is most critical for the distal extension prosthesis, as functional stability and comfort often relate directly to the ability for this transfer of forces to occur without undue movement. Although its primary purpose is related to masticatory function, the denture base also may add to the cosmetic effect of the replacement, particularly when techniques for tinting and reproducing natural-looking contours are used. Most of the techniques for creating a natural appearance in complete denture bases are applicable equally well to partial denture bases. Still another function of the denture base is stimulation of the underlying tissues of the residual ridge. Some vertical movement occurs with any denture base, even those sup-ported entirely by abutment teeth, because of the physiologic movement of those teeth under function. It is clearly evident that oral tissues placed under functional stress within their physiologic tolerance maintain their form and tone better than similar tissues suffering from disuse. The term disuse atrophy is applicable to both periodontal tissues and the tissues of a residual ridge. Tooth-Supported Partial Denture Base Denture bases differ in functional purpose and may differ in terms of the material of which they are made. In a tooth-supported prosthesis, the denture base is primarily a span between two abutments supporting artificial teeth. Thus occlusal forces are transferred directly to the abutments through rests. Also, the denture base and the supplied teeth serve to prevent horizontal migration of all abutment teeth in the partially edentulous arch and vertical migration of teeth in the opposing arch. 104 Part I General Concepts/Treatment Planning provide support for the distal extension base, as the distance from the abutment increases, support from the underlying ridge tissues becomes increasingly important. Maximum support from the residual ridge may be obtained by using broad, accurate denture bases, which spread the occlusal load equitably over the entire area available for such support. The space available for a denture base is determined by the structures surrounding the space and by their movement during function. Maximum support for the denture base therefore can be accomplished only by using knowledge of the limiting anatomic structures and of the histologic nature of the basal seat areas, accuracy of the impression, and accu-racy of the denture base (Figure 9-2). The first two of these support features relate to the gross size and cellular charac-teristics of the residual ridge tissues. These are highly vari-able between patients, and consequently not all residual ridges can provide the same quality of support. Therefore the ability to control functional displacement of the distal extension base is a determination that is unique for indi-vidual patients. When only posterior teeth are being replaced, esthetics is usually only a secondary consideration. On the other hand, when anterior teeth are replaced, esthetics may be of primary importance. Theoretically, the tooth-supported partial denture base that replaces anterior teeth must perform the following functions: (1) provide desirable esthetics; (2) support and retain the artificial teeth in such a way that they provide masticatory efficiency and assist in transferring occlusal forces directly to abutment teeth through rests; (3) prevent vertical and horizontal migration of remaining natural teeth; (4) eliminate undesirable food traps (oral cleanliness); and (5) stimulate the underlying tissues. Distal Extension Partial Denture Base In a distal extension partial denture, denture bases other than those in tooth-supported modifications must contrib-ute to the support of the denture. Such support is critical to the goal of minimizing functional movement and improving stability of the prosthesis. Although the abutment teeth A B C D Figure 9-1 A, Class I maxillary distal extension removable partial denture showing tissue side (intaglio) of denture bases. B, Occlusal side of maxillary prosthesis: posterior artificial teeth are attached to the bases. C, Class II modification 1 mandibular distal extension removable partial denture shows intaglio of both extension and modification bases. D, Occlusal side of mandibular prosthesis: posterior artificial teeth are attached to the bases. For both prostheses, the bases are extended within the limits of physiologic activity of sur-rounding oral structures. 105 Chapter 9 Denture Base Considerations Retention of denture bases has been described as the result of the following forces: (1) adhesion, which is the attraction of saliva to the denture and tissues; (2) cohesion, which is the attraction of the molecules of saliva to each other; (3) atmospheric pressure, which is dependent on a border seal and results in a partial vacuum beneath the denture base when a dislodging force is applied; (4) physi-ologic molding of the tissues around the polished surfaces of the denture; and (5) the effects of gravity on the mandibu-lar denture. Boucher, writing on the subject of complete denture impressions, described these forces as follows: Adhesion and cohesion are effective when there is perfect apposition of the impressioned surface of the denture to the mucous membrane surfaces. These forces lose their effec-tiveness if any horizontal displacement of the dentures breaks the continuity of this contact. Atmospheric pressure is effective primarily as a rescue force when extreme dislodg-ing forces are applied to the denture. It depends on a perfect border seal to keep the pressure applied on only one side of the denture. The presence of air on the impression surface would neutralize the pressure of the air against the polished surface. Because each of these forces is directly proportional to the area covered by the dentures, the dentures should be extended to the limits of the supporting structures. The molding of the soft tissues around the polished sur-faces of denture bases helps to perfect the border seal. Also, it forms a mechanical lock at certain locations on the den-tures, provided these surfaces are properly prepared. This lock is developed automatically and without effort by the patient if the impression is made with an understanding of the anatomic possibilities. The snowshoe principle, which suggests that broad cover-age furnishes the best support with the least load per unit area, is the principle of choice for providing maximum support. Therefore support should be the primary consider-ation in selecting, designing, and fabricating a distal exten-sion partial denture base. Of secondary importance (but to be considered nevertheless) are esthetics, stimulation of the underlying tissues, and oral cleanliness. Methods used to accomplish maximum support of the restoration through its base(s) are presented in Chapters 15 and 16. In addition to their difference in functional purposes, denture bases vary in material of fabrication. This difference is directly related to their function because of the need for some dentures to be relined. Because the tooth-supported base has an abutment tooth at each end on which a rest has been placed, future relining or rebasing may not be necessary to reestablish support. Relining is necessary only when tissue changes have occurred beneath the tooth-supported base to the point that poor esthetics or accumulation of debris results. For these reasons alone, tooth-supported bases made soon after extractions should be of a material that permits later relining. Such materials are the denture resins, the most common of which are copolymer and methyl methacrylate resins. Primary retention for the removable partial denture is accomplished mechanically by placing retaining elements on the abutment teeth. Secondary retention is provided by the intimate relationship of denture bases and major con-nectors (maxillary) with the underlying tissues. The latter is similar to the retention of complete dentures and is propor-tionate to the accuracy of the impression registration, the accuracy of the fit of the denture bases, and the total area of contact involved. B A Figure 9-2 Maxillary and mandibular distal extension removable partial dentures with resin denture bases. Bases are extended buc-cally within the physiologic tolerance of border structures. A, Maxillary denture bases cover both the maxillary tuberosities, extend into the pterygomaxillary notches, and provide for adaptation along the posterior border, taking care not to extend beyond the soft palatal flexure. B, Mandibular bilateral distal extension removable partial denture bases cover the retromolar pads and extend into the retro-mylohyoid fossae. The impression procedure used established buccal shelves as primary stress-bearing areas of basal seats. From Boucher CO: Complete denture impression based upon the anatomy of the mouth, J Am Dent Assoc 31:117-1181, 1994. 106 Part I General Concepts/Treatment Planning Figure 9-4 Unlike the minor connector designs in Figure 9-3, the designs used for this prosthesis have a plastic mesh pattern. Although such designs can be reinforced to be more rigid, the bulk of the connector itself may contribute to weakening of the resin base. A more open type of minor connector seems preferable. Figure 9-3 Mandibular Class II, modification 1 wax pattern developed on an investment cast. Adequate provision is made for attaching the resin base to the major connector on the eden-tulous side by way of a ladderlike minor connector and a butt-type joint. A similar minor connector design will be used for modification space. Note: Relief space beneath the minor con-nectors is established by relief wax placed on the original master cast and duplicated in this refractory cast. This allows processed resin to surround the minor connectors in creating the denture base. Few partial dentures are made without some mechanical retention. Retention from the denture bases may contribute significantly to the overall retention of the partial denture and therefore must not be discounted. Denture bases should be designed and fabricated so that they will contrib-ute as much retention to the partial denture as possible. However, it is questionable whether atmospheric pressure plays as important a role in retention of removable partial dentures because a border seal cannot be obtained as readily as it can be with complete dentures. Therefore adhesion and cohesion gained by excellent apposition of the denture base and soft tissues of the basal seat play an important retentive role. Methods of Attaching Denture Bases Acrylic-resin bases are attached to the partial denture frame-work by means of a minor connector designed so that a space exists between the framework and the underlying tissues of the residual ridge (Figure 9-3). Relief of at least a 20-gauge thickness over the basal seat areas of the master cast is used to create a raised platform on the investment cast on which the pattern for the retentive frame is formed (Figure 9-4). Thus after casting, the portion of the retentive framework to which the acrylic-resin base will be attached will stand away from the tissue surface sufficiently to permit a flow of acrylic-resin base material beneath its surface. The retentive framework for the base should be embed-ded in the base material with sufficient thickness of resin (1.5 mm) to allow for relieving if this becomes necessary during the denture adjustment period or during relining procedures. Thickness is also necessary to avoid weakness and subsequent fracture of the acrylic-resin base material surrounding the metal framework. The use of plastic mesh patterns in forming the retentive framework is generally less satisfactory than a more open framework (see Figure 9-4). Less weakening of the resin by the embedded framework results from use of the more open form. Pieces of 12- or 14-gauge half-round wax and 18-gauge round wax are used to form a ladderlike framework rather than the finer latticework of the mesh pattern. The precise design of the retentive framework, other than that it should be located both buccally and lingually, is not as important as its effective rigidity and strength when it is embedded in the acrylic resin base. It should also be free of interference with future adjustment, should not interfere with arrange-ment of artificial teeth, and should be open enough to avoid weakening any portion of the attached acrylic-resin. Design-ing the retentive framework for denture bases by having elements located buccal and lingual to the residual ridge not only will strengthen the acrylic-resin base but also will mini-mize distortion of the base created by the release of inherent strains in the acrylic-resin base during use or storage of the restoration (Figure 9-5). Metal bases are usually cast as integral parts of the partial denture framework. Mandibular metal bases may be assem-bled and attached to the framework with acrylic-resin (Figure 9-6). Ideal Denture Base Material The requirements for an ideal denture base are as follows: 1. Accuracy of adaptation to the tissues, with minimal volume change 107 Chapter 9 Denture Base Considerations 2. Dense, nonirritating surface capable of receiving and maintaining a good finish 3. Thermal conductivity 4. Low specific gravity; lightweight in the mouth 5. Sufficient strength; resistance to fracture or distortion 6. Easily kept clean 7. Esthetic acceptability 8. Potential for future relining 9. Low initial cost Such an ideal denture base material does not exist, nor is it likely to be developed in the near future. However, any denture base, whether of resin or metal and regardless of the method of fabrication, should come as close to this ideal as possible. Figure 9-5 Note that minor connectors by which resin denture bases will be attached to the framework are open, lad-derlike configurations that extend on both buccal and lingual surfaces. This not only provides excellent attachment of the resin bases, but minimizes warping of bases resulting from the release of inherent strains in compression-molded resin. Figure 9-6 Cast distal extension base of a maxillary remov-able partial denture. Cast bases are an integral part of the frame-work and not only provide support to the prosthetic dentition but reinforce framework rigidity. Advantages of Metal Bases Except for those edentulous ridges with recent extrac-tions, metal can be used for tooth-supported bases and is thought to provide several advantages. Its principal dis-advantages are that it is difficult to adjust and reline. A commonly stated advantage is that the stimulation it gives to the underlying tissues is so beneficial that it pre-vents some alveolar atrophy that would otherwise occur under a resin base and thereby prolongs the health of the tissues that it contacts. Some of the advantages of a metal base are discussed in the following sections. Accuracy and Permanence of Form Cast metal bases, whether of gold, chrome, or titanium alloys, not only may be cast more accurately than denture resins but also can maintain their accuracy of form without changes in the mouth. Internal strains that may be released later to cause distortion are not present. Although some resins and some processing techniques are superior to others in accuracy and permanence of form, modern cast alloys are generally superior in this respect. Evidence of this fact is that an additional posterior palatal seal may be eliminated entirely when a cast palate is used for a com-plete denture, as compared with the need for a definite post-dam when the palate is made of acrylic-resin. Distor-tion of an acrylic-resin base is manifest in the maxillary denture by a distortion away from the palate in the midline and toward the tuberosities on the buccal flanges. The greater the curvature of the tissues, the greater is this dis-108 Part I General Concepts/Treatment Planning tortion. Similar distortions occur in a mandibular denture but are more difficult to detect. Accurate metal castings are not subject to distortion by the release of internal strains, as are most denture resins. Because of its accuracy, the metal base provides an intimacy of contact that contributes considerably to the retention of a denture. Sometimes called interfacial surface tension, direct retention from a cast denture base is sig-nificant in proportion to the area involved. This has been mentioned previously as an important factor in both direct and direct-indirect retention of maxillary restora-tions. Such intimacy of contact is not possible with acrylic-resin bases. Permanence of form of the cast base is also ensured because of its resistance to abrasion from denture clean-ing agents. Cleanliness of the denture base should be stressed; however, constant brushing of the tissue side of the acrylic-resin denture base inevitably causes some loss of accuracy by abrasion. Intimacy of contact, which is never as great with an acrylic-resin base as with a metal base, is therefore jeopardized further by cleaning habits. The metal bases, particularly the harder chrome alloys, withstand repeated cleaning without significant changes in surface accuracy. Comparative Tissue Response Clinical observations have demonstrated that the inher-ent cleanliness of the cast metal base contributes to the health of oral tissues when compared with an acrylic-resin base. Perhaps some of the reasons for this are the greater density and the bacteriostatic activity contributed by ionization and oxidation of the metal base. Acrylic-resin bases tend to accumulate mucinous deposits con-taining food particles, as well as calcareous deposits. Unfavorable tissue reaction to decomposing food parti-cles and bacterial enzymes and mechanical irritation from calculus may result if the denture is not kept meticulously clean. Although calculus, which must be removed peri-odically, can precipitate on a cast metal base, other depos-its do not accumulate as they do on an acrylic-resin base. For this reason, a metal base is naturally cleaner than an acrylic-resin base. Thermal Conductivity Temperature changes are transmitted through the metal base to the underlying tissues, thereby helping to main-tain the health of those tissues. Freedom of interchange of temperature between the tissues covered and the sur-rounding external influences (temperature of liquids, solid foods, and inspired air) contributes much to the patient’s acceptance of a denture and may help avoid the feeling of the presence of a foreign body. Conversely, denture acrylic resins have insulating properties that prevent interchange of temperature between the inside and the outside of the denture base. Weight and Bulk Metal alloy may be cast much thinner than acrylic-resin and still have adequate strength and rigidity. Cast gold must be given slightly more bulk to provide the same amount of rigidity but still may be made with less thick-ness than acrylic-resin materials. Even less weight and bulk are possible when the denture bases are made of chrome or titanium alloys. At times, however, both weight and thickness may be used to advantage in denture bases. In the mandibular arch, the weight of the denture may be an asset with regard to retention, and for this reason a cast gold base may be preferred. On the other hand, extreme loss of residual alveolar bone may make it necessary to add full-ness to the denture base to restore normal facial contours and to fill out the buccal vestibule to prevent food from being trapped in the vestibule beneath the denture. In such situations, an acrylic-resin base may be preferred to the thinner metal base. In the maxillary arch, an acrylic-resin base may be preferred to the thinner metal base to provide fullness in the buccal flanges or to fill a maxillary buccal vestibule. Acrylic-resin may also be preferred over the thinner metal base for esthetic reasons. In these instances, the thinness of the metal base may be of no advantage, but in areas where the tongue and cheek need maximum room, thin-ness may be desirable. Denture base contours for functional tongue and cheek contact can best be accomplished with acrylic-resin. Metal bases are usually made thin to minimize bulk and weight, whereas acrylic-resin bases may be contoured to provide ideal polished surfaces that contrib-ute to retention of the denture, to restore facial contours, and to avoid the accumulation of food at denture borders. Lingual surfaces usually are made concave except in the distal palatal area. Buccal surfaces are made convex at gingival margins, over root prominences, and at the border to fill the area recorded in the impression. Between the border and the gingival contours, the base can be made convex to aid in retention and to facilitate return of the food bolus to the occlusal table during mastication. Such contours prevent food from being entrapped in the cheek and from working under the denture. This usually cannot be accomplished with metal bases. However, the advantages of a metal base need not be sacrificed for the sake of esthetics or desirable denture contours when the use of such a base is indicated. Denture bases may be designed to provide almost total metallic coverage, yet they have resin borders to avoid a display of metal and to add buccal fullness when needed (Figure 9-7). The advantages of thermal conductivity are not nec-essarily lost by covering a portion of the metal base as long as other parts of the denture are exposed to effect temperature changes through conduction. 109 Chapter 9 Denture Base Considerations Methods of Attaching Artificial Teeth Selection of artificial teeth for form, color, and material must precede attachment to the denture. Artificial teeth may be attached to denture bases by the several means that follow: with acrylic resin, with cement, processed directly to metal, cast with the framework, and by chemical bonding. Use of acrylic-resin to attach artificial teeth to a denture base is the most common method. Porcelain or Acrylic-Resin Artificial Teeth Attached With Acrylic-Resin Artificial porcelain teeth are mechanically retained. The pos-terior teeth are retained by acrylic-resin in their diatoric holes. The anterior porcelain teeth are retained by acrylic-resin surrounding their lingually placed retention pins. Arti-ficial resin teeth are retained by a chemical union with the acrylic-resin of the denture base that occurs during labora-tory processing procedures. Attachment of acrylic-resin to the metal base may be accomplished by nailhead retention, retention loops, or diagonal spurs placed at random. Attachment mecha-nisms should be placed so that they will not interfere with placement of the teeth on the metal base (see Figure 9-7). Any junction of acrylic-resin with metal should occur at an undercut finishing line or should be associated with some retentive undercut. Because only a mechanical attachment exists between metal and acrylic-resin, every attempt should be made to avoid separation and seepage, which result in discoloration and uncleanliness. Denture odors are fre-quently caused by accretions at the junction of the acrylic-resin with metal when only a mechanical union exists. Separation that occurs between the acrylic-resin and the metal can eventually lead to some loosening of the acrylic-resin base. Figure 9-7 Partial metal bases used with a palatal strap and resin denture teeth attached directly to the cast metal bases. If needed, a buccal flange of resin could be added to such a base region; however, for these small spans no such flange was needed. Porcelain or Resin Tube Teeth and Facings Cemented Directly to Metal Bases (Figure 9-8) Some disadvantages of this type of attachment are the difficulties involved in obtaining satisfactory occlusion, the lack of adequate contours for functional tongue and cheek contact, and the unesthetic display of metal at gin-gival margins. The latter is avoided when the tooth is butted directly to the residual ridge, but then the reten-tion for the tooth frequently becomes inadequate. A modification of this method is the attachment of ready-made acrylic-resin teeth to the metal base with acrylic-resin of the same shade. This is called pressing on a resin tooth and is not the same as using acrylic-resin for cementation. It is particularly applicable to anterior replacements, because it is desirable to know in advance of making the casting that the shade and contours of the selected tooth will be acceptable (see Figure 9-7). After a labial index of the position of the teeth is made, the lingual portion of the tooth may be cut away or a post-hole prepared in the tooth for retention on the casting. Subsequently, the tooth is attached to the denture with acrylic-resin of the same shade. Because this is done under pressure, the acrylic-resin attachment is compa-rable with the manufactured tooth in hardness and strength. Tube or side-groove teeth must be selected in advance of waxing the denture framework (Figure 9-9). However, for best occlusal relationships, jaw relation records always should be made with the denture casting in the mouth. This problem may be solved by selecting tube teeth for width but with occlusal surfaces slightly higher than will be necessary. The teeth are ground to fit the ridge with sufficient clearance beneath, for a thin metal base, and beveled to accommodate a boxing of metal. If an acrylic-resin tube tooth is used, the diatoric hole should be made slightly larger than that provided. The casting is com-pleted and tried in, occlusal relationships are recorded, and then the teeth are ground to harmonious occlusion with the opposing dentition. As is discussed in Chapter 17, artificial posterior teeth on partial dentures can hardly ever be used unaltered but rather should be considered material from which occlusal forms may be created to function harmoniously with the remaining natural occlusion. Resin Teeth Processed Directly to Metal Bases Modern cross-linked copolymers enable the dentist or technician to process acrylic-resin teeth that have satis-factory hardness and abrasion resistance for many situa-tions. Thus occlusion may be created without resorting to the modification of ready-made artificial teeth (Figure 9-10). Recesses in the denture pattern may be carved by hand or may be created around manufactured teeth that are used only to form the recess in the pattern. Occlusal 110 Part I General Concepts/Treatment Planning B A C Figure 9-8 Tissue surface of Class IV removable partial denture in which artificial dentition was added to a metal base. A, Teeth were set before completion of the framework to allow the modification space design to incorporate altered teeth. B, Anterior teeth were adjusted to the ridge, creating a ridge lap; then the framework was waxed to accommodate a custom tooth position. C, Metal reinforcement adds strength to the artificial teeth and protects against dislodgment. relationships may be established in the mouth on the denture framework and transferred to an articulator. The teeth can then be carved and processed in acrylic-resin of the proper shade to fit the opposing occlusal record. Better attachment to the metal base than by cementation is thus possible. In addition, unusually long, short, wide, or narrow teeth may be created when necessary to fill spaces not easily filled by the limited selection of com-mercially available teeth. Occlusion on acrylic-resin teeth may be reestablished to compensate for wear or settling by reprocessing new acrylic-resin or using light-activated acrylic-resin when this becomes necessary. A distinction always should be made between the need for relining to reestablish occlu-sion (on a distal extension partial denture) and the need for rebuilding occlusal surfaces on an otherwise satisfac-tory base (on a tooth-supported or a tooth and tissue– supported partial denture). Reestablishment of occlusion may also be accom-plished by placing cast gold or other suitable cast alloy restorations on existing resin teeth. Although this may be done on porcelain teeth as well, it is difficult to cut prepa-rations in porcelain teeth unless air abrasive methods are used. Therefore, if later additions to occlusal surfaces are anticipated, acrylic-resin teeth should be used, thereby facilitating the addition of new resin or cast gold surfaces. A simple technique that can be used to fabricate cast gold occlusal surfaces and attach them to resin teeth is described in Chapter 18. Metal Teeth Occasionally, a second molar tooth may be replaced as part of the partial denture casting (Figure 9-11). This is usually done when space is too limited for the attachment of an artificial tooth and yet the addition of a second molar is desirable to prevent extrusion of an opposing second molar. Because the occlusal surface must be waxed before casting, perfect occlusion is not possible. Because metal, particularly a chrome alloy, is abrasion resistant, the area of occlusal contact should be held to a minimum to avoid damage to the periodontium of the opposing tooth and associated discomfort to the patient. Occlusal adjustment on gold occlusal surfaces is readily accom-plished, whereas metal teeth made of chrome alloys are difficult to adjust and are objectionably hard for use as occlusal surfaces. Therefore they should be used only to fill a space and to prevent tooth extrusion. Chemical Bonding Recent developments in resin bonding have provided a means of direct chemical bonding of acrylic-resin to metal frameworks. The investing alveolar and gingival tissue replacement components can be attached without the use of loops, mesh, or surface mechanical locks. Sec-111 Chapter 9 Denture Base Considerations Tapered hole Lingual collar 45-degree bevel Figure 9-9 Stock porcelain or resin tube tooth, or artificial tooth used as tube tooth, should be ground to accommodate cast coping as illustrated. A hole is drilled from the underside of the tooth, or, if a hole is already present, it is made larger. Then the tooth is ground to fit the ridge with enough clearance for minimum thickness of metal. A 45-degree bevel then is formed around the base of the tooth; finally, a collar is created on the lingual side, extending to the interproximal area. The tooth is then lubricated, and a wax pattern for the denture base is formed around it. B A Figure 9-10 Direct attachment of resin teeth to metal bases. A, Anterior modification space prepared to receive a denture tooth, which will be reinforced with a post. B, Posterior molar attached to a previously waxed base to specifically receive the replacement tooth. tions of a metal framework that are used to support replacement teeth can be roughened with abrasives and then treated with a vaporized silica coating. On this surface, an acrylic-resin bonding agent is applied, fol-lowed by a thin film of acrylic-resin that acts as a substrate for later attachment of replacement acrylic-resin teeth or for processing of acrylic-resin tissue replacements (Figure 9-12). A second method of fusing a microscopic layer of ceramic to the metal is accomplished by a process referred to as tribochemical coating. This system involves sandblasting the metal framework with a special silica particle material, Rocatec-Plus (3M Espe Dental Prod-ucts, Irvine, CA). Silica from these particles is attached to the framework by impact. A silane is added to this ceramic-like film to form a chemical bond between the silicate layer and the denture base acrylic-resin. Denture base acrylic-resins formulated with 4-meta are also avail-able and provide a mechanism for bonding acrylic-resin to metal. 112 Part I General Concepts/Treatment Planning green or blue casting wax is generally used, although the thinner 30-gauge or the thicker 26-gauge wax may be used for better evaluation of the clearance between areas not in contact. Loss of support for a distal extension base will result in loss of occlusal contact between the prosthetically supplied Need for Relining The distal extension base differs from the tooth-supported base in several respects, one of which is that it must be made of a material that can be relined or rebased when it becomes necessary to reestablish tissue support for the distal exten-sion base. Therefore acrylic-resin denture base materials that can be relined are generally used. Although satisfactory techniques for making distal exten-sion partial denture bases of cast metal are available, the fact that metal bases are difficult if not impossible to reline limits their use to stable ridges that will change little over a long period. Loss of support for distal extension bases results from changes in residual ridge form over time. These changes may not be readily visible; however, manifestations of this change can be assessed. One of these is loss of occlusion between the distal extension denture base and the opposing dentition, which increases as the distance from the abutment increases (Figure 9-13). This change is proved by having the patient close on strips of 28-gauge green casting wax, or any similar wax, and tapping in centric relation only. Indentations in a wax strip of known thickness are quantitative, whereas marks made with articulating ribbon are only qualitative. In other words, indentations in the wax may be interpreted as light, medium, or heavy, whereas it is difficult if not impos-sible to interpret a mark made with articulating ribbon as light or heavy. In fact, the heaviest occlusal contact may perforate paper-articulating ribbon and make a lesser mark than areas of lighter contact. Therefore the use of any articu-lating ribbon is of limited value in checking occlusion intra-orally. In making occlusal adjustments, articulating ribbon should be used only to indicate where relief should be pro-vided after the need for relief has been established by using wax strips of known thickness. For this purpose, 28-gauge Figure 9-11 Maxillary cast molar designed as an integral part of the framework. Interocclusal space limitation necessitated the use of metal rather than another form of artificial posterior teeth. Note overlays on premolar and molar teeth were used to resist tooth wear. (Courtesy Dr. C.J. Andres, Indianapolis, IN.) B A C Figure 9-12 Coating of metal framework with silica facilitates improved application and seal of resin or composite for denture base regions. Example shows a maxillary minor with beads for added retention of the anterior denture base (A), after air abra-sion preparation of the surface (B), and after sili-coating (C). 113 Chapter 9 Denture Base Considerations ored acrylic-resin, tooth-colored composite, cast occlusal surfaces, or new teeth. In any event, a new occlusion should be established on the existing bases. Relining in this instance would be the wrong solution to the problem. Loss of support may also be assessed clinically by other methods. A layer of rather free-flowing irreversible hydro-colloid, wax, or tissue-conditioning material can be spread on the basal seat portion of the dried denture base(s), and the restoration returned to the patient’s mouth. Care is exer-cised to ensure that the framework is correctly seated (rests and indirect retainers in planned positions). The restoration is removed when the material has set. Significant thicknesses of material remaining under the bases indicate lack of inti-mate contact of the bases with the residual ridges, suggesting a need for relining. More often, however, loss of occlusion is accompanied by settling of the denture base to the extent that rotation about the fulcrum line is manifest. Because relining is the only remedy short of making completely new bases, use of an acrylic-resin base originally facilitates later relining. For this reason, acrylic-resin bases are generally preferred for distal extension partial dentures. The question remains as to when, if ever, metal bases with their several advantages may be used for distal extension partial dentures. It is debatable as to what type of ridge will most likely remain stable under functional loading without apparent change. Certainly the age and general health of the patient will influence the ability of a residual ridge to support function. Minimum and harmonious occlusion and the accuracy with which the base fits the underlying tissues will influence the amount of trauma that will occur under func-tion. The absence of trauma plays a big part in the ability of the ridge to maintain its original form. The best indication for the use of metal distal extension bases is a ridge that has supported a previous partial denture without having become narrowed or flat or without consist-ing primarily of easily displaceable tissues. When such changes have occurred under a previous denture, further change may be anticipated because of the possibility that the oral tissues in question are not capable of supporting a denture base without further change. Despite every advan-tage in their favor, some individuals have ridges that respond unfavorably when called on to support any denture base. In other instances, such as when a new partial denture is to be made because of the loss of additional teeth, the ridges may still be firm and healthy. Because the ridges have previ-ously supported a denture base and have sustained occlu-sion, bony trabeculae will have become arranged to best support vertical and horizontal loading, cortical bone may have been formed, and tissues will have become favorable for continued support of a denture base. In only a relatively few situations is the need for relining a distal extension base not anticipated; metal bases may be considered. However, many instances may be considered borderline. In these cases, metal bases may be used with full understanding on the part of the patient that a new denture teeth and the opposing dentition and a return to heavy occlusal contact between the remaining natural teeth. Usually this is an indication that relining is needed to rees-tablish the original occlusion by reestablishing supporting contact with the residual ridge. It must be remembered, however, that occlusion on a distal extension base is some-times maintained at the expense of extrusion of the opposing natural teeth. In such a situation, checking the occlusion alone will not show that settling of the extension base has occurred because changes in the supporting ridge may have also taken place. A second manifestation of change also must be observ-able to justify relining. This second manifestation of change in the supporting ridge is evidence of rotation about the fulcrum line with the indirect retainers lifting from their seats as the distal extension base is pressed against the ridge tissues. Originally, if the distal extension base was made to fit the supporting form of the residual ridge, rotation about the fulcrum line is not visible. At the time the denture is initially placed, no anteroposterior rotational movement should occur when alternating finger pressure is applied to the indirect retainer and the distal end of a distal extension base or bases. After changes in the ridge form have occurred, which cause some loss of support, rotation occurs about the fulcrum line when alternating finger pressure is applied. This indicates changes in the supporting ridge that must be com-pensated for by relining or rebasing. If occlusal contact has been lost and rotation about the fulcrum line is evident, relining is indicated. On the other hand, if occlusal contact has been lost without any evidence of denture rotation, and if stability of the denture base is otherwise satisfactory, reestablishing the occlusion is the remedy, rather than relining. For the latter, the original denture base may be used in much the same manner as the original trial base was used to record the occlusal relation. Teeth may then be reoriented to an opposing cast or to an occlusal template with the use of light-activated tooth-col-Figure 9-13 28-guage soft green wax used to identify occlusal contacts between a mandibular C1 I distal extension base and an opposing complete denture. 114 Part I General Concepts/Treatment Planning may become necessary if unforeseen tissue changes occur. The technique shown in Figure 9-6 permits replacement of metal distal extension bases without the need to remake the entire denture. This method should be seriously considered any time a distal extension partial denture is to be made with a metal base or bases. For reasons previously outlined, the possibility that tissues will remain healthier beneath a metal base than they will beneath an acrylic-resin base may justify wider use of metal bases for distal extension partial dentures. Through careful treatment planning, better patient education on the problems involved in making a distal extension base partial denture, and greater care taken in the fabrication of denture bases, metal may be used to advantage in some situations in which acrylic-resin bases are ordinarily used. Stress-Breakers (Stress Equalizers) The previous chapters describing component parts of a partial denture have presumed absolute rigidity of all parts of the partial denture framework except the reten-tive arm of the direct retainer. All vertical and horizontal forces applied to the artificial teeth are thus distributed throughout the supporting portions of the dental arch. Broad distribution of stress is accomplished through the rigidity of the major and minor connectors. The effects of the stabilizing components are also made possible by the rigidity of the connectors. In distal extension situations, the use of a rigid con-nection between the denture base and supporting teeth must account for base movement without causing tooth or tissue damage. In such situations, stress on the abut-ment teeth and residual ridge is minimized through the use of functional basing, broad coverage, harmonious occlusion, and correct choice of direct retainers. Gener-ally, two major types of clasp assemblies are used for distal extensions because of their stress-breaking design. Retentive clasp arms may be cast only if they engage undercuts on the abutment teeth in such a manner that tissue-ward movement of the extension base transmits only minimum leverage to the abutment. Otherwise, tapered wrought-wire retentive clasp arms should be used because of their greater flexibility. The tapered, round wrought-wire clasp arm acts somewhat as a stress-breaker between the denture base and the abutment tooth by reducing the effects of denture base movement on the tooth through its flexibility. Another concept of stress-breaking insists on separat-ing the actions of the retaining elements from the move-ment of the denture base by allowing independent movement of the denture base (or its supporting frame-work) and the direct retainers. This form of stress-breaker, also referred to as a stress equalizer, has been used as a means of compensating for inappropriately designed removable partial dentures. Figure 9-14 shows an example of a split bar major connector, which is a commonly used stress breaker. Regardless of their design, most all stress-breakers effectively dissipate vertical stresses, which is the purpose for which they are used. However, this occurs at the expense of horizontal stability and the harmful effects of reduced horizontal stability (excessive ridge resorption, tissue impingement, inefficient mastication), which far outweigh the benefits of vertical stress-breaking. It is the rigid nature of the more conventional removable partial denture that allows satisfaction of all requirements for support, stability, and retention without overemphasis on only one principle to the detriment of the oral tissues. The student is referred to two textbooks that describe in detail the use of stress-breakers and articulated partial denture designs: (1) Precision Attachments in Dentistry, ed 3, by H.W. Preiskel; and (2) Theory and Practice of Precision Attachment Removable Partial Dentures, by J.L. Baker and R.J. Goodkind. B A Figure 9-14 With occlusal loading of posterior distal extensions, the superior edge of the mandibular lingual plate major connector is displaced from the planned contact. If appropriate contact of the major connector and teeth does not return after release of the distal load, this suggests ridge support is poor and occlusal function is suboptimum. The denture base should be considered for a reline to re-establish occlusion contact, ensuring functional use. 115 CHAPTER 10 Principles of Removable Partial Denture Design Chapter Outline Difference in Prosthesis Support and Influence on Design Differentiation Between Two Main Types of Removable Partial Dentures Differences in support Impression registration Differences in clasp design Essentials of Partial Denture Design Components of Partial Denture Design Tooth support Ridge support Major and minor connectors Direct retainers for tooth-supported partial dentures Direct retainers for distal extension partial dentures Stabilizing components Guiding plane Indirect retainers Implant Considerations in Design Examples of Systematic Approach to Design Class III removable partial denture Kennedy Class I, bilateral, distal extension removable partial dentures Kennedy Class II removable partial dentures Additional Considerations Influencing Design Use of a splint bar for denture support Internal clip attachment Overlay abutment as support for a denture base Use of a component partial to gain support Difference in Prosthesis Support and Influence on Design Some of the biomechanical considerations of removable partial denture design were presented in Chapter 4. The strategy of selecting component parts for a partial denture to help control movement of the prosthesis under functional load has been highlighted as a method to be considered for logical partial denture design. The requirements for move-ment control are generally functions of whether the prosthe-sis will be tooth supported or tooth-tissue supported. For a tooth-supported prosthesis, the movement poten-tial is less because resistance to functional loading is pro-vided by the teeth. Teeth do not vary widely in their ability to provide this support; consequently, designs for prostheses are less variable. This is the case even though the amount of supporting bone, the crown-to-root ratios, the crown and root morphologies, and the tooth number and position in the arch relative to edentulous spaces are well established and may be variable for tooth- and tooth-tissue–supported removable partial dentures (RPDs). For a tooth-tissue–sup-ported prosthesis, the residual ridge (remaining alveolar bone and overlying connective tissue covered with mucosa) presents with variable potential for support. Not only does the underlying alveolar bone demonstrate a highly variable form following extraction, it continues to change with time. As alveolar bone responds to the loss of teeth, the overlying connective tissue and mucosa undergo change that places the soft tissue at risk for pressure-induced inflammatory changes. This variable tissue support potential adds com-plexity to design considerations when one is dealing with tooth-tissue–supported prostheses. This occurs because unlike the efficient support provided by teeth, which results in limited prosthesis movement, the reaction of the ridge tissue to functional forces can be highly variable, leading to variable amounts of prosthesis movement. An understand-116 Part I General Concepts/Treatment Planning A B Figure 10-1 A, Kennedy Class I partially edentulous arch. Major support for denture bases must come from residual ridges, tooth support from occlusal rests being effective only at the anterior portion of each base. B, Kennedy Class III, modification 1 partially eden-tulous arch provides total tooth support for the prosthesis. A removable partial denture made for this arch is totally supported by rests on properly prepared occlusal rest seats on four abutment teeth. Mucosa/mucoperiosteum [2.0 mm] Periodontal ligament [0.25 0.1 mm] ] Figure 10-2 Distortion of tissues over the edentulous ridge will be approximately 500 µm under 4 newtons of force, whereas abutment teeth will demonstrate approximately 20 µm of intru-sion under the same load. ing of the potential sources of functional force from the opposing arch that can have an effect on the movement potential of the prosthesis is helpful. Factors related to the opposing arch tooth position, the existence and nature of prosthesis support in the opposing arch, and the potential for establishing a harmonious occlu-sion can greatly influence the partial denture design. Oppos-ing tooth positions that apply forces outside the primary support of the prosthesis can introduce leverage forces that act to dislodge the prosthesis. Such an effect is variable and is based on the nature of the opposing occlusion, because the forces of occlusion differ between natural teeth, remov-able partial dentures, and complete dentures. In general, removable partial dentures opposing natural teeth will require greater support and stabilization over time because of the greater functional load demands. Therefore, occlusal relationships at maximum intercuspation should be broadly dissipated to the supporting units. Differentiation Between Two Main Types of Removable Partial Dentures On the basis of the previous discussion, it is clear that two distinctly different types of RPDs exist. Certain points of difference are present between Kennedy Class I and Class II types of partial dentures on the one hand and the Class III type of partial denture on the other. The first consideration is the manner in which each is supported. The Class I type and the distal extension side of the Class II type derive their primary support from tissues underlying the base and sec-ondary support from the abutment teeth (Figure 10-1, A and Figure 10-2). The Class III type derives all of its support from the abutment teeth (Figure 10-1, B and Figure 10-2). Second, for reasons directly related to the manner of support, the method of impression registration and the jaw record required for each type will vary. Third, the need for some kind of indirect retention exists in the distal extension type of partial denture, whereas in the tooth-supported, Class III type, no extension base is present to lift away from the supporting tissues because of the action of sticky foods and the movements of tissues of the mouth against the borders of the denture. This is so because each end of each denture base is secured by a direct retainer on an abutment tooth. Therefore the tooth-supported partial denture does not rotate about a fulcrum, as does the distal extension partial denture. Fourth, the manner in which the distal extension type of partial denture is supported often necessitates the use of a base material that can be relined to compensate for tissue changes. Acrylic-resin is generally used as a base material for distal extension bases. The Class III partial denture, on the 117 Chapter 10 Principles of Removable Partial Denture Design Impression Registration An impression registration for the fabrication of a partial denture must fulfill the following two requirements: 1. The anatomic form and the relationship of the remaining teeth in the dental arch, as well as the surrounding soft tissues, must be recorded accurately so the denture will not exert pressure on those structures beyond their physi-ologic limits. A type of impression material that can be removed from undercut areas without permanent distor-tion must be used to fulfill this requirement. Elastic impression materials such as irreversible hydrocolloid (alginate), mercaptan rubber base (Thiokol), silicone impression materials (both condensation and addition reaction), and the polyethers are best suited for this purpose. 2. The supporting form of the soft tissues underlying the distal extension base of the partial denture should be recorded so firm areas are used as primary stress–bearing areas and readily displaceable tissues are not overloaded. Only in this way can maximum support of the partial other hand, which is entirely tooth supported, does not require relining except when it is advisable to eliminate an unhygienic, unesthetic, or uncomfortable condition resulting from loss of tissue contact. Metal bases therefore are more frequently used in tooth-supported restorations, because relining is not as likely to be necessary with them. Differences in Support The distal extension partial denture derives its major support from the residual ridge with its fibrous connective tissue covering. The length and contour of the residual ridge sig-nificantly influence the amount of available support and stability (Figure 10-3). Some areas of this residual ridge are firm, with limited displaceability, whereas other areas are displaceable, depending on the thickness and structural character of the tissues overlying the residual alveolar bone. The movement of the base under function determines the occlusal efficiency of the partial denture and also the degree to which the abutment teeth are subjected to torque and tipping stresses. Figure 10-3 A, The longer the edentulous area covered by the denture base, the greater the potential lever action on the abutment teeth. If an extension base area is 30 mm (ac) and tissue displacement is 2 mm (ab), the amount of movement of the proximal plate on the guiding plane will be approximately 0.25 mm: [α = √ (ab)2 + (ac)2]; arc of the tangent ab/ad = x/cd (2/30 = x/3.75 = 0.25 mm). B, The flat ridge will provide good support, poor stability. C, The sharp spiny ridge will provide poor support, poor to fair stability. D, Displaceable tissue on the ridge will provide poor support and poor stability. 118 Part I General Concepts/Treatment Planning sion (as is often seen with a distal rest) must be able to flex sufficiently to dissipate stresses that otherwise would be transmitted directly to the abutment tooth as leverage. On the other hand, a clasp used in conjunction with a mesial rest may not transmit as much stress to the abutment tooth because of the reduction in leverage forces that results from a change in the fulcrum position. This serves the purpose of reducing or “breaking” the stress, hence the term stress-breakers, and is a strategy that is often incorporated into partial denture designs through various means. Some den-tists strongly believe that a stress-breaker is the best means of preventing leverage from being transmitted to the abut-ment teeth. Others believe just as strongly that a wrought-wire or bar-type retentive arm more effectively accomplishes this purpose with greater simplicity and ease of application. A retentive clasp arm made of wrought wire can flex more readily in all directions than can the cast half-round clasp arm. Thereby, it may more effectively dissipate those stresses that would otherwise be transmitted to the abutment tooth. A discussion of the limitations of stress-breakers has been presented in Chapter 9. Only the retentive arm of the circumferential clasp, however, should be made of wrought metal. Reciprocation and stabilization against lateral and torquing movement must be obtained through use of the rigid cast elements that make up the remainder of the clasp. This is called a combina-tion clasp because it is a combination of cast and wrought materials incorporated into one direct retainer. It is fre-quently used on the terminal abutment for the distal exten-sion partial denture and is indicated where a mesiobuccal but no distobuccal undercut exists, or where a gross tissue undercut, cervical and buccal to the abutment tooth, exists. It must always be remembered that the factors of length and material contribute to the flexibility of clasp arms. From a materials physical property standpoint, a short wrought-wire arm may be a destructive element because of its reduced ability to flex compared with a longer wrought-wire arm. However, in addition to its greater flexibility compared with the cast circumferential clasp, the combination clasp offers the advantages of adjustability, minimum tooth contact, and better esthetics, which justify its occasional use in tooth-supported designs. The amount of stress transferred to the supporting eden-tulous ridge(s) and the abutment teeth will depend on: (1) the direction and magnitude of the force; (2) the length of the denture base lever arm(s); (3) the quality of resistance (support from the edentulous ridges and remaining natural teeth); and (4) the design characteristics of the partial denture. As was stated in Chapter 7, the location of the rest, the design of the minor connector as it relates to its corre-sponding guiding plane, and the location of the retentive arm are all factors that influence how a clasp system func-tions. The greater the surface area contact of each minor connector to its corresponding guiding plane, the more horizontal the distribution of force (Figure 10-4). denture base be obtained. An impression material capable of displacing tissue sufficiently to register the supporting form of the ridge will fulfill this second requirement. A fluid mouth-temperature wax or any of the readily flowing impression materials (rubber base, the silicones, or the polyethers in an individual, corrected tray) may be employed for registering the supporting form. Zinc oxide–eugenol impression paste can also be used when only the extension base area is being impressed (see Chapter 15). No single impression material can satisfactorily fulfill both of the previously mentioned requirements. Recording the anatomic form of both teeth and supporting tissues will result in inadequate support for the distal extension base. This is so because the cast will not represent the optimum coordinating forms, which require that the ridge must be related to the teeth in a supportive form. This coordination of support maximizes the support capacity for the arch and minimizes movement of the partial denture under function. Differences in Clasp Design A fifth point of difference between the two main types of removable partial dentures lies in their requirements for direct retention. The tooth-supported partial denture, which is totally supported by abutment teeth, is retained and stabilized by a clasp at each end of each edentulous space. Because this type of prosthesis does not move under function (other than within the physiologic limitations of tooth support units), the only requirement for such clasps is that they flex suffi-ciently during placement and removal of the denture to pass over the height of contour of the teeth in approaching or escaping from an undercut area. While in its terminal posi-tion on the tooth, a retentive clasp should be passive and should not flex except when one is engaging the undercut area of the tooth for resisting a vertical dislodging force. Cast retentive arms are generally used for this purpose. These may be of the circumferential type, arising from the body of the clasp and approaching the undercut from an occlusal direction, or of the bar type, arising from the base of the denture and approaching the undercut area from a gingival direction. Each of these two types of cast clasps has its advantages and disadvantages. In the combination tooth and tissue–supported RPD, because of the anticipated functional movement of the distal extension base, the direct retainer adjacent to the distal extension base must perform still another function, in addi-tion to resisting vertical displacement. Because of the lack of tooth support distally, the denture base will move tissue-ward under function proportionate to the quality (displace-ability) of the supporting soft tissues, the accuracy of the denture base, and the total occlusal load applied. Because of this tissue-ward movement, those elements of a clasp that lie in an undercut area mesial to the fulcrum for a distal exten-119 Chapter 10 Principles of Removable Partial Denture Design edentulous ridge areas. In evaluating the potential support available from edentulous ridge areas, consideration must be given to (1) the quality of the residual ridge, which includes contour and quality of the supporting bone (how the bone has responded to previous stress) and quality of the support-ing mucosa; (2) the extent to which the residual ridge will be covered by the denture base; (3) the type and accuracy of the impression registration; (4) the accuracy of the denture base; (5) the design characteristics of the component parts of the partial denture framework; and (6) the anticipated occlusal load. A full explanation of tissue support for exten-sion base partial dentures is found in Chapter 16. Denture base areas adjacent to abutment teeth are primarily tooth supported. As one proceeds away from the abutment teeth, they become more tissue supported. Therefore it is necessary to incorporate characteristics in the partial denture design that will distribute the functional load equitably between the abutment teeth and the support-ing tissues of the edentulous ridge. Locating tooth support units (rests) on the principal abutment teeth and designing the minor connectors that are adjacent to the edentulous areas to contact the guiding planes in such a manner that the functional load is dispersed equitably between the available tooth and tissue supporting units will provide designs with controlled distribution of support (see Figure 10-4). The second step in systematic development of the design for any removable partial denture is to connect the tooth and tissue support units. This connection is facilitated by design-Essentials of Partial Denture Design The design of the partial denture framework should be sys-tematically developed and outlined on an accurate diagnos-tic cast based on the following prosthesis concepts: where the prosthesis is supported, how the support is connected, how the prosthesis is retained, how the retention and support are connected, and how edentulous base support is connected. In developing the design, it is first necessary to determine how the partial denture is to be supported. In an entirely tooth-supported partial denture, the most ideal location for the support units (rests) is on prepared rest seats on the occlusal, cingulum, or incisal surface of the abutment adja-cent to each edentulous space (see Figure 10-1, B). The type of rest and amount of support required must be based on interpretation of the diagnostic data collected from the patient. In evaluating the potential support that an abutment tooth can provide, consideration should be given to (1) peri-odontal health; (2) crown and root morphologies; (3) crown-to-root ratio; (4) bone index area (how tooth has responded to previous stress); (5) location of the tooth in the arch; (6) relationship of the tooth to other support units (length of edentulous span); and (7) the opposing dentition. (For a more in-depth understanding of these considerations, review Chapters 6 and 12.) In a tooth and tissue–supported partial denture, attention to these same considerations must be given to the abutment teeth. However, equitable support must come from the Figure 10-4 1, Maximum contact of the proximal plate minor connector with the guiding plane produces a more horizontal distribu-tion of stress to the abutment teeth. 2, Minimum contact or disengagement of the minor connector with the guiding plane allows rotation around the fulcrum located on the mesio-occlusal rest, producing a more vertical distribution of stress to the ridge area. 3, Minor connector contact with the guiding plane from the marginal ridge to the junction of the middle and gingival thirds of the abutment tooth distributes load vertically to the ridge and horizontally to the abutment tooth. F is the location of the fulcrum of move-ment for the distal extension base. 120 Part I General Concepts/Treatment Planning necessary to ensure rigidity of the base material without interfering with tooth placement. Components of Partial Denture Design All partial dentures have two things in common: (1) they must be supported by oral structures, and (2) they must be retained against reasonable dislodging forces. In the Kennedy Class III partial denture, three compo-nents are necessary: support provided by rests, the connec-tors (stabilizing components), and the retainers. The partial denture that does not have the advantage of tooth support at each end of each edentulous space still must be supported, but in this situation, the support comes from both the teeth and the underlying ridge tissues rather than from the teeth alone. This is a composite support, and the prosthesis must be fabricated so that the resilient support provided by the edentulous ridge is coordinated with the more stable support offered by the abutment teeth. The essentials—support, connectors, and retainers—must be designed and executed more carefully because of the move-ment of tissue-supported denture base areas. In addition, provision must be made for three other factors, as follows: 1. The best possible support must be obtained from the resilient tissues that cover the edentulous ridges. This is accomplished by the impression technique more than by the partial denture design, although the area covered by the partial denture base is a contributing factor in such support. 2. The method of direct retention must take into account the inevitable tissue-ward movement of the distal exten-sion base(s) under the stresses of mastication and occlu-sion. Direct retainers must be designed so that occlusal loading will result in direct transmission of this load to the long axis of the abutment teeth instead of as leverage. 3. The partial denture, with one or more distal extension denture bases, must be designed so that movement of the extension base away from the tissues will be minimized. This is often referred to as indirect retention and is best described in relation to an axis of rotation through the rest areas of the principal abutments (see Chapter 8). However, retention from the removable partial denture base itself frequently can be made to help prevent this movement and, in such instances, may be discussed as direct-indirect retention. Tooth Support Support of the removable partial denture by the abutment teeth is dependent on the alveolar support of those teeth, the crown and root morphology, the rigidity of the partial denture framework, and the design of the occlusal rests. Through clinical and roentgenographic interpretation, the dentist may evaluate the abutment teeth and decide whether they will provide adequate support. In some instances, the splinting of two or more teeth, either by using fixed partial ing and locating major and minor connectors in compliance with the basic principles and concepts presented in Chapter 5. Major connectors must be rigid so that forces applied to any portion of the denture can be effectively distributed to the supporting structures. Minor connectors arising from the major connector make it possible to transfer functional stress to each abutment tooth through its connection to the corresponding rest and also to transfer the effects of the retainers, rests, and stabilizing components to the remainder of the denture and throughout the dental arch. The third step is to determine how the removable partial denture is to be retained. The retention must be sufficient to resist reasonable dislodging forces. As was stated in Chapter 7, retention is accomplished by placement of mechanical retaining elements (clasps) on the abutment teeth and by the intimate relationship of the denture bases and major con-nectors (maxillary) with the underlying tissues. The key to selecting a successful clasp design for any given situation is to choose one that will (1) avoid direct transmission of tipping or torquing forces to the abutment; (2) accommo-date the basic principles of clasp design by definitive location of component parts correctly positioned on abutment tooth surfaces; (3) provide retention against reasonable dislodging forces (with consideration for indirect retention); and (4) be compatible with undercut location, tissue contour, and esthetic desires of the patient. Location of the undercut is the most important single factor in selection of a clasp. Undercut location, however, can be modified by recontour-ing or restoring the abutment tooth to accommodate a clasp design better suited to satisfy the criteria for clasp selection. The relative importance of retention is highlighted by the results from a clinical trial investigating prosthesis designs. A 5-year randomized clinical trial of two basic removable partial denture designs—one with rest, proximal plate, and I-bar (RPI) design and one with circumferential clasp design—demonstrated no discernible changes after 60 months in nine periodontal health components of the abut-ment teeth with either of the two designs. The overall results indicate that the two designs did not differ in terms of success rates, maintenance, or effects on abutment teeth. Therefore, a well-constructed removable partial denture that is supported by favorable abutments and good residual ridges that are properly prepared and maintained in a patient who exhibits good oral hygiene offers the best opportunity for satisfactory treatment. The fourth step is to connect the retention units to the support units. If direct and indirect retainers are to function as designed, each must be rigidly attached to the major con-nector. The criteria for selection, location, and design are the same as those indicated for connecting the tooth and tissue support units. The fifth and last step in this systematic approach to design is to outline and join the edentulous area to the already established design components. Strict attention to details of the design characteristics outlined in Chapter 9 is 121 Chapter 10 Principles of Removable Partial Denture Design The total occlusal load applied to the residual ridge may be influenced by reducing the area of occlusal contact. This is done with the use of fewer, narrower, and more effectively shaped artificial teeth (Figure 10-5). The distal extension removable partial denture is unique in that its support is derived from abutment teeth, which are comparatively unyielding, and from soft tissues overly-ing bone, which may be comparatively yielding under occlu-sal forces. Resilient tissues, which are distorted or displaced by occlusal load, are unable to provide support for the denture base comparable with that offered by the abutment teeth. This problem of support is further complicated by the fact that the patient may have natural teeth remaining that can exert far greater occlusal force on the supporting tissues than would result if the patient were completely edentulous. This fact is clearly evident from the damage that often occurs to an edentulous ridge when it is opposed by a few remaining anterior teeth in the opposing arch, and espe-cially when the opposing occlusion of anterior teeth has been arranged so that contact occurs in both centric and eccentric positions. Ridge tissues recorded in their resting or nonfunctioning form are incapable of providing the composite support needed for a denture that derives its support from both hard and soft tissue. Three factors must be considered in the acceptance of an impression technique for distal extension removable partial dentures: (1) the material should record the tissues covering the primary stress–bearing areas in their supporting form; (2) tissues within the basal seat area other than primary stress–bearing areas must be recorded in their anatomic form; and (3) the total area covered by the impres-sion should be sufficient to distribute the load over as large an area as can be tolerated by the border tissues. This is an application of the principle of the snowshoe. Anyone who has had the opportunity to compare two master casts for the same partially edentulous arch—one cast having the distal extension area recorded in its anatomic or resting form, and the other cast having the distal exten-sion area recorded in its functional form—has been impressed by the differences in topography (Figure 10-6). A denture base processed to the functional form is generally less irregular and provides greater area coverage than does a denture base processed to the anatomic or resting form. Moreover, and of far greater significance, a denture base made to anatomic form exhibits less stability under rotating and/or torquing forces than does a denture base processed to functional form and thus fails to maintain its occlusal relation with opposing teeth. When the patient is asked to close onto strips of soft wax, it is evident that occlusion is maintained at a point of equilibrium over a longer period of time when the denture base has been made to the functional form. In contrast, evidence indicates that rapid “settling” of the denture base occurs when it has been made to the ana-tomic form, with an early return of the occlusion to natural tooth contact only. Such a denture not only fails to distribute the occlusal load equitably but also allows rotational move-dentures or by soldering two or more individual restorations together, is advisable. In other instances, a tooth may be deemed too weak to be used as an abutment, and extraction is indicated in favor of obtaining better support from an adjacent tooth. Having decided on the abutments, the dentist is respon-sible for preparation and restoration of the abutment teeth to accommodate the most ideal design of the partial denture. This includes the form of the occlusal rest seats. These modi-fications may be prepared in sound tooth enamel or in restorative materials that will withstand the functional stress and wear of the component parts of the removable partial denture. The technician cannot be blamed for inadequate abutment tooth preparation, such as occlusal rest support. On the other hand, the technician is solely to blame if he or she extends the casting beyond, or fails to include, the total prepared areas. If the dentist has sufficiently reduced the marginal ridge area of the rest seat to avoid interference from opposing teeth, and if a definite occlusal rest seat is faithfully recorded in the master cast and delineated in the penciled design, then no excuse can be made for poor occlusal rest form on the partial denture. Ridge Support Support for the tooth-supported removable partial denture or the tooth-supported modification space comes entirely from the abutment teeth by means of rests. Support for the distal extension denture base comes primarily from the over-lying soft tissues and the residual alveolar bone of the distal extension base area. In the latter, rest support is effective only at the abutment end of the denture base. The effectiveness of tissue support depends on six things: (1) the quality of the residual ridge; (2) the extent to which the residual ridge will be covered by the denture base; (3) the accuracy and type of impression registration; (4) the accuracy of the denture bases; (5) the design characteristics of component parts of the partial denture framework; and (6) the occlusal load applied. The quality of the residual ridge cannot be influenced, except that it can be improved by tissue conditioning, or it can be modified by surgical intervention. Such modifica-tions are almost always needed but are not frequently done. The accuracy of the impression technique is entirely in the hands of the dentist. Maximum tissue coverage for support that encompasses the primary stress–bearing areas should be the primary objective in any partial denture impression technique. The manner in which this is accom-plished should be based on biological comprehension of what happens beneath a distal extension denture base when an occlusal load is applied. The accuracy of the denture base is influenced by the choice of materials and by the exactness of the processing techniques. Inaccurate and warped denture bases adversely influence the support of the partial denture. Materials and techniques that will ensure the greatest dimensional stability should be selected. 122 Part I General Concepts/Treatment Planning Major and Minor Connectors Major connectors are the units of a partial denture that connect the parts of the prosthesis located on one side of the arch with those on the opposite side. Minor connectors arise from the major connector and join it with other parts of the denture; thus they serve to connect the tooth and tissue support units. A major connector should be properly located ment, which is damaging to the abutment teeth and their investing structures. An implant can efficiently serve to improve ridge support by replacing the tissue compression seen on functional loading with the stiff resistance offered by bone supporting an implant. The benefit for movement control is achieved as if a change was made from a tooth-tissue–borne prosthesis to a tooth-tooth–borne prosthesis. Figure 10-5 A, The total occlusal load applied may be reduced by using comparatively smaller posterior teeth represented by the right-hand illustration. B, Less muscular force will be required to penetrate a food bolus with a reduced occlusal table, thereby reducing forces to supporting oral structures. Figure 10-6 A, Cast of partially edentulous arch representing anatomic form of residual ridges. An impression was made in the stock tray by using irreversible hydrocolloid. B, An impression recording the functional or supporting form of residual ridges was made in an individualized impression tray, permitting placement of tissues and definitive border molding. 123 Chapter 10 Principles of Removable Partial Denture Design may provide direct-indirect retention that may sometimes, but rarely, eliminate the need for separate indirect retainers. Direct Retainers for Tooth-Supported Partial Dentures Retainers for tooth-supported partial dentures have only two functions: to retain the prosthesis against reasonable dislodging forces without damage to the abutment teeth, and to aid in resisting any tendency of the denture to be displaced in a horizontal plane. The prosthesis cannot move tissue-ward because the retentive components of the clasp assem-bly are supported by the rest. No movement away from the tissues should occur, and therefore no rotation about a fulcrum, because the retentive component is secured by a direct retainer. Any type of direct retainer is acceptable as long as the abutment tooth is not jeopardized by its presence. Intra-coronal (frictional) retainers are ideal for tooth-supported restorations and offer esthetic advantages that are not pos-sible with extracoronal (clasp) retainers. Nevertheless, cir-cumferential and bar-type clasp retainers are mechanically effective and are more economically constructed than are intracoronal retainers. Therefore they are more universally used. Vulnerable areas on the abutment teeth must be pro-tected by restorations with either type of retainer. The clasp retainer must not impinge on gingival tissues. The clasp must not exert excessive torque on the abutment tooth during placement and removal. It must be located the least distance into the tooth undercut for adequate retention, and it must be designed with a minimum of bulk and tooth contact. The bar clasp arm should be used only when the area for retention lies close to the gingival margin of the tooth and little tissue blockout is necessary. If the clasp must be placed high, if the vestibule is extremely shallow, or if an objectionable space would exist beneath the bar clasp arm because of blockout of tissue undercuts, the bar clasp arm should not be used. In the event of an excessive tissue undercut, consideration should be given to recontouring the abutment and using some type of circumferential direct retainer. Direct Retainers for Distal Extension Partial Dentures Retainers for distal extension partial dentures, while retain-ing the prosthesis, must also be able to flex or disengage when the denture base moves tissue-ward under function. Thus the retainer may act as a stress-breaker. Mechanical stress-breakers accomplish the same thing, but they do so at the expense of horizontal stabilization. When some kind of mechanical stress-breaker is used, the denture flange must be able to prevent horizontal movement. Clasp designs that allow flexing of the retentive clasp arm may accomplish the same purpose as that of mechanical stress-breakers, without in relation to gingival and moving tissues and should be designed to be rigid. Rigidity in a major connector is neces-sary to provide proper distribution of forces to and from the supporting components. A lingual bar connector should be tapered superiorly with a half-pear shape in cross section and should be relieved sufficiently but not excessively over the underlying tissues when such relief is indicated. The addition of a continuous bar retainer or a lingual apron does not alter the basic design of the lingual bar. These are added solely for support, stabi-lization, rigidity, and protection of the anterior teeth and are neither connectors nor indirect retainers. The finished infe-rior border of a lingual bar or a linguoplate should be gently rounded to avoid irritation to subjacent tissues when the restoration moves even slightly in function. The use of a linguoplate is indicated when the lower ante-rior teeth are weakened by periodontal disease. It is also indicated in Kennedy Class I partially edentulous arches when additional resistance to horizontal rotation of the denture is required because of excessively resorbed residual ridges. Still another indication is seen in those situations in which the floor of the mouth so closely approximates the lingual gingiva of anterior teeth that an adequately inflexible lingual bar cannot be positioned without impinging on gin-gival tissues. Experience with the linguoplate has shown that with good oral hygiene, the underlying tissues remain healthy and no harmful effects to the tissues result from the metallic coverage per se. However, adequate relief must be provided whenever a metal component crosses the gingival margins and the adjacent gingivae. Excessive relief should be avoided because tissues tend to fill a void, resulting in the overgrowth of abnormal tissue. The amount of relief used, therefore, should be only the minimum necessary to avoid gingival impingement. It does not seem that there are many advantages to be found in the use of the continuous bar retainer versus the linguoplate. In rare instances, when a linguoplate would be visible through multiple interproximal embrasures, the con-tinuous bar retainer may be preferred for esthetic reasons only. In other instances, when a single diastema exists, a linguoplate may be cut out in this area to avoid the display of metal, without sacrificing its use when otherwise indicated. The rigidity of a palatal major connector is just as impor-tant and its location and design just as critical as for a lingual bar. A U-shaped palatal connector is rarely justified except to avoid an inoperable palatal torus that extends to the junc-tion of the hard and soft palates. Neither can the routine use of a narrow, single palatal bar be justified. The combination anterior-posterior palatal strap–type major connector is mechanically and biologically sound if it is located so that it does not impinge on tissues. The broad, anatomic palatal major connector is frequently preferred because of its rigid-ity, better acceptance by the patient, and greater stability without tissue damage. In addition, this type of connector 124 Part I General Concepts/Treatment Planning sible. This means that they should be confined to interdental embrasures whenever possible. When minor connectors are located on vertical tooth surfaces, it is best that these surfaces be parallel to the path of placement. When cast restorations are used, these surfaces of the wax patterns should be made parallel on the surveyor before casting. A modification of minor connector design has been pro-posed that places the minor connector in the center of the lingual surface of the abutment tooth (Figure 10-7). Propo-nents of this design claim that it reduces the amount of gingival tissue coverage and provides enhanced bracing and guidance during placement. Disadvantages may include increased encroachment on the tongue space, more obvious borders, and potentially greater space between the connector and the abutment tooth. This proposed variation, however, when combined with thoughtful design principles, may provide some benefit to the periodontal health of the abut-ment teeth and may be acceptable to some patients. Reciprocal clasp arms also must be rigid, and they must be placed occlusally to the height of contour of the abutment teeth, where they will be nonretentive. By their rigidity, these clasp arms reciprocate the opposing retentive clasp; they also prevent horizontal movement of the prosthesis under func-tional stresses. For a reciprocal clasp arm to be placed favor-ably, some reduction of the tooth surfaces involved is frequently necessary to increase the suprabulge area. When crown restorations are used, a lingual reciprocal clasp arm may be inset into the tooth contour by providing a ledge on the crown on which the clasp arm may rest. This permits the use of a wider clasp arm and restores a more nearly normal tooth contour, at the same time maintaining its strength and rigidity (see Chapter 14). sacrificing horizontal stabilization and with less complicated techniques. In evaluating the ability of a clasp arm to act as a stress-breaker, one must realize that flexing in one plane is not enough. The clasp arm must be freely flexible in any direc-tion, as dictated by the stresses applied. Bulky, half-round clasp arms cannot do this, and neither can a bar clasp engag-ing an undercut on the side of the tooth away from the denture base. Round, tapered clasp forms offer the advan-tages of greater and more universal flexibility, less tooth contact, and better esthetics. Either the combination cir-cumferential clasp with its tapered wrought-wire retentive arm or the carefully located and properly designed circum-ferential or bar clasp can be considered for use on all abut-ment teeth adjacent to extension denture bases if the abutment teeth have been properly prepared and tissue support effectively achieved, and if the patient exercises good oral hygiene. Stabilizing Components Stabilizing components of the removable partial denture framework are those rigid components that assist in stabiliz-ing the denture against horizontal movement. The purpose of all stabilizing components should be to distribute stresses equally to all supporting teeth without overworking any one tooth. The minor connectors that join the rests and the clasp assemblies to the major connector serve as stabilizing components. All minor connectors that contact vertical tooth surfaces (and all reciprocal clasp arms) act as stabilizing components. It is necessary that minor connectors have sufficient bulk to be rigid and yet present as little bulk to the tongue as pos-Figure 10-7 Prospective guiding plane surfaces are indicated by lines located on the respective surfaces of abutment teeth. These surfaces, when used, can be made vertically parallel to the path of placement. However, by including guiding plane surfaces, which are not in the same parallel plane horizontally (arrows) but are divergent, cross-arch resistance to horizontal rotation of the denture is enhanced. 125 Chapter 10 Principles of Removable Partial Denture Design Guiding Plane The term guiding plane is defined as two or more parallel, vertical surfaces of abutment teeth, so shaped to direct a prosthesis during placement and removal. After the most favorable path of placement has been ascertained, axial sur-faces of abutment teeth are prepared parallel to the path of placement, and therefore become parallel to each other. Guiding planes may be contacted by various components of the partial denture—the body of an extracoronal direct retainer, the stabilizing arm of a direct retainer, the minor connector portion of an indirect retainer—or by a minor connector specifically designed to contact the guiding plane surface. The functions of guiding plane surfaces are as follows: (1) to provide for one path of placement and removal of the restoration (to eliminate detrimental strain to abutment teeth and framework components during placement and removal); (2) to ensure the intended actions of reciprocal, stabilizing, and retentive components (to provide retention against dislodgment of the restoration when the dislodging force is directed other than parallel to the path of removal and also to provide stabilization against horizontal rotation of the denture); and (3) to eliminate gross food traps between abutment teeth and components of the denture. Guiding plane surfaces need to be created so that they are as nearly parallel to the long axes of abutment teeth as pos-sible. Establishing guiding planes on several abutment teeth (preferably more than two teeth), located at widely separated positions in the dental arch, provides for more effective use of these surfaces. The effectiveness of guiding plane surfaces is enhanced if these surfaces are prepared on more than one common axial surface of the abutment teeth (see Figure 10-7). As a rule, proximal guiding plane surfaces should be about one-half the width of the distance between the tips of adjacent buccal and lingual cusps, or about one-third the buccal lingual width of the tooth, and should extend verti-cally about two thirds of the length of the enamel crown portion of the tooth from the marginal ridge cervically. In the preparation of guiding plane surfaces, care must be exer-cised to avoid creating buccal or lingual line angles (Figure 10-8). If it is assumed that the stabilizing or retentive arm of a direct retainer may originate in the guiding plane region, a line angle preparation would weaken either or both com-ponents of the clasp assembly. A guiding plane should be located on the abutment surface adjacent to an edentulous area. However, excess torquing is inevitable if the guiding planes squarely facing each other on a lone standing abutment adjacent to an extension area are used (Figure 10-9). Indirect Retainers An indirect retainer must be placed as far anterior from the fulcrum line as adequate tooth support permits, if it is to function with the direct retainer to restrict movement of a Figure 10-8 A, The guiding plane surface should be like an area on a cylindrical object. It should be a continuous surface unbounded by even, rounded line angles. B, Minor connector contacting the guiding plane surface has the same curvature as that surface. From an occlusal view, it tapers buccally from the thicker lingual portion, thus permitting closer contact of the abutment tooth and the prosthetically supplied tooth. Viewed from the buccal aspect, the minor connector contacts the enamel of the tooth on its proximal surface about two-thirds its length. Figure 10-9 Guiding planes squarely facing each other should not be prepared on a lone standing abutment. Minor connectors of the framework (gray areas) would place undue strain on the abutment when the denture if rotated vertically, either superiorly or inferiorly. distal extension base away from the basal seat tissues. It must be placed on a rest seat prepared in an abutment tooth that is capable of withstanding the forces placed on it. An indirect retainer cannot function effectively on an inclined tooth surface, nor can a single weak incisor tooth be used for this 126 Part I General Concepts/Treatment Planning fit the prepared surfaces of the anatomic form of the teeth and surrounding structures. It does not require an impres-sion of the functional form of the ridge tissues, nor does it require indirect retention. Cast clasps of the circumferential variety, the bar type, or the combination clasp may be used, depending on how one can modify the surfaces of the abut-ment teeth (guiding planes, rests, contours for proper loca-tion of clasp arms). Unless the need for later relining is anticipated, as in the situation of recently extracted teeth, the denture base may be made of metal, which offers several advantages. The Class III partial denture can frequently be purpose. A canine or premolar tooth should be used for the support of an indirect retainer, and the rest seat must be prepared with as much care as is given any other rest seat. An incisal rest or a lingual rest may be used on an anterior tooth, provided a definite seat can be obtained in sound enamel or on a suitable restoration. A second purpose that indirect retainers serve in partial denture design is that of support for major connectors. A long lingual bar or an anterior palatal major connector is thereby prevented from settling into the tissues. Even in the absence of a need for indirect retention, provision for such auxiliary support is sometimes indicated. Contrary to common use, a cingulum bar or a linguoplate does not in itself act as an indirect retainer. Because these are located on inclined tooth surfaces, they serve more as orthodontic appliances than as support for the partial denture. When a linguoplate or a cingulum bar is used, terminal rests should always be provided at either end to stabilize the denture and to prevent orthodontic movement of the teeth contacted. Such terminal rests may function as indirect retainers, but they would function equally well in that capacity without the continuous bar retainer or linguoplate. Implant Considerations in Design As was mentioned in Chapter 4, the objectives of RPD design are to replace missing teeth with a prosthesis that exhibits limited movement under the influence of functional forces, and to ensure that movement is within physiologic toler-ance. Physiologic tolerance would include tissue tolerance as well as a patient’s physiologic ability to accommodate to the prosthesis. The Kennedy Class III tooth–supported RPD presents less of a challenge to oral tissues and patient accommodation than does the Kennedy Class I or II tooth-tissue prosthesis. The challenge is chiefly related to prosthesis movement to an extent allowed by tissue displaceability under an applied force. Use of dental implants to reduce this displacement can significantly benefit tissue tolerability and reduce any chal-lenge to accommodation presented by prosthesis movement. Use of implants can also assist other worthwhile goals such as improved stability and retention when these aspects are needed because of anatomic deficiencies or related factors. The clinician must consider potential movements of the prosthesis and the ability to control movement given the existing oral tissues, teeth, and occlusion. Selective applica-tion of dental implants can provide needed movement control. Examples of Systematic Approach to Design Class III Removable Partial Denture The Kennedy Class III removable partial denture (Figures 10-10 and 10-11), entirely tooth supported, may be made to Figure 10-10 Removable partial denture in maxillary Class III arch. The design consists of anterior and posterior palatal bar major connectors, resin-supported artificial teeth, and bar clasp arms throughout. Figure 10-11 Removable partial denture framework in maxil-lary Class III arch. The design consists of a single palatal strap major connector, bar and circumferential clasp arms, and a means to attach resin-supported artificial teeth. 127 Chapter 10 Principles of Removable Partial Denture Design is not adequate. It seems rational under these circumstances to minimize these effects through optimum denture base adaptation, to reduce movement or to provide implant support (see Figure 10-13). Should the bar-type retainer be contraindicated because of a severe tissue undercut or the existence of only a mesiobuccal undercut on the anterior abutment, then a combination direct retainer with the reten-tive arm made of tapered wrought wire should be used. A thorough understanding of the advantages and disadvan-tages of various clasp designs is necessary in determining the type of direct retainer that is to be used for each abutment tooth. Steps included in fabrication of the Class II partial denture closely follow those used with the Class I partial denture, used as a valuable aid in periodontal treatment because of its stabilizing influence on the remaining teeth. Kennedy Class I, Bilateral, Distal Extension Removable Partial Dentures The Class I, bilateral, distal extension partial denture is as different from the Class III type as any two dental restora-tions could be (see Figure 10-1). Because it derives its prin-cipal support from the tissues underlying its base, a Class I partial denture made to anatomic ridge form cannot provide uniform and adequate support. Yet, unfortunately, many Class I mandibular removable partial dentures are made from a single irreversible hydrocolloid impression. In such situations, both the abutment teeth and the residual ridges suffer because the occlusal load placed on the remaining teeth is increased by the lack of adequate posterior support. Many dentists, recognizing the need for some type of impression registration that will record the supporting form of the residual ridge, attempt to record this form with a metallic oxide, a rubber base, or one of the silicone impres-sion materials. Such materials actually record only the ana-tomic form of the ridge, except when the special design of impression trays permits recording of the primary stress– bearing areas under a simulated load. Others prefer to place a base, made to fit the anatomic form of the ridge, under some pressure at the time that it is related to the remaining teeth, thus obtaining functional support. Still others, who believe that a properly compounded mouth-temperature wax will displace only those tissues that are incapable of providing support to the denture base, use a wax secondary impression to record the supporting, or functional, form of the edentulous ridge. Any impression record will be influ-enced by the consistency of the impression material and the amount of hydraulic pressure exerted by its confinement within the impression tray. Kennedy Class II Removable Partial Dentures The Kennedy Class II partial denture (Figures 10-12 and 10-13) actually may be a combination of tissue-supported and tooth-supported restorations. The distal extension base must have adequate tissue support, whereas tooth-supported bases elsewhere in the arch may be made to fit the anatomic form of the underlying ridge. Indirect retention must be provided for; however, occasionally the anterior abutment on the tooth-supported side will satisfy this requirement. If additional indirect retention is needed, provisions must be made for it. Cast clasps are generally used on the tooth-supported side; however, a clasp design in which wrought wire is used may reduce the application of torque on the abutment tooth adjacent to the distal extension and should be considered. The use of a cast circumferential clasp engaging a mesiobuc-cal undercut on the anterior abutment of the tooth-sup-ported modification space may result in a Class I leverlike action if the abutment teeth have not been properly pre-pared, and/or if tissue support from the extension base area Figure 10-12 Mandibular Class II removable partial denture with distal extension base. Because of tissue undercut cervical to the buccal surface of the right second premolar and lack of dis-tobuccal undercut, a wrought-wire (tapered) retainer arm was used. Figure 10-13 Mandibular Class II, modification 1 partially edentulous arch. Note that bar-type retentive arms are used on both premolar abutments, engaging distobuccal undercuts at their terminal ends. Leverlike forces may not be as readily imparted to the right premolar, as opposed to the cast circum-ferential direct retainer engaging the mesiobuccal undercut. 128 Part I General Concepts/Treatment Planning except that the distal extension base is usually made of an acrylic-resin material, whereas the base for any tooth-sup-ported area can be made of metal. This is permissible because the residual ridge beneath tooth-supported bases is not called on to provide support for the denture, and later rebas-ing is not as likely to be necessary. Additional Considerations Influencing Design Every effort should be made to gain the greatest support possible for removable prostheses with the use of abut-ments bounding edentulous spaces. This not only will relieve the residual ridges of some of their obligation for support but also may allow the design of the framework to be greatly simplified. To this end, use of splint bars, internal clip attachments, overlay abutments, overlay attachments, a component partial, and implants should be considered. Use of a Splint Bar for Denture Support In the Chapter 14 discussion of missing anterior teeth, mention is made of the fact that missing anterior teeth are best replaced with a fixed partial denture. The follow-ing is quoted from that chapter: “From a biomechanical standpoint … a removable partial denture should replace only the missing posterior teeth after the remainder of the arch has been made intact by fixed restorations.” Occasionally, a situation is found in which it is neces-sary for several missing anterior teeth to be replaced by the removable partial denture rather than by fixed resto-rations. This may be caused by the length of the edentu-lous span or by the loss of a large amount of the residual ridge due to resorption, accident, or surgery, or it may result from a situation in which too much vertical space prevents the use of a fixed partial denture, or in which esthetic requirements can better be met through the use of teeth added to the denture framework. In such instances it is necessary that the best possible support for the replaced anterior teeth be provided. Ordinarily this is done through the placement of occlusal or lingual rests, or both, on the adjacent natural teeth, but when the eden-tulous span is too large to ensure adequate support from the adjacent teeth, other methods must be used. This is included here only because it influences the design of the major connector that must then be used. An anterior splint bar may be attached to the adjacent abutment teeth in such a manner that fixed splinting of the abutment teeth results, with a smooth, contoured bar resting lightly on the gingival tissues to support the removable partial denture. As with any fixed partial denture, the type of abutment retainers and the decision to use multiple abutments will depend on the length of the span and the available support and stability of the teeth being used as abutments. Regardless of the type of abutment retainers used, the connecting bar may be cast of a rigid alloy, or a commercially available bar may be used and cast to the abutments or attached to the abut-ments by soldering. The length of the span influences the size of a splint bar. Long spans require more rigid bars (10-gauge) than short spans (13-gauge). If the bar is to be soldered, it is best that recesses be formed in the proximal surfaces of the abutments and that the connecting bar, which rests lightly on the tissues, be cast or made to fit into these recesses and then attached by soldering. Because of the greater rigidity of chromium-cobalt alloys, the splint bar is preferably cast in one of these materials and then is attached to the abutments by solder-ing. The complete assembly (abutments and connecting bar) is then cemented permanently to the abutment teeth, in the same way as for a fixed partial denture. The impres-sion for the partial denture is then made, and a master cast is obtained that accurately reproduces the contours of the abutments and the splint bar. The denture frame-work then is made to fit the abutments and the bar by extending the major connector or minor connectors to cover and rest upon the splint bar. Retention for the attachment of a resin base, or any other acceptable means of attaching the replaced anterior teeth, is incorporated into the denture design. In those situations wherein the removable partial denture will be tooth supported, the splint bar may be curved to follow the crest of the residual ridge. However, in a distal extension situation, because of the vertical rotation of the denture, caution must be exer-cised to form the splint bar so that excessive torque will not accrue to its supporting abutments (Figure 10-14). The proximal contours of abutments adjacent to splint bars should be parallel to the path of placement. This serves three purposes: (1) it permits a desirable arrange-ment of artificial teeth; (2) it aids in resisting horizontal rotation of the restoration; and (3) these components act as guiding planes to direct the partial denture to and from its terminal position. The splint bar must be positioned anteroposteriorly just lingual to the residual ridge to allow an esthetic arrangement of artificial teeth. The resulting partial denture offers esthetic advantages of removable anterior replacements and positive support, retention, and stabil-ity from the underlying splint bar (Figure 10-15). Internal Clip Attachment The internal clip attachment differs from the splint bar in that the internal clip attachment provides both support and retention from the connecting bar. Several preformed connecting bars are commercially available in plastic patterns. These can be customized for length and cast in the metal alloy of choice. Internal clip attachments are also commercially available in various metal alloys and durable nylon. When a custom-made 129 Chapter 10 Principles of Removable Partial Denture Design Figure 10-14 A, Insofar as is possible, the splint bar should be round or ovoid. Provisions must be made in the construction and location of the bar so that dental floss may be threaded underneath the bar to allow proper cleaning by the patient. B, As viewed from above, the bar is in a straight line between abut-ments. This is especially critical for distal extension removable partial dentures to avoid excess torque on abutments as the denture rotates in function. C, Sagittal section through the bar demonstrates the rounded form of the bar making point contact with the residual ridge. The entire tissue surface of the bar is easily accessible for cleaning with dental floss. A pear-shaped bar (in cross section) will permit rotation of the removable partial denture without appreciable resistance or torque. Figure 10-15 Lower canines splinted together with a splint bar. The longevity of these teeth is greatly enhanced by splinting. Tissue surfaces are minimally contacted by the rounded form of the lower portion of the bar. Anterior and posterior slopes of the splint bar must be compatible with the path of placement of the denture. Floss will be used by the patient to clean the inferior portion of the splint bar. connecting bar and clip is fabricated, the bar should be cast from 10- or 13-gauge sprue wax. The cast bar should rest lightly or should be located slightly above the tissues. Retention is provided by one of the commercial pre-formed metal or nylon clips, which is contoured to fit the bar and is retained in a preformed metal housing or par-tially embedded by means of retention spurs or loops into the overlying resin denture base. The internal clip attachment thus provides support, stability, and retention for the anterior modification area and may serve to eliminate both occlusal rests and reten-tive clasps on adjacent abutment teeth. Overlay Abutment as Support for a Denture Base Every consideration should be directed toward prevent-ing the need for a distal extension removable partial denture. In many instances, it is possible to salvage the roots and a portion of the crown of a badly broken-down molar through endodontic treatment. A periodontally involved molar, otherwise indicated for extraction, some-times may be salvaged by periodontal and endodontic treatment accompanied by reduction of the clinical crown almost level with the gingival tissues. In another situation, an unopposed molar may have extruded to such an extent that restoring the tooth with a crown is inadequate to develop a harmonious occlusion. Then too, it is not unusual to encounter a molar that is so grossly tipped anteriorly that it cannot serve as an abutment unless the clinical crown is reduced drastically. Such teeth should be considered for possible support for an otherwise distal extension denture base. Endodon-tic treatment and preparation of the coronal portion of the tooth as a slightly elevated dome-shaped abutment often offer an alternative to a distal extension base. The student is referred to the “Selected Reading Resources” section (textbooks; abutment retainers) for sources of information on overdenture abutments and overlay-type prostheses. Use of a Component Partial to Gain Support A component partial is a removable partial denture in which the framework is designed and fabricated in sepa-rate parts. The tooth support and tissue-supported com-ponents are individually fabricated, and the two are joined with a high-impact acrylic-resin to become a single rigid functioning unit. 130 CHAPTER 11 Surveying Chapter Outline Description of Dental Surveyor Purposes of the Surveyor Surveying the diagnostic cast Contouring wax patterns Surveying ceramic veneer crowns Placement of intracoronal retainers (internal attachments) Placement of internal rest seats Machining cast restorations Surveying the master cast Factors That Determine Path of Placement and Removal Guiding planes Retentive areas Interference Esthetics Step-by-Step Procedures in Surveying a Diagnostic Cast Guiding planes Retentive areas Interference Esthetics Final Path of Placement Recording Relation of Cast to Surveyor Surveying the Master Cast Measuring Retention Blocking Out the Master Cast Relieving the Master Cast Paralleled Blockout, Shaped Blockout, Arbitrary Blockout, and Relief When a fixed partial denture (FPD) is prepared, the orientation of the diamond bur is controlled to remove an amount of tooth structure necessary to satisfy the require-ments of the path of insertion for the prosthesis. Accom-plishment of parallel preparations is ultimately verified by complete seating of the prosthesis, but could be verified on the master cast or dies by the use of the surveyor. Once the FPD is fabricated and completely seated, it is ensured full engagement of the entire circumference of and occlusal support from the abutment retainers. If adequate resistance form and fit of the prosthesis are provided, the chance for functional stability equivalent to natural teeth is good. This could not be ensured unless the relationship of the fixed prosthesis and the prepared teeth were carefully controlled. For a removable prosthesis, the necessity for appropri-ately planned and executed tooth preparation, followed by verification of a well-fitting prosthesis that engages the teeth as planned, is equally important. As was briefly mentioned in Chapter 7, a dental surveyor is vitally important to the planning, execution, and verification of appropriate mouth modifications for a removable partial denture. Although it does not necessarily affect the occlusal rest preparations on abutment teeth, use of the surveyor is critical for planning the modifications of all tooth surfaces that will be involved in support, stabilization, and retention of the prosthesis. In this role, the use of a surveyor to determine the needed mouth preparation is vitally important in helping to provide stable and comfortable removable prostheses. A dental surveyor has been defined as an instrument used to determine the relative parallelism of two or more surfaces of the teeth or other parts of the cast of a dental arch. There-fore the primary purpose of surveying is to identify the modifications of oral structures that are necessary to fabri-cate a removable partial denture that will have a successful prognosis. It is the modification of tooth surfaces to accom-modate placement of the component parts of the partial denture in their designated ideal positions on abutment teeth that facilitates this prognosis. 131 Chapter 11 Surveying The principal parts of the Jelenko surveyor are essentially the same as those of the Ney surveyor except that when the nut at the top of the vertical arm is loosened, the horizontal arm may be made to swivel. The objective of this feature, originally designed by Dr. Noble Wills, is to permit freedom of movement of the arm in a horizontal plane rather than to depend entirely on the horizontal movement of the cast. To some this is confusing because two horizontal movements must thus be coordinated. For those who prefer to move the cast only in horizontal relationship to a fixed vertical arm, the nut may be tightened and the horizontal arm used in a fixed position. Another difference between Ney and Jelenko surveyors is that the vertical arm on the Ney surveyor is retained by fric-tion within a fixed bearing. The shaft may be moved up or down within this bearing but remains in a vertical position until again moved. The shaft may be fixed in any vertical position desired by tightening a setscrew. In contrast, the vertical arm of the Jelenko surveyor is spring mounted and returns to the top position when it is released. It must be held down against spring tension while it is in use, which to some is a disadvantage. The spring may be removed, but the friction of the two bearings supporting the arm does not hold it in position as securely as does a bearing designed for that purpose. These minor differences in the two survey-ors lead to personal preference but do not detract from the effectiveness of either surveyor when each is properly used. Any one of several moderately priced surveyors on the market will adequately accomplish the procedures necessary to design and construct a partial denture. In addition, these surveyors may be used to parallel internal rests and intra-coronal retainers. With a handpiece holder added, they may be used to machine internal rests and to make the guiding-plane surfaces of abutment restorations parallel. Description of Dental Surveyor The most widely used surveyors are the Ney (Figure 11-1) and the Jelenko (Figure 11-2). Both of these are precision-made instruments. They differ principally in that the Jelenko arm swivels, whereas the Ney arm is fixed. The technique for surveying and trimming blockout is therefore somewhat dif-ferent. Other surveyors also differ in this respect, and the dentist may prefer one over another for this reason. The principal parts of the Ney surveyor are as follows: 1. Platform on which the base is moved 2. Vertical arm that supports the superstructure 3. Horizontal arm from which the surveying tool suspends 4. Table to which the cast is attached 5. Base on which the table swivels 6. Paralleling tool or guideline marker (this tool contacts the convex surface to be studied in a tangential manner; the relative parallelism of one surface to another may thus be determined; with substitution of a carbon marker, the height of contour may be delineated on the surfaces of the abutment teeth and on areas of interference requiring reduction on blockout) 7. Mandrel for holding special tools (Figure 11-3) Figure 11-1 The Ney surveyor is widely used because of its simplicity and durability. Dental students should be required to own such a surveyor. By becoming familiar with and dependent on its use, they are more likely to continue using the surveyor in practice as a necessary piece of equipment toward more ade-quate diagnosis, effective treatment planning, and performance of many other aspects of prosthodontic treatment. Figure 11-2 The Jelenko surveyor. Note the spring-mounted paralleling tool and swivel at the top of the vertical arm. The horizontal arm may be fixed in any position by tightening the nut at the top of the vertical arm. 132 Part I General Concepts/Treatment Planning Several other types of surveyors have been designed and are in use today. Many of these are elaborate and costly, yet provide little advantage over simpler types of surveyors. Purposes of the Surveyor The surveyor may be used for surveying the diagnostic cast, recontouring abutment teeth on the diagnostic cast, con-Because the shaft on the Ney surveyor is stable in any vertical position—yet may be moved vertically with ease—it lends itself well for use as a drill press when a handpiece holder is added (Figure 11-4). The handpiece may thus be used to cut recesses in cast restorations with precision with burs or carborundum points of various sizes in a dental handpiece. A B C D F G E Ney undercut gauge Ney carbon marker and sheath Jelenko carbon marker Jelenko undercut gauge 0.03 0.03 0.02 0.02 0.01 0.01 Path of placement Ney wax trimmer Blockout wax 2 taper tool 6 taper tool Ney surveyor blade Figure 11-3 Various tools that may be used with a dental surveyor. A, Ney undercut gauges. B, Jelenko undercut gauge. C, Ney carbon marker with metal reinforcement sheath. D, Jelenko carbon marker. E, Tapered tools, 2- and 6-degree, for trimming blockout when some nonparallelism is desired. F, Ney wax trimmer for paralleling blockout. G, Surveying blade used for trimming blockout. 133 Chapter 11 Surveying tooth surfaces (guiding planes) made parallel to that path of placement. 2. To identify proximal tooth surfaces that are or need to be made parallel, so they act as guiding planes during place-ment and removal. 3. To locate and measure areas of the teeth that may be used for retention. 4. To determine whether teeth and bony areas of interfer-ence will need to be eliminated surgically or by selection of a different path of placement. 5. To determine the most suitable path of placement that will permit locating retainers and artificial teeth to provide the best esthetic advantage. 6. To permit accurate charting of the mouth preparations to be made. This includes the preparation of proximal tooth surfaces to provide guiding planes and the reduc-tion of excessive tooth contours to eliminate interference and to permit a more acceptable location of reciprocal and retentive clasp arms. By marking these areas on the diagnostic cast in red, using an undercut gauge to esti-mate the amount of tooth structure that may safely (without exposing dentin) be removed, and then trim-ming the marked areas on the stone cast with the surveyor blade, the angulation and extent of tooth reduction may be established before the teeth are prepared in the mouth (Figure 11-6). With the diagnostic cast on the surveyor at the time of mouth preparation, reduction of tooth con-tours may be accomplished with acceptable accuracy. 7. To delineate the height of contour on abutment teeth and to locate areas of undesirable tooth undercut that are to touring wax patterns, measuring a specific depth of under-cut, surveying ceramic veneer crowns, placing intracoronal retainers, placing internal rests, machining cast restorations, and surveying and blocking out the master cast. Surveying the Diagnostic Cast Surveying the diagnostic cast is essential to effective diagno-sis and treatment planning. The objectives are as follows: 1. To determine the most desirable path of placement that will eliminate or minimize interference to placement and removal (Figure 11-5). The path of placement is the direction in which a restoration moves from the point of initial contact of its rigid parts with the supporting teeth to its terminal resting position, with rests seated and the denture base in contact with the tissues. The path of removal is exactly the reverse because it is the direction of restoration movement from its terminal resting posi-tion to the last contact of its rigid parts with the support-ing teeth. When the restoration is properly designed to have positive guiding planes, the patient may place and remove the restoration with ease in only one direction. This is possible only because of the guiding influence of Figure 11-4 Lab handpiece clamp. Handpiece holders attach to the vertical spindle of surveyors and may be used to create and refine any parallel surface on a surveyed crown, as a drill press to prepare internal rests and recesses in patterns and/or castings, and to establish lingual surfaces above the ledge that are parallel to the path of placement in abutment restorations. Path of placement Figure 11-5 Tilt of the cast on the adjustable table of a sur-veyor in relation to the vertical arm establishes the path of place-ment and removal that the removable partial denture will take. All mouth preparations must be made to conform to this deter-mined path of placement, which has been recorded by scoring the base of the cast or by tripoding. 134 Part I General Concepts/Treatment Planning Contouring Wax Patterns The surveyor blade is used as a wax carver during this phase of mouth preparation, so that the proposed path of place-ment may be maintained throughout the preparation of cast restorations for abutment teeth (Figure 11-7). Guiding planes on all proximal surfaces of wax patterns adjacent to edentulous areas should be made parallel to the previously determined path of placement. Similarly, all other tooth contours that will be contacted by rigid components be avoided, eliminated, or blocked out. This will include areas of the teeth to be contacted by rigid connectors, the location of nonretentive reciprocal and stabilizing arms, and the location of retentive clasp terminals. 8. To record the cast position in relation to the selected path of placement for future reference. This may be done by locating three dots or parallel lines on the cast, thus estab-lishing the horizontal plane in relation to the vertical arm of the surveyor (see Figures 11-6 and 11-16). B C A Figure 11-6 A, Solid line represents the height of contour on the abutment at selected orientation of the diagnostic cast to the verti-cal spindle of the surveyor. Dotted line represents the desirable height of the contour for optimally locating components of the direct retainer assembly. A 0.01-inch (0.25-mm) undercut gauge is used to mark the location of the tip of the retentive arm of the direct retainer. B, By reducing the axial contour of the tooth by only 0.01 inch, the optimum height of contour can be achieved without expos-ing the dentin. C, Stone tooth is trimmed with a surveyor blade to the desired height of contour. The trimmed area is marked in red pencil and serves as a blueprint for similar recontouring in the mouth. If one can safely assume that the enamel is 1 to 1.5 mm thick in the area of contemplated reduction, only 0.25 mm of enamel needs to be removed to achieve the optimum height of contour. A B Figure 11-7 After the cast has been oriented to the surveyor at the predetermined path of placement, designated axial surfaces of the wax pattern are altered with the surveyor blade to meet specific requirements for placement of framework components. A, Wax pattern is carved with the surveyor blade to produce a distal guide-plane surface parallel to the selected path of insertion. B, Same pattern is modified from the distal guide plane along the buccal surface to align the surface with the height of contour most favorable to the direct retainer specifications. 135 Chapter 11 Surveying should be made parallel. The surfaces of restorations on which reciprocal and stabilizing components will be placed should be contoured to permit their location well below occlusal surfaces and on nonretentive areas. Those surfaces of restorations that are to provide retention for clasp arms should be contoured so that retentive clasps may be placed in the cervical third of the crown and to the best esthetic advantage. Generally, a small amount of undercut from 0.01 to 0.02 inch (0.250 to 0.50 mm) or less is sufficient for reten-tive purposes. Surveying Ceramic Veneer Crowns Ceramic veneer crowns are often used to restore abutment teeth on which extracoronal direct retainers will be placed. The surveyor is used to contour all areas of the wax pattern for the veneer crown except the buccal or labial surface. It must be remembered that one of the principal goals in using a porcelain veneer restoration is to develop an esthetic replica of a natural tooth. It is unlikely that the ceramic veneer portion can be fabricated exactly to the form required for the planned placement of retentive clasp arms without some reshaping with stones. Before the final glaze is accom-plished, the abutment crowns should be returned to the surveyor on a full arch cast to ensure the correct contour of the veneered portions or to locate those areas that need recontouring (Figure 11-8). The final glaze is accomplished only after the crowns have been recontoured. Figure 11-8 Resultant metal-ceramic surveyed crown from Figure 11-7, which is being refined to maintain the distal guide plane and buccal height of the contour previously designed. Final glaze has not been placed on the veneer crown and required alterations of surfaces to conform to ideal placement of the retainer (solid line) can be performed by machining. Final glaze is produced only after necessary recontouring is accomplished. Placement of Intracoronal Retainers (Internal Attachments) In the placement of intracoronal retainers, the surveyor is used as follows: 1. To select a path of placement in relation to the long axes of the abutment teeth that will avoid areas of interference elsewhere in the arch 2. To cut recesses in the stone teeth on the diagnostic cast for estimating the proximity of the recess to the pulp (used in conjunction with roentgenographic informa-tion to estimate pulp size and location) and to facili-tate the fabrication of metal or resin jigs to guide preparations of the recesses in the mouth 3. To carve recesses in wax patterns, to place internal attachment trays in wax patterns, or to cut recesses in castings with the handpiece holder (whichever method is preferred) 4. To place the keyway portion of the attachment in the casting before investing and soldering; each keyway must be located parallel to the other keyways else-where in the arch The student is referred to the “Selected Reading Resources” section of the textbook for sources of information on intracoronal retainers (internal attachments). Placement of Internal Rest Seats The surveyor may be used as a drill press, with a dental handpiece attached to the vertical arm by a handpiece holder. Internal rest seats may be carved in the wax pat-terns and further refined with the handpiece after casting, or the entire rest seat may be cut in the cast restoration with the handpiece. It is best to carve the outline form of the rest seat in wax and merely refine the casting with the handpiece. An internal rest differs from an internal attachment in that some portion of the prosthesis framework is waxed and cast to fit into the rest seat rather than a matched key and keyway attachment used (see Figures 6-13 and 6-14). The former is usually nonretentive but provides a definite seat for a removable partial denture or a cantilever rest for a broken-stress fixed partial denture. When they are used with fixed partial dentures, nonparallel abutment pieces may be placed separately. The internal rest in partial denture construction pro-vides a positive occlusal support that is more favorably located in relation to the rotational axis of the abutment tooth than the conventional spoon-shaped occlusal rest. It also provides horizontal stabilization through the par-allelism of the vertical walls, thereby serving the same purpose as stabilizing and reciprocal arms placed extra-coronally. Because of the movement of a distal extension base, more torque may be applied to the abutment tooth by an interlocking type of rest, and for this reason its use 136 Part I General Concepts/Treatment Planning machined with a true-running cylindrical carborundum point. Although machined parallelism may be considered ideal and beyond the realm of everyday application, its merits more than justify the additional steps required to accomplish it. When such parallelism is accomplished and reproduced in a master cast, it is essential that subsequent laboratory steps be directed toward the use of these parallel guiding plane surfaces. Surveying the Master Cast Because surveying the master cast follows mouth prepara-tions, the path of placement, the location of retentive areas, and the location of remaining interference must be known before the final design of the denture framework is com-pleted. The objectives of surveying the master cast are as follows: 1. To select the most suitable path of placement by following mouth preparations that satisfy the requirements of guiding planes, retention, noninterference, and esthetics. 2. To permit measurement of retentive areas and to identify the location of clasp terminals in proportion to the flex-ibility of the clasp arm being used; flexibility will depend on many of the following factors: the alloy used for the clasp, the design and type of clasp, whether its form is round or half-round, whether it is of cast or wrought material, and the length of the clasp arm from its point of origin to its terminal end; retention then will depend on (a) the flexibility of the clasp arm, (b) the magnitude of the tooth undercut, and (c) the depth the clasp termi-nal is placed into this undercut. 3. To locate undesirable undercut areas that will be crossed by rigid parts of the restoration during placement and removal; these must be eliminated by blockout. 4. To trim blockout material parallel to the path of place-ment before duplication (Figure 11-9). Machining Cast Restorations With a handpiece holder attached (see Figure 11-4), the axial surfaces of cast and ceramic restorations may be refined by machining with a suitable cylindrical carborundum point. Proximal surfaces of crowns and inlays, which will serve as guiding planes, and vertical surfaces above crown ledges may be improved by machining, but only if the relationship of one crown to another is correct (see Figure 14-9). Unless the seating of removable dies is accurate and they are held in place with additional stone or plaster, cast restorations should first be tried in the mouth and then transferred, by means of a plaster or acrylic-resin index impression, to a reinforced stone cast for machining purposes. The new cast is then positioned on the surveyor, conforming to the path of placement of the partial denture, and vertical surfaces are A B Figure 11-9 Master casts are modified by the addition of wax relief to nonbearing regions and by placement of blockout wax parallel to the path of insertion at regions beneath the height of contour where framework contact is not planned (i.e., all areas except retentive clasp tips). A, Blockout wax is provided for tooth contours beneath the height of contour on teeth #21 and #28. B, Similar blockout was accomplished for mandibular molar #31. Blockout is carved with a straight surveyor blade to ensure parallelism with the identified path of insertion. in conjunction with a distal extension partial denture is considered to be contraindicated. The ball-and-socket, spoon-shaped occlusal, or noninterlocking, rest should be used in distal extension partial denture designs. Use of the dovetailed or interlocking internal rest should be limited to tooth-supported removable restorations, except when it is used in conjunction with some kind of stress-breaker between the abutments and the movable base. The use of stress-breakers has been discussed in Chapter 9. Internal rest seats may be made in the form of a non-retentive box, a retentive box fashioned after the internal attachment, or a semiretentive box. In the latter, the walls are usually parallel and nonretentive, but a recess in the floor of the box prevents proximal movement of the male portion. Internal rest seats are cut with dental burs of various sizes and shapes. Tapered or cylindrical fissure burs are used to form the vertical walls, and small round burs are used to cut recesses in the floor of the rest seat. 137 Chapter 11 Surveying retentive surfaces of abutment teeth. The other is to alter the flexibility of the clasp arm by changing its design, its size and length, or the material of which it is made. Interference The prosthesis must be designed so that it may be placed and removed without encountering tooth or soft tissue interfer-ence. A path of placement may be selected that encounters interference only if the interference can be eliminated during mouth preparations or on the master cast by a reasonable amount of blockout. Interference may be eliminated during mouth preparations by surgery, extraction, modification of interfering tooth surfaces, or alteration of tooth contours with restorations. Generally, interference that cannot be eliminated for one reason or another will take precedence over the factors of retention and guiding planes. Sometimes certain areas can be made noninterfering only by selecting a different path of placement at the expense of existing retentive areas and guiding planes. These must then be modified with restora-tions that are in harmony with the path dictated by the existing interference. On the other hand, if areas of interfer-ence can be eliminated by various reasonable means, this should be done. When this occurs, the axial contours of existing abutments may frequently be used with little alteration. Esthetics By one path of placement, the most esthetic location of artificial teeth is made possible, and less clasp metal and base material may be displayed. The location of retentive areas may influence the path of placement selected; therefore retentive areas should always be selected with the most esthetic locations of clasps in mind. When restorations are to be made for other reasons, they should be contoured to permit the least display of clasp metal. Generally, less metal will be displayed if the retentive clasp is placed at a more distogingival area of tooth surface, made possible by the path of placement selected or by the contour of the restorations. Esthetics also may dictate the choice of path selected when missing anterior teeth must be replaced with the partial denture. In such situations, a more vertical path of placement is often necessary so that neither the artificial teeth nor the adjacent natural teeth will have to be modified excessively (Figure 11-10). In this instance, esthetics may take precedence over other factors. This necessitates the preparation of abutment teeth to eliminate interferences and to provide guiding planes and retention in harmony with that path of placement dictated by esthetic factors. Because the primary consideration should be the preser-vation of remaining oral tissues, esthetics should not be allowed to jeopardize the success of the partial denture. The replacement of missing anterior teeth therefore should be accomplished by means of fixed partial dentures whenever possible, especially if the mechanical and functional effec-The partial denture must be designed so that (1) it will not stress abutment teeth beyond their physiologic toler-ance; (2) it can be easily placed and removed by the patient; (3) it will be retained against reasonable dislodging forces; and (4) it will not create an unfavorable appearance. It is necessary that the diagnostic cast be surveyed with these principles in mind. Mouth preparation should therefore be planned in accordance with certain factors that will influ-ence the path of placement and removal. Factors that Determine Path of Placement and Removal The factors that will determine the path of placement and removal are guiding planes, retentive areas, interference, and esthetics. Guiding Planes Proximal tooth surfaces that bear a parallel relationship to one another must be found or must be created to act as guiding planes during placement and removal of the pros-thesis. Guiding planes are necessary to ensure the passage of rigid parts of the prosthesis past existing areas of interfer-ence. Thus the denture can be easily placed and removed by the patient without strain on the teeth contacted or on the denture itself and without damage to the underlying soft tissues. Guiding planes are also necessary to ensure predictable clasp assembly function, including retention and stabiliza-tion. For a clasp to be retentive, its retentive arm must be forced to flex. Hence, guiding planes are necessary to give a positive direction to the movement of the restoration to and from its terminal position. Retentive Areas Retentive areas must exist for a given path of placement and must be contacted by retentive clasp arms that are forced to flex over a convex surface during placement and removal. Satisfactory clasp retention is no more than the resistance of metal to deformation. For a clasp to be retentive, its path of escapement must be other than parallel to the path of removal of the denture itself; otherwise it would not be forced to flex and thereby generate the resistance known as retention. Clasp retention therefore depends on the exis-tence of a definite path of placement and removal. Although desirable, retention at each principal abutment may not be balanced in relation to the tooth on the opposite side of the arch (exactly equal and opposite in magnitude and relative location); however, positive cross-arch recipro-cation to retentive elements must be present. Retention should be sufficient only to resist reasonable dislodging forces. In other words, it should be the minimum acceptable for adequate retention against reasonable dislodging forces. Fairly even retention may be obtained by one of two means. One is to change the path of placement to increase or decrease the angle of cervical convergence of opposing 138 Part I General Concepts/Treatment Planning A B C Figure 11-10 A, When anterior teeth must be replaced with the removable partial denture, the selected vertical path should consider the junction of the natural tooth and the denture tooth. The path of insertion that requires the least amount of alteration of the natural teeth and the denture tooth (maximizing natural embrasure contours) is desirable. B, Distal of canine slightly altered to accommodate the path that optimizes anterior denture tooth placement. C, Canine crowns required to satisfy desired contours of the natural tooth and the denture teeth. tiveness of the partial denture will require significant tooth preparation. Step-By-Step Procedures in Surveying A Diagnostic Cast Attach the cast to the adjustable surveyor table by means of the clamp provided. Position the adjustable table so that the occlusal surfaces of the teeth are approximately parallel to the platform (Figure 11-11). Such an orientation is a tenta-tive but practical way to start considering the factors that influence the path of placement and removal. Guiding Planes Determine the relative parallelism of proximal surfaces of all of the potential abutment teeth by contacting the proximal tooth surfaces with the surveyor blade or diagnostic stylus. Alter the cast position anteroposteriorly until these proximal surfaces are in as close to a parallel relation to one another as possible, or near enough that they can be made parallel by recontouring. For posterior modification spaces, this will determine the anteroposterior tilt of the cast in relation to Figure 11-11 Recommended method for manipulating the dental surveyor. The right hand is braced on the horizontal arm of the surveyor, and the fingers are used, as illustrated, to raise and lower the vertical shaft in its spindle. The left hand holding the cast on the adjustable table slides horizontally on the plat-form in relation to the vertical arm. The right hand must be used to loosen and tighten the tilting mechanism as a suitable antero-posterior and lateral tilt of the cast in relation to the surveyor is being determined. 139 Chapter 11 Surveying ing below their height of convexity may be determined. This is best accomplished by directing a small source of light toward the cast from the side away from the dentist. The angle of cervical convergence is best observed as a triangle of light between the surveyor blade and the apical portion of the tooth surface being studied (see Figure 7-41). Alter the cast position by tilting it laterally until similar retentive areas exist on the principal abutment teeth. If only two abutment teeth are involved, as in a Kennedy Class I partially edentulous arch, they are both principal abutments. However, if four abutment teeth are involved (as they are in a Kennedy Class III, modification 1 arch), they are all prin-cipal abutments, and retentive areas should be located on all four. But if three abutment teeth are involved (as they are in a Kennedy Class II, modification 1 arch), the posterior abut-ment on the tooth-supported side and the abutment on the distal extension side are considered to be the principal abut-ments, and retention needs to be equalized accordingly. The third abutment may be considered to be secondary, and less retention is expected from it than from the other two. An exception is when the posterior abutment on the tooth-supported side has a poor prognosis and the denture is designed to ultimately be a Class I. In such a situation, the two stronger abutments are considered to be principal abutments. When the cast is tilted laterally to establish reasonable uniformity of retention, it is necessary that the table be rotated about an imaginary longitudinal axis without dis-turbing the anteroposterior tilt previously established. The resulting position is one that provides or makes possible parallel guiding planes and provides for acceptable retention on the abutment teeth. It should be noted that this most desirable position will always require some tooth modifica-tion to be achieved. Note that possible interference to this tentative path of placement has not, as yet, been taken into consideration. the vertical arm of the surveyor (Figure 11-12). Although the surveyor table is universally adjustable, it should be thought of as having only two axes, thus allowing only anteroposte-rior and lateral tilting. When a choice is made between having contact with a proximal surface at the cervical area only or contact at the marginal ridge only, the latter is preferred because a plane may then be established by recontouring (Figure 11-13). It is obvious that when only gingival contact exists, a restora-tion is the only means of establishing a guiding plane. There-fore, if a tilt that does not provide proximal contact is apparent, the proximal surface must be established with some kind of restoration. The end result of selecting a suitable anteroposterior tilt should be to provide the greatest combined areas of parallel proximal surfaces that may act as guiding planes. Other axial surfaces of abutment teeth may also be used as guiding planes. This is realized most often by having the stabilizing component of the direct retainer assembly contacting in its entirety the axial surface of the abutment, which has been found or made parallel to the path of placement (see Figures 14-7 to 14-9). Therefore a lateral tilt of the cast to the vertical arm of the surveyor must also be considered, as well as the anteroposterior tilt, when guiding planes are used. Retentive Areas Through contact of buccal and lingual surfaces of abutment teeth with the surveyor blade, the amount of retention exist-Figure 11-12 Relative parallelism of proximal tooth surfaces will determine anteroposterior tilt of the cast in relation to the vertical arm of the surveyor. A B Figure 11-13 When the most desirable anteroposterior tilt of the cast in relation to the surveyor blade is determined, a choice must be made between positions illustrated in A and B. In A, the distal surface of the left premolar abutment would have to be extended by means of a restoration. In B, the right premolar could be altered slightly to provide an acceptably parallel guiding plane. Unless restorations are necessary for other reasons, the tilt shown in B is preferred. 140 Part I General Concepts/Treatment Planning Tooth surfaces on which reciprocal and stabilizing clasp arms will be placed should be studied to see whether suffi-cient areas exist above the height of convexity for the place-ment of these components. The addition of a clasp arm to the occlusal third of an abutment tooth adds to its occlusal dimension and therefore to the occlusal loading of that tooth. Nonretentive and stabilizing clasp arms are best located between the middle third and the gingival third of the crown rather than on the occlusal third. Areas of interference to proper placement of clasp arms can be eliminated by reshaping tooth surfaces during mouth preparations. These areas should be indicated on the diag-nostic cast. Areas of interference to the placement of clasps may necessitate minor changes in the path of placement or changes in the clasp design. For example, a bar clasp arm originating mesially from the major connector to provide reciprocation and stabilization might be substituted for a distally originating circumferential arm. Areas of interference often overlooked are the distal line angles of premolar abutment teeth and the mesial line angles of molar abutments. These areas frequently offer interfer-ence to the origin of circumferential clasp arms. If not detected at the time of initial survey, they are not included in the plan for mouth preparations. When such an undercut exists, the following three alternatives may be considered: 1. They may be blocked out the same as any other area of interference. This is by far the least satisfactory method because the origin of the clasp must then stand away from the tooth in proportion to the amount of blockout used. Although this is perhaps less objectionable than its being placed occlusally, it may be objectionable to the tongue and the cheek and may create a food trap. 2. They may be circumvented by approaching the retentive area from a gingival direction with a bar clasp arm. This is often a satisfactory solution to the problem if other contraindications to the use of a bar clasp arm, such as a severe tissue undercut or a retentive area that is too high on the tooth, are not present. 3. They may be eliminated by reducing the offending tooth contour during mouth preparation. This permits the use of a circumferential clasp arm originating well below the occlusal surface in a satisfactory manner. If the tooth is to be modified during mouth preparations, it should be indicated on the diagnostic cast with a colored pencil. When the retentive area is located objectionably high on the abutment tooth or the undercut is too severe, interfer-ence may also exist on tooth surfaces that support retentive clasps. Such areas of extreme or high convexity must be considered as areas of interference and should be reduced accordingly. These areas are likewise indicated on the diag-nostic cast for reduction during mouth preparations. Esthetics The path of placement thus established must still be consid-ered from the standpoint of esthetics, as to both the location of clasps and the arrangement of artificial teeth. Interference If a mandibular cast is being surveyed, check the lingual surfaces that will be crossed by a lingual bar major connector during placement and removal. Bony prominences and lin-gually inclined premolar teeth are the most common causes of interference to a lingual bar connector. If the interference is bilateral, surgery or recontouring of lingual tooth surfaces, or both, may be unavoidable. If the interference is only unilateral, a change in the lateral tilt may avoid an area of tooth or tissue interference. In changing the path of placement to prevent interference, previously estab-lished guiding planes and an ideal location for retentive ele-ments may be lost. Then the decision must be made whether to remove the existing interference by whatever means nec-essary or to resort to restorations on the abutment teeth, thereby changing the proximal and retentive areas to conform to the new path of placement. In a like manner, bony undercuts that will offer interfer-ence to the seating of denture bases must be evaluated and the decision made to remove them surgically; to change the path of placement at the expense of modifying or restoring teeth to achieve guiding planes and retention; or to design denture bases to avoid such undercut areas. The latter may be done by shortening buccal and labial flanges and disto-lingual extension of the denture bases. However, it should be remembered that the maximum area available for support of the denture base should be used whenever possible. Interference to major connectors rarely exists in the maxil-lary arch. Areas of interference are usually found on buccally inclined posterior teeth and those bony areas on the buccal aspect of edentulous spaces. As with the mandibular cast, the decision must be made whether to eliminate them, to change the path of placement at the expense of modifying or restor-ing teeth to achieve the required guiding planes and reten-tion, or to design the connectors and bases to avoid them. Other areas of possible interference to be evaluated are those surfaces of abutment teeth that will support or be crossed by minor connectors and clasp arms. Although interference to vertical minor connectors may be blocked out, doing so may cause discomfort to the patient’s tongue and may create objectionable spaces, which could result in the trapping of food. Also it is desirable that tooth surfaces contacted by vertical connectors be used as auxiliary guiding planes whenever possible. Too much relief is perhaps better than too little because of the possibility of irritation to soft tissues. It is always better that the relief be placed with some objective in mind. If possible, a minor connector should pass vertically along a tooth surface that is parallel to the path of placement (which is considered ideal) or tapered occlusally. If tooth undercuts that necessitate the use of an objection-able amount of blockout exist, they may be eliminated or minimized by slight changes in the path of placement and/ or eliminated during mouth preparations. The need for such alteration should be indicated on the diagnostic cast in red pencil after final acceptance of a path of placement. 141 Chapter 11 Surveying done, which consist of the preparation of proximal surfaces, the reduction of buccal and lingual surfaces, and the prepa-ration of rest seats. Except when they are placed in the wax pattern for a cast restoration, preparation of rest seats should always be deferred until all other mouth preparations have been completed. The actual locations of rests will be determined by the proposed design of the denture framework. Therefore the tentative design should be sketched on the diagnostic cast in pencil after the path of placement has been decided. This is done not only to locate rest areas but also to record graphi-cally the plan of treatment before mouth preparations. In the intervening time between patient visits, other partial denture restorations may have been considered. The dentist should have the plan of treatment readily available at each succeeding appointment to avoid confusion and to keep a reminder about that which is to be done and the sequence that will be required. The plan for treatment should include (1) the diagnostic cast with the mouth preparations and the denture design marked on it; (2) a chart showing the proposed design and the planned treatment for each abutment; (3) a working chart showing the total treatment involved that will permit a quick review and a check-off of each step as the work progresses; and (4) a record of the fee quoted for each phase of treatment that can be checked off as it is recorded on the patient’s permanent record. Red pencil marks on the diagnostic cast are used to indi-cate the locations of areas to be modified as well as the loca-tions of rests (Figure 11-14). Although it is not necessary that rest areas be prepared on the diagnostic cast, it is advisable for the beginning student to have done this before proceed-ing to alter the abutment teeth. This applies equally to crown and inlay preparations on abutment teeth. It is advisable, however, for even the most experienced dentist to have trimmed the stone teeth with the surveyor blade wherever tooth reduction is to be done. This identifies not only the Clasp designs that will provide satisfactory esthetics for any given path of placement usually may be selected. In some instances, gingivally placed bar clasp arms may be used to advantage; in others, circumferential clasp arms located cervically may be used. This is especially true when other abutment teeth located more posteriorly may bear the major responsibility for retention. In still other instances, a tapered wrought-wire retentive clasp arm may be placed to better esthetic advantage than a cast clasp arm. The placement of clasp arms for esthetic reasons does not ordinarily justify altering the path of placement at the expense of mechanical factors. However, it should be considered concurrently with other factors, and if a choice between two paths of insertion of equal merit permits a more esthetic placement of clasp arms by one path than the other, that path should be given preference. When anterior replacements are involved, the choice of path is limited to a more vertical one for reasons previously stated. In this instance alone, esthetics must be given primary consideration, even at the expense of altering the path of placement and making all other factors conform. This factor should be remembered when the other three factors are con-sidered, so that compromises can be made at the time other factors are being considered. Final Path of Placement The final path of placement will be the anteroposterior and lateral position of the cast, in relation to the vertical arm of the surveyor, that best satisfies all four factors: guiding planes, retention, interference, and esthetics. All proposed mouth changes should be indicated on the diagnostic cast in red pencil, with the exception of restora-tions to be done. These may also be indicated on an accom-panying chart if desired. Extractions and surgery are given priority to allow time for healing. The remaining red marks represent actual modifications of the teeth that remain to be A B Figure 11-14 Diagnostic casts can serve as a visual guide for tooth preparation. A, Surveyed cast shows areas requiring tooth reduc-tion in red (mesio-occlusal rest and distal guide plane #28, cingulum rest #27), as well as path of insertion tripod marks. B, This mesially tipped molar has been diagnosed to have a ring clasp. Red markings show the necessary mesio-occlusal and disto-occlusal rests required, as well as the mesial guide plane. Also shown is the reduction necessary to lower the lingual height of contour at the mesiolingual line angle. All required axial contour adjustments are determined through the appropriate use of a surveyor. 142 Part I General Concepts/Treatment Planning cially during mouth preparations. The same applies to the need for returning any working cast to the surveyor for shaping wax patterns, trimming blockout on the master cast, or locating clasp arms in relation to undercut areas. Obviously the trimmed base will vary with each cast; therefore recording the position of the surveyor table is of no value. If it were, calibrations could be incorporated on the surveyor table that would allow the same position to be reestablished. Instead, the position of each cast must be established separately, and any positional record applies only to that cast. Of several methods, two seem to be the most convenient and accurate. One method is to place three widely divergent dots on the tissue surface of the cast with the tip of a carbon marker with the vertical arm of the surveyor in a locked position. Preferably these dots should not be placed on areas of the cast involved in the framework design. The dots should be encircled with a colored pencil for easy identifica-tion. When the cast is returned to the surveyor, it may be tilted until the tip of the surveyor blade or diagnostic stylus again contacts the three dots in the same plane. This approach, which will produce the original position of the cast and therefore the original path of placement, is known as tripoding the cast (Figure 11-15). Some dentists prefer to amount to be removed in a given area but also the plane in which the tooth is to be prepared. For example, a proximal surface may need to be recontoured in only the upper third or the middle third to establish a guiding plane that will be parallel to the path of placement. This is not usually parallel to the long axis of the tooth, and if the rotary instrument is laid against the side of the tooth, the existing surface angle will be maintained, thus avoiding the need to establish a new plane that is parallel to the path of placement. The surveyor blade, which represents the path of place-ment, may be used to advantage to trim the surface of the abutment tooth whenever a red mark appears. The resulting surface represents the amount of tooth to be removed in the mouth and indicates the angle at which the handpiece must be held. The cut surface on the stone tooth is not marked with red pencil again, but it is outlined in red pencil to posi-tively locate the area that is to be prepared. Recording Relation of Cast to Surveyor Some method of recording the relation of the cast to the vertical arm of the surveyor must be used so that the cast may be returned to the surveyor for future reference, espe-Carbon rod substituted to mark height of contour A D C B Figure 11-15 A-B, The path of placement is determined, and the base of the cast is scored to record its relation to the surveyor for future repositioning. C, An alternate method of recording the relation of the cast to the surveyor is known as tripoding. A carbon marker is placed in the vertical arm of the surveyor, and the arm is adjusted to the height by which the cast can be contacted in three divergent locations. The vertical arm is locked in position, and the cast is brought into contact with the tip of the carbon marker. Three resultant marks are encircled with colored lead pencil for ease of identification. Reorientation of the cast to the surveyor is accomplished by tilting the cast until the plane created by three marks is at a right angle to the vertical arm of the surveyor. D, Height of contour is then delineated by a carbon marker. 143 Chapter 11 Surveying cuted, the undercuts remaining to be blocked out should be minimal. The base of the cast is now scored, or the cast is tripoded as described previously. The surveyor blade or diagnostic stylus then may be replaced with a carbon marker, and the height of convexity of each abutment tooth and soft tissue contours may be delineated. Similarly, any areas of interfer-ence to the rigid parts of the framework during seating and removal should be indicated with the carbon marker so areas to be blocked out or relieved can be located. Carbon markers that become the slightest bit worn from use should be discarded. A worn (tapered) carbon marker will indicate heights of contour more occlusally located than those that actually exist. The carbon marker must be parallel to the vertical spindle of the surveyor (Figure 11-16). The diagnostic stylus should always be checked to ensure that it is not bent or distorted. Measuring Retention The surveyor is used with the master cast for two purposes: (1) to delineate the height of contour of the abutment teeth both to locate clasp arms and to identify the location and magnitude of retentive undercuts; and (2) to trim blockout of any remaining interference to placement and removal of the denture. The areas involved are those that will be crossed by rigid parts of the denture framework. The exact undercut that retentive clasp terminals will occupy must be measured and marked on the master cast (Figure 11-17). Undercuts may be measured with an undercut gauge, such as those provided with the Ney and Jelenko surveyors. The amount of undercut is measured in hundredths of an inch, with the gauges allowing measurements up to 0.03 inch. Theoretically the amount of undercut used may vary with the clasp to be used, up to a full 0.03 inch. However, undercuts of 0.01 inch are often adequate for retention by cast retainers. Tapered make tiny pits in the cast at the location of the tripoding dots to preserve the orientation of the cast and to transfer this relationship to the refractory cast. A second method is to score two sides and the dorsal aspect of the base of the cast with a sharp instrument held against the surveyor blade (see Figure 11-15). When the cast is tilted until all three lines are again parallel to the surveyor blade, the original cast position can be reestablished. Fortu-nately, the scratch lines will be reproduced in any duplica-tion, thereby permitting any duplicate cast to be related to the surveyor in a similar manner. Whereas a diagnostic cast and a master cast cannot be made to be interchangeable, a refractory cast, which is a duplicate of the master cast, can be repositioned on the surveyor at any time. The technician must be cautioned not to trim the sides of the cast on the cast trimmer and thereby lose the reference marks for repositioning. It must be remembered that repositioning a cast on a surveyor at any time can involve a certain amount of human error. It has been estimated that an error of 0.2 mm can be anticipated when a cast with three reference points on its base is reoriented. This reorientation error can influence the placement of appropriate blockout wax and may result in ineffective placement of direct retainers into prescribed undercuts and improper contacts of minor connectors with guiding planes. Therefore reorientation of the cast to the surveyor by any method must be done with great care. Surveying the Master Cast The master cast must be surveyed as a new cast, but the prepared proximal guiding plane surfaces will indicate the correct anteroposterior tilt. Some compromises may be nec-essary, but the amount of guiding plane surface remaining after blockout should be the maximum for each tooth. Areas above the point of contact with the surveyor blade are not considered as part of the guiding plane area, and neither are gingival undercut areas, which will be blocked out. The lateral tilt will be the position that provides equal retentive areas on all principal abutments in relation to the planned clasp design. Factors of flexibility, including the need for extra flexibility on distal extension abutments, must be considered when one is deciding what will provide equal retention on all abutment teeth. For example, cast circum-ferential or cast bar retention on the tooth-supported side of a Class II design should be balanced against the 18-gauge wrought-wire retention on a distal abutment only if the more rigid cast clasp engages a lesser undercut than the wrought-wire clasp arm. Therefore the degree of undercut alone does not ensure relatively equal retention unless clasp arms of equal length, diameter, form, and material are used. Gross interference will have been eliminated during mouth preparation. Thus for a given path of placement that is providing guiding planes and balanced retention, any remaining interference must be eliminated with blockout. If mouth preparations have been adequately planned and exe-Figure 11-16 A worn carbon marker (left) should be dis-carded because it will invariably misleadingly mark the height of contour for a given orientation of the cast to the vertical spindle of the surveyor. Unworn carbon (right) with an angled end is preferable for marking heights of contour on abutment teeth and performing surveys of soft tissue areas. 144 Part I General Concepts/Treatment Planning wrought-wire retention may safely use up to 0.02 inch without inducing undesirable torque on the abutment tooth, provided the wire retentive arm is long enough (at least 8 mm). The use of 0.03 inch is rarely, if ever, justified with any clasp. When greater retention is required, such as when abutment teeth remain on only one side of the arch, multiple abutments should be used, rather than increased retention on any one tooth. When a source of light is directed toward the tooth being surveyed, a triangle of light is visible. This triangle is bounded by the surface of the abutment tooth on one side and the blade of the surveyor on the other, the apex being the point of contact at the height of convexity and the base of the triangle being the gingival tissues (Figure 11-18). Retention will be determined by (1) the magnitude of the angle of cervical convergence below the point of convexity; (2) the depth at which the clasp terminal is placed in the angle; and (3) the flexibility of the clasp arm. The intelligent application of various clasp designs with their relative flexibility is of greater importance than the ability to measure an undercut with precise accuracy. The final design may now be drawn on the master cast with a fine crayon pencil, preferably one that will not come off during duplication. Graphite is usually lifted in duplica-tion, but some crayon pencil marks will withstand duplica-tion without blurring or transfer. Sizing or spraying the master cast to protect such pencil marks is usually not advis-able unless it is done with extreme care to avoid obliterating the surface detail. A B Figure 11-17 A, Undercut gauge will measure the depth of undercut below the height of contour. I-bar direct retainer will contact the tooth from the point of the undercut to the height of contour. The depth to which the retentive clasp arm can be placed depends not only on its length, taper, and diameter and the alloy from which it is made, but also on the type of clasp. A circumferential clasp arm is more flexible than a bar clasp arm of the same length (see Chapter 7). B, Specific measurement of the undercut gingiva to the height of contour may be ascertained with the use of an undercut gauge attached to the surveyor. Simultaneous contact of the shank of the undercut gauge at the height of contour and of the lip of a specific undercut gauge on a tooth in the infrabulge area establishes definitively the degree and location of undercut. Therefore the tip of the retentive arm of the direct retainer may be placed at the planned depth of the undercut. Figure 11-18 Tooth undercut is best viewed against a good source of light passing through the triangle bounded by the surface of the abutment tooth, the surveyor blade, and the gin-gival tissues. 145 Chapter 11 Surveying B A Figure 11-19 The wax ledge on the buccal surface of the molar abutment will be duplicated in a refractory cast for exact placement of the clasp pattern. Note that the ledge has been carved slightly below the penciled outline of the clasp arm. This will allow the gingival edge of the clasp arm to be polished and still remain in its planned relationship to the tooth when the denture is seated. It should also be noted that the wax ledge definitively establishes planned placement of the direct retainer tip into the measured undercut. Figure 11-20 All guiding-plane areas must be parallel to the path of placement, and all other areas that will be contacted by rigid parts of the denture framework must be made free of the undercut by parallel blockout. Relief must also be provided for the gingival crevice and gingival margin. Black regions designate parallel blockout at proximal guide-plane surfaces and relief along the palatal marginal gingiva. Blocking Out the Master Cast After the path of placement and the location of undercut areas have been established on the master cast, any under-cut areas that will be crossed by rigid parts of the denture (which is every part of the denture framework but the retentive clasp terminals) must be eliminated by blockout. In the broader sense of the term, blockout includes not only the areas crossed by the denture framework during seating and removal but also (1) those areas not involved that are blocked out for convenience; (2) ledges on which clasp patterns are to be placed; (3) relief beneath connec-tors to avoid tissue impingement; and (4) relief to provide for attachment of the denture base to the framework. Ledges or shelves (shaped blockout) for locating clasp patterns may or may not be used (Figure 11-19). However, this should not be confused with the actual blocking out of undercut areas that would offer interference to the placement of the denture framework. Only the latter is made on the surveyor, with the surveyor blade or diag-nostic stylus used as a paralleling device. Hard inlay wax may be used satisfactorily as a blockout material. It is easily applied and is easily trimmed with the surveyor blade. Trimming is facilitated by slight warming of the surveyor blade with an alcohol torch. Whereas it is true that any wax will melt more readily than a wax-clay mixture if the temperature of the dupli-cating material is too high, it should be presumed that the duplicating material will not be used at such an elevated temperature. If the temperature of the duplicating mate-rial is high enough to damage a wax blockout, other dis-tortions resulting in an inaccurate duplication will likely occur. Paralleled blockout is necessary for areas that are cer-vical to guiding-plane surfaces and over all undercut areas that will be crossed by major or minor connectors. Other areas that are to be blocked out for convenience and for avoidance of difficulties in duplication should be blocked out with hard baseplate wax or oil-base modeling clay (artist’s modeling clay). Such areas include the labial surfaces and labial undercuts not involved in the denture design and the sublingual and distolingual areas beyond the limits of the denture design. These are blocked out arbitrarily with hard baseplate wax or clay, but because they have no relation to the path of placement, they do not require the use of the surveyor. Modeling clay that is water soluble should not be used when duplication pro-cedures are involved. Areas to be crossed by rigid connectors, on the other hand, should be trimmed with the surveyor blade or some other surveyor tool parallel to the path of placement (Figure 11-20). This imposes a considerable responsibil-ity on the technician. If the blockout is not sufficiently trimmed to expose guiding-plane surfaces, the effects of 146 Part I General Concepts/Treatment Planning Figure 11-21 Parallel blockout (A, labial surfaces of all teeth and gingivae to the retentive clasp on tooth #2) and relief (B, marginal gingivae of palatal surfaces of teeth and at distal minor connector) in preparation for framework casting. These spaces allow planned seating of the framework without tissue trauma while accommodating the addition of acrylic-resin beneath the distal extension minor connector for base support without metal contact. A B Figure 11-22 Relief and blockout of the master cast before duplication. All undercuts involved in the denture design have been blocked out parallel to the path of placement, except the retentive tips of the retainer clasps. Residual ridges have been provided 20-gauge relief for denture base material. these guiding planes, which were carefully established by the dentist, will be nullified. If, on the other hand, the technician is overzealous in paralleling the blockout, the stone cast may be abraded by heavy contact with the surveyor blade. Although the resulting cast framework would seat back onto the master cast without interfer-ence, interference to placement in the mouth would result. This would necessitate relieving the casting at the chair, which is not only an embarrassing and time- consuming operation but also one that may have the effect of obliterating guiding plane surfaces. Relieving the Master Cast Tissue undercuts that must be blocked out are paralleled in much the same manner as tooth undercuts. The dif-ference between blockout and relief must be clearly understood (Figures 11-21 and 11-22). For example, tissue undercuts that would offer interference to the seating of a lingual bar connector are blocked out with blockout wax and trimmed parallel to the path of place-ment. This does not in itself necessarily afford relief to avoid tissue impingement. In addition to such blockout, a relief of varying thickness must sometimes be used, depending on the location of the connector, the relative slope of the alveolar ridge, and the predictable effect of denture rotation. It must be assumed that indirect retain-ers, as such, or indirect retention is provided in the design of the denture to prevent rotation of the lingual bar infe-riorly. A vertical downward rotation of the denture bases around posterior abutments places the bar increasingly farther from the lingual aspect of the alveolar ridge when this surface slopes inferiorly and posteriorly (Figure 11-23). Adequate relief of soft tissues adjacent to the lingual bar is obtained by the initial finishing and polish-ing of the framework in these instances. However, exces-sive upward vertical rotation of a lingual bar will impinge on lingual tissues if the alveolar ridge is nearly vertical or undercut to the path of placement (Figure 11-24). The 147 Chapter 11 Surveying Figure 11-23 Sagittal section of the cast and denture frame-work. The lingual alveolar ridge slopes inferiorly and posteriorly (upper figure). When the force is directed to displace the denture base downward, the lingual bar rotates forward and upward but does not impinge on the soft tissue of the alveolar ridge (lower figure). Therefore in such instances, adequate relief to avoid impingement is gained when the tissue side of the lingual bar is highly polished during the finishing process. Figure 11-24 An undercut alveolar ridge was blocked out parallel to the path of placement in fabricating the lingual bar (upper figure). Application of vertical force to cause rotation of the lingual bar upward can cause impingement of lingual tissue on the alveolar ridge (lower figure). To avoid impingement in these instances, not only should the master cast be blocked out parallel to the path of placement, but an additional relief of 32-gauge sheet wax should be used to block out the cast in such undercut areas. region of the cast involving proposed placement of the lingual bar should, in this situation, be relieved first by parallel blockout and then by a 32-gauge wax strip. Low-fusing casting wax such as Kerr’s green casting wax should not be used for this purpose: it is too easily thinned during adapting and may be affected by the temperature of the duplicating material. Pink casting wax should be used, even though it is difficult to adapt uniformly. A pressure-sensitive, adhesive-coated casting wax is pre-ferred because it adapts readily and adheres to the cast surface. Any wax, even the adhesive type, should be sealed all around its borders with a hot spatula to prevent its lifting when the cast is moistened before or during duplication. Horizontal rotational tendencies of mandibular distal extension removable partial dentures account for many of the tissue irritations seen adjacent to a lingual man-dibular major connector. These irritations can usually be avoided by blocking out all undercuts adjacent to the bar parallel to the path of placement and then including adequate stabilizing components in the design of the framework to resist horizontal rotation. Judicious relief of the tissue side of the lingual bar with rubber wheels at the site of the irritation most often will correct the dis-crepancy. Under no circumstances should the rigidity of the major connector be jeopardized by grinding any portion of it. Still other areas requiring relief are the areas where component parts cross the gingiva and gingival crevices. All gingival areas bridged by the denture framework should be protected from possible impingement resulting from rotation of the denture framework. Hard inlay wax may be used to block out gingival crevices (see Figure 11-21). Paralleled Blockout, Shaped Blockout, Arbitrary Blockout, and Relief Table 11-1 differentiates between paralleled blockout, shaped blockout, arbitrary blockout, and relief. The same factors apply to both maxillary and mandibular arches, except that relief is ordinarily not used beneath palatal major connectors, as it is with mandibular lingual bar connectors, except when maxillary tori cannot be circum-vented, or when resistive median palatal raphes are encountered. 148 Part I General Concepts/Treatment Planning Table 11-1 Differentiation Between Parallel Blockout, Shaped Blockout, Arbitrary Blockout, and Relief Site Material Thickness Parallel Blockout Proximal tooth surfaces to be used as guiding planes Hard baseplate wax or blockout material Only undercut remaining below contact of the surveyor blade with tooth surface Beneath all minor connectors Hard baseplate wax or blockout material Only undercut remaining below contact of the surveyor blade with tooth surface Tissue undercuts to be crossed by rigid connectors Hard baseplate wax or blockout material Only undercut remaining below contact of the surveyor blade with surface of the cast Tissue undercuts to be crossed by the origin of bar clasps Hard baseplate wax or blockout material Only undercut remaining below contact of the surveyor blade with surface of the cast Deep interproximal spaces to be covered by minor connectors or linguoplates Hard baseplate wax or blockout material Only undercut remaining below contact of the surveyor blade with surface of the cast Beneath bar clasp arms to gingival crevice Hard baseplate wax or blockout material Only undercut area involved in attachment of the clasp arm to the minor connector Shaped Blockout On buccal and lingual surfaces to locate plastic or wax patterns for clasp arms Hard baseplate wax Ledges for location of reciprocal clasp arms to follow height or convexity so that they may be placed as cervical as possible without becoming retentive Ledges for location of retentive clasp arms to be placed as cervical as tooth contour permits; point of origin of clasp to be occlusal or incisal to height of the convexity, crossing the survey line at fourth terminal, and to include undercut area previously selected in keeping with flexibility of the clasp type being used Arbitrary Blockout All gingival crevices Hard baseplate wax Enough to just eliminate gingival crevice Gross tissue undercuts situated below areas involved in the design of denture framework Hard baseplate wax or oil-based clay Leveled arbitrarily with a wax spatula Tissue undercuts distal to the cast framework Hard baseplate wax or oil-based clay Smoothed arbitrarily with a wax spatula Labial and buccal tooth and tissue undercuts not involved in denture design Hard baseplate wax or oil-based clay Filled and tapered with spatula to within the upper third or crown Relief Beneath lingual bar connectors or the bar portion of the linguoplates when indicated (see text) Adhesive wax sealed to the cast; should be wider than the major connector to be placed on it 32-Gauge wax if the slope of the lingual alveolar ridge is parallel to the path of placement; 32-gauge wax after parallel blockout of undercuts if the slope of the lingual alveolar ridge is undercut to the path of placement Areas in which major connectors will contact thin tissue, such as hard areas so frequently found on lingual or mandibular ridges and elevated palatal raphes Hard baseplate wax Thin layer flowed on with hot wax spatula; however, if the maxillary torus must be covered, the thickness of the relief must represent the difference in the degree of displacement of the tissues covering the torus and the tissues covering the residual ridges Beneath framework extensions onto ridge areas for attachment of resin bases Adhesive wax, well adapted to and sealed to the cast beyond the involved area 20-Gauge wax 150 CHAPTER 12 Diagnosis and Treatment Planning Chapter Outline Purpose and Uniqueness of Treatment Patient Interview Shared Decision Making Clinical Examination Objectives of Prosthodontic Treatment Oral Examination Sequence of oral examination Diagnostic Casts Purposes of diagnostic casts Mounting diagnostic casts Sequence for mounting maxillary cast to axis-orbital plane Jaw relationship records for diagnostic casts Materials and methods for recording centric relation Diagnostic Findings Interpretation of Examination Data Radiographic interpretation Periodontal considerations Caries risk assessment considerations Evaluation of the prosthesis foundation—teeth and residual ridge Surgical preparation Analysis of occlusal factors Fixed restorations Orthodontic treatment Need for determining type of mandibular major connector Need for reshaping remaining teeth Infection Control Differential Diagnosis: Fixed or Removable Partial Dentures Indications for use of fixed restorations Indications for removable partial dentures Choice Between Complete Dentures and Removable Partial Dentures Clinical Factors in Selecting Metal Alloys for Removable Partial Denture Frameworks Comparative physical properties of gold and chromium-cobalt Wrought wire: selection and quality control Summary Purpose and Uniqueness of Treatment The purpose of dental treatment is to respond to a patient’s needs, both the needs perceived by the patient and those demonstrated through a clinical examination and patient interview. Although similarities have been noted between partially edentulous patients (such as Classification designa-tions), significant differences exist, making each patient, and the ultimate treatment, unique. The delineation of each patient’s uniqueness occurs through the patient interview and diagnostic clinical exami-nation process. This includes four distinct processes: (1) understanding the patient’s desires or chief concerns/com-plaints regarding his or her condition (including its history) through a systematic interview process, (2) ascertaining the patient’s dental needs through a diagnostic clinical examina-tion, (3) developing a treatment plan that reflects the best management of desires and needs (with influences unique to the medical condition or oral environment), and (4) exe-cuting appropriately sequenced treatment with planned follow-up. The ultimate treatment is individualized to address disease management and the coordinated restorative and prosthetic needs that are unique to the patient. Provi-sion of the best care for a patient may involve no treatment, limited treatment, or extensive treatment, and the dentist must be prepared to help patients decide the best treatment option given his or her individual circumstances. Patient Interview Although oral health is an important aspect of overall health, it is an elective health pursuit for most individuals. Conse-151 Chapter 12 Diagnosis and Treatment Planning nance of the prosthesis can be improved to provide a more positive experience. Shared Decision Making When helping patients understand their oral health status, comprising both disease and deficit considerations, and the means to address both, we should carefully consider what it is they need to hear from us. For most partially edentulous patients, the discussion may involve fairly complex rehabili-tation options for addressing their missing teeth. Because of this complexity, our responsibility is to help them sort through the options in an attempt to help them come to the best decision for them. Using a communication model termed shared decision making gives structure to a process where the provider and the patient identify together the best course of care. This process recognizes that there may be complex “trade-offs” in care choice, and it addresses the need to fully inform patients about risks and benefits of care options, as well as ensuring that patient values and prefer-ences play a prominent role in the process. Although it is clear that not all patients desire to participate equally in care decisions, because the options can vary significantly (some are more invasive, have greater risks, are accompanied by higher treatment burden than others; there are often varying maintenance needs between options), we should actively engage them in the process. This is more important given the fact that the tooth replacement is often an elective pursuit, and because of this, there is seldom great urgency involved in making a decision. Clinical Examination Objectives of Prosthodontic Treatment The objectives of any prosthodontic treatment may be stated as follows: (1) the elimination of disease; (2) the preserva-tion, restoration, and maintenance of the health of the remaining teeth and oral tissues (which will enhance the removable partial denture design); and (3) the selected replacement of lost teeth; for the purpose of (4) restoration of function in a manner that ensures optimum stability and comfort in an esthetically pleasing manner. Preservation is a principle that protects from decisions that place too high a premium on cosmetic concerns, and it is the dentist’s obligation to emphasize the importance of restoring the total mouth to a state of health and of preserving the remaining teeth and surrounding tissues. Diagnosis and treatment planning for oral rehabilitation of partially edentulous mouths must take into consideration the following: control of caries and periodontal disease, res-toration of individual teeth, provision of harmonious occlu-sal relationships, and the replacement of missing teeth by fixed (using natural teeth and/or implants) or removable quently, the patient presents for professional evaluation (1) to address some perception of an abnormality that requires correction, or (2) to maintain optimum oral health. In either situation, but especially for the patient presenting with some chief complaint (often with an important history related to that complaint), it is mandatory that the dentist clearly understand what brings the patient to this evaluation. Failure to do so leads to the chance that the patient will be unhappy with the treatment result, as it might not address the very reason he or she came for help. With experience, this subtle point becomes a major component of a clinician’s manage-ment focus. A fundamental objective of the patient interview, which accompanies the diagnostic examination, is to gain a clear understanding of why the patient is presenting for evalua-tion; this involves having the patient describe the history related to the chief complaint. For complicated clinical problems, the interview and diagnostic examination require two appointments to allow complete gathering of all diag-nostic information needed to formulate a complete plan of treatment. The interview, an opportunity to develop rapport with the patient, involves listening to and understanding the patient’s chief complaint or concern about his or her oral health. This can include clinical symptoms of pain (pro-voked or unprovoked), difficulty with function, concern about appearance, problems with an existing prosthesis, or any combination of symptoms related to the teeth, peri-odontium, jaws, or previous dental treatment. It is impor-tant to listen carefully to what the patient has stated is the reason for presenting for evaluation; this is because all sub-sequent information gathered will be used to discuss these concerns and to relate whether the proposed treatment will affect the patient in any way. Such a discussion at the outset of patient care helps to outline realistic expectations. Although formats for sequencing the patient interview (and clinical examination) vary, to ensure thoroughness the dentist should follow a sequence that includes: 1. Chief complaint and its history 2. Medical history review 3. Dental history review, especially related to previous pros-thetic experience(s) 4. Patient expectations It is from the above interaction that patient uniqueness, as mentioned earlier, is best defined. The expectations described by the patient are critical to an understanding of whether a removable partial denture will satisfy the stated treatment goal(s). The fact that removable partial dentures by necessity require material bulk and often use oral soft tissues for support may be hard to comprehend by patients with no such prosthetic history. Helping the patient under-stand the normal phase of accommodation to such a pros-thesis is an important discussion point in selection of a prosthesis. For those patients with a negative past prosthesis experience, it is necessary to determine before treatment is started whether the design, fit, occlusion, or lack of mainte-152 Part II Clinical and Laboratory accomplished by an evaluation of factors that generate func-tional forces and those that resist them. The stability of tooth and prosthesis position is the goal of such an evaluation. The following sequence of examination allows attention to be paid to aspects of each of these critical features of evaluation for removable partial denture service. Sequence for Oral Examination An oral examination should be accomplished in the follow-ing sequence: visual examination, pain relief and temporary restorations, radiographs, oral prophylaxis, evaluation of teeth and periodontium, vitality tests of individual teeth, determination of the floor of the mouth position, and impressions of each arch. 1. Relief of pain and discomfort and caries control by place-ment of temporary restorations. A preliminary examina-tion is conducted to determine the need for management of acute needs and whether a prophylaxis is required to conduct a thorough oral examination. It is advisable not only to relieve discomfort arising from tooth defects but also to determine as early as possible the extent of caries, and to arrest further caries activity until definitive treat-ment can be instituted. If tooth contours are restored with temporary restorations, the impression will not be torn on removal from the mouth, and a more accurate diagnostic cast may be obtained. 2. A thorough and complete oral prophylaxis. An adequate examination can be accomplished best with the teeth free of accumulated calculus and debris. Also, accurate diag-nostic casts of the dental arches can be obtained only if the teeth are clean; otherwise the teeth reproduced on the diagnostic casts are not a true representation of tooth and gingival contours. Cursory examination may precede an oral prophylaxis, but a complete oral examination should be deferred until the teeth have been thoroughly cleaned. 3. Complete intraoral radiographic survey (Figure 12-1). The objectives of a radiographic examination are (a) to locate areas of infection and other pathosis that may be present; (b) to reveal the presence of root fragments, foreign objects, bone spicules, and irregular ridge formations; (c) to reveal the presence and extent of caries and the relation prostheses. Because these procedures are integrally related, the appropriate selection and sequencing of treatment should precede all irreversible procedures. The treatment plan for the removable partial denture, which is often the final step in a lengthy sequence of treat-ment, should precede all but emergency treatment. This allows abutment teeth and other areas in the mouth to be properly prepared to support, stabilize, and retain the removable partial denture. This means that diagnostic casts, for designing and planning removable partial denture treat-ment, must be made before definitive treatment is under-taken. After the major factors that create functional forces are evaluated and those that resist it are understood, the removable partial denture design is drawn on the diagnostic cast, along with a detailed chart of mouth conditions and proposed treatment. This becomes the master plan for the mouth preparations and the design of the removable partial denture to follow. As was pointed out in Chapter 1, failure of removable partial dentures can usually be attributed to factors that result in poor stability. These can result from inadequate diagnosis and failure to properly evaluate the conditions present. This results in failure to prepare the patient and the oral tissues properly before the master cast is fabricated. The importance of the examination, the consideration of favor-able and unfavorable aspects relative to movement control, and the importance of planning the elimination of unfavor-able influences cannot be overemphasized (see Chapter 2). As was mentioned earlier, for complex treatment, two appointments are often required. The first will likely include a preliminary oral examination (to determine the need for management of acute needs), a prophylaxis, full-mouth radiographs, diagnostic casts, and mounting records if base-plates are not required. The follow-up appointment includes mounting of the diagnostic casts (when baseplates and occlusion rims are needed), a definitive oral examination, review of the radiographs to augment and correlate with clinical findings, and arrangement of additional consulta-tions when required. Following collection and synthesis of all patient and clinical information, including surveying of the casts, a treatment plan (often with options) is presented. Oral Examination A complete oral examination should precede any treatment decisions. It should include visual and digital examination of the teeth and surrounding tissues with a mouth mirror, explorer, and periodontal probe, vitality tests of critical teeth, and examination of casts correctly oriented on a suit-able articulator. Clinical findings are augmented by and cor-related with a complete intraoral radiographic survey. During the examination, the objective to be kept fore-most in mind should be the consideration of possibilities for restoring and maintaining the remaining oral structures in a state of health for the longest period of time. This is best Figure 12-1 Complete intraoral radiographic survey of rem aining teeth and adjacent edentulous areas reveals much information vital to effective diagnosis and treatment planning. The response of bone to previous stress is of particular value in establishing the prognosis of teeth that are to be used as abutments. 153 Chapter 12 Diagnosis and Treatment Planning 4. Impressions for making accurate diagnostic casts to be mounted for occlusal examination. The casts preferably will be articulated on a suitable instrument. The impor-tance of accurate diagnostic casts and their use will be discussed later in this chapter. 5. Examination of teeth, investing structures, and residual ridges. The teeth, periodontium, and residual ridges can be explored by instrumentation and visual means. Recording of relevant patient history and clinical data on diagnosis charts is important to document features important to clinical presentation. These can be recorded on electronic or paper charts for future reference (Figures 12-2 and 12-3). Visual examination will reveal many of the signs of dental disease. Consideration of caries susceptibility is of primary of carious lesions to the pulp and periodontal attach-ment; (d) to permit evaluation of existing restorations as to evidence of recurrent caries, marginal leakage, and overhanging gingival margins; (e) to reveal the presence of root canal fillings and to permit their evaluation as to future prognosis (the design of the removable partial denture may hinge on the decision to retain or extract an endodontically treated tooth); (f) to permit evaluation of periodontal conditions present and to establish the need and possibilities for treatment; and (g) to evaluate the alveolar support of abutment teeth, their number, the supporting length and morphology of their roots, the relative amount of alveolar bone loss suffered through pathogenic processes, and the amount of alveolar support remaining. REMOVABLE PARTIAL PROSTHODONTICS PATIENT NAME PATIENT NUMBER TREATMENT PLAN COLOR CODE: BLUE:---- CAST METAL RED: ----- RESIN BASE AND WROUGHT WIRE GREEN:-- AREAS TO BE CONTOURED LABORATORY INSTRUCTIONS DESIGN SPECIFICATIONS: 1. RESTS 2. RETENTION 3. RECIPROCATION 4. MAJOR CONNECTOR 5. INDIRECT RETENTION 6. GUIDE PLANES 7. BASE RETENTION 8. AREAS TO BE MODIFIED OR CONTOURED INSTRUCTOR: APPROVAL TO SEND TO LABORATORY: DATE: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 R L R L A Figure 12-2 A, Diagnosis record for recording pertinent data. Continued 154 Part II Clinical and Laboratory B B, Treatment record chart for recording treatment plan and treatment progress. Figure 12-2, cont’d 155 Chapter 12 Diagnosis and Treatment Planning 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Right Left Name: Summary Plan: Procedures: John Doe Cr Cr Red line each completed unit Follow-up care July 1, 1999 Date: Porcelain fused to metal abutment Crowns #18, #29 Maxillary Conventional Complete denture Mandibular Class II mod. 1 removable partial denture Tissue conditioning maxillary arch Primary impression-both arches; make impression trays Preparations. Contour and rest seats #21, #28. Abutment crown preparations #18, #29 Try-in abutment crowns; pick-up for cast for framework casting Fluid wax functional impression: Make altered cast Jaw relation records; shade and mold selection Tooth arrangement Try-in abutment crowns and framework Try-in trial set-up; verify jaw relations Placement: Mandibular crowns, partial denture, maxillary complete denture Figure 12-3 Simple working chart. Restorations for individual teeth, crowns, and fixed partial dentures to be made may be marked on the chart and checked off as completed during mouth preparations. importance. The number of restored teeth present, signs of recurrent caries, and evidence of decalcification should be noted. Only those patients with demonstrated good oral hygiene habits and low caries susceptibility should be con-sidered good risks without resorting to prophylactic mea-sures such as the restoration of abutment teeth. At the time of the initial examination, periodontal disease, gingival inflammation, the degree of gingival recession, and muco-gingival relationships should be observed. Such an examina-tion will not provide sufficient information to allow a definitive diagnosis and treatment plan. For this purpose, complete periodontal charting that includes pocket depths, assessment of attachment levels, furcation involvement, mucogingival problems, and tooth mobility should be per-formed. The extent of periodontal destruction must be determined with appropriate radiographs and use of the periodontal probe. The number of teeth remaining, the location of the eden-tulous areas, and the quality of the residual ridge will have a definite bearing on the proportionate amount of support that the removable partial denture will receive from the teeth and edentulous ridges. Tissue contours may appear to present a well-formed edentulous residual ridge; however, palpation often indicates that supporting bone has been resorbed and has been replaced by displaceable, fibrous con-nective tissue. Such a situation is common in maxillary tuberosity regions. The removable partial denture cannot be supported adequately by tissues that are easily displaced. When the mouth is prepared, this tissue should be recon-toured or removed surgically, unless otherwise contraindi-cated. A small but stable residual ridge is preferable to a larger unstable ridge for providing support for the denture. The presence of tori or other bony exostoses must be detected and their presence in relation to framework design must be evaluated. Failure to palpate the tissue over the median palatal raphe to ascertain the difference in its displaceability as compared with the displaceability of the soft tissues cover-ing the residual ridges can lead to a rocking, unstable, uncomfortable denture and to a dissatisfied patient. Ade-quate relief of the palatal major connectors must be planned, and the amount of relief required is directly proportionate to the difference in displaceability of the tissues over the midline of the palate and the tissues covering the residual ridges. During the examination, not only each arch but also its occlusal relationship with the opposing arch must be con-sidered separately. A situation that looks simple when the teeth are apart may be complicated when the teeth are in 156 Part II Clinical and Laboratory is dental plaster. Generally the improved dental stones (die stones) are not used for diagnostic casts because of their cost. Their greater resistance to abrasion does, however, justify their use for master casts. The impression for the diagnostic cast is usually made with an irreversible hydrocolloid (alginate) in a stock (per-forated or rim lock) impression tray. The size of the arch will determine the size of the tray to be used. The tray should be sufficiently oversized to ensure an optimum thickness of impression material to avoid distortion or tearing on removal from the mouth. The technique for making impressions is covered in more detail in Chapter 15. Purposes of Diagnostic Casts Diagnostic casts serve several purposes as an aid to diagnosis and treatment planning. Some of these are as follows: 1. Diagnostic casts are used to supplement the oral exami-nation by permitting a view of the occlusion from the lingual, as well as from the buccal, aspect. Analysis of the existing occlusion is made possible when opposing casts are occluded, as is a study of the possibilities for improve-ment by occlusal adjustment, occlusal reconstruction, or both. The degree of overclosure, the amount of interoc-clusal space available, and the possibilities of interference with the location of rests may also be determined. As was stated previously, opportunities for improvement of the occlusal scheme, by occlusal adjustment or occlusal reconstruction, are best evaluated by analysis and modi-fication of mounted diagnostic casts. Such procedures often include diagnostic waxing to determine the possi-bility of enhancing the occlusion before definitive treat-ment is begun (Figure 12-4). In other words, diagnostic casts permit the dentist to plan ahead to avoid undesir-able compromises in the treatment being given a patient. 2. Diagnostic casts are used to permit a topographic survey of the dental arch that is to be restored by means of a removable partial denture. The cast of the arch in ques-tion may be surveyed individually with a cast surveyor to determine the parallelism or lack of parallelism of tooth surfaces involved, and to establish their influence on the design of the removable partial denture. The principal considerations in studying the parallelism of tooth and tissue surfaces of each dental arch are to determine the need for mouth preparation: (a) proximal tooth surfaces, which can be made parallel to serve as guiding planes; (b) retentive and nonretentive areas of the abutment teeth; (c) areas of interference with placement and removal; and (d) esthetic effects of the selected path of insertion. From such a survey, a path of placement may be selected that will satisfy requirements for parallelism and retention to the best mechanical, functional, and esthetic advantage. Then mouth preparations may be planned accordingly. 3. Diagnostic casts are used to permit a logical and compre-hensive presentation to the patient of present and future restorative needs, as well as of the hazards of future neglect. Occluded and individual diagnostic casts can be occlusion. For example, an extreme vertical overlap may complicate the attachment of anterior teeth to a maxillary denture. Extrusion of a tooth or teeth into an opposing edentulous area may complicate the replacement of teeth in the edentulous area or may create occlusal interference, which will complicate the location and design of clasp retain-ers and occlusal rests. Such findings subsequently will be evaluated further by careful analysis of mounted diagnostic casts. A breakdown of the fee may be recorded on the back of this chart for easy reference if adjustments or substitutions become necessary because of changes in diagnosis as the work progresses. 6. Vitality tests of remaining teeth. Vitality tests should be given particularly to teeth to be used as abutments and those having deep restorations or deep carious lesions. This should be done through both thermal and electronic means. 7. Determination of the height of the floor of the mouth to locate inferior borders of lingual mandibular major connec-tors. Mouth preparation procedures are influenced by the choice of major connectors (see Figure 5-6). This deter-mination must precede altering contours of abutment teeth. The fee for examination, which should include the cost of the radiographic survey and the examination of articu-lated diagnostic casts, should be established before the exam-ination is performed and should not be related to the cost of treatment. It should be understood that the fee for exami-nation is based on the time involved and the service ren-dered, and that the material value of the radiograph and diagnostic casts is incidental to the effectiveness of the examination. The examination record should always be available in the office for future consultation. If consultation with another dentist is requested, respect for the hazards of unnecessary radiation justifies loaning the dentist the radiograph for this purpose. However, duplicate films should be retained in the dentist’s files. Diagnostic Casts A diagnostic cast should be an accurate reproduction of all the potential features that aid diagnosis. These include the teeth locations, contours, and occlusal plane relationship; the residual ridge contour, size, and mucosal consistency; and the oral anatomy delineating the prosthesis extensions (vestibules, retromolar pads, pterygomaxillary notch, hard/soft palatal junction, floor of the mouth, and frena). Additional information provided by appropriate cast mount-ing includes occlusal plane orientation and the impact on the opposing arch; tooth-to-palatal soft tissue relation-ship; and tooth-to-ridge relationships both vertically and horizontally. A diagnostic cast is usually made of dental stone because of its strength and the fact that it is less easily abraded than 157 Chapter 12 Diagnosis and Treatment Planning impression. If wax blockout is to be used in the fabrica-tion of individual trays, a duplicate cast made from an irreversible hydrocolloid (alginate) impression of the diagnostic cast should be used for this purpose. The diag-nostic cast is too valuable for purposes of future reference to risk damage resulting from the making of an impres-sion tray. On the other hand, if oil-based clay blockout is used, the diagnostic cast may be used without fear of damage. 5. Diagnostic casts may be used as a constant reference as the work progresses. Penciled marks indicating the type of restoration, the areas of tooth surfaces to be modified, the location of rests, and the design of the removable partial denture framework, as well as the path of place-ment and removal, all may be recorded on the diagnostic cast for future reference (Figure 12-5). Then these steps may be checked off the worksheet as they are completed. Areas of abutment teeth to be modified may first be changed on the duplicate diagnostic cast by trimming the used to point out to the patient (a) evidence of tooth migration and the existing results of such migration; (b) effects of further tooth migration; (c) loss of occlusal support and its consequences; (d) hazards of traumatic occlusal contacts; and (e) cariogenic and periodontal implications of further neglect. Treatment planning actu-ally may be accomplished with the patient present, so that economic considerations may be discussed. Such use of diagnostic casts permits justification of the proposed fee through the patient’s understanding of the problems involved and of the treatment needed. Inasmuch as mouth rehabilitation procedures are frequently lengthy and often irreversible, there must be complete accord between dentist and patient before extensive treatment is begun, and financial arrangements must be consum-mated during the planning phase. 4. Individual impression trays may be fabricated on the diagnostic casts, or the diagnostic cast may be used in selecting and fitting a stock impression tray for the final A B C Figure 12-4 A, Following mounting of the diagnostic casts, tooth arrangement for the mandibular occlusal plane requirements can be accomplished. B, Following placement of the maxillary anterior teeth in an ideal position, diagnostic arrangement of occlusion results in a space posterior to surveyed crown #27. If such a finding were objectionable, alternative arrangements could be investigated. This is not possible unless a diagnostic workup is completed. C, Occlusion of the mandibular removable partial denture will be enhanced by improving the maxillary posterior occlusal plane of the super-erupted molars. 158 Part II Clinical and Laboratory placement of the maxillary cast in a position relative to the opening axis on the articulator, which is similar to the posi-tion of the maxilla in relation to the temporomandibular joint of the patient (Figure 12-6). The mandibular cast is then placed beneath the maxillary cast in a horizontal posi-tion dictated by mandibular rotation without tooth contact, at a minimal vertical opening. The Glossary of Prosthodontic Terms describes an articu-lator as a mechanical device that represents the temporo-mandibular joints and jaw members, to which maxillary and mandibular casts may be attached. Because the dominant influence on mandibular movement in a partially edentu-lous mouth is the occlusal plane and the cusps of the remain-ing teeth, an anatomic reproduction of condylar paths is probably not necessary. Still, movement of the casts in relation to one another as influenced by the occlusal plane and the cusps of the remaining teeth, when mounted at a reasonably accurate distance from the axis of condylar rotation, permits a relatively valid analysis of occlusal relations. This is more anatomically accurate than a simple hinge mounting. stone cast with the surveyor blade. A record is thus made of the location and degree of modification to be done in the mouth. This must be done in relation to a definite path of placement. Any mouth preparations to be accom-plished with new restorations require that restored teeth be shaped in accordance with a previously determined path of placement. Even so, the shaping of abutment teeth on the duplicate diagnostic cast serves as a guide to the form of the abutment. This is particularly true if the contouring of wax patterns is to be delegated to the tech-nician, as it may be in a busy practice. 6. Unaltered diagnostic casts should become a permanent part of the patient’s record because records of conditions existing before treatment are just as important as are preoperative radiographs. Therefore diagnostic casts should be duplicated, with one cast serving as a perma-nent record and the duplicate cast used in situations that may require alterations to it. Mounting Diagnostic Casts For diagnostic purposes, casts should be related on an ana-tomically appropriate articulator to best understand the role occlusion may have in the design and functional stability of the removable partial denture. This becomes increasingly important as the prosthesis replaces more teeth. If the patient presents with a harmonious occlusion and the edentulous span is a tooth-bound space, simple hand articulation is generally all that is required. However, when the natural dentition is not harmonious and/or when the replacement teeth must be positioned within the normal movement pat-terns of the jaws, the diagnostic casts must be related in an anatomically appropriate manner for diagnosis. This means Figure 12-5 Proposed mouth changes and design of the removable partial denture framework are indicated in pencil on the diagnostic cast in relation to the previously determined path of placement. This serves as a means of communicating with the patient and as a chair-side guide to tooth modification. The Journal of Prosthetic Dentistry, Vol. 94, No. 1, The Glossary of Prosthodontic Terms, 8th edition, 2005, pp. 10-81. Available at: http:// www.journals.elsevierhealth.com/periodicals/ympr/article/ PIIS0022391305001757/fulltext It is better that the casts be mounted in relation to the axis-orbital plane to permit better interpretation of the plane of occlusion in relation to the horizontal plane. Although it is true that an axis-orbital mounting has no functional value on a nonarcon instrument because that plane ceases to exist when opposing casts are separated, the value of such a mounting lies in the orientation of the casts in occlusion. (An arcon articulator is one in which the condyles are attached to the lower member as they are in nature, the term being a derivation coined by Berg-ström from the words articulation and condyle. Many of the more widely used articulators such as the Hanau H series, Dentatus, and improved Gysi have the condyles attached to the upper member and are therefore nonar-con instruments.) Sequence for Mounting Maxillary Cast to Axis-Orbital Plane The initial steps allow recording of the maxilla–temporo-mandibular joint (TMJ) relationship: 1. Identify the anterior and posterior reference points for the facebow (e.g., external auditory meatus, orbitale). 2. Prepare the bite fork and occlusion rim. 3. Place the bite fork centered to the arch, indexing it to the teeth with wax or elastomer. 4. Place the facebow over the bite fork rod anteriorly. 159 Chapter 12 Diagnosis and Treatment Planning 5. Place the bow evenly into the ears posteriorly. 6. Secure the bow anteriorly. 7. Position the bow anteriorly to the third point of refer-ence (establishes the horizontal plane). 8. Secure the bite fork vertical rod, then the horizontal rod (holding the bow securely to prevent torque). 9. Release the bow anteriorly to allow spread, and disen-gage from the ears. 10. Remove the fork downward and out of the mouth with the attached bow. 11. Carefully check the security of the attachments. The next steps allow transfer of the recorded relationship to the articulator: 1. Position the posterior reference points on the articulator (usually a posterior attachment point). 2. Secure the posterior points by securing the bow anteriorly. 3. Vertically relate the secured bow to the articulator ante-rior reference point. 4. Seat the maxillary cast into the bite fork registration (wax or elastomer). 5. Close the articulator and check clearance for mounting plaster (modify the cast as needed). 6. Mount with low-expansion plaster. Figure 12-6 Use of the facebow makes possible the recording of the spatial relationship of the maxillae to some anatomic reference points and transference of this relationship to an articulator. The facebow is a relatively simple device used to obtain a transfer record for orienting a maxillary cast on an articu-lating instrument. Originally, the facebow was used only to transfer a radius from condyle reference points, so that a given point on the cast would be the same distance from the condyle as it is on the patient. The addition of an adjustable infraorbital pointer to the facebow and the addition of an orbital plane indicator to the articulator make possible the transfer of the elevation of the cast in relation to the axis-orbital plane. This permits the maxil-lary cast to be correctly oriented in the articulator space comparable with the relationship of the maxilla to the axis-orbital plane on the patient. To accommodate this orientation of the maxillary cast and still have room for the mandibular cast, the posts of the conventional articu-lator must be lengthened. The older Hanau model H articulator usually will not permit a facebow transfer with an infraorbital pointer. A facebow may be used to transfer a comparable radius from arbitrary reference points, or it may be designed so that the transfer can be made from hinge axis points. The latter type of transfer requires that a hinge-bow attached to the mandible should be used initially to determine the 160 Part II Clinical and Laboratory Figure 12-7 The base of the cast has been prepared for mounting by placing three triangular grooves to allow indexing when mounted. The grooves are prepared with a 3-inch stone mounted in a laboratory lathe. hinge axis points, to which the facebow is then adjusted for making the hinge axis transfer. A facebow transfer of the maxillary cast, which is ori-ented to the axis-orbital plane in a suitable articulator, is an uncomplicated procedure. The Hanau series Wide-Vue 183-2, all 96H2-0 models, the Whip-Mix articulator (Whip-Mix Corp, Louisville, KY), and the Dentatus model ARH (Dentatus USA, New York, NY) will accept this transfer. The Hanau earpiece facebow models 153 and 158, the Hanau fascia facebow 132-2SM, and the Dentatus facebow type AEB incorporate the infraorbital plane to the articulator. None of these are hinge axis bows; they are used instead at an arbitrary point. The location of the arbitrary point or axis has long been the subject of controversy. Gysi and others have placed it 11 to 13 mm anterior to the upper third of the tragus of the ear on a line extending from the upper margin of the external auditory meatus to the outer canthus of the eye. Others have placed it 13 mm anterior to the posterior margin of the center of the tragus of the ear on a line extending to the corner of the eye. Bergström has located the arbitrary axis 10 mm anterior to the center of a spherical insert for the external auditory meatus and 7 mm below the Frankfort horizontal plane. In a series of experiments reported by Beck, it was shown that the arbitrary axis suggested by Bergström falls consistently closer to the kinematic axis than do the other two. It is desirable that an arbitrary axis is placed as close as possible to the kinematic axis. Although most authori-ties agree that any of the three axes will permit transfer of the maxillary cast with reasonable accuracy, it would seem that the Bergström point compares most favorably with the kinematic axis. The lowest point on the inferior orbital margin is taken as the third point of reference for establishing the axis-orbital plane. Some authorities use the point on the lower margin of the bony orbit in line with the center of the pupil of the eye. For the sake of consistency, the right infraorbital point is generally used and the facebow assembled in this relationship. All three points (right and left axes and infraorbital point) are marked on the face with an ink dot before the transfer is made. Casts are prepared for mounting on an articulator by placing three index grooves in the base of the casts. Two V-shaped grooves are placed in the posterior section of the cast and one groove in the anterior portion (Figure 12-7). An occlusion rim properly oriented on a well-fitting record base should be used in facebow procedures involv-ing the transfer of casts representative of the Class I and II partially edentulous situations. Without occlusion rims, such casts cannot be located accurately in the imprints of the wax covering the facebow fork. Tissues covering the residual ridges may be displaced grossly when the patient closes into the wax on the facebow fork. Therefore the wax imprints of the soft tissues will not be true negatives of the edentulous regions of the diagnostic casts. For purposes of illustration, a facebow using the exter-nal auditory meatus as the posterior reference point, the Whip-Mix Facebow technique (DB 2000, Whip-Mix Corp, Louisville, KY), will be shown. The facebow fork is covered with a polyether, polyvinyl siloxane or a roll of softened baseplate wax with the material distributed equally on the top and on the underneath side of the facebow fork. Then the fork should be pressed lightly on the diagnostic casts with the midline of the facebow fork corresponding to the midline of the central incisors (Figure 12-8). This will leave imprints of the occlusal and incisal surfaces of the maxillary casts and occlusion rim on the softened baseplate wax and is an aid in correctly orienting the facebow fork in the patient’s mouth. The facebow fork is placed in position in the mouth, and the patient is asked to close the lower teeth into the wax to stabilize it in position. It is removed from the mouth and chilled in cold water and then replaced in position in the patient’s mouth. An alternative method of stabilizing the facebow fork and recording bases is to enlist the assis-tance of the patient. If an earpiece facebow is to be used, the patient should be reminded that the plastic earpieces in the auditory canals will greatly amplify noise. With the facebow fork in position, the facebow toggle is slipped over the anterior projection of the facebow fork (Figure 12-9). The patient 161 Chapter 12 Diagnosis and Treatment Planning Figure 12-8 Orienting the facebow fork to the maxillary cast and occlusion rims will avoid displacing the occlusion rim in the mouth through patient closure or another uneven force. Polyvinyl siloxane material has been evenly distributed around the facebow fork, and care is exercised to position the fork to be centered at the mid-incisal position without any fork extension posterior to the record base, which could cause discomfort. 1 4 2 3 Figure 12-9 The horizontal toggle clamp of the Whip-Mix earpiece facebow (1) is slid onto the shaft of the facebow fork protruding from the patient’s mouth. The patient then helps guide plastic earpieces into the external auditory meatus and holds them in place while the operator tightens three thumb screws (2) and centers the plastic nosepiece (3) securely on the nasion. The horizontal toggle clamp is positioned and secured near (but not touching) the lip. The T screw (4) on the vertical bar is tightened. Note: Extreme care should be exercised not to tilt the facebow out of position when tightening. can assist in guiding the plastic earpieces into the external auditory meatus. The patient can then hold the arms of the facebow in place with firm pressure while the opera-tor secures the bite fork to the facebow. This accom-plishes the radius aspect of the facebow transfer. If an infraorbital pointer is used, it is placed on the extreme right side of the facebow and angled toward the infraorbital point previously identified with an ink dot. It is then locked into position with its tip lightly touching the skin at the dot. This establishes the elevation of the facebow in relation to the axis-orbital plane. Extreme care must be taken to avoid any slip that might injure the patient’s eye. With all elements tightened securely, the patient is asked to open, and the entire assembly is removed intact, rinsed with cold water, and set aside. The facebow records not only the radius from the condyles to the incisal contacts of the upper central incisors, but also the angular relationship of the occlusal plane to the axis-orbital plane. The facebow must be positioned on the articulator in the same axis-orbital relation as on the patient. If an arbitrary-type facebow is used, the calibrated condyle rods of the facebow ordinarily will not fit the condyle shafts of the articulator unless the width between the condyles just happens to be the same. With a Hanau model 132-25M facebow, the calibrations must be reequalized when in position on the articulator. For example, they have read 74 (mm) on each side of the patient but must be adjusted to read 69 (mm) on each side of the articulator. Some later model articulators have adjustable condyle rods and may be adjusted to fit the facebow. It is necessary that the facebow be centered in either case. Some facebows are self-centering, as is the Hanau Spring-Bow (Whip-Mix Corp, Louisville, KY). The third point of reference is the orbital plane indica-tor, which must be swung to the right so that it will be above the tip of the infraorbital pointer. The entire facebow with maxillary cast in place must be raised until the tip of the pointer contacts the orbital plane indicated. The elevation having thus been established, for all practi-cal purposes the orbital plane indicator and the pointer may now be removed because they may interfere with placing the mounting stone. An auxiliary device called a cast support is available; it is used to support the facebow fork and the maxillary cast during the mounting operation (Figure 12-10). With this device, the weight of the cast and the mounting stone are supported separately from the facebow, thus prevent-ing possible downward movement resulting from their combined weight. The cast support is raised to support-ing contact with the facebow fork after the facebow height has been adjusted to the level of the orbital plane. Use of some type of cast support is highly recommended as an adjunct to facebow mounting. The keyed and lubricated maxillary cast is now attached to the upper arm of the articulator with the mounting stone, thus completing the facebow transfer (Figure 12-11). Not only will the facebow have permitted the upper cast to be mounted with reasonable accuracy, it also will have served as a convenient means of support-ing the cast during mounting. Once mastered, its use 162 Part II Clinical and Laboratory Jaw Relationship Records for Diagnostic Casts One of the first critical decisions that must be made in a removable partial denture service involves selection of the horizontal jaw relationship to which the removable partial denture will be fabricated (centric relation or the maximum intercuspal position). All mouth-preparation procedures depend on this analysis. Failure to make this decision cor-rectly may result in poor prosthesis stability, discomfort, and deterioration of the residual ridges and supporting teeth. It is recommended that deflective occlusal contacts in the maximum intercuspal and eccentric positions be corrected as a preventive measure. Not all dentists agree that centric relation and the maximum intercuspal position must be har-monious in the natural dentition. Many dentitions function satisfactorily with the opposing teeth maximally intercusped in an eccentric position without either diagnosable or sub-jective indications of temporomandibular joint dysfunction, muscle dysfunction, or disease of the supporting structures of the teeth. In many such situations, no attempt should be made to alter the occlusion. It is not a requirement to inter-fere with an occlusion simply because it does not completely conform to a relationship that is considered ideal. If most natural posterior teeth remain and if no evidence of temporomandibular joint disturbances, neuromuscular dysfunction, or periodontal disturbances related to occlusal factors exists, the proposed restorations may be fabricated safely with maximum intercuspation of remaining teeth. However, when most natural centric stops are missing, the proposed prosthesis should be fabricated so that the maximum intercuspal position is in harmony with centric relation. By far the greater majority of removable partial dentures should be fabricated in the horizontal jaw relation-ship of centric relation. In most instances in which edentu-lous spaces have not been restored, the remaining posterior teeth will have assumed malaligned positions through drift-ing, tipping, or extrusion. Correction of the remaining natural occlusion to create a coincidence of centric relation and the maximum intercuspal position is indicated in such situations. Regardless of the method used in creating a harmonious functional occlusion, an evaluation of the existing relation-ships of the opposing natural teeth must be made and is accomplished with a diagnostic mounting. This evaluation is performed in addition to, and in conjunction with, other diagnostic procedures that contribute to an adequate diag-nosis and treatment plan. Diagnostic casts provide an opportunity to evaluate the relationships of remaining oral structures when correctly mounted on a semiadjustable articulator with use of a facebow transfer and interocclusal records. Diagnostic casts are mounted in centric relation (most retruded relation of the mandible to the maxillae) so that deflective occlusal con-tacts can be correlated with those observed in the mouth. Deflective contacts of opposing teeth are usually destructive to the supporting structures involved and should be elimi-Figure 12-11 Facebow mounting is complete. The relation-ship of the maxillary cast to the articulator condylar components is anatomically similar to that between the patient’s maxilla and the bilateral temporomandibular joint (TMJ) complex. Any sub-sequent tooth arrangement and occlusal contact development will represent the mouth more accurately than more arbitrary mountings. The benefits of the anatomic similarity are seen in more accurate occlusion for the finalized prosthesis (i.e., less intraoral adjustment required). Figure 12-10 Facebow fork support used to maintain the fork and cast in position while mounting. becomes a great convenience rather than a time-consum-ing nuisance. It is preferable that the maxillary cast be mounted while the patient is still present, thus eliminating a pos-sible reappointment if the facebow record is unacceptable for some reason. Not too infrequently, the facebow record has to be redone with the offset-type facebow fork repo-sitioned to avoid interference with some part of the articulator. 163 Chapter 12 Diagnosis and Treatment Planning the arch. If occlusion rims are necessary to correctly orient casts on an articulator, a centric relation should usually be the horizontal jaw relationship to which the removable partial denture will be constructed. Materials and Methods for Recording Centric Relation Materials available for recording centric relation are (1) wax; (2) modeling plastic; (3) quick-setting impression plaster; (4) metallic oxide bite registration paste; (5) polyether impression materials; and (6) silicone impression materials. Of these, wax is likely to be least satisfactory unless carefully handled. If not uniformly softened when introduced into the mouth, it can record a position with unequal tissue place-ment. Also, it does not remain rigid and dimensionally stable after removal unless carefully chilled and handled upon removal (Figure 12-12). Modeling plastic is a satisfactory record medium because it can be flamed and tempered until uniformly soft before it is placed into the mouth. After modeling plastic is chilled, it is sufficiently stable to permit the mounting of casts with accuracy. For these reasons, it is a satisfactory medium for recording occlusal relations for complete or partial dentures. It also can be used with opposing natural teeth. Impression plaster has advantages of softness when intro-duced and rigidity when set, which make it a satisfactory material for recording jaw relations. Its use is highly recom-mended when occlusion rims are used to mount casts cor-rectly or to adjust articulators with interocclusal eccentric records. Metallic oxide bite registration paste offers many of the advantages of plaster, with less friability. Although not strong enough to be used alone, when supported by a gauze mesh attached to a metal frame, it is a satisfactory recording medium. Also it may be used with occlusion rims. After the paste sets, the frame is removed from the mouth and the buccal side of the gauze released where it was secured with sticky wax. The tube on the lingual side may then be slid off the lingual extension of the frame. The frame is not needed when casts are mounted with this type of registration because the tube alone lends sufficient support to the interocclusal record. Elastomeric materials are excellent for recording interoc-clusal relationships (Figure 12-13). Some are specially for-mulated for this purpose and have the qualities of extremely low viscosity, minimal resistance to closure, rapid set, low rebound, lack of distortion, and stability after removal from the mouth. Care should be exercised to ensure that no elastic rebound results when the record is related to the cast during the mounting procedure. The mandibular cast should be mounted on the lower arm of the articulator, with the articulator inverted (Figure 12-14A). The articulator is first locked in centric position, and the incisal pin is adjusted so that the anterior distance between the upper and lower arms of the articulator will be increased 2 to 3 mm greater than the normal parallel rela-nated. Diagnostic casts demonstrate the presence and loca-tion of such interfering tooth contacts and permit visualization of the treatment that would be necessary for their correction. Necessary alteration of teeth to harmonize the occlusion can be performed initially on duplicates of the mounted diagnostic casts to act as guides for similar neces-sary corrections in the mouth. In many instances the degree of alteration required will indicate the need for crowns or onlays to be fabricated, or for recontouring, repositioning, or elimination of extruded teeth. As was previously mentioned, the maxillary cast is cor-rectly oriented to the opening axis of the articulator by means of the facebow transfer and becomes spatially related to the upper member of the articulator in the same relation-ship that the maxilla has to the hinge axis and the Frankfort plane. Similarly, when a centric relation record is made at an established vertical dimension, the mandible is in its most retruded relation to the maxilla. Therefore when the maxil-lary cast is correctly oriented to the axis of the articulator, the mandibular cast automatically becomes correctly ori-ented to the opening axis, when attached to and mounted with an accurate centric relation record. Unlike recording the fixed relationship of the maxilla to the mandibular opening axis (using the facebow transfer record), the mandibular position is recorded in space and is not a fixed point. Consequently, it is necessary to prove that the relationship of the mounted casts is correct. This can be done simply by making another interocclusal record, at centric relation, fitting the casts into the record, and check-ing to see that the condylar elements of the articulator are snug against the condylar housings. If this is not seen, another record is made until duplicate records are produced. Because centric relation is the only jaw position that can be repeated by the patient, mountings in this position can be replicated and verified for correctness. A straightforward protrusive record is made to adjust the horizontal condylar inclines on the articulator. Lateral eccentric records are made so that the lateral condylar incli-nations can be properly adjusted. All interocclusal records should be made as near the vertical relation of occlusion as possible. Opposing teeth or occlusion rims must not be allowed to contact when the records are made. A contact of the inclined planes of opposing teeth will invalidate an inter-occlusal record. In some instances, a mounting of a duplicate diagnostic cast in the maximum intercuspal position may also be desir-able to definitively study this relationship on the articulator. Because articulators simulate only jaw movements, it is not unreasonable to assume that the relationship of the casts mounted in centric relation may differ minutely from the maximum intercuspal position seen on the articulator and observed in the mouth. When diagnostic casts are hand related by maximum intercuspation for purposes of mount-ing on an articulator, it is essential that three (preferably four) positive contacts of opposing posterior teeth are present, having widespread molar contacts on each side of 164 Part II Clinical and Laboratory An articulator mounting thus made will have related the casts in centric relation (Figure 12-14B). The dentist then can proceed to make an occlusal analysis by observing the influence of cusps in relation to one another after the articu-lator has been adjusted by using eccentric interocclusal records. After an occlusal analysis has been made, the casts may be removed from their mounting for the purpose of survey-ing them individually and for other purposes as outlined previously. The indexed mounting ring record also should be retained throughout the course of treatment in the event that further study should be needed. It is advisable that the mounting be identified with the articulator that is used, so that it may always be placed back onto the same articulator. Diagnostic Findings The information gathered in the patient interview and clini-cal examination provides the basis for establishing whether treatment is indicated, and if so, what specific treatment should be considered. More than one treatment option can be considered, and financial implications need to be consid-ered against long-term expectations if the best decision is to be reached. Provision of a removable partial denture does not often preclude future consideration for other treat-ments, a fact that is not often the case for alternative treat-ments. The patient interview can reveal medical considerations that influence the decision to provide any prosthesis. When it is felt that general medical health is being neglected, tionship of the arms. This is done to compensate for the thickness of the interocclusal record so that the arms of the articulator will again be nearly parallel when the interoc-clusal record is removed and the opposing casts come into contact. The base of the cast should be keyed and lightly lubricated for future removal. With the diagnostic casts accurately seated and secured in the occlusal record, the mandibular cast is affixed with stone to the lower member of the inverted articulator. A B Figure 12-12 A wax interocclusal record made on a cast framework. The modification spaces first had baseplate wax added; these were adjusted intraorally to provide space at the occlusal vertical dimension for recording wax, the wax was softened using a wax spatula and a hot water bath, the framework was placed in the mouth, and care was exercised to guide the patient to close into a previously verified (and practiced) interocclusal position deemed appropriate (in this instance, centric relation position). The record was recovered from the mouth, excess wax was removed with a warm scalpel, and the wax was chilled and replaced in the mouth to verify the record. If not verified, the wax was resoftened (with additional wax added as needed) and the procedure was repeated. B, Immediately after verification, the framework with interocclusal registration was replaced on the mandibular cast and inverted on the maxillary cast for mounting. Figure 12-13 Elastomeric interocclusal registration material used to record mandibular position. 165 Chapter 12 Diagnosis and Treatment Planning is required if one is to have a chance to correct it with a similar prosthesis. If examination does not confirm any such relationship, it would be difficult to proceed without some concern for repeating the patient response to therapy unless a different form of therapy is selected (e.g., replacing a prob-lematic removable partial denture with an implant- supported prosthesis). Interpretation of Examination Data As a result of the oral examination, several diagnoses are made that are related to the various tissues, conditions, and clinical information gathered. The integration of these diag-noses serves as the basis for decisions that will ultimately identify the suggested treatment. The treatment decision reflects a confluence of several aspects of the patient’s past, present, and potential oral health status. It is helpful to consider how the various diagnoses are integrated; consequently, a suggested framework is provided that highlights aspects of disease management, followed by reconstruction considerations for (1) prosthesis support, and (2) prosthesis design-specific aspects. Disease management takes into account findings from the radiographic examination, periodontal disease and caries assessments, and pathology requiring endodontic consider-ations. Reconstruction considerations include diagnoses relative to prosthesis support (teeth and residual ridges) and prosthesis-specific design elements. Prosthesis support related to the remaining teeth requires radiographic exami-nation of alveolar support and root morphology, endodon-tic evaluation, analysis of occlusal factors, assessment of the patients should be strongly encouraged to seek a general medical examination. Alternatively, patients who regularly see their physician may be found to take multiple medica-tions that can contribute to a dry mouth and, potentially, an altered oral microflora with some increased risk for plaque-induced disease. Although such a condition can influence any prosthodontic care, given the unique features of remov-able partial denture service relative to the need for increased hygiene awareness and care, any factor that places an addi-tional risk for plaque-induced disease should be emphasized with the patient and corrected if possible. Health conditions that negatively affect oral mucosal health (e.g., diabetes mel-litus, Sjögren’s syndrome, lupus, atrophic changes) may pose a risk for patient comfort for a tissue-supported pros-thesis and factor into a treatment decision. For the patient who has had previous experience with some form of prosthesis, the patient interview provides additional information that can influence treatment deci-sions. Identifying possible reasons (or more importantly, a lack of any reason) for both positive and negative past pros-thesis experiences is important for determining whether a patient can predictably be helped. Although the clinical examination will point out the oral tissue responses to such therapy, the interview will highlight the subjective patient response to therapy and provides significant information that should be pursued. As was mentioned previously, a patient complaint regarding the prosthesis needs to be con-firmed through evaluation. The patient generally expresses concern about a symptom that can be related to support, stability, retention, and/or appearance. Confirmation of a design feature or oral condition that can explain the symptom A B Figure 12-14 A, Mandibular cast inverted on the mounted maxillary cast, making sure that the cast is fully seated into the interoc-clusal record and stabilized to the opposing cast. It is important to check the posterior occlusion rim contact to ensure that no interfering contact has altered the record. Space should be observed between the opposing record bases (or record base and opposing occlusion). B, Mounted casts demonstrating the occlusal plane as found in the mouth. The Frankfort plane of the patient is oriented parallel to the articulator base and the floor. Also, inspection of the posterior rims demonstrates space between the rims, which ensures that the recorded position was registered without influence from rigid contacting components, only from softened wax. 166 Part II Clinical and Laboratory forces but to torque as well because of movement of the tissue-supported base. Vertical support and stabilization against horizontal movement with rigid connectors are just as important as they are with a tooth-supported prosthesis, and the removable partial denture must be designed accordingly. In addition, the abutment tooth adja-cent to the extension base will be subjected to torque in proportion to the design of the retainers, the size of the denture base, the tissue support received by the base, and the total occlusal load applied. With this in mind, each abut-ment tooth must be evaluated carefully as to the alveolar bone support present and the past reaction of that bone to occlusal stress. It is important to judge whether the teeth and their respective periodontium can favorably respond to the demands of a prosthesis. Can radiographic interpretation provide clues to predicting tooth response to increased loading from prostheses? Assessment of regions within the mouth that have been subjected to increased loading can provide some clues as to the predictability of future similar response. An understanding of bone density, index areas, and lamina dura response is helpful for these judgments. Bone Density The quality and quantity of bone in any part of the body are often evaluated by radiographic means. A detailed treatise concerning bone support of the abutment tooth should include many considerations not possible to include in this text because of space limitations. The reader should realize that subclinical variations in bone may exist but may not be observed because of the limitations inherent in technical methods and equipment. Of importance to the dentist in evaluating the quality and quantity of the alveolar bone are the height and the quality of remaining bone. In estimating bone height, care must be taken to avoid interpretive errors resulting from angulation factors. Technically, when a radiographic exposure is made, the central ray should be directed at right angles to both the tooth and the film. The short-cone technique does not follow this principle; instead the ray is directed through the root of the tooth at a predetermined angle. This technique invariably causes the buccal bone to be projected higher on the crown than the lingual or palatal bone. Therefore in interpreting bone height, it is imperative to follow the line of the lamina dura from the apex toward the crown of the tooth until the opacity of the lamina materially decreases. At this point of opacity change, a less dense bone extends farther toward the tooth crown. This additional amount of bone represents false bone height. Thus the true height of the bone is ordinarily where the lamina shows a marked decrease in opacity. At this point, the trabecular pattern of bone superimposed on the tooth root is lost. The portion of the root between the cementoenamel junction and the true bone height has the appearance of being bare or devoid of covering. benefit for fixed prostheses or orthodontics, and evaluation of the need for extraction. Residual ridge support involves radiographic examination of ridge contours and height, and evaluation of the need for pre-prosthetic surgical interven-tion. Prosthesis-specific design considerations include deter-mination of anatomic relationships related to mandibular major connector design, the need for tooth modification to facilitate prosthesis function, and analysis of the occlusion. Each of these is considered in the following sections. Radiographic Interpretation Many of the reasons for radiographic interpretation during oral examination are outlined herein and are considered in greater detail in other texts. Aspects of such interpretation that are most pertinent to removable partial denture con-struction are those relative to the prognosis of remaining teeth that may be used as abutments. Disease Validation It is important to verify by clinical examination disease found through radiographic interpretation. Also, if the clini-cal examination reveals dental caries and/or periodontal disease, its severity can be confirmed by radiographic inter-pretation. It would be important to delineate caries severity, in terms of numbers of lesions and dentin/pulpal involve-ment, to gain insight as to level of disease risk associated with the patient, as well as to identify what therapy is required to maintain teeth. The same is true for periodontal disease risk and severity, as such a diagnosis affects both current and future tooth prognosis for prosthesis support. Radiographic interpretation allows diagnosis of bone lesions associated with both the jaws and the teeth. The implications for tooth stability and ridge support are important to factor into prosthesis prognosis. Surgical and postoperative management of such lesions can vary sig-nificantly with diagnosis (benign vs. malignant), and defini-tive prosthesis treatment is often complicated by resective procedures. Tooth Support The quality of the alveolar support of an abutment tooth is of primary importance because the tooth will have to with-stand greater stress loads when supporting a dental prosthe-sis. Abutment teeth providing total abutment support to the prosthesis, whether fixed or removable, will have to with-stand a greater load and especially greater horizontal forces. The latter may be minimized by establishing a harmonious occlusion and by distributing the horizontal forces among several teeth through the use of rigid connectors. Bilateral stabilization against horizontal forces is one of the attributes of a properly designed tooth-supported removable prosthe-sis. In many instances, abutment teeth may be aided more than weakened by the presence of a bilaterally rigid remov-able partial denture. In contrast, abutment teeth adjacent to distal extension bases are subjected not only to vertical and horizontal 167 Chapter 12 Diagnosis and Treatment Planning Index Areas Index areas are those areas of alveolar support that disclose the reaction of bone to additional stress. Favorable reaction to such stress may be taken as an indication of future reac-tion to an added stress load. Teeth that have been subjected to abnormal loading because of the loss of adjacent teeth or that have withstood tipping forces in addition to occlusal loading may be better risks as abutment teeth than those that have not been called on to carry an extra occlusal load (Figures 12-15 and 12-16). If occlusal harmony can be improved and unfavorable forces minimized by the reshap-ing of occlusal surfaces and the favorable distribution of occlusal loading, such teeth may be expected to support the prosthesis without difficulty. At the same time, other teeth, although not at present carrying an extra load, may be expected to react favorably because of the favorable reaction of alveolar bone to abnormal loading elsewhere in the same arch. Radiographic evaluation of bone quality is hazardous but is often necessary. It is essential to emphasize that changes in bone mineralization up to 25% often cannot be recognized by ordinary radiographic means. Optimum bone qualities are ordinarily expressed by normal-sized interdental trabecular spaces that tend to decrease slightly in size as examination of the bone proceeds from the root apex toward the coronal portion. The normal interproxi-mal crest is ordinarily shown by a relatively thin white line crossing from the lamina dura of one tooth to the lamina dura of the adjacent tooth. Considerable variation in the size of trabecular spaces may exist within normal limits, and the radiographic appearance of crestal alveolar bone may vary considerably, depending on its shape and the direction that the ray takes as it passes through the bone. Normal bone usually responds favorably to ordinary stresses. Abnormal stresses, however, may create a reduc-tion in the size of the trabecular pattern, particularly in that area of bone directly adjacent to the lamina dura of the affected tooth. This decrease in size of the trabecular pattern (i.e., so-called bone condensation) is often regarded as a favorable bone response, indicative of an improvement in bone quality. This is not necessarily an accurate interpretation. Such bone changes usually indicate stresses that should be relieved because if the resistance of the patient decreases, the bone may exhibit a progressively less favorable response on future radiographs. Increased thickness of the periodontal space ordinarily suggests varying degrees of tooth mobility. This should be evaluated clinically. Radiographic evidence coupled with clinical findings may suggest to the dentist the inad-visability of using such a tooth as an abutment. Further-more, an irregular intercrestal bone surface should make the dentist suspicious of active bone deterioration. It is essential that the dentist realize that radiographic evidence shows the result of changes that have taken place and may not necessarily represent the present condition. For example, periodontal disease may have progressed beyond the stage visibly demonstrated on the radiograph. As was pointed out earlier, radiographic changes are not observed until approximately 25% of the mineral content has been depleted. On the other hand, bone condensation probably does represent the current situation. Radiographic findings should serve the dentist as an adjunct to clinical observations. Too often the radio-graphic appearance alone is used to arrive at a diagnosis; therefore radiographic findings should always be con-firmed by clinical examination. Radiographic interpreta-tion will also serve an important function if used periodically after the prosthesis has been placed. Future bone changes of any type suggest traumatic interference from some source. The nature of such interference should be determined and corrective measures taken. Other index areas are those around teeth that have been subjected to abnormal occlusal loading; that have been subjected to diagonal occlusal loading caused by tooth migration; and that have reacted to additional loading, such as around existing fixed partial denture abutments. The reaction of the bone to additional stresses in these areas may be either positive or negative, with evidence of a supporting trabecular pattern, a heavy cortical layer, and a dense lamina dura, or the reverse response. With the former, the patient is said to have a positive bone factor, which means the ability to build additional support wherever needed. With the latter, the patient is said to have a negative bone factor, which means the inability to respond favorably to stress. Alveolar Lamina Dura The alveolar lamina dura is also considered in a radiographic interpretation of abutment teeth. The lamina dura is the thin layer of hard cortical bone that normally lines the sockets of all teeth. It affords attachment for the fibers of the periodon-tal membrane, and, as with all cortical bone, its function is to withstand mechanical strain. In a roentgenogram, the lamina dura is shown as a radiopaque white line around the radiolucent dark line that represents the periodontal membrane. When a tooth is in the process of being tipped, the center of rotation is not at the apex of the root, but in the apical third. Resorption of bone occurs where there is pressure, and apposition occurs where there is tension. Therefore during the active tipping process, the lamina dura is uneven, with evidence of both pressure and tension on the same side of the root. For example, in a mesially tipping lower molar, the lamina dura will be thinner on the coronal mesial and apicodistal aspects and thicker on the apicomesial and 168 Part II Clinical and Laboratory Figure 12-15 The reaction of bone adjacent to teeth that have been subjected to abnormal stress serves as an indication of probable reactions of that bone when such teeth are used as abutments for fixed or removable restorations. Such areas are called index areas. A B Figure 12-16 A, The canine has provided support for the distal extension removable partial denture for 10 years. There has obviously been positive bone response (arrow) to increased stress generated by the removable partial denture. B, The mandibular first premolar has provided support for the distal extension denture for 3 years. Bone response (arrow) to past additional stress has been unfavorable. 169 Chapter 12 Diagnosis and Treatment Planning A B Figure 12-17 A, The prognosis for abutment service is more favorable for a molar with divergent roots (shaded) than for the same tooth if its roots were fused and conical. B, Evidence that prospective abutment has conical and fused roots indicates the necessity for formulating a framework design that will minimize additional stresses placed on the tooth by the abutment service. Figure 12-18 First and second molars have been lost by this 18-year-old patient. A distal extension removable partial denture may be constructed until the third molar erupts and is fully formed. A tooth-supported restoration may then be considered. coronal distal aspects because the axis of rotation is not at the root apex but is above it. When the tooth has been tipped into an edentulous space by some change in the occlusion and becomes set in its new position, the effects of leverage are discontinued. The lamina dura on the side to which the tooth is sloping becomes uniformly heavier, which is nature’s reinforcement against abnormal stresses. The bone trabecu-lations are most often arranged at right angles to the heavier lamina dura. Thus it is possible to say that for a given individual, nature is able to build support where it is needed and on this basis to predict future reactions elsewhere in the arch to additional loading of teeth used as abutments. However, because bone is approximately 30% organic, and this mostly protein, and because the body is not able to store a protein reserve in large amounts, any change in body health may be reflected in the patient’s ability to maintain this support permanently. When systemic disease is associated with faulty protein metabolism and when the ability to repair is dimin-ished, bone is resorbed and the lamina dura is disturbed. Therefore the loading of any abutment tooth must be kept to a minimum inasmuch as the patient’s future health status and the eventualities of aging are unpredictable. Root Morphology The morphologic characteristics of the roots determine to a great extent the ability of prospective abutment teeth to resist successfully additional rotational forces that may be placed on them. Teeth with multiple and divergent roots will resist stresses better than teeth with fused and conical roots, because the resultant forces are distributed through a greater number of periodontal fibers to a larger amount of support-ing bone (Figure 12-17). Third Molars Unerupted third molars should be considered as prospective future abutments to eliminate the need for a distal extension removable partial denture (Figure 12-18). The increased sta-bility of a tooth-supported denture is most desirable to enhance the health of the oral environment. Periodontal Considerations An assessment of the periodontium in general and abutment teeth in particular must be made before prosthetic restora-tion. One must evaluate the condition of the gingiva, looking for adequate zones of attached gingiva and the presence or absence of periodontal pockets. The ideal periodontal condi-tion is a disease-free periodontium with adequate attached mucosa in regions at or adjacent to removable partial denture component parts that cross the gingival margins to best resist the mechanical challenges posed as the result of func-tion and use. The condition of the supporting bone must be evaluated, with specific attention to reduced bone support and mobility patterns recorded. If mucogingival involve-ments, osseous defects, or mobility patterns are recorded, the causes and potential treatment must be determined. Oral hygiene habits of the patient must be determined, and efforts made to educate the patient relative to plaque control. The most decisive evidence of oral hygiene habits is the condition of the mouth before the initial prophylaxis. Good or bad oral hygiene is basic to the patient’s nature, and although it may be influenced somewhat by patient educa-tion, the long-range view must be taken. It is reasonably fair 170 Part II Clinical and Laboratory methylcellulose base, which can be enriched with fluoride in an effort to counteract caries. Frequent use provides an excellent means of maintaining high fluoride intraorally for long periods of time, thus enhancing the remineralization of incipient caries. Although providing instructions for improvement of oral hygiene is a duty of the dental team, suspected problems of dietary deficiencies should be referred to a nutritionist. Evaluation of the Prosthesis Foundation—Teeth and Residual Ridge An evaluation of the prosthesis foundation is required to ensure that an appropriately stable base of sound teeth and/ or residual ridge(s) is provided to maximize prosthesis func-tion and patient comfort. To that end, the evaluation focuses on the identification of conditions that are inconsistent with sound support and predictably stable function. Surgical Preparation The need for pre-prosthetic surgery or extractions must be evaluated. The same criteria apply to surgical intervention in the partially edentulous arch as in the completely edentu-lous arch. Grossly displaceable soft tissues covering basal seat areas and hyperplastic tissue should be removed to provide a firm denture foundation. Mandibular tori should be removed if they will interfere with the optimum location of a lingual bar connector or a favorable path of placement. Any other areas of bone prominence that will interfere with the path of placement should be removed also. The path of placement will be dictated primarily by the guiding plane of the abutment teeth. Therefore some areas may present inter-ference to the path of placement of the removable partial denture by reason of the fact that other unalterable factors such as retention and esthetics must take precedence in selecting that path. Clinical research in pre-prosthetic surgical concepts has contributed significant developments to management of the compromised partially edentulous patient. Bone augmenta-tion and guided bone regeneration procedures have been used with varying degrees of success as an alternative method of improving ridge support for the denture base areas. Skill and judgment must be exercised in patient selection, proce-dural planning, and surgical and prosthetic management to optimize clinical results. Use of osseointegrated implants can provide a foundation for developing suitable abutment support for removable partial dentures. As in any surgical procedure, results depend on careful treatment planning and cautious surgical management. Extraction of teeth may be indicated for one of the fol-lowing three reasons: 1. If the tooth cannot be restored to a state of health, extrac-tion may be unavoidable. Modern advancements in the treatment of periodontal disease and in restorative pro-cedures, including endodontic therapy, have resulted in the saving of teeth that were once considered untreatable. All reasonable avenues of treatment should be considered to assume that the patient will do little more in the long-term future than he has done in the past. In making deci-sions as to the method of treatment based on oral hygiene, the future in years, rather than in weeks and months, must be considered. It is probably best not to give the patient the benefit of any doubt as to future oral hygiene habits. Rather, the benefit should come from protective measures where any doubt exists about future oral hygiene habits. Therefore for patients at greatest risk, an oral prophylaxis with continued oral hygiene instructions should be scheduled for 3- to 4-month intervals. In addition, the patient must be advised of the importance of regular maintenance appointments for tissue-supported prostheses to maintain occlusal relation-ships. When these ongoing observations and prophylactic requirements are described, the patient is faced with the realization that he or she must be willing to share responsi-bility for maintaining the health of the mouth after restor-ative and prosthodontic treatment. The remaining teeth and prosthesis will require meticu-lous plaque control after placement of a removable partial denture. Because of the nature of material coverage of oral tissues, the oral microflora can change with the use of a removable prosthesis. Coupled with this microbial change is the potential for a mechanical challenge to tissue integrity if the appropriate relationship of the prosthesis and soft tissues of the residual ridge, as well as the marginal gingival, is not maintained. Caries Risk Assessment Considerations Caries activity in the mouth, past and present, and the need for protective restorations must be considered. The decision to use full coverage is based on a need to reshape abutment teeth to accommodate the components of the removable partial denture, prevention of restoration breakdown when abutments have large direct restorations, or evidence of recurrent caries risk. Occasionally, three-quarter crowns may be used where buccal or lingual surfaces are completely sound, but intracoronal restorations (inlays) are seldom indicated in any mouth with evidence of past extensive caries or precarious areas of decalcification, erosion, or exposed cementum. Frequent consumption of sugars can lead to carious involvement of roots, caries around restorations, or caries associated with clasps of removable partial dentures. Intel-ligent consumption of sweets (smaller amounts and less fre-quent consumption) and frequent plaque removal are the recommended countermeasures. Excellent protection from caries can be provided by fluoride applications via tooth-pastes, mouth rinses, or (in extreme cases, such as postradia-tion xerostomia) 1% NaF gels applied daily with plastic trays. Xerostomia, caused by degeneration of salivary glands (Sjögren’s syndrome) or various medications, will enhance the occurrence and severity of caries, as well as contribute to irritation of the oral mucosa. A possible way to alleviate xerostomia is the use of synthetic saliva, with a carboxy-171 Chapter 12 Diagnosis and Treatment Planning ments, and consequently do not increase the functional burden on the natural dentition. Although the predictability of contemporary implant procedures (surgery and prosth-odontics) makes them a consideration for short span pros-theses, the main advantage is the opportunity to provide replacement teeth without involving adjacent teeth in the reconstruction. Therefore, when the adjacent teeth are in need of restoration, a conventional prosthesis should be considered. Longer Modification Spaces Longer span modification spaces (≥4 missing teeth) present a greater challenge for natural tooth–supported fixed pros-theses. Consequently, options for treatment include the removable partial denture and the implant-supported pros-thesis. An implant prosthesis has the same bone volume requirements as stated above, and for an increased span will likely require additional implants. Because residual ridge resorption can be greater with longer spans, the need for augmentation may also be greater. Both of these character-istics of longer spans cause implant use to be more costly and can significantly increase the cost difference between treatment options. The increased morbidity associated with augmentation procedures can also limit universal applica-tion. Because the removable partial denture remains largely tooth supported (unless the span includes anterior and pos-terior segments that may cause it to function similar to a distal extension), the functional stability requirements should be efficiently met through the tooth support. Distal Extension Spaces Without tooth support at each end of the missing teeth, the removable partial denture and the implant-supported pros-thesis are the primary treatment considerations (double-abutted cantilevered fixed prostheses opposing maxillary complete dentures have been suggested to be a reasonable option for some patients). It then becomes obvious that when anatomic limitations to implant placement exist and surgical measures cannot be taken to correct this, the remov-able partial denture is the only option (unless no treatment is elected). Current surgical options are available to correct most anatomic limitations, yet frequently implant therapy is not elected because of patient medical factors, concerns for the risk of surgical morbidity, increased time required for treatment, and costs. It is important to note that a compari-son of long-term maintenance requirements between these two options may demonstrate little cost difference over time. This is related to the effects of continued residual ridge resorption acting on the removable prosthesis and not the implant prosthesis. Endodontic Treatment Abutments for removable partial dentures are required to withstand various forces depending on the classification. The requirement for a distal extension abutment is different than that of a tooth-supported prosthesis in that torsional from both prognostic and economic standpoints before extraction is recommended. 2. A tooth may be removed if its absence will permit a more serviceable and less complicated removable partial denture design. Teeth in extreme malposition (lingually inclined mandibular teeth, buccally inclined maxillary teeth, and mesially inclined teeth posterior to an edentu-lous space) may be removed if an adjacent tooth is in good alignment and if good support is available for use as an abutment. Justification for extraction lies in the decision that a suitable restoration, which will provide satisfactory contour and support, cannot be fabricated, or that orthodontic treatment to realign the tooth is not feasible. An exception to the arbitrary removal of a mal-posed tooth occurs when a distal extension removable partial denture base would have to be made rather than using the more desirable tooth-supported base of the tooth in question. If alveolar support is adequate, a pos-terior abutment should be retained if at all possible in preference to a tissue-supported extension base. Teeth deemed to have insufficient alveolar support may be extracted if their prognosis is poor and if other adjacent teeth may be used to better advantage as abutments. The decision to extract such a tooth should be based on the degree of mobility and other periodontal considerations and on the number, length, and shape of the roots con-tributing to its support. 3. A tooth may be extracted if it is so unesthetically located as to justify its removal to improve appearance. In this situation, a veneer crown should be considered in prefer-ence to removal if the crown can satisfy the esthetic needs. If removal is advisable because of unesthetic tooth posi-tion, the biomechanical problems involved in replacing anterior teeth with a removable partial denture must be weighed against the problems involved in making an esthetically acceptable fixed restoration. Admittedly, the removable replacement is commonly the more esthetic of the two, despite modern advancements in retainers and pontics. However, the mechanical disadvantage of the removable restoration often makes the fixed replacement of missing anterior teeth preferable. Another consideration for pre-prosthetic surgery involves the decision between use of a removable partial denture and an implant-supported prosthesis. The following categories of tooth loss are presented with comparative comments germane to such decisions. Short Modification Spaces For short spans (≤3 missing teeth), natural tooth– and implant-supported fixed prostheses as well as removable partial dentures can generally be considered. Implant place-ment requires the decision that ample bone volume exists, or can be provided with minimal morbidity, to adequately house sufficient implants to support prosthetic teeth. Implant prostheses have the advantage of not requiring the use of teeth for support, stability, and retention require-172 Part II Clinical and Laboratory with isolated abutment teeth and distal extension bases. Bio-mechanical considerations and the future health of the remaining teeth should be given preference over economic considerations when such a choice is possible. Orthodontic Treatment Occasionally, orthodontic movement of malposed teeth fol-lowed by retention through the use of fixed partial dentures makes possible a better removable partial denture design mechanically and esthetically than could otherwise be used. Although adequate anchorage for tooth movement can be a major limitation in partially edentulous arches, carefully placed implants that subsequently can be used for prosthesis support have been used to expand orthodontic applications for this patient group. Need for Determining Type of Mandibular Major Connector As was discussed in Chapter 5, one of the criteria used to determine the use of the lingual bar or linguoplate is the height of the floor of the patient’s mouth when the tongue is elevated. Because the inferior borders of the lingual bar and the linguoplate are placed at the same vertical level, and because subsequent mouth preparations depend in part on the design of the mandibular major connector, determina-tion of the type of major connector must be made during the oral examination. This determination is facilitated by measuring the height of the elevated floor of the patient’s mouth in relation to the lingual gingiva with a periodontal probe and recording the measurement for later transfer to diagnostic and master casts. It is most difficult to make a determination of the type of mandibular major connector to be used solely from a stone cast that may or may not accu-rately indicate the active range of movement of the floor of the patient’s mouth. Too many mandibular major connec-tors are ruined or made flexible because subsequent grinding of the inferior border is necessary to relieve impingement of the sensitive tissues of the floor of the mouth. Need for Reshaping Remaining Teeth The clinical crown shapes of anterior and posterior teeth are not capable of supporting a removable partial denture framework without appropriate modification. Without the required modifications, the prosthesis does not adequately benefit from the support and stability offered by the teeth and consequently will not be comfortable to the patient. Many failures of removable partial dentures can be attrib-uted to the fact that the teeth were not reshaped properly to establish guiding planes or to receive clasp arms and occlusal rests before the impression for the master cast was made. Of particular importance are the paralleling of proximal tooth surfaces to act as guiding planes, the preparation of adequate rest areas, and the reduction of unfavorable tooth contours (Figure 12-19). To neglect planning such mouth prepara-tions in advance is inexcusable and leads to unsuccessful removable prosthesis service. forces exist in the distal extension situation. For this reason, an abutment for a distal extension that is endodontically treated carries a greater risk for complications than a similar tooth not involved in removable partial denture function. Because tooth support helps control prosthesis move-ment, the need for endodontic treatment should include assessment of overdenture abutments for removable partial dentures, especially to control movement of distal extensions. Analysis of Occlusal Factors From the occlusal analysis made by evaluating the mounted diagnostic casts, the dentist must decide whether it is best to accept and maintain the existing occlusion or to attempt to improve on it by means of occlusal adjustment and/or res-toration of occlusal surfaces. It must be remembered that the removable partial denture can supplement the occlusion that exists only at the time the prosthesis is constructed. The dominant force that dictates the occlusal pattern will be the cuspal harmony or disharmony of the remaining teeth and their proprioceptive influence on mandibular movement. The goal of artificial tooth placement is to harmonize with the functional parameters of the existing occlusion provid-ing bilateral, simultaneous functional contact. Chapter 17 identifies schemes of occlusion recommended for partially edentulous configurations. A review of these recommendations will provide a guide for modifying the existing occlusion or developing the appropriate occlusal scheme for each partially edentulous configuration. Improvements in the natural occlusion must be accom-plished before the prosthesis is fabricated, not subsequent to its fabrication. The objective of occlusal reconstruction by any means should be occlusal harmony of the restored denti-tion in relation to the natural forces already present or estab-lished. Therefore one of the earliest decisions in planning reconstructive treatment must be whether to accept or reject the existing vertical dimension of occlusion and the occlusal contact relationships in centric and eccentric positions. If occlusal adjustment is indicated, cuspal analysis always should precede any corrective procedures in the mouth by selective grinding. On the other hand, if reconstruction is to be the means of correction, the manner and sequence should be outlined as part of the overall treatment plan. Fixed Restorations There may be a need to restore modification spaces with fixed restorations rather than include them in the removable partial denture, especially when dealing with isolated abut-ment teeth. The advantage of splinting must be weighed against the total cost, with the weight of experience always in favor of using fixed restorations for tooth-bounded spaces unless the space will facilitate simplification of the remov-able partial denture design without jeopardizing the abut-ment teeth. One of the least successful removable partial denture designs is seen when multiple tooth-bounded areas are replaced with removable partial dentures in conjunction 173 Chapter 12 Diagnosis and Treatment Planning restoration is fully seated. Such a lingual bar will be located so that it will interfere with tongue comfort and function. These are only some of the objectionable consequences of inadequate mouth preparations. The amount of reduction of tooth contours should be kept to a minimum, and all modified tooth surfaces not only should be repolished after reduction but also should be sub-jected to fluoride treatment to lessen the incidence of caries. If it is not possible to produce the contour desired without perforating the enamel, then the teeth should be recon-toured with an acceptable restorative material. The age of the patient, caries activity evidenced elsewhere in the mouth, and apparent oral hygiene habits must be taken into consid-eration when one is deciding between reducing the enamel or modifying tooth contours with protective restorations. Some of the areas that frequently need correction are the lingual surfaces of mandibular premolars, the mesial and lingual surfaces of mandibular molars, the distobuccal line The design of clasps is dependent on the location of the retentive, stabilizing, reciprocal, and supporting areas in relation to a definite path of placement and removal. Failure to reshape unfavorably inclined tooth surfaces and, if neces-sary, to place restorations with suitable contours not only complicates the design and location of clasp retainers but also often leads to failure of the removable partial denture because of poor clasp design. A malaligned tooth or one that is inclined unfavorably may make it necessary to place certain parts of the clasp so that they interfere with the opposing teeth. Unparallel proxi-mal tooth surfaces not only will fail to provide needed guiding planes during placement and removal but also will result in excessive blockout. This inevitably results in place-ment of the connectors so far out of contact with tooth surfaces that food traps are created. To pass lingually inclined lower teeth, clearance for a lingual bar major connector may have to be so great that a food trap will result when the A B C Figure 12-19 A, Unmodified buccal surface of the mandibular premolar illustrates a typical height of contour location natural for this tooth (middle and occlusal thirds of the tooth). B, Proximal surface modification is required (hatched region) to produce a guide-plane surface. C, Buccal surface modification is needed to position the height of contour for favorable clasp location. The tooth modi-fication is a continuation of the proximal surface modification onto the buccal surface, and generally requires less than 0.5 mm of tooth removal. 174 Part II Clinical and Laboratory (contaminated instruments, operatory equipment, or envi-ronmental surfaces), and contact with airborne contami-nants present in droplet spatter or in aerosols of oral and respiratory fluids. For infection to occur via any of these routes, the “chain of infection” must be present. This includes a susceptible host, a pathogen with sufficient infec-tivity and numbers to cause infection, and a portal through which the pathogen may enter the host. For infection control procedures to be effective, one or more of these “links” in the chain must be broken. Studies from the CDC report that clothing exposed to the acquired immunodeficiency syndrome (AIDS) virus may be safely used after a normal laundry cycle. A high-temperature (140° F to 160° F, 60° C to 70 °C) wash cycle with normal bleach concentrations, followed by machine drying (212° F, 100° C, or higher), is preferable if clothing is visibly soiled with blood or other body fluids. Dry cleaning and steam pressing will also kill the AIDS virus, according to these studies. Patients with oral lesions suggestive of infectious disease and patients with a known history of hepatitis B, AIDS, AIDS-related complex, or other infectious diseases should be referred for appropriate medical care. In addition to environmental surface and equipment disinfection, all instruments, stones, burs, and other reusable items should be disinfected in 2% glutaraldehyde for 10 minutes, cleaned of debris, rinsed, and patted dry before the sterilizing process is initiated. Heat-sensitive items can be sterilized with the use of ethylene oxide (gas). For items that have been used in the mouth, including laboratory materials (e.g., impressions, bite registrations, fixed and removable prostheses, orthodontic appliances), cleaning and disinfection are required before they are manipulated in the laboratory (whether on-site or at a remote location). Any item manipulated in the laboratory should also be cleaned and disinfected before placement in the patient’s mouth. Fresh pumice with iodophor should be used for each polishing procedure, and the pumice pan should be washed, rinsed, and dried after each procedure. Because materials are constantly evolving, DHCWs are advised to follow manufacturers’ suggested procedures for specific materials relative to disinfection procedures. As a guide, use of a chemical germicide that has at least an inter-mediate level of activity (i.e., “tuberculocidal hospital disin-fectant”) is appropriate for such disinfection. Careful communication between dental office and dental laboratory regarding the specific protocol for handling and decontami-nation of supplies and materials is important to prevent any cross contamination. Differential Diagnosis: Fixed or Removable Partial Dentures Total oral rehabilitation (disease management, defective tooth restoration, and tooth replacement) is an objective in treating the partially edentulous patient. Although replace-ment of missing teeth by means of fixed partial dentures, angle of maxillary premolars, and the mesiobuccal line angle of maxillary molars. The actual degree of inclination of teeth in relation to the path of placement and the location of retentive and supportive areas are not readily interpretable during visual examination. These are established during comprehensive analysis of the diagnostic cast with a sur-veyor, which should follow the visual examination. Infection Control The American Dental Association follows the Centers for Disease Control (CDC)–recommended infection control procedures for dentistry. The most recent recommendations were made in 2003 and included updates from the previous 1993 guidelines. Most of the updates will be familiar to prac-titioners and are already largely practiced routinely. They are designed to prevent or reduce the potential for disease transmission from patient to DHCW (Dental Health Care Worker), from DHCW to patient, and from patient to patient. The document emphasizes the use of “standard pre-cautions” (which replaces the term “universal precautions”) for the prevention of exposure to and transmission of not only bloodborne pathogens, but also other pathogens encountered in oral health care settings. Major updates and additions include application of stan-dard precautions rather than universal precautions, work restrictions for health care personnel infected with, or occu-pationally exposed to, infectious diseases; management of occupational exposure to bloodborne pathogens, including postexposure prophylaxis for work exposures to hepatitis B virus (HBV), hepatitis C virus (HCV), and human immu-nodeficiency virus (HIV); selection and use of devices with features designed to prevent sharps injury, contact dermati-tis, and latex hypersensitivity; hand hygiene; dental unit waterlines; and biofilm and water quality; special consider-ations include dental handpieces and other devices attached to air lines and waterlines, saliva ejectors, radiology, paren-teral medications, single-use or disposable devices, pre- procedural mouth rinses, oral surgical procedures, handling of biopsy specimens and extracted teeth, laser/electrosurgery plumes, Mycobacterium tuberculosis, Creutzfeldt-Jakob disease and other prion diseases, program evaluation, and research considerations. The recommendations provide guidance for measures to be taken that will reduce the risks of disease transmission, among both dental health care workers (DHCWs) and their patients. Dental patients and DHCWs potentially may be exposed to a variety of microorganisms. Exposure can occur via blood and/or oral or respiratory secretions. The microor-ganisms may include viruses and bacteria that infect the upper respiratory tract in general, as well as cytomegalovi-rus, HBV, HCV, herpes simplex virus types 1 and 2, HIV, Mycobacterium tuberculosis, staphylococci, and streptococci. The transmission of infection in the dental operatory can occur through several routes. These include direct contact (blood, oral fluids, or other secretions), indirect contact 175 Chapter 12 Diagnosis and Treatment Planning support does not place additional functional demands on adjacent teeth likely contributes to their preservation, although this has not been universally demonstrated. For conventional fixed prostheses, lack of parallelism of the abutment teeth may be counteracted with copings or locking connectors to provide parallel sectional placement. Sound abutment teeth make possible the use of more con-servative retainers, such as partial-veneer crowns, or resin-bonded-to-metal restorations, rather than full crowns. The age of the patient, evidence of caries activity, oral hygiene habits, and the soundness of remaining tooth structure must be considered in any decision to use less than full coverage for abutment teeth. Two specific contraindications for the use of unilateral fixed restorations are known. One is a long edentulous span with abutment teeth that would not be able to withstand the trauma of nonaxial occlusal forces. The other is abutment teeth, which exhibit reduced periodontal support due to periodontal disease, which would benefit from cross-arch stabilization. In either situation, a bilateral removable resto-ration can be used more effectively to replace the missing teeth. Modification Spaces A removable partial denture for a Class III arch is better supported and stabilized when a modification area on the opposite side of the arch is present. A fixed partial denture need not be used to restore such an edentulous area because its inclusion may simplify the design of the removable partial denture. However, when a modification space is bound by a lone-standing single-rooted abutment, it is better restored by means of a fixed partial denture. This acts to stabilize the at-risk tooth, and the denture is made less complicated by not having to include other abutment teeth for the support and retention of an additional edentulous space or spaces. When an edentulous space that is a modification of a Class I or Class II arch exists anterior to a lone-standing abutment tooth, this tooth is subjected to trauma by the movements of a distal extension removable partial denture far in excess of its ability to withstand such stresses. The splinting of the lone abutment to the nearest tooth is manda-tory. The abutment crowns should be contoured for support and retention of the removable partial denture; in addition, a means of supporting a stabilizing component on the ante-rior abutment of the fixed partial denture or on the occlusal surface of the pontic usually should be provided. Anterior Modification Spaces Usually any missing anterior teeth in a partially edentulous arch, except in a Kennedy Class IV arch in which only ante-rior teeth are missing, are best replaced by means of a fixed restoration. There are exceptions. Sometimes a better esthetic result is obtainable when the anterior replacements are sup-plied by a removable partial denture, at other times treat-ment is simplified by inclusion of an anterior modification space into the removable partial denture (Figure 12-21). either tooth or implant supported, is generally the method of choice, there are many reasons why a removable partial denture may be the better method of treatment for a specific patient. The dentist must follow the best procedure for the welfare of the patient, who is always free to seek more than one opinion. Ultimately, the choice of treatment must meet the economic limitations and personal desires of the patient. The exception to this guideline is the Class III arch with a modification space on the opposite side of the arch, which will provide better cross-arch stabilization and a simpler design for the removable partial denture (Figure 12-20). Although uncommon, unilateral tooth loss is sometimes inappropriately treated with a unilateral removable partial denture in place of a fixed partial denture. This type of pros-thesis is not enhanced by cross-arch stabilization and places excessive stress on abutment teeth. Possibly more important, the risk for aspiration is significant if such a prosthesis is dislodged during use. For these reasons, use of the unilateral removable partial denture is strongly discouraged. Indications for Use of Fixed Restorations Tooth-Bounded Edentulous Regions Generally any unilateral edentulous space bounded by teeth suitable for use as abutments should be restored with a fixed partial denture cemented to one or more abutment teeth at either end. The length of the span and the periodontal support of the abutment teeth will determine the number of abutments required. As was mentioned earlier, such a span could be managed with the use of dental implants if deemed feasible and elected by the patient. The fact that implant A A Figure 12-20 Class III, modification 2 arch, in which modifi-cation spaces on the patient’s left (spaces designated at A) will be included in the design of the removable partial denture rather than restored with a long-span fixed partial denture. The design for a removable restoration is greatly simplified, resulting in significantly enhanced stability. 176 Part II Clinical and Laboratory A B C Figure 12-21 A, Diagnostic waxing of this complex case revealed the best means to manage replacement of tooth #’s 6 and 7 was with a fixed prosthesis, especially since the ridge defect was not severe and the adjacent teeth offered good retainer support. B, In contrast, this complex situation requires the maxillary anterior to be repositioned palatally to address an esthetic concern caused by the condition of the maxillary canines and the need to replace the posterior teeth as well. C, The anterior teeth will be more easily managed as part of the removable partial denture. (Courtesy Dr. M. Alfaro, Columbus, OH.) This is also true when excessive tissue and bone resorption necessitates placement of the pontics in a fixed partial denture too far palatally for good esthetics or for an accept-able relation with the opposing teeth. However, in most instances, from mechanical and biological standpoints, ante-rior replacements are best accomplished with fixed restora-tions. The replacement of missing posterior teeth with a removable partial denture is then made much less compli-cated and gives more satisfactory results. Replacement of Unilaterally Missing Molars (Shortened Dental Arch) Often the decision must be made to replace unilaterally missing molars (Figure 12-22). The decision must balance the impact of the treatment on the remaining oral structures with the potential benefit to the patient long term. To restore the missing molars with a fixed partial denture would require a cantilever prosthesis or the use of dental implants. A can-tilever-fixed prosthesis is most applicable if the second molar is to be ignored, then only first molar occlusion need be supplied with the use of a cantilever-type fixed partial denture. Occlusion need be only minimal to maintain occlu-sal relations between the natural first molar in the one arch and the prosthetic molar in the opposite arch. The cantile-vered pontic should be narrow buccolingually and need not occlude with more than one half to two thirds of the oppos-ing tooth. Often such a restoration is the preferred method of treatment. However, at least two abutments should be used to support a cantilevered molar opposed by a natural molar. To replace unilaterally missing molars with a removable partial denture necessitates the use of a distal extension pros-thesis. This involves the major connector joining the eden-tulous side to retentive and stabilizing components located on the non-edentulous side of the arch. Leverage factors are frequently unfavorable, and the retainers used on the non-177 Chapter 12 Diagnosis and Treatment Planning A B Figure 12-22 A, Unilaterally missing molars. If the patient exhibits opposing contacts to the remaining six posterior teeth (bilateral premolars, right first and second molars), functional gain attained by replacing the left molars may be minimal. B, By contrast, the functional gain resulting from replacement of the posterior occlusion in this patient is likely significant. of a multiple-abutment cantilevered fixed restoration or an implant-supported prosthesis. The most common partially edentulous situations are the Kennedy Class I and Class II. With the latter, an edentulous space on the opposite side of the arch is often conveniently present to aid in required retention and stabilization of the removable partial denture. If no space is present, selected abutment teeth can be modi-fied to accommodate appropriate clasp assemblies, or intra-coronal retainers can be used. As was previously stated, all other edentulous areas are best replaced with fixed partial dentures. After Recent Extractions The replacement of teeth after recent extractions often cannot be accomplished satisfactorily with a fixed restora-tion. When relining will be required later, or when a fixed restoration using natural teeth or implants will be con-structed later, a temporary removable partial denture can be used. If an all-resin denture is used rather than a cast frame-work removable partial denture, the immediate cost to the patient is much less, and the resin denture lends itself best to future temporary modifications, including those required after implant placement and before restoration. Tissue changes are inevitable following extractions. Tooth-bounded edentulous areas (as a result of extractions) are best initially restored with removable partial dentures. Relining of a tooth-supported resin denture base is then possible. This is usually done to improve esthetics, oral cleanliness, or patient comfort. Support for such a restora-tion is supplied by occlusal rests on the abutment teeth at each end of the edentulous space. Long Span A long span may be totally tooth supported if the abutments and the means of transferring the support to the denture are adequate, and if the denture framework is rigid. There is edentulous side are often unsatisfactory. Two factors impor-tant to consider in making the decision to provide a unilateral, distal extension removable partial denture include the opposing teeth and the future effect of the maxillary tuberosity. First, the opposing teeth must be considered if it is con-sidered important to prevent extrusion and migration. This influences replacement of the missing molars far more than any improvement in masticating efficiency that might result. Replacement of missing molars on one side is seldom neces-sary for reasons of mastication alone. Second, the future effect of a maxillary tuberosity must be considered if concern exists for tuberosity enlargement. Often when left uncovered, the tuberosity increases in size, making future occlusal treatment difficult. However, cover-ing the tuberosity with a removable partial denture base, in combination with the stimulating effect of the intermittent occlusion, helps maintain tuberosity size and position. In such an instance, it may be better to make a removable partial denture with cross-arch stabilization and retention than to leave a maxillary tuberosity uncovered. Indications for Removable Partial Dentures Although a removable partial denture should be considered only when a fixed restoration is contraindicated, there are several specific indications for the use of a removable restoration. Distal Extension Situations Replacement of missing posterior teeth is often best accom-plished with a removable partial denture (see Figure 12-22B), especially when implant treatment is not feasible for the patient. The exception to this includes situations in which the replacement of missing second (and third) molars is inadvisable or unnecessary, or in which unilateral replace-ment of a missing first molar can be accomplished by means 178 Part II Clinical and Laboratory position of the natural dentition for normal tongue and cheek contacts. This is particularly true of a maxillary denture. Anteriorly, loss of residual bone occurs from the labial aspect. Often the incisive papilla lies at the crest of the resid-ual ridge. Because the central incisors are normally located anterior to this landmark, any other location of artificial central incisors is unnatural. An anterior fixed partial denture made for such a mouth will have pontics resting on the labial aspect of this resorbed ridge and will be too far lingual to provide desirable lip support. Often the only way the incisal edges of the pontics can be made to occlude with the opposing lower anterior teeth is to use a labial inclination that is excessive and unnatural, and both esthetics and lip support suffer. Because the same condition exists with a removable partial denture in which the anterior teeth are abutted on the residual ridge, a labial flange must be used to permit the teeth to be located closer to their natural position. The same method of treatment applies to the replacement of missing mandibular anterior teeth. Sometimes a man-dibular anterior fixed partial denture is made six or more units in length, in which the remaining space necessitates leaving out one anterior tooth or using the original number of teeth but with all of them too narrow for esthetics. In either instance, the denture is nearly in a straight line because the pontics follow the form of the resorbed ridge. A remov-able partial denture will permit the location of the replaced teeth in a favorable relation to the lip and opposing dentition regardless of the shape of the residual ridge. When such a removable prosthesis is made, however, positive support must be obtained from the adjacent abutments. little if any difference between the support afforded a remov-able partial denture and that afforded a fixed restoration by the adjacent abutment teeth. However, in the absence of cross-arch stabilization, the torque and leverage on the two abutment teeth would be excessive. Instead, a removable denture that derives retention, support, and stabilization from abutment teeth on the opposite side of the arch is indicated as the logical means of replacing the missing teeth. Need for Effect of Bilateral Stabilization In a mouth weakened by periodontal disease, a fixed restora-tion may jeopardize the future of the involved abutment teeth unless the splinting effect of multiple abutments is used. The removable partial denture, on the other hand, may act as a periodontal splint through its effective cross-arch stabilizing of teeth weakened by periodontal disease. When abutment teeth throughout the arch are properly prepared and restored, the beneficial effect of a removable partial denture can be far greater than that of a unilateral fixed partial denture. Excessive Loss of Residual Bone The pontic of a fixed partial denture must be correctly related to the residual ridge and in such a manner that the contact with the mucosa is minimal. Whenever excessive resorption has occurred, teeth supported by a denture base may be arranged in a more acceptable buccolingual position than is possible with a fixed partial denture (Figure 12-23). Unlike a fixed partial denture, the artificial teeth sup-ported by a denture base can be located without regard for the crest of the residual ridge and more nearly in the A B Figure 12-23 A, Occlusal view of the anterior ridge defect (Kennedy Class IV) shows the palatal position of the ridge crest. Incisal edges of opposing dentition require a more labial position, which would create a difficult pontic form. B, Labial view of the same cast shows the significance of the vertical bone loss. Replacement of the teeth and ridge anatomy is best accomplished with a removable partial denture. 179 Chapter 12 Diagnosis and Treatment Planning tated by these considerations must be described clearly to the patient as a compromise and not as representative of the best that modern dentistry has to offer. A prosthesis that is made to satisfy economic considerations alone may provide only limited success and result in more costly treatment in the future. Unusually Sound Abutment Teeth Sometimes the reasoning for making a removable restora-tion is the desire to see sound teeth preserved in their natural state and not prepared for restorations. As was mentioned previously, if this decision is made because it is felt that no tooth modification is necessary for removable partial den-tures, then the prosthesis will lack tooth-derived stability and support. When this condition exists, the dentist should not hesitate to reshape and modify existing enamel surfaces to provide proximal guiding planes, occlusal rest areas, optimum retentive areas, and surfaces on which nonreten-tive stabilizing components may be placed. Continued dura-bility of the natural teeth is best ensured if the modifications that optimize prosthesis function are provided. This is due to the fact that such modifications also ensure the most harmonious use of the natural dentition. Abutments With Guarded Prognoses If the prognosis of an abutment tooth is questionable or if it becomes unfavorable while under treatment, it might be possible to compensate for its impending loss by a change in denture design. The questionable or condemned tooth or teeth may then be included in the original design and, if subsequently lost, the removable partial denture can be modified or remade (Figure 12-24). Most removable partial denture designs do not lend themselves well to later addi-tions, although this eventuality should be considered in the design of the denture. When the tooth in question will be used as an abutment, every diagnostic aid should be used to determine its prog-nosis as a prospective abutment. It is usually not as difficult to add a tooth or teeth to a removable partial denture as it is to add a retaining unit when the original abutment is lost and the next adjacent tooth must be used for that purpose. It is sometimes possible to design a removable partial denture so that a single posterior abutment, about which there is some doubt, can be retained and used at one end of the tooth-supported base. Then if the posterior abutment is lost, it could be replaced by adding an extension base to the existing denture framework. Such an original design must include provisions for future indirect retention, flexible clasping of the future abutment, and provisions for estab-lishing tissue support. Anterior abutments that are consid-ered poor risks may not be so freely used because of the problems involved in adding a new abutment retainer when the original one is lost. It is rational that such questionable teeth should be condemned in favor of more suitable abut-ments, even though the original treatment plan must be modified accordingly. Economic Considerations Economics should not be the sole criterion used to arrive at a method of treatment. When, for economic reasons, com-plete treatment is out of the question and yet replacement of missing teeth is indicated, the restorative procedures dic-C A B Figure 12-24 Kennedy Class II, modification 1, where the molar abutment has a guarded prognosis. A, Anterior abutment of the modification space has a clasp assembly that accommo-dates for potential future loss of the distal molar while currently providing adequate support, stability, and retention. B, Premolar clasp assembly comprises a mesial rest, a distal guide plane, and a wrought-wire retainer design, which will accommodate future distal extension movement. C, Buccal view shows guide-plane contact and a wrought-wire location that is appropriate for a distal extension. 180 Part II Clinical and Laboratory decision to maintain teeth is again based on risk assessment, costs for use of the teeth, added benefit to prosthesis func-tional stability, and comparative functional expectations between a mucosal-borne denture and a removable partial denture that uses teeth for some support, stability, and retention. The remaining tooth location and distribution can also affect the decision to maintain teeth. It makes a difference whether the remaining teeth are located on only one side of the arch. Having bilateral teeth remaining, especially if they are in similar locations (canines-canines, canines/ premolars-canines/premolars), offers advantages for pros-thesis design and occlusal development compared with asymmetrical tooth locations. Some teeth may not serve well as a stabilizing component for a removable partial denture and should not be maintained. If the remaining terminal tooth adjacent to a distal extension base is an incisor, the likelihood of long-term support, stability, and retention is poor. An additional factor for consideration when one is decid-ing between a complete denture and a removable partial denture is whether there is a strong patient desire to main-tain teeth. As was mentioned previously, because the change to a complete denture is a significant transformation, suffi-cient discussion must take place before this decision is made. The dentist must be very clear that the patient understands the functional differences between a mucosal-borne pros-thesis for all aspects of function (i.e., chewing, talking, etc.) and the natural dentition or a removable partial denture. The uniqueness of the patient is again appreciated these issues are discussed with the patient. One patient may prefer complete dentures rather than complete oral rehabilitation, regardless of ability to pay. Another may be so determined to keep his own teeth that he will make great financial sac-rifice if given a reasonable assurance of success of oral reha-bilitation. Listening to the patient during the examination and the diagnostic procedures pays off significantly when the treatment options differ so vastly as complete and removable partial dentures often do. During the presentation of perti-nent facts, time should be allowed for patients to express themselves freely as to their desires in retaining and restoring their natural teeth. At this time, a treatment plan may be influenced or even drastically changed to conform to the expressed and implied wishes of the patient. For example, there may be a reasonable possibility of saving teeth in both arches through the use of removable partial dentures. With only anterior teeth remaining, a removable partial denture can be made to replace the posterior teeth with the use of good abutment support and, in the maxillary arch, use of full palatal coverage for retention and stability. If patients express a desire to retain their anterior teeth at any cost, and if the remaining teeth are esthetically acceptable and functionally sound, the dentist should make every effort to provide successful treatment. If patients prefer a mandibular removable partial denture because of fear of difficulty in wearing a mandibular complete denture, then, all factors Choice Between Complete Dentures and Removable Partial Dentures One of the more difficult decisions to make for the partially edentulous patient involves making the choice of a complete denture over a removable partial denture. Many factors need to be considered when one is making such a decision; these generally fall under the categories of tooth-related factors, factors of comparative functional expectations between pros-theses, and patient-specific factors. Because the difference between a tooth-tissue–born prosthesis and a tissue-born prosthesis can be significant, especially since it is difficult for the partially edentulous patient to conceptualize the tissue-born situation, such an irreversible decision is not trivial. An evaluation of the remaining teeth will determine whether any caries or periodontal disease exists. The deci-sion as to whether a tooth is useful for inclusion in a pros-thetic treatment plan can be made on the basis of an understanding that with appropriate disease management, the tooth provides a reasonable 5-year prognosis for sur-vival. This takes into account the added functional demand by the prosthesis and a risk assessment for recurrent disease. Because this scenario concerns teeth with disease, the expec-tation is that tooth structure and/or support is compro-mised. The added functional burden, along with a potentially increased risk for disease, is an important concern when one is determining the long-term benefit for retaining teeth with a removable partial denture. If the teeth can be maintained with a reasonable progno-sis, the next questions to ask are “Do they require restoration with surveyed crowns?” and “How much improvement to the prosthesis support, stability, and retention do they provide?” If the expected prognosis for a given tooth is ques-tionable, the costs associated with restoration high, and the added benefit to the prosthesis low, the tooth should likely not be maintained unless the patient strongly desires to maintain all teeth. However, if the same scenario exists and the long-term impact on the support, stability, and retention of the prosthesis is great, the decision strongly favors keeping the tooth. The question of whether retained teeth offer a significant advantage to the prosthesis from a support, stability, and retention standpoint requires comparative evaluation of potential denture-bearing foundations. If the expectation is that an edentulous arch would have unfavorable physical features (poor ridge form, poor arch configuration, displace-able mucosa, high frena attachments, minimum denture bearing area, and/or an unfavorable jaw relationship), then retention of teeth is likely to provide a more significant benefit. If retention of teeth can help to prevent or delay age-related denture-bearing foundation changes seen with complete denture use, then retention of teeth can be of sig-nificant benefit. When evaluation demonstrates that the remaining teeth have no active disease, then the often-negative impact of disease management on prognosis is not a concern. The 181 Chapter 12 Diagnosis and Treatment Planning Clinical Factors Related to Metal Alloys Used for Removable Partial Denture Frameworks The cast framework offers significant advantages over the all acrylic-resin removable partial denture. In general, the ability to predictably utilize the remaining teeth for support, stability, and retention over time is best assured when the interface between prosthesis and teeth consists of a cast structure and not a polymer. Although the utility of all acrylic-resin prostheses can be extended if wire “rests” are provided, typical polymer properties do not allow for a durable interface, which is required if one is to take advantage of the stabilizing effects of tooth contact. Expecta-tions of how the metal framework improves functional per-formance are related to the properties of the metal alloy. Various alloys can be considered for use. Following is a discussion of the most common framework alloys in use today. Practically all cast frameworks for removable partial den-tures are made from a chromium-cobalt (Cr-Co) alloy. The popularity of Cr-Co alloys has been attributed to their low density (weight), high modulus of elasticity (stiffness), low material cost, and resistance to tarnish. The term stellite alloy historically has referred to this class of alloy. Today the more common alloys contain 60% to 63% Co, 29% to 31.5% Cr, and 5% to 6% Mo, with the balance including Si, Mn, Fe, N, and C. The addition of controlled amounts of nitrogen (<.5%) is reported to improve physical properties. Titanium is also used as a removable partial denture (RPD) frame material; however, production difficulties continue to hinder its widespread use. The dentist should become familiar with the alloy used by her/his laboratory and should closely monitor fit, density, and rigidity. The following are comparable characteristics of gold alloys and chromium-cobalt alloys: (1) each is well tolerated by oral tissues; (2) they are equally acceptable esthetically; (3) enamel abrasion by either alloy is insignificant on vertical tooth surfaces; (4) a low-fusing chrome-cobalt alloy or gold alloy can be cast to wrought wire, and wrought-wire components may be soldered to either gold or chrome-cobalt alloys (these characteristics are important in overcoming the objection by some dentists to the increased stiffness of chromium-cobalt alloys for the por-tions of direct retainers that must engage an undercut of the abutment tooth); (5) the accuracy obtainable in casting either alloy is clinically acceptable under strictly controlled investing and casting procedures; and (6) soldering proce-dures for the repair of frameworks can be performed on each alloy. Comparative Physical Properties of Gold and Chromium-Cobalt Chromium-cobalt alloys generally have less yield strength when compared with gold alloys used for removable partial dentures. Yield strength is the greatest amount of stress an being acceptable, their wishes should be respected and treat-ment should be planned accordingly. The professional obli-gation to present the facts and then do the best that can be done in accordance with the patients’ expressed desires still applies. Other patients may wish to retain remaining teeth for an indefinite but relatively short period of time, with eventual complete dentures a foregone conclusion. In this instance, the professional obligation may be to recommend interim removable partial dentures without extensive mouth prepa-ration. Such dentures will aid in mastication and will provide esthetic replacements, at the same time serving as condition-ing restorations, which will make the later transition to com-plete dentures somewhat easier. Such removable partial dentures should be designed and fabricated with care, but the total cost of removable partial denture service should be considerably less. An expressed desire on the part of patients to retain only six mandibular anterior teeth must be considered carefully before this is agreed to as the planned treatment. The advan-tages for patients are obvious: they may retain six esthetically acceptable teeth; they do not become totally edentulous; and they have the advantages of direct retention for the removable partial denture that would not be possible if they were com-pletely edentulous. Retaining even the mandibular canine teeth would accomplish the latter two objectives. Potential disadvantages relate directly to the patient keeping up with prosthesis maintenance procedures. The disadvantages relate to the poor response of the anterior maxilla to functional stress concentrated from the opposing natural dentition. If the functional forces of occlusion are not well distributed, the natural anterior can concentrate stress to the anterior maxil-lary arch. The possible result of such poorly distributed func-tional force includes the loss of residual maxillary bone, loosening of the maxillary denture caused by the tripping influence of the natural mandibular teeth, and loss of the basal foundation for the support of future prostheses. However, if the maxillary anterior teeth are arranged to contact in balanced eccentric positions and if patients comply with periodic recall to maintain these relationships, these problems are minimized. Prevention of this sequence of events lies in the maintenance of positive occlusal support posteriorly and the continual elimination of traumatic influ-ence from the remaining anterior teeth. Such support is sometimes impossible to maintain without frequent relining or remaking of the lower removable partial denture base. The presence of inflamed hyperplastic tissue is a frequent sequela to continued loss of support and denture movement. Although some patients are able to successfully function with a lower removable partial denture supported only by anterior teeth against a complete maxillary denture, it is likely that undesirable consequences will result unless the patient faithfully follows the instructions of the dentist. In no other situation in treatment planning are the general health of the patient and the quality of residual alveolar bone as critical as they are in this situation. 182 Part II Clinical and Laboratory is indicated for the bilateral distal extension removable partial denture. Weight is a factor that must be considered when the force of gravity must be overcome, so that usually passive direct retainers will not be activated constantly to the detriment of abutment teeth. The hardness of chromium-cobalt alloys presents a dis-advantage when a component of the framework, such as a rest, is opposed by a natural tooth or by one that has been restored. We have observed more wear of natural teeth opposed by some of the various chromium-cobalt alloys as contrasted with type IV gold alloys. It has been observed that gold frameworks for removable partial dentures are more prone to produce uncomfortable galvanic shock to abutment teeth restored with silver amalgam than are frameworks made of chromium-cobalt alloy. This may not be a valid criterion for the selection of a particular alloy when the dentist has complete control over the choice of restorative materials. Commercially pure (CP) titanium and titanium in alloys containing aluminum and vanadium, or palladium (Ti-O Pd), should be considered potential future materials for removable partial denture frameworks. Their versatility and well-known biocompatibility are promising; however, long-term clinical trials are needed to validate their potential use-fulness. Currently, when CP titanium is cast under dental conditions, the material properties change dramatically. During the casting procedure, the high affinity of the liquid metal for elements such as oxygen, nitrogen, and hydrogen results in their incorporation from the atmosphere. As interstitial alloying elements, their deleterious effect on mechanical properties is a problem. Also, reactions between molten titanium metal and the investment refractory produce gases, which cause porosity. With alpha-beta alloys, such as Ti-6Al-4V, a surface skin of alpha titanium can form (alpha-case zone), which has a tremendous effect on electrochemical behavior and mechanical properties. This could be important for small thin structures, such as clasp assemblies and major and minor connectors. The CP grades of titanium have yield strengths that are too low for clinical use as clasps (450 MPa minimum), although the ductility is high. The much higher yield strengths of the Ti-6Al-4V alloys are the same as that of a typical bench-cooled cobalt-chromium alloy, but with far superior ductil-ity. The typical Young’s modulus of elasticity of titanium alloy is half that of cobalt-chromium and just slightly higher than that of type IV gold alloys. This would require a differ-ent approach to clasp design than is used with cobalt- chromium alloys, and would present some advantages. Wrought titanium alloy wires are also flexible because of the same low elastic modulus. Beta alloys, which are used in orthodontics, have two-thirds the elastic modulus of CP titanium and Ti-6Al-4V. The joining of titanium by brazing is a problem because like-casting inert atmospheres must be used. The corrosion and fatigue behavior of brazed joints has yet to be tested for long-term corrosion resistance and alloy will withstand and still return to its original shape in an unweakened condition. Possessing a lower proportional limit, the chromium-cobalt alloys will deform permanently at lower loads than gold alloys. Therefore the dentist must design the chromium-cobalt framework so that the degree of deformation expected in a direct retainer is less than a comparable degree of deformation for a gold compo-nent. The modulus of elasticity refers to the stiffness of an alloy. Gold alloys have a modulus of elasticity approxi-mately one-half that for chromium-cobalt alloys for similar uses. The greater stiffness of the chromium-cobalt alloy is advantageous but at the same time offers disadvantages. Greater rigidity can be obtained with the chromium-cobalt alloy in reduced sections in which cross-arch stabilization is required, thereby eliminating an appreciable bulk of the framework. Its greater rigidity is also an advantage when the greatest undercut that can be found on an abutment tooth is in the nature of 0.05 inch. A gold retentive element would not be as efficient in retaining the restoration under such conditions as would the chromium-cobalt clasp arm. A high yield strength and a low modulus of elasticity produce greater flexibility. The gold alloys are approximately twice as flexible as the chromium-cobalt alloys; in many instances, this provides a distinct advantage in the optimum location of retentive elements of the framework. The greater flexibility of the gold alloys usually permits location of the tips of retainer arms in the gingival third of the abutment tooth. The stiffness of chromium-cobalt alloys can be over-come by including wrought-wire retentive elements in the framework. The bulk of a retentive clasp arm for a removable partial denture is often reduced for greater flexibility when chro-mium-cobalt alloys are used as opposed to gold alloys. This, however, is inadvisable because the grain size of chromium-cobalt alloys is usually larger and is associated with a lower proportional limit, and so a decrease in the bulk of chro-mium-cobalt cast clasps increases the likelihood of fracture or permanent deformation. The retentive clasp arms for both alloys should be approximately the same size, but the depth of undercut used for retention must be reduced by one half when chromium-cobalt is the choice of alloys. Chromium-cobalt alloys are reported to work/harden more rapidly than gold alloys, and this, associated with coarse grain size, may lead to failure in service. When adjustments by bending are necessary, they must be executed with extreme caution and limited optimism. Chromium-cobalt alloys have a lower density (weight) than gold alloys in comparable sections and therefore are about one half as heavy as gold alloys. The weight of the alloy in most instances is not a valid criterion for selection of one metal over another because after placement of a removable partial denture, the patient seldom notices the weight of the restoration. The comparable lightness of chromium-cobalt alloys, however, is an advantage when full palatal coverage 183 Chapter 12 Diagnosis and Treatment Planning Figure 12-25 The wrought-wire retainer arm has been con-toured to design and incorporated into the wax pattern of this frame, where it will become an integral part of the framework. The wire is contoured in two planes and will be mechanically retained in the casting. The tensile strength of the wrought structure is approxi-mately 25% greater than that of the cast alloy from which it was made. The wrought structure’s hardness and strength are also greater. This means that a wrought structure that has a smaller cross section than a cast structure may be used as a retainer arm (retentive) to perform the same function. It has been suggested that a minimum yield strength of 60,000 psi is required for the retentive element of a direct retainer. A percentage elongation of less than 6% is indicative that a wrought wire may not be amenable to contouring without attendant undesirable changes in microstructure. Regardless of the method of attaching the wrought-wire retainer that is used, that is, embedding, soldering, or cast-to, tapering the wrought arm seems most rational. A retainer arm is in essence a cantilever that can be made more service-able and efficient by tapering. Tapering to 0.8 mm permits more uniform distribution of service stresses throughout the length of the arm, being readily demonstrated by photoelas-tic stress analysis. Uniform tapering of an 18-gauge, round wire arm can be accomplished by rapidly rotating the wire in angled contact with an abrasive disk in the dental lathe. It is then polished by rotating the wire in angled contact with a mildly abrasive rubber disk in the dental lathe. The appro-priate taper is shown in Figure 12-26. clinical efficacy. Clinical use has demonstrated reasonable short-term results, but laboratory fabrication difficulties need to be addressed, and long-term advantages over exist-ing alloys must be demonstrated before titanium will gain broad clinical use. Wrought Wire: Selection and Quality Control Wrought-wire direct retainer arms may be attached to the restoration by embedding a portion of the wire in a resin denture base, by soldering to the fabricated framework, or by casting the framework to a wire embedded in the wax pattern (Figure 12-25). The physical (mechanical) proper-ties of available wrought wires are most important consid-erations when a proper wire for the desired method of attachment is selected. These properties include yield strength or proportional limit, percentage elongation, tensile strength, and fusion temperature. After the wire is selected, the procedures to which the wire is subjected in fabricating the restoration become critical. Improper laboratory proce-dures can diminish certain desirable physical properties of the wrought structure, rendering it relatively useless for its intended purpose. For example, when wrought wire is heated, as in a cast-to or soldering procedure, its physical properties and microstructure may be considerably altered, depending on temperature, heating time, and cooling opera-tion. All manufacturers of wrought forms for dental applica-tions furnish charts listing their products and the physical properties of each product. The percentage of noble metals is given. In addition, most manufacturers designate wires that may be used in a cast-to procedure. American Dental Association (ADA) Specification No. 7 addresses itself to wrought gold wire in terms of both content and minimum physical properties (Table 12-1). Table 12-1 Comparative Specifications Contained in ADA Specification No. 7 Type I Type II Content of metals of the gold, platinum group (minimum) 75% 65% Minimum fusion temperature 1742° F 1898°F Minimum yield point value (hardened or oven cooled) 125,000 psi 95,000 psi Minimum elongation (hardened) 4% 2% Minimum elongation (softened) 15% 15% ADA, American Dental Association. D ½ D Figure 12-26 Round, 18-gauge wrought wire for the retentive component of the direct retainer assembly (clasp) is uniformly tapered to 0.8 mm from its full diameter to its terminus. Taper-ing should precede contouring of the wire for the retainer arm. 184 Part II Clinical and Laboratory Summary In selecting materials, it must be remembered that funda-mentals do not change. These are inviolable. It is only methods, procedures, and substances—by which the dentist effects the best possible end result—that change. The respon-sibility of the decision still rests with the dentist, who must evaluate all factors in relation to the results desired. In any instance therefore, the dentist must weigh the problems involved, compare and evaluate the characteristics of differ-ent potential materials, and then make a decision that leads to delivery of the greatest possible service to the patient. 185 CHAPTER 13 Preparation of the Mouth for Removable Partial Dentures Chapter Outline Oral Surgical Preparation Extractions Removal of residual roots Impacted teeth Malposed teeth Cysts and odontogenic tumors Exostoses and tori Hyperplastic tissue Muscle attachments and frena Bony spines and knife-edge ridges Polyps, papillomas, and traumatic hemangiomas Hyperkeratoses, erythroplasia, and ulcerations Dentofacial deformity Osseointegrated devices Augmentation of alveolar bone Conditioning of Abused and Irritated Tissues Use of tissue conditioning materials Periodontal Preparation Objectives of periodontal therapy Periodontal diagnosis and treatment planning Initial disease control therapy (phase 1) Definitive periodontal surgery (phase 2) Recall maintenance (phase 3) Advantages of periodontal therapy Abutment Teeth Preparation Abutment restorations Contouring wax patterns Rest seats The preparation of the mouth is fundamental to a successful removable partial denture service. Mouth preparation, perhaps more than any other single factor, contributes to the philosophy that the prescribed prosthesis not only must replace what is missing but also must preserve the remaining tissues and structures that will enhance the removable partial denture. Mouth preparation follows the preliminary diagnosis and the development of a tentative treatment plan. Final treat-ment planning may be deferred until the response to the preparatory procedures can be ascertained. In general, mouth preparation includes procedures in four categories: oral surgical preparation, conditioning of abused and irri-tated tissues, periodontal preparation, and preparation of abutment teeth. The objectives of the procedures involved in all four areas are to return the mouth to optimum health and to eliminate any condition that would be detrimental to the success of the removable partial denture. Naturally, mouth preparation must be accomplished before the impression procedures are performed that will produce the master cast on which the removable partial denture will be fabricated. Oral surgical and periodontal procedures should precede abutment tooth preparation and should be completed far enough in advance to allow the necessary healing period. If at all possible, at least 6 weeks, and preferably 3 to 6 months, should be provided between surgical and restorative dentistry procedures. This depends on the extent of the surgery and its impact on the overall support, stability, and retention of the proposed prosthesis. Oral Surgical Preparation As a rule, all pre-prosthetic surgical treatment for the remov-able partial denture patient should be completed as early as possible. When possible, necessary endodontic surgery, peri-186 Part II Clinical and Laboratory alveolar ridge height or endangering adjacent teeth (Figure 13-2). Impacted Teeth All impacted teeth, including those in edentulous areas, as well as those adjacent to abutment teeth, should be consid-ered for removal. The periodontal implications of impacted teeth adjacent to abutments are similar to those for retained roots. These teeth are often neglected until serious periodon-tal implications arise. A B Figure 13-1 Diagnostic mounting allows confirmation of the need for extraction after clinical examination. A, Anterior tooth position and chronic periodontal disease status require extrac-tion to address the patient’s concern of malpositioned and painful teeth. B, Root tips require immediate extraction to allow ridge healing to begin. The status of the molar (#15) requires additional workup to determine pulpal involvement of the carious lesion and the extent of occlusal reduction required to optimize the occlusal plane. The decision to maintain this tooth, although potentially costly, must consider the stabilizing effect it will have on the posterior left functional occlusion. odontal surgery, and oral surgery should be planned, so that they can be completed during the same time frame. The longer the interval between the surgery and the impression procedure, the more complete the healing and consequently the more stable the denture-bearing areas. A variety of oral surgical techniques can prove beneficial to the clinician in preparing the patient for prosthetic replacements. However, it is not the purpose of this section to present the details of surgical correction. Rather, attention is called to some of the more common oral conditions or changes in which surgical intervention is indicated as an aid to removable partial denture design and fabrication, and as an aid to the successful function of the restoration. Addi-tional information regarding the techniques used is available in oral surgery texts and journal publications. It is important to emphasize, however, that the dentist who is providing the removable partial denture treatment bears the responsibility for ensuring that the necessary surgical procedures are accomplished in accordance with the treatment plan. Mea-sures to control apprehension, including the use of intrave-nous and inhalation agents, have made the most extensive surgery acceptable to patients. Whether the dentist chooses to perform these procedures or elects to refer the patient to someone more qualified is immaterial. The important con-sideration is that the patient should not be deprived of any treatment that would enhance the success of the removable partial denture. Extractions Planned extractions should occur early in the treatment regimen but not before a careful and thorough evaluation of each remaining tooth in the dental arch is completed (Figure 13-1). Regardless of its condition, each tooth must be evalu-ated in terms of its strategic importance and its potential contribution to the success of the removable partial denture. With the knowledge and technical capability available in dentistry today, almost any tooth may be salvaged if its retention is sufficiently important to warrant the procedures necessary. On the other hand, heroic attempts to salvage seriously involved teeth or those of doubtful prognosis for which retention would contribute little if anything, even if successfully treated and maintained, are contraindicated. Extraction of nonstrategic teeth that would present compli-cations or those that may be detrimental to the design of the removable partial denture is a necessary part of the overall treatment plan. Removal of Residual Roots Generally, all retained roots or root fragments should be removed. This is particularly true if they are in close proxim-ity to the tissue surface, or if associated pathologic findings are evident. Residual roots adjacent to the abutment teeth may contribute to the progression of periodontal pockets and compromise the results of subsequent periodontal therapy. Removal of root tips can be accomplished from the facial or palatal surfaces without resulting in a reduction of 187 Chapter 13 Preparation of the Mouth for Removable Partial Dentures The skeletal structure of the body changes with age. Asymptomatic impacted teeth in the elderly that are covered with bone, with no evidence of a pathologic condition, should be left to preserve the arch morphology. If an impacted tooth is left, this should be recorded in the patient’s record, and the patient should be informed of its presence. Roentgenograms should be taken at reasonable intervals to ensure that no adverse changes occur. Alterations that affect the jaws can result in minute expo-sures of impacted teeth to the oral cavity via sinus tracts. Resultant infections can cause considerable bone destruc-tion and serious illness for persons who are elderly and not physically able to tolerate the debilitation. Early elective removal of impactions prevents later serious acute and chronic infection with extensive bone loss. Any impacted teeth that can be reached with a periodontal probe must be removed to treat the periodontal pocket and prevent more extensive damage (Figure 13-3). Malposed Teeth The loss of individual teeth or groups of teeth may lead to extrusion, drifting, or combinations of malpositioning of remaining teeth (Figure 13-4). In most instances, the alveo-lar bone supporting extruded teeth will be carried occlusally as the teeth continue to erupt. Orthodontics may be useful in correcting many occlusal discrepancies, but for some patients, such treatment may not be practical because of lack of teeth for anchorage of the orthodontic appliances or for other reasons. In such situations, individual teeth or groups of teeth and their supporting alveolar bone can be surgically repositioned. This type of surgery can be accomplished in an outpatient setting and should be given serious consideration before additional teeth are condemned or the design of removable partial dentures is compromised. Cysts and Odontogenic Tumors Panoramic roentgenograms of the jaws are recommended to survey the jaws for unsuspected pathologic conditions. Figure 13-2 Retained root with associated bone resorption. (From Costich ER, White RP Jr: Fundamentals of oral surgery, Philadelphia, 1971, Saunders.) Figure 13-3 Lateral oblique roentgenogram showing uner upted maxillary third molar and impacted mandibular second and third molars. Maxillary third molar and mandibular second molar could be contacted by periodontal probe. (From Costich ER, White RP Jr: Fundamentals of oral surgery, Philadelphia, 1971, Saunders.) A B Figure 13-4 A, Malpositioned maxillary dentition due to loss of posterior occlusion and excessive wear of opposing mandibu-lar anterior teeth. B, Restored dentition made possible by a com-bination of endodontics, periodontics, and fixed and removable partial prosthodontics. (Courtesy Dr. M. Alfaro, Columbus, OH.) 188 Part II Clinical and Laboratory strain on the supporting teeth and tissues, and in many instances will provide a more favorable orientation of the occlusal plane and arch form for the arrangement of the artificial teeth. Appropriate surgical approaches should not reduce vestibular depth. Hyperplastic tissue can be removed with any preferred combination of scalpel, curette, electro-surgery, or laser. Some form of surgical stent should always be considered for these patients, so that the period of healing is more comfortable. An old removable partial denture properly modified can serve as a surgical stent. Although hyperplastic tissue has no great malignant propensity, all such excised tissue should be sent to an oral pathologist for microscopic study. Muscle Attachments and Frena As a result of the loss of bone height, muscle attachments may insert on or near the residual ridge crest. The mylohy-oid, buccinator, mentalis, and genioglossus muscles are most likely to introduce problems of this nature. In addition to the problem of the attachments of the muscles themselves, the mentalis and genioglossus muscles occasionally produce When a suspicious area appears on the survey film, a periapi-cal roentgenogram should be taken to confirm or deny the presence of a lesion. All radiolucencies or radiopacities observed in the jaws should be investigated. Although the diagnosis may appear obvious from clinical and roentgeno-graphic examinations, the dentist should confirm the diag-nosis through appropriate consultation and, if necessary, perform a biopsy of the area and submit the specimens to a pathologist for microscopic study. The patient should be informed of the diagnosis and provided with various options for resolution of the abnormality as confirmed by the pathologist’s report. Exostoses and Tori The existence of abnormal bony enlargements should not be allowed to compromise the design of the removable partial denture (Figure 13-5). Although modification of denture design can, at times, accommodate for exostoses, more fre-quently this results in additional stress to the supporting elements and compromised function. The removal of exos-toses and tori is not a complex procedure, and the advan-tages to be realized from such removal are great in contrast to the deleterious effects that their continued presence can create. Ordinarily the mucosa covering bony protuberances is extremely thin and friable. Removable partial denture components in proximity to this type of tissue may cause irritation and chronic ulceration. Also, exostoses approxi-mating gingival margins may complicate the maintenance of periodontal health and lead to the eventual loss of strategic abutment teeth. Hyperplastic Tissue Hyperplastic tissues are seen in the form of fibrous tuberosi-ties, soft flabby ridges, folds of redundant tissue in the ves-tibule or floor of the mouth, and palatal papillomatosis (Figure 13-6). All these forms of excess tissue should be removed to provide a firm base for the denture. This removal will produce a more stable denture, will reduce stress and A B Figure 13-5 Tori and exostoses. Figure 13-6 Hyperplasia fibrous tuberosities. 189 Chapter 13 Preparation of the Mouth for Removable Partial Dentures Dentofacial Deformity Patients with a dentofacial deformity often have multiple missing teeth as part of their problem. Correction of the jaw deformity can simplify the dental rehabilitation. Before spe-cific problems with the dentition can be corrected, the patient’s overall problem must be evaluated thoroughly. Several dental professionals (prosthodontist, oral surgeon, periodontist, orthodontist, general dentist) may play a role in the patient’s treatment. These individuals must be involved in producing the diagnostic database and in planning treat-ment for the patient. Information obtained from a general patient evaluation done to determine the patient’s health status, a clinical evaluation directed toward facial esthetics and the status of the teeth and oral soft tissues, and analysis of appropriate diagnostic records can be used to produce a database. From this database, the patient’s problems can be enumerated, with the most severe problem being placed at the top of the list. Other identified problems would follow in order of their severity. It is only after this step that input from several dentists can provide a correctly sequenced final treatment plan for the patient. Surgical correction of a jaw deformity can be made in horizontal, sagittal, or frontal planes. The mandible and maxillae may be positioned anteriorly or posteriorly, and their relationship to the facial planes may be surgically altered to achieve improved appearance. Replacement of missing teeth and development of a harmonious occlusion are almost always major problems in treating these patients. Osseointegrated Devices A number of implant devices to support the replacement of teeth have been introduced to the dental profession. These devices offer a significant stabilizing effect on dental prostheses through a rigid connection to living bone. The system that pioneered clinical prosthodontic applications with the use of commercially pure (CP) titanium endosseous implants is that of Brånemark and coworkers (Figure 13-7). This titanium implant was designed to provide a direct titanium-to-bone interface (osseointegrated), with basic laboratory and clinical results supporting the value of this procedure. Implants are carefully placed using controlled surgical procedures and, in general, bone healing to the device is allowed to occur before a dental prosthesis is fabricated. Long-term clinical research has demonstrated good results for the treatment of complete and partially edentulous dental patients using dental implants. Although research on implant applications with removable partial dentures has been very limited, the inclusion of strategically placed implants can significantly control prosthesis movement (See Chapter 25, Figures 13-8 through 13-10). Augmentation of Alveolar Bone Considerable attention has been devoted to ridge augmenta-tion with the use of autogenous and alloplastic materials, bony protuberances at their attachments that may also inter-fere with removable partial denture design. Appropriate ridge extension procedures can reposition attachments and remove bony spines, which will enhance the comfort and function of the removable partial denture. Repositioning of the mylohyoid muscle is successfully achieved by several methods. The genioglossus muscle is more difficult to reposition, but careful surgery can reduce the prominence of the genial tubercles, as well as provide some sulcus depth in the anterior lingual area. Surgical procedures that use skin or mucosal grafts have largely replaced secondary epithelialization procedures for the facial aspect of the mandible. Mucosal grafts that use the palate as a donor site offer the best possibility for success; transplanted skin can be used when large areas must be grafted. The maxillary labial and mandibular lingual frena are the most common sources of frenum interference with denture design. These can be modified easily through any of several surgical procedures. Under no circumstances should a frenum be allowed to compromise the design or comfort of a removable partial denture. Bony Spines and Knife-Edge Ridges Sharp bony spicules should be removed and knifelike crests gently rounded. These procedures should be carried out with minimum bone loss. If, however, correction of a knife-edge residual crest results in insufficient ridge support for the denture base, the dentist should resort to vestibular deepen-ing for correction of the deficiency or insertion of the various bone grafting materials that have demonstrated successful clinical trials. Polyps, Papillomas, and Traumatic Hemangiomas All abnormal soft tissue lesions should be excised and sub-mitted for pathologic examination before a removable partial denture is fabricated. Even though the patient may relate a history of the condition’s having been present for an indefinite period, its removal is indicated. New or additional stimulation to the area introduced by the prosthesis may produce discomfort or even malignant changes in the tumor. Hyperkeratoses, Erythroplasia, and Ulcerations All abnormal white, red, or ulcerative lesions should be investigated, regardless of their relationship to the proposed denture base or framework. A biopsy of areas larger than 5 mm should be completed, and if the lesions are large (over 2 cm in diameter), multiple biopsies should be taken. The biopsy report will determine whether the margins of the tissue to be excised can be wide or narrow. The lesions should be removed and healing accomplished before the removable partial denture is fabricated. On occasion, such as after irradiation treatment or the excoriation of erosive lichen planus, the removable partial denture design will have to be radically modified to avoid areas of possible sensitivity. 190 Part II Clinical and Laboratory B 1 2 3 4 5 6 7 A C Figure 13-7 A, Brånemark system components. From lower to upper: implant, cover screws, abutment, abutment screw, gold cyl-inder, and gold screw. B, Basic procedures in second-stage surgery: (1) exploration to locate cover screw; (2) removal of soft tissue; (3) removal of bony tissue; (4) removal of cover screw; (5) use of depth gauge to measure the amount of soft tissue; (6) abutment con-nection; and (7) placement of healing cap. C, Diagram of freestanding three-unit fixed partial denture supported by two osseointegrated implants that restore the extension base area, which would have been restored with a Class II removable partial denture if implants had not been used. (A and C redrawn from Hobo S, Ichida E, Garcia LT: Osseointegration and occlusal rehabilitation, Tokyo, Japan Quintessence, 1989.) especially in preparation for implant placement. Larger ridge volume gains necessitate consideration of autogenous grafts; however, these procedures are accompanied by concerns for surgical morbidity. Alloplastic materials have displayed short-term success; however, no randomized controlled trials have been conducted to provide evidence of long-term increases in ridge width and height for removable prostheses. Clinical results depend on careful evaluation of the need for augmentation, the projected volume of required mate-191 Chapter 13 Preparation of the Mouth for Removable Partial Dentures Conditioning of Abused and Irritated Tissues Many removable partial denture patients require some con-ditioning of supporting tissues in edentulous areas before the final impression phase of treatment begins. Patients who require conditioning treatment often demonstrate the following symptoms: 1. Inflammation and irritation of the mucosa covering denture-bearing areas (Figure 13-11) 2. Distortion of normal anatomic structures, such as incisive papillae, rugae, and retromolar pads rial, and the site and method of placement. Considerable emphasis must be placed on a sound clinical understanding that some of the alloplastic materials can migrate or be dis-placed under occlusal loads if not appropriately supported by underlying bone and contained by buttressing soft tissues. Careful clinical judgment with sound surgical and prosthetic principles must be exercised. A B C Figure 13-8 A, Implant bar and natural tooth copings used to support and retain this maxillary prosthesis. B, Tissue side of prosthesis showing the implant bar space, which when fitted will derive both support and stability from the implants while reten-tion is gained through resilient O-rings on the natural tooth copings. C, Maxillary prosthesis seated and in occlusion. (Cour-tesy Dr. N. Van Roekel, Monterey, CA.) C A B Figure 13-9 A, An anterior implant-supported bar demon-strating excellent access for hygiene and a parallel relationship to opposing occlusion. B, Prosthesis with implant bar space (housing three retentive male components for retention and a flat surface for bar contact and support) and bilateral posterior embrasure clasps. C, Prosthesis seated and in occlusion. (Cour-tesy Dr. N. Van Roekel, Monterey, CA.) 192 Part II Clinical and Laboratory C A B Figure 13-10 A, A Class II, modification 1 maxillary arch with a posterior implant at the distal location of the extension base. B, Maxillary gold framework with broad palatal coverage, maximum stabilization through palatal contacts of multiple maxillary teeth, and implant position at the distal extension base. A single implant should be protected from excessive occlusal forces; consequently the broad palatal coverage and maximum bracing are important features of the overall design. The ball attachment abutment was used for retentive purposes. C, Occlusal view of the prosthesis with implant (see A), which provides improved retention to the distal exten-sion base. (Courtesy Dr. James Taylor, Ottawa, Ontario.) A B Figure 13-11 A, Inflamed and distorted denture bearing mucosa due to an ill-fitting prosthesis that is worn 24 hours a day. B, After the tissue abuse is treated via modification of the denture base with a tissue conditioning resilient liner material, the prosthesis is removed for portions of the day, and the abused tissue is massaged, the denture bearing foundation is healthy again. 193 Chapter 13 Preparation of the Mouth for Removable Partial Dentures 3. A burning sensation in residual ridge areas, the tongue, and the cheeks and lips These conditions are usually associated with ill-fitting or poorly occluding removable partial dentures. However, nutritional deficiencies, endocrine imbalances, severe health problems (diabetes or blood dyscrasias), and bruxism must be considered in a differential diagnosis. If the use of a new removable partial denture or the relining of a present denture is attempted without first correcting these conditions, the chances for successful treatment will be compromised because the same old prob-lems will be perpetuated. The patient must be made to realize that fabrication of a new prosthesis should be delayed until the oral tissues can be returned to a healthy state. If there are unresolved systemic problems, removable partial denture treatment will usually result in failure or limited success. The first treatment procedure should consist of immedi-ate institution of a good home care program. A suggested home care program includes rinsing the mouth three times a day with a prescribed saline solution; massaging the residual ridge areas, palate, and tongue with a soft tooth-brush; removing the prosthesis at night; and using a prescribed therapeutic multiple vitamin along with a prescribed high-protein, low-carbohydrate diet. Some inflammatory oral conditions caused by ill-fitting dentures can be resolved by removing the dentures for extended periods. However, few patients are willing to undergo such inconveniences. Use of Tissue Conditioning Materials The tissue conditioning materials are elastopolymers that continue to flow for an extended period, permitting dis-torted tissues to rebound and assume their normal form. These soft materials apparently have a massaging effect on irritated mucosa, and because they are soft, occlusal forces are probably more evenly distributed. A B Figure 13-12 A, Mandibular removable partial denture with underextended bases, which contributed to tissue irritation. B, Denture bases properly extended to enhance support, stability, and retention. Maximum benefit from using tissue conditioning materi-als may be obtained by (1) eliminating deflective or interfer-ing occlusal contacts of old dentures (by remounting in an articulator if necessary); (2) extending denture bases to proper form to enhance support, retention, and stability (Figure 13-12); (3) relieving the tissue side of denture bases sufficiently (2 mm) to provide space for even thickness and distribution of conditioning material; (4) applying the mate-rial in amounts sufficient to provide support and a cushion-ing effect (Figure 13-13); and (5) following the manufacturer’s directions for manipulation and placement of the condition-ing material. The conditioning procedure should be repeated until the supporting tissues display an undistorted and healthy appearance. Many dentists find that intervals of 4 to 7 days between changes of the conditioning material are clinically acceptable. Improvement in irritated and distorted tissues is usually noted within a few visits, and in some patients a dramatic improvement will be seen. Usually three or four changes of the conditioning material are adequate, but in some instances additional changes are required. If positive results are not seen within 3 to 4 weeks, one should suspect more serious health problems and request a consultation from a physician. Figure 13-13 Tissue conditioning should be of sufficient thickness to be resilient and not place undue stress on the soft tissue. 194 Part II Clinical and Laboratory Complete periodontal charting that includes the record-ing of pocket depths, assessment of attachment levels, and recording of furcation involvements, mucogingival prob-lems, and tooth mobility should be performed. Determining the severity of periodontal disease should also include the use of appropriate radiographs. The dentist who is consider-ing removable partial denture fabrication must be certain that these criteria have been satisfied before continuing with impression procedures for the master cast. Periodontal Diagnosis and Treatment Planning Diagnosis The diagnosis of periodontal diseases is based on a system-atic and carefully accomplished examination of the peri-odontium. It follows the procurement of the health history of the patient and is performed with direct vision, palpation, a periodontal probe, a mouth mirror, and other auxiliary aids, such as curved explorers, furcation probes, diagnostic casts, and appropriate radiographs. In the examination procedure, nothing is as important as careful exploration of the gingival sulcus and recording of the probing pocket depth and sites that bleed on probing with a suitably designed periodontal probe. Under no cir-cumstances should removable partial denture fabrication begin without an accurate appraisal of sulcus/pocket depth and health. The probe is positioned as close to parallel to the long axis of the tooth as possible and is inserted gently between the gingival margin and the tooth surface, and the depth of the sulcus/pocket is determined circumferentially around each tooth. At least six probing depth readings are recorded on the patient’s chart for each tooth. Usually depths are recorded for the distobuccal, buccal, mesiobuccal, distolingual, lingual, and mesiolingual aspects of each tooth. Sulcular health can also be assessed by the presence or absence of bleeding upon probing. Dental radiographs can be used to supplement the clinical examination but should not be used as a substitute for it. A critical evaluation of the following factors should be made: (1) type, location, and severity of bone loss; (2) location, severity, and distribution of furcation involvements; (3) alterations of the periodontal ligament space; (4) alterations of the lamina dura; (5) the presence of calcified deposits; (6) the location and conformity of restorative margins; (7) eval-uation of crown and root morphologies; (8) root proximity; (9) caries; and (10) evaluation of other associated anatomic features, such as the mandibular canal or sinus proximity. This information serves to substantiate the impression gained from the clinical examination. Each tooth should be evaluated carefully for mobility. Unfortunately, there is no universally accepted standard for mobility. In general, mobility is graded according to the ease and extent of tooth movement. Normal mobility is in the order of 0.05 to 0.10 mm. Grade I mobility is present when less than 1 mm of movement occurs in a bucco-lingual direction; grade II is present when mobility in the bucco-lingual direction is between 1 and 2 mm; grade III is present Periodontal Preparation Periodontal preparation of the mouth usually follows any oral surgical procedure and is performed simultaneously with tissue conditioning procedures. Ordinarily, tooth extraction and removal of impacted teeth and retained roots or fragments are accomplished before definitive periodontal therapy is provided. However, it is strongly recommended that a gross debridement be performed before tooth extrac-tion when patients present with significant calculus accumu-lation. This helps limit the possibility of accidentally dislodging a piece of calculus into the extraction socket, which could lead to an infection. Elimination of exostoses, tori, hyperplastic tissue, muscle attachments, and frena, on the other hand, can be incorporated with periodontal surgi-cal techniques. In any situation, periodontal therapy should be completed before restorative dentistry procedures are begun for any dental patient. This is particularly true when a removable partial denture is contemplated because the ultimate success of this restoration depends directly on the health and integrity of the supporting structures of the remaining teeth. The periodontal health of the remaining teeth, especially those to be used as abutments, must be evaluated carefully by the dentist and corrective measures instituted before a removable partial denture is fabricated. It has been demonstrated that following periodontal therapy and with a good recall and oral hygiene program, properly designed removable partial dentures will not adversely affect the progression of periodontal disease or carious lesions. This discussion attempts to demonstrate how periodontal procedures affect diagnosis and treatment planning in a removable partial denture service rather than how the pro-cedures are actually accomplished. For technical details, the reader is referred to any of several excellent textbooks on periodontics. Objectives of Periodontal Therapy The objective of periodontal therapy is the return to health of supporting structures of the teeth, creating an environ-ment in which the periodontium may be maintained. The specific criteria for satisfying this objective are as follows: 1. Removal and control of all etiologic factors contributing to periodontal disease along with reduction or elimina-tion of bleeding on probing 2. Elimination of, or reduction in, the pocket depth of all pockets with the establishment of healthy gingival sulci whenever possible 3. Establishment of functional atraumatic occlusal relation-ships and tooth stability 4. Development of a personalized plaque control program and a definitive maintenance schedule Edited by Vanchit John, DDS, Associate Professor and Chair, Depart-ment of Periodontics and Allied Dental Programs, Indiana University School of Dentistry, Indianapolis, Indiana. 195 Chapter 13 Preparation of the Mouth for Removable Partial Dentures patient’s acceptance and compliance with the prescribed procedure, as evidenced by improved oral hygiene, will provide the dentist with a valuable means of evaluating that patient’s interest and the long-term prognosis of treatment. For the oral hygiene routine to be successful, the patient must be convinced to follow the prescribed procedure regu-larly and conscientiously. The most effective motivation techniques require a good understanding by the patient of his/her periodontal condition. Only then can the benefits of routine treatment become evident. Hence, an explanation of dental/periodontal disease, including its causes, initiation, and progression, is an important component of oral hygiene instruction. After this discussion, the patient should be instructed on the use of disclosing wafers/tablets, a soft/ medium-bristle toothbrush, and unwaxed/waxed dental floss. At subsequent appointments, oral hygiene can be eval-uated carefully, and other oral hygiene aids such as an inter-dental and or sulcular brushes can be incorporated as needed. Further treatment should be withheld until a satis-factory level of plaque control has been achieved. This is a particularly critical point for the patient who requires exten-sive restorative dentistry or a removable partial denture. Without good oral hygiene, any dental procedure, regardless of how well it is performed, is ultimately doomed to failure. The informed dentist insists that acceptable oral hygiene is demonstrated and maintained before embarking on an extensive restorative dentistry treatment plan. Scaling and Root Planing One of the most important services rendered to the patient is the removal of calculus and plaque deposits from the coronal and root surfaces of the teeth. Careful scaling and root planing are fundamental to the reestablishment of peri-odontal health. Without meticulous removal of calculus, plaque, and toxic material in the cementum, other forms of periodontal therapy cannot be successful. The use of ultrasonic instrumentation for calculus removal followed by root planing with sharp periodontal curettes is recommended. The curette is designed specifically for root planing and, when used correctly in combination with ultrasonic instrumentation, results in calculus removal and root surface decontamination. Thorough scaling and root planing should precede definitive surgical periodontal procedures that may be indicated before removable partial denture fabrication. Elimination of Local Irritating Factors Other Than Calculus Overhanging restoration margins and open contacts that allow food impaction should be corrected before definitive prosthetic treatment is begun. Although periodontal health predisposes to a much better environment for restorative procedures, it is not always possible or prudent to delay all restorative procedures until complete periodontal therapy and healing have occurred. This is especially true for patients when greater than 2 mm of mobility occurs in the bucco-lingual direction and/or the tooth is vertically depressible. Tooth mobility is an indication of the condition of the supporting structures, namely, the periodontium, and usually is caused by inflammatory changes in the periodon-tal ligament, traumatic occlusion, loss of attachment, or a combination of the three factors. The degree of mobility present, coupled with a determination of the causative factors responsible, provides additional information that is invaluable in planning for the removable partial denture. If the causative factor can be removed, many grade I and grade II mobile teeth can become stable and may be used success-fully to help support, stabilize, and retain the removable partial denture. Mobility in itself is not an indication for extraction unless the mobile tooth cannot aid in support or stability of the removable partial denture, or mobility cannot be reduced. (Grade III usually cannot be reversed and will not provide support or stability.) Treatment Planning Depending on the extent and severity of the periodontal changes present, a variety of therapeutic procedures ranging from simple to relatively complex may be indicated. As was the situation with the previously discussed oral surgical pro-cedures, it is the responsibility of the dentist rendering the removable partial denture treatment to see that the required periodontal care is accomplished for the patient. Periodontal treatment planning can usually be divided into three phases. The first phase is considered disease control or initial therapy because the objective is to essentially eliminate or reduce local causative factors before any periodontal surgical pro-cedures are accomplished. The procedures that are accom-plished as part of the initial preparation phase include oral hygiene instruction, scaling, and root planing and polishing, as well as endodontics, occlusal adjustment, and temporary splinting, if indicated. In many instances, carefully per-formed scaling and root planing combined with excellent patient compliance may negate the need for periodontal surgery. During the second, or periodontal, surgical phase, any needed periodontal surgery, such as free gingival grafts, osseous grafts, or pocket reduction, is accomplished. It is advisable to discuss the possible need for these treatment procedures with the patient at the initial examination appointment or during the initial phase of therapy, as this will likely involve referral of the patient to a periodontist. The maintenance of periodontal health is accomplished in phase 3 and is always ongoing. A definitive recall schedule should be established with the patient and is usually kept at 3- to 4-month intervals. Initial Disease Control Therapy (Phase 1) Oral Hygiene Instruction Ordinarily, dental treatment should be introduced to the patient through instruction provided in a carefully devised oral hygiene regimen. The cooperation witnessed by the 196 Part II Clinical and Laboratory eccentric functional relations. Occlusion can be coordinated only by selective spot grinding. Ground tooth surfaces should be subsequently smoothed and polished. 1. A static coordinated occlusal contact of the maximum number of teeth (maximum intercuspal position) when the mandible is in centric relation to the maxilla should be the first objective. a. A prematurely contacting cusp should be reduced only if the cusp point is in premature contact in both centric and eccentric relations. If a cusp point is in premature contact in centric relation only, the oppos-ing sulcus should be deepened. b. When anterior teeth are in premature contact in centric relation, or in both centric and eccentric rela-tions, corrections should be made by grinding the incisal edges of the mandibular teeth. If premature contact occurs only in the eccentric relation, correc-tion must be made by grinding the lingual inclines of the maxillary teeth. c. Usually, premature contacts in centric relation are relieved by grinding the buccal cusps of mandibular teeth, the lingual cusps of maxillary teeth, and the incisal edges of mandibular anterior teeth. Deepening the fossa of a posterior tooth or the lingual contact area in the centric relation of a maxillary anterior tooth changes and increases the steepness of the eccen-tric guiding inclines of the tooth; although this relieves trauma in centric relation, it may predispose the tooth to trauma in eccentric relations. 2. After establishing a static, even distribution of stress over the maximum number of teeth in centric relation, we are ready to evaluate opposing tooth contact or lack of contact in eccentric functional relations. Our attention is directed first to balancing side contacts. In extreme cases of pathologic balancing contacts, relief may be needed even before corrective procedures in centric relation are performed. Where balancing contacts exist, it is extremely difficult to differentiate the harmless from the destructive because we cannot visualize the influence of these fulcrum contacts on the functional movements of the condyle in the articular fossa. Subluxation, pain, lack of normal functional movement of the joint, or loss of alveolar support of the teeth involved may be evidence of exces-sive balancing contacts. Balancing side contacts receive less frictional wear than working side contacts, and pre-mature contacts may develop progressively with wear. A reduction in the steepness of guiding tooth inclines on the working side will increase the proximity of the teeth on the balancing side and may contribute to destructive prematurities. In all corrective grinding to relieve prema-ture or excessive contacts in eccentric relations, care must be exercised to avoid the loss of a static supporting contact in centric relation. This static support in centric relation may exist with the mandibular buccal cusp fitting into the central fossa of the maxillary tooth or with the maxillary lingual cusp fitting into the central fossa of the mandibu-with severe carious lesions in which pulpal involvement is likely. Excavation of these areas and placement of adequate restorations must be incorporated early in treatment. The placement of temporary or treatment fillings must not, in itself, become a local causative factor. Elimination of Gross Occlusal Interferences Bacterial plaque accumulations and calculus deposits are the primary factors involved in the initiation and progression of inflammatory periodontal disease. However, poor restor-ative dentistry can contribute to damage of the periodon-tium, and poor occlusal relationships may act as another factor that contributes to more rapid loss of periodontal attachment. Although occlusal interferences may be elimi-nated through a variety of techniques, at this stage of treat-ment, selective grinding is the procedure generally applied. Particular attention is directed to the occlusal relationships of mobile teeth. Traumatic cuspal interferences are removed by a selective grinding procedure. An attempt is made to establish a positive planned intercuspal position that coin-cides with centric relation. Deflective contacts in the centric path of closure are removed, eliminating mandibular dis-placement from the closing pattern. After this, the relation-ship of the teeth in various excursive movements of the mandible is observed, with special attention to cuspal contact, wear, mobility, and roentgenographic changes in the periodontium. The presence of working and nonwork-ing interferences should be evaluated; if present, they should be removed. The mere presence of occlusal abnormalities, in the absence of demonstrable pathologic change associated with the occlusion, does not necessarily constitute an indication for selective grinding. The indication for occlusal adjustment is based on the presence of a pathologic condition rather than on a preconceived articulation pattern. In the natural dentition, attempts to create bilateral balance, in the pros-thetic sense, have no place in the occlusal adjustment pro-cedure. Bilateral balanced occlusion not only is difficult to obtain in a natural dentition but also is apparently unneces-sary in view of its absence in most normal healthy mouths. Occlusion on natural teeth needs to be perfected only to the point at which cuspal interference within the patient’s func-tional range of contact is eliminated and normal physiologic function can occur. Guide to Occlusal Adjustment Schuyler has provided the following guide to occlusal adjust-ment by selective grinding: In the study or evaluation of occlusal disharmony of the natural dentition, accurately mounted diagnostic casts are extremely helpful, if not essential, in determining static cusp-to-fossa contacts of opposing teeth and as a guide in the correction of occlusal anomalies in both centric and Courtesy Dr. C.H. Schuyler, Montclair, NJ. 197 Chapter 13 Preparation of the Mouth for Removable Partial Dentures ations, however, the teeth must be stabilized because of loss of supporting structure from the periodontal process. Teeth may be immobilized during periodontal treatment by acid etching teeth with composite resin, with fiber-rein-forced resins, with cast removable splints, or with intracoro-nal attachments. The latter, an example of which is the A-splint, necessitates cutting tooth surfaces and embedding a ridge connector between adjacent teeth. After periodontal treatment is performed, splinting may be accomplished with cast removable restorations or cast cemented restorations. The preferred form of permanent splinting uses two or more cast restorations soldered or cast together. They may be cemented with permanent (zinc oxy-phosphate or resin) cements or temporary (zinc oxide–euge-nol) cements. A properly designed removable partial denture can also stabilize mobile teeth if provision for such immo-bilization is planned as the denture is designed. Use of a Nightguard The removable acrylic-resin splint, originally designed as an aid in eliminating the deleterious effects of nocturnal clench-ing and grinding, has been used to advantage for the remov-able partial denture patient. The nightguard may prove helpful as a form of temporary splinting if worn at night when the removable partial denture has been removed. The flat occlusal surface prevents intercuspation of the teeth, which eliminates lateral occlusal forces (Figure 13-14). The nightguard is particularly useful before fabrication of a removable partial denture when one of the abutment teeth has been unopposed for an extended period. The periodon-tal ligament of a tooth without an antagonist undergoes changes characterized by loss of orientation of periodontal ligament fibers, loss of supporting bone, and narrowing of the periodontal ligament space. If such a tooth is suddenly returned to full function when it is carrying an increased burden, pain and prolonged sensitivity may result. However lar tooth, or it may exist in both situations. Although both the maxillary lingual cusp and the mandibular buccal cusp may sometimes have a static centric contact in the sulcus of the opposing tooth, often only one of these cusps has this static contact. In such instances, the con-tacting cusp must be left untouched to maintain this essential support in the planned intercuspal position, and all corrective grinding to relieve premature contacts in eccentric positions would be done on the opposing tooth inclines. The mandibular buccal cusp is in a static central contact in the maxillary sulcus more often than the max-illary lingual cusp is in a static contact in its opposing mandibular sulcus. Therefore corrective grinding to relieve premature balancing contacts is more often done on the maxillary lingual cusps. 3. To obtain maximum function and distribution of func-tional stress in eccentric positions on the working side, necessary grinding must be done on the lingual surfaces of the maxillary anterior teeth. Corrective grinding on the posterior teeth at this time should always be done on the buccal cusp of the maxillary premolars and molars and on the lingual cusp of the mandibular premolars and molars. The grinding of mandibular buccal cusps or max-illary lingual cusps at this time would rob these cusps of their static contact in the opposing central sulci in centric relation. 4. Corrective grinding to relieve premature protrusive con-tacts of one or more anterior teeth should be accom-plished by grinding the lingual surface of the maxillary anterior teeth. Anterior teeth should never be ground to bring the posterior teeth into contact in either protrusive position or on the balancing side. In the elimination of premature protrusive contacts of posterior teeth, neither the maxillary lingual cusps nor the mandibular buccal cusps should be ground. Corrective grinding should be done on the surfaces of the opposing teeth on which these cusps function in the eccentric position, leaving the centric contact undisturbed. 5. Any sharp edges left by grinding should be rounded off. Temporary Splinting Teeth that are mobile at the time of the initial examination frequently present a diagnostic problem for the dentist. The cause of the mobility must be determined and then a decision made for elimination of the causative factors. The response of these teeth to temporary immobilization fol-lowed by appropriate treatment may be helpful in establish-ing a prognosis for them and may lead to a rational decision as to whether they should be retained or sacrificed. Second-ary mobility resulting from the presence of an inflammatory lesion may be reversible if the disease process has not destroyed too much of the attachment apparatus. Primary mobility caused by occlusal interference also may disappear after selective grinding. In instances of angular types of osseous defects, one should consider guided tissue regenera-tion as a means of increasing attachment levels. In some situ-Figure 13-14 The removable acrylic-resin splint with a flat occlusal plane can be used effectively as a form of temporary stabilization and as a means of eliminating excessive lateral forces created by clenching and grinding habits. 198 Part II Clinical and Laboratory surgery based on the anatomy of defects following the removal of diseased granulation tissue. Osseous resection involves the use of both osteoplasty and ostectomy proce-dures. Osteoplasty refers to reshaping the bone without removing tooth-supporting bone; ostectomy includes the removal of tooth-supporting bone. Consequently, the flap is widely applied in the treatment of periodontal disease. Guided Tissue Regeneration. Guided tissue regenera-tion (GTR) has been defined as those procedures that attempt regeneration of lost periodontal structures through differing tissue responses. The rationale for GTR is based on the physiologic healing response of the tissues after peri-odontal surgery. After periodontal surgery, a race to repopu-late the root surface begins among the four tissue types of the periodontium, namely, epithelium, connective tissue, periodontal ligament (PDL), and bone. Epithelium, which migrates at a rate of 0.5 mm per day, typically migrates first along the root surface, preventing new attachment. There-fore to allow the undifferentiated mesenchymal cells from the PDL and the endosteum of bone to repopulate the root against surfaces, the epithelial cells and the gingival connec-tive tissue cells should be isolated. This isolation during initial healing enables periodontal structures to become reestablished and may lead to better long-term health of the tooth. The GTR procedure commonly involves the use of an osseous graft along with a resorbable membrane (Figure 13-15). This technique has the potential to lead to substan-tial improvement of the periodontal condition when used around carefully selected two- and three-walled osseous defects and mandibular furcation involvements. Periodontal Plastic Surgery. Periodontal plastic surgery, which was previously referred to as “mucogingival surgery,” is applied to those procedures used to resolve problems involving the interrelationship between the gingiva and the alveolar mucosa. Mucogingival surgery consists of plastic surgical procedures that are used for correction of gingiva– mucous membrane relationships that complicate periodon-tal disease and may interfere with the success of periodontal treatment. The objectives of periodontal plastic surgery are several and include elimination of pockets that transverse the mucogingival junction, creation of an adequate zone of attached gingiva, correction of gingival recession by root coverage techniques, relief of the pull of frena and muscle attachments on the gingival margin, and correction of defor-mities of edentulous ridges, done to permit access to the underlying alveolar process and correction of osseous defor-mities when there is sufficient or insufficient attached gingiva, to deepen a shallow vestibule, and to assist in orth-odontic therapy. Commonly used periodontal plastic surgi-cal procedures include lateral sliding flaps, free gingival grafts, pedicle grafts, coronally positioned grafts, double papilla flaps, semilunar coronally positioned flaps, subepi-thelial connective tissue grafts, and edentulous ridge aug-mentation using one of the above techniques. In addition, GTR has been used for periodontal plastic surgical if a nightguard is used to return some functional stimulation to the tooth, the periodontal ligament changes are reversed and an uneventful course can be experienced when the tooth is returned to full function. Minor Tooth Movement The increased use of orthodontic procedures in conjunction with restorative and prosthetic dentistry has contributed to the success of many restorations by altering the periodontal environment in which they are placed. Malposed teeth that were once doomed to extraction should be considered now for repositioning and retention. The additional stability pro-vided for a removable partial denture by uprighting a tilted or drifted tooth may mean much in terms of comfort to the patient. The techniques employed are not difficult to master, and the rewards in terms of a better restorative dentistry service are great. Definitive Periodontal Surgery (Phase 2) Periodontal Surgery After initial therapy is completed, the patient is reevaluated for the surgical phase. If oral hygiene is at an optimum level, yet pockets with inflammation and osseous defects are still present, a variety of periodontal surgical techniques should be considered to improve periodontal health. The proce-dures selected should have the potential to enhance the results obtained during Phase 1 therapy. Pocket reduction or elimination may be achieved by root planing when the cause of pocket depth is edema caused by gingival inflammation. Apically positioned flap surgery or occasionally a gingivectomy may be considered for reduc-tion of suprabony pockets. Osseous resection or regenera-tion using a flap approach is a form of surgical therapy that is commonly employed to help with treatment of the dis-eased periodontium. It must be noted that elimination of the inflammatory disease process and restoration of the peri-odontal attachment apparatus are the major objectives of periodontal therapy. Periodontal Flaps. Today, use of one of the various flap procedures is the surgical approach that offers the greatest versatility. Periodontal flap surgery involves the elevation of either mucosa alone or both the mucosa and the periosteum. Although there are several indications for flap elevation, the most important goal of flap elevation is to allow access to the bone and the root surfaces for complete instrumenta-tion. Other goals of the flap approach include access for pocket elimination, caries control, crown lengthening to allow for optimum restorative dental treatment, root ampu-tation or hemisection, as required and access to the furcation of the tooth. A decision is made before surgery is performed if the aim is resection of osseous tissue to allow for a more physiologic osseous anatomy and subsequently gingival contour, or to regenerate some of the lost periodontal attachment appara-tus. However, sometimes changes have to be made during 199 Chapter 13 Preparation of the Mouth for Removable Partial Dentures A B C D E Figure 13-15 Guided tissue regeneration (GTR) procedure performed to address a furcation involvement. A, Tooth #30 presented with a grade 2 furcation involvement with the probe entering 3 mm in a horizontal direction. A GTR procedure using a combination of a bone graft and a nonresorbable membrane was planned. B, Following hand and ultrasonic instrumentation, decalcified freeze-dried bone allograft was grafted around the furcation. C, A nonresorbable membrane was placed over the bone graft. D, The flap was then sutured with a nonresorbable expanded polytetraethylene suture. E, Two months following surgery, the membrane was removed. Note the presence of red rubbery tissue filling the previously exposed furcation site. This tissue has the potential to form osseous tissue and close the access to the furcation entrance. 200 Part II Clinical and Laboratory normal gingival contour at a stable position on the tooth surface. Thus the optimum position for gingival margins of individual restorations can be established with accuracy. The coronal contours of these restorations can also be developed in correct relationships to the gingival margin, ensuring the proper degree of protection and functional stimulation to gingival tissues. Third, the response of strategic but ques-tionable teeth to periodontal therapy provides an important opportunity for reevaluating their prognosis before the final decision is made to include (or exclude) them in the remov-able partial denture design. And last, the overall reaction of the patient to periodontal procedures provides the dentist with an excellent indication of the degree of cooperation to be expected in the future. Even in the absence of periodontal disease, certain peri-odontal procedures may be an invaluable aid in removable partial denture construction. Through periodontal surgical techniques, the environment of potential abutment teeth may be altered to the point of making an otherwise unac-ceptable tooth a most satisfactory retainer for a removable partial denture. Abutment Teeth Preparation Abutment Restorations Equipped with the diagnostic casts on which a tentative removable partial denture design has been drawn, the dentist is able to accomplish preparation of abutment teeth with accuracy. The information at hand should include the pro-posed path of placement, the areas of teeth to be altered and tooth contours to be changed, and the locations of rest seats and guiding planes (see Figure 12-5). During examination and subsequent treatment planning, in conjunction with a survey of diagnostic casts, each abut-procedures. Recently, use of the commercially available acel-lular dermal graft has gained popularity. However, the most commonly used procedure is the subepithelial connective tissue graft (Figure 13-16). These plastic surgical procedures should be considered whenever an abutment tooth lacks adequate attached kera-tinized gingiva and requires root coverage to facilitate removable partial denture construction and maintenance. Recall Maintenance (Phase 3) Several longitudinal studies have now demonstrated the increasing importance of maintenance for all patients who have undergone any periodontal therapy. This includes not only reinforcement of plaque control measures but also thorough debridement of all root surfaces of supragingival and subgingival calculus and plaque by the dentist or an auxiliary. The frequency of recall appointments should be custom-ized for the patient, depending on the susceptibility and severity of periodontal disease. It is now understood that patients with a history of moderate to severe periodontitis should be placed on a 3- to 4-month recall system to main-tain results achieved by nonsurgical and surgical therapy. Advantages of Periodontal Therapy Periodontal therapy done before a removable prosthesis is fabricated has several advantages. First, the elimination of periodontal disease removes a primary causative factor in tooth loss. The long-term success of dental treatment depends on the maintenance of the remaining oral struc-tures, and periodontal health is mandatory if further loss is to be avoided. Second, a periodontium free of disease pres-ents a much better environment for restorative correction. Elimination of periodontal pockets with the associated return of a physiologic architectural pattern establishes a A B Figure 13-16 Gingival recession addressed with subepithelial connective tissue graft procedure. A, The patient presents with evi-dence of severe gingival recession associated with teeth #6, 7, and 8. This was an esthetic problem. The patient also complained of hypersensitivity associated with these teeth. A subepithelial connective tissue graft was planned to help correct the gingival recession. B, Clinical appearance 6 months following treatment with a subepithelial connective tissue graft on teeth #6, 7, and 8. The patient was very satisfied with the postoperative appearance, and clinically the symptom of hypersensitivity was no longer significant. 201 Chapter 13 Preparation of the Mouth for Removable Partial Dentures ment tooth is considered individually as to what type of restoration is indicated. Abutment teeth presenting sound enamel surfaces in a mouth in which good oral hygiene habits are evident may be considered a fair risk for use as removable partial denture abutments. One should not be misled, however, by a patient’s promise to do better as far as oral hygiene habits are concerned. Good or bad oral hygiene is a habit of long standing and is not likely to be changed appreciably because a removable partial denture is being worn. Therefore one must be conservative in evaluating the oral hygiene habits of the patient in the future. Remember that clasps as such do not cause teeth to decay, and if the individual will keep the teeth and the removable partial denture clean, one need not condemn clasps from a cario-genic standpoint. On the other hand, more removable partial dentures have been condemned as cariogenic because the dentist did not provide for the protection of abutment teeth rather than because of inadequate care on the part of the patient. Esthetic veneer types of crowns should be used when a canine or premolar abutment is to be restored or protected. Less frequently, the molar will have to be treated in such a manner, and except for maxillary first molars, the full cast crown is usually acceptable. When there is proximal caries on abutment teeth with sound buccal and lingual enamel surfaces, in a mouth exhib-iting average oral hygiene and low caries activity, a gold inlay may be indicated. However, silver amalgam or composite for the restoration of those teeth with proximal caries should not be condemned, although one must admit that an inlay cast of a hard type of gold will provide the best possible support for occlusal rests, at the same time giving an estheti-cally pleasing restoration. However, an amalgam restoration, properly condensed, is capable of supporting an occlusal rest without appreciable flow over a long period. The most vulnerable area on the abutment tooth is the proximal gingival area, which lies beneath the minor con-nector of the removable partial denture framework and is therefore subject to accumulation of debris in an area most susceptible to caries. Even when the removable partial denture is removed, these areas are often missed by the toothbrush, which allows bacterial plaque and debris to remain for long periods. Because of this unique removable partial denture concern, special attention should be paid to these areas during patient education and follow-up. Even when a complete crown restoration is placed in this most vulnerable area, recurrent caries can occur. Caries risk is best managed through effective home care and professional fol-low-up procedures, rather than through the placement of restorations. All proximal abutment surfaces that are to serve as guiding planes for the removable partial denture should be prepared so that they will be made as nearly parallel as pos-sible to the path of placement. Preparations may include modifying the contour of existing ceramic restorations, if necessary. This may be accomplished with abrasive stones or diamond finishing stones. A polished surface for the altered ceramic restoration may be restored by using any of several polishing kits supplied by manufacturers. When preparing abutments that will receive surveyed crowns, it is important to plan for the tooth reduction neces-sary to allow placement of sufficient restorative material for durability, contour, and esthetics, as well as the contours prescribed for the desired clasp assembly (Figure 13-17). This can be accomplished by first modifying the axial con-tours of the abutments to those required of the completed crown, then starting controlled tooth reduction (prepara-tion) to accommodate the thickness of the materials for durability, contour, and esthetics. This ensures that the wax patterns and resultant crowns can be restored to the desired form. Contouring Wax Patterns Modern indirect techniques permit the contouring of wax patterns on the master cast with the aid of the surveyor blade. All abutment teeth to be restored with castings can be prepared at one time and an impression made that will provide an accurate stone replica of the prepared arch. Wax patterns may then be refined on separated individual dies or removable dies. All abutment surfaces facing edentulous areas should be made parallel to the path of placement by the use of the surveyor blade (Figure 13-18). This technique will provide proximal surfaces that will be parallel without any further alteration in the mouth, will permit the most positive seating of the removable partial denture along the path of placement, and will provide the least amount of undesirable space beneath minor connectors for the lodge-ment of debris. Rest Seats After the proximal surfaces of the wax patterns have been made parallel, and buccal and lingual contours have been established to satisfy the requirements of stability and reten-tion with the best possible esthetic placement of clasp arms, the occlusal rest seats should be prepared in the wax pattern rather than in the finished restoration. The placement of occlusal rests should be considered at the time the teeth are prepared to receive cast restorations, so there will be suffi-cient clearance beneath the floor of the occlusal rest seat. Too many times, a completed cast restoration is cemented in the mouth for a removable partial denture abutment without any provision for the occlusal rest having been made in the wax pattern. The dentist then proceeds to prepare an occlusal rest seat in the cast restoration, while ever conscious of the fact that he or she may perforate the casting during the process of forming the rest seat. The unfortunate result is usually a poorly formed rest seat that is too shallow. If tooth structure has been removed to provide placement of the occlusal rest seat, it may be ideally placed in the wax pattern by using a No. 8 round bur to lower the marginal ridge and establish the outline form of the rest and then using a No. 6 round bur to slightly deepen the floor of the 202 Part II Clinical and Laboratory abutment tooth. A ball-and-socket type of relationship between occlusal rest and abutment tooth is the most desir-able. At the same time, the marginal ridge must be lowered so that the angle formed by the occlusal rest and the minor connector will stand above the occlusal surface of the abut-ment tooth as little as possible and avoid interference with the opposing teeth. Simultaneously, sufficient bulk must be provided to prevent weakness in the occlusal rest at the marginal ridge. The marginal ridge must be lowered and yet not be the deepest part of the rest preparation. To permit occlusal stresses to be directed toward the center of the abut-ment tooth, the angle formed by the floor of the occlusal rest with the minor connector should be less than 90 degrees. In other words, the floor of the occlusal rest should incline slightly from the lowered marginal ridge toward the center of the tooth. This proper form can be readily accomplished in the wax pattern, if care is taken during crown or inlay prepara-rest seat inside this lowered marginal ridge. This approach provides an occlusal rest that best satisfies the requirements that it be placed so that any occlusal force will be directed axially and that there will be the least possible interference to occlusion with the opposing teeth. Perhaps the most important function of a rest is the divi-sion of stress loads from the removable partial denture to provide the greatest efficiency with the least damaging effect to the supporting abutment teeth. For a distal extension removable partial denture, the rest must be able to transmit occlusal forces to the abutment teeth in a vertical direction only, thereby permitting the least possible lateral stresses to be transmitted to the abutment teeth. For this reason, the floor of the rest seat should incline toward the center of the tooth so that the occlusal forces, insofar as possible, are centered over the root apex. Any other form but that of a spoon shape can permit locking of the occlusal rest and the transmission of tipping forces to the A B C D Figure 13-17 A, Diagnostic cast at an orientation best for all abutments considered. The buccal survey line is too close to the mar-ginal gingival and the distal surface does not lend itself to guide-plane preparation. A surveyed crown is indicated. B, Abutment contours appropriate to clasp design (distal guide plane and mid-buccal 0.01 inch undercut) are produced in wax. C, Cast of abutment prepara-tion provides buccal surface reduction adequate to replace with metal ceramic material at the required contour. Without careful con-sideration of survey line placement needs before and during preparation, it is easy to reproduce incorrect contours in finished crowns. D, Cast of a seated surveyed crown demonstrates desired contours for the clasp design chosen. 203 Chapter 13 Preparation of the Mouth for Removable Partial Dentures preparation of proximal tooth surfaces should be done first because if the occlusal portion of the rest seat is placed first and the proximal tooth surface is altered later, the outline form of the rest seat is sometimes irreparably altered. Following proximal surface recontouring (guided plane preparation), the larger round bur is used to lower the mar-ginal ridge 1.5 to 2.0 mm while at the same time creating the relative outline form of the rest seat. The result is a rest seat preparation with marginal ridge lowered and gross outline form established but without sufficient deepening of the rest seat preparation toward the center of the tooth. A smaller tion to provide the location of the rest. If direct restorations are used, sufficient bulk must be present in this area to allow proper occlusal rest seat form without weakening the restoration. There is insufficient evidence to show that direct restorations used as rest seats perform equally to enamel. When the rest seat is placed in sound enamel, this is best accomplished by the use of round carbide burs (No. 4, 6, and 8 sizes) that leave a smooth enamel surface. Rest seat preparations in sound enamel (or in existing restorations that are not to be replaced) should always follow the recontouring of proximal tooth surfaces. The A B C D E Figure 13-18 A, Occlusal view of full contour wax patterns, which will be splinted between crowns and across the midline with a 13-gauge splint bar. Rests are evident on the lingual surfaces of abutment wax patterns. B, Wax patterns showing labial cut-back for porcelain. Bilateral guide-plane surfaces will be reproduced in metal and are parallel to the path of insertion. C, An abutment veneered crown with an appropriate height of contour and a 0.02-inch under-cut for the anticipated wrought-wire retainer. D, Completed pros-thesis splinted between retainer crowns and across the midline. Splint bar with added vertical support provides indirect retention. E, Prosthesis inserted intraorally. 204 Part II Clinical and Laboratory The success or failure of a removable partial denture depends on how well the mouth preparations were accom-plished. It is only through intelligent planning and compe-tent execution of mouth preparations that the denture can satisfactorily restore lost dental functions and contribute to the health of the remaining oral tissues. round bur (a No. 4 or 6) may then be used to deepen the floor of the rest seat to a gradual incline toward the center of the tooth. Enamel rods are then smoothed by the planing action of a round bur revolving with little pressure. Abrasive rubber points are sufficient to complete the polishing of the rest seat preparation. 205 Chapter 14 Preparation of Abutment Teeth 205 CHAPTER 14 Preparation of Abutment Teeth Chapter Outline Classification of Abutment Teeth Sequence of Abutment Preparations on Sound Enamel or Existing Restorations Abutment Preparations Using Conservative Restorations Abutment Preparations Using Crowns Ledges on abutment crowns Spark erosion Veneer crowns for support of clasp arms Splinting of Abutment Teeth Use of Isolated Teeth as Abutments Missing Anterior Teeth Temporary Crowns When a Removable Partial Denture Is Being Worn Cementation of temporary crowns Fabricating Restorations to Fit Existing Denture Retainers After surgery, periodontal treatment, endodontic treatment, and tissue conditioning of the arch involved, the abutment teeth may be prepared to provide support, stabilization, reciprocation, and retention for the removable partial denture. Rarely, if ever, is the situation encountered in which alterations of the abutment are not indicated because teeth do not develop with guiding planes, rests, and contours to accommodate clasp assemblies. A favorable response to any deep restorations, endodon-tic therapy, and the results of periodontal treatment should be established before the removable partial denture is fabricated. If the prognosis of a tooth under treatment becomes unfavorable, its loss can be compensated for by a change in the removable partial denture design. If teeth are lost after the removable partial denture is fabricated, then the removable partial denture must be added to or replaced. Most removable partial denture designs do not lend them-selves well to later additions, although this possibility should be considered in the original design of a denture. Every diagnostic aid should be used to determine which teeth are to be used as abutments or are potential abutments for future designs. When an original abutment is lost, it is extremely difficult to effectively modify the removable partial denture to use the next adjacent tooth as a retaining unit. It is sometimes possible to design a removable partial denture so that a single posterior abutment that is question-able can be retained and used to support one end of a tooth-supported base. Then, if that posterior abutment was lost, it could be replaced with a distal extension base (see Figure 12-25). Such a design must include provision for future indirect retention, flexible clasping on the remaining termi-nal abutment, and provision for establishing tissue support by a secondary impression. Anterior abutments, which are considered poor risks, may not be so freely used because of the problems involved in adding a new abutment retainer when the original one is lost. Such questionable teeth should be treatment planned for extraction in favor of a better abut-ment in the original treatment plan. 206 Part II Clinical and Laboratory the occlusal surface, preferably at the junction of the middle and gingival thirds; (b) retentive clasp terminals may be placed in the gingival third of the crown for better esthetics and better mechanical advantage; and (c) recip-rocal clasp arms may be placed on and above a height of contour that is no higher than the cervical portion of the middle third of the crown of the abutment tooth. 3. After alterations of axial contours are accomplished and before rest seat preparations are instituted, an impression of the arch should be made in irreversible hydrocolloid and a cast formed in a fast-setting stone. This cast can be returned to the surveyor to determine the adequacy of axial alterations before proceeding with rest seat prepara-tions. If axial surfaces require additional axial recontour-ing, this can be performed during the same appointment and without compromise. 4. Occlusal rest areas should be prepared that will direct occlusal forces along the long axis of the abutment tooth (Figure 14-1D). Mouth preparation should follow the removable partial denture design that was outlined on the diagnostic cast at the time the cast was surveyed and the treatment plan confirmed. Proposed changes to abut-ment teeth should be made on the diagnostic cast and outlined in colored pencil to indicate the area, amount, and angulation of the modification to be done (see Chapter 12). Although occlusal rest seats may also be prepared on the diagnostic cast, indication of their loca-tion in colored pencil is usually sufficient for the experi-enced dentist because rest preparations follow a definite pattern (see Chapter 6). Abutment Preparations Using Conservative Restorations Conventional inlay preparations are permissible on the proximal surface of a tooth not to be contacted by a minor connector of the removable partial denture. On the other hand, proximal and occlusal surfaces that support minor connectors and occlusal rests require somewhat different treatment. The extent of occlusal coverage (i.e., whether cusps are covered) will be governed by the usual factors, such as the extent of caries, the presence of unsupported enamel walls, and the extent of occlusal abrasion and attrition. When an inlay is the restoration of choice for an abut-ment tooth, certain modifications of the outline form are necessary. To prevent the buccal and lingual proximal margins from lying at or near the minor connector or the occlusal rest, these margins must be extended well beyond the line angles of the tooth. This additional extension may be accomplished by widening the conventional box prepara-tion. However, the margin of a cast restoration produced for such a preparation may be quite thin and may be damaged by the clasp during placement or removal of the removable partial denture. This hazard may be avoided by extending the outline of the box beyond the line angle, thus producing a strong restoration-to-tooth junction. Classification of Abutment Teeth The subject of abutment preparations may be grouped as follows: (1) those abutment teeth that require only minor modifications to their coronal portions, (2) those that are to have restorations other than complete coverage crowns, and (3) those that are to have crowns (complete coverage). Abutment teeth that require only minor modifications include teeth with sound enamel, those with small restora-tions not involved in the removable partial denture design, those with acceptable restorations that will be involved in the removable partial denture design, and those that have existing crown restorations requiring minor modification that will not jeopardize the integrity of the crown. The latter may exist as an individual crown or as the abutment of a fixed partial denture. The use of unprotected abutments has been discussed previously. Although complete coverage of all abutments may be desirable, it is not always possible or practical. The decision to use unprotected abutments involves certain risks of which the patient must be advised and includes responsi-bility for maintaining oral hygiene and caries control. Making crown restorations fit existing denture clasps is a difficult task; however, the fact that it is possible to do may influence the decision to use uncrowned but otherwise sound teeth as abutments. Complete coverage restorations provide the best possible support for occlusal rests. If the patient’s economic status or other factors beyond the control of the dentist prevent the use of complete coverage restorations, then an amalgam alloy restoration, if properly condensed, is capable of sup-porting an occlusal rest without appreciable flow for a long period. Any existing silver amalgam alloy restoration about which there is any doubt should be replaced with new amalgam restorations. This should be done before guiding planes and occlusal rest seats are prepared, to allow the res-toration to reach maximum strength and be polished. Continued improvement in dimensional stability, strength, and wear resistance of composite resin restorations will add another dimension to the preparation and modifi-cation of abutment teeth for removable partial dentures that should be less invasive than placement of complete coverage restorations and more economical. Sequence of Abutment Preparations on Sound Enamel or Existing Restorations Abutment preparations on sound enamel or on existing res-torations that have been judged as acceptable should be done in the following order: 1. Proximal surfaces parallel to the path of placement should be prepared to provide guiding planes (Figure 14-1A). 2. Tooth contours should be modified (Figure 14-1B and C), lowering the height of contour so that (a) the origin of circumferential clasp arms may be placed well below 207 Chapter 14 Preparation of Abutment Teeth Every effort should be made to provide the restoration with maximum resistance and retention, as well as with clinically imperceptible margins. The first requisite can be satisfied by preparing opposing cavity walls 5 degrees or less from paral-lel and producing flat floors and sharp, clean line angles. It is sometimes necessary to use an inlay on a mandibular first premolar for the support of an indirect retainer. The narrow occlusal width bucco-lingually and the lingual incli-nation of the occlusal surface of such a tooth often compli-cate the two-surface inlay preparation. Even the most In this type of preparation, the pulp is particularly vulner-able unless the axial wall is curved to conform to the external proximal curvature of the tooth. When caries is of minimal depth, the gingival seat should have an axial depth at all points about the width of a No. 559 fissure bur. It is of utmost importance that the gingival seat be placed where it can be easily accessed to maintain good oral hygiene. The proximal contour necessary to produce the proper guiding plane surface and the close proximity of the minor connec-tor render this area particularly vulnerable to future caries. A B C D Figure 14-1 Abutment contours should be altered during mouth preparations in the following sequence. A, The proximal surface is prepared parallel to the path of placement to create a guiding plane. B, Height of contour on the buccal and lingual surfaces is lowered when necessary to permit the retentive clasp terminus to be located within the gingival third of the crown, bracing part of the retentive arm at the junction of the middle and gingival thirds of the crown, and the reciprocal clasp arm on the opposite side of tooth to be placed no higher than the cervical portion of the middle third of the crown. C, The area of the tooth at which the retentive clasp arm originates should be altered if necessary to permit a more direct approach to the gingival third of the tooth: (1) incorrect position of retentive clasp arm; (2) area of tooth modified to accommodate better position of retentive clasp arm; (3) more ideal position of reten-tive clasp arm. D, Occlusal rest preparation that will direct occlusal forces along the long axis of the tooth should be the final step in mouth preparations. 208 Part II Clinical and Laboratory surveyor for proximal surface refinement. This can be done accurately with the aid of a handpiece holder attached to the vertical spindle of the surveyor or some similar machining device. One of the advantages of making cast restorations for abutment teeth is that mouth preparations that would oth-erwise have to be done in the mouth may be done on the surveyor with far greater accuracy. It is generally impossible to make several proximal surfaces parallel to one another when preparing them intraorally. The opportunity for con-touring wax patterns and making them parallel on the sur-veyor in relation to a path of placement should be used to its full advantage whenever cast restorations are being made. The ideal crown restoration for a removable partial denture abutment is the complete coverage crown, which can be carved, cast, and finished to ideally satisfy all require-ments for support, stabilization, and retention without com-promise for cosmetic reasons (Figure 14-3). Porcelain veneer crowns can be made equally satisfactory but only by the added step of contouring the veneered surface on the sur-veyor before the final glaze. If this is not done, retentive contours may be excessive or inadequate. The three-quarter crown does not permit creation of retentive areas as does the complete coverage crown. However, if buccal or labial surfaces are sound and retentive areas are acceptable or can be made so by slight modification of tooth surfaces, the three-quarter crown is a conservative restoration of merit. The same criteria apply in the decision to leave a portion of an abutment unprotected, as in the decision to leave any tooth unprotected that is to serve as a removable partial denture abutment. Regardless of the type of crown used, preparation should be made to provide the appropriate depth for the occlusal rest seat. This is best accomplished by altering the axial con-tours of the tooth to the ideal before preparing the tooth and exacting occlusal cavity preparation often leaves a thin and weak lingual cusp remaining. Abutment Preparations Using Crowns When multiple crowns are to be restored as removable partial denture abutments, it is best that all wax patterns be made at the same time. A cast of the arch with removable dies may be used if they are stable and sufficiently keyed for accuracy. If preferred, contouring wax patterns and making them parallel may be done on a solid cast of the arch (Figure 14-2), with individual dies used to refine margins. Modern impression materials and indirect techniques make either method equally satisfactory. The same sequence for preparing teeth in the mouth applies to the contouring of wax patterns. After the cast has been placed on the surveyor to conform to the selected path of placement and after the wax patterns have been prelimi-narily carved for occlusion and contact, proximal surfaces that are to act as guiding planes are carved parallel to the path of placement with a surveyor blade. Guiding planes are extended from the marginal ridge to the junction of the middle and gingival thirds of the tooth surface involved. One must be careful not to extend the guiding plane to the gin-gival margin because the minor connector must be relieved when it crosses the gingivae. A guiding plane that includes the occlusal two thirds or even one third of the proximal area is usually adequate without endangering gingival tissues. After the guiding planes are parallel and any other con-touring to accommodate the removable partial denture design is accomplished, occlusal rest seats are carved in the wax pattern. This method has been outlined in Chapter 6. It should be emphasized that critical areas prepared in wax should not be destroyed by careless spruing or polish-ing. The wax pattern should be sprued to preserve paralleled surfaces and rest areas. Polishing should consist of little more than burnishing. Rest seat areas should need only refining with round finishing burs. If some interference by spruing is unavoidable, the casting must be returned to the Figure 14-2 Solid cast of multiple abutment crowns for a removable partial denture. Wax patterns for crown #21, #28, #30, and #31 can be completed at the same time using the identi-cal cast orientation. This allows control of the path of insertion features on all fitting surfaces of the removable prostheses. Figure 14-3 Metal ceramic crowns for teeth #4 and #5 dem-onstrating occlusal rests in metal and evidence of palatal finish-ing procedures. The distal surface of #4 provides a guide-plane surface that is continued onto a portion of the lingual surface for maximum stabilization. 209 Chapter 14 Preparation of Abutment Teeth creating a depression in the prepared tooth at the occlusal rest area (Figure 14-4). Because the location of occlusal rests is established during treatment planning, this information will be known in advance of any tooth preparations. If, for example, double occlusal rests are to be used, this will be known so that the tooth can be prepared to accommodate the depth of both rests. It is inexcusable when waxing a pattern to find that a rest seat has to be made shallower than is desirable because of post-treatment planning. It can also create serious problems when a rest seat has to be made shallow in an existing crown or inlay because its thickness is not known. The opportunity for creating an ideal rest seat (if it has been properly treatment planned) depends only on the few seconds it takes to create a space for it. Figure 14-4 Metal-ceramic crown preparation on tooth #21 shows mesial-occlusal (MO) rest space provided in the crown preparation at the mesial. Inset picture gives a perspective of the vertical height this provides for the rest to be prepared in the wax pattern. Path of placement Height of contour Height of contour A B Figure 14-5 A, Incorrect relationship of retentive and recipro-cal clasp arms to each other when the removable partial denture framework is fully seated. As the retentive clasp arm flexes over the height of contour during placement and removal, the recipro-cal clasp arm cannot be effective because it is not in contact with the tooth until the denture framework is fully seated. B, Horizon-tal forces applied to the abutment tooth as the retentive clasp flexes over the height of contour during placement and removal. Open circles at the top and bottom illustrate that the retentive clasp is passive only at its first contact with the tooth during placement and when in its terminal position with the denture fully seated. During placement and removal, a rigid clasp arm placed on the opposite side of the tooth cannot provide resis-tance against these horizontal forces. See Figure 14-6 for a method to ensure true reciprocation. Ledges on Abutment Crowns In addition to providing abutment protection, more ideal retentive contours, definite guiding planes, and optimum occlusal rest support, complete coverage restorations on teeth used as removable partial denture abutments offer still another advantage not obtainable on natural teeth. This is the crown ledge or shoulder, which provides effec-tive stabilization and reciprocation. The functions of the reciprocal clasp arm have been stated in Chapter 6. Briefly, these are reciprocation, stabilization, and auxiliary indirect retention. Any rigid reciprocal arm may provide horizontal stabilization if it is located on axial surfaces parallel to the path of place-ment. To a large extent, because it is placed at the height of convexity, a rigid reciprocal arm may also act as an auxiliary indirect retainer. However, its function as a reciprocating arm against the action of the retentive clasp arm is limited to stabilization against possible orthodon-tic movement when the denture framework is in its ter-minal position. Such reciprocation is needed when the retentive clasp produces an active orthodontic force because of accidental distortion or improper design. Reciprocation, to prevent transient horizontal forces that may be detrimental to abutment stability, is most needed when the restoration is placed or when a dislodging force is applied. Perhaps the term orthodontic force is incorrect, because the term signifies a slight but continuous influ-ence that would logically reach equilibrium when the tooth is orthodontically moved. Instead, the transient forces of placement and removal are intermittent but forceful, which can lead to periodontal destruction and eventual instability rather than to orthodontic movement. True reciprocation is not possible with a clasp arm that is placed on an occlusally inclined tooth surface because it does not become effective until the prosthesis is fully seated. When a dislodging force is applied, the reciprocal clasp arm, along with the occlusal rest, breaks contact with the supporting tooth surfaces, and they are no longer effective. Thus, as the retentive clasp flexes over the height of contour and exerts a horizontal force on the abutment, reciprocation is nonexistent just when it is needed most (Figure 14-5). True reciprocation can be obtained only by creating a path of placement for the reciprocal clasp arm that is parallel to other guiding planes. In this manner, the infe-rior border of the reciprocal clasp makes contact with its guiding surface before the retentive clasp on the other side of the tooth begins to flex (Figure 14-6). Thus 210 Part II Clinical and Laboratory Lingual view Surveyor blade Veneered area A B C D E Figure 14-6 A, Preparation of the ledge in a wax pattern with a surveyor blade parallel to the path of placement. B, Refinement of the ledge on casting, using a suitable stone or milling device in a handpiece attached to the dental surveyor or a specialized drill press for the same purpose. C, Approximate width and depth of the ledge formed on the abutment crown, which will permit the reciprocal clasp arm to be inlaid within the normal contours of the tooth. D, True reciprocation throughout the full path of placement and removal is possible when the reciprocal clasp arm is inlaid onto the ledge on the abutment crown. E, Direct retainer assembly is fully seated. The reciprocal arm restores the lingual contour of the abutment. reciprocation exists during the entire path of placement and removal. A ledge on the abutment crown acts as a terminal stop for the reciprocal clasp arm. It also aug-ments the occlusal rest and provides indirect retention for a distal extension removable partial denture. A ledge on an abutment crown has still another advan-tage. The usual reciprocal clasp arm is half-round, and therefore convex, and is superimposed on and increases the bulk of an already convex surface. A reciprocal clasp arm built on a crown ledge is actually inlayed into the crown and reproduces more normal crown contours (see Figure 14-6). The patient’s tongue then contacts a con-tinuously convex surface rather than the projection of a clasp arm. Unfortunately, the enamel is not thick enough nor the tooth so shaped that an effective ledge can be created on an unrestored tooth. Narrow enamel shoul-ders are sometimes used as rest seats on anterior teeth, but these do not provide the parallelism that is essential to reciprocation during placement and removal. The crown ledge may be used on any complete or three-quarter crown restored surface that is opposite the retentive side of an abutment tooth. It is used most fre-quently on premolars and molars but also may be used on canine restorations. It is not ordinarily used on buccal surfaces for reciprocation against lingual retention because of the excessive display of metal, but it may be used just as effectively on posterior abutments when esthetics is not a factor. The fact that a crown ledge is to be used should be known in advance of crown preparation to ensure suffi-cient removal of tooth structure in this area. Although a shoulder or ledge is not included in the preparation itself, adequate space must be provided so that the ledge may be made sufficiently wide and the surface above it made parallel to the path of placement. The ledge should be placed at the junction of the gingival and middle thirds of the tooth, curving slightly to follow the curvature of the gingival tissues. On the side of the tooth where the clasp arm will originate, the ledge must be kept low enough to allow the origin of the clasp arm to be wide enough for sufficient strength and rigidity. In forming the crown ledge, which is usually located on the lingual surface, the wax pattern of the crown is completed except for refinement of the margins before the ledge is carved. After the proximal guiding planes and the occlusal rests and retentive contours are formed, the ledge is carved with the surveyor blade so that the surface above is parallel to the path of placement. Thus a con-tinuous guiding plane surface will exist from the proxi-mal surface around the lingual surface. 211 Chapter 14 Preparation of Abutment Teeth A B Figure 14-7 A milling machine used to prepare parallel surfaces, internal rest seats, lingual grooves, and ledges in cast restorations. Such a device permits more precise milling than is possible with a dental handpiece attached to the dental surveyor. To be effective, the cast must be positioned on the drill in such a manner that the previously established path of placement is maintained. A movable stage or base therefore should be adjustable until the relation of the cast to the axis of the drill has been made the same as that obtained when the cast was on the dental surveyor. The full effectiveness of the crown ledge can be achieved only when the crown is returned to the surveyor for refinement after casting. To afford true reciprocation, the crown casting must have a surface above the ledge that is parallel to the path of placement. This can be accomplished with precision only by machining the casting parallel to the path of placement with a handpiece holder in the surveyor or some other suitable machining device (Figure 14-7). Similarly, the parallelism of proximal guiding planes needs to be perfected after casting and polishing. Although it is possible to approximate parallelism and, at the same time, form the crown ledge on the wax pattern with a surveyor blade, some of its accuracy is lost in casting and polishing. The use of suitable burs such as No. 557, 558, and 559 fissure burs and true cylindrical carborundum stones in the handpiece holder permits the paralleling of all guiding planes on the finished casting with the accu-racy necessary for the effectiveness of those guiding plane surfaces. The reciprocal clasp arm is ultimately waxed on the investment cast so that it is continuous with the ledge inferiorly and contoured superiorly to restore the crown contour, including the tip of the cusp. It is obvious that polishing must be controlled so as not to destroy the form of the shoulder that was prepared in wax or the parallel-ism of the guiding plane surface. It is equally vital that the removable partial denture casting be finished with great care so that the accuracy of the counterpart is not destroyed. Modern investments, casting alloys, and pol-ishing techniques make this degree of accuracy possible. Spark Erosion Spark erosion technology is a highly advanced system for producing the ultimate in precision fit of the reciprocal arm to the ledge on the casting. This technology uses a tool system that permits repositioning of the casting with great accuracy and an electric discharge machine that is programmed to erode minute metal particles through periodic spark intervals. Regardless of the method or technique used, it is imperative that the predetermined cast orientation be maintained to ensure that ledges and proximal guide planes remain parallel. 212 Part II Clinical and Laboratory porcelain should be used to ensure the future retentiveness of the veneered surface. Present-day acrylic-resins, which are cross-linked copolymers, will withstand abrasion for consid-erable time, but not nearly to the same degree as porcelain. Therefore acrylic-resin veneers are best used in conjunction with metal that supports the half-round clasp terminal. Splinting of Abutment Teeth Often, a tooth is considered too weak to use alone as a removable partial denture abutment because of the short length or excessive taper of a single root, or because of bone loss resulting in an unfavorable crown-to-root ratio. In such instances, splinting to the adjacent tooth or teeth can be used as a means of improving abutment support. Thus, two sin-gle-rooted teeth serve as a multi-rooted abutment. Splinting should not be used to retain a tooth that would otherwise be condemned for periodontal reasons. When the length of service of a restoration depends on the service-ability of an abutment, any periodontally questionable tooth should be condemned in favor of using an adjacent healthy tooth as the abutment, even though the span is increased one tooth by doing so. The most common application of the use of multiple abutments is the splinting of two premolars or a first pre-molar and a canine (Figure 14-9). Mandibular premolars generally have round and tapered roots, which are easily loosened by rotational, as well as by tipping, forces. They are the weakest of the posterior abutments. Maxillary premolars also often have tapered roots, which may make them poor risks as abutments, particularly when they will be called on to resist the leverage of a distal extension base. Such teeth are best splinted by casting or soldering two crowns together. When a first premolar to be used as an abutment has poor root form or support, it is best that it be splinted to the stronger canine. Anterior teeth on which lingual rests are to be placed often must be splinted together to avoid orthodontic move-Veneer Crowns for Support of Clasp Arms For cosmetic reasons, resin and porcelain veneer crowns are used on abutment teeth that would otherwise display an objectionable amount of metal. They may be present in the form of porcelain veneers retained by pins and cemented to the crown; porcelain fused directly to a cast metal substruc-ture; porcelain fused to a machined coping; cast porcelain; pressed ceramic crowns; computer-assisted designed and machined ceramic restorations; or acrylic-resin processed directly to a cast crown. The development of abrasion-resis-tant composites offers materials suitable for veneering that can withstand clasp contact, thereby eliminating an undesir-able display of metal. Veneer crowns must be contoured to provide suitable retention. This means that the veneer must be slightly over-contoured and then shaped to provide the desired undercut for the location of the retentive clasp arm (Figure 14-8). If the veneer is of porcelain, this procedure must precede glazing, and if it is of resin, it must precede final polishing. If this important step in making veneered abutments is neglected or omitted, excessive or inadequate retentive con-tours may result. In limited clinical trials, porcelain laminates demon-strated resistance to wear of 5-year equivalence. The porce-lain, however, resulted in slight wear on the clasps. The flat underside of the cast clasp makes sufficient contact with the surface of the veneer so that abrasion of a resin veneer may result. Although the underside of the clasp may be polished (with some loss in accuracy of fit), abrasion results from the trapping and holding of food debris against the tooth surface as the clasp moves during function. There-fore, unless the retentive clasp terminal rests on metal, glazed Figure 14-9 First premolars and canines have been splinted in this Class I, modification 1 partially edentulous arch. The splint bar was added to provide cross-arch stabilization for splinted abutments and to support and retain the anterior segment of the removable restoration. The prospective longevity of the abut-ments has been enhanced. Figure 14-8 A porcelain veneer crown is resurveyed following adjustment, glazing, and polishing. It is important to survey crowns returned from the laboratory before cementation. The best time to ensure control of all abutment contours for a remov-able partial denture is when surveyed crowns are used and they are resurveyed before permanent placement. 213 Chapter 14 Preparation of Abutment Teeth there may be notable exceptions when a molar abutment would benefit from the effect of splinting, as in a hemi-sected molar root (Figure 14-10). Use of Isolated Teeth as Abutments The average abutment tooth is subjected to some distal tipping, rotation, torquing, and horizontal movement, all of which must be held to a minimum by the quality of tissue support and the design of the removable partial denture. The isolated abutment tooth, however, is subjected also to mesial tipping caused by lack of proximal contact. Despite indirect retention, some lifting of the distal extension base is inevi-table, causing torque to the abutment. In a tooth-supported prosthesis, an isolated tooth may be used as an abutment by including a fifth abutment for addi-tional support. Thus rotational and horizontal forces are resisted by the additional stabilization obtained from the fifth abutment. When two such isolated abutments exist, a sixth abutment should be included for the same reason. Thus the two canines, the two isolated premolars, and two posterior teeth are used as abutments. In contrast, an isolated anterior abutment adjacent to a distal extension base usually should be splinted to the nearest tooth by means of a fixed partial denture. The effect is ment of individual teeth. Mandibular anterior teeth are seldom used for support, but if they are, splinting of the teeth involved is advisable. When splinting is impossible, individual lingual rests on cast restorations may be slightly inclined apically to avoid possible tooth displacement, or lingual rests may be used in conjunction with incisal rests, slightly engaging the labial surface of the teeth. Lingual rests should always be placed as low on the cin-gulum as possible, and single anterior teeth, other than canines, should not be used for occlusal support. Where lingual rests are used on central and lateral incisors, as many teeth as possible should be included to distribute the load, thereby minimizing the force on any one tooth. Even so, some movement of individual teeth is likely to occur, par-ticularly when they are subjected to the forces of indirect retention or when bone support is compromised. This is best avoided by splinting several teeth with united cast restora-tions. The condition of the teeth and cosmetic consider-ations will dictate whether complete crowns, three-quarter crowns, pin ledge inlays, resin-bonded retainers, or compos-ite restorations will be used for this purpose. Splinting of molar teeth for multiple abutment support is less frequently used because they are generally multi-rooted. A two- or three-rooted tooth that is not strong enough alone is probably a poor abutment risk. However, C A B Figure 14-10 A, Periodontal disease required removal of #30 distal and #31 mesial roots. B, The first premolar and hemi-sected roots were splinted using a five-unit fixed partial denture. C, Fixed prosthesis provided cross-arch support, stability, and retention to a Kennedy Class II removable partial denture. 214 Part II Clinical and Laboratory assume responsibility for use of the isolated tooth as an abutment. The economic aspect of the use of fixed restorations as part of the mouth preparations for a removable partial denture is essentially the same as that for any other splinting procedure: the best design of the fixed partial denture that will ensure the longevity of its service makes the additional procedure and expense necessary. Although it must be rec-ognized that economic considerations, combined with a particularly favorable prognosis of an isolated tooth, may influence the decision to forego the advantages of using a fixed partial denture, the original treatment plan should include this provision, even though the alternative method may be accepted for economic reasons. Missing Anterior Teeth When a removable partial denture is used to replace missing posterior teeth, especially in the absence of distal abutments, any additional missing anterior teeth are best replaced by means of fixed restorations rather than included in the removable partial denture. In any distal extension situation, some anteroposterior rotational action will result from the addition of an anterior segment to the denture. The ideal treatment plan, which would consider the anterior edentu-lous space separately, may result in conflict with economic and esthetic realities. Each situation must be treated accord-ing to its own merits. Often the best esthetic result can be obtained by replacing missing anterior teeth and tissues with the removable partial denture rather than with a fixed res-toration. From a biomechanical standpoint, however, it is generally advisable that a removable partial denture should replace the missing posterior teeth only after the remainder of the anterior arch has been made intact by fixed restorations. Although the need for compromise is recognized, the decision to include an anterior segment on the denture depends largely on the support available for that part of the removable partial denture. The greater the number of twofold: (1) the anterior edentulous segment is eliminated, thereby creating an intact dental arch anterior to the eden-tulous space; and (2) the isolated tooth is splinted to the other abutment of the fixed partial denture, thereby provid-ing multiple abutment support. Splinting should be used here only to gain multiple abutment support rather than to support an otherwise weak abutment tooth. Although splinting is advocated for abutment teeth that are considered too weak to risk being used alone, a single abutment standing alone in the dental arch anterior to a distal extension basal seat generally requires the splinting effect of a fixed partial denture (Figures 14-11 and 14-12). Even though the form and length of the root and the sup-porting bone seem to be adequate for an ordinary abutment, the fact that the tooth lacks proximal contact endangers the tooth when it is used to support a distal extension base removable partial denture. A second factor that may influence the decision to use an isolated tooth as an abutment is an esthetic consideration. However, neither esthetics nor economics should deter the dentist from recommending to the patient that an iso-lated tooth to be used as a terminal abutment should be given the advantage of splinting by means of a fixed partial denture. If compromises are necessary, the patient must A B Figure 14-12 A, Isolated abutments have been splinted using splint bars. B, The removable partial denture is more adequately sup-ported by the splinting mechanism shown in A than could be realized with isolated abutments. Figure 14-11 Lone-standing premolar should be splinted to the canine with a fixed partial denture. Not only will the design of the removable partial denture be simplified, but the longevity of abutment service by the premolar will be greatly extended. 215 Chapter 14 Preparation of Abutment Teeth Acrylic-resin temporary crowns that duplicate the original form of the abutment teeth must be made. The technique for making temporary crowns to fit direct retainers is similar to that used for other types of acrylic-resin temporary crowns. The principal difference is that an impression, made with an elastic impression material, must be made of the entire arch with the existing removable partial denture in place. It is necessary that the removable partial denture remain in the impression when it is removed from the mouth. If it remains in the mouth, it must be removed and inserted into the impression in its designated position. The impression with the removable partial denture in place is disinfected, wrapped in a wet paper towel (if irreversible hydrocolloid was used as the impression mate-rial) or placed in a plastic bag, and set aside while the tooth or teeth are being prepared for new crowns. After the preparations are completed and the impressions and jaw relation records have been made, the prepared teeth are dried and lubricated. The original impression is trimmed to eliminate any excess, undercuts, and interproximal pro-jections that would interfere with the replacement of the impression in the mouth. The methyl methacrylate acrylic-resins, composites, copolymers, and fiber-reinforced resins may serve as excel-lent materials for temporary crowns in conjunction with removable partial dentures. Making temporary crowns requires a small mixing cup or dappen dish; a cement spatula; and a small, disposable, plastic syringe. Autopoly-merizing acrylic-resin of the appropriate tooth color is placed in the cup or dappen dish, and monomer is added to make a slightly viscous mix. The volume should be slightly in excess of the amount estimated to fabricate the temporary restorations. The mix should be spatulated to a smooth con-sistency and the mix immediately poured into the barrel of the disposable syringe. A small amount of the mix should be injected over and around the margins of the prepared teeth. The remaining material should be injected into the impres-sion of the prepared teeth. The impression is seated into the mouth, where the dentist holds it in place until sufficient time has elapsed for it to reach a stiff, rubbery stage, or a consistency recommended by the manufacturer. This again must be based on experience with the particular resin used. At this time, the impression is removed. The crowns may remain in the impression. If so, they are stripped out of the impression, all excess is trimmed away with scissors, and the crowns are reseated on the prepared abutments. The remov-able partial denture is then removed from the impression and reseated in the mouth onto the temporary crowns, which should be in a stiff-rubbery state. The patient may bring the teeth into occlusion to reestablish the former posi-tion and occlusal relationship of the existing removable partial denture. After the resin crown or crowns have polymerized, the removable partial denture is removed and the crowns remain on the teeth. These are then carefully removed, contoured to accommodate oral hygiene access, trimmed, polished, natural anterior teeth remaining, the better is the available support for the edentulous segment. If definite rest seats can be prepared on multiple abutments, the anterior segment may be treated as any other tooth-bounded modification space. Sound principles of rest support apply just as much as elsewhere in the arch. Inclined tooth surfaces should not be used for occlusal support, nor should rests be placed on unprepared lingual surfaces. The best possible support for an anterior segment is multiple support extending, if possible, posteriorly across prepared lingual rest seats on the canine teeth to mesio-occlusal rest seats on the first premolars. Such support would permit the missing anterior teeth to be included in the removable partial denture, often with some cosmetic advantages over fixed restorations. In some instances, the replacement of anterior teeth by means of a removable partial denture cannot be avoided. However, without adequate tooth support, any such pros-thesis will lack the stability that would result from replacing only the posterior teeth with the removable partial denture and the anterior teeth with fixed restorations. When anterior teeth have been lost through accident or have been missing for some time, resorption of the anterior residual ridge may have progressed to the point that neither fixed nor remov-able pontics may be butted to the residual ridge. In such instances, for reasons of esthetics and orofacial tissue support, the missing teeth must be replaced with a denture base supporting teeth that are more nearly in their original position, considerably forward from the residual ridge. Although such teeth may be positioned to better cosmetic advantage, the contouring and coloring of a denture base to be esthetically pleasing require the maximum artistic effort of both the dentist and the technician. Such a removable partial denture, both from an esthetic and a biomechanical standpoint, is one of the most difficult of all prosthetic res-torations. However, a splint bar, connected by abutments on both sides of the edentulous space, will provide much-needed support and retention to the anterior segment of the removable partial denture. Because the splint bar will provide vertical support, rest seats on abutments adjacent to the edentulous area need not be prepared, thus simplifying an anterior restoration to some extent. The concept of a dual path of placement to enhance the esthetic replacement of missing anterior teeth with a remov-able partial denture is recognized. Sources of information on this concept are made available in the “Selected Reading Resources” of this text under “Partial Denture Design.” Temporary Crowns When A Removable Partial Denture Is Being Worn Occasionally, an existing removable partial denture must remain serviceable while the mouth is being prepared for a new prosthesis. In such situations, temporary crowns must be made that will support the old removable partial denture and will not interfere with its placement and removal. 216 Part II Clinical and Laboratory and temporarily cemented. The result is a temporary crown that restores the original abutment contours and allows the removable partial denture to be placed and removed without interference, while temporarily providing the same support to the denture that existed before the teeth were prepared. Cementation of Temporary Crowns Cementation of temporary crowns may require slight relief of the internal surface of the crowns to accommodate the temporary cement and to facilitate removal. The temporary cement should be thin and applied only to the inside gingival margin of the crowns to ensure complete seating. As soon as the temporary cement has hardened, the occlusion should be checked and adjusted accordingly. Regardless of the type of temporary cement used, any excess that might irritate the gingivae should be removed. Fabricating Restorations to Fit Existing Denture Retainers It is often necessary that an abutment tooth be restored with a complete crown (or other restoration) that will fit the inside of the clasp of an otherwise serviceable remov-able partial denture. One technique for doing so is simple enough, but it requires that an indirect-direct pattern be made and therefore justifies a fee for service above that required for the usual restoration. The technique for making a crown to fit the inside of a clasp is as follows: An irreversible hydrocolloid impres-sion of the mouth is made with the removable partial denture in place. This impression, which is used to make the temporary crown, is wrapped in a wet paper towel or placed in a plastic bag and set aside while the tooth is being prepared. Even though several abutment teeth are to be restored, it is usually necessary that each temporary restoration be completed before the next one is begun. This is necessary so that the original support and occlusal relationship of the removable partial denture can be maintained as each new temporary crown is being made. During preparation of the abutment tooth, the removable partial denture is replaced frequently to ascertain that sufficient tooth structure has been removed to allow for the thickness of the casting. When the preparation is completed, an individual impression of the tooth is obtained from which a stone die is made. A temporary crown is then made in the original irreversible hydrocol-loid impression, as outlined in the preceding paragraphs. It is trimmed, polished, and temporarily cemented, and the removable partial denture is returned to the mouth. The patient is dismissed after the excess cement has been removed. On the stone die made from the individual impression, a thin, autopolymerizing resin coping will be formed with a brush technique. The stone die should first be trimmed to the finishing line of the preparation, which is then delineated with a pencil, and the die painted with a tinfoil substitute. A separating material, such as a tinfoil substi-tute, should be used and will form a thin film on a cold, dry surface. Not all tinfoil substitutes are suitable for this purpose. With autopolymerizing resin powder and liquid in separate dappen dishes and a fine brush, a coping of resin of uniform thickness is painted onto the die. This should extend not quite to the pencil line representing the limit of the crown preparation. After hardening, the resin coping may be removed, inspected, and trimmed if neces-sary. The thin film of foil substitute should be removed before the coping is reseated onto the die. The wax pattern buildup on the resin coping is usually not begun until the patient returns. A sequence using a functional chew-in technique for occlusion would be followed establishing proximal contact and contours appropriate for the clasp assembly as outlined below. First, the occlusal portion of the wax pattern is estab-lished by having the patient close into maximum inter-cuspation, followed by excursive movements (Figure 14-13A). The wax pattern is returned to the cast, and additions are made to dull areas as required. The process is repeated until a smooth occlusal registration has been obtained. Except for narrowing of the occlusal surface and carving of grooves and spillways, this will be the occlusal anatomy of the finished restoration. The second step is the addition of sufficient wax to establish contact relations with the adjacent tooth. At this time, the occlusal relation of the marginal ridges also must be established. Next, wax is added to buccal and lingual surfaces where the clasp arms will contact the crown, and the wax pattern is again reseated in the mouth. The clasp arms, minor connectors, and occlusal rests involved in the removable partial denture are carefully warmed with a needlepoint flame, carefully avoiding any adjacent resin, and the removable partial denture is posi-tioned in the mouth and onto the wax pattern (Figure 14-13B). Several attempts may be necessary until the removable partial denture is fully seated and the compo-nents of the clasp are clearly recorded in the wax pattern. Each time the removable partial denture is removed, the pattern will draw with it and must be teased out of the clasp. When contact with the clasp arms and the occlusal relation of the removable partial denture have been estab-lished satisfactorily, the temporary crown may be replaced and the patient dismissed. The crown pattern is com-pleted on the die by narrowing the occlusal surface bucco-lingually, adding grooves and spillways, and refin-ing the margins. Any wax ledge remaining below the reciprocal clasp arm may be left to provide some of the advantages of a crown ledge that were described earlier in this chapter. Excess wax remaining below the retentive clasp arm, however, must be removed to permit the adding of a retentive undercut later (Figure 14-13 C). 217 Chapter 14 Preparation of Abutment Teeth A B C Resin coping Inlay wax Inlay wax Lingual ledge Resin coping Figure 14-13 Making of the cast crown to fit an existing removable partial denture clasp. A, Thin acrylic-resin coping is made first on the individual die of a prepared tooth. Inlay wax is then added and coping placed onto the prepared tooth where occlusal surfaces and contact relations are established directly in the mouth. The clasp assembly is warmed with a needlepoint flame only enough to soften the inlay wax, and the removable partial denture is placed into the mouth, where it is guided gently into place by the opposing occlusion. This step must be repeated several times and excess wax removed or wax added until full supporting contact with the under-side of the clasp assembly has been established, with the denture fully seated. Usually, the wax pattern withdraws with the denture and must be gently teased out of the clasp each time. B, The wax pattern is then placed back onto the individual die to complete the occlusal anatomy and refine the margins. Excess wax remaining below the impression of the retentive clasp arm must be removed, but the wax ledge may be left below the reciprocal clasp arm. C, Finished casting in the mouth. The terminus of the retentive clasp is then readapted to engage the undercut. It is frequently necessary to remove some interference from casting, as indicated by articulating paper placed between the clasp and the crown, until the clasp is fully seated. If a veneer material is to be added, the veneer space must now be carved in the wax pattern. In such situa-tions, the contour of the veneer may be recorded by making a stone matrix of the buccal surface, which can be repositioned on the completed casting to ensure the proper contour of the composite veneer. The wax pattern must be sprued with care so that essen-tial areas on the pattern are not destroyed. After casting, the crown should be subjected to a minimum of polishing because the exact form of the axial and occlusal surfaces must be maintained. Because it is impossible to withdraw a clasp arm from a retentive undercut on the wax pattern, the casting must be made without any provision for clasp retention. After the crown has been tried in the mouth with the denture in place, the location of the retentive clasp terminal is identified by scoring the crown with a sharp instrument. Then the crown may be ground and polished slightly in this area to create a retentive under-cut. The clasp terminal then may be carefully adapted into this undercut, thereby creating clasp retention on the new crown. An alternate method for making crowns to fit existing retainers uses mounted casts with the removable prosthe-sis adapted to the working cast to develop occlusal sur-faces for the involved crowns. Ideally, all abutment teeth would best be protected with complete crowns before the removable partial denture is fabricated. Except for the possibility of recur-rent caries caused by defective crown margins or gingival recession, abutment teeth so protected may be expected to give many years of satisfactory service in support, sta-bilization, and retention of the removable partial denture. Economically, a policy of insisting on complete coverage for all abutment teeth may well be justified from the long-term viewpoint. It must be recognized, however, that in practice, complete coverage of all abutment teeth is not 218 Part II Clinical and Laboratory always possible at the time of treatment planning. Many factors influence the future health status of an abutment tooth, some of which cannot be foreseen. It is necessary that the dentist be able to treat abutment teeth that later become defective so that their service as abutments may be restored and the serviceability of the removable partial denture maintained. Although not part of the original mouth preparations, this service accomplishes much the same objective by providing support, stability, and reten-tion, and the dentist must be technically capable of pro-viding this removable partial denture service when it becomes necessary. 219 Chapter 15 Impression Materials and Procedures for Removable Partial Dentures 219 CHAPTER 15 Impression Materials and Procedures for Removable Partial Dentures Chapter Outline Rigid Materials Plaster of paris Metallic oxide paste Thermoplastic Materials Modeling plastic Impression waxes and natural resins Elastic Materials Reversible hydrocolloids Irreversible hydrocolloids Mercaptan rubber–base impression materials Polyether impression materials Silicone impression materials Impressions of the Partially Edentulous Arch Important precautions to be observed in the handling of hydrocolloid impressions Step-by-step procedure for making a hydrocolloid impression Step-by-step procedure for making a stone cast from a hydrocolloid impression Possible causes of an inaccurate and/or a weak cast of a dental arch Individual Impression Trays Technique for making individual acrylic-resin impression trays Impression materials used in the various phases of partial denture fabrication may be classified as rigid, thermoplastic, or elastic substances. Rigid impression materials are those that set to a rigid consistency. Thermoplastic impression materials are those that become plastic at higher tempera-tures and resume their original form when cooled. Elastic impression materials are those that remain in an elastic or flexible state after they have set and have been removed from the mouth. Although rigid impression materials may be capable of recording tooth and tissue details accurately, they cannot be removed from the mouth without fracture and reassembly. Thermoplastic materials cannot record minute details accu-rately because they undergo permanent distortion during withdrawal from tooth and tissue undercuts. Elastic materi-als are the only ones that can be withdrawn from tooth and tissue undercuts without permanent deformation and there-fore are used generally for making impressions for remov-able partial dentures, immediate dentures, crowns, and fixed partial dentures when tooth and tissue undercuts and surface detail must be recorded with accuracy. Some of the historical parts of this discussion have been quoted or paraphrased from McCracken WL: Impression materials in prosthetic dentistry, Dent Clin North Am 2:671-684, 1958. Rigid Materials Plaster of Paris One type of rigid impression material is plaster of Paris, which has been used in dentistry for over 200 years. Although all plaster of Paris impression materials are handled in approximately the same manner, the setting and flow characteristics of each manufacturer’s product will vary. Some are pure and finely ground with only an 220 Part II Clinical and Laboratory be used successfully for this purpose if the original denture base has been relieved sufficiently to allow the material to flow without displacement of the denture or the underlying tissues. Thermoplastic Materials Modeling Plastic Like plaster of Paris, modeling plastic is among the oldest impression materials used in prosthetic dentistry. This mate-rial is most often used for border correction (border molding) of custom impression trays for Kennedy Class I and Class II removable partial denture bases. It is manufactured in several different colors, each color being an indication of the temperature range at which the material is plastic and work-able. A common error in the use of modeling plastic is that it is often subjected to higher temperatures than intended by the manufacturer. It then becomes too soft and loses some of its favorable working characteristics. If a temperature-controlled water bath is not used, a thermometer should be used to maintain the water temperature. If modeling plastic is softened at a temperature above that intended by the manufacturer, the material becomes brittle and unpredict-able. Also, there is the ever-present danger of burning the patient when the temperature used in softening the model-ing plastic is too high. The most commonly used modeling plastic for corrected impressions of extension base areas is the red (red-brown) material, in cake form, that softens at about 132° F. It should never be softened at temperatures much above this. Neither it nor any other modeling plastic should be immersed in the water bath for an indefinite period. It should be dipped and kneaded until soft and subjected to no more heat than neces-sary before the tray is loaded and it is placed in the mouth. Then it may be flamed with an alcohol torch for the purpose of border molding, but it should always be tempered by being dipped back into the water bath before its return to the mouth to avoid burning the patient. The modeling plastic then may be chilled using a water spray before removal from the mouth, although this is not necessary if care is used in removing the impression. During sectional flaming and border molding, the modeling plastic should be chilled in ice water after each removal from the mouth; then it may be trimmed with a sharp knife without danger of fracture or distortion. Red, gray, and green modeling plastics are obtainable in stick form for use in border molding an impression or an impression tray. The green material is the lowest fusing of the modeling plastics. The red and gray sticks have a higher and broader working range than do the cakes of like color so they may be flamed without harming the material. The Metallic Oxide Paste A second type of rigid impression material is metallic oxide paste, which is usually some form of a zinc oxide–eugenol combination. A number of these pastes are available; however, they are not used as primary impression materials and should never be used for impressions that include remaining natural teeth. They also are not to be used in stock impression trays. Metallic oxide pastes are manufactured with a wide varia-tion of consistencies and setting characteristics. For conve-nience, most of them are dispensed from two tubes; this enables the dentist to dispense and mix the correct propor-tion from each tube on a mixing slab. The previously pre-pared tray for the edentulous ridge segments is loaded and positioned in the mouth with or without any attempt at border molding. Border molding with metallic oxide impres-sion pastes is not advisable because wrinkles will occur if movement is permitted at the time the material reaches its setting state. As with all impression techniques, the accuracy of the primary impression and of the impression tray has a great influence on the final impression. Some metallic oxide pastes remain fluid for a longer period than do others, and some manufacturers claim that border molding is possible. In general, however, all metallic oxide pastes have one thing in common with plaster of Paris impression materials: they all have a setting time during which they should not be disturbed and after which no further border molding is effective. Metallic oxide pastes, which are rigid substances, can be used as secondary impression materials for complete den-tures and for extension base edentulous ridge areas of a removable partial denture if a custom impression tray has been properly designed and attached to the removable partial denture framework (see Chapter 16). Metallic oxide pastes can also be used as an impression material for relining distal extension denture bases and may accelerator added to expedite setting within reasonable working limits. Others are modified impression plasters to which binders and plasticizers have been added to permit limited border manipulation while the material is setting. These do not set as hard or fracture as cleanly as pure plaster of Paris and therefore cannot be reassembled with as much accuracy if fracture occurs. Plaster of Paris was once the only material available for removable partial denture impressions, but now elastic materials have completely replaced the impression plas-ters in this phase of prosthetic dentistry. It can be used for making accurate transfers of abutment castings or copings in the fabrication of fixed restorations and inter-nal attachment dentures and for making rigid indexes and matrices for various purposes in prosthetic dentistry. Modified impression plasters may be used to record max-illomandibular relationships. Some of the historical parts of this discussion have been quoted or paraphrased from McCracken WL: Impression materials in prosthetic dentistry, Dent Clin North Am 2:671-684, 1958. 221 Chapter 15 Impression Materials and Procedures for Removable Partial Dentures functional activity. These mouth-temperature materials also may be used successfully in open-mouth impression tech-niques. Iowa wax will not distort after removal from the mouth at ordinary room temperatures, but the more resin-ous waxes must be stored at much lower temperatures to avoid flow when they are out of the mouth. Resinous waxes are not ordinarily used in removable partial denture impres-sion techniques except for secondary impressions. Elastic Materials Reversible Hydrocolloids Reversible (agar-agar) hydrocolloids, which are fluid at higher temperatures and gel on reduction in temperature, are used primarily as impression materials for fixed restora-tions. They demonstrate acceptable accuracy when properly used; however, the reversible hydrocolloid impression mate-rials offer few advantages over the irreversible (alginate) hydrocolloids when used as a removable partial denture impression material. Present-day irreversible hydrocolloids are sufficiently accurate for making master casts for remov-able partial dentures. However, border control of impres-sions made with these materials is difficult. Irreversible Hydrocolloids Irreversible hydrocolloids are used for making diagnostic casts, orthodontic treatment casts, and master casts for removable partial denture procedures. Because they are made of colloid materials, neither reversible nor irreversible hydrocolloid impressions can be stored for any length of time, but must be poured immediately. These materials have low tear strength, provide less surface detail than other materials (e.g., mercaptan rubber base), and are not as dimensionally stable as other materials. They can, however, be used in the presence of moisture (saliva); are hydrophilic; pour well with stone; have a pleasant taste and odor; and are nontoxic, nonstaining, and inexpensive. The combination reversible-irreversible hydrocolloids have demonstrated a tendency to separate and should be used with that understanding. The hydrocolloids can be acceptably disinfected with a spray solution of 2% acid glutaraldehyde, stored in 100% humidity, and poured within 1 hour. Mercaptan Rubber–Base Impression Materials The mercaptan rubber–base (Thiokol) impression materials can also be used for removable partial denture impressions and especially for secondary corrected or altered cast impres-sions. To be accurate, the impression must have a uniform thickness that does not exceed 3 mm (18 inch). This neces-sitates the use of a carefully made individual impression tray gray material in stick form is preferred by some dentists for border molding because of its contrasting lighter color. The choice between the use of green and gray sticks is purely optional and entirely up to the dentist. Some dentists still prefer to use modeling plastic as a secondary impression material to record edentulous ridges in removable partial denture fabrication. When this is done, it is generally used only as a means of building up the under-side of the denture before the tissues are recorded with some secondary impression material (see Chapter 16). Impression Waxes and Natural Resins A second group of thermoplastic impression materials con-sists of those impression waxes and resins commonly spoken of as mouth-temperature waxes. The most familiar of these have been the Iowa wax (Kerr Co., Romulus, MI) and the Korecta waxes (D-R Miner Dental, Concord, CA), all of which were developed for specific techniques. Knowledge of the characteristics of mouth-temperature waxes is important if they are to be used correctly. The Iowa wax was developed for use in recording the functional or supporting form of an edentulous ridge. It may be used as a secondary impression material or as an impres-sion material for relining the finished removable partial denture to obtain support from the underlying tissues. The mouth-temperature waxes lend themselves well to all relin-ing techniques as they will flow sufficiently in the mouth to avoid displacement of tissues. As with any relining tech-nique, it is necessary that sufficient relief and venting be provided to give the material the opportunity to flow. The difference between impression wax and modeling plastic is that impression waxes have the ability to flow as long as they are in the mouth and thereby permit equaliza-tion of pressure and prevent displacement. The modeling plastics flow only in proportion to the amount of flaming and tempering that can be done outside of the mouth; this does not continue after the plastic has approached mouth temperature. The principal advantage of mouth-tempera-ture waxes is that, given sufficient time, they permit a rebound of those tissues that may have been forcibly displaced. The impression waxes also may be used to correct the borders of impressions made of more rigid materials, thereby establishing optimum contact at the border of the denture. All mouth-temperature wax impressions have the ability to record border detail accurately and include the correct width of the denture border. They also have the advantage of being correctable. Mouth-temperature waxes vary in their working charac-teristics. They are designed primarily for impression tech-niques that attempt to record the tissues under an occlusal load. In such techniques, the occlusion rim or the arrange-ment of artificial teeth is completed first. Mouth- temperature wax is then applied to the tissue side of the denture base, and the final impression is made under functional loading by using various movements to simulate Some of the historical parts of this discussion have been quoted or paraphrased from McCracken WL: Impression materials in prosthetic dentistry, Dent Clin North Am 2:671-684, 1958. 222 Part II Clinical and Laboratory dures. The stiffness of the material can result in cast breakage when removal of the cast from a custom tray is attempted. These materials have a higher permanent deformation than the addition reaction silicones. Some have an unpleasant taste, and because the material will absorb moisture, it cannot be immersed in disinfecting solutions or stored in high humidity for any extended period of time. The materi-als should be poured within 2 hours; however, manufactur-ers claim that if the impression is kept dry, clinically acceptable casts can be poured for up to 7 days. Silicone Impression Materials The silicone impression materials are more accurate and easier to use than the other elastic impression materials. The condensation silicones have a moderate (5 to 7 minutes) working time that can be altered by adjusting the amount of the accelerator. They have a pleasant odor, moderately high tear strength, and excellent recovery from deformation. These materials can be used with a compatible putty material to form fit a custom tray. Silicone impression materials are hydrophobic, which can make cast formation a problem. These materials can be disinfected in any of the disinfecting solutions with no alteration in accuracy. Ideally, these mate-rials should be poured within 1 hour. The addition reaction silicones are the most accurate of the elastic impression materials. They have less polymeriza-tion shrinkage, low distortion, fast recovery from deforma-tion, and moderately high tear strength. These materials have a working time of 3 to 5 minutes, which can be easily modified with the use of retardants and temperature con-trols. They are available in both hydrophilic and hydropho-bic forms, have no smell or taste, and also come in putty form, to assist in form fitting the impression tray at chair-side. Most of the addition reaction silicones are available in automixing devices, can be poured up to 1 week after impres-sion making with acceptable clinical results, and are stable in most sterilizing solutions. Sulfur in latex gloves and in ferric and aluminum sulfate retraction solution may inhibit polymerization. Many of the hydrophobic types are difficult to pour with stone, and adhesion to acrylic-resin trays is not good. The putties for these materials have a relatively short shelf life, and they are more expensive than the other elastic impression materials. Impressions of the Partially Edentulous Arch An impression of the partially edentulous arch must record accurately the anatomic form of the teeth and surrounding tissues. This is necessary so that the prosthesis may be designed to follow a definite path of placement and removal and so that support, stability, and retention derived from the abutment teeth may be precise and accurate. Materials that could be permanently deformed by removal from tooth or tissue undercuts should not be used. The thermoplastic impression materials and metallic oxide pastes of acrylic-resin or some other material possessing adequate rigidity and stability. Those materials that are highly cross-linked (medium and heavy body) do not recover well from deformation and should not be used when large or multiple undercuts are present. For example, when large numbers of teeth with natural tooth contours that display multiple undercuts remain, these materials will be subjected to clini-cally significant distortion upon withdrawal. The long-term dimensional stability of these materials is poor because of water loss after setting. The material must be held still during the impression-making procedure because it does not have a snap set; it should be allowed to rebound for 7 to 15 minutes after it is removed from the mouth and should then be poured immediately. Many of these materials have an unpleasant odor and can stain clothes. These materials are moderately inexpensive, have high tear strength and long working and setting times (8 to 10 minutes), and can be disinfected in liquid, cold-sterilizing solutions. The accuracy of mercaptan rubber base is acceptable for making impres-sions for removable partial dentures; however, as with hydrocolloid impression materials, certain precautions must be taken to avoid distortion of the impression. Mercaptan rubber–base impression materials do have an advantage over hydrocolloid materials in that the surface of an artificial stone poured against them is of a smoother texture and therefore appears to be smoother and harder than one poured against a hydrocolloid material. This is probably so because the rubber material does not have the ability to retard or etch the surface of the setting stone. Despite their accuracy, this has always been a disadvantage of all hydrocol-loid impression materials. The fact that a smoother surface results does not, however, preclude the possibility of a grossly inaccurate impression and stone cast resulting from other causes. Rubber-base impression materials possess a longer setting time than the irreversible hydrocolloid mate-rials and lend themselves better to border molding in ade-quate supporting trays. Polyether Impression Materials Polyether impression material is an elastic-type material, as are the polysulfide and silicone materials. These materials have demonstrated good accuracy in clinical evaluations and are thixotropic, which provides good surface detail and makes them useful as a border molding material. It should be noted, however, that these materials are not compatible with the addition reaction silicone impression materials and should not be used to border mold custom trays when the silicone impression materials are to be used as the final impression material. The polyethers are also hydrophilic, which produces good wetability for easy cast forming. The polyethers have low to moderate tear strength and much shorter working and setting times, which can limit the usefulness of the material. The flow characteristics and flex-ibility of the polyether materials are the lowest of any of the elastic materials. These characteristics can limit the use of polyethers in removable partial denture impression proce-223 Chapter 15 Impression Materials and Procedures for Removable Partial Dentures curacy. All modern irreversible hydrocolloid impression materials have an accelerator incorporated into the powder and no longer need to be treated with a fixing solution. Because no heat is used in the preparation of irreversible hydrocolloid, there is no danger of burning the patient. For this reason, the patient should be more relaxed and coopera-tive during the positioning of the tray. However, some dis-advantages are associated with the use of irreversible hydrocolloid. This material gels by means of a chemical reac-tion that is accelerated by the warmth of the tissues, whereas reversible hydrocolloid gels from the tray in toward the tissues, because of the cooling action of the water that cir-culates through the tray. In the irreversible hydrocolloid, gelation first takes place next to the tissues, and any move-ment of the tray during gelation will result in internal stresses that are released on removal of the impression from the mouth. A distorted and therefore inaccurate impression results from an irreversible hydrocolloid impression that is not held immobile during gelation. Another disadvantage of irreversible hydrocolloid is that it must be introduced into the mouth at approximately 70° F, which results in an immediate increase in the viscosity and surface tension of the material. Air bubbles are therefore harder to dispel, and it is inevitable that more air will be trapped in an irreversible impression than in a reversible impression. Every precaution must be taken to avoid the entrapment of air in critical areas. Important Precautions to Be Observed in the Handling of Hydrocolloid Impressions Some important precautions to be observed in the handling of hydrocolloid are as follows: 1. The impression should not be exposed to air because some dehydration will inevitably occur and result in shrinkage. 2. The impression should not be immersed in water or dis-infectants because some imbibition will inevitably result, with an accompanying expansion. 3. The impression should be protected from dehydration by placing it in a humid atmosphere or wrapping it in a damp paper towel until a cast can be poured. To avoid volume change, this should be done within 15 minutes after removal of the impression from the mouth. 4. Exudate from the hydrocolloid has a retarding effect on the chemical reaction of gypsum products and results in a chalky cast surface. This can be prevented by pouring the cast immediately or by first immersing the impression in a solution of accelerator if an accelerator is not included in the formula. Step-by-Step Procedure for Making a Hydrocolloid Impression The step-by-step procedure and important points to observe in the making of a hydrocolloid impression are as follows: 1. Select a suitable, sterilized, perforated or rim-lock impression tray that is large enough to provide a 4- to are therefore excluded for recording the anatomic form of the dental arch. Rubber-base materials that are highly cross-linked should not be used when large or multiple undercuts are present because these materials will be subjected to con-siderable distortion upon withdrawal. Plaster of Paris and modeling plastic are capable of recording tissue detail accu-rately, but they must be sectioned for removal and subse-quently reassembled, which often leads to permanent deformation. The introduction of hydrocolloids as impression materi-als was a giant step forward in dentistry. For the first time, impressions could be made of undercut areas with a material that was elastic enough to be withdrawn from those under-cuts without permanent distortion. It permitted the making of a one-piece impression, which did not require the use of a separating medium, and was and still is an acceptably accurate material when handled properly. The principal differences between reversible and irrevers-ible hydrocolloids are as follows: 1. Reversible hydrocolloid converts from the gel form to a sol by the application of heat. It may be reverted to gel form by a reduction in temperature. This physical change is reversible. 2. Irreversible hydrocolloid becomes a gel via a chemical reaction as a result of mixing alginate powder with water. This physical change is irreversible. Reversible hydrocolloid does have some disadvantages. It must be introduced into the mouth while warm enough to be a sol, and then it converts to an elastic gel on cooling. Therefore there is an ever-present danger of burning the tissues of the mouth—a burn that is painful and slow to heal. It requires warming and tempering equipment that is ther-mostatically controlled and necessitates the use of water-jacketed impression trays for cooling. All hydrocolloids are dimensionally stable only during a brief period after removal from the mouth. If exposed to the air, they rapidly lose water content, with resulting shrinkage and other dimensional changes. If immersed in water, they imbibe water, with accompanying swelling and dimensional changes. All hydrocolloid impressions should be poured immediately, but if they must be stored for a brief period, they should be in a saturated atmosphere rather than immersed in water. This can be accomplished simply by wrapping the impression in a damp paper towel or sealing it in a plastic bag. Hydrocolloids also exhibit a phenomenon known as syn-eresis, which is associated with the giving off of a mucinous exudate. This mucinous exudate has a retarding effect on any gypsum material, which results in a soft or chalky cast surface. Sometimes this is detected only by close examina-tion of the impression after removal from the cast. Neverthe-less, such a cast surface is inaccurate and ultimately will result in an inaccurate removable partial denture frame-work. Pouring the cast immediately and using some chemi-cal accelerator, such as potassium sulfate, to counteract the retarding effect of the hydrocolloid can prevent this inac-224 Part II Clinical and Laboratory mixing bowl (600-mL capacity). Add the correct measure of powder. Spatulate rapidly against the side of the bowl with a short, stiff spatula. This should be accomplished in less than 1 minute. The patient should rinse his or her mouth with cool water to eliminate excess saliva while the impression material is being mixed and the tray is being loaded. 6. In placing the material in the tray, avoid entrapping air. Have the first layer of material lock through the perfora-tions of the tray or rim-lock to prevent any possible dislodgment after gelation. 7. After loading the tray, remove the gauze with the topical anesthetic and quickly place (rub) some of the impression material on any critical areas using your finger (areas such as rest preparations and abutment teeth). If a maxillary impression is being made, place the material in the highest aspect of the palate and over the rugae. 8. Use a mouth mirror or index finger to retract the cheek on the side away from you as the tray is rotated into the mouth from the near side. 9. Seat the tray first on the side away from you, next on the anterior area, while reflecting the lip, and then on the near side, with the mouth mirror or finger for cheek retraction. Finally, make sure that the lip is draping naturally over the tray. 10. Be careful not to seat the tray too deeply, leaving room for a thickness of material over the occlusal and incisal surfaces. 11. Hold the tray immobile for 3 minutes with light finger pressure over the left and right premolar areas. To avoid internal stresses in the finished impression, do not allow the tray to move during gelation. Any movement of the tray during gelation will produce an inaccurate impres-sion. If, for example, you allow the patient or the assis-tant to hold the tray in position at any time during the impression procedure, some movement of the tray will be inevitable during the transfer and the impression will probably be inaccurate. Do not remove the impression from the mouth until the impression material has com-pletely set. 12. After releasing the surface tension, remove the impres-sion quickly in line with the long axis of the teeth to avoid tearing or other distortion. 13. Rinse the impression free of saliva with slurry water, or dust it with plaster, and rinse gently; then examine it critically. Spray the impression thoroughly with a suit-able disinfectant and cover it immediately with a damp paper towel. A cast should be poured immediately into a disinfected hydrocolloid impression to avoid dimensional changes and syneresis. Circumstances often necessitate some delay, but this time lapse should be kept to a minimum. A delay of 15 minutes will satisfy the disinfection requirements and should not be deleterious if the impression is kept in a humid atmosphere. 5-mm thickness of the impression material between the teeth and tissues and the tray. 2. Build up the palatal portion of the maxillary impression tray with wax or modeling plastic to ensure even distri-bution of the impression material and to prevent the material from slumping away from the palatal surface (Figure 15-1A). At this time, it is also helpful to pack the palate with gauze that has been sprayed with a topical anesthetic. This will serve to anesthetize the minor sali-vary glands and mucous glands of the palate and thus prevent secretions as a response to smell or taste or to the physical presence of the impression material. If gela-tion occurs next to the tissues while the deeper portion is still fluid, a distorted impression of the palate may result, which cannot be detected in the finished impres-sion. This may result in the major connector of the finished casting not being in contact with the underlying tissues. The maxillary tray frequently has to be extended posteriorly to include the tuberosities and the vibrating line region of the palate. Such an extension also aids in correctly orienting the tray in the patient’s mouth when the impression is made. 3. The lingual flange of the mandibular tray may need to be lengthened with wax in the retromylohyoid area or to be extended posteriorly, but it rarely ever needs to be lengthened elsewhere. Wax may need to be added inside the distolingual flange to prevent the tissues of the floor of the mouth from rising inside the tray (Figure 15-1B). 4. Place the patient in an upright position, with the arch to be impressed nearly parallel to the floor. 5. When irreversible hydrocolloid is used, place the mea-sured amount of water (at 70°F) in a clean, dry, rubber A B Figure 15-1 A, Maxillary impression tray with palatal portion built up with baseplate wax to prevent impression material from sagging away from palatal surface. B, Mandibular impression tray with periphery wax added to lingual flanges to prevent tissues of the floor of the mouth from rising inside the tray. The posterior end of the tray is extended with periphery wax to cover the retromolar pad regions. 225 Chapter 15 Impression Materials and Procedures for Removable Partial Dentures to add small increments of material at this same distal area, with each portion of added stone pushing the mass ahead of it. This avoids the entrapment of air. The weight of the material causes any excess water to be pushed around the arch and to be expelled ultimately at the opposite end of the impression. Discard this fluid mate-rial. When the impressions of all teeth have been filled, continue to add artificial stone in larger portions until the impression is completely filled. 5. The filled impression should be placed so that its weight does not distort the hydocolloid impression material. The base of the cast can be completed with the same mix of stone. The base of the cast should be 16 to 18 mm ( 23 to 34 inch) at its thinnest portion and should be extended beyond the borders of the impression so that buccal, labial, and lingual borders will be recorded correctly in the finished cast. A distorted cast may result from an inverted impression. 6. As soon as the cast material has developed sufficient body, trim the excess from the sides of the cast. Wrap the impression and cast in a wet paper towel, or place it in a humidor, until the initial set of the stone has taken place. The impression is thus prevented from losing water by evaporation, which might deprive the cast material of sufficient water for crystallization. Chalky cast surfaces around the teeth are often the result of the hydrocolloid’s acting as a sponge and robbing the cast material of its necessary water for crystallization. 7. After the cast and impression have been in the humid atmosphere for 30 minutes, separate the impression from the cast. Thirty minutes is sufficient for initial setting. Step-by-Step Procedure for Making a Stone Cast From a Hydrocolloid Impression The step-by-step procedure for making a stone cast from the impression is as follows: 1. A more abrasive-resistant type IV stone should be used to form removable partial denture casts. Have the mea-sured dental stone at hand, along with the designated quantity of room temperature water, as recommended by the manufacturer. A clean 600-mL rubber mixing bowl, a stiff spatula, and a vibrator complete the preparations. A No. 7 spatula also should be within reach. 2. First, pour the measure of water into the mixing bowl and then add the measure of stone. Spatulate thoroughly for 1 minute, remembering that a weak and porous stone cast may result from insufficient spatulation. Mechanical spatulation under vacuum is preferred. After any spatula-tion other than in a vacuum, place the mixing bowl on the vibrator and knead the material to permit the escape of any trapped air. 3. After removing the impression from the damp towel, gently shake out surplus moisture and hold the impression over the vibrator, impression side up, with only the handle of the tray contacting the vibrator. The impression material must not be placed in contact with the vibrator because of possible distortion of the impression. 4. With a small spatula, add the first cast material to the distal area away from you. Allow this first material to be vibrated around the arch from tooth to tooth toward the anterior part of the impression (Figure 15-2). Continue A B Figure 15-2 A, Stone is introduced at one posterior region of the impression, with care taken to trace the stone moving into each tooth as it rounds the arch. B, Additional stone is not needed until the stone has reached the opposite-most posterior tooth. 226 Part II Clinical and Laboratory dering tissues; otherwise an individual tray made of some acrylic-resin tray material should be used for the final ana-tomic impression. Most stock or disposable removable partial denture trays are of the rim-lock or perforated varieties. Both are made in a limited selection of sizes and shapes. Wide selections of trays are available that can be used for partially edentulous patients, including trays for both bilateral and unilateral edentulous areas. All of these trays have reinforced borders. Although a complete denture impression tray is, or should be, made of material that permits trimming and shaping to fit the mouth, the existence of a beaded border and the rigidity of a stock removable partial denture tray allow no trimming and little shaping. The resulting impression is often a record of border tissues distorted by an ill-fitting tray rather than an impres-sion of tissues draping naturally over a slightly underex-tended impression tray. An individual acrylic-resin tray, on the other hand, can be made with sufficient clearance for the impression mate-rial and can be trimmed just short of the vestibular reflec-tions to allow the tissues to drape naturally without distortion. The removable partial denture borders may then be made as accurately as complete denture borders with equal advantages. Although techniques have been proposed for making individual impression trays that incorporate plastic tubing for water-cooling reversible hydrocolloid impressions, the final anatomic impression usually will be made with irre-versible hydrocolloid, mercaptan rubber, or silicone impres-sion materials. Technique for Making Individual Acrylic-Resin Impression Trays The diagnostic cast is often adequate for preparation of the individual tray. However, if extensive surgery or extractions were performed after the diagnostic cast was made, a new impression in a rigid stock tray and a new cast must be made. The procedures for making the new cast are identical to those described previously. A duplicate of the diagnostic cast, on which the individual tray can be fabricated, should be made because the cast on which an individual tray is made is often damaged or must be mutilated to separate the tray from the cast. Obviously the original diagnostic cast must be retained as a permanent record in the patient’s file. Several techniques may be used to make individual impression trays. One technique for making an individual maxillary tray is described in Figures 15-3 and 15-4. This format could be used for both autopo-lymerizing acrylic-resin and visible light-cured (VLC) acrylic-resin. The VLC custom tray materials are premixed sheet materials that, when polymerized, provide a highly stable, distortion-free custom impression tray that is ready to use in minutes. These materials are provided by the man-ufacturers in sheet forms of various sizes, thicknesses, and colors. Any stone that interferes with separation from the tray must be trimmed away with a knife. 8. Clean the impression tray immediately while the used impression material is still elastic. 9. Trimming of the cast should be deferred until final setting has occurred. The sides of the cast then may be trimmed to be parallel, and any blebs or defects resulting from air bubbles in the impression may be removed. If this is a cast for a permanent record, it may be trimmed to orth-odontic specification to present a neat appearance for demonstration purposes. Master casts and other working casts are ordinarily trimmed only to remove excess stone. Possible Causes of an Inaccurate and/or a Weak Cast of a Dental Arch The possible causes of an inaccurate cast are as follows: 1. Distortion of the hydrocolloid impression (a) by use of an impression tray that is not rigid; (b) by partial dislodg-ment from the tray; (c) by shrinkage caused by dehydra-tion; (d) by expansion caused by imbibition (this will be toward the teeth and will result in an undersized rather than oversized cast); and (e) by attempting to pour the cast with stone that has already begun to set. 2. A ratio of water to powder that is too high. Although this may not cause volumetric changes in the size of the cast, it will result in a weak cast. 3. Improper mixing. This also results in a weak cast or one with a chalky surface. 4. Trapping of air, either in the mix or in pouring, because of insufficient vibration. 5. Soft or chalky cast surface that results from the retarding action of the hydrocolloid or the absorption of necessary water for crystallization by the dehydrating hydrocolloid. 6. Premature separation of the cast from the impression. 7. Failure to separate the cast from the impression for an extended period. Individual Impression Trays This chapter has previously dealt with making an impression in a rigid stock tray of the anatomic form of a dental arch for making a diagnostic cast, a working cast for restorations, or a master cast. There are times, however, when a stock tray is not suitable for making the final anatomic impression of the dental arch. Most tooth-supported removable partial dentures may be made on a master cast from such an impres-sion. Some maxillary distal extension removable partial den-tures with broad palatal coverage, particularly those for a Kennedy Class I arch, may also be made on an anatomic cast, but usually these necessitate the use of an individually made tray. A stock tray must be sufficiently rigid to avoid distortion during the impression and cast forming procedures and should fit the mouth with about 4 to 5 mm clearance for the impression material without interfering with teeth or bor-227 Chapter 15 Impression Materials and Procedures for Removable Partial Dentures A B C D Figure 15-3 A, Desired outline of the tray is drawn on the diagnostic cast. The tray must include all teeth and tissues that will be involved in the removable partial denture. B, One thickness of baseplate wax is adapted to the cast and is trimmed to the penciled outline, which is 2 to 3 mm short of the desired border. The posterior palatal seal region is not covered by wax but will be included in finished tray. Two thicknesses of baseplate wax cover the teeth. A window is created in the wax spacer over the incisal edges. C, A model release agent is painted on the stone surfaces of the cast that will be contacted by the resin. D, The visible light-cured (VLC) resin tray material is removed from the light-proof wrap and shaped to the desired outline in a uniform manner. A technique for making an individual maxillary tray with light-polymerized resin is as follows: 1. Outline the extent of the tray on the cast with a pencil. The tray must include all teeth and tissues that will be involved in the removable partial denture. 2. Adapt one layer of baseplate wax over the tissue surfaces and two layers over the teeth of the cast to serve as a spacer for impression material. The wax spacer should be trimmed 2 to 3 mm short of the outline drawn on the diagnostic cast. Wax covering the posterior palatal seal area should be removed so that intimate contact of the tray and tissue in this region may serve as an aid in correctly orienting the tray when the impression is made. Expose portions of the incisal edges of the central incisors to serve as anterior stops when placing the tray in the mouth. Bevel the wax so that the completed tray will have a guiding incline that will help position the tray on the anterior stop. Other cast undercuts should be blocked out with wax or modeling compound. note: Adapt an additional layer of baseplate wax over the teeth if the impression is to be made in irreversible hydrocol-loid. This step is not necessary if the choice of impres-sion material is a rubber-base or silicone type of material. 3. Paint the exposed surfaces of the cast that may be con-tacted by the light-polymerized resin tray material with a model release agent (MRA) to facilitate separation of the polymerized tray from the cast. 4. Remove the VLC tray material from the light-proof pouch and carefully cut the desired length with a knife or scalpel. Adapt the VLC material to the cast and trim it with a knife. Be sure not to thin the material over the teeth or the posterior border area. 5. Attach a handle by molding excess VLC material into the desired shape and blend it into the tray material in the cast. With some materials, a paper clip or similar wire may be shaped and used to reinforce the handle. Alternatively, some manufacturers make pre-fabricated metal custom tray handles that may be easily adapted. Continued 228 Part II Clinical and Laboratory E F G E, A handle is added to provide a means to place and remove the tray, as well as to pass the tray from assistant to dentist. Its form should consider the lip length and need to manipulate the perioral region. F, Before the tray is placed in the curing oven, an air barrier coating is painted on the surface. The tray is then polymerized as per manufacturers’recommendations. G, As soon as the tray material has hardened, the tray is removed from the cast, and the wax spacer is removed from the rough tray. An acrylic-resin trimmer in the lathe is used to rough finish the tray. Holes are drilled through the tray, spaced approximately 4.5 mm apart. These holes will serve to lock the impression material in the tray. In addition, excess impression material is forced out of the holes when the impression is made, thereby minimally displacement of soft oral tissues. These two features will assist in correctly orienting the indi-vidualized impression tray in the mouth. Figure 15-3, cont’d 6. Place the cast with the adapted tray in the light polymer-izing unit and process according to the manufacturer’s directions—usually a maximum of 1 minute. 7. Remove the cast from the unit and gently remove the tray from the cast. Peel the softened wax out of the tray while the wax is still warm. 8. Paint the entire impression tray with the manufacturer’s air barrier coating material and return the tray to the unit turnstile for additional polymerizing, tissue side up. 9. When the polymerizing cycle is completed, remove the tray from the unit and clean it with a brush and water. 10. Perfect the borders of the tray with rotary instruments (vulcanite burs, acrylic-resin trimmers, etc.), and slightly polish the external surface of the tray. 11. Place perforations (No. 8 bur size) in the VLC resin tray at 5-mm (316-inch) intervals, with the exception of the alveolar groove areas, if an irreversible hydrocolloid 229 Chapter 15 Impression Materials and Procedures for Removable Partial Dentures A B C D E Figure 15-4 A similar technique to the one used for fabrication of the maxillary tray in Figure 15-3 is used for the mandibular tray. A, Outline of the tray is penciled on a duplicate mandibular diagnostic cast. B, A single sheet of baseplate wax is adapted to the outline of the tray, and another sheet of baseplate wax is adapted over the teeth. A window is cut in the spacer to expose the incisal edges of the lower central incisors to serve as a stop in seating the tray. C, A model release agent is painted on regions of the cast to be in contact with the resin. D, A visible light-cured (VLC) tray material wafer is adapted over the cast and spacer. E, A handle is formed with excess tray material, as previously described. impression material is to be used (see Figures 15-3 and 4). 12. The finished tray must be sanitized and tried in the mouth so that any necessary corrections to the tray can be accomplished before the impression is made. The technique for making an individual mandibular VLC resin tray follows the same procedures. The buccal shelf regions on the mandibular cast are not covered by the wax spacer because these areas provide the primary support for the mandibular removable partial denture (RPD) and Continued 230 Part II Clinical and Laboratory F, An air barrier coating is painted on the tray material and it is processed as described in Figure 15-3. G, Fol-lowing processing, multiple holes are placed throughout the tray. F G Figure 15-3, cont’d serve as posterior stops in orienting the tray in the patient’s mouth. During impression making, these areas will permit selective placement of tissues in the mandibular stress- bearing areas. If mercaptan rubber is to be used, perforations are not usually necessary to lock the material in the tray, as the adhesive provided by the manufacturer provides reliable retention, and some confinement of these materials is desir-able. However, a series of perforations are necessary in the median palatal raphe and incisive papilla areas of the maxil-lary tray so that excess impression material will escape through them, thus providing relief of the tissues in this area. For the same reasons, perforations are placed in the alveolar groove of the mandibular tray. With the use of adhesives, the impression material is not easily removed from the tray should a faulty impression have to be remade, but this is an inconvenience common to all newer elastic materials and does not prevent reuse of the impression tray. Opaque elastic impression materials and adhesives can prevent the detec-tion of undesirable pressure areas when an impression is evaluated. Master casts made from impressions in individual acrylic-resin trays are generally more accurate than those made in rigid stock trays. The use of individual trays should be considered a necessary step in making the majority of removable partial dentures when a secondary impression technique is not to be used. Reasons and methods for making a secondary impression will be con-sidered in Chapter 16. Final impressions for maxillary tooth–supported remov-able partial dentures often may be made in carefully selected and recontoured rigid stock impression trays. However, an individual acrylic-resin tray is preferred in those situations in the mandibular arch when the floor of the mouth closely approximates the lingual gingiva of the remaining anterior teeth. Recording the floor of the mouth at the elevation it assumes when the lips are licked is important in selecting the type of major connector to be used (see Chapter 5). Modification of the borders of an individual tray to fulfill the requirements of an adequate tray is much easier than is the modification of a metal stock tray. 231 Chapter 16 Support for the Distal Extension Denture Base 231 CHAPTER 16 Support for the Distal Extension Denture Base Chapter Outline Distal Extension Removable Partial Denture Factors Influencing the Support of a Distal Extension Base Contour and quality of the residual ridge Extent of residual ridge coverage by the denture base Type and accuracy of the impression registration Accuracy of the fit of the denture base Design of the removable partial denture framework Total occlusal load applied Anatomic Form Impression Methods for Obtaining Functional Support for the Distal Extension Base Selective tissue placement impression method Functional impression technique In a tooth-supported removable partial denture, a metal base or the framework that supports an acrylic-resin base is connected to and is part of a rigid framework that permits the direct transfer of occlusal forces to the abutment teeth through the occlusal rests. Even though the denture base of the modification space(s) in a Kennedy Class III removable partial denture provides support for the supplied teeth, the residual ridge beneath the base is not called on to aid in the support of the removable partial denture. Therefore the resiliency of the ridge tissues, the ridge configuration, and the type of bone that supports these tissues are not factors in denture support. Regardless of the length of the edentu-lous spans, if the framework is rigid, the abutment teeth are sound enough to carry the additional load, and the occlusal rests are properly formed, support comes entirely from the abutment teeth at either end of that span. Support may be augmented by splinting and by the use of additional abut-ments, but in any event the abutments are the sole support of the removable restoration. An impression (and resultant stone cast) records the ana-tomic form of the teeth and their surrounding structures and is needed to make a tooth-supported removable partial denture. The impression should also record the moving tissues that will border the denture in an unstrained posi-tion, so the relationship of the denture base to those tissues may be as accurate as possible. Although underextension of the denture base in a tooth-supported prosthesis is the lesser of two evils, an underextended base may lead to food entrap-ment and inadequate facial contours, particularly on the buccal and labial sides. To accurately record the moving tissues of the floor of the mouth, an individual impression tray should be used, rather than an ill-fitting or overex-tended stock tray. This has been discussed at length in Chap-ters 5 and 15. 232 Part II Clinical and Laboratory Distal Extension Removable Partial Denture The distal extension removable partial denture does not have the advantage of total tooth support because one or more bases are extensions covering the residual ridge distal to the last abutment. It therefore is dependent on the residual ridge for a portion of its support. The distal extension removable partial denture must depend on the residual ridge for some support, stability, and retention. Indirect retention, to prevent the denture from lifting away from the residual ridge, should also be incorporated in the design. The tooth-supported base is secured at either end by the action of a direct retainer and is supported at either end by a rest, whereas this degree of support and direct retention is lacking in the distal exten-sion prosthesis. For this reason, a distal abutment should be preserved whenever possible. In the event of the loss or absence of a distal abutment tooth, the patient must be made aware of the movements to be expected with a distal extension removable partial denture and the limitations imposed on the dentist when the residual ridge must be used for support, stability, and retention for that part of the prosthesis. Factors Influencing the Support of A Distal Extension Base Because one of the stated objectives of prosthodontic treat-ment is the restoration of function and comfort in an esthet-ically pleasing manner, maintenance of occlusal contact in distal extension removable partial dentures demands an understanding of the factors that influence residual ridge support. Support from the residual ridge becomes more important as the distance from the last abutment increases and will depend on the following several factors: 1. Contour and quality of the residual ridge 2. Extent of residual ridge coverage by the denture base 3. Type and accuracy of the impression registration 4. Accuracy of the fit of the denture base 5. Design of the removable partial denture framework 6. Total occlusal load applied Contour and Quality of the Residual Ridge The ideal residual ridge to support a denture base would consist of cortical bone that covers relatively dense cancel-lous bone, with a broad rounded crest with high vertical slopes, and is covered by firm, dense, fibrous connective tissue. Such a residual ridge would optimally support vertical and horizontal stresses placed on it by denture bases. Unfor-tunately this ideal is seldom encountered. Easily displaceable tissue will not adequately support a denture base, and tissues that are interposed between a sharp, bony residual ridge and a denture base will not remain in a healthy state. Not only must the nature of the bone of the residual ridge be considered in developing optimum Figure 16-1 The dotted portion outlines the crest of the resid-ual ridge, which should be recorded in its anatomic form in impression procedures. Similarly, retromolar pads should not be displaced by impression. Buccal shelf regions are outlined by a herringbone pattern, and selected additional pressures may be placed on these regions for vertical support of the denture base. Lingual slopes of the residual ridge (cross-hatched) may furnish some vertical support to the restoration; however, these regions principally resist the horizontal rotational tendencies of the denture base and should be recorded by the impression in undis-placed form. support for the denture base, but also its positional relation-ship to the direction of forces that will be placed on it. The crest of the bony mandibular residual ridge is most often cancellous. Because lining mucosa restricts both the buccal and lingual mucosae adjacent to teeth in the man-dible, loss of firm mucosa overlying the residual ridge is common following tooth extraction in the posterior man-dible. Pressures placed on tissues overlying the crest of the mandibular residual ridge usually result in irritation of these tissues, accompanied by the sequelae of chronic inflamma-tion. Therefore the crest of the mandibular residual ridge cannot be a primary stress-bearing region. The buccal shelf region (bounded by the external oblique line and the crest of the alveolar ridge) seems to be better suited for a primary stress-bearing role because it is covered by relatively firm, dense, fibrous connective tissue supported by cortical bone. In most instances this region bears more of a horizontal relationship to vertical forces than do other regions of the residual ridge (Figure 16-1). The slopes of the residual ridge then would become the primary stress-bearing region for resistance of horizontal and off-vertical forces. The immediate crest of the bone of the maxillary residual ridge may consist primarily of cancellous bone. Unlike in the mandible, oral tissues that overlie the maxillary residual alveolar bone are usually of a firm, dense nature (similar to the mucosa of the hard palate) or can be surgically prepared to support a denture base. The topography of a partially edentulous maxillary arch imposes a restriction on selection of a primary stress-bearing area. In spite of impression pro-233 Chapter 16 Support for the Distal Extension Denture Base Figure 16-2 The crest of the maxillary residual ridge (herring-bone pattern) is the primary supporting region for the maxillary distal extension denture base. Buccal and lingual slopes may furnish limited vertical support to the denture base. It seems logical that their primary role is to counteract the horizontal rotational tendencies of the denture base. The dotted portion outlines the incisive papilla and the median palatal raphe. Relief must be provided for these regions, especially if tissues covering the palatal raphe are less displaceable than those covering the crest of the residual ridge. cedures, the crestal area of the residual ridge will become the primary stress-bearing area for vertically directed forces. Some resistance to these forces may be obtained by the immediate buccal and lingual slopes of the ridge. Palatal tissues between the medial palatal raphe and the lingual slope of the posterior edentulous ridge are readily displace-able and cannot be considered as primary stress-bearing sites (Figure 16-2). The tissues covering the crest of the maxillary residual ridge must be less displaceable than the tissues that cover palatal areas, or relief of palatal tissues must be pro-vided in the denture bases or for palatal major connectors. A B Figure 16-3 Comparison of two removable partial dentures for the same patient. The denture on the right has severely underextended bases. Its replacement, with properly extended bases, is on the left. Occlusal forces are more readily distributed to denture-bearing areas by the replacement denture. Extent of Residual Ridge Coverage by the Denture Base The broader the residual ridge coverage, the greater is the distribution of the load, which results in less load per unit area (Figure 16-3). A denture base should cover as much of the residual ridge as possible and should be extended the maximum amount within the physiologic tolerance of the limiting border structures or tissues. Knowledge of these border tissues and the structures that influence their movement is paramount to the development of broad coverage denture bases. In a series of experiments, Kaires has shown that “maximum coverage of denture-bearing areas with large, wide denture bases is of the utmost importance in withstanding both vertical and horizontal stresses.” It is not within the scope of this text to review the ana-tomic considerations related to denture bases. The student is referred to several articles listed in the “Selected Reading Resources” regarding this subject. Type and Accuracy of the Impression Registration The residual ridge may be said to have two forms: the ana-tomic form and the functional form (Figure 16-4). The ana-tomic form is the surface contour of the ridge when it is not supporting an occlusal load. The functional form of the residual ridge is the surface contour of the ridge when it is supporting a functional load. The anatomic form is recorded by a soft impression material, such as a metallic oxide impression paste, if the entire impression tray is uniformly relieved. Depending on the viscosity of the particular impression material used and the rigidity of the impression tray, it is also the form that can be recorded by mercaptan rubber, silicone, and hydrocolloid impression materials. Distortion and tissue displacement by pressure may result from confinement of the impression material within the tray and from insufficient thickness of 234 Part II Clinical and Laboratory tional form of the ridge itself, may provide acceptable support for the removable partial denture. On the other hand, those who use the anatomic ridge form for the remov-able partial denture should seriously consider the need for some mechanical stress-breaker to avoid the possible canti-lever action of the distal extension base against the abutment teeth. Steffel has classified advocates of the various methods for treating the distal extension removable partial denture as follows: 1. Those who believe that ridge and tooth supports can best be equalized by the use of stress-breakers or resilient equalizers. 2. Those who insist on bringing about the equalization of ridge and tooth support by physiologic basing, which is accomplished by a pressure impression or by relining of the denture under functional stresses. 3. Those who uphold the idea of extensive stress distribu-tion for stress reduction at any one point. It would seem that there is little difference in the philoso-phy behind methods 2 and 3 as given by Steffel, for both the equalization of tooth and tissue support and stress distribu-tion over the greatest area are objectives of the functional type of impression. Many of the requirements and advan-tages that are associated with the distributed stress denture apply equally well to the functionally or physiologically based denture. Some of these requirements are (1) positive occlusal rests; (2) an all-rigid, nonflexible framework; (3) indirect retainers to add stability; and (4) well-adapted, broad coverage bases. Those who do not accept the theory of physiologic basing, for one reason or another, should use some form of stress-breaker between the abutment and the distal extension base. The advantages and disadvantages of doing so have been given in Chapter 9. impression material between the tray and the tissues, as well as from the viscosity of the impression material; however, none of these factors is selective or physiologic in its action. These accidental distortions of the tissues occur because of faulty technique. Use of the anatomic form of the residual ridge in fabricating complete dentures is quite common because of a belief that this is the most physiologic form for support of the dentures. However, many other dentists believe that certain regions of the residual ridge(s) in a partially edentulous patient are more capable of supporting dentures than other regions. Their impression methods are directed to place more stress on primary stress-bearing regions with specially constructed individual trays and at the same time record the anatomic form of other basal seat tissues, which cannot assume a stress-bearing role. The form of the residual ridge recorded under some loading, whether by occlusal loading, finger loading, specially designed individual trays, or the consis-tency of the recording medium, is called the functional form. This is the surface contour of the ridge when it is supporting a functional load. How much it will differ from the anatomic form will depend on the thickness and structural character-istics of the soft tissues overlying the residual bone. It will also differ from the anatomic form in proportion to the total load applied to the denture base. Of the two philosophies, the latter seems to be more logical. McLean and others recognized the need to record the tissues that support a distal extension removable partial denture base in their functional form, or supporting state, and then relate them to the remainder of the arch by means of a secondary impression. This was called a functional impression because it recorded the ridge relation under sim-ulated function. Any method, whether it records the functional relation-ship of the ridge to the remainder of the arch, or the func-A B Figure 16-4 Comparison of anatomic and functional ridge forms. A, Original master cast with the edentulous area recorded in its anatomic form, using elastic impression material. B, Same cast after the edentulous area has been repoured to its functional form as recorded by the secondary impression. 235 Chapter 16 Support for the Distal Extension Denture Base denture base (Figure 16-6). They have suggested that occlu-sal rests may be moved anteriorly to better use the residual ridge for support without jeopardizing either vertical or horizontal support of the denture by occlusal rests and guiding planes (Figure 16-7). It is possible, however, that the proximal plate minor connector adjacent to the edentulous area will not disengage from or break contact with the guiding plane. When one considers impression and cast formation variables; waxing, investing, and casting discrepancies; and finishing and pol-ishing procedures, it may be somewhat philosophical to assume that the minor connector proximal plate will have contact, and if it does, will disengage its contact with the guiding plane, especially if the tissues that support the exten-sion denture base are healthy and demonstrate favorable contour. It is more likely that the abutment tooth will move physiologically to contact the minor connector and that the disengagement will depend on the amount of tissue displace-ment. Geometrically, the actual amount of tissue displace-ment in the extension base area under occlusal load may not be enough to cause the minor connector to break contact with the guiding plane. Total Occlusal Load Applied Patients with distal extension removable partial dentures generally orient the food bolus over natural teeth rather than prosthetic teeth. This is likely due to the more stable nature of the natural dentition, the proprioceptive feedback it pro-vides for chewing, and the possible nocioceptive feedback from the supporting mucosa. This has an effect on the direc-tion and magnitude of the occlusal load to the removable partial denture, and thus on the load transferred to the abut-ments. Given this, the support from the residual ridge should be optimized and shared appropriately with the remaining natural dentition. The number of artificial teeth, the width of their occlusal surfaces, and their occlusal efficiency influence the total occlusal load applied to the removable partial denture. Kaires conducted an investigation under laboratory condi-tions and concluded that “the reduction of the size of the Accuracy of the Fit of the Denture Base Support of the distal extension base is enhanced by intimacy of contact of the tissue surface of the base and the tissues that cover the residual ridge. The tissue surface of the denture base must optimally represent a true negative of the basal seat regions of the master cast. Denture bases have been discussed in Chapter 9. In addition, the denture base must be related to the removable partial denture framework in the same manner as the basal seat tissues were related to the abutment teeth when the impression was made. Every precaution must be taken to ensure this relationship when the altered cast tech-nique of making a master cast is used. Design of the Removable Partial Denture Framework Some rotation movement of a distal extension base at the distal abutment is inevitable under functional loading. It must be remembered that the extent to which abutments are subjected to rotational and torquing forces that result from masticatory function is directly related to the position and resistance of the food bolus. The greatest movement takes place at the most posterior extent of the denture base. The retromolar pad region of the mandibular residual ridge and the tuberosity region of the maxillary residual ridge there-fore are subjected to the greatest movement of the denture base (Figure 16-5). Steffel and Kratochvil have suggested that as the rotational axis is moved from a disto-occlusal rest to a more anterior location, more of the residual ridge receives vertically directed occlusal forces to support the A C B Figure 16-5 Acute dip of the short denture base is compared with that of the long one in the upper figure. In the lower figure, when the point of rotation is changed from C to B by the loss of more teeth, it can be seen that a proportionally greater area of the residual ridge is used to support the denture base than occurs when the fulcrum line passes through C. The amount of movement is directly related to the quality of tissue support. Line AC represents the length of the denture base. (See also Figure 10-3.) Figure 16-6 If rotation of the distal extension base occurs around the nearest rest, as the rest is moved anteriorly more of the residual ridge will be used to resist rotation. Compare the vertical arcs of the long-dash broken line with the arcs of the solid line. (See also Figure 10-4.) 236 Part II Clinical and Laboratory of the base near the abutment tooth from transmitting the load to the underlying anatomic structures. The distal end of the base(s), however, that is able to move more freely, will transmit more of the masticatory load to the underlying extension base tissues and will transmit more torque to the abutment teeth through the rigid removable partial denture framework. It is obvious that the soft tissues that cover the ridge cannot by themselves carry any load applied to them. They act as a protective padding for the bone, which in the final analysis is the structure that receives and resists the mastica-tory load. Distribution of this load over a maximum area of bone is a prime requisite in preventing trauma both to the tissues of the extension base areas and to the abutment teeth. A removable partial denture fabricated from a one-stage impression, which records only the anatomic form of basal seat tissues, places more of the masticatory load on the abut-ment teeth and that part of the bone that underlies the distal end of the extension base. The balance of the bony ridge will not function in carrying its share of the load. The result will be a traumatic load to the bone underlying the distal end of the base and to the abutment tooth, which in turn can result in bone loss and loosening of the abutment tooth. A properly prepared, individualized impression tray can be used to record the primary stress-bearing areas in a functional form and the non–stress-bearing areas in an anatomic form, just as is often accomplished in making impressions for complete dentures. Some dentists believe that every removable partial denture should be relined before its final placement in the mouth. Some believe that tissue can be evenly displaced and use impression materials of heavy consistency. This latter prac-tice can introduce traumatic stresses to the underlying tissues. Some dentists use free-flowing pastes that produce an impression of the soft tissues at rest. A removable partial denture made according to this technique will be similar to a removable partial denture fabricated from a one-piece impression. The occlusal rest will act as a stop and prevent an even distribution of the masticatory load by the base to the edentulous ridge. Methods for Obtaining Functional Support for the Distal Extension Base The objective of any functional impression technique is to provide maximum support for the removable partial denture bases. This allows for the maintenance of occlusal contact between natural and artificial dentition and, at the same time, minimal movement of the base, which would create leverage on the abutment teeth. Although some tissue-ward movement of the distal extension base is unavoidable and is dependent on the six factors listed previously, it can be mini-mized by providing the best possible support for the denture base. A thorough understanding of the characteristics of each of the impression materials and impression methods leads occlusal table reduces the vertical and horizontal forces that act on the removable partial dentures and lessens the stress on the abutment teeth and supporting tissues.” Anatomic Form Impression The anatomic form impression is a one-stage impression method using an elastic impression material that will produce a cast that does not represent a functional relation-ship between the various supporting structures of the par-tially edentulous mouth. It will represent only the hard and soft tissues at rest. With the removable partial denture in position in the dental arch, the occlusal rest(s) will fit the rest seat(s) of the abutment teeth, while the denture base(s) will fit the surface of the mucosa at rest. When a masticatory load is applied to the extension base(s) with a food bolus, the rest(s) will act as a definite stop, which will limit the part Figure 16-7 A, The occlusal rest is placed on the mesio-occlusal surface of the lower first premolar; this will move the point of rotation anterior to the conventionally placed disto-occlusal rest if contact of the proximal minor connector on the distal guiding plane is designed to release under function. The occlusal rest is connected to the lingual bar by a minor connec-tor, which contacts the small mesiolingual prepared guiding plane. Note the vertical extension of the denture base minor connector contacting the distal guiding-plane surface. The lingual guiding plane is prepared to extend from the occlusal surface inferiorly to approximately one third of the height of the lingual surface and is as broad as the contacting minor connector. The distal guiding plane extends from the distal marginal ridge gin-givally about two thirds of the height of the distal surface. Such preparations must be designed not to lock the tooth in a viselike grip when the denture base moves toward the residual ridge. From Kaires AK: Effect of partial denture design on bilateral force distribution, J Prosthet Dent 6:373-389, 1956. 237 Chapter 16 Support for the Distal Extension Denture Base displaced (placed) by impression procedures for definitive border control respond favorably to the additional pressures placed on them by the resultant denture bases if these pres-sures are intermittent rather than continuous. The selective tissue placement impression method is based on these clinical observations, the histologic nature of tissues that cover the residual alveolar bone, the nature of the residual ridge bone, and its positional relationship to the direction of stresses that will be placed on it. It is further believed that by use of specially designed individual trays for impressions, denture bases can be developed that will use those portions of the residual ridge that can withstand addi-tional stress and at the same time relieve the tissues of the residual ridge that cannot withstand functional loading and remain healthy. There should be no philosophical difference between the requirement of support and coverage by bases of distal extension removable partial dentures and that of complete dentures, either maxillary or mandibular, because the objec-tive of maximum support is the same. The tray is unques-tionably the most important part of an impression. However, a tray must be so formed and modified that the impression philosophy of the dentist can be carried out. Making indi-vidualized acrylic-resin impression trays is described in Chapter 15, and a method of attaching custom trays to a removable partial denture framework is illustrated in Figure 16-8. Because the goal is to maximize soft tissue support while using the teeth to their supportive advantage, the framework fitted to the teeth while soft tissue support is registered pro-vides a means of coordinating both. This means that before the trays are attached, the framework must be fitted in the mouth as illustrated in Figure 16-9. Fitting the framework involves the following steps: to the conclusion that no single material can record the anatomic form of the teeth and tissues in the dental arch and, at the same time, the functional form of the residual ridge. Therefore, some secondary or corrected impression method must be used. The method selected is greatly influenced by determina-tion of the support potential of the residual ridge mucosa. Mucosa that is firm and minimally displaceable provides a different support potential than mucosa that is more easily displaced. Methods for obtaining functional support for either should satisfy the two requirements for providing adequate support to the distal extension removable partial denture base. These are (1) that it records and relates the supporting soft tissue under some loading, and (2) that it distributes the load over as large an area as possible. Selective Tissue Placement Impression Method Soft tissues that cover basal seat areas may be placed, displaced, or recorded in their resting or anatomic form. Placed and displaced tissues differ in the degree of alteration from their resting form and in their physiologic reaction to the amount of displacement. For example, the palatal tissues in the vicinity of the vibrating line can be slightly displaced to develop a posterior palatal seal for the maxillary complete denture and will remain in a healthy state for extended periods. On the other hand, these tissues develop an immediate inflammatory response when they have been overly displaced in developing the posterior palatal seal. Oral tissues that have been overly displaced or distorted attempt to regain their anatomic form. When they are not permitted to do this by the denture bases, the tissues become inflamed and their physiologic functions become impaired, accompanied by bone resorption. Tissues that are minimally A B Figure 16-8 A secondary impression for the distal extension mandibular removable partial denture is made in individual trays attached to the denture framework. A, The framework has been tried in the mouth and fits the mouth and master cast as planned. B, The outline of acrylic-resin trays is penciled on the cast. Continued 238 Part II Clinical and Laboratory E F C D G C, One thickness of baseplate wax is adapted to outlines to act as spacers so that room for the impression material exists in the finished trays. Windows are cut in the wax spacers corresponding to regions on the cast contacted by minor con-nectors (tissue stops) for denture bases. D, The framework is warmed and pressed to position on the relieved master cast. All regions of the cast that will be contacted by autopolymerizing acrylic-resin or visible light-cured (VLC) resin are painted with tinfoil substitute (Alcote) or model release agent (MRA). E, Acrylic-resin material is adapted to the cast and over the framework with finger pressure as described in Chapter 15. Excess material over the borders of the cast is removed with a sharp knife while the material is still soft. F, Polymerized acrylic-resin trays and framework are removed from the cast, and trays are trimmed to outline the wax spacer. G, Borders of the trays will be adjusted to extend 2 mm short of the tissue reflections. Holes will be placed in the trays corresponding to the crest of the residual ridge and retromolar pads to allow escape of excess impression material when an impression is made. Figure 16-8, cont’d 239 Chapter 16 Support for the Distal Extension Denture Base B A C D E Figure 16-9 The framework must be evaluated to assure complete seating, full contact with the remaining dentition for stabilization, support, and retention as planned, and to allow full natural tooth contact. A, Several types of disclosing media may be used, such as stencil correction fluid, rouge, and chloroform, and disclosing fluids, pastes, and waxes. Here, a spray disclosing medium has been applied and the framework is placed with mild pressure. Incomplete seating is seen when the framework binds. It is imperative that the framework not be forced to place at this initial seating. B, A portion of the proximal plate is preventing complete seating. C, The framework is carefully adjusted as over-adjustment can result in a poorly adapted framework. D, The framework seats completely after adequate adjustment. This may require repeated disclosing and careful adjustment; however, if improve-ment is not seen with each framework modification there should be concern regard-ing frame accuracy. E, Following complete seating and verification of appropriate tooth contacts by component parts (i.e., rests, proximal plates, stabilizing compo-nents) the occlusion must be checked and the framework adjusted until natural tooth contacts that exist without the framework seated are achieved with the frame-work in place. All adjusted regions can be carefully polished with rotary rubber points. 240 Part II Clinical and Laboratory C D A B E Figure 16-10 Selective tissue placement impression technique. A, Tray attached at the frame try-in, which in B is seen incompletely seated. C, Completed border molding, which defines the primary bearing areas of the buccal shelves bilaterally. This bearing area and the lingual extension are seen in the final impression (D), which can be seen to be in contact with use of the pressure indicating paste (E). 1. Use of disclosing media to identify interferences to com-pletely seating the removable partial denture framework. 2. Use of disclosing media to identify the appropriate contact(s) of the component parts of the framework during seating of the framework and when the framework is completely seated in its designated terminal position. 3. Adjusting the seated framework to the opposing occlusion. If there are opposing frameworks, the maxillary frame-work is removed from the mouth and the mandibular framework is adjusted to the natural maxillary dentition; then the maxillary framework is replaced and it is adjusted to the mandibular dentition with its framework in place. It is important to remember that the metal frameworks must allow all of the natural dentition to maintain the same designed contact relationship with the opposing arch as when the frameworks are out of the mouth. After the frame-work has been fitted and the custom trays have been attached, selective tissue placement impression and cast formation can be accomplished as described in Figure 16-10. 241 Chapter 16 Support for the Distal Extension Denture Base tray be formed to provide proper space for the particular impression materials, but provision must be made so that the functional form of primary stress-bearing areas can be recorded. Such an impression procedure, properly executed, can be used when metal bases are to be incorporated in the design of the restoration. There is little difference, if any, between recording the basal seats in the partially edentulous arch and recording like areas for complete dentures on an edentulous arch. A secondary impression made in custom trays attached to the framework only makes definitive border control and tissue placement a bit easier, compared with the individualized complete arch tray. The altered cast method of impression making is most commonly used for the mandibular distal extension partially edentulous arch (Kennedy Class I and Class II arch forms). A common clinical finding in these situations is greater variation in tissue mobility and tissue distortion or displace-ability, which requires some selective tissue placement to obtain the desired support from these tissues. This variabil-ity in tissue mobility is probably related to the pattern of mandibular residual ridge resorption. Altered cast impres-sion methods are seldom used in the maxillary arch because of the nature of the masticatory mucosa and the amount of firm palatal tissue present to provide soft tissue support. These tissues seldom require placement to provide the required support. If excessive tissue mobility is present, it is often best managed by surgical resection, as this is a primary supporting area. Support is obtained from the primary support areas in the manner by which the flow of the impression material is controlled during the impression-making procedure. Restricting the flow of the material in the primary stress-bearing areas (by minimizing the amount of relief over the area when the custom tray was made) causes greater pressure to be exerted on the tissues in this area (compared with other areas of unrestricted flow where a greater amount of relief or venting of the impression tray was provided). This is often referred to as the “selected pressure” or “dynamic” impres-sion method. By controlling the flow of the impression material with wax relief and venting, one can place or dis-place the soft tissues over the primary support areas so that they are the primary areas to provide support to the denture base when a removable partial denture is functionally loaded. An impression for a mandibular distal extension partially edentulous arch may also be adequately made in an indi-vidualized, complete arch tray. To do so, not only must the Functional Impression Technique When the residual ridge mucosa demonstrates a uni-formly firm consistency, an impression technique that involves capturing the tissue form while the patient is in occlusion can be considered. Such a technique records the mucosal position and shape under the influence of a static closure force, similar to functional masticatory forces. The more the mucosa displaces under function, the more rebound there is likely to be. Because the pros-thesis will be under occlusal load for only a portion of a day, minimal rebound is desired so as to maintain the clasp assembly–tooth relationship. When such a tech-nique is applied to firm, minimally displaceable mucosa, a minimal rebound effect is seen on the prosthesis posi-tion. The selective pressure technique described earlier can be applied to all varieties of residual ridges because it is customized to mucosal conditions, whereas the func-tional impression technique has limited application to a uniformly firm ridge consistency. 242 CHAPTER 17 Occlusal Relationships for Removable Partial Dentures Chapter Outline Desirable Occlusal Contact Relationships for Removable Partial Dentures Methods for Establishing Occlusal Relationships Direct apposition of casts Interocclusal records with posterior teeth remaining Occlusal relations using occlusion rims on record bases Jaw relation records made entirely on occlusion rims Establishing occlusion by the recording of occlusal pathways Materials for Artificial Posterior Teeth Arranging teeth to an occluding template Establishing Jaw Relations for a Mandibular Removable Partial Denture Opposing a Maxillary Complete Denture The fourth phase in the treatment of patients with remov-able partial dentures is the establishment of a functional and harmonious occlusion. Occlusal harmony between a remov-able partial denture and the remaining natural teeth is a major factor in the preservation of the health of their sur-rounding structures. In the treatment of patients with com-plete dentures, the inclination of the condyle path is the only factor not within the control of the dentist. All other factors may be altered to obtain occlusal balance and harmony in eccentric positions to conform to a particular concept and philosophy of occlusion. Balanced occlusion is desirable with complete dentures because unbalanced occlusal stresses may cause instability of the dentures and trauma to the supporting structures. These stresses can reach a point that causes movement of the denture bases. In removable partial dentures, however, because of the attachment of the removable partial denture to the abutment teeth, occlusal stresses can be transmitted directly to the abutment teeth and other supporting struc-tures, resulting in sustained stresses that may be more dam-aging than those transient stresses found in complete dentures. Failure to provide and maintain adequate occlu-sion on the removable partial denture is primarily a result of (1) lack of support for the denture base, (2) the fallacy of establishing occlusion to a single static jaw relation record, and (3) an unacceptable occlusal plane. In establishing occlusion on a removable partial denture, the influence of the remaining natural teeth is usually such that the occlusal forms of the teeth on the removable partial denture must be made to conform to an already established occlusal pattern. Occlusal adjustment or restoration may have altered this pattern. However, the pattern present at the time the removable partial denture is made dictates the See Chapter 2, under discussion on the six phases of removable partial denture service. 243 Chapter 17 Occlusal Relationships for Removable Partial Dentures 5. Simultaneous working and balancing contacts should be formulated for the maxillary bilateral distal extension removable partial denture whenever possible (Figure 17-3). Such an arrangement will compensate in part for the unfavorable position the maxillary artificial teeth must occupy in relation to the residual ridge, which is usually lateral to the crest of the ridge. However, this desirable relationship often must be compromised when the patient’s anterior teeth have an excessively steep vertical overlap with little or no horizontal overlap. Even in this situation, working side contacts can be obtained without resorting to excessively steep cuspal inclinations. 6. Only working contacts need to be formulated for the maxillary or mandibular unilateral distal extension removable partial denture (Figure 17-4). Balancing side contacts would not enhance the stability of the denture Figure 17-1 Posterior occlusion of a maxillary complete denture opposing a Class I mandibular removable partial denture. The stability of the maxillary complete denture can be promoted by developing balanced occlusion as shown. Figure 17-2 Bilateral distal extension mandibular removable partial denture opposed by natural dentition in the maxillary arch. Working contacts are achieved, balancing contacts are purpose-fully avoided because they would not enhance the stability of the restoration, and protrusive balance is avoided in favor of an acceptable appearance and a favorable occlusal plane. occlusion on the removable partial denture. The only excep-tions are those in which an opposing complete denture can be made to function harmoniously with the removable partial denture, or in which only anterior teeth remain in both arches and the incisal relationship can be made so that tooth contacts do not disturb denture stability or retention. In these situations, jaw relation records and the arrangement of the teeth may proceed in the same manner as with com-plete dentures, and the same general principles apply. With all other types of removable partial dentures, the remaining teeth dictate the occlusion. The dentist should strive for planned contacts in centric occlusion and no inter-ferences in lateral excursions. Although a functional rela-tionship of the removable partial denture to the natural dentition sometimes may be adjusted satisfactorily in the mouth, extraoral adjustment is often easier for both dentist and patient, is more accurate, and can be accomplished in a more comprehensive manner. Establishment of a satisfactory occlusion for the remov-able partial denture patient should include the following: (1) analysis of the existing occlusion; (2) correction of existing occlusal disharmony; (3) recording of centric relation or an adjusted centric occlusion; (4) harmonizing of eccentric jaw movements for a functional eccentric occlusion; and (5) correction of occlusal discrepancies created by the fit of the framework and during processing of the removable partial denture. Desirable Occlusal Contact Relationships for Removable Partial Dentures The following occlusal arrangements are recommended to develop a harmonious occlusal relationship among remov-able partial dentures and to enhance stability of the remov-able partial dentures: 1. Simultaneous bilateral contacts of opposing posterior teeth must occur in centric occlusion. 2. Occlusion for tooth-supported removable partial den-tures may be arranged similarly to the occlusion seen in a harmonious natural dentition, because stability of such removable partial dentures results from the effects of direct retainers at both ends of the denture base. 3. Bilateral balanced occlusion in eccentric positions should be formulated when a maxillary complete denture (Figure 17-1) opposes the removable partial denture. This is accomplished primarily to promote the stability of the complete denture. However, simultaneous contacts in a protrusive relationship do not receive priority over appearance, phonetics, and/or a favorable occlusal plane. 4. Working side contacts should be obtained for the man-dibular distal extension denture (Figure 17-2). These contacts should occur simultaneously with working side contacts of the natural teeth to distribute the stress over the greatest possible area. Masticatory function of the denture is improved by such an arrangement. 244 Part II Clinical and Laboratory A B Figure 17-3 Opposing Class I partially edentulous arches arranged to allow working side contacts of opposing posterior teeth with balancing contact arranged to minimize tipping of the maxillary removable partial denture and to broadly distribute forces accruing to its supporting structures (abutments and residual ridges). Working contacts Figure 17-4 When occlusion is developed for a Class II removable partial denture (maxillary or mandibular), only working side contacts are necessary, as the cross-arch framework stability gained from tooth engagement provides resistance to move-ment. Balancing side contacts do not enhance stability beyond that provided by the contralateral teeth. Figure 17-5 Mandibular posterior teeth should not be arranged distal to the upward incline (ascending ramus) of the residual ridge. The molar tooth has been placed just anterior to a mark on the cast land area designating the beginning incline. because it is entirely tooth supported by the framework on the balancing side. 7. In the Kennedy Class IV removable partial denture con-figuration, contact of opposing anterior teeth in the planned intercuspal position is desired to prevent con-tinuous eruption of the opposing natural incisors, unless they are otherwise prevented from extrusion by means of a lingual plate or auxiliary bar, or by splinting. Contact of the opposing anterior teeth in eccentric positions can be developed to enhance incisive function but should be arranged to permit balanced occlusion without excursive interferences. 8. Artificial posterior teeth should not be arranged farther distally than the beginning of a sharp upward incline of the mandibular residual ridge or over the retromolar pad (Figure 17-5). To do so would have the effect of shunting the denture anteriorly. A harmonious relationship of opposing occlusal and incisal surfaces alone is not adequate to ensure stability of distal extension removable partial dentures. In addition, the relationship of the teeth to the residual ridges must be con-sidered. Bilateral eccentric contact of the mandibular distal extension removable partial denture need not be formulated to stabilize the denture. The buccal cusps, however, must be favorably placed to direct stress toward the buccal shelf, which is the primary support area in the mandibular arch. In such positions, the denture is not subjected to excessive tilting forces (Figure 17-6). On the other hand, the artificial teeth of the bilateral, distal extension, maxillary removable partial denture often must be placed lateral to the crest of the residual ridge (Figure 17-7). Such an unfavorable posi-tion can cause tipping of the denture, which is restrained only by direct retainer action on the balancing side. To enhance the stability of the denture, it seems logical to provide simultaneous working and balancing contacts in these situations if possible. 245 Chapter 17 Occlusal Relationships for Removable Partial Dentures Although a hinge axis mounting may be desirable for complete oral rehabilitation procedures, any of the common types of facebow will facilitate mounting of the maxillary cast in relation to the condylar axis in the articulating instru-ment with reasonable accuracy and are acceptable for a removable partial denture. As was suggested in Chapter 12, it is still better that the plane of occlusion be related to the axis-orbital plane. Because the dominant factor in remov-able partial denture occlusion is the remaining natural teeth and their proprioceptive influence on occlusion, a compa-rable radius at the oriented plane of occlusion in an accept-able instrument will allow reasonably valid mandibular movements to be reproduced. Semiadjustable articulators can simulate but not dupli-cate jaw movement. Realization of the limitations of a spe-cific instrument and knowledge of the procedures that can overcome these limitations are necessary if an adequate occlusion is to be created. The recording of occlusal relationships for the partially edentulous arch may vary from the simple apposition of opposing casts (by occluding sufficient remaining natural teeth) to the recording of jaw relations in the same manner as for a completely edentulous patient. As long as some natural teeth remain in contact, however, the cuspal influ-ence that those teeth will have on functional jaw movements dictates the placement of the artificial teeth and the occlusal scheme. The horizontal jaw relation (planned intercuspal position or centric relation) in which the restoration is to be fabri-Methods for Establishing Occlusal Relationships Five methods of establishing interocclusal relations for removable partial dentures will be briefly described. Before any of these is described, it is necessary that the use of a facebow mounting of the maxillary cast and pertinent factors in removable partial denture occlusion be considered. The technique for applying the facebow has been described briefly in Chapter 12. Figure 17-6 The posterior teeth in this distal extension with a narrower buccal-lingual width than the original teeth have been selected, and they are placed relative to the primary support (buccal shelf) to distribute the functional load to the most ana-tomically favorable location in a manner that reduces leverage effects. A B Figure 17-7 A, Maxillary molar occluded in a normal horizontal relationship to the opposing molar. B, The resultant position is lateral to the supporting crest of the residual ridge. This position is functionally unfavorable because of the potentially unstable leverage effects; however, stability can be improved by arranging simultaneous working and balancing contacts in the occlusal scheme. 246 Part II Clinical and Laboratory will influence the successful recording of centric relation with an interocclusal wax record after chilling. Excess wax that contacts the mucosal surfaces may distort soft tissue, thereby preventing accurate seating of the wax record onto the stone casts. Distortion of wax during or after removal from the mouth may also interfere with accurate seating. Therefore a definite procedure for making interocclusal wax records is given as follows: • A uniformly softened, metal-reinforced wafer of baseplate or set-up wax is placed between the teeth, and the patient is guided to close in centric relation (Figure 17-8). Correct closure should have been rehearsed before placement of the wax so that the patient will not hesitate or deviate in closing. The wax then is removed and immediately chilled thoroughly in room-temperature water. It should be replaced a second time to correct the distortion that results from chilling and then again chilled after removal. • All excess wax should now be removed with a sharp knife. It is most important at this time that all wax that contacts mucosal surfaces be trimmed free of contact. The chilled wax record again should be replaced to make sure that no contact with soft tissue occurs. A wax record can be further corrected with a freely flowing occlusal registration material, such as a metallic oxide paste, which is used as the final recording medium. In making such a corrected wax record, the opposing teeth (and also the patient’s lips and the dentist’s gloves) should first be lightly coated with petroleum jelly or a silicone prep-aration. The occlusal registration material then is mixed and applied to both sides of the metal-reinforced wax record. It is quickly placed, and the patient is assisted with closing in the rehearsed path, which will be guided by the previous wax record. After the occlusal registration material has set, the corrected wax record is removed and inspected for accuracy. Any excess projecting beyond the wax matrix should be removed with a sharp knife until only the registration of the cusp tips remains. Such a record should seat on accurate cated should have been determined during diagnosis and treatment planning. Mouth preparations also should have been accomplished in keeping with this determination, including occlusal adjustment of the natural dentition, if such was indicated. Therefore one of the following condi-tions should exist: (1) centric relation and planned intercus-pal position coincide with no evidence of occlusal pathologic conditions, therefore the decision should be to fabricate the restoration in centric relation; (2) centric relation and the planned intercuspal position do not coincide, but the planned intercuspal position is clearly defined and the deci-sion has been made to fabricate the restoration in the planned intercuspal position; (3) centric relation and the planned intercuspal position do not coincide and the intercuspal position is not clearly defined, therefore the decision should be made to fabricate the restoration in centric relation; and (4) posterior teeth are not present in one or both arches, and the denture will be fabricated in centric relation. Occlusal relationships may be established by using the most appropriate of the following methods to fit a particular partially edentulous situation. Direct Apposition of Casts The first method is used when sufficient opposing teeth remain in contact to make the existing jaw relationship obvious, or when only a few teeth are to be replaced on short denture bases and no evidence of occlusal abnormalities is found. With this method, opposing casts may be occluded by hand. The occluded casts should be held in apposition with rigid supports attached with sticky wax to the bases of the casts until they are securely mounted in the articulator. At best, this method can only perpetuate the existing occlusal vertical dimension and any existing occlusal dishar-mony present between the natural dentition. Occlusal analy-sis and the correction of any existing occlusal disharmony should precede the acceptance of such a jaw relation record. The limitations of such a method should be obvious. Yet, such a jaw relation record is better than an inaccurate inter-occlusal record between the remaining natural teeth. Unless a record is made that does not influence the closing path of the mandible because of its bulk and/or the consistency of the recording medium, direct apposition of opposing casts at least eliminates the possibility that the patient may have a faulty jaw relationship. Interocclusal Records With Posterior Teeth Remaining A second method, which is a modification of the first, is used when sufficient natural teeth remain to support the remov-able partial denture (Kennedy Class III or IV) but the rela-tion of opposing natural teeth does not permit the occluding of casts by hand. In such situations, jaw relations must be established as for fixed restorations with some type of inter-occlusal record. The least accurate of these methods is the interocclusal wax record. The bulk, consistency, and accuracy of the wax Figure 17-8 A wax wafer softened in a water bath is used to register interocclusal position. Once removed and cooled, it is corrected with a more rigid and accurate registration material. Before it is used for mounting, the registration is trimmed to allow complete seating of the cast without distortion of the material. 247 Chapter 17 Occlusal Relationships for Removable Partial Dentures is large, or when opposing teeth do not meet. In these instances, occlusion rims on accurate jaw relation record bases must be used. Simple wax records of edentulous areas are never acceptable. Any wax, however soft, will displace soft tissue. It is impossible to accurately seat such a wax record on a stone cast of the arch. With this method, the recording proceeds much the same as with the second method, except that occlusion rims are substituted for missing teeth (Figure 17-9). It is essential that accurate bases be used to help support the occlusal relation-ship. Visible light-cured (VLC) bases may be adapted to the casts through the technique described in Chapter 15 for making impression trays. Utilizing the master cast, block out undesirable undercuts. However, do not add any wax for spacing. Paint a thin layer of the model release agent on the cast and the relief wax to aid in removal of the base from the cast after the designated time of polymerization recom-mended by the manufacturer. Carefully, adapt the VLC base material to the cast. Do not thin the material, and do not adapt the VLC base material over the remaining teeth. Process the base in the polymerization to set the material, then separate the record base from the cast and remove any remaining blockout wax. Clean the base and apply the air barrier coating over the entire record base, and process (tissue side up) as directed. Record bases may also be made entirely of autopolymer-izing acrylic-resin. Those materials used in dough form lack sufficient accuracy for this purpose unless they are corrected by relining. An acrylic-resin base may be formed by sprin-kling monomer and polymer into a shallow matrix of wax or clay after any undercuts are blocked out. If the matrix and blockout have been formed with care, interference with removal will not occur, and little trimming will be necessary. When the sprinkling method is used and sufficient time is allowed for progressive polymerization to occur, such bases casts without discrepancy or interference and will provide an accurate interocclusal record. When an intact opposing arch is present, use of an opposing cast may not be necessary. Instead a hard stone may be poured directly into the occlusal registration material record to serve as an opposing cast. However, although this may be an acceptable procedure in the fabrication of a unilateral fixed partial denture, the advantages of having casts properly oriented in a suitable articulator contraindicate the practice. The only exception to this occurs if the maxillary cast on which the removable partial denture is to be fabricated has been mounted previ-ously with the aid of a facebow. In such an instance, an intact mandibular arch may be reproduced in stone by pouring a cast directly into the interocclusal record. Some of the advantages of using a metallic oxide paste over wax as a recording medium for occlusal records include (1) uniformity of consistency; (2) ease of displacement on closure; (3) accuracy of occlusal surface reproduction; (4) dimensional stability; (5) the possibility of some modifica-tion in occlusal relationship after closure, if it is made before the material sets; and (6) reduced likelihood of distortion during mounting procedures. Three important details to be observed when one uses such a material are as follows: 1. Make sure that the occlusion is satisfactory before making the interocclusal record. 2. Be sure that the casts are accurate reproductions of the teeth being recorded. 3. Trim the record with a sharp knife wherever it engages undercuts, soft tissue, or deep grooves. Occlusal Relations Using Occlusion Rims on Record Bases A third method is used when one or more distal extension areas are present, when a tooth-supported edentulous space A B Figure 17-9 Relationship and distribution of remaining teeth for this patient requires that record bases and occlusion rims be used for accurate mounting of casts. A, Acrylic-resin record bases and hard baseplate wax occlusion rims for an edentulous maxilla and Kennedy Class I mandibular arch. These record bases are stable and were formed by sprinkling autopolymerizing acrylic-resin. B, Occlu-sion rims substitute for missing posterior teeth and provide an opportunity for posterior support when interocclusal records are made; this is most critical for the longer edentulous span on the mandibular right. 248 Part II Clinical and Laboratory movements. Jaw relation records are made entirely on occlu-sion rims when either arch has only anterior teeth present (Figure 17-10). In any of these situations, jaw relation records are made entirely on occlusion rims. The occlusion rims must be sup-ported by accurate jaw relation record bases. Here, the choice of method for recording jaw relations is much the same as that for complete dentures. Either some direct inter-occlusal method or a stylus tracing may be used. As with complete denture fabrication, the use of a facebow, the choice of articulator, the choice of method for recording jaw relations, and the use of eccentric positional records are optional, based on the training, ability, and desires of the individual dentist. are stable and accurate. Other record bases include the use of cast metal and compression molded or processed acrylic-resin bases for jaw relation records. Relative to the third method, some mention must be made of the ridge on which the record bases are formed. If the prosthesis is to be tooth supported or if a distal extension base is to be made on the anatomic ridge form, the bases will be made to fit that form of the residual ridge. But if a distal extension base is to be supported by the functional form of the residual ridge, it is necessary that the recording of jaw relations be deferred until the master cast has been corrected to that functional form. Record bases must be as nearly identical as possible to the bases of the finished prosthesis. Jaw relation record bases are useless unless they are made on the same cast or a duplicate cast on which the denture will be processed, or are them-selves the final denture bases. The latter may be a cast alloy or a processed acrylic-resin base. Jaw relation records made by this method accomplish essentially the same purpose as the two previous methods. The fact that record bases are used to support edentulous areas does not alter the effect. In any method, the skill and care used by the dentist in making occlusal adjustments on the finished prosthesis will govern the accuracy of the result-ing occlusion. Methods for Recording Centric Relation on Record Bases Centric relation may be recorded in many ways when record bases are used. The least accurate is the use of softened wax occlusion rims. Modeling plastic occlusion rims, on the other hand, may be uniformly softened by flaming and tem-pering, resulting in a generally acceptable occlusal record. This method is time proved, and when competently done, it is equal in accuracy to any other method. When wax occlusion rims are used, they should be reduced in height until just out of occlusal contact at the desired vertical dimension of occlusion. A single stop is then added to maintain their terminal position as a jaw relation record is made in some uniformly soft material, which sets to a hard state. Quick-setting impression plaster, bite regis-tration paste, or autopolymerizing resin may be used. With any of these materials, opposing teeth must be lubricated to facilitate easy separation. Whatever the recording medium, it must permit normal closure into centric relation without resistance and must be transferable with accuracy to the casts for mounting purposes. Jaw Relation Records Made Entirely on Occlusion Rims The fourth method is used when no occlusal contact occurs between the remaining natural teeth, such as when an oppos-ing maxillary complete denture is to be made concurrently with a mandibular removable partial denture. It may also be used in those rare situations in which the few remaining teeth do not occlude and will not influence eccentric jaw Figure 17-10 Opposing Kennedy Class I dental arches with remaining anterior teeth only. Recording of maxillomandibular relations was accomplished by using stable record bases and occlusion rims. Establishing Occlusion by the Recording of Occlusal Pathways The fifth method of establishing occlusion on the remov-able partial denture is the registration of occlusal path-ways and the use of an occluding template rather than a cast of the opposing arch. When a static jaw relation record is used, with or without eccentric articulatory movements, the prosthetically supplied teeth are arranged to occlude according to a specific concept of occlusion. On the other hand, when a functional occlusal record is used, the teeth are modified to accept every recorded eccentric jaw movement. These movements are made more complicated by the influence of the remaining natural teeth. Occlusal harmony on complete dentures and in complete mouth rehabilitation may be obtained by the use of several dif-ferent instruments and techniques. Schuyler has empha-sized the importance of establishing first the anterior tooth relation and incisal guidance before proceeding with any complete oral rehabilitation. Others have shown 249 Chapter 17 Occlusal Relationships for Removable Partial Dentures the advantages of establishing canine guidance as a key to functional occlusion before proceeding with any func-tional registration against an opposing prosthetically restored arch. This is done on the theory that the canine teeth serve to guide the mandible during eccentric move-ments when the opposing teeth come into functional contact. It also has been pointed out that the canine teeth transmit periodontal proprioceptor impulses to the muscles of mastication and thus have an influence on mandibular movement even without actual contact guid-ance. However, as long as the occlusal surfaces of unre-stored natural teeth remain in contact, as in many a partially dentulous mouth, these teeth will always provide the primary influence on mandibular movement. The degree of occlusal harmony that can be obtained on a fixed or removable restoration will depend on the occlu-sal harmony that exists between these teeth. Regarding occlusion, Thompson has written, “Observ-ing the occlusion with the teeth in static relations and then moving the mandible into various eccentric posi-tions is not sufficient. A dynamic concept is necessary to produce an occlusion that is in functional harmony with the facial skeleton, the musculature, and the temporo-mandibular joints.” By adding “and with the remaining natural teeth,” the requirements for removable partial denture occlusion are more completely defined. Some of the methods described previously may be applied to the fabrication of removable partial dentures in both arches simultaneously, whereas the registration of occlusal pathways necessitates that an opposing arch be intact or restored to the extent of planned treatment. A diagnostic wax-up of both maxillary and mandibular arches will facilitate visualization of the proposed mouth preparation and restorative procedures required to accommodate the planned occlusal scheme, correct ori-entation of the occlusal plane, correct arch form, and complete tooth modifications to accommodate the removable partial denture design—all at the desired ver-tical dimension of occlusion. If removable partial den-tures are planned for both arches, a decision is necessary as to which denture is to be made first and which is to bear a functional occlusal relation to the opposing arch. Generally the mandibular arch is restored first and the maxillary removable partial denture is occluded to that restored arch. If the maxillary arch is to be restored with a complete denture or a fixed partial denture or crowns, a full diagnostic wax-up must be done before occlusion is established on the opposing removable partial denture. If opposing fixed partial dentures or opposing occluding crowns are to be fabricated, it may be advantageous to From Thompson JR: Temporomandibular disorders: diagnosis and dental treatment in the temporomandibular joint. In Sarnat B, editor: The temporomandibular joint, ed 2, Springfield, Ill, 1964, Bernard G. Sarnat and Charles C. Thomas, pp 146–184. develop the occlusion and fabricate them simultaneously to ensure optimal positioning, cuspal relationship, and functional integrity. Regardless of the method used for recording jaw rela-tions, when one arch is completely restored first, that arch is treated as an intact arch even though it is wholly or partially restored by prosthetic means. The dentist must consider at the time of treatment planning the possible advantages of establishing the final occlusion to an intact arch. Step-by-Step Procedure for Registering Occlusal Pathways After the framework has been adjusted to fit the mouth, the technique used for the registration of occlusal path-ways is as follows: 1. Support the wax occlusion rim with a denture base that has the same degree of accuracy and stability as the finished denture base. Ideally, this would be the final denture base, which is one of the advantages of making the denture with a metal base. Otherwise, make a temporary base of VLC resin or sprinkled autopolymerizing acrylic-resin (see Figs. 18-34 through 18-38), either of which is essentially identical to the final acrylic-resin base. In any distal extension removable partial denture, make this base on a cast that has been corrected to desired functional or sup-porting forms of the edentulous ridge. Place a film of hard sticky wax on the base before the wax occlusion rim is secured to it. The wax used for the occlusion rim should be hard enough to support biting stress and should be tough enough to resist fracture. Hard inlay wax has proved to be suitable for most patients. However, some individuals with weak musculature or tender mouths may have difficulty in reducing this wax. In such situations, use a slightly less hard wax. Make the occlusion rim wide enough to record all extremes of mandibular movement. 2. Inform the patient that the occlusion rim must be worn for 24 hours or longer. It should be worn con-stantly, including at nighttime, except for removal during meals. With wearing and biting into a hard wax occlusion rim, a record is made of all extremes of jaw movement. The wax occlusion rim must maintain positive contact with the opposing dentition in all excursions and must be left high enough to ensure that a record of the functional path of each cusp will be carved in wax. This record should include not only voluntary excursive movements but also involun-tary movements and changes in jaw movement caused by changes in posture. Extreme jaw positions and habitual movements during sleep should also be recorded. The occlusal paths, thus recorded, will represent each tooth in its three-dimensional aspect. Although the cast 250 Part II Clinical and Laboratory poured against this will resemble the opposing teeth, it will be wider than the teeth that carved it because it rep-resents those teeth in all extremes of movement. The recording of occlusal paths in this manner eliminates entirely the need to reproduce mandibular movement on an instrument. Instruct the patient in the removal and placement of the removable partial denture that supports the occlusion rim, and explain that by chewing in all direc-tions, the wax will be carved by the opposing teeth. The opposing teeth must be cleaned occasionally of accumu-lated wax particles. It is necessary that the patient com-prehend what is to be accomplished and understand that both voluntary and involuntary movements must be recorded. Before dismissing the patient, add or remove wax where indicated to provide continuous contact through-out the chewing range. To accomplish this, repeatedly warm the wax with a hot spatula and have the patient close and chew into the warmed wax rim with the oppos-ing dentition, each time adding to any area that is defi-cient. Any area left unsupported by its flow under occlusal forces must be reinforced with additional wax. It is important that the wax rim be absolutely dry and free of saliva before additional wax is applied. Each addition of wax must be made homogeneous with the larger mass to prevent separation or fracture of the occlusion rim during the time it is being worn. Leave the wax occlusion rim from 1 to 3 mm high, depending on whether the occlusal vertical dimension is to be increased. 3. After 24 hours, the occlusal surface of the wax rim should show a continuous gloss, which indicates func-tional contact with the opposing teeth in all extremes of movement. Any areas deficient in contact should be added to at this time. The reasons for maintaining positive occlusal contact throughout the time the occlusion rim is being worn are that (a) all opposing teeth may be placed in function; (b) an opposing denture, if present, will become fully seated; and (c) vertical dimension of occlusion in the molar region will be increased, thus the head of the mandibular condyle will be repositioned and temporomandibular tissue can return to a normal relationship. If during this period the wax occlusion rim has not been reduced to natural tooth contact, warm it by direct-ing air from the air syringe through a flame onto the surface of the wax. If the wax rim is held with the fingers during warming, a gradual softening process will result, rather than a melting of the surfaces already established. Repeatedly warm the occlusion rim and replace it in the mouth until the occlusal height has been reduced and lateral excursions have been recorded. At this time, use additional wax to support those areas left unsupported by the flow of wax to the buccal or lingual surfaces. Trim the areas obviously not involved, thus narrowing the occlu-sion rim as much as possible. Remove those areas project-ing above the occlusal surface, which by their presence might limit functional movement. When seating of the denture and changes in mandibu-lar position have been accomplished by the previous period of wear, it is possible to complete the occlusal registration in an operatory. However, if all involuntary movements and those caused by changes in posture are to be recorded, the patient should again wear the occlu-sion rim for a period of time. 4. After a second 24- to 48-hour period of wear, the registration should be complete and acceptable. The remaining teeth that serve as vertical stops should be in contact, and the occlusion rim should show an intact glossy surface representing each cusp in all extremes of movement. Natural teeth formerly in contact will not necessarily be in contact on completion of the occlusal registration. Those teeth that have been depressed over a period of years and those that have been moved to accommodate overclosure or mandibular rotation may not be in contact upon reestablishment of mandibular equilibrium. Such teeth may possibly return to occlusal contact in the future or may have to be restored to occlusal contact after initial placement of the denture. Because the mandibular posi-tion may have been changed during the process of occlu-sal registration, the cuspal relation of some of the natural teeth may be different from before. This fact must be recognized in determining the correct restored vertical dimension of occlusion. Occlusion thus established on the removable partial denture will have more complete harmony with the opposing natural or artificial teeth than can be obtained by adjustments in the mouth alone, because occlusal adjustment to accommodate voluntary movement does not necessarily prevent occlusal disharmony in all pos-tural positions or during periods of stress. Furthermore, occlusal adjustment in the mouth without occlusal analy-sis is limited by the dentist’s ability to correctly interpret occlusal markings made intraorally, whether by articulat-ing ribbon or by other means. The registration of occlusal pathways has additional advantages. It makes obtaining jaw relations possible under actual working conditions, with the denture frame-work in its terminal position, the opposing teeth in func-tion, and an opposing denture, if present, fully seated. In some instances, it also makes possible the recovery of the lost vertical dimension of occlusion, unilaterally or bilat-erally, when overclosure or mandibular rotation has occurred. The completed registration is now ready for conver-sion to an occluding template. This is usually done by boxing the occlusal registration with modeling clay after it has been reseated and secured onto the master or pro-cessing cast. Only the wax registration and areas for verti-251 Chapter 17 Occlusal Relationships for Removable Partial Dentures Establishing Jaw Relations for a Mandibular Removable Partial Denture Opposing a Maxillary Complete Denture It is common for a mandibular removable partial denture to be made to occlude with an opposing maxillary complete denture. The maxillary denture may already be present, or it may be made concurrently with the opposing removable partial denture. In any event, the establishment of jaw rela-tions in this situation may be accomplished by one of several previously outlined methods. If an existing maxillary complete denture is satisfactory and the occlusal plane is oriented to an acceptable anatomic, functional, and esthetic position (which rarely occurs), then the complete denture need not be replaced and the opposing arch is treated as an intact arch as though natural teeth were present. A facebow transfer is made of that arch, and the cast is mounted on the articulator. To accomplish this, a facebow record is made with the complete denture in place. After the facebow apparatus has been removed from the patient, the complete denture is removed and an irreversible hydrocol-loid impression of the denture is made. When the impres-sion material has set, the denture is removed, cleaned, and Materials for Artificial Posterior Teeth Contemporary acrylic-resin teeth are generally preferred to porcelain teeth, because they are more readily modified and are thought to more nearly resemble enamel in their abra-sion potential against opposing teeth. Acrylic-resin teeth with gold occlusal surfaces are preferably used in opposition to natural teeth restored with gold occlusal surfaces, whereas porcelain teeth are generally used in opposition to other porcelain teeth. Acrylic-resin tooth surfaces, however, may in time become impregnated with abrasive particles, thereby becom-ing an abrasive substance themselves. This may explain why acrylic-resin teeth are sometimes capable of wearing oppos-ing gold surfaces. An evaluation of occlusal contact or lack of contact, however, should be meticulously accomplished at each 6-month recall appointment, regardless of the choice of material for posterior tooth forms. Although some controversy may continue with regard to the use of porcelain or acrylic-resin artificial teeth, there is broad agreement that narrow (reduced bucco-lingual) occlusal surfaces are desirable. Posterior teeth that will satisfy this requirement should be selected, and the use of tooth forms that have excessive bucco-lingual dimension should be avoided. Acrylic-resin teeth are easily modified and readily lend themselves to construction of cast gold surfaces on their occlusal portions. A simple procedure for fabricating gold occlusal surfaces and attaching them to acrylic-resin teeth is described in Chapter 18 under posterior tooth forms. cal stops are left exposed. It is then filled with a hard die stone to form an occluding template (see Chapter 18). It is necessary that stone stops are used to maintain the vertical relation rather than relying on some adjust-able part of the articulating instrument, which might be changed accidentally. Also, if stone stops are used and both the denture cast and the template are mounted before they are separated, a simple hinge instrument may be used. Arranging Teeth to an Occluding Template The occlusal surface of the artificial teeth, porcelain or resin, must be modified to occlude with the template. With this method, they are actually only raw materials from which an occlusal surface is developed that is in harmony with an existing occlusal pattern. Therefore the teeth must be occluded too high and then modified to fit the template at the established vertical dimension of occlusion. Teeth arranged to an occluding template ordinarily should be placed in the center of the functional range. Whenever possible, the teeth should be arranged bucco-lingually in the center of the template. When natural teeth have registered the functional occlusion, this may be con-sidered the normal physiologic position of the artificial dentition, regardless of its relation to the residual ridge. On the other hand, if some artificial occlusion in the opposing arch has been recorded, such as that of an opposing denture, the teeth should be arranged in a favorable relation to their foundation, even if this means arranging them slightly buccally or lingually from the center of the template. The teeth are usually arranged for intercuspation with the opposing teeth in a normal cuspal relationship. Whenever possible, the mesiobuccal cusp of the maxillary first molar should be located in relation to the buccal groove of the mandibular first molar, and all other teeth arranged accordingly. With a functionally generated occlusion, however, it is not absolutely necessary that normal opposing tooth relationships be reestablished. In the first place, the opposing teeth in a dental arch that is not contiguous may not be in normal alignment, and intercuspation may be difficult to accomplish. In the second place, the occlusal surfaces will need to be modi-fied so that they will function favorably regardless of their anteroposterior position. Because cusps modified to fit an occlusal template will be in harmony with the opposing dentition, it is not necessary that the teeth themselves be arranged to conform to the usual concept of what consti-tutes a normal anteroposterior relationship. 252 Part II Clinical and Laboratory posterior teeth arranged close to the residual ridge without regard for the interarch relationship and with an occlusal plane that is too low. Usually, however, a new maxillary denture must be made concurrently with the mandibular removable partial denture, and jaw relations may be estab-lished in one of two ways. If the mandibular removable partial denture will be tooth supported (a Kennedy Class III arch accommodating a bilat-eral removable prosthesis), the mandibular arch is restored first. The same applies to a mandibular arch that is restored with fixed partial dentures. In either situation, the mandibu-lar arch is completely restored first, and jaw relations are established, as they would be to a full complement of oppos-ing teeth. Thus the maxillary complete denture is made to occlude with an intact arch. On the other hand, as is more often the situation, the mandibular removable partial denture may have one or more distal extension bases. The situation then necessitates that the occlusion be established on both dentures simultaneously. All mouth preparations and restorative procedures required to correctly orient the occlusal plane, correct the arch form, accommodate the desired occlusal scheme, and accommodate the removable partial denture design must be accomplished on the remaining natural teeth. In addition, all supporting tissue must be in an acceptable state of health before the final impression is made. After final impressions, which include the alter cast impression or the corrected cast impression, are made, the maxillary occlusion rim is con-toured, occlusal vertical relation with the remaining lower teeth is established, and a facebow transfer of the maxillary arch is made. The maxillomandibular relations may be recorded by any of the several methods previously outlined and the articulator mounting completed. Occlusion may be established as for complete dentures, with care taken to establish a favorable tooth-to-ridge relationship in both arches, an optimum occlusal plane, and cuspal harmony between all occluding teeth. After try-in, several methods may be used to complete the restorations. Both dentures may be processed concurrently and remounted for occlusal correction, or the removable partial denture may be processed first. After the dentures are completed and remounted, the teeth—still in wax on the complete denture—are adjusted to any discrepancies that are occurring. Occlusal discrepancies created during processing must be corrected before the patient is permitted to use the denture(s). Methods by which these discrepancies may be corrected are discussed in Chapter 18. returned to the patient. A cast is formed in the impression and then is mounted on the articulator with the facebow record. Maxillomandibular relations may be recorded on accurate record bases attached to the mandibular removable partial denture framework with the use of a suitable record-ing medium. Centric relation is recorded and is transferred to the articulator. Eccentric records can then be made to program the articulator. In rare instances, when the mandibular removable partial denture replaces all posterior teeth and the anterior teeth are noninterfering, a central bearing point tracer may be mounted in the palate of the maxillary denture, and centric relation recorded by means of an intraoral stylus tracing against a stable mandibular base. If a stylus jaw relation recording method is used, the stylus must be carefully removed from the denture and attached to the same palatal location of the stone cast that was transferred to the articula-tor via the facebow. The mandibular cast can then be ori-ented by way of the horizontal jaw relation record and attached to the articulator. When an existing complete denture is opposing the arch on which a removable partial denture is fabricated, a cast of the complete denture may be used during the fabrication procedures. However, when the occlusion is corrected after processing, and the removable partial denture is finalized during initial placement, the complete denture should be retrieved and mounted on the articulator with a centric rela-tion record at the desired vertical dimension of occlusion. This will ensure a more accurate cuspal relationship and will prevent abrasion of the cusp contacts that would occur if a stone cast of the denture is used. This procedure is com-pleted when the patient is in the office so as not to deprive the patient of use of the existing complete denture. If the relationship of the posterior teeth on the maxillary denture to the mandibular ridge is favorable and the com-plete denture is stable, jaw relations may be established by recording occlusal pathways in the mandibular arch just as for any opposing intact arch. The success of this method depends on the stability of the denture bases, the quality of tissue support, the relation of the opposing teeth to the mandibular ridge, and the interrelation of existing artificial and natural teeth. More often than not, the existing maxillary complete denture will not be acceptable to use because of poor tooth position. The denture will have been made to occlude with malpositioned mandibular teeth, which have since been lost, or the teeth will have been arranged without consideration for the future occlusal relation with a mandibular removable partial denture. Too often, one sees a maxillary denture with 253 Chapter 18 Laboratory Procedures 253 CHAPTER 18 Laboratory Procedures Chapter Outline Duplicating a Stone Cast Duplicating materials and flasks Duplicating procedure Waxing the Removable Partial Denture Framework Forming the wax pattern for a mandibular class II removable partial denture framework Attaching wrought-wire retainer arms by soldering Waxing metal bases Spruing, Investing, Burnout, Casting, and Finishing of the Removable Partial Denture Framework Spruing Investing the sprued pattern Burnout Casting Removing the casting from the investment Finishing and polishing Making Record Bases Technique for making a sprinkled acrylic-resin record base Occlusion Rims Making a Stone Occlusal Template From a Functional Occlusal Record Arranging Posterior Teeth to an Opposing Cast or Template Posterior tooth forms Arranging teeth to an occluding surface Types of Anterior Teeth Waxing and Investing the Removable Partial Denture Before Processing Acrylic-Resin Bases Waxing the removable partial denture base Investing the removable partial denture Processing the Denture Remounting and Occlusal Correction to an Occlusal Template Precautions to be taken in remounting Polishing the Denture Denture borders Facial surfaces Finishing gingival and interproximal areas This chapter covers only those phases of dental laboratory procedures that are directly related to removable partial denture fabrication. Familiarity with laboratory procedures relative to making fixed restorations and complete dentures is presumed. Such information is available in numerous textbooks on those subjects and is not duplicated here. For example, the principles and techniques involved in the waxing, casting, and finishing of single inlays, crowns, and fixed partial dentures are adequately covered in lecture material and textbooks and in manuals available to the dental student, the dental laboratory technician, and the practicing dentist. Similarly, knowledge of the principles and techniques for mounting casts, articulating teeth, and waxing, processing, and polishing complete dentures is pre-sumed as a necessary background for the laboratory phases of removable partial denture fabrication. Therefore this chapter is directed specifically toward the laboratory proce-dures involved in the fabrication of a removable partial denture. Duplicating a Stone Cast A stone cast may be duplicated for several purposes. One is the duplication in stone of the original or corrected master cast to preserve the original. One use of such a cast is for fitting a removable partial denture framework without danger of fracture or abrading the surface of the original master cast. Another use might be for processing a tempo-rary prosthesis where wax relief and blockout on the original 254 Part II Clinical and Laboratory cast allow production of a duplicate, which following pro-cessing will make insertion of the prosthesis easier. Another purpose of duplicating a cast is to allow an investment cast to be formed for framework fabrication. Careful preparation of the master cast for production of this investment cast involves consideration of the defined path of insertion, heights of contour, and retentive and stabiliza-tion areas designed into the mouth preparations. The frame-work produced should be carefully evaluated on the cast for fit, just as a fixed partial denture is evaluated on a working cast with dies (Figure 18-1). Blockout should be accomplished on the master cast before an investment cast is made. On this investment cast, the wax or plastic pattern is formed. The use of preformed plastic patterns eliminates some of the danger of damaging the surface of the investment cast in the process of forming the pattern. With freehand waxing, considerable care must be taken not to score or abrade the investment cast. The metal framework is ultimately cast against its surface, and the finished casting and the original cast are then returned to the dentist after all fitting has been completed on the duplicate cast. A B Figure 18-1 A, Framework returned from the laboratory with the master cast. B, Replacement on the cast reveals areas of contact that can erode the cast surface. Careful fitting to the cast will allow determination of potential framework adjustment regions before fitting in the mouth. If the framework is indiscriminately finished in the laboratory or before intraoral placement, critical tooth contact regions may be lost, resulting in a poorly retained and stabilized prosthesis. Most often, such an overly adjusted framework is poorly stabilized, not poorly retained. Waxing the Removable Partial Denture Framework When preformed plastic patterns are used (Figure 18-7), parts of the denture framework must be waxed freehand to prevent excessive bulk and to create the desired contours for a satisfactory custom-made removable partial denture framework (Figure 18-8). Although most dentists do not fabricate their own remov-able partial denture castings, it is essential that they have an understanding of the dental laboratory procedures involved. This enables them to design the removable partial denture Duplicating Materials and Flasks Duplicating materials include both colloidal and silicone materials. Colloidal materials are made fluid by heating and return to a gel while cooling. The cast to be dupli-cated must be placed in the bottom of a suitable flask, called a duplicating flask. A duplicating flask should contain the fluid material to facilitate its cooling, to support the mold while it is being filled with the cast material, and to facilitate removal of the cast from the mold without permanent deformation or damage to the mold. Numerous duplicating flasks are available on the market. The technique for duplicating is the same for any cast, whether or not blockout is present. However, if wax or clay blockout is present, the temperature of the duplicat-ing material must not be any higher than that recom-mended by the manufacturer, to prevent melting and distortion of the blockout material. Ordinary baseplate wax may be used for paralleled blockout and ledges, but care must be taken that the temperature of the duplicating material is not too high because it will then melt the wax. The use of prepared blockout material, such as Ney blockout wax or Wills undercut material, may be preferred. Duplicating Procedure The equipment needed and the steps required for dupli-cation using a silicone material are described in Figures 18-2 through 18-6. 255 Chapter 18 Laboratory Procedures Figure 18-2 Duplicating material with a vacuum curing unit in the background, which is used to ensure a dense mold. Figure 18-3 Duplicating mold following removal of the relieved and blockout cast. Figure 18-4 Refractory cast (right) following removal from the duplication mold (left). This cast incorporates all the features of the relieved and blockout master cast generated from the clinical framework impression. A B Figure 18-5 Close up of the refractory cast (A) with initial wax added to rests and the posterior bead line (B). framework, complete a laboratory work authorization that communicates the desired design and authorizes its fabrica-tion, and evaluate the quality of the framework (Figure 18-9). Understanding and evaluation of the key features required in a completed removable partial denture frame-work ensure the patient of a chance to function comfortably with the finished product (Figure 18-10). The contrary is also true. Forming the Wax Pattern for a Mandibular Class II Removable Partial Denture Framework Examples that illustrate many of the essentials of waxing a removable partial denture framework are presented in Figures 18-11 through 18-20. This exercise includes the 256 Part II Clinical and Laboratory A B Figure 18-6 Example of a completed maxillary waxed framework on a refractory cast with the sprue access positioned. A, From occlusal. B, From anterior. Figure 18-7 Preformed plastic patterns are available in many shapes and sizes for clasps, for minor connectors connecting to teeth, or for distal extension bases, bars, mesh, and palatal cover-age. Because they are made of soft plastic material, they tend to stretch on removal from their backing. Therefore care must be exercised when one is removing patterns. Their use generally requires that tacky liquid be applied first to the investment cast at their area of placement. waxing of three types of direct retainers (circumferential, combination, and bar type), a mandibular lingual bar major connector, a maxillary anterior-posterior palatal strap major connector, both distal extension and tooth-bound modification spaces, and adaptation of round, 18-gauge, wrought-wire retentive arms for combination-type direct retainers. Attaching Wrought-Wire Retainer Arms by Soldering Wrought-wire retainers may be attached to a removable partial denture framework after it has been cast and finished (Figures 18-21 and 18-22). The soldering procedure may be accomplished by electric soldering or by a direct-heat method with an oxygen-gas flame. With either method, care must be taken to use compatible alloys, appropriate solder, and flux in conjunction with the careful application of con-trolled heat. Students are encouraged to review the Chapter 12 discus-sion of the selection of metal alloys to enhance their under-standing of the properties of solder, flux, the effects of heat on metal alloys, and the necessity for quality control in sol-dering procedures. Waxing Metal Bases A technique for forming the retentive framework for the attachment of acrylic-resin bases has been given. Two basic types of metal bases may be used instead of the resin base. The advantages of cast metal bases in preference to acrylic-resin bases have been discussed in Chapter 9. The type of base to be used must be determined before block-out and duplication, so that the relief over each edentu-lous ridge may be provided or eliminated as required. For an acrylic-resin base, relief for the retentive frame must be provided. No relief over the residual ridge is used for a complete metal base. For a partial metal base, the junction between metal and acrylic-resin must be clearly defined by trimming the relief along a definite, previously determined line. One type of metal base is the complete base with a metal border to which tube teeth, cast copings, or an 257 Chapter 18 Laboratory Procedures A B C Figure 18-8 Three steps are involved in making a denture framework by using relief, blockout ledges, and ready-made pattern forms. A, Master cast with relief, blockout of undercuts at posterior right and anterior lingual regions, and shaped blockout ledges for location of retentive and nonretentive clasp arms. B, Completed pattern using lingual bar major connector pattern, plastic clasp forms resting on investment ledges, wrought wire, and open retention mesh. C, Finished casting returned to master cast. acrylic-resin superstructure may be attached. If a porce-lain or plastic tube or grooved teeth are used, they must be positioned first and the pattern waxed around them to form a coping. The teeth are then attached to the metal base by cementation or, with the use of resin teeth, are attached with additional acrylic-resin under pressure, a so-called pressed-on method of attaching acrylic-resin teeth to a metal base. Another method of attaching teeth is to wax the base to form a coping for each tooth, either by carving recesses in the wax or by waxing around arti-ficial teeth. Rather than attaching a stock tooth, the full tooth may be waxed into occlusion, the base invested, and the wax patterns replaced with heat-polymerized acrylic-resin. This method permits some variation in the dimen-sion and form of the supplied teeth that is not possible with stock teeth. It is particularly applicable to abnor-mally long or short spaces or when a stock tooth of desired width is not available. With modern cross-linked acrylic-resins, such processed teeth are fairly durable; however, the addition of gold occlusal surfaces is an option. When artificial teeth are to be arranged to occlude with an opposing cast or an opposing template, the metal base must be formed for attachment of the tissue-colored denture resin supporting the teeth. This is the most common method of attaching teeth to a metal base. The wax pattern for the base is formed from one thickness of 24-gauge casting wax, which is then reinforced at the 258 Part II Clinical and Laboratory A B Figure 18-10 A, Design of a mandibular removable partial denture framework is outlined on the master cast for the techni-cian to follow in waxing and casting the framework. B, Cast framework as returned from the laboratory is evaluated intra-orally, revealing sufficiently accurate adaptation and design iden-tical to the embrasure clasp and mesio-occlusal rest #28, as shown on the design cast in A. Figure 18-9 Evaluation of the framework by the clinician is as important for removable partial denture frameworks as it is for implant and/or fixed partial denture castings. Careful scrutiny of attention to detail in following the design specifications is neces-sary, as is evaluation of the fit of the framework to the cast. On this cast, initial seating of the framework reveals interference with complete seating bilaterally from the embrasure clasps. Only minor adjustment should be required to completely seat the framework. If major adjustment is needed, or if after com-plete seating of the framework the clasp assemblies do not contact the teeth as prescribed, the framework should be remade. border and forms the retention of the acrylic-resin super-structure. Because metal borders are more difficult to adjust than acrylic-resin, they are usually made somewhat short of the area normally covered with an acrylic-resin base. Also, because border thickness adds objectionable weight to the denture, it is made with only a slight border roll. This is one disadvantage of the complete metal base, because the border accuracy of the impression registra-tion cannot be used to fullest advantage, and contouring of facial and lingual surfaces cannot be as effective as with an acrylic-resin base, in which added bulk can sometimes be used to advantage. The border is first penciled lightly on the investment cast, and then the 24-gauge sheet casting wax is smoothly adapted. Considerable care must be taken not to stretch and thin the sheet wax in adapting it to the cast. To prevent wrinkling, the wax should be adapted in at least two longitudinal pieces and joined and sealed together at the ridge crest. The wax is then trimmed along the pen-ciled outline with a warm, dull instrument to prevent scoring of the investment cast. A single piece of 14-gauge round wax is now adapted around the border over the sheet wax. With a hot spatula, the round wax must be sealed to the cast along its outer border. Sufficient wax is flowed onto the round wax to blend it smoothly into the sheet wax, thus completing a border roll. The inner half of the round wax form remains untouched. Wax is added when needed to facilitate carving without trimming the original 24-gauge thick-ness. The result should be a rounded border that blends smoothly into the sheet wax. The boxing for the resin, which will in turn support the artificial teeth, is now added, again with 14-gauge round wax. The proposed outline for the boxing is identi-fied by lightly scoring the sheet wax. On this scored line, the 14-gauge round wax is adapted, thus forming the outline of the boxing. With additional wax, the ditch between the sheet wax and the outer border of the round wax is filled in and 259 Chapter 18 Laboratory Procedures Figure 18-11 Occlusal view of mandibular Kennedy Class I, modification 1 wax pattern on a refractory cast. The lingual bar major connector joins three clasp assemblies (RPA: rest, proxi-mal plate and Akers clasp, wrought wire, and cast circumferential). A Figure 18-12 Buccal view of left side of the pattern shown in Figure 18-11. A tapered retainer clasp pattern is placed on the refractory cast ledge produced by duplication of shaped blockout ledges. The cast illustrates relief provided beneath the minor connector, which retains the denture base resin. The tissue index at the gingival region of the proximal plate will provide an easily identifiable finish line for resin finishing and future trimming of altered cast impression. Figure 18-13 Buccal view of modification space of the pattern shown in Figure 18-11. A wrought-wire clasp is contoured at the anterior extent of the modification space. Such a retainer will not overly stress the tooth if opposite denture base movement causes rotation across the fulcrum (see Figure 18-11, occlusal rests at #21 and #31). A tapered cast circumferential clasp follows the shaped ledge to a distal buccal 0.01-inch undercut. Both proximal plates have been waxed with sufficient bulk to provide a complete casting. If thickness presents a problem with tooth placement, some finishing can be accomplished at a later time. Figure 18-14 Lingual view of the pattern shown in Figure 18-11. The lingual bar major connector rigidly connects compo-nents cross-arch. The minor connector for resin retention on the left has buccal and lingual struts to allow unimpeded tooth place-ment. The tapered proximal plate on the right (tooth #28) is thicker on the lingual and thinner on the buccal, to allow close placement of the buccal surface of the denture tooth to #28. The finish line of the modification space is positioned below adjacent tooth gingival margins to allow normal lingual contouring of the resin matrix. blended smoothly onto the sheet wax. This is done in the same manner as the border, with sufficient wax added to allow for smoothing and carving. As was mentioned pre-viously, the pattern should not be flamed or polished with a cloth. Instead the pattern must be smoothed by carving. The result thus far should be a pattern reinforced at the border and at the boxing and slightly concave in between, with some of the original sheet wax exposed. The inside of the boxing is not sealed to the sheet wax, thus leaving a slight undercut for the attachment of the acrylic-resin. With a sharp blade, the margins of the boxing are then carved to a knife-edge finishing line. 260 Part II Clinical and Laboratory Figure 18-15 Another example of a wax pattern for a man-dibular framework. A wrought wire has been adapted to tooth #27. The clasp demonstrates an appropriate circlet shape, allow-ing maximum length. The circlet shape also allows the undercut to be approached from below; this improves retention—push retention is more efficient than pull retention. Ridge relief is evident beneath the resin minor connector, and the proximal plate housing the wire has been waxed with bulk to facilitate casting. The plate may require adjustment at a later time. Figure 18-16 Same pattern as shown in Figure 18-15. Includes a lingual plate major connector design. The external finishing line at the anterior modification space is lingual to the anticipated tooth and resin position, allowing a natural contour to be devel-oped in the completed prosthesis. The posterior denture base minor connector patterns have been reinforced with additional wax to add stiffness at the junction with the proximal plate and major connector. Because of the potential for repeated flexure in this region under function, such reinforcement is critical to long-term success with patterns such as this. Because the wrought-wire retainers are not cast, it is important that they be contoured to contact the teeth as accurately as possible, as shown. Figure 18-17 Buccal view of the pattern shown in Figure 18-16. Double wrought wires are contoured to follow the ledges, the tissue stop at the posterior of the ridge relief is shown, and an occlusal rest distal of #20 will require finishing following completion of casting. Figure 18-18 Occlusal view of the maxillary refractory cast with a wax pattern. An anatomic replica has been used to develop an anterior-posterior palatal strap major connector with beading evident at the anterior, posterior, and internal edges. Bilateral, tapered I-bar retainers have been waxed as extensions of the resin-retaining minor connectors, and double rests were incor-porated into the design on each side. 261 Chapter 18 Laboratory Procedures Figure 18-19 Buccal view of right I-bar shown in Figure 18-18. The tissue index is demonstrated gingival to the proximal plate, tissue relief is provided beneath the denture base minor connec-tor, and the I-bar shows a taper as it approaches the mid-buccal 0.01-inch undercut. Figure 18-20 Buccal view of I-bar opposite to that shown in Figure 18-19. Many of the same features seen in Figure 18-19 are evident in this example. The tissue stop distal to the ridge relief is shown and the palatal external finishing line continues distal to the junction of the hard and soft palate (which is the terminus of the posterior palatal strap). Figure 18-21 Wrought wire can be soldered to cast frame-works as an alternative method to incorporation in the wax pattern and casting procedure. Electric soldering is a common method for soldering and does not require heating of the entire framework. Following preparation of the minor connector where the wire will be soldered, an 18-gauge, round, wrought-wire clasp is carefully adapted and then is secured to the framework and the duplicate stone cast with the use of refractory investment. In this situation, a distolingual molar undercut will be used bilater-ally for retention. The flux is placed at the proximal plate, followed by placement of sufficient solder, and the electric heat element tip is placed into contact with the solder while the frame is grounded at another location. With the backside of the large end of the No. 7 wax spatula, this margin may be lifted slightly, further deep-ening the undercut beneath the finishing line. In addition to the undercut finishing line, retention spurs, loops, or nailheads are added for retention of the acrylic-resin to be added later. Spurs are usually made of 18-gauge or smaller round wax attached at one end only at random acute angles to the sheet wax. Loops are small-gauge round (wax, resin, or metal) circles attached verti-cally or horizontally with space beneath for the acrylic-resin attachment. Nailheads are made of short pieces of 18-gauge round wax attached vertically to the sheet wax, with the head flattened by a slightly warmed spatula. Any method of providing retention is acceptable if it permits positive attachment of the acrylic-resin and will not interfere with the placement of artificial teeth. A metal base waxed as described will provide optimum contours with a minimum of bulk and weight, and with adequate provision for the attachment of artificial teeth to the metal base. If properly designed, the more visible portions of the metal base will be covered with the acrylic-resin supporting the supplied teeth. Spruing, Investing, Burnout, Casting, and Finishing of the Removable Partial Denture Framework Brumfield has listed factors that influence the excellence of a dental casting (Box 18-1). 262 Part II Clinical and Laboratory A B Figure 18-22 A, An electric soldering technique has been used to solder the wrought wire. The solder has flowed into the space between the framework and the wire as well as around the wire to ensure its secure attachment to the framework. B, Finished and polished wrought-wire retainer. Box 18-1 FACTORS THAT INFLUENCE THE EXCELLENCE OF A DENTAL CASTING 1. Care and accuracy with which the cast is reproduced. 2. Intelligence with which the framework is designed and proportioned. 3. Care and cleanliness in waxing up the cast. 4. Consideration of the expansion of the wax caused by temperature. 5. Size, length, configuration, points of attachment, and manner of attachment of the sprues. 6. Choice of investment. 7. Location of the pattern in the mold. 8. Mixing water: amount, temperature, and impurities. 9. Spatulation of the investment during mixing. 10. Restraint offered to expansion of the investment caused by the investment ring. 11. Setting time. 12. Burnout temperature and time. 13. Method of casting. 14. Gases: adhered, entrapped, and absorbed. 15. Force used in throwing the metal into the mold. 16. Shrinkage on cooling. 17. Removal from the investment after casting. 18. Scrubbing, pickling, and so on. 19. Polishing and finishing. 20. Heat handling. Spruing Brumfield described the function of sprues as follows: The sprue channel is the opening that leads from the crucible to the cavity in which the framework is to be cast. Sprues have the purpose of leading the molten metal from the crucible into the mold cavity. For this purpose, they should be large enough to accommodate the entering stream and of the proper shape to lead the metal into the mold cavity as quickly as possible, but with the least amount of turbulence. Sprues have the further purpose of providing a reservoir of molten metal from which the casting may draw during solidification, thus preventing porosity caused by shrinkage. Spruing of the cast may be roughly summarized in the following three general rules. 1. Sprues should be large enough that the molten metal in them will not solidify until after the metal in the casting proper has frozen (8- to 12-gauge round wax is usually used for multiple spruing of removable partial denture castings). 2. Sprues should lead into the mold cavity as directly as possible and still permit a configuration that will induce a minimal amount of turbulence in the stream of molten metal. 3. Sprues should leave the crucible from a common point and be attached to the wax pattern at its bulkier sec-tions, that is, no thin sections of casting should inter-vene between two bulky, unsprued portions. The configuration of the sprues, from their point of attachment at the crucible until they reach the mold cavity, may be influential in reducing turbulence. One of the more important sources of difficulty in casting is the entrapment of gases in the mold cavity before they have a chance to escape. If the sprue channels contain sharp right-angle turns, great turbulence is induced, which is calculated to entrap such gases and so lead to faulty cast-ings. Sprue channels should make long radii and easy turns, and should enter the mold cavity from a direction designed to prevent splashing at this point. As was previously stated, the sprues should be attached to the bulky points of the mold pattern. If two bulky points exist with a thin section between them, each of the bulky spots should be sprued. The points of attachment should be flared out and local constrictions avoided. If this practice is followed, the sprue, which is bulky enough to freeze after the case framework has frozen, will con-tinue to feed molten metal to the framework until it has entirely solidified. The intent is to provide sound metal in the casting proper with all shrinkage porosity forced into the sprue rod, which is later discarded. The laboratory procedure for multiple spruing is essentially the same for all mandibular and maxillary cast-ings, except those with a palatal plate. Typical examples are illustrated in Figures 18-23 and 18-24. Two basic types of sprues are used: multiple and single. Most removable partial denture castings require multiple 263 Chapter 18 Laboratory Procedures Figure 18-23 Mandibular cast framework with sprue intact, showing three 8-gauge sprues attached to the lingual bar. If one is concerned about completely filling distal extension base minor connectors, 12-gauge sprues can be attached to the minor connector. Figure 18-24 For a maxillary framework, multiple 8-gauge sprues or a single main sprue located posteriorly can be used. When a single main sprue is used, additional sprues can feed critical regions for casting completeness. spruing, with the use of 8- to 12-gauge round wax shapes for main sprues and 12- to 18-gauge round wax shapes for auxiliary sprues. Occasionally, however, a single sprue is preferred for cast palates and cast metal bases for the mandibular arch when these are used as complete denture bases. With removable partial dentures, the use of a single sprue is limited to those maxillary frameworks in which— because of the presence of a palatal plate—it is impossible to locate multiple sprues centrally. In such situations, the single sprue may be used advantageously. A single sprue must be attached to the wax pattern so the direction of flow of the molten metal will be parallel to the long axis of the single sprue. In some instances, the investment cast may have to be cut away anteriorly to make room for the attachment of the sprue; in others, the sprue may be attached posteriorly. One disadvantage of using a single sprue for large castings is that an extra long investment ring must be used. Some important points to remember in multiple spruing are as follows: 1. Use a few sprues of large diameter rather than several smaller sprues. 2. Keep all sprues as short and direct as possible. 3. Avoid abrupt changes in direction; avoid T-shaped junctions as much as possible. 4. Reinforce all junctions with additional wax to prevent constrictions in the sprue channel and to prevent V-shaped sections of investment that might break away and be carried into the casting. Investing the Sprued Pattern The investment for a removable partial denture casting consists of two parts: the investment cast on which the pattern is formed, and the outer investment surrounding the cast and pattern (see Figure 18-41). The latter is con-fined within a metal ring, which may or may not be removed after the outer investment has set. If the metal ring is not to be removed, it must be lined with a layer of cellulose, asbestos substitute, or ceramic fiber paper to allow for both setting and thermal expansion of the mold in all directions. The investment must conform accurately to the shape of the pattern and must preserve the configuration of the pattern as a cavity after the pattern itself has been elimi-nated through vaporization and oxidation. Brumfield has listed the purposes of the investment as follows: 1. It provides the strength necessary to hold the forces exerted by the entering stream of molten metal until this metal has solidified into the form of the pattern. 2. It provides a smooth surface for the mold cavity so that the final casting will require as little finishing as pos-sible; in some situations, a deoxidizing agent is used to keep surfaces bright. 3. It provides an avenue of escape for most of the gases entrapped in the mold cavity by the entering stream of molten metal. 4. It, together with other factors, provides necessary compensation for the dimensional changes of the alloy from the molten to the solid, cold state. Note the substitution of the word alloy, as the same principles apply whether the metal is a precious metal alloy or a chromium-cobalt alloy. In some of the latter alloys, the cobalt is partially replaced by nickel; such alloys are sometimes described as Stellite alloys. 264 Part II Clinical and Laboratory Figure 18-25 Wax pattern sprued and ready to invest. Figure 18-26 Applying the investment with a soft moistened brush to ensure complete adaptation of the investment. This will help reduce the number of bubbles and will make for a smoother cast surface. The investment for casting gold alloys is a plaster-bound silica material, so compounded that the total mold expansion will offset the casting shrinkage of the alloy, which varies from 1% to 1.74% (the highest figure being the shrinkage of pure gold). Generally, the higher the percentage of gold in the alloy, the greater is the contrac-tion of the casting on solidifying. Only one chromium-cobalt alloy has a sufficiently low melting temperature to be cast into a plaster-silica invest-ment mold. According to Peyton, for the other alloys that have a higher melting temperature, an investment that contains quartz powder held together by an ethyl silicate or sodium silicate binder is generally used. Expansion to offset casting shrinkage for chromium-cobalt alloys is accomplished primarily through thermal expansion of the mold and must be sufficient to offset their greater casting shrinkage, which is on the order of 2.3%. For this reason, the casting ring is usually removed after the mold has hardened to allow for the greater mold expansion necessary with these alloys. Because the investments for chromium-cobalt alloys are generally less porous, there is greater danger that gases will be entrapped in the mold cavity by the molten metal. Spruing must be done with greater care, and in some instances, provision for venting the mold is necessary to prevent defective castings. Step-by-Step Procedure The technique for applying the outer investment is usually referred to as investing the pattern. Actually the cast on which the pattern is formed is part of the investment also. This investment technique is presented in Figures 18-25 through 18-27. Burnout The burnout operation serves three purposes: it drives off moisture in the mold; it vaporizes and thus eliminates the pattern, leaving a cavity in the mold; and it expands the mold to compensate for contraction of the metal on cooling. For the investment to heat uniformly, it should be moist at the start of the burnout cycle. Steam will then carry the heat into the investment during the early stages of the burnout. Therefore if the investment is not burned out on the same day that it is poured, it should be soaked in water for a few minutes before it is placed in the burnout furnace. Just before it is placed in the furnace, the mold should be placed in the casting machine to balance the weight against the weight of the mold. At this time, the mold should be properly oriented to the machine and its cru-cible, and a scratch line should be made at the top for later repositioning of the hot mold. The mold should be placed in the oven with the sprue hole down and the orientation mark forward. Burnout should be started with a cold oven, or nearly so. Then the temperature of the oven should be increased slowly to a temperature recommended by the manufacturer. This temperature should be maintained for the period recom-mended by the manufacturer to ensure uniform heat penetration. More time must be allowed for plastic pat-terns, particularly palatal anatomic replica patterns. It is important that the peak temperature recommended by the manufacturer not be exceeded during the burnout period. (When a high-heat investment is used, follow the manufacturer’s instructions as to burnout temperature.) 265 Chapter 18 Laboratory Procedures Figure 18-27 Investment mold has been trimmed to prepare the sprue end for burnout and subsequent complete adaptation to the casting machine. A B Figure 18-28 A, An invested pattern is removed from the burnout oven following complete elimination of the wax pattern. B, An investment mold is placed into the casting machine. The induction casting process provides consistency to the casting procedure. Casting The method of casting will vary widely with the alloy and equipment used. All methods use force to quickly inject the molten metal into the mold cavity. This force may be centrifugal or air pressure; the former is more commonly used. In any case, too much or too little force is undesir-able. If too little force is used, the mold is not completely filled before the metal begins to freeze. If too much force is used, excessive turbulence may result in the entrap-ment of gases in the casting. With centrifugal casting machines, this is regulated by the number of turns put on the actuating spring. The metal may be melted with a gas-oxygen blowtorch or by an electric muffle surrounding the metal. In some commercial casting procedures and in some dental labo-ratories, the induction method may be used (Figure 18-28); this provides a rapid and accurate method of melting the metal. Currently available casting machines include those that are electrically controlled to heat alloys to a specified temperature and to release the molten metal at precisely the correct casting temperature. These machines are relatively expensive and are primarily located in commercial laboratories or institutions where casting volume is high. Removing the Casting From the Investment Chromium-cobalt alloys are usually allowed to cool in the mold, are divested (Figure 18-29), and are not cleaned by pickling. Finishing (Figure 18-30) and polishing (Figure 18-31), which are done with special high-speed equipment, require technical skill in the use of bench lathes. Just before they are polished (high-shine), chro-mium-cobalt castings are electropolished; this is a con-trolled deplating process (Figure 18-32). Finishing and Polishing Some authorities hold that the sprues should not be removed from the casting until most of the polishing is 266 Part II Clinical and Laboratory A B Figure 18-29 A, Divesting of the framework involves bulk removal of the investment. B, The framework is divested with aluminum oxide. Figure 18-30 Gross finishing is accomplished with metal fin-ishing burs, abrasive stones, or sintered diamonds (except at contact areas). completed. Although it is true that this policy may prevent accidental distortion, it is difficult to adhere to and is therefore somewhat impractical. Instead, reasonable care should be exercised to prevent any distortion resulting from careless handling. A specific precaution is that a cast clasp arm should not be indiscriminately polished and then pliered into place. The waxing should have been done in such a manner that a minimum of finishing is necessary and the intended relationship of the clasp to the abutment is maintained (Figure 18-33). Actual polishing procedures may vary widely accord-ing to personal preference for certain abrasive shapes and sizes. However, the following several rules for finishing the casting are important: 1. High speeds are preferable to low speeds. Not only are they effective but in experienced hands there is less danger of the casting being caught and thrown out of the hands by the rotating instrument. 2. The wheels or points and the speed of their rotation should do the cutting. Excessive pressure heats the work, crushes the abrasive particles, causes the wheels to clog and glaze, and slows the cutting. 3. A definite sequence for finishing should be adopted and followed for every framework. 4. Clean polishing wheels should be used. If contaminated wheels are used, foreign particles may become embed-ded in the surface, which will lead to later discoloration. 5. Be sure that each finishing operation completely removes all scratches left by the preceding one. Remember that each successive finishing step uses a finer abrasive and therefore cuts more slowly and requires more time to accomplish. 267 Chapter 18 Laboratory Procedures tage, as do acrylic-resin bases that are processed directly to the master cast, thus becoming the permanent denture base. When undercuts are present, the master cast will be destroyed during removal of the base. Then existing undercuts must be blocked out inside the denture base before dental stone is poured into it to make a cast for articulator mounting. A second cast, which includes the undercuts, must be poured against the entire base to support it when the overlying acrylic-resin is processed. With both the processed base and the overlying acrylic-resin, some care must be taken to prevent visible junction lines between the original acrylic-resin base and the acrylic-resin that supports the teeth and establishes facial contours. Some autopolymerizing acrylic-resin materials are suffi-ciently accurate for use as jaw relation bases. These are used with a sprinkling technique, which, when properly done, permits formation of a base that compares favorably with a processed base or a visible light polymerized base. A material must be selected that will polymerize in a reasonable time (usually a 12-minute monomer) and will retain its form during the sprinkling process. Because polymerization with typical shrinkage toward the cast begins immediately, alter-nate addition of monomer and polymer in small increments results in reduced overall shrinkage and greater accuracy. Another option is to use visible light-cured (VLC) denture base materials, which employ a technique similar to that described in Chapter 15 for making custom impression trays. Technique for Making a Sprinkled Acrylic-Resin Record Base The technique for sprinkling a record base will be given here because the techniques used for the VLC impression tray and record bases were presented in Chapter 15 and elsewhere in this chapter. Some blockout and lubrication of the cast are necessary (Figure 18-34). Relief of undercut areas on the cast is best accomplished with a water-soluble modeling clay or Making Record Bases Bases for jaw relation records should be made of materials that possess accuracy or those that can be relined to provide such accuracy. Relining may be accomplished by seating the previously adapted base onto the tinfoil or lubricant in the cast with an intervening mix of zinc oxide–eugenol paste or with autopolymerizing acrylic-resin. Some use has been made of mercaptan and silicone impression materials for this purpose, but the wisdom of using an elastic lining mate-rial for jaw relation record bases is questionable. When rigid setting materials are used for this purpose, any undercuts on the cast must be blocked out with wax or clay to facilitate their removal without damage to the cast. The ideal jaw relation record base is one that is processed (or cast) to the form of the master cast, becoming the per-manent base of the completed prosthesis. Cast metal bases for complete or removable partial dentures offer this advan-A B Figure 18-31 The framework is finished with a rubber polishing point before final “high-shine” polish in a region adjacent to the teeth (A) and at the periphery of a major connector (B). Figure 18-32 Electropolish unit used for the final surface polish of frameworks in a heated polishing liquid. 268 Part II Clinical and Laboratory as little of the surface of the cast as possible. A close-fitting base may then be made that will have the necessary accuracy and stability and yet may be lifted from and returned to the master cast without abrading it. The cast and the blockout or relief are coated with a sepa-rating medium. Following this, the cast is wet with the monomer from a dropper bottle. After the surface has been wetted with the monomer, the polymer is sprinkled or dusted onto the wet surface until all the monomer has been absorbed. Sprinkling is best accomplished with a large-mouthed bottle with a single hole in the lid near the rim. This facilitates placement of the polymer without excess in any one area. A flexible bottle with a suitable applicator tip also may be used. The objective should be to apply polymer baseplate wax. Modeling clay is easily formed and shaped on the cast and is easily removed from the cast or the base with a natural-bristle toothbrush under warm running water. Wax must be flushed off the cast with hot water and possibly removed from the inside of the base before use. Bases for jaw relation records must have maximum contact with the supporting tissue. The accuracy of the base will proportionate to the contact provided to the total area of intimate tissue. Those areas are most often undercut and require blockout of the distolingual and retromylohyoid areas of the mandibular cast, the distobuccal and labial aspects of the maxillary cast, and, frequently, small multiple undercuts in the palatal rugae. These areas and any others are blocked out with a minimum of clay or wax, to obliterate A B Figure 18-33 Attention to detail in forming the wax pattern not only ensures the quality of the framework, it also saves time in finishing and polishing the resultant casting. A, Wax pattern of the mandibular framework, which incorporates three wrought-wire retainers. B, The same framework following casting shows a smooth casting surface and attention to detail regarding casting complete-ness and clasp form. A B Figure 18-34 A, Preparation of Kennedy Class I, modification 1 distal extension for sprinkle-on record base technique. Cast undercuts are blocked out with wax, and the cast is coated with separator. B, Peripheral extent of the base (the same as peripheral extent of the prosthesis captured during border molding and an impression-making procedure for this altered cast) is outlined with rope wax to contain resin. 269 Chapter 18 Laboratory Procedures added. Then the monomer may be added in excess and is immediately absorbed by the application of more polymer, as before. This process is repeated selectively until a uniform layer has been built up that is just thick enough so that none of the underlying cast or relief may be seen. The final step in sprinkling is the addition of monomer sufficient to leave a wet surface. Immediately, the cast should be placed in a covered glass dish or covered with an inverted bowl. This permits final polymerization in a saturated atmo-sphere of monomer and prevents evaporation of surface monomer. The cast should not be placed in water nor any attempt made to accelerate polymerization. Slow polymer-ization is necessary so that shrinkage toward the cast will occur. Only then will overall shrinkage be negligible and accuracy of fit ensured. This is of little consequence in making an impression tray, but it is most essential in making a sprinkled acrylic-resin base. Although polymerization will be about 90% complete within an hour and an impression tray may even be lifted within a half-hour, it is critical that a sprinkled denture base should be left overnight before it is separated from the cast. It should be lifted dry or under lukewarm tap water. It should not be immersed in hot water because some warping may occur. A sprinkled acrylic-resin base made with the precautions outlined previously will retain its accuracy for days, or even for an indefinite period, similar to a heat-polymerized resin base (Figure 18-37) or a VLC base. uniformly over the entire ridge rather than allowing excess to accumulate at the border to be trimmed later. An auto-polymerizing acrylic-resin material must be used that will retain its form during the sprinkling procedure without objectionable flow into low areas (Figures 18-35 and 18-36). Once the polymer has been sprinkled in slight excess, the monomer is again added. Flooding must be prevented; therefore the monomer must be directed over the entire surface gradually until the polymer has just absorbed the monomer. A few seconds delay before the addition of excess monomer will allow the mass to reach a tacky consistency and will prevent it from flowing when more monomer is Figure 18-35 The cast is wetted with monomer, and polymer resin is added in increments to uniformly control thickness. Use of a typical eyedropper may be difficult if the monomer addition cannot be controlled. Use of peripheral rope wax helps in this regard. Figure 18-36 The record base is complete when a uniform thickness is created that provides strength and accuracy. The completed unpolymerized resin is covered to ensure polymeriza-tion without loss of surface monomer. Figure 18-37 Once completely polymerized, the record base and the framework can be removed from the cast, finished, and prepared for addition of the record base. The tissue side of the record base (intaglio) should possess similar accuracy and stabil-ity as are seen with the completed prosthesis. Such a record base provides a significant advantage for jaw relation records when minimal teeth remain and ridge configurations along with exten-sive base areas place a premium on base accuracy and stability. 270 Part II Clinical and Laboratory the shape of the palatal vault and the arch form of the man-dibular arch, will crowd the patient’s tongue, will have an unwelcome effect on the patient, and will offer more resis-tance to jaw relation recording media than will a correctly shaped occlusion rim. Modeling plastic (compound) has several advantages and may be used rather than wax for occlusion rims. It may be softened uniformly by flaming, yet when chilled it becomes rigid and sufficiently accurate. It may be trimmed with a sharp knife to expose the tips of the opposing cusps to recheck or position an opposing cast into the record rim. Opposing occlusion rims of modeling plastic may be keyed with greater accuracy than opposing wax rims. Preferably, however, even those should be trimmed short of contact at the vertical dimension of occlusion, and bite registration paste should be interposed for the final record. As with wax rims, an adjustable frame may be used to support the final record. Occlusion rims made of extra hard baseplate wax or mod-eling plastic may be used to support intraoral central bearing devices, intraoral tracing devices, or both. Because of its greater stability, modeling plastic is preferable for this purpose when the edentulous situation permits the use of flat plane tracings. An example of such a situation occurs when an opposing complete denture is made concurrently with the removable partial denture. In such a situation, modeling plastic occlusion rims provide greater stability than wax rims, with corresponding improvement in the pre-dictable accuracy of such a jaw relation record. Although sealing opposing occlusion rims or using clips for complete denture jaw relation records may be acceptable, particularly for an initial articulator mounting, the existence of a remov-able partial denture framework makes this practice hazard-ous. The framework with attached base should be seated accurately on its cast before the opposing cast is repositioned in occlusion to it, because it is necessary for the dentist to be able to see the removable partial denture framework is in its designed terminal relation to the supporting teeth before articulating the casts. Occlusion Rims It has been explained that jaw relation records for removable partial dentures always should be made on accurate bases that may be part of the denture casting itself or may be attached to it in exactly the same relation as the final denture base will be. Further, it has been stated that although use of the final denture base is best for jaw relation records, a sprinkled or corrected acrylic-resin base may be used satis-factorily. In any case, accuracy of the base supporting a max-illomandibular record must be ensured before the function of occlusion rims is considered. Occlusion rims may be made of several materials. The material that is most commonly used to establish static occlusal relationships is the hard baseplate wax rim. However, use of a wax occlusion rim can be inaccurate when the occlusal portion of the rim is mishandled. When some soft material that sets to a rigid state, such as impression plaster or bite registration paste, is used in conjunction with wax rims to record static occlusal relations, many of the errors common to wax rims are eliminated—provided some space for the material exists between the occlusion rims, the opposing teeth, or both, at the desired vertical dimension to be recorded. Registration made on wax occlusion rims with the use of a wax registration material must be handled care-fully and mounted immediately. Occlusion rims for static jaw relation records should be so shaped that they represent the lost teeth and their sup-porting structures (Figure 18-38). An occlusion rim that is too broad and is extended beyond where prosthetic teeth will be located is inexcusable. Such rims will substantially alter Figure 18-38 Occlusion rims are added to allow recording of jaw relation records. Placement of the wax record is dictated by the opposing tooth position and the supporting ridge character. When possible, the occlusion rim should allow recording of the jaw position within the primary bearing area of the ridge. Occlusion rims for recording functional, or dynamic, occlusion must be made of a hard wax that can be carved by the opposing dentition. This method, outlined in Chapter 17, presumes that the opposing arch is intact or has been restored. Functional occlusion records cannot be made by this method when both arches are restored simultaneously. Rather, an opposing arch must be as intact as the treatment plan calls for, or it must be restored by whatever prosthetic means the situation dictates. Opposing removable partial dentures or an opposing complete denture may be carried concurrently up to the final occlusal record. One denture is then completed and the functional record is made in opposition to it. Often 271 Chapter 18 Laboratory Procedures this necessitates that all opposing teeth are articulated first in wax to establish optimum ridge relations and the correct occlusal plane. One denture is then carried to completion, and the teeth that remain in wax on the opposing denture are removed while the functional occlusal record is being made. The laboratory steps required for establishing an occlusion with the use of functional occlusal records were described in Chapter 17; however, more detail is provided in the following section. Some inlay waxes are used for this purpose because they can be carved by the opposing dentition, and because most of them are hard enough to support occlusion over a period of hours or days. A wax for recording functional crown and bridge occlusion, because it is established entirely in the dental office, is selected on the basis of how well it may be carved by the opposing dentition in a relatively short time. Therefore a softer wax may be used than is required for the recording of occlusal paths over 24 hours or longer. For this latter purpose, hard inlay wax seems to satisfy best the require-ments for a wax that is durable yet capable of recording a functional occlusal pattern. This wax is packaged in the form of sticks. A layer of hard, sticky wax is first flowed onto the surface of the denture base. Two sticks of the inlay wax are then laid parallel along the longitudinal center of the denture base and are secured to it with a hot spatula. This is the only preparation needed before the dental appointment. Because neither the height nor the width of the occlusion rim can be known in advance, and because deep warming of a chilled wax rim is difficult, the rim is not completed before the appointment. With the patient in the chair, a hot spatula is inserted into the crevice between the two sticks of wax, making the center portion fluid between two supporting walls. Some transfer of heat to the supporting walls occurs, resulting uniform softening of the occlusion rims. The patient is asked to close into this wax rim until the natural teeth are in contact; this establishes both the height and the width of the occlusion rim. Wax is added or carved away as indicated, and the patient is asked to make lateral excursions. Any excess wax is then removed, and any unsupported wax is supported by addition. Finally, wax is added to increase the occlusal vertical dimension suf-ficiently to allow for (1) denture settling, (2) changes in jaw relations brought about by the reestablishment of posterior support, and (3) carving in all mandibular excursions. When sufficient height and width have been established to accommodate all excursive movements, the patient is given instructions for chewing in the functional record and is then dismissed. Although this discussion has been included in the chapter on laboratory procedures, the entire procedure of establishing occlusion rims for recording functional occlusion should be considered a chairside procedure rather than a laboratory procedure. It is necessary that the purpose of a functional occlusal record be clearly under-stood, so that subsequent laboratory steps may be accom-plished in such a manner that the effect of the occlusal record may be reproduced on the finished denture. Making a Stone Occlusal Template from a Functional Occlusal Record After final acceptance of the occlusal record developed by the patient, the effectiveness of this method for establish-ing functional occlusion on the removable partial denture will depend on how accurately the following procedures are carried out. For this reason, it will be given as a step-by-step procedure. If the base of the master cast (or pro-cessing cast) has not been keyed previously, do this before proceeding. Reduce the thickness and width of the base if it is so large that difficulty will be encountered in flask-ing. The base may not be reduced after removal from the articulator because the mounting record would be lost. Keying may be done in several ways, but a method whereby the keyed portions are visible on the articulator mounting eliminates some possibility of remounting error. According to the preferred method, form a 45-degree bevel on the base of the cast by hand or with the model trimmer, and then add three V-shaped grooves on the anterior and posterior aspects of the base of the cast at the bevel. The bevel serves to facilitate reseating of the cast on the articulator mounting, and the mounting surfaces are made still more definite by the triangular grooves. Once placed at the beveled margin, the triangu-lar grooves are visible at all times, and any discrepancy may be clearly seen. Inspect the underside of the cast framework and the denture bases, removing any particles of wax or other debris. Similarly, inspect and clean the master cast of any particles of stone, wax, or blockout material, or any other debris that might prevent the casting from being seated accurately on it. Now seat the denture framework on the cast in its original terminal position. This is the position that is maintained by securing it with sticky wax with all occlusal rests seated while the trial denture base is made. It is also the position that the casting assumed in the mouth while the occlusal record was being made, and that must be duplicated on return of the denture framework to the master cast. While holding the framework in this termi-nal position, secure it again with sticky wax. (If a process-ing cast is used in place of the master cast, the denture base will have been made on that cast, and the same pre-cautions used in returning the framework to its original position apply.) With the denture framework and the occlusal record in position, form a matrix of clay around the occlusal record to confine the hard stone, which will form the stone occlusal template. The clay matrix is the same for a 272 Part II Clinical and Laboratory metal or electroplated surface as for the wax record. The clay matrix should rise at a 45-degree angle from the buccal and lingual limits of the occlusal registration. Then arch the clay or a sheet of wax across from one side to the other, forming a vault that will permit lingual access when the teeth are articulated. Leave the occlusal surfaces of a processing cast exposed so that they may act as vertical stops. This will serve to maintain the vertical relation in the articulator. Unless such stone-to-stone stops are used, the technician may alter the vertical relation in the articulator, accidentally or otherwise. Any change in vertical relations is incom-patible with a concept of dynamic conclusion because the occlusal pattern is directly related to the degree of jaw separation. Although it may be true that occlusal vertical dimension may be changed when casts are mounted in relation to the opening axis of the mandible, as long as natural cusps remain to influence mandibular move-ment, the occlusal vertical relation established with a functional occlusal record must not be changed in the articulator. Cover the surfaces of the adjacent abutment teeth left exposed with sodium silicate, microfilm, or some other separating medium to ensure separation of the stone ver-tical stops. If the wax record has not been electroplated, use a hard dental stone to form the opposing template. This may be an improved stone, but the use of a stone die material is preferred. Only the occluding surface needs to be poured in the harder stone, with a less costly labora-tory stone used to back it up. If this is done, add the second layer to the first before the former takes its initial set to prevent any possibility of accidental separation between the two materials. Vibrate the stone only into the wax registration and against the stone stops. Pile on the rest of the stone, and leave it uneven to facilitate attachment to the mounting stone. Attach the occlusal template to the artic-ulator without provision for removal or remounting because only the working cast needs to be keyed for remounting. After the stone template has set, attach the occluded casts to both arms of the articulator before separating the casts. The type of articulating instrument used is of little importance because all eccentric positions are recorded on the template, and whatever instrument is used acts purely as a simple hinge or a tripod. Therefore any labo-ratory articulator or tripod may be used. Casts should be attached to the articulating instru-ment with stone rather than with plaster. Mounting stones are available that have been especially formulated and prepared to minimize the setting expansion inherent in most gypsum products. The least amount of setting expansion of the mounting medium is most desirable to maintain the intended relationship of the opposing casts. One must remember which arch will be movable as a working cast, and the articulator mounting should be made accordingly. For example, for a mandibular denture, the template is attached to the upper arm of the articula-tor. For a maxillary denture, the template is mounted upside down on the lower arm. The keyed base of the working cast attached to the opposing arm must be coated with a light coat of microfilm, mineral oil, or petroleum jelly to facilitate its separation from the mount-ing stone. After the mounting has been completed, separate the casts and remove the clay. The template, with its mount-ing, may be removed from the articulator if a mounting ring or mounting stud permits; otherwise trimming must be done on the articulator. With a pencil, outline the limits of the occlusal registration and any excess stone around its borders. With a knife, trim the vertical stops to a sharp edge on the buccal surface where they contact the working cast. Remove any overhanging stone, leaving the occluding template and vertical stops clearly visible and accessible. Remove the wax registration before arranging artificial teeth to the occluding template. Arranging Posterior Teeth to an Opposing Cast or Template Whether posterior teeth are to be arranged to occlude with an opposing cast or with an occlusal template, the denture base on which the jaw relation record has been made must first be removed and discarded unless metal bases are part of the denture framework, or heat- polymerized acrylic-resin bases were used. This statement is based on the assumption that where an adjustable articulator has been used to develop the occlusion, the trial dentures have been evaluated, the articulator mount-ing has been proved, and the articulator has been pro-grammed for eccentric positions. Because record bases that are entirely tissue supported have no place in record-ing occlusal relations for removable partial dentures, the bases must be attached to the denture framework. Metal bases that are part of the prosthesis present no problem. The teeth may be arranged in wax or replaced on the metal base, depending on the type of posterior tooth used, and these must be occluded directly to the opposing cast or template. Unless occlusal relations are recorded on final acrylic-resin bases, autopolymerizing acrylic-resin bases attained by the sprinkling method or VLC bases are the most accurate and stable of bases that may be used for this purpose. (An alternate method is relining of the original impression bases, thus accomplishing the same purpose.) Although static relations may be recorded successfully on corrected bases, functional registrations are best accom-plished on new acrylic-resin bases made for that purpose. 273 Chapter 18 Laboratory Procedures Arranging Teeth to an Occluding Surface The procedure for arranging teeth to a static relationship with an opposing cast is essentially the same as for arranging teeth to an occluding template. On the other hand, articula-tion of artificial teeth on an adjustable instrument, which reproduces to some extent mandibular movement, will follow more closely the customary pattern for complete denture occlusion. The procedure for arranging posterior teeth to an occluding template was presented in Chapter 17. Types of Anterior Teeth Anterior teeth on removable partial dentures are concerned primarily with esthetics and the function of incising. These are best arranged when the patient is present because an added appointment for try-in would be necessary anyway. They may be arranged arbitrarily on the cast and then tried in, but a stone index of their labial surfaces should be made on the master cast after the final arrangement has been established to preserve the arrangement that the patient saw and approved. From a purely mechanical standpoint, all missing ante-rior teeth are best replaced with fixed restorations rather than with the removable partial denture. However, for cos-metic or economic reasons, or in situations in which several missing anterior teeth are involved—such as in a Class IV partially edentulous arch—their replacement with the removable partial denture may be unavoidable. Posterior Tooth Forms Posterior tooth forms for removable partial dentures should not be selected arbitrarily. One should bear in mind that the objective in removable partial denture occlusion is harmony between natural and artificial dentition. Whether the teeth are arranged to occlude with an opposing cast or to an occlu-sal template, they should be modified to harmonize with the existing dentition. In this respect, removable partial denture occlusion may differ from complete denture occlusion. In the latter case, posterior teeth may be selected and articu-lated according to the dentist’s concept of what constitutes the most favorable complete denture occlusion, whereas removable partial denture occlusion must be made to har-monize with an existing occlusal pattern. Thus the occlusal surfaces on the finished removable partial denture may bear little resemblance to the original occlusal surfaces of the teeth as manufactured. Artificial tooth forms should be selected to restore the space and fulfill the esthetic demands of the missing denti-tion. Manufactured tooth forms usually require modifica-tion to satisfactorily articulate with an opposing dentition. The original occlusal form, therefore, is of little importance in forming the posterior occlusion for the removable partial denture. The posterior teeth may be made of porcelain or resin (including all resin forms—composite, interpolymer network, cross-link, double cross-link, etc.). An advantage of resin teeth is that they are more easily modified and sub-sequently reshaped for masticating efficiency by the addition of grooves and spillways. Resin teeth are also more easily narrowed bucco-lingually to reduce the size of the occlusal table without sacrificing strength or esthetics. They may be more easily ground to fit minor connectors and irregular spaces and to avoid retentive elements of the removable partial denture framework. However, when resin teeth are used, the occlusion must be evaluated periodically to make sure that the occlusal surfaces have not worn out of contact. It seems that the best combinations of opposing occlusal surfaces used to maintain the established occlusion and to prevent deleterious abrasion are porcelain-to-porcelain sur-faces, gold surfaces to natural or restored natural teeth, and gold surfaces to gold surfaces. In either situation, the denture cannot be completed on these bases; neither can the bases be removed conve-niently from the retentive framework during the boil out after flasking. Therefore the metal framework must be lifted from the cast, and the original record base removed by flaming its underside. Care must be taken not to allow the acrylic-resin to burn, or the cast framework will become discolored with carbon. The framework is repol-ished and is then returned to its original position on the master cast and secured there with sticky wax before the artificial teeth are arranged. Some types of anterior teeth used in removable partial dentures are as follows: 1. Porcelain or resin teeth, attached to the framework with acrylic-resin. 2. Ready-made resin teeth processed directly to retentive elements on the metal framework with a matching resin. This is called a pressed-on method and has the advan-tage of permitting previous selection and evaluation of the anterior teeth, plus the advantage of the use of ready-made resin teeth for labial surfaces. These are then hollowed out on the lingual surface to facilitate their permanent attachment to the denture framework with the resin of the same shade. 3. Resin teeth processed to a metal framework in the labora-tory. Tooth forms of wax may be carved on the remov-able partial denture framework and tried in the mouth, adjusted for esthetics and occlusion, and then pro-cessed in an acrylic-resin of a suitable shade. There is some question as to whether the shade and durability of such teeth are comparable with those of manufac-tured resin teeth, but improvements in materials have led to improved quality and appearance of laboratory-made teeth. Moreover, such teeth may often be shaped and characterized to better blend with adjacent natural teeth. 274 Part II Clinical and Laboratory ishing the acrylic-resin back to the metal finishing line with finishing burs. Abrasive wheels and disks should not be used for this purpose, because they will cut into the metal and may burn the acrylic-resin. Pumice and a rag wheel should be used sparingly for polishing, because they will abrade the acrylic-resin more rapidly than the metal and will leave the finishing line elevated above the adjacent acrylic-resin. When waxing to polished metal parts that do not possess a finishing line is done, it must be remembered that no attachment will exist, and that over a period of time, some seepage, separation, and discoloration of the acrylic-resin in this area are inevitable. This may be prevented to some extent by roughening the metal whenever possible to effect some mechanical attachment by silicoating the attachment or by using one of the resin adhesives. The wax should be left 1.5 to 2.0 mm thick so that the acrylic-resin will have some bulk at its junction with the polished metal. Thin films of acrylic-resin over metal should be avoided. In finishing, these should be cut back to an area of bulk with finishing burs. Otherwise, any thin acrylic-resin film eventually will separate and become discolored and unclean as a result of marginal seepage. Gingival forms should be waxed in accordance with modern concepts of esthetics and should be made to prevent entrapment of food particles. Dental students should become familiar with normal gingival architecture as found on diag-nostic casts of natural dentitions, beginning in basic tech-nique exercises with the casts of each other’s mouths. In this manner, they may acquire a better concept of gingival con-tours to be reproduced on prosthetic restorations. Artificial teeth should be uncovered fully to expose the entire anatomic crown and beyond when gingival recession is simulated. Adjacent or contralateral tooth-gingival rela-tionships should be used as a guide to facilitate the harmoni-ous esthetic presentation of the gingival contours. Relatively few prosthodontic patients are in an age bracket in which some gingival recession and exposed cementum would not normally be present; this should be simulated on prosth-odontic restorations in proportion to the patient’s age. With removable partial dentures, gingival contours around the remaining natural teeth should be used as a guide to the gingival contours to be reproduced on the prosthesis. However, interproximal spaces are almost always filled, par-ticularly between posterior artificial teeth. Waxing and Investing the Removable Partial Denture Before Processing Acrylic-Resin Bases Waxing the Removable Partial Denture Base Waxing the removable partial denture base before investing differs little from waxing a complete denture. The only dif-ference is the waxing of and around exposed parts of the metal framework. At the framework–denture base junction, undercut finishing lines should be provided whenever pos-sible. Then the waxing is merely butted to the finishing line with a little excess to allow for finishing. Otherwise, small voids in the wax may become filled with investing plaster, or fine edges of the investment may break off during boil out and packing. In either situation, small pieces of investment may become embedded in the acrylic-resin at the finishing lines. This is prevented by slightly overwaxing and then fin-4. Porcelain or acrylic-resin facings cemented to the denture framework. These may be tried in the mouth on a baseplate wax base and adjusted for esthetics. Ready-made plastic backings may be used, which become part of the pattern for the removable partial denture framework. The teeth are then ultimately cemented to the framework. Esthetically, these are less satisfactory than other types of anterior teeth, but because the plastic backing is cast as part of the removable partial denture framework, they have the advantage of greater strength and are replaced easily. A record of the mold and shade of each tooth should be kept, and only the ridge lap of the replacement teeth needs to be ground to fit. When replaceability is the main reason for its use, the stock facing should not be beveled, or diffi-culty will be encountered in replacing it. Replacement also may be accomplished by waxing and processing a resin tooth that directly faces the metal backing. Stock tube or side-groove teeth are not ordinarily used for anterior teeth on removable partial dentures because of the horizontal forces that tend to dislodge them. 5. Anterior resin denture teeth can be modified to be used as resin veneers, the same as for veneer crowns and veneer pontics on fixed partial dentures. This is most applicable when the removable partial denture framework is to be cast in an alloy. Then labial surfaces may be waxed and the final carving for esthetics done in the mouth. A modification of this method is the waxing of the veneer coping on a previously cast metal base. These then are cast separately and are attached to the frame-work by soldering. Esthetically, the result is compa-rable with that obtained with resin veneer crowns. This method is particularly applicable when there is a desire to make the replaced teeth match adjacent veneered abutment crowns. Frush has listed the following rules for varying the height of the gingival tissue at the cervical portion of the teeth: 1. Slightly below the high lip line at the central incisors. 2. Lower than the central incisor gingival margin at the lateral incisors. 3. Higher than the central or lateral incisor gingival margin at the canine. 275 Chapter 18 Laboratory Procedures Investing the Removable Partial Denture In investing a removable partial denture for processing an acrylic-resin base, it must be remembered that the denture cast must be recovered from the flask intact for remounting. The practice of cutting the teeth off the cast to expose the connectors and retainers, which are then embedded in the upper half of the flask, is permissible only when an existing denture base is being relined and no provision has been made for remounting. (In such a situation, it seems that this practice offers no advantage over investing the denture that is being so relined upside down in the lower half of the flask.) Because in the past some increase in occlusal vertical dimen-sion has been inevitable in any split-mold processing tech-nique, this method results in raising of the removable partial denture framework from the supporting teeth by the amount of increase. Occlusal adjustment in the mouth may tempo-rarily reestablish a harmonious occlusal relation with the opposing teeth, whereas the removable partial denture framework will then settle into supporting contact with the abutment teeth at the expense of the underlying ridge. The denture should be waxed and carved as for a cast restoration, which it actually is, regardless of the material or the method of processing used. The fact that a split-mold technique is used for processing does not alter the fact that the form of the denture base is to be reproduced by a casting procedure. Therefore the denture pattern should be waxed with care in the same form as that desired for the finished restoration rather than attempting to shape facial contours on the prosthesis during the polishing phase (Figure 18-39). Polishing should consist primarily of trimming away the flash, stippling polished surfaces when desired, and polish-ing lightly with brush wheels and pumice, followed by final polishing with a soft brush wheel and a nonabrasive shining agent, such as whiting. Gross trimming and polishing with pumice should not be necessary if the denture has been properly waxed before investing. Because the polished surfaces of any denture play an important part in both retention and control of the food bolus, buccal and lingual contours generally should be made concave. In most situations, the border thickness of the denture should be left as recorded in the impression. The only exceptions are the distolingual aspect of the mandibular denture base, to prevent interference with the tongue, and the distobuccal aspect of the maxillary denture base, to prevent interference with the coronoid process of the man-dible. These are the only areas that cannot ordinarily be waxed to final contour before investing and may need to be thinned by the dentist at the time of final polishing. Figure 18-39 Waxing of the removable partial denture base should reproduce anatomic contours normal for the specific patient’s characteristics. This is especially important for regions where the junction of denture tooth and resin will be visible. Healthy interdental papillae are convex and extend to contact points of the teeth. Root prominence can be carved into wax to create a natural appearance. Stippling is typically accomplished after processing done with the use of an eccentric round bur. All of these features should be considered against the specific oral environment in which the prosthesis will be placed. A finished and polished prosthesis that demonstrates esthetic features not found in the patient’s mouth may be considered objectionable. 4. Slightly lower than the canine at the premolar and variable for both premolars and molars. The correctly formed papilla should be shaped so that it will be self-cleansing. It should be carved so that it is in harmony with the interpretation of age and will be the deciding factor in the visible outline form of the tooth. As Frush has pointed out, even a drop of wax properly placed can change the appearance of a square tooth to a tapering or ovoid appearance. A properly formed papilla further enhances the natural appearance by increasing the color in this area. The rules for forming the papilla were given by Frush as follows: 1. The papilla must extend to the point of tooth contact for cleanliness. 2. The papillae must be of various lengths. 3. The papilla must be convex in all directions. 4. The papilla must be shaped according to the age of the patient. 5. The papilla must end near the labial face of the tooth and must never slope inward to terminate toward the lingual portion of the interproximal surface. From Frush JP: Dentogenic restorations and dynesthetics, Los Angeles, 1957, Swissdent Foundation. Changes in occlusal vertical dimension may be held to a minimum by the use of acrylic denture resins that can be placed in the mold in a fluid rather than in a doughy 276 Part II Clinical and Laboratory A B C D Figure 18-40 Prostheses invested in processing flasks. A, Mandibular Class I master cast with exposed minor connectors of the distal extension bases. B, Denture teeth for framework shown in A. C, Maxillary distal extension minor connector embedded in investing stone. D, Opposing flask unit with denture teeth for distal extension shown in C. state or those that may be injected in a fluid state into a closed mold. Dimensional changes that occur during relining may also be held to a minimum by the use of autopolymerizing acrylic-resins for this purpose, thus preventing thermal expansion of a mold subjected to elevated temperatures. When two opposing removable partial dentures are being made concurrently, one is sometimes processed and placed first, and then the final occlusion is established on the second denture to a fully restored arch. In such a situation, when no natural teeth are in opposition, it is not necessary for the first denture to be remounted after processing. In all other situations, remounting to correct for errors in occlusion is absolutely necessary. Flasking must be accomplished so that the cast may be recovered from the flask undamaged. Minute voids in the base of the cast will have been reproduced in the stone mounting, and although the obvious larger blebs may be trimmed away, smaller blebs will remain. If the voids in the cast become filled with investing material, the effect is that two particles are trying to occupy the same space. Covering the base with tinfoil substitute before investing can prevent this. Coating the base and sides of the cast with petroleum jelly not only keeps the base of the cast isolated from investing material but also may allow the cast to be more easily recovered from the surrounding investment. The entire cast, except for the wax and teeth, may then be invested in the lower half of the flask (Figure 18-40). As with a complete denture, only the supplied teeth and wax are left exposed to be invested in the upper half. Also, as with a complete denture, the investment in the lower half must be smooth and free of undercuts and must be coated with a separator to facilitate separation of the two halves of the flask. An alternate and preferred procedure is to invest the cast only to the top of the tinfoil on the base, smoothing the investment and applying a reliable separator. Then a second layer of investment placed around the anatomic portion of the cast covers the natural teeth and the exposed parts of the denture framework. This is likewise smoothed and made free of undercuts and coated with a separator before the top half of the flask is poured. Recov-ery of the cast is thus made easier by having a shell of investment over the anatomic portion of the cast, which may be removed separately. When the denture base is to be characterized by apply-ing tinted acrylic-resins to the mold, care should be taken not to embed the wax border in the lower half of the flask. Bennett has pointed out the need for investing only to the border of the wax, leaving the entire surface that is to be tinted reproduced in the upper half of the flask. With this precaution, tinting may be carried all the way to the border, and later removal of the flask will not mar the 277 Chapter 18 Laboratory Procedures tinted surface. If tinting is not to be done or is to be done only at the cervical margins of the teeth and the papillae, the wax border should be embedded in the lower half, where it may be faithfully reproduced and preserved during polishing. The use of acrylic-resin materials that require trial packing is complicated by the presence of the retentive framework of the removable partial denture. With their use, trial packing must be done with two sheets of cello-phane between two layers of the resin dough; otherwise the flask could not be opened without pulling the resin away from the teeth in one half of the flask or the metal framework in the other. Acrylic-resin dough is placed in each half of the flask, the sheets of cellophane are placed between them, and the flask is closed for trial packing. The flask is then opened, the cellophane is removed, and the excess flash is trimmed away. Trial packing should be repeated until no excess is visible. Final closure is accom-plished without the intervening sheets of cellophane. Acrylic-resin materials have been developed that require no trial packing. These are mixed as usual but may be poured into the mold or placed into the mold in a soft state. They offer little or no resistance to closure of the flask, yet the finished product is comparable with acrylic-resin materials packed in a doughy state. They must be used in some excess, with the excess escaping between the halves of the flask. Although they are soft enough to allow the escape of gross excess, the use of a land space is advisable to prevent a thin film from forming on the land area. Any film that exists on the land area after deflasking may be interpreted as an opening of the flask by that amount, hence the need for some provision for an intervening space to accommodate the excess and to facilitate its escape as the flask is closed. To provide such a land space, the land area on the lower half of the flask may be painted with melted base-plate wax before the top half is poured. After wax elimina-tion, a space then remains to accommodate any excess acrylic-resin remaining after the flask has been com-pletely closed. It is necessary that no plaster or wax is allowed to remain on the rim of the flask, and that the flask makes metal-to-metal contact before the second half is poured. Only in this way is it possible to see that the flask is completely closed before it is placed in the curing unit. Pouring of the top half of the flask follows the same procedure as that used with a complete denture. Although it is not absolutely necessary for the entire top half to be poured in stone, it is necessary for a stone cap of some type to be used to prevent tooth movement in an occlusal direction. This is so because of the inability of plaster to withstand closing pressures. All plaster remaining on the occlusal surfaces of the teeth should be removed and a separator added before the stone cap is poured to facili-tate its removal during deflasking. If the use of stone investment is preferred, a shell of improved stone or die stone may be painted or applied with the fingers onto the wax and teeth and allowed to harden before the remain-der of the flask is filled with plaster. If a full stone invest-ment is preferred, some provision should be made for easy separation during deflasking. Not only should a separator stone cap be used, but metal separators or knife cuts radiating out to the walls of the flask should be placed on the partially set stone. Deflasking then is easily accomplished by removing the stone cap and inserting a knife blade between the sections of stone. Boil out should be deferred until the investing material has set for several hours or preferably overnight. Boil out must effectively eliminate all wax residue; an adequate source of clean hot water must be available. Immersion of a flask containing the invested denture in boiling water for 5 minutes will adequately soften the wax supporting the artificial teeth so that flask halves may be separated and the remaining wax flushed out. After wax elimination with boiling water, the invested denture should be flushed with a solution of grease-dissolving detergent and again with clean boiling water. Immediately after boil out, the warm mold should be painted with a thin film of tinfoil substitute, with care taken to not allow it to collect around the cervical por-tions of the teeth. No tinfoil substitute should be used on any part of the denture teeth. A second coat should be applied after the first coat has reasonably dried, and packing of the mold should proceed immediately after this film has dried to the touch. When the master cast for a distal extension removable partial denture has been repoured from a secondary impression, the supporting foot on the retention frame may not necessarily be in contact with the cast. Closing pressure within the flask may distort the unsupported extension of the metal framework, with subsequent rebound on deflasking. The finished denture base will then lack contact with the supporting tissue, resulting in denture rotation about the fulcrum line, similar to that which occurs after tissue resorption. To provide support for the distal extension of the metal framework during flask closure, autopolymerizing acrylic-resin should be sprinkled or painted around the tissue stop at the distal end of the framework and allowed to harden before the denture resin continues to be packed (see Figure 5-39). Processing the Denture Processing follows the same procedure as that used for a complete denture. Denture base characterization tinting may be added just before packing. This is most desirable if denture base material will be visible when in the mouth. Posterior acrylic-resin bases alone ordinarily do not require characterization, but the dentist should select a denture base 278 Part II Clinical and Laboratory material that closely resembles the color of the surrounding tissue. The ideal acrylic-resin base material for removable partial dentures therefore is one that (1) may be used without trial packing; (2) possesses a shade that is compatible with surrounding tissue; (3) is dimensionally stable and accurate; (4) is dense and lends itself to polishing; and (5) polymerizes completely. There has never been any question concerning the merits of placing tinfoil over the denture before investing; this results in a tinfoil-lined matrix and eliminates the need for a separating film. The fact remains, however, that the use of a tinfoil substitute has become universal. At best, any tinfoil substitute creates an undesirable film at the gingival margins of the teeth, resulting in microscopic separation between the teeth and the sur-rounding acrylic-resin. This may be shown by sectioning a finished denture and by observing the marginal discol-oration around the cervical portions of the teeth after several months in the mouth. To some extent, injection molding obviates this objection to the use of a tinfoil substitute; this is one of the principal advantages of injec-tion molding over compression molding. Because the use of compression molding is widespread and is likely to continue, methods are needed that elimi-nate the use of a tinfoil substitute. The layered silicone rubber method results in more complete adaptation of the acrylic-resin around the cervical portions of porcelain teeth and more complete bonding to acrylic-resin teeth. In addition, denture base tints may be applied directly to the mold without first applying a separating film. A room-temperature polymerizing silicone rubber that has sufficient body and toughness for the purpose is applied to the wax surface of the denture and over the teeth. To prevent movement of the teeth during process-ing, the occlusal surfaces should be exposed before the upper half of the flask is poured. The manufacturer’s instructions must be followed as to mixing and time elapsed before the outer stone investment is added, to ensure polymerization and bonding to the overlying investment. Boil out then is completed in the usual way. A further advantage of the layered silicone rubber method is the ease of accomplishing deflasking. If the wax carving of the denture has been completed with care before flasking, denture tints remain unaltered by unnec-essary trimming and polishing of the processed denture. All resin base materials available up to the present time exhibit some dimensional change, both during process-ing and in the mouth. The fit of the denture is therefore dependent on the accuracy of the denture base material because impression and cast materials in use today are themselves reasonably accurate. In an attempt to mini-mize dimensional changes in the denture base, materials and techniques are being improved constantly. Denture base materials that may be poured into the mold or placed into the mold in a soft state are also popular. This technique eliminates trial packing and excessive pressures, which lead to open flasks and altered occlusal vertical dimension as is sometimes experienced with compression molding of acrylic-resin base materials. Activated, or autopolymerizing, acrylic-resins are com-monly used to prevent mold expansion at higher tem-peratures. Materials other than acrylic-resins are used with various techniques, some of which include styrene, vinyl, and experimentally epoxy resins. The main objec-tive behind the development of newer techniques and materials is greater dimensional accuracy and stability, combined with improved strength and better appearance. The use of injection molding or poured materials to process removable partial denture base materials com-bines accuracy and efficiency to help create a well-fitting denture base. The Success Injection System by Dentsply Trubyte (York, PA) combines the accuracy of injection molding with Lucitone 199 (Dentsply Trubyte, York, PA). The hardware consists of the injection unit, alumi-num alloy flasks, and associated system flask compo-nents. The flasks are numbered; specifically, they are mated halves that need to be matched for more accurate results. The investment and processing techniques are as follows: 1. The cast with the completed wax-up is embedded in the designated side “1” of the flask in the usual manner, placing the cast as close as possible to the back of the flask. After any undercuts on the cast are eliminated, flat-tened wax sticks (approximately 7 mm in diameter) are used to build the injection sprues. For maxillary removable partial dentures, attach the sprue to the posterior border, ensuring that the sprue is sufficiently wide. For mandibular removable partial dentures, position one sprue for each base extension. Apply a separator to the investment, and place the top half of the flask on the bottom half, ensuring complete inti-mate metal contact and closure of the halves. Secure the flask brackets to the flask and tighten. Place the flask on the leveler with side “2” up, and complete the investing procedure in the usual manner. 2. When the investment has set, loosen the bolts and remove the metal flask brackets. Place the flask in boiling water (8 to 10 minutes) and complete the boil out proce-dure. Place the metal injection insert into the back of the flask, and slide the plastic injection socket into the metal insert as far as possible. The plastic injection socket lip should rest flush against the trim of the metal injection insert. Close the flask, position the metal flask brackets, and tighten the bolts. 3. Use the powder/liquid vials to measure sufficient resin for the removable partial denture. Note that the maximum powder and/or liquid that the injection car-279 Chapter 18 Laboratory Procedures tridge can hold is 38 g (56 mL) powder and/or 17.5 mL liquid. Stir the powder and/or liquid for approxi-mately 15 seconds. Do not excessively mix. Cover the mixing jar until the material reaches the “soft pack” stage (approximately 6 minutes). Do not allow the material to reach the “snap set” stage. 4. Place the resin material into the plastic injection car-tridge and insert the blue plastic cartridge plug into the large open end, ribbed side out. Push the blue cartridge plug in as far as possible to compress the material, and insert the cartridge nozzle into the plastic injection socket until it seats on the injection socket’s lip. Place the metal protective sleeve over the cartridge and place the flask in the injection unit, ensuring that the bolts and the flask brackets face to the operator’s right side. Position the open slots on the cartridge sleeve facing out, then push the sleeve up toward the unit’s cross-head, securing the sleeve around the blue rubber O-ring. Tighten the unit’s hand wheel to secure the flask. 5. Complete the injection process, ensuring that the mold is completely filled by viewing the blue cartridge plug through the sleeve slots until the plug stops moving. When completed, remove the flask from the unit, remove the cartridge sleeve, and pull the plastic car-tridge out of the flask with a slight twist. Keep the injection socket in place inside the metal injection insert. Fit the small, blue plastic piston cap onto the end of the pressing device piston. Place the piston of the processing device into the plastic injection socket at the back of the flask, and screw the pressing device onto the metal injection insert until the etched groove is visible on the pin at the top of the pressing device. 6. Allow the flask to sit for 30 minutes before heat-polym-erizing to ensure a good bond between the denture resin and the teeth. Submerge the closed flask in water at 163 ± 2°F for 112 hours. Follow with an additional 30-minute boil. An alternate polymerization method is 9 hours in a water bath of 163 ± 2°F with no boil. Remove the flask from the polymerization tank and allow to air cool for approximately 30 minutes. Place the flask in a lukewarm water bath to cool completely. 7. Unscrew the pressing device and loosen the bolts on the flask. Remove the flask brackets and separate the flask. Remove the investment and divest the removable partial denture and cast. Cut off the injection sprue(s), and finish and polish the removable partial denture in a conventional manner. The use of VLC denture base materials is claimed to save considerable time in providing a processed base. Manufacturers recommend the use of a light-colored cast to enhance the polymerization of the VLC material. A stone matrix must be fabricated so that the denture teeth can be positively relocated in the same position during the polymerizing process. Once the stone matrix has been completed and verified, the wax and teeth can be removed from the framework, and they can be cleaned and sili-coated or coated with the resin bonding agent. The ideal thickness for the VLC material requires a 1.5-mm space between the edentulous ridge and the retentive component of the framework. The ridge lap of the denture teeth should also be 1.5 mm above the reten-tive component of the framework, and the tissue side finishing line of the framework should be even with or slightly higher than the palatal side finishing line. If VLC denture base material is to be used, these requirements must be considered at the diagnosis and treatment plan-ning stages. Without the framework in place, a thin coat of model release agent (MRA) is applied to the denture base areas of the cast, and the VLC material is adapted to the eden-tulous denture base area and is trimmed to the general outline with a sharp blade. The framework is then seated firmly by being embedded into the uncured VLC mate-rial. Make certain that the rests, tissue stops, and other components of the removable partial denture framework are correctly positioned in their designated terminal posi-tions on the cast. Remove any excess material that may interfere with articulation of the casts or positioning of the teeth, and check to ensure that the VLC material is adapted into the tissue side finishing line of the frame-work on the cast. This will necessitate returning the cast with the framework to the articulator. Once this has been verified, remove the cast from the articulator and process it in the light-polymerizing unit for 2 minutes. The denture teeth can then be secured to the stone matrix and related to the cast with the framework and the removable partial denture base. The denture teeth may have to be adjusted before the removable partial denture base can be fitted. To do this, trim the ridge lap areas of the teeth enough to provide the required 1.5 mm of space between the tooth and the denture base material. Remem-ber that thin areas should be avoided between the frame-work and the cast (minimum of 1.5 mm), and between the framework and the denture teeth (minimum of 1.5 mm). The denture tooth surfaces that are to be bonded to the VLC denture base must be lightly ground and cleaned. The teeth may then be placed in modeling compound or putty to hold them while the bonding agent is applied to all of the designated surfaces to be bonded to the VLC denture base. These coated surfaces should be allowed to sit for 2 minutes and then are processed in the light polymerization unit for 1 minute. To secure the teeth into their designated position on the VLC denture base, apply a small piece of VLC mate-rial to each tooth with the modeling tool provided by the manufacturer. With the aid of the stone matrix and by 280 Part II Clinical and Laboratory ferred to an instrument and how closely the instrument is capable of reproducing functional occlusion. But even though the articulator is capable of reproducing only a static centric relation, that relation at least should be reestablished before the denture is placed. Although it is admitted that there are limitations to the perfection of eccentric occlusion in the mouth, some believe that it can be done with more accuracy than on an instru-ment that is incapable of reproducing eccentric positions. Correction for errors in centric occlusion, however, should not be included in a concept that presumes that centric occlusion may be established satisfactorily by intraoral adjustment, followed then by perfecting of eccentric occlu-sion. Because of denture instability and the inaccessibility of the occlusion for analysis, accurate intraoral corrections are not possible. Practically, even the occlusal adjustment of natural dentition in which each tooth has its own support can best be done when preceded by an analysis of articulated diagnostic casts. One cardinal premise must be accepted if prosthetic den-tistry is to be anything more than a haphazard procedure. It is possible to transfer centric jaw relation to an instrument with accuracy and to maintain this relation throughout fab-rication of the prosthesis. If this is true, then centric occlu-sion, coinciding with centric jaw relation, with centric occlusion of the remaining natural teeth, or with both, must have been established before initial placement of the prosthe-sis. This means that occlusal correction to reestablish centric relation by remounting after final processing is an absolute necessity for the success of the restoration. Remounting after processing is accomplished by returning the cast to a keyed relationship with the articulator mounting. Precautions to Be Taken in Remounting The following precautions should be taken to ensure the accuracy of remounting to make final occlusal adjustment before polishing and initial placement of the denture. These apply to all types of occlusal relationship records but are directed particularly at remounting an occlusal template when stone vertical stops are used: 1. Make sure that the base of the cast has been reduced to fit the flask before keying and mounting, so it will not have to be altered later. 2. Bevel the margins of the base of the cast so it will seat in a definite boxlike manner in the articulator mounting. 3. Notch the posterior and anterior aspects of the base to ensure its return to its original position. Notches at the margins are preferable to depressions within the base because the former permit a visual check of the accuracy of the remounting. 4. Lubricate lightly the base and sides of the cast before it is mounted to facilitate its easy removal from the mounting stone. 5. Add tinfoil or lightly lubricate the base and sides of the cast before flasking it, so that traces of investment will not be present to interfere with remounting. Remounting and Occlusal Correction to an Occlusal Template Even with improved denture base materials and processing techniques, some movement of artificial teeth will still occur because of the dimensional instability of the wax in which the artificial teeth were arranged. Until sources of error can be eliminated, remounting will continue to be necessary. How well the occlusion may be perfected by remounting will depend on the manner in which jaw relations were trans-using a high-intensity light, tack each individual tooth into position on the VLC denture base. After tacking the teeth to the VLC denture base, paint a narrow band of bonding agent on the junction line between the teeth and the denture base material. The bonding agent acts as a sealer to prevent leakage and aids bonding of the teeth to the denture base material. Allow the bonding agent to set for 2 minutes and then process in the light unit for 1 minute. You can now complete the buccal and lingual contouring of the denture base with additional VLC material. Paint all of the exposed surfaces of the VLC denture base material with the air barrier coating material, and process in the light polymerization unit for 2 minutes. After processing, carefully remove the removable partial denture from the cast. Do not attempt to pry it off. Paint the tissue surfaces of the edentulous areas with the air barrier coating material, and place in the light polymerization unit for 6 minutes with the tissue side up. Clean the removable partial denture with water and a brush to remove all traces of the air barrier coating. Trim and polish in a routine manner to complete the removable partial denture. The study of the history of denture base materials is a most interesting one that has been covered elsewhere in dental literature. The future of denture base materials promises to be just as fascinating a study, but such a discussion cannot be included within the scope of this book. With newer materials, the future of methyl meth-acrylate as a denture base material is uncertain despite its acceptance as the best material available since its intro-duction in 1937. Although it has made possible the simu-lation of natural tissue color and contours combined with ease of manipulation, the fact remains that it leaves much to be desired as far as accuracy and dimensional stability are concerned. Whether other and newer materials will eventually supplant methyl methacrylate as a denture base material remains to be seen. The fact is that the denture base of the future (1) must be capable of accu-rately reproducing natural tissue tones faithfully through the use of characterizing stains and customizing proce-dures, and (2) must not require elaborate processing pro-cedures and equipment, which would make the cost prohibitive for general use. 281 Chapter 18 Laboratory Procedures Porcelain teeth may be reshaped with abrasive or dia-mond-mounted points. Resin teeth lend themselves better to reshaping with small burs to restore functional anatomy. Either type should be repolished judiciously to prevent reduction of cuspal contacts. Although cusps may be nar-rowed, spillways added, and the total area of contact reduced to improve masticating efficiency, critical areas of contact, both vertical and horizontal, must always be preserved. The term remounting is also applied to the mounting of a completed prosthetic restoration back into an instrument by using some kind of interocclusal records. Discrepancies in occlusion resulting from processing of tooth-supported dentures may be corrected by reattaching the indexed pro-cessing cast and denture to the same instrument on which the occlusion was formulated. However, because of some instability inherent in distal extension removable partial dentures, such dentures should be retrieved from processing investment, finished, and polished and prepared for perfor-mance of occlusal corrections with the use of new intraoral records. The dentist must make a remounting cast before occlusal corrections can be accomplished. This is done simply first by placing the denture in the mouth and making an irreversible hydrocolloid (alginate) impression of the denture and the remaining teeth in the arch (Figure 18-41). When the impression is removed, the denture usually will remain in the impression or can be accurately replaced. Undercuts in the denture bases are blocked out, the retentive elements of the framework are covered with a thin layer of 6. When remounting the cast, secure it in the articulator with sticky wax, a hot glue gun, or modeling plastic, followed by stone over both the mounting and the sides of the cast. 7. Before adjusting the occlusion, make certain that no traces of investment remain on the vertical stops. 8. Take care to not abrade the opposing occlusal surface during occlusal adjustment. The use of marking tape or inked ribbon is preferable to articulating paper. The arti-ficial tooth is less likely to cut through and mar the opposing surface, and ink or dye will not build up a false opposing surface, as will wax from articulating paper. 9. Occlusal readjustment to an occlusal template is complete when the stone vertical stops are again in contact. With other types of articulator mountings, readjustment is complete when the vertical pin is again in contact and any valid horizontal excursions are freed of interference. Occlusal readjustment, as with the original articulation, is done at the expense of the original tooth anatomy. Occlu-sal surfaces should be reshaped by adding grooves and spill-ways and by reducing the area of the occlusal table, thus improving the masticating efficiency of the artificial tooth. Although this may be done immediately after occlusal read-justment and before initial placement of the denture, it may be deferred until completion of the final adjustment. In any event, it is a necessary step in the completion of any remov-able prosthesis. A B Figure 18-41 A, A stock, perforated tray is used to make an irreversible hydrocolloid (alginate) impression of the denture and dental arch. Blockout of undercuts in the denture base and of tips of direct retainers is necessary, so the denture can be readily removed and replaced on the resultant remounting cast, as illustrated in B. B, Remounting cast poured in stone with the prosthesis in place and interocclusal registration for mounting against a maxillary cast. Wax placed at the retainer tip (A) allows the prosthesis to be readily removed and replaced on the cast for occlusal correction procedures that use an articulator. 282 Part II Clinical and Laboratory The dentist must correct any overextension remaining after arbitrarily trimming the border from cast landmarks in the mouth. It is preferable for the dentist to finish the borders of dentures, having painstakingly developed them during impression procedures. Facial Surfaces The facial surfaces of the denture base are those polished surfaces lying between the buccal borders and the supplied teeth. Methods have been proposed for making sectional impression records of buccal contours, thereby permitting the denture base to be made to conform to the facial mus-culature. These have never received wide acceptance and may be considered impractical in removable partial prosthodontics. Facial surfaces may be established in wax or may be carved into the denture base after processing. Generally, it is desirable that it be done in wax as part of the wax pattern because it is easier to do so, and because contours can best be established at a time when modifications can be made if desired. Buccal surfaces should be contoured to aid in reten-tion of the denture by border molding, to preserve the border roll and thereby prevent food impaction, and to facilitate return of the food bolus back onto the masticating table. Lingual surfaces should be made concave to provide tongue room and to aid in retention of the denture. Polish-ing of concave surfaces is always more difficult than polish-ing of flat and convex surfaces. If such contours are established previously in wax, not only is finishing more easily accomplished, but border and gingival areas are less likely to be inadvertently altered. Finishing Gingival and Interproximal Areas The contouring of gingival and interproximal areas after processing is difficult and generally unsatisfactory (Figure 18-42). The practice of doing so dates back to the days when vulcanite rubber was trimmed and shaped with Pearson-type chisels, and a trimming block was a necessary piece of equipment in any dental laboratory. Finishing was done with vulcanite burs and with brush wheels and pumice, creating the vertical interproximal grooves that for many years were typical of the denture look. Not only is this contrary to modern concepts of denture esthetics, but also, gingival and interproximal carving of the denture resin around plastic teeth may not be done without causing some damage to the teeth themselves. Modern cosmetic considerations demand that gingival carving be done around each tooth individually, with varia-tions in the height of the gingival curve and in the length of the papillae. Interproximally, the papillae should be convex rather than concave. The gingival attachment should be free of grooves and ditches that would accumulate debris and stain and should be as free for cleansing as possible. All this precludes gross shaping and trimming of gingival areas after processing. Gingival carving should be done in wax, and investing should be done with care to prevent blebs and molten wax, and a remounting cast is poured in the impres-sion. The remounting casts are then oriented in the articula-tor by the same type of interocclusal records that were used to orient the casts to formulate the occlusion. These proce-dures will be covered in Chapter 20 as an integral part of the initial placement appointment. Occlusal harmony must exist before the patient is given possession of the dentures. Delaying the correction of occlu-sal discrepancies until the dentures have had a chance to settle is not justifiable. Polishing the Denture Areas to be considered in the polishing of a removable partial denture are (1) the borders of the denture bases, (2) the facial surfaces, and (3) the teeth and adjacent areas. The borders on complete metal bases will have been established previously. On partial metal bases and complete acrylic-resin bases, the accuracy with which the border may be finished will depend on the accuracy of the impression record and how well this was preserved on the stone cast. Edentulous areas recorded from impressions in stock trays generally lack the accuracy at the borders that is found on casts made from impressions in individualized trays and by secondary impression methods. Border accuracy is deter-mined also by whether or not the impression recorded a functional or a static relationship of the bordering tissue attachments. Denture Borders The principal objectives to be considered in making an impression of edentulous areas of a partially edentulous arch include (1) maximal support for the edentulous removable partial denture base, and (2) extension of the borders to obtain maximum coverage compatible with moving tissue. Although this second objective may be met with an adequate individualized impression tray, it is best accomplished with a secondary impression method. Not only the extent of the border but also its width should be recorded accurately. Both extent and width as recorded should be preserved on the stone cast. With the exception of certain areas that are arbitrarily thinned by polishing (mentioned previously in this chapter), finishing and polishing the denture borders should consist only of removing any flash and artifact blebs. Otherwise, borders should be left as recorded in the impression. When the impression is made in a stock tray, the tray itself will have influenced both the extent and the width of the border. Some areas will be left short of the total area available for denture support, whereas others will be extended beyond functional limits by overextension of the tray. Unfortunately, the technician may attempt to interpret the anatomy of the mouth and arbitrarily trim the denture borders. This presumes that the technician has an intimate knowledge of the anatomy of the mouth of the patient for whom the restoration is being made, which is most unlikely. 283 Chapter 18 Laboratory Procedures pumice, and finally with a soft brush wheel and a nonabra-sive polishing agent specially made for this purpose. Pumicing of gingival areas can only serve to polish the high spots; although it may be done lightly, its use should be limited to light buffing of areas already made as smooth as in the waxing phase. Heavy pumicing of the denture resin not only creates a typical denture look but also alters the delicately carved wax surface and any plastic teeth present. If pumicing must be done, plastic teeth should be protected with adhesive masking tape during the process. Any polishing operation on a removable partial denture done on a lathe is made hazardous by the presence of direct retainers, which can easily become caught in the polishing wheel. Although the least damage that might occur is the distortion of a clasp arm, there is a greater possibility that the denture may be thrown forcibly into the lathe pan, with serious damage to the framework or other parts of the denture. The technician must ever be conscious of this possibility and must always cover any projecting clasp with the finger while it is near the polishing wheel. In addition, it is wise to keep a pumice pan well filled with wet pumice to cushion the shock should an accident occur. Any other lathe pan used in polishing should be lined with a towel or with a resilient material for the same reason. The potential hazard of retainers catching gloved fingers or hands that hold a removable partial denture is always present. Serious injury can result, and extreme caution must be taken to prevent such an occurrence. The risk of infection is increased if the removable partial denture has been in place intraorally. artifacts. Finishing should consist only of trimming around the teeth and the papillae with small round burs to create a more natural simulation of living tissue, plus light stippling with an off-center round bur for the same reason. Polishing should consist only of light buffing with brush wheels and A B Figure 18-42 Attention to adjacent tooth-tissue contours can facilitate the production of natural-appearing prostheses. The interproximal papilla at A demonstrates both vertical and hori-zontal components. In general, the horizontal component increases with age. The prosthetic interproximal papilla at B exhibits only a vertical component and appears artificial. The anterior border () contour is blunted and provides an obvious and abrupt contour change. Contouring the border to bevel into the interproximal region will reduce its objectionable appearance. 284 CHAPTER 19 Work Authorizations for Removable Partial Dentures Chapter Outline Work Authorization Content Function Characteristics Definitive Instructions by Work Authorizations Legal Aspects of Work Authorizations Delineation of Responsibilities by Work Authorizations Work Authorization A work authorization contains the written directions for laboratory procedures to be performed for fabrication of dental restorations. The responsibility of a dentist to the public and to the dental profession to safeguard the quality of prosthodontic services is controlled in part through meaningful work authorizations. If work authorizations are properly completed, they provide a means for increased pro-fessional quality assurance and satisfaction in a removable partial denture service. A work authorization completed by a dentist is similar to granting power of attorney. It grants authority for others to act on the dentist’s behalf and specifically prescribes what is authorized. When properly executed, work authorizations are effec-tive channels of communication. They enhance the quality of the completed restorations by providing instructions for individually and scientifically considered prostheses. Content Information contained in a work authorization should include the following: (1) the name and address of the dental laboratory; (2) the name and address of the dentist who initiates the work authorization; (3) the identification of the patient; (4) the date of work authorization; (5) the desired completion date of the request; (6) specific instructions; (7) the signature of the dentist; and (8) the registered license number of the dentist. All these requirements can be accom-modated in a simply designed form (Figure 19-1). Function The following four important functions are performed by a work authorization: 1. It furnishes definite instructions for laboratory proce-dures to be accomplished and implies an expectation of a level of acceptable quality for the services rendered. 285 Chapter 19 Work Authorizations for Removable Partial Dentures tory technicians are decoding experts. Sufficient information must be included in a work authorization to enable the technician to understand and execute the request. Many dentists are overly presumptive in assuming that a request can be acceptably fulfilled without proper directions. It is sound practice to provide the dental laboratory tech-nician with adequate written instructions for each laboratory service required in the fabrication of a restoration. Therefore a new work authorization should accompany the material returned to the laboratory for continuing progress to 2. It provides a means of protecting the public from the illegal practice of dentistry. 3. It is a protective legal document for both the dentist and the dental laboratory technician. 4. It completely delineates the responsibilities of the dentist and the dental laboratory technician. Characteristics A work authorization must be legible, clear, concise, and readily understood. It is unreasonable to assume that labora-1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 R R L L Copyright Attachments International, Inc. Treatment plan 1. Rests 2. Retention 3. Reciprocation 4. Major connector 5. Indirect retention 6. Guide planes 7. Base retention 8. Areas to be modified or contoured Instructor: Approval to send to laboratory: Date: Color code: Blue: Cast metal Red: Resin base and wrought wire Green: Areas to be contoured Removable Partial Prosthodontics Laboratory instructions Design specifications: Patient name: Student name: Patient number: Student number: Figure 19-1 Work authorization form, designed specifically for removable partial dentures, to furnish detailed information to the laboratory technician. Form used to specifically plan framework design and to designate mouth alterations and preparations. 286 Part II Clinical and Laboratory differences in the laboratory phases necessary for their fab-rication establish a requirement for individual work autho-rization forms. Definitive Instructions by Work Authorizations Work authorization forms may be designed so that only a minimum of writing is necessary to provide thorough instructions (Figure 19-2). The form can complete the restoration. In a modern dental practice, it is highly improbable that a one-trip laboratory service will be adequate to provide a truly professional removable restoration. No single work authorization form is adequate to furnish detailed instructions for accomplishing the laboratory phases in the fabrication of removable partial dentures, crowns, and fixed partial dentures, or complete dentures, or for accom-plishing orthodontic laboratory procedures. Inherent differ-ences in the many types of restorations themselves and 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 R R L L Copyright Attachments International, Inc. John Doe Joe Smith 158390 Cingulum rests, #22, #27 MO occlusal rest #20 MO occlusal rest #28 18 ga. loops, 12 ga. inferior border Metal tooth to replace #21 Distal #20, #22, #27 Mesial #20 Cingulum rest #22, #27 Lingual plate #22, #27, #28 Lingual bar I-bar clasp #28 I-bar clasp #22 1234 12 ga 18 ga loops 18 ga loops Cast I-bar Cast I-bar 2/15/09 Treatment plan Design specifications: 1. Rests 2. Retention 3. Reciprocation 4. Major connector 5. Indirect retention 6. Guide planes 7. Base retention 8. Areas to be modified or contoured Patient name: Student name: Instructor: Approval to send to laboratory: Date: Patient number: Student number: Color code: Blue: Cast metal Red: Resin base and wrought wire Green: Areas to be contoured Removable Partial Prosthodontics Laboratory instructions Figure 19-2 This work authorization accompanies a master cast on which the dentist has designed and drawn an outline for a removable partial denture framework. It is simple and is not time consuming to execute, yet it furnishes detailed information so the request can be properly fulfilled. 287 Chapter 19 Work Authorizations for Removable Partial Dentures of a removable partial denture service. Instructions should leave no doubt about the dentist’s requirements in a request for laboratory services. It is foolish to use undercut dimen-sions of 0.01 or 0.02 inch when a master cast is surveyed, unless directions are included to incorporate these dimen-sions into the finished framework. Work authorization blanks should be available in such a manner that a duplicate can be conveniently made and thus a copy can be supplied for both the dentist and the dental laboratory technician. The original may be a different color than the duplicate for ready identification. Legal Aspects of Work Authorizations Although the National Association of Dental Laboratories (NADL) provides guidelines for statutory regulation, it is the inherent right of each state to implement its own regulation. Fortunately, all states exercise this control. Interpretations of acts that constitute the practice of dentistry are moder-ately uniform. However, statutory restrictions on dental laboratory operations vary from state to state. Properly exe-cuted work authorizations serve to document communica-tion and protect the professional relationship between the dentist and the dental laboratory. Many states require that work authorizations be made in duplicate and that both the dentist and the dental laboratory technician retain a copy for a specified period from the date of work authorization. Thus documents are available to sub-stantiate or refute claims and counterclaims that concern the illegal practice of dentistry, or to aid in the settlement of misunderstandings between a dentist and a dental labora-tory technician. Delineation of Responsibilities by Work Authorizations The dentist is responsible for all phases of a removable partial denture service in the strict sense of the word, although the dental laboratory technician may be requested to perform certain technical phases of the service. However, the laboratory technician is responsible only to the dentist and never to the patient. A dentist who relegates the design of a removable partial denture to a less qualified individual accepts the possibility of an inferior removable partial denture service. A dentist who imposes on auxiliary personnel those responsibilities that legally and morally belong with the dentist does a great injustice to the patients, the technicians, and the dental profession. There is little doubt that the illegal practice of dentistry and the presently existing impasse between dentist and some dental laboratory technicians are in part a result of many individual dentists who impose unrealistic responsibility on laboratory technicians. Further-more, this unwelcome relationship may have been caused by the submission of poor impressions, casts, records, and instructions to the laboratory technician, with the demand contain printed listings of materials and specifications that require either a checkmark or a fill-in for authorization of their use. A reminder space is included to designate the choice of metal for the framework. Frameworks for removable partial dentures are usually cast in type IV gold, chromium-cobalt alloy, or a titanium alloy. The nature of the material of the denture base may also be indicated by a checkmark. It is difficult to elicit this information from the markings on master casts. Space is reserved on the work authorization form to furnish the technician with information on the dentist’s selection of teeth. The responsibility for tooth selection must remain with the dentist. The success of the removable partial denture depends in part on the consideration given to the size, number, and placement of the artificial teeth, and to the material from which they are made. A display of courtesy deserved by and a demonstration of respect for the laboratory technician are indicated. The general request is prefaced by “Please” and the specific instructions are ended with “Thank you.” Do any other three words promote better relations? A good work authorization form not only ensures clarity but also simplifies correct execution. Figures can be pro-vided on which diagrams may be drawn to enhance written descriptions when necessary. These diagrams may show the occlusal and lingual surfaces of the posterior teeth and the lingual surfaces of the anterior teeth. The palatal region of the maxillary arch and the lingual slopes of the mandibular alveolar ridge can also be included. These features allow a clear, diagrammatic representation of the locations of major connectors, which will complement the outline of the frame-work on the master cast. A color-code index can be used to explain the markings on the master cast when it is submitted to the laboratory for the fabrication of a framework. For example, a green pencil can be used to outline the framework; red to designate the desired location of finishing lines on the framework; and black lines to denote the height of contour on teeth and soft tissue created during the survey of the cast. The color code eliminates confusion in interpreting the markings on the master cast. Specifications for waxing the framework components for gold, chromium-cobalt, or titanium alloy castings must be furnished for the technician and are an integral part of the work authorization form. Specifications that are adequate for most removable partial denture frameworks may be listed. This feature alone saves time and effort in preparing the work authorization and serves as a handy reference for the laboratory technician. The listing of average specifica-tions does not preclude altering a specification when the situation necessitates other characteristics in a given component. The specific instructions provided in a work authoriza-tion must be so constructed that they will be a constant source of direction and supervision for the laboratory phases 288 Part II Clinical and Laboratory of impossible quality in the returned restoration under threat of economic boycott. Most dental laboratory technicians are ethical and ear-nestly desire to contribute their talents to the dental profes-sion. The dental profession is vitally interested in increasing the number of serious-minded dental auxiliary personnel to share in providing oral health care. However, until the dental profession elevates itself in the eyes of laboratory technicians and also elevates the stature of dental laboratory technology, greater availability of responsible auxiliary personnel is more fancied than real. The dental laboratory technician is a member of a team whose objectives are the prevention of oral disease and the maintenance of oral health as adjuncts to the physical and mental well-being of the public. A good dental laboratory technician is a valuable team member working with the dentist and contributes much to the team effort in providing oral health care for patients. The degree and quality of the team effort are the responsibility of the dentist and depend on the dentist’s knowledge, experience, technical skill, administrative ability, integrity, and ability to communicate effectively. A dentist may delegate much of the laboratory phase of a removable partial denture service. Work authorizations help fulfill the moral obligation to supervise and direct those technical phases that can be accomplished by dental laboratory technicians. Substantial indications suggest that many members of the dental profession either are not cognizant of the rewards of writing good work authoriza-tions or are not proficient in their execution. It is not a secret that some dentists do not submit instructions when availing themselves of commercial dental laboratory services. If the practice of prosthodontics is to remain in the control of dentists, each member of the dental profession must avoid delegating responsibility to those who are less qualified to accept the responsibility. Movements to allow denturism are seemingly becoming more prevalent and pos-sibly are related to poor laboratory communication regard-ing removable prosthodontics. 289 Chapter 20 Initial Placement, Adjustment, and Servicing of the Removable Partial Denture 289 CHAPTER 20 Initial Placement, Adjustment, and Servicing of the Removable Partial Denture Chapter Outline Adjustments to Bearing Surfaces of Denture Bases Occlusal Interference From Denture Framework Adjustment of Occlusion in Harmony With Natural and Artificial Dentition Instructions to the Patient Follow-up Services Initial placement of the completed removable partial denture, the fifth of six essential phases of removable partial denture service mentioned in Chapter 2, should be a routinely sched-uled appointment. All too often the prosthesis is quickly placed and the patient dismissed with instructions to return when soreness or discomfort develops. Patients should not be given possession of removable prostheses until denture bases have been initially adjusted as required, occlusal dis-crepancies have been eliminated, and patient education pro-cedures have been continued. Although it is true that some accommodation is a neces-sary part of adjusting to new dentures, many other factors are also pertinent. Among these are how well the patient has been informed of the mechanical and biological problems involved in the fabrication and wearing of a removable pros-thetic restoration, and how much confidence the patient has acquired in the excellence of the finished product. Knowing in advance that every step has been carefully planned and executed with skill, and having acquired confidence in both the dentist and the excellence of the prosthesis, the patient is better able to accept the adjustment period as a necessary but transient step in learning to wear the prosthesis. This confidence could be lost if the dentist does not approach the insertion and postinsertion phases as equally important for the success of the treatment. The term adjustment has two connotations, each of which must be considered separately. The first is adjustment of the denture bearing and occlusal surfaces of the denture made by the dentist at the time of initial placement and thereafter. The second is the adjustment or accommodation by the patient, both psychologically and biologically, to the new prosthesis. 290 Part II Clinical and Laboratory shows through the film of indicator paste may be errone-ously interpreted as a pressure spot, when actually the paste had adhered to the tissue in that area. Therefore only those areas that show through an intact film of indicator paste should be interpreted as pressure areas and relieved accord-ingly. The decision to relieve an area of pressure must con-sider whether the pressure is in a primary, secondary, or nonsupportive denture bearing area. The primary denture bearing areas should be expected to show greater contact than other areas. Pressure areas most commonly encountered are as follows: in the mandibular arch—(1) the lingual slope of the mandibular ridge in the premolar area, (2) the mylohyoid ridge, (3) the border extension into the retromylohyoid space, and (4) the distobuccal border in the vicinity of the ascending ramus and the external oblique ridge; in the max-illary arch—(1) the inside of the buccal flange of the denture over the tuberosities, (2) the border of the denture lying at the malar prominence, and (3) the point at the pterygomax-illary notch where the denture may impinge on the pterygo-mandibular raphe or the pterygoid hamulus. In addition, bony spicules or irregularities in the denture base that will require specific relief may be found in either arch. The amount of relief necessary will depend on the accu-racy of the impression, the master cast, and the denture base. Despite the accuracy of modern impression and cast materi-als, many denture base materials leave much to be desired in this regard, and the element of technical error is always present. It is therefore essential that discrepancies in the denture base are detected and corrected before the tissues of the mouth are subjected to the stress of supporting a pros-thetic restoration. One of our major responsibilities to the patient is that trauma should always be held to a minimum. Therefore the appointment time for initial placement of the denture must be adequate to permit such adjustment. Occlusal Interference from Denture Framework Any occlusal interference from occlusal rests and other parts of the denture framework should have been eliminated before or during the establishment of occlusal relations. The denture framework should have been tried in the mouth before a final jaw relation is established, and any such inter-ference should have been detected and eliminated. Much of this need not occur if mouth preparations and the design of the removable partial denture framework are carried out with a specific treatment plan in mind. In any event, occlusal interference from the framework should not ordinarily require further adjustment at the time the finished denture is initially placed. For the dentist to have sent an impression or casts of the patient’s mouth to the laboratory and to receive a finished removable partial denture prosthesis without having tried the cast framework in the mouth is a dereliction of responsibility to the patient and the profession. After the resin bases have been processed and before den-tures are separated from the casts, the occluding teeth must be altered to perfect the occlusal relationship between oppos-ing artificial dentition or between artificial dentition and an opposing cast or template. Denture bases must be finished to eliminate excess and perfect the contours of polished sur-faces for the best functional and esthetic results. This is made necessary by the inadequacies of casting procedures, because both the metal and resin parts of a prosthetic restoration are produced by casting methods. Unfortunately, such proce-dures in the laboratory rarely eliminate the need for final adjustment in the mouth to perfect the fit of the restoration to the oral tissue. Included in this final step in a long sequence of finishing procedures necessary to produce a biologically acceptable prosthetic restoration are the following: (1) adjustment of the bearing surfaces of the denture bases to be in harmony with the supporting soft tissue; (2) adjustment of the occlu-sion to accommodate the occlusal rests and other metal parts of the denture; and (3) final adjustment of the occlusion on the artificial dentition to harmonize with natural occlusion in all mandibular positions. Adjustments to Bearing Surfaces of Denture Bases Altering bearing surfaces to perfect the fit of the denture to the supporting tissue should be accomplished with the use of some kind of indicator paste (Figure 20-1). The paste must be one that will be readily displaced by positive tissue contact and that will not adhere to the tissue of the mouth. Several pressure indicator pastes are commercially available. However, equal parts of a vegetable shortening and USP zinc oxide powder can be combined to make an acceptable paste. The components must be thoroughly spatulated to a homo-geneous mixture. A quantity sufficient to fill several small ointment jars may be mixed at one time. Rather than dismissing the patient with instructions to return when soreness develops and then overrelieving the denture for a traumatized area to restore patient comfort, use a pressure indicator paste with any tissue bearing pros-thetic restoration. The paste should be applied by the dentist in a thin layer over the bearing surfaces. The material should be rinsed in water so it will not stick to the soft tissue, and then digital pressure should be applied to the denture in a tissue-ward direction. The patient cannot be expected to apply a heavy enough force to the new denture bases to register all of the pressure areas present. The dentist should apply both vertical and horizontal forces with the fingers in excess of what might be expected of the patient. The denture is then removed and inspected. Any areas where pressure has been heavy enough to displace a thin film of indicator paste should be relieved and the procedure repeated with a new film of indicator until excessive pressure areas have been eliminated. This is particularly difficult to interpret when patients exhibit xerostomia. An area of the denture base that 291 Chapter 20 Initial Placement, Adjustment, and Servicing of the Removable Partial Denture A B C Figure 20-1 A, Tissue side of finished bases of a Kennedy Class I modification 1 removable partial denture, where pressure indicates that paste has been applied. Paste was applied following careful inspection of the tissue surface for irregularities or sharp projections, which must be eliminated before fitting in the mouth. The entire tissue surface of the bases was dried before it was coated with a thin coat of pressure indicator paste using a stiff-bristle brush. Brush marks are evident, and it is the change in the pattern of brush marks that guides adjustment. It is important to avoid thick application of indicator paste, which can hide the presence of significant pressure. B, The prosthesis can be dipped in cold water or sprayed with a provided release agent before placement in the patient’s mouth, to prevent paste from sticking to oral tissues. After careful seating of the denture, the patient can close firmly on cotton rolls for a few seconds, or the dentist can alternately apply a tissue-ward pressure over the bases to simulate functional movement. The presence of tissue contact is evident in the pattern of the paste, which is different from the brushed pattern. There is no suggestion of excessive pressure in this tissue contact pattern. However, it is not uncommon to relieve the area adjacent to the abutment sparingly. Several placements of the denture with indicator paste are usually necessary for evaluation of the accuracy of the bases. C, A different denture base recovered from the mouth after manipulation simulating function. The tissue contact reveals excessive pressure at the region lingual to the retromolar pad. Adjustment of Occlusion in Harmony with Natural and Artificial Dentition The final step in the adjustment of the removable partial denture at the time of initial placement is adjustment of the occlusion to harmonize with the natural occlusion in all mandibular excursions. When opposing removable partial dentures are placed concurrently, the adjustment of the occlusion will parallel to some extent the adjustment of occlusion on complete dentures. This is particularly true when the few remaining natural teeth are out of occlusion. But where one or more natural teeth may occlude in any mandibular position, those teeth will influence mandibular movement to some extent. It is necessary therefore that the artificial dentition on the removable partial denture be made to harmonize with whatever natural occlusion remains. Occlusal adjustment of tooth-supported removable partial dentures may be performed accurately by any of several intraoral methods. Occlusal adjustment of distal extension removable partial dentures is accomplished more 292 Part II Clinical and Laboratory A B Figure 20-2 Sequence of laboratory and clinical procedures performed for correction of occlusal discrepancies caused by processing of removable partial dentures. The arch with the prosthesis will require a new cast and record to provide occlusal correction. If it is a maxillary prosthesis, this involves preserving the facebow record by replacing the processed maxillary removable partial denture and cast on the articulator and indexing the occlusal surfaces using a remount jig, or making another facebow record at the try-in appoint-ment. If it is a mandibular prosthesis, the opposing arch can be prepared before insertion. A, In this example, the opposing arch is a complete denture that is not to be altered. To produce a cast for use in correcting the occlusion on the articulator, a pick-up impression of the mandibular prosthesis is made. The prosthesis stays within the irreversible hydrocolloid impression; the clasps, proximal plates, and any undercut or parallel surface are carefully blocked out with wax before the remount cast is formed. B, The remount cast that is formed is then inverted and positioned with the use of an interocclusal record. accurately with the use of an articulator than by any intraoral method. Because distal extension denture bases will exhibit some movement under a closing force, intraoral indications of occlusal discrepancies, whether produced by articulating paper or disclosing waxes, are difficult to interpret. Distal extension dentures positioned on remounting casts can con-veniently be related in the articulator with new, nonpressure interocclusal records, and the occlusion can be adjusted accurately at the appointment for initial placement of the dentures (Figure 20-2). The methods by which occlusal relations may be estab-lished and recorded have been discussed in Chapter 17. In this chapter, the advantages of establishing a functional occlusal relationship with an intact opposing arch have been discussed, along with the limitations that exist to perfecting harmonious occlusion on the finished prosthesis by intraoral adjustment alone. Even when the occlusion on two opposing removable partial dentures is adjusted, it is best that one arch be considered an intact arch and the other one adjusted to it. This is accomplished by first eliminating any occlusal interference with mandibular movement imposed by one denture and adjusting any opposing natural dentition to accommodate the prosthetically supplied teeth. Then the opposing removable partial denture is placed, and occlusal adjustments are made to harmonize with both the natural dentition and the opposing denture, which is now consid-ered part of an intact dental arch. Which removable partial denture is adjusted first, and which is made to occlude with it is somewhat arbitrary, with the following exceptions: If one removable partial denture is entirely tooth supported and the other has a tissue-supported base, the tooth-sup-ported denture is adjusted to final occlusion with any oppos-ing natural teeth. That arch is then treated as an intact arch and the opposing denture adjusted to occlude with it. If both removable partial dentures are entirely tooth supported, the one that occludes with the most natural teeth is adjusted first and the second denture is then adjusted to occlude with an intact arch. Tooth-supported segments of a tooth and tis-sue–supported removable partial denture are likewise adjusted first to harmonize with any opposing natural denti-tion. The final adjustment of occlusion on opposing tissue-supported bases is usually done on the mandibular removable partial denture because this is the moving member, and the occlusion is made to harmonize with the maxillary remov-able partial denture, which is treated as part of an intact arch. Intraoral occlusal adjustment is accomplished with the use of some kind of indicator and suitable mounted points and burs. Diamond or other abrasive points must be used to reduce enamel, porcelain, and metal contacts. These also may be used to reduce plastic tooth surfaces, but burs may be used for plastic with greater effectiveness. Articulation paper may be used as an indicator if one recognizes that heavy interocclusal contacts may become perforated, leaving only a light mark. Secondary contacts, which are lighter and 293 Chapter 20 Initial Placement, Adjustment, and Servicing of the Removable Partial Denture C D E C, The mounted mandibular cast and the interocclusal record showing that the record was made without tooth contact. This allows the position recorded to not be influenced by the teeth, which could alter the closure path and introduce error. D, This example shows a maxillary complete denture that was mounted before the patient visit with the use of a remount index (preserved facebow by indexing before recovery from the processed cast) and the mandibular remount cast and interocclusal record (as in C). The record is removed and occlusal correction accomplished to control the postprocess occlusion. Use of the completed prostheses provides the best chance to obtain an accurate and reliable interocclusal record, given the fact that the bases are very accu-rate and stable. E, The goal of the remounting procedure is to provide the occlusal position prescribed by the arrangement of prosthesis teeth. It would be inappropriate to allow the patient to attempt to accommodate to a new prosthesis in which the occlusion is not optimized. Figure 20-2, cont’d frequently sliding, may make a heavier mark. Although articulation ribbon does not become perforated, it is not easy to use in the mouth, and differentiation between primary and secondary contacts is difficult, if not impossi-ble, to ascertain. In general, occlusal adjustment of multiple contacts between natural and artificial dentition when tooth-sup-ported removable partial dentures are involved follows the same principles as those for natural dentition alone. This is because removable partial dentures are retained by devices attached to the abutment teeth, whereas no mechanical retainers are present with complete dentures. The use of more than one color of articulation paper or ribbon to record and differentiate between centric and eccentric con-tacts is just as helpful in adjusting removable partial denture occlusion as natural occlusion, and this method may be used for the initial adjustment. For final adjustment, because one denture will be adjusted to occlude with a predetermined arch, the use of an occlusal wax may be necessary to establish points of excessive contact and interference. This cannot be done by articulation paper alone. An occlusal indicating wax (Figure 20-3) that is adhe-294 Part II Clinical and Laboratory than expecting the patient to chew without food actually being present. The small bolus of banana promotes normal functional activity of the chewing mechanism, yet by its soft consistency does not in itself cause indentations in the soft wax. Any interfering contacts encountered during the chewing stroke are thus detected as perforations in the wax, which are marked with pencil and relieved accordingly. After adjustment of the occlusion, the anatomy of the artificial teeth should be restored to maximal efficiency by restoring grooves and spillways (food escapeways) and by narrowing the teeth bucco-lingually to increase the sharp-ness of the cusps and reduce the width of the food table. Mandibular buccal and maxillary lingual surfaces in particu-lar should be narrowed to ensure that these areas will not interfere with closure into the opposing sulci, because arti-ficial teeth used on removable partial dentures that oppose natural or restored dentition should always be considered material out of which a harmonious occlusal surface is created. Final adjustment of the occlusion should always be followed by meticulous restoration of the most functional occlusal anatomy possible. Although this may be done after a subsequent occlusal adjustment is made at a later date, the possibility that the patient may fail to return on schedule is always present; in the meantime, broad and inefficient occlusal surfaces may cause overloading of the supporting structures, which would be traumatogenic. Therefore resto-ration of an efficient occlusal anatomy is an essential part of the denture adjustment at the time of placement. Again, this necessitates that sufficient time is allotted for initial place-ment of the removable partial denture to permit accom-plishment of all necessary occlusal corrections. Adjustments to occlusion should be repeated at a reason-able interval after the dentures have reached a point of equi-librium and the musculature has become adjusted to the changes brought about by restoration of occlusal contacts. This second occlusal adjustment usually may be considered sufficient until such time as tissue-supported denture bases no longer support the occlusion, and corrective measures— either reoccluding the teeth or relining the denture—must be used. However, a periodic recheck of occlusion at inter-vals of 6 months is advisable to prevent traumatic interfer-ence resulting from changes in denture support or tooth migration. Instructions to the Patient Finally, before the patient is dismissed, he or she should be reminded of the chronic nature of the missing tooth condi-tion and of the fact that treatment solutions, such as a removable partial denture (RPD), require monitoring to ensure that they continue to provide optimum function without harming the mouth. Patients should be instructed in the proper placement and removal of the removable partial denture. They should demonstrate that they can place and remove the prosthesis themselves. Clasp breakage can be avoided by instructing sive on one side, or strips of 28-gauge casting wax or other similar soft wax, may be used. This should always be done bilaterally, with two strips folded together at the midline. Thus the patient is not as likely to deviate to one side as when wax is introduced unilaterally. For centric contacts, the patient is guided to tap into the wax. Then the wax is removed and inspected for perforations under transillumination. Premature contacts or excessive contacts appear as perforated areas and must be adjusted. One of two methods may be used to locate specific areas to be relieved. Articulation ribbon may be used to mark the occlusion; then those marks that represent areas of excessive contact are identified by referring to the wax record and are relieved accordingly. A second method is to introduce the wax strips a second time, this time adapting them to the buccal and lingual surfaces for retention. After the patient has tapped into the wax, perforated areas are marked with a waterproof pencil. The wax is then stripped off and the pen-ciled areas are relieved. Whichever method is used, it must be repeated until occlusal balance in the planned intercuspal position has been established and uniform contacts without perforations are evident from a final interocclusal wax record. After adjust-ment has been completed, any remaining areas of interfer-ence are reduced, thus ensuring that there is no interference during the chewing stroke. Adjustments to relieve interfer-ence during the chewing stroke should be confined to buccal surfaces of mandibular teeth and lingual surfaces of maxil-lary teeth. This serves to narrow the cusps so that they will go all the way into the opposing sulci without wedging as they travel into the planned intercuspal contact. Skinner proposed giving a small bite of soft banana to chew rather Figure 20-3 Occlusal indicator wax. (Courtesy Kerr Corp., Orange, CA.) 295 Chapter 20 Initial Placement, Adjustment, and Servicing of the Removable Partial Denture debris should be avoided as much as possible, particularly around abutment teeth and beneath minor connectors. Fur-thermore, inflammation of gingival tissue is prevented by removing accumulated debris and substituting toothbrush massage for the normal stimulation of tongue and food contact with areas that will be covered by the denture framework. The mouth and the removable partial denture should be cleaned after eating and before retiring. Brushing before breakfast also may be effective in reducing the bacterial count, which may help to lessen acid formation after eating in the caries-susceptible individual. A removable partial denture may be effectively cleaned with the use of a small, soft-bristle brush. Debris may be effectively removed through the use of nonabrasive dentifrices, because they contain the essential elements for cleaning. Household cleaners and toothpastes should not be used, because they are too abrasive for use on acrylic-resin surfaces. The patient—the elderly or handicapped patient in particular—should be advised to clean the denture over a basin partially filled with water so that denture impact will be less if the denture is dropped accidentally during cleaning. Along with brushing with a dentifrice, additional cleaning may be accomplished with the use of a proprietary denture cleaning solution. The patient should be advised to soak the dentures in the solution for 15 minutes once daily, followed by thorough brushing with a dentifrice. Although hypochlo-rite solutions are effective denture cleansers, they have a tendency to tarnish chromium-cobalt frameworks and should be avoided. In some mouths, the precipitation of salivary calculus on the removable partial denture necessitates taking extra mea-sures for its removal. Thorough daily brushing of the denture will prevent deposits of calculus for many patients. However, any buildup of calculus noted by the patient between sched-uled recall appointments should be removed in the dental office. This can be quickly and readily accomplished with an ultrasonic cleaner. Because many patients may dine away from home, the informed patient should provide some means of carrying out midday oral hygiene. Simply rinsing the removable partial denture and the mouth with water after eating is beneficial if brushing is not possible. Opinion is divided on the question of whether or not a removable partial denture should be worn during sleep. Conditions should determine the advice given the patient, although generally the tissue should be allowed to rest by removal of the denture at night. The denture should be placed in a container and covered with water to prevent its dehydration and subsequent dimensional change. About the only situation that possibly justifies wearing removable partial dentures at night is when stresses generated by bruxism would be more destructive because they would be concentrated on fewer teeth. Broader distribution of the stress load, plus the splinting effect of the removable partial denture, may make wearing the denture at night advisable. patients to remove the removable partial denture by the bases and not by repeated lifting of the clasp arms away from the teeth with the fingernails. Patients should be advised that some discomfort or minor annoyance might be experienced initially. To some extent, this may be caused by the bulk of the prosthesis to which the tongue must become accustomed. Patients must be advised of the possibility of the develop-ment of soreness despite every attempt on the part of the dentist to prevent its occurrence. Because patients vary widely in their ability to tolerate discomfort, it is best to advise every patient that needed adjustments will be made. On the other hand, the dentist should be aware that some patients are unable to accommodate the presence of a removable prosthesis. Fortunately, these are few in any prac-tice. However, the dentist must avoid any statements that might be interpreted or construed by the patient to be posi-tive assurance tantamount to a guarantee that the patient will be able to use the prosthesis with comfort and accep-tance. Too much depends on the patient’s ability to accept a foreign object and to tolerate reasonable pressure to make such assurance possible. Discussing phonetics with the patient in regard to the new dentures may indicate that this is a unique problem to be overcome because of the influence of the prosthesis on speech. With few exceptions, which usually result from excessive and preventable bulk in the denture design, contour of denture bases, or improper placement of teeth, the average patient will experience little difficulty in wearing the remov-able partial denture. Most hindrances to normal speech will disappear in a few days. Similarly, perhaps little or nothing should be said to the patient about the possibility of gagging or the tongue’s reac-tion to a foreign object. Most patients will experience little or no difficulty in this regard, and the tongue will normally accept smooth, nonbulky contours without objection. Con-tours that are too thick, too bulky, or improperly placed should be avoided in the construction of the denture, but if present, these should be detected and eliminated at the time of placement of the denture. The dentist should palpate the prosthesis in the mouth and reduce excessive bulk accord-ingly before the patient has an opportunity to object to it. The area that most often needs thinning is the distolingual flange of the mandibular denture. Here the denture flange should always be thinned during finishing and polishing of the denture base. Sublingually, the denture flange should be reproduced as recorded in the impression, but distal to the second molar the flange should be trimmed somewhat thinner. Then, when the denture is placed, the dentist should palpate this area to ascertain that a minimum of bulk exists that might be encountered by the side and base of the tongue. If this needs further reduction, it should be done and the denture repolished before the patient is dismissed. The patient should be advised of the need to keep the dentures and the abutment teeth meticulously clean. If car-iogenic processes are to be prevented, the accumulation of 296 Part II Clinical and Laboratory to a passive relationship with the abutment tooth. Unfortu-nately, this is almost the only adjustment that can be made to a half-round cast clasp arm. On the other hand, the round wrought-wire clasp arm may be cervically adjusted and brought into a deeper part of the retentive undercut. Thus the passivity of the clasp arm in its terminal position is maintained but retention is increased, because it is forced to flex more to withdraw from the deeper undercut. The patient should be advised that the abutment tooth and the clasp will serve longer if the retention is held minimally, which is only that amount necessary to resist reasonable dislodging forces. The future development of denture rocking or looseness may be the result of a change in the form of the supporting ridges rather than lack of retention. This should be detected as early as possible after it occurs and corrected by relining or rebasing. The loss of tissue support is usually so gradual that the patient may be unable to detect the need for relining. This usually must be determined by the dentist at subse-quent examinations as evidenced by rotation of the distal extension denture about the fulcrum line. If the removable partial denture is opposed by natural dentition, the loss of base support causes loss of occlusal contact, which may be detected by having the patient close on wax or Mylar strips placed bilaterally. If, however, a complete denture or a distal extension removable partial denture opposes the removable partial denture, the interocclusal wax test is not dependable because posterior closure, changes in the temporomandibu-lar joint, or migration of the opposing denture may have maintained occlusal contact. In such cases, evidence of loss of ridge support is determined solely by the indirect retainer leaving its seat as the distal extension denture rotates about the fulcrum. No assurance can be given to the patient that crowned or uncrowned abutment teeth will not decay at some future time. The patient can be assured, however, that prophylactic measures in the form of meticulous oral hygiene, coupled with routine care by the dentist, will be rewarded by greater health and longevity of the remaining teeth. The patient should be advised that maximal service may be expected from the removable partial denture if the fol-lowing rules are observed: 1. Avoid careless handling of the denture, which may lead to distortion or breakage. Damage to the removable partial denture occurs while it is out of the mouth as a result of dropping it during cleaning or an accident that occurs when the denture is not worn. Fractured teeth and denture bases and broken clasp arms can be repaired, but a distorted framework can rarely if ever be satisfactorily readapted or repaired. 2. Protect teeth from caries with proper oral hygiene, proper diet, and frequent dental care. The teeth will be no less susceptible to caries when a removable partial denture is being worn but may be more so because of the retention of debris. At the same time, the remaining teeth have become all the more important because of oral rehabilita-tion, and abutment teeth have become even more valu-However, an individual mouth protector should be worn at night until the cause of the bruxism is eliminated. Often the question arises whether an opposing complete denture should be worn when a removable partial denture in the other arch is out of the mouth. The answer is that if the removable partial denture is to be removed at night, the opposing complete denture should not be left in the mouth. There is no more certain way of destroying the alveolar ridge, which supports a maxillary complete denture, than to have it occlude with a few remaining anterior mandibular teeth. The patient with a removable partial denture should not be dismissed as completed without at least one subsequent appointment for evaluation of the response of oral struc-tures to the restorations and minor adjustment if needed. This should be made at an interval of 24 hours after initial placement of the denture. It need not be a lengthy appoint-ment but should be made as a definite rather than a drop-in appointment. This not only gives the patient assurance that any necessary adjustments will be made and provides the dentist with an opportunity to check on the patient’s accep-tance of the prosthesis but also avoids giving the patient any idea that the dentist’s schedule may be interrupted at will and serves to give notice that an appointment is necessary for future adjustments. Follow-Up Services The patient must understand the sixth and final phase of removable partial denture service (periodic recall) and its rationale. Patients need to understand that the support for a prosthesis (Kennedy Class I and II) may change with time. Patients may experience only limited success with the treat-ment and prostheses so meticulously accomplished by the dentist, unless they return for periodic oral evaluations. After all necessary adjustments have been made to the removable partial denture and the patient has been instructed on proper care of the denture, the patient must also be advised as to future care of the mouth to ensure health and longevity of the remaining structures. How often the dentist should examine the mouth and the denture depends on the oral and physical condition of the patient. Patients who are caries susceptible or who have tendencies toward periodon-tal disease or alveolar atrophy should be examined more often. Every 6 months should be the rule if conditions are normal. The need to increase retention on clasp arms to make the denture more secure will depend on the type of clasp that has been used. Increasing retention should be accomplished by contouring the clasp arm to engage a deeper part of the retentive undercut rather than by forcing the clasp in toward the tooth. The latter creates only frictional retention, which violates the principle of clasp retention. As an active force, such retention contributes to tooth or restoration move-ment, or both, in a horizontal direction, disappearing only when the tooth has been moved or the clasp arm has returned 297 Chapter 20 Initial Placement, Adjustment, and Servicing of the Removable Partial Denture corrected by relining or whatever procedure is indicated. 4. Accept removable partial denture treatment as something that cannot be considered permanent, but partial dentures must receive regular and continuous care by both the patient and the dentist. The obligations for maintaining caries control and for returning at stated intervals for treatment must be clearly understood, along with the fact that regular charges will be made by the dentist for whatever treatment is rendered. able because of their importance to the success of the removable partial denture. Therefore a rigid regimen of oral hygiene, diet control, and periodic clinical observa-tion and treatment is essential for the future health of the entire mouth. Also the patient must be more conscien-tious about returning periodically for examination and necessary treatment at intervals stated by the dentist. 3. Prevent periodontal damage to the abutment teeth by main-taining tissue support of any distal extension bases. As a result of periodic evaluation, this can be detected and 299 Chapter 21 Relining and Rebasing the Removable Partial Denture 299 CHAPTER 21 Relining and Rebasing the Removable Partial Denture Chapter Outline Relining Tooth-Supported Denture Bases Relining Distal Extension Denture Bases Methods of Reestablishing Occlusion on a Relined Removable Partial Denture Differentiation between relining and rebasing the removable partial denture has been discussed previously in Chapter 1. Briefly, relining is the resurfacing of the tissue of a denture base with new material to make it fit the underlying tissue more accurately. Rebasing is the replacement of the entire denture base with new material. The artificial teeth may need to be replaced in a rebase procedure. Relin-ing removable partial dentures is a common occurrence in many dental practices; however, rebasing is not indicated as often. In either situation, a new impression is necessary and uses the existing denture base with modifications (Figure 21-1) as an impression tray for a closed-mouth or an open-mouth impression procedure. One of several types of impression materials may be used, such as metallic oxide impression paste, rubber-base or silicone elastomers, tissue condition-ing materials, or mouth-temperature wax. With a tooth-supported prosthesis, the impression method (open- or closed-mouth) is not as critical. In deciding between a closed-mouth and an open-mouth impression method for relining a distal extension removable partial denture, a major consideration is the resiliency of the mucosa covering the residual ridge. As with secondary impression techniques, a firm mucosal foundation can likely accommodate a closed-mouth functional impression technique or an open-mouth selective pressure technique. However, when the mucosa is easily displaced, the open-mouth selective pressure technique is preferable. Both techniques should guard against framework movement during the impression procedure. Before relining or rebasing is undertaken, the oral tissue must be returned to an acceptable state of health (Figure 21-2). For more information, refer to the Chapter 13 discussion about conditioning abused and irritated tissue. 300 Part III Maintenance of occlusal rests, boxlike internal rests, internal attachments, or supporting ledges on abutment restorations. Except for intrusion of abutment teeth under functional stress, the sup-porting abutments prevent settling of the restoration toward the tissue of the residual ridge. Tissue changes that occur beneath tooth-supported denture bases do not affect the support of the denture; therefore relining or rebasing is usually done for reasons that include (1) unhygienic condi-tions and the trapping of debris between the denture base and the residual ridge; (2) an unsightly condition that results from the space that has developed; or (3) patient discomfort associated with lack of tissue contact that arises from open spaces between the denture base and the tissue. Anteriorly, loss of support beneath a denture base may lead to some denture movement, despite occlusal support and direct retainers located posteriorly. Rebasing would be the treat-ment of choice if the artificial teeth are to be replaced or rearranged, or if the denture base needs to be replaced for esthetic reasons, or because it has become defective. To accomplish relining or rebasing, the original denture base must have been made of a resin material that can be relined or replaced. Commonly, tooth-supported removable partial denture bases are made of metal as part of the cast framework. These generally cannot be satisfactorily relined, although they sometimes may be altered by drastic grinding to provide mechanical retention for the attachment of an entirely new resin base, or some of the new resin bonding agent may be used. Ordinarily, a metal base, with its several advantages, is not used in a tooth-supported area in which early tissue changes are anticipated. A metal base should not be used after recent extractions or other surgery, or for a long span when relining is anticipated to provide secondary tissue support. A distal extension metal base ordinarily is used only when a removable partial denture is made over tissue that has become conditioned to supporting a previous denture base. Because the tooth-supported denture base cannot be depressed beyond its terminal position with the occlusal rests seated and the teeth in occlusion, and because it cannot rotate about a fulcrum, a closed-mouth impression method is used. Virtually any impression material may be used, pro-vided sufficient space is allowed beneath the denture base to permit the excess material to flow to the borders—where it may be turned by the bordering tissue, or, as in the palate, may be allowed to escape through venting holes without undue displacement of the underlying tissue. The qualities of each type of impression material must be kept in mind when the material to be used is selected. Ordinarily, an impression material is used that will record the anatomic form of the oral tissue. A word of caution should be mentioned when a tooth-supported resin base is relined with autopolymerizing resin as an intraoral procedure. When one or more relatively short spans are to be relined, making an impression for relining purposes necessitates that the denture be flasked and pro-cessed. The possibilities that the vertical dimension of occlu-Relining Tooth-Supported Denture Bases When total abutment support is available, but for one reason or another a removable partial denture has been the restora-tion of choice, and support for that restoration is derived entirely from the abutment teeth at each end of each eden-tulous span. This support may be effective through the use Figure 21-1 Use of an existing Kennedy Class I removable partial denture base as a tray during a reline impression. The selective pressure impression philosophy requires space for the impression material that is greater over the ridge crest (second-ary stress bearing area) than at the buccal shelf region (primary stress bearing area). A pear-shaped laboratory bur is used to provide general relief (0.5 to 1.0 mm) of the denture base, with additional relief (1.0 mm) obtained over the ridge crest with a #8 round straight shank laboratory bur. Care must be taken to ensure that the tissue surface is relieved of all undercuts that could cause cast fracture when one is recovering the cast from the impression. Figure 21-2 Kennedy Class I modification 1 arch with a removable partial denture that requires relining. Tissue abuse evident at the left buccal shelf region must be corrected before the reline impression is made. Management requires a period of function without the prosthesis or relief of the prosthesis in the affected region along with placement of a tissue resilient liner in an effort to reduce the traumatic effects of pressure. 301 Chapter 21 Relining and Rebasing the Removable Partial Denture rial and any material that has flowed onto proximal tooth surfaces and other components of the removable partial denture framework. While doing this, have the patient again rinse the mouth with cold water. Then replace the denture in its terminal position to bring the teeth into occlusion. Repeat the border movements with the patient’s mouth open. By this time, or soon thereafter, the material will have become firm enough to maintain its form out of the mouth. 7. Remove the denture, quickly rinse it in water, and dry the relined surface with compressed air. Apply a generous coat of glycerin with a brush or cotton pellet to prevent frosting of the surface caused by evaporation of the monomer. Allow the material to polymerize in a con-tainer of cold water. This will eliminate any patient dis-comfort and tissue damage that could have resulted from exothermic heat or prolonged contact of the tissue with unreacted monomer. Although it is preferable for 20 to 30 minutes to elapse before trimming and polishing, it may be done as soon as the material hardens. Polymeriza-tion may be expedited and made denser by placing the denture in warm water in a pressure pot for 15 minutes at 20 psi. The masking tape must be removed before trimming is done but should be replaced over the teeth and polished surfaces below the junction of the new and old materials to protect those surfaces during final polishing. When properly done, a direct reline is entirely acceptable for most tooth-supported removable partial denture bases made of a resin material, except when some tissue support may be obtained for long spans between abutment teeth. In the latter situation, a reline impression in tissue conditioning material or other suitable elastic impression material may be accomplished. The denture may then be flasked, and a pro-cessed reline may be added for optimal tissue contact and support. Relining Distal Extension Denture Bases A distal extension removable partial denture, which derives its major support from the tissue of the residual ridge, requires relining much more often than does a tooth-sup-ported denture. Because of this, distal extension bases are usually made of a resin material that can be relined to com-pensate for loss of support caused by tissue changes. Although tooth-supported areas are relined for other reasons, the primary reason for relining a distal extension base is to rees-tablish tissue support for that base. The need for relining a distal extension base is deter-mined by evaluating the stability and occlusion at reasonable intervals after initial placement of the denture. Before initial placement of the denture, the patient must be advised that (1) periodic examination and also relining, when it becomes necessary, are imperative; (2) the success of the removable partial denture and the health of the remaining tissue and sion may be increased and that the denture may be distorted during laboratory procedures must be weighed against the disadvantages of using a direct-reline material. Fortunately, these materials are constantly improved with greater pre-dictability and color stability. The possibility that the origi-nal denture base will become crazed or distorted by the action of the activated monomer is minimal when the base is made of modern cross-linked resin. However, caution should be exercised to ensure that the older types of resin bases are compatible when one is relining with direct-reline resins. When relining in the mouth with a resin reline material is done with an appropriate technique, the results can be highly satisfactory, with complete bonding to the existing denture base, good color stability, permanence, and accu-racy. The procedure for applying a direct reline of an existing resin base is as follows: 1. Generously relieve the tissue side of the denture base. Lightly relieve the borders. This not only provides space for an adequate thickness of new material but also elimi-nates the possibility of tissue impingement caused by confinement of the material. 2. Apply lubricant or tape over the polished surfaces from the relieved border to the occlusal surfaces of the teeth to prevent new resin from adhering to the preserved bases and teeth. 3. Mix the powder and the liquid in a suitable container according to the proportions recommended by the manufacturer. 4. While the material is reaching the desired consistency, have the patient rinse the mouth with cold water. At the same time, wipe the fresh surfaces of the dried denture base with a cotton pellet or small brush saturated with some of the reline resin monomer. This facilitates bonding and ensures that the surface is free of any contamination. 5. When the material has first begun to thicken, but while it is still quite fluid, apply it to the tissue side of the denture base and over the borders. Immediately place the removable partial denture in the mouth in its terminal position, and have the patient lightly close into occlusion. Be sure that no material flows over the occlusal surfaces or alters the established vertical dimension of occlusion. Then, with the patient’s mouth open, manipulate the cheeks to turn the excess at the border and establish harmony with bordering attachments. If a mandibular removable partial denture is being relined, have the patient move the tongue into each cheek and then against the anterior teeth to establish a functional lingual border. It is necessary that the direct retainers be effective to prevent displacement of the denture while molding of the borders is accomplished. Otherwise the denture must be held in its terminal position with finger pressure on the occlusal surfaces while border molding is in progress. 6. Immediately remove the denture from the mouth and, with fine curved iris scissors, trim away gross excess mate-302 Part III Maintenance The best way to ensure framework orientation through-out the impression procedure for a distal extension remov-able partial denture is with an open-mouth procedure done in exactly the same manner as the original secondary impres-sion (see Figure 16-10, D). The denture to be relined first is relieved generously on the tissue side (see Figure 21-1) and then is treated in the same way as the original impression base for a functional impression. The step-by-step proce-dure is the same, with the dentist’s three fingers placed on the two principal occlusal rests and at a third point between, preferably at an indirect retainer farthest from the axis of rotation. The framework thus is returned to its original ter-minal position, with all tooth-supported components fully seated. The tissue beneath the distal extension base then is registered in a relationship to the original position of the denture that will ensure that (1) the denture framework will be returned to its intended relationship with the supporting teeth; (2) optimum tissue support will be reestablished for the distal extension base; and (3) the original occlusal rela-tionship with the opposing teeth will be restored. Although it is true that the teeth are not allowed to come into occlusion during an open-mouth impression proce-dure, the original position of the denture is positively deter-mined by its relationship with the supporting abutment teeth. Because this is the relationship on which the original occlusion was established, returning the denture to this posi-tion should bring about a return to the original occlusal relationship if two conditions are satisfied. The first condi-tion is that laboratory procedures during relining must be done accurately without any increase in the vertical dimen-sion of occlusion. This is essential to any reline procedure, but it is a particular necessity with a removable partial denture because any change in occlusal vertical dimension will prevent occlusal rests from seating and will result in overloading and trauma to the underlying tissue. The second condition is that the opposing teeth have not extruded or migrated, or that the position of an opposing denture has not become altered irreversibly. In the latter situation, some adjustment of the occlusion will be necessary, but this should be deferred until the opposing teeth or denture and the structures associated with the temporomandibular joint have had a chance to return to their original position before denture settling occurs. One of the greatest satisfactions of a job well done is seen in the execution of an open-mouth reline procedure as described in the previous paragraph, which results in the restoration not only of the original denture relationship and of tissue support but also of the original occlusal relationship (Figure 21-3). Methods of Reestablishing Occlusion on a Relined Removable Partial Denture Occlusion on a relined removable partial denture may be reestablished by several methods, depending on whether the relining results in an increase in the vertical dimension of abutment teeth depend on periodic examination and servic-ing of both the denture and the abutment teeth; and (3) a charge will be made for these visits in proportion to the required treatment. There are two indications of the need for relining a distal extension removable partial denture base. First, loss of occlu-sal contact between opposing dentures or between the denture and the opposing natural dentition may be evident (see Figures 9-13 and 9-14). This is determined by having the patient close on two strips of 28-gauge, soft green or blue (casting) wax or Mylar matrix strips. If occlusal contact between artificial dentition is weak or lacking while the remaining natural teeth in opposition are making firm contact, the distal extension denture needs to have occlusion reestablished on the present base by altering the occlusion, thereby reestablishing the original position of the denture framework and base, or sometimes both. In most instances, reestablishing the original relationship of the denture is nec-essary, and the occlusion will automatically be reestablished. Second, loss of tissue support that causes rotation and settling of the distal extension base or bases is obvious when alternate finger pressure is applied on either side of the fulcrum line. Although checking for occlusal contact alone may be misleading, such rotation is positive proof that relin-ing is necessary. If occlusal inadequacy is detected without any evidence of denture rotation toward the residual ridge, all that needs to be done is to reestablish occlusal contact by rearranging the teeth or by adding to the occlusal surfaces with resin or cast gold onlays. On the other hand, if occlusal contact is adequate but denture rotation can be demon-strated, this is usually the result of migration or extrusion of opposing teeth or a shift in position of an opposing maxillary denture, thus maintaining occlusal contact at the expense of the stability and tissue support of that denture. This is often the situation when a maxillary complete denture opposes a removable partial denture. It is not unusual for a patient to complain of looseness of the maxillary complete denture and to request relining of that denture when actually it is the removable partial denture that needs relining. Relining and thus repositioning of the removable partial denture result in repositioning of the maxillary complete denture with a return of stability and retention in that denture. Therefore evidence of rotation of a distal extension removable partial denture about the fulcrum line must be the deciding factor as to whether relining needs to be done. Rotation tissue-ward about the fulcrum line always results in lifting of the indirect retainer(s). The framework of any distal extension removable partial denture must be in its original terminal position with indirect retainers fully seated during and at the end of any relining procedure. Any possibility of rotation about the fulcrum line caused by occlusal influence must be prevented, and therefore the framework must be held in its original terminal position during the time the impression is being made. This all but eliminates the use of a closed-mouth impression procedure when unilateral or bilateral distal extension bases are relined. 303 Chapter 21 Relining and Rebasing the Removable Partial Denture A B Figure 21-4 A, An irreversible hydrocolloid pick-up impression made following reline of the mandibular removable partial denture. All regions in the framework that are below the height of contour will be coated with a thin layer of baseplate wax to allow recovery of the prosthesis from the remount cast without damage to the cast. This allows replacement of the prosthesis in the mouth for an inter-occlusal registration. B, Mounted mandibular prosthesis against a maxillary complete denture. The maxillary prosthesis was mounted using a remount cast made after finishing and polishing of the prosthesis. It is not uncommon to see a significant occlusal problem when one is relining a mandibular distal extension removable partial denture when the residual ridge resorption has been left unchecked for a long time. The maxillary prosthesis can often assume a more inferior position, and when the mandibular prosthesis is reoriented to the natural dentition, this raises the mandibular occlusal plane, resulting in an occlusal relationship as shown. The best resolution of this requires addressing the maxillary arch at the same time and requires a repositioning of the maxillary occlusal plane. occlusion or in lack of opposing occlusal contacts. In either instance, it is usually necessary to make a remounting cast for the relined removable partial denture so the denture can be correctly related to an opposing cast or prosthesis in an articulator (Figure 21-4). In rare instances, after a distal extension removable partial denture has been relined by the method previously described, the occlusion is found to be negative rather than positive, or the same as it was before relining. This may be the result of wear of occlusal surfaces over a period of time, a high position of the original occlusion with resulting depression of opposing teeth, or other reasons. In such a situation, occlusion on the denture must be restored to rees-tablish an even distribution of occlusal loading over both A B Figure 21-3 A, Mandibular reline impression accomplished with an open-mouth impression technique. B, Following processing of the reline and occlusal correction with a clinical remount procedure (see Chapter 20), the occlusion is restored to the original relationship. natural and artificial dentition. Otherwise the natural denti-tion must carry the burden of mastication unaided, and the denture becomes only a space-filling or cosmetic device. If the artificial teeth to be corrected are resin, the occlu-sion can be reestablished by adding autopolymerizing or light-activated resin to occlusal surfaces, or by fabricating gold occlusal surfaces, which can be attached to the original replaced teeth. The original teeth may also be removed from the denture base and replaced by new teeth arranged to harmonize with the opposing occlusal surfaces. Baseplate wax may be used to support the teeth as they are arranged. This wax should be carved to restore the lingual anatomy of the teeth and the portion of the denture base that was elimi-nated when the original teeth were removed. A stone matrix 304 Part III Maintenance is made that covers the occlusal and lingual surfaces of the teeth and denture flange. Then wax may be removed from the denture base and teeth and the tissue-bearing surface coated with a bonding agent. Those areas on the stone matrix intimate to the new resin to be added should be painted with a tinfoil substitute or an air barrier coating material if a visible light-cured (VLC) material is to be used. The new teeth are placed in the stone matrix, and the matrix is accurately attached to the denture base with sticky wax or a hot glue gun. VLC material or an autopolymerizing resin is then used to attach the teeth. If an autopolymerizing mate-rial is used, it can be conveniently sprinkled on by a buccal approach. The buccal surface of the denture base adjacent to the teeth should be slightly overfilled so the correct shape may be restored to this portion of the base during finishing and polishing procedures. Occlusal discrepancies caused by this procedure should be corrected in the articulator by new jaw relation records if the denture has a distal extension base. A second method involves removing the original teeth and replacing them with a hard inlay wax occlusion rim on which a functional registration of occlusal pathways is then established (see Chapter 17). The original teeth or new teeth may then be arranged to occlude with the template thus obtained and subsequently attached to the denture base with processed VLC material or autopolymerizing resin. If the latter is used, the need for flasking may be eliminated by securing the teeth to a stone matrix while the resin attach-ment is applied with a brush technique. Regardless of the method used for reattaching the teeth, the occlusion thus established should require little adjustment in the mouth and should exhibit the occlusal harmony that is possible to attain by this method. 305 Chapter 22 Repairs and Additions to Removable Partial Dentures 305 CHAPTER 22 Repairs and Additions to Removable Partial Dentures Chapter Outline Broken Clasp Arms Fractured Occlusal Rests Distortion or Breakage of Other Components—Major and Minor Connectors Loss of a Tooth or Teeth Not Involved in Support or Retention of the Restoration Loss of an Abutment Tooth Necessitating Its Replacement and Making a New Direct Retainer Other Types of Repairs Repair by Soldering The need for repairing or adding to a removable partial denture will occasionally arise. However, the frequency of this occurrence should be held to a minimum by careful diagnosis, intelligent treatment planning, adequate mouth preparations, and the carrying out of an effective removable partial denture design with proper fabrication of all compo-nent parts. Any need for repairs or additions will then be the result of unforeseen complications that arise in abutment or other teeth in the arch, breakage or distortion of the denture through accident, or careless handling by the patient, rather than faulty design or fabrication. It is important that the patient is instructed in proper placement and removal of the prosthesis so that undue strain is not placed on clasp arms, on other parts of the denture, or on contacted abutment teeth. The patient also should be advised that care must be given to the prosthesis when it is out of the mouth, and that any distortion may be irrepara-ble. It should be made clear that there can be no guarantee against breakage or distortion from causes other than obvious structural defects. Broken Clasp Arms The following are several reasons for breakage of clasp arms: 1. Breakage may result from repeated flexure into and out of too severe an undercut. If the periodontal support is greater than the fatigue limit of the clasp arm, failure of the metal occurs first. Otherwise the abutment tooth is loosened and eventually is lost because of the persistent strain that is placed on it. Locating clasp arms only where an acceptable minimum of retention exists, as deter-mined by an accurate survey of the master cast, can prevent this type of breakage. 2. Breakage may occur as a result of structural failure of the clasp arm itself. A cast clasp arm that is not properly 306 Part III Maintenance ance of burnout temperatures exceeding 1300°F, and avoidance of excessive casting temperatures when a cast-to method is used. When wrought wire is attached to the framework by soldering, the soldering technique must avoid recrystallization of the wire. For this reason, it is best that soldering be done electrically to prevent the wrought wire from overheating. A low-fusing (1420°F to 1500°F), triple-thick, color-matching gold solder should be used rather than a solder that possesses a higher fusing temperature. 3. Breakage may occur because of careless handling by the patient. Any clasp arm will become distorted or will break if subjected to excessive abuse by the patient. The most common cause of failure of a cast clasp arm is distortion caused by accidental dropping of the removable partial denture. A broken retentive clasp arm, regardless of its type, may be replaced with a wrought-wire retentive arm embedded in a resin base (see Figure 22-1C and D) or attached to a metal base by electric soldering. Often this avoids the necessity of fabricating an entirely new clasp arm. formed or is subject to careless finishing and polishing eventually will break at its weakest point. This can be prevented by providing the appropriate taper to flexible retentive clasp arms and uniform bulk to all rigid nonre-tentive clasp arms. Wrought-wire clasp arms may eventu-ally fail because of repeated flexure at the region where it exits from the resin base (Figure 22-1), or at a point at which a nick or constriction occurred as a result of care-less use of contouring pliers. They also may break at the point of origin from the casting as a result of excessive manipulation during initial adaptation to the tooth or subsequent readaptation. Clasp breakage can best be pre-vented by cautioning the patient against removing the removable partial denture by sliding the clasp arm away from the tooth with the fingernails. A wrought-wire clasp arm can normally be adjusted several times over a period of years without failure. It is only when the number of adjustments is excessive that breakage is likely to occur. Wrought-wire clasp arms also may break at the point of origin because of recrystallization of the metal. This can be prevented by proper selection of wrought wire, avoid-A B C D Figure 22-1 Fractured direct retainer on canine abutment. The reason for breakage is likely the long-term repeated flexure from movement associated with this 8-year-old distal extension prosthesis. The denture must be evaluated for prospective serviceability if the retainer arm is repaired. Often, the patient will best be served by replacing the denture with a new restoration. A, The cast produced from an irreversible hydrocolloid pick-up impression. The height of contour is shown in pencil, with a red line illustrating to the labora-tory the location of repair wire (18-gauge). B, Clasp adapted to the designated line on the canine and fitted into the resin trough distal to the canine and palatal to the first and second premolars. Note the curvature placed at the end of the wire to prevent movement within the polymerized resin. C, Finished and polished wire repair from the buccal. D, Palatal view. 307 Chapter 22 Repairs and Additions to Removable Partial Dentures removed from the cast, and platinum foil is adapted to the rest seat and the marginal ridge and overlaps the guiding plane. The removable partial denture is returned to the cast and, with a fluoride flux, gold solder is electrically fused to the platinum foil and the minor connector in sufficient bulk to form an occlusal rest. An alternative solder to use is an industrial brazing alloy, which is higher fusing but responds excellently to electric soldering and does not tarnish. Distortion or Breakage of Other Components—Major and Minor Connectors Assuming that major and minor connectors were originally made with adequate bulk, distortion usually occurs from abuse by the patient (Figure 22-2). All such components should be designed and fabricated with sufficient bulk to ensure their rigidity and permanence of form under normal circumstances. Major and minor connectors occasionally become weak-ened by adjustment to prevent or eliminate tissue impinge-ment. Such adjustment at the time of initial placement may Fractured Occlusal Rests Breakage of an occlusal rest almost always occurs where it crosses the marginal ridge. Improperly prepared occlusal rest seats are the usual cause of such weakness: an occlusal rest that crosses a marginal ridge that was not lowered suf-ficiently during mouth preparations may be made too thin or may be thinned by adjustment in the mouth to prevent occlusal interference. Failure of an occlusal rest rarely results from a structural defect in the metal and rarely if ever is caused by accidental distortion. Therefore the blame for such failure must often be assumed by the dentist for not having provided sufficient space for the rest during mouth preparations. Soldering may repair broken occlusal rests. In prepara-tion for the repair, it may be necessary to alter the rest seat of the broken rest or to relieve occlusal interferences. With the removable partial denture in its terminal position, an impression is made in irreversible hydrocolloid and then is removed, with the removable partial denture remaining in the impression. The dental stone is poured into the impres-sion and is allowed to set. The removable partial denture is A B C D Figure 22-2 A, The maxillary juncture between major and minor connectors at the distalmost posterior molar has fractured. Thin platinum foil has been adapted to the cast beneath the fracture, the clasp assembly has been stabilized on the cast with fast-set plaster, the remainder of the prosthesis has been positioned on the cast in full contact with teeth and tissues, and the solder has been posi-tioned for the electric tip to be placed. B, The electric soldering tip and ground are in place. C, Immediately following solder flow, the fracture has been eliminated by the solder connecting the two segments. D, The polished solder repair is ready to be cleaned and returned to the patient. The patient is told that this repair is not as strong as the original, and although it is difficult to know how long it could serve the patient, careful handling of the prosthesis is mandatory. 308 Part III Maintenance nents. A new clasp assembly may be cast for this tooth and the denture reassembled with the new replacement tooth added. Other Types of Repairs Other types of repairs may include the replacement of a broken or lost prosthetic tooth, the repair of a broken resin base, or the reattachment of a loosened resin base to the metal framework. Breakage is sometimes the result of poor design, faulty fabrication, or use of the wrong material for a given situation. Other times, it results from an accident that will not necessarily repeat itself. If the latter occurs, repair or replacement usually suffices. On the other hand, if frac-ture has occurred because of structural defects, or if it occurs a second time after the denture has been repaired once before, then some change in the design—by modification of the original denture or with a new denture—may be necessary. Repair by Soldering Approximately 80% of all soldering in dentistry can be done electrically. Electric soldering units are available for this purpose, and most dental laboratories are so equipped. Elec-tric soldering permits soldering close to a resin base without removing that base because of rapid localization of heat at the electrode. The resin base needs only to be protected with a wet casting ring liner during soldering. result from inadequate survey of the master cast or from faulty design or fabrication of the casting. This is inexcusable and reflects on the dentist. Such a restoration should be remade instead of further weakening the restoration by attempting to compensate for its inadequacies by relieving the metal. Similarly, tissue impingement that arises from inadequately relieved components results from faulty plan-ning, and the casting should be remade with enough relief to prevent impingement. Failure of any component that was weakened by adjustment at the time of initial placement is the responsibility of the dentist. However, adjustment made necessary by settling of the restoration because abutment teeth have become intruded under functional loading may be unavoidable. Subsequent failure that results from the weakening effect of such an adjustment may necessitate making a new restoration as a consequence of tissue changes. Commonly, repeated adjustment to a major or minor con-nector results in loss of rigidity to the point that the connec-tor can no longer function effectively. In such situations, a new restoration must be made, or that part must be replaced by casting a new section and then reassembling the denture by soldering. This occasionally requires disassembly of denture bases and artificial teeth. The cost and probable success must then be weighed against the cost of a new res-toration. Generally the new restoration is advisable. Loss of a Tooth or Teeth not Involved in Support or Retention of the Restoration Additions to a removable partial denture are usually simply made when the bases are made of resin (Figure 22-3). The addition of teeth to metal bases is more complex and neces-sitates casting a new component and attaching it by solder-ing or creating retentive elements for the attachment of a resin extension. In most instances when a distal extension denture base is extended, the need should be considered for subsequent relining of the entire base. After the denture base has been extended, a relining procedure for both the new and the old base should be carried out to provide optimum tissue support for the restoration. Loss of an Abutment Tooth Necessitating its Replacement and Making a New Direct Retainer In the event of a lost abutment, the next adjacent tooth is usually selected as a retaining abutment, and it generally will require modification or a restoration. Any new restoration should be made to conform to the original path of place-ment, with proximal guiding plane, rest seat, and suitable retentive area. Otherwise, modifications to the existing tooth should be done the same as during any other mouth prepa-rations, with proximal recontouring, preparation of an ade-quate occlusal rest seat, and any reduction in tooth contours necessary to accommodate retentive and stabilizing compo-Color-matching gold solder may be used for soldering both gold and chromium-cobalt alloys. A solder for gold alloys that melts at between 1420°F and 1500°F is entirely adequate for soldering gold alloys to chromium-cobalt alloys, thereby lessening the chance of recrystallizing gold wrought wire via excessive and prolonged heat. For elec-tric soldering, triple-thick solder should be used so the additional bulk of the solder will retard melting momen-tarily, while the carbon electrode conducts heat to the area to be soldered. For soldering chromium-cobalt alloys, a color-matching white 19K gold solder—which melts at about 1676°F—is used. An application of flux is essential for the success of any soldering operation to prevent oxidation of the parts to be joined and the solder itself. A borax-type flux is used when gold alloys are sol-dered. Fluoride-type flux must be used when chromium-cobalt alloys are soldered. When a gold alloy is to be soldered to a chromium-cobalt alloy, a fluoride-type flux should be chosen. The following is a procedure for electric soldering: 1. Roughen both sections to be joined. 2. Adapt platinum foil to the master cast beneath the framework to serve as a backing on which the solder will flow. Lift the edges of the foil to form a trough to confine the flow of the solder. 309 Chapter 22 Repairs and Additions to Removable Partial Dentures A B D C E Retention groove #10 Figure 22-3 This patient presented with an asymptomatic fractured lateral incisor. A, Clinical presentation of the fractured tooth and prosthesis. Evaluation of the prosthesis revealed it to be adequately fitting, stable, and retentive. B, Pick-up impression of the prosthesis. C, Cast formed from the pick-up impression, showing a fully seated prosthesis. D, Preparation of the prosthesis included a mechanical means for retention (which was provided by creating a recess in the resin adjacent to the missing tooth) and creating a trough at the external finishing line to repair an area of marginal breakdown. E, The finished repair, which will be taken to the mouth and checked for occlusal clearance lingual to the maxillary anterior teeth. 3. Seat the pieces to be soldered onto the master cast and secure them temporarily with sticky wax. Over each piece, add enough soldering investment to secure them after the sticky wax has been eliminated, but leave as much metal exposed as possible. 4. After flushing off the sticky wax with hot water, secure the cast to the soldering stand. Cut sufficient solder and place conveniently nearby. 5. Flux both sections. Put sufficient triple-thick solder on or in the joint to complete the soldering in one opera-tion, always starting with enough solder to complete the job. 6. Wet the carbon tip with water to aid conduction of the current, and then touch the carbon tip to the solder (be sure the solder is held firmly in place). Place the other electrode on any portion of the framework to com-310 Part III Maintenance plete the electric circuit and heat the carbon electrode. Do not push the solder with the carbon tip, but let the heat alone make the solder flow. Do not remove the carbon tip from the solder while the soldering opera-tion is in progress; this will cause surface pitting due to arcing. After the solder has flowed, remove the elec-trodes, removing the carbon tip electrode last, and proceed to remove the work from the cast for finishing. Torch soldering requires an entirely different approach. It is used when the solder joint is long or unusually bulky, and when a larger quantity of solder has to be used. Torch soldering cannot be undertaken to repair a removable partial denture framework that has resin denture bases or artificial teeth supported by resin. The procedure for torch soldering is as follows: 1. Roughen both sections to be joined. 2. Adapt platinum foil to the master cast so it extends under both sections. 3. Seat the sections on the master cast in the correct relationship, and secure them temporarily with sticky wax. Also flow sticky wax into the joint to be soldered. 4. Attach a dental bur or nail over the two sections with a liberal amount of sticky wax. Attach a second and even a third nail or bur across other areas to lend additional support. Never use pieces of wood for this purpose because the wood will swell if it gets wet, thus distorting the relationship of the two sections. 5. Carefully remove the assembled casting from the master cast. Adapt a stock of utility wax directly under each section on either side of the platinum foil. After boil out is done, investment will remain in the center to support the platinum foil. 6. Invest the casting in sufficient soldering investment to secure it, and expose as much of the area to be soldered as possible. When the investment has set, boil out the sticky and utility waxes. Then place the investment in a drying oven at a temperature not exceeding 200°F until the contained moisture has been eliminated. Do not dry or preheat the investment with a torch because oxides then will be formed that will interfere with the flow of the solder. 7. Use the reducing part of the flame, which is that feath-ery part just outside the blue inner cone. Flux the joint thoroughly and dry out the flux with the outer part of the flame until it has a powdery appearance. Heat the casting until it is dull red, and then, while holding a strip of solder in the soldering tweezers, dip it into the flux and feed it into the joint, while the casting is being held at a dull-red heat with the torch. Once the solder-ing operation has begun, do not remove the flame, because any cooling will cause oxides to form. The heat from the casting should be sufficient to melt the solder; therefore do not put the flame directly on the solder because it will become overheated and pitting will result. 8. After the soldering has been completed, allow the investment to cool slowly before quenching and pro-ceeding with finishing. Remember that any soldering operation that heats the entire casting is in effect a softening heat-treating operation, and heat-hardening of a repaired gold alloy casting is desirable to restore its optimal physical properties. 311 Chapter 23 Interim Removable Partial Dentures 311 CHAPTER 23 Interim Removable Partial Dentures Chapter Outline Appearance Space Maintenance Reestablishing Occlusal Relationships Conditioning Teeth and Residual Ridges Interim Restoration During Treatment Conditioning the Patient for Wearing a Prosthesis Clinical Procedure for Placement Tooth replacement is required for a variety of reasons. Sometimes replacements may be necessary for shorter periods of time that serve alternative purposes than perma-nent replacement, such as while tissue is healing or related treatment is being provided. When such applications require the temporary use of removable partial dentures, their fab-rication and use must be incorporated into a total prosth-odontic treatment plan. These various uses of interim prostheses for the partially edentulous mouth strive to achieve temporary goals with minimum time and expense. These prostheses are typically resin with wire retention and may include components to provide tooth support. The difficulty in achieving and main-taining strategic tooth support and stability with such pros-theses makes it important that patients be made aware that these prostheses are temporary and may jeopardize the integrity of adjacent teeth and the health of supporting tissue if worn for extended periods without supportive care. Interim prostheses may be indicated as a part of total treatment for the following: 1. Sake of appearance 2. Maintenance of a space 3. Reestablishment of occlusal relationships 4. Conditioning of teeth and residual ridges 5. Interim restoration during treatment 6. Conditioning the patient for wearing a prosthesis Appearance For the sake of appearance, an interim removable partial denture may replace one or more missing anterior teeth, or it may replace several teeth, both anterior and posterior. Such a restoration is usually made of resin, which may be produced by a sprinkling method, by the visible light-cured (VLC) method, or by waxing, flasking, and processing with autopolymerizing or heat polymerizing resin (Figure 23-1). It may be retained by circumferential wrought-wire clasps, Crozat-type clasps, interproximal spurs, or wire loops. 312 Part III Maintenance A B Figure 23-1 A, Although loss of mandibular teeth does not always have a significant esthetic impact, this interim removable partial denture was needed because of the visibility of the mandibular incisors. Provision of the mandibular left molars allowed early accom-modation of the residual ridge during the temporary prosthesis period. B, Tissue surface of the interim prosthesis revealing a rounded lingual flange and a tapered labial flange. The latter was needed to improve lip movement and reduce the feeling of bulk, both of which enhance normal lip activity. A B Figure 23-2 A, Malpositioned maxillary anterior teeth require extraction. Following a period of healing, the patient will decide on a definitive treatment option, which may include a removable partial denture or an implant-supported prosthesis. Because the length of time until definitive care will be provided is not known, a temporary interim prosthesis not only replaces esthetically important teeth but provides stabilization of adjacent and opposing dentition as well. B, Occlusal view of the interim prosthesis showing clasp place-ment at the most posterior locations without crossing of the occlusion and anterior positions bilaterally. Full palatal coverage allows less stress to the remaining maxillary dentition and may prevent prosthesis-induced gingival trauma as well as tooth movement. Space Maintenance When a space results from recent extractions or traumatic loss of teeth, it is usually prudent to maintain the space while the tissue heals. In younger patients, the space should be maintained until the adjacent teeth have reached sufficient maturity to be used as abutments for fixed restorations, or so that an implant can be placed. In adult patients, maintenance of the space can prevent unde-sirable migration and extrusion of adjacent or opposing teeth until definitive treatment can be accomplished (Figure 23-2). 313 Chapter 23 Interim Removable Partial Dentures Interim Restoration During Treatment In some instances, an existing removable partial denture can be used with modifications as an interim removable partial denture. Such modifications may include relining and adding teeth and clasps to an existing denture. In other instances, an existing removable partial denture may be con-verted to a transitional complete denture for immediate placement while the tissue heals and an opposing arch is prepared to receive a removable partial denture. Sometimes a temporary interim removable partial denture must be made to replace missing anterior teeth in a partially edentu-lous arch, which are ultimately to be replaced with fixed restorations. On occasion, the anterior portion of the resto-ration is cut away when the fixed restorations are placed; this leaves the remainder of the denture to be worn while poste-rior abutment teeth are prepared. Still another type of interim removable partial denture is one in which missing posterior teeth are replaced temporarily with a resin occlu-sion rim rather than with occluding teeth. Conditioning the Patient for Wearing a Prosthesis A temporary restoration may be made to aid the patient in making a transition to complete dentures when the total loss Reestablishing Occlusal Relationships Interim removable partial dentures are used for the follow-ing reasons: (1) to establish a new occlusal relationship or occlusal vertical dimension; and (2) to condition teeth and ridge tissue for optimum support of the definitive removable partial denture that will follow. Interim removable partial dentures may be used as occlu-sal splints in much the same manner as cast or resin occlusal splints are used on natural teeth. When total tooth support is available, there is little difference between a fixed and a removable occlusal splint, except that a removable splint is likely to be left out of the mouth unless the patient is actually made more comfortable by its presence. This is usually true when the wearing of an occlusal splint alleviates a temporo-mandibular joint condition. In other situations, it may be advisable to cement the removable restoration to the teeth until such time as the patient has become accustomed to, and dependent on, the jaw relationship provided by the splint. Both fixed and removable tooth-supported occlusal splints have much in common. Either of them may be elimi-nated in sections as restorative treatment is being done, thus maintaining the established jaw relation until all restorative treatment has been completed. The dentist decides whether these are to be fixed or removable, and whether they are made of a cast alloy, a composite, or a resin material. When one or more distal extension bases exist on an occlusal splint, a different situation occurs. The establish-ment of a new occlusal relation depends on the quality of support and stability the splint receives from the denture support. Both broad coverage and functional basing of tis-sue-supported bases are desirable, along with some type of occlusal rest on the nearest abutments. Any tissue-supported occlusal splint should be at least relined in the mouth with an autopolymerizing reline resin to afford optimal coverage and support for the distal extension base. Conditioning Teeth and Residual Ridges O.C. Applegate, in an article on the choice of partial or complete denture treatment, emphasized the advantages of conditioning edentulous areas to provide stable support for distal extension removable partial dentures. This is accomplished by having the patient wear an interim removable partial denture for a period of time before the final base is fabricated. In the absence of opposing occlusion, stimulating the underlying tissue by applying intermittent finger pressure to the denture base is advised. Whether the stimulation results from occlusal or finger pressure, there seems to be little doubt that the tissues of the residual ridge become more capable of sup-porting a distal extension removable partial denture when they have been previously conditioned by wearing of a restoration. Abutment teeth also benefit from wearing of an interim restoration when such a restoration applies an occlusal load to those teeth, either through occlusal cov-erage or through occlusal rests. Commonly a tooth that is to be used as an abutment for a removable partial denture has been out of occlusion for some time. Imme-diately on applying an occlusal load to that tooth suffi-cient to support any type of removable prosthesis, some intrusion of the tooth will occur. If such intrusion is allowed to occur after initial placement of the final pros-thesis, the occlusal relationship of the prosthesis and its relation to the adjacent gingival tissue will be altered. Perhaps this is one reason for gingival impingement, which occurs after the prosthesis has been worn for some time, even though seemingly adequate relief was provided initially. When an interim removable partial denture is worn, such abutment teeth have an opportunity to become stabilized under the loading of the temporary restoration, and intrusion will have occurred before the impression for the master cast is made. There is sufficient reason to believe that both abutment teeth and support-ing ridge tissue are more capable of providing continued support for the removable partial denture when they have been previously conditioned by the wearing of a tempo-rary interim restoration. 314 Part III Maintenance of teeth is inevitable. Such a removable partial denture also may be considered a valid part of the treatment, because the patient is at the same time being conditioned to wear a removable prosthesis. It should be considered strictly a tem-porary measure that provides the patient with a restoration for the remaining life of the natural teeth when further restorative treatment of those teeth is impractical or eco-nomically or technically impossible. This type of a removable partial denture may be worn for prolonged periods, in the meantime undergoing revision, modification to include additional teeth lost, or relining when such becomes necessary or advisable. The dentist should agree to provide such a removable partial denture only under the following conditions: (1) that a definite fee for the treatment is appropriate and that the fee will depend on the servicing necessary; and (2) that when further wearing of the transitional denture is unwise and jeopardizes the health of remaining tissue, the transition to complete den-tures will proceed. It is imperative that a distinction is made between tem-porary restorations and a true removable partial denture service, and that the patient is advised of the purposes and limitations of such restorations. Clinical Procedure for Placement It is important to consider proper fitting of the prosthesis to ensure comfortable use during the temporary phase of treat-ment. Careful attention to planned use of the teeth for support, stability, and retention without undue stress from gingival tissue contact or improper occlusal loading will ensure more comfortable use. To ensure proper use of the remaining natural teeth, the prosthesis must be completely seated in the arch. Common areas requiring adjustment to ensure complete seating include interproximal extensions, regions where clasps exit from the acrylic-resin base, tissue undercuts (labial under-cuts from recent extractions or the lingual/retromylohyoid region), and any portion of the prosthesis that lies inferior to the height of contour, especially if bilaterally opposed (Figure 23-3). Once seated, it is important to check that no undue pres-sure to the marginal gingival region is present. To facilitate this step, and to help with the complete seating requirement, it is possible to have the laboratory block out the marginal gingival region and infrabulge regions to reduce seating problems (Figure 23-4). Infrabulge regions may include lingual and palatal tooth surfaces, as well as modification space regions. Because temporary prostheses are generally fabricated on unprepared teeth, these regions often require correction. However, if the blockout is not accomplished carefully, the prosthesis may seat easily, but it may not be as stable as possible because of insufficient tooth contact. Sta-bility and retention are improved when it is possible to have the prosthesis contact portions of the teeth superior to the A B B C A B Figure 23-3 A, Maxillary interim removable partial denture. Areas that require adjustment commonly include interproximal extensions (A), the region where the clasp exits from the resin (B), and tissue undercuts of prosthesis extensions (C). B, Inter-proximal tooth extensions and regions where marginal gingival is crossed by the prosthesis should be carefully adjusted (outlined in red). height of contour because of tooth-dictated control of movement. It is possible to create occlusal imbalance if the lingual/ palatal portion of the prosthesis is too bulky. Consequently, an opposing cast should be provided to allow placement of clasps and acrylic-resin contours that do not cause occlusal interference (Figure 23-5). Once fully seated and relieved appropriately, the occlusion should contribute to the remaining natural dentition (as in a definitive prosthesis) and harmonize with natural tooth–dictated function. Typi-cally, the prosthesis should not be the sole source of occlusal contact. In such situations, the functional forces are concen-trated at the acrylic-resin-to-tooth junction, and predictably a change in orientation occurs, allowing tissue-ward move-ment and a change in occlusion with an increase in soft tissue contact. 315 Chapter 23 Interim Removable Partial Dentures A B Figure 23-4 A, Maxillary cast with an outline of the prosthesis design and with the region at the marginal gingival crossing outlined. If the cast is not relieved at the marginal gingival, the prosthesis should be corrected before insertion. B, Cast showing gingival regions relieved with baseplate wax. A duplicate of this cast will provide the necessary relief for use as a processing cast. If interproximal regions have undercuts, these can also be blocked out to allow easier insertion of a temporary prosthesis. A B Figure 23-5 A, Evaluation of mounted casts allows determination of the occlusal impact on maxillary temporary prosthesis clasp selection and placement. Interocclusal space allows placement of a ball clasp distal to the canine and a circumferential clasp from the distal of the second molar, both without occlusal interference if the clasps are carefully contoured. B, Anterior cast modification in anticipation of postextraction ridge contours for this immediate temporary prosthesis. Diagnosis of the limited space at the anterior region before surgery allows presurgical discussion of the possible need for mandibular incisor modification to improve the interim and final prosthesis occlusion. 316 CHAPTER 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics Chapter Outline Maxillofacial Prosthetics Maxillofacial classification Timing of Dental and Maxillofacial Prosthetic Care for Acquired Defects Preoperative and intraoperative care Interim care Potential complications Defect and oral hygiene Definitive care Intraoral Prostheses: Design Considerations Surgical Preservation for Prosthesis Benefit Maxillary defects Mandibular defects Mandibular reconstruction—bone grafts Maxillary Prostheses Obturator prostheses Speech aid prostheses Palatal lift prostheses Palatal augmentation prostheses Mandibular Prostheses Evolution of mandibular surgical resection Type I resection Type II resection Type III resection Type IV resection Type V resection Mandibular guide flange prosthesis Jaw Relation Records for Mandibular Resection Patients Summary Maxillofacial Prosthetics The preceding chapters have dealt with prosthesis consider-ations for partially edentulous individuals. In these patients, the extent of loss includes teeth and a varying degree of residual ridge bone, yet the remaining anatomy of the jaws and adjacent regions is functionally and physically intact. For these patients, the major distinguishing feature that affects removable partial denture design is whether the prosthesis will be tooth supported or tooth and tissue supported. The maxillofacial patient can experience unique altera-tions in the normal oral/craniofacial environment; these result from surgical resections (Figure 24-1), maxillofacial trauma, congenital defects, developmental anomalies, or neuromuscular disease. In contrast to the above, when removable partial dentures are considered for these indi-viduals, not only are tooth and tissue support considerations important, but the design must also take into account what impact the altered environment will have on prosthesis support, stability, and retention. In general, environmental changes reduce the capacity for residual teeth and tissue to provide optimum cross-arch support, stability, and retention. As a subspecialty of prosthodontics, maxillofacial pros-thetics is concerned with the restoration and/or replacement of the stomatognathic system and associated facial structures with prostheses that may or may not be removed on a regular or elective basis. This chapter discusses important back-ground information related to maxillofacial prostheses and the principles involved in removable partial denture design for the maxillofacial patient. 317 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics Maxillofacial Classification Patients can be categorized by maxillofacial defects that are acquired, congenital, or developmental. Acquired defects include those that are the result of trauma, or of disease and its treatment. These may include a soft and/or hard palate defect resulting from removal of a squamous cell carcinoma of the region (Figure 24-2). Congenital defects are typically craniofacial defects that are present from birth. The most common of these include cleft defects of the palate that may include the premaxillary alveolus. Developmental defects are those defects that occur because of some genetic predisposi-tion that is expressed during growth and development (Figure 24-3). Such a classification order is helpful as patients within each category share similar characteristics (beyond those features specifically related to the prosthesis design), which become part of the total management plan. For example, prosthetic management of an adult who has expe-A B Figure 24-1 A unilateral arrangement of maxillary teeth (A), no remaining horizontal hard palate, and a surgical defect, which includes nasal and sinus cavities (B). This unique environment, which is the result of a surgical resection, requires careful application of remov-able prosthodontic principles modified for maxillofacial needs. Figure 24-2 Large squamous cell carcinoma involving the maxillary tuberosity region, which will result in an acquired defect following surgical removal. rienced a maxillectomy procedure can be quite different from management of a patient with an unrepaired cleft palate. Another helpful way to classify maxillofacial patients is by the type of prosthesis under consideration. Consequently, prostheses are said to be extraoral (cranial or facial replace-ment) or intraoral (involving the oral cavity); interim (for short periods of time, often perioperative) or definitive (more permanent); and treatment (used as a component of management, such as a splint or stent) prostheses. Timing of Dental and Maxillofacial Prosthetic Care for Acquired Defects Acquired defects are the most common maxillofacial defects managed by using removable prostheses. A conceptual framework for the timing of dental/oral care that best emphasizes the initial important surgical requirements, fol-lowed later by the important prosthetic requirements, is helpful to consider regarding the coordination of care for patients with acquired defects. Such a framework considers preoperative and intraoperative interim care and definitive care. Although it may seem unrelated initially, it is included in this discussion of removable partial dentures for maxil-lofacial applications because of the important impact that decisions made at all stages of management can have on prosthesis function and patient outcomes. Preoperative and Intraoperative Care The planning of prosthetic treatment for acquired oral defects should begin before surgery. For the patient facing head and neck surgery, consideration should be given to dental needs that will improve the immediate postoperative course. Consequently, the prosthodontist who will help with management of the patient’s care should see the patient before surgery (Figure 24-4). The dental objectives of the preoperative and intraoperative care stages are to remove 318 Part III Maintenance A B Figure 24-3 A functional jaw position developed because of a combination of tooth loss and growth discrepancy. This developmental defect is illustrated by a protruded and overclosed mandibular position (A), which has created a significantly irregular maxillary occlusal plane (B). Figure 24-4 Presurgical presentation of a patient with a max-illary malignant melanoma. The benefits of having such a visit before surgery are both psychological and functional. The psy-chological benefits include the chance to discuss functional defi-cits associated with the anticipated surgical procedure and to describe how and to what extent the stages of prosthetic manage-ment will address them. The functional benefit from a prosthesis standpoint is that strategically important teeth, for definitive and/or interim prosthesis use, can be discussed with the surgical team and treatment planned for preservation. potential dental postoperative complications, to plan for the subsequent prosthetic treatment, and to make recommenda-tions for surgical site preparation that improve structural integrity. Important patient benefits of such a preoperative consultation include the opportunity to develop the patient-clinician relationship, to discuss the functional deficits associated with the anticipated surgical procedure, and to describe how and to what extent the stages of prosthetic management will address them. The benefit from a prosthe-sis standpoint is that strategically important teeth, for defini-tive and/or interim prosthesis use, can be discussed with the surgical team and treatment planned for preservation. The immediate postoperative period will be significantly challenging to the patient. If preexisting dental disease is severe enough to potentially create symptoms during the immediate postoperative period, treatment should be pro-vided to remove such a complication. Large carious lesions, which could create pain, can be temporarily restored by endodontic therapy if they offer some advantage for postop-erative prosthetic function. Teeth exhibiting acute periodon-tal disease (such as acute necrotizing ulcerative gingivitis) should be treated, as should any periodontal condition that could potentially cause postoperative pain because of exces-sive mobility or oral infection. Any tooth deemed nonrestor-able because of advanced caries or periodontal disease, and not critical for temporary use during the interim care period following temporary treatment, should be removed before, or at the time of, surgical resection. Teeth that may appear to have a limited long-term prognosis may significantly enhance prosthetic service during the initial postsurgical period and should be maintained until the initiation of definitive care. Impressions are made of the maxillary and mandibular arches to provide a record of existing conditions and occlu-sion to allow fabrication of immediate or interim prostheses (Figure 24-5) and to assess the need for immediate and delayed modification of the teeth or adjacent structures to optimize prosthetic care. It is important at this stage to begin planning for the definitive prosthesis because the greatest impact on the success of the maxillofacial prosthesis stems from the integrity of the remaining teeth and surrounding structures. Interim Care The major emphasis during this stage of care is the surgical (and adjunctive) management needs of the patient. In today’s environment of appropriately aggressive mandibular surgical reconstruction, mandibular discontinuity defects are seldom a surgical outcome. When discontinuity defects 319 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics surgical dressing and split-thickness skin graft during the immediate postsurgical period. Such prostheses are best sta-bilized by appropriate wiring to remaining teeth or alveolar bone, or they may be suspended from superior skeletal struc-tures. For some patients who have teeth remaining, such an immediate surgical prosthesis could be retained by wires in the prosthesis that engage undercuts on the teeth and would be removable; however, the ability to control the surgical dressing may be less predictable with such an approach. Immediate placement of a prosthesis has been suggested to improve patient acceptance of the surgical defect, although no measure of this psychological impact has been shown; this method offers greater assurance of adequate nourish-ment by mouth—potentially precluding the use of a naso-gastric tube. It may be preferable to stabilize the surgical dressing by suturing a sponge bolster to provide stabilization to the split-thickness skin graft. Following the primary healing stage, the sponge with packing (or the immediate prosthesis if used) is removed by the surgeon and an interim obturator prosthesis can be placed (Figure 24-7). For the patient who has been provided with bolster obturation, the presurgical prosth-odontic evaluation is very important to ensure that the patient is prepared for the transition from bolster to pros-thesis, and to ensure that plans for the prosthesis are made, especially if an interim prosthesis is to be fabricated. Interim prostheses are wire-retained resin prostheses that generally do not have teeth initially but may be modified with the addition of teeth after an initial period of accommodation (Figure 24-8). When surgical defects become large, as in a near-total maxillectomy defect, prosthesis support, stability, and reten-tion are not likely to be satisfactory unless extension into the defect can be accomplished. When teeth remain, the impact of the defect size is somewhat lessened. But when the remain-ing teeth are few or are located unilaterally in a straight line (see Figure 24-1), the mechanical advantage for prosthesis in the mandible result following surgery, interim prosthetic care is not indicated and the discussion will be directed to the maxillary defect. The typical maxillary acquired defect results in oral com-munication with the nose and/or maxillary sinus, although the composition of the surgical defect may vary widely (Figure 24-6). This creates physiologic and functional defi-ciencies in mastication, deglutition, and speech. Such defects have a negative impact on the psychological disposition of patients, especially if the defect also affects cosmetic appear-ance. The major deficiencies directly addressed by prosthetic management at this interim care time are deglutition and speech. This immediate postsurgical time is very challenging for patients, and it is important that they have been mentally prepared for it during the preoperative period. However, even with preliminary discussion, the impact of the surgery is often very distressing. An initial focus on improvement in swallowing and speech with the interim prosthesis can help boost the rehabilitation process significantly. The patient is counseled that chewing on the defect side is not allowed because of its effect on prosthesis movement. The objective of this interim obturator prosthesis is to sepa-rate the oral and nasal cavities by obturating the communi-cation. Such obturator prostheses most commonly refer to obturation of a hard palatal defect but conceptually can be considered the same for soft palatal defects at this stage of management, because both attempt to artificially block the free transfer of speech sounds and foods/liquids between the oral and nasal cavities. The advantages of having the ability to take nourishment by mouth without nasal reflux (allow-ing for nasogastric tube removal) and to communicate with family members are a significant component of early pros-thetic management. How immediately such care should be provided depends on a number of factors. A prosthesis can be provided at surgery (see Figure 24-5B). Such a surgical obturator prosthesis is placed at the time of surgical access closure and serves to control the A B Figure 24-5 A, A maxillary cast of the presurgical oral condition, which allows consultation with the surgeon regarding resection margins and the benefits of preservation of teeth. B, Another maxillary cast altered, following consultation with the head and neck surgeon, to allow fabrication of a surgical stent. Perforations are made to allow fixation to the remaining teeth and to superior anatomic regions with the use of wires. 320 Part III Maintenance A B C Figure 24-6 Maxillary defects. A, A resection that resulted in a small communication with the sinus, with some hard palate remain-ing, and adjacent mucosa typical of the oral cavity. B, A resection that did not follow classic maxillectomy technique; however, the midline resection was made through the socket of tooth #9 preserving its alveolar housing. C, A resection along the palatal midline that did not preserve oral mucosa at the resection margin, which allows chronic ulceration at this point of prosthesis fulcrum. Notice the split-thickness skin graft in the superior-lateral region. Engagement of this region can provide support to the obturator extension, minimizing movement with function. Figure 24-7 An interim obturator prosthesis fabricated of resin, retained by wires, and provided following surgical pack removal. stability is less. The ability of the defect tissue to offer the needed mechanical characteristics to the interim prosthesis is unpredictable at best. It is this patient who benefits the most from a well-planned surgery that preserves oral and defective anatomy to the advantage of the prosthesis. Potential Complications The interim phase of prosthetic management can last for three or more months. The primary objective is to allow the patient to pass from an active surgical (and adjunctive treat-ment) phase to an observational phase of management with minimal complications. During the transition, the patient recovers from the systemic effects of the treatment, deals with the psychological impact of the defect using his or her own coping strategy, and becomes more aware of the func-tional deficits associated with the surgical defect(s). Mini-mizing potential complications during the transition, which 321 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics tongue, opposing dentition, and cheek/lips place force on the prosthesis that must be resisted over a large area to prevent movement. Because the defect is least likely to be able to resist movement, the relative size and structural integrity of the defect compared with the remaining teeth and/or edentulous ridge determine the potential prosthesis movement and most affect the discomfort related to such movement. When teeth are available (and especially if located both close to and far away from the defect), retention is enhanced by engaging them with prosthetic clasps. Clasp retention is the most efficient means of effectively resisting dislodgment. The clasps will require periodic adjustment to maintain their effectiveness as the movement of the prosthesis flexes the clasps beyond their elastic recovery capacity. For edentulous patients, because the surgical defect allows communication between cavities, the fitting surface of the prosthesis can no longer create a closed environment to develop a seal for resisting dislodgment. Consequently, during the interim phase, when complete engagement of the defect is not pos-sible because of tissue sensitivity, the careful use of denture adhesives is required to facilitate retention. The patient should be instructed that adhesives can alter the prosthesis fit and disrupt the close adaptation of the prosthesis to the remaining tissue. Used adhesive must be removed before new adhesive is reapplied, to maintain fit and hygiene. Also related to retention is the inability to completely place the prosthesis, which for maxillectomy patients can be due to contracture of the scar band. When the maxillary resection leaves the cheek unsupported by bone, the prosthesis pro-vides the necessary support for wound maturation. If the patient removes the interim obturator prosthesis for a period sufficient to allow contraction, the prosthesis will be more difficult to place. Once placed, however, the scar band will relax and subsequent removal and placement will be more easily accomplished. The discomfort associated with this phenomenon is mostly due to patient anxiety and can be includes preparing the patient for those anticipated to occur, facilitates the process for the patient and family. Common interim prosthetic complications are related to tissue trauma and the associated discomfort; inadequate retention (loose-ness) of the maxillary prosthesis; incomplete obturation with leakage of air, food, and liquid around the obturator portion of the prosthesis; and the tissue effects of chemo-therapy and radiation therapy. Discomfort related to the use of interim prostheses can be due to surgical wound healing dynamics, defect condi-tions, mucosal effects of adjunctive treatment, and/or pros-thetic fit. Common areas of surgical wound pain include junctions of the oral and lip/cheek mucosa, especially at the anterior alveolar region for maxillectomy patients. The lateral scar band produced when the skin graft heals to the oral mucosa can be the site of discomfort in some patients. When a split-thickness skin graft is not placed, discomfort caused by the prosthesis fit within the defect can be a con-sistent and long-term problem. The hard palate surgical margin when not covered with surgically reflected oral mucosa most often will be covered by nasal epithelium, which is also very prone to discomfort. Alveolar bone cuts that have not been rounded will perforate the oral mucosa and will be painful whether or not a prosthesis is worn. This is most frequently a finding for mandibular resection supe-rior alveolar margins when the reconstruction has restored the lower and labial contour to the mandible, but the intra-oral mucosa at the superior surface is under tension because of a difference in height. The prosthesis can create discomfort via excessive static pressure from the internal surfaces or from overextension into the vestibular tissue. The prosthesis can also create dis-comfort caused by functional movement associated with swallowing and speech. As was discussed previously, pros-thesis movement is dependent on the quality of the support-ing structures. Teeth offer the best support, followed by firm edentulous ridges, and lastly, surgical defect structures. The A B Figure 24-8 A-B, An interim obturator prosthesis fabricated of resin, retained by wires, and including artificial teeth for cosmesis during an extended period of recovery. The superior and lateral surfaces may need modification to improve stability and retention as the surgical site matures and allows more aggressive engagement. 322 Part III Maintenance become more familiar with the surgical defect, patients should be encouraged to clean the defect of food debris and mucous secretions routinely. Defect hygiene will allow quicker healing and will improve the ability to adequately fit a prosthesis. Common defect hygiene practices include (1) lavage procedures, which include rinsing of the defect during normal showering, (2) rinsing of the defect using a bulb syringe or a modified oral irrigating device (modified to provide a multiple orifice “shower” effect), and (3) manual cleaning procedures, such as the use of a sponge-handled cleaning aid. Frequently, dried mucous secretions are diffi-cult to remove and require adequate hydration before mechanical removal. Following surgical pack removal, the patient may be reluctant to begin oral hygiene practices because of oral dis-comfort. As patients use the interim prosthesis, which requires daily removal and cleaning at a minimum, they will realize the need for and benefit of normal oral hygiene prac-tices because of improved prosthesis fit and tolerance. When teeth are remaining, it is important to the success of long-term prosthesis care to maintain a high level of oral hygiene. This is more critical for patients who exhibit xerostomia and have increased risk of caries. For these patients, daily appli-cation of fluoride in custom-formed carriers is prescribed along with frequent professional cleanings. The successful use of maxillofacial prostheses is enhanced greatly by the support provided by natural teeth. Consequently, during the interim prosthetic period, periodontal management proce-dures are begun in anticipation of the definitive treatment to allow a smooth transition from the interim to definitive prosthetic stages. Definitive Care When the active treatment phase has been completed, defin-itive prosthetic management can be initiated for as long as it takes the defect tissue to mature sufficiently to tolerate more aggressive manipulation and obturation. This phase can be considered a transition for the patient-physician rela-tionship, in which the primary emphasis shifts from active treatment to observation. The primary emphasis from the patient’s standpoint shifts to prosthetic management, and the goals and design of the prosthesis differ from those of the interim prosthesis (Figure 24-9). However, for some patients, more definitive prostheses are delayed because of general health concerns, questionable tumor prognosis or control, or failure of the patient to reach a level of oral and/ or defect hygiene that warrants more sophisticated treat-ment. Although this phase of management can be consid-ered elective, without definitive prostheses patients are not afforded the opportunity for complete rehabilitation. It is the extended use of temporary prostheses beyond their ser-viceable life span that has given a poor impression of prosthetic service to many surgeons and patients. Every opportunity should be provided to the patient for the most complete rehabilitation possible, and this necessitates con-sideration of more definitive prostheses. effectively addressed by reassuring the patient that this is an easily handled complication. During the immediate postoperative healing stage, the surgical defect will undergo a change in dimension that affects the prosthesis fit and seal. If space is created with the change, speech will be altered (increase in nasality) and nasal reflux with swallowing will occur. The interim prosthesis is made of easily adjustable material to allow accommodation for such changes. The most common manner of adjustment is through the use of temporary resilient denture lining materials, which offer the ability to mold to the tissue directly and reduce the mechanical effects of movement by virtue of their viscoelastic nature. Leakage can occur quite easily when swallowing unless the patient follows certain instructions. Because the prosthesis cannot offer a watertight seal that matches the presurgical state, patients will be instructed not to swallow large quantities at one time, and to hold their heads horizontal when swallowing. When the head posture is forward, as when one is taking soup from a spoon, leakage easily occurs around the obturator component of a prosthe-sis. Another difficult condition that presents difficulty in controlling leakage on swallowing is the midline soft palatal resection. The functional movement of the remaining soft palate is often very difficult to retain with a prosthesis. It is also difficult to provide an adequate seal during the interim prosthesis stage. When combination treatment is prescribed for the patient, it is commonly provided during the postsurgical phase, when the patient is using an interim prosthesis. The major intraoral complication associated with both radiation therapy and chemotherapy, which affects interim prosthetic service, is mucositis. A careful balance between comfort and adequate fit for speech and swallowing needs must be deter-mined with input from the patient. If prosthesis adjustment can offer relief to ensure completion of treatment and the patient understands the impact adjustment may have on speech and swallowing, then it should be accomplished. The long-term effects of radiation therapy, especially radiation-induced xerostomia and capillary bed changes (obliterative endarteritis) within the mandible, present a potentially significant threat to any remaining dentition and to the development of osteoradionecrosis. During the interim prosthesis stage, the patient will begin to notice the xerostomic effects, which include development of thick, ropy saliva that makes swallowing more difficult, and an increase in discomfort associated with removable prostheses. Defect and Oral Hygiene Following surgical pack removal, the defect site will mature with time and exposure to the external environment. Initial loss of incompletely consolidated skin graft, mucous secre-tions mixed with blood, and residual food debris within the cavity are common oral findings for the patient with a maxil-lary defect. These cause concern for patients who are unpre-pared and unfamiliar with these new oral findings. As they 323 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics tive prosthesis that is acceptable in appearance and exhibits minimal movement under function, thereby preserving the maximum amount of supporting tissue. A strategy for achieving these goals includes maximum coverage of the edentulous ridge within the movement capacity of the mus-cular attachments, maximum engagement of the remaining teeth to help control retention and movement under func-tion, and placement of artificial teeth to facilitate mainte-nance of this prescribed tooth-tissue contact during normal functional contacts. Maintaining these basic concepts within an otherwise normal anatomic environment (relative to food control and deglutition) has provided reasonable success for patients requiring replacement of missing teeth. The challenges faced in doing so for removable maxillofacial prostheses are quite different. Normal resistance to functional loads is achieved by the highly sophisticated periodontal attachment of the natural dentition, which provides support and stability to teeth. When the dentition is partially depleted and is replaced by prostheses that are tooth supported, the support and stabil-ity of the replacement teeth remain to be provided by the natural attachment. When tooth loss includes several poste-rior teeth, replacements are placed over the residual edentu-lous ridge, and the prosthesis receives support and stability from both teeth and mucosa. When all teeth are lost, support and stability are totally provided by the mucosa covering the residual edentulous ridges. Finally, when surgical removal of tumors results in tooth and supporting structure loss, support and stability are provided by combinations of remaining teeth and/or residual ridges and areas within the surgical defect. For partial and complete tissue-supported prostheses, the mechanism of functional load support—as provided by the mucosa—is unsuited to the task from a biological standpoint. Given this understanding, when a maxillofacial prosthesis is required to involve a surgical defect for support and stability, it is obvious that the envi-ronment within the surgical defect is even less suited to the task. Surgical Preservation for Prosthesis Benefit Maxillary Defects Surgical outcomes that influence prosthesis success can be considered as those that determine the number of maxillary structures removed (Figure 24-10) and/or those that affect the structural integrity and quality of the defect. For surgical defects of the hard and/or soft palate, the primary prosthetic objectives include restoration of physical separation of the oral and nasal cavities in a manner that restores mastication, deglutition, speech, and facial contour to as near a normal state as possible. Typical prostheses used to achieve these objectives include the obturator prosthesis (Figure 24-11A and B), typically referring to prostheses that obturate defects within the bony palate, and the speech aid prosthesis (Figure From the previous discussion regarding removable pros-thetic physiology, the inability of these static artificial replacements to mimic their natural counterparts results in less than ideal functional measures. Factors related to the structural integrity of the surgical defect and associated reconstructions as they affect this already compromised functional capacity are important considerations, especially when few teeth remain. As was stated previously, the fact that control of removable maxillofacial prostheses has a large skilled performance requirement of patients suggests that oral and defect structures adjacent to the prostheses are important for successful performance. This is crucial to an understanding of the impact that postsurgical defect charac-teristics and soft tissue reconstructions have on maxillofacial prosthesis management. The reasons for this are twofold: (1) the opportunity for maximal prosthetic benefit necessitates consideration of surgical site characteristics that are separate from classic tumor control approaches, and (2) the ability of the patient to biomechanically control large removable pros-theses following surgery may be notably hindered by surgical closure/reconstruction options. Surgical outcomes that can improve prosthetic function without adversely affecting tumor control should be considered and will be described for the more common surgical defects and associated prostheses. Intraoral Prostheses: Design Considerations Maxillofacial prosthetics is largely a removable prosthetic discipline, with the exception of dental implant–retained prostheses for some applications. For maxillofacial recon-struction with removable partial denture prostheses, typical goals of treatment consist of a well-supported, stable, reten-Figure 24-9 A definitive (left) and interim (right) obturator prosthesis contrasting the materials used and the obturator bulb contour. Clasp retention is more stabilizing with the definitive prosthesis because of the cast half-round clasp configuration, the use of embrasure clasps, and the opportunity for guide-plane use. Also, because the surgical site is more mature, prosthesis extension into this region to augment support, stability, and retention when necessary is possible. 324 Part III Maintenance A B Figure 24-10 A, A maxillary defect where a tooth distal to the resection was maintained. The tooth will significantly stabilize the prosthesis by preventing movement of the obturator bulb into the defect at the distal resection margin. B, A maxillary defect that dem-onstrates preservation of the anterior arch curvature, providing enhanced stability through a tripod effect. Also evident is the use of a split-thickness skin graft in the superior-lateral region, which improves the opportunity for useful support. A B D C Figure 24-11 A, Superior view of an obturator prosthesis demonstrating the cast framework, three posterior cast half-round clasps and an anterior I-bar clasp, and a superior obturator surface contoured to encourage secretions to flow posteriorly. B, The same pros-thesis seated intraorally. C, A speech aid prosthesis with posterior retention and anterior indirect retention, and a resin speech bulb. D, The same prosthesis showing bilateral embrasure clasps and obturation of the palatopharyngeal defect. 325 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics Figure 24-12 A maxillary obturator prosthesis demonstrating a distal extension, which engages a soft palatal remnant for added retention. 24-11C and D), which typically refers to prostheses that restore palatopharyngeal function for defects of the soft palate. Current preoperative diagnostic procedures have improved the ability to discern the location and regional bone involvement of tumors of the maxilla and associated paranasal sinuses. Relative to prosthetically important surgi-cal modifications, if it can be determined that tumor control does not require a classic radical maxillectomy approach, or that the inferior sinus floor, hard palate, and alveolus are uninvolved, preservation of as much hard palate and alveo-lar bone and as many teeth as possible should be considered. Tooth preservation has the greatest impact on success because of its stabilizing effect on prosthetic movement. When teeth can be retained in the premaxilla for more pos-terior tumors, or in the posterior molar region for more anterior tumors, control of prosthesis movement is more easily accomplished and prosthetic success can be consider-ably improved (see Figure 24-10). Because the classic midline maxillectomy defect is significantly more debilitating for the average patient than a defect whereby preservation of the premaxillary component was accomplished, inclusion of the anterior premaxillary component should be an individ-ual decision based on tumor control, classic resection technique. For resections in patients with teeth, the tooth adjacent to the defect is subjected to significant force from prosthesis movement. When the surgical alveolar ostectomy cut is planned, the resection should be made through the extrac-tion site of the adjacent tooth to provide the most favorable prognosis for this supportive tooth (see Figure 24-6). This procedure ensures adequate alveolar support for the adja-cent tooth, which is a critical tooth for prosthesis success, and improves the tooth’s prognosis for long-term survival. The midline of the hard palate is a common area of remov-able prosthesis pressure because of movement of the pros-thesis into the defect under functional forces of swallowing and mastication. To provide the best surgical preparation for this area, when the hard palate is resected, the vertical surface of the bone cut should be covered with an advancement flap of palatal mucosa, to provide a firm and resistant mucosal covering to this region, where the prosthesis can notably act as a fulcrum. The soft palate owes its normal function to the bilateral sling nature of the musculature, which provides the shape and movement capacity specific for speech and deglutition requirements. When this is altered because of surgery, there appears to be a variable response in the ability to continue to provide palatopharyngeal competence based on the amount of continuous band of posterior tissue remaining. Often an insufficient band of palatal tissue fails to provide palatopharyngeal competence and hinders prosthetic man-agement of the problem. To serve as a guide for decision making in surgery, it has been suggested that if the required resection leaves less than one third of the posterior aspect of the soft palate, the entire soft palate should be removed. The exception to this would be the edentulous patient who is undergoing a radical maxillectomy. Without teeth to provide the necessary retention for one side of a prosthesis, the patient benefits from the ability to place the prosthesis above the posterior soft tissue band for retention (Figure 24-12). Preparation of the maxillary surgical site can improve prosthesis tolerance through the use of a split-thickness skin graft (see Figure 24-1). Lining the reflected cheek flap and posterior denuded structures with a graft improves tissue response by decreasing the pain associated with functional contact often seen when this surface is left to heal second-arily. If the posterior structures, pterygoid plates, or anterior temporal bone can provide a firm supportive base for the prosthesis, a skin graft covering is extremely helpful. Later-ally, the junction of the skin and oral mucosa creates a scar contracture, which provides a natural retentive region for the obturator portion of the prosthesis. Careful attention is given to this region in fabricating a prosthesis, to maximize support, stability, and retention of the prosthesis. In general, the need to extend a prosthesis into the defect is greater for edentulous patients than patients with teeth. When teeth remain, they are used to a greater extent to stabilize and retain the obturator component of the prosthe-sis, and the defect region is not required for such objectives. However, all patients with maxillary defects should have sufficient access to the lateral-posterior region of the defect to seal the defect at a minimum. In the edentulous patient, for maximum ability to obturate a maxillary defect, access to the regions superior to the defect opening is required. Nasal turbinates and mucosal connections that do not allow full extension into the necessary retentive and supportive areas of the defect compromise function. The function of turbinates in the newly externalized environment is not 326 Part III Maintenance large tongue resection that may require augmentation of the palatal contours to facilitate speech production. Such a palatal augmentation prosthesis is most beneficial when coordinated speech therapy can guide the optimal prosthesis configuration. Other resections may appear to require palatal augmentation for speech, yet the functional problem is tongue immobility secondary to tension created by the reconstructive tissue. Consideration should be given to soft tissue reconstructions that are of sufficient size and mobility, are less prone to contracture tension, and can produce a more normal alveololingual sulcus because these charac-teristics have been shown to have a significant influence on tongue mobility. Other desirable characteristics, such as sensation and lubrication, are also possible but necessitate a choice of which one is most required given the goals desired. beneficial for breathed air humidification or warming, and consequently may not warrant preservation. Surgical reconstruction of maxillary defects should be undertaken when restoration of the functional goals of speech, deglutition, and mastication is better served by such procedures. Surgical reconstruction of a maxillary hard palatal defect in a manner that provides separation of the oral and nasal cavities without consideration for oral space requirements for speech, or for the supportive requirements of replacement teeth, is not only incomplete management but can preclude subsequent prosthetic management. When surgical defects measure 3 cm or less and can be recon-structed to normal contours without compromising adja-cent tissue function, surgical management is an appropriate consideration. Larger defects are very difficult to surgically reconstruct and, without careful planning for subsequent functional needs, could create an environment incapable of supporting a prosthesis. For partial soft palatal reconstruc-tions, it is very difficult to provide functional tissue replace-ment without compromising palatal function. In light of this unpredictability, predictable prosthetic management of such defects is most often the treatment of choice. Mandibular Defects The functions of mastication, deglutition, speech, and oral competence (saliva control) are made possible through coordinated efforts of separate anatomic regions, which include the oral sphincter, alveololingual and buccal sulci, alveolar ridges, floor of the mouth, mobile tongue, base of the tongue, tonsillar pillars, soft palate, hard palate, and buccal mucosa. The more regions that are involved in a surgical procedure, the greater is the demand on surgical reconstructive efforts. When the mandible is also involved, the complexity of the reconstructive procedure is dependent on the location and amount of mandible to be included in the resection and the decision to maintain continuity with normal mandibular position and contour (Figure 24-13). For disease involving the functional anatomy around the mandible, surgical outcomes that influence prosthesis success are based on decisions to take mandibular portions or segments and decisions regarding reconstruction. The primary prosthetic objectives for mandibular defects are to restore mastication and cosmesis by the replacement of teeth. Achieving the mastication goal requires an under-standing that regardless of the manner of prosthesis support (natural teeth, reconstructed soft tissue, or implants), the impact of the prosthetic device on success is dependent on appropriate surgical management of both soft tissue and bone. Disease involving soft tissue structures adjacent to and enveloping the mandible necessitates consideration of a mandibular resection to ensure control. When the soft tissue disease is clearly separate from the mandible and does not require bone removal, surgical defects involving these struc-tures should be surgically reconstructed and therefore do not require prosthetic management. The exception to this is the A B Figure 24-13 A, Marginal (left) and segmental (right) resec-tions of the mandible. When a segmental resection is not stabi-lized with a reconstruction bar or bone graft, the continuity of the mandible is lost. Such a defect is a discontinuity defect of the mandible. B, When not stabilized, the discontinuous man-dible deviates toward the defect and presents significant prob-lems with mastication restoration. 327 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics The cosmetic deformity associated with mandibular resection is improved through the use of reconstruction plates to maintain the presurgical contour to the lower jaw. This form of mandibular contour and position maintenance should be considered the minimum standard of care for mandibular resection patients from a functional standpoint. Use of reconstruction plates can maintain cosmetic appear-ance and preserve the bilateral nature of mandibular move-ment. However, the use of reconstruction plates alone precludes replacement of teeth in the region of resection. Prosthetic replacement of teeth cannot be provided for regions superior to the reconstruction bar because of the potential for mucosal perforation and exposure of the bar from functional loading of the soft tissue. From a mastica-tory function standpoint, this may not be a significant nega-tive impact for some patients because of the maintenance of sufficient numbers of occlusal contacts postsurgically. Mandibular Reconstruction—Bone Grafts The evolution of head and neck reconstructive surgery has been dramatic over the past three decades. The vascularized tissue options of the forehead and deltopectoral regions gave way to the more popular pedicled myocutaneous flaps from the 1960s to the 1970s. By the 1980s, numerous osteomyo-cutaneous free-flap donor sites had been identified and were being used for mandibular reconstruction and particulate cancellous bone marrow in formed allogeneic frames. Equally important to the functional outcome of mastication was the development of the science and the clinical applica-tion of the osseointegration phenomenon in the area of dental implants. The ideal prosthetic characteristics of the replacement mandible include a stable union to proximal and distal seg-ments, restoration of contour to the lower third of the face, a rounded ridge contour with attached mucosa of 2 to 3 mm thickness, and adjacent sulci providing free movement of When tumors are primary to the mandible, as an amelo-blastoma is, or when they involve the mandible from adja-cent regions, surgical resection of segments of the mandible is required for tumor control. It may be difficult to always predict the functional deficit and the exact plan of recon-struction because the surgeon determines the extent of the resection based on presurgical and surgical findings. However, common anatomically based mandibular resec-tions include the lateral mandibular resection, the anterior mandibular resection, and the hemimandibular resection. From the standpoint of the surviving mandibular resection patient, the most significant decision regarding his or her management is the decision to maintain mandibular conti-nuity, which allows maintenance of position for adjacent intraoral and extraoral soft tissue. Surgical evolution of procedures that maintain continuity for the mandible has significantly improved the opportunity for functional restoration of mastication, deglutition, and speech. The debilitating effects of the discontinuity defect include a significant cosmetic deformity to the lower third of the face, decreased masticatory function secondary to uni-lateral closure, compromised coordination of tongue and teeth, altered speech ability, and impaired deglutition (Figure 24-14). Given an appreciation of the decreased performance seen with conventional mucosa-borne denture prostheses, it should be obvious that masticatory rehabilitation for the resection patient without mandibular continuity is unpre-dictable at best and is never achieved for most patients. Even for patients with remaining teeth, the altered mandibular position created in time presents a significant functional and cosmetic handicap. From a prosthetic rehabilitation stand-point, the most significantly handicapped postsurgical head and neck condition is the discontinuous mandible. Conse-quently, such a postsurgical condition should be the rare exception (typically because of reconstruction plate failure) and should not be the planned surgical outcome. A B Figure 24-14 A, A deviated mandibular position following segmental resection without reconstruction. The mandibular midline is left of the maxillary midline by two teeth. B, With mandibular and maxillary prostheses in place, the patient closes to a functional posi-tion that is unique to the unilateral closure pattern. 328 Part III Maintenance nasal regions (Figure 24-15). Aramany developed a classifi-cation for partially edentulous maxillectomy dental arches (Figure 24-16). The various defects resulting from resection contain and are bounded by anatomic structures and an epithelial lining (either transplanted skin and/or native mucosa) that are quite different from normal partially eden-tulous arch anatomic features. The expectation for this altered region to contribute significantly to prosthesis support, stability, or retention is infrequently met. Conse-quently, prosthesis support and stabilization are largely dependent on the ability to aggressively engage the remain-ing teeth and residual ridge structures. In comparison with partially edentulous arches, the movement potential for the prosthesis extension into the defect can be significant. When engagement of the distobuc-cal temporal bone is possible, upward movement of the obturator bulb can be greatly minimized. Movement poten-tial increases as the remaining tooth number decreases and their arrangement becomes more linear (Figure 24-17). This illustrates the importance of maintaining teeth when possi-ble, which allows for greater prosthesis stabilization through direct tooth engagement and through cross-arch stabiliza-tion that increases with nonlinear tooth configurations (Figure 24-18). To help control potential movement, various suggestions have been made relative to prosthesis design. The basic prin-ciple of placing support, stabilization, and retention imme-diately adjacent to and as far from the defect as possible acts to distribute the tooth effect on prosthesis performance to the greatest mechanical advantage. Because the teeth adja-cent to the anterior resection margin are often incisors, it may be necessary to consider splinting them to improve the long-term prognosis. This region is critical for prosthesis performance, and the requirement for a cingulum rest and labial retention is often difficult to optimize without crowns. Distally, it is often necessary to incorporate an embrasure clasp to provide maximum retention and stabilization. Such a clasp assembly must have sufficient room for occlusal clearance, and it is not uncommon for the opposing occlu-sion to need adjustment to accommodate such a rest complex. When possible, the palatal surfaces of the maxillary teeth should be surveyed to determine whether guide-plane surfaces can be produced to impart a stabilizing effect. When accomplished, the prosthesis benefits from improved move-ment resistance, and it does so with more teeth contributing to the effect, thereby distributing the stress more appropri-ately. Brown described how the vertical height of the lateral portion of the obturator above the buccal scar band can contribute to prosthesis movement control by helping to prevent vertical displacement (see Figure 24-15). Speech Aid Prostheses The defining characteristics of speech aid prostheses are that they are functionally shaped to the palatopharyngeal mus-culature to restore or compensate for areas of the soft palate that are deficient because of surgery or congenital anomaly buccal and lingual soft tissues for food control. Regardless of the type of prosthesis to be used, the appropriate place-ment of the bone relative to the opposing arch is vital to the intended functional use. If a removable prosthesis is planned and is expected to cover the bone reconstruction, the contour of the developed ridge should provide a surface covered with firm, thin soft tissue, and a rounded superior contour with buccal and lingual slopes approaching parallel to each other and with sufficient vestibular depth to provide horizontal stability. Such a ridge condition is the surgical analog of a minimally resorbed edentulous ridge. With adequate cheek and tongue movement, this should provide a reasonable prognosis for prosthetic success, provided sufficient numbers of teeth remain on the nonresected side. For the optimum chance of prosthetic function, dental implants should be considered, and with sufficient bulk of bone and the same characteristics listed for the removable prosthesis, the prog-nosis for success is the greatest. To reiterate, the major deter-mining factor for improved function will be the quality of the soft tissue reconstruction. The major complications seen with mandibular recon-struction are related to the bulk of the soft tissue component and lack of mobility of the tongue. When these factors are controlled for, complications are caused most often by bone placement and size. The common use of free flaps, including bone from other regions of the body that do not possess the native mandibular shape, presents a significant degree of technical difficulty associated with the procedure. The fibula, which is a popular choice for mandibular replacement, pres-ents some challenges in meeting the ideal requirements mentioned previously. Because of the straight nature of the bone, it is easy to err in both horizontal and vertical posi-tioning, especially for reconstructions that span to the midline. Lingual positioning requires prosthetic placement at a position that may become functionally unstable over time. Such a location requires implant positions that create a mechanical cantilever that can be detrimental to the long-term success of the implant-supported prosthesis. Posteri-orly, the inability to re-create the natural ascending curve of the mandible can restrict placement of teeth and preclude restoration of complete occlusion on the resected side. It is common to have a mismatch in height at the anterior junc-tion of the graft with the resident mandible. For implant-supported prostheses, this area can present significant challenges in terms of adequate hygiene of the implants, and over time, this can compromise implant health. For remov-able prostheses, this can become a source of irritation if fulcrumlike action occurs with movement. Maxillary Prostheses Obturator Prostheses The defining characteristic of an obturator prosthesis is that it serves to restore separation of the oral and adjacent cavities following surgical resection of tumors of the nasal and para-329 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics A pediatric speech aid is a temporary prosthesis used to improve voice quality during the growing years. It is made of materials that are easily modified as growth or orthodon-tic treatment progresses. Because a speech aid has a signifi-cant posterior extension into the pharyngeal region, torque is evident from the long moment arm. A basic principle of posterior retention with anterior indirect retention must be applied to the design of such a maxillary prosthesis. Poste-rior retention is gained by the use of wrought-wire clasps around the most distal maxillary molars, whereas the ante-rior extension of the prosthesis onto the hard palate provides indirect retention. If clinical crown length and undercut are adequate to provide retention, orthodontic bands with buccal tie wings can be used in conjunction with the wrought wires. This design facilitates the maintenance of the pharyn-geal part of the pediatric speech aid in the proper position in the palatopharyngeal opening. (see Figure 24-11). Such a prosthesis consists of a palatal component, which contacts the teeth to provide stability and anchorage for retention; a palatal extension, which crosses the residual soft palate; and a pharyngeal component, which fills the palatopharyngeal port during muscular function, serving to restore the speech valve of the palatopharyngeal region. Because the typical speech aid prosthesis does not provide tooth replacement, the patient should expect only minimum functional movement. Movement of the pharyngeal exten-sion imposed by the residual palatopharyngeal musculature is generally undesired and is a sign that modification is required. Common reasons for such movement include a low position, causing tongue encroachment; superior exten-sion that does not account for head flexure; or impression procedures that do not accurately record residual soft palatal position or movement. A B C D Vertical displacement Less Greater Long radius sweep Short radius sweep Given horizontal flexure Figure 24-15 A, Coronal view of proposed maxillary resection. Bold lines designate typical area to be resected. B, Demonstrates the value of lateral wall height in the design of a removable partial denture obturator. As the defect side of the prosthesis is displaced, the lateral wall of the obturator will engage the scar band and aid in retaining the prosthesis. C, Coronal section with surgical obturator in place. With the prosthesis in place, the relation of the scar band (arrow) to the lateral portion of the obturator can be seen. A buccal scar band will develop at the height of the previous vestibule where the buccal mucosa and the skin graft in the surgical defect join. D, Axial view of the resected area illustrates the defect. Dotted lines indicate areas available for intraoral retention. 330 Part III Maintenance Palatal Lift Prostheses The defining characteristic of a palatal lift is that it positions a flaccid soft palate posteriorly and superiorly to narrow the palatopharyngeal opening for the purpose of improving oral air pressure and therefore speech. Patients who exhibit a structurally normal soft palate and pharyngeal port can In the adult whose palatopharyngeal insufficiency is the result of a cleft palate or palatal surgery, an adult speech aid prosthesis can be constructed of more definitive materials because growth changes will not have to be accommodated. If teeth are missing, the prosthesis will incorporate a reten-tive partial denture framework. The basic design should include posterior retention and anterior indirect retention. I II III IV V VI Figure 24-16 Aramany’s classification for partially edentulous maxillectomy dental arches: Class I—midline resection. Class II— unilateral resection. Class III—central resection. Class IV—bilateral anteroposterior resection. Class V—posterior resection. Class VI— anterior resection. A B Figure 24-17 A, A maxillary obturator prosthesis in which remaining teeth provide significant stabilization to the obturator extension because of their number and location, which allows cross-arch prosthesis engagement. B, Another obturator prosthesis, which benefits from teeth in a linear arrangement and therefore does not have any cross-arch tooth stabilization. Obturator movement in B is likely to be significantly greater than in A. The requirement for using the defect to provide support where possible is therefore greater in B than in A. 331 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics mentation. Bilateral rests and direct retainers should be positioned to facilitate the design for the acrylic retention because stability needs related to functional force are not a significant design concern. Mandibular Prostheses Resection prostheses are those prostheses provided to patients who have acquired mandibular defects that result in loss of teeth and significant portions of the mandible. Man-dibular resection results in defects that may preserve man-dibular continuity or may result in discontinuity defects. These are further subclassified by Cantor and Curtis (Figure 24-19) and provide a meaningful foundation for a discussion of removable prosthesis design considerations. Evolution of Mandibular Surgical Resection When mandibular continuity is preserved, as in a marginal resection (type I mandibular defect, see Figure 24-19), func-tion is least affected and the major prosthesis concern is related to the soft tissue potential for support. With good remaining dental support, near-normal function can often be achieved with prosthodontic rehabilitation. Although it is not as common an outcome as in the past, when continuity of the mandible is lost because of a segmen-tal resection that was not reconstructed, the bilateral joint complex no longer controls the remaining mandibular segment. Consequently, the function of the remaining man-dibular segment is severely compromised because of loss of coordinated bilateral muscular action functioning across a bilateral joint. The resulting segmental movement is an uncoordinated action dictated by the remaining unilateral muscular activity within a surgical environment that changes with healing dynamics and patient rehabilitation efforts. Successful removable prosthodontic intervention for these situations necessitates a combination of clinician knowledge demonstrate hypernasal speech caused by paralysis of the regional musculature. This condition is referred to as pala-topharyngeal incompetence because the failure lies in func-tion, not in anatomic deficiency. The paralysis can result from a variety of neuromuscular conditions (flaccid paraly-sis of the soft palate from closed head injuries, cerebral palsy, muscular dystrophy, or myasthenia gravis) that have varying clinical courses. The palatal lift prosthesis must physically position the soft palate to redirect air pressure orally. In placing the soft palate, any tissue resistance met acts as a dislodging force on the prosthesis. This dislodging force must be resisted by adequate direct and indirect retention. To efficiently maintain prosthesis position, the dislodging force is best resisted by bilateral direct retainers placed close to the posterior lift and anteriorly placed indirect retention. Success with a palatal lift prosthesis depends upon the pres-ence of a number of maxillary posterior teeth, which can provide retention for the prosthesis, coupled with an easily placed flaccid soft palate. Palatal Augmentation Prostheses When surgical resection involving the tongue and/or floor of the mouth limits tongue mobility, it affects both speech and deglutition. With tongue mobility limitations, the contour of the palate can be augmented by a prosthesis to modify the space of Donder to allow food manipulation to be more easily transferred posteriorly into the oropharynx. Prosthesis movement potential is low because the func-tional forces involved are those imparted by the tongue during deglutition and speech, neither of which creates force similar to mastication. It is common to use a diagnostic resin augmentation prosthesis retained with wire clasps to plan the necessary contour needs. Once the appropriate palatal contour has been determined, a definitive augmentation prosthesis can be constructed of cast metal with appropri-ately placed minor connectors for attaching the resin aug-A B Figure 24-18 A, Tooth arrangement that offers cross-arch stability (as in Figure 24-17, A) because of the arch curvature of the remaining tooth distribution and the tripod effect it allows. B, More linear arrangement of teeth does not provide cross-arch stability and places greater demand on the defect integrity for prosthesis performance. 332 Part III Maintenance Type I Resection In a type I resection of the mandible, the inferior border is intact and normal movements can be expected to occur. The major difference between this situation and a typical eden-tulous span is the nature of the soft tissue foundation. For type I resections, the denture-bearing area may be compro-mised by closure of the defect with the use of adjacent lining mucosa (which can reduce the bucco-lingual width), or by the presence of a split-thickness skin graft. Ideally, one would like to see a firm, nonmovable tissue bed with normal buccal and lingual vestibular extension. If the defect is unilateral and posterior, the framework would be typical of a Kennedy Class II design, taking into account whatever modification spaces may be present. When the marginal resection is in the anterior area, the design may be more typical of a Kennedy Class IV design (Figure 24-20). Anterior marginal resections sometimes include part of the anterior tongue and floor of the mouth. With loss of normal tongue function, the remaining teeth are no longer retained in a neutral zone, and as a result, they often collapse lingually because of lip pressure. If this occurs, the use of a labial bar major connector may be necessary. Corrected cast impression procedures provide a major advantage for fabrication of removable partial dentures in partial mandibulectomy patients. Capture of the unique buccal, lingual, and labial functional contours in the final prosthesis can contribute significantly to stabilization of the prosthesis, especially in discontinuity defects. Type II Resection In the type II resection, the mandible is often resected in the region of the second premolar and first molar. If no other teeth in the arch are missing, a prosthesis usually is not indicated. In some situations, however, a prosthesis may have to be fabricated to support the buccal tissue and to help fill the space between the tongue and the cheek to prevent food and saliva from collecting in the region. Framework design should be similar to a Kennedy Class II design, with extension into the vestibular areas of the resection. This area would be considered nonfunctional and should not be required to support mastication. It must be remembered that extension into the defect area can place significant stress on the remaining abutment teeth; therefore occlusal rests should be placed near the defect, and an attempt should be made to gain tripod support from remain-ing teeth and tissue where possible. An example of a framework design for a type II mandibu-lar resection with missing molars on the nonsurgical side is illustrated in Figure 24-21. The choice of major connector depends on the height of the floor of the mouth as it relates to the position of the attached gingival margins during func-tion. An extension base with artificial teeth can be used on the surgical side if space is available. The extent of this base is determined by a functional impression, and determina-of the functional movements of the remaining residual man-dible and concerted effort and persistence of the patient. Historically, mandibular stabilization by bone grafts or reconstruction bars was not always a surgical goal. The major exception was the anterior defect (type V), which was recognized to pose significant airway risks if not managed. Currently, most lateral segmental mandibulectomies are also reconstructed surgically. When the mandible is not stabi-lized following resection and a discontinuity defect results, a mandibular resection prosthesis should be provided to restore mastication within the unique movement capabili-ties of the residual functioning mandible. The following discussion highlights design considerations for the major defect classifications outlined. A common feature among all removable resection prostheses is that all framework designs should be dictated by basic prosthodon-tic principles of design. These include broad stress distribu-tion, cross-arch stabilization with use of a rigid major connector, stabilizing and retaining components at locations within the arch to best minimize dislodging functional forces, and replacement tooth positions that optimize pros-thesis stability and functional needs. Modifications to these principles are determined on an individual basis and are greatly influenced by unique residual tissue characteristics and mandibular movement dynamics. I II V III IV Figure 24-19 Cantor and Curtis classification of partial man-dibulectomy. (Redrawn from Cantor R, Curtis TA: J Prosthet Dent 25:446-457, 1971.) 333 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics mesial proximal surfaces of the anterior abutments. Lingual retention with buccal reciprocation on the remaining poste-rior teeth should be considered. The longitudinal axis of rotation in this design should be considered to be a straight line through the remaining teeth. Depression of the prosthe-sis on the edentulous side will have less of a chance to dis-lodge the prosthesis if retention is on the lingual surfaces than if on the buccal. Suggested framework designs for this patient group are illustrated in Figure 24-22. Physiologic relief of minor connectors is always recom-mended. When the remaining teeth are in a straight line, a Swing-Lock major connector design (Swing-Lock, Inc, Milford, TX) may be used to take advantage of as many buccal and/or labial undercuts as possible. Because elderly patients often complain of difficulty manipulating Swing-Lock mechanisms, in straight-line situations it may be pos-sible to use alternate buccal and lingual retention effectively (Figure 24-23). In the type II resection with anterior and posterior missing teeth on the resected side and posterior missing tion should be cautious of the potential for bone exposure at the superior margin of the resection. Retention can be achieved through the use of various types of clasp assemblies on the distal abutments. Indirect retention can be derived from rests prepared in the mesial fossae of the first premolars and/or the lingual surfaces of the canines. Unlike the result in Figure 24-21, use of an infrabulge retainer on the surgical side may be difficult if a shallow vestibule results from surgical closure. The locations of minor connectors should be physiologically determined to minimize stress on the abutment teeth and to enhance resistance to reasonable dislodging forces. Wrought-wire circumferential retainers are acceptable alternatives. In a type II mandibular resection, where posterior and anterior teeth are missing on the defect side, the remaining teeth on the intact side of the arch are often present in a straight-line configuration. Embrasure clasps may be used on the posterior teeth, with an infrabulge retainer on the anterior abutment. In some situations, a rotational path design may be used to engage the natural undercuts on the B A C D Figure 24-20 A type I resection of the anterior mandible. A, Bilateral molars remain to stabilize an anterior extension removable partial denture. A split-thickness skin graft has been used to reconstruct the denture bearing area. B, The prosthesis showing cast clasps and the anterior extension base. C, The prosthesis in place and covering the skin graft with a configuration produced through a correc-tive cast impression technique. D, The resection prosthesis in occlusion. It is critical to have the remaining natural teeth occlude at the same vertical dimension as the prosthetic teeth to ensure comfortable function. 334 Part III Maintenance Type IV Resection A type IV resection (see Figure 24-19) would use the same design concepts as type II or III resections with the corre-sponding edentulous areas. If the graft does not provide an articulation and the soft tissue covering the graft is not firmly attached to the bone graft, the movement potential will be dictated by functional forces of movement coupled with soft tissue supportive capacities. If a type IV resection extends to the midline with the extension of a graft into the defect area, but does not include temporomandibular joint reconstruction on the surgical side, the design will be similar to the type III resection with an extension base on the surgical side. If the type IV resection extends beyond the midline, with less than half of the mandible remaining, the design will be similar to the type II resection that has an extension base into the surgical defect area. Type V Resection In the type V resected mandible, when the anterior or posterior denture-bearing area of the mandible has been teeth on the nonresected side, the prosthesis will have three denture base regions. This prosthesis may have a straight-line longitudinal axis of rotation, as previously discussed. Rests should be placed on as many teeth as possible, minor connectors should be placed to enhance stability, and wrought-wire retainers represent an acceptable alternative to the bar clasps. Type III Resection A type III resection (see Figure 24-19) produces a defect to the midline or farther toward the intact side, leaving half or less of the mandible remaining. The importance of retaining as many teeth as possible in this situation cannot be overemphasized. The design of a framework for this situation would be similar to the type II resection. The longitudinal axis of rotation is again consid-ered to be a straight line through the remaining teeth. This resection provides a greater chance of prosthesis dislodg-ment caused by lack of support under the anterior extension. Alternating buccal and lingual retention in a rigid design or the Swing-Lock design should be considered. B A C Figure 24-21 Type II resection and prosthesis. A, Clinical presentation of the mandibular right resection and missing mandibular left molars. B, Resection prosthesis with a cast lingual plate major connector and wrought-wire clasps. C, Resection prosthesis in place demonstrating the two-tooth extension on the defect side (patient’s right). (Courtesy Dr. Ron Desjardins, Rochester, MN.) 335 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics surgically reconstructed, the removable partial denture design is similar to the type I resection design. The principal difference between a type V resected man-dible and the intact mandible with the same tooth loss pattern lies in the management of soft tissue at the graft site. For design purposes, one should consider the residual man-dibles of the type I and V resections to be similar to nonsur-gical mandibles with the same tooth-loss pattern. Mandibular Guide Flange Prosthesis As was mentioned earlier, in a discontinuity defect, the movement of the residual mandibular segment is an unco-ordinated action dictated by two features unique to the spe-cific defect and patient. The first is the remaining unilateral muscular activity that will be specific to the surgical resec-tion and that will have a characteristic resting posture to the defect side with a diagonal movement on “closure.” The second is that the surgical environment will change as healing progresses, and patient efforts to train movement A B C D Figure 24-22 A, Frame design for type II resection, no teeth missing on the nonresected side. Note the provision for extension into the resection space between tongue and cheek. B, Type II design, with missing posterior teeth on the nonresected side. C, Type II design, with missing anterior teeth. D, Type II design, with missing anterior and posterior teeth. A R R R R Figure 24-23 Conventional clasping with the use of alternat-ing buccal and lingual retention (arrows). 336 Part III Maintenance ation, allowing the patient to make unassisted masticatory contact. The components of the guide flange prosthesis include the major and minor connectors needed to support, stabi-lize, and retain the prosthesis and the guiding mechanism. This may include a cast buccal guide bar and guide flange, or simply a resin flange, which engages the opposing arch buccal tooth surfaces. In either case, the opposing arch must provide a stable foundation to resist any forces needed to guide the deviated mandibular segment into maximum occlusal contact. The buccal guide bar is placed as close as possible to the buccal occlusal line angle of the remaining natural teeth to allow maximum opening. The lateral position of the bar must be adequate to prevent the guide from contacting the buccal mucosa of the maxillary alveolus. The length of the bar should overlie the premolars and the first molar where possible. Retention of the maxillary frame should not be problematic because the force directed on the bar is in a palatal direction. The guide flange is attached to the man-dibular major connector by two generous interproximal minor connectors. As with the maxillary frame, significant during this healing period will help maintain both position and movement range. To facilitate training of the mandibu-lar segment to maintain a more midline closure pattern, clinicians have used a guide flange prosthesis. The mandibular guide flange prosthesis is used primarily as an interim training device. When no missing teeth are supplied, it may be considered a training appliance rather than a prosthesis. This appliance is used in dentulous patients with nonreconstructed lateral discontinuity defects who have severe deviation of the mandible toward the surgical side and are unable to achieve unassisted intercuspation on the nonsurgical side (Figure 24-24). Generally these patients have had a significant amount of soft tissue removed along with the resected mandibular segment and have attained surgical closure by suturing of the lateral surface of the tongue to the buccal mucosa, which causes a deviation toward the defect side. Scarring also occurs and is worse for patients who have not been placed on an exercise program during the healing period. The guide flange pros-thesis is designed to restrict the patient to vertical opening and closing movements into maximum occlusal contacts. Over time, this guided function should promote scar relax-B A C D Figure 24-24 A mandibular guide flange prosthesis. A, Flange extension is incorporated into a mandibular type II resection prosthesis using a resin extension. B, Resection prosthesis inserted. C, Opposing maxillary prosthesis designed to engage palatal surfaces of all remaining teeth for maximal stability against flange-induced forces. D, Flange extending to the buccal region of the opposing prosthesis and teeth. Upon closure, the flange will guide the mandible to maximum intercuspation, at which time the flange extension will reside in the maxillary left buccal vestibule. (Courtesy Dr. Ron Desjardins, Rochester, MN.) 337 Chapter 24 Removable Partial Denture Considerations in Maxillofacial Prosthetics Jaw Relation Records for Mandibular Resection Patients Interocclusal records must be made using verbal guidance only for resection patients with discontinuity defects. A hands-on approach, like that used for conventional eden-tulous jaw relation records, will lead to unnatural rotation of the mandible and an inaccurate record. The patient should be instructed to move the mandible toward the non-surgical side and close into a nonresistant recording medium at the preestablished occlusal vertical dimension, which will be the occlusal contact position. If the surgical side is significantly deficient, an occlusion rim may have to be extended into the defect area to support the recording medium. Head position is of extreme importance during registration of jaw relation records. If the patient is in a semirecumbent position in the dental chair during the recording procedure, the mandible may be retracted and deviated toward the surgical side, preventing movement toward the intact side. To minimize this problem, the recording should be made with the patient seated in a normal upright postural position. Most patients with lateral discontinuity defects can make lateral movements toward the nonsurgical side, even without the presence of a lateral pterygoid muscle functioning on the balancing (surgical) side. This is possible because of the compensatory effects of the horizontal fibers of the tempo-ralis and the lateral pterygoid muscle on the normal side, causing a rotational effect on the remaining condyle. Summary Maxillofacial prosthetic treatment of the patient with an oral defect is among the most challenging treatments in den-tistry. Defects are highly individual and require the clinician to call upon all knowledge and experience to fabricate a functional prosthesis. The basic principles and concepts described throughout this text will help the clinician to suc-cessfully design maxillofacial removable partial dentures. The interested reader is encouraged to pursue maxillofacial texts for more information regarding prosthesis design for this patient group. interproximal tooth structure must be cleared to provide the necessary bulk for the minor connectors. The height of the guide flange is determined by the depth of the buccal vesti-bule. A small hook is placed at the middle of the top of the guide to prevent disengagement on wide opening. Because the mandibular segment has a constant medial force, the flange acts as a powerful lever with a strong lateral force on the teeth. Therefore extra rests and additional stabilization and retention on multiple teeth must be considered to prevent overstressing of individual teeth. Retention on the tooth adjacent to the defect is critical for resistance to lifting of the frame. Lingual retention in the premolar area may be considered as an aid in resistance to displacement. When necessary, missing teeth can be added to a guide flange pros-thesis. Flange prostheses can be provisionally designed for modification into definitive removable partial dentures after guidance is no longer necessary. This is accomplished by removal of the buccal flange and buccal guide bar compo-nents after the patient is able to make occlusal contacts without use of the guide. However, many patients with man-dibular resections have difficulty making repeated occlusal contacts—a fact described in several studies. Occlusal con-siderations in mandibulectomy patients have been discussed extensively by Desjardins. Palatal occlusal ramps have been used to guide those patients with less severe deviation than those who require a guide flange into a more stable intercuspal contact position. These prostheses incorporate a palatal ramp that simulates the function of the guide flange prosthesis. This inclination of the palatal ramp is determined by the severity of the devia-tion of the remaining mandible. Some patients have the ability to move laterally into occlusion but have a tendency to close medially and palatally rather than close into an acceptable cuspal relationship. These patients can benefit from a palatal ramp, which can be functionally generated in wax at the try-in stage. This provides a platform for occlusal contact in the entire bucco-lingual range of movement. A supplemental row of prosthetic teeth may be arranged, then removed at the boil out stage, and processed in pink acrylic resin for esthetics. Patients who have experienced both smooth and tooth-form ramps usually prefer the tooth form if the width is adequate. 338 CHAPTER 25 Considerations for the Use of Dental Implants With Removable Partial Dentures CHAPTER OUTLINE Physiologic Distinction Between Prostheses Replacing Anatomy and Functional Ability Strategically Placed Implants for RPD Stability and Improved Patient Accommodation Movement Control With Selective Implant Placement Placement for support vs. retention Anatomic concerns for Class I and II arches Benefits of implant vs. surveyed crown Strategically lost tooth and implant use Use of abutment for support and/or retention Influence of opposing occlusion on implant location and implant abutment design Treatment Planning Survey considerations: path of insertion for teeth and implants Location influenced by anatomic and opposing occlusal factors Clasp assembly requirements if an implant is used adjacent to tooth Clinical Examples Summary Dental implants are increasingly being used in a variety of ways in the replacement of teeth. Providing implants to support all teeth needing replacement is often preferable if indicated and if the patient can afford to do so. If the patient is unable to pursue an implant-only supported prosthesis, this should not keep him or her from considering an implant, because the patient still may benefit from a carefully selected implant used for critical clinical performance advantage when removable partial dentures are pursued. Additionally, implants can be used for removable partial dentures to allow future implant-only treatment options. This chapter presents basic considerations in the selection of implants to improve prosthesis performance by increasing functional stability. Prosthesis instability, a commonly stated problem with removable prostheses, is presented as a poten tially confounding and unique component of oral sensory-motor function at issue when patients are challenged with removable prostheses. When all the relevant information necessary to adequately address how best to manage a patient’s tooth loss is considered, one aspect that should be kept in mind is that among the various prostheses from which we choose (conventional fixed partial denture, implant-supported partial denture, and removable partial denture), the removable partial denture is the most unique. Exactly how we describe this uniqueness to our patients is critical to their appropriate prosthesis selection and provides a clue to us as to how dental implants can best be used in conjunction with removable partial dentures (RPDs). Patient factors—clinical factors (dentition and residual ridge condi tions), functional factors, psychological factors; prosthesis factors— implant, fixed, or removable partial dentures. 339 Chapter 25 Considerations for the Use of Dental Implants With Removable Partial Dentures related dynamics. This notion of variable prosthesis impact suggests that either the prostheses are not the same or patients may vary in terms of neuromuscular ability. What is meant by the term neuromuscular ability? This is the task-specific sensory-motor performance “ability” that a specific patient possesses for the task/act of mastication. It involves multiple factors, including the selection of a pros thesis that may vary in qualitative neuromuscular ability compared with the gold standard—natural teeth. This pros thesis influence can be favorable when replacements are more like teeth (in their neuromuscular sensory input nature), or it can be unfavorable if they are unlike teeth and potentially cause confounding sensory input that diminishes function (e.g., implant fixed partial denture vs. denture base). This therapeutic variability suggests that prostheses differ in the influence they have on oral sensory receptors; therefore this may affect our patient’s perspective on how much we have improved the rehabilitation of oral function (i.e., neurophysiologic ability rehabilitation). Stability under function is critical for limiting confound ing sensory input because instability is a significant con founder of normal function. Whether a prosthesis is stable or unstable influences whether it provides a positive or a negative contribution to an already reduced peri pheral receptor influence. This is compounded by the inherent neuromuscular variability seen between patients. PHYSIOLOGIC DISTINCTION BETWEEN PROSTHESES Although physiologic aspects of replacement prostheses have not been adequately researched, the fact that removable partial dentures are less like teeth than fixed prostheses strongly suggests that we should delineate this to our patients. The most obvious difference between treatment options is the manner in which they are supported—conventional fixed partial dentures using teeth with periodontal ligaments versus implant prostheses using osseointegrated implants versus RPDs using teeth, or teeth and mucosa. An additional difference is the rigidity associated with the prosthesis-tooth interface connection—cemented prostheses rigidly con nected to teeth versus screw-retained implant prostheses rigidly connected between implants versus clasp/attach ment-retained removable partial denture prostheses. Given this support and rigid connection difference, it is easy to recognize that one distinguishing difference between fixed prostheses (tooth or implant supported) and tooth-tissue–supported RPDs is the potential for movement under function. The impact that such movement will have on related oral sensory and oral functional expectations of patients is potentially critical to understand. These patient-specific perspectives are variable because some of our patients are better at distinguishing oral sensory input than others and some are more functionally capable than others. The importance of sensory-functional considerations is evident when it is realized that the oral cavity is unique from a sensory perception standpoint, because the sensory input from the mouth regions involved in the task of mastication represents one of the largest collections of sensory informa tion processed for a single functional act in the body, rivaled closely by the multifunctional hand (Figure 25-1). REPLACING ANATOMY AND FUNCTIONAL ABILITY With this relative importance of oral sensory function asso ciated with mastication in mind, it would be helpful to con sider what exactly we are attempting to replace when we discuss with our patients the management of missing teeth. This involves both the physical tools for mastication and the oral ability for precise and stable neuromuscular functions to manipulate a bolus of food. Analysis of chewing studies shows that the oral receptor feedback that guides movement of the mandible comes from various sources. The most sen sitive input (i.e., the input that provides the most refined and precisely controlled movement) comes from the periodontal mechanoreceptors (PMRs), with additional input coming from the gingiva, mucosa, periosteum/bone, and temporo mandibular joint (TMJ) complex. Loss of teeth leads to loss of guidance precision, which may be caused by the deficit in PMRs. No artificial replacements reestablish the same ability inherent in the PMR guidance. Yet prostheses may vary in their ability to affect the deficit on the basis of sensory-Foot Hip Genitalia Toes Leg Trunk Head Lips Tongue Pharynx Intra-abdominal Upper lip Face Nose Eye Thumb Index Middle Ring Hand Wrist Forearm Elbow Arm Little Lower lip Teeth, gums, and jaw Shoulder Neck Figure 25-1 Somatosensory cortex. In this discussion, neurophysiologic and neuromuscular both refer to the relationship between oral sensory input and the resultant oral func tional movement. 340 Part III Maintenance well-supported prosthesis is likely to impart less displace ment to the soft tissue; less displacement likely is associated with a more comfortable prosthesis. If the major concern is comfort, then support is more often the best treatment target for an implant. The most effective location for resist ing force would be the posterior occlusal contact position— that position where the forces of occlusion are likely the greatest. Anatomic Concerns for Class I and II Arches It is clear that placement of implants within the alveolar ridge requires adequate bone volume. Tooth loss commonly leads to residual ridge resorption; the longer from the time of extraction that a patient is seen, the more alveolar resorp tion is expected. When this occurs in the posterior jaw, vertical loss of volume at the ridge crest is complicated by the maxillary sinus and inferior alveolar nerve at the apical location. These anatomic boundaries require careful consideration of surgi cal manipulation and must be considered as potentially more burdensome to the patient (i.e., they increase surgical risk). It is unclear at this time whether shortened implants (≤ 8 mm) in posterior edentulous regions are adequate for the necessary resistance. Certainly, the opposing occlusion contributes to this decision and natural teeth would be expected to generate more forceful occlusal function demanding greater implant resistance, compared with an opposing complete denture prosthesis. Benefits of Implant versus Surveyed Crown Surveyed crowns have long been used to maximize support, stability, and retention for removable partial dentures. The ability to control expected movement in the removable pros thesis can be improved by the use of such a carefully con toured crown. The same can be said for an implant, which presents a logical comparison as to which is a better choice for patients. As an example, if a patient is deciding between a crown on tooth #6 versus an implant at site #5, what should he or she be considering? Both can be used to provide support, stability, and retention. If the crown is designed to accom modate an attachment, it can also be used without a visible clasp. This is a more technically demanding intervention and carries with it more challenging maintenance requirements. Also, the typical consideration regarding caries risk should be discussed. This suggests the need to think about long-term needs when deciding between options. The implant at site #5 can reduce the need for a clasp, while at the same time providing a reasonably simple main tenance requirement for managing attachment needs. The implant attachment can be easily accommodated within a simple RPD framework that accommodates the #5 replace ment tooth and available prosthetic space for attachment needs. Variable retention is available based on the type of attachment selected—an advantage over the use of a clasp, Consequently, in some of our patients, replacement teeth may have less of a “margin” for sensory contribution without creating functional impediments. STRATEGICALLY PLACED IMPLANTS FOR RPD STABILITY AND IMPROVED PATIENT ACCOMMODATION To decide which treatment option is best from a patient accommodation perspective, it must be determined whether this specific patient possesses the ability to meet the chal lenge posed by the neuromuscular deficit. If the answer is yes, then the patient may be an appropriate candidate for an RPD. However, if the answer is no and for good reasons an RPD is chosen, the use of strategically located implants to assist the patient in meeting the challenge should be discussed. Stated differently, the success of any prosthesis may be compromised by a significant level of sensory loss. In addi tion, if the patient also possesses an inherently poor neuro physiologic ability, the choice of a prosthesis that creates an additional sensory “burden” may make the situation too challenging for the patient. The greatest contributing factor to the challenge is instability, and stability is most efficiently improved by the use of dental implants. MOVEMENT CONTROL WITH SELECTIVE IMPLANT PLACEMENT The most beneficial use of an implant with a removable partial denture is to reduce the negative impact of any sensory input resulting from prosthesis movement. The greatest potential for movement is seen with Kennedy Class I and II prostheses. Consequently, the greatest benefits in the use of implants with Class I and II removable partial den tures are to gain stability, control prosthesis movement, and reduce unnatural sensations that are movement related. The clinician who is deciding exactly where to place an implant for the greatest benefit should take into consider ation many factors. The ultimate decision will likely be influ enced by a combination of important clinical issues. Basic considerations include factors associated with support, sta bility, and retention; residual ridge anatomy; strategic tooth loss; opposing occlusal influence; and the condition of a terminal tooth. Placement for Support versus Retention When considering where to place an implant to assist an RPD, we should ask, “What will best serve the patient over time?” Certainly, we want the prostheses to be adequately retained, but as we consider what the retentive need for our removable partial denture is, we should also ask, “Is it the most important issue that patients face?” Support for prostheses is the characteristic that resists the greatest functional forces—those associated with chewing. A 341 Chapter 25 Considerations for the Use of Dental Implants With Removable Partial Dentures TREATMENT PLANNING Survey Considerations: Path of Insertion for Teeth and Implants Engagement of implant abutments requires consideration of the prosthesis path of insertion (POI). An implant placed coincident with the POI allows greater prosthetic flexibility in abutment selection and a more favorable removable partial denture connection (Figure 25-2). Location Influenced by Anatomic and Opposing Occlusal Factors Implant use requires interocclusal space sufficient for pros thetic materials: denture base with framework minor con nector, attachment housing, and denture tooth. Therefore the opposing occlusion should be evaluated preoperatively. This becomes even more critical when less residual ridge resorption occurs. With excessive maxillary residual ridge resorption, the implant location may be more palatal/lingual, requiring consideration of the impact that such a location may have on palatal or lingual contours. Clasp Assembly Requirements If an Implant Is Used Adjacent to Tooth Engagement of an implant abutment to provide horizontal and vertical resistance supplements the resistance derived from tooth engagement. Consequently, less tooth coverage may be necessary for regions of the arch that have an implant. If implant use is directed primarily to retention, tooth engagement for maximum resistance remains necessary to improve stability. If this is not achieved, deterioration in retention may be more rapid because of excessive forces directed to the retentive element, which could result in more rapid material deformation. CLINICAL EXAMPLES Figures 25-3 to 25-6 present clinical examples. Primary use—support Figure 25-3: Kennedy Class I mandibular Figure 25-4: Kennedy Class II Combined use—support and retention Figure 25-5: Kennedy Class III Figure 25-6: Kennedy Class II SUMMARY Removable partial dentures are unique compared with pros theses supported by natural teeth. Their uniqueness largely relates to their potentially significant negative impact on oral sensory input during function, as well as on the prosthetic bulk required. Because of this, RPDs may present a challenge to accommodation, of which patients should be made aware during the treatment planning stage. which carries the risk of fracture with repeated adjustment over time. Strategically Lost Tooth and Implant Use Loss of a canine or molar abutment in an existing removable prosthesis can have a significantly negative impact on the prosthesis. Resistance to movement toward the tissue and across the tissue (aspects of support and stability) is greatly affected. Previous interventions for strategic tooth loss, such as the Swing-Lock prosthesis, have been based on recognition of the need to use more teeth to meet the demands for comfortable, stable function. Use of an implant in this situation can directly address the loss of a strategic component that controls functional movement and may be able to utilize the existing prosthesis. Use of Abutment for Support and/or Retention As an implant is placed more distally (in a distal extension), it serves more of a support function than a retention function. In a situation where support and retention are desired, the connecting design (the actual attachment male-female components) must allow adequate resistance to supporting forces without premature deformation of the retentive component and reduction of the retaining properties. Generally a component designed to primarily provide retention may not serve a supportive role for the desired period of time before replacement. Use of a more resistant retentive element to better resist supportive forces may provide retention in excess of the patient’s ability to com fortably remove the prosthesis for daily hygiene. Influence of Opposing Occlusion on Implant Location and Implant Abutment Design Distal extension removable partial dentures are significantly influenced by the type and plane of orientation of the oppos ing dentition. Opposing complete dentures, which exhibit a regular occlusal plane, encourage equal force distribution associated with occlusion, allowing an optimum scenario for comfort. Irregular occlusal planes created by super-erupted oppos ing natural teeth make control of functional forces more challenging at the residual alveolar ridge level. Use of implants for such oral conditions may improve prosthesis comfort, especially if support is fully utilized (i.e., no retainer function). In such a scenario, implant abutment design should serve to control movement toward and across the denture bearing area. This places a premium on nonresilient and nonretentive abutment designs. Care should be taken to closely follow such designs during the early postinsertion phase because the potential for excessive occlusal loads could have detrimental effects on abutment connections and implant stability. 342 Part III Maintenance A B C D E Figure 25-2 Parallel path of insertion established with tooth guide planes (distal of the canines and mesial of the molar) and adjacent implants. 343 Chapter 25 Considerations for the Use of Dental Implants With Removable Partial Dentures A B D C Figure 25-3 Significant residual ridge resorption with a strategically missing tooth #22. The implant provided sufficient support to improve functional comfort and occlusal function. $ % & Figure 25-4 A long span modification space across the mandibular midline. The implant provided support and retention improved function and comfort while reducing functional stress on the mandibular incisors. 344 Part III Maintenance A B C Figure 25-5 Support provided by the cingulum and mesial-occlusal rests of the teeth, and by the implant at site #3. Implants at sites #5 and #12 provide retention. Guide-plane surfaces augment implants for sufficient retention without clasps. 345 Chapter 25 Considerations for the Use of Dental Implants With Removable Partial Dentures A B E C D Figure 25-6 An implant-assisted Class III removable partial denture with minimum interocclusal space in an elderly patient desiring no prosthesis clasps. Denture teeth were shaped and positioned for optimum esthetics, allowing creation of occlusal contours by acrylic-resin, attaching the teeth to the frame and attachment housings. Patient sensory perception and neuromuscular variability may influence prosthesis selection. The sensory deficit asso ciated with tooth loss can be complicated by unstable pros theses; for challenging RPDs, the use of implants can significantly contribute to stability. ACKNOWLEDGMENTS The authors would like to acknowledge Dr. Tom Salinas for his helpful input during the writing of this chapter. 346 APPENDIX A Glossary Abutment A tooth, a portion of a tooth, or a portion of an implant that serves to support and/or retain a prosthesis Acrylic Formed from acrylic acid (e.g., acrylic resin) Acrylic resin 1: Pertaining to polymers of acrylic acid, methacrylic acid, or acrylonitrile; for example, acrylic fibers or acrylic resins; 2: any of a group of thermoplastic resins made by polymerizing Anatomic ridge form Surface form of the edentulous ridge when at rest or when not supporting a functional load; often recorded in a soft impression material such as hydrocolloid or metallic oxide impression paste, and results when an impression tray is uniformly relieved. See also static form. Angle of cervical convergence Angle viewed between a ver-tical rod contacting an abutment tooth and the axial surface of the abutment cervical to the height of contour Appliances Devices (such as splints, orthodontic appli-ances, and space maintainers) worn by the patient in the course of treatment Balanced occlusion The simultaneous contacting of maxil-lary and mandibular teeth on the right and left in the anterior and posterior occlusal areas in centric or any eccentric position within the functional range Bar clasp Type of extracoronal retainer that originates from the denture base or framework, traverses soft tissue, and approaches the tooth undercut area from a gingival direction Basal seat Oral tissues and structures of the residual ridge supporting a denture base. See also denture foundation area. Cast An accurate and positive reproduction of a maxillary or mandibular dental arch made from an impression of that arch; further designated according to the purpose for which it is made, such as diagnostic cast, master cast, or investment cast; also may be used as an infinitive (to cast) or as an adjective (cast framework, or cast metal base) Casting A metal object shaped by being poured into a mold to harden; used primarily to designate the cast metal framework of a partial denture but also may be used to describe a molded metal denture base that is actually cast into a mold Centric jaw relation See Centric relation. Centric occlusion The occlusion of opposing teeth when the mandible is in centric relation. This may or may not coincide with the maximal intercuspal position Centric relation 1: The maxillomandibular relationship in which the condyles articulate with the thinnest avascular portion of their respective disks with the complex in the anterior-superior position against the shapes of the artic-ular eminences. This position is independent of tooth contact. This position is clinically discernible when the mandible is directed superior and anteriorly. It is restricted to a purely rotary movement about the trans-verse horizontal axis (GPT-5); 2: the most retruded phys-iologic relation of the mandible to the maxillae to and from which the individual can make lateral movements. It is a condition that can exist at various degrees of jaw separation. It occurs around the terminal hinge axis (GPT-3); 3: the most retruded relation of the mandible to the maxillae when the condyles are in the most poste-rior unstrained position in the glenoid fossae from which lateral movement can be made at any given degree of jaw separation (GPT-1); 4: the most posterior relation of the lower to the upper jaw from which lateral movements can be made at a given vertical dimension (Boucher); 5: a maxilla-to-mandible relationship in which the condyles and disks are thought to be in the midmost, uppermost position. The position has been difficult to define ana-Certain terms in this glossary have been adapted from The Glossary of Prosthodontic Terms. Additional terminology can be reviewed in The Glossary of Prosthodontic Terms, The Journal of Prosthetic Dentistry, Vol. 94, No. 1, and The Glossary of Prosthodontic Terms, 8th edition, pp. 10-81, © 2005, ympr/article/PIIS0022391305001757/fulltext. 347 Appendix A Glossary tomically but is determined clinically by assessing when the jaw can hinge on a fixed terminal axis (up to 25 mm). It is a clinically determined relationship of the mandible to the maxilla when the condyle disk assemblies are posi-tioned in their most superior position in the mandibular fossae and against the distal slope of the articular emi-nence (Ash); 6: the relation of the mandible to the maxil-lae when the condyles are in the uppermost and rearmost position in the glenoid fossae. This position may not be able to be recorded in the presence of dysfunction of the masticatory system; 7: a clinically determined position of the mandible placing both condyles into their anterior uppermost position. This can be determined in patients without pain or derangement in the temporomandibular joint (TMJ) (Ramsfjord). See also centric jaw relation. Circumferential clasp Term used to designate a clasp arm that originates above the height of contour and approaches the tooth undercut from an occlusal direction Clasp (or direct retainer) Component of the clasp assembly that engages a portion of the tooth surface and either enters an undercut for retention or remains entirely above the height of contour to act as a reciprocating element; generally used to stabilize or retain a removable prosthesis Clasp assembly Part of a removable partial denture that acts as a direct retainer and/or stabilizer for a prosthesis by partially encompassing or contacting an abutment Complete denture Dental prosthesis that replaces all of the natural dentition and associated structures of the maxilla or mandible. It is entirely supported by tissues (mucous membrane, connective tissues, and underlying bone). Dental cast surveyor Instrument used to determine the relative parallelism of two or more axial surfaces of teeth or other parts of a cast of a dental arch; also used to locate and delineate the contours and relative positions of abut-ment teeth and associated structures Dental stones Used to form an artificial stone reproduction from an impression, and used as an investment or for mounting purposes; all dental stones are gypsum products Denture base Part of a denture (whether it is metal or is made of a resinous material) that rests on the residual bone covered by soft tissue and to which the teeth are attached Denture foundation area See Basal seat. Direct retainer Component of a removable partial denture used to retain or prevent dislodgment; consists of a clasp assembly or precision attachment Edentulous ridge See Residual ridge. Functional impression Impression and resulting cast of the supporting form of the edentulous ridge; artificially created by means of a specially molded (individualized) impression tray or an impression material, or both, that displaces those tissues that can be readily displaced and that would be incapable of rendering support to the denture base when it is supporting functional load. See also functional ridge form. Functional occlusal registration Used to designate a dynamic registration of opposing dentition rather than the recording of a static relationship of one jaw to another Functional ridge form See Functional impression. Guiding planes Two or more vertically parallel surfaces of abutment teeth shaped to direct a prosthesis during placement and removal; surfaces are parallel to the path of the placement and parallel to each other; preferably these surfaces are made parallel to the long axes of abut-ment teeth Height of contour Line encircling a tooth, designating its greatest circumference at a selected position determined by a dental surveyor Incisal rest A rest placed on the incisal edge of an anterior abutment tooth Indirect retainer Part of a removable partial denture that assists direct retainers in preventing displacement of distal extension denture bases by resisting lever action from the opposite side of the fulcrum line Interim denture (or provisional denture) Dental prosthe-sis used for a short time for reasons of esthetics, mastica-tion, occlusal support, or convenience, or for conditioning the patient to accept an artificial substitute for missing natural teeth until a more definite prosthetic dental treat-ment can be provided Internal attachment See Precision attachment. Investment cast Cast compounded to withstand high tem-peratures without disintegrating and, incidentally, to perform certain functions relative to burnout and expan-sion of the mold. See also refractory cast. Lingual bar connector Component of the partial denture framework located lingual to the dental arch and above the moving tissues of the floor of the mouth but as far below the gingival tissues as possible Lingual rest A rest that occupies a position on the lingual surface of an anterior tooth Linguoplate Designation given when the lingual bar major connector is attached to a thin, contoured apron adjacent to the lingual surfaces of the anterior teeth in the mandible Major connector Part of a removable partial denture that connects the components on one side of the arch to the components on the opposite side of the arch Mold Word used to indicate either the cavity into which a casting is made or the shape of an artificial tooth; use of the term is incorrect when referring to a reproduction of a dental arch or a portion thereof Boucher CO: Occlusion in prosthodontics, J Prosthet Dent 3:633-656, 1953; Ash MM: Personal communication, July 1993; Lang BR, Kelsey CC: International prosthodontic workshop on complete denture occlu-sion, Ann Arbor, 1973, The University of Michigan School of Dentistry; Ramsfjord SP: Personal communication, July 1993. 348 Appendix A Glossary Occlusal rest A rest placed on the occlusal surface of a pos-terior tooth Palatal bar Thin, broad palatal coverage with a width of less than 8 mm used as a major connector Palatal major connector Any thin, broad palatal coverage that is used as a major connector Palatal strap Proportionally thinner and broader than a palatal bar, although differentiation is somewhat objective Plastic Refers to any of various substances that harden and retain their shape after being molded Precision attachment Refers to an interlocking device, one component of which is fixed to an abutment or abut-ments, while the other is integrated into a removable prosthesis to stabilize and/or retain it. See also internal attachment. Prosthesis A denture, an obturator, a fixed partial denture, or a crown Rebasing Process that goes beyond relining and involves refitting a denture by replacing the entire denture base with new material without changing the occlusal relations of the teeth Refractory cast See investment cast. Refractory investment Investment material that can with-stand the high temperatures of casting or soldering; plaster of Paris and artificial stone may be considered investment if either is used to invest any part of a dental restoration for processing Relining Resurfacing of a denture base with new material to make it fit the underlying tissue Removable partial denture Prosthesis that replaces some teeth in a partially dentate arch, and can be removed from the mouth and replaced at will Residual ridge Residual bone with its soft tissue that covers the underlying area of the denture base; the exact char-acter of the soft tissue covering may vary, but it includes the mucous membrane and underlying fibrous connec-tive tissue. See also edentulous ridge. Resin Used broadly for substances named according to their chemical composition, physical structure, and means for activation or curing, such as acrylic resin Rest Any component of the partial denture that is placed on an abutment tooth, ideally in a prepared rest seat, so that it limits movement of the denture in a gingival direc-tion and transmits functional forces to the tooth Retainer Any type of clasp, attachment, device, etc., used for fixation, stabilization, or retention of a prosthesis; may be intracoronal or extracoronal and can be used as a means of retaining a removable or a fixed restoration Retention Quality inherent in the denture that resists the vertical forces of dislodgment (e.g., the force of gravity, the adhesiveness of foods, the forces associated with opening of the jaws) Semiprecision rest Rigid metallic extension of a fixed or removable partial denture that fits into an intracoronal preparation in a cast restoration Stability Quality of a prosthesis of being firm, stable, or constant and resisting displacement by functional, hori-zontal, or rotational stresses Static form See Anatomic ridge form. Stone Use of the word in dentistry should be limited to those gypsum materials that are employed for their hard-ness, accuracy, or abrasion resistance Support Foundation on which a dental prosthesis rests, or to hold up and serve as a foundation Undercut When used in reference to an abutment tooth, that portion of a tooth that lies between the height of contour and the gingiva; when used in reference to other oral structures, the contour or cross section of a residual ridge or dental arch that would prevent placement of a denture Wax pattern Converted to a casting via elimination of the pattern by heat, leaving a mold into which the molten metal is forced by centrifugal force or other means 349 Appendix B Selected Reading Resources 349 APPENDIX B Selected Reading Resources A textbook is rarely found to be all-inclusive in subject matter related to a dental clinical discipline or subdiscipline. Therefore this section, which lists other textbooks and arti-cles from dental periodical literature, may assist in broaden-ing a student’s perspective on principles and concepts of removable partial prosthodontics. Some of the articles have historic significance and are considered classics. Contemporary selections are also included. The serious student of dentistry may extract from this section background information and details related to the progress of removable partial prosthodontics over the years. We do not imply that sources have been exhausted in compiling these lists of textbooks and articles. We have attempted to correctly classify listed articles for ready refer-ence; however, the length of the “Miscellaneous” section attests to the difficulties encountered. Textbooks Alberktsson T, Zarb GA: The Brånemark osseointegrated implant, 1989, Chicago, Quintessence. Anusavice KJ: Phillips’ science of dental materials, ed 11, St Louis, 2003, Saunders. Applegate OC: Essentials of removable partial denture prosthesis, ed 3, Philadelphia, 1965, WB Saunders. Babbush CA et al: Dental implants: the art and science, ed 2, St. Louis, 2011, Saunders. Beumer J, Curtis TA, Firtell DN: Maxillofacial prosthetics, St Louis, 1979, Mosby. Block MS: Color atlas of dental implant surgery, ed 3, St. Louis, 2011, Saunders. Brand RW, Isselhard DE: Anatomy of orofacial structures, ed 7, St Louis, 2003, Mosby. Brewer AA, Morrow RM: Overdentures, ed 2, St Louis, 1980, Mosby. Brunette DM: Critical thinking: understanding and evaluating dental research, ed 2, Chicago, 2007, Quintessence. Dawson PE: Functional occlusion: from TMJ to smile design, St Louis, 2007, Mosby. Dolder EJ, Durrer GT: The bar-joint denture, Chicago, 1978, Quintessence. Dubrul EL: Sicher and Dubrul’s oral anatomy, ed 8, St Louis/Tokyo, 1988, Ishiyaku EuroAmerica. Dykema RW, Cunningham DM, Johnston JF: Modern practice in removable partial prosthodontics, Philadelphia, 1969, WB Saunders. Fonseca RJ, Davis WH: Reconstructive preprosthetic oral and maxil-lofacial surgery, ed 2, Philadelphia, 1995, WB Saunders. Graber G: Removable partial dentures, Stuttgart, Germany, 1988, Thieme Medical. Graber DA, Goldstein RE, Feinman RA: Porcelain laminate veneers, Chicago, 1988, Quintessence. Grasso JE, Miller EL: Partial prosthodontics, ed 3, St Louis, 1991, Mosby. Hoag PM, Pawlak EA: Essentials of periodontics, ed 4, St Louis, 1990, Mosby. Johnson DL, Stratton RJ: Fundamentals of removable prosthodontics, Chicago, 1980, Quintessence. Johnston JF et al: Modern practice in crown and bridge prosthodontics, ed 4, Philadelphia, 1986, WB Saunders. Jordon RE: Esthetic composite bonding, ed 2, St Louis, 1993, Mosby. Kratochvil FJ: Partial removable prosthodontics, Philadelphia, 1988, WB Saunders. Krol AJ: Removable partial denture design outline syllabus, ed 5, San Francisco, 1999, University of the Pacific School of Dentistry. Laney WR et al: Maxillofacial prosthetics, Littleton, Mass, 1979, PSG. Laney WR, Gibilisco JA: Diagnosis and treatment in prosthodontics, Philadelphia, 1983, Lea & Febiger. Little JW, Falace DA, Miller C, Rhodus NL: Dental management of the medically compromised patient, ed 7, St Louis, 2008, Mosby. Malamed SF: Medical emergencies in the dental office, ed 6, St Louis, 2007, Mosby. Malone WPF, Koth DC: Tylman’s theory and practice of fixed prosth-odontics, ed 8, Tokyo, 1989, IEA. Miller CH, Palenik CJ: Infection control and management of hazardous materials for the dental team, ed 4, St Louis, 2010, Mosby. Morrow RM, Rudd KD, Rhoads JE: Dental laboratory procedures: complete dentures, vol 1, St Louis, 1985, Mosby. Mosby’s dental dictionary, ed 2, St Louis, 2008, Mosby. Nelson SJ: Wheeler’s dental anatomy, physiology, and occlusion, ed 9, St Louis, 2010, Saunders. Nevins M, Mellonig JT: Periodontal therapy: clinical approaches and evidence of success, vol 1, Chicago, 1998, Quintessence. O’Brien WJ: Dental materials and their selection, ed 4, Chicago, 2009, Quintessence. 350 Appendix B Selected Reading Resources Okeson JP: Management of temporomandibular disorders and occlu-sion, ed 6, St Louis, 2008, Mosby. Okeson JP: Orofacial pain: guidelines for assessment, diagnosis, and management, Chicago, 1996, Quintessence. Osborne J: Osborne and Lammie’s partial dentures, ed 5, Oxford, 1986, Blackwell Scientific. Phoenix RD, Cagna DR, DeFreest CF: Stewart’s clinical removable partial prosthodontics, ed 4, Chicago, 2008, Quintessence. Powers JM, Sakaguchi RL: Craig’s restorative dental materials, ed 12, St Louis, 2006, Mosby. Powers JM, Wataha JC: Dental materials: properties and manipulation, ed 9, St Louis, 2008, Mosby. Preiskel HW: Precision attachments in prosthodontics, Chicago, 1996, Quintessence. Rahn AO, Ivanhoe JR, Plummer KD: Textbook of complete dentures, ed 6, Shelton, CT, 2009, People’s Medical Publishing House. Ramfjord SP, Ash MM Jr: Occlusion, ed 4, Philadelphia, 1995, WB Saunders. Renner RP, Boucher LJ: Removable partial prosthodontics, Chicago, 1987, Quintessence. Rosenstiel SF, Land MF, Fujimoto J: Contemporary fixed prosthodon-tics, ed 4, St Louis, 2006, Mosby. Rudd KD, Rhoads JE, Morrow RM: Dental laboratory procedures, vol 3, ed 2, St Louis, 1986, Mosby. Sarnat BG, Laskin DM: The temporomandibular joint: a biological basis for clinical practice, ed 4, Philadelphia, 1992, WB Saunders. Shillingberg HT et al: Fundamentals of fixed prosthodontics, ed 3, Chicago, 1997, Quintessence. Singh P, Cranin AN: Atlas of oral implantology, ed 3, St. Louis, 2010, Mosby. Stratton RP, Wiebelt FJ: An atlas of removable partial denture design, Chicago, 1988, Quintessence. Watt DM, MacGregor AR: Designing partial dentures, Littleton, MA, 1985, PSG. Winkler S: Essentials of complete denture prosthodontics, ed 2, Little-ton, MA, 1988, PSG. Wood NK: Differential diagnosis of oral and maxillofacial lesions, ed 5, St Louis, 1997, Mosby. Wood NK: Review of diagnosis, oral medicine, radiology, and treat-ment planning, ed 4, St Louis, 1999, Mosby. Yalisove IL, Dietz JB Jr: Telescopic prosthetic therapy, Philadelphia, 1979, George F Stickley. Zarb GA: Temporomandibular joint and masticatory muscle disorders, St Louis, 1995, Mosby. Zarb GA et al: Prosthodontic treatment for edentulous patients: com-plete dentures and implant-supported prostheses, ed 12, St Louis, 2004, Mosby. Abutment Retainers: External and Internal Attachments Adisman IK: The internal clip attachment in fixed removable partial denture prosthesis, N Y J Dent 32:125-129, 1962. Ainamo J: Precision removable partial dentures with pontic abutments, J Prosthet Dent 23:289-295, 1970. Augsburger RH: The Gilmore attachment, J Prosthet Dent 16:1090-1102, 1966. Becker CM, Campbell MC, Williams DL: The Thompson dowel-rest system modified for chrome-cobalt removable partial denture frameworks, J Prosthet Dent 39:384-391, 1978. Ben-Ur Z, Aviv I, Cardash HS: A modified direct retainer design for distal extension removable partial dentures, J Prosthet Dent 70:342-344, 1988. Benson D, Spolsky VW: A clinical evaluation of removable partial dentures with I-bar retainers. Part I, J Prosthet Dent 41:246, 1979. Berg T Jr: I-bar: myth and counter myth, Dent Clin North Am 23:1, 65-75, 1979. Blatterfein L: Design and positional arrangement of clasps for partial dentures, N Y J Dent 22:305-306, 1952. Blatterfein L: Study of partial denture clasping, J Am Dent Assoc 43:169-185, 1951. Brodbelt RHW: A simple paralleling template for precision attach-ments, J Prosthet Dent 27:285-288, 1972. Brudvik JS, Morris HF: Stress-relaxation testing. Part III: influence of wire alloys, gauges, and lengths on clasp behavior, J Prosthet Dent 46:374, 1981. Brudvik JS, Wormley JH: Construction techniques for wrought-wire retentive clasp arms as related to clasp flexibility, J Prosthet Dent 30:769-774, 1973. Chandler JA, Brudvik JS: Clinical evaluation of patients eight to nine years after placement of removable partial dentures, J Prosthet Dent 51:736, 1984. Chou TM et al: Photoelastic analysis and comparison of force-trans-mission characteristics of intracoronal attachments with clasp distal-extension removable partial dentures, J Prosthet Dent 62:313-319, 1989. Chou TM et al: Stereophotogrammetric analysis of abutment tooth movement in distal-extension removable partial dentures with intracoronal attachments and clasps, J Prosthet Dent 66:343-349, 1991. Clayton JA: A stable base precision attachment removable partial denture (PARPD): theories and principles, Dent Clin North Am 24:3-29, 1980. Cooper H: Practice management related to precision attachment pros-theses, Dent Clin North Am 24:45-61, 1980. DeVan MM: Preserving natural teeth through the use of clasps, J Pros-thet Dent 5:208-214, 1955. Dietz WH: Modified abutments for removable and fixed prosthodon-tics, J Prosthet Dent 11:1112-1116, 1961. Dixon DL et al: Wear of I-bar clasps and porcelain laminate restora-tions, Int J Prosthet 5:28-33, 1992. Dolder EJ: The bar joint mandibular denture, J Prosthet Dent 11:689-707, 1961. Eliason CM: RPA clasp design for distal-extension removable partial dentures, J Prosthet Dent 49:25-27, 1983. Frank RP, Brudvik JS, Nicholls JI: A comparison of the flexibility of wrought-wire and cast circumferential clasps, J Prosthet Dent 49:471-476, 1983. Getz II: Making a full-coverage restoration on an abutment to fit an existing removable partial denture, J Prosthet Dent 54:335-336, 1985. Gilson TD: A fixable-removable prosthetic attachment, J Prosthet Dent 9:247-255, 1959. Gindea AE: A retentive device for removable dentures, J Prosthet Dent 27:501-508, 1972. Grasso JE: A new removable partial denture clasp assembly, J Prosthet Dent 43:618-621, 1980. Green JH: The hinge-lock abutment attachment, J Am Dent Assoc 47:175-180, 1953. Hebel KS, Graser GN, Featherstone JD: Abrasion of enamel and com-posite resin by removable partial denture clasps, J Prosthet Dent 52:389, 1984. Highton R, Caputo AA, Matyas J: Retention and stress characteristics for a magnetically retained partial denture, J Dent Res (IADR abstract 279) 62(entire issue), 1982. Isaacson GO: Telescope crown retainers for removable partial dentures, J Prosthet Dent 22:436-448, 1969. Ivanhoe JR: Alternative cingulum rest seat, J Prosthet Dent 54:395-396, 1985. James AG: Self-locking posterior attachment for removable tooth-supported partial dentures, J Prosthet Dent 5:200-205, 1955. 351 Appendix B Selected Reading Resources Johnson DL, Stratton RJ, Duncanson MG Jr: The effect of single plane curvature on half-round cast clasps, J Dent Res 62:833-836, 1983. Johnson JF: The application and construction of the pinledge retainer, J Prosthet Dent 3:559-567, 1953. Kapur KK et al: A randomized clinical trial of two basic removable partial denture designs. Part I: comparisons of five-year success rates and periodontal health, J Prosthet Dent 72:268-282, 1994. Knodle JM: Experimental overlay and pin partial denture, J Prosthet Dent 17:472-478, 1967. Knowles LE: A dowel attachment removable partial denture, J Prosthet Dent 13:679-687, 1963. Koper A: Retainer for removable partial dentures: the Thompson dowel, J Prosthet Dent 30:759-768, 1973. Kotowicz WE: Clinical procedures in precision attachment removable partial denture construction, Dent Clin North Am 24:143-164, 1980. Kotowicz WE et al: The combination clasp and the distal extension removable partial denture, Dent Clin North Am 17:651-660, 1973. Kratochvil FJ, Davidson PN, Tandarts JG: Five-year study of treatment with removable partial dentures. Part I, J Prosthet Dent 48:237, 1982. Krol AJ: Clasp design for extension base removable partial dentures, J Prosthet Dent 29:408-415, 1973. Krol AJ: RPI clasp retainer and its modifications, Dent Clin North Am 17:631-649, 1973. Langer A: Combinations of diverse retainers in removable partial den-tures, J Prosthet Dent 40:378-384, 1978. LaVere AM: Analysis of facial surface undercuts to determine use of RPI or RPA clasps, J Prosthet Dent 56:741-743, 1986. Leupold RJ, Faraone KL: Etched castings as an adjunct to mouth prepa-ration for removable partial dentures, J Prosthet Dent 53:655-658, 1985. Lubovich RP, Peterson T: The fabrication of a ceramic-metal crown to fit an existing removable partial denture clasp, J Prosthet Dent 37:610-614, 1977. Marinello CP et al: Resin-bonded etched castings with extracoronal attachments for removable partial dentures, J Prosthet Dent 66:52-55, 1991. Maroso DJ, Schmidt JR, Blustein R: A preliminary study of wear of porcelain when subjected to functional movements of retentive clasp arms, J Prosthet Dent 45:14, 1981. McLeod NS: Improved design for the Thompson dowel rest semipreci-sion intracoronal retainer, J Prosthet Dent 40:513-516, 1978. McLeod NS: A theoretical analysis of the mechanics of the Thompson dowel semiprecision intracoronal retainer, J Prosthet Dent 37:19-27, 1977. Mensor MC Jr: Attachment fixation for overdentures. Part I, J Prosthet Dent 37:366-373, 1977. Mensor MC Jr: Attachment fixation of the overdentures. Part II, J Prosthet Dent 39:16-20, 1978. Morris HF et al: Stress distribution within circumferential clasp arms, J Oral Rehabil 3:387-391, 1976. Morris HF et al: Stress-relaxation testing. Part II. Comparison of bending profiles, microstructures, microhardness, and surface characteristics of several wrought-wires, J Prosthet Dent 46:256, 1981. Morris HF et al: Stress-relaxation testing. Part IV. Clasp pattern dimen-sions and their influence on clasp behavior, J Prosthet Dent 50:319, 1983. Morrison ML: Internal precision attachment retainers for partial den-tures, J Am Dent Assoc 64:209-215, 1962. Morrow RM: Tooth-supported complete dentures: an approach to pre-ventive prosthodontics, J Prosthet Dent 21:513-522, 1969. Oddo VJ Jr: The movable-arm clasp for complete passivity in partial denture construction, J Am Dent Assoc 74:1009-1015, 1967. Plotnik IJ: Internal attachment for fixed removable partial dentures, J Prosthet Dent 8:85-93, 1958. Pound E: Cross-arch splinting vs. premature extractions, J Prosthet Dent 16:1058-1068, 1966. Preiskel H: Precision attachments for free-end saddle prostheses, Br Dent J 127:462, 468, 1969. Preiskel H: Screw retained telescopic prosthesis, Br Dent J 130:107-112, 1971. Prince IB: Conservation of the supportive mechanism, J Prosthet Dent 15:327-338, 1965. Sato Y et al: Effect of friction coefficient on Akers clasp retention, J Prosthet Dent 78:22-27, 1997. Seto BG, Avera S, Kagawa T: Resin bonded etched cast cingulum rest retainers for removable partial dentures, Quintessence Int 16:757-760, 1985. Singer F: Improvements in precision: attached removable partial den-tures, J Prosthet Dent 17:69-72, 1967. Smith RA, Rymarz FP: Cast clasp transitional removable partial den-tures, J Prosthet Dent 22:381-385, 1969. Snyder HA, Duncanson MG, Johnson D: Effect of clasp flexure on a 4-meta adhered light-polymerized composite resin, Int J Prostho-dont 4:364-370, 1991. Soderfeldt B et al: A multilevel analysis of factors affecting the longevity of fixed partial dentures, retainers and abutments, J Oral Rehabil 25:245-252, 1998. Spielberger MC et al: Effect of retentive clasp design on gingival health: a feasibility study, J Prosthet Dent 52:397, 1984. Stankewitz CG, Gardner FM, Butler GV: Adjustment of cast clasps for direct retention, J Prosthet Dent 45:344, 1981. Stansbury BE: A retentive attachment for overdentures, J Prosthet Dent 35:228-230, 1976. Stern MA, Brudvik JS, Frank RP: Clinical evaluation of removable partial denture rest seat adaptation, J Prosthet Dent 53:658-662, 1985. Stewart BL, Edwards RO: Retention and wear of precision-type attach-ments, J Prosthet Dent 49:28-34, 1983. Strohaver RA, Trovillion HM: Removable partial overdentures, J Pros-thet Dent 35:624-629, 1976. Symposium on semiprecision attachments in removable partial den-tures, Dent Clin North Am 29:1-237, 1985. Tautin FS: Abutment stabilization using a nonresilient gingival bar connector, J Am Dent Assoc 99:988-989, 1979. Tietge JD et al: Wear of composite resins and cast direct retainers, Int J Prosthet 5:145-153, 1992. Vig RG: Splinting bars and maxillary indirect retainers for removable partial dentures, J Prosthet Dent 13:125-129, 1963. Walter JD: Anchor attachments used as locking devices in two-part removable prostheses, J Prosthet Dent 33:628-632, 1975. Waltz ME: Ceka extracoronal attachments, J Prosthet Dent 29:167-171, 1973. White JT: Visualization of stress and strain related to removable partial denture abutments, J Prosthet Dent 40:143-151, 1978. Wiebelt FJ, Shillingburg HT Jr: Abutment preparation modifications for removable partial denture rest seats, Quintessence Dent Technol 9:449-451, 1985. Williams AG: Technique for provisional splint with attachment, J Pros-thet Dent 21:555-559, 1969. Willis LM, Swoope CC: Precision attachment partial dentures. In Clark JW, editor: Clinical dentistry, vol 5, New York, 1976, Harper & Row. Wright SM: Use of spring-loaded attachments for retention of remov-able partial dentures, J Prosthet Dent 51:605-610, 1984. Zakler JM: Intracoronal precision attachments, Dent Clin North Am 24:131-141, 1980. Zinner ID, Miller RD, Panno FV: Clinical management of abutments with intracoronal attachments, J Prosthet Dent 67:761-767, 1992. Zinner ID, Miller RD, Panno FV: Precision attachments, Dent Clin North Am 31:395-416, 1987. Zinner ID, Miller RD, Panno FV: Semiprecision rest system for distal extension removable partial denture design, J Prosthet Dent 42:131-134, 1979. 352 Appendix B Selected Reading Resources Anatomy Bennett NG: A contribution to the study of the movements of the mandible, J Prosthet Dent 8:41-54, 1958. Boucher CO: Complete denture impressions based upon the anatomy of the mouth, J Am Dent Assoc 31:1174-1181, 1944. Brodie AG: Anatomy and physiology of head and neck musculature, Am J Orthod 36:831-844, 1950. Casey DM: Palatopharyngeal anatomy and physiology, J Prosthet Dent 49:371-378, 1983. Craddock FW: Retromolar region of the mandible, J Am Dent Assoc 47:453-455, 1953. Haines RW, Barnett SG: The structure of the mouth in the mandibular molar region, J Prosthet Dent 9:962-974, 1959. Martone AL et al: Anatomy of the mouth and related structures: I, J Prosthet Dent 11:1009-1018, 1961; II, 12:4-27, 1962; III, 12:206-219, 1962; IV, 12:409-419, 1962; V, 12:629-636, 1962; VI, 12:817-834, 1962; VII, 13:4-33, 1963; VIII, 13:204-228, 1963. Merkeley HJ: The labial and buccal accessory muscles of mastication, J Prosthet Dent 4:327-334, 1954. Merkeley HJ: Mandibular rearmament. I. Anatomic considerations, J Prosthet Dent 9:559-566, 1959. Monteith BD: Management of loading forces on the mandibular exten-sion prosthesis. Part II: Classification for matching modalities to clinical situations, J Prosthet Dent 52:832-835, 1984. Pendleton EC: Anatomy of the face and mouth from the standpoint of the denture prosthetist, J Am Dent Assoc 33:219-234, 1946. Pendleton EC: Changes in the denture supporting tissues, J Am Dent Assoc 42:1-15, 1951. Pietrokovski J: The bony residual ridge in man, J Prosthet Dent 34:456-462, 1975. Pietrokovski J, Sorin S, Zvia H: The residual ridge in partially edentu-lous patients, J Prosthet Dent 36:150-158, 1976. Preti G, Bruscagin C, Fava C: Anatomic and statistical study to deter-mine the inclination of the condylar long axis, J Prosthet Dent 49:572-575, 1983. Roche AF: Functional anatomy of the muscles of mastication, J Pros-thet Dent 13:548-570, 1963. Silverman SI: Denture prosthesis and the functional anatomy of the maxillofacial structures, J Prosthet Dent 6:305-331, 1956. Biomechanics Asher ML: Application of the rotational path design concept to a removable partial denture with a distal-extension base, J Prosthet Dent 68:641-643, 1992. Augthun M et al: The influence of spruing technique on the develop-ment of tension in a cast partial denture framework, Int J Prostho-dont 7:72-76, 1994. Avant WE: Factors that influence retention of removable partial den-tures, J Prosthet Dent 25:265-270, 1971. Avant WE: Fulcrum and retention lines in planning removable partial dentures, J Prosthet Dent 25:162-166, 1971. Aviv I, Ben-Ur Z, Cardash HS: An analysis of rotational movement of asymmetrical distal-extension removable partial dentures, J Prosthet Dent 61:211-214, 1989. Aydinlik E, Akay HU: Effect of a resilient layer in a removable partial denture base on stress distribution to the mandible, J Prosthet Dent 44:17-20, 1980. Ben-Ur Z et al: Designing clasps for the asymmetric distal extension removable partial denture, Int J Prosthodont 9:374-378, 1996. Berg TE, Caputo AA: Comparison of load transfer by maxillary distal extension removable partial dentures with a spring-loaded plunger attachment and I-bar retainer, J Prosthet Dent 68:492-499, 1992. Berg TE, Caputo AA: Load transfer by a maxillary distal-extension removable partial denture with extracoronal attachments, J Prosthet Dent 68:784-789, 1992. Bezzon OL et al: Surveying removable partial dentures: the importance of guiding planes and path of insertion for stability, J Prosthet Dent 78:412-418, 1997. Bridgeman JT et al: Comparison of titanium and cobalt-chromium removable partial denture clasps, J Prosthet Dent 78:187-193, 1997. Browning JD, Eick JD, McGarrah HE: Abutment tooth movement measured in vivo by using stereophotogrammetry, J Prosthet Dent 57:323-328, 1987. Brudvik JS, Morris HF: Stress-relaxation testing. Part III: Influence of wire alloys, gauges, and lengths on clasp behavior, J Prosthet Dent 46:374-379, 1981. Byron R Jr, Frazer RQ, Herren MC: Rotational path removable partial denture: an esthetic alternative. Gen Dent 55:245-250, 2007; quizzes 251, 264. Cecconi BT: Effect of rest design on transmission of forces to abutment teeth, J Prosthet Dent 32:141-151, 1974. Cecconi BT, Asgar K, Dootz E: Clasp assembly modifications and their effect on abutment tooth movement, J Prosthet Dent 27:160-167, 1972. Cecconi BT, Asgar K, Dootz E: The effect of partial denture clasp design on abutment tooth movement, J Prosthet Dent 25:44-56, 1971. Cecconi BT, Asgar K, Dootz E: Removable partial denture abutment tooth movement as affected by inclination of residual ridges and types of loading, J Prosthet Dent 25:375-381, 1971. Chou TM et al: Photoelastic analysis and comparison of force-trans-mission characteristics of intracoronal attachments with clasp distal-extension removable partial dentures, J Prosthet Dent 62:313-319, 1989. Chou TM et al: Stereophotogrammetric analysis of abutment tooth movement in distal-extension removable partial dentures with intracoronal attachments and clasps, J Prosthet Dent 66:343-349, 1991. Clayton JA, Jaslow C: A measurement of clasp forces on teeth, J Pros-thet Dent 25:21-43, 1971. Craig RG, Farah JW: Stresses from loading distal extension removable partial dentures, J Prosthet Dent 39:274-277, 1978. DeVan MM: The nature of the partial denture foundation: suggestions for its preservation, J Prosthet Dent 2:210-218, 1952. el Charkawi HG et al: The effect of the resilient-layer distal-extension partial denture on movement of the abutment teeth: a new method-ology, J Prosthet Dent 60:622-630, 1988. Fisher RL: Factors that influence the base stability of mandibular distal-extension removable partial dentures: a longitudinal study, J Pros-thet Dent 50:167-171, 1983. Frank RP, Nicholls JI: A study of the flexibility of wrought-wire clasps, J Prosthet Dent 45:259-267, 1981. Frechette AR: The influence of partial denture design on distribution of force to abutment teeth, J Prosthet Dent 6:195-212, 1956. Goodkind RJ: The effects of removable partial dentures on abutment tooth mobility, J Prosthet Dent 30:139-146, 1973. Goodman JJ, Goodman HW: Balance of force in precision free-end restorations, J Prosthet Dent 13:302-308, 1963. Hall WA: Variations in registering interarch transfers in removable partial denture construction, J Prosthet Dent 30:548-553, 1973. Harrop J, Javid N: Reciprocal arms of direct retainers in removable partial dentures, J Can Dent Assoc 4:208-211, 1976. Henderson D, Seward TE: Design and force distribution with remov-able partial dentures: a progress report, J Prosthet Dent 17:350-364, 1967. Henriques GE et al: Soldering and remelting influence on fatigue strength of cobalt-chromium alloy, J Prosthet Dent 78:146-152, 1997. Hindels GW: Stress analysis in distal extension partial dentures, J Pros-thet Dent 7:197-205, 1957. Iwama CY et al: Cobalt-chromium-titanium alloy for removable partial dentures, Int J Prosthodont 10:309-317, 1997. 353 Appendix B Selected Reading Resources Johnson DL, Stratton RJ, Duncanson MGJ: The effect of single plane curvature on half-round cast clasps, J Dent Res 62:833-836, 1983. Kaires AK: Partial denture design and its relation to force distribution and masticatory performance, J Prosthet Dent 6:672-683, 1956. Knowles LE: The biomechanics of removable partial dentures and its relationship to fixed prosthesis, J Prosthet Dent 8:426-430, 1958. Kratochvil FJ: Influence of occlusal rest position and clasp design on movement of abutment teeth, J Prosthet Dent 13:114-124, 1963. Kratochvil FJ, Caputo AA: Photoelastic analysis of pressure on teeth and bone supporting removable partial dentures, J Prosthet Dent 3:52, 1975. Kratochvil FJ, Thompson WD, Caputo AA: Photoelastic analysis of stress patterns on teeth and bone with attachment retainers for removable partial dentures, J Prosthet Dent 46:21-28, 1981. Lofbers PG, Ericson G, Eliasson S: A clinical and radiographic evalua-tion of removable partial dentures retained by attachments to alveo-lar bars, J Prosthet Dent 47:126-132, 1982. Lowe RD, Kydd WL, Smith DE: Swallowing and resting forces related to lingual flange thickness in removable partial dentures, J Prosthet Dent 23:279-288, 1970. MacGregor AR, Miller TPG, Farah JW: Stress analysis of partial den-tures, J Dent 6:125-132, 1978. Marei MK: Measurement (in vitro) of the amount of force required to dislodge specific clasps from different depths of undercut, J Prosthet Dent 74:258-263, 1995. Maroso DJ, Schmidt JR, Blustein R: A preliminary study of wear of porcelain when subjected to functional movements of retentive clasp arms, J Prosthet Dent 45:14-17, 1981. Matheson GR, Brudvik JS, Nicholls JI: Behavior of wrought-wire clasps after repeated permanent deformation, J Prosthet Dent 55:226-231, 1986. Maxfield JB, Nicholls JI, Smith DE: The measurement of forces trans-mitted to abutment teeth of removable partial dentures, J Prosthet Dent 41:134, 1979. McCartney JW: Motion vector analysis of an abutment for a distal-extension removable partial denture, J Prosthet Dent 43:15-21, 1980. McDowell GC: Force transmission by indirect retainers during unilat-eral loading, J Prosthet Dent 39:616-621, 1978. McDowell GC, Fisher RL: Force transmission by indirect retainers when a unilateral dislodging force is applied, J Prosthet Dent 47:360-365, 1982. McLeod NS: An analysis of the rotational axes of semiprecision and precision distal-extension removable partial dentures, J Prosthet Dent 48:130-134, 1982. Morris HF, Asgar K, Tillitson E: Stress-relaxation testing. I. A new approach to the testing of removable partial denture alloys, wrought-wires, and clasp behavior, J Prosthet Dent 46:133-141, 1981. Morris HF, Brudvik JS: Influence of polishing on cast clasp properties, J Prosthet Dent 55:75-77, 1986. Morris HF et al: Stress-relaxation testing. IV. Clasp pattern dimensions and their influence on clasp behavior, J Prosthet Dent 50:319-326, 1983. NaBadalung DP et al: Comparison of bond strengths of denture base resins to nickel-chromium-beryllium removable partial denture alloy, J Prosthet Dent 78:566-573, 1997. NaBadalung DP et al: Frictional resistance of removable partial den-tures with retrofitted resin composite guide planes, Int J Prosthodont 10:116-122, 1997. NaBadalung DP et al: Laser welding of a cobalt-chromium removable partial denture alloy, J Prosthet Dent 79:285-290, 1998. Ogata K, Shimigu K: Longitudinal study of forces transmitted from denture base to retainers of free-end saddle dentures with Akers clasps, J Oral Rehabil 18:471-480, 1991. Plotnick IJ, Beresin VE, Simkins AB: The effects of variations in the opposing dentition on changes in the partially edentulous mandible, J Prosthet Dent; I, 33:278-286, 1975; II, 33:403-406, 1975; III, 33:529-534, 1975. Sansom BP et al: Rest seat designs for inclined posterior abutments: a photoelastic comparison, J Prosthet Dent 58:57-62, 1987. Shohet H: Relative magnitudes of stress on abutment teeth with differ-ent retainers, J Prosthet Dent 21:267-282, 1969. Smith BH: Changes in occlusal face height with removable partial den-tures, J Prosthet Dent 34:278-285, 1975. Smith BJ, Turner CH: The use of crowns to modify abutment teeth of removable partial dentures, J Dent 7:52-56, 1979. Smyd ES: Biomechanics of prosthetic dentistry, J Prosthet Dent 4:368-383, 1954. Stern WJ: Guiding planes in clasp reciprocation and retention, J Pros-thet Dent 34:408-414, 1975. Swoope CC, Frank RP: Stress control and design. In Clark JW, editor: Clinical dentistry, vol 5, New York, 1976, Harper & Row. Taylor DT, Pflushoeft FA, McGivney GP: Effect of two clasping assem-blies on arch integrity as modified by base adaptation, J Prosthet Dent 47:120-125, 1982. Tebrock OC et al: The effect of various clasping systems on the mobility of abutment teeth for distal-extension removable partial dentures, J Prosthet Dent 41:511, 1979. Thompson WD, Kratochvil FJ, Caputo AA: Evaluation of photoelastic stress patterns produced by various designs of bilateral distal-exten-sion removable partial dentures, J Prosthet Dent 38:261, 1977. Toth RW et al: Shear strength of lingual rest seats prepared in bonded composite, J Prosthet Dent 56:99-104, 1986. Vallittu PK: Comparison of the in vitro fatigue resistance of an acrylic resin removable partial denture reinforced with continuous glass fibers or metal wires, J Prosthodont 5:115-121, 1996. Vallittu PK: Deflection fatigue of cobalt-chromium, titanium, and gold alloy cast denture clasp, J Prosthet Dent 74:412-419, 1995. Waldmeier MD et al: Bend testing of wrought-wire removable partial denture alloys, J Prosthet Dent 76:559-565, 1996. Wills DJ, Manderson RD: Biomechanical aspects of the support of partial dentures, J Dent 5:310-318, 1977. Yurkstas A, Fridley HH, Manly RS: A functional evaluation of fixed and removable bridgework, J Prosthet Dent 1:570-577, 1951. Zoeller GN, Kelly WJ Jr: Block form stability in removable partial prosthodontics, J Prosthet Dent 25:515-519, 1971. Classification Applegate OC: The rationale of partial denture choice, J Prosthet Dent 10:891-907, 1960. Avant WE: A universal classification for removable partial denture situ-ations, J Prosthet Dent 16:533-539, 1966. Bailyn M: Tissue support in partial denture construction, Dent Cosmos 70:988-997, 1928. Beckett LS: The influence of saddle classification on the design of partial removable restoration, J Prosthet Dent 3:506-516, 1953. Costa E: A simplified system for identifying partially edentulous arches, J Prosthet Dent 32:639-645, 1974. Cummer WE: Partial denture service. In Anthony LP, editor: American textbook of prosthetic dentistry, Philadelphia, 1942, Lea & Febiger. Friedman J: The ABC classification of partial denture segments, J Pros-thet Dent 3:517-524, 1953. Godfrey RJ: Classification of removable partial dentures, J Am Coll Dent 18:5-13, 1951. Kennedy E: Partial denture construction, Dental Items of Interest 3-8, 1928. Mensor MC Jr: Classification and selection of attachments, J Prosthet Dent 29:494-497, 1973. Miller EL: Systems for classifying partially dentulous arches, J Prosthet Dent 24:25-40, 1970. Skinner CN: A classification of removable partial dentures based upon the principles of anatomy and physiology, J Prosthet Dent 9:240-246, 1959. 354 Appendix B Selected Reading Resources Cleft Palate Aram A, Subtelny JD: Velopharyngeal function and cleft palate pros-theses, J Prosthet Dent 9:149-158, 1959. Baden E: Fundamental principles of orofacial prosthetic therapy in congenital cleft palate, J Prosthet Dent 4:420-433, 1954. Bixler D: Heritability of clefts of the lips and palate, J Prosthet Dent 33:100-108, 1975. Buckner H: Construction of a denture with hollow obturator, lid and soft acrylic lining, J Prosthet Dent 31:95-99, 1974. Calvan J: The error of Gustan Passavant, Plast Reconstr Surg 13:275-289, 1954. Cooper HK: Integration of service in the treatment of cleft lip and cleft palate, J Am Dent Assoc 47:27-32, 1953. Dalston RM: Prosthodontic management of the cleft palate patient: a speech pathologist’s view, J Prosthet Dent 37:327-329, 1978. Ettinger RL: Use of teeth with a poor prognosis in cleft palate prosth-odontics, J Am Dent Assoc 94:910-914, 1977. Fox A: Prosthetic correction of a severe acquired cleft palate, J Prosthet Dent 8:542-546, 1958. Gibbons P, Bloomer H: A supportive-type prosthetic speech aid, J Prosthet Dent 8:362-369, 1958. Graber TM: Oral and nasal structures in cleft palate speech, J Am Dent Assoc 53:693-706, 1956. Harkins CS: Modern concepts in the prosthetic rehabilitation of cleft palate patients, J Oral Surg 10:298-312, 1952. Harkins CS, Ivy RH: Surgery and prosthesis in the rehabilitation of cleft palate patients, J South Calif Dent Assoc 19:16-24, 1951. Immekus JE, Aramany MA: A fixed-removable partial denture for cleft palate patients, J Prosthet Dent 34:286-291, 1975. Landa JS: The prosthodontist views the rehabilitation of the cleft palate patient, J Prosthet Dent 6:421-427, 1956. Lavelle WE, Zach GE: The tissue bar and Ceka anchor as aids in cleft palate rehabilitation, J Prosthet Dent 30:321-325, 1973. Lloyd RS, Pruzansky S, Subtelny JD: Prosthetic rehabilitation of a cleft palate patient subsequent to multiple surgical and prosthetic failures, J Prosthet Dent 7:216-230, 1957. Merkeley HJ: Cleft palate prosthesis, J Prosthet Dent 9:506-513, 1959. Minsley GE, Warren DW, Hairfield WM: The effect of cleft palate speech aid prostheses on the nasopharyngeal airway and breathing, J Prosthet Dent 65:122-126, 1991. Nidiffer TJ, Shipmon TH: The hollow-bulb obturator for acquired palatal openings, J Prosthet Dent 7:126-134, 1957. Olinger NA: Cleft palate prosthesis rehabilitation, J Prosthet Dent 2:117-135, 1952. Rosen MS: Prosthetics for the cleft palate patient, J Am Dent Assoc 60:715-721, 1960. Rothenberg LIA: Overlay dentures for the cleft-palate patient, J Pros-thet Dent 37:190-195, 1977. Schneiderman CR, Maun MB: Air flow and intelligibility of speech of normal speakers and speakers with a prosthodontically repaired cleft palate, J Prosthet Dent 39:193-199, 1978. Sharry JJ: The meatus obturator in cleft palate prosthesis, Oral Surg 7:852-855, 1954. Sharry JJ: Meatus obturator in particular and pharyngeal impressions in general, J Prosthet Dent 8:893-896, 1958. Tautin FS, Schaaf NA: Superiorly based obturator, J Prosthet Dent 33:96-99, 1975. Walter JD: Palatopharyngeal activity in cleft palate subjects, J Prosthet Dent 63:187-192, 1990. Complete Mouth and Occlusal Rehabilitation Brewer AA, Fenton AH: The overdenture, Dent Clin North Am 17:723-746, 1973. Bronstein BR: Rationale and technique of biomechanical occlusal reha-bilitation, J Prosthet Dent 4:352-367, 1954. Cohn LA: Occluso-rehabilitation, principles of diagnosis and treatment planning, Dent Clin North Am 6:281, 1962. Curtis SR: Integrating fixed and removable provisional restorations, J Prosthet Dent 70:374-377, 1993. Dubin NA: Advances in functional occlusal rehabilitation, J Prosthet Dent 6:252-258, 1956. Ferencz JL: Splinting, Dent Clin North Am 31:383-393, 1987. Kazis H: Functional aspects of complete mouth rehabilitation, J Pros-thet Dent 4:833-841, 1954. Kornfeld M: The problem of function in restorative dentistry, J Pros-thet Dent 5:670-676, 1955. Landa JS: An analysis of current practices in mouth rehabilitation, J Prosthet Dent 5:527-537, 1955. Lang BR: Complete denture occlusion, Dent Clin North Am 40:85-101, 1996. Mann AW, Pankey LD: Oral rehabilitation. I. Use of the P-M instru-ment in treatment planning and restoring the lower posterior teeth, J Prosthet Dent 10:135-150, 1960. Mann AW, Pankey LD: Oral rehabilitation. II. Reconstruction of the upper teeth using a functionally generated path technique, J Prosthet Dent 10:151-162, 1960. Mann AW, Pankey LD: Oral rehabilitation utilizing the Pankey-Mann instrument and a functional bite technique, Dent Clin North Am March:215-230, 1959. McCartney JW: Occlusal reconstruction and rebase procedure for distal extension removable partial dentures, J Prosthet Dent 43:695-698, 1980. Schuyler CH: An evaluation of incisal guidance and its influence on restorative dentistry, J Prosthet Dent 9:374-378, 1959. Schweitzer JM: Open bite from the prosthetic point of view, Dent Clin North Am 1:269-283, 1957. Crowns and Fixed Partial Dentures Alexander PC: Analysis of the cuspid protective occlusion, J Prosthet Dent 13:309-317, 1963. Bader JD et al: Effect of crown margins on periodontal conditions in regularly attending patients, J Prosthet Dent 65:75-79, 1991. Beeson PE: The use of acrylic resins as an aid in the development of patterns for two types of crowns, J Prosthet Dent 13:493-498, 1963. Binkley TK, Binkley C: Porcelain-fused-to-metal crowns as replace-ments for denture teeth in removable partial denture construction, J Prosthet Dent 58:124-125, 1987. Blackman R, Baeg R, Barghi N: Marginal accuracy and geometry of cast titanium copings, J Prosthet Dent 67:435-440, 1992. Budtz-Jorgenson E, Isidor F: A five-year longitudinal study of cantile-ver fixed partial dentures compared with removable partial dentures in a geriatric population, J Prosthet Dent 64:42-47, 1990. Caplan J: Maintenance of full coverage fixed-abutment bridges, J Pros-thet Dent 5:852-854, 1955. Cheug SP, Dimmer A: Management of worn dentition with resin-bonded cast metal lingual veneering, J Prosthet Dent 63:122-123, 1990. Coelho DH: Criteria for the use of fixed prosthesis, Dent Clin North Am 1:299-311, 1957. Cooper TM et al: Effect of venting on cast gold full crowns, J Prosthet Dent 26:621-626, 1971. Cowgen GT: Retention, resistance and esthetics of the anterior three-quarter crown, J Am Dent Assoc 62:167-171, 1961. Culpepper WD, Moulton PS: Considerations in fixed prosthodontics, Dent Clin North Am 23:21-35, 1979. Dental technology standards, J Dent Technol 14:26-31, 1997. Ekfeldt A et al: Changes of masticatory movement characteristics after prosthodontic rehabilitation of individuals with extensive tooth wear, Int J Prosthodont 9(6):539-546, 1996. 355 Appendix B Selected Reading Resources Elledge DA, Schorr BL: A provisional and new crown to fit with a clasp of an existing removable partial denture, J Prosthet Dent 63:541-544, 1990. Felton DA et al: Effect of in vivo crown margin discrepancies on peri-odontal health, J Prosthet Dent 65:357-364, 1991. Glantz PO et al: The devitalized tooth as an abutment in dentitions with reduced but healthy periodontium, Periodontol 2000 4:52-57, 1994. Goldberg A, Jones RD: Constructing cast crowns to fit existing remov-able partial denture clasps, J Prosthet Dent 36:382-386, 1976. Goodacre CJ et al: The prosthodontic management of endodontically treated teeth: a literature review. Part I. Success and failure data, treatment concepts, J Prosthodont 3:243-250, 1994. Guyer SE: Nonrigid subocclusal connector for fixed partial dentures, J Prosthet Dent 26:433-436, 1971. Hansen CA, Cook PA, Nelson DF: Pin-modified facial inlay to enhance retentive contours on a removable partial denture abutment, J Pros-thet Dent 55:480-481, 1986. Henderson D et al: The cantilever type of posterior fixed partial den-tures: a laboratory study, J Prosthet Dent 24:47-67, 1970. Johnson EA Jr: Combination of fixed and removable partial dentures, J Prosthet Dent 14:1099-1106, 1964. Johnston JF et al: Construction and assembly of porcelain veneer gold crowns and pontics, J Prosthet Dent 12:1125-1137, 1962. Kapur KK et al: Veterans Administration Cooperative Dental Implant Study: Comparisons between fixed partial dentures supported by blade-vent implants and removable partial dentures. Part II. Com-parison of success rates and periodontal health between two treat-ment modalities, J Prosthet Dent 62:685-703, 1992. Kapur KK et al: Veterans Administration Cooperation Dental Implant Study: Comparisons between fixed partial dentures supported by blade-vent implants and removable partial dentures. Part IV. Com-parisons of patient satisfaction between two treatment modalities, J Prosthet Dent 66:517-530, 1991. Kunisch WH, Dodd J: A conversion alternative to ceramics in a crown-and-sleeve coping prosthesis, J Prosthet Dent 49:581-582, 1983. Leff A: New concepts in the preparation of teeth for full coverage, J Prosthet Dent 5:392-400, 1955. Leff A: Reproduction of tooth anatomy and positional relationship in full cast or veneer crowns, J Prosthet Dent 6:550-557, 1956. Libby G et al: Longevity of fixed partial dentures, J Prosthet Dent 78:127-131, 1997. Malson TS: Anatomic cast crown reproduction, J Prosthet Dent 9:106-112, 1959. Marinello CP, Scharer P: Resin-bonded etched cast extracoronal attachments for removable partial dentures: clinical experiences, Int J Periodont Res Dent 7:36-49, 1987. McArthur DR: Fabrication of full coverage restorations for existing removable partial dentures, J Prosthet Dent 51:574-576, 1984. Mojon P et al: Relationship between prosthodontic status, caries and periodontal disease in a geriatric population, Int J Prosthodont 26:564-571, 1995. Morris HF et al: Department of Veterans Affairs Cooperative Studies Project No. 242: Quantitative and qualitative evaluation of the mar-ginal fit of cast ceramic, porcelain-shoulder, and cast metal full crown margins, J Prosthet Dent 67:198-204, 1992. Moulding MB, Holland GA, Sulik WD: An alternative orientation of nonrigid connectors in fixed partial dentures, J Prosthet Dent 6:236-238, 1992. Mueninghoff LA, Johnson MH: Fixed-removable partial dentures, J Prosthet Dent 48:547-550, 1982. Palmquist S et al: Multivariate analyses of factors influencing the lon-gevity of fixed partial dentures, retainers and abutments, J Prosthet Dent 71: 245-250, 1994. Patur B: The role of occlusion and the periodontium in restorative procedures, J Prosthet Dent 21:371-379, 1969. Pezzoli M et al: Magnetizable abutment crowns for distal-extension removable partial dentures, J Prosthet Dent 55:475-480, 1986. Phillips RW, Biggs DH: Distortion of wax patterns as influenced by storage time, storage temperature, and temperature of wax manipu-lation, J Am Dent Assoc 41:28-37, 1950. Phillips RW, Price RR: Some factors which influence the surface of stone dies poured in alginate impressions, J Prosthet Dent 5:72-79, 1955. Phillips RW, Swartz ML: A study of adaptation of veneers to cast gold crowns, J Prosthet Dent 7:817-822, 1957. Pound E: The problem of the lower anterior bridge, J Prosthet Dent 5:543-545, 1955. Preston JD: Preventing ceramic failures when integrating fixed and removable prostheses, Dent Clin North Am 23:37-52, 1979. Pruden KC: A hydrocolloid technique for pinledge bridge abutments, J Prosthet Dent 6:65-71, 1956. Pruden WH: Full coverage, partial coverage, and the role of pins, J Prosthet Dent 26:302-306, 1971. Rhoads JE: The fixed-removable partial denture, J Prosthet Dent 48:122-129, 1982. Rubin MK: Full coverage: the provisional and final restorations made easier, J Prosthet Dent 8:664-672, 1958. Schorr BL, Peregrina AM, Elledge DA: Alternatives to posterior com-plete crowns: integrating foundations with cuspal protection, J Pros-thet Dent 69:165-170, 1993. Seals RR Jr, Stratton RJ: Surveyed crowns: a key for integrating fixed and removable prosthodontics, Quintessence Dent Technol 11:43-49, 1987. Sheets CE: Dowel and core foundations, J Prosthet Dent 23:58-65, 1970. Shooshan ED: The reverse pin-porcelain facing, J Prosthet Dent 9:284-301, 1959. Smith GP: The marginal fit of the full cast shoulderless crown, J Pros-thet Dent 7:231-243, 1957. Smith GP: Objectives of a fixed partial denture, J Prosthet Dent 11:463-473, 1961. Staffanou RS, Thayer KE: Reverse pin-porcelain veneer and pontic technique, J Prosthet Dent 12:1138, 1145, 1962. Thurgood BW, Thayer KE, Lee RE: Complete crowns constructed for an existing partial denture, J Prosthet Dent 29:507-512, 1973. Treppo KW, Smith FW: A technique for restoring abutments for removable partial dentures, J Prosthet Dent 40:398-401, 1978. Troxell RR: The polishing of gold castings, J Prosthet Dent 9:668-675, 1959. Turner KA, Messirlian DM: Restoration of the extremely worn denti-tion, J Prosthet Dent 52:464-474, 1984. Wagman SS: Tissue management for full cast veneer crowns, J Prosthet Dent 15:106-117, 1965. Wagner AW, Burkhart JW, Fayle HE Jr: Contouring abutment teeth with cast gold inlays for removable partial dentures, J Prosthet Dent 201:330-334, 1968. Wallace FH: Resin transfer copings, J Prosthet Dent 8:289-292, 1958. Wang CJ, Millstein PL, Nathanson D: Effects of cement, cement space, marginal design, seating aid materials, and seating force on crown cementation, J Prosthet Dent 67:786-790, 1992. Welsh SL: Complete crown construction for a clasp-bearing abutment, J Prosthet Dent 34:320-323, 1975. Wheeler RC: Complete crown form and the periodontium, J Prosthet Dent 11:722-734, 1961. Yalisove IL: Crown and sleeve-coping retainers for removable partial prostheses, J Prosthet Dent 16:1069-1085, 1966. Dental Laboratory Procedures ADA Council on Scientific Affairs and ADA Council on Dental Prac-tice: Infection control recommendations for the dental office and the dental laboratory, J Am Dent Assoc 11:395-399, 1996. 356 Appendix B Selected Reading Resources Asgar K, Peyton FA: Casting dental alloys to embedded wires, J Prosthet Dent 15:312-321, 1965. Becker CM, Smith EE, Nicholls JI: The comparison of denture-base processing techniques. I. Material characteristics, J Prosthet Dent 37:330-338, 1977. Berg E et al: Mechanical properties of laser-welded cast and wrought titanium, J Prosthet Dent 74:250-257, 1995. Blanchard CH: Filling undercuts on refractory casts with investment, J Prosthet Dent 3:417-418, 1953. Bolouri A, Hilger TC, Gowrylok MD: Modified flasking technique for removable partial dentures, J Prosthet Dent 34:221-223, 1975. Brudvik JS, Nicholls JI: Soldering of removable partial dentures, J Pros-thet Dent 49:762-765, 1983. Burnett CA et al: Sprue design in removable partial denture casting, J Dent 24:99-103, 1996. Calverley MJ, Moergeli JR Jr: Effect on the fit of removable partial denture frameworks when master casts are treated with cyanoacry-late resin, J Prosthet Dent 58:327-329, 1987. Casey DM, Crowther DS, Lauciello FR: Strengthening abutment or isolated teeth on removable partial denture master casts, J Prosthet Dent 46:105-106, 1981. Dirksen LC, Campagna SJ: Mat surface and rugae reproduction for upper partial denture castings, J Prosthet Dent 4:67-72, 1954. Dootz ER, Craig RG, Peyton FA: Influence of investments and duplicat-ing procedures on the accuracy of partial denture castings, J Prosthet Dent 15:679-690, 1965. Dootz ER, Craig RG, Peyton FA: Simplification of the chrome-cobalt partial denture casting procedure, J Prosthet Dent 17:464-471, 1967. Elbert CA, Ryge G: The effect of heat treatment on hardness of a chrome-cobalt alloy, J Prosthet Dent 15:873-879, 1965. Elliott RW: The effects of heat on gold partial denture castings, J Pros-thet Dent 13:688-698, 1963. Enright CM: Dentist-dental laboratory harmony, J Prosthet Dent 11:393-394, 1961. Fiebiger GE, Parr GR, Goldman BM: Remount casts for removable partial dentures, J Prosthet Dent 48:106-107, 1982. Firtell DN, Muncheryan AM, Green AJ: Laboratory accuracy in casting removable partial denture frameworks, J Prosthet Dent 54:856-862, 1985. Fowler JA Jr, Kuebker WA, Escobedo JJ: Laboratory procedures for the maintenance of a removable partial overdenture, J Prosthet Dent 50:121-126, 1983. Garver DG: Updated laboratory procedure for the subpontic clasping system, J Prosthet Dent 48:734-735, 1982. Gay WD: Laboratory procedures for fitting removable partial denture frameworks, J Prosthet Dent 40:227-229, 1978. Gilson TD, Asgar K, Peyton FA: The quality of union formed in casting gold to embedded attachment metals, J Prosthet Dent 15:464-473, 1965. Grunewald AH, Paffenbarger GC, Dickson G: Dentist, dental labora-tory, and the patient, J Prosthet Dent 8:55-60, 1958. Grunewald AH, Paffenbarger GC, Dickson G: The effect of molding processes on some properties of denture resins, J Am Dent Assoc 44:269-284, 1952. Grunewald AH, Paffenbarger GC, Dickson G: The role of the dental technician in a prosthetic service, Dent Clin North Am 4:359-370, 1960. Hanson JG et al: Effect on dimensional accuracy when reattaching fractured lone standing teeth of a cast, J Prosthet Dent 47:488-492, 1982. Johnson HB: Technique for packing and staining complete or partial denture bases, J Prosthet Dent 6:154-159, 1956. Jones DW: Thermal analysis and stability of refractory investments, J Prosthet Dent 18:234-241, 1967. Jordan RD, Turner KA, Taylor TD: Multiple crowns fabricated for an existing removable partial denture, J Prosthet Dent 48:102-105, 1982. Kazanoglu A, Smith EH: Replacement technique for a broken occlusal rest, J Prosthet Dent 48:621-623, 1982. Krand M et al: Study on the surface of resins that burn without residues in the lost-wax procedure, J Prosthodont 5:259-265, Dec 1996. Lanier BR, Rudd KD, Strunk RR: Making chromium-cobalt removable partial dentures: a modified technique, J Prosthet Dent 25:197-205, 1971. Lauciello FR: Technique for remounting removable partial dentures opposing maxillary complete dentures, J Prosthet Dent 45:336-340, 1981. Mahler DB, Ady AB: The influence of various factors on the effective setting expansion of casting investments, J Prosthet Dent 13:365-373, 1963. Maxson BB et al: Quality assurance for the laboratory aspects of prosth-odontic treatment, J Prosthodont 6:204-209, 1997. May KB, Razzoog ME: Silane to enhance the bond between polymethyl methacrylate and titanium, J Prosthet Dent 73:428-431, 1995. McCartney JW: The acrylic resin base maxillary removable partial denture: technical considerations, J Prosthet Dent 43:467-468, 1980. Mohammed H et al: Button versus buttonless castings for removable partial denture frameworks, J Prosthet Dent 72:433-444, 1994. Moreno de Delgado M, Garcia LT, Rudd KD: Camouflaging partial denture clasps, J Prosthet Dent 55:656-660, 1986. Mori T et al: Titanium for removable dentures. I. Laboratory proce-dures, J Oral Rehabil 24:238-341, 1997. Morris HF et al: The influence of heat treatments on several types of base-metal removable partial denture alloys, J Prosthet Dent 41:388-395, 1979. NaBadalung DP et al: Comparison of bond strengths of denture base resins to nickel-chromium-beryllium removable partial denture alloy, J Prosthet Dent 78:566-573, 1997. NaBadalung DP et al: Effectiveness of adhesive systems for Co-Cr removable partial denture alloy, J Prosthet Dent 7:17-25, Mar 1998. Nelson DR et al: Expediting the fabrication of a nickel-chromium casting, J Prosthet Dent 55:400-403, 1986. Nelson DR, von Gonten AS, Kelly TW Jr: The cast round RPA clasp, J Prosthet Dent 54:307-309, 1985. Palmer BL, Coffey KW: Investing and packing removable partial denture bases to minimize vertical processing error, J Prosthet Dent 56:123-124, 1986. Parr FR, Gardner LK: The removable partial denture design template, Compendium 8:594, 596, 598-600, 1987. Perry CK: Transfer base for removable partial dentures, J Prosthet Dent 31:582-584, 1974. Peyton FA, Anthony DH: Evaluation of dentures processed by different techniques, J Prosthet Dent 13:269-281, 1963. Quinlivan JT: Fabrication of a simple ball-socket attachment, J Prosthet Dent 32:222-225, 1974. Radue JT, Unser JW: Constructing stable record bases for removable partial dentures, J Prosthet Dent 46:463, 1981. Rantanen T, Eerikainen E: Accuracy of the palatal plate of removable partial dentures, and influence of laboratory handling of the invest-ment on the accuracy, Dent Mater 2:28-31, 1986. Raskin ER: An indirect technique for fabricating a crown under an existing clasp, J Prosthet Dent 50:580-581, 1983. Ring M: Rest seats in existing crowns, Dent Lab Rev 60:24-25, 1985. Ryge G, Kozak SF, Fairhurst CW: Porosities in dental gold castings, J Am Dent Assoc 54:746-754, 1957. Sarnat AE, Klugman RS: A method to record the path of insertion of a removable partial denture, J Prosthet Dent 46:222-223, 1981. Scandrett FR, Hanson JG, Unsicker RL: Layered silicone rubber tech-nique for flasking removable partial dentures, J Prosthet Dent 40:349-350, 1978. Schmidt AH: Repairing chrome-cobalt castings, J Prosthet Dent 5:385-387, 1955. 357 Appendix B Selected Reading Resources Schmitt SM, Chance DA, Cronin RJ: Refining cast implant-retained restorations by electrical discharge machining, J Prosthet Dent 73:280-283, 1995. Schneider R: Metals used to fabricate removable partial denture frame-works, J Dent Technol 13:35-42, 1996. Schneider RL: Adapting ceramometal restorations to existing remov-able partial dentures, J Prosthet Dent 49:279-281, 1983. Schneider RL: Custom metal occlusal surfaces for acrylic resin denture teeth, J Prosthet Dent 46:98-101, 1981. Schwalm CA, LaSpina FY: Fabricating swinglock removable partial denture frameworks, J Prosthet Dent 45:216-220, 1981. Schwedhelm ER et al: Fracture strength of type IV and type V die stone as a function of time, J Prosthet Dent 78:554-559, 1997. Shay JS, Mattingly SL: Technique for the immediate repair of removable partial denture facings, J Prosthet Dent 47:104-106, 1982. Smith GP: The responsibility of the dentist toward laboratory proce-dures in fixed and removable partial denture prostheses, J Prosthet Dent 13:295-301, 1963. Smith RA: Clasp repair for removable partial dentures, J Prosthet Dent 29:231-234, 1973. Stade EH et al: Influence of fabrication technique on wrought-wire clasp flexibility, J Prosthet Dent 54:538-543, 1985. Stankewitz CG: Acrylic resin blockout for interim removable partial dentures, J Prosthet Dent 40:470-471, 1978. Swoope CC, Frank RP: Fabrication procedures. In Clark JW, editor: Clinical dentistry, vol 5, New York, 1976, Harper & Row. Sykora O: A new tripoding technique, J Prosthet Dent 44:463-464, 1980. Sykora O: Removable partial denture design by Canadian laboratories: a retrospective study, J Can Dent Assoc 61:615-621, 1995. Tambasco J et al: Laser welding in the dental laboratory: an alternative to soldering, J Dent Technol 13:23-31, May 1996. Teppo KW, Smith FW: A method of immediate clasp repair, J Prosthet Dent 30:77-80, 1975. Tran CD, Sherraden DR, Curtis TA: A review of techniques of crown fabrication for existing removable partial dentures, J Prosthet Dent 55:671-673, 1986. Tuccillo JJ, Nielsen JP: Compatibility of alginate impression materials and dental stones, J Prosthet Dent 25:556-566, 1971. Ulmer FC, Ward JE: Simplified technique for production of a distal-extension removable partial denture remounting cast, J Prosthet Dent 41:473-474, 1979. von Gonten AS, Nelson DR: Laboratory pitfalls that contribute to embrasure clasp failure, J Prosthet Dent 53:136-138, 1985. Williams HN, Falkler WA Jr, Hasler JF: Acinetobacter contamination of laboratory dental pumice, J Dent Res 62:1073-1075, 1983. Zalkind M, Avital R, Rehany A: Fabrication of a replacement for a broken attachment, J Prosthet Dent 51:714-716, 1984. Denture Esthetics: Tooth Selection and Arrangement Askinas SW: Facings in removable partial dentures, J Prosthet Dent 33:633-636, 1975. Culpepper WD: A comparative study of shade-matching procedures, J Prosthet Dent 24:166-173, 1971. DeVan MM: The appearance phase of denture construction, Dent Clin North Am 1:255-268, 1957. Engelmeier RL: Complete-denture esthetics, Dent Clin North Am 40:71-84, 1996. Fields H Jr, Birtles JT, Shay J: Combination prosthesis for optimum esthetic appearance, J Am Dent Assoc 101:276-279, 1980. French FA: The selection and arrangement of the anterior teeth in prosthetic dentures, J Prosthet Dent 1:587-593, 1951. Frush JP, Fisher RD: How dentogenic restorations interpret the sex factor, J Prosthet Dent 6:160-172, 1956. Frush JP, Fisher RD: How dentogenics interprets the personality factor, J Prosthet Dent 6:441-449, 1956. Frush JP, Fisher RD: Introduction to dentogenic restorations, J Pros-thet Dent 5:586-595, 1955. Hughes GA: Facial types and tooth arrangement, J Prosthet Dent 1:82-95, 1951. Krajicek DD: Natural appearance for the individual denture patient, J Prosthet Dent 10:205-214, 1960. Lang BR: Complete denture occlusion, Dent Clin North Am 40:85-101, 1996. Levin EI: Dental esthetics and the golden proportion, J Prosthet Dent 40:244-252, 1978. Lombardi RE: Factors mediating against excellence in dental esthetics, J Prosthet Dent 38:243-248, 1977. Myerson RL: The use of porcelain and plastic teeth in opposing com-plete dentures, J Prosthet Dent 7:625-633, 1957. Payne AGL: Factors influencing the position of artificial upper anterior teeth, J Prosthet Dent 26:26-32, 1971. Pound E: Applying harmony in selecting and arranging teeth, Dent Clin North Am 6:241-258, 1962. Pound E: Lost—fine arts in the fallacy of the ridges, J Prosthet Dent 4:6-16, 1954. Pound E: Recapturing esthetic tooth position in the edentulous patient, J Am Dent Assoc 55:181-191, 1957. Roraff AR: Instant photographs for developing esthetics, J Prosthet Dent 26:21-25, 1971. Smith BJ: Esthetic factors in removable partial prosthodontics, Dent Clin North Am 23:53-63, 1979. Sykora O: Fabrication of a posterior shade guide for removable partial dentures, J Prosthet Dent 50:287-288, 1983. Tillman EJ: Molding and staining acrylic resin anterior teeth, J Prosthet Dent 5:497-507, 1955; Dent Abstr 1:111, 1956. Van Victor A: The mold guide cast: its significance in denture esthetics, J Prosthet Dent 13:406-415, 1963. Van Victor A: Positive duplication of anterior teeth for immediate dentures, J Prosthet Dent 3:165-177, 1953. Vig RG: The denture look, J Prosthet Dent 11:9-15, 1961. Wallace DH: The use of gold occlusal surfaces in complete and partial dentures, J Prosthet Dent 14:326-333, 1964. Weiner S, Krause AS, Nicholas W: Esthetic modification of removable partial denture teeth with light-cured composites, J Prosthet Dent 57:381-384, 1987. Wolfson E: Staining and characterization of acrylic teeth, Dent Abstr 1:41, 1956. Young HA: Denture esthetics, J Prosthet Dent 6:748-755, 1956. Zarb GA, MacKay HF: Cosmetics and removable partial dentures: the Class IV partially edentulous patient, J Prosthet Dent 46:360-368, 1981. Diagnosis and Treatment Planning Academy of Prosthodontics: Principles, concepts and practices in prosthodontics, J Prosthet Dent 73:73-94, 1995. Applegate OC: Evaluating oral structures for removable partial den-tures, J Prosthet Dent 11:882-885, 1961. Bartels JC: Diagnosis and treatment planning, J Prosthet Dent 7:657-662, 1957. Beaumont AJ: An overview of esthetics with removable partial den-tures, Quintessence Int 33:745-755, 2002. Bezzon OL et al: Surveying removable partial dentures: the importance of guiding planes and path of insertion for stability, J Prosthet Dent 78:412-418, 1997. Blatterfein L, Kaufman EG: Prevention of problems with removable partial dentures: Council on Dental Materials, Instruments, and Equipment, J Am Dent Assoc 100:919-921, 1980. 358 Appendix B Selected Reading Resources Bolender CL, Swenson RD, Yamane C: Evaluation of treatment of inflammatory papillary hyperplasia of the palate, J Prosthet Dent 15:1013-1022, 1965. Budtz-Jorgensen E: Restoration of the partially edentulous mouth: a comparison of overdentures, removable partial dentures, fixed partial dentures and implant treatment, J Dent 24:237-244, July 1996. Casey DM, Lauciello FR: A review of the submerged-root concept, J Prosthet Dent 43:128-132, 1980. Contino RM, Stallard H: Instruments essential for obtaining data needed in making a functional diagnosis of the human mouth, J Prosthet Dent 7:66-77, 1957. Dreizen S: Nutritional changes in the oral cavity, J Prosthet Dent 16:1144-1150, 1966. Dummer PMH, Cidden J: The upper anterior sectional denture, J Pros-thet Dent 41:146-152, 1979. Dunn BW: Treatment planning for removable partial dentures, J Pros-thet Dent 11:247-255, 1961. Faine MP: Dietary factors related to preservation of oral and skeletal bone mass in women, J Prosthet Dent 73:65-72, 1995. Foster TD: The use of the face-bow in making permanent study casts, J Prosthet Dent 9:717-721, 1959. Frechette AR: Partial denture planning with special reference to stress distribution, J Prosthet Dent 1:700-707 (disc, 208-209), 1951. Friedman S: Effective use of diagnostic data, J Prosthet Dent 9:729-737, 1959. Garver DC et al: Vital root retention in humans: a preliminary report, J Prosthet Dent 40:23-28, 1978. Garver DC, Fenster RK: Vital root retention in humans: a final report, J Prosthet Dent 43:368-373, 1980. Guyer SE: Selectively retained vital roots for partial support of over-dentures: a patient report, J Prosthet Dent 33:258-263, 1975. Harvey WL: A transitional prosthetic appliance, J Prosthet Dent 14:60-70, 1964. Heintz WD: Treatment planning and design: prevention of errors of omission and commission, Dent Clin North Am 23:3-12, 1979. Henderson D, Hickey JC, Wehner PJ: Prevention and preservation: the challenge of removable partial denture service, Dent Clin North Am 9:459-473, 1965. House MM: The relationship of oral examination to dental diagnosis, J Prosthet Dent 8:208-219, 1958. Kabcenell JL: Planning for individualized prosthetic treatment, J Pros-thet Dent 34:405-407, 1975. Kaldahl WB, Becher CM: Prosthetic contingencies for future tooth loss, J Prosthet Dent 54:1-6, 1985. Kanno T, Carlsson GE: A review of the shortened dental arch concept focusing on the work by the Käyser/Nijmegen group, J Oral Rehabil 33:850-862, 2006. Kayser AF: Limited treatment goals: shortened dental arches, Periodon-tol 2000 4:7-14, 1994. Killebrew RF: Crown construction and splinting of mobile partial denture abutments, J Am Dent Assoc 70:334-338, 1965. Krikos AA: Preparing guide planes for removable partial dentures, J Prosthet Dent 34:152-155, 1975. Lambson GO: Papillary hyperplasia of the palate, J Prosthet Dent 16:636-645, 1966. Langer Y et al: Modalities of treatment for the combination syndrome, J Prosthodont 4:76-81, June 1995. Lopes I, Norlau LA: Specific mechanics for abutment uprighting, Aust Dent J 25:273-278, 1980. McCracken WL: Differential diagnosis: fixed or removable partial den-tures, J Am Dent Assoc 63:767-775, 1961. McGill WJ: Acquiring space for partial dentures, J Prosthet Dent 17:163-165, 1967. Miller EL: Critical factors in selecting removable prosthesis, J Prosthet Dent 34:486-490, 1975. Miller EL: Planning partial denture construction, Dent Clin North Am 17:571-584, 1973. Mopsik ER et al: Surgical intervention to reestablish adequate inter-maxillary space before fixed or removable prosthodontics, J Am Dent Assoc 95:957-960, 1977. Moulton GH: The importance of centric occlusion in diagnosis and treatment planning, J Prosthet Dent 10:921-926, 1960. Nassif J, Blumenfeld WL: Joint consultation services by the periodon-tist and prosthodontist, J Prosthet Dent 29:55-60, 1973. Nassif J, Blumenfeld WL, Tarsitano JT: Dialogue—a treatment modal-ity, J Prosthet Dent 33:696-700, 1975. Payne SH: Diagnostic factors which influence the choice of posterior occlusion, Dent Clin North Am 1:203-213, 1957. Rudd KD, Dunn BW: Accurate removable partial dentures, J Prosthet Dent 18:559-570, 1967. Saunders TR, Gillis RE, Desjardins RP: The maxillary complete denture opposing the mandibular bilateral distal-extension partial denture: treatment considerations, J Prosthet Dent 41:124-128, 1979. Sauser CW: Pretreatment evaluation of partially edentulous arches, J Prosthet Dent 11:886-893, 1961. Seiden A: Occlusal rests and rest seats, J Prosthet Dent 8:431-440, 1958. Silverman SI: Differential diagnosis: fixed or removable prosthesis, Dent Clin North Am 31:347-362, 1987. Swoope CC, Frank RP: Removable partial dentures indications and planning. In Clark JE, editor: Clinical dentistry, vol 5, New York, 1976, Harper & Row. Turner CE, Shaffer FW: Planning the treatment of the complex prosth-odontic case, J Am Dent Assoc 97:992-993, 1978. Uccellani EL: Evaluating the mucous membranes of the edentulous mouth, J Prosthet Dent 15:295-303, 1965. Vahidi F: The provisional restoration, Dent Clin North Am 31:363-381, 1987. Wagner AG: Instructions for the use and care of removable partial dentures, J Prosthet Dent 26:481-490, 1971. Waldron CA: Oral leukoplakia, carcinoma, and the prosthodontist, J Prosthet Dent 15:367-376, 1965. Welker WA, Kramer DC: Claspless chrome-cobalt transitional remov-able partial dentures, J Am Dent Assoc 96:814-818, 1978. Wöstmann B, Budtz-Jørgensen E, Jepson N, Mushimoto E, Palmqvist S, Sofou A, Owall B: Indications for removable partial dentures: a literature review, Int J Prosthodont 18:139-145, 2005. Wright P, Hellyer PH: Gingival recession related to removable partial dentures in older patients, J Prosthet Dent 74:602-607, 1995. Young HA: Diagnostic survey of edentulous patients, J Prosthet Dent 5:5-14, 1955. Implants and RPDs Grossmann Y, Levin L, Sadan A: A retrospective case series of implants used to restore partially edentulous patients with implant-supported removable partial dentures: 31-month mean follow-up results, Quintessence Int 39:665-671, 2008. Grossmann Y, Nissan J, Levin L: Clinical effectiveness of implant-supported removable partial dentures: a review of the literature and retrospective case evaluation, J Oral Maxillofac Surg 67:1941-1946, 2009. Kaufmann R, Friedli M, Hug S, Mericske-Stern R: Removable dentures with implant support in strategic positions followed for up to 8 years, Int J Prosthodont 22:233-241, 2009. Mijiritsky E: Implants in conjunction with removable partial dentures: a literature review, Implant Dent 16:146-154, 2007. Strassburger C, Kerschbaum T, Heydecke G: Influence of implant and conventional prostheses on satisfaction and quality of life: a litera-ture review. Part 2. Qualitative analysis and evaluation of the studies, Int J Prosthodont 19:339-348, 2006. 359 Appendix B Selected Reading Resources Impression Materials and Methods: The Partial Denture Base Akerly WB: A combination impression and occlusal registration tech-nique for extension-base removable partial dentures, J Prosthet Dent 39:226-229, 1978. Appleby DC et al: The combined reversible hydrocolloid/ irreversible hydrocolloid impression system: clinical application, J Prosthet Dent 46:48-58, 1981. Applegate OC: An evaluation of the support for the removable partial denture, J Prosthet Dent 10:112-123, 1960. Applegate OC: The partial denture base, J Prosthet Dent 5:636-648, 1955. Bailey LR: Rubber base impression techniques, Dent Clin North Am 1:156-166, 1957. Bauman R, DeBoer J: A modification of the altered cast technique, J Prosthet Dent 47:212-213, 1982. Beaumont AJ: Sectional impression for maxillary Class I removable partial dentures and maxillary immediate dentures, J Prosthet Dent 49:438-441, 1983. Berkey D, Berg R: Geriatric oral health issues in the United States, Int Dent J 51:254-264, 2001. Beyerle MP et al: Immersion disinfection of irreversible hydrocolloid impressions. Part I. Microbiology, Int J Prosthodont 7:234-238, May 1994. Birnbach S: Impression technique for maxillary removable partial den-tures, J Prosthet Dent 51:286, 1984. Blatterfein L, Klein IE, Miglino JC: A loading impression technique for semiprecision and precision removable partial dentures, J Prosthet Dent 43:9-14, 1980. Boretti G, Bickel M, Geering AH: A review of masticatory ability and efficiency, J Prosthet Dent 74:400-403, 1995. Carlsson GE: Masticatory efficiency: the effect of age, the loss of teeth and prosthetic rehabilitation. Int Dent J 34:93-97, 1984. Chaffee NR et al: Dimensional accuracy of improved dental stone and epoxy resin die materials. Part I. Single die, J Prosthet Dent 77:131-135, 1997. Chaffee NR et al: Dimensional accuracy of improved dental stone and epoxy resin die materials. Part II. Complete arch form, J Prosthet Dent 77:235-238, 1997. Chai J et al: Clinically relevant mechanical properties of elastomeric impression materials, Int J Prosthodont 11:219-223, 1998. Chase WW: Adaptation of rubber-base impression materials to removable denture prosthetics, J Prosthet Dent 10:1043-1050, 1960. Chau VB et al: In-depth disinfection of acrylic resin, J Prosthet Dent 74:309-313, 1995. Chen MS et al: An altered-cast impression technique that eliminates conventional cast dissecting and impression boxing, J Prosthet Dent 57:471-474, 1987. Cho GC et al: Tensile bond strength of polyvinyl siloxane impressions bonded to a custom tray as a function of drying time. Part I, J Pros-thet Dent 73:419-423, 1995. Chong MP et al: The tear test as a means of evaluating the resistance to rupture of alginate impression materials, Aust Dent J 16:145-151, 1971. Clark RJ, Phillips RW: Flow studies of certain dental impression mate-rials, J Prosthet Dent 7:259-266, 1957. Cohen BI et al: Dimensional accuracy of three different alginate impres-sion materials, J Prosthodont 4:195-199, 1995. Corso M et al: The effect of temperature changes on the dimensional stability of polyvinyl siloxane and polyether impression materials, J Prosthet Dent 79:626-631, 1998. Cserna A et al: Irreversible hydrocolloids: a comparison of antimicro-bial efficacy, J Prosthet Dent 71:387-389, 1994. Davidson CL, Boere G: Liquid-supported dentures. Part I. Theoretical and technical considerations, J Prosthet Dent 68:303-306, 1990. Davidson CL, Boere G: Liquid-supported dentures. Part II. Clinical study: a preliminary report, J Prosthet Dent 68:434-436, 1990. Davis BA et al: Effect of immersion disinfection on properties of impression materials, J Prosthodont 3:31-34, 1994. DeFreitas JF: Potential toxicants in alginate powders, Aust Dent J 25:224-228, 1980. Dixon DL, Breeding LC, Ekstrand KG: Linear dimensional variability of three denture base resins after processing and in water storage, J Prosthet Dent 68:196-200, 1992. Dixon DL, Ekstrand KG, Breeding LC: The transverse strengths of three denture base resins, J Prosthet Dent 66:510-513, 1991. Dootz ER: Fabricating non-precious metal bases, Dent Clin North Am 24:113-122, 1980. Dootz ER, Craig RG: Comparison of the physical properties of eleven soft denture liners, J Prosthet Dent 67:707-712, 1992. Douglas CW, Shih A, Ostry L: Will there be a need for complete den-tures in the United States in 2020? J Prosthet Dent 87:5-8, 2002. Douglas CW, Watson AJ: Future needs for fixed and removable partial dentures in the United States, J Prosthet Dent 87:9-14, 2002. Drennon DG, Johnson GH: The effect of immersion disinfection of elastomeric impressions on the surface detail reproduction of improved gypsum casts, J Prosthet Dent 63:233-241, 1990. Fitzloff RA: Functional impressions with thermoplastic materials for reline procedures, J Prosthet Dent 52:25-27, 1984. Frank RP: Analysis of pressures produced during maxillary edentulous impression procedures, J Prosthet Dent 22:400-403, 1969. Fusayama T, Nakazato M: The design of stock trays and the retention of irreversible hydrocolloid impressions, J Prosthet Dent 21:136-142, 1969. Gelbard S et al: Effect of impression materials and techniques on the marginal fit of metal castings, J Prosthet Dent 71:1-6, 1994. Gilmore WH, Schnell RJ, Phillips RW: Factors influencing the accuracy of silicone impression materials, J Prosthet Dent 9:304-314, 1959. Hans S, Gunne J: Masticatory efficiency and dental state: a comparison between two methods, Acta Odont Scand 43:139-146, 1985. Harris WT Jr: Water temperature and accuracy of alginate impressions, J Prosthet Dent 21:613-617, 1969. Harrison JD: Prevention of failures in making impressions and dies, Dent Clin North Am 23:13-20, 1979. Heartwell CM Jr et al: Comparison of impressions made in perforated and nonperforated rimlock trays, J Prosthet Dent 27:494-500, 1972. Helkimo E, Carlsson GE, Helkimo M: Chewing efficiency and state of the dentition, Acta Odont Scand 36:33-41, 1978. Herfort TW et al: Viscosity of elastomeric impression materials, J Pros-thet Dent 38:396-404, 1977. Hesby RM et al: Effects of radiofrequency glow discharge on impres-sion material surface wettability, J Prosthet Dent 77:414-422, 1997. Holmes JB: Influence of impression procedures and occlusal loading on partial denture movement, J Prosthet Dent 15:474-481, 1965. Hondrum SO et al: Effects of long-term storage on properties of an alginate impression material, J Prosthet Dent 77:601-606, 1997. Hudson WC: Clinical uses of rubber impression materials and electro-forming of casts and dies in pure silver, J Prosthet Dent 8:107-114, 1958. Huggett R et al: Dimensional accuracy and stability of acrylic resin denture bases, J Prosthet Dent 68:634-640, 1992. Iglesias A et al: Accuracy of wax, autopolymerized, and light-polymer-ized resin pattern materials, J Prosthodont 5:193-200, 1996. Ivanovski S et al: Disinfection of dental stone casts: antimicrobial effects and physical property alterations, Dent Mater 11:19-23, 1995. James JS: A simplified alternative to the altered-cast impression tech-nique for removable partial dentures, J Prosthet Dent 53:598, 1985. Jasim FA, Brudvik JS, Nicholls JI: Impression distortion from abutment tooth inclination in removable partial dentures, J Prosthet Dent 54:532-538, 1985. 360 Appendix B Selected Reading Resources Johnson GH et al: Dimensional stability and detail reproduction of irreversible hydrocolloid and elastomeric impressions disinfected by immersion, J Prosthet Dent 79:446-453, 1998. Johnston JF, Cunningham DM, Bogan RG: The dentist, the patient, and ridge preservation, J Prosthet Dent 10:288-295, 1960. Jones RH et al: Effect of provisional luting agents on polyvinyl siloxane impression materials, J Prosthet Dent 75:360-363, 1996. Kawamura Y: Recent concepts of the physiology of mastication. Adv Oral Biol 1:77-109, 1964. Kawano F et al: Comparison of bond strength of six soft denture liners to denture base resin, J Prosthet Dent 68:368-371, 1992. Koran A III: Impression materials for recording the denture bearing mucosa, Dent Clin North Am 24:97-111, 1980. Kramer HM: Impression technique for removable partial dentures, J Prosthet Dent 11:84-92, 1961. Landesman HM, Wright WE: A technique for making impressions on patients requiring complete and removable partial dentures, CDA J 14:20-24, 1986. Langenwalter EM, Aquilino SA: The dimensional stability of elasto-meric impression materials following disinfection, J Prosthet Dent 63:270-276, 1990. Leach CD, Donovan TE: Impression technique for maxillary removable partial dentures, J Prosthet Dent 50:283-286, 1983. Leake JL, Hawkins R, Locker D: Social and functional impact of reduced posterior dental units in older adults, J Oral Rehab 21:1-10, 1994. Lee IK et al: Evaluation of factors affecting the accuracy of impressions using quantitative surface analysis, Oper Dent 20:246-252, 1995. Lee RE: Mucostatics, Dent Clin North Am 24:81-96, 1980. Lepe X et al: Accuracy of polyether and addition silicone after long-term immersion disinfection, J Prosthet Dent 78:245-249, 1997. Leupold RJ: A comparative study of impression procedures for distal extension removable partial dentures, J Prosthet Dent 16:708-720, 1966. Leupold RJ, Flinton RJ, Pfeifer DL: Comparison of vertical movement occurring during loading of distal-extension removable partial denture bases made by three impression techniques, J Prosthet Dent 68:290-293, 1992. Leupold RJ, Kratochvil FJ: An altered-cast procedure to improve support for removable partial dentures, J Prosthet Dent 15:672-678, 1965. Liedberg B, Spiechowicz E, Owall B: Mastication with and without removable partial dentures: an intraindividual study, Dysphagia 10:107-112, 1995. Loh PL et al: An evaluation of microwave-polymerized resin bases for removable partial dentures, J Prosthet Dent 79:389-392, 1998. Lucas W, Luke H: The processes of selection and breakage in mastica-tion, Arch Oral Biol 28:813-818, 1983. Lund PS, Aquilino SA: Prefabricated custom impression trays for the altered cast technique, J Prosthet Dent 66:782-783, 1991. Manly RS, Vinton P: A survey of the chewing ability of denture wearers, J Dent Res 30:314-321, 1951. Matis BA et al: The effect of the use of dental gloves on mixing vinyl polysiloxane putties, J Prosthodont 6:189-192, 1997. Millar BJ et al: The effect of surface wetting agent on void formation in impressions, J Prosthet Dent 77:54-56, 1997. Millar BJ et al: In vitro study of the number of surface defects in mono-phase and two-phase addition silicone impressions, J Prosthet Dent 80:32-35, 1998. Mitchell JV, Damele JJ: Influence of tray design upon elastic impression materials, J Prosthet Dent 23:51-57, 1970. Mitchener RW, Omori MD: Putty materials for stable removable partial denture bases, J Prosthet Dent 53:435-436, 1985. Morrow RM et al: Compatibility of alginate impression materials and dental stones, J Prosthet Dent 25:556-566, 1971. Myers GE: Electroformed die technique for rubber base impressions, J Prosthet Dent 8:531-535, 1958. Myers GE, Wepfer GG, Peyton FA: The Thiokol rubber base impres-sion materials, J Prosthet Dent 8:330-339, 1958. Nishigawa G et al: Efficacy of tray adhesives for the adhesion of elasto-mer rubber impression materials to impression modeling plastics for border molding, J Prosthet Dent 79:140-144, 1998. O’Brien WJ: Base retention, Dent Clin North Am 24:123-130, 1980. Olin PS et al: The effects of sterilization on addition silicone impres-sions in custom and stock metal trays, J Prosthet Dent 71:625-630, 1994. Oosterhaven SP et al: Social and psychological implications of missing teeth for chewing ability, Comm Dent Oral Epid 16:79-82, 1988. Parker MH et al: Comparison of occlusal contacts in maximum inter-cuspation for two impression techniques, J Prosthet Dent 78:255-259, 1997. Pfeiffer KA: Clinical problems in the use of alginate hydrocolloid, Dent Abstr 2:82, 1957. Phillips RW: Factors affecting the surface of stone dies poured in hydrocolloid impressions, J Prosthet Dent 2:390-400, 1952. Phillips RW: Factors influencing the accuracy of reversible hydrocol-loid impressions, J Am Dent Assoc 43:1-17, 1951. Phillips RW: Physical properties and manipulation of rubber impres-sion materials, J Am Dent Assoc 59:454-458, 1959. Pratten DH, Covey DA, Sheats RD: Effect of disinfectant solutions on the wettability of elastomeric impression materials, J Prosthet Dent 63:223-227, 1990. Prieskel HW: Impression techniques for attachment retained distal extension removable partial dentures, J Prosthet Dent 25:620-628, 1971. Prinz JF, Lucas PW: Swallow thresholds in human mastication, Arch Oral Biol 40:401-403, 1995. Rapuano JA: Single tray dual-impression technique for distal extension partial dentures, J Prosthet Dent 24:41-46, 1970. Redford M et al: Denture use and the technical quality of dental pros-theses among persons 18-74 years of age: United States, 1988-1991, J Dent Res 75:714-725, 1996. Render PJ: An impression technique to make a new master cast for an existing removable partial denture, J Prosthet Dent 67:488-490, 1992. Rudd KD et al: Comparison of effects of tap water and slurry water on gypsum casts, J Prosthet Dent 24:563-570, 1970. Rudd KD, Morrow RM, Bange AA: Accurate casts, J Prosthet Dent 21:545-554, 1969. Rudd KD, Morrow RM, Strunk RR: Accurate alginate impressions, J Prosthet Dent 22:294-300, 1969. Samadzadeh A et al: Fracture strength of provisional restorations rein-forced with plasma-treated woven polyethylene fiber, J Prosthet Dent 5:447-450, 1997. Scott GK et al: Check bite impressions using irreversible alginate/ reversible hydrocolloid combination, J Prosthet Dent 77:83-85, 1997. Sherfudhin H et al: Preparation of void-free casts from vinyl polysilox-ane impressions, J Dent 24:95-98, 1996. Silver M: Impressions and silver-plated dies from a rubber impression material, J Prosthet Dent 6:543-549, 1956. Smith RA: Secondary palatal impressions for major connector adapta-tion, J Prosthet Dent 24:108-110, 1970. Steffel VL: Relining removable partial dentures for fit and function, J Prosthet Dent 4:496-509, 1954; J Tenn Dent Assoc 36:35-43, 1956. Taylor TD, Morton TJ Jr: Ulcerative lesions of the palate associated with removable partial denture castings, J Prosthet Dent 66:213-221, 1991. Thompson GA et al: Effects of disinfection of custom tray materials on adhesive properties of several impression material systems, J Pros-thet Dent 72:651-656, 1994. Tjan AH et al: Marginal fidelity of crowns fabricated from six propri-etary provisional materials, J Prosthet Dent 77:482-485, 1997. 361 Appendix B Selected Reading Resources Vahidi F: Vertical displacement of distal-extension ridges by different impression techniques, J Prosthet Dent 40:374-377, 1978. van Waas M, et al: Relationship between wearing a removable partial denture and satisfaction in the elderly, Comm Dent Oral Epid 22:315-318, 1994. Vandewalle KS et al: Immersion disinfection of irreversible hydrocol-loid impressions with sodium hypochlorite. Part II. Effect on gypsum, Int J Prosthodont 7:315-322, 1994. Verran J et al: Microbiological study of selected risk areas in dental technology laboratories, J Dent 24:77-80, 1996. Wang HY et al: Vertical distortion on distal extension ridges and palatal area of casts made by different techniques, J Prosthet Dent 75:302-308, 1996. Wang RR, Nguyen T, Boyle AM: The effect of tray material and surface condition on the shear bond strength of impression materials, J Prosthet Dent 74:449-454, 1995. Wilson JH: Partial dentures: relining the saddle supported by the mucosa and alveolar bone, J Prosthet Dent 3:807-813, 1953. Young JM: Surface characteristics of dental stone: impression orienta-tion, J Prosthet Dent 33:336-341, 1975. Yurkstas AA: The masticatory act, J Prosthet Dent 15:248-260, 1965. Zinner ID: Impression procedures for the removable component of a combination fixed and removable prosthesis, Dent Clin North Am 31:417-440, 1987. Maxillofacial Prosthesis Ackerman AJ: Maxillofacial prosthesis, Oral Surg 6:176-200, 1953. Ackerman AJ: The prosthetic management of oral and facial defects following cancer surgery, J Prosthet Dent 5:413-432, 1955. Adams D: A cantilevered swinglock removable partial denture design for the treatment of the partial mandibulectomy patient, J Oral Rehabil 12:113-118, 1985. Brown KE: Fabrication of a hollow-bulb obturator, J Prosthet Dent 21:97-103, 1969. Brown KE: Reconstruction considerations for severe dental attrition, J Prosthet Dent 44:384-388, 1980. Cantor R et al: Methods for evaluating prosthetic facial materials, J Prosthet Dent 21:324-332, 1969. Curtis TA, Cantor R: The forgotten patient in maxillofacial prosthetics, J Prosthet Dent 31:662-680, 1974. Desjardins RP: Prosthodontic management of the cleft palate patient, J Prosthet Dent 33:655-665, 1975. Firtell DN, Curtis TA: Removable partial denture design for the man-dibular resection patient, J Prosthet Dent 48:437-443, 1982. Firtell DN, Grisius RJ: Retention of obturator: removable partial den-tures: a comparison of buccal and lingual retention, J Prosthet Dent 43:211-217, 1980. Gay WD, King GE: Applying basic prosthodontic principles in the dentulous maxillectomy patient, J Prosthet Dent 43:433-435, 1980. Goll G: Design for maximal retention of obturator prosthesis for hemi-maxillectomy patients (letter), J Prosthet Dent 48:108-109, 1982. Immekus JE, Aramy M: Adverse effects of resilient denture liners in overlay dentures, J Prosthet Dent 32:178-181, 1974. Kelley EK: Partial denture design applicable to the maxillofacial patient, J Prosthet Dent 15:168-173, 1965. King GE, Martin JW: Cast circumferential and wire clasps for obturator retention, J Prosthet Dent 49:799-802, 1983. Metz HH: Mandibular staple implant for an atrophic mandibular ridge: solving retention difficulties of a denture, J Prosthet Dent 32:572-578, 1974. Monteith GG: The partially edentulous patient with special problems, Dent Clin North Am 23:107-115, 1979. Moore DJ: Cervical esophagus prosthesis, J Prosthet Dent 30:442-445, 1973. Myers RE, Mitchell DL: A photoelastic study of stress induced by framework design in a maxillary resection, J Prosthet Dent 61:590-594, 1989. Nethery WJ, Delclos L: Prosthetic stent for gold-grain implant to the floor of the mouth, J Prosthet Dent 23:81-87, 1970. Shifman A, Lepley JB: Prosthodontic management of postsurgical soft tissue deformities associated with marginal mandibulectomy. Part I. Loss of the vestibule, J Prosthet Dent 48:178-183, 1982. Smith EH Jr: Prosthetic treatment of maxillofacial injuries, J Prosthet Dent 5:112-128, 1955. Strain JC: A mechanical device for duplicating a mirror image of a cast or moulage in three dimensions, J Prosthet Dent 5:129-132, 1955. Toremalm NG: A disposable obturator for maxillary defects, J Prosthet Dent 29:94-96, 1973. Weintraub GS, Yalisove IL: Prosthodontic therapy for cleidocranial dysostosis: report of case, J Am Dent Assoc 96:301-305, 1978. Wright SM, Pullen-Warner EA, LeTissier DR: Design for maximal retention of obturator prosthesis for hemimaxillectomy patients, J Prosthet Dent 47:88-91, 1982. Young JM: The prosthodontist’s role in total treatment of patients, J Prosthet Dent 27:399-412, 1972. Miscellaneous Abere DJ: Post-placement care of complete and removable partial den-tures, Dent Clin North Am 23:143-151, 1979. Academy of Denture Prosthetics: Principles, concepts and practices in prosthodontics, J Prosthet Dent 61:88-109, 1989. Adisman IK: What a prosthodontist should know, J Prosthet Dent 21:409-416, 1969. American Association of Dental Schools: Curricular guidelines for removable prosthodontics, J Dent Educ 44:343-346, 1980. Applegate OC: Conditions which may influence the choice of partial or complete denture service, J Prosthet Dent 7:182-196, 1957. Applegate OC: Factors to be considered in choosing an alloy, Dent Clin North Am 4:583-590, 1960. Asgar K, Techow BO, Jacobson JM: A new alloy for partial dentures, J Prosthet Dent 23:36-43, 1970. Atwood DA: Practice of prosthodontics: past, present, and future, J Prosthet Dent 21:393-401, 1970. Augsburger RH: Evaluating removable partial dentures by mathemati-cal equations, J Prosthet Dent 22:528-543, 1969. Backenstose WM, Wells JG: Side effects of immersion-type cleansers on the metal components of dentures, J Prosthet Dent 37:615-621, 1977. Baker CR: Difficulties in evaluating removable partial dentures, J Pros-thet Dent 17:60-62, 1967. Baker CR: Occlusal reactive prosthodontics, J Prosthet Dent 17:566-569, 1967. Barrett DA, Pilling LO: The restoration of carious clasp-bearing teeth, J Prosthet Dent 15:309-311, 1965. Bates JF: Studies related to fracture of partial dentures, Br Dent J 120:79-83, 1966. Beck HO: Alloys for removable partial dentures, Dent Clin North Am 4:591-596, 1960. Beck HO: A clinical evaluation of the arcon concept of articulation, J Prosthet Dent 9:409-421, 1959. Beck HO, Morrison WE: Investigation of an arcon articulator, J Pros-thet Dent 6:359-372, 1956. Becker CM, Bolender CL: Designing swinglock partial dentures, J Pros-thet Dent 46:126-132, 1981. Bergman B, Hugoson A, Olsson CO: Caries, periodontal and prosthetic findings in patients with removable partial dentures: a ten-year lon-gitudinal study, J Prosthet Dent 48:506-514, 1982. Blanco-Dalmau L: The nickel problem, J Prosthet Dent 48:99-101, 1982. 362 Appendix B Selected Reading Resources Blatterfein L, Pearce RL, Jackota JT: Minimum acceptable procedures for satisfactory removable partial denture service, J Prosthet Dent 27:84-87, 1972. Bolender CL, Becker CM: Swinglock removable partial dentures: where and when, J Prosthet Dent 45:4-10, 1981. Boucher CO: Writing as a means for learning, J Prosthet Dent 27:229-234, 1972. Budtz-Jorgensen E, Isidor F: Cantilever bridges or removable partial dentures in geriatric patients: a two-year study, J Oral Rehabil 14:239-249, 1987. Cavalaris CJ: Pathologic considerations associated with partial den-tures, Dent Clin North Am 17:585-600, 1973. Chandler JA, Brudvik JS: Clinical evaluation of patients eight to nine years after placement of removable partial dentures, J Prosthet Dent 51:736-743, 1984. Chen MS et al: Simplicity in interim tooth-supported removable partial denture construction, J Prosthet Dent 54:740-744, 1985. Cotmore JM et al: Removable partial denture survey: clinical practice today, J Prosthet Dent 49:321-327, 1983. Coy RE, Arnold PD: Survey and design of diagnostic casts for removable partial dentures, J Prosthet Dent 32:103-106, 1974. Cunningham DM: Comparison of base metal alloys and type IV gold alloys for removable partial denture frameworks, Dent Clin North Am 17:719-722, 1973. Diaz-Arnold AM, Langenwalter EM, Hatch LK: Cast restorations made to existing removable partial dentures, J Prosthet Dent 61:414-417, 1989. Dukes BS, Fields H Jr: Comparison of disclosing media used for adjust-ment of removable partial denture frameworks, J Prosthet Dent 45:380-382, 1981. Elliott RW: The effects of heat on gold partial denture castings, J Pros-thet Dent 13:688-698, 1963. Ettinger RL: The acrylic removable partial denture, J Am Dent Assoc 95:945-949, 1977. Ettinger RL, Beck JD, Jakobsen J: Removable prosthodontic treatment needs: a survey, J Prosthet Dent 51:419-427, 1984. Ewing JE: The construction of accurate full crown restorations for an existing clasp by using a direct metal pattern technique, J Prosthet Dent 15:889-899, 1965. Farah JW, MacGregor AR, Miller TPG: Stress analysis of disjunct removable partial dentures, J Prosthet Dent 42:271-275, 1979. Federation of Prosthodontic Organizations: Guidelines for evaluation of completed prosthodontic treatment for removable partial den-tures, J Prosthet Dent 27:326-328, 1972. Fenton AH, Zarb GA, MacKay HF: Overdenture oversights, Dent Clin North Am 23:117-130, 1979. Fields H, Campfield RW: Removable partial prosthesis partially sup-ported by an endosseous blade implant, J Prosthet Dent 31:273-278, 1974. Firtell DN, Kouyoumdjian JH, Holmes JB: Attitudes toward abutment preparation for removable partial dentures, J Prosthet Dent 55:131-133, 1986. Fish SF: Partial dentures, Br Dent J 128:243-246, 289-293, 339-344, 398-402, 446-453, 495-502, 547-551, 590-592, 1970. Fisher R: Relation of removable partial denture base stability to sex, age, and other factors, J Dent Res (IADR abstract 613) 59:entire issue, 1980. Frank RP: Evaluating refractory cast wax-ups for removable partial dentures, J Prosthet Dent 35:388-392, 1976. Girardot RL: The physiologic aspects of partial denture restorations, J Prosthet Dent 3:689-698, 1953. Gordon SR: Measurement of oral status and treatment need among subjects with dental prostheses: are the measures less reliable than the prostheses? Part I. Oral status in removable prosthodontics, J Prosthet Dent 65:664-668, 1991. Harrison WM, Stansbury BE: The effect of joint surface contours on the transverse strength of repaired acrylic resin, J Prosthet Dent 23:464-472, 1970. Heintz WD: Principles, planning, and practice for prevention, Dent Clin North Am 17:705-718, 1973. Helel KS, Graser GN, Featherstone JD: Abrasion of enamel and com-posite resin by removable partial denture clasps, J Prosthet Dent 52:389-397, 1984. Henderson CW et al: Evaluation of the barrier system: an infection control system for the dental laboratory, J Prosthet Dent 58:517-521, 1987. Izikowitz L: A long-term prognosis for the free-end saddle-bridge, J Oral Rehabil 12:247-262, 1985. Jankelson BH: Adjustment of dentures at time of insertion and altera-tions to compensate for tissue changes, J Am Dent Assoc 64:521-531, 1962. Jones RR: The lower partial denture, J Prosthet Dent 2:219-229, 1952. Kaaber S: Twelve year changes in mandibular bone level in free end saddle denture wearers, J Dent Res (IADR abstract 1367) 60:entire issue, 1981. Kaires AK: A study of partial denture design and masticatory pressures in a mandibular bilateral distal extension case, J Prosthet Dent 8:340-350, 1958. Kelly E: Changes caused by a mandibular removable partial denture opposing a maxillary complete denture, J Prosthet Dent 27:140-150, 1972. Kelly E: Fatigue failure in denture base polymers, J Prosthet Dent 21:257-266, 1969. Kelly EK: The physiologic approach to partial denture design, J Pros-thet Dent 3:699-710, 1953. Kessler B: An analysis of the tongue factor and its functioning areas in dental prosthesis, J Prosthet Dent 5:629-635, 1955. Klein IE, Blatterfein L, Kaufman EG: Minimum clinical procedures for satisfactory complete denture, removable partial denture, and fixed partial denture services, J Prosthet Dent 22:4-10, 1969. Kratochvil FJ: Maintaining supporting structures with a removable partial prosthesis, J Prosthet Dent 25:167-174, 1971. Kratochvil FJ, Caputo AA: Photoelastic analysis of pressure on teeth and bone supporting removable partial dentures, J Prosthet Dent 32:52-61, 1974. Kratochvil FJ, Davidson PN, Guijt J: Five-year survey of treatment with removable partial dentures. Part I, J Prosthet Dent 48:237-244, 1982. Landa JS: The troublesome transition from a partial lower to a com-plete lower denture, J Prosthet Dent 4:42-51, 1954. Lanser A: Tooth-supported telescope restorations, J Prosthet Dent 45:515-520, 1981. Lechner SK: A longitudinal survey of removable partial dentures. I. Patient assessment of dentures, Aust Dent J 30:104-111, 1985. Lechner SK: A longitudinal survey of removable partial dentures. II. Clinical evaluation of dentures, Aust Dent J 30:194-197, 1985. Lechner SK: A longitudinal survey of removable partial dentures. III. Tissue reactions to various denture components, Aust Dent J 30:291-295, 1985. Lee MW et al: O-ring coping attachments for removable partial den-tures, J Prosthet Dent 74:235-241, 1995. Lewis AJ: Failure of removable partial denture castings during service, J Prosthet Dent 39:147-149, 1978. Lewis AJ: Radiographic evaluation of porosities in removable partial denture castings, J Prosthet Dent 39:278-281, 1978. Lopuck SE, Reitz PV, Altadonna J: Hinge for a unilateral maxillary arch prosthesis, J Prosthet Dent 45:446-448, 1981. Lorton L: A method of stabilizing removable partial denture castings during clinical laboratory procedures, J Prosthet Dent 39:344-345, 1978. MacEntee MI, Hawbolt EB, Zahel JI: The tensile and shear strength of a base metal weld joint used in dentistry, J Dent Res 60:154-158, 1981. 363 Appendix B Selected Reading Resources Maetani T et al: Effect of TFE coating on plaque accumulation on dental castings, J Dent Res (IADR abstract 1359) 60:entire issue, 1981. Maison WG: Instructions to denture patients, J Prosthet Dent 9:825-831, 1959. Makrauer FL, Davis JS: Gastroscopic removal of a partial denture, J Am Dent Assoc 94:904-906, 1977. Marcus SE et al: The retention and tooth loss in the permanent denti-tion of adults: United States, 1988-1991, J Dent Res 75:684-695, 1996. Martone AL: The challenge of the partially edentulous mouth, J Pros-thet Dent 8:942-954, 1958. Martone AL: The fallacy of saving time at the chair, J Prosthet Dent 7:416-419, 1957. Massler M: Geriatric nutrition: the role of taste and smell in appetite, J Prosthet Dent 32:247-250, 1980. McCracken WL: Auxiliary uses of cold-curing acrylic resins in pros-thetic dentistry, J Am Dent Assoc 47:298-304, 1953. McCracken WL: A comparison of tooth-borne and tooth-tissue-borne removable partial dentures, J Prosthet Dent 3:375-381, 1953. McCracken WL: A philosophy of partial denture treatment, J Prosthet Dent 13:889-900, 1963. Means CR, Flenniken IE: Gagging: a problem in prosthetic dentistry, J Prosthet Dent 23:614-620, 1970. Mehringer EJ: The saliva as it is related to the wearing of dentures, J Prosthet Dent 4:312-318, 1954. Michell DL, Wilke ND: Articulators through the years. I. Up to 1940, J Prosthet Dent 39:330-338, 1978; II, from 1940, 39:451-458, 1978. Miller EL: Clinical management of denture-induced inflammations, J Prosthet Dent 38:362-365, 1977. Mohamed SE, Schmidt JR, Harrison JD: Articulators in dental educa-tion and practice, J Prosthet Dent 36:319-325, 1976. Morris HF, Asgar K: Physical properties and microstructure of four new commercial partial denture alloys, J Prosthet Dent 33:36-46, 1975. Neufeld JO: Changes in the trabecular pattern of the mandible follow-ing the loss of teeth, J Prosthet Dent 8:685-697, 1958. Oatlund SG: Saliva and denture retention, J Prosthet Dent 10:658-663, 1960. Ogle RE, Sorensen SE, Lewis EA: A new visible light-cured resin system applied to removable prosthodontics, J Prosthet Dent 56:497-506, 1986. Osborne J, Lammie GA: The bilateral free-end saddle lower denture, J Prosthet Dent 4:640-652, 1954. Overton RG, Bramblett RM: Prosthodontic services: a study of need and availability in the United States, J Prosthet Dent 27:329-339, 1972. Pascoe DF, Wimmer J: A radiographic technique for the detection of internal defects in dental castings, J Prosthet Dent 39:150-157, 1978. Phillips RW, Leonard LJ: A study of enamel abrasion as related to partial denture clasps, J Prosthet Dent 6:657-671, 1956. Plainfield S: Communication distortion: the language of patients and practitioners of dentistry, J Prosthet Dent 22:11-19, 1969. Prieskel HW: The distal extension prosthesis reappraised, J Dent 5:217-230, 1977. Ramsey WO: The relation of emotional factors to prosthodontic service, J Prosthet Dent 23:4-10, 1970. Raybin NH: The polished surface of complete dentures, J Prosthet Dent 13:236-239, 1963. Removable prosthodontics, Dent Clin North Am 28:entire issue, 1984. Renggli HH, Allet B, Spanauf AJ: Splinting of teeth with fixed bridges: biological effect, J Oral Rehabil 11:535-537, 1984. Reynolds JM: Crown construction for abutments of existing removable partial dentures, J Am Dent Assoc 69:423-426, 1964. Rissen L et al: Effect of fixed and removable partial dentures on the alveolar bone of abutment teeth, J Dent Res (IADR abstract 1368) 60:entire issue, 1981. Rissen L et al: Six-year report of the periodontal health of fixed and removable partial denture abutment teeth, J Prosthet Dent 54:461-467, 1985. Rothman R: Phonetic considerations in denture prosthesis, J Prosthet Dent 11:214-223, 1961. Rudd KD, Dunn BW: Accurate removable partial dentures, J Prosthet Dent 18:559-570, 1967. Rushford CB: A technique for precision removable partial denture construction, J Prosthet Dent 31:377-383, 1974. Ruyter IE, Svendsen SA: Flexural properties of denture base polymers, J Prosthet Dent 43:95-104, 1980. Sadig W, Fahmi F: The modified swing-lock: a new approach, J Pros-thet Dent 74:428-431, 1995. Savage RD, MacGregor AR: Behavior therapy in prosthodontics, J Prosthet Dent 24:126-132, 1970. Schabel RW: Dentist-patient communication: a major factor in treat-ment prognosis, J Prosthet Dent 21:3-5, 1969. Schabel RW: The psychology of aging, J Prosthet Dent 27:569-573, 1972. Schmitt SM: Combination syndrome: a treatment approach, J Prosthet Dent 54:307-309, 1985. Schole ML: Management of the gagging patient, J Prosthet Dent 9:578-583, 1959. Schopper AF: Loss of vertical dimension: causes and effects: diagnosis and various recommended treatments, J Prosthet Dent 9:428-431, 1959. Schopper AF: Removable appliances for the preservation of the teeth, J Prosthet Dent 4:634-639, 1954. Schulte JK, Smith DE: Clinical evaluation of swinglock removable partial dentures, J Prosthet Dent 44:595-603, 1980. Schuyler CH: Stress distribution as the prime requisite to the success of a partial denture, J Am Dent Assoc 20:2148-2154, 1963. Schwarz WD, Barsby MJ: Design of partial dentures in dental practice, J Dent 6:166-170, 1978. Sears VH: Comprehensive denture service, J Am Dent Assoc 64:531-552, 1962. Skinner EW, Gordon CC: Some experiments on the surface hardness of dental stones, J Prosthet Dent 6:94-100, 1956. Skinner EW, Jones PM: Dimensional stability of self-curing denture base acrylic resin, J Am Dent Assoc 51:426-431, 1955. Smith DE: Removable prosthodontics research—quo vadis? J Prosthet Dent 62:707-711, 1989. Smith FW, Applegate OC: Roentgenographic study of bone changes during exercise stimulation of edentulous areas, J Prosthet Dent 11:1086-1097, 1961. Stendahl CG, Grob DJ: Detection of binding areas on removable partial denture frameworks, Dent Clin North Am 23:101-106, 1979. Swoope CC, Frank RP: Insertion and post-insertion care. In Clark J, editor: Clinical dentistry, vol 5, New York, 1976, Harper & Row. Sykora O: Definitive immediate cast removable partial dentures, Can Dent Assoc J 51:767-769, 1985. Sykora O: Extracoronal removable partial denture service in Canada, J Prosthet Dent 39:37-41, 1978. Tallgren A: Alveolar bone loss in denture wearers as related to facial morphology, Acta Odontol Scand 28:251-270, 1970. Taylor TD et al: Prosthodontic survey. I. Removable prosthodontic laboratory survey, J Prosthet Dent 52:598-601, 1984. Taylor TD et al: Prosthodontic survey. II. Removable prosthodontic curriculum survey, J Prosthet Dent 52:747-749, 1984. Teppo KW, Smith FW: A method of immediate clasp repair, J Prosthet Dent 34:77-80, 1975. Trainor JE, Elliott RW Jr: Removable partial dentures designed by dentists before and after graduate level instruction: a comparative study, J Prosthet Dent 27:509-514, 1972. von Gonten AS, Nelson DR: Laboratory pitfalls that contribute to embrasure clasp failure, J Prosthet Dent 53:136-138, 1985. 364 Appendix B Selected Reading Resources von Gonten AS, Palik JF: Tooth preparation guide for embrasure clasp designs, J Prosthet Dent 53:281-282, 1985. Wagner AG: Maintenance of the partially edentulous mouth and care of the denture, Dent Clin North Am 17:755-768, 1973. Wagner AG, Forgue EG: A study of four methods of recording the path of insertion of removable partial dentures, J Prosthet Dent 35:267-272, 1976. Wallace DH: The use of gold occlusal surfaces in complete and partial dentures, J Prosthet Dent 14:326-333, 1964. Walter JD: Partial denture technique. I. Introduction, Br Dent J 147:241-243, 1979; II. The purpose of the denture: choice of mate-rial, 147:302-304, 1979; III. Supporting the denture, 148:13-16, 1980; IV. Guide planes, 148:70-72, 1980. Weaver RE, Goebel WM: Reactions to acrylic resin dental prostheses, J Prosthet Dent 43:138-142, 1980. Whitsitt JA, Battle LW, Jarosz CJ: Enhanced retention for the distal extension-base removable partial denture using a heat-cured resil-ient soft liner, J Prosthet Dent 52:447-448, 1984. Williams EO, Hartman GE: Instructional aid for teaching removable partial denture design, J Prosthet Dent 48:222, 1982. Wise HB, Kaiser DA: A radiographic technique for examination of internal defects in metal frameworks, J Prosthet Dent 42:594-595, 1979. Young HA: Factors contributory to success in prosthodontic practice, J Prosthet Dent 5:354-360, 1955. Young L Jr: Try-in of the removable partial denture framework, J Prosthet Dent 46:579-580, 1981. Zach GA: Advantages of mesial rests for removable partial dentures, J Prosthet Dent 33:32-35, 1975. Zerosi C: A new type of removable splint: its indications and function, Dent Abstr 1:451-452, 1956. Zurasky JE, Duke ES: Improved adhesion of denture acrylic resins to base metal alloys, J Prosthet Dent 57:520-524, 1987. Mouth Preparations Alexander JM, Van Sickels JE: Posterior maxillary osteotomies: an aid for a difficult prosthodontic problem, J Prosthet Dent 41:614-617, 1979. Atwood DA: Reduction of residual ridges in the partially edentulous patient, Dent Clin North Am 17:745-754, 1973. Axinn S: Preparation of retentive areas for clasps in enamel, J Prosthet Dent 34:405-407, 1975. Belinfante LS, Abney JM Jr: A teamwork approach to correct a severe prosthodontic problem, J Am Dent Assoc 91:357-359, 1975. Dixon DL, Breeding LC, Swift EJ: Use of a partial coverage porcelain laminate to enhance clasp retention, J Prosthet Dent 63:55-58, 1990. Glann GW, Appleby RC: Mouth preparations for removable partial dentures, J Prosthet Dent 10:698-706, 1960. Johnston JF: Preparation of mouths for fixed and removable partial dentures, J Prosthet Dent 11:456-462, 1961. Jones RM: Dentin exposed and decay incidence in removable partial denture rest seats, Int J Prosthodont 5:227-236, 1992. Kahn AE: Partial versus full coverage, J Prosthet Dent 10:167-178, 1960. Kapur KK et al: A randomized clinical trial of two basic removable partial denture designs. Part I. Comparisons of five-year success rates and periodontal health, J Prosthet Dent 72:268-282, 1994. Laney WR, Desjardins RP: Comparison of base metal alloys and type IV gold alloys for removable partial denture framework, Dent Clin North Am 17:611-630, 1973. Lorey RE: Abutment considerations, Dent Clin North Am 24:63-79, 1980. Marquardt GL: Dolder bar joint mandibular overdenture: a technique for nonparallel abutment teeth, J Prosthet Dent 36:101-111, 1976. McArthur DR, Turvey TA: Maxillary segmental osteotomies for man-dibular removable partial denture patients, J Prosthet Dent 41:381-387, 1979. McCarthy JA, Moser JB: Mechanical properties of tissue conditioners. I. Theoretical considerations, behavioral characteristics and tensile properties, J Prosthet Dent 40:89-97, 1978. McCarthy JA, Moser JB: Mechanical properties of tissue conditioners. II. Creep characteristics, J Prosthet Dent 40:334-342, 1978. McCracken WL: Mouth preparations for partial dentures, J Prosthet Dent 6:39-52, 1956. Mills M: Mouth preparation for removable partial denture, J Am Dent Assoc 60:154-159, 1960. Mopsik ER et al: Surgical intervention to reestablish adequate inter-maxillary space before fixed or removable prosthodontics, J Am Dent Assoc 95:957-960, 1977. Nishimura RD: Etched metal cingulum rest retainer, J Am Dent Assoc 112:177-179, 1986. Phillips RJ Jr: Design sequence and mouth preparation for the remov-able partial denture, J Calif Dent Assoc 25:363-370, 1997. Phillips RW: Report of the Committee on Scientific Investigation of the Academy of Restorative Dentistry, J Prosthet Dent 13:515-535, 1963. Schorr L, Clayman LH: Reshaping abutment teeth for reception of partial denture clasps, J Prosthet Dent 4:625-633, 1954. Stamps JT, Tanquist RA: Restoration of removable partial denture rest seats using dental amalgam, J Prosthet Dent 41:224-227, 1979. Stern WJ: Guiding planes in clasp reciprocation and retention, J Pros-thet Dent 34:408-414, 1975. Swoope CC, Frank RP: Mouth preparation. In Clark JW, editor: Clini-cal dentistry, vol 5, New York, 1976, Harper & Row. Tiege JD et al: In vitro investigation of the wear of resin composite materials and cast direct retainers during removable partial denture placement and removal, Int J Prosthodont 5:145-153, 1992. Tucker KM, Heget HS: The incidence of inflammatory papillary hyper-plasia, J Am Dent Assoc 93:610-613, 1976. Wong R, Nicholls JI, Smith DE: Evaluation of prefabricated lingual rest seats for removable partial dentures, J Prosthet Dent 48:521-526, 1982. Occlusion, Jaw Relation Records, Transfer Methods Applegate OC: Loss of posterior occlusion, J Prosthet Dent 4:197-199, 1954. Baraban DJ: Establishing centric relation and vertical dimension in occlusal rehabilitation, J Prosthet Dent 12:1157-1165, 1962. Bauman R: Minimizing postinsertion problems: a procedure for removable partial denture placement, J Prosthet Dent 42:381-385, 1979. Beck HO: Choosing the articulator, J Am Dent Assoc 64:468-475, 1962. Beck HO: A clinical evaluation of the arcon concept of articulation, J Prosthet Dent 9:409-421, 1959. Beck HO: Selection of an articulator and jaw registration, J Prosthet Dent 10:878-886, 1960. Beckett LS: Accurate occlusal relations in partial denture construction, J Prosthet Dent 4:487-495, 1954. Berke JD, Moleres I: A removable appliance for the correction of maxil-lomandibular disproportion, J Prosthet Dent 17:172-177, 1967. Berman MH: Accurate interocclusal records, J Prosthet Dent 10:620-630, 1960. Beyron HL: Characteristics of functionally optimal occlusion and prin-ciples of occlusal rehabilitation, J Am Dent Assoc 48:648-656, 1954. Beyron HL: Occlusal changes in adult dentition, J Am Dent Assoc 48:674-686, 1954. Beyron HL: Occlusal relationship, Int Dent J 2:467-496, 1952. Block LS: Preparing and conditioning the patient for intermaxillary relations, J Prosthet Dent 2:599-603, 1952. 365 Appendix B Selected Reading Resources Block LS: Tensions and intermaxillary relations, J Prosthet Dent 4:204-207, 1954. Boos RH: Basic anatomic factors of jaw position, J Prosthet Dent 4:200-203, 1954. Boos RH: Maxillomandibular relations, occlusion, and the temporo-mandibular joint, Dent Clin North Am, 6:19-35, 1962. Boos RH: Occlusion from rest position, J Prosthet Dent 2:575-588, 1952. Borgh O, Posselt U: Hinge axis registration: experiments on the articu-lator, J Prosthet Dent 8:35-40, 1958. Boucher CO: Occlusion in prosthodontics, J Prosthet Dent 3:633-656, 1953. Braly BV: Occlusal analysis and treatment planning for restorative den-tistry, J Prosthet Dent 27:168-171, 1972. Breeding LC et al: Accuracy of three interocclusal recording materials used to mount a working cast, J Prosthet Dent 71:265-270, 1994. Cerveris AR: Vibracentric equilibration of centric occlusion, J Am Dent Assoc 63:476-483, 1961. Christensen PB: Accurate casts and positional relation records, J Pros-thet Dent 8:475-482, 1958. Clayton JA, Kotowicz WE, Zahler JM: Pantographic tracings of man-dibular movements and occlusion, J Prosthet Dent 25:389-396, 1971. Cohn LA: Factors of dental occlusion pertinent to the restorative and prosthetic problem, J Prosthet Dent 9:256-257, 1959. Collett HA: Balancing the occlusion of partial dentures, J Am Dent Assoc 42:162-168, 1951. Colman AJ: Occlusal requirements for removable partial dentures, J Prosthet Dent 17:155-162, 1967. D’Amico A: Functional occlusion of the natural teeth of man, J Prosthet Dent 11:899-915, 1961. Draper DH: Forward trends in occlusion, J Prosthet Dent 13:724-731, 1963. Emmert JH: A method for registering occlusion in semiedentulous mouths, J Prosthet Dent 8:94-99, 1958. Farmer JB, Connelly ME: Treatment of open occlusions with onlay and overlay removable partial dentures, J Prosthet Dent 51:300-303, 1984. Fedi PF: Cardinal differences in occlusion of natural teeth and that of artificial teeth, J Am Dent Assoc 62:482-485, 1926. Fountain HW: Seating the condyles for centric relation records, J Pros-thet Dent 11:1050-1058, 1961. Freilich MA, Altieri JW, Wahle JJ: Principles of selecting interocclusal records for articulation of dentate and partially dentate casts, J Pros-thet Dent 68:361-367, 1992. Gilson TD: Theory of centric correction in natural teeth, J Prosthet Dent 8:468-474, 1958. Granger ER: The articulator and the patient, Dent Clin North Am 4:527-539, 1960. Hansen CA et al: Simplified procedure for making gold occlusal sur-faces on denture teeth, J Prosthet Dent 71:413-416, 1994. Hausman M: Interceptive and pivotal occlusal contacts, J Am Dent Assoc 66:165-171, 1963. Henderson D: Occlusion in removable partial prosthodontics, J Pros-thet Dent 27:151-159, 1971. Hindels GW: Occlusion in removable partial denture prosthesis, Dent Clin North Am 6:137-146, 1962. Hughes GA, Regli CP: What is centric relation? J Prosthet Dent 11:16-22, 1961. Ivanhoe JR, Vaught RD: Occlusion in the combination fixed removable prosthodontic patient, Dent Clin North Am 31:305-322, 1987. Jankelson B: Considerations of occlusion on fixed partial dentures, Dent Clin North Am 3:187-203, 1959. Jeffreys FE, Platner RL: Occlusion in removable partial dentures, J Prosthet Dent 10:912-920, 1960. Kapur KK et al: A randomized clinical trial of two basic removable partial denture designs. Part I. Comparisons of five-year success rates and periodontal health, J Prosthet Dent 72:268-282, 1994. Lang BR: Complete denture occlusion, Dent Clin North Am 40:85-101, 1996. Lauritzen AG, Bodner GH: Variations in location of arbitrary and true hinge axis points, J Prosthet Dent 11:224-229, 1961. Lay LS et al: Making the framework try-in, altered cast impression and occlusal registration in one appointment, J Prosthet Dent 75:446-448, Apr 1996. Lindblom G: Balanced occlusion with partial reconstructions, Int Dent J 1:84-98, 1951. Lindblom G: The value of bite analysis, J Am Dent Assoc 48:657-664, 1954. Long JH Jr: Location of the terminal hinge axis by intraoral means, J Prosthet Dent 23:11-24, 1970. Lucia VO: Centric relation theory and practice, J Prosthet Dent 10:849-956, 1960. Lucia VO: The gnathological concept of articulation, Dent Clin North Am 6:183-197, 1962. Lundquist DO, Fiebiger GE: Registration for relating the mandibular cast to the maxillary cast based on Kennedy’s classification system, J Prosthet Dent 35:371-375, 1976. Mann AW, Pankey LD: The PM philosophy of occlusal rehabilitation, Dent Clin North Am 7:621-636, 1963. McCollum BB: The mandibular hinge axis and a method of locating it, J Prosthet Dent 10:428-435, 1960. McCracken WL: Functional occlusion in removable partial denture construction, J Prosthet Dent 8:955-963, 1958. McCracken WL: Occlusion in partial denture prosthesis, Dent Clin North Am 6:109-119, 1962. Mehta JD, Joglekar AP: Vertical jaw relations as a factor in partial dentures, J Prosthet Dent 21:618-625, 1969. Meyer FS: The generated path technique in reconstruction dentistry. I and II, J Prosthet Dent 9:354-366, 432-440, 1959. Millstein PL, Kronman JH, Clark RE: Determination of the accuracy of wax interocclusal registrations, J Prosthet Dent 25:189-196, 1971. Moore AW: Ideal versus adequate dental occlusion, J Am Dent Assoc 55:51-56, 1957. Moulton GH: The importance of centric occlusion in diagnosis and treatment planning, J Prosthet Dent 10:921-926, 1960. Nayyar A, Bill JA Jr, Twiggs SW: Comparison of interocclusal recording materials for mounting a working cast, J Dent Res (IADR abstract 1216) 60:entire issue, 1981. Nuttall EB: Establishing posterior functional occlusion for fixed partial dentures, J Am Dent Assoc 66:341-348, 1963. O’Leary TJ, Shanley DB, Drake RB: Tooth mobility in cuspid-protected and group-function occlusions, J Prosthet Dent 27:21-25, 1972. Olsson A, Posselt U: Relationship of various skull reference lines, J Prosthet Dent 11:1045-1049, 1961. Reitz PV: Technique for mounting removable partial dentures on an articulator, J Prosthet Dent 22:490-494, 1969. Reynolds JM: Occlusal wear facets, J Prosthet Dent 24:367-372, 1970. Ricketts RM: Occlusion: the medium of dentistry, J Prosthet Dent 21:39-60, 1969. Robinson MJ: Centric position, J Prosthet Dent 1:384-386, 1951. Scaife RR Jr, Holt JE: Natural occurrence of cuspid guidance, J Prosthet Dent 22:225-229, 1969. Scandrett FR, Hanson JC: Technique for attaching the master cast to its split mounting index, J Prosthet Dent 40:467-469, 1978. Schireson S: Grinding teeth for masticatory efficiency and gingival health, J Prosthet Dent 13:337-345, 1963. Schuyler CH: An evaluation of incisal guidance and its influence in restorative dentistry, J Prosthet Dent 9:374-378, 1959. Schuyler CH: Factors contributing to traumatic occlusion, J Prosthet Dent 11:708-715, 1961. Schuyler CH: Factors of occlusion applicable to restorative dentistry, J Prosthet Dent 3:772-782, 1953. 366 Appendix B Selected Reading Resources Schuyler CH: Fundamental principles in the correction of occlusal disharmony: natural and artificial (grinding), J Am Dent Assoc 22:1193-1202, 1935. Sears VH: Centric and eccentric occlusions, J Prosthet Dent 10:1029-1036, 1960. Sears VH: Occlusal pivots, J Prosthet Dent 6:332-338, 1956. Sears VH: Mandibular equilibration, J Am Dent Assoc 65:45-55, 1962. Sears VH: Occlusion: the common meeting ground in dentistry, J Pros-thet Dent 2:15-21, 1952. Shanahan TEJ, Leff A: Interocclusal records, J Prosthet Dent 10:842-848, 1960. Silverman MM: Determination of vertical dimension by phonetics, J Prosthet Dent 6:465-471, 1956; Dent Abstr 2:221, 1957. Skurnik H: Accurate interocclusal records, J Prosthet Dent 21:154-165, 1969. Stuart CE: Accuracy in measuring functional dimensions and relations in oral prosthesis, J Prosthet Dent 9:220-236, 1959. Teteruck WR, Lundeen HC: The accuracy of an ear face-bow, J Prosthet Dent 16:1039-1046, 1966. Trushkowsky RD, Guiv B: Restoration of occlusal vertical dimension by means of a silica-coated onlay removable partial denture in con-junction with dentin bonding: a clinical report, J Prosthet Dent 66:283-286, 1991. Wagner AG: A technique to record jaw relations for distally edentulous dental arches, J Prosthet Dent 29:405-407, 1973. Weinberg LA: Arcon principle in the condylar mechanism of adjustable articulators, J Prosthet Dent 13:263-268, 1963. Weinberg LA: An evaluation of basic articulators and their concepts. I and II, J Prosthet Dent 13:622-863, 1963. Weinberg LA: An evaluation of the face-bow mounting, J Prosthet Dent 11:32-42, 1961. Weinberg LA: The transverse hinge axis: real or imaginary, J Prosthet Dent 9:775-787, 1959. Partial Denture Design Akagawa Y, Seo T, Ohkawa S, Tsuru H: A new telescopic crown system using a soldered horizontal pin for removable partial dentures, J Prosthet Dent 69:228-231, 1993. Antos EW Jr, Tenner RP, Foerth D: The swinglock partial denture: an alternative approach to conventional removable partial denture service, J Prosthet Dent 40:257-262, 1978. Avant EW: Indirect retention in partial denture design, J Prosthet Dent 16:1103-1110, 1966. Axinn S, O’Connor RP Jr, Kopp EN: Immediate removable partial denture frameworks, J Am Dent Assoc 95:583-585, 1977. Beaumont AJ Jr, Bianco HJ: Microcomputer-aided removable partial denture design: the next evolution, J Prosthet Dent 62:551-556, 1989. Becker CW, Bolender CL: Designing swinglock partial dentures, J Pros-thet Dent 46:126-132, 1981. Becker CM, Kaiser DA, Goldfogel MH: Evolution of removable partial denture design, J Prosthodont 3:158-166, 1994. Ben-Ur Z et al: Designing clasps for the asymmetric distal extension removable partial denture, Int J Prosthodont 9:374-378, July 1996. Ben-Ur Z et al: Rigidity of major connectors when subjected to bending and torsion forces, J Prosthet Dent 62:557-562, 1989. Berg E: Periodontal problems associated with use of distal extension removable partial dentures: a matter of construction? J Oral Rehabil 12:369-379, 1985. Berg T Jr: I-bar: myth and counter-myth, Dent Clin North Am 23:65-75, 1979. Berg T Jr, Caputo AA: Anterior rests for maxillary removable partial dentures, J Prosthet Dent 39:139-146, 1978. Blatterfein L: A systematic method of designing upper partial denture bases, J Am Dent Assoc 46:510-525, 1953. Blatterfein L: The use of the semiprecision rest in removable partial dentures, J Prosthet Dent 22:301-306, 1969. Bolouri A: Removable partial denture design for a few remaining natural teeth, J Prosthet Dent 39:346-348, 1978. Breeding L, Dixon DL: Prosthetic restoration of the anterior edentulous space, J Prosthet Dent 67:144-148, 1992. Bridgeman JT et al: Comparison of titanium and cobalt-chromium removable partial denture clasps, J Prosthet Dent 78:187-193, 1997. Brown DT, Desjardins RP, Chao EY: Fatigue failure in acrylic resin retaining minor connectors, J Prosthet Dent 58:329-335, 1987. Browning JD et al: Effect of positional loading of three removable partial denture clasp assemblies on movement of abutment teeth, J Prosthet Dent 55:347-351, 1986. Browning JD, Meadors LW, Eick JD: Movement of three removable partial denture clasp assemblies under occlusal loading, J Prosthet Dent 55:69-74, 1986. Brudvik J, Reimers D: The tooth-removable partial denture interface, J Prosthet Dent 68:924-927, 1992. Budtz-Jorgensen E et al: Alternate framework designs for removable partial dentures, J Prosthet Dent 80:58-66, July 1998. Burns DR, Ward JE, Nance GL: Removable partial denture design and fabrication survey of the prosthodontic specialist, J Prosthet Dent 62:303-307, 1989. Campbell LD: Subjective reactions to major connector designs for removable partial dentures, J Prosthet Dent 36:507-516, 1977. Campbell SD, Weiner H: The hinged-clasp assembly removable partial denture, J Prosthet Dent 63:59-61, 1990. Casey DM, Lauciello FR: A method for marking the functional depth of the floor of the mouth, J Prosthet Dent 43:108-111, 1980. Cecconi BT: Lingual bar design, J Prosthet Dent 28:635-639, 1973. Cecconi BT et al: The component partial: a new RPD construction system, J Calif Dent Assoc 25:363-370, 1997. Cowles KR: Partial denture design: a simple teaching aid, J Prosthet Dent 47:219, 1982. Davenport JC et al: The acquisition and validation of removable partial denture design knowledge. I. Methodology and overview, J Oral Rehabil 23:152-157, 1996. Davenport JC et al: The acquisition and validation of removable partial denture design knowledge. II. Design rules and expert reaction, J Oral Rehabil 23:811-824, 1996. Demer WJ: An analysis of mesial rest-I-bar clasps designs, J Prosthet Dent 36:243-253, 1976. Dunny JA, King GE: Minor connector designs for anterior acrylic resin bases: a preliminary study, J Prosthet Dent 34:496-497, 1975. Eick JD et al: Abutment tooth movement related to fit of a removable partial denture, J Prosthet Dent 57:66-72, 1987. Ettinger RL: The acrylic removable partial denture, J Am Dent Assoc 85:945-949, 1977. Farmer JB et al: Interim removable partial dentures: a modified tech-nique, Quintessence Dent Technol 8:511-516, 1985. Feingold FM, Grant AA, Johnson W: The effect of partial denture design on abutment tooth and saddle movement, J Oral Rehabil 13:549-557, 1986. Firtell DN: Effect of clasp design upon retention of removable partial dentures, J Prosthet Dent 20:43-52, 1968. Firtell DN, Grisius RJ, Muncheryan AM: Reaction of the anterior abut-ment of a Kennedy Class II removable partial denture to various clasp arm designs: an in vitro study, J Prosthet Dent 53:77-82, 1985. Fisher RL, Jaslow C: The efficiency of an indirect retainer, J Prosthet Dent 33:24-30, 1975. Fisher RL, McDowell GC: Removable partial denture design and poten-tial stress to the periodontium, Int J Periodont Restor Dent 4:34-47, 1984. Frank RP: Direct retainers for distal-extension removable partial den-tures, J Prosthet Dent 56:562-567, 1986. Frank RP: An investigation of the effectiveness of indirect retainers, J Prosthet Dent 38:494-506, 1977. 367 Appendix B Selected Reading Resources Frantz WR: Variations in a removable maxillary partial denture design by dentists, J Prosthet Dent 34:625-633, 1975. Ghamrawy E: Oral ecologic response caused by removable partial den-tures, J Dent Res (IADR abstract 2898) 61:entire issue, 1982. Ghamrawy E: Plaque formation and crevicular temperature relation to minor connector position, J Dent Res (IADR abstract 387) 61:entire issue, 1982. Giradot RL: History and development of partial denture design, J Am Dent Assoc 28:1399-1408, 1941. Hansen CA: Metal minibases in removable prosthodontics, J Prosthet Dent 54:442-446, 1985. Hansen CA, Campbell DJ: Clinical comparison of two mandibular major connector designs: the sublingual bar and the lingual plate, J Prosthet Dent 54:805-809, 1985. Henderson D: Major connectors for mandibular removable partial dentures, J Prosthet Dent 30:532-548, 1973. Henderson D: Major connectors: united it stands, Dent Clin North Am 17:661-668, 1973. Hero H et al: Ductility and structure of some cobalt-base dental casting alloys, Biomaterials 5:201-208, 1984. Highton R, Caputo AA, Rhodes S: Force transmission and retentive capabilities utilizing labial and palatal I-bar partial dentures, J Dent Res (IADR abstract 1214) 60:entire issue, 1981. Iwama CY et al: Cobalt-chromium-titanium alloy for removable partial dentures, Int J Prosthodont 10:309-317, 1997. Jacobson TE: Rotational path partial denture design: a 10-year clinical follow-up. Parts I and II, J Prosthet Dent 71:271-282, 1994. Jacobson TE, Krol AJ: Rotational path removable partial denture design, J Prosthet Dent 48:370-376, 1982. Jordan LG: Designing removable partial dentures with external attach-ments (clasps), J Prosthet Dent 2:716-722, 1952. Kapur KK et al: A randomized clinical trial of two basic removable partial denture designs. Part I. Comparisons of five-year success rates and periodontal health, J Prosthet Dent 72:268-282, 1994. Kelly EK: The physiologic approach to partial denture design, J Pros-thet Dent 3:699-710, 1953. King GE: Dual-path design for removable partial dentures, J Prosthet Dent 39:392-395, 1978. King GE, Barco MT, Olson RJ: Inconspicuous retention for removable partial dentures, J Prosthet Dent 39:505-507, 1978. Knodle JM: Experimental overlay and pin partial denture, J Prosthet Dent 17:472-478, 1967. Ko SH, McDowell GC, Kotowicz WE: Photoelastic stress analysis of mandibular removable partial dentures with mesial and distal occlu-sal rests, J Prosthet Dent 56:454-460, 1986. Krikos AA: Artificial undercuts for teeth which have unfavorable shapes for clasping, J Prosthet Dent 22:301-306, 1969. Lanser A: Telescope retainers for removable partial dentures, J Prosthet Dent 45:37-43, 1981. Latta GH et al: Wear of visible light-cured restorative materials and removable partial denture direct retainers, J Prosthodont 6:104-109, 1997. LaVere AM, Freda AL: A simplified procedure for survey and design of diagnostic casts, J Prosthet Dent 37:680-683, 1977. LaVere AM, Krol AJ: Selection of a major connector for the extension base removable partial denture, J Prosthet Dent 30:102-105, 1973. LaVere AM, Smith RC, Serka RJ: Cross-arch bar splint, J Prosthet Dent 67:82-84, 1992. Lindquist TJ et al: Effectiveness of computer-aided partial denture design, J Prosthodont 6:122-127, 1997. Lorencki SF: Planning precision attachment restorations, J Prosthet Dent 21:506-508, 1969. Luk K et al: Unilateral rotational path removable partial dentures for tilted mandibular molars, J Prosthet Dent 78:102-105, 1997. Marxkors R: Mastering the removable partial denture. Part I. Basic reflections about construction, J Dent Technol 14:34-39, 1997. Marxkors R: Mastering the removable partial denture. Part II. Connec-tion of partial dentures to the abutment teeth, J Dent Technol 14:24-30, 1997. Maxfield JB, Nicholls JE, Smith DE: The measurement of forces trans-mitted to abutment teeth of removable partial dentures, J Prosthet Dent 41:134-142, 1979. McCartney JW: Lingual plating for reciprocation, J Prosthet Dent 42:624-625, 1979. McCracken WL: Contemporary partial denture designs, J Prosthet Dent 8:71-84, 1958. McCracken WL: Survey of partial denture designs by commercial dental laboratories, J Prosthet Dent 12:1089-1110, 1962. McHenry KR, Johansson DE, Christensson LA: The effect of removable partial denture framework design on gingival inflammation: a clini-cal model, J Prosthet Dent 68:799-803, 1992. Meeuwissen R, Keltjens HM, Battistugzi PG: Cingulum bar as a major connector for mandibular removable partial dentures, J Prosthet Dent 66:221-223, 1991. Monteith BD: Management of loading forces on mandibular distal-extension prostheses. I. Evaluation of concepts for design, J Prosthet Dent 52:673-681, 1984. Monteith BD: Management of loading forces on mandibular distal-extension prostheses. II. Classification for matching modalities to clinical situations, J Prosthet Dent 52:832-836, 1984. Myers RE et al: A photoelastic study of rests on solitary abutments for distal-extension removable partial dentures, J Prosthet Dent 56:702-707, 1986. NaBadalung DP et al: Frictional resistance of removable partial den-tures with retrofitted resin composite guide planes, Int J Prosthodont 10:116-122, 1997. Naim RI: The problem of free-end denture bases, J Prosthet Dent 16:522-532, 1966. Navas MTR, del Campo ML: A new free-end removable partial denture design, J Prosthet Dent 70:176-179, 1993. Pardo-Mindan S, Ruiz-Villandiego JC: A flexible lingual clasp as an esthetic alternative: a clinical report, J Prosthet Dent 69:245-246, 1993. Perry C: Philosophy of partial denture design, J Prosthet Dent 6:775-784, 1956. Pienkos TE, Morris WJ, Gronet PM, Cameron SM, Looney SW: The strength of multiple major connector designs under simulated func-tional loading, J Prosthet Dent 97:299-304, 2007. Pipko DJ: Combinations in fixed-removable prostheses, J Prosthet Dent 26:481-490, 1971. Potter RB, Appleby RC, Adams CD: Removable partial denture design: a review and a challenge, J Prosthet Dent 17:63-68, 1967. Radford DR, Walter JD: A variation in minor connector design for partial denture, Int J Prosthet 6:50-53, 1993. Russell MD, Tumer P: A three-part sectional design for an upper removable partial denture with an anterior modification, Br Dent J 162:24-26, 1987. Rybeck SA Jr: Simplicity in a distal extension partial denture, J Prosthet Dent 4:87-92, 1954. Schmidt AH: Planning and designing removable partial dentures, J Prosthet Dent 3:783-806, 1953. Schuyler CH: The partial denture as a means of stabilizing abutment teeth, J Am Dent Assoc 28:1121-1125, 1941. Schwartz RS, Murchison DG: Design variations of the rotational path removable partial denture, J Prosthet Dent 58:336-338, 1987. Seals RR Jr, Schwartz IS: Successful integration of fixed and removable prosthodontics, J Prosthet Dent 53:763-766, 1985. Shifman A: Use of an Adam’s clasp for a cast unilateral removable partial denture, J Prosthet Dent 61:703-705, 1989. Shohet H: Relative magnitudes of stress on abutment teeth with differ-ent retainers, J Prosthet Dent 21:267-282, 1969. Steffel VL: Fundamental principles involved in partial denture design, J Am Dent Assoc 42:534-544, 1951. 368 Appendix B Selected Reading Resources Steffel VL: Simplified clasp partial dentures designed for maximum function, J Am Dent Assoc 32:1093-1100, 1945. Sykora O: Removable partial denture design by Canadian dental labo-ratories: a retrospective study, J Can Dent Assoc 61:615-621, 1995. Sykora O, Calikkocaoglu S: Maxillary removable partial denture designs by commercial dental laboratories, J Prosthet Dent 22:633-640, 1970. Tautin FS: Abutment stabilization using a nonresilient gingival bar connector, J Am Dent Assoc 99:988-998, 1979. Thompson WD, Kratochvil FJ, Caputo AA: Evaluation of photoelastic stress patterns produced by various designs of bilateral distal-exten-sion removable partial dentures, J Prosthet Dent 38:261-273, 1977. Tsao DH: Designing occlusal rests using mathematical principles, J Prosthet Dent 23:154-163, 1970. Unger JW, Badr SE: Esthetic placement of bar-clasp direct retainers, J Prosthet Dent 56:381-382, 1986. Vallittu PK: Comparison of the in vitro fatigue resistance of an acrylic resin removable partial denture reinforced with continuous fibers or metal wires. J Prosthodont 5:115-121, 1996. Vofa M, Kotowicz WE: Plaque retention with lingual bar and lingual plate major connectors, J Dent Res (AADR abstract 609) 59:entire issue, 1980. Wagner AC, Traweek FC: Comparison of major connectors for remov-able partial dentures, J Prosthet Dent 47:242-245, 1982. Waller NI: The root rest and the removable partial denture, J Prosthet Dent 33:16-23, 1975. Walter JD: Alternative major connectors for mandibular partial den-tures, Restorative Dent 2:80, 82-84, 1986. Warren AB, Caputo AA: Load transfer to alveolar bone as influenced by abutment design for tooth supported dentures, J Prosthet Dent 33:137-148, 1975. Weinberg LA: Lateral force in relation to the denture base and clasp design, J Prosthet Dent 6:785-800, 1956. Williams RJ et al: Use of a cast flexible plate as a hinge substitute in a hinge-lock design removable partial denture, J Prosthet Dent 80:220-223, 1998. Zach GA: Advantages of mesial rests for removable partial dentures, J Prosthet Dent 33:32-35, 1975. Zoller GN et al: Technique to improve surveying in confined areas, J Prosthet Dent 73:223-224, 1995. Periodontal Considerations Amsterdam M, Fox L: Provisional splinting: principles and technics, Dent Clin North Am 3:73-99, 1959. App GR: Periodontal treatment for the removable partial prosthesis patient: another half century, Dent Clin North Am 17:601-610, 1973. Applegate OC: The interdependence of periodontics and removable partial denture prosthesis, J Prosthet Dent 8:269-281, 1958. Aydinlik E, Dayangac B, Celik E: Effect of splintings on abutment tooth movement, J Prosthet Dent 49:477-480, 1983. Bates JF, Addy M: Partial dentures and plaque accumulation, J Dent 6:285-293, 1978. Bazirgan MK, Bates JF: Effect of clasp design on gingival health, J Oral Rehabil 14:271-281, 1987. Becker CM, Kaldahl WB: Using removable partial dentures to stabilize teeth with secondary occlusal traumatism, J Prosthet Dent 47:587-594, 1982. Berg TE, Caputo AA: Maxillary distal extension removable partial denture abutments with reduced periodontal support, J Prosthet Dent 70:245-250, 1993. Bergman B: Periodontal reactions related to removable partial den-tures: a literature review, J Prosthet Dent 58:454-458, 1987. Bergman B, Ericson G: Cross-sectional study of the periodontal status of removable partial denture patients, J Prosthet Dent 61:208-211, 1989. Brill N et al: Ecologic changes in the oral cavity caused by removable partial dentures, J Prosthet Dent 38:138-148, 1977. Clarke NG: Treatment planning for fixed and removable partial den-tures: a periodontal view, J Prosthet Dent 36:44-50, 1976. Dello Russo NM: Gingival autografts as an adjunct to removable partial dentures, J Am Dent Assoc 104:179-181, 1982. Erperstein H: The role of the prosthodontist in the treatment of peri-odontal disease, Int Dent J 36:18-29, 1986. Fisher RL, McDowell GC: Removable partial denture design and poten-tial stress to the periodontium, Int J Periodont Res Dent 4:34-47, 1984. Garfield RE: A prosthetic solution to the periodontally compromised/ furcation involved abutment tooth. I, Quintessence Int 15:805-813, 1984. Gilson CM: Periodontal considerations, Dent Clin North Am 24:31-44, 1980. Gomes BC et al: A clinical study of the periodontal status of abutment teeth supporting swinglock removable partial dentures: a pilot study, J Prosthet Dent 46:7-13, 1981. Gomes BC, Renner RP, Bauer PN: Periodontal considerations in removable partial dentures, J Am Dent Assoc 101:496-498, 1980. Hall WB: Periodontal preparation of the mouth for restoration, Dent Clin North Am 24:195-213, 1980. Hirschfeld Z et al: New sustained release dosage form of chlorhexidine for dental use: use for plaque control in partial denture wearers, J Oral Rehabil 11:477-482, 1984. Isidor F, Budtz-Jorgensen E: Periodontal conditions following treat-ment with cantilever bridges or removable partial dentures in geri-atric patients: a 2-year study, Gerodontics 3:117-121, 1987. Ivancie GP: Interrelationship between restorative dentistry and peri-odontics, J Prosthet Dent 8:819-830, 1958. Jacobson TE: Periodontal considerations in removable partial denture design, Compendium 8:530-534, 536-539, 1987. Jordan LG: Treatment of advanced periodontal disease by prosthodon-tic procedures, J Prosthet Dent 10:908-911, 1960. Kimball HD: The role of periodontia in prosthetic dentistry, J Prosthet Dent 1:286-294, 1951. Krogh-Poulsen W: Partial denture design in relation to occlusal trauma in periodontal breakdown, Int Dent J 4:847-867, 1954; also Acad Rev 3:18-23, 1955. McKenzie JS: Mutual problems of the periodontist and prosthodontist, J Prosthet Dent 5:37-42, 1955. Morris ML: Artificial crown contours and gingival health, J Prosthet Dent 12:1146-1155, 1962. Nevin RB: Periodontal aspects of partial denture prosthesis, J Prosthet Dent 5:215-219, 1955. Orban BS: Biologic principles in correction of occlusal disharmonies, J Prosthet Dent 6:637-641, 1956. Overby GE: Esthetic splinting of mobile periodontally involved teeth by vertical pinning, J Prosthet Dent 11:112-118, 1961. Perel ML: Periodontal consideration of crown contours, J Prosthet Dent 26:627-630, 1971. Picton DCA, Wills DJ: Viscoelastic properties of the periodontal liga-ment and mucous membrane, J Prosthet Dent 40:263-272, 1978. Rissin L et al: Effect of age and removable partial dentures on gingivitis and periodontal disease, J Prosthet Dent 42:217-223, 1979. Rudd KD, O’Leary TJ: Stabilizing periodontally weakened teeth by using guide plane removable partial dentures: a preliminary report, J Prosthet Dent 16:721-727, 1966. Schuyler CH: The partial denture and a means of stabilizing abutment teeth, J Am Dent Assoc 28:1121-1125, 1941. Schwalm CA, Smith DE, Erickson JD: A clinical study of patients 1 to 2 years after placement of removable partial dentures, J Prosthet Dent 38:380-391, 1977. Seibert JS, Cohen DW: Periodontal considerations in preparation for fixed and removable prosthodontics, Dent Clin North Am 31:529-555, 1987. 369 Appendix B Selected Reading Resources Spiekermann H: Prosthetic and periodontal considerations of free-end removable partial dentures, Int J Periodont Restor Dent 6:148-163, 1986. Sternlicht HC: Prosthetic treatment planning for the periodontal patient, Dent Abstr 2:81-82, 1957. Stipho HDK, Murphy WM, Adams D: Effect of oral prostheses on plaque accumulation, Br Dent J 145:47-50, 1978. Talkov L: Survey for complete periodontal prosthesis, J Prosthet Dent 11:124-131, 1961. Tebrock OC et al: The effect of various clasping systems on the mobility of abutment teeth for distal-extension removable partial dentures, J Prosthet Dent 41:511-516, 1979. Thayer HH, Kratochvil FJ: Periodontal considerations with removable partial dentures, Dent Clin North Am 24:195-213, 1980. Thomas BOA, Gallager JW: Practical management of occlusal dysfunc-tions in periodontal therapy, J Am Dent Assoc 46:18-31, 1953. Trapozzano VR, Winter CR: Periodontal aspects of partial denture design, J Prosthet Dent 2:101-107, 1952. Waerhaug J: Justification for splinting in periodontal therapy, J Pros-thet Dent 22:201-208, 1969. Ward HL, Weinberg LA: An evaluation of periodontal splinting, J Am Dent Assoc 63:48-54, 1961. Physiology: Mandibular Movement Brekke CA: Jaw function. I. Hinge rotation, J Prosthet Dent 9:600-606, 1959; II. Hinge axis, hinge axes, 9:936-940, 1959; III. Condylar place-ment and condylar retrusion, 10:78-85, 1960. Brotman DN: Contemporary concepts of articulation, J Prosthet Dent 10:221-230, 1960. Budtz-Jorgensen E: Restoration of the occlusal face height by remov-able partial dentures in elderly patients, Gerodontics 2:67-71, 1986. Emig GE: The physiology of the muscles of mastication, J Prosthet Dent 1:700-707, 1951. Fountain HW: The temporomandibular joints: a fulcrum, J Prosthet Dent 25:78-84, 1971. Gibbs CH et al: Functional movements of the mandible, J Prosthet Dent 26:604-620, 1971. Jankelson B: Physiology of human dental occlusion, J Am Dent Assoc 50:664-680, 1955. Jemt T, Hedegard B, Wickberg K: Chewing patterns before and after treatment with complete maxillary and bilateral distal-extension mandibular removable partial dentures, J Prosthet Dent 50:566-569, 1983. Kurth LE: Centric relation and mandibular movement, J Am Dent Assoc 50:309-315, 1955. Kurth LE: Mandibular movement and articulator occlusion, J Am Dent Assoc 39:37-46, 1949. McMillen LB: Border movements of the human mandible, J Prosthet Dent 27:524-532, 1972. Messerman T: A concept of jaw function with a related clinical applica-tion, J Prosthet Dent 13:130-140, 1963. Naylor JG: Role of the external pterygoid muscles in temporomandibu-lar articulation, J Prosthet Dent 10:1037-1042, 1960. Plotnick IJ, Beresin VE, Simkins AB: The effects of variations in the opposing dentition on changes in the partially edentulous mandible. I. Bone changes observed in serial radiographs, J Prosthet Dent 33:278-286, 1975. Plotnick IJ, Beresin VE, Simkins AB: The effects of variations in the opposing dentition on changes in the partially edentulous mandible. III. Tooth mobility and chewing efficiency with various maxillary dentitions, J Prosthet Dent 33:529-534, 1975. Posselt U: Movement areas of the mandible, J Prosthet Dent 7:375-385, 1957. Posselt U: Studies in the mobility of the human mandible, Acta Odontol Scand 10:19-160, 1952. Posselt U: Terminal hinge movement of the mandible, J Prosthet Dent 7:787-797, 1957. Saizar P: Centric relation and condylar movement, J Prosthet Dent 26:581-591, 1971. Schweitzer JM: Masticatory function in man, J Prosthet Dent 11:625-647, 1961. Shanahan TEJ: Dental physiology for dentures: the direct application of the masticatory cycle to denture occlusion, J Prosthet Dent 2:3, 1952. Shore NA: Educational program for patients with temporomandibular joint dysfunction (ligaments), J Prosthet Dent 23:691-695, 1970. Sicher H: Positions and movements of the mandible, J Am Dent Assoc 48:620-625, 1954. Skinner CN: Physiology of the occlusal coordination of natural teeth, complete dentures, and partial dentures, J Prosthet Dent 17:559-565, 1967. Sostenbo HR: CE Luce’s recordings of mandibular movement, J Pros-thet Dent 11:1068-1073, 1961. Tallgren A, Mizutani H, Tryda G: A two-year kinesiograph, study of mandibular movement patterns in denture wearers, J Prosthet Dent 62:594-600, 1989. Ulrich J: The human temporomandibular joint: kinematics and actions of the masticatory muscles, J Prosthet Dent 9:399-406, 1959. Vaughan HC: The external pterygoid mechanism, J Prosthet Dent 5:80-92, 1955. Rebasing and Relining Blatterfein L: Rebasing procedures for removable partial dentures, J Prosthet Dent 8:441-467, 1958. Bolouri A et al: A procedure for relining a complete or removable partial denture without the use of wax, J Prosthet Dent 79:604-606, May 1998. Breeding LC, Dixon DL, Lund TS: Dimensional changes of processed denture bases after relining with three resins, J Prosthet Dent 66:650-656, 1991. Grady RD: Objective criteria for relining distal-extension removable partial dentures: a preliminary report, J Prosthet Dent 49:178-181, 1983. McGivney GP: A reline technique for extension base removable partial dentures. In Lefkowitz W, editor: Proceedings of the Second Inter-national Prosthodontic Congress, St Louis, 1979, Mosby. Steffel VL: Relining removable partial dentures for fit and function, J Prosthet Dent 4:496-509, 1954. Turck MD, Richards MW: Microwave processing for denture relines, repairs, and rebases, J Prosthet Dent 69:340-343, 1993. Wilson JH: Partial dentures: relining the saddle supported by the mucosa and alveolar bone, J Prosthet Dent 3:807-813, 1953. Yasuda N et al: New adhesive resin to metal in removable prosthodon-tics field, J Dent Res (IADR abstract 213) 59:entire issue, 1980. Stress-Breaker Designs Bartlett AA: Duplication of precision attachment partial dentures, J Prosthet Dent 16:1111-1115, 1966. Bickley RW: Combined splint-stress breaker removable partial denture, J Prosthet Dent 21:509-512, 1969. Cecconi BT, Kaiser C, Rahe A: Stress-breakers and the removable partial denture, J Prosthet Dent 34:145-151, 1975. Hansen CA, Singer MT: The segmented framework removable partial denture, J Prosthet Dent 47:765-768, 1987. Hirschtritt E: Removable partial dentures with stress-broken extension bases, J Prosthet Dent 7:318-324, 1957. 370 Appendix B Selected Reading Resources James AC: Stress-breakers which automatically return the saddle to rest position following displacement: mandibular distal extension partial dentures, J Prosthet Dent 4:73-81, 1954. Kabcenell JL: Stress-breaking for partial dentures, J Am Dent Assoc 63:593-602, 1961. Kane BE: Buoyant stress equalizer, J Prosthet Dent 14:698-704, 1964. Kane BE: Improved buoyant stress equalizer, J Prosthet Dent 17:365-371, 1967. Levin B: Stress-breakers: a practical approach, Dent Clin North Am 23:77-86, 1979. Levitch HC: Physiologic stress-equalizer, J Prosthet Dent 3:232-238, 1953. MacGregor AR: Stress-breaking in partial dentures, Aust Prosthodont Soc Bull 16:65-70, 1986. Marris FN: The precision dowel rest attachment, J Prosthet Dent 5:43-48, 1955. Neill DJ: The problem of the lower free-end removable partial denture, J Prosthet Dent 8:623-634, 1958. Plotnik IJ: Stress regulator for complete and partial dentures, J Prosthet Dent 17:166-171, 1967. Reitz PV, Caputo AA: A photoelastic study of stress distribution by a mandibular split major connector, J Prosthet Dent 54:220-225, 1985. Reitz PV, Sanders JL, Caputo AA: A photoelastic study of a split palatal major connector, J Prosthet Dent 51:19-23, 1984. Simpson DH: Considerations for abutments, J Prosthet Dent 5:375-384, 1955. Terrell WH: Split bar technique applicable to both precision attach-ment and clasp cases, J South Calif Dent Assoc 9:10-14, 1942. Zinner ID: A modification of the Thompson Dowel rest for distal-extension removable partial dentures, J Prosthet Dent 61:374-378, 1989. Surveying Applegate OC: Use of paralleling surveyor in modern partial denture construction, J Am Dent Assoc 27:1317-1407, 1940. Atkinson HF: Partial denture problems: surveyors and surveying, Aust J Dent 59:28-31, 1955. Bezzon OL et al: Surveying removable partial dentures: the importance of guiding planes and path of insertion for stability, J Prosthet Dent 78:412-418, 1997. Chestner SC: A methodical approach to the analysis of study cases, J Prosthet Dent 4:622-624, 1954. Hanson JC: Surveying, J Am Dent Assoc 91:826-828, 1975. Katulski EM, Appleyard WN: Biological concepts of the use of the mechanical cast surveyor, J Prosthet Dent 7:627-634, 1959. Knapp JC, Shotwell JL, Kotowicz WE: Technique for recording dental cast-surveyor relations, J Prosthet Dent 41:352-354, 1979. McCarthy MF: An intraoral surveyor, J Prosthet Dent 61:462-464, 1989. Solle W: An improved dental surveyor, J Am Dent Assoc 60:727-731, 1960. Wagner AC, Forque EC: A study of four methods of recording the path of insertion of removable partial dentures, J Prosthet Dent 35:267-272, 1976. Yilmaz C: Optical surveying of casts for removable partial dentures, J Prosthet Dent 34:292-296, 1975. Zoller GN et al: Technique to improve surveying in confined areas, J Prosthet Dent 73:223-224, Feb 1995. Work Authorizations Brown ET: The dentist, the laboratory technician, and the prescription law, J Prosthet Dent 15:1132-1138, 1965. Dutton DA: Standard abbreviations (and definitions) for use in dental laboratory work authorizations, J Prosthet Dent 27:94-95, 1972. Gehl DH: Investment in the future, J Prosthet Dent 18:190-201, 1968. Henderson D: Writing work authorizations for removable partial den-tures, J Prosthet Dent 16:696-707, 1966. Henderson D, Frazier Q: Communicating with dental laboratory tech-nicians, Dent Clin North Am 14:603-615, 1970. Leeper SH: Dentist and laboratory: a love-hate relationship, Dent Clin North Am 23:87-99, 1979. Quinn L: Status of the dental laboratory work authorization, J Am Dent Assoc 79:1189-1190, 1969. Travaglini EA, Jannetto LB: A work authorization format for remov-able partial dentures, J Am Dent Assoc 6:429-431, 1978. 371 Index 371 Index A abutment teeth application of fluoride gel to, 32 artificial teeth as, 48 avoiding damage to, 297 benefits of interim dentures to, 313 cast restorations for, 208 classification of, 206 contour adjustments to, 141f contour of, 173 crowns for, 208-212, 209f decay of, 296 with guarded prognosis, 179 isolated, 214f ledges on abutment crowns, 209-211 lever action on, 117f loss of, 308, 341 with missing anterior teeth, 214-215 modification of natural, 60f occlusal force on, 59f occlusal rests on secondary, 101 preparation of, 120-121, 200-204 contouring, 28f, 207f, 283f contouring wax patterns, 201 on sound enamel, 206 prognosis for service of, 39, 169f protection of, 83, 123 rest seats, 201-204 restorations, 200-201 retention of, 91 retention on terminal, 38f soundness of, 179 spark erosion, 211 splinting of, 212-214 strain on, 10 support characteristics of, 23 temporary crowns, 215-216 tipped/tilted, 87f cast evaluation of, 59f contour adjustments with, 141f rest preparation with, 59 tooth-tissue contours, 283f unprotected, 206 use of isolated teeth as, 213-214 using conservative restorations, 206-208 veneer crowns on, 212 vulnerable areas of, 201 acrylic-resin facings, 274 acrylic-resin material, 277 for artificial teeth, 251 for bonding, 111 for denture bases, 106-108, 215, 256, 274-277 for impression trays, 226-230 for teeth, occlusion on, 109-110 acrylic-resin record bases, 247f ADA specifications, No. 7, 183t additions to dentures, 308, 309f adhesion forces, 105 adjustment to dentures by patients, 289-290, 297 adjustments to bearing area/surfaces, 290 breakage during, 307-308 final/finishing, 290 to interim partial dentures, 314f occlusal (See occlusal adjustments) to tipped/tilted abutment teeth, 141f aesthetics. See esthetics agar-agar, 221 age, tooth loss patterns and, 2-3 AIDS infection control, 174 allografts, bone, 327-328 alloplastic materials, 189-190 alloys. See metal alloys alveolar bone, augmentation of, 189-191 alveolar lamina dura, 167-169 alveolar ostectomies, 325 anatomic consequences of tooth loss, 5 anatomic form impressions, 236 anatomic references, 159f anatomic ridge, 14f angle of cervical convergence, 87-91 anterior ridge defects, 178f anterior teeth age and loss of, 3 replacement of, 215 types of, 273-274 appearance. See esthetics Applegate’s rules for applying Kennedy classifications, 20 arbitrary axis, 160 arbitrary blockouts, 147, 148t arches, Class I, II, III, IV, 97f arm assemblies, 71 cross-sectional form of, 91 design of, 71 diameter of arm, 91 flexing action of, 68f, 88 length of arm, 90 material used for, 91 position of, 57 reciprocal arms, 79f counteraction of reciprocation, 68f functions of, 69-70, 209 incorrect relationship to retentive arms of, 209f rigidity of, 92f, 124 retentive arms, 73f, 80f-81f cast clasp, 90f casting of, 114 engagement of undercut with, 79f incorrect relationship to reciprocal arms of, 209f placement of, 91-92 soldering, 261f wrought-wire, 183f stabilizing-reciprocal cast clasp arms, 92 types of, 71-72 uniformity of retention, 91-92 wrought-wire clasp arms distortion of, 80 location of, 179f arm designs, 27f arm functions, 30f, 69-70 arrangement of teeth, establishment and verification of, 13 articulation ribbon, 294 articulators, 158, 245, 291-292 artificial teeth as abutments, 48 arrangement/placement of, 244f-245f, 251, 257-258, 323 attachment to denture bases, 106, 109-111, 111f with acrylic-resin, 109 chemical bonding, 110-111 metal, 110 porcelain/acrylic-resin, 109 resin, 109-110 attachments to, 126f esthetic considerations for, 137-138 framework for supporting, 110-111 functionally unfavorable positioning of, 245f implants (See implants) in interim obturator prostheses, 321f materials for posterior teeth, 251 porcelain, 109, 111f, 281 porcelain facings, 274 porcelain veneer crowns, 212f ready-made, 109-110 reattaching, 304 reorientation of, 113 resin, 303-304 stock porcelain or resin, 111f types of anterior, 273-274 uncovering of, 274 using smaller, 122f assessment. See diagnosis/assessment atmospheric pressure, 105 atrophy of tissues, 103 attachments of artificial teeth to bases (See under artificial teeth) f indicates figure; b, box; t, table. arm assemblies (Continued) 372 Index of facings, 109 internal (See internal attachments) internal clip, 128-129 locking internal, 94 muscle, 188-189 position for Classification II, modification 1 maxillary removable partial dentures, 95f of resin denture bases, 106f of resin teeth, 273 resin tube teeth, 109 rigid, 26f types of extracoronal, 71 wrought-wire retainer arms, 256 augmentation of alveolar bone, 189-191 authorizations, work. See work authorizations autopolymerizing resin, 300-301 auxiliary occlusal rests, 98f axial surfaces, recontouring of, 62f axis of rotation, 94f, 96, 98f, 235 longitudinal, 24, 25f vertical, 24-26, 25f axles, 22, 23f B back-action clasps, 83-84, 84f ball type rests, 34 ball-and-socket relationships, 202 bar retainers, 77f bar-type clasp assemblies, 73-76, 75f-76f, 126f contraindications for, 78f design of, 77f errors and corrections in, 78f I-bars, 80f inability to use, 74 indications for, 74-75 length of, 90 one-half T-type, 80f permissible flexibilities of, 91t placement of, 76f use of, 77f, 123 use of bar clasp arm, 77f bar-type major connectors anterior/posterior palatal, 40f, 45, 45f, 54-55 single palatal, 40f, 45, 54 baseplate wax, thickness of, 237f-238f bases. See denture bases beading, 45, 45f-46f beading lines, 46f bearing areas/surfaces, 43f bearing area/surfaces, 290 biomechanics, 22 cantilevers, 26f considerations of, 22-23 design solutions and, 21-22 framework designs, 27f impact of implants on movement, 28 lever length, 26f machinery, 23f possible movements of partial dentures, 24-28 removable partial dentures, 22 cantilevers, 26f considerations of, 22-23 framework designs, 27f impact of implants on movement, 28 lever length, 26f possible movements of partial dentures, 24-28 rotational forces, 25f blocking out the master cast, 145-146 blockout wax, 136f blockouts, 123, 146f, 254, 268-269 parallel, 145f-146f shaped, 147, 148t summary of types of, 148t boil out procedure, 277 bonding acrylic-resin, 111 chemical, 110-111 bone augmentation, 189-191 bone density, 166-167, 331-332 bone grafts, 199f, 327-328 bone loss, 5, 178, 178f bone resorption, 187f bony enlargements exostoses and tori, 188 spicules, 189 bony ridge, 236 bony spines, 189 bony undercuts, 140 Brånemark system, 189, 190f breakage. See repairs buccal guide bars, 336-337 buccal retention, versus lingual, 80 buccal surface modification, 173f bulk, sensation of, 312f bulk of denture bases, 108 burnout operation, 264, 265f C calculus, 196 on acrylic-resin bases, 108 removal of, 195 salivary, 295 cancers of oral cavity, 317f-318f, 327 canine abutments, 213 canine crowns, 138f canine extensions, 100, 100f canine rests, 63-64, 100 cantilever designs, 23, 26f carbon markers, 132f caries beneath clasp components, 10 control of, 151-152 rests and, 32 risk assessment for, 5t susceptibility evaluation of, 153-155 carious lesions, 195-196 cast metal bases, 107-108. See also gold/ gold alloys; metal alloys cast molar designs, maxillary, 112f cast supports, 161 cast veneer crowns, lingual rests on, 34 casts/casting, See also laboratory procedures. impression-related entries abutment teeth, diagnostic cast evaluation of, 59f accurate mounting of, 247f anatomic and functional ridge forms, 234f casting methods, 265 dental cast surveyors, 12, 13f diagnostic, 157f, 186f, 202f, 227f-228f diagnostic casts, surveying, 133-134, 137-138, 141f-142f factors influencing excellence of, 262b framework design, 258f injection molding, 278-279 for interim partial dentures, 315f machining, 136 making the cast crown, 217f master casts, 254f beading lines, 46f examples of, 256f minor connectors, 50f refractory, 46f repouring of, 277 maxillary casts, beading of, 45, 46f milling of, 211f mounted maxillary, 165f, 315f preparation of base, 160f reciprocal arms, 79f refractory casts, 259f-260f remounting, 281-282, 281f of residual ridges, 122f solid cast example, 208f trimming, 229f-230f voids in casts, 276 cementation, of temporary crowns, 216 Centers for Disease Control (CDC), infection control procedures, 174 centric relations on record bases, 163-164, 248 ceramic veneer crowns, surveying, 135 cervical convergence angle of, 86f size of/distance into angle of, 88-90 chain of infection, 174 Chayes, Herman E. S., 71, 93 chewing strokes, 294 chewing studies, 5, 339. See also mastication chrome alloys, 110 chromium-cobalt alloy rest seats, 34-35 chromium-cobalt alloys, 91t, 128-129 cingulum (continuous) bars, 31f, 33f, 36-37, 37f, 53 functions of, 126 linguoplates and, 100 cingulums, shape of, 32 circumferential clasps, 71f, 83f back-action, 84f improper applications of, 81f permissible flexibilities of, 91t types of cast, 85 attachments (Continued) biomechanics (Continued) 373 Index clasp arm assemblies, 71 arm designs, 27f arm functions, 30f, 69-70 cross-sectional form of, 91 design of, 71 diameter of arm, 91 flexing action of, 68f, 88 length of arm, 90 material used for, 91 position of, 57 reasons for breakage of, 305-306 relationship of retentive and reciprocal, 209f repairs to clasp arms, 305-306 retentive clasp arms, 80f-81f stabilizing-reciprocal cast clasp arms, 92 as stress breakers, 124 types of, 71-72 uniformity of retention, 91-92 veneers for support of, 212 clasp retention amount of, 88f importance of, 67-68 primary/secondary, 68 reasons for failure of, 14-15 on terminal abutments, 38f uniformity of, 91-92 clasps/clasp assemblies, 60f with alternating buccal and lingual retention, 335f amount of retention of, 88-92 analysis of tooth contours for, 86-87 assembly types, 30f, 47f back-action, 83-84 bar-type, 68f, 70f, 75f bilateral opposition of retentive clasps, 69f cast, 127 circumferential clasps (See circumferential clasps) design of, 68-70 embrasure, 82-83 fitting crowns inside clasps, 216 functions and positions of, 86t hairpin clasps, 80f half-and-half, 84-85 lack of edentulous space for clasping, 82-83 with maxillary defects, 328 multiple, 84 placement of, 28f, 145f preformed plastic patterns, 256f retention without, 344f reverse-action, 85 ring-type, 81-82, 83f RPI, RPA, and bar clasps, 73-80 selecting a design, 71-72, 92-93 tapered circumferential clasps, 259f types of, 72-85 undesirable, 85 vertical occlusal forces on, 93f Class I, II, III, IV partial dentures. See under Kennedy classifications classifications of partially edentulous arches Applegate’s rules for applying Kennedy classifications, 20 confusion/disagreement about, 16 examples of, 17f, 19f Kennedy method (See Kennedy classifications) requirements for acceptable, 17 clenching/grinding, 197f clinical examinations, 151-164 clinical procedures occlusal adjustments, 196-197, 281f, 292f-293f initial, 290 with natural and artificial dentition, 291-294 placing interim partial dentures, 314 clip attachments, 129 cohesion, 105 colloidal materials, duplicating, 254 combination clasps, 76-80, 79f, 118 combined techniques, 187f commercially pure (CP) titanium, 182-183 comfort issues, 44f abused/irritated tissues, 191-193 adjustments to bearing surfaces for, 290 impingement (See impingement problems) implant use and, 343f provision of tissue relief, 261f relief of pain/discomfort, 152 surgical procedures for, 325 tongue interference, 45f transfer of forces for, 103 complete coverage restoration, 252 complete coverage restorations, 206, 252 complete palatal coverage major connectors, 42f, 55 component parts, adding, 35-36 composite materials, 215 composite support, 120 conditioning of abused/irritated tissue, 191-193, 193f conditioning teeth, with abutment teeth, 313 connector areas, outlining, 44 connectors major (See major connectors) minor (See minor connectors) tissue reaction to metallic coverage, 49-50 conservative restorations, 206-208 continuous bars. See cingulum (continuous) bars contour height, 86-88, 87f, 134f, 142f, 173f, 306f contouring of abutment teeth. See abutment teeth; recontouring contouring of major connectors, 31-32, 34-35 contouring wax patterns, 134-135, 134f, 201, 203f contours of denture bases for contact with tongue, 108 occlusal, 345f palatal, 41f-42f residual ridge, 232-233 tooth analysis of, 86-87 ineffective, 89f unfavorable, 38 tooth-tissue, 283f contraindications for bar-type clasps, 78f for hinged labial bars, 39 cosmetic considerations. See esthetics CP (commercially pure) titanium, 182-183 cross-arch distribution of stress, 45 cross-arch stability, 29 crown restorations, 124 for abutment teeth, 208-212, 209f canine crowns, 138f ceramic veneer crowns, 135 crowns remaining in impressions, 215 fabricating to accommodate rests, 32 fitting crowns inside clasps, 216 ledges on abutment crowns, 209-211 making the cast crown, 217f metal-ceramic, 135f, 209f metal-ceramic crowns, 208f restoration of multiple, 208 surveyed crowns vs. implants, 340-341 temporary crowns, 152, 215-216 three-quarter crowns, 208 veneer crowns for esthetics, 201 lingual rests on, 34 porcelain, 212f for support of clasp arms, 212 custom impression trays, 34f cyst removal, 187-188 D decision making choice of major connectors, 332-333 for implant use in stability, 340 influence of patient interviews on, 165 position choices for surveying, 139f shared, 8-9, 151, 158f using anterior segments on dentures, 214-215 using isolated teeth as abutments, 214 deformities, dentofacial, 189 deglutition, 326 dental arch, shortened, 6, 176-177 dental cast surveyors, 12, 13f Dental Health Care Workers (DHCWs), 174 dental surveyors. See surveyors/surveying dentist-patient relationships failure of, 15 rapport, 151 shared decision making, 151, 158f dentofacial deformities, 189 classifications of partially edentulous arches (Continued) 374 Index denture bases accuracy of form of, 107-108, 121 attaching artificial teeth, 106, 109-111, 111f clasp-retained, 15 contours for tongue/cheek contact, 108 distal extension partial (See distal extension denture bases) function in movement control of, 103-106 ideal material for, 106-108 irritation to tissue by, 192f-193f materials for, 278 need for relining, 112-114 overlay abutments for support of, 129 purpose/function of, 103 stress-breakers (stress equalizers), 112-114 tooth-supported, 103-104 weight of, 108 denture bearing foundation, 192f design considerations and principles articulated partial dentures, 114 biomechanics and, 21-22 clasps, 89f, 118, 141 for clasps, 71-72 component partials for support, 129 components of design, 120-126 for controlling movement, 28 for denture bases, 106f differentiation of types of removable partial dentures, 116-118 direct retainer assemblies, 60f for distal extension removable partial dentures, 26f essentials of developing the design, 119-120 for esthetics, 36 first premolar design, 27f frameworks, 235, 258f, 332 implant considerations, 126 influence of prosthesis support differences on, 115-116 interim partial dentures, 315f internal clip attachments, 128-129 joining of components to supports, 120 Kennedy classifications (See Kennedy classifications) major connectors, 37f for maxillary connectors, 42-44 for maxillofacial prosthetics, 323 for minor connectors, 48, 49f of minor connectors, 124 objectionable, 44f overlay abutments, 129 with questionable abutments, 205 rationale for, 22 for retention, 106 splint bar use, 128 systematic approach to (examples), 126-128 DHCWs (Dental Health Care Workers), 174 diagnosis/assessment. See also treatment planning for clasp-retained partial dentures, 14-15 clinical examination, 151-164 oral examinations, 152-156 treatment objectives for, 151-152 criteria for classification of partially edentulous arches, 16 diagnosis phase of service, 12-13 diagnostic casts, 156-164, 157f jaw relations records for, 162-163 mounting, 158-162 purposes of, 156-158 recording centric relation, 163-164 surveying, 137-138, 141f differential diagnosis, 174-179 indications for removable partial dentures, 177-179 use of fixed restorations, 175-177 findings, 164-184 infection control procedures, 174 interpretation of data, 165-174 caries risk assessment, 170 determination of mandibular major connectors, 172 distal extension spaces, 171 endodontic treatment need, 171-172 fixed restorations, 172 longer modification spaces, 171 occlusal factors, 172 orthodontic treatment need, 172 periodontal considerations, 169-170 prosthesis foundation, 170 radiographic, 166-169 reshaping of remaining teeth, 172-174 short modification spaces, 171 surgical preparation need, 170-172 of loss of occlusion, 112 need for relining, 113 for oral rehabilitation, 151-152 patient interviews, 150-151 periodontal, 194-195 preoperative, 325 for pulpal involvement, 186f recording data (sample form), 153f-154f waxing for management of replacement teeth, 176f working chart for (sample form), 155f diatoric holes, 109 direct apposition of casts, 246 direct retainers, 4f, 24, 60f amount of retention of, 88-92 analysis of tooth contours for, 86-87 arm functions, 30f, 69-70 clasp-type (See clasps/clasp assemblies) description of, 67-68 design of, 68-70, 120, 123-124 extracoronal, 73f function of, 120 implants as, 92 internal attachments, 10, 71, 93-95, 95f advantages/disadvantages of, 93-95 nonlocking, 94f principle of, 93 types of, 94 mechanical laws of, 69f need for new, 308 repairs to, 306f role in movement control of, 67-68 selecting a design, 71-72 types of, 70-85, 80f, 92-93 direction of effort (force), 22, 24f discontinuous mandibles, 327 disease management, 165-166 diagnosis of periodontal disease, 194-195 initial disease control therapy phases, 198-200 periodontal disease, 213f radiographic validation of disease, 166 soft tissue disease, 327 displaceability of periodontal ligament, 25f displacement, limiting the freedom of, 68 distal extension denture bases, 13, 97f, 104-106, 177, 252 accuracy of fit of, 235 anatomic form impressions, 236 description of, 232 designs for, 26f evaluation of framework, 239f horizontal rotational tendencies of, 147 retention from, 101-102 rotation of, 23, 25f, 235f support for, 13, 232-241 underextended, 233f distal extension obturators, 325f distal extension partial dentures direct retainers for, 123-124 methods for treating, 234 support for, 121 support from, 104 disto-occlusal rest, 26f distortion of components, 307-308 disuse atrophy, 103 divesting of the framework, 266f double occlusal rests, 80f double-strap connectors, 44 dovetail internal attachments, 94 dual placement paths, 215 duplication molds, 255f duration of force, 22 E economic considerations, 179, 214 education, patient. See patient instructions effort arms, 26f, 73 elastic impression materials, 221-222 irreversible hydrocolloids, 221 mercaptan rubber-base, 221-222 polyether, 222 reversible hydrocolloids, 221 silicone, 222 elastic materials for impressions, polyether, 222 elastomeric interocclusal registration material, 163, 164f electric soldering, 308-310 direct retainers (Continued) 375 Index electropolishing units, 267f embrasure clasps, 82-83, 83f-84f embrasure spaces, 47f enamel defects encountered in, 32 rest seats in, 32 sound, 32, 206 enamelplasty procedures, 62f encirclement principle, 68 endodontic treatment, 171-172 errors in bar clasp designs, 74-75 in class assembly design, 92-93 improper applications of circumferential clasps, 81f improperly designed ring clasps, 82f inaccurate/weak casts, 226 incorrect relationship of retentive and reciprocal clasp arms, 209f when repositioning casts on surveyors, 143 erythroplasia, 189 esthetics artificial teeth and, 137-138, 321f of clasps, 10 cosmetic deformities, mandibular, 327 denture bases and, 104 gingival carving, 282-283 impact of tooth loss and, 5 interim partial dentures for, 311, 312f with metal bases, 108 path of placement considerations for, 140-141 placement of clasp arms and, 91-92 of prosthesis, 9 of resin teeth, 274 reverse-action clasp arms, 85f of ring-type clasps, 82 shape and position of denture teeth, 345f simulation of living tissue, 282-283 veneer crowns for, 201 wrought-wire clasp arms, 80 evaluations. See diagnosis/assessment examination, clinical. See clinical examinations; diagnosis/assessment exostoses, 155, 188, 188f extended occlusal rest seats, 59 extended occlusal rests, 59, 59f extension base removable partial dentures, 10 extension bases, 10 external rests, preferred sites for, 32 extracoronal (clasp-type) retainers, 10, 11f, 70, 73f. See also clasps/clasp assemblies; direct retainers advantages of, 92 benefits/disadvantages of, 10 failure of, 14-15 forms of, 71 misuse of, 86 types of, 80f visibility of clasps, 14 extractions, 170-171, 177, 186, 186f, 205 F fabrication of components, 129 fabrication of removable partial dentures, 249 facebows, 159, 159f, 251-252 fork supports used with, 162f mounting the cast in, 162f orienting fork of, 161f positioning of, 161f facial surfaces, 282 facings, attaching, 109 failure of removable partial dentures component failures, 307-308 reasons for, 152 fees for examinations, 156 finger extensions, 100 finish lines, 48f-49f, 49, 52f finishing/polishing casts, 265-266 electropolishing units, 267f esthetic features, 275f gross finishing, 266f polishing the framework, 267f first-class levers, 26f-27f fixation to teeth, 319f fixed partial dentures, 3f, 187f. See also fixed partial dentures Brånemark system, 190f indications for, 175-177 modification spaces, 175-176 replacement of unilaterally missing molars, 176-177, 177f tooth-bounded edentulous regions, 175 preparation of, 130 flap procedures, 198 flask units, 276f flasking, 277-278, 311 flasks duplicating, 254 processing, 276f flexibilities of clasp arms chromium-cobalt alloys, 91t Type IV gold alloys, 91t floor height, 34, 156 floor taper for rest seats, 63f fluoride gel, 32 follow-up treatment, 296 denture service (periodic recall), 296-297 periodic recalls, 14 food bolus size, 6 food debris, 49, 74-75 food entrapment, 37f food reduction index, 6 food traps, avoiding, 36 forces on adjacent teeth, 325 Boucher’s description of, 105 on denture base, 27f direction of effort (force), 24f, 77f dislodging, 89f, 330-331 displacement, 67-68 distribution of, 46, 244f downward, 147f functional, 9-10, 314 gravity, 24 horizontal, 26 lifting, 88f, 119f locations for resisting, 340 masticatory, 73f occlusal, 93f, 119f resistance to, 28f for retention, 105 rotational, 25f toleration of, 22-23 transfer of, 103 fork supports, 162f form of occlusal rests and rest seats, 58-59 form/location of minor connectors, 47-48 frameworks/framework design biomechanics and, 27f clasp-retained partial dentures, 15 design considerations and principles, 235, 258f, 332 evaluation of framework, 239f gold frameworks, 192f laboratory procedures divesting of the framework, 266f evaluation of framework, 258f frameworks from master casts, 254f, 256f polishing the framework, 267f steps in making frameworks, 257f wax patterns, 254-261, 260f, 268f for major connectors, 30f, 45f mandibular framework, 4f maxillary framework, 4f for minor connectors, 30f supporting artificial teeth, 110-111 using maxillary teeth, 61f Frankfort plane, 165f freedom of displacement, limiting the, 68 frena, 188-189 frequency of force, 22 frictional resistance, 72f, 81 fulcrum, 23, 23f-24f change in fulcrum point, 27f rotation around, 25f fulcrum lines, 24, 96, 97f-98f inaccessible, 100 location for indirect retainer on, 100f rotation around, 113 full dentures, transitioning patients to, 313-314 function gain of, 153f-154f loss of, 5, 7 functional forces, 9-10 concentration of, 314 transfer of, 29 functional impression technique, 241 functional load, 115 distribution of, 245f normal resistance to, 323 functional movement, clasp designs to accommodate, 73-80 without accommodating, 80-85 functional restorations, prostheses with, 5-6 forces (Continued) 376 Index functional stress, 343f functional support, for distal extension denture bases, methods for obtaining, 236-241 furcation entrance, closing, 199f fusing layers of denture bases, 111 G gagging reflex, 295 geometry, preparation, 67 gingiva anatomic changes to, 5 recession of, 200f gingival areas, finishing, 282-283 gingival carving, 282-283 gingival forms, 274 gingival tissue, height of, 274-275 Glossary of Prosthodontic Terms, 2-3, 158 gold/gold alloys, 181. See also metal alloys cast gold, 91 gold frameworks, 192f gold solder, 308 occlusal adjustments with, 110 physical properties of, 181-183 Type IV, 91t grafts bone, 199f, 327-328 mucosal, 189 skin, 189, 319, 320f, 324f, 325-326, 329f, 332, 333f, 334-335 gravity, 24 grinding/clenching, 197f GTR (guided tissue regeneration), 197-198, 199f guided tissue regeneration (GTR), 197-198, 199f guiding planes, 4f, 76f of bar-type clasp assemblies, 76f contacts, 179f design/function of, 125 facing, 125f full use of, 48 functions of, 125 location of, 125 opposing, 9-10 parallelism and, 90, 145f, 211 preparation of, 201 purpose of, 137 surfaces of, 124f-125f surveying casts, 138-139 H hairpin clasps, 80f, 83f, 85f half-and-half clasps, 84-85, 85f half-round retentive arms, 27f, 80f hand hygiene, 174 hard palatal defects, 326 hard palate defects, 317f health, major connectors and, 32b height of contour, 86-88, 87f height of floor of mouth, 34, 34f hemangiomas, traumatic, 189 hepatitis virus control, 174 hinge axis mountings, 245 hinges, 38f HIV infection control, 174 horizontal forces, 26 hydrocolloid impression materials irreversible hydrocolloids, 32, 122f, 221, 303f precautions for handling, 223 reversible hydrocolloids, 221 stone casts from, 225-226 hygiene, oral. See oral hygiene hyperkeratoses, 189 hyperplasia fibrous tuberosities, 188f hyperplastic tissue, 188 I I-bars, 261f impacted teeth, 186-187 impingement problems avoiding, 147f by bar clasp assemblies, 76f by major connectors, 31-33 implants. See also artificial teeth anatomic concerns for Class I and II arches, 340 Brånemark system, 190f clinical examples of, 341 implant-assisted Class III denture, 345f long span modifications across mandibular midline, 343f parallel path of insertion, 342f residual ridge resorption with strategically missing tooth, 343f support from cingulum and mesial-occlusal rests, 344f as direct retainers, 92 excessive force on, 192f functional ability and replacing anatomy, 339-340 implant bars/natural tooth coping, 191f implant-only support, 338 for improving ridge support, 122 influence of opposing occlusion on, 341 movement control with selective placement of, 340-341 physiologic distinction between prostheses and, 339 placement of, 92, 340 as rests, 57, 66 roles for, 28 strategic placement of, 340 strategically lost teeth and, 341 for support vs. retention, 340-341 surveyed crowns vs., 340-341 titanium endosseous, 189 treatment planning, 341 clasp assembly requirements for using adjacent tooth with, 341 influence of anatomic and opposing occlusal factors on placement of, 341 survey considerations, 341 impression materials, 225-226, 233-234. See also casts/casting elastic, 221-222 mercaptan rubber-base, 221-222 polyether, 222 silicone, 222 hydrocolloid (See hydrocolloid impression materials) mercaptan rubber, 230 rigid, 219-220 metallic oxide paste, 220 plaster of Paris, 163, 219-220 thermoplastic, 220-221 modeling plastic, 220-221 waxes and natural resins, 221 impression procedures, 222-226. See also laboratory procedures adding handles, 227f-228f air barrier coatings, 229f-230f anatomic forms, 236 causes of inaccurate/weak casts, 226 crowns remaining in, 215 for distal extension removable partial dentures, 302 functional impression technique, 241 open-mouth technique, 303f registration, 117-118, 127 elastomeric interocclusal registration material, 164f type/accuracy of, 233-234 secondary impressions, 237f-238f selective tissue placement, 237-241, 240f use of existing dentures as base for relining, 300f impression trays acrylic-resin, 226-230 individual, 157 stock, 226, 281f, 282 stock versus custom, 34f impressions anatomic form, 236 distorted, 223 final, 230 purpose of, 231 incisal rest seats, 64-65 incisal rests, 64-65 lingual versus, 32 placement of, 35 incisors, lingual rests on, 63-64 inclined plane, 22 index areas, 167 indications for fixed restorations, 175-177 modification spaces, 175-176 replacement of unilaterally missing molars, 176-177, 177f tooth-bounded edentulous regions, 175 indications for removable partial dentures, 177-179 abutments with guarded prognosis, 179 after recent extractions, 177 distal extension situations, 177 economic considerations, 179 excess loss of residual bone, 178 long spans, 177-178 need for bilateral stabilization, 178 unusually sound abutment teeth, 179 indicator paste, 290 indicator wax, 293-294, 294f indirect retainer principle, 98f, 120 impression materials (Continued) 377 Index indirect retainers, 4f, 24, 41f auxiliary functions of, 99 design/function of, 125-126 effectiveness of, 99 forms of, 99-102 functions of, 120, 126 with palatal plate-type major connector, 98f placement of components of, 96-99, 125-126 planning the location of, 98f rests for, 35 role in movement control of, 96-99 individual impression trays, 157, 226-230 infection control procedures, 174 infrabulge region, 87f, 88 initial placement procedures, 13-14 injection molding, 278-279 inlays, 207-208 insertion paths, 138f instability, of removable partial dentures, 11 instructions to patients. See patient instructions intaglio surface of occlusal rests, 60f interfacial surface tension, 108 interference designing to avoid, 137 eliminating gross, 143 from frena, 189 surveying for, 140 interim obturator prostheses, 320f-321f, 323f with distal extension, 325f with palatopharyngeal defects, 324f interim partial dentures adjustments to, 314f for appearance, 311, 312f clinical procedure for placing, 314 conditioning teeth and residual ridges with, 313 conditioning the patient with, 313-314 design of, 315f indications for, 311 reestablishing occlusal relationships with, 313 restoration during treatment with, 313 space maintenance with, 312, 312f types of, 313 internal attachments direct retainers, 10, 71, 95f advantages/disadvantages of, 93-95 nonlocking, 94f principle of, 93 types of, 94 placement of, 135 internal clip attachments, 128-129 internal occlusal rests, 61, 61f internal rest seats, 32 placement of, 135-136 internal rests, 31-32 depth of, 32 lingual retention with, 92-93 interocclusal positions, 164f, 341 interocclusal records, 246-247, 337 interocclusal space, 60f interpretation of diagnostic data, 165-174 caries risk assessment, 170 determination of mandibular major connectors, 172 distal extension spaces, 171 endodontic treatment need, 171-172 fixed restorations, 172 longer modification spaces, 171 occlusal factors, 172 orthodontic treatment need, 172 periodontal considerations, 169-170 prosthesis foundation, 170 radiographic, 166-169 reshaping of remaining teeth, 172-174 short modification spaces, 171 surgical preparation need, 170-172 interproximal areas, finishing, 282-283 interproximal occlusal rest seats, 59-61 interproximal ring clasp assemblies, 80f interviews, patient. See patient interviews intracoronal rests, 31-32 intracoronal retainers, 70, 72f. See also direct retainers fabrication of, 71 placement of, 135 investing/investments applying, 264f investing before processing, 275-277 investing sprued patterns, 263-264 preparing, 265f removing casting from investments, 265 tinfoil-lined matrixes, 278 wax burnout procedure, 265f irreversible hydrocolloids, 32, 122f, 221, 303f irritating factors, removal of, 195-196 J jaw deformities, 189 jaw relation records, 162-163, 245-246, 248, 252, 270 jaw relations, 318f establishing, 270f interocclusal records for, 337 mandibular removables opposing maxillary complete dentures, 251-252 Jelenko surveyors, 131, 131f-132f K Kennedy classifications, 17-20 advantages of, 20 Applegate’s rules for applying, 20 Class I and II arches, 126, 175, 241 Class I-IV, 17-20 Class I, 27-28, 99, 123, 244f, 248f, 300f, 340 anatomic concerns of implants with, 340 bilateral, distal extension, 127 levers, 76-78 mandibular, 35f maxillary distal extension, 104f modification 1, 259f, 291f, 300f modifications to, 4f palatal plate major connectors, 42f partially edentulous arches, 116f Class II, 27-28, 79f, 127-128, 244f, 332, 340 anatomic concerns of implants with, 340 indirect retainers for, 99-100 mandibular, 100f, 127f maxillary, 101f modification 1, 95f, 101f, 104f, 106f, 139, 179f, 192f examples of, 17f-18f, 126-128 extended occlusal rests with, 59 follow-up support service for, 296-297 maxillary, 95f rules governing, 19b modifications to, 4f Class III, 27-28, 126-128 components of, 120 design of, 126f modification 1, 116f modification 2, 173f Class IV, 27-28, 175-176 contact of opposing anterior teeth, 244 defects, 178f tissue surface of, 110f kinematic axis, 160 knife-edge ridges, 189 L labial bars, 33f, 37-39, 38f, 54 laboratory procedures. See also casts/ casting; work authorizations; impression-related entries adding monomer, 269f arranging posterior teeth to opposing casts/templates, 272-273 to an occluding surface, 273 posterior tooth forms for, 273 attaching wrought-wire retainer arms, 256 burnout operation, 264 casting methods, 265 for clasp-retained partial dentures, 15 divesting of the framework, 266f duplicating stone casts, 253-254 materials for, 255f molds, 255f evaluation of framework, 258f factors influencing excellence in casting, 262b finishing/polishing casts, 265-266 electropolishing units, 267f esthetic features, 275f gross finishing, 266f polishing the framework, 267f forming wax patterns, mandibular Class II frameworks, 255-256 frameworks from master casts, 254f examples of, 256f gold solder for repairs, 308 I-bars, 261f investments applying, 264f preparing, 265f Kennedy classifications (Continued) 378 Index before processing, 275-277 removing casting from, 265 wax burnout procedure, 265f jaw relation records, 270f making record bases, 267-269 occlusion rims, 270-271 polishing the denture, 282-283 denture borders, 282 facial surfaces, 282 gingival and interproximal areas, 282-283 preformed plastic patterns, 256f processing flasks, 276f processing the denture, 277-280 record bases, sprinkle-on technique, 268f refractory casts, 255f remounting and occlusal corrections to occlusal templates, 280-282 precautions in remounting, 280-282 sequence of, 292f-293f soldering wrought-wire, 261f-262f spruing, 262-264, 263f-264f steps in making denture frameworks, 257f stone occlusal templates from functional occlusal records, 271-272 types of anterior teeth used, 273-274 wax patterns, 259f waxing of base, 275f the framework, 254-261 metal bases, 256-261 before processing, 274-275 lamina dura, alveolar, 167 land space, 277 latching mechanisms, 38f lateral discontinuity defects, 337 ledges on abutment crowns, 209-211 forming the crown ledge, 210 preparation of, 210f as terminal stops, 210 wax, 145f legal aspects of work authorizations, 287 lever action, on abutment teeth, 117f lever systems, 79f leverage effects, reducing, 245f leverage principle, 29 levers, 22-23, 23f classes of, 24f first-class, 26f length of, 26f lifting forces, 119f lingual aprons adapting, 35f adding, to clasp arms, 71-72 lingual bars, 30f-31f, 32-34, 33f, 37f-38f, 52 location of, 31 purpose of, 123 lingual finishing lines, 52f lingual rest seats, 63f lingual rests on canines and incisors, 63-64 chromium-cobalt, 65f incisal versus, 32 tooth preparation for receiving, 32 lingual retention, 80, 92-93 linguoplates, 31, 33f, 34-36, 35f-36f, 42f-43f, 52-53, 123 cingulum bars and, 100 functions of, 126 lip movement, 312f load-displacement character, 22 loading, abnormal, 167 locking internal attachments, 94 long spans, 177-178 longitudinal axis, rotation around, 24, 25f luting agents, 67 M magnitude of force, 22 major connectors, 4f, 41f anterior/posterior palatal bar-type, 40f, 45, 45f, 54-55 beading of maxillary cast, 45, 46f characteristics of, 32b, 43f choice of, 332-333 cingulum (continuous) bars, 31f, 33f, 36-37, 37f, 53 complete palatal coverage, 42f, 55 design considerations, 37f design considerations for, 122-123 framework for, 30f, 45f height of floor of mouth, 34f interference to, 140 junction of minor connectors and, 49, 52f labial bars, 33f, 37-39, 38f, 54 lingual bars, 30f-31f, 32-34, 33f, 37f-38f, 52 lingual finishing lines, 52f linguoplates, 33f, 34-36, 35f-36f, 42f-43f, 52-53 mandibular, 32-39, 33f, 35f maxillary, 39-45, 43f palatal plate-type, 40-42, 42f palatal strap-type, 39-40, 40f-41f, 47f, 54 pleuralatal plate-type, 32f repairs to, 307-308 rigidity of, 123 role in control of prosthesis movement, 30-45 single palatal bar-type, 40f, 45, 54 sublingual bar-type, 31f, 33f, 36, 53 U-shaped palatal, 40f, 44-45, 44f, 55 malpositioned teeth, 187, 187f management of partial edentulism, 2, 8 mandible resections, 326f, 327 mandibular anterior teeth, replacement of, 178 mandibular bilateral distal extension dentures, 3f mandibular bilateral distal extension partial dentures, 30f mandibular Class I design, 100f mandibular Class II, modification 1 partially edentulous arches, 127f mandibular Class II design, 100f, 127f mandibular defects, 326 Cantor and Curtis classifications of partial mandibulectomy, 332f Type I, 331 Type I resections, 332, 333f Type II resections, 332-334, 334f-335f Type III resections, 334 Type IV resections, 334 Type V resections, 334-335 mandibular denture bases, 105f mandibular framework, 4f mandibular lingual bars, 52 mandibular major connectors, 32-39, 33f, 35f mandibular prostheses, occlusal view, 104f mandibular reconstruction, 327-328 complications of, 328 mandibular ridge, 252 mandibulectomies, 332f marginal ridge, 60f master casts, 50f, 234f. See also casts/ casting; laboratory procedures accuracy of, 230 beading lines, 46f blocking out, 145-146 color-coded markings on, 287 with fluoride gel use, 32 modification of, 136f refractory, 46f repouring of, 277 surveying, 136-137, 143, 146-147 mastication chewing studies, 5 food reduction issues of, 6 function of, 6, 326 interference adjustments for chewing strokes, 294 loss of function of, 5 process of, 6 restoration of, 326f support for, 340 masticatory forces, 73f, 236 materials for arm assemblies, 91 artificial teeth porcelain, 109, 274, 281 porcelain veneer crowns, 212f stock porcelain or resin, 111f for artificial teeth, 251 for bone augmentation, 189-190 cast metal base, 108 colloidal, 254 for conditioning of abused/irritated tissue, 193 deformation of, 341 for denture bases, 106-107, 278 for diagnostic casts, 156 for duplicating stone casts, 255f elastomeric interocclusal registration, 164f history of, 280 impression (See impression materials) indicator paste, 290 for interim obturator prostheses, 323f for interim partial dentures, 311 laboratory procedures (Continued) 379 Index metal advantages of, 107-108 attachment to denture bases, 110 esthetics with, 108 for palatal strap-type connectors, 109f tissue reaction to metallic coverage, 49-50 metal-ceramic crowns, 135f, 208f for relining tooth-supported bases, 300-301 for resin denture bases, 221 for temporary crowns, 215 tinfoil, 216 VLC, 279 maxillary arch, edentulism in, 3-5 maxillary casts, 107f mounting to axis-orbital plane, 158-162 maxillary Class III arches, 126f maxillary defects, 323-326 surgical reconstruction of, 326 maxillary denture bases, 105f maxillary framework, 4f maxillary major connectors, 39-45, 43f maxillary removable partial dentures, 30f maxillary teeth, framework design using, 61f maxilla-temporo-mandibular joint relationship, 158-159 maxillofacial prosthetics, 336f acquired defects following surgical removal, 317f Aramany’s classification for, 330f classification of defects, 317 defects and oral hygiene with, 322 description/need for, 316-317 design considerations for intraoral, 323, 329f clasping, 335f frame designs, 335f interim obturator prostheses, 320f-321f, 323f with distal extension, 325f with palatopharyngeal defects, 324f jaw relation records, 337 lack of hard palate, 317f mandible resections, 326f for mandibular defects, 326-327 bone grafts, 327-328 mandibular guide flange prostheses, 335-337, 336f mandibular prostheses, 331-337 evolution of, 331-332 potential complications, 320-322 maxillary prostheses, 328-331 obturator, 328, 330f palatal augmentation, 331 palatal lift, 330-331 speech aid, 328-330 obturation of defects by, 325-326 potential complications with, 320-322 resections affecting sinus and hard palate, 320f with segmental reconstruction, 327f support/stabilization issues with, 324f surgical preservation for benefit of, 323-328 surgical stents, 319f timing of dental care with, 317-323 interim care, 318-320 oral hygiene, 322-323 preoperative and intraoperative care, 317-318 Type I resections, 332 anterior mandible, 333f Type II resection and prosthesis, 334f Type II resections, 332-334 Type III resections, 334 Type IV resections, 334 Type V resections, 334-335 maxillomandibular relations, 252 mechanical forces, 22 of direct retainer assemblies, 69f mechanical locks, 105 melanoma, 318f mercaptan rubber-base impression material, 221-222, 230 mesial rest concept, 27f mesio-occlusal rest seats, 63f metal alloy, chromium-cobalt, 128-129 metal alloys, 79-80, 93, 181-183 cast metal base, 108 chrome, 110 chromium-cobalt, 91, 181 gold (See gold/gold alloys) retention characteristics of, 88 titanium, 182-183 metal materials advantages of, 107-108 attachment to denture bases, 110 esthetics with, 108 metal bases advantages of, 110f casting of, 106 coating with silica of, 112f for distal extension partial dentures, 113 partial, 109f waxing, 256-261 metal-ceramic crowns, 135f, 208f tissue reaction to metallic coverage, 49-50 metallic oxide bite registration paste, 163 metallic oxide paste, 220 methyl methacrylates, 215 microorganisms, exposure to, 174 milling machines, 211f minor connectors, 4f, 38f, 46-49 avoiding wedging effects with, 74f clasp assemblies, 47f curvature of, 125f in denture bases, 107f design considerations for, 122-123 design of, 48, 49f disengagement from guiding plane by, 235 embedment of, 276f embrasure spaces for, 47f finishing lines, 48f-49f, 49 form/location of, 47-48 framework for, 30f functions of, 46 junction of major connector and, 49 junction of major connectors and, 52f master casts, 50f occlusal forces on, 93f physiologic relief of, 333 proximal plate, 47f relief spaces under, 106f repairs to, 307-308 as stabilizing components, 124 terminal portion of, 51f tissue stops, 48-49, 50f-51f vertical rotation of, 125f modeling plastic, 163, 220-221, 270 modification areas Class II partial dentures, 101 support for, 101 modification spaces, 175-176 modifications to dentures, 48 molars maxillary, cast molar designs, 112f splinting of, 213 unerupted third, 169, 169f, 187f unilaterally missing, 176-177, 177f molding of tissues, 105 molds/molding (casts) duplicating stone casts, 255f duplication molds, 255f injection molding, 278-279 monomers, adding, 269f Mosby’s Dental Dictionary, 2-3, 12 mouth preparation, 12-13 abutment teeth (See under abutment teeth) clasp-retained partial dentures, 14 conditioning of abused/irritated tissue, 191-193, 193f jaw relation records, 162-163 for occlusal adjustment, 245-246 oral surgical preparation augmentation of alveolar bone, 189-191 bony spines and knife-edge ridges, 189 combined techniques, 187f cyst/tumor removal, 187-188 dentofacial deformities, 189 exostoses and tori, 188, 188f extractions, 186, 186f hyperkeratoses, erythroplasia, ulcerations, 189 hyperplastic tissue, 188, 188f impacted teeth, 186-187 malpositioned teeth, 187 muscle attachments and frena, 188-189 osseointegrated devices, 189 polyps, papillomas, traumatic hemangiomas, 189 removal of residual roots, 186 periodontal preparation, 194-200 advantages of, 200 diagnosis of periodontal disease, 194-195 initial disease control therapy phases, 198-200 objectives of, 194 treatment planning, 195 sequencing of, 12-13 materials (Continued) maxillofacial prosthetics (Continued) minor connectors (Continued) 380 Index movement (anatomical) jaws, 163 of tissues, 32, 166 of tongue, 326, 331 tooth mobility, 167, 194-195, 197-198 movement (of prostheses). See also stability/stabilization allowing independent, 114 biomechanical issues, 24-28 breakage caused by, 306f calculating amount of, 117f control of, 9-12, 126 distribution of stress for, 114 function of denture bases in, 103-106 impact of, 339 internal attachments and, 93-94 planes of movement, 96 reasons for, 329 of removable partial dentures, 21-22 role of indirect retainers in, 96-99 role of major connectors in, 30-45 role of rests in, 56-57 three-plane movement, 27 tissue support and amount of, 235f with tooth-supported dentures, 26-27 understanding of, 24 mucosa displacement of, 14f over bony protuberances, 188 residual ridge, 10, 241 mucosal grafts, 189 multiple clasps, 84, 84f muscle attachments, 188-189 muscles, repositioning of, 189 N National Association of Dental Laboratories (NADL), 287 natural teeth modification of, 60f occlusal adjustments to, 291-294 occlusal relationships between dentures and, 243f retention of, 180-181 support for prostheses by, 9-10 neuromuscular ability, 339 Ney surveyors, 131, 131f-132f nightguards, 197-198, 197f nonbearing areas, 43f outlining, 42 nonlocking internal attachments, 94, 94f nonsupporting teeth, 97f O obturator extensions, 320f obturators. See interim obturator prostheses occluding surfaces, arranging teeth to, 273 occlusal adjustments, 275, 281f, 291-292, 292f-293f for created discrepancies, 252 final, 293-294 with gold alloys, 110 indicator ribbon for, 294 indicator wax for, 293-294, 294f initial, 290 intraoral, 292-293 with natural and artificial dentition, 291-294 on relined removable partial dentures, 302-304 removal of interferences, 196, 292 Schuyler’s guide to, 196-197 occlusal contact areas of, 6, 13-14, 113f desirable arrangements for, 243-244 loss of, 296 occlusal contours, 345f occlusal disharmony, 246 occlusal forces, 59f, 93f, 119f, 233f occlusal function, 343f occlusal harmony, 15, 167, 242, 291-294 occlusal imbalance, 314 occlusal load, 27f, 113f, 122f, 167 distribution of, 57 intrusion to teeth of, 313 total applied, 121, 235-236 occlusal pathways, 248-251 occlusal plane, irregular, 318f occlusal records, 13 occlusal relationships accurate mounting of casts, 247f balancing side contacts, 244f Class I arches, 244f desirable, 243-244 development of, for Class II dentures, 244f establishment and verification of, 13 evaluation of existing, 155-156 functionally unfavorable positions, 245f jaw relation records, 248, 251-252 maxillomandibular, 248f methods for establishing direct apposition of casts, 246 interocclusal records with posterior teeth remaining, 246-247 occlusion rims on record bases, 247-248 recording occlusal pathways, 248-251 between natural dentition and removable partial dentures, 243f occlusal rest seats extended, 59 interproximal, 59-61 preparation of, 206 occlusal rests, 305-306 auxiliary, 74f canine extensions from, 100 double, 80f extended, 59, 59f form of, 58-59 internal, 61, 61f placement of, 236f preparation of, 58f repairs to, 307 occlusal splints, 313 occlusal vertical dimensions, 275-276 occlusion on acrylic-resin teeth, 109-110 enhancing, 157f establishing, 242-243 failure to maintain adequate, 242 inadequate, 302 loss of, 112 problems, with clasp-retained partial dentures, 15 reestablishment of, 110 occlusion rims, 247f, 270-271, 270f occlusion rims on record bases, 247-248 odontogenic tumors, 187-188 one-half T-type bar clasps, 80f open-mouth impression procedures, 302 opposing arch tooth positions, 116 opposing guide planes, 9-10 oral cavity, forces inherent in, 22 oral competence, 326 oral examinations, 152-156 sequence for, 152-156 oral function, measures of, 6 oral hygiene, 200-201 brushing of acrylic-resin dentures, 108 with maxillofacial defects and prosthetics, 322 patient instruction for, 195, 296-297 patient instructions for, 295 tissue reaction and poor, 49 oral prophylaxis, 152 oral receptor feedback, 339 oral rehabilitation, 151-152 oral structures, preservation of, 12 oral surgical preparation augmentation of alveolar bone, 189-191 bony spines and knife-edge ridges, 189 combined techniques, 187f cyst/tumor removal, 187-188 dentofacial deformities, 189 exostoses and tori, 188, 188f extractions, 186, 186f hyperkeratoses, erythroplasia, ulcerations, 189 hyperplastic tissue, 188, 188f impacted teeth, 186-187 malpositioned teeth, 187 muscle attachments and frena, 188-189 osseointegrated devices, 189 polyps, papillomas, traumatic hemangiomas, 189 removal of residual roots, 186 orthodontic procedures, 198 osseointegrated devices, 189 osseous tissue formation, 199f osteoplasty, 198 outcomes adverse, clasp assembly-related, 72 of current removable partial dentures, 7 patient’s expected, 8-9 P palatal connectors, 126f. See also under major connectors location of, 42 palatal contours, 41f-42f palatal coverage, absence of full, 101-102 palatal finish lines, 52f palatal major connectors, 41f palatal obturators. See interim obturator prostheses palatal occlusal ramps, 337 occlusal adjustments (Continued) occlusion (Continued) 381 Index palatal plate-type major connectors, 40-42, 42f palatal strap-type connectors, 54, 109f palatopharyngeal competence, 325, 328-329 palatopharyngeal defects, 324f papilla, rules for forming, 275 papillomas, 189 paralleled blockouts, 145f, 147, 148t partial denture service phases, 12 path of insertion (POI), 341 path of placement. See placement paths paths of insertion, 138f patient instructions, 12 after repairs, 307f care of dentures, 295 compliance with, 181 conditioning with interim partial dentures, 313-314 guide flange prosthesis use, 336 home care program, 193 for oral hygiene, 195 for placement/removal/care, 294-296 prevention of breakage, 305 rules for handling dentures, 296 temporary restorations versus removable partial dentures, 314 patient interviews, 150-151 formats for sequence of, 151 influence on treatment decisions from, 165 objectives of, 151 patient-dentist relationships failure of, 15 rapport, 151 shared decision making, 151, 158f pediatric speech aids, 329 periodic recall evaluations, 14 periodontal disease, 213f periodontal mechanoreceptors (PMRs), 5, 339 periodontal preparation, 194-200 advantages of, 200 diagnosis of periodontal disease, 194-195 initial disease control therapy phases, 198-200 objectives of, 194 treatment planning, 195 periodontal surgery, 198-200 guided tissue regeneration (GTR), 198, 199f periodontal flaps, 198 plastic surgery, 198-200 permanence of form of denture bases, 107-108 phases of partial denture service, 12-14 physiologic basing theory, 234 physiologic consequences of tooth loss, 5 placement of interim partial dentures, 314 placement paths, 140 determining, 137-138, 142f dual, 215 for esthetics, 140-141 final, 141-142 recontouring for, 89-90 recording of, 133f for true reciprocation, 209-210 vertical, 138f plaque, 196 plaster of Paris, 163, 219-220 plastic mesh designs, 106, 106f plastic patterns, preformed, 256f plate-type major connectors, 32f, 40-42, 42f pleuralatal plate-type major connectors, 32f pocket depth, 194 POI (path of insertion), 341 polishing the denture, 275, 282-283 denture borders, 282 facial surfaces, 282 gingival and interproximal areas, 282-283 polyether, 222 polymer resin record bases, 269f polyps, 189 polyvinyl siloxane, 161f porcelain artificial teeth, 251, 281 attaching, 109 porcelain facings, 274 porcelain veneer crowns, 212f stock porcelain or resin, 111f positive vertical support, 64f posterior teeth age and loss of, 3 arranging, to opposing casts/templates, 272-273 relationships of, 252 premolars design considerations, 27f splinting of lone-standing, 214f preparation geometry, 67 preparation of mouth. See mouth preparation preparation of rests and rest seats. See rest seats; rests pressing on a resin tooth, 109 pressure common areas of, 290 traumatic effects of, 300f pressure indicator paste, 291f primary retention, 68, 105 primary support areas, 241 probing, pocket-depth, 194 prognoses abutment teeth with guarded, 179, 205 for service of abutment teeth, 39, 169f prophylaxis before examination, 152 prostheses design of, 9-10 desirability of, 8-9 distribution of types of, 7t with functional restorations, 5-6 internal attachment, 10-11, 11f options for, 9 patients’ understanding of different, 9 tooth- and tissue-supported, 10-12 tooth-supported, 9-10 prosthodontic treatment, objectives of, 2 proximal plates, 47f, 76f proximal surface modification, 173f proximal tooth surfaces, relative parallelism of, 139f psychological issues, 318f pulleys, 22, 23f Q quality concerns, 7t, 183, 268f R radiographic examination, short-cone technique, 166 radiographic examinations/interpretation, 152, 166-169 alveolar lamina dura, 167-169 bone density, 166-167 bone response to previous stress, 152f, 168f disease validation, 166 index areas, 167, 168f for periodontal diagnosis, 194 root morphology, 169 tooth support, 166 unerupted third molars, 169, 169f radiographic interpretation retained roots, 187f root morphology, 213f unerupted third molars, 187f rebasing. See relining/rebasing denture bases recall maintenance, 200 reciprocal arms, 79f counteraction of reciprocation, 68f functions of, 69-70, 209 incorrect relationship to retentive arms of, 209f rigidity of, 92f, 124 reciprocation during placement/removal, 69 true, 209-210, 210f reconstruction, surgical. See surgical procedures/preparation reconstruction considerations, 165 recontouring of abutting teeth, 35f burs/points for, 62f for path of placement, 89-90 record bases, 127, 248f acrylic-resin, 247f centric relations on, 248 completed, 269f interocclusal, 164f occlusal, 13 occlusion rims on, 247-248 techniques for making, 267-269 sprinkle-on, 267-269, 268f recording data (sample form), 153f-154f refractory casts, 46f, 255f-256f, 260f for Class I, modification 1 designs, 259f registering interocclusal positions, 246f registration. See also casts/casting; impression-related entries impression procedures, 117-118, 127 elastomeric interocclusal registration material, 164f type/accuracy of, 233-234 interocclusal positions, 246f placement paths (Continued) 382 Index materials for elastomeric interocclusal, 163, 164f elastomeric interocclusal registration, 164f metallic oxide bite registration paste, 163 relief/relief spaces, 106f, 146f-147f, 147, 148t with interim partial dentures, 315f of minor connectors, 333 physiologic relief of, 333 relief spaces under, 106f for soft tissue, 31f supporting regions, 233f reliefs, wax, 136f, 253-254 relined removable partial dentures, 236, 302-304 relining/rebasing denture bases anticipation of, 126-127 distal extension, 301-302 indications for, 112-114, 302 need for, 112-114 open-mouth technique, 303f preparing for need for, 105 procedure for relining, 301 tooth-supported, 300-301 using existing dentures as base for, 300f remounting, 303f definition of, 281-282 goals of, 292f-293f precautions for, 280-282 remounting and occlusal corrections to occlusal templates, 280-282 precautions in remounting, 280-282 removable partial dentures current use of, 7 daily removal of, 49-50 impact of, 5 need for, 7 technical quality concerns for, 7t tooth-supported, 26-27 removal of prostheses, frictional resistance to, 72f removal of residual roots, 186 removal paths, determining, 137-138 repairs broken clasp arms, 305-306 direct retainers, 306f evaluation of denture for replacement versus, 306f major and minor connectors, 307-308 occlusal rests, 307 soldering, 307f, 308-310 repairs to clasp arms, 305-306 repouring of master casts, 277 research, proximal plate, Akers (RPA) clasp assemblies, 73-74 resections. See mandible resections; mandibular defects residual bone, 178 residual ridge. See also ridge-related entries to ascending ramus, 244f casting of, 122f conditioning of, 313 contour and quality of, 232-233 evaluation of, 170 extent covered by denture base, 233 recording anatomic form of, 232f resorption of, 341, 343f support from, 235-236 support issues for, 235f support quality of, 121 residual ridge crest, 233f residual roots, removal of, 186 resin denture bases, 30f, 105f attachment of, 106f materials for, 221 precautions for, 300-301 resin interim dentures, 311 resin teeth anterior, 274 attachment of, 273 pressing on, 109 processing of, 273 resin tube teeth, 111f attaching, 109 resistance, 24f direction of, 100f frictional, 72f, 81 resistance arms, 26f, 76-78 rest, proximal plate, I-bar (RPI) clasp assemblies, 73-76, 76f-77f rest seats, 100f chromium-cobalt alloy, 34-35 extended occlusal, 59 form of, 58-59 incisal, 64-65, 65f-66f incline angle of, 58f internal, creation of, 32 interproximal occlusal, 59-61 for Kennedy Class I arches, 116f mesio-occlusal, 63f occlusal, 57f size/shape of, 32 in sound enamel, 32 placement of, 35 positive vertical support by, 64f preparation of, 58f, 60f, 64f, 66f, 201-204 selecting type of, 36 restoration during treatment, with interim partial dentures, 313 restorations abutment, 200-201 fabricating for existing denture retainers, 216-218 perforations to existing, 32 sequence of preparation on, 206 rests, 4f, 38f adjacent/joined, 59-60 auxiliary occlusal, 99-100 ball type, 34 bilateral, 99 canine, 100 determining locations of, 141 extended occlusal, 59 form of occlusal, 58-59 implants as, 66 incisal, 64-65 internal occlusal, 61 intracoronal, 31-32 lingual, on canines and incisors, 63-64 occlusal, 74f auxiliary, 98f canine extensions from, 100 double, 80f intaglio surface of, 60f internal, 61f placement of, 236f preparation of, 58f-59f on secondary abutments, 101 shape of, 58f preparation of, 9-10 role in movement control of, 56-57 support for, 61-63, 116f terminal, 100 retainers direct (See direct retainers) indirect (See indirect retainers) for interim dentures, 311 retention achievement of, 333 amount of, 88f causes of, 105 design considerations for, 120 implant use for, 341 importance of, 67-68 measuring, 143-144 mechanical, 105-106 primary, 68, 105 secondary, 68, 105 on terminal abutments, 38f uniformity of, 91-92 weight/bulk and, 108 without clasps, 344f retentive areas, 89f purpose of, 137 surveying, 139 retentive arms, 80f-81f retentive terminals, 88f retromolar pads, 105f, 232f reverse-action clasps, 85, 85f reversible hydrocolloids, 221 ridge anatomic, 14f, 121 residual (See residual ridge) ridge crest, shortening of, 340 ridge forms, anatomic and functional, 234f ridge mucosa, residual, 10, 51f ridge support design considerations for, 121-122 implants for improving, 122 ridge volume, 5 ridges, knife-edge, 189 rigid clasp arms, 209f rigid connections, 114 rigid impression materials, 219-220 metallic oxide paste, 220 plaster of Paris, 163 rigidity of major connectors, 36 ring-type clasps, 81-82, 82f Roche clasp arms, 74 root morphology, 169 root planing/scaling, 195 root tips, extraction of, 186f registration (Continued) residual ridge (Continued) rests (Continued) 383 Index roots removal of residual, 186 retained, 187f split, 213f rotation around fulcrum lines, 113, 302 rotational axes, 94f rotational forces, 23, 25f, 28f RPA (research, proximal plate, Akers) clasp assemblies, 73-74 RPI (rest, proximal plate, I-bar) clasp assemblies, 73-76, 76f-77f rugae support, for indirect retainers, 101-102 S saliva control, 326 scaling, 195 screws, 22, 23f seating the denture, 291f, 314 seating the framework, 258f secondary retention, 68, 105 selective pressure technique, 241 selective tissue placement impressions, 237-241 sensory-functional considerations, 339 sequencing of mouth preparation, 12-13 settling of denture base, 121-122 shaped blockouts, 147, 148t shared decision making, 151, 158f shortened dental arches, 176-177 side contacts, balancing, 244f side-groove teeth, 109 silicone, 222 Sjögren’s syndrome, 170 skin grafts, 189, 319, 320f, 324f, 325-326, 329f, 332, 333f, 334-335 sleeping with dentures, 295-296 slippage, prevention of, 74f snowshoe principle, 105 soft palate, 325 normal function of, 325 surgical reconstruction of, 326 soft tissue diseases/reconstruction of, 327 over basal seat areas, 237 protective function of, 236 relief for, 31f surgical reconstruction of, 326 soldering of occlusal rests, 307 for repairs, 307f, 308-310 techniques for, 262f wrought-wire retainer arms, 256, 261f soldering wrought-wire, 261f-262f somatosensory cortex, 339f sound enamel abutment preparations on, 206 occlusal rests on, 32 soundness of abutment teeth, 179 space maintenance, 312, 312f spark erosion technology, 211 spatial relationships of maxillae, 159f specifications, ADA No. 7, 183t speech, 295, 326, 330-331 speech aid prostheses, 328-330 spicules, 189 splint bars, 203f for cross-arch stabilization, 212f design of, 128 shape of, 129f for tooth longevity, 129f splints/splinting of abutment teeth, 212-213 interim dentures as occlusal, 313 of lone-standing premolars, 214f problems for, 43f temporary, 197, 197f split roots, 213f sprinkle-on technique, 267-269, 268f, 311 sprues/spruing, 262-264, 263f-264f example of, 263f function of, 262 sprue access, 256f squamous cell carcinoma, 317f stability/stabilization, 9-10, 38f. See also movement entries; reciprocal arms by auxiliary occlusal rests, 98f balancing side contacts for, 244f cross-arch stability, 29, 92 design of stabilizing components, 124 against flange-induced forces, 336f implant use for, 340 need for bilateral, 178 reasons for poor, 152 stability under function, 339-340 stabilizing components, 28f of weak anterior teeth, 100 without mandibular canines, 38f stabilizing (reciprocal) clasp arms, 92 standard precautions, 174 stock impression trays, 34f, 281f, 282 stone casts, 225-226, 225f. See also casts/ casting; laboratory procedures duplicating, 253-254 materials for, 255f molds, 255f from hydrocolloid impressions, 225-226 stone dies, 216 stone occlusal templates from functional occlusal records, 271-272 strap-type major connectors palatal, 39-40, 40f-41f, 47f single palatal, 39, 40f-41f strategic tooth loss, 341 strategically tooth loss, 341 stress breakers mechanical, 123-124 use of, 10-11 stress distribution, 45 stress equalizers, 114 stress-bearing areas, 14f, 105f, 241 stresses horizontal, 69 horizontal distribution of, 119f transference of, 118 transient, 242 sublingual bar-type major connectors, 31f, 33f, 36, 53 Success Injection System, 278-279 support, 38f chewing and, 340 for clasp-retained partial dentures, 15 component partials for gaining, 129 design considerations for, 119-120, 128-129 differences in, 117 for/from distal extension denture bases (See distal extension denture bases) with maxillary defects, 325-326 overlay abutments for, 129 positive vertical, 64f for rests, 36, 61-63 splint bars for, 128 suprabulge region, 87f, 88, 90 surface relationships, 87f surgical procedures/preparation augmentation of alveolar bone, 189-191 bony spines and knife-edge ridges, 189 combined techniques, 187f cyst/tumor removal, 187-188 dentofacial deformities, 189 exostoses and tori, 188, 188f extractions, 186, 186f hyperkeratoses, erythroplasia, ulcerations, 189 hyperplastic tissue, 188, 188f impacted teeth, 186-187 for improved prosthesis tolerance, 325 indications for, 326 malpositioned teeth, 187 on maxillary defects, 323-328 muscle attachments and frena, 188-189 osseointegrated devices, 189 polyps, papillomas, traumatic hemangiomas, 189 presurgical interim partial dentures, 315f removal of residual roots, 186 surveyors/surveying, 85f, 86-87 ceramic veneer crowns, 135 contouring wax patterns with, 134-135, 134f contouring with, 134-135, 134f definition/description of, 130-132 dental casts, 12, 13f, 85f description/types of surveyors, 13f, 131-132 determining placement and removal paths with, 137-138 diagnostic casts, 133-134, 137-138, 141f-142f, 156 errors when repositioning casts on surveyors, 143 final path of placement, 141-142 guiding planes, 138-139 for interference, 140 Jelenko surveyors, 131, 131f-132f master casts, 136-137, 143, 145-147 matching cast restorations with, 136 measuring retention, 143-144 method for manipulating, 138f Ney surveyors, 131, 131f-132f parallel/shaped/arbitrary blockout and relief, 146f, 147, 148t for placement of intracoronal retainers, 135 384 Index position choices for, 139f purposes of, 132-137 recording placement paths, 133f relation of cast to surveyor, 142-143 relative parallelism of proximal tooth surfaces, 139f retentive areas, 139 survey line placement, 202f surveyed crowns vs. implants, 340-341 tools used with, 132f-133f, 143f, 211f for treatment planning, 341 undercut gauges, 144f swallowing, 6, 326 Swing-Lock design, 38, 333-334, 341 syneresis, 223 T tapered wrought-wire retentive arm, 27f temperature, freedom of interchange of, 108 temporary crowns, 152, 215-216 temporary restorations, versus removable partial dentures, 314 temporary splinting, 197, 197f temporomandibular joint (TMJ) complex, 5, 158-159, 339 tensile strength of wrought structures, 183 terminal portion of minor connectors, 51f terminal rests, 100 terminology, 2-3 thermal conductivity, through metal denture bases, 108 thermoplastic impression materials, 220-221 modeling plastic, 220-221 waxes and natural resins, 221 third molars, unerupted, 169, 169f three-quarter crowns, 208 tinfoil, 216, 278 tipped/tilted abutment teeth, 87f cast evaluation of, 59f contour adjustments with, 141f rest preparation with, 59 tipping forces, 28f, 69, 167-169, 244f tissue atrophy, 103 tissue conditioning, 193f conditioning of abused/irritated tissue, 191-193 with interim dentures, 312 tissue contact areas, 291f tissue grafting, 200f tissue index contacts/stops, 51f tissue molding, 105 tissue records, 222-223 tissue relief, provision of, 261f tissue stops, 48-49, 50f-51f tissue support, loss of, 296, 302 tissues changes to, 300 conditioning of abused/irritated, 191-193, 193f damage to causes of, 192f-193f from major connectors, 31 management of, 300f distortion of, 116f, 237 health of, 252, 299 reaction to metallic coverage by, 49-50, 108 stimulation of, 103 types of bases and health of, 114 tissue-supported prostheses design of ridge support with, 121-122 of extension base, 27f tooth-supported and, 10-12 titanium, 182-183 titanium endosseous implants, 189 tongue contours of denture bases for contact with, 108 functional movement of, 34f mobility of, 326, 331 surgical defects of, 326 tools for surveying, 132f-133f, 143f tooth arrangements to ascending ramus of residual ridge, 244f cross-arch stability with, 331f establishment and verification of, 13 tooth contours analysis of, 86-87 unfavorable, 38 tooth foundation, evaluation of, 170 tooth loss additions to dentures due to, 308 age and, 3-5 consequences of, 5 current management of partial, 2 in existing prostheses, 341 tooth mobility, 167, 194-195, 197-198 tooth replacements, patient perspectives on, 8-9 tooth restorations, 129 tooth-supported prostheses, 9-10, 252, 300 copings, 191f design considerations for, 120-121 direct retainers for, 123 movement of, 26-27 tissue-supported and, 10-12 tooth-tissue contours, 283f tooth-to-ridge relationships, 252 torch soldering, 310 tori, 155, 188, 188f trabecular pattern, 166 trauma to teeth, 90 traumatic effects of pressure, 300f traumatic hemangiomas, 189 trays. See impression trays treatment limitations of, 12 objectives of, 151-164 purpose/uniqueness of, 150 treatment planning. See also diagnosis/ assessment choosing complete versus removable partial dentures, 180-181 for clasp-retained partial dentures, 14-15 clinical factors with metal alloys, 181-183 gold and chromium-cobalt, 181-183 wrought-wire selection, 183 extractions, 205 periodontal surgery, 195 proposed mouth changes, 158f treatment planning phase of service, 12-13 treatment sequencing phase of service, 12-13 trial packing, 277 tribochemical coating, 111 true reciprocation, 209-210 tube teeth, 109 tumors, removal of, 187-188 Type I resections, 331-332, 333f Type II resections, 332-334, 334f-335f Type III resections, 334 Type IV gold alloys, 91t Type IV resections, 334 Type V resections, 334-335 U ulcerations, 189 undercut gauges, 132f, 143-144, 144f undercuts approach to, 90 blocking out, 147f depth of, 90 as limiting factors, 74-75 tissue, 89-90, 146-147 unilateral tooth loss, 175 unilaterally missing molars, 176-177, 177f U-shaped palatal major connectors, 40f, 44-45, 44f, 55 V veneer crowns, 201 lingual rests on, 34 porcelain, 212f for support of clasp arms, 212 surveying ceramic, 135 veneers, resin, 274 vertical axis, rotation around, 24-26, 25f vertical movement, 24 vertical path selection, 138f vertical stresses, 114 vertical support, 57f. See also rest seats; rests visible light-cured (VLC) methods, 279-280, 311 artificial teeth, 304 bases, 247 custom trays, 226 vitality tests of remaining teeth, 156 voids in casts, 276 V-shaped grooves, 45f W wax baseplates, 227f-228f, 237f-238f wax blockouts, 136f wax burnout procedure, 265f wax impression material, 221 wax indicators for occlusal adjustments, 293-294, 294f wax interocclusal records, 164f, 246 wax ledges, 145f surveyors/surveying (Continued) tissues (Continued) treatment planning (Continued) 385 Index wax occlusion rims, 270 wax patterns, 259f contouring, 134-135, 134f, 201, 203f forming, 255-256 for mandibular frameworks, 260f peripheral extent of base, 268f quality of framework and, 268f veneer spaces in, 217 wax reliefs, 136f, 253-254 wax trimmers, 132f waxes and natural resins, 221 waxing denture bases, 274-275, 275f the framework, 254-261 for management of replacement teeth, 176f metal bases, 256-261 to polished metal parts, 274 before processing, 274-275 wedges, 22, 23f wedging effects, 47 avoiding, 74f prevention of, 59-60 well-being, major connectors and, 32b wheels, 23f Whip-Mix facebows, 160, 161f work authorizations, 284-286. See also laboratory procedures characteristics of, 285-286 content of, 284 definitive instructions by, 286-287 delineation of responsibilities by, 287-288 function of, 284-285 legal aspects of, 287 sample forms, 285f-286f working chart for (sample form), 155f wrought-titanium alloy wires, 182-183 wrought-wire clasp arms, tensile strength of, 91 wrought-wire retainer arms, 79f-80f attaching, 256 X xerostomia, 170 Y Young’s modulus of elasticity of titanium alloy, 182-183 Z zinc oxide-eugenol impression paste, 118
1607	https://www.teacherspayteachers.com/Product/Compound-Probability-using-Tree-Diagrams-and-Charts-Grade-7-7SPC8-4684732	Compound Probability using Tree Diagrams and Charts - Grade 7 (7.SP.C.8) Description Excellent lesson and practice problems to teach how to determine the entire sample space for compound events! Students will learn to use either TREE DIAGRAMS or CHARTS to generate a list of the sample space for events that are combined into one outcome (i.e. rolling 2 die). Students will practice expressing various probabilities based on the compound sample space, and make predictions if an event is repeated multiple times. Students also learn the FUNDAMENTAL COUNTING PRINCIPLE which lets them quickly calculate how many outcomes there will be in a compound event. High quality, professional lessons, practice/homework sheets. Easy to follow, guided lesson layout allows you to just print and teach. Full solution sets to all lessons and practice problems are provided. Scaffolded notes with practice consolidation for each topic. Are you pressed for time and want lessons that teach the most important skills? No need to search further, or worry about creating your own lessons. My Grade 7 Probability Lessons are excellent to teach and review the most important and fundamental probability skills. WANT MORE PROBABILITY LESSONS? LOOK AT THE OTHER LESSONS IN MY STORE! The Hungry Games - Experimental Probability Game Describing Probability (FREE) Theoretical and Experimental Probability Making Predictions Using Probability Compound Probability The Prize Wheel Unit Problem Odds and Evens Experiment Probability Challenge Problems Probability Unit Test (Editable Download) BUNDLE and SAVE! Buy all of these products together in a fantastic Thematic Unit Bundle! See the Bundle Link above the Product Description. ONTARIO MATH CURRICULUM: Data Management and Probability Grade 7 To Earn Credit for Future Purchases, Go to My Purchases page (you may need to login). Beside each purchase you'll see a Provide Feedback Link. Simply click it and you will be taken to a page where you can give a quick rating and leave a short comment for the resource. Each time you give feedback, you receive feedback credits that you can use to lower the cost of your future purchases. LICENSING/COPYRIGHT TERMS: This purchase includes a license for one teacher only for personal use and is non-transferable. No part of this resource is to be shared with colleagues or used by whole departments, schools, or districts without purchasing the proper number of licenses. If you have questions about licensing more than one copy, please email me at mathytechy@gmail.com. This resource may not be uploaded to the internet in any form, unless the site is password protected and can only be accessed by students. Compound Probability using Tree Diagrams and Charts - Grade 7 (7.SP.C.8) Save even more with bundles Reviews Questions & Answers Standards
1608	https://www.archbronconeumol.org/en-pulmonary-mucormycosis-at-onset-diabetes-articulo-S157921291730215X	Pulmonary Mucormycosis at Onset of Diabetes in a Young Patient \| Archivos de Bronconeumología Advanced Search Share Share Access Enter your user name and password Access I have forgotten my password Contact Us Access Required fields Enter your email address HomeSeptember 2017Pulmonary Mucormycosis at Onset of Diabetes in a Young Patient Home All contents Ahead of print Latest issue All issues Supplements Covers gallery Podcasts Related journal - Open Respiratory Archives Subscribe to our newsletter Publish your article Guide for authors Submit an article Ethics in publishing Open Access Option Language Editing services About the journal Aims and scope Editorial Board Contact Advertising Metrics Most often read Most cited Most popular All metrics Archivos de Bronconeumología ISSN: 0300-2896 Archivos de Bronconeumologia is an international journal that publishes original studies whose content is based upon results of research initiatives dealing with several aspects of respiratory medicine including epidemiology, respiratory physiology, pathophysiology of respiratory diseases, clinical management, thoracic surgery, pediatric lung diseases, respiratory critical care, respiratory allergy and translational research. Other types of articles such as editorials, reviews, and different types of letters are also published in the journal. Additionally, the journal expresses the voice of the following scientific societies: the Spanish Respiratory Society of Pneumology and Thoracic Surgery (SEPAR; the Latin American Thoracic Society (ALAT; and the Iberian American Association of Thoracic Surgery (AIACT; It is a monthly journal in which all manuscripts are sent to peer-review and handled by the editor or an associate editor from the team and the final decision is made on the basis of the comments from the expert reviewers and the editors. The journal is published solely in English. All the published data is composed of novel manuscripts not previously published in any other journal and not being in consideration for publication in any other journal.. The journal is indexed at Science Citation Index Expanded, Medline/Pubmed, Embase and SCOPUS. 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See more SJR 2024 0.466 SNIP SNIP measures contextual citation impact by wighting citations based on the total number of citations in a subject field. See more SNIP 2024 0.505 View more metrics Open Access Option Hide Journal Information Previous article \| Next article Vol. 53. Issue 9. Pages 531-533(September 2017) Lee este artículo en Español Export reference Share Share Twitter Facebook Linkedin whatsapp E-mail Print Download PDF More article options Statistics Vol. 53. Issue 9. Pages 531-533(September 2017) Scientific Letter DOI: 10.1016/j.arbr.2017.07.001 Full text access Pulmonary Mucormycosis at Onset of Diabetes in a Young Patient Mucormicosis pulmonar en paciente joven con inicio de diabetes mellitus Visits 5980 Download PDF Javier Espíldora-Hernándeza,, Carmen Pérez-Lópeza, Manuel Abarca-Costalagoa, Enrique Nuño-Álvarezb aServicio de Medicina Interna, Hospital Universitario Virgen de la Victoria, Málaga, Spain bServicio de Enfermedades Infecciosas, Hospital Universitario Virgen de la Victoria, Málaga, Spain This item has received 5980 Visits 2 Cites Article information Full Text Bibliography Download PDFStatistics Figures (1) Full Text To the Editor, Mucormycosis is an infection caused by filamentous fungi that presents in different forms: rhinocerebral, pulmonary, renal, cutaneous, and gastrointestinal. The species Rhizopus oryzae, responsible for 70% of cases, is the most frequently isolated organism.1 Risk factors for developing mucormycosis include blood diseases, diabetes mellitus with poor metabolic control, solid organ or hematopoietic transplantation, neutropenia, injury, iron overload, and severe burns. It is unclear whether the chronic use of corticosteroids predisposes patients to developing mucormycosis. In recent years, we have witnessed an increase in the incidence of this entity due to population aging, which goes hand in hand with an increase in the above-mentioned risk factors.2,3 We report the case of a 29-year-old woman, smoker of 10 pack-years, with recent onset of diabetes mellitus type 1 (ketoacidosis the week before presentation of this clinical episode). She consulted due to a few hours history of dyspnea, fever 38°C, pain in the right flank, cough and rust-colored expectoration. Auscultation revealed crackles in the right lung base. Clinical laboratory tests showed significant leukocytosis (30 100/μl) and elevated CRP (224 mg/l). Consolidation of the right lower lobe was observed on chest radiograph. The patient was diagnosed with community-acquired pneumonia, and empirical antibiotic therapy was started. The chest computed tomography (CT) scan showed consolidation of the pulmonary parenchyma in the right lower lobe, with the formation of a thick-walled hypodense lesion containing air bubbles, with axial diameters measuring 6.1 cm×4.2 cm, consistent with an abscess (Fig. 1A). During hospitalization in the general ward, the patient had several episodes of hemoptysis, so fiberoptic bronchoscopy was performed, revealing total stenosis of the right anterior basal bronchus (B8) and partial stenosis of the right basal-lateral bronchus (B9) with necrotic tissue on solid tissue. Given the lack of clinical improvement, the chest CT was repeated, showing, in addition to the necrotizing consolidation, aneurysmal dilation of the exit of the segmental artery of right bronchial segment 10, consistent with mycotic aneurysm, measuring 1.4 cm in length and 5 cm in diameter (Fig. 1B). Arteriography (Fig. 1C) confirmed the diagnosis. The aneurysm was successfully embolized with a 14-mm type 2 Amplatzer® plug and hemoptysis was controlled. The source of the aneurysm and the right lower truncal branch were subsequently embolized with 10-mm coils and an 8-mm Amplatzer® plug. Bronchial biopsy obtained during the bronchoscopy showed the presence of hyphae consistent with mucormycosis, so treatment began with liposomal amphotericin and caspofungin. Following this, the patient showed slow but clear clinical improvement. Lower right lobectomy was performed by posterolateral thoracotomy, revealing pleural adhesions throughout the lung surface, particularly between the lower right lobe and the diaphragm. The postoperative period was free of complications, recovery was favorable, and cure was achieved. Fig. 1. (A) Chest CT with contrast medium (parenchymal window), showing a lung abscess associated with necrotizing pneumonia of the right lower lobe. (B) Sagittal chest CT slice (mediastinal window), showing a mycotic aneurysm in the infero-medial segmental branch (asterisk). (C) Pulmonary arteriography performed before embolization of the aneurysm. A defining characteristic of pulmonary mucormycosis is its rapid progress and marked angioinvasive capacity. Invaded tissue becomes necrotized and occupied by hyphae, causing infarction and fostering the development of cavitating pneumonias. Mucormycosis can also invade adjacent structures such as the mediastinum, the heart, or the bloodstream (fungemia). Hemoptysis is a common complication and can be massive. Diagnosing mucormycosis is a complex process, since the presentation is similar to that of community-acquired pneumonia: the most common symptoms are fever, pleuritic pain, and cough with purulent expectoration. Imaging tests are nonspecific, since no characteristics signs exist that can be distinguished from other processes. Identification of the organism in the tissue is necessary to reach a safe diagnosis of invasive fungal infection. The species can be confirmed on culture. Bronchoalveolar lavage (BAL) can be used in immunosuppressed patients, although a positive result for fungi is only orientative. However, it is considered highly suggestive if hyphae are visualized on optical microscopy. Medical treatment of choice is intravenous liposomal amphotericin B at a dose of 5 mg/kg/day, which should continue until clinical and radiological resolution of the process. Resistance to voriconazole is a significant feature of these organisms. For drug treatment to be effective, existing necrotic tissue must be removed, either by debridement or by lobectomy, so that the antifungals can penetrate well-perfused tissues.4,5 Mucormycosis is an emerging disease that must be considered in daily clinical practice, and therein lies the interest in this case: necrotizing pneumonia in a patient with predisposing factors, presence of hemoptysis, and necrotic areas in the bronchial mucosa are key data that point toward a diagnosis of invasive fungal infection. References A.S. Ibrahim, B. Spellberg, T.J. Walsh, D.P. Kontoyiannis. Pathogenesis of mucormycosis. Clin Infect Dis, 54 (2012), pp. S16-S22 \| Medline G. Petrikkos, A. Skiada, O. Lortholary, E. Reoilides, T.J. Walsh, D.P. Kontoyiannis. Epidemiology and clinical manifestations of mucormycosis. Clin Infect Dis, 54 (2012), pp. S23-S34 \| Medline C.A. Kauffman, A.N. Malani. Zygomycosis: an emerging fungal infection with new options for management. Curr Infect Dis Rep, 9 (2007), pp. 435-440 Medline R.N. Greenberg, L.J. Scott, H.H. Vaughn, J.A. Ribes. Zygomycosis (mucormycosis): emerging clinical importance and new treatments. Curr Opin Infect Dis, 17 (2004), pp. 517-525 Medline D.P. Kontoyiannis, R.E. Lewis. How I treat mucormycosis. Blood, 118 (2011), pp. 1216-1224 \| Medline ☆ Please cite this article as: Espíldora-Hernández J, Pérez-López C, Abarca-Costalago M, Nuño-Álvarez E. Mucormicosis pulmonar en paciente joven con inicio de diabetes mellitus. Arch Bronconeumol. 2017;53:531–533. Copyright © 2016. 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1609	https://www.youtube.com/watch?v=XE-tkMD5N0M	Astronomy - Chapter 1: Introduction (7 of 10) How to Measure Angular Size? Michel van Biezen 1130000 subscribers 1626 likes Description 86503 views Posted: 27 Aug 2014 Visit for more math and science lectures! In this video I will discuss angular measure and angular size. 127 comments Transcript: welcome to alect on line and now we're going to take a look at angular measure or angular size in astronomy it is very difficult to figure out how far things are and we'll talk a lot about that in our future videos but one thing we can do is measure the angular size of object for example let's say that we are looking at the moon and we're wondering well how big does the Moon look and you look at well it looks about that big but really the way to measure it is to say okay if I draw a line from where I'm looking to the top of the Moon and I draw a line from when I'm looking to the bottom of the Moon notice that this then makes up an angle let's call the angle Theta and so we measure things in astronomy we measure things in the in the universe by how big they appear angle wise how much of an angle does this subtend and for the moon it's about 1 half a degree so 0.5° would be the angular size of the moon now how big is the sun well we know the sun is much much bigger than the moon but since the sun is also much farther away than on the moon it turns out in the sky they look at about they look about the same size that's why when there's a solar eclipse the the uh the moon travels in front of the Sun the moon disc the size of disc just barely blocks this disc of the Sun and when we have a solar eclipse that's because the disc of the Moon just passes past the disc of the Sun and since they're about the same size the the moon's disc covers up the sun completely and so it turns out the angular size of the sun is roughly about the same size of the angular size of the Moon and it's about a half degree so they have the same angular size even though of course knowing that the sun is much bigger than the moon let's say we look at a constellation here's a part of the constellation called Orion here is Orion's Bel of course you would see the the great nebble of Orion would be right about there and then if you also take a look with a telescope right here would be the horse head nebble and so forth well that's for another story but again how big does the constellation look when you go out there and you finally see it you go wow look how big it looks but how do you put a measure to that again what you can do is you can go ahead and draw lines like this and like that and you see that this would be roughly an angular size of about 5° if we take the whole constellation which goes up like that and then another con another part of constellation so Orion actually is a little bit bigger if you take all of the stars into account in the constellation but if you just take it this part that most everybody recognizes that makes an angular size about 5° so that gives you kind of a feel of how big things are so what we have to keep in mind with astronomy and angle measurements is that of course if you have a small circle here and you make an angle Theta this big you can see that on the edge of the circle it makes for an arc length about that size but if the circle is bigger so the edge of circle is farther away for the same angle you have a much longer um a much longer what Arc Length along the circle and you can see so two different objects of different size can have the same angle measure depending upon where they're located so in astronomy it's easy to measure angle and so the way we measure angles is of course we can say that if you go all the way around the circle that would be a 360° angle if you go a quarter circle like this that's called a 90° angle so typically in astronomy we're looking at things that are much smaller than 90 degrees a few degrees or a fraction of a degree matter of fact in astronomy most things are so small because they're so far away not that they are small but they appear to us they appear as very small because they're enormous distances we tend to think of size in terms of less than a degree so let's talk about that so we have of course one degree that's equal to 1360 of a circle so 1° angle is typically a very very tiny angle so imagine that there's 360° in a circle so a 1° angle is just a very small angle relative to the size of an entire circle like that but since in astronomy even a one de angle is quite a large angle notice that the Moon is only half degree in in angular size and so is the sun and stars of course far away much smaller than that we have smaller angular sizes for example we have what we call one arc minute which is equal to one with a single Dash around it like that which is equal to 1 16th of a degree so there are 60 arc minutes in one degree imagine a small angular SES like this and chopping it up into 60 equal angles and each one of those tiny little slices would be one arc minute and then we have even smaller angle sides that we use in Astron we call it 1 AR second and noce that is equal to 160th or maybe I'll write it like this first first of all the notation would be one with two double dashes like that and so this is equal to to um uh 1 160th of an arc minute which is 1 160th of an arc minutes so that's just a notation I'll just write it out so you can see that which since there's 60 arc minutes in a in a degree this is equal to 1 over 3,600 of 1 Dee I don't know if that makes sense to you when I write it like this so maybe I'll write it like that again 1 over 3,600 of a degree okay this little circle right here up there that simply is a degree symbol so when you see that one degree that means one degree or we can write out one degree or 1 over 3,600 of a degree like that so NC second is a tiny T tiny little sliver of an angle and yes in a lot of cases we use ARC seconds because things in astronomy are so far away and they tend to be so tiny only very powerful telescopes can see them that we need to use those very small angular measurements so an arcc is a common common measurement in astronomy matter of fact most stars have an diameter an angular diameter of an arcc or even less that's how tiny they are that's why they look at little Points of Light even though they're spheres and then larger things in astronomy such as nebulas and planets that are a little bit nearby and so forth well they have a slightly larger angle measure of course than the moon the Sun and large constellations they have much larger angle of measurement which can typically be measured in terms of a fraction of a degree or several degrees in size so now you have an idea if someone says oh look at that star is .5 AR seconds in angular size you know you're looking for a very tiny dot in the sky and that's how we use angular measure
1610	https://math.stackexchange.com/questions/1732977/showing-psix-thetaxo-sqrt-x-log-x-for-chebyshevs-function-psi	number theory - Showing $\psi(x)=\theta(x)+O(\sqrt x\log x)$ for Chebyshev's function $\psi$ - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Showing ψ(x)=θ(x)+O(x−−√log x)ψ(x)=θ(x)+O(x log⁡x) for Chebyshev's function ψ ψ Ask Question Asked 9 years, 5 months ago Modified9 years, 5 months ago Viewed 1k times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. In my textbook, there is the following theorem: For all x>0 x>0, we have ψ(x)=∑α=1∞θ(x 1/α)ψ(x)=∑α=1∞θ(x 1/α) and hence ψ(x)=θ(x)+O(x−−√log x).ψ(x)=θ(x)+O(x log⁡x). Here θ(x)=∑p≤x log p θ(x)=∑p≤x log⁡p and ψ(x)=∑p α≤x log p ψ(x)=∑p α≤x log⁡p are the Chebyshev functions. The first formula is proven with a little sum manipulations, but the second formula is not explained. My guess is the author is writing ∑∞α=1 θ(x 1/α)=θ(x)+∑∞α=2 θ(x 1/α)∑α=1∞θ(x 1/α)=θ(x)+∑α=2∞θ(x 1/α), and the leading term of the remainder is θ(x−−√)θ(x). Since this is introductory material we don't have the PNT, so we don't know θ(x)∼x θ(x)∼x, but a simple estimate gives θ(x)≤x log x θ(x)≤x log⁡x so the leading term is indeed x−−√log(x−−√)=1 2 x−−√log x x log⁡(x)=1 2 x log⁡x. But it is not obvious to me that the other terms, of order x 1/3 log x x 1/3 log⁡x, x 1/4 log x x 1/4 log⁡x etc. don't necessarily overwhelm the first term, if there are enough of them. Again, a naive bound on α α gives α≤log 2 x α≤log 2⁡x, but this is much too weak if you just take log x log⁡x copies of x 1/2 log x x 1/2 log⁡x, and I don't think ∑α x 1/α∑α x 1/α converges. number-theory prime-numbers asymptotics Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Apr 8, 2016 at 5:28 Mario CarneiroMario Carneiro asked Apr 8, 2016 at 5:11 Mario CarneiroMario Carneiro 28.3k 4 4 gold badges 73 73 silver badges 160 160 bronze badges Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. We can do the following elementary estimate: ψ(x)−θ(x)=∑p m≤x log(p)−∑p≤x log(p)=∑p≤x√log(p)∑2≤m≤log(x)log(p)1≤∑p≤x√log(p)[log(x)log(p)]≤x−−√log(x).ψ(x)−θ(x)=∑p m≤x log⁡(p)−∑p≤x log⁡(p)=∑p≤x log⁡(p)∑2≤m≤log⁡(x)log⁡(p)1≤∑p≤x log⁡(p)[log⁡(x)log⁡(p)]≤x log⁡(x). Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Apr 8, 2016 at 8:40 Dietrich BurdeDietrich Burde 142k 8 8 gold badges 99 99 silver badges 178 178 bronze badges 1 Hm, I suppose this works, although since it goes from a completely different starting point I wonder what the author meant by "and hence".Mario Carneiro –Mario Carneiro 2016-04-08 14:32:11 +00:00 Commented Apr 8, 2016 at 14:32 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions number-theory prime-numbers asymptotics See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Related 6Chebyshev's first ϑ(x)ϑ(x) function question 11Looking for help understanding the Möbius Inversion Formula 0Prime number theorem lemma: prove that ψ(x)∼π(x)log(x)ψ(x)∼π(x)log⁡(x) 1An upper bound for the Chebyshev function? 1Prove that log(n n)=Θ(log(n!))log⁡(n n)=Θ(log⁡(n!)) 2Prove the equivalent form of the Selberg's Formula ψ(x)log x+∑p≤x ψ(x p)log p=2 x log x+O(x).ψ(x)log⁡x+∑p≤x ψ(x p)log⁡p=2 x log⁡x+O(x). Hot Network Questions how do I remove a item from the applications menu With line sustain pedal markings, do I release the pedal at the beginning or end of the last note? Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish? 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1611	https://www.ck12.org/flexi/cbse-math/points-in-the-coordinate-plane/how-can-one-graph-a-piecewise-function-on-desmos/	How can one graph a piecewise function on Desmos? Flexi Says: Graphing piecewise functions on Desmos involves the following steps: Go to the Desmos online graphing calculator (www.desmos.com/calculator). Click on the "+" button on the upper left side to add a new equation. Type in your piecewise function. A piecewise function consists of multiple sub-functions, each defined over a different domain. For example, let's consider this piecewise function: @$\begin{align}f(x) = \begin{cases} 2x+3 & x\leq 1 \ -x+2 & x>1 \end{cases}\end{align}@$ On Desmos, you would type this as follows: (y=2x+3(x \leq 1)) (y=-x+2(x>1)) The notation "(x <= 1)" after the first equation means that this equation is valid for (x \leq 1). Similarly, the notation "(x > 1)" after the second equation means that this equation is valid for (x > 1). Desmos will graph each piece over its specified domain and show the full graph of the piecewise function. Try Asking: How can you locate points on the cartesian plane?If x = 0 and y > 0, where is the point (x, y) located?Point lies By messaging Flexi, you agree to our Terms and Privacy Policy
1612	https://pmc.ncbi.nlm.nih.gov/articles/PMC10118814/	Hypoparathyroidism: update of guidelines from the 2022 International Task Force - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer \| PMC Copyright Notice Arch Endocrinol Metab . 2022 Nov 10;66(5):604–610. doi: 10.20945/2359-3997000000549 Search in PMC Search in PubMed View in NLM Catalog Add to search Hypoparathyroidism: update of guidelines from the 2022 International Task Force Bart L Clarke Bart L Clarke 1 Mayo Clinic Division of Endocrinology, Diabetes, Metabolism, and Nutrition, Rochester Minnesota, USA, Mayo Clinic Division of Endocrinology, Diabetes, Metabolism, and Nutrition, Rochester, Minnesota, USA Find articles by Bart L Clarke 1,✉ Author information Article notes Copyright and License information 1 Mayo Clinic Division of Endocrinology, Diabetes, Metabolism, and Nutrition, Rochester Minnesota, USA, Mayo Clinic Division of Endocrinology, Diabetes, Metabolism, and Nutrition, Rochester, Minnesota, USA ✉ Correspondence to: Bart L. Clarke. Mayo Clinic Division of Endocrinology, Diabetes, Metabolism, and Nutrition East 18-A1, 200 1st Street SW Rochester, Minnesota 55905 USA. clarke.bart@mayo.edu Received 2022 Jun 29; Accepted 2022 Jul 18; Collection date 2022. This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. PMC Copyright notice PMCID: PMC10118814 PMID: 36382749 ABSTRACT The 2022 International Task Force guidelines for chronic hypoparathyroidism will be published within several months in the Journal of Bone and Mineral Research. These guidelines update the original guidelines published in 2016, and include new information from literature published since then. Chronic postsurgical hypoparathyroidism is now defined as lasting for at least 12 months after surgery, rather than 6 months. Chronic postsurgical hypoparathyroidism may be predicted by serum PTH <10 pg/mL in the first 12-24 hours after surgery. The most common symptoms and complications of chronic hypoparathyroidism based on the literature are summarized in detail. How to monitor and manage patients with hypoparathyroidism is described in detail where recommendations can be given. These guidelines are intended to frame the diagnosis and care of patients with chronic hypoparathyroidism for at least the next five years. Keywords: Hypoparathyroidism, hypocalcemia, nephrolithiasis, nephrocalcinosis, increased bone density INTRODUCTION Care of patients with parathyroid disorders is often challenging due to complexities in the diagnosis, imaging, and surgical or medical management of primary hyperparathyroidism, and inadequacies with conventional management of hypoparathyroidism. This review summarizes recommendations from the 2 nd International Guidelines for Hypoparathyroidism (1) that will soon be published in the Journal of Bone and Mineral Research, along with the supporting systematic and narrative reviews (2–5). These guidelines summarize information published in the medical literature back as far as 1940, with particular focus on papers published between 1970 and 2020, and emphasizing new information published between 2015 and 2020. The previous 1 st International Guidelines on Hypoparathyroidism were published in 2016 (6). MATERIALS AND METHODS The 2 nd International Guidelines on Hypoparathyroidism will be published in a separate upcoming issue from the 5 th International Guidelines on Primary Hyperparathyroidism in the Journal of Bone and Mineral Research. The 2 nd International Guidelines discuss the prevention, diagnosis, and management of hypoparathyroidism (HypoPT), and provide evidence-based recommendations for care of patients with this rare disorder. The four HypoPT Task Forces included a total of 50 international experts, including representatives of sponsoring international, national, and regional endocrine and medical societies. Dr. Guyatt and his team supported these taskforces and conducted the systematic reviews (7). A formal process following GRADE methodology and the systematic reviews provided the structure for 7 of the guideline recommendations. The Task Forces used a less structured approach based on narrative reviews with non-GRADEd recommendations for 20 of the recommendations. Co-chairs of four Task Forces for HypoPT worked entirely virtually over an 18-month period due to the COVID-19 pandemic. Meetings were held regularly with specific tasks designated to individuals or subsets of Task Force members. A comprehensive review of the literature was undertaken by each of the Task Forces using the search engines described in the papers, including PubMed, Medline, Embase, and Cochrane. Searches were conducted for systematic reviews, meta-analyses, and original publications. References to this field extended to 1940 for historical reference, but more recently between 1970 and 2020 for all other aspects of HypoPT, with particular emphasis on papers published between 2015-2020. For questions specifically related to medical management of HypoPT, GRADE methodology was employed (8). GRADEd recommendations followed a structured process that included framing questions in patient/intervention/comparator/outcome format; conduct of a systematic evidence search and associated summary; specification of values and preferences; and classifying and presenting recommendations as strong or weak with the corresponding quality of evidence. A strong recommendation was made when the desirable effects were much greater than undesirable effects or vice versa. The word “recommend” was applied to systematic reviews that reached this conclusion. A weak recommendation was made if there was low certainty of evidence or a close balance between desirable and undesirable effects. The word “suggest” was applied to systematic reviews that reached this conclusion. Recognizing that this rigorous approach to evidence-based review may have necessarily omitted worthy observations due to the strict screening criteria for selection, the evidence from the GRADE methodology was amplified to incorporate other noteworthy observations. When recommendations based upon narrative reviews were made, the terms “the Panel recommends” or “the Panel proposes” or “Panel Recommendations (ungraded)” clearly distinguish ungraded recommendations from recommendations based upon the systematic reviews. The Steering Committee organized the Task Forces with two co-chairs designated for each of the workshops. In collaboration with these Task Force co-chairs, 9-12 Task Force members were appointed to each Task Force. After drafts of each paper were prepared, a virtual meeting was held between the PHPT and HypoPT Task Force members. A second virtual meeting was then held, attended by representatives from all societies, organizations, and patient advocacy groups that expressed interest in this work with a view towards endorsing the guidelines. Recommendations from both meetings, each attended by about 100 individuals, were considered in revising and finalizing the papers that form the basis for the summary statements and their integration into this summary statement. The document was circulated to all organizations during a 6-8-week comment period. The organizations endorsing these guidelines will be listed at the end of the reference sections of the guidelines. Summary of recommendations for hypoparathyroidism The recommendations summarized below are intended to guide practice and not intended to be used for the development of reimbursement policies for patients treated for hypoparathyroidism. Other than panel statements 3, 5, and 6 regarding surgical management of patients with primary hyperparathyroidism, which were based upon GRADE analysis, all the other panel statements either could not be analyzed by GRADE methodology (statements 1-2) or the available data were not of sufficiently high quality to permit GRADE analysis (statements 4, 7, 8) (7). 1. How should chronic HypoPT be diagnosed? Hypocalcemia is defined as low ionized serum calcium or total serum calcium adjusted for albumin in the presence of an undetectable or inappropriately low intact PTH measured with either a 2 nd or 3 rd generation assay on two occasions at least two weeks apart (6). This definition will eliminate misdiagnosis caused by a single measurement of low calcium due to other causes. Additional abnormalities caused by low PTH supporting the diagnosis include increased serum phosphorus, decreased 1,25(OH)2 D, and increased urinary fractional excretion of calcium (6). Based on discussions with surgeons on the task forces, postsurgical HypoPT is now considered permanent if the HypoPT persists > 12 months after surgery, rather than 6 months after surgery. 2. How can the risks of chronic postsurgical HypoPT be minimized? The infrequent occurrence of postsurgical HypoPT is a major source of morbidity to patients undergoing anterior neck surgery. Avoidance of parathyroid autotransplantation during anterior neck surgery may reduce the risk of chronic postsurgical hypoparathyroidism (9). Autotransplantation is recommended only in the setting of inadvertent parathyroidectomy (3). 3. What is the value of determining serum calcium and PTH post-thyroidectomy to predict future permanent postsurgical HypoPT? (GRADEd Recommendation) Serum PTH measurement after total thyroidectomy may be used to predict which patients will not develop permanent postsurgical HypoPT (strong recommendation, moderate quality evidence) (3). If PTH values are > 10 pg/mL (1.05 pmol/L) 12-24 hours after surgery, the development of permanent HypoPT is unlikely and, therefore, there is no need for long-term treatment with active vitamin D and calcium supplements above the recommended daily allowance. Many patients with PTH values < 10 pg/mL (1.05 pmol/L) 12-24 hours after surgery may recover from their temporary HypoPT. 4. What is the role of genetic testing in the diagnosis and evaluation of chronic HypoPT? In patients with nonsurgical HypoPT who have a positive family history of nonsurgical HypoPT or present with syndromic features, or are younger than 40 years, genetic testing is recommended (3,4,10). In patients with nonsurgical HypoPT who have other clinical features of autoimmune polyendocrinopathy-candidiasis-ectodermal dystrophy (APECED) syndrome, genetic testing for AIRE variants is recommended. The designation of “autoimmune HypoPT” for patients who do not have APECED syndrome should be avoided, as there are no definitive diagnostic tests for polygenic autoimmune HypPT (3). 5. What are the most common symptoms and complications of chronic HypoPT reported in the literature? (GRADed recommendation) Observational studies comparing patients with HypoPT to controls with normal parathyroid function have identified multiple complications associated with HypoPT (11–17). Percentages listed here represent the median among published studies. Cataracts are reported in 24% of patients, infections in 18%, nephrocalcinosis/nephrolithiasis in 15%, renal insufficiency in 13%, seizures in 12%, depression in 11%, ischemic heart disease in 9%, and arrhythmias in 7% (11). Other complications commonly associated with HypoPT, such as basal ganglia calcification, are not frequently reported in published case series, so are not included in the eight most common complications listed here. 6. What is the optimal monitoring strategy for chronic HypoPT? All panel members were surveyed regarding their current clinical practice, with 70% of respondents reported monitoring 70% of the time as described below (18). These recommendations are graded as low-quality since they are based on expert experience and opinion. The panel agreed that new patients should always be assessed for serum ionized or albumin-adjusted serum calcium, phosphorus, magnesium, creatinine, eGFR if creatinine clearance is not measured separately, 25OHD, and 24-hour urine calcium and creatinine. The panel also agreed that stable patients should be assessed for serum ionized or albumin-adjusted calcium, phosphorus, magnesium, creatinine, and eGFR every 3-12 months. Serum 25OHD should be remeasured every 6-12 months. 24-hour urine calcium and creatinine should be remeasured every 6-24 months. The consensus was that unstable patients should be assessed with frequent serum calcium and phosphorus measurements as clinically indicated. The panel also proposed that patients should have baseline assessment for the presence of renal calcification or stones with renal imaging (11), and that serum ionized or albumin-adjusted calcium should be reassessed within several days of a significant change in medical treatment. These latter two recommendations were based on expert opinion and not survey results. 7. How are patients with HypoPT managed? (GRADed recommendations) In patients with chronic HypoPT, the panel suggests conventional therapy as first line therapy (weak recommendation, low quality evidence) (5,19,20). When conventional therapy is deemed unsatisfactory, the panel considers use of PTH (23). An insufficient number of placebo-controlled trials with PTH analogues has been published to date to justify a stronger recommendation. 8. Recommendations for management: In patients with chronic HypoPT, the panel proposed treatment with calcium and an active vitamin D analogue, with the goal of raising serum calcium into the target range, i.e., the lower half of the normal reference range or just below the normal reference range (5,19,20). At this time, it is not clear how to best balance the doses of calcium relative to those of the active vitamin D analogue. The goal should be to alleviate symptomatic hypocalcemia while avoiding hypercalciuria, and to avoid hypercalciuria when titrating calcium and active vitamin D analogue therapy, aiming for low-normal serum calcium levels. The panel proposed achieving 24-hour urinary calcium of < 6.25 mmol or 250 mg for adult women and < 7.5 mmol or 300 mg for adult men, respectively. Data from the general population has shown a relationship between hypercalciuria and the development of renal stones, but such data does not exist yet in patients with HypoPT. However, panel members inferred that hypercalciuria may be associated with a higher risk of renal stones in patients with HypoPT as well, and thus proposed avoiding hypercalciuria. Hypercalciuria may be treated with thiazide diuretics in conjunction with a low sodium diet with careful monitoring of blood pressure, serum magnesium, potassium, and renal function (10). Panel members proposed avoiding hyperphosphatemia. Panel members prescribe calcium supplements with meals to reduce phosphorus absorption after the meal, implement a low-phosphate diet in adults if needed, and judiciously use active vitamin D therapy that may increase phosphorus absorption. No data are available on the use of non-calcium phosphate binders, such as sevelamer, lanthanum, or others in HypoPT. Hyperphosphatemia may be associated with an increased incidence of ectopic calcification in other populations, but currently there is no evidence of this in HypoPT. Panel members recommend treating to normalize serum magnesium levels. Magnesium supplements may be used as tolerated by the patient. The serum 25OHD level should be kept in the normal reference range of the laboratory used (e.g., 20 to 50 ng/mL or 75-125 nmol/L). PTH replacement therapy should be considered in patients who are not adequately controlled on conventional therapy. Inadequate control is considered to be defined when any one of the following are present despite maximal effort with conventional therapy: symptomatic hypocalcemia, hyperphosphatemia, renal insufficiency, hypercalciuria, or poor quality of life (21–31). Individuals with poor compliance, malabsorption, or who are intolerant of large doses of calcium and active vitamin D may also benefit from PTH therapy. Individuals requiring high doses of conventional therapy, such as calcium supplement of > 2 mg/day, or active vitamin D > 2 mcg/day, may also benefit from PTH therapy (21). 9. Ungraded Consensus Management Recommendations for Hypoparathyroidism During Pregnancy and Lactation In pregnant women with HypoPT, the panel proposed that serum ionized or albumin-adjusted calcium should be maintained in the mid- to low-normal reference range throughout pregnancy (32–37). Serum phosphorus, magnesium, and 25OHD levels should be maintained in the normal reference range. Serum ionized or albumin-adjusted calcium should be monitored every 3-4 weeks during pregnancy and lactation, with increased frequency in the month preceding and following parturition, as well as in the presence of symptoms of hypercalcemia or hypocalcemia. Monitoring physicians should work closely with the obstetrician to optimize pregnancy outcomes, and coordinate with the neonatal pediatric team to ensure appropriate post-natal monitoring for transient hypo- or hypercalcemia in the infant. Thiazide diuretics, high-dose vitamin D2 or D3, and PTH or PTH analogues should be avoided during pregnancy. DISCUSSION The 2 nd International Guidelines for Hypoparathyroidism represent the most recent evidence-based update to the previously published 1 st International Guidelines in 2016. These guidelines summarize the evidence published since 1940, but focus on more recent evidence published since 2000, and emphasize newly published research since 2016. The 90 or so international experts in parathyroid disorders who formed the four task forces addressing various aspects of hypoparathyroid disease were drawn from a wide pool of clinicians and scientists who have published in this area over the last 20 years. For the first time, the recommendations generated in response to important questions felt to be relevant to current management of this disorder were evaluated using GRADE methodology where possible. Several systematic reviews were written describing the latest evidence in support of the new guidelines, and multiple narrative reviews written to evaluate recent evidence in topic areas without sufficient evidence to use GRADE methodology. These guidelines, summarizing multiple manuscripts written in support of the guidelines, are under review for eventual publication in the Journal of Bone and Mineral Research in the Fall of 2022, and have been presented at several national meetings in the U.S. and elsewhere. New evidence has not changed the definition of HypoPT but has led to recommendation for biochemical assessment on at least two occasions using a 2 nd or 3 rd generation PTH assay. Most clinical hospital laboratories currently use a 2 nd generation assay. Patients with HypoPT are mostly moderately to severely symptomatic at presentation, but milder cases or those with longer-standing disease may be relatively asymptomatic. Those who are symptomatic commonly have had anterior neck surgery in the recent past, mostly for thyroid cancer, other head or neck cancer, or PHPT. Surveys have shown that about 75-80% of patients with chronic hypoparathyroidism have postsurgical hypoparathyroidism. Most of the postsurgical symptomatic patients will not have had end-organ involvement yet due to the short duration of their disease. Those with nonsurgical chronic hypoparathyroidism frequently have end-organ involvement due to the longer duration of their disease, sometimes for their entire lifetime if they have genetic disease. Patients with normocalcemic hypoparathyroidism have been described, but these clearly compose a small subset of the relatively asymptomatic group that may not have organ involvement and, by definition, have normal serum ionized and albumin-adjusted total calcium, but inappropriately low PTH levels. It is not yet clear that these patients benefit from specific treatment, and the guidelines do not make a recommendation for treatment of these mildly affected patients. Recommendations for biochemical evaluation have not changed significantly. Measurement of serum calcium, phosphorous, creatinine, magnesium, intact PTH, and 25OHD, and 24-hour urine calcium and creatinine are advised. The 2 nd International Guidelines for Hypoparathyroidism give more precise recommendations for maintenance of 24-hour urine calcium in patients with this disorder at < 250 mg in women and < 300 mg in men. Earlier guidelines either did not specify a threshold for maintenance of 24-hour urine calcium or recommended keeping 24-hour urine calcium less than 400 mg to prevent kidney stones. The earlier threshold of 400 mg was recommended based on guidelines for prevention of kidney stones in the general population, and not specifically in patients with PHPT. Skeletal assessment of patients with chronic HypoPT by bone density testing by DXA, vertebral fracture analysis, spine x-rays, or trabecular bone score, if available, are not routinely recommended because these patients usually have high bone mass due to absence or lack of circulating PTH. This leads to low bone turnover and relative preservation of bone mass. If DXA is to be used to assess bone density, it should be measured at the lumbar spine, total hip, and femur neck, and may be used at the 1/3 distal radius, to quantify the severity of bone loss. Patients with chronic HypoPT may have comorbidities that cause bone loss independent of their low serum PTH. Postmenopausal women may sometimes lose bone density despite being relatively protected against this. Renal imaging by kidney x-ray, ultrasound, or CT are recommended to detect kidney stones or nephrocalcinosis, if present. Calcium-containing kidney stones may develop in patients with chronic HypoPT due to hypercalciuria that frequently develops due to high-dose calcium and/or active vitamin D supplements given over many years of treatment. Medical management options are limited but may be useful in particular situations. Oral or intravenous bisphosphonates, denosumab, estrogen, and raloxifene should not be routinely used in patients with chronic HypoPT because these antiresorptive agents will cause the already-present low bone turnover to worsen. Cinacalcet should never be used as this prevents residual PTH secretion by the remaining parathyroid glands if present. Thiazide-type diuretics may be used to reduce urinary calcium excretion and prevent hypercalciuria and kidney stones. These guidelines summarize current concepts regarding management of pregnancy and lactation in patients with HypoPT. Patients with HypoPT are at greater risk of adverse outcomes for themselves and their babies. Babies may develop postpartum hypercalcemia due to low maternal serum calcium if this is not adequately managed during pregnancy. Babies may also develop postpartum hypocalcemia if their mothers are over-treated with calcium and active vitamin D supplements during pregnancy. New features of the 2 nd International Guidelines on Hypoparathyroidism include defining chronic postsurgical HypoPT as persisting for >12 months after anterior neck surgery. The previous guidelines (6) required that low serum calcium and parathyroid levels be present for at least 6 months to define chronic hypoparathyroidism. The duration of hypocalcemia and hypoparathyroidism was changed because occasional patients are able to recover from their postsurgical hypoparathyroidism for up to 12 months after surgery. Surgeons on the HypoPT task forces indicated that recovery from hypoparathyroidism may rarely occur later than 12 months after surgery. To predict development of permanent postsurgical HypoPT, the guidelines recommend measuring serum PTH within 12-24 hours post-total thyroidectomy (strong recommendation, moderate quality evidence). PTH > 10 pg/mL (>1.05 pmol/L) is considered to virtually exclude long-term postsurgical HypoPT. Patients with nonsurgical HypoPT may benefit from genetic testing in the presence of a positive family history of nonsurgical HypoPT, syndromic features, or in individuals younger than 40 years. The AIRE gene is frequently mutated in patients with nonsurgical chronic HypoPT. A variety of very rare mutations have been described that limit parathyroid function or prevent parathyroid gland development. Commercial laboratories offer gene panels in patients with idiopathic hypoparathyroidism. HypoPT may be associated with multiple complications including nephrocalcinosis, nephrolithiasis, renal insufficiency, cataracts, seizures, cardiac arrhythmias, ischemic heart disease, depression. and increased risk of infection. Other complications are described in the literature, but some complications are less common or not routinely assessed for, and therefore not captured in the systematic review. Minimizing or preventing complications of HypoPT requires careful evaluation and routine monitoring of laboratory indices and imaging studies. Patients with chronic HypoPT should be treated with conventional therapy with calcium and active vitamin D metabolites as first line therapy (weak recommendation, low quality evidence). Clinical experience has shown that many patients with chronic HypoPT are not adequately controlled with conventional therapy, despite the best efforts of knowledgeable and engaged clinicians. When conventional therapy is inadequate, use of recombinant human PTH is recommended. Multiple new agents are coming that will expand therapeutic options for treating HypoPT. These include TransCon PTH, other long-acting forms of PTH, oral teriparatide, PTH1 receptor agonists, and calcium-sensing receptor antagonists. These guidelines also include ungraded consensus recommendations for management of pregnancy and lactation in patients with HypoPT. These are based on expert opinion due to lack of evidence beyond what has been published in small case series and case reports of women during pregnancy or lactation. No clinical trials have been conducted in women with HypoPT who are pregnant or lactating. In conclusion: the 2 nd International Guidelines on Hypoparathyroidism provide updated evidence-based management recommendations for patients with chronic HypoPT, evaluated using GRADE methodology where possible. The 18-month effort behind these guidelines involved around 50 international experts in this disorder, as well as Dr. Gordon Guyatt and his methodology team from McMaster University in Canada. These guidelines will hopefully help advance the care of patients with HypoPT throughout the world Acknowledgements: BLC performed design/conceptualization of this review; data acquisition, review, and analysis; project administration; drafting and preparing the manuscript; and review/editing of the manuscript. Footnotes Funding: no funding was received in support of this manuscript. Disclosure: the author has received institutional research support for clinical trials from Takeda/Shire, Ascendis, and Chugai, and consulted for Takeda/Shire, Ascendis, Amolyt, Entera-Bio, Extend-Bio, and Calcilytix. REFERENCES 1.Khan AA, Bilezikian JP, Brandi ML, Clarke BL, Gittoes NJ, Pasieka JL, et al. Evaluation and management of hypoparathyroidism. Summary statement and guidelines from the 2nd Internation Workshop. J Bone Miner Res. 2022 doi: 10.1002/jbmr.4691. [DOI] [PubMed] [Google Scholar] 2.Bjornsdottir S, Ing S, Mitchell DM, Sikjaer T, Underbjerg L, Hassan-Smith Z, et al. Epidemiology and financial burden of adult chronic hypoparathyroidism. J Bone Miner Res. 2022 doi: 10.1002/jbmr.4675. [DOI] [PMC free article] [PubMed] [Google Scholar] 3.Pasieka JL, Wentworth K, Yeo CT, Cremers S, Dempster DL, Fukumoto S, et al. Etiology and pathophysiology of hypoparathyroidism: A narrative review. 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[DOI] [PubMed] [Google Scholar] Articles from Archives of Endocrinology and Metabolism are provided here courtesy of Sociedade Brasileira de Endocrinologia e Metabologia ACTIONS View on publisher site PDF (131.6 KB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page ABSTRACT INTRODUCTION MATERIALS AND METHODS DISCUSSION Acknowledgements: Footnotes REFERENCES Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
1613	https://spark.iop.org/de-broglie-wavelength	De Broglie wavelength \| IOPSpark Skip to main content Menu Menu Search CPD Collections Close Take a browse Established resources Practical Physics Supporting Physics Teaching Teaching Advanced Physics Marvin and Milo Popular resources Quick demonstrations and activities Stories from Physics Energy in the New Curriculum Glossary Perspectives Teaching Medical Physics Teaching Exoplanet Physics Teaching Football and Physics Teaching Astronomy and Space Teaching Radioactivity Inclusive teaching resources Inside Story: Physics in Medicine Do Try This at Home Casgliad o Adnoddau yn y Gymraeg More... Misconceptions Events Classroom Physics Search Close Travel to De Broglie Wavelength Quantum and Nuclear De Broglie wavelength Glossary Definition for 16-19 Description All particles can show wave-like properties. The de Broglie wavelength of a particle indicates the length scale at which wave-like properties are important for that particle. De Broglie wavelength is usually represented by the symbol λ or λ dB. For a particle with momentum p, the de Broglie wavelength is defined as: _λ_ dB=h p where h is the Planck constant. Discussion If a particle is significantly larger than its own de Broglie wavelength, or if it is interacting with other objects on a scale significantly larger than its de Broglie wavelength, then its wave-like properties are not noticeable. For everyday objects at normal speeds, λ dB is far too small for us to see any observable quantum effects. A car of 1,000 kg travelling at 30 m s–1, has a de Broglie wavelength _λ_ dB=2×10–38 m, many orders of magnitude smaller than the sizes of atomic nuclei. A typical electron in a metal has a de Broglie wavelength is of order ~10 nm. Therefore, we see quantum-mechanical effects in the properties of a metal when the width of the sample is around that value. SI unit metre, m Expressed in SI base units m Other commonly-used unit(s) nm Mathematical expressions _λ_ dB=h p where h is the Planck constant and p is the momentum of the particle. Related entries Wavelength In context We can infer the wave-like nature of matter by observing the diffraction pattern produced when electrons pass through a crystalline material. The pattern occurs when the de Broglie wavelength of the electrons is comparable with the spacing between the atoms of the crystals. For a material such as graphite, where the interatomic spacing is 0.1–0.2 nm, electrons need to be travelling at speeds of the order of ~10 6 m s–1 for this to be the case. Appears in these Collections Show me Filter Apply filters Apply Filter Thinking About Teaching Collection Glossary of physical quantities ------------------------------- We have been working with the National Physical Laboratory (NPL) to jointly develop... For 16-19 33 Resources Other resources on De Broglie Wavelength Wave-Particle Duality Quantum and Nuclear Episode 506: Particles as waves ------------------------------- This episode introduces an important phenomenon: wave – particle duality. In studying the photoelectric effect, students have... Lesson 16-19 Neutron Quantum and Nuclear Episode 531: Neutron diffraction -------------------------------- This episode takes a very brief look at the topic of neutron diffraction. Lesson 16-19 De Broglie Wavelength Follow us on Instagram @IOPSPARK Discover easy classroom activities, quick video explainers, and fresh teaching ideas with CPD support. Explore resources from IOPSpark on Instagram – one scroll at a time. Follow us About IOPSpark Supporting Physics Teaching Teaching Advanced Physics Practical Physics Accessibility Copyright Disclaimer Privacy Terms & Conditions © 2025 IOP All rights reserved. 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1614	https://googology.fandom.com/wiki/List_of_powers_of_10	Skip to content Googology Wiki 35,611 pages Contents 1 0 to 10-1 2 100 to 1049 3 1050 to 1099 4 10100 to 10199 5 10200 to 10299 6 10300 to 10999 7 101,000 to 109,999 8 1010,000 to 10999,999 9 101,000,000 to 109,999,999,999 10 1010,000,000,000 to 10 10 20 − 1 11 10 10 20 to 10 10 50 − 1 12 10 10 50 to 10 10 100 − 1 13 10 10 100 to 10 10 1 , 000 − 1 14 10 10 1 , 000 to 10 10 1 , 000 , 000 − 1 15 10 10 1 , 000 , 000 to 10 10 10 30 − 1 16 10 10 10 30 to 10 10 10 100 − 1 17 10 10 10 100 to 10 10 10 10 , 000 18 10 10 10 10 , 000 to E10#4 = 10 10 10 10 10 19 E10#4 = 10 10 10 10 10 to E100#4 = 10 10 10 10 100 20 E100#4 to 10↑↑7 21 10↑↑7 to 10↑↑10 22 10↑↑10 ~ 10↑↑20 23 10↑↑20 ~ 10↑↑100 24 10 ↑↑ 100 to 10 ↑↑ 1 , 000 , 000 25 10↑↑1,000,000 ~ 10 ↑↑ 10 1 , 000 , 000 26 10 ↑↑ 10 1 , 000 , 000 to 10 ↑↑ 10 ↑↑ 10 27 10 ↑↑↑ 3 to 10 ↑↑↑ 10 28 10 ↑↑↑ 10 to 10 ↑↑↑↑ 10 in: Lists, Powers of 10, Numbers List of powers of 10 Sign in to edit History Purge Talk (10) View full site to see MathJax equation This is a list of powers of 10 in ascending order that have articles on the Googology Wiki. All numbers with the "-illion" suffix are in short scale. Contents 1 0 to 10-1 2 100 to 1049 3 1050 to 1099 4 10100 to 10199 5 10200 to 10299 6 10300 to 10999 7 101,000 to 109,999 8 1010,000 to 10999,999 9 101,000,000 to 109,999,999,999 10 1010,000,000,000 to 10 10 20 − 1 11 10 10 20 to 10 10 50 − 1 12 10 10 50 to 10 10 100 − 1 13 10 10 100 to 10 10 1 , 000 − 1 14 10 10 1 , 000 to 10 10 1 , 000 , 000 − 1 15 10 10 1 , 000 , 000 to 10 10 10 30 − 1 16 10 10 10 30 to 10 10 10 100 − 1 17 10 10 10 100 to 10 10 10 10 , 000 18 10 10 10 10 , 000 to E10#4 = 10 10 10 10 10 19 E10#4 = 10 10 10 10 10 to E100#4 = 10 10 10 10 100 20 E100#4 to 10↑↑7 21 10↑↑7 to 10↑↑10 22 10↑↑10 ~ 10↑↑20 23 10↑↑20 ~ 10↑↑100 24 10 ↑↑ 100 to 10 ↑↑ 1 , 000 , 000 25 10↑↑1,000,000 ~ 10 ↑↑ 10 1 , 000 , 000 26 10 ↑↑ 10 1 , 000 , 000 to 10 ↑↑ 10 ↑↑ 10 27 10 ↑↑↑ 3 to 10 ↑↑↑ 10 28 10 ↑↑↑ 10 to 10 ↑↑↑↑ 10 0 to 10-1[] Googolplexminex, Googolminex, Googol-minutia-speck, 10-110 Googol-minutia, 10-100 Ogol-minutia, 10-80 Gogol-minutia, 10-50 Goby-minutia, 10-35 Minnow-minutia, 10-25 Guppy-minutia, 10-20 100 to 1049[] One, 100 Ten, 101 Hundred, 102 Thousand, 103 Myriad, 104 Lakh, 105 Million, 106 Crore, 107 Myllion, 108 Billion, 109 Dialogue / Guppyspeck, 1010 Undoocol, 1011 Trillion, 1012 Trdoocol, 1013 Qdrdoocol, 1014 Quadrillion / Minnowspeck / Guppycrumb, 1015 Byllion, 1016 Sptdoocol, 1017 Quintillion, 1018 Guppychunk, 1019 Guppy / Minnowcrumb, 1020 Sextillion / Guppybunch, 1021 Goonrol, 1022 Septillion / Minnowchunk, 1024 Minnow / Guppycrowd / Gobyspeck, 1025 Octillion, 1027 Rang, 1028 Nonillion / Minnowcrowd / Guppyswarm / Minnowcrumb / Gobycrumb, 1030 Tryllion, 1032 Decillion, 1033 Gobychunk, 1034 Goby / Minnowswarm, 1035 Undecillion / Gobybunch, 1036 Duodecillion, 1039 Gogolspeck / Gobycrowd, 1040 Tredecillion, 1042 Zai, 1044 Quattuordecillion / Gobyswarm, 1045 Quindecillion, 1048 Gogolchunk, 1049 1050 to 1099[] Lcillion / Gogol, 1050 Sexdecillion / Gogolbunch, 1051 Gougasha, 1052 Tallakshana, 1053 Septendecillion, 1054 Asougi, 1056 Octodecillion, 1057 Novemdecillion / Gogolswarm / Jumbo shrimp-crumb, 1060 Vigintillion, 1063 Quadryllion / Jumbo shrimp-chunk, 1064 Jumbo shrimp / Lightweight-speck, 1065 Unvigintillion / Jumbo shrimp-bunch, 1066 Muryoutaisuu, 1068 Duovigintillion, 1069 Ogolspeck / Lightweight-crumb / Jumbo shrimp-crowd, 1070 Tresvigintillion, 1072 Gazillion / Lightweight-chunk, 1074 Quattuorvigintillion / Ogolcrumb / Lightweight / Jumbo shrimp-swarm, 1075 Lightweight-bunch, 1076 Quinvigintillion, 1078 Ogolchunk, 1079 Ogol / Lightweight-crowd, 1080 Sesvigintillion / Ogolbunch, 1081 Septemvigintillion, 1084 Tiny twerpuloid / Ogolcrowd / Lightweight-swarm, 1085 Octovigintillion, 1087 Novemvigintillion / Ogolswarm / Googolspeck, 1090 Trigintillion, 1093 Googolcrumb, 1095 Untrigintillion, 1096 Uppala, 1098 Duotrigintillion, 1099 10100 to 10199[] Googol, 10100 Googolty / Googolbunch, 10101 Trestrigintillion, 10102 Quattuortrigintillion / Googolcrowd, 10105 Quintrigintillion, 10108 Eleventyplex / Googolswarm, 10110 Sestrigintillion, 10111 Pundarika, 10112 Septentrigintillion, 10114 Octotrigintillion, 10117 Noventrigintillion, 10120 Quadragintillion, 10123 Unquadragintillion, 10126 Quintyllion, 10128 Duoquadragintillion, 10129 Tresquadragintillion, 10132 Mahakathana, 10133 Quattuorquadragintillion, 10135 Quinquadragintillion, 10138 Asankhyeya, 10140 Sesquadragintillion, 10141 Septenquadragintillion, 10144 Dvajagranisamani, 10145 Octoquadragintillion, 10147 Novenquadragintillion, 10150 Quinquagintillion, 10153 Unquinquagintillion, 10156 Duoquinquagintillion, 10159 Tresquinquagintillion, 10162 Quattuorquinquagintillion, 10165 Quinquinquagintillion, 10168 Sesquinquagintillion, 10171 Septenquinquagintillion, 10174 Octoquinquagintillion, 10177 Novenquinquagintillion, 10180 Sexagintillion, 10183 Unsexagintillion, 10186 Duosexagintillion, 10189 Vahanaprajnapti, 10191 Tresexagintillion, 10192 Quattuorsexagintillion, 10195 Quinsexagintillion, 10198 10200 to 10299[] Gargoogol, 10200 Sesexagintillion, 10201 Septensexagintillion, 10204 Octosexagintillion, 10207 Novensexagintillion, 10210 Septuagintillion, 10213 Unseptuagintillion, 10216 Duoseptuagintillion, 10219 Treseptuagintillion, 10222 Quattuorseptuagintillion, 10225 Quinseptuagintillion, 10228 Seseptuagintillion, 10231 Septenseptuagintillion, 10234 Octoseptuagintillion, 10237 Novenseptuagintillion, 10240 Octogintillion, 10243 Unoctogintillion, 10246 Duooctogintillion, 10249 Gogolding, 10250 Tresoctogintillion, 10252 Quattuoroctogintillion, 10255 Quinoctogintillion, 10258 Sexoctogintillion, 10261 Septemoctogintillion, 10264 Octooctogintillion, 10267 Novemoctogintillion, 10270 Nonagintillion, 10273 Unnonagintillion, 10276 Duononagintillion, 10279 Trenonagintillion, 10282 Kuruta, 10283 Quattuornonagintillion, 10285 Quinnonagintillion, 10288 Senonagintillion, 10291 Ecetonspeck, 10293 Septenonagintillion, 10294 Octononagintillion, 10297 Ecetoncrumb, 10298 10300 to 10999[] Novenonagintillion, 10300 Ecetonchunk, 10302 Centillion, 10303 Ecetonbunch, 10304 Uncentillion / cenuntillion, 10306 Ecetoncrowd, 10308 Duocentillion, 10309 Trescentillion / centretillion, 10312 Ecetonswarm, 10313 Quattuorcentillion, 10315 Quincentillion / quinquacentillion, 10318 Sexcentillion, 10321 Septencentillion, 10324 Octocentillion / cenoctotillion, 10327 Sarvanikshepa, 10329 Novencentillion, 10330 Decicentillion, 10333 Unviginticentillion / primo-vigesimo-centillion / cenunvigintillion, 10366 Quattuorviginticentillion / agrasara, 10375 Octoviginticentillion / epi-bitillion, 10387 Ogolding, 10400 Noventrigintacentillion / bentaggoogol, 10420 Uttaraparamanurajahpravesa, 10421 Googolding, 10500 Novemoctogintacentillion, 10570 Novenonagintacentillion / sxoohol, 10600 Ducentillion, 10603 Novenducentillion / Tentaggoogol, 10630 Unvigintiducentillion, 10666 Soophol, 10700 Sesquinquagintaducentillion / bi-exitillion, 10771 Ogolchime/ooohol/ 800-noogol, 10800 Novenonagintaducentillion / duocennovemnonagintillion / tricentillion / noohol / 900-noogol, 10900 Trecentillion, 10903 Duotrigintatrecentillion, 10999 101,000 to 109,999[] Googolchime, 101,000 Novenonagintatrecentillion, 101,200 Quadringentillion, 101,203 Duotrigintaquadringentillion, 101,299 Quingentillion, 101,503 Quattuorquingentillion / ecetonding, 101,515 Duodeciquingentillion / en-bitillion, 101,539 Conflurfin, 101,651 Doug's Zillion, 101,722 Sescentillion, 101,803 Duotrigintatrecentillion / tresexagintasescentillion, 101,992 Guppytoll / doothol / twoohol / 2000-noogol / gargoogolchime, 102,000 Septuagintasescentillion, 102,013 Septingentillion, 102,103 Ex-tritillion, 102,190 Octingentillion, 102,403 Gogolbell, 102,500 Nongentillion, 102,703 Novenongentillion, 102,730 Quinquagintanongentillion / quinquagintinongentillion, 102,853 Trenonagintanongentillion, 102,982 Octononagintanongentillion / enneahectaenneacontaennillion / ennicennigiennillion, 102,997 Novenonagintanongentillion / googood, 103,000 Millillion, 103,003 Milliauntillion, 103,006 Milliadotillion, 103,009 Ecetonchime, 103,030 New vision, 103,050 Dek-bitillion, 103,075 Ogolbell, 104,000 Decyllion, 104,096 Googolbell, 105,000 Platillion / millianongennovemnonagintillion, 106,000 Dumillillion, 106,003 Epi-tritillion, 106,564 Ogoltoll, 108,000 Undecyllion, 108,192 Duomillianongennovemnonagintillion, 109,000 Trimillillion, 109,003 Trimilliduotrigintatrecentillion, 109,999 1010,000 to 10999,999[] Googoltoll, 1010,000 Mevillion, 1010,002 Quadrimillillion, 1012,003 Tre-exitillion, 1012,291 Marioplex, 1012,431 Quinmillillion, 1015,003 Ecetonbell, 1015,150 Sexmillillion, 1018,003 Oct-tritillion, 1019,686 Guppygong, 1020,000 Septimillillion, 1021,003 Octimillillion, 1024,003 Nonimillillion, 1027,003 Myrillion, 1030,003 Ecetontoll, 1030,300 Gogolgong, 1050,000 En-tritillion, 1059,052 Dumyrillion, 1060,003 Ogolgong, 1080,000 Greg's gazillion, 1086,340 Trigintimillillion, 1090,003 Tremyritrimilliduotrigintatrecentillion, 1099,999 Googolgong, 10100,000 Ter-exitillion, 10196,611 Googolbar, 100,00050,000 = 10250,000 Nonagintimillillion, 10270,003 Centimillillion, 10300,003 Ecetongong, 10303,000 Trecentimillitremyritrimilliduotrigintatrecentillion, 10999,999 101,000,000 to 109,999,999,999[] Milliplexion, 101,000,000 Ekatommyrillion, 103,000,000 Milli-millillion, 103,000,003 Blastomlastillion, 103,003,000 Pen-exitillion, 103,145,731 Vigintyllion, 104,194,304 Fzmillion / biblastillion, 106,000,000 Goospolplex, 1010,000,000 Guppybong, 1020,000,000 Gogolbong, 1050,000,000 Pixul, 1075,075,080 Ogolbong, 1080,000,000 Googolbong, 10100,000,000 Yearillion, 10100,000,005 Ecetonbong, 10303,000,000 Maximusbillion / billionplex / billiplexion, 101,000,000,000 Disekatommyrillion, 103,000,000,000 Nanillion, 103,000,000,003 Trigintyllion, 104,294,967,296 Duonanillion, 106,000,000,003 Superior-billion, (109)(109) = 109,000,000,000 Trenanillion, 109,000,000,003 Tremilliamilliamilliatrecentretriginmilliamilliatrecentretriginmilliatrecendotrigintillion, 109,999,999,999 1010,000,000,000 to [] Trialogue, 1010,000,000,000 Guppythrong, 1020,000,000,000 Gogolthrong, 1050,000,000,000 Ogolthrong, 1080,000,000,000 Googolthrong / googoltrine, 10100,000,000,000 Ecetonthrong, 10303,000,000,000 Trecentretriginmilliamilliamilliatrecentretriginmilliamilliatrecentretriginmilliatrecenduotrigintillion, 10999,999,999,999 Maximustrillion / trillionplex / trilliplexion, 101,000,000,000,000 Tetlastillion, 103,000,000,000,000 Picillion, 103,000,000,000,003 Guppygandingan, 1020,000,000,000,000 Gogolgandingan, 1050,000,000,000,000 Ogolgandingan, 1080,000,000,000,000 Googolgandingan, 10100,000,000,000,000 Ecetongandingan, 10303,000,000,000,000 Maximusquadrillion, 101,000,000,000,000,000 Pentlastillion, 103,000,000,000,000,000 Femtillion, 103,000,000,000,000,003 Googolquintigong, 10100,000,000,000,000,000 Hetlastillion, 103,000,000,000,000,000,000 Attillion, 103,000,000,000,000,000,003 Hyper-exitillion, to [] Guppyplex, Heptlastillion, Zeptillion, Googolseptigong, Heptakisekatommyrillion, Yoctillion, Qoofolplex, Googoloctigong, Hyper-icositillion, Little foot, Oktakisekatommyrillion, Xonillion, Googolnonigong, Vecillion, Centyllion, Asankhyeya (Avatamsaka Sutra), Googoldecigong, Mecillion, Googol-undecigong, Duecillion, Nirabhilapya nirabhilapya parivarta, Googol-duodecigong, Trecillion, Googol-tredecigong, Tetrecillion, Googol-quattuordecigong, Pentecillion, Googol-quindecigong, Hexecillion, to [] Gogolplex, Heptecillion, Googol-septendecigong, Octecillion, Googol-octodecigong, Ennecillion, Googol-novemdecigong, Icosillion, Treicosillion, Googolvigintigong, Meicosillion, Dueicosillion, Trioicosillion, Tetreicosillion, Penteicosillion, Hexeicosillion, Hepteicosillion, Octeicosillion, Enneicosillion, Triacontillion, Googoltrigintigong, Metriacontillion, Duetriacontillion, Triotriacontillion, to [] Googolplex, Great googolplex, Gargoogolplex, Fzgoogol, Tetretriacontillion, Tetracontillion, Googolquadragintigong, Pentacontillion, Googolquinquagintigong, Hexacontillion, Googolsexagintigong, Fugahundred, = Gargoogol-plexed, Heptacontillion, Googolseptuagintigong, Octacontillion, Googoloctogintigong, Ennacontillion, Googolnonagintigong, Hectillion, Googolcentigong, Ecetonplex, Mehectillion, Duehectillion, Hyper-bi-exitillion, Triohectillion, Tetrehectillion, Pentehectillion, Hexehectillion, Heptehectillion, Octehectillion, Ennehectillion, Dohectillion, Medohectillion, Triahectillion, to [] Googolplexichime / goomolplex, Tetrahectillion, Pentahectillion, Hexahectillion, Heptahectillion, Octahectillion, Ennahectillion, Enneennaconteennahectillion, Mlastlastilion, Killillion, Bimlastlastillion, Googolmilligong, Hyper-tre-exitillion, Biomlastlastillion, Micrekillillion, Nanekillillion, Googolplexitoll, Picekillillion, Femtekillillion, Attekillillion, Zeptekillillion, Yoctekillillion, Xonekillillion, Vecekillillion, Mecekillillion, Duecekillillion, Icosekillillion, Hyper-ter-exitillion, Triacontekillillion, Googolplexigong, Tetrecontekillillion, Hectekillillion, to [] Millionduplex, Fzmilliplexion, Hyper-pen-exitillion, Megillion, Googolmilli-milligong, Micremegillion, Nanemegillion, Picemegillion, Hyper-ex-exitillion, Googolplexibong / googolpleximine, Hyper-epi-exitillion, Billiduplexion, Gigillion, Hyper-oct-exitillion, Tetralogue, Hyper-en-exitillion, Googolplexithrong, Trilliduplexion, Hyper-dek-exitillion, Terillion, Petillion, Exillion, Treexillion, Guppyduplex, Zettillion, Yoctyllion, Yottillion, Enneenneconteennahecteyotta-enneenneconteennahectezetta-enneenneconteennahecteexa-enneenneconteennahectepeta-enneenneconteennahectetera-enneenneconteennahectegiga-enneenneconteennahectemega-enneenneconteennahectekilla-enneenneconteennahectillion, Xennillion, to [] Dakillion, Hendillion, Dokillion, Tradakillion, Tedakillion, Pedakillion, Exdakillion, Gogolduplex, Zedakillion, Exitrillion, , where Yodakillion, Nedakillion, Ikillion, Onillion, Ikenillion, Icodillion, Ictrillion, Icterillion, Icpetillion, Ikectillion, Iczetillion, Ikyotillion, Icxenillion, Trakillion, Trakenillion, Tracodillion, Tractrillion, to [] Googolduplex, Fzgoogolplex, Googolbangplex, or Fugagoogol, Tracterillion, Tracpetillion, Trakectillion, Traczetillion, Trakyotillion, Tracxenillion, Tekillion, Pekillion, Exakillion, Zakillion, Yokillion, Nekillion, Hotillion, Ecetonduplex, E303#3 = Botillion, Trotillion, Googolduplexichime, E3#4 = E1000#3 = Totillion, Potillion, Exotillion, Zotillion, Yootillion, Notillion, Nonekillion, Nonecxenillion, Kalillion, Dalillion, Tralillion, to E10#4 = [] Googolduplexitoll, E4#4 = E10,000#3 = Talillion, Palillion, Exalillion, Zalillion, Yalillion, Nalillion, Dakalillion, Hendakalillion, Dokalillion, Tradakalillion, Tedakalillion, Pedakalillion, Exadakalillion, Zedakalillion, Yodakalillion, Nedakalillion, Ikalillion, Googolduplexigong, E5#4 = E100,000#3 = Zacalillion, Hotalillion, Millitriplexion, E6#4 = E1,000,000#3 = Totalillion, Mejillion, Dejillion, Dakejillion or crorillion, Googolduplexibong, E8#4 = E100,000,000#3 = Hotejillion, Billitriplexion, E9#4 = E1,000,000,000#3 = Gijillion, E10#4 = to E100#4 = [] Pentalogue, E10#4 = Dakijillion, Googolduplexithrong, E11#4 = Hotijillion, Astillion, Dakastillion, Hotastillion, Lunillion, Dakunillion, Hotunillion, Fermillion, Dakermillion, Guppytriplex, E20#4 = Hotermillion, Jovillion, Dakovillion, Hotovillion, Solillion, Dakolillion, Hotolillion, Betillion, Daketillion, Hotetillion, Glocillion, Dakocillion, Hotocillion, Gaxillion, Dakaxillion, Hotaxillion, Supillion, Dakupillion, Hotupillion, Versillion, Dakersillion, Hotersillion, Multillion, Dakultillion, Hotultillion, Nonecxenultillion, Nonecxenulti-nonecxenersi-nonecxenupi-nonecxenaxi-nonecxenoci-nonecxeneti-nonecxenoli-nonecxenovi-nonecxenermi-nonecxenuni-nonecxenasti-nonecxeniji-nonecxeneji-nonecxenali-nonecxenillion, Gogoltriplex, E50#4 = Exiterillion, , where E100#4 to 10↑↑7[] Googoltriplex, E100#4 = Fzgargoogolplex, Googolplexbangplex, Fugagoogolplex, Googolbangbangplex, Ecetontriplex, E303#4 = Googoltriplexichime, E1000#4 = E3#5 = Googoltriplexitoll, E10,000#4 = E4#5 = Googoltriplexigong, E100,000#4 = E5#5 = googoltripleximine, E8#5 = Bentrizillion, Hexalogue, E10#5 = E1#6 = Googoltriplexithrong, E100,000,000,000#4 = E11#5 = Guppyquadriplex, E20#5 = Gogolquadriplex, E50#5 = Exipetillion, , where Googolquadriplex, E100#5 = E2#6 = Fzgargantugoogolplex, Fugagargoogolplex, Ecetonquadriplex, E303#5 = Googolquadriplexichime, E1000#5 = E3#6 = Hydrillion, 103103103103103,003+3 Googolquadriplexitoll, E10,000#5 = E4#6 = Googolquadriplexigong, E100,000#5 = E5#6 = 10↑↑7 to 10↑↑10[] Heptalogue, E10#6 = E1#7 = Alumillion, 1031031031031031018+3 Guppyquintiplex, E20#6 = Gogolquintiplex, E50#6 = Exitexillion, H(H(H(H(H(H(16)))))) = , where Googolquinplex / googolquintiplex, E100#6 = E2#7 = Ecetonquintiplex, E303#6 = Googolquintiplexichime, E1000#6 = E3#7 = Googolquintiplexitoll, E10,000#6 = E4#7 = Googolquintiplexigong, E100,000#6 = E5#7 = Millionsextiplex, E6#7 = E1,000,000#6 Octalogue, E10#7 = E1#8 = Latinlatinlatinbyllionyllionyllionyllion, Guppysextiplex, E20#7 = Gogolsextiplex, E50#7 = Exitepillion, , where Googolsextiplex, E100#7 = E2#8 = Ecetonsextiplex, E303#7 = Googolsextiplexichime, E1000#7 = E3#8 = Googolsextiplexitoll, E10,000#7 = E4#8 = Googolsextiplexigong, E100,000#7 = E5#8 = Millionseptiplex, E6#8 or E1,000,000#7 Ennalogue, E10#8 = E1#9 = Guppyseptiplex, E20#8 = Gogolseptiplex, E50#8 = Exitokillion, , where H(x) = Googolseptiplex, E100#8 = E2#9 = Ecetonseptiplex, E303#8 = Googolseptiplexichime, E1000#8 = E3#9 = Googolseptiplexitoll, E10,000#8 = E4#9 = Googolseptiplexigong, E100,000#8 = E5#9 = 10↑↑10 ~ 10↑↑20[] Decker / dekalogue, E10#9 = E1#10 = Guppyoctiplex, E20#9 Gogoloctiplex, E50#9 Exinovillion, , where H(x) = Googoloctiplex, E100#9 Ecetonoctiplex, E303#9 Googoloctiplexichime, E1000#9 Googoloctiplexitoll, E10,000#9 Googoloctiplexigong, E100,000#9 Endekalogue / equinoxal, E10#10 = E1#11 = 10↑↑11 Guppynoniplex, E20#10 Hyperillion, , where H(x) = Gogolnoniplex, E50#10 Googolnoniplex, E100#10 Ecetonnoniplex, E303#10 Googolnoniplexichime, E1000#10 Googolnoniplexitoll, E10,000#10 Googolnoniplexigong, E100,000#10 Dodekalogue, E10#11 = E1#12 = 10↑↑12 Guppydeciplex, E20#11 Gogoldeciplex, E50#11 Googoldeciplex, E100#11 Ecetondeciplex, E303#11 Googoldeciplexichime, E1000#11 Googoldeciplexitoll, E10,000#11 Googoldeciplexigong, E100,000#11 Googoldeciplexibong, E100,000,000#11 = E8#12 Triadekalogue, E10#12 = E1#13 = 10↑↑13 Googoldeciplexithrong, E100,000,000,000#11 = E11#12 Tetradekalogue, E10#13 = E1#14 = 10↑↑14 Pentadekalogue, E10#14 = E1#15 = 10↑↑15 Hexadekalogue, E10#15 = E1#16 = 10↑↑16 Heptadekalogue or petistron, E10#16 = E1#17 = 10↑↑17 Exi-hyperillion, where H(x) = Octadekalogue, E10#17 = E1#18 = 10↑↑18 Ennadekalogue, E10#18 = E1#19 = 10↑↑19 10↑↑20 ~ 10↑↑100[] Icosalogue, = Icosi-hyperillion, where H(x) = Guppyvigintiplex, Gogolvigintiplex, Eceton-vigintiplex, E303#21 or 10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^(110^303) Googolvigintiplexichime, Googolvigintiplexitoll, Googolvigintiplexigong, Yottistron, 10↑↑23 or 10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10,000,000,000 Qigfol, = Triantalogue, = Guppytrigintiplex, Gogoltrigintiplex, Eceton-trigintiplex, E303#31 Googoltrigintiplexichime, Googoltrigintiplexitoll, Googoltrigintiplexigong, Terantalogue, = Guppyquadragintiplex, Gogolquadragintiplex, Eceton-quadragintiplex, E303#41 Googolquadragintiplexichime, Googolquadragintiplexitoll, Googolquadragintiplexigong, Penantalogue, = Guppyquinquagintiplex, Gogolquinquagintiplex, Eceton-quinquagintiplex, E303#51 Googolquinquagintiplexichime, Googolquinquagintiplexitoll, Googolquinquagintiplexigong, Exatalogue, = Guppysexagintiplex, Gogolsexagintiplex, Eceton-sexagintiplex, E303#61 Googolsexagintiplexichime, Googolsexagintiplexitoll, Googolsexagintiplexigong, Eptatalogue, = Guppyseptuagintiplex, Gogolseptuagintiplex, Eceton-septuagintiplex, E303#71 Googolseptuagintiplexichime, E1,000#71 Googolseptuagintiplexitoll, E10,000#71 Googolseptuagintiplexigong, E100,000#71 Ogdatalogue, = Guppyoctogintiplex, Gogoloctogintiplex, Eceton-octogintiplex, E303#81 Googoloctogintiplexichime, E1,000#81 Googoloctogintiplexitoll, E10,000#81 Googoloctogintiplexigong, E100,000#81 Entatalogue, = Guppynonagintiplex, Gogolnonagintiplex, Eceton-nonagintiplex, E303#91 Googolnonagintiplexichime, E1,000#91 Googolnonagintiplexitoll, E10,000#91 Googolnonagintiplexigong, E100,000#91 to [] Giggol / hectalogue, = Gargiggol, (10↑↑100)2 Megafugahundred, ~ Giggolunex, = Fzgiggol, Grangol, Big Box, Cent-hyperillion, where H(x) = Giggolduunex, = Fugagiggol, (10↑↑100)↓↓(10↑↑100) Guppycentiplex, Gogolcentiplex, Grangolplex / googolcentiplex, Eceton-centiplex, E303#101 Googolcentiplexichime, E1,000#101 Googolcentiplexitoll, E10,000#101 Googolcentiplexigong, E100,000#101 Googolcentiplexibong, E100,000,000#101 = E8#102 Giggoltreunex, = Googolcentiplexithrong, E100,000,000,000#101 Giggolquadruunex, = Giggolquinunex, = Giggolsexunex, = Giggolseptenunex, = Giggoloctounex, 10↑↑108 Giggolnovemunex, 10↑↑109 Giggoldecaunex, 10↑↑110 Bighol or dooducol, = Chilialogue / giggolchime, = Thousalogue, L((3)) Grangolchime, Great Big Box, ~ E3,003#1,000 Guppymilliplex, E20#1,001 Gogolmilliplex, Googolmilliplex, Eceton-milliplex, E303#1,001 Googolmilliplexichime, E1,000#1,001 Googolmilliplexitoll, E10,000#1,001 Googolmilliplexigong, E100,000#1,001 Googolmilliplexibong, E100,000,000#1,001 Googolmilliplexithrong, E100,000,000,000#1,001 Myrialogue / giggoltoll, = Grangoltoll, Giggolgong, = Grangolgong, 10↑↑1,000,000 ~ [] Chilia-chilialogue, = Millilogue, L((6)) Megafugamillion, 1,000,000↑↑1,000,000 Guppymilli-milliplex, Gogolmilli-milliplex, Googolmegaplex, Eceton-millimilliplex, E303#1,000,001 Googolmillimilliplexichime, Googolmillimilliplexitoll, E10,000#1,000,001 Googolmillimilliplexigong, E100,000#1,000,001 Googolmillimilliplexibong, E100,000,000#1,000,001 Googolmillimilliplexithrong, E100,000,000,000#1,000,001 Chilia-myrialogue / joycian gygol, = Myria-myrialogue / giggolbong, = Grangolbong, Latinmyllionalatinmyllionlatinmyllionayllion, \approx Joycian gagol, Billilogue, L((9)) Googolgigaplex, Dialogialogue, = Giggolthrong, = Grangolthrong, Googolteraplex, E100#1012+1 Googolpetaplex, E100#1015+1 Sedeniadalogue, 10↑↑1016 Googolexaplex, E100#1018+1 Guppylogue, Googolzettaplex, Googolyottaplex, Minnowlogue, Gobylogue, Gogologue, Ogologue, Zootzoot, or googol!1 in hyperfactorial array notation. Googol-stack / googologue, Googolstackplex, 10↑↑(10100+1) Googoldex, E100#10100 = E100#1#2 Megafugagoogol, Googolstackplexplex, 10↑↑(10100+2) Googoldexiplex, Googoldexiduplex, Googolecettaplex, Ecetondex, (E303#10^{303}) Mevalkillion, 103[3,3]3+3 ~ 10↑↑10333 Faxulogue, 10↑↑7.8865710374 Goomolduex, 10↑↑101,000 Goomyrolduex, 10↑↑1010,000 Gooqnolplexiduex, 10↑↑10100,000 to [] Trialogialogue, Zootzootplex, Googolplexstack, Googolplexidex, Googolplexidexiplex, E100#(1010100+1) Megafugagoogolplex, Googolplexidexiduplex, Googolbangstack, Goomolplexiduex, 10↑↑10101,000 Tetralogialogue / Goodcolduplexiduex, Zootzootduplex, Googolduplexilogue / googolduplexiduex, 10↑↑101010100 Googolduplexidex, Megafugagargoogolplex, Googolplexbangstack, 10↑↑(1010100!) Googolbangplexstack, 10↑↑(1010100!) Pentalogialogue, Zootzoottriplex, Googoltriplexilogue / Googoltriplexiduex, 10↑↑(E100#4) Googoltriplexidex, Megafugagargantugoogolplex, Hexalogialogue, Googolquadriplexiduex, 10↑↑(E100#5) Googolquadriplexidex, Heptalogialogue, Googolquintiplexiduex, 10↑↑(E100#6) Googolquintiplexidex, Octalogialogue, 10↑↑10↑↑8 or E1#8#2 Googolsextiplexidex, Ennalogialogue, 10↑↑10↑↑9 or E1#9#2 Googolseptiplexidex, to [] Tria-taxis / tria-teraksys / dekalogialogue, E1#1#3 = 10↑↑10↑↑10 Googoloctiplexidex, Equiduoxal, 10(≡≡) Grand Hyperillion, H(10,2) Googolnoniplexidex, Googoldeciplexidex, Grand-exi-hyperillion, H(16,2) Doovolplex, 10↑↑10↑↑20 Grand-icosi-hyperillion, H(20,2) Dooqtolplex, 10↑↑10↑↑25 Doohfolplex, 10↑↑10↑↑50 Giggolplex/ hectalogialogue / giggolduex / doogolplex / doogolduex, Grangoldex, Chilialogialogue, Grangoldexichime, Myrialogialogue, Grangoldexitoll, Grangoldexigong, Octadialogialogue, Grangoldexibong, Latinlatinmyllionalatinmyllionlatinmyllionayllionalatinmyllionlatinlatinmyllionalatinmyllionlatinmyllionayllionayllion, Sedeniadialogialogue, Googolstackstack / googolgoogolduplex / googolduduex, Googolstackstackplex, 10↑↑(1+10↑↑10100) Googoldudex, E100#1#3 Ecetondudex, E303#1#3 Goomolduduex, 10↑↑10↑↑101,000 Trialogialogialogue / Goodcolplexiduduex, E1#3#3 = Zootzootzootplex / triplezootplex, Googolplexiduduex, 10↑↑10↑↑1010100 Googolplexidudex, E100#2#3 Tetralogialogialogue, Googolduplexiduduex, 10↑↑10↑↑101010100 Googolduplexidudex, E100#3#3 Pentalogialogialogue, Googoltriplexiduduex, 10↑↑10↑↑(E100#4) Googoltriplexidudex, E100#4#3 Hexalogialogialogue, Googolquadriplexiduduex, 10↑↑10↑↑(E100#5) Googolquadriplexidudex, E100#5#3 Heptalogialogialogue, Googolquintiplexidudex, E100#6#3 Octalogialogialogue, Googolsextiplexidudex, E100#7#3 Ennalogialogialogue, Googolseptiplexidudex, E100#8#3 Tetra-taxis, Googoloctiplexidudex, E100#9#3 Equitrioxal, 10(≡≡≡) Bigrand Hyperillion, H(10,3) Googolnoniplexidudex, E100#10#3 Googoldeciplexidudex, E100#11#3 Bigrand-exi-hyperillion, H(16,3) Bigrand-icosi-hyperillion, H(20,3) Giggolduplex/ hectalogialogialogue / doogolduplex, {10,giggolplex,2} = Grangoldudex, Chilialogialogialogue / Doomolduplex, Grangoldudexichime, Myrialogialogialogue / Doomyrolduplex, Grangoldudexitoll, Grangoldudexigong, Ekatommyriaplo ekatommyriaplekatommyriakis ekatommyriaplo ekatommyriaplekatommyriakis ekatommyriaplo ekatommyriaplekatommyriakis ekatommyrio, E(E(E(1,000,000) ~ Octadialogialogialogue, Grangoldudexibong, Grangoldudexithrong, Sedeniadialogialogialogue, Googolgoogoltriplex / googolstackstackstack / googoltreduex / googoltriduex, 10↑↑10↑↑10↑↑10100 Googoltridex, E100#1#4 Ecetontridex, E303#1#4 Googolplexitriduex, 10↑↑10↑↑10↑↑1010100 Googolplexitridex, E100#2#4 Googolduplexitriduex, 10↑↑10↑↑10↑↑101010100 Googolduplexitridex, E100#3#4 Penta-taxis, Giggoltreplex / doogoltriplex, E1#100#4 Grangoltridex, E100#100#4 Grangoltridexichime, E1,000#1,000#4 Grangoltridexitoll, E10,000#10,000#4 Grangoltridexigong, E100,000#100,000#4 Googolquadruduex / googolquadriduex, 10↑↑10↑↑10↑↑10↑↑10100 Googolquadridex, E100#1#5 Ecetonquadridex, E303#1#5 Googolplexiquadriduex, 10↑↑10↑↑10↑↑10↑↑1010100 Googolplexiquadridex, E100#2#5 Googolduplexiquadridex, E100#3#5 Hexa-taxis, Giggolquadruplex / doogolquadriplex, E1#100#5 Grangolquadridex, E100#100#5 Grangolquadridexichime, E1,000#1,000#5 Grangolquadridexitoll, E10,000#10,000#5 Grangolquadridexigong, E100,000#100,000#5 Googolquinduex / googolquintiduex, E1#(10100)#5 Googolquintidex, E100#1#6 Ecetonquintidex, E303#1#6 Googolplexiquintidex, E100#2#6 Googolduplexiquintidex, E100#3#6 Hepta-taxis, Grangolquintidex, E100#100#6 Grangolquintidexichime, Grangolquintidexitoll, E10,000#10,000#6 Grangolquintidexigong, Googolsexduex, E1#(10100)#6 Googolsextidex, E100#1#7 Ecetonsextidex, E303#1#7 Googolplexisextidex, E100#2#7 Googolduplexisextidex, E100#3#7 Octa-taxis, 10↑↑↑8 Giggolsexplex / doogolsextiplex, E1#100#7 Grangolsextidex, E100#100#7 Grangolsextidexichime, Grangolsextidexitoll, Grangolsextidexigong, Googolseptenduex, E1#(10100)#7 Googolseptidex, E100#1#8 Ecetonseptidex, E303#1#8 Googolplexiseptidex, E100#2#8 Googolduplexiseptidex, E100#3#8 Enna-taxis, 10↑↑↑9 Giggolseptenplex / doogolseptiplex, E1#100#8 Grangolseptidex, E100#100#8 Grangolseptidexichime, Grangolseptidexitoll, Grangolseptidexigong, Googoloctoduex, E1#(10100)#8 Googoloctidex, E100#1#9 Googolplexioctidex, E100#2#9 Googolduplexioctidex, E100#3#9 to [] Deka-taxis, Grangoloctidex, Grangoloctidexichime, Grangoloctidexitoll, E10,000#10,000#9 Grangoloctidexigong, E100,000#100,000#9 Googolnovemduex, E1#(10100)#9 Googolnonidex, E100#1#10 Ecetonnonidex, E303#1#10 Googolplexinonidex, E100#2#10 Googolduplexinonidex, E100#3#10 Megiston, Giggolnovemplex / doogolnoniplex, E1#100#10 Grangolnonidex, E100#100#10 Grangolnonidexichime, E1,000#1,000#10 Grangolnonidexitoll, E10,000#10,000#10 Grangolnonidexigong, Googoldecaduex, E1#(10100)#10 Googoldecidex, E100#1#11 Ecetondecidex, E303#1#11 Googolplexidecidex, E100#2#11 Googolduplexidecidex, E100#3#11 Giggoldecaplex / doogoldeciplex, E1#100#11 Grangoldecidex, Grangoldecidexichime, Grangoldecidexitoll, Grangoldecidexigong, Toovol, 10↑↑↑20 Qagfol, 10↑↑↑25 Tootrol, 10↑↑↑30 Tooqrol, 10↑↑↑40 Hagfol, Gaggol, 10↑↑↑100 Gaggolunex / toogolunex, 1010↑↑↑100 Gaggolduunex / toogolduunex, 101010↑↑↑100 Greagol, Greagolplex, Greagoldex, Tooducol, Chilia-taxis, Greagolchime, Myria-taxis, Greagoltoll, Gaggolgong, Greagolgong, Chilia-chilia-taxis, Chilia-myria-taxis, Myria-myria-taxis / gaggolbong, Greagolbong, Gaggolthrong, Greagolthrong, Sedeniadia-taxis, Googol-3-flex, Googolthrex, ~ Ecetonthrex, ~ Trialogia-taxis, Tetralogia-taxis, Pentalogia-taxis, Hexalogia-taxis, Heptalogia-taxis, Octalogia-taxis, Ennalogia-taxis, Dekalogia-taxis, Giggolthrex, Grangolthrex, Triataxia-taxis, Tria-petaxis, Gaggolplex, Greagolthrex, Greagolthrexichime, Greagolthrexitoll, Greagolthrexigong, Googolduthrex, Triataxiataxia-taxis, Tetra-petaxis, Gaggolduplex, Greagolduthrex, Greagolduthrexichime, Greagolduthrexitoll, Greagolduthrexigong, Googoltrethrex / googoltrithrex, Penta-petaxis, Greagoltrithrex, Greagoltrithrexichime, Greagoltrithrexitoll, Greagoltrithrexigong, Googolquadruthrex, Hexa-petaxis, Greagolquadrithrex, Greagolquadrithrexichime, Greagolquadrithrexitoll, Greagolquadrithrexigong, Hepta-petaxis, Greagolquintithrex, Greagolquintithrexichime, Greagolquintithrexitoll, Greagolquintithrexigong, Octa-petaxis, Greagolsextithrex, Enna-petaxis, Greagolseptithrex, Deka-petaxis, ↑ no reference Categories Community content is available under CC-BY-SA unless otherwise noted. 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1615	https://www.youtube.com/watch?v=SiQbegnH1NA	Converting Between Fractions and Decimals \| Fractions to Decimals \| Decimals to Fractions Math with Mr. J 1730000 subscribers 308 likes Description 22596 views Posted: 23 Sep 2024 Welcome to Converting Between Fractions and Decimals with Mr. J! Need help with converting fractions to decimals or decimals to fractions? You're in the right place! Whether you're just starting out, or need a quick refresher, this is the video for you if you're looking for help with how to convert between fractions and decimals. Mr. J will go through two examples of how to convert fractions to decimals and two examples of how to convert decimals to fractions. ✅ Check out more videos on fractions, decimals, and percents here: ✅ Chapters and Timestamps 00:00 = Converting Fractions to Decimals 05:34 = Converting Decimals to Fractions About : Math with Mr. J is a math education channel that offers instructional math videos to anyone looking for a little extra help with math! Email : math5.mrj@gmail.com Music : Hopefully this video is what you're looking for when it comes to fractions and decimals. Have a great rest of your day and thanks again for watching! ✌️✌️✌️ 27 comments Transcript: Converting Fractions to Decimals [Music] welcome to math with Mr J in this video I'm going to cover how to convert between fractions and decimals so fractions to decimals and decimals to fractions let's start with fractions to decimals starting with number one where we have 1/4 now when we go from fractions to decimals we can divide the numerator the top number by the denominator the bottom number so here we need to do 1 / 4 remember fractions are a representation of division and if we do the division we get the decimal form of the fraction and they are equal one form is just a fraction and the other a decimal so again 1 / 4 here and I'm going to do the division by hand in this video so let's set this up 1 / 4 so let's start with 1 / 4 how many whole groups of four in 1 how many fours in one well we can't do that so we need to put a decimal after one and then a zero to the right of the decimal and I'm going to extend that division bar there now remember Zer to the right of a decimal or to the right of decimal digits do not change the value of the number so we're able to do this in order to work through the division problem now we bring the decimal straight up into the quotient the answer now we can think of this as 10 / 4 so how many whole groups of four in 10 how many fours in 10 well two so let's put the two above the 10 and make sure it's above the zero now we multiply 2 4 is 8 subtract 10 - 8 is 2 we don't have that cleancut zero at the bottom here so we can continue on let's use another zero that we can bring down so now we have 20 20 / 4 which is 5 5 4 is 20 20 - 20 is 0 nothing else to bring down within our problem and we have that clean cut zero we are done with the division and we can write this over here and I'm going to start with a zero and then a decimal to show that we are working with a decimal here it helps us recognize we have a decimal helps us see the decimal so 025 1/4 equal 2500s let's move on to number two where we have 516 so we need to do five divided by 16 so as far as 5 ID 16 how many whole groups of 16 in five well we can't do that so we need a decimal and then a zero let's extend this division bar here and bring the decimal straight up into the quotient the answer now we can think of this as 50 / 16 how many whole groups of 16 and 50 well three that gets us to 48 3 16 16 48 subtract 50 - 48 is 2 so we don't have that clean cut zero quite yet so let's use another zero that we can bring down now we have 20 20 divided by 16 how many whole groups of 16 in 20 well 1 1 16 is 16 subtract 20 - 16 is 4 so let's continue on by using another zero that we can bring down so now we have 40 40 ID 16 how many whole groups of 16 and 40 two that gets us to 32 2 16 32 subtract 40 - 32 is 8 so let's continue on so another zero bring that down and now we have 80 80 / 16 how many whole groups of 16 in 80 well five and that's going to hit 80 exactly 5 16 is 80 80us 80 gives us that clean cut zero we are done so 516 equals 3125 3,125 10,000 so let's write this over here and we will start with the zero and the decimal remember that zero helps us recognize we have a decimal here so 0.3125 so 516 equals 3,125 10,000 now before we move on I do want to mention what happens if we don't get to that clean cut zero what happens if that decimal keeps going in that case we need to round so for example go to the 10,000 place and round to the thousandths or go to the thousand place and round to the hundredths do whatever works best for you now let's Converting Decimals to Fractions move on to going from decimals to fractions and all we need to do here is use place value to determine our denominator and then we can simplify if possible for example number three we have 0.7 so 7/10 here this decimal ends in the 10th place that means our denominator is 10 10 so we write this as 71th again our decimal ends in the 10's place and that tells us what our denominator is 10 and that's it we can read the decimal as 710 and the fraction as 7/10 as well they have the same exact value one is just a decimal and the other a fraction now as far as simplifying 710 is in simplest form the only common factor between 7 and 10 is one so we are done moving on to number four this looks similar but we have 0.07 so 7 hundredths here the seven is in the hundredths place so that's what we're going to use for the denominator so for number four we have 7 over 100 7 H hundredths the only common factor between 7 and 100 is 1 so we are are in simplest form let's move on to number five where we have 0.55 55 hundredths now this decimal ends in the hundred's place so that's going to be our denominator so we have 55 over 155 hundredths now this fraction can be simplified we have a greatest common factor of five that we can divide the numerator and denominator by in order to simplify 55 / 5 gives us 11 and 100 divided 5 gives us 20 now the only common factor between 11 and 20 is 1 so we are in simplest form 11 20ths lastly moving on to number seven we have 2.4 2 and 410 now don't let the whole number hold us up on this all we need to do is write the whole number and then we worry about the decimal here we have 4410 so the decimal ends in the 10th place so that's going to be our denominator so 4 10ths so this is 2 and 4/10 and this is 2 and 4/10 now we can simplify the fractional part of this mixed number we have a greatest common factor of two that we can divide the numerator and denominator by so we have a whole number of two and then 4 / two gives us 2 and 10 / 2 gives us 5 so this simplifies to 2 and 2 fths so there you have it there's how to go from fractions to decimals and decimals to fractions I hope that helped thanks so much for watching until next next time peace
1616	https://www.chegg.com/homework-help/questions-and-answers/suppose-lim-n-1-lim-n-bn-1-0-bn-n-evaluate-following-limits-state-limit-exist-state-enough-q113947719	Solved Suppose that lim n → ∞ an = 1, lim n → ∞ bn = −1, and \| Chegg.com Skip to main content Books Rent/Buy Read Return Sell Study Tasks Homework help Understand a topic Writing & citations Tools Expert Q&A Math Solver Citations Plagiarism checker Grammar checker Expert proofreading Career For educators Help Sign in Paste Copy Cut Options Upload Image Math Mode ÷ ≤ ≥ o π ∞ ∩ ∪           √  ∫              Math Math Geometry Physics Greek Alphabet Math Calculus Calculus questions and answers Suppose that lim n → ∞ an = 1, lim n → ∞ bn = −1, and 0 < −bn < an for all n. Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists. Solve 23 and 25 please. Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. See Answer See Answer See Answer done loading Question: Suppose that lim n → ∞ an = 1, lim n → ∞ bn = −1, and 0 < −bn < an for all n. Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists. Solve 23 and 25 please. Suppose that lim n → ∞ an = 1, lim n → ∞ bn = −1, and 0 < −bn < an for all n. Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists. Solve 23 and 25 please. Show transcribed image text There are 2 steps to solve this one.Solution Share Share Share done loading Copy link Step 1 Given that lim n→∞a n=1,lim n→∞b n=−1 and 0<−b n<a n. View the full answer Step 2 UnlockAnswer Unlock Previous questionNext question Transcribed image text: Suppose that lim n→∞a n=1,lim n→∞b n=−1, and 0<−b n<a n for all n. Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists. 23. lim n→∞3 a n−4 b n 25. lim n→∞a n−b na n+b n Not the question you’re looking for? Post any question and get expert help quickly. Start learning Chegg Products & Services Chegg Study Help Citation Generator Grammar Checker Math Solver Mobile Apps Plagiarism Checker Chegg Perks Company Company About Chegg Chegg For Good Advertise with us Investor Relations Jobs Join Our Affiliate Program Media Center Chegg Network Chegg Network Busuu Citation Machine EasyBib Mathway Customer Service Customer Service Give Us Feedback Customer Service Manage Subscription Educators Educators Academic Integrity Honor Shield Institute of Digital Learning © 2003-2025 Chegg Inc. 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1617	http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/radfrac.html	Blackbody Radiation Radiated Power from Blackbody ============================= When the temperature of a blackbody radiator increases, the overall radiated energy increases and the peak of the radiation curve moves to shorter wavelengths. When the maximum is evaluated from the Planck radiation formula, the product of the peak wavelength and the temperature is found to be a constant. Calculation The total power radiated by a blackbody is given by the Stefan-Boltzmann equation, but it is often interesting to know the fraction of power which is emitted in the visible or some other wavelength range. Temperature T = K = °C Area A = cm 2 = x10^ m 2 Emissivity = (e = 1 for ideal radiator) The total power radiated is P = watts = x10^ watts. Finding the power radiated within a given wavelength range requires integration of the Planck radiation formula over that range. The radiated power per unit area is the Planck energy density multiplied by c/4. It can be approximated numerically by taking a sum of values of the Planck radiation density times a wavelength interval. The result of dividing the wavelength interval into 100 steps is as follows. For the wavelength range λ 1 = nm to λ 2 = nm, The radiated power is P interval = watts = x10^ watts. This is % of the total radiated power. The radiated power in a given wavelength interval Δλ at wavelength λ can be approximated by The above approximate calculation for the radiated power in a chosen wavelength range is a brute force sum over 100 terms formed by dividing the specified wavelength range into 100 parts. This is then compared to the total radiated power calculated from the Stefan-Boltzmann equation. You can check out the calculation by choosing the wavelength range so that it covers essentially all the radiated energy. You will find that the calculation diverges if you pick a starting wavelength too close to zero, and if you put in too large a wavelength. But by examining the wavelength range for which the radiated power appears to be significant, you can choose reasonable limits on wavelength and confirm that you get essentially all the radiated power.Index Blackbody radiation concepts HyperPhysics Quantum PhysicsR NaveGo Back
1618	https://artofproblemsolving.com/wiki/index.php/Cyclotomic_polynomial?srsltid=AfmBOoohZZcLCSLQZO-lzpxgn1N5yTZzNlqJRgrxPRD5Ly-epmz802ie	Art of Problem Solving Cyclotomic polynomial - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Cyclotomic polynomial Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Cyclotomic polynomial The Cyclotomic Polynomials are a family of polynomials that are observed frequently in number theory and algebra. While most sources on the internet introduce them on a preliminary level, there is far more rigor and connections to different disciplines which will all be shared here. Contents [hide] 1 Motivation 2 The Cyclotomic Field 3 The Cyclotomic Polynomial at First Glance 4 The Cyclotomic Polynomials Fully Defined Motivation The main reason why one even cares about the Cyclotomic polynomials begins with the study of splitting fields. Definition: The extension of a field is a splitting field for if can be written as the product of irreducible factors in , and does not factor into a product of irreducible factors in any other proper subfield of containing . To ease any worries, any field is guaranteed an extension , so the existence of such a splitting field is not problematic. An example to consider would be the splitting field of over the field . This, of course, would just be since its roots in are both in . So this raises the question: what is the splitting field of over based on this definition? The Cyclotomic Field Recall that over there are distinct solutions of the equation , which are for . These are known as the th roots of unity. These harken back to the language of generators, since we see that In fact, the collection of the th roots of unity forms a cyclic group under multiplication, which we will call . Now we are in the language of groups. Since is cyclic, it certainly has generators, so we shall define them. We call the generators of the primitive th roots of unity, which we denote as . The other primitive roots of unity are of course where where , so it follows that there are primitive roots of unity. This makes sense because if we let then . Back to the language of fields. As we saw in the case of , we can view this splitting field for over as a field generated over by the field . Specifically, our splitting field is going to be generated by . Definition: The splitting field of over is called the cyclotomic field of the th roots of unity, which we denote by . The Cyclotomic Polynomial at First Glance The Cyclotomic polynomials come into play when we look at . More specifically, over we get the following factorization: but since for any , we see that must be a root of . This is such a special property that we give this polynomial a special name. The Cyclotomic Polynomials of order are given by This would just be a mathematical novelty if not for this crucial theorem. Theorem: The Cyclotomic Polynomial is irreducible over for all . Proof: The proof is rather iconic. Let be prime. We can write the cyclotomic polynomial in a more helpful form: Recall by that by the Binomial Theorem we have where for . By considering we have and dividing the right hand side by gives which is irreducible over by eisenstein's criterion. The implication that being irreducible in can easily be seen with , so by Gauss' lemma the result follows. Since is irreducible over , it follows that is the splitting field for . Hence, . But this is only for . What happens if we generalize this for all ? The Cyclotomic Polynomials Fully Defined We define the th cyclotomic polynomial as the polynomial whose roots are the th roots of unity. Let be a subgroup of of order . By definition we see that This allows us to compute the Cyclotomic Polynomial of order recursively. However, there is a nicer way that requires less expansion. Theorem: . Proof: For context, the Möbius Inversion Formula states that if and are arithmetic functions then where denotes the Möbius Function. We begin with the following formula that states We now take the logarithm of both sides of this equation. We see that Now we apply Möbius. By the formula we see that and unexponentiating by considering this equation in gives which is exactly what we wanted to show. As a corollary to the previous theorem, this implies that rather easily. This helps aid in computation, and once again ties this strange polynomial to number theory. The Cyclotomic Polynomial has some ties to algebraic number theory in particular, and also has some nice results in Galois theory. Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
1619	https://people.richland.edu/james/ictcm/2004/twoway.html	Two-Way Analysis of Variance Introduction The two-way ANOVA is an extension of the one-way ANOVA. The "two-way" comes because each item is classified in two ways, as opposed to one way. For example, one way classifications might be: gender, political party, religion, or race. Two way classifications might be by gender and political party, gender and race, or religion and race. Each classification variable is a called a factor and so there are two factors, each having several levels within that factor. The factors are called the "row factor" and the "column factor" because the data is usually arranged into table format. Each combination of a row level and a column level is called a treatment. The two-way ANOVA that we're going to discuss requires a balanced design. The balanced design is where each treatment has the same sample size. Example: Drug Testing A pharmaceutical company is testing a new drug to see if it helps reduce the time to recover from a fever. They decide to test the drug on three different races (Caucasian, African American, and Hispanic) and both genders (male and female). This makes six treatments (3 races × 2 genders = 6 treatments).They randomly select five test subjects from each of those six treatments, so all together, they have 3 × 2 × 5 = 30 test subjects. The response variable is the time in minutes after taking the medicine before the fever is reduced. The data might look something like this. Data \| \| Male \| Female \| --- \| Caucasian \| 54, 49, 59, 39, 55 \| 25, 29, 47, 26, 28 \| \| African American \| 53, 72, 43, 56, 52 \| 46, 51, 33, 47, 41 \| \| Hispanic \| 33, 30, 26, 25, 29 \| 18, 21, 34, 40, 24 \| Summary Statistics Here are the summary statistics for each of the six treatments as well as for each level within each factor. \| \| Male \| Female \| All \| --- --- \| \| Caucasian \| mean = 51.2 stdev = 7.694 \| mean = 31.0 stdev = 9.083 \| mean = 49.4 stdev = 10.405 \| \| African American \| mean = 55.2 stdev = 10.569 \| mean = 43.6 stdev = 6.914 \| mean = 41.1 stdev = 13.279 \| \| Hispanic \| mean = 28.6 stdev = 3.209 \| mean = 27.4 stdev = 9.263 \| mean = 28.0 stdev = 6.566 \| \| All \| mean = 45.0 stdev = 14.097 \| mean = 34.0 stdev = 10.650 \| mean = 39.5 stdev = 13.490 \| Hypothesis Tests There are three sets of hypothesis tests for the Two-Way ANOVA. H0: The means of each row (race) are equal H1: The mean of at least one row (race) is different H0: The means of each column (gender) are equal H1: The mean of at least one column (gender) is different H0: There is no interaction between the row (race) and column (gender) H1: There is interaction between the row (race) and column (gender) The first two hypotheses are essentially one-way ANOVAs for the row (race) or column (gender) variables. The third hypothesis is similar to a chi-squared test for independence where no interaction means they are not related to each other. This test is also similar to the test for independence in the way that the degrees of freedom are calculated, the df here is the df(Row) × df(Column). Two-Way ANOVA Table There are extra rows in the Two-Way ANOVA table. There are three sources besides the error (unexplained), so we have a row for each of those sources. The variations (SS) are best found using technology. \| Source \| SS \| df \| MS \| F \| --- --- \| Row (race) \| 2328.2 \| \| \| \| \| Column (gender) \| 907.5 \| \| \| \| \| Interaction (race × gender) \| 452.6 \| \| \| \| \| Error \| 1589.2 \| \| \| \| \| Total \| 5277.5 \| \| \| \| How to find the degrees of freedom We're back to that "one less than" rule for df. There are 3 races, so there are 2 df for the races There are 2 genders, so there is 1 df for the gender Interaction is race × gender and so we multiple df(row) × df(column) = 2 × 1 = 2 df there The error df is the sum of the individual df's for each treatment. There were 5 in each treatment group and so there are 4 df for each. There are 6 treatment groups of 4 df each, so there are 24 df for the error term. The total df is one less than the sample size. \| Source \| SS \| df \| MS \| F \| --- --- \| Row (race) \| 2328.2 \| 2 \| \| \| \| Column (gender) \| 907.5 \| 1 \| \| \| \| Interaction (race × gender) \| 452.6 \| 2 \| \| \| \| Error \| 1589.2 \| 24 \| \| \| \| Total \| 5277.5 \| 29 \| \| \| Finishing the Table Once you have the df, the table works like it did before. The MS = SS / df. MS(race) = 2328.2 / 2 = 1164.1 MS(gender) = 907.5 / 1 = 907.5 MS(interaction) = 452.6 / 2 = 226.3 MS(error) = 1589.2 / 24 = 66.22 [this is the pooled estimate of the variance] MS(total) = 5277.5 / 29 = 181.98 [this is technically not part of the table, but it is the variance on the response variable] The F test statistic is found by dividing the MS for each row by the MS for the error source. F(race) = 1164.1 / 66.22 = 17.58 F(gender) = 907.5 / 66.22 = 13.71 F(interaction) = 226.3 / 66.22 = 3.42 There is no F for the error or total sources. \| Source \| SS \| df \| MS \| F \| --- --- \| Row (race) \| 2328.2 \| 2 \| 1164.10 \| 17.58 \| \| Column (gender) \| 907.5 \| 1 \| 907.50 \| 13.71 \| \| Interaction (race × gender) \| 452.6 \| 2 \| 226.30 \| 3.42 \| \| Error \| 1589.2 \| 24 \| 66.22 \| \| \| Total \| 5277.5 \| 29 \| 181.98 \| \| The table is complete. Finding the p-values To make a decision about the hypothesis test, you really need a p-value. These test statistics have F distributions. The numerator df is the df for the source and the denominator df is the df for the error. Race: F has 2 numerator and 24 denominator df Gender: F has 1 numerator and 24 denominator df Interaction: F has 2 numerator and 24 denominator df The p-value is the area to the right of the test statistic since this is always a right tail test. The p-value for the Race factor is the area to the right F = 17.58 using 2 numerator and 24 denominator df. The p-value for the Race factor is the area to the right F = 13.71 using 1 numerator and 24 denominator df. The p-value for the Race factor is the area to the right F = 3.42 using 2 numerator and 24 denominator df. \| Source \| SS \| df \| MS \| F \| P \| --- --- --- \| \| Row (race) \| 2328.2 \| 2 \| 1164.10 \| 17.58 \| 0.000 \| \| Column (gender) \| 907.5 \| 1 \| 907.50 \| 13.71 \| 0.001 \| \| Interaction (race × gender) \| 452.6 \| 2 \| 226.30 \| 3.42 \| 0.049 \| \| Error \| 1589.2 \| 24 \| 66.22 \| \| \| \| Total \| 5277.5 \| 29 \| 181.98 \| \| \| Each test statistic and p-value are used to answer the hypothesis dealing with the appropriate source. That is, the F = 17.58 is for the race source, so it would be used to determine if there is a difference in the mean reaction times of the different races. F = 13.71 is for the gender source, so it would be used to determine if there is a difference in the mean reaction times of the different genders. F = 3.42 is for the interaction source, so it would be used to determine if there is interaction between the race and gender. Last modified December 9, 2008 11:25 PM Return to ICTCM 2004 Short Course page Return to James Jones homepage
1620	https://openstax.org/books/elementary-algebra-2e/pages/6-7-integer-exponents-and-scientific-notation	Skip to ContentGo to accessibility pageKeyboard shortcuts menu Elementary Algebra 2e 6.7 Integer Exponents and Scientific Notation Elementary Algebra 2e6.7 Integer Exponents and Scientific Notation Search for key terms or text. Learning Objectives By the end of this section, you will be able to: Use the definition of a negative exponent Simplify expressions with integer exponents Convert from decimal notation to scientific notation Convert scientific notation to decimal form Multiply and divide using scientific notation Be Prepared 6.15 Before you get started, take this readiness quiz. What is the place value of the in the number ? If you missed this problem, review Example 1.1. Be Prepared 6.16 Name the decimal: If you missed this problem, review Example 1.91. Be Prepared 6.17 Subtract: If you missed this problem, review Example 1.42. Use the Definition of a Negative Exponent We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator. Quotient Property for Exponents If is a real number, , and are whole numbers, then What if we just subtract exponents regardless of which is larger? Let’s consider . We subtract the exponent in the denominator from the exponent in the numerator. We can also simplify by dividing out common factors: This implies that and it leads us to the definition of a negative exponent. Negative Exponent If is an integer and , then . The negative exponent tells us we can re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents. For example, if after simplifying an expression we end up with the expression , we will take one more step and write . The answer is considered to be in simplest form when it has only positive exponents. Example 6.89 Simplify: ⓐ ⓑ Solution \| \| \| --- \| \| ⓐ \| \| \| Use the definition of a negative exponent, . \| \| \| Simplify. \| \| \| ⓑ \| \| \| Use the definition of a negative exponent, . \| \| \| Simplify. \| \| Try It 6.177 Simplify: ⓐ ⓑ Try It 6.178 Simplify: ⓐ ⓑ In Example 6.89 we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent. \| \| \| --- \| \| \| \| \| Use the definition of a negative exponent, . \| \| \| Simplify the complex fraction. \| \| \| Multiply. \| \| This leads to the Property of Negative Exponents. Property of Negative Exponents If is an integer and , then . Example 6.90 Simplify: ⓐ ⓑ Solution \| \| \| --- \| \| ⓐ \| \| \| Use the property of a negative exponent, . \| \| \| ⓑ \| \| \| Use the property of a negative exponent, . \| \| \| Simplify. \| \| Try It 6.179 Simplify: ⓐ ⓑ Try It 6.180 Simplify: ⓐ ⓑ Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property. \| \| \| --- \| \| \| \| \| Use the definition of a negative exponent, . \| \| \| Simplify the denominator. \| \| \| Simplify the complex fraction. \| \| \| But we know that is . \| \| \| This tells us that: \| \| Table 6.4 To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent. This leads us to the Quotient to a Negative Power Property. Quotient to a Negative Exponent Property If are real numbers, and is an integer, then . Example 6.91 Simplify: ⓐ ⓑ Solution \| \| \| --- \| \| ⓐ \| \| \| Use the Quotient to a Negative Exponent Property, . \| \| \| Take the reciprocal of the fraction and change the sign of the exponent. \| \| \| Simplify. \| \| \| ⓑ \| \| \| Use the Quotient to a Negative Exponent Property, . \| \| \| Take the reciprocal of the fraction and change the sign of the exponent. \| \| \| Simplify. \| \| Try It 6.181 Simplify: ⓐ ⓑ Try It 6.182 Simplify: ⓐ ⓑ When simplifying an expression with exponents, we must be careful to correctly identify the base. Example 6.92 Simplify: ⓐ ⓑ ⓒ ⓓ Solution \| \| \| --- \| \| ⓐ Here the exponent applies to the base . \| \| \| Take the reciprocal of the base and change the sign of the exponent. \| \| \| Simplify. \| \| \| ⓑ The expression means "find the opposite of ." Here the exponent applies to the base . \| \| \| Rewrite as a product with . \| \| \| Take the reciprocal of the base and change the sign of the exponent. \| \| \| Simplify. \| \| \| ⓒ Here the exponent applies to the base . \| \| \| Take the reciprocal of the base and change the sign of the exponent. \| \| \| Simplify. \| \| \| ⓓ The expression means "find the opposite of ." Here the exponent applies to the base . \| \| Rewrite as a product with . \| \| \| Take the reciprocal of the base and change the sign of the exponent. \| \| \| Simplify. \| \| Try It 6.183 Simplify: ⓐ ⓑ ⓒ ⓓ Try It 6.184 Simplify: ⓐ ⓑ , ⓒ ⓓ We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different. Example 6.93 Simplify: ⓐ ⓑ Solution \| \| \| --- \| \| ⓐ Do exponents before multiplication. \| \| \| Use . \| \| \| Simplify. \| \| \| ⓑ \| \| \| Simplify inside the parentheses first. \| \| \| Use . \| \| \| Simplify. \| \| Try It 6.185 Simplify: ⓐ ⓑ Try It 6.186 Simplify: ⓐ ⓑ When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are non-zero. Example 6.94 Simplify: ⓐ ⓑ Solution ⓐ ⓑ Try It 6.187 Simplify: ⓐ ⓑ Try It 6.188 Simplify: ⓐ ⓑ When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example. Example 6.95 Simplify: ⓐ ⓑ ⓒ Solution ⓐ \| \| \| --- \| \| \| \| \| Notice the exponent applies to just the base y.Take the reciprocal of y and change the sign of the exponent. \| \| \| Simplify. \| \| ⓑ \| \| \| --- \| \| \| \| \| Here the parentheses make the exponent apply to the base 5y.Take the reciprocal of 5y and change the sign of the exponent. \| \| \| Simplify. \| \| ⓒ \| \| \| --- \| \| \| \| \| The base here is −5y.Take the reciprocal of −5y and change the sign of the exponent. \| \| \| Simplify. \| \| \| Use \| \| Try It 6.189 Simplify: ⓐ ⓑ ⓒ Try It 6.190 Simplify: ⓐ ⓑ ⓒ With negative exponents, the Quotient Rule needs only one form , for . When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. Simplify Expressions with Integer Exponents All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference. Summary of Exponent Properties If are real numbers, and are integers, then Example 6.96 Simplify: ⓐ ⓑ ⓒ Solution ⓐ \| \| \| --- \| \| \| \| \| Use the Product Property, \| \| \| Simplify. \| \| ⓑ \| \| \| --- \| \| \| \| \| Notice the same bases, so add the exponents. \| \| \| Simplify. \| \| \| Use the definition of a negative exponent, \| \| ⓒ \| \| \| --- \| \| \| \| \| Add the exponents, since the bases are the same. \| \| \| Simplify. \| \| \| Take the reciprocal and change the sign of the exponent,using the definition of a negative exponent. \| \| Try It 6.191 Simplify: ⓐ ⓑ ⓒ Try It 6.192 Simplify: ⓐ ⓑ ⓒ In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property. Example 6.97 Simplify: Solution \| \| \| --- \| \| \| \| \| Use the Commutative Property to get like bases together. \| \| \| Add the exponents for each base. \| \| \| Take reciprocals and change the signs of the exponents. \| \| \| Simplify. \| \| Try It 6.193 Simplify: Try It 6.194 Simplify: Try It 6.195 Simplify: In the next two examples, we’ll use the Power Property and the Product to a Power Property. Example 6.98 Simplify: Solution \| \| \| --- \| \| \| \| \| Use the Product to a Power Property, \| \| \| Use the Power Property, \| \| \| Use the Definition of a Negative Exponent, \| \| \| Simplify. \| \| Try It 6.196 Simplify: Try It 6.197 Simplify: Example 6.99 Simplify: Solution \| \| \| --- \| \| \| \| \| Use the Product to a Power Property, \| \| \| Simplify 52 and multiply the exponents of x using the PowerProperty, \| \| \| Rewrite x−6 by using the Definition of a Negative Exponent, \| \| \| Simplify. \| \| Try It 6.198 Simplify: Try It 6.199 Simplify: To simplify a fraction, we use the Quotient Property and subtract the exponents. Example 6.100 Simplify: Solution \| \| \| --- \| \| \| \| \| Use the Quotient Property, \| \| \| Simplify. \| \| Try It 6.200 Simplify: Try It 6.201 Simplify: Convert from Decimal Notation to Scientific Notation Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and . We know that 4,000 means and 0.004 means . If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way: When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation. Scientific Notation A number is expressed in scientific notation when it is of the form It is customary in scientific notation to use as the multiplication sign, even though we avoid using this sign elsewhere in algebra. If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation. In both cases, the decimal was moved 3 places to get the first factor between 1 and 10. Example 6.101 How to Convert from Decimal Notation to Scientific Notation Write in scientific notation: 37,000. Solution Try It 6.202 Write in scientific notation: Try It 6.203 Write in scientific notation: How To Convert from decimal notation to scientific notation Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. Step 2. Count the number of decimal places, n, that the decimal point was moved. Step 3. Write the number as a product with a power of 10.If the original number is: greater than 1, the power of 10 will be 10n. between 0 and 1, the power of 10 will be 10−n. 4. Step 4. Check. Example 6.102 Write in scientific notation: Solution The original number, , is between 0 and 1 so we will have a negative power of 10. \| \| \| \| \| Move the decimal point to get 5.2, a number between 1 and 10. \| \| Count the number of decimal places the point was moved. \| \| Write as a product with a power of 10. \| \| Check. \| \| \| \| \| \| \| Try It 6.204 Write in scientific notation: Try It 6.205 Write in scientific notation: Convert Scientific Notation to Decimal Form How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see. If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form. In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left. Example 6.103 How to Convert Scientific Notation to Decimal Form Convert to decimal form: Solution Try It 6.206 Convert to decimal form: Try It 6.207 Convert to decimal form: The steps are summarized below. How To Convert scientific notation to decimal form. To convert scientific notation to decimal form: Step 1. Determine the exponent, , on the factor 10. Step 2. Move the decimal places, adding zeros if needed. If the exponent is positive, move the decimal point places to the right. If the exponent is negative, move the decimal point places to the left. 3. Step 3. Check. Example 6.104 Convert to decimal form: Solution \| \| \| \| \| Determine the exponent, n, on the factor 10. \| \| Since the exponent is negative, move the decimal point 2 places to the left. \| \| Add zeros as needed for placeholders. \| Try It 6.208 Convert to decimal form: Try It 6.209 Convert to decimal form: Multiply and Divide Using Scientific Notation Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation. Example 6.105 Multiply. Write answers in decimal form: Solution Try It 6.210 Multiply . Write answers in decimal form. Try It 6.211 Multiply . Write answers in decimal form. Example 6.106 Divide. Write answers in decimal form: Solution \| \| \| --- \| \| \| \| \| Separate the factors, rewriting as the product of two fractions. \| \| \| Divide. \| \| \| Change to decimal form by moving the decimal five places right. \| 300,000 \| Try It 6.212 Divide . Write answers in decimal form. Try It 6.213 Divide . Write answers in decimal form. Media Access these online resources for additional instruction and practice with integer exponents and scientific notation: Negative Exponents Scientific Notation Scientific Notation 2 Section 6.7 Exercises Practice Makes Perfect Use the Definition of a Negative Exponent In the following exercises, simplify. ⓐ ⓑ 501. ⓐ ⓑ ⓐ ⓑ 503. ⓐ ⓑ ⓐ ⓑ 505. ⓐ ⓑ ⓐ ⓑ 507. ⓐ ⓑ ⓐ ⓑ 509. ⓐ ⓑ ⓐ ⓑ 511. ⓐ ⓑ ⓐ ⓑ ⓒ ⓓ 513. ⓐ ⓑ ⓒ ⓓ ⓐ ⓑ ⓒ ⓓ 515. ⓐ ⓑ ⓒ ⓓ ⓐ ⓑ 517. ⓐ ⓑ ⓐ ⓑ 519. ⓐ ⓑ ⓐ ⓑ 521. ⓐ ⓑ ⓐ ⓑ 523. ⓐ ⓑ ⓐ ⓑ ⓒ 525. ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ 527. ⓐ ⓑ ⓒ Simplify Expressions with Integer Exponents In the following exercises, simplify. ⓐ ⓑ ⓒ 529. ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ 531. ⓐ ⓑ ⓒ 532. 533. 534. 535. 536. 537. 538. 539. 540. 541. 542. 543. 544. 545. 546. 547. 548. 549. Convert from Decimal Notation to Scientific Notation In the following exercises, write each number in scientific notation. 57,000 551. 340,000 8,750,000 553. 1,290,000 0.026 555. 0.041 0.00000871 557. 0.00000103 Convert Scientific Notation to Decimal Form In the following exercises, convert each number to decimal form. 558. 559. 560. 561. 562. 563. 564. 565. Multiply and Divide Using Scientific Notation In the following exercises, multiply. Write your answer in decimal form. 566. 567. 568. 569. In the following exercises, divide. Write your answer in decimal form. 570. 571. 572. 573. Everyday Math The population of the United States on July 4, 2010 was almost 310,000,000. Write the number in scientific notation. 575. The population of the world on July 4, 2010 was more than 6,850,000,000. Write the number in scientific notation The average width of a human hair is 0.0018 centimeters. Write the number in scientific notation. 577. The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation. In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was . Convert this number to decimal form. 579. At the start of 2012, the US federal budget had a deficit of more than . Convert this number to decimal form. The concentration of carbon dioxide in the atmosphere is . Convert this number to decimal form. 581. The width of a proton is of the width of an atom. Convert this number to decimal form. Health care costs The Centers for Medicare and Medicaid projects that consumers will spend more than $4 trillion on health care by 2017. ⓐ Write 4 trillion in decimal notation. ⓑ Write 4 trillion in scientific notation. 583. Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation. Distance The distance between Earth and one of the brightest stars in the night star is 33.7 light years. One light year is about 6,000,000,000,000 (6 trillion), miles. ⓐ Write the number of miles in one light year in scientific notation. ⓑ Use scientific notation to find the distance between Earth and the star in miles. Write the answer in scientific notation. 585. Debt At the end of fiscal year 2015 the gross United States federal government debt was estimated to be approximately $18,600,000,000,000 ($18.6 trillion), according to the Federal Budget. The population of the United States was approximately 300,000,000 people at the end of fiscal year 2015. ⓐ Write the debt in scientific notation. ⓑ Write the population in scientific notation. ⓒ Find the amount of debt per person by using scientific notation to divide the debt by the population. Write the answer in scientific notation. Writing Exercises ⓐ Explain the meaning of the exponent in the expression . ⓑ Explain the meaning of the exponent in the expression . 587. When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative? Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not? PreviousNext Order a print copy Citation/Attribution This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax. 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1621	https://zhuanlan.zhihu.com/p/550574852	证明函数单调性的方法 - 知乎关注推荐热榜专栏圈子 New付费咨询知学堂直答切换模式登录/注册证明函数单调性的方法首发于高中数学切换模式证明函数单调性的方法秦思涯更新速度慢，喜欢文章点关注容易查，莫收藏免吃灰。收录于 · 高中数学 49 人赞同了该文章目录 0. 写在开头更多数学问题私信小秦为你解决。本篇文章介绍函数单调性的几种基本证明方法的深度提高，作差法，作商法，斜率式f(x 1)−f(x 2)x 1−x 2\frac{f(x_1)-f(x_2)}{x_1-x_2} ，放缩法，导数法。每一种方法层层递进，挖掘思维本质，拓宽思路。例如在介绍作差法时，由最基础的两数作差与0比较，挖掘方法思维本质，提出采用因式分解变换式子，乘以某个因子变换式子，最终推向一般的变换方式，即利用函数思想将 f(x 1)−f(x 2)f(x_1)-f(x_2) 变换，寻找的变换法则对于 (−∞,0),(0,+∞)(-\infty, 0),(0,+\infty) 映射后得到的值是交集为空的两区间，如 (−∞,a),(a,+∞)(-\infty,a),(a,+\infty) 。关键词证单调性变换 1. 作差法在《如何理解函数单调性》中，相信读者已经理解了函数单调性的概念，并初步认识到使用作差法证明函数单调性。这里，进一步指出，在单调性证明的过程中，实际我们更多地在于关心 f(x 1),f(x 2)f(x_1),f(x_2) 之间的相对大小，也就是比较两数大小，而比较两数大小最基本的方式就是作差法。如果采用作差法，那么我们后续实质上就是要关注 f(x 1)−f(x 2)f(x_1)-f(x_2) 这个式子的整体是正号，还是负号。认识到这一点，那么作差法比较两数的思路便不难开拓了。当然，还是提醒大家，最终确定单调性，还需结合原先设定的 x 1,x 2 x_1,x_2 之间的相对大小，这一点详细内容，已经在《如何理解函数单调性》中解释了。例如《如何理解函数单调性》中提到的因式分解，如果 f(x 1)−f(x 2)f(x_1)-f(x_2) 这个式子可以进行因式分解，就将式子分解成多个整式乘积的形式，分解直到能够证明每个整式的符号为止。我们采用例1演示说明例1.1 证明函数 y=x 4 y=x^4 的单调性证明记 f(x)=x 4,x∈R f(x)=x^4,x\in R 取 ∀x 1,x 2∈R 且 x 1<x 2\forall x_1,x_2\in R且x_1\lt x_2 那么 f(x 1)−f(x 2)=x 1 4−x 2 4=(x 1 2+x 2 2)(x 1 2−x 2 2)f(x_1)-f(x_2)=x_1^4-x_2^4=(x_1^2+x_2^2)(x_1^2-x_2^2) 因为已经证明 y=x 2 y=x^2 在 x>0 x>0 上单调递增， x<0 x<0 上单调递减所以当 0<x 1<x 2 0<x_1<x_2 时， 0<x 1 2<x 2 2 0<x_1^2<x_2^2 ，即 x 1 2−x 2 2<0 x_1^2-x_2^2<0 ， x 1 2+x 2 2>0 x_1^2+x_2^2\gt 0 （备注：也就是说分解后的式子，第一个整式的符号为＋，第二个整式的符号是-，由初中数学知识就可以知道，正负得负，于是分解后的符号当然是负号，而分解前后并无改变式子符号，所以分解前的式子当然也是负的了。在初中就已经学习了，负数是小于0的数，于是有了下面的式子。）那么 f(x 1)−f(x 2)<0 f(x_1)-f(x_2)\lt 0 ，即 f(x 1)<f(x 2)f(x_1)\lt f(x_2) （备注：结合不等式性质对不等式进行变换。）所以， f(x)f(x) 在 x>0 x>0 上单调递增当 x_1\lt x_2 \lt 0 时， x_1^2\gt x_2^2\gt 0 ，即 x_1^2-x_2^2\gt 0,x_1^2+x_2^2\gt 0 （备注：类似前面，看每个整式的符号）那么 f(x_1)-f(x_2)\gt 0 ，即 f(x_1)\gt f(x_2) 所以， f(x) 在 x\lt 0 上单调递减综上， f(x) 在 (0,+\infty) 上单调递增，在 (-\infty,0) 上单调递减。备注：读者也可进一步因式分解成(x_1^2+x_2^2)(x_1+x_2)(x_1-x_2) 现在，读者一定深深体会到，利用作差证明函数单调性，对于 f(x_1),f(x_2) 具体值是多少，我们是不关心的，对于 f(x_1)-f(x_2) 具体的差值是多少，也是不怎么关注的，更多地关注 f(x_1)-f(x_2) 的符号到底是＋还是 - 。进一步地，思路就更广阔了，认识到如果能够确定 f(x_1)-f(x_2) 的符号，那么我们进行的任何变换都可以。非常容易想到的，就是让这个式子，乘以某个能够确定式子的符号，得到的乘积也能够确定符号，那么结合初中数学的运算，不难明白原先的式子 f(x_1)-f(x_2) 的符号肯定能够确定。就是说，令式子 A 与式子 f(x_1)-f(x_2) 作乘运算，得到 A(f(x_1)-f(x_2)) ，只要确定 A 的符号，以及 A(f(x_1)-f(x_2)) 的符号，当然可以确定 f(x_1)-f(x_2) 的符号。具体例题可以参考《如何理解函数单调性》的例题1。更加推广的来说，如果有对应法则 g ，定义域中的数在其作用后得到的结果，不改变原有的符号，那么我们可以有通过操作 g(f(x_1)-f(x_2)) ，变换 f(x_1)-f(x_2) ，与0进行比较。或者，g 有这样的特点，原先定义域的数 x 的符号是正的，在 g 作用后得到的 g(x) 是负的，原先的 x 是负的，在 g 作用后得到的 g(x) 是正的。这样的函数 g 可以有很多，其中非常简单的一种就是奇函数。关于奇函数的构造可以参考后续文章《常见的奇奇偶函数》。例题1.2 证明函数 \ln x 的单调性证明记 f(x)=\ln x,x\in (0,+\infty) 构造函数 g(x)=e^x-e^{-x} 易证 g(x) 为奇函数，且 x>0 时 g(x)>0 ， x<0 时 g(x)<0 取 \forall x_1,x_2\in (0,+\infty)且x_1\lt x_2 f(x_1)-f(x_2)=\ln x_1 - \ln x_2=\ln \frac{x_1}{x_2} 现有 g(\ln \frac{x_1}{x_2})=e^{\ln\frac{x_1}{x_2}}-e^{-\ln \frac{x_1}{x_2}}=\frac{x_1}{x_2}-\frac{x_2}{x_1}=\frac{x_1^2-x_2^2}{x_2x_1} 已知 y=x^2 在 (0,+\infty) 上单调递增，所以 x_1^2\lt x_2^2 所以 g(\ln \frac{x_1}{x_2})\lt 0 即 f(x_1)\lt f(x_2) 故 \ln x 在 (0,+\infty) 上单调递增。实际上，我们可以将这样的变换推向更一般。假如找到的 g 如下图图1 函数g的图像可以看到，当 x\lt 0 时， g(x)\lt 2 ；当 x>0 时， g(x)\gt 2 。一般地，用 x=0 这条线切分 g(x) 定义域，如果符合 g(x) 符合这样的特征： x<0 时 g(x)\lt(\gt) a ， x\gt 0 时 g(x)\gt(\lt) a ，那么 g(x) 便可以用于对 f(x_1)-f(x_2) 进行变换，协助求证函数 f(x) 的单调性。 2. 作商法在小学时，我们就已经认识了真分数与假分数的概念：分子比分母大的数是假分数，分子比分母小的分数是真分数；假分数是比1大的分数，真分数是比1小的分数。由此，我们不难想到，如果要比较 f(x_1),f(x_2) 二者的相对大小，可以进行除法运算 \frac{f(x_1)}{f(x_2)} 最终的结果是与1比较。但是，需要注意的是，我们小学所学的真分数与假分数的概念，是基于分子、分母都是正数的情况。进入初中之后，我们认识的数便不止正数了。这里，需要读者引起注意，采用作商法的几个限制。首先，分数的分母不能为0，也就是说除数不能为0。这就意味着，在进行这样的操作之前，我们需要证明 f(x) 恒不为0，或者将 f(x) 为0的情况提取出来额外讨论。另外，如果 f(x_1),f(x_2) 的符号是不同的，那么显然 \frac{f(x_1)}{f(x_2)}<1 是恒成立的，而且无法判断负号到底是在分子还是在分母，也就无法判断到底是分子大还是分母大了。所以，作商比较时，解答者需要注意额外讨论分母为0的情况，需要确保 f(x_1),f(x_2) 的符号同样是正号或者同样是负号。当然，如果同样是负号时，根据不等式性质，进行不等式变换得到 f(x_1)?f(x_2) 注意要变号。类似地，我们可以像作差法那样作变换。但不同的是，作差法看的分割点是 x=0 ，而作商看的水平线则是 x=1 。也就是说，需要找到的 g ，符合这样的特征，当 0\lt x\lt 1 时， g(x)\lt (\gt )a ，当 x\gt 1 时， g(x)\gt(\lt) a 。例题2.1 利用作商法证明 \sqrt{x} 的单调性证明记 f(x)=\sqrt{x},x\in (0,+\infty) 取 \forall x_1,x_2\in (0,+\infty)且x_1\lt x_2 易知 \sqrt{x}>0 （备注：这里就确保了能够采用作商法）于是 \frac{f(x_1)}{f(x_2)}=\frac{\sqrt{x_1}}{\sqrt{x_2}}=\sqrt{\frac{x_1}{x_2}} 现有函数 g(x)=x^2 在 (0,+\infty) 上单调递增，且有 g(1)=1 所以当 0<x<1 时， g(x)\lt g(1)=1 ，即 g(x)\lt 1 当 x\gt 1 时， g(x)\gt g(1)=1 ，即 g(x)\gt 1 现有 g(\sqrt{\frac{x_1}{x_2}})=\frac{x_1}{x_2}<1 说明 0\lt \sqrt{\frac{x_1}{x_2}}\lt 1 ，于是 0\lt \frac{f(x_1)}{f(x_2)}\lt 1 ，即 f(x_1)\lt f(x_2) 综上 f(x) 在 (0,+\infty) 上单调递增。 3. 斜率式\frac{f(x_1)-f(x_2)}{x_1-x_2} 在方法2的作商法中，就已经认识到这个方法的局限性。但同时也发现作商运算的一个特点，一个分数如果大于0，那么分子分母是同号的，如果小于0，那么分子分母就是异号的。而我们的单调性定义中，同时要比较 x_1?x_2,\quad f(x_1)?f(x_2) 的 ? 是否同为 < 或 > ，又是否一个 < 另一个 > 。由不等式性质，实质可转换为，判断 x_1-x_2 与 f(x_1)-f(x_2) 的符号是否相同，于是不难想到，可以构造这样的式子 \frac{f(x_1)-f(x_2)}{x_1-x_2} 或者 \frac{f(x_2)-f(x_1)}{x_2-x_1} 如果大于0，说明分子分母同号，于是函数 f 单调递增，如果小于0，说明分子分母异号，于是函数 f 单调递减。构造这样的式子，也解决了前面作商法的局限，一个是分母不能为0，这个是肯定的，因为取得的 x_1,x_2 必然不同；另一方面利用分式的正负号判断函数单调性，而非分子分母的相对大小，从而不必排除异号情况；还要特别指出，因为我们把 x_1,x_2 的相对大小的比较已经在这个式子中有所体现了，所以在将两个自变量从定义域中取出时，可以不用设定两数的相对大小。这个方法为解答者减少证明函数单调性的许多前期工作。例题3.1 证明函数 y=\sqrt{x} 的单调性证明记 f(x)=\sqrt{x},x\in (0,+\infty) \forall x_1,x_2\in (0,+\infty) \frac{f(x_1)-f(x_2)}{x_1-x_2}=\frac{\sqrt{x_1}-\sqrt{x_2}}{x_1-x_2}\ =\frac{\sqrt{x_1}-\sqrt{x_2}}{(\sqrt{x_1}-\sqrt{x_2})(\sqrt{x_1}+\sqrt{x_2})} =\frac{1}{\sqrt{x_1}+\sqrt{x_2}}\gt 0 所以，函数 f(x) 在 (0,+\infty) 上单调递增。当然，我们还会使用常用的恒等变换，若 a=b 则 0=a-b，常用处理方式无非是加上这个数的同时减去这个数实现式子的恒等变换。还有若 a=b\neq 0 则 \frac{a}{b}=\frac{b}{a}=1 例题3.2 证明函数 y=\sqrt{k+x}-\sqrt{x} 的单调性证明记 f(x)=\sqrt{k+x}-\sqrt{x},x\in D 取 \forall x_1,x_2\in D \frac{f(x_1)-f(x_2)}{x_1-x_2}=\frac{(\sqrt{x_1+k}-\sqrt{x_1}) - (\sqrt{x_2+k}-\sqrt{x_2})}{x_1-x_2}\ =\frac{\sqrt{x_1+k}-\sqrt{x_2+k}}{(k+x_1)-(k+x_2)}-\frac{\sqrt{x_1}-\sqrt{x_2}}{x_1-x_2}\ =\frac{1}{\sqrt{k+x_1}+\sqrt{k+x_2}}-\frac{1}{\sqrt{x_1}+\sqrt{x_2}} 由例题3.1已证 y=\sqrt{x} 是单调递增的。当 k>0 时，函数的定义域为 (0,+\infty) ，有 x+k\gt x ，则有 \sqrt{x+k}\gt \sqrt{x} ，由不等式性质不难得出 \ \frac{1}{\sqrt{k+x_1}+\sqrt{k+x_2}}-\frac{1}{\sqrt{x_1}+\sqrt{x_2}}\lt 0 此时函数 f(x) 在 (0,+\infty) 上单调递减。当 k\lt 0 时，函数的定义域为 (-k,+\infty) ，有 x+k\lt x ，则有 \sqrt{x+k}\lt \sqrt{x} ，由不等式性质不难得出 \ \frac{1}{\sqrt{k+x_1}+\sqrt{k+x_2}}-\frac{1}{\sqrt{x_1}+\sqrt{x_2}}\gt 0 此时函数 f(x) 在 (-k,+\infty) 上单调递增。 4. 放缩法这里我们关注放缩在证明函数单调性中的应用，如果读者希望明白放缩的原理是什么，以及常见的放缩方式，可以参考后续作品《细解放缩》，《放缩的常见的方式》。由前面的文章《如何理解函数单调性》，就已经认识到函数的单调性具有局限性。于是，我们不妨设想，将函数的定义域切分成若干个区间，讨论每个区间的单调性，从而求证函数的单调性。回顾利用作差法求证函数单调性，不难看出， f(x_1)-f(x_2)>0 或 f(x_1)- f(x_2)\lt 0 可以看成是一个关于 x_1,x_2 的二元不等式，但二元不等式显然要比一元不等式难解一些，此时会有这样的想法，如果有某种方式能够将二元不等式变换成一元不等式，那么解一元不等式则是一件比较轻松的事情了。以取函数单调递增区间为例，设 f(x) 是定义在 D 上的函数，取 \forall x_1,x_2\in D且x_1\lt x_2 。如果要取函数单调递增区间，那么不难明白需要有 f(x_1)-f(x_2)\lt 0 。但这是一个二元不等式，现在如果能够找到函数 g(x) 使得 f(x_1)-f(x_2)\lt g(x) 在 x\in D 上恒成立，那么后续如果能够证明 g(x)\lt 0 ，那么由不等式的传递性不难得知， f(x_1)-f(x_2) 是成立。但需要注意的是， g(x)\lt 0 这个不等式的解集，例如我们记为 E ，如果 D\subseteq E 那么函数 f 在其定义域上是单调递增，但如果 E\subseteq D ，则显然我们只是求证了函数 f 在其定义域的某部分上的单调性，这部分是 E ，而 D 削去 E 剩下的区间，函数的单调性仍旧不确定，这部分我们还是需要根据要求进一步求证的。另外，更一般来说，如果 f(x_1)- f(x_2)\lt g(x) 在 x\in F \subseteq D 成立的，那么最终求证的单调区间是取 E\cap F 最简单的放缩情景，无非就是依据现有以设定的 x_1\lt x_2 这一关系进行放缩，将 f(x_1)-f(x_2) 放缩成只关于 x_1 或 x_2 的一元不等式。对于作商法，斜率式也是类似讨论的。例题4.1 证明 y=x^2-2x 的单调性证明记 f(x)=x^2-2x 取 \forall x_1,x_2\in R,且x_1\lt x_2 \frac{f(x_1)-f(x_2)}{x_1-x_2}=\frac{x_1^2-2x_1-(x_2^2-2x_2)}{x_1-x_2} =x_1+x_2-2 结合 x_1\lt x_2 ，不难得出 x_1+x_2-2\lt 2x_2-2 令 2x_2-2\lt 0 ，得 x_2\lt 1 由此可知，当 x_1\lt x_2\lt 1 时， \frac{f(x_1)-f(x_2)}{x_1-x_2}\lt 0 ， f(x) 单调递减再次结合 x_1\gt x_2 ，不难得出 x_1+x_2-2\gt 2x_1-2 令 2x_1-2\gt 0 ，得 x_1\gt 1 由此可知，当 1\lt x_1\lt x_2 时， \frac{f(x_1)-f(x_2)}{x_1-x_2}\gt 0 ， f(x) 单调递增（备注：以上我们已经把函数的整个定义域都讨论了）综上，函数在 (1,+\infty) 上单调递增，在 (-\infty, 1) 上单调递减 5. 导数法关于导数法的基础认识，以及常见的函数导数方式将在后续作品《关于导数的基础认识》中详细介绍，这里并不赘述，而是介绍导数在证明函数单调性过程中的一些想法和思维。导数实际上是前面介绍的斜率式 \frac{f(x_1)-f(x_2)}{x_1-x_2} 的极限，所以对函数进行求导后得到其导函数 f'(x) ，要用它来判断原函数的单调性，自然是要让 f'(x) 与0进行比较的。也就是说，我们如下描述的想法。设 f(x) 是定义在 D 的函数， f'(x) 是其导函数，集合 I\subseteq D 。如果 \forall x\in I 有 f'(x)\lt 0 ，那么 f(x) 在 I 上单调递减；如果 f'(x)\gt 0 ，那么 f(x) 在 I 上单调递增。通俗地来讲，我们需要求出导函数 f'(x) 在定义域 D 的哪些区域上是小于0的，从而确定原函数 f(x) 在定义域 D 的哪些区域上是单调递减的。这句话“我们需要求出导函数 f'(x) 在定义域 D 的哪些区域上是小于0的”再说明白一点，也就是说解释这句话的含义，我们需要找出定义域 D 上的哪些数，在导函数 f' 作用后得到的结果是小于0的。读者自行类似理解利用导函数求函数单调递增的区域。导数法和斜率式非常相似，但导数法要求函数是连续可导的，当然，对于连续可导这一点高中阶段只需要知道是一条没有尖点的连续不断曲线。而斜率式呢，则要求没那么苛刻，但导数法因此苛刻条件具备了非常突出的优点，就是求导后是一个一元函数。关于一元函数的处理则是我们当前阶段非常熟悉的，讨论一元函数与0相比较的问题也是常见的基础问题。至此，我们实际已经介绍了利用导函数求证原函数单调性的框架思路。图2 证明函数单调性的过程关于一元函数与0相比较的讨论，详细内容可以参考文章《单调性的应用》。需要指出的是，为了确定导函数 f'(x) 的单调性，我们可以对导函数再次求导得到 f''(x) ，但它确定的是 f'(x) 的单调性，而不是 f(x) 的单调性，如有必要，可以一直求导下去，只要你得到的导函数是初等函数的组合，就高中阶段来说。另外，还需要注意，图2所示的过程，只是展示了求一元函数与0比较的简单情景，更多过程建议参考文章《单调性的应用》中的相关介绍。例题5.1 证明函数 y=\sqrt{x} 的单调性证明记 f(x)=\sqrt{x},x\in [0,+\infty) 那么 f'(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2}\cdot \frac{1}{\sqrt{x}} 易知 \sqrt{x}\gt 0 所以 f'(x)\gt 0 在 [0,+\infty) 恒成立所以 y=\sqrt{x} 在 [0,+\infty) 上单调递增。例题5.2 证明 y=x^2-2x 的单调性证明令 f(x)=x^2-2x 那么 f'(x)=2x-2 令 f'(x)=0 得 x_0=1 又 f''(x)=2\gt 0 ，所以 f'(x) 单调递增那么，当 x\lt x_0=1 时， f'(x)\lt 0 ， f(x) 单调递减当 x\gt x_0=1 时， f'(x)\gt 0 ， f(x) 单调递增综上， f(x) 在 (1,+\infty) 上单调递增，在 (-\infty, 1) 上单调递减例题5.3 求 y=x^3-13x^2+14x 的单调性解记 f(x)=x^3-8x^2+3x 那么 f'(x)=3x^2-8x+3=(3x+1)(x-3) 令 f'(x)=0 得 x_1=-\frac{1}{3},x_2=3 （备注：结合二次函数其实已经可以完成了，但这里让读者感受一下导数并不一定导一次或者两次，只要可导且有必要那么就一直导下去）又 f''(x)=6x-8 ，令 f''(x)=0 得 x_0=\frac{3}{4} 又 f'''(x)=6\gt 0 ，则 f''(x) 单调递增（备注： f'''(x) 判断的是 f''(x) 的单调性）所以当 x\lt x_0=\frac{3}{4} 时， f''(x)\lt 0,f'(x) 单调递减当 x\gt x_0=\frac{3}{4} 时， f''(x)\gt 0,f'(x) 单调递增于是，当 x\lt x_1 时， f'(x)\gt f'(x_1)=0 ， f(x) 单调递增当 x_1\lt x\lt \frac{3}{4} 时， 0=f'(x_1)\gt f'(x) ， f(x) 单调递减当 \frac{3}{4}\lt x\lt x_2 时， f'(x)\lt f'(x_2)=0 ， f(x) 单调递减当 x\gt x_2 时， f'(x)\gt f'(x_2)=0 ， f(x) 单调递增由 f(x)的单调区间可知 x_0=\frac{3}{4} 是 f(x) 在 (x_1,\frac{3}{4}) 的最小值，也是 f(x) 在 (\frac{3}{4},x_2) 的最大值所以 f(x) 在 (x_1,x_2) 上是单调递减的综上， f(x) 在 (-\infty,-\frac{1}{3}),(3,+\infty) 上单调递增，在 (-\frac{1}{3},3) 上单调递减。例题5.4 求函数 f(x)=\ln x+e^x 的单调性解 f(x)=\ln x+e^x,x\in (0,+\infty) 那么 f'(x)=\frac{1}{x}+e^x\gt e^x 易证 y=e^x 在 (0,+\infty) 上单调递增，所以 e^x\gt e^0=1>0 所以 f'(x)\gt 0 在 (0,+\infty) 上恒成立所以 f(x) 在 (0,+\infty) 上单调递增由以上例题，读者应当能够感受到，实际上利用导数求证函数单调性，就是比较 f'(x) 与0的相对大小从而确定原函数的单调区间。于是，只要处理实际就落在了 f'(x) 上，我们可以对它进行因式分解(如果可以的话)，也可以乘以某个因子或因式，也可以对这个式子进行某种变换，就像作差法时介绍的那样，进行各种各样的处理，最终的目的就在于确定 f'(x) 与0的相对大小，从而确定 f(x) 的单调区间。当然确定 f'(x) 的单调性实际一定是要有的，不管零点是否可求还是恒成立，因为我们要确定的是一个区间，而不是某一个值。也就是需要借助单调性，以当前值为基础（例如例题5.2的 x_0 ），比较定义域上的其它数在 f 作用后的结果是否也小于0或大于0。只是有些特别明显就能看出在某个区间 f'(x) 与0的相对大小的，我们就省略了求证 f'(x) 单调性的过程，例如例题5.1。 6. 其他方法在《单调性的基础结论》一文中，详细介绍了初等函数的单调性，并结合函数运算对函数单调性进行判断。这里只是为了“证明函数单调性的方法”这一话题的完整性，提出根据某个函数各自的特点，结合函数图像简略地判断函数单调性。参考 ^abcd如何理解函数单调性 ^常见的奇偶函数 ^例题3.2来源 ^细解放缩 ^放缩的常见方式 ^导数的基础认识 ^单调性的应用 ^单调性的基础结论编辑于 2022-08-11 01:15 函数的单调性函数方法学赞同 491 条评论分享喜欢收藏申请转载写下你的评论... 1 条评论默认最新羊羔第4点的"再次结合x1＞x2”写错了，应该是x2＞x1 2023-06-23 · 广东回复喜欢关于作者秦思涯更新速度慢，喜欢文章点关注容易查，莫收藏免吃灰。回答 38文章 77关注者 130 关注他发私信推荐阅读函数的性质：单调性与奇偶性（重难点专项突破教师版） ========================= 刀神李流水发表于高中数学习...如何理解函数单调性 ========= 0. 写在开头更多数学问题可以私信小秦为你解决。文章简介：一篇能够让你清楚明白函数单调性含义，愉快应用函数单调性解决问题的文章。文章思路介绍：单调性的一个简单作用->定义域只有… 秦思涯发表于高中数学（实变函数论）如何证明单调函数是几乎处处可微的？ ======================== 运飞美少女发表于大学数学系...解析函数与调和函数 ========= 一、调和函数实际问题中，常碰到一种特殊的二元函数，调和函数，例如：流速场的流函数和势函数、静电场的力函数和势函数、热流场的流函数等。它们与解析函数有密切关系。定义3.8.1 如果… 几件芬格斯玛尔的家具想来知乎工作？请发送邮件到 jobs@zhihu.com 打开知乎App 在「我的页」右上角打开扫一扫其他扫码方式：微信下载知乎App 无障碍模式验证码登录密码登录开通机构号中国 +86 获取短信验证码获取语音验证码登录/注册其他方式登录未注册手机验证后自动登录，注册即代表同意《知乎协议》《隐私保护指引》扫码下载知乎 App 关闭二维码打开知乎App 在「我的页」右上角打开扫一扫其他扫码方式：微信下载知乎App 无障碍模式验证码登录密码登录开通机构号中国 +86 获取短信验证码获取语音验证码登录/注册其他方式登录未注册手机验证后自动登录，注册即代表同意《知乎协议》《隐私保护指引》扫码下载知乎 App 关闭二维码
1622	https://www.sciencedaily.com/releases/1997/11/971107070528.htm	Great Lakes Intensify Ferocity Of Passing Storms, Scientists Say \| ScienceDaily Skip to main content Your source for the latest research news Follow:FacebookX/TwitterSubscribe:RSS FeedsNewsletter New! Sign up for our free email newsletter. Science News from research organizations Great Lakes Intensify Ferocity Of Passing Storms, Scientists Say Date:November 7, 1997 Source:University Of Illinois At Urbana-Champaign Summary:The Great Lakes exert a significant influence on passing cyclones, causing storms to speed up and grow in strength, say researchers at the University of Illinois and the Illinois State Water Survey. Also, the number of potentially dangerous storms is on the rise, they report. Share: FacebookTwitterPinterestLinkedINEmail FULL STORY CHAMPAIGN, Ill. -- The Great Lakes exert a significant influence on passing cyclones, causing storms to speed up and grow in strength, say researchers at the University of Illinois and the Illinois State Water Survey. Also, the number of potentially dangerous storms is on the rise, they report. "Cyclones that traverse the Great Lakes have important impacts on the physical environment and human habitation in the region," said James Angel, a climatologist with the Survey. "There is a lot of development along the lakes, and when the water level is high -- as it is now -- the area becomes extremely vulnerable to shoreline damage from these storms. A better understanding of how the Great Lakes affect passing cyclones may allow better forecasting of these storms and their potential effects." Cyclones are low-pressure storm centers, "often accompanied by high winds and heavy precipitation," said Scott Isard, a U. of I. professor of geography. "The ensuing storms can be huge, ranging in size from 800 to 1,500 miles in diameter." To study the effect the Great Lakes have on passing cyclones, Angel and Isard examined the rates of movement and the changes in intensity for 583 cyclones that passed over the region between the years 1965 to 1990. The researchers' findings, published in the September issue of Monthly Weather Review, identify several important features regarding the lakes' influence on these storm systems. "In general, we found that cyclones accelerated as they approached the Great Lakes region and increased in intensity over the lakes," Angel said. "This effect was most pronounced from September to November, when the surface waters of the lakes are warmer than the surrounding air and can provide a major source of both moisture and heat that energizes passing storms." From January to March, when broken ice cover is generally present on the lakes, cyclones accelerated less and did not intensify, Angel said. However, cyclones that traversed the region during May and June did speed up and grow in strength. "This surprised us, because the lakes are usually cooler than the overriding air mass during spring and summer, and have not generally been considered as an important energy source for cyclones at that time," Angel said. "We don't yet have a satisfactory explanation for this phenomenon." In another study (to appear in the journal Climate), Angel and Isard analyzed trends in storm strength for the years 1900 to 1990. "We are seeing evidence of an increase in the number of stronger storms, particularly in the months of November and December," Angel said. Historically, some of these cyclones have produced hurricane-force winds and caused extensive damage to shipping. The "great storm of 1913," for example, sank a dozen ships and claimed more than 250 lives. More recently, the ore carrier Edmund Fitzgerald -- popularized in a ballad by Canadian singer and songwriter Gordon Lightfoot -- sank in Lake Superior during a major storm on Nov. 10, 1975. All hands were lost. RELATED TOPICS Earth & Climate Severe Weather Storms Hurricanes and Cyclones Geography Water Weather Geomagnetic Storms Tornadoes RELATED TERMS Water pollution Great Lakes Effects of global warming Hurricane preparedness Hurricane Lake Weather Supercell Story Source: Materials provided by University Of Illinois At Urbana-Champaign. Note: Content may be edited for style and length. Cite This Page: MLA APA Chicago University Of Illinois At Urbana-Champaign. "Great Lakes Intensify Ferocity Of Passing Storms, Scientists Say." ScienceDaily. ScienceDaily, 7 November 1997. . University Of Illinois At Urbana-Champaign. (1997, November 7). Great Lakes Intensify Ferocity Of Passing Storms, Scientists Say. ScienceDaily. Retrieved June 3, 2025 from www.sciencedaily.com/releases/1997/11/971107070528.htm University Of Illinois At Urbana-Champaign. "Great Lakes Intensify Ferocity Of Passing Storms, Scientists Say." ScienceDaily. www.sciencedaily.com/releases/1997/11/971107070528.htm (accessed June 3, 2025). Explore More from ScienceDaily RELATED STORIES Research on Past Hurricanes Aims to Reduce Future Risk Jan. 17, 2025 — New research emphasizes that studying the impacts of past tropical storms can help communities better prepare for future storms. A key part of the study is analyzing the types and quantities of ... 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1623	https://www.ck12.org/flexi/cbse-math/venn-diagrams-and-operations-of-sets/what-are-the-laws-of-union-of-sets/	What are the laws of union of sets? Flexi Says: The key laws or properties of the Union of Sets are as follows: Idempotent Law: For any set @$\begin{align}A,\end{align}@$ the union of the set with itself is the set itself. Mathematically, this can be represented as @$\begin{align}A \cup A = A\end{align}@$. Commutative Law: The union of set @$\begin{align}A\end{align}@$ with set @$\begin{align}B\end{align}@$ is the same as the union of set @$\begin{align}B\end{align}@$ with set @$\begin{align}A.\end{align}@$ Mathematically, this can be represented as @$\begin{align}A \cup B = B \cup A\end{align}@$. Associative Law: The union of set @$\begin{align}A\end{align}@$ with the union of sets @$\begin{align}B\end{align}@$ and @$\begin{align}C\end{align}@$ is the same as the union of the union of sets @$\begin{align}A\end{align}@$ and@$\begin{align}B\end{align}@$ with set @$\begin{align}C.\end{align}@$ Mathematically, this can be represented as @$\begin{align}A \cup (B \cup C) = (A \cup B) \cup C\end{align}@$. Distributive Law: The union of set A with the intersection of sets @$\begin{align}B\end{align}@$and @$\begin{align}C\end{align}@$ is the same as the intersection of the union of sets @$\begin{align}A\end{align}@$ and @$\begin{align}B\end{align}@$ with the union of sets @$\begin{align}A\end{align}@$ and @$\begin{align}C.\end{align}@$ Mathematically, this can be represented as @$\begin{align}A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\end{align}@$. Identity Law: The union of any set @$\begin{align}A\end{align}@$ with the empty set is the set @$\begin{align}A\end{align}@$ itself. Mathematically, this can be represented as @$\begin{align}A \cup \emptyset = A\end{align}@$. Universal Law: The union of any set @$\begin{align}A\end{align}@$ with the universal set is the universal set itself. Mathematically, this can be represented as @$\begin{align}A \cup U = U\end{align}@$, where @$\begin{align}U\end{align}@$ is the universal set. Complement Law: The union of any set @$\begin{align}A\end{align}@$ with its complement is the universal set. Mathematically, this can be represented as @$\begin{align}A \cup A' = U\end{align}@$, where @$\begin{align}A'\end{align}@$ is the complement of @$\begin{align}A\end{align}@$ and @$\begin{align}U\end{align}@$ is the universal set. Try Asking: What are the properties of the complement of a set?Define the Property of Ⲫ/ Identity Law in set theory.What is the range of a set? By messaging Flexi, you agree to our Terms and Privacy Policy
1624	https://math.stackexchange.com/questions/1300509/lagrange-multiplier-method-on-linear-equation-set	Skip to main content Lagrange Multiplier Method On Linear Equation Set Ask Question Asked Modified 6 years, 9 months ago Viewed 4k times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. I am trying to perform a Lagrange constraint problem for a simple set of linear equations (I realize this can be solved by substitution) but I'm curious why/how the Lagrange method is failing and I'm getting a conflicting Lagrange multiplier (terminology?). In my example, x and y are constants, S represents the constraint, Psi represents the function. Ψ(a,b)=(a+b)∗x+b∗y S(a,b)=a+b=0.5 ∇Ψ(a,b)=λ∇S(a,b) ∂Ψ∂a=λ∂S∂a x=λ∗1 ∂Ψ∂b=λ∂S∂b x+y=λ∗1 lagrange-multiplier Share CC BY-SA 3.0 Follow this question to receive notifications asked May 27, 2015 at 4:01 Carl JohnsonCarl Johnson 3311 silver badge33 bronze badges 5 Are there other constraints (non-negative?) on a and b? Else, the infimum value of the objective function is −∞. – Michael Commented May 27, 2015 at 4:12 Thank you all for the helpful responses. How would the problem change if I added the constraint that 0<a<0.5 and 0<b<0.5? Does this reduce down to checking those end corners (0, 0.5), (0.5, 0)? – Carl Johnson Commented May 27, 2015 at 14:34 It depends on what type of Lagrange multiplier theorem you are looking for. For example, you can compute the dual function d(λ)=inf(a,b)∈[0,1/2]2[Ψ(a,b)+λ(a+b−1/2)]. Indeed maximizing d(λ) over λ∈R gives you the optimal objective function value. – Michael Commented May 27, 2015 at 15:41 1 Or, you can use Kuhn-Tucker gradient conditions, in which case you need 4 additional constraints to ensure a≥0, a≤1/2, b≥0, b≤1/2. – Michael Commented May 27, 2015 at 15:42 Either way, most statements about Lagrange multipliers start by assuming existence of an optimal solution x∗, which is not true if the optimal objective value is −∞. – Michael Commented May 27, 2015 at 15:43 Add a comment \| 2 Answers 2 Reset to default This answer is useful 3 Save this answer. Show activity on this post. Part of the problem is that your constraint is a line, this isn't a compact set (closed and bounded), so you aren't guaranteed that your function will have an extreme value on the set. Furthermore your function, being linear, is just going to grow/decay monotonically as you traverse the constraint. So there won't be any critical points for the method to detect. Another way to think of this, The gradient of your function is a constant vector, i.e., independent of a and b. The gradient of the constraint function is also constant. The method of Lagrange multipliers answers the question "when are these vectors parallel". If the two constant vectors are not parallel then the method cannot give you an answer. Share CC BY-SA 3.0 Follow this answer to receive notifications edited May 27, 2015 at 4:31 answered May 27, 2015 at 4:13 SpencerSpencer 12.7k33 gold badges3838 silver badges6767 bronze badges Add a comment \| This answer is useful 0 Save this answer. Show activity on this post. The equivalent problem is to minimize x/2+by over the plane (x,y), which clearly is driven to −∞ when both x,y→−∞. Share CC BY-SA 3.0 Follow this answer to receive notifications answered May 27, 2015 at 4:25 gt6989bgt6989b 55k33 gold badges3939 silver badges7575 bronze badges 0 Add a comment \| You must log in to answer this question. 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1625	https://math.stackexchange.com/questions/4985652/tiling-rectangles-with-i-and-t-tetrominoes	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Tiling rectangles with I and T tetrominoes Ask Question Asked Modified 11 months ago Viewed 150 times 1 $\begingroup$ I wonder which $m \times n$ rectangles can be tiled by $I$ (straight line of length $4$) and $T$ tetrominoes (rotations allowed). I am mostly looking for references, but any help is welcome. So far, I know the following : An $m \times n$ rectangle can be tiled by $I$ tetrominoes if and only if one of $m$ or $n$ is a multiple of $4$ One direction is trivial, for the other direction one can use a coloring by ${1, 2, 3, 4}$. An $m \times n$ rectangle can be tiled by $T$ tetrominoes if and only if both m and n are multiples of 4. One direction is trivial, the other one is due to Walkup If an $m \times n$ rectangle can be tiled by $I$ and $T$ tetrominoes, then $mn$ is a multiple of $4$. So in view of the above, we only need to look at the case where both $m$ and $n$ are congruent to $2$ modulo $4$. It is easy to show that $2 \times (4k+2)$ is not doable. So the smallest next possibilities are $6 \times 6$ and $6 \times 10$. In both case, we can use : If an $m \times n$ rectangle can be tiled by $I$ and $T$ tetrominoes and both $m$ and $n$ are even, then both the number of horizontal $T$ and the number of vertical $T$ are even. To show this, color the rows (the column) alternatively in black and white. If I believe this website $6 \times 6$ is not tileable, while $6 \times 10$ is. (Be aware that on the above website, tiling by ${I,T}$ means that at least one $I$ and one $T$ must be used.). However, the website provide no proof for the impossibility of $6 \times 6$ and the image for the $6 \times 10$ tiling is broken. I tried some coloring arguments for $6 \times 6$ as well as to have a lucky guess for $6 \times 10$, but without any luck so far. combinatorics discrete-mathematics solution-verification recreational-mathematics tiling Share edited Oct 17, 2024 at 4:43 Prem 15.2k22 gold badges1919 silver badges3939 bronze badges asked Oct 17, 2024 at 3:36 PHLPHL 29411 silver badge66 bronze badges $\endgroup$ Add a comment \| 1 Answer 1 Reset to default 2 $\begingroup$ I think OP is asking for 2-3 things : (A) Impossibility for $6 \times 6$ When we use one $I$ tetromino at a corner , we get $6 \times 5$ left over , with one $2 \times 1$ corner. That corner either will require 2 $I$ tetrominoes (which will give a new $2 \times 1$ corner) or will require 1 $T$ tetromino with 1 $I$ tetromino. Continuing that way , we will eventually run out without getting the Solution. When we alternately use one $T$ at the corner , there are only a few ways to continue tiling , which will eventually run out without getting the Solution. Hence $6 \times 6$ is impossible. (B) Solution for $6 \times 10$ The website gives this : (C) Checking higher values We can easily get $6 \times 14$ using $6 \times 10$ with $6 \times 4$ Attaching $4 \times 14$ will give $10 \times 14$ Attaching $4 \times 14$ twice will give $14 \times 14$ That way , we can easily add $4$ to either Direction to get the Solutions for $(n+4) \times (m)$ , $(n) \times (m+4)$ , $(n+4) \times (m+4)$ in terms of $(n) \times (m)$ We can thus always tile $(4n+2) \times (4m+2)$ , except for $6 \times 6$ which is impossible. Share edited Oct 17, 2024 at 9:47 answered Oct 17, 2024 at 4:37 PremPrem 15.2k22 gold badges1919 silver badges3939 bronze badges $\endgroup$ 4 $\begingroup$ Thanks for the picture. I was not able to see it correctly. Apparently, something is wrong with my version of Firefox. However, I do not understand the proof for A. Using one I will not leave a 6 x5 left over. $\endgroup$ PHL – PHL 2024-10-17 05:32:43 +00:00 Commented Oct 17, 2024 at 5:32 $\begingroup$ Oh , I was thinking something else entirely when I wrongly wrote that ! Nice Catch , @PHL , I have rectified that now. $\endgroup$ Prem – Prem 2024-10-17 09:50:07 +00:00 Commented Oct 17, 2024 at 9:50 1 $\begingroup$ I had a similar proof for the impossibility of 6x6, depending on the position of an I in the rectangle (corner, side, middle). But then I used Walkup result to get ride of the all Ts cases. Your version is more direct. I hoped for something using colorings, but it is probably too much to ask for. $\endgroup$ PHL – PHL 2024-10-21 09:34:04 +00:00 Commented Oct 21, 2024 at 9:34 $\begingroup$ I think coloring approach is hard here. While $T$ has a "center" , which we can color to then count the total colorings , there is no "center" for the $I$ , hence coloring it will not be very easy ! We will have to check multiple ways to color , @PHL , ending up with more work ! $\endgroup$ Prem – Prem 2024-10-21 15:12:38 +00:00 Commented Oct 21, 2024 at 15:12 Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions combinatorics discrete-mathematics solution-verification recreational-mathematics tiling See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Related 6 Squaring rectangles 6 Prime number proof for tiling a rectangle 1 Tiling $41$ unit squares with $L$ tetrominoes and $L$ trominoes 7 is it possible to divide $9\times 11$ rectangle into one $1\times 3$ rectangle and N tetrominoes? 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1626	https://thestoryreadingapeblog.com/2018/09/13/a-short-analysis-of-the-thirty-days-hath-september-rhyme/	A Short Analysis of the ‘Thirty Days Hath September’ Rhyme \| Chris The Story Reading Ape's Blog Advertisement Search for: Menu Skip to content Home ABOUT TSRA HALLs of FAME AUTHORS NAVIGATION AWARDS FOLLOW ME ALSO AT SPOTLIGHTS Hello from Honduras Honduras Addendum IF YOU LOVE ME… a child’s plea to a parent by Marta Merajver-Kurlat Littlest Druid celebrates Lughnassad by ElfKat Ode to the Librarian’s Cousin by Jo Robinson Shock and Possible Horror by Dianne Horsfield THE MAKAPANSGAT FACE – A short story by Aura Burrows – based on an article she found online The Random Thoughts of a Wandering Mind by M J Jones Untitled, Dedicated to a special friend by deb84writings Victoria (Tori) Zigler –Author When is a Book a BOOK by Dianne Horsfield TSRA TALES A Short Halloween Tale from Chris Graham, The Story Reading Ape A Writing Challenge – My Response… Artie, The Time Traveling Chimp… Does That Lettuce Look HUNGRY? – A TSRA Tale… Experiment Successful – a VERY short story by Chris Graham… I wish I knew then what I know now! McKinley – by Chris Graham… Once Upon a Halloween… (A short story by Chris Graham) Send an Extra Card this Christmas The First Halloween – by, Chris Graham… The Hunt – by Chris Graham… Trouble at the North Pole – by Chris Graham ‘The EVENT’ – A short tale by Chris Graham (aka The Story Reading Ape) WORLDREADER AFRICAN AUTHORS PRIVACY POLICY RECOMMENDED WRITER’S RESOURCES COPYRIGHT INFRINGEMENT? eBOOK FORMATTING EDITING 101: EDITORS GRAMMAR – ENGLISH HOW TO – BUILD A BOOK TRAILER HOW TO – MAKE AUDIOBOOKS HOW TO 101: INFORMATION-BUSINESS MARKETING PROOFREADERS PUBLISHERS FacebookLinkedInTumblrTwitter Chris The Story Reading Ape's Blog READER – WRITER – CURATED RESOURCES – & MORE A Short Analysis of the ‘Thirty Days Hath September’Rhyme Posted on September 13, 2018 September 13, 2018 by Chris The Story Reading Ape Interesting Literature As Groucho Marx once said, ‘My favourite poem is the one that starts “Thirty Days Hath September”, because it actually means something.’ The meaning of ‘Thirty Days Hath September’ is self-evident and straightforward. But what are the origins of this famous rhyme? ‘Thirty Days Hath September’ runs, of course: Thirty days hath September, April, June and November; All the rest have thirty-one, Excepting February alone. Which only has but twenty-eight days clear And twenty-nine in each leap year. One early reference to ‘Thirty Days Hath September’, from William Harrison in 1577, actually begins, er … ‘Thirty days hath November’: View original post229 more words If you enjoyed this article, why not share it with all your friends, online connections and groups - GO ON - you KNOW you WANT to! 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Related Writing a Narrative Poem: Everything You Need to Know (A Step by Step Guide) – by The Write Life Team… When we think of poetry, the first thing that usually comes to people’s mind is rhymes (a close second: the other assumption that all poems are short!). But of course, this doesn’t capture the entire poetry genre. Narrative poetry is one of the most unique forms of literature because of… In "Informative" Types of Rhymes in Poetry – by Melissa Donovan… We’re all familiar with rhyming poems. After all, these are the first poems most of us encounter as children, from the delightful stories of Dr. Seuss to the hilarious poetry of Shel Silverstein. People often think rhyming poems are rigid, conforming to strict meter and perfect rhymes at the end… In "Informative" Tee shirt or T-shirt – Which is Correct? – by Writing Explained… Most humans alive today have worn a light, short-sleeved shirt with no collar and no buttons. Such garments are nearly ubiquitous in modern life and are now worn while relaxing at home, in the boardrooms of some tech start-ups, and everywhere in between. Is this popular garment a tee shirt… In "Informative" This entry was posted in Informative and tagged Informative. Bookmark the permalink. Post navigation ← GP: Self-publishing: Now what? by Nicolette Stephens How to Reach Forgotten Markets for Self-published Books – by Anne-Catherine De Fombelle… → DON'T BE SHY - LEAVE A REPLY Cancel reply Δ This site uses Akismet to reduce spam. Learn how your comment data is processed. FIND A TOPIC ON THIS BLOG Search for: TRANSLATE THIS POST / PAGE Powered by Translate Categories Categories Archives Except where otherwisenoted, content on this site is licensed to Christopher Graham (aka, The Story Reading Ape), under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. 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1627	https://brainly.com/question/14806468	[FREE] Is there a real number whose square is -1? a. Is there a real number x such that x^2 = -1? b. Does there - brainly.com 4 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +64,4k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +20,8k Ace exams faster, with practice that adapts to you Practice Worksheets +8,1k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified Is there a real number whose square is −1? a. Is there a real number x such that x 2=−1? b. Does there exist x such that x 2=−1? 2 See answers Explain with Learning Companion NEW Asked by sofiaday47 • 02/17/2020 0:00 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 59837417 people 59M 4.0 3 Upload your school material for a more relevant answer a.x 2=−1 b.a real number x Explanation We are given that statement. We have to rewrite the given statement using variable or variables. Statement:Is there a real number whose square is -1. a.Let x bet the real number The square of real number x written as x 2 According to question x 2=−1 Therefore, Is there a real number x such that x 2=−1 b.Does there exist a real number x such that x 2=−1 Answered by lublana •4.1K answers•59.8M people helped Thanks 3 4.0 (4 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 59837417 people 59M 3.7 3 Calculus-Based Physics - Jeffrey W. Schnick Fields and Circuits - Benjamin Crowell Mathematics for Biomedical Physics - Jogindra M Wadehra Upload your school material for a more relevant answer There is no real number whose square is -1, as the square of any real number is non-negative. The equation x 2=−1 does not have solutions in the real number set. Solutions exist in complex numbers, specifically with the imaginary unit i. Explanation To determine if there is a real number whose square is −1, we can express this mathematically as follows: Understanding the equation: We want to find a real number x such that x 2=−1. Analyzing the equation: The square of any real number is always non-negative. This means that: x 2≥0 for all real numbers x. Conclusion about real numbers: Since −1 is negative, there cannot be any real number whose square equals −1. This shows that the equation x 2=−1 has no solutions in the set of real numbers. Complex numbers: To find a solution to the equation, we look to complex numbers, specifically the imaginary unit i which is defined such that i 2=−1. Therefore, in the realm of complex numbers, x=i and x=−i satisfy the equation, but they are not real numbers. Hence we can conclude: a. There is no real number x such that x 2=−1. b. There does not exist a real number x such that x 2=−1. Examples & Evidence For instance, if we take the numbers 1 and -1, both yield squares of 1. Therefore, there are no real numbers that yield a square of -1, confirming our conclusion. On the other hand, using the number i, we see that i 2=−1, showcasing the transition from real to complex numbers. Mathematically, the property of squares states that any real number squared results in a non-negative value. This property is foundational and applies universally within real number arithmetic. Thanks 3 3.7 (3 votes) Advertisement Community Answer This answer helped 9474405 people 9M 5.0 0 There is no real number whose square is -1. However, in the domain of complex numbers, 'i' is defined as the square root of -1. Complex numbers include both real and imaginary parts. Explanation In the realm of real numbers, there isn't a real number whose square is -1. In the context of complex numbers, however, 'i' is defined to be the square root of -1. In other words, i2 = -1. It's important to note that complex numbers consist of a real part and an imaginary part (where 'i' is the basis for the imaginary part), and are beyond the usual scope of real numbers. Learn more about Complex Numbers here: brainly.com/question/33170548 SPJ3 Answered by MonteBlue •15.9K answers•9.5M people helped Thanks 0 5.0 (5 votes) Advertisement ### Free Mathematics solutions and answers Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four? Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish? Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit? Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. Community Answer 4 Click an Item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Express In simplified exponential notation. 18a^3b^2/ 2ab New questions in Mathematics Consider two events, A and B, with the following probabilities: P(A)=0.5, P(B)=0.37, P(A∩B)=0.36. Use the Additive Rule of Probability to find the probability of A∪B. Round to the nearest hundredth as needed. Factor by grouping. 2 y 3+3 y 2+10 y+15 The value, in dollars, of a stock t hours into a day of trading is modeled by V(t)=1800 t+7000. Find the rate at which the stock's value is increasing or decreasing after 3 hours. (Round your answer to the nearest cent.) Rationalize the denominator. 12 x9 Factor. x 2+15 x y+26 y 2 Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
1628	https://revisionmaths.com/gcse-maths/ratio-proportion-and-rates-change/conversions-area-and-volume	Conversions of Area and Volume - Maths GCSE Revision Revision Maths - Revision World Networks Ltd. If you wish to opt-out of the sale, sharing to third parties, or processing of your personal or sensitive information for targeted advertising by us, please use the below opt-out section to confirm your selection. Please note that after your opt-out request is processed you may continue seeing interest-based ads based on personal information utilized by us or personal information disclosed to third parties prior to your opt-out. You may separately opt-out of the further disclosure of your personal information by third parties on the IAB’s list of downstream participants. This information may also be disclosed by us to third parties on the IAB’s List of Downstream Participants that may further disclose it to other third parties. Personal Data Processing Opt Outs Child Sensitive Data Processing Opt Outs CONFIRM Data DeletionData AccessPrivacy Policy Skip to main content Revision Maths GCSE Maths Number Algebra Ratio, Proportion and Rates of Change Geometry and Measures Trigonometry Statistics and Probability GCSE Maths Past Papers GCSE Maths (9-1) Revision Advice GCSE Maths Exam Tips Other GCSE Subjects How to Achieve a Grade 9 in GCSE Statistics How to Prepare for the Non-Calculator GCSE Maths Exam Maths GCSE Questions and Quizzes How to Achieve a Grade 9 in GCSE Maths A-Level Maths Study Calendar Revision Science Revision World Revision Videos Student Advice Search User account menu Sign up Log in GCSE Maths Number Algebra Ratio, Proportion and Rates of Change Geometry and Measures Trigonometry Statistics and Probability GCSE Maths Past Papers GCSE Maths (9-1) Revision Advice GCSE Maths Exam Tips Other GCSE Subjects How to Achieve a Grade 9 in GCSE Statistics How to Prepare for the Non-Calculator GCSE Maths Exam Maths GCSE Questions and Quizzes How to Achieve a Grade 9 in GCSE Maths A-Level Maths Study Calendar Revision Science Revision World Revision Videos Student Advice Breadcrumb Home GCSE Maths Ratio, Proportion and Rates of Change Conversions of Area and Volume Converting area and volume measurements are an essential skill in Mathematics. These types of calculations are straightforward once you understand the basic metric units of measurement. Below is a guide to help you navigate through these conversions with clarity and examples. Converting Area Area is a measurement of the surface of a two-dimensional shape. When converting area, remember that the measurement is always in square units (e.g. cm², m²), as it represents a 2D space. The key principle to remember is that when you are converting between different units for area, you are working with squared values. Example 1: Converting m² to cm² Let’s look at how you would convert 1 square metre (1m²) to square centimetres (cm²). Start with the definition of 1m². It represents an area of 1 metre by 1 metre. In equation form: 1m² =1m×1m Now, we know that 1 metre is equal to 100 centimetres. Therefore, to convert 1 metre to centimetres: 1m=100cm Next, substitute the value of 1m with 100cm into the equation: 1m² =100cm×100cm Perform the multiplication: 1m² =10,000cm² So, 1 square metre is equal to 10,000 square centimetres (1m² = 10,000cm²). Example 2: Converting 3m² to cm² To convert 3 square metres to square centimetres: First, recall that 1m² = 10,000cm². Multiply 3 by 10,000: 3m² = 3×10,000cm² =30,000cm² Thus, 3 square metres is equal to 30,000 square centimetres. Converting Volume Volume measures the amount of space occupied by a three-dimensional object. When converting volume, remember that you are working with cubic units (e.g. cm³, m³), as volume involves three dimensions. Example 1: Converting m³ to cm³ Let’s now look at how to convert 1 cubic metre (1m³) into cubic centimetres (cm³). Start with the definition of 1m³. It represents a cube with each side measuring 1 metre. In equation form: 1m³ = 1m x 1m x 1m Since we know that 1 metre equals 100 centimetres, substitute 100cm for each metre: 1m³ = 100m x 100m x 100m Multiply the three values together: 1m³ = 1,000,000cm³ Therefore, 1 cubic metre is equal to 1,000,000 cubic centimetres (1m³ = 1,000,000cm³). Example 2: Converting 2m³ to cm³ To convert 2 cubic metres to cubic centimetres: Recall that 1m³ = 1,000,000cm³. Multiply 2 by 1,000,000: 2m³ = 2×1,000,000cm³ = 2,000,000cm³ Thus, 2 cubic metres is equal to 2,000,000 cubic centimetres. General Conversion Formula To generalise, when converting between different units of area or volume, remember: For area: 1m² =10,000cm² (since 1m=100cm) For volume: 1m³ = 1,000,000cm³ (since 1m=100cm) Additional Examples Example 3: Converting 5m² to mm² We know that 1m = 1000mm. So, 1m² = 1000mm × 1000mm = 1,000,000mm². To convert 5m², multiply 5 by 1,000,000: 5m² = 5×1,000,000mm² = 5,000,000mm² Thus, 5 square metres is equal to 5,000,000 square millimetres. Example 4: Converting 0.5m³ to cm³ Recall that 1m³ = 1,000,000cm³. Multiply 0.5 by 1,000,000: 0.5m³ = 0.5 x 1,000,000cm³ = 500,000cm³ Thus, 0.5 cubic metres is equal to 500,000 cubic centimetres. By understanding the principles of converting area and volume and practising these types of conversions, you'll find it much easier to solve problems that involve changing between metric units. Keep these conversions and the key differences between square and cubic units in mind, and practice with a variety of examples to solidify your skills. For more information about area and shapes visit ourGeometry and measures section. Revision Maths hierarchy navigation Breadcrumb Home GCSE Maths Ratio, Proportion and Rates of Change Log in Username or email address Password Forgot your password? Create new account Slot Discover more Science Maths maths mathematics Mathematics Home About Us Advertise with us Cookies Policy Privacy Policy Copyright©2007-2025 Revision World Networks Ltd.
1629	https://artofproblemsolving.com/wiki/index.php/Recursion?srsltid=AfmBOoqQ6puSh-uNmGUPQA6dzKiDnyeVydIRfXtQBK_GBLggxLXyApkm	Art of Problem Solving Recursion - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Recursion Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Recursion Recursion is a method of defining something (usually a sequence or function) in terms of previously defined values. The most famous example of a recursive definition is that of the Fibonacci sequence. If we let be the th Fibonacci number, the sequence is defined recursively by the relations and . (That is, each term is the sum of the previous two terms.) Then we can easily calculate early values of the sequence in terms of previous values: , and so on. Often, it is convenient to convert a recursive definition into a closed-form definition. For instance, the sequence defined recursively by and for also has the closed-form definition . In computer science, recursion also refers to the technique of having a function repeatedly call itself. The concept is very similar to recursively defined mathematical functions, but can also be used to simplify the implementation of a variety of other computing tasks. Examples Mock AIME 2 2006-2007 Problem 8 (number theory) 1994 AIME Problem 9 A combinatorial use of recursion: 2006 AIME I Problem 11 Another combinatorial use of recursion: 2001 AIME I Problem 14 Use of recursion to compute an explicit formula: 2006 AIME I Problem 13 Use of recursion to count a type of number: 2007 AMC 12A Problem 25 Yet another use in combinatorics 2008 AIME I Problem 11 2015 AMC 12A Problem 22 2019 AMC 10B Problem 25 2004 AIME I Problem 15 See also Combinatorics Sequence Induction Recursion Retrieved from " Categories: Combinatorics Definition Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
1630	https://www.scribd.com/document/330881780/Apostrophe-and-Possessives	The Apostrophe: Purdue University Online Writing Lab \| PDF \| Plural \| Noun Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 0 ratings 0% found this document useful (0 votes) 48 views 3 pages The Apostrophe: Purdue University Online Writing Lab The document summarizes the three uses of the apostrophe: 1) to form possessives of nouns, 2) to show the omission of letters, and 3) to indicate certain plurals of lowercase letters. It pro… Full description Uploaded by delsolarster6988 AI-enhanced title and description Go to previous items Go to next items Download Save Save Apostrophe and Possessives For Later Share 0%0% found this document useful, undefined 0%, undefined Print Embed Ask AI Report Download Save Apostrophe and Possessives For Later You are on page 1/ 3 Search Fullscreen The Apostrophe Brought to you by the Purdue University Online Writing LabThe apostrophe has three uses: 1) to form possessives of nouns2) to sho the omission of letters!) to indi"ate "ertain plurals of loer"ase letters# Forming possessives of nouns To see if you need to ma$e a possessive% turn the phrase around and ma$e it an &of the###& phrase# 'or e(ample:the boys hat the hat of the boy three days +ourney +ourney of three days,f the noun after &of& is a building% an ob+e"t% or a pie"e of furniture% then no apostrophe is needed-room of the hotel hotel room door of the "ar "ar door leg of the table table leg On"e youve determined hether you need to ma$e a possessive% follo these rules to "reate one# . add 's to the singular form of the word (even if it ends in -s): the oners "ar /amess hat. add 's to the plural forms that do not end in -s: the "hildrens game the geeses hon$ing adDownload to read ad-free . add 'to the end of plural nouns that end in -s: houses roofs three friends letters. add 's to the end of compound words: my brother0in0las money. add 's to the last noun to show joint possession of an object: To dd a nd nnes apartment a$e the possessive form of the nouns given# Eample: y brother's house is in London# 3brother) !" y shirt is purple# 3friend) " The boo$s are on the des$# 3girls) $" ,ts birthday on onday# 3/ohn) %" 4o you have nespaper5 3today) &" The bags are in the bedroom# 3"hildren) " y offi"e is ne(t to mine# 3boss) " 6our un"le is your brother# 3father) " 6our aunt is your sister# 3mothe r) " y birthdays are ne(t month# 3friends) !+"  tail is long# 3mon$ey)0000000000 ,core  adDownload to read ad-free Share this document Share on Facebook, opens a new window Share on LinkedIn, opens a new window Share with Email, opens mail client Copy link Millions of documents at your fingertips, ad-free Subscribe with a free trial You might also like Long Beach Wharf Design Guide No ratings yet Long Beach Wharf Design Guide 103 pages Possessive Nouns: Singular vs. Plural No ratings yet Possessive Nouns: Singular vs. Plural 3 pages Performance Measurament No ratings yet Performance Measurament 42 pages Group 2 NOUNS No ratings yet Group 2 NOUNS 34 pages S14 Zenki ECU Pinout Guide No ratings yet S14 Zenki ECU Pinout Guide 1 page pOSSESSIVE nOUNS 100% (3) pOSSESSIVE nOUNS 59 pages Manufacturing Process I Diploma in Mechanical Engineering 3 RD Semester No ratings yet Manufacturing Process I Diploma in Mechanical Engineering 3 RD Semester 18 pages Possessive Nouns: by Lyka P. 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1631	https://www.quora.com/How-do-you-convert-40-C-to-F	How to convert -40°C to °F - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Fahrenheit Temperature Conversion Method Science Units and Measurements Physics Degree Conversion Arithmetic Celsius Scale 5 How do you convert -40°C to °F? All related (53) Sort Recommended Philip Lovelace BA in English (Creative Writing), Idaho State University (Graduated 2006) · Author has 2.8K answers and 6.7M answer views ·6y Related How in the world is it possible that -40 degrees Celsius is the same as -40 degrees Fahrenheit? Fahrenheit is sort of arbitrary, and places the freezing point of pure water at standard pressure at 32 degrees, and the boiling point at 212. That means there are 180 degrees F between boiling and freezing. Celsius is a different scale, with freezing at 0 degrees, and boiling at 100. There are exactly 100 degrees C between freezing and boiling. Degrees Celsuis are thus a little “larger”, such that one degree C = 9/5 of a degree F. As a result, the conversion looks like this: Where degrees F = f and degrees C = c c = (f-32) 5/9 f = (c9/5) + 32 I don’t remember enough algebra to show you anymore, but Continue Reading Fahrenheit is sort of arbitrary, and places the freezing point of pure water at standard pressure at 32 degrees, and the boiling point at 212. That means there are 180 degrees F between boiling and freezing. Celsius is a different scale, with freezing at 0 degrees, and boiling at 100. There are exactly 100 degrees C between freezing and boiling. Degrees Celsuis are thus a little “larger”, such that one degree C = 9/5 of a degree F. As a result, the conversion looks like this: Where degrees F = f and degrees C = c c = (f-32) 5/9 f = (c9/5) + 32 I don’t remember enough algebra to show you anymore, but basically, -40 is the point where the output of either function equals the input. Edit: No, wait. I remembered. You put both equations in slope-intercept form and solve the system. y=(x-32)5/9 y=(9/5x) + 32 (x-32)5/9=(9/5x) + 32 5/9x - 160/9=(9/5x) + 32 5/9x - 9/5x = 32 + 160/9 25/45x - 81/45 x = 448/9 x=448/9 (-45/56) x=-40 Upvote · 99 27 9 3 Sponsored by Amazon Web Services (AWS) Reliability you can trust with AWS Databases. Your databases should be secure, reliable, and built for performance. Explore how to build powerful apps. Sign Up 99 13 Related questions More answers below How does -40°C = -40°f? How do I convert negative 40 degrees Celsius to Fahrenheit? What is the reason -40 Fahrenheit the same as -40 Celsius? How does the conversion work to get the same number at -40? Is -40 degrees Fahrenheit equal to -40 degrees Celsius? What temperature feels colder -40 °F or -40 °C? Avirat Dhodare Jr.Executive - Exam Services at Coempt Eduteck (2022–present) · Author has 3.4K answers and 7.2M answer views ·4y Originally Answered: How do I convert negative 40 degrees Celsius to Fahrenheit? · Formula (°C × 9/5) + 32 = °F ∴ (-40°C × 9/5) + 32 = -40°F It is the only point on the Fahrenheit and Celsius scales where the te... Upvote · 9 2 John (GrabCash.xyz) Marketing Specialist at Hewlett-Packard (company) (2022–present) · Author has 709 answers and 76.2M answer views ·1y To convert temperatures from Celsius to Fahrenheit, you can use the following formula: °F = (°C × 9/5) + 32 Let's apply this formula to the given temperature of -40°C: °F = (-40 × 9/5) + 32 °F = (-72) + 32 °F = -40 So, -40°C is equal to -40°F. Upvote · 9 6 Assistant Bot · 1y To convert Celsius to Fahrenheit, you can use the formula: °F=(°C×9 5)+32°F=(°C×9 5)+32 For -40°C: °F=(−40×9 5)+32°F=(−40×9 5)+32 Calculating it step by step: Multiply -40 by 9 5 9 5: −40×9 5=−72−40×9 5=−72 2. Add 32: −72+32=−40−72+32=−40 So, -40°C is equal to -40°F. Upvote · Related questions More answers below Is 40 degrees Celsius the same as 40 degrees Fahrenheit? Is 50 degrees Fahrenheit cold or warm? What is 35 degrees Fahrenheit equal to in Celsius? How can I convert ℃ into °F in simple steps? How many watts are required to raise one liter of water by one degree centigrade? Derek Wood Data Engineer, Solution Architect ·4y Originally Answered: How do I convert negative 40 degrees Celsius to Fahrenheit? · Trick question. They are equal at -40. That is -40 F = -40 C. Upvote · 99 13 9 2 Promoted by Grammarly Grammarly Great Writing, Simplified ·Aug 18 Which are the best AI tools for students? There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you? That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners. Here are five AI tools inside Grammarly’s document editor that are worth checking out: Do Continue Reading There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you? That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners. Here are five AI tools inside Grammarly’s document editor that are worth checking out: Docs – Your all-in-one writing surface Think of docs as your smart notebook meets your favorite editor. It’s a writing surface where you can brainstorm, draft, organize your thoughts, and edit—all in one place. It comes with a panel of smart tools to help you refine your work at every step of the writing process and even includes AI Chat to help you get started or unstuck. Expert Review – Your built-in subject expert Need to make sure your ideas land with credibility? Expert Review gives you tailored, discipline-aware feedback grounded in your field—whether you're writing about a specific topic, looking for historical context, or looking for some extra back-up on a point. It’s like having the leading expert on the topic read your paper before you submit it. AI Grader – Your predictive professor preview Curious what your instructor might think? Now, you can get a better idea before you hit send. AI Grader simulates feedback based on your rubric and course context, so you can get a realistic sense of how your paper measures up. It helps you catch weak points and revise with confidence before the official grade rolls in. Citation Finder – Your research sidekick Not sure if you’ve backed up your claims properly? Citation Finder scans your paper and identifies where you need sources—then suggests credible ones to help you tighten your argument. Think fact-checker and librarian rolled into one, working alongside your draft. Reader Reactions – Your clarity compass Writing well is one thing. Writing that resonates with the person reading it is another. Reader Reactions helps you predict how your audience (whether that’s your professor, a TA, recruiter, or classmate) will respond to your writing. With this tool, easily identify what’s clear, what might confuse your reader, and what’s most likely to be remembered. All five tools work together inside Grammarly’s document editor to help you grow your skills and get your writing across the finish line—whether you’re just starting out or fine-tuning your final draft. The best part? It’s built for school, and it’s ready when you are. Try these features and more for free at Grammarly.com and get started today! Upvote · 999 201 99 34 9 3 Larry Scholnick Senior Accountant at LS Capital (2013–present) · Author has 9.6K answers and 6.6M answer views ·Updated 1y Originally Answered: How do I convert negative 40 degrees Celsius to Fahrenheit? · The same process applies, regardless of the Celsius starting point: 1st, multiply times 1.8, 2nd, add 32. Let's try 3 starting points, 100, 0, and -40. 100°C 1.8 = 180 + 32 = 212°F. (Boiling point-water). 0°C 1.8 = 0 + 32 = 32°F. (Freezing point-water). -40°C 1.8 = -72 + 32 = -40°F. (-40 is the only matching point). Upvote · 9 3 9 2 Gopinatha Das Asst Professor at Orissa Engineering College (2009–present) ·2y Related Are -40 Celsius and -40 Fahrenheit the same temperature? The Fahrenheit scale is used primarily in the United States, while Celsius is used throughout the world. The two scales have different zero points and the Celsius degree is bigger than the Fahrenheit However, there is one point on the Fahrenheit and Celsius scales where the temperatures in degrees are equal. This is -40 °C and -40 °F. There is a simple algebraic method to find the answer °F = (°C × 9/5) + 32 °C = (°F - 32) × 5/9 It does not matter which equation you use; simply use x instead of degrees Celsius and Fahrenheit. You can solve this problem by solving for x: °C = 5/9 × (°F - 32) x = 5/9 Continue Reading The Fahrenheit scale is used primarily in the United States, while Celsius is used throughout the world. The two scales have different zero points and the Celsius degree is bigger than the Fahrenheit However, there is one point on the Fahrenheit and Celsius scales where the temperatures in degrees are equal. This is -40 °C and -40 °F. There is a simple algebraic method to find the answer °F = (°C × 9/5) + 32 °C = (°F - 32) × 5/9 It does not matter which equation you use; simply use x instead of degrees Celsius and Fahrenheit. You can solve this problem by solving for x: °C = 5/9 × (°F - 32) x = 5/9 × (x - 32) x = (5/9)x - 17.778 1x - (5/9)x = -17.778 0.444x = -17.778 x = - 40 degrees Celsius or Fahrenheit Working using the other equation, you get the same answer: °F = (°C × 9/5) + 32 °x - (°x × 9/5) = 32 -4/5 × °x = 32 °x = -32 × 5/4 x = - 40° Upvote · 9 4 Promoted by HP HP Tech Takes Tech Enthusiast \| Insights, Tips & Guidance ·Updated Sep 4 What are the all the functions of a printer? Modern printers have evolved far beyond basic document reproduction. Today’s models offer a range of built-in features that streamline both personal and professional tasks. Depending on the device, a printer can serve as a multifunctional hub for printing, scanning, copying and even faxing. Enhanced connectivity allows seamless operation from mobile devices and cloud-based platforms, while efficiency-focused options such as duplex printing and high-yield cartridges help reduce both cost and waste. For users balancing home office duties with general everyday printing, choosing the right combina Continue Reading Modern printers have evolved far beyond basic document reproduction. Today’s models offer a range of built-in features that streamline both personal and professional tasks. Depending on the device, a printer can serve as a multifunctional hub for printing, scanning, copying and even faxing. Enhanced connectivity allows seamless operation from mobile devices and cloud-based platforms, while efficiency-focused options such as duplex printing and high-yield cartridges help reduce both cost and waste. For users balancing home office duties with general everyday printing, choosing the right combination of features matters just as much as print quality itself. For those who need consistent performance and advanced functionality, the HP LaserJet Pro MFP 3302fdw offers robust print, scan and copy features in a compact all-in-one design. It delivers fast output with sharp resolution and supports duplex printing for efficient paper use. Integrated Wi-Fi and mobile compatibility make remote document handling straightforward. The control panel is intuitive and the device is built for moderate-to-heavy workloads, making it a dependable option for home offices with daily print demands. LaserJet Printers - Black & White or Color Document Printers If colour reproduction is a priority for photos, marketing materials or vivid documents, the HP Color Laser MFP 179fnw is a versatile alternative. It supports full-colour printing along with scanning, copying and faxing, and offers borderless output to create professional-grade visuals. Though it has a smaller footprint and lower print speeds compared to the 3302fdw, it remains suitable for light-to-medium use in creative or business settings. Its wireless features ensure flexible access from laptops and smartphones, enhancing convenience in dynamic work environments. Color Laser Printers - HP LaserJet Pro A printer’s value lies not only in what it prints but in how effectively it adapts to your workflow. HP’s all-in-one models like the HP LaserJet Pro MFP 3302fdw and Color Laser MFP 179fnw deliver a thoughtful mix of performance, reliability and smart features tailored to a variety of personal and professional needs. Selecting the right model comes down to the volume, content and style of your typical print tasks. I hope this clarifies the different functions of printers for you, and check out the Printer Buying Guide linked below to choose the right model for your printing needs! Inkjet vs LaserJet vs OfficeJet: HP Printer Buying Guide By Henry - HP Tech Expert Upvote · 99 15 9 2 Stanley Sartell Former Geographer/Cartographer at US ARMY TOPOCOM · Author has 178 answers and 46.8K answer views ·4y Originally Answered: How do I convert negative 40 degrees Celsius to Fahrenheit? · Minus 40 degrees Celsius is also minus 40 degrees Fahrenheit. That is only temperature where the two scales readings are the same. Upvote · 9 8 Berl Herzenberg Retired Chief Curmudgeon and Loving It. (2011–present) · Author has 53 answers and 48.9K answer views ·6y Originally Answered: How does -40°C = -40°f? · The formula for F to C conversion is: Subtract 32 from the Fahrenheit value and then multiply by 5/9 or. 5556 -40 F - 32 = - 72 -72 0. 5556 = - 40.0032 Rounding answer gives 40 C Proving that - 40C = -40F Upvote · 9 1 Sponsored by Avnet Silica We're at the Pulse of the Market. Tap into emerging tech and market insights to stay informed, reduce risk, and make better decisions. Learn More Douglas Stout 6y Originally Answered: How does -40°C = -40°f? · A standard conversion from Fahrenheit (F) to Celsius (C) is given by the equation F - 32 = 1.8 x C. If one solves the equation for F = -40, then we have: -40–32 = 1.8 x C or -72 = 1.8 x C or -72/1.8 = C or C = -40 So, -40 degrees C = -40 degrees F. Upvote · Daryl Baker, P. Eng BaSC in Chemical Engineering, University of Waterloo (Graduated 1974) · Author has 1.8K answers and 2.3M answer views ·4y Originally Answered: How do I convert negative 40 degrees Celsius to Fahrenheit? · Ah, a little homework problem. Minus 40 degrees C = -40 degree C x 1.8 + 32 degrees F = - 72+32 = -40 degrees F. Thus minus 40 C = minus 40 F, the point where the two temperature scales are equal. Upvote · 9 4 Doug White I tend to retain what I read and enjoy reading of science · Author has 12.7K answers and 19M answer views ·3y Put the cursor after the C and use the destructive backspace to remove it. Then type F. Nine-fifths of —40 = —72. Add 32 to get —40. Upvote · 9 2 Michael Smith MS in Chemistry, Eastern Washington University (Graduated 1988) · Author has 457 answers and 420.8K answer views ·Updated 3y The number of degrees from the freezing point of water to the boiling point is 180 in the Fahrenheit scale and 100 in the Celsius. In addition there is an offset of 32 degrees F so the equations to convert are F = (180 / 100) x C + 32 and C = (F - 32) x (100 / 180) or F=9/5 x C + 32 and C=(F-32)x5/9 -40 is the one point where the two scales are equal thus -40 degrees C = -40 degrees F (use the equations to prove this by solving for the common data point by replacing one unit with the other in one of the equations and then solving for that variable.) Upvote · Related questions How does -40°C = -40°f? How do I convert negative 40 degrees Celsius to Fahrenheit? What is the reason -40 Fahrenheit the same as -40 Celsius? How does the conversion work to get the same number at -40? Is -40 degrees Fahrenheit equal to -40 degrees Celsius? What temperature feels colder -40 °F or -40 °C? Is 40 degrees Celsius the same as 40 degrees Fahrenheit? Is 50 degrees Fahrenheit cold or warm? What is 35 degrees Fahrenheit equal to in Celsius? How can I convert ℃ into °F in simple steps? How many watts are required to raise one liter of water by one degree centigrade? Why is Celsius preferred over Fahrenheit? Fahrenheit units are smaller and thus more precise. Which is cooler -10° or -2°? What temperature in Fahrenheit feels cold or hot to human beings? 1 celsius is equal to how many fahrenheit? Is 2 to 30 degrees Celsius temperature warm or cold? Related questions How does -40°C = -40°f? How do I convert negative 40 degrees Celsius to Fahrenheit? What is the reason -40 Fahrenheit the same as -40 Celsius? How does the conversion work to get the same number at -40? Is -40 degrees Fahrenheit equal to -40 degrees Celsius? What temperature feels colder -40 °F or -40 °C? Is 40 degrees Celsius the same as 40 degrees Fahrenheit? 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1632	https://mathcenter.oxford.emory.edu/site/math125/powerSetCardinality/	The Cardinality of the Power Set About Statistics Number Theory Java Data Structures Cornerstones Calculus The Cardinality of the Power Set Theorem: The power set of a set S (i.e., the set of all subsets of S) always has higher cardinality than the set S, itself. Proof: Suppose we denote the power set of S S by P(S)P(S). First note that it can't possibly happen that P(S)P(S) has smaller cardinality than S S, as for every element x x of S, {x}{x} is a member of P(S)P(S). It remains to show that the cardinality of P(S)P(S) is not equal to that of S S. We argue indirectly. Suppose they were equal. Then there would be a bijection f(x)f(x) (i.e., a pairing) that takes elements of S S to elements of P(S)P(S). To think metaphorically, suppose the elements of S S are people. Then there is a one-to-one, and onto pairing between the people of S S and all possible subsets of people from S S. Perhaps one might imagine the subset related to any given person, say Fred for example, as those people (in S S) that Fred believes should earn an award this year for their good work. It is possible that Fred believes he, himself, should earn an award this year. With this in mind, let us call Fred (and others like him) "cocky". Alternatively, if a person does not believe they should earn an award this year, let us call him or her "humble". To put this mathematically, we have just defined sets C C (for "cocky") and H H (for "humble") C={x∈S\|x∈f(x)}and H={x∈S\|x∉f(x)}C={x∈S\|x∈f(x)}and H={x∈S\|x∉f(x)} Now, suppose we consider H H, the set of all humble people in S S. This is, of course, a subset of S S, and given our assumption of a pairing (coming from the bijection) between the elements of S S and P(S)P(S), there must be some person, say Abby, in S S that believes only the humble people in H H deserve an award this year. In other words, given that f(x)f(x) is a bijection, we can find some a∈S a∈S such that f(a)=H f(a)=H. Certainly, Abby must be either humble or cocky, but not both. However, If Abby is humble, she must be in H H. But then she believes she deserves an award this year, which makes her cocky. If Abby is cocky, she can't be in H H. But then she doesn't believe she deserves an award this year, which makes her humble. More precisely, a∈H a∈H implies a∉f(a)a∉f(a) which implies a∉H a∉H, while a∉H a∉H implies a∉f(a)a∉f(a), which implies a∈H a∈H. Either way, we get a contradiction. The only assumption we made was the existence of a bijection, so such a bijection must not exist. Hence, the cardinality of the power set of S S is necessarily larger than that of S S, itself.
1633	https://www.youtube.com/watch?v=H5qap5Ktrlk	Thumb Opposition Baptist Health 28400 subscribers Description 22665 views Posted: 22 Jul 2020 - Thumb opposition. To perform this exercise, begin with your affected hand in a neutral position. Gently touch the tip of the thumb to the tip of each finger in succession from the index finger to the little finger. Perform 1 set of 10 repetitions twice a day, five days a week. Transcript: Thumb opposition. To perform this exercise, begin with your affected hand in a neutral position. Gently touch the tip of the thumb to the tip of each finger in succession from the index finger to the little finger. Perform 1 set of 10 repetitions twice a day, five days a week.
1634	https://math.stackexchange.com/questions/804089/formula-to-find-the-angle-between-two-slopes	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Formula to find the Angle between two slopes Ask Question Asked Modified 7 years ago Viewed 54k times 6 $\begingroup$ I have given two slopes $m_1 = \frac{1}{2}$ and $m_2 = 1$ While finding the angle I made use of the formula $\tan(\theta) = \frac{m_1-m_2}{1+m_1m_2}$ answer is : $\theta = \arctan(\frac{-1}{3})$ But in book the answer is $\theta = \arctan(\frac{1}{3})$. what would be the right answer? geometry Share edited May 21, 2014 at 13:47 Peter Woolfitt 21.5k66 gold badges5959 silver badges9191 bronze badges asked May 21, 2014 at 13:38 zonniezonnie 14511 gold badge33 silver badges99 bronze badges $\endgroup$ 5 $\begingroup$ Note that there are two angles between the lines, an acute one and an obtuse one. Using $\tan \theta = \left\|\dfrac{ m_1 - m_2}{1+m_1m_2} \right\|$ gives you the acute angle. $\endgroup$ Ragib Zaman – Ragib Zaman 2014-05-21 13:43:00 +00:00 Commented May 21, 2014 at 13:43 3 $\begingroup$ It depends on which angle you are looking at. It's a simple matter of "angle of slope 1 to slope 2" versus "angle of slope 2 to slope 1". If you look at it graphically, you probably want a positive angle. $\endgroup$ orion – orion 2014-05-21 13:43:20 +00:00 Commented May 21, 2014 at 13:43 $\begingroup$ So in above question what would be the right answer..? as i found these slopes from two curves. $\endgroup$ zonnie – zonnie 2014-05-21 13:46:54 +00:00 Commented May 21, 2014 at 13:46 $\begingroup$ Look at the two lines (tangent lines) and the point where they intersect. You want an acute angle measure -- going counterclockwise around the point of intersection, which line do you get to first before the acute angle? That slope needs to be $m_1$. If you go clockwise, you get the other angle measure, still "acute" but negative measure because clockwise, or equivalently (as far as arctan can tell) the obtuse angle counterclockwise. $\endgroup$ user128390 – user128390 2014-05-21 20:29:40 +00:00 Commented May 21, 2014 at 20:29 $\begingroup$ Its not clear yet.. Can you guide me. If the point of intersection of two curves is (1,1) and I have found the slope m1 = 1/2 and m2 = 1. Now what would be the angle? $\endgroup$ zonnie – zonnie 2014-05-22 14:06:55 +00:00 Commented May 22, 2014 at 14:06 Add a comment \| 4 Answers 4 Reset to default 1 $\begingroup$ $$ \frac{1-\frac12}{1+1\cdot\frac 12} = \frac 1 3, \qquad \frac{\frac12-1}{1+\frac12\cdot1} =-\frac13. $$ Either of these is the angle between those two lines. Where two lines intersect they form angles with measures adding up to $180^\circ$. If $\alpha+\beta=180^\circ$ then $\tan\alpha = - \tan\beta$. Share answered Dec 3, 2015 at 19:18 Michael HardyMichael Hardy 1 $\endgroup$ Add a comment \| 1 $\begingroup$ Correct formula with sign convention positive for $\theta$ counter-clockwise rotation from direction of radius vector $1$ to $2$ is $$\tan(\theta) = \dfrac{m_2-m_1}{1+m_1m_2}.$$ Share answered Dec 3, 2015 at 19:26 NarasimhamNarasimham 42.5k77 gold badges4646 silver badges112112 bronze badges $\endgroup$ Add a comment \| 1 $\begingroup$ You might not observed the modules in the formula. They remove the negative signs. So the answer you get \| -1/3 \| becomes +1/3. So the answer in the book is correct. Share answered Apr 9, 2017 at 12:17 Narendra ReddyNarendra Reddy 1111 bronze badge $\endgroup$ 0 Add a comment \| 1 $\begingroup$ The solution given in the book is correct. The actual formula says, one needs to take the absolute value of (m1-m2)/(1+m1.m2), before you perform a tan inverse (arctan), in order to evaluate the final value of angle between two line segments. Magnitude OR Absolute value OR Modulus of any number is always positive (it is without regard to it's sign, negative or positive). So, it ALWAYS will be arctan of some positive number. Share edited Sep 26, 2018 at 19:12 answered Sep 26, 2018 at 19:05 Shruti MadanShruti Madan 3633 bronze badges $\endgroup$ Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions geometry See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Related Angle between two straight lines (tangent formula) Prove that these two formulas to calculate the angle between two 2D lines are equivalent 2 Slopes and lines 0 The acute angle formed by intersecting lines is equal to the acute angle formed by their normals 3 How to determine whether a triangle is obtuse angled or not from the equation of its sides? 1 Given a triangle and equation of all its sides, how do I understand if an internal angle is obtuse or acute? 2 Find the angle between two tangents drawn from point $(0, -2)$ to the curve $y=x^2$ 1 Finding the angle between two lines meeting at one point. 2 Understanding $\tan\theta=\left\|\frac{m_1-m_2}{1+m_1m_2}\right\|$ Hot Network Questions Verify a Chinese ID Number How to locate a leak in an irrigation system? 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1635	https://www.sketchy.com/mcat-lessons/doppler-effect	Opens in a new window Opens an external website Opens an external website in a new window We use cookies to provide necessary functionality and improve your experience. By remaining on this website you indicate your consent. Cookie Policy Try for FreeLogin GET 20% OFF SKETCHY MCAT WITH CODE REG20 \| REGISTRATION DAY SALE MCAT Curriculum / Physics / Sound / Doppler Effect Doppler Effect Tags: No items found. Physics The Doppler Effect is a phenomenon that occurs when there is relative movement between a source of waves and an observer, causing a change in the perceived frequency of the waves. This effect applies to all types of waves, including sound and light. In terms of sound, the frequency of a sound wave determines its pitch. Due to the Doppler Effect, the compressions of a moving wave source reach an observer at different frequencies than when the source is stationary. As a result, the perceived sound has a higher pitch when the source moves towards the observer and a lower pitch when it moves away. The mathematical equation for the Doppler Effect, represented by the perceived frequency (f'), is f' = f (V ± VD) / (V ∓ VS). In this equation, f is the actual frequency, V is the speed of sound in the medium, VD is the velocity of the detector (observer), and VS is the velocity of the source. When an object exceeds the speed of sound in a medium (Mach 1), it creates a shock wave or sonic boom, which can be heard by an observer after the object has passed by. Lesson Outline Introduction to the Doppler Effect Examples: sound of sirens, whistle of dropped object from a plane Applies to all waves, including light Understanding the Doppler Effect with sound Frequency determines pitch Movement introduces Doppler Effect Observed sound changes due to movement Variance between perceived frequency and actual frequency due to movement is the Doppler Effect Doppler Effect with light waves Blue shift: increase in perceived frequency of light waves Red shift: decrease in perceived frequency of light waves Calculating perceived frequency (f') Equation: f' = f (V + VD) / (V - VS) f' = perceived frequency f = actual frequency V = speed of sound in the medium VD = velocity of the detector (observer) VS = velocity of the source Shock waves Wave fronts buildup as object's speed approaches speed of sound (Mach 1) Shockwave, sonic boom, or supersonic boom occurs when object passes observer at or above Mach 1 Don't stop here! Get access to 29 more Physics lessons & 8 more full MCAT courses with one subscription! Try 7 Days Free FAQs What is the Doppler Effect and how does it relate to sound waves and frequency? The Doppler Effect is a change in frequency and wavelength of a wave in relation to an observer who is moving relative to the wave source. In the case of sound waves, a listener perceives a change in pitch (higher or lower), which is a result of the change in frequency of the sound waves. The perceived frequency increases when the source of the sound waves is moving toward the observer and decreases when the source is moving away from the observer. How does the Doppler Effect apply to light waves, and what are blue shift and redshift? The Doppler Effect also applies to light waves, where the observed frequency and wavelength are affected by the relative motion between the observer and the source of light. Blue shift occurs when the light source is moving toward the observer, causing the observed frequency of light to increase and the wavelength to decrease, thus appearing more blue. Redshift occurs when the light source is moving away from the observer, causing the observed frequency to decrease and the wavelength to increase, making the light appear more red. Both blue shift and redshift are used in astronomy to study the movement of celestial bodies. How can the Doppler Effect be used to measure the speed of sound? The Doppler Effect can help in measuring the speed of sound by observing the change in the frequency of a sound wave produced by a known moving source with a known frequency. When the source of sound is in motion relative to the observer, the observed frequency will differ from the emitted frequency. By knowing the speed and direction of the moving source and the change in the perceived frequency, the speed of sound in the medium can be calculated using a set of equations known as the Doppler Shift Equations. What is a sonic boom and how is it related to the Doppler Effect? A sonic boom is a loud noise created when an object, usually an aircraft, travels through the air faster than the speed of sound. In this situation, the sound waves emitted by the object are compressed together due to the object's fast movement, causing a shock wave. This shock wave is experienced by an observer on the ground as a sudden increase in sound pressure, resulting in the characteristic loud "boom" sound. The Doppler Effect plays a role in this phenomenon, as the sudden change in the perceived frequency of the sound waves creates the startling noise of a sonic boom. Programs MedicalMCATPANP CompanyBlogCareersPrivacy PolicyTerms of UseCookie Policy
1636	https://www.michaelperess.com/teaching/MLE.pdf	Lecture Notes on Maximum Likelihood Estimation Michael Peress December 30, 2024 Department of Political Science, SUNY-Stony Brook. michael.peress@stonybrook.edu Contents 1 Overview and Logit 5 1.1 What Makes a Model Nonlinear? . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Types of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Nonlinear Model Complications . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Logit 8 2.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Marginal Eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Measures of Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 When is Using a Logit Model Necessary . . . . . . . . . . . . . . . . . . . . 16 2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Numerical Optimization 18 3.1 The Need for Numerical Optimization . . . . . . . . . . . . . . . . . . . . . . 18 3.2 One-Dimensional Root-Finding . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 One-Dimensional Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Multi-Dimensional Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.5 Checking for an Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.6 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Theory of MLE 25 4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 Identiﬁcation of a Statistical Model . . . . . . . . . . . . . . . . . . . . . . . 31 4.4 Asymptotic Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.5 Eﬃciency and the Cramer Rao Lower Bound . . . . . . . . . . . . . . . . . . 39 4.6 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.6.2 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1 5 Probit, More on Logit, and Ordered Probit 44 5.1 The Probit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.2 Perfect Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Testing Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.4 Interactions in Binary Choice Models . . . . . . . . . . . . . . . . . . . . . . 46 5.5 Ordered Probit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.6 Ordered Logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.8 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 Multinomial Choice 61 6.1 Multinomial Logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2 Conditional Logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.3 The IIA Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.4 Multinomial Probit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.5 Substantive Eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.6 Measures of Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.7 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7 Count Models 69 7.1 Poisson Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.2 Negative Binomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.3 Zero-Inﬂated Poisson Regression . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.4 Semi-parametric Analysis of the Count Regression Model . . . . . . . . . . . 73 7.5 Model Fit for Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . 75 7.6 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 8 Censoring, Selection, and Truncation 76 8.1 The Tobit Model (Censored Regression) . . . . . . . . . . . . . . . . . . . . 76 2 8.2 Truncated Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.3 The Heckman Selection Model . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 8.5 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 9 Duration Models 87 9.1 Survival Functions and Hazard Rates . . . . . . . . . . . . . . . . . . . . . . 88 9.2 Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 9.3 The Cox Proportional Hazard Model . . . . . . . . . . . . . . . . . . . . . . 90 9.4 Discrete Duration Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9.5 Time Varying Covariates in Continuous Models . . . . . . . . . . . . . . . . 91 9.6 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9.6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 10 Monte Carlo Simulation and the Bootstrap 93 10.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 10.2 The Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 10.3 Recommended Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 10.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 10.3.2 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 11 Nonlinear Panel Data 95 11.1 Clustered Standard Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 11.2 Nonlinear Fixed Eﬀects Models . . . . . . . . . . . . . . . . . . . . . . . . . 97 11.3 Conditional Fixed Eﬀects Estimators . . . . . . . . . . . . . . . . . . . . . . 98 11.4 Random Eﬀects Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 11.5 Generalized Estimating Equations . . . . . . . . . . . . . . . . . . . . . . . . 100 11.6 Recommended Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 11.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 11.6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 11.6.3 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3 12 Time Series Dependence in Nonlinear Models 106 12.1 Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 12.2 Semiparametric Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 12.3 Recommended Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 12.3.1 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 13 References 107 A Appendix 111 A.1 Proof that Asymptotic Normality Implies Asymptotic Unbiasedness . . . . . 111 A.2 Derivation of the Expected Value of yn in the Heckman Selection Model . . . 112 4 1 Overview and Logit 1.1 What Makes a Model Nonlinear? Consider the classical linear regression model, yn = β′ 0xn + εn (1) with E[εn\|xn] = 0. Here, we have E[yn\|xn] = β′ 0xn, or that the expectation of the dependent variable is linear in the independent variables. Consider alternatively the logistic regression model, Pr(yn = 1\|xn) = eβ′ 0xn 1 + eβ′ 0xn (2) We have that, E[yn\|xn] = Pr(yn = 1\|xn) = eβ′ 0xn 1 + eβ′ 0xn (3) so that the expectation of the dependent variable is not linear in the independent variables. 1.2 Types of Data An important motivation for considering nonlinear models involves dealing with special types of data. In the classical linear model, the regressor is continuous, having an interval scale, ranging from negative inﬁnity to inﬁnity. The linear regression framework can handle devi-ations from this—it can often be applied as long as the expectation of the DV conditional on the IV is linear in the IV, but this approach has some limitations. We can consider the following alternative forms of dependent variables: Logit and Probit: The dependent variable is binary Ordered Probit: The dependent variable is ordered, but does not have an interval level scale Multinomial Logit, Multinomial Probit, and Conditional Logit: The dependent vari-able is discrete and unordered Tobit: The dependent variable is censored at 0 (or possibly other values). The depen-dent variable in this case is neither discrete nor continuous—it is “mixed” 5 Poisson and Negative Binomial Regression: The dependent variable is discrete and weakly positive (i.e. count data) For each of these types of variables, we cannot avoid nonlinearity of the expectation. To see this, consider the case of a binary dependent variable. If E[yn\|xn] = Pr(yn = 1\|xn) = β′ 0xn, then it must be the case that for xn such that β′ 0xn is small (large), Pr(yn = 1\|xn) must dip below zero (above one) and is thus not a proper probability model unless the range of xn is restricted. For each of the above models, we specify the probability distribution of the dependent variable conditional on the independent variables. When the dependent variable is discrete, we specify the probability mass function of the DV given the IVs. When the dependent variable is continuous, we specify the density of the DV given the IVs. The case where the dependent variable is mixed is somewhat more complicated to deal with, but we can characterize the dependent variable using the cumulative distribution function. We write the probability model of the as follows: pY \|X(y\|x; θ) (y is discrete) fY \|X(y\|x; θ) (y is continuous) FY \|X(y\|x; θ) (y is mixed) In most cases, we will assume that (yn, xn) are independent, in which case we have, pY \|X(y\|x; θ) = QN n=1 pYn\|Xn(yn\|xn; θ) (y is discrete) fY \|X(y\|x; θ) = QN n=1 fYn\|Xn(yn\|xn; θ) (y is continuous) FY \|X(y\|x; θ) = QN n=1 FYn\|Xn(yn\|xn; θ) (y is mixed) 1.3 Maximum Likelihood Estimation Given the types of models described above, maximum likelihood estimation is a procedure for deriving an estimator from a probability model. The MLE is given by,1 ˆ θMLE = arg max θ∈Θ pY \|X(y\|x; θ) (y is discrete) 1The deﬁnition of the MLE for mixed random variables is more complicated, so we will deal with it when we consider the Tobit model. 6 ˆ θMLE = arg max θ∈Θ fY \|X(y\|x; θ) (y is continuous) Here, L(θ) = pY \|X(y\|x; θ) for the discrete case and L(θ) = fY \|X(y\|x; θ) for the continuous case is called the “likelihood”, which is the probability of observing the data in the discrete case and the density for observing the data in the continuous case. The MLE selects the parameters that make it most likely that we would observe the data we actually observed. When (yn, xn) are independent, we have, L(θ) = QN n=1 pYn\|Xn(yn\|xn; θ) (y is discrete) L(θ) = QN n=1 fYn\|Xn(yn\|xn; θ) (y is continuous) We deﬁne the log-likelihood l(θ) as the log of the likelihood, l(θ) = PN n=1 log pYn\|Xn(yn\|xn; θ) (y is discrete) l(θ) = PN n=1 log fYn\|Xn(yn\|xn; θ) (y is continuous) Maximizing the log-likelihood is equivalent to maximizing the likelihood (since log is a mono-tonically increasing transformation) and it is often more convenient to maximize the log likelihood because the derivatives of the log-likelihood are easier to compute (the product in the likelihood means we would have to apply the product rule N times) and because the product in the deﬁnition of the likelihood can often lead to having to work with very small numbers. Under general conditions, the MLE is guaranteed to have several “nice” properties. The MLE is not the only such procedure—Method of Moments, Generalized Method of Moments, and Bayesian estimation are alternative procedures for automatically generating an estimator for a statistical model that is guaranteed to have nice properties. Under speciﬁed conditions, MLEs can be shown to have the following nice properties: 1. Invariance 2. Consistency 3. Asymptotic Normality 4. Asymptotic Eﬃciency 5. The Information Equality 7 We can compare this to the list of nice properties of the ordinary least squares estimator for linear regression: 1. Unbiasedness 2. Known ﬁnite sample distribution (normal, chi-squared, t, etc.) 3. Gauss-Markov Theorem 4. Rao-Blackwell Theorem 5. Consistency 6. Asymptotic Normality 7. Eﬃciency Besides invariance (which is a very weak property), the nice properties of MLEs only hold in large samples. The ordinary least squares estimator has nice properties that hold in ﬁnite samples. 1.4 Nonlinear Model Complications Nonlinear models present a number of complications. In general, there will not be an ana-lytical solution to the maximization problem that deﬁnes the MLE.2 The lack of analytical expressions for the estimator means we cannot use simple math to derive the properties of MLEs. Marginal eﬀects are also more complicated to calculate in nonlinear models. 2 Logit Consider a binary dependent variable yn ∈{0, 1}. We can specify a probability model for y\|x by assuming that (yn, xn) are independent and specifying Pr(yn = 1\|xn)—this would complete the model since Pr(yn = 0\|xn) = 1 −Pr(yn = 1\|xn). We would like the model to have the property that 0 ≤Pr(yn = 1\|xn) ≤1 for all xn and for the model to depend on xn only through β′xn. There are many possibilities—one possibility is, 2We have already encountered this problem for some estimators of the linear model such as the MLEs for the ARMA model and the linear model with heteroskedastic errors. 8 Pr(yn = 1\|xn) = eβ′ 0xn 1 + eβ′ 0xn (4) Note that ez 1+ez is monotonically increasing in z with lim z→−∞ ez 1+ez = 0 and lim z→∞ ez 1+ez = 1. One way to motivate the logistic regression model is as a model where a latent variable is linear. We have, y∗ n = β′xn + εn (5) where yn = 1{y∗ n ≥0}. The logistic distribution has density function, f(z; µ, s) = e−z−µ s s(1 + e−z−µ s )2 (6) and cumulative distribution function, F(z; µ, s) = 1 1 + e−z−µ s (7) where µ is a location parameter and s is a scale parameter. The mean, median, and mode of the logistic distribution are all µ and the variance is s2 π2 3 . If we take εn to have the Logistic(0, 1) distribution, we have that Fεn(z) = 1 1+e−z . Consequently, Pr(yn = 1\|xn) = Pr(y∗ n ≥0\|xn) = Pr(β′xn + εn ≥0\|xn) = Pr(εn ≥−β′xn\|xn) (8) = 1−Pr(εn ≤−β′xn\|xn) = 1−Fεn(−β′xn) = 1− 1 1 + eβ′xn = 1 + eβ′xn 1 + eβ′xn − 1 1 + eβ′xn = eβ′xn 1 + eβ′xn Alternatively, selecting εn ∼N(0, 1) yields the (binomial) probit model, Pr(yn = 1\|xn) = Φ(β′ 0xn) (9) Other choices of εn will lead to alternative models, but these are not common. In many cases, we can think of yn as a censored random variable. For example, let y∗ n be the two-party vote share in excess of 50% of the incumbent candidate for governor. We could imagine a variable yn = 1{y∗ n ≥0}, in which case yn = 1 if the incumbent governor is 9 reelected. Alternatively, we could let y∗ n denote the preference for voting for a Republican candidate. In this case, yn = 1 could denote a survey respondent indicating a preference for the Republican candidate. The variable y∗ n is not observable, but we can still think of elections as censoring a latent preference between the two parties. For almost any binary DV, we can think of a theoretically continuous DV that is censored—for civil war, we can imagine a continuum under which certain events or characteristics of countries make civil wars more likely. When a threshold is reached, a civil war is observed. 2.1 Estimation To estimate the logit model, we start with the deﬁnition of the MLE, ˆ βMLE = arg max β∈B l(β) = N X n=1 log pYn\|Xn(yn\|xn; β) (10) = arg max β∈B N X n=1 log Pr(yn = 1\|xn)yn Pr(yn = 0\|xn)1−yn = arg max β∈B N X n=1 yn log Pr(yn = 1\|xn) + (1 −yn) log Pr(yn = 0\|xn) = arg max β∈B N X n=1 yn log eβ′xn 1 + eβ′xn + (1 −yn) log 1 1 + eβ′xn It is tempting to solve this problem by taking ﬁrst-order conditions, but this approach does not lead anywhere good. Instead, we use numerical optimization to maximize l(β). To compute standard errors for ˆ βMLE, we rely on the following result, √ N(ˆ βMLE −β0) dist. − →N(0, V0) (11) where, V0 = E[− ∂2 ∂β∂β′ log pYn\|Xn(yn\|xn; β0)]−1 (12) To estimate V0, we use ˆ V = (−1 N lββ(ˆ βMLE))−1, where lββ is the matrix of second derivatives of the log-likelihood function evaluated at the MLE estimate ˆ βMLE. To calculate lββ, we again typically would use numerical methods rather than computing it by hand. 10 2.2 Marginal Eﬀects Consider the linear probability model, Pr(yn = 1\|xn) = β′xn. In this case, marginal eﬀects are very easy to calculate, ∂ ∂xnk Pr(yn = 1\|xn) = βk (13) The marginal eﬀect of an IV on the DV is given by a single number, which is just the estimated coeﬃcient. For the logit model, things are more complicated, ∂ ∂xnk eβ′xn 1 + eβ′xn = (1 + eβ′xn)eβ′xnβk −eβ′xneβ′xnβk (1 + eβ′xn)2 = eβ′xn (1 + eβ′xn)2βk = πn(1 −πn)βk (14) where πn = eβ′xn 1+eβ′xn . Since 0 < πn < 1, the sign of the marginal eﬀect is the same as the sign of the estimated coeﬃcient, but the actual marginal eﬀect depends on the entire vector xn. If we start with a value of β′xn such that πn = eβ′xn 1+eβ′xn is far from 1 2, the marginal eﬀect will be smaller. We can compute the marginal eﬀect for a given individual using ˆ πn(1 −ˆ πn)ˆ βk where ˆ πn = e ˆ β′xn 1+e ˆ β′xn , but then we have a separate marginal eﬀect from each point in our sample, or a marginal eﬀect for each hypothetical value of xn. Two alternative approaches are used. In the ﬁrst approach, we compute the marginal eﬀect for the mean value of xn in the sample, ¯ x. This marginal eﬀect is given by, d ME1 = e ˆ β′¯ x (1 + eˆ β′¯ x)2 ˆ βk (15) This can be interpreted as the marginal eﬀect for an individual whose independent variables are the average in the sample. An alternative is to compute the marginal eﬀect for each unit in the sample and report the average, d ME2 = 1 N N X n=1 e ˆ β′xn (1 + eˆ β′xn)2 ˆ βk (16) This can be interpreted as the average marginal eﬀect in the sample. Instead of computing the marginal eﬀect, we can compute the change in the predicted probability for a speciﬁc change in one of the independent variables. We could consider, 11 e ˆ β′ −kxn,−k+ˆ βkδ1 1 + e ˆ β′ −kxn,−k+ˆ βkδ1 − e ˆ β′ −kxn,−k+ˆ βkδ0 1 + e ˆ β′ −kxn,−k+ˆ βkδ0 (17) Again, this would take on a diﬀerent value for each individual in our sample. We could report the average value in the sample, 1 N N X n=1 e ˆ β′ −kxn,−k+ˆ βkδ1 1 + e ˆ β′ −kxn,−k+ˆ βkδ1 − e ˆ β′ −kxn,−k+ˆ βkδ0 1 + e ˆ β′ −kxn,−k+ˆ βkδ0 (18) We could think of xnk as representing the growth rate and yn as indicating whether a survey respondent votes for the incumbent candidate. If we set δ1 = 3 and δ0 = 1, the quantity above would indicate the increase in the probability of voting for incumbent candidates when growth increases from 1 percent to 3 percent. We could get a diﬀerent answer if we selected δ1 = 4 and δ0 = 2. To resolve this indeterminacy, it is common to consider a change between one standard deviation below the mean and one standard deviation above the mean: d CP 1 = e ˆ β′¯ x+ˆ βksxk 1 + e ˆ β′¯ x+ˆ βksxk − e ˆ β′¯ x−ˆ βksxk 1 + e ˆ β′¯ x−ˆ βksxk (19) Alternatively, we can consider the average magnitude of change in an individuals probability of a success when every individual’s covariate k is changed by two standard deviations: d CP 2 = 1 N N X n=1 e ˆ β′xn+ˆ βksxk 1 + e ˆ β′xn+ˆ βksxk − e ˆ β′xn−ˆ βksxk 1 + e ˆ β′xn−ˆ βksxk (20) For each type of marginal eﬀect, we may also wish to compute a measure of uncertainty. The delta method provides one way of doing this. Each marginal eﬀect can be expressed as a function of the parameter β, c(β). In each case above, c(β) is a diﬀerentiable function. In this case, the delta method tells us that, √ N(c(ˆ β) −c(β0)) dist. − →cβ(β0) √ N(ˆ β −β0) (21) where cβ is the matrix of derivatives of the vector-valued function c(β). Using the fact that √ N(ˆ β −β0) dist. − →N(0, V0), we have that, √ N(c(ˆ β) −c(β0)) dist. − →N(0, ˆ cβ ˆ V ˆ c′ β) (22) where ˆ cβ = cβ(ˆ β) and ˆ V is an estimate of the asymptotic covariance matrix of ˆ β. Consider ﬁrst d ME1 where, 12 c(β) = eβ′¯ x (1 + eβ′¯ x)2βk (23) In principle, we could take the derivatives by hand to obtain: [cβ(β)]j =    (1+eβ′ ¯ x)(1+¯ xkβk)−2eβ′ ¯ xβ2 k (1+eβ′ ¯ x)3 eβ′¯ x, j = k (1−eβ′ ¯ x)eβ′ ¯ x (1+eβ′ ¯ x)3 βk¯ xj, j ̸= k (24) Even this very simple example is very tedious. A simpler approach is to apply numerical diﬀerentiation. Speciﬁcally, recall the deﬁnition of the derivative, f ′(x) = lim h→∞ f(x + h) −f(x) h (25) If we select a “small” value of h (for example, h = 0.000001), we can approximate the derivative using, f ′(x) ≈f(x + h) −f(x) h (26) To compute ˆ cβ, we could use, ˆ cβ,j ≈c(ˆ β + ejh) −c(ˆ β) h (27) where ej is a unit vector. Next, we mention one simple shortcut for interpreting the coeﬃcients. Since the marginal eﬀect is ˆ πn(1−ˆ πn)ˆ βk and 0 < ˆ πn(1−ˆ πn) < 1 4, the largest possible marginal eﬀect is 1 4 ˆ βk. We can therefore obtain some idea of the marginal eﬀect just by dividing the coeﬃcient by 4, though this can be seriously misleading in certain circumstances. If most of the observations have ˆ πn close to 0.5, this will give us a sense of the average marginal eﬀect, but having most of the data with ˆ πn close to 0.5 corresponds to a model with little predictive power. If many observations have ˆ πn close to zero or one, the average marginal eﬀect will be much smaller. One more alternative approach for interpreting logistic regression estimates is given by the odds ratio. The odds ratio for xnk = a vs. xnk = b is given by, 13 Pr(yn=1\|xnk=a,xn,−k) Pr(yn=0\|xnk=a,xn,−k) Pr(yn=1\|xnk=b,xn,−k) Pr(yn=0\|xnk=b,xn,−k) = eβ′ −kx−k+βka 1+e β′ −kx−k+βka 1 1+e β′ −kx−k+βka eβ′ −kx−k+βkb 1+e β′ −kx−k+βkb 1 1+e β′ −kx−k+βkb = eβ′ −kx−k+βka eβ′ −kx−k+βkb = eβk(a−b) (28) Suppose that yn is voter turnout (0 or 1) and xn is whether someone is assigned to a GOTV call (a = 1 and b = 0). Then eβk is an estimate of the ratio of the odds of voting to the odds of nonvoting. If 75% of people vote when receiving no GOTV call, the odds of voting is 3 to 1. If when receiving a phone call, 80% of people vote, the odds of voting are 4 to 1. In this case, the odds ratio would be 4/3=1.333, indicating a 33.33% increase in the odds of voting from receiving a GOTV call. The odds ratio would typically be used for a dummy IV (in which case a = 1 and b = 0). More generally, the odds ratio would indicate the increase in the odds of observing a success when changing an IV from a to b. Odds ratios can be viewed as more interpretable than lo-gistic regression coeﬃcients, however, predicted probabilities are arguably more interpretable than odds ratios. The example above suggests why odds ratios are sometimes used—they make eﬀects seem bigger. A 33.33% increase in the odds of voting sounds bigger than a 5 percentage point increase in voter turnout. In some cases, odds ratios may be reported instead of regression coeﬃcients. In this case, to determine whether a variable has a positive eﬀect, it must be compared to 1 rather than 0. We can compute standard errors for odds ratios using the delta method. Speciﬁcally, since c(β) = eβk, cβ(β) = eβk in which case we can compute the standard error of the odds ratio e ˆ βk using e ˆ βkse(ˆ βk) where se(ˆ βk) is the standard error for the logistic regression coeﬃcient. We can also reverse this process to get coeﬃcient estimates and their standard errors from odds ratios and their standard errors using ˆ βk = log d ORk and se(ˆ βk) = se(d ORk) d ORk . 2.3 Measures of Model Fit A number of diﬀerent measures of model ﬁt are available for the logistic regression model. The basic is the percent correctly predicted, PCP = 1 N N X n=1 yn1{ˆ β′xn > 0} + (1 −yn)1{ˆ β′xn < 0} (29) Alternatively, one might desire a measure more closely related to the R-squared from the 14 linear regression model. There are many diﬀerent variants of the “pseudo R-squared” for the logistic regression model, but below two are presented. The McKelvey and Zavoina pseudo R-squared starts with the latent variable formulation, y∗ n = β′ 0xn + εn, which is very closely related to the linear regression model, yn = β′ 0xn + εn. In the linear regression model, we could calculate, R2 = ESS TSS = PN n=1(ˆ yn −¯ y)2 PN n=1(yn −¯ y)2 = 1 N−1 PN n=1(ˆ yn −¯ y)2 1 N−1 PN n=1(yn −¯ y)2 ≈V ar(ˆ yn) V ar(yn) (30) = V ar(ˆ β′xn) V ar(β′ 0xn + εn) = V ar(ˆ β′xn) V ar(β′ 0xn) + V ar(εn) = ˆ β′V ar(xn)ˆ β β′ 0V ar(xn)β0 + V ar(εn) ≈ ˆ β′Sx ˆ β ˆ β′Sx ˆ β + V ar(εn) (31) where Sx = 1 N−1 PN n=1(xn−¯ x)(xn−¯ x)′ is the sample covariance matrix. For the Logistic(0, 1) distribution, we have V ar(εn) = π2 3 . Analogizing to the linear regression model, we could select, pseudo −R2 ≈ ˆ β′Sx ˆ β ˆ β′Sx ˆ β + π2 3 (32) Note that if the model has no explanatory power (ˆ β = 0), pseudo −R2 = 0. If ˆ β′Sx ˆ β is very large relative to the error variance, pseudo −R2 will be very close to 1. This particular measure has the advantage that it is, apart from sampling error, the R-squared one would obtain if the latent variable were not censored. This version of the R-squared can be applied to other models that can be interpreted as censored versions of the linear regression model, such as probit, ordered logit, ordered probit, and tobit models. An alternative measure is McFadden’s R-squared, which is deﬁned by, pseudo −R2 = 1 −l l0 (33) where l is the log-likelihood and l0 is the log-likelihood from a model that only includes an intercept. The idea is that a model with only an intercept term does not explain any variance around the mean and will have likelihood L0 ∈(0, 1]. A perfect model will have a likelihood of L = 1. All models will have L0 ≤L ≤1 or log L0 ≤log L ≤0 or l0 ≤l ≤0. The worst case is, pseudo −R2 = 1 −l0 l0 = 0 while the best case if pseudo −R2 = 1 −0 l0 = 1. All in 15 between models will have 0 < pseudo −R2 < 1. This version of the R-squared be applied to other models of discrete dependent variables such as probit, ordered logit, ordered probit, multinomial logit, multinomial probit, and poisson regression models. In addition, AIC and BIC are alternative measures of model ﬁt which we can use to compare between models. We have, AIC = −2l + 2k (34) BIC = −2l + k log n (35) where we select models where the AIC an BIC are small. These measures have the disadvan-tage that their magnitudes are not interpretable, but have the advantage that they penalize models for having many parameters that don’t contribute to improved model ﬁt. 2.4 When is Using a Logit Model Necessary If the goal is simply to identify the signs of coeﬃcients, than the logit model will rarely produce estimates that diﬀer substantially from the linear probability model. If the goal is to report average marginal eﬀects, again the logit model will rarely produce estimates that diﬀer substantially from the linear probability model. The logit model should be used when the goal to obtain predictions for individual observations because the predictions from the linear probability model can fall outside the zero-one interval. In addition, if the goal is to consider more speciﬁc or complicated counterfactuals, the logit model should be used. There is rarely any drawback to estimating a logit model though—while the logit model maybe slightly more computationally complex, it would be very rare to ﬁnd a dataset large enough for this to be an issue. The logit model makes instrumental variable estimation more diﬃcult. The logit model also makes it more diﬃcult to include ﬁxed eﬀects, since ﬁxed eﬀects logit will require the number of observations per ﬁxed eﬀect to go to inﬁnity for consistency (while the linear probability model with ﬁxed eﬀects will only requires the total number of observations to go to inﬁnity for consistent estimation of the main eﬀects). This issue will be considered in 11. 2.5 Applications Application 2.1 (Individual Level Model of the Economic Vote). 16 In this application, we consider an individual level model of the economic vote. The data come from the Comparative Study of Electoral Systems, modules 1 and 2, which merges survey data from multiple elections in multiple countries. The dependent variable is equal to 1 if the individual voted for the prime minister’s party and 0 if the individual voted for someone else. Table 1 presents the estimates of the model, along with odds ratios and four measures of substantive eﬀect sizes. Table 2 compares a basic model (with only growth included as an explanatory variable) with a model that includes additional controls. Model Odds Ratio ME1 ME2 CP1 CP2 Independent Variables: Constant -0.231 (0.085) Distance -0.361 0.697ˆ -0.072 -0.070 -0.066 -0.065 (0.008) (0.005) (0.002) (0.001) (0.001) (0.001) Education -0.111 0.895ˆ -0.022 -0.022 -0.022 -0.021 (0.007) (0.007) (0.001) (0.001) (0.001) (0.001) Age 0.006 1.006ˆ 0.001 0.001 0.001 0.001 (0.001) (0.001) (0.000) (0.000) (0.000) (0.000) Gender 0.021 1.021 0.004 0.004 0.004 0.004 (0.023) (0.024) (0.005) (0.004) (0.005) (0.004) Income 0.070 1.072ˆ 0.014 0.014 0.014 0.014 (0.009) (0.010) (0.002) (0.002) (0.002) (0.002) Growth 0.132 1.141ˆ 0.027 0.026 0.027 0.026 (0.008) (0.009) (0.002) (0.002) (0.002) (0.002) Unemployment -0.038 0.963ˆ -0.008 -0.007 -0.008 -0.007 (0.005) (0.004) (0.001) (0.001) (0.001) (0.001) N 39550 Table 1: Binomial Logit Model and Substantive Eﬀects — Standard errors in parentheses. +p < .10,∗p < .05,∗∗p < .01, and ∗∗∗p < .001 indicate that the coeﬃcient or marginal eﬀect is statistically signiﬁcantly diﬀerent from 0. ˆp < .05 indicates that the odds ratio is statistically signiﬁcantly diﬀerent from 1. Application 2.2 (Predicting the Likelihood of Civil War). In this application, we consider predicting the likelihood of a civil war. The data come from the Correlates of War project, with each observation being a country-year. The de-pending variable is equal to 1 if a country was experiencing a civil war in that year and is equal to zero otherwise. Table 3 reports a binomial logit model, four types of associated substantive eﬀects, and a linear probability model. 2.6 Suggested Reading 2.6.1 Background Greene (2000) 17 (1) (2) Independent Variables: Constant -1.258 -0.231 (0.026) (0.085) Growth 0.140 0.132 (0.008) (0.008) Distance -0.361 (0.008) Education -0.111 (0.007) Age 0.006 (0.001) Gender 0.021 (0.023) Income 0.070 (0.009) Unemployment -0.038 (0.005) N 39550 39550 AIC 47783 45123 BIC 47800 45192 McKelvey R2 0.012 0.121 McFadden R2 0.007 0.062 Table 2: Binomial Logit Models and Fit Statistics — Standard errors in parentheses. +p < .10,∗p < .05,∗∗p < .01,∗∗∗p < .001. Kennedy (1992) King (1998) 3 Numerical Optimization 3.1 The Need for Numerical Optimization Consider the following very simple equation, ex + .3x = .7. Since ex + .3x is monotonically increasing, there must be a unique x that solves ex + .3x = .7, however, we cannot solve this equation for x by hand, so we say that it has no analytical solution. Consider optimization problem, min x ex + .15x2 −.7x (36) To ﬁnd the minimum, we can attempt to take ﬁrst-order conditions, FOC = ex + .3x −.7 = 0 (37) or equivalently, 18 Logit ME1 ME2 CP1 CP2 OLS Independent Variables: Constant -6.623 -0.381 (0.359) (0.034) Log(Population) 0.519 0.035 0.048 0.044 0.055 0.053 (0.029) (0.002) (0.003) (0.003) (0.003) (0.003) Western Democracies -2.201 -0.150 -0.204 -0.065 -0.111 -0.206 (0.275) (0.018) (0.025) (0.004) (0.006) (0.024) Eastern Europe -1.945 -0.133 -0.180 -0.063 -0.105 -0.117 (0.341) (0.023) (0.031) (0.005) (0.009) (0.023) Latin America -0.195 -0.013 -0.018 -0.012 -0.017 -0.002 (0.222) (0.015) (0.021) (0.013) (0.018) (0.022) Sub-Saharan Africa -0.055 -0.004 -0.005 -0.004 -0.005 0.028 (0.206) (0.014) (0.019) (0.013) (0.019) (0.021) Asia -0.329 -0.022 -0.030 -0.020+ -0.028+ 0.057 (0.207) (0.014) (0.019) (0.011) (0.016) (0.024) Polity Score 0.045 0.003 0.004 0.003 0.004 0.005 (0.007) (0.000) (0.001) (0.001) (0.001) (0.001) GDP per Capita -0.199 -0.014 -0.018 -0.012 -0.017 -0.006 (0.027) (0.002) (0.003) (0.001) (0.002) (0.001) Former British Colony -0.017 -0.001 -0.002 -0.001 -0.002 -0.013 (0.124) (0.008) (0.011) (0.008) (0.011) (0.012) Former French Colony 0.237 0.016 0.022 0.018 0.023 -0.015 (0.145) (0.010) (0.013) (0.012) (0.015) (0.015) Per. Mountainous Terrain 0.012 0.001 0.001 0.001 0.001 0.001 (0.002) (0.000) (0.000) (0.000) (0.000) (0.000) Num. of Contiguous Countries 1.386 0.095 0.128 0.168 0.176 0.135 (0.135) (0.009) (0.012) (0.024) (0.021) (0.016) Ethnic Fractionalization 0.667 0.046 0.062 0.061 0.073 0.081 (0.183) (0.013) (0.017) (0.021) (0.023) (0.020) Per. Muslim 0.002 0.000 0.000 0.000 0.000 0.001 (0.002) (0.000) (0.000) (0.000) (0.000) (0.000) N 6214 6214 r.squared 0.174 Table 3: Binomial Logit Model and Substantive Eﬀects — Standard errors in parentheses. +p < .10,∗p < .05,∗∗p < .01, and ∗∗∗p < .001. 19 ex + .3x = .7 (38) but we already found that this has a unique solution that cannot be computed by hand. We can take second-order conditions, SOC = ex + .3 > 0 (39) The second-order conditions indicate that ex + .15x2 −.7 is globally concave and hence the solution to ex + .3x = .7 is the unique minimizer of ex + .15x2 −.7x, but we still need a way to compute the value of the minimizer. 3.2 One-Dimensional Root-Finding Consider solving f(x) = 0. Suppose that we know that f(x) < 0 and f(¯ x) > 0 where x < ¯ x. Suppose further that f is continuous. From the we have x < ¯ x with f(x) < 0 and f(¯ x) > 0, we say that we bracket a solution to f(x) = 0. From the intermediate value theorem, it follows that there is an x ∈(x, ¯ x) such that f(x) = 0. To ﬁnd the solution to f(x) = 0, we could consider the following algorithm. We set y = 1 2x + 1 2 ¯ x . If f(y) = 0, we have found a solution. If f(y) < 0, we now bracket a solution between y and ¯ x, so we set the new x = y. If f(y) > 0, we now bracket a solution between x and y, so we set the new ¯ x = y. In either case, we bracket a new solution, but x and ¯ x are half as far apart. Each time we do this process, x and ¯ x get closer and closer and we close in on the solution. We can stop this procedure when f(y) is closer enough to zero (say, within 0.00000001). This algorithm is called the bisection method. An alternative algorithm assumes that f is diﬀerentiable. Let us start with the guess x0. We take a Taylor expansion of f around x0. We have, f(x) ≈f(x0) + f ′(x0)(x −x0). We want to set f(x) = 0, so we solve x = x0 −f(x0) f′(x0). We use this so set up the following iteration, xk = xk−1 −f(xk−1) f ′(xk−1) (40) We stop this process when f(xk) is close enough to zero. This process is called Newton’s method. In some cases, we can calculate f ′(xk−1) by hand. In other cases, we might approximate it using numerical diﬀerentiation, 20 f ′(xk−1) ≈f(xk−1 + h) −f(xk−1) h (41) A third option is the secant method. The secant method starts with two iterates, x1 and x0. It approximates the derivative in the formula for Newton’s method using, f ′(xk−1) ≈f(xk−1) −f(xk−2) xk−1 −xk−2 (42) The secant method algorithm becomes, xk = xk−1 −f(xk−1)(xk−1 −xk−2) f(xk−1) −f(xk−2) (43) 3.3 One-Dimensional Optimization Consider the problem of ﬁnding the minimum of the function f(x). Suppose that we have 3 points, a < b < c such that f(a) > f(b) and f(c) > f(b). Suppose that the function f(x) is continuous, then f must have a local minimum between a and c. Since f goes down between a and b, it must turn around to reach f(c) at c. At some point the slope must go from negative to positive, and this point will be a local minimum. The algorithm selects an intermediate point between a and c, d. If this new point has f(d) < f(b), it replaces b. Otherwise, it replaces one of the boundary points. Continuing this procedure, we will obtain smaller and smaller intervals and converge to a local minimum. This process is called golden-section search. An alternative procedure is Newton’s method. Newton’s method uses a quadratic ap-proximation for f(x) around x0, f(x) ≈f(x0) + f ′(x0)(x −x0) + 1 2f ′′(x0)(x −x0)2 (44) If x is the minimum, we must have, f ′(x) ≈f ′(x0) + f ′′(x0)(x −x0) = 0 (45) Solving this for x, we obtain, x = x0 −f ′(x0) f ′′(x0) (46) This suggest the following iteration procedure, 21 xk = xk−1 −f ′(xk−1) f ′′(xk−1) (47) Note that we now must know both f ′ and f ′′. It is possible to approximate f ′′ by applying the ﬁnite diﬀerence formula twice, f ′(x + h) ≈f(x + h) −f(x) h (48) f ′(x) ≈f(x) −f(x −h) h (49) f ′′(x) ≈f ′(x + h) −f ′(x) h ≈ f(x+h)−f(x) h −f(x)−f(x−h) h h (50) = f(x + h) −2f(x) + f(x −h) h2 However, in practice approximating the second derivative at each step of Newton’s method is rarely done. Instead, the second derivative is typically approximated using secant-like methods. 3.4 Multi-Dimensional Optimization The ﬁrst set of algorithms we consider are based on Newton’s method. In particular, we approximate the objective function f(x) with a quadratic function, f(x∗) ≈f(x) + f ′(x)(x∗−x) + 1 2(x∗−x)′f ′′(x)(x∗−x) (51) Now, suppose that f ′′(x) is negative deﬁnite. If x∗is the minimum of this function, then we must have, f ′(x) + f ′′(x)(x∗−x) = 0 (52) which is obtained by taking the ﬁrst-derivative of the right hand side with respect to x∗. We can solve this for x∗to obtain, x∗≈x −f ′′(x)−1f ′(x), which suggests at iterative approach for ﬁnding the minimum of f. Let xk denote the current iterate, let gk = ∇f(xk), and let Hk = ∇2f(xk). We can generate a new iterate using the scheme, 22 xk+1 = xk −H−1 k gk (53) Notice that if the current iterate is approximately a minimum, then gk = ∇f(xk) ≈0, indicating that our algorithm will stop near a local minimum (it will also stop when exactly at a local maximum). In practice, Newton’s method is unlikely to converge without medication. We can achieve better results if we use, xk+1 = xk −λH−1 k gk (54) for some 0 < λ ≤1. The question becomes, how can we choose λ? A back-tracking procedure can be eﬀective. First, we try the full Newton’s step, with λ = 1. We then see if this decreases the function value. If it does not, we reduce the step by some factor (usually one half) and continue, until we reach the point where the function decreases. Two drawbacks still exists with Newton’s method. The ﬁrst is that we need to supply the Hessian, which may be expensive to compute. To see why the Hessian is diﬃcult to compute, note that H is a matrix with components Hij = ∂2f ∂xi∂xj (x). We can compute, ∂f ∂xi ≈f(x + eih) −f(x) h (55) where ei is a unit vector with the ith element equal to 1. We then have, ∂2f ∂xi∂xj (x) ≈ ∂f ∂xi(x + ejh) −∂f ∂xi(x) h ≈ f(x+ejh+eih)−f(x+ejh) h −f(x+eih)−f(x) h h (56) = f(x + ejh + eih) −f(x + ejh) −f(x + eih) + f(x) h2 Computing the Hessian H using numerical derivatives thus requires computing the objective function f at J(J + 1)/2 points above what is needed to compute the gradient. The second drawback to using the Hessian in Newton’s methods is that the Hessian will fail to be negative deﬁnite far from the minimum in many applications. We can actually solve both problems simultaneously by considering quasi-Newton methods. These methods extend on the idea of secant methods, and approximate the Hessian using recent values of the objection function and its’ gradient. The most widely used approximation to the Hessian is the BFGS update, 23 Hk+1 = Hk + (gk −gk−1)(gk −gk−1)′ (gk −gk−1)′(xk −xk−1) −Hk(xk −xk−1)(xk −xk−1)′Hk (xk −xk−1)′Hk(xk −xk−1) (57) This process is guaranteed to produce a symmetric deﬁnite updated Hessian as long as (gk −gk−1)′(xk −xk−1) > 0. The BFGS algorithm then proceeds by only updating when this condition is met. To summarize, we have the following algorithm, 1. Initialize x0, f0 = f(x0), g0 = f ′(x0), and H0 = I (or some other diagonal matrix). 2. In iteration k, (a) If gk is small enough, terminate successfully. (b) Set xk+1 = xk −H−1 k gk and compute fk+1 = f(xk+1) (c) If fk+1 > fk −δ (with δ > 0), then initialize λ = 1 and start backtracking. Otherwise, go to (d). i. Set λ = 1 2λ ii. Set xk+1 = xk −λH−1 k gk and compute fk+1 = f(xk+1). iii. If fk+1 > fk −δ , then go to (i), otherwise go to (d). (d) Compute gk+1 = f ′(xk+1) (e) Update, Hk+1 = Hk + (gk−gk−1)(gk−gk−1)′ (gk−gk−1)′(xk−xk−1) −Hk(xk−xk−1)(xk−xk−1)′Hk (xk−xk−1)′Hk(xk−xk−1) (f) Go to (a) Quasi-Newton methods are most eﬀective in problems that are suﬃciently smooth. The Nelder Mead simplex method does not rely on derivatives. The basic idea behind the Nelder Mead simplex method is to maintain a simplex of dimension n + 1 (for an n-dimensional minimization problem) that spans the n-dimensional space. We start by reﬂecting the worst point in the simplex through one of the vertices of the simplex. If the new function value is an improvement over the best function value, then we take a double step. Alternatively, if the reelected point was worse than the second highest, we contract by a factor of one-half. We continue this process until our simplex is really small. Since contraction will only occur when there are no nearby points with lower function values, this procedure will tend to converge to a local minimum. 24 3.5 Checking for an Optimum A necessary condition for a stationary point is that f ′(x∗) = 0. A suﬃcient condition for a local minimum is that f ′′(x∗) is positive deﬁnite. Determining whether x∗is a global minimum is more diﬃcult. If f is globally concave, then a local minimum is also a global minimum, but the objective function we work with are typically not globally concave. If we use a quasi-Newton method to minimize f(x), it is fairly likely to ﬁnd a point where f ′(x∗) = 0. It is also very unlikely to converge to a point that is not a local minimum. Nonetheless, we will typically check that f ′′(x∗) is positive deﬁnite because computing f ′′(x∗) is necessary for computing the asymptotic variance of the parameter estimates. We can check whether f ′′(x∗) is positive deﬁnite by checking that its eigenvalues are strictly positive. The most typical way in which f ′′(x∗) will fail to be positive deﬁnite is the case where is has zero eigenvalues, which can happen for an unidentiﬁed model (see Section 4.3). This often means that we either speciﬁed the model incorrectly or that some of the variables are perfect linear combinations of the others. 3.6 Suggested Reading 3.6.1 Background Greene (2000) King (1998) 4 Theory of MLE In developing the theory of MLE, we will assume that observations are independent over n. We will assume that the data are described by a probability mass function (in the case of discrete data) or a probability density function (in the case of continuous data). The model may be model for a data vector xn or a model for the dependent variable conditional on a vector of independent variables yn\|xn. Using independence, we have that the log-likelihood function is given by, l(θ) = PN n=1 log p(xn; θ) (discrete case) l(θ) = PN n=1 log f(xn; θ) (continuous case) 25 We can deﬁne the MLE using, ˆ θMLE = arg max θ∈Θ l(θ) (58) 4.1 Examples Example 4.1 (Poisson Distribution MLE). Consider the Poisson distribution, with probability mass function, p(x; λ) = e−λλx x! , for x ∈{0, 1, 2, ...} and λ ≥0. We can write the log-likelihood as, l(λ) = N X n=1 log(e−λ) + log(λxn) −log(xn!) = −Nλ + log(λ) N X n=1 xn − N X n=1 log(xn!) (59) The ﬁrst-order condition yields, lλ(ˆ λ) = −N + ˆ λ−1 N X n=1 xn = 0 (60) which implies ˆ λ = ¯ x. The second-order condition yields, lλλ(ˆ λ) = −ˆ λ−2 N X n=1 xn = −N ¯ x−1 < 0 (61) indicating that ˆ λ = ¯ x is a local maximum. Since it is the only local maximum and lim λ→0 l(λ) = ∞, it is also the global maximum, indicating that ˆ λ = ¯ x is the unique MLE. Example 4.2 (Exponential Distribution MLE). Suppose that xn has the exponential distribution, with probability density function, f(x; λ) = λeλx for x ≥0 and λ ≥0. The log-likelihood function is given by, l(λ) = N X n=1 log(λ) −λxn (62) The ﬁrst-order condition yields, lλ(ˆ λ) = Nˆ λ−1 − N X n=1 xn = 0 (63) 26 which implies ˆ λ = ¯ x−1. The second-order condition yields, lλλ(ˆ λ) = −Nˆ λ−2 = −N ¯ x2 < 0 (64) indicating that ˆ λ = ¯ x−1 is a local maximum. Since it is the only local maximum and lim λ→0 l(λ) = ∞, it is also the global maximum, indicating that ˆ λ = ¯ x−1 is the unique MLE. Example 4.3 (Normal Distribution MLE). Suppose xn has the normal distribution, with probability density function, f(x; µ, σ2) = 1 σ √ 2πe−1 2σ2 (x−µ)2 (65) where ∞< x < ∞and σ > 0. We can form the log-likelihood function as, l(µ, σ) = N X n=1 log 1 σ √ 2πe−1 2σ2 (xn−µ)2 = −N log σ −1 2N log(2π) − 1 2σ2 N X n=1 (xn −µ)2 (66) We can take ﬁrst order conditions to ﬁnd the MLE, lµ(ˆ µ, ˆ σ) = 1 ˆ σ2 N X n=1 (xn −ˆ µ) = 0 (67) lσ(X; ˆ µ, ˆ σ) = −N ˆ σ + 1 ˆ σ3 N X n=1 (xn −ˆ µ)2 = 0 (68) The ﬁrst equation indicates that, ˆ µ = 1 N N X n=1 xn = ¯ x (69) while the second equation yields, ˆ σ2 = 1 N N X n=1 (xn −ˆ µ)2 = N N−1s2 x (70) The second-order condition yields, lµµ(ˆ µ, ˆ σ) = −N ˆ σ2 (71) 27 lµσ(ˆ µ, ˆ σ) = lσµ(µ, σ) = −2 ˆ σ3 N X n=1 (xn −ˆ µ) = 0 (72) lσσ(ˆ µ, ˆ σ) = −N ˆ σ2 − 3 ˆ σ4 N X n=1 (xn −ˆ µ)2 = −4N ˆ σ2 (73) Since the matrix of second derivatives is negative deﬁnite, ˆ µ = ¯ X and ˆ σ = N N−1s2 x is a local maximum. Since it is the only local maximum, if is also a global maximum, indicate that ˆ µ = ¯ X and ˆ σ = N N−1s2 x is the unique MLE. Example 4.4 (Uniform Distribution MLE). Suppose that xn is uniformly distributed on the interval [0, θ]. The probability density function is given by f(x; θ) = 1 θ1{0 ≤x ≤θ}. The likelihood function is given by, L(θ) = N Y n=1 1 θ1{xn ≤θ} (74) Notice that this likelihood function is zero whenever there exists an xn > θ, so it must by the case that L(θ) is maximized at a point for which xn ≤θ for all n. Notice further that among such points, decreasing θ always increases the likelihood. Thus, the maximum will occur when ˆ θ = max 1≤n≤N{xn}. 4.2 Consistency We will derive the consistency of MLEs from the consistency of M-estimators. An M-estimator is deﬁned by, ˆ θ = arg max θ∈Θ 1 N N X n=1 ψ(xn; θ) (75) MLEs are a special case of M-estimators, where ψ(xn; θ) = log p(xn; θ) or ψ(xn; θ) = log f(xn; θ). MLEs inherit the properties of M-estimators, but have some additional proper-ties (see subsection 4.4). The basic idea of the proof of consistency is as follows. First, we show that the objec-tive function 1 N PN n=1 ψ(xn; θ) converges to its expectation E[ψ(xn; θ)]. From this, we can determine that ˆ θ converges to the minimizer of E[ψ(xn; θ)]. Finally, we show that θ0 is the unique maximizer of E[ψ(xn; θ)] (this is called the identiﬁcation condition). 28 Theorem 4.1 (Consistency of M-Estimators). Suppose that xn are independent identically distributed, and that θ0 is the unique maximizer of E[ψ(xn; θ)]. Suppose that Θ is compact and ψ(x; θ) is continuous at each θ ∈Θ with probability 1. Suppose that E sup θ∈Θ \|ψ(xn; θ)\| < ∞. Deﬁne ˆ θ = arg max θ∈Θ 1 N PN n=1 ψ(xn; θ). Then ˆ θ prob. − →θ0.3 Recall that maximum likelihood estimators are a special case of M-estimators. In order for maximum likelihood estimators to be consistent, it must be the case that certain reg-ularity conditions are met and that the MLE objective function identiﬁes the population parameters. We can show that the condition that the MLE objective function identiﬁes θ0 holds automatically. Suppose that ˆ θ is a maximum likelihood estimator and consider the discrete case (the continuous case is very similar). We have ψ(xn; θ) = log p(xn; θ) where xn has probability mass function p(xn; θ0). We would like to show that E[log p(xn; θ)] has a unique maximum at θ = θ0. Theorem 4.2 (Identiﬁcation of Maximum Likelihood Estimators). . (i) Suppose that there does not exist a θ ̸= θ0 such that p(x; θ) = p(x; θ0) for all x. Then θ0 is the unique maximizer of E[log p(xn; θ)]. (discrete case) (ii) Suppose that there does not exist a θ ̸= θ0 such that f(x; θ) = f(x; θ0) for all x ∈ X where R x∈X f(x; θ0)dx = 1. Then θ0 is the unique maximizer of E[log f(xn; θ)].4 (continuous case) Proof. We prove the discrete case. The proof for the continuous case is nearly identical. We want to show that for all θ ̸= θ0, E[log p(xn; θ)] < E[log p(xn; θ0)] (76) Notice that −1 + z ≥log z and −1 + z > log z for z ̸= 1. Setting z = p(x;θ) p(x;θ0), we have, −1 + p(x; θ) p(x; θ0) ≥log p(x; θ) p(x; θ0) (77) Multiplying both sides by p(x; θ0), we obtain, 3Adapted from Newey and McFadden (1994). 4Adapted from Gallant (1997). 29 −p(x; θ0) + p(x; θ) ≥p(x; θ0) log p(x; θ) p(x; θ0) (78) Summing both sides over x, we obtain, − X x p(x; θ0) + X x p(x; θ) ≥ X x p(x; θ0) log p(x; θ) p(x; θ0) (79) Since p(x; θ0) and p(x; θ) are probability mass functions, they sum to 1. Moreover, the left hand side is the expectation of log p(x;θ) p(x;θ0) , so we have, 0 ≥E log p(xn; θ) p(xn; θ0) = E[log(p(xn; θ)] −E[log(p(xn; θ)] (80) This can be rearranged to give, E[log(p(xn; θ)] ≥E[log(p(xn; θ)] ∀θ ∈Θ (81) To complete the proof, we must show that equation (81) hold strictly when θ ̸= θ0. By assumption, there is at least one value of x such that p(x; θ) ̸= p(x; θ0) for θ ̸= θ0. Therefore, −1 + z > log z for at least one value of z, and when we sum equation (78) for θ ̸= θ0, the inequality is strict, proving the result. The condition that there does not exist a θ ̸= θ0 such that p(x; θ) = p(x; θ0) for all x is called identiﬁcation of the statistical model. We will study it in detail in the next subsection. For now, we state a consistency result for maximum likelihood estimators. Theorem 4.3 (Consistency of Maximum Likelihood Estimators). . (i) Suppose that xn are independent identically distributed with probability mass function p(x; θ0). Suppose that there does not exists a θ ̸= θ0 such that p(x; θ) ̸= p(x; θ0) for all x. Suppose that Θ is compact and log p(x; θ) is continuous at each θ ∈Θ with probability 1. Suppose that E sup θ∈Θ \| log p(xn; θ)\| < ∞. Then ˆ θMLE prob. − →θ0. (discrete case) (ii) Suppose that xn are independent identically distributed with probability density function f(x; θ0). Suppose that there does not exist a θ ̸= θ0 such that f(x; θ) ̸= f(x; θ0) for all x ∈X where R x∈X f(x; θ0)dx = 1. Suppose that Θ is compact and log f(x; θ) is 30 continuous at each θ ∈Θ with probability 1. Suppose that E sup θ∈Θ \| log f(xn; θ)\| < ∞. Then ˆ θMLE prob. − →θ0.5 (continuous case) 4.3 Identiﬁcation of a Statistical Model Suppose that the data xn are i.i.d. Then the statistical model can be described by a cdf F(x; θ0) where θ0 represents the true parameter value. The true parameter varies in the class θ ∈Θ. Identiﬁcation answers the question, “are the data informative about the parameter of interest”. Deﬁnition 4.1 (Identiﬁcation of a Statistical Model). We say that θ0 is identiﬁed if there does not exists a θ ̸= θ0 such that F(x; θ) = F(x; θ0) for all x.6 Identiﬁcation is a necessary condition for a consistent estimator to exist (Gabrielsen, 1978; Rao, 1992; Mart´ ın and Quintana, 2002). Since identiﬁcation of the underlying statis-tical model is a necessary condition, we will spend some time on describing how to check identiﬁcation. We have the following deﬁnitions of identiﬁcation, Deﬁnition 4.2 (Identiﬁcation of a Statistical Model in the Continuous Case). Let X be the set of x such that f(x; θ0) > 0. We say that θ0 is identiﬁed if there does not exists a θ ̸= θ0 such that f(x; θ) ̸= f(x; θ0) for all x ∈X. Deﬁnition 4.3 (Identiﬁcation of a Statistical Model in the Discrete Case). Let X be the set of x such that p(x; θ0) > 0. We say that θ0 is identiﬁed if there does not exists a θ ̸= θ0 such that p(x; θ) ̸= p(x; θ0) for all x ∈X. Example 4.1 (continued). We wish to show that there does not exists a λ > 0 with λ ̸= λ0 such that p(x; λ) = p(x, λ0) for all x ∈{0, 1, 2, ...}. Consider a λ such that p(0; λ) = p(0, λ0). In this case, we have p(0; λ) = e−λ and p(0; λ0) = e−λ0. Equating these two implies that λ = λ0, so there does not exist a λ ̸= λ0 such that p(x; λ) = p(x, λ0) for all x. Example 4.2 (continued). We wish to show that there does not exists a λ > 0 with λ ̸= λ0 such that f(x; λ) = f(x, λ0) for all x ≥0. Consider a λ such that f(0; λ) = f(0, λ0). In this case, we have f(0; λ) = λ and f(0; λ0) = e−λ0. Equating these two implies that λ = λ0, so there does not exist a λ ̸= λ0 such that f(x; λ) = f(x, λ0) for all x. 5Adapted from Newey and McFadden (1994). 6To be technically correct, we would replace “for all x ” with “for x on a set of measure 1”. 31 Example 4.3 (continued). Suppose that xn ∼N(µ0, σ2 0) and we observe xn, then (µ0, σ2 0) are identiﬁed. To see that this is the case, we can show that there is no (µ, σ2) ̸= (µ0, σ2 0) such that φ(x; µ, σ2) = φ(x; µ0, σ2 0) for all x. We will show that φ(x; µ, σ2) = φ(x; µ0, σ2 0) for x = 0, x = µ0, and x = 2µ0 implies that (µ, σ2) = (µ0, σ2 0). The three conditions imply that, 1 σ √ 2πe−1 2 µ2/σ2 = 1 σ0 √ 2πe−1 2 µ2 0/σ2 0 (82) 1 σ √ 2πe−1 2 (µ0−µ)2/σ2 = 1 σ0 √ 2π (83) 1 σ √ 2πe−1 2 (2µ0−µ)2/σ2 = 1 σ0 √ 2πe−1 2 µ2 0/σ2 0 (84) The ﬁrst and third conditions imply that, 1 σ √ 2πe−1 2 µ2/σ2 = 1 σ √ 2πe−1 2 (2µ0−µ)2/σ2 (85) which implies that µ = µ0. Plugging this into the second condition gives, 1 σ √ 2π = 1 σ0 √ 2π (86) or that σ = σ0. Hence, a contradiction implies that the model is identiﬁed. Example 4.5 (Identiﬁcation Failure). Suppose that xn ∼N(0, σ2 0), but that we observe, yn = 1{xn ≥0}. Hence, we only observe the number of positive occurrences. In this case, the distribution of yn is given by P(yn = 0) = Φ −0 σ0 = Φ (0) = 1 2 and P(yn = 1) = 1 2. Since these probabilities hold for all σ ̸= σ0 with σ > 0, the model is not identiﬁed. Example 4.6 (Identiﬁcation of the Uniform Distribution). Suppose that xn ∼U(a0, b0) and that we observe xn. Then the parameters (a0, b0) are identiﬁed. To see that this is the case, we have, f(x; a, b) = 1 b−a1{a ≤x ≤b} (87) We wish to show that for any (a0, b0) ̸= (a, b), we have f(x; a, b) ̸= f(x; a0, b0) for some x. Suppose that a < a0. When a0 < x < a, 32 f(x; a, b) −f(x; a0, b0) = 1 b−a1{a ≤x ≤b} − 1 b0−a01{a0 ≤x ≤b0} = 1 b−a ̸= 0 (88) Now suppose that a > a0. When a < x < a0, f(x; a, b) −f(x; a0, b0) = 1 b−a1{a ≤x ≤b} − 1 b0−a01{a0 ≤x ≤b0} = − 1 b0−a0 ̸= 0 (89) Now suppose that b > b0. When b < x < b0, we have, f(x; a, b) −f(x; a0, b0) = 1 b−a1{a ≤x ≤b} − 1 b0−a01{a0 ≤x ≤b0} = − 1 b0−a0 ̸= 0 (90) Finally, suppose that b < b0. When b0 < x < b, we have, f(x; a, b) −f(x; a0, b0) = 1 b−a1{a ≤x ≤b} − 1 b0−a01{a0 ≤x ≤b0} = 1 b−a ̸= 0 (91) Example 4.7 (Identiﬁcation of the (Overspeciﬁed) Logit Model). Consider the logit model, but assume that the error term does not have scale parameter 1. That is, suppose that, y∗ n = β′xn + εn, εn ∼Logistic(0, s2), and yn = 1{y∗ n ≥0}. We have that, Pr(yn = 1\|xn; β, s2) = Pr(yn∗≥0\|xn; β, s2) = Pr(β′xn + εn ≥0\|xn; β, s2) (92) = Pr(εn ≥−β′xn\|xn; β, s2) = 1 −Pr(εn < −β′xn\|xn; β, s2) = 1 − 1 1 + e −β′xn s = e −β′xn s 1 + e −β′xn s Does there exist a (β, s2) ̸= (β0, s2 0) such that, e β′xn s 1 + e β′xn s = e β′ 0xn s0 1 + e β′ 0xn s0 (93) for all xn? Consider (β, s2) = (2β0, 2s2 0). We have, 33 e 2β′ 0xn 2s0 1 + e 2β′ 0xn 2s0 = e β′ 0xn s0 1 + e β′ 0xn s0 (94) which is true for all xn, so (β0, s2 0) is not identiﬁed. Example 4.8 (Identiﬁcation of the (Properly Speciﬁed) Logit Model). Consider the standard logit model, with y∗ n = β′xn + εn, εn ∼Logistic(0, 1), and yn = 1{y∗ n ≥0}. We have that, Pr(yn = 1\|xn; β, σ2) = eβ′xn 1 + eβ′xn (95) We will check whether there exists a β ̸= β0 such that, eβ′xn 1 + eβ′xn = eβ′ 0xn 1 + eβ′ 0xn (96) We can rearrange this to obtain, eβ′xn = eβ′ 0xn (97) Taking logs, we have, β′xn = β′ 0xn (98) or, (β −β0)′xn = 0 (99) Does there exists β ̸= β0 such that (β −β0)′xn = 0 for all xn? All long as xn has full support, the answer is no. This conditional can fail if some of the xn’s are perfect linear combinations of the other xn’s. 4.4 Asymptotic Normality Consider an M-estimator deﬁned by ˆ θ = arg max θ∈Θ 1 N PN n=1 ψ(xn; θ). Suppose that ˆ θ prob. − →θ0. Deﬁne Q(x; θ) = 1 N PN n=1 ψ(xn; θ) and suppose that Q is twice continuously diﬀerentiable (notice that Qθ(x; θ) = 1 N PN n=1 ψθ(xn; θ) and Qθθ(x; θ) = 1 N PN n=1 ψθθ(xn; θ)). We have that Qθ(x; ˆ θ) = 0. Taking a Taylor expansion of Qθ around θ0, we have, 34 Qθ(x; θ) = Qθ(x; θ0) + Qθθ(x; ¯ θ)(θ −θ0) (100) where ¯ θ = λθ + (1 −λ)θ0 and λ ∈(0, 1). Plugging in ˆ θ for θ, we obtain, 0 = Qθ(x; ˆ θ) = Qθ(x; θ0) + Qθθ(x; ¯ θ)(ˆ θ −θ0) (101) We can rearrange this to obtain, (ˆ θ −θ0) = −Qθθ(x; ¯ θ)−1Qθ(x; θ0) (102) or, √ N(ˆ θ −θ0) = −Qθθ(x; ¯ θ)−1√ NQθ(x; θ0) (103) We can write, √ N(ˆ θ −θ0) = − " 1 N N X n=1 ψθθ(xn; ¯ θ) #−1 1 √ N N X n=1 ψθ(xn; θ0) (104) A central limit theorem implies that, 1 √ N N X n=1 (ψθ(xn; θ0) −E[ψθ(xn; θ0)]) dist. − →N(0, V ar(ψθ(xn; θ0)) (105) Since θ0 = arg max θ∈Θ E[ψ(xn; θ)], we have E[ψθ(xn; θ0)] = 0 which implies that, 1 √ N N X n=1 ψθ(xn; θ0) dist. − →N(0, V ar(ψθ(xn; θ0)) (106) We further have that, V ar(ψθ(xn; θ0)) = E[ψθ(xn; θ0)ψθ(xn; θ0)′] −E[ψθ(xn; θ0)]E[ψθ(xn; θ0)]′ (107) = E[ψθ(xn; θ0)ψθ(xn; θ0)′] so that, 1 √ N N X n=1 ψθ(xn; θ0) dist. − →N(0, E[ψθ(xn; θ0)ψθ(xn; θ0)′]) (108) 35 Using the fact that ¯ θ = λˆ θ + (1 −λ)θ0 prob. − →θ0, we have, 1 N N X n=1 ψθθ(xn; ¯ θ) prob. − →1 N N X n=1 ψθθ(xn; θ0) (109) A law of large numbers implies that, 1 N N X n=1 ψθθ(xn; θ0) prob. − →E[ψθθ(xn; θ0)] (110) which together implies that, 1 N N X n=1 ψθθ(xn; ¯ θ) prob. − →E[ψθθ(xn; θ0)] (111) Slutsky’s theorem then implies that, √ N(ˆ θ −θ0) dist. − →N(0, E[ψθθ(xn; θ0)]−1E[ψθ(xn; θ0)ψθ(xn; θ0)′]E[ψθθ(xn; θ0)]−1) (112) We state this result formally below, Theorem 4.4 (Asymptotic Normality of M-Estimators). Suppose that the conditions of Theorem 4.1 hold, θ0 ∈int(Θ), ψ(xn; θ) is twice continuous diﬀerentiable in a neighborhood N of θ0, V ar(ψθ(xn; θ0)) = E[ψθ(xn; θ0)ψθ(xn; θ0)′] = B0 is ﬁnite, C(θ) = E[ψθθ(xn; θ)] is continuous at θ0 and sup θ∈N \|\| 1 N PN n=1 ψθθ(xn; θ)−C(θ)\|\| prob. − →0 where C0 = C(θ0) is nonsingular. Deﬁne ˆ θ as ˆ θ ∈arg max θ∈Θ 1 N PN n=1 ψ(xn; θ). Then √ N(ˆ θ −θ0) dist. − →N(0, C−1 0 B0C−1 0 ).7 To apply this result, we need estimators for B0 and C0. The obvious estimators are, ˆ B = 1 N N X n=1 ψθ(xn; ˆ θ)ψθ(xn; ˆ θ)′ (113) ˆ C = 1 N N X n=1 ψθθ(xn; ˆ θ) (114) An important property of maximum likelihood estimators is the information equality. Here, we demonstrate the result in the discrete case (the proof for the continuous case is 7Adapted from Newey and McFadden (1994). 36 nearly identical. We begin with P x p(x; θ)dx = 1, which we diﬀerentiate on both sides to yield, X x ∂ ∂θp(x; θ)dx = 0 (115) Notice that, ∂ ∂θ log p(x; θ) = ∂ ∂θp(x; θ) p(x; θ) (116) so that, p(x; θ)[ ∂ ∂θ log p(x; θ)] = ∂ ∂θp(x; θ) (117) We sum both sides over x to yield, X x [ ∂ ∂θ log p(x; θ)]p(x; θ) = X x ∂ ∂θp(x; θ) = 0 (118) We can diﬀerentiate these one more time to give, X x [ ∂2 ∂θ∂θ′ log p(x; θ)]p(x; θ) + X x [ ∂ ∂θ log p(x; θ)][ ∂ ∂θp(x; θ)dx]′ (119) = X x [ ∂2 ∂θ∂θ′ log p(x; θ)]p(x; θ) + X x [ ∂ ∂θ log p(x; θ)][ ∂ ∂θ log p(x; θ)]′p(x; θ) = 0 We therefore have, E h ∂2 ∂θ∂θ′ log p(x; θ) i = −E [ ∂ ∂θ log p(x; θ)][ ∂ ∂θ log p(x; θ)]′ (120) This last result is known as the information equality. We deﬁne the information matrix by J0 = E [ ∂ ∂θ log p(x; θ0)][ ∂ ∂θ log p(x; θ0)]′ in the discrete case and J0 = E [ ∂ ∂θ log f(x; θ0)][ ∂ ∂θ log f(x; θ0)]′ in the continuous case. Theorem 4.5 (Information Equality for Maximum Likelihood Estimators). We have, E h ∂2 ∂θ∂θ′ log p(x; θ0) i = −E [ ∂ ∂θ log p(x; θ0)][ ∂ ∂θ log p(x; θ0)]′ (discrete case) E h ∂2 ∂θ∂θ′ log f(x; θ0) i = −E [ ∂ ∂θ log f(x; θ0)][ ∂ ∂θ log f(x; θ0)]′ (continuous case) In the special case where ˆ θ is a MLE, 37 B0 = E[ψθ(xn; θ0)ψθ(xn; θ0)′] = E[ ∂ ∂θ log p(xn; θ0) ∂ ∂θ log p(xn; θ0)′] (121) C0 = E[ψθθ(xn; θ0)] = E[ ∂2 ∂θ∂θ′ log p(xn; θ0)] (122) The information equality states that, E h ∂2 ∂θ∂θ′ log p(x; θ0) i = −E [ ∂ ∂θ log p(x; θ0)][ ∂ ∂θ log p(x; θ0)]′ (123) or that C0 = −B0, which yields the following 3 equivalent formulas for the asymptotic variance of the MLE, V0 = E[ ∂2 ∂θ∂θ′ log f(xn; θ0)]−1E[ ∂ ∂θ log p(xn; θ0) ∂ ∂θ log p(xn; θ0)′]E[ ∂2 ∂θ∂θ′ log f(xn; θ0)]−1 (124) V0 = −E[ ∂2 ∂θ∂θ′ log p(xn; θ0)]−1 (125) V0 = E[ ∂ ∂θ log p(xn; θ0) ∂ ∂θ log p(xn; θ0)′]−1 (126) Applying Theorem 4.4 yields the following result, Theorem 4.6 (Asymptotic Normality of Maximum Likelihood Estimators). . (i) Suppose that the conditions of Theorem 4.3 are satisﬁed, θ0 ∈int(Θ), p(xn; θ) is twice continuously diﬀerentiable, p(xn; θ) > 0 in a neighborhood N of θ0, J0 = −E[ ∂2 ∂θ∂θ′ log p(xn; θ0)] exists and is non-singular, and E sup θ∈N \|\| ∂2 ∂θ∂θ′ log p(xn; θ)\|\| < ∞. Then √ N(ˆ θ−θ0) dist. − → N(0, V0) where V0 = J−1 0 . (discrete case) (ii) Suppose that the conditions of Theorem 4.3 are satisﬁed, θ0 ∈int(Θ), f(xn; θ) is twice continuously diﬀerentiable, f(xn; θ) > 0 in a neighborhood N of θ0, R sup θ∈N \|\|fθ(xn; θ)\|\|dx < ∞, R sup θ∈N \|\|fθθ(xn; θ)\|\|dx < ∞, J0 = −E[ ∂2 ∂θ∂θ′ log f(xn; θ0)] exists and is non-singular, and E sup θ∈N \|\| ∂2 ∂θ∂θ′ log f(xn; θ)\|\| < ∞. Then √ N(ˆ θ−θ0) dist. − →N(0, V0) where V0 = J−1 0 .8 (continuous case) 8Adapted from Newey and McFadden (1994). 38 To estimate V0, we can use one of the following three formulas, ˆ V = 1 N N X n=1 ∂2 ∂θ∂θ′ log f(xn; ˆ θ) !−1 1 N N X n=1 ∂ ∂θ log f(xn; ˆ θ) ∂ ∂θ log f(xn; ˆ θ)′ ! 1 N N X n=1 ∂2 ∂θ∂θ′ log f(xn; ˆ θ) !−1 (127) ˆ V = − 1 N N X n=1 ∂2 ∂θ∂θ′ log f(xn; ˆ θ) !−1 (128) ˆ V = 1 N N X n=1 ∂ ∂θ log f(xn; ˆ θ) ∂ ∂θ log f(xn; ˆ θ)′ !−1 (129) Note that, ∂2 ∂θ∂θ′ 1 N l(ˆ θ) = ∂2 ∂θ∂θ′ 1 N N X n=1 log f(xn; ˆ θ) = 1 N N X n=1 ∂2 ∂θ∂θ′ log f(xn; ˆ θ) (130) Since we must compute ∂2 ∂θ∂θ′ 1 N l(ˆ θ) in the course of checking whether ˆ θ satisﬁed the SOC, the simplest estimator is ˆ V = − ∂2 ∂θ∂θ′ 1 N l(ˆ θ) −1 . The ﬁrst of the three estimators above is called the sandwich estimator and is the estimator we would use if specifying the “, robust” option in stata. It is important to note that in general, the standard errors will not be robust to departures from the assumed model since consistency of ˆ θ is only guaranteed when the assumed model is correct. Exceptions to this are the linear model, where the MLE will be consistent even if the error terms of heteroskedastic or not normally distributed. In this case, the information equality will fail to hold and the sandwich estimator for the variance must be used. For most MLEs, there is no beneﬁt (but also no harm) to using robust standard errors. 4.5 Eﬃciency and the Cramer Rao Lower Bound Why do people like maximum likelihood estimators so much? To guarantee a consistent estimator, the limiting objective function must satisfy the identiﬁcation condition. Finding an objective function that satisﬁes this condition is hard however, and maximum likelihood gives an automatic way to satisfy the identiﬁcation condition. There is another important reason to prefer maximum likelihood estimators over alter-native consistent and asymptotically normal estimators—they are eﬃcient. Consider any 39 estimator ˆ θ (not even, necessarily, an M-estimator) such that √ N(ˆ θ −θ0) dist. − →N(0, V0) where V0 is positive deﬁnite. The Cramer Rao Lower Bound implies that V0 −J−1 0 is a posi-tive semi-deﬁnite matrix. Since the maximum likelihood estimator achieves the lower bound J−1 0 , we say that the maximum likelihood estimator is eﬃcient. There are other estimators that achieve this lower bound—for example, a properly constructed Bayesian estimator. Theorem 4.7 (Cramer Rao Lower Bound). Let ˆ θ be an estimator of θ0 such that √ N(ˆ θ − θ0) dist. − →N(0, V0) where V0 is positive deﬁnite. Then V0 −J−1 0 is a positive semi-deﬁnite matrix. Proof. Suppose that √ N(ˆ θ −θ0) dist. − →N(0, V0). This implies that lim N→∞E[ˆ θ −θ0] = 0.9 Consider, E[ˆ θ −θ] = Z x (ˆ θ −θ)f(x; θ)dx (131) Diﬀerentiating with respect to θ′, we obtain, ∂ ∂θ′E[ˆ θ −θ] = −I Z x f(x; θ)dx + Z x (ˆ θ −θ)fθ(x; θ)′dx (132) Note that ∂ ∂θ′ log f(x; θ) = fθ(x;θ)′ f(x;θ) , or f(x; θ) ∂ ∂θ′ log f(x; θ) = fθ(x; θ)′, so that, ∂ ∂θ′E[ˆ θ −θ] = −I + Z x (ˆ θ −θ) ∂ ∂θ log f(x; θ)′f(x; θ)dx (133) Evaluating this at θ = θ0 and taking limits of both sides, we obtain, 0 = lim N→∞ ∂ ∂θ′E[ˆ θ −θ0] = −I + lim N→∞ Z x (ˆ θ −θ0) ∂ ∂θ log f(x; θ0)′f(x; θ0)dx (134) From this, we can obtain, I = lim N→∞ Z x √ N(ˆ θ −θ0) 1 √ N N X n=1 ∂ ∂θ log f(xn; θ0)′f(x; θ0)dx (135) or, I = lim N→∞E " √ N(ˆ θ −θ0) 1 √ N N X n=1 ∂ ∂θ log f(xn; θ0)′ # (136) Deﬁne, 9See A.1 40 V0 = lim N→∞V ar( √ N(ˆ θ −θ0)) (137) Note that, V ar( 1 √ N N X n=1 ∂ ∂θ log f(xn; θ0)′) = V ar( ∂ ∂θ log f(xn; θ0)′) (138) = E[ ∂ ∂θ log f(x; θ0) ∂ ∂θ log f(x; θ0)′] lim N→∞Cov( √ N(ˆ θ −θ0), 1 √ N ∂ ∂θ0 log f(x; θ0) (139) = lim N→∞E " √ N(ˆ θ −θ0) 1 √ N N X n=1 ∂ ∂θ log f(xn; θ0)′ # = I Combined, we have, lim N→∞V ar √ N(ˆ θ −θ0) 1 √ N PN n=1 ∂ ∂θ log f(xn; θ0) ! = " V0 I I J0 # (140) Since the variance must be positive semi-deﬁnite, we have that for all z, z′ " V0 I I J0 # z ≥0 (141) We have, z′ 1V0z1 +z′ 2J0z2 +2z′ 1Iz2 ≥0. Setting z1 = α and z2 = −J0−1α for any α, we obtain α′(V0 −J0−1)α ≥0 for all α. It follows that V0 −J0−1 is positive semi-deﬁnite. It is possible that an alternative estimator could be “supereﬃcient” (to have lower variance than the Cramer Rao lower bound). This is ruled out by the requirement that √ N(ˆ θ −θ0) dist. − →N(0, V0) where V0 is positive deﬁnite. Supereﬃcient estimators converge at a rate faster than √ N. van der Vaart (1997), for example, argues that while such estimators exist, they are supereﬃcient for θ0 on a set of measure zero. It is also possible for an esti-mator that is asymptotically biased to be “supereﬃcient”. Both of these are technicalities that are not worth worrying about because they do not lead to estimators that are better in practice. Example 4.9 (An Ineﬃcient Estimator of the Mean of the Normal Distribution). 41 Suppose that Xn ∼N(µ, 1). We can show that the MLE is ˆ µ = ¯ X. Consider the alterna-tive estimator, ˆ µ2 = 2 N (X1 + X3 + ... + X2N−1), which is also consistent and asymptotically normal. We have that, V ar(ˆ µ) = V ar 1 N N X n=1 Xn ! = 1 N2 N X n=1 V ar(Xn) = 1 N2 N X n=1 1 = 1 N (142) V ar(ˆ µ2) = V ar( 2 N (X1 + X3 + ... + X2N−1)) (143) = 4 N2(V ar(X1) + V ar(X3) + ... + V ar(X2N−1)) = 4 N2 N 2 1 = 2 N Note that V ar(ˆ µ2) > V ar(ˆ µ), or that the alternative estimator has larger variance than the MLE. Example 4.10 (Another Ineﬃcient Estimator of the Mean of the Normal Distribution). Suppose that Xn ∼N(µ, σ2 n) where σ2 n is known. The log-likelihood is given by, l(µ) = N X n=1 log 1 σn √ 2πe−1 2 (Xn−µ)2/σ2 n = N X n=1 −log σn −1 2 log(2π) −1 2(Xn −µ)2/σ2 n (144) Using ﬁrst-order conditions, we can determine that the MLE is given by, ˆ µ = PN n=1 Xn σ2 n PN n=1 1 σ2 n (145) Consider the alternative estimator ¯ X which is also consistent and asymptotically normal. We can show that, V ar(ˆ µ) = PN n=1 V ar( Xn σ2 n ) PN n=1 1 σ2 n 2 = PN n=1 1 σ4 nσ2 n PN n=1 1 σ2 n 2 = 1 PN n=1 1 σ2 n = 1 N 1 N PN n=1 1 σ2 n (146) V ar( ¯ X) = 1 N2 N X n=1 σ2 n (147) Consider the special case where σ2 n = (1, 2, 1, 2, ...). We have, 42 V ar(ˆ µ) = 1 N 1 N PN n=1(1 + 1 2 + 1 + 1 2 + ...) = 1 N 1 N 3 2 N 2 = 4 3N (148) V ar( ¯ X) = 1 N2 N X n=1 (1 + 2 + 1 + 2 + ...) = 1 N23 N 2 = 3 2N (149) Since 3 2N > 4 3N , we have that the alternative estimator has smaller variance than the MLE. More generally, let φ(x) = x−2. We have φ′′(x) = 6x−4 > 0 indicating that φ is convex. Jensen’s inequality states that for a convex function, φ(λ1x1 + ... + λNxN) ≤λ1φ(x1) + ... + λNφ(xN) for λ1 + ... + λN = 1 where equality holds if and only if x1 = ... = xN. Applying this, we have, 1 N N X n=1 1 σ2 n > 1 1 N PN n=1 σ2 n (150) or, 1 N N X n=1 σ2 n > 1 1 N PN n=1 1 σ2 n (151) which further implies, 1 N2 N X n=1 σ2 n > 1 N 1 N PN n=1 1 σ2 n (152) Hence, V ar( ¯ X) > V ar(ˆ µ), i.e. the alternative estimator has greater variance than the MLE. 4.6 Suggested Reading 4.6.1 Background Gallant (1997) King (1998) Greene (2000) Wooldridge (2002) 43 4.6.2 Advanced Newey and McFadden (1994) White (1984) 5 Probit, More on Logit, and Ordered Probit 5.1 The Probit Model The Probit model is an alternative binary choice model where, Pr(yn = 1\|xn; β) = Φ(β′xn) (153) We can derive the Probit model as a latent variable model. Suppose that y∗ n = β′xn + εn and εn ∼N(0, 1). In this case, we have, Pr(yn = 1\|xn; β) = Pr(y∗ n ≥0\|xn; β) = Pr(β′xn + εn ≥0\|xn; β) (154) = Pr(εn ≥−β′xn\|xn; β) = 1 −Pr(εn < −β′xn\|xn; β) = 1 −Fε(−β′xn) = 1 −Φ(−β′xn) = Φ(β′xn) There is a third interpretation of the probit model, which is the random utility interpre-tation. Let u0 n and u1 n denote the utilities of option 0 and option 1. We can specify, u0 n = 0 (155) u1 n = β′xn + εn (156) where εn ∼N(0, 1). We “normalize” u0 n = 0 since only diﬀerence in utilities matter. We observe the individual choosing 1 over 0 if u1 n ≥u0 n or if u1 n −u0 n = β′xn + εn. Notice that this is equivalent to the latent variable formulation, so we can interpret y∗ n as the latent utility diﬀerence between option 1 and option 0 and β as indicating whether covariate xn is associated with increasing the utility diﬀerence between option 1 and option 0. For the Probit model, we can compute marginal eﬀects as follows, 44 ∂ ∂xn,k Pr(yn = 1\|xn; β) = φ(β′xn)βk = 1 √ 2πe−1 2 (β′xn)2βk (157) An alternative approach for computing substantive eﬀects is to rely on diﬀerences in predicted probabilities Φ(β′a) −Φ(β′b), for two vectors of covariates a and b. It is possible to compare logit and probit coeﬃcients, but a scale adjustment must be made. It is conventional to assume that V ar(εn) = 1 for a probit model and V ar(εn) = π2 3 for a logit model. For this reasons, to compare the logit and probit models, we would use βlogit ≈ π √ 3βprobit. 5.2 Perfect Separation Consider a logit model Pr(Yn = 1; β) = Λ(β1 + β2Xn) and suppose that the data are such that Yn = 1 for Xn = −0.8, −0.6, −0.5, −0.3 and Yn = 0 for Xn = 0.2, 0.3, 1.5, 8.1. We can write the log likelihood as, l(β) = log Λ(β1−β2∗0.8)+log Λ(β1−β2∗0.6)+log Λ(β1−β2∗0.5)+log Λ(β1−β2∗0.3) (158) + log(1 −Λ(β1 + β2 ∗0.2)) + log(1 −Λ(β1 + β2 ∗0.3)) + log(1 −Λ(β1 + β2 ∗1.5)) + log(1 −Λ(β1 + β2 ∗8.1)) If we select β1 = 0, we can make the log-likelihood arbitrarily close to zero by making β2 arbitrarily close to negative inﬁnity. We can say that the maximum will occur when β2 = −∞ and β1 is any ﬁnite number. In this case, there are multiple optima that are only obtained asymptotically. This type of situation is called perfect separation. When perfect separation is present, there is no unique optimum. This is diﬀerent than an identiﬁcation problem because if the underlying probit model is correct, this cannot occur in large samples. Since the error distribution has full support, there is always some probability of observing a zero and a one for every combination of independent variables. 45 5.3 Testing Hypotheses Suppose that we want to test the null hypothesis H0 : c(β0) = 0 and suppose that √ N(ˆ β − β0) dist. − →N(0, V0). The same derivation used by the delta method implies that √ N(c(ˆ β) − c(β0)) dist. − →N(0, cβV0c′ β). Under the null hypothesis, we have, √ Nc(ˆ β) dist. − →N(0, cβV0c′ β) (159) or, (cβV0c′ β)−1/2√ Nc(ˆ β) dist. − →N(0, I) (160) or, W = Nc(ˆ β)′(cβV0c′ β)−1c(ˆ β) dist. − →χ2 R (161) where R = dim(c). We can use this test (called the Wald test) to test potentially nonlinear hypotheses about multiple parameter values. 5.4 Interactions in Binary Choice Models Consider a linear regression model with two independent variables and no interaction term. We have, E[yn\|xn, zn; β] = β1 + β2xn + β3zn (162) ∂2 ∂xn∂znE[yn\|xn, zn; β] = ∂ ∂znβ2 = 0 (163) Consider alternatively a binary choice model with two independent variables and no inter-action term (this can be either a logit or probit model), Pr(yn = 1\|xn, zn; β) = G(β1 + β2xn + β3zn) (164) where G(x) = Λ(x) = ex 1+ex or G(x) = Φ(x). We have, ∂2 ∂xn∂zn Pr(yn = 1\|xn, zn; β) = ∂ ∂zng(β1 + β2xn + β3zn)β2 = g′(β1 + β2xn + β3zn)β2β3 (165) 46 We can calculate g and g′ for the logit and probit models as follows. If G(x) = ex 1+ex, then g(x) = ex (1+ex)2 and g′(x) = (1−ex)ex (1+ex)3 . If G(x) = Φ(x), then g(x) = φ(x) = 1 √ 2πe−1 2 x2 and g′(x) = − 1 √ 2πxe−1 2 x2. The marginal eﬀect of xn depends on zn even though no interaction term is explicitly included in the model. Does this mean that we can study interaction eﬀects without an interaction term? In general, this is not a good idea because even though the interaction eﬀects are non-zero, their form is not ﬂexibly estimated. Consider the same calculation with an interaction term, ∂2 ∂xn∂zn Pr(yn = 1\|xn, zn; β) = ∂ ∂zng(β1 + β2xn + β3zn + β4xnzn)(β2 + β4zn) (166) = g′(β1 + β2xn + β3zn + β4xnzn)(β2 + β4zn)(β3 + β4xn) + g(β1 + β2xn + β3zn + β4xnzn)β4 Even in this case, it turns out that regardless of the coeﬃcient vector β, there will always exists values of (xn, zn) such that the interaction is positive and other values of (xn, zn) such that the interaction is negative. For example, consider the probit model. Suppose that (xn, zn) = (B, B) for large B > 0. We have, ∂2 ∂xn∂zn Pr(yn = 1\|xn, zn; β) (167) = g′(β1 + β2B + β3B + β4BB)(β2 + β4B)(β3 + β4B) + g(β1 + β2B + β3B + β4BB)β4 →g′(β4BB)(β4B)2 + g(β4B2)β4 = 1 √ 2πe−1 2 (β4B2)2 −(β4B)2 + β4 →− 1 √ 2πe−1 2 (β4B2)2(β4B)2 < 0 If we instead suppose that (xn, zn) = (B, −B) for large B > 0, we have, ∂2 ∂xn∂zn Pr(yn = 1\|xn, zn; β) = (168) 47 = g′(β1 + β2B −β3B −β4BB)(β2 −β4B)(β3 + β4B) + g(β1 + β2B −β3B −β4BB)β4 →−g′(−β4BB)(β4B)2 + g(−β4B2)β4 = 1 √ 2πe−1 2 (β4B2)2 (β4B)2 + β4 → 1 √ 2πe−1 2 (β4B2)2(β4B)2 > 0 There are a few consequences of this though—it may not make sense to theorize about interaction eﬀects for binary variables since interaction eﬀects on the probability of observing a 1 will always be present and both signs will always be possible. So for example, it does not make sense to develop a theory that says that the positions of candidates and the economy have a positive interactive eﬀect on the probability of voting for the incumbent, since there will always be covariates values for which this is true and covariate values for which this is false. We can however develop theories about interaction eﬀects on the latent propensity to vote for the incumbent or about the latent utility for voting for the incumbent. However, if we develop these theories, we would test them based on the coeﬃcient β4 rather than some measure of the marginal eﬀect ∂2 ∂xn∂zn Pr(yn = 1\|xn, zn; β) and the diﬀerence in predicted probabilities. 5.5 Ordered Probit The ordered probit model is typically motivated using a latent variable approach. Suppose that y∗ n = β′xn + εn where εn ∼N(0, 1). Suppose that yn ∈{1, 2, ..., J} where we observe, yn = 1 ⇔y∗ n < τ1 (169) yn = 2 ⇔τ1 < y∗ n < τ2 (170) ... (171) yn = J −1 ⇔τJ−2 < y∗ n < τJ−1 (172) 48 yn = J ⇔y∗ n > τJ−1 (173) where τ1 < τ2 < ... < τJ−1. From this, we obtain, Pr(yn = 1) = Φ(τ1 −β′xn) (174) Pr(yn = 2) = Φ(τ2 −β′xn) −Φ(τ1 −β′xn) (175) ... (176) Pr(yn = J −1) = Φ(τJ−1 −β′xn) −Φ(τJ−2 −β′xn) (177) Pr(yn = J) = 1 −Φ(τJ−1 −β′xn) (178) To identify the model, we must omit a constant term (or alternatively, normalize τJ−1 = 0). To simplify writing the likelihood, we deﬁne τ0 = −∞and τJ = ∞so that Pr(yn = j) = Φ(τj −β′xn) −Φ(τj−1 −β′xn). This allows us to write the log-likelihood as, l(β, τ) = N X n=1 log [Φ(τyn −β′xn) −Φ(τyn−1 −β′xn)] (179) or equivalently, l(β, τ) = N X n=1 J X j=1 1{yn = j} log [Φ(τj −β′xn) −Φ(τj−1 −β′xn)] (180) Marginal eﬀects are fairly complicated with the ordered probit model. Consider, Pr(yn = 2\|xn; β) = Φ(τ2 −β′xn) −Φ(τ1 −β′xn) (181) ∂ ∂xnk Pr(yn = 2\|xn; β) = −(φ(τ2 −β′xn) −φ(τ1 −β′xn))βk (182) = 1 √ 2π(e−1 2 (−β′xn+τ1)2 −e−1 2 (−β′xn+τ2)2)βk = 1 √ 2π(e−1 2 (β′xn−τ1)2 −e−1 2 (β′xn−τ2)2)βk 49 = 1 √ 2π(e−1 2 (z−τ1)2 −e−1 2 (z−τ2)2)βk For z < 1 2(τ1+τ2), we have (z−τ2)2 > (z−τ1)2 which implies −1 2(z−τ2)2 < −1 2(z−τ1)2 which further implies e−1 2 (z−τ2)2 < e−1 2 (z−τ1)2. Hence, sign(ME) = sign(βk). For z > 1 2(τ1 + τ2), we have (z −τ2)2 < (z −τ1)2 which implies −1 2(z −τ2)2 > −1 2(z −τ1)2 which further implies e−1 2 (z−τ2)2 > e−1 2 (z−τ1)2. Hence, sign(ME) ̸= sign(βk). The sign of the marginal eﬀect will reverse depending on how large z is, for any intermediate category (for the end categories, it will be monotonic). In general, marginal eﬀects will not be useful summaries of eﬀect sizes because they will generally not be monotonic. Instead, the best approach is to predict the probabilities of each value of the dependent variable for diﬀerent values of x. When interpreting the coeﬃcients directly, we can use the sign of a coeﬃcient to determine the eﬀect of the IV on the latent variable. From this, a positive (negative) coeﬃcient implies means that as the IV increases, we will observe higher (lower) values of the DV. However, we cannot say anything about the eﬀect of the IV on observing particular, intermediate, values of the DV, without computing substantive eﬀects. 5.6 Ordered Logit The ordered logit is nearly identical to the ordered probit model. We set y∗ n = β′xn + εn where εn ∼Logistic(0, 1). We can write, Pr(yn = j) = Λ(τj −β′xn) −Λ(τj−1 −β′xn) (183) where Λ(z) = ez 1+ez is the CDF of the standard logistic distribution. Besides replacing the normal CDF Φ with the logistic CDF Λ in the various expression, the only diﬀerence between the ordered logit and ordered probit models is that you can interpret eﬀects sizes using odds ratios. Note that, Pr(yn > j) = 1 −Λ(τj −β′xn) = 1 − eτj−β′xn 1 + eτj−β′xn = 1 1 + eτj−β′xn (184) Pr(yn ≤j) = 1 −Pr(yn > j) = 1 − 1 1 + eτj−β′xn = eτj−β′xn 1 + eτj−β′xn (185) 50 Thus, if we calculate the odds of yn > j over yn ≤j at xn = (xn,k = a, xn,−k), and deﬁne β−k = (β1, · · · , βk−1, βk+1, · · · , βK), then, Pr(yn > j) Pr(yn ≤j) = 1 1 + eτj−β′xn eτj−β′xn 1 + eτj−β′xn = 1 eτj−β′xn = e−(τj−β′xn) (186) We can calculate the odds ratio when xn,k = a + 1 over xn,k = a, keeping the rest of the variables at their mean value (that is, xn,−k), Pr(yn > j\|xnk = a + 1) Pr(yn ≤j\|xnk = a + 1) Pr(yn > j\|xnk = a) Pr(yn ≤j\|xnk = a) = e−τj+β′ −kxn,−k+βk·(a+1) e−τj+β′ −kxn,−k+βk·a = eβk (187) This implies that if we exponentiate βk, we have the odds ratio of an outcome greater than j to an outcome less than or equal to j, for a one unit increase in xnk. 5.7 Applications Application 2.1: Individual Level Model of the Economic Vote (continued). In Table 4, we compare the results of a logit and probit model. As can be see, the coeﬃcient generally have the same signs, but are of diﬀerent magnitudes. The third column examines the ratios of the logit to probit coeﬃcients and the fourth column compares these directly to p π 3. Table 5 compares the marginal eﬀects of the logit and probit models. These are nearly identical, suggesting that there is little consequence of the choice between a logit and probit model. Application 5.1 (The Eﬀect of Election Day Registration on Voter Turnout). This application is drawn from Berry, Demeritt and Esaray (2010), who compare speciﬁ-cations from Wolﬁnger and Rosenstone (1980) and Nagler (1991). The main eﬀect of interest is the eﬀect of the closing date for registration on voter turnout. Voters turnout is modeled using a probit model. Closing date is measured in terms of days to an election before reg-istration closes (with a value of zero indicating election day registration). Table 6 presents speciﬁcation with and without interaction terms. Figure 1 plots the eﬀect sizes using both models. Application 5.2 (The Eﬀect of Anti-war Speeches on Support for War). 51 Logit Probit Ratio Ratio / q π 3 Independent Variables: Constant -0.231 -0.157 1.467 0.809 (0.085) (0.050) Distance -0.361 -0.214 1.686 0.929 (0.008) (0.004) Education -0.111 -0.067 1.670 0.921 (0.007) (0.004) Age 0.006 0.004 1.680 0.926 (0.001) (0.000) Gender 0.021 0.014 1.550 0.855 (0.023) (0.014) Income 0.070 0.040 1.739 0.959 (0.009) (0.005) Growth 0.132 0.078 1.683 0.928 (0.008) (0.005) Unemployment -0.038 -0.021 1.827 1.007 (0.005) (0.003) N 39550 39550 Table 4: Comparing Logit and Probit — Standard errors in parentheses. +p < .10,∗p < .05,∗∗p < .01, and ∗∗∗p < .001. ME1 ME2 CP1 CP2 Logit Probit Logit Probit Logit Probit Logit Probit Independent Variables: Distance -0.072 -0.072 -0.070 -0.069 -0.066 -0.067 -0.065 -0.065 (0.002) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) Education -0.022 -0.023 -0.022 -0.022 -0.022 -0.022 -0.021 -0.021 (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) Age 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Gender 0.004 0.005 0.004 0.004 0.004 0.005 0.004 0.004 (0.005) (0.005) (0.004) (0.004) (0.005) (0.005) (0.004) (0.004) Income 0.014 0.014 0.014 0.013 0.014 0.014 0.014 0.013 (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) Growth 0.027 0.026 0.026 0.025 0.027 0.027 0.026 0.026 (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) Unemployment -0.008 -0.007 -0.007 -0.007 -0.008 -0.007 -0.007 -0.007 (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) Table 5: Comparing Logit and Probit Marginal Eﬀects — Standard errors in parentheses. +p < .10,∗p < .05,∗∗p < .01, and ∗∗∗p < .001. 52 (1) (2) Independent Variables: Constant -2.523 -2.743 (0.048) (0.109) Closing Date -0.008 0.001 (0.000) (0.004) Education 0.182 0.265 (0.015) (0.043) Education Sq. 0.012 0.005 (0.001) (0.004) Age 0.070 0.070 (0.001) (0.001) Age Sq. -0.001 -0.001 (0.000) (0.000) South -0.116 -0.115 (0.011) (0.011) Governor Race 0.003 0.003 (0.012) (0.012) Closing Date Education -0.003 (0.002) Closing Date Education Sq. 0.000+ (0.000) N 99676 99676 AIC 111652 111651 BIC 111728 111746 McFadden R2 0.117 0.117 McKelvey R2 0.233 0.233 P-Value for Joint Test of Interaction Terms 0.066+ Eﬀect of Moving Closing Date to Election Day 0.060 0.059 Table 6: The Eﬀect of Registration Closing Dates on Voter Turnout — Standard errors in parentheses. +p < .10,∗p < .05,∗∗p < .01, and ∗∗∗p < .001. 53 0 2 4 6 8 −0.10 −0.05 0.00 0.05 0.10 No Interaction Model Education Level Effect of Moving Closing Date from Mean to Zero 0 2 4 6 8 −0.10 −0.05 0.00 0.05 0.10 Interaction Model Education Level Effect of Moving Closing Date from Mean to Zero Figure 1 54 G G G G G G G −0.2 0.0 0.2 0.4 0.6 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 Ordered Probit Coefficients OLS Coefficients G G G G G G G −0.2 0.0 0.2 0.4 0.6 0.0 0.5 1.0 Ordered Probit Coefficients Ordered Logit Coefficients Figure 2: Comparing OLS, Ordered Probit, and Ordered Logit Coeﬃcients OLS Ordered Probit Ordered Logit Independent Variables: Constant 1.603 (0.149) Anti-war Speeches -0.047 -0.067 -0.107 (0.015) (0.023) (0.040) Republican 0.482 0.712 1.326 (0.059) (0.093) (0.167) Democrat -0.195 -0.283 -0.425 (0.064) (0.092) (0.161) Age 0.004 0.006 0.011 (0.002) (0.002) (0.004) Education 0.087 0.120 0.225 (0.028) (0.042) (0.077) Male 0.287 0.426 0.721 (0.049) (0.073) (0.126) White 0.228 0.319 0.584 (0.077) (0.113) (0.192) Cutpoint 1\|2 0.001 0.196 (0.223) (0.389) Cutpoint 2\|3 1.185 2.213 (0.228) (0.395) Cutpoint 3\|4 2.840 5.147 (0.248) (0.442) N 956 956 956 AIC 2157 2112 2105 BIC 2201 2160 2153 R2 0.209 McKelvey R2 0.238 0.228 McFadden R2 0.095 0.098 Table 7: Comparing OLS, Ordered Probit, and Ordered Logit — Standard errors in parentheses. +p < .10,∗p < .05,∗∗p < .01, and ∗∗∗p < .001. 55 0 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 Anti−war Speaches Prob. Stay the Course:1 Stay the Course:2 Stay the Course:3 Stay the Course:4 Figure 3: Substantive Eﬀects of Anti-War Speeches 56 −40 −20 0 20 40 0.0 0.2 0.4 0.6 0.8 1.0 Anti−war Speaches Prob. Stay the Course:1 Stay the Course:2 Stay the Course:3 Stay the Course:4 Figure 4: Substantive Eﬀects of Anti-War Speeches, Extended Range 57 0 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 Ordered Probit Anti−war Speaches Prob. Stay the Course:1 Stay the Course:2 Stay the Course:3 Stay the Course:4 0 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 Ordered Logit Anti−war Speaches Prob. Figure 5: Substantive Eﬀects of Anti-War Speeches, Ordered Probit vs. Ordered Logit 0 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 High Info Anti−war Speaches Prob. Stay the Course:1 Stay the Course:2 Stay the Course:3 Stay the Course:4 0 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 Low Info Anti−war Speaches Prob. Stay the Course:1 Stay the Course:2 Stay the Course:3 Stay the Course:4 Figure 6: Substantive Eﬀects of Anti-War Speeches, Interactive Model 58 (1) (2) Independent Variables: Anti-war Speeches -0.067 -0.025 (0.023) (0.024) Know Party 0.014 (0.077) Anti-war Speeches Know Party -0.092+ (0.050) Republican 0.712 0.708 (0.093) (0.094) Democrat -0.283 -0.281 (0.092) (0.092) Age 0.006 0.006 (0.002) (0.002) Education 0.120 0.123 (0.042) (0.043) Male 0.426 0.430 (0.073) (0.074) White 0.319 0.323 (0.113) (0.113) Cutpoint 1\|2 0.001 0.020 (0.223) (0.223) Cutpoint 2\|3 1.185 1.208 (0.228) (0.227) Cutpoint 3\|4 2.840 2.864 (0.248) (0.248) N 956 956 AIC 2112 2112 BIC 2160 2171 McKelvey R2 0.238 0.241 McFadden R2 0.095 0.097 Table 8: Ordered Probit Models — Standard errors in parentheses. +p < .10,∗p < .05,∗∗p < .01,∗∗∗p < .001. 59 Application 5.3 (Measuring the Locations of National Policy). This application is drawn from Richman (2011). −10 −5 0 5 Status Quo Estimates Prior to 2000 Congressionl Election W−Nominate Score G Spend: Nat. Def. G Spend: Arts G Spend: Env. Prog. G Spend: Welfare G Spend: Inter. Aid G Spend: Educ. G Spend: Nat. Parks G Spend: Army Training G Spend: Covert Intel. G Spend: Defense Plant Conv. G Spend: Mili. Hardware G Spend: Mili. Space Shuttle G Spend: Weapon Modern. G Spend: Mis. Defense G Spend: R&D G Spend: Army Readiness G Tax: Retiree Inc. > 40k G Tax: Fam. Inc. < 25k G Tax: Fam. Inc. 25−75k G Tax: Fam. Inc. 75−150k G Tax: Fam. Inc. > 150k G Tax: Alcohol G Tax: Cap. Gains G Tax: Char. Deduct. G Tax: Cigarette G Tax: Corporate G Tax: Earned Inc. Tax. Cred. G Tax: Estate G Tax: Med. Exp. Deduct. G Tax: Mort. Deduct. G Tax: Gasoline G Tax: Stud. Loan Tax Cred. Figure 7: Status Quo Policy Outcomes 5.8 Suggested Reading 5.8.1 Background Berry, Demeritt and Esaray (2010) Greene (2000) Kennedy (1992) 5.8.2 Examples Kriner and Shen (2014) Richman (2011) 60 6 Multinomial Choice 6.1 Multinomial Logit Consider an unordered categorical variable yn ∈{1, 2, ..., J}. We will motivate the model using the random utility formulation. Let unj denote the utility individual n gets from choice j. We specify, unj = β′ jxn + εnj (188) where we normalize β1 = 0. We will let εnj ∼EV1(0, 1) and we assume that the errors are independent across choices. Here, EV1(0, 1) denotes the type-I extreme value distribution, with PDF fX(x) ∼e−xe−e−x and CDF FX(x) ∼e−e−x. We observe a choice of yn = j if unj ≥unk for all k ̸= j. We can write, Pr(yn = j\|xn) = Pr(unj ≥unk∀k ̸= j) = Pr(β′ jxn + εnj ≥β′ kxn + εnk∀k ̸= j) (189) = Z εn:β′ jxn+εnj≥β′ kxn+εnk∀k̸=j fεn(εn)dεn = Z εn:β′ jxn+εnj≥β′ kxn+εnk∀k̸=j fεn(εn,−j\|εnj)fεnj(εnj)dεn = Z ∞ εnj=−∞ fεnj(εnj) Z εn,−j:εnk≤(βj−βk)′xn+εnj∀k̸=j fεn(εn,−j\|εnj)dεn,−jdεnj = Z ∞ εnj=−∞ fεnj(εnj) Y k̸=j Fεnk((βj−βk)′xn+εnj)dεnj = Z ∞ εnj=−∞ e−εnje−e−εnj Y k̸=j e−e−εnj−(βj−βk)′xndεnj = R ∞ εnj=−∞e−εnje−e−εnj (1+P k̸=j e−(βj−βk)′xn)dεnj = R ∞ u=0 e−u(1+P k̸=j e−βj−βk)′xn)du = e−u(1+P k̸=j e−(βj−βk)′xn) −(1 + P k̸=j e−(βj−βk)′xn) ∞ u=0 = 1 1 + P k̸=j e−(βj−βk)′xn = 1 1 + P k̸=j e−β′ jxneβ′ kxn = eβ′ jxn PJ k=1 eβ′ kxn 61 We thus have that, Pr(yn = j\|xn) = eβ′ jxn PJ k=1 eβ′ kxn (190) We can write the log-likelihood for the multinomial logit model as, l(β) = N X n=1 log eβ′ ynxn PJ k=1 eβ′ kxn (191) For identiﬁcation purposes, we will normalize β1 = 0. To see why this is necessary, consider the system of equations, eβ′ jx PJ k=1 eβ′ kx = eβ′ j0x PJ k=1 eβ′ k0x∀j, x ∈X (192) Consider βk = βk0 + a with a ̸= 0. We have, e(βj0+a)′x PJ k=1 e(βk0+a)′x = eβ′ j0xea′x PJ k=1 eβ′ k0xea′x = eβ′ j0x PJ k=1 eβ′ k0x∀j, x ∈X (193) Hence, β0 cannot be identiﬁed. Instead, consider β1 = 0. Suppose that, eβ′ jx PJ k=1 eβ′ kx = eβ′ j0x PJ k=1 eβ′ k0x∀j, x ∈X (194) We can divide each equation for j > 1 by each equation for j = 1 to obtain, eβ′ jx = eβ′ j0x∀j > 1, x ∈X (195) which is equivalent to, e(βj−βj0)′x = 1∀j > 1, x ∈X (196) which is equivalent to, (βj −βj0)′x = 0∀j > 1, x ∈X (197) This will hold as long as X has full support. Hence, the model is identiﬁed once the restriction is made. With the multinomial logit model, we can report odds ratios. For j > 1, we have, 62 Pr(yn=j\|xnl=a,xn,−l) Pr(yn=1\|xnl=a,xn,−l) Pr(yn=j\|xnl=b,xn,−l) Pr(yn=1\|xnl=b,xn,−l) = eβ′ j,−lx−l+βj,la PJ k=1 e β′ k,−lx−l+βk,la 1 PJ k=1 e β′ k,−lx−l+βk,la e β′ j,−lx−l+βj,lb PJ k=1 e β′ k,−lx−l+βk,lb 1 PJ k=1 e β′ k,−lx−l+βk,lb = eβ′ j,−lx−l+βj,la eβ′ j,−lx−l+βj,lb = eβj,l(a−b) (198) If we exponentiate the coeﬃcient on choice j > 1, we can interpret this as the ratio of odds of observing choice j relative to choice 1, from a one unit increase in the covariate. 6.2 Conditional Logit Under the conditional logit model, unj = β′xnj + εnj, where εnj are independent EV1(0, 1) random variables. As before, we assume that we observe yn = j if unj ≥unk for all k ̸= j. This slight diﬀerence between these models is that the covariates here are choice speciﬁc. A similar derivation as before leads to, Pr(yn = j\|xn) = eβ′xnj PJ k=1 eβ′xnk (199) In the multinomial logit model, we always restrict the choice set to be {1, 2, ..., J}. With the conditional logit model, we can generalize the choice set to Jn ⊂{1, 2, ..., J} in which case we have, Pr(yn = j\|xn) = eβ′xnj P k∈Jn eβ′xnk ∀j ∈Jn (200) We can write the log-likelihood as, l(β) = N X n=1 log eβ′xn,yn P k∈Jn eβ′xnk (201) The identiﬁcation of the conditional logit model is slightly more tricky. Note that, eβ′xj P k∈Jn eβ′xk = eβ′ 0xj P k∈Jn eβ′ 0xk ∀j, x ∈X (202) which holds if and only if, eβ′xj eβ′xl = eβ′ 0xj eβ′ 0xl ∀j, l, x ∈X (203) 63 which holds if and only if, (β −β0)′(xj −xl) = 0∀j, l, x ∈X (204) If x has some common covariates, the coeﬃcients on those common covariates cannot be identiﬁed because if xj = (z, wj), we have xj −xl = (0, wj −wl). We can interact a common variable with a choice-speciﬁc dummy, but we cannot interact it with every choice speciﬁc dummy. We also cannot include a constant term. 6.3 The IIA Property The multinomial logit model has a potentially unattractive property. Consider the proba-bility of choosing j over l. We have, Pr(yn = j\|xn) Pr(yn = l\|xn) = eβ′ jxn P k eβ′ kxn eβ′ lxn P k eβ′ kxn = eβ′ jxn eβ′ lxn = e(βj−βl)′xn (205) Note that the probability does not depend on the choices in the choice set. This is called the Independence of Irrelevant Alternative (IIA) property. A similar problem occurs with the conditional logit model. Consider the choice of three transportation options: Bus, Fancy Bus, Car. Suppose that the Fancy Bus is eliminated. Suppose that Pr(Bus) / Pr(Car) = 2 (i.e. twice as many people take the regular bus than the car. Suppose that the fancy bus is eliminated. We would expect that that Pr(Bus) / Pr(Car) > 2 since the people who take the fancy bus probably don’t have a car. This is impossible in the multinomial logit framework (at least conditional on the x’s). Consider another example. Suppose that George W. Bush, Al Gore, and Ralph Nader are running. Suppose that Bush gets 48% of the vote, Gore gets 49% of the vote, and Nader gets 3%. This means that Gore’s two-party vote share would be .49 / (.49 + .48). If Nader were to leave the race, the multinomial logit model would predict that Gore’s 2 party vote share would stay exactly the same. In reality, we would expect that far more of the Nader voters would go to Gore than Bush. This limitation motivates the multinomial probit model, considered in the next subsection. 64 6.4 Multinomial Probit A solution to this problem is the multinomial probit model. As with the multinomial logit model, we specify unj = β′ jxn + εnj, but we now assume that εn ∼N(0, Ω). As before, we assume that we observe yn = j if unj ≥unk for all k ̸= j. We have, Pr(yn = j\|xn) = Pr(unj ≥unk∀k ̸= j) = Pr(β′ jxn + εnj ≥β′ kxn + εnk∀k ̸= j) (206) = Z εn:β′ jxn+εnj≥β′ kxn+εnk∀k̸=j fεn(εn)dεn = Z εn:β′ jxn+εnj≥β′ kxn+εnk∀k̸=j φ(εn; 0, Ω)dεn The integral above has no closed form solution. The log-likelihood for the multinomial probit model is given by, l(β, Ω) = N X n=1 log Z εn:β′ ynxn+εn,yn≥β′ kxn+εnk∀k̸=yn φ(εn; 0, Ω)dεn (207) In order to compute the MLE, we need a way of approximating the integral above. Older software had typically used Gaussian quadrature to compute the integral, but this approach is not very eﬀective, especially when there are more than three choices. Two approaches are now used—the ﬁrst is to approximate the integral above using the GHK simulator. The second approach is to abandon maximum likelihood estimation and used a Bayesian approach, which also relies of simulation (Markov Chain Monte Carlo methods in particular). The GHK simulator can be used to compute integrals of the form, Pr(a ≤x ≤b) = Z a≤x≤b φ(x; µ, Ω)dx (208) which are called rectangles of the normal distribution. One approach for computing such integrals would use the fact that, Pr(a ≤x ≤b) = Z a≤x≤b φ(x; µ, Ω)dx (209) = Z x 1{a ≤x ≤b}φ(x; µ, Ω)dx = E[1{a ≤x ≤b}] 65 Suppose that {˜ vr}R r=1 are independent draws from φ(x; 0, I). We can form ˜ xr = µ + Ω1/2˜ vr, which will be independent draws from the N(µ, Ω) distribution. A law of large numbers would then imply that, 1 R R X r=1 g(˜ xr) prob. − →E[g(x)] (210) Applying this to rectangles of the normal distribution, we have, 1 R R X r=1 1{a ≤µ + Ω1/2˜ vr ≤b} prob. − → Z a≤x≤b φ(x; µ, Ω)dx (211) This approach, while eﬀective for compute rectangles of the normal distribution for particu-lar values of (a, b, µ, Ω) is problematic when applied as an approximation within a likelihood function that is being optimized. This is because 1 R PR r=1 1{a ≤µ+Ω1/2˜ vr ≤b} varies discon-tinuously in the parameters (a, b, µ, Ω). The numerical optimizers that are used to compute maximum likelihood estimators rely on continuity and diﬀerentiability of the likelihood func-tion and will not function correctly when applied to a discontinuous and non-diﬀerentiable approximation to the likelihood function. The GHK simulator is designed to approximate rectangles of the normal distribution in a way that is continuous and diﬀerentiable in (a, b, µ, Ω). We can obtain the following using a change of variables, Pr(a ≤x ≤b) = Z a≤x≤b φ(x; µ, Ω)dx = Z a≤µ+Lv≤b φ(v; 0, I)dv (212) = Pr(a ≤µ + Lv ≤b) where L is the (lower triangular) Cholesky decomposition of Ω. We illustrate the GHK simulator in the two dimensional case. We have, Pr(a ≤µ + Lv ≤b) (213) = Pr(a1 ≤µ1 + L11v1 ≤b1, a2 ≤µ2 + L21v1 + L22v2 ≤b2) = Pr(a1 ≤µ1 + L11v1 ≤b1) Pr(a2 ≤µ2 + L21v1 + L22v2 ≤b2\|a1 ≤µ1 + L11v1 ≤b1) 66 = Pr a1 −µ1 L11 ≤v1 ≤b1 −µ1 L11 ∗Pr(a2 −µ2 −L21v1 L22 ≤v2 ≤b2 −µ2 −L21v1 L22 \|a1 −µ1 L11 ≤v1 ≤b1 −µ1 L11 ) Note ﬁrst that, Pr a1 −µ1 L11 ≤v1 ≤b1 −µ1 L11 = Φ b1 −µ1 L11 −Φ a1 −µ1 L11 (214) For the second probability, note that v1 ∼TN 0, 1, a1−µ1 L11 , b1−µ1 L11 where TN denotes the truncated normal distribution. Let {˜ v1r}R r=1 denote samples from this truncated normal distribution. We can approximate the second probability using, Pr(a2 −µ2 −L21v1 L22 ≤v2 ≤b2 −µ2 −L21v1 L22 \|a1 −µ1 L11 ≤v1 ≤b1µ1 L11 ) (215) ≈1 R R X r=1 Φ b2 −µ2 −L21˜ v1r L22 −Φ a2 −µ2 −L21˜ v1r L22 Putting this together, we have, Pr(a ≤µ + Lv ≤b) ≈ (216) ≈1 R R X r=1 Φ b1 −µ1 L11 −Φ a1 −µ1 L11 Φ b2 −µ2 −L21˜ v1r L22 −Φ a2 −µ2 −L21˜ v1r L22 Note that unlike the expression in equation (211), equation (216) is continuous and diﬀer-entiable in (µ, Ω, a, b). To identify the multinomial probit model, we typically normalize the β1 = 0. We also assume that εn1 = 0 (so that Ωjk = 0 if j = 1 or k = 1) and we assume that Ω22 = 1. A slight variation of the multinomial probit model allows for choice speciﬁc covariates, unj = β′ jxnj + εnj , and similar to the conditional logit model, there are diﬀerent considerations for identiﬁcation of β0. 67 6.5 Substantive Eﬀects When computing eﬀect sizes for the multinomial logit and conditional logit models, the easiest approach is to compute Pr(y = j\|x) for diﬀerent possible values of x and observe the probabilities of all the choices. Like the ordered probit model, marginal eﬀects will often not be very informative summaries because marginal eﬀects will be non-monotonic in the covariates. To compute substantive eﬀects for the multinomial probit model, we have to use simu-lation. Here, two approaches are possible—the ﬁrst based on equation (211) and the second based on the GHK simulator. To illustrate the ﬁrst approach, let (ˆ β, ˆ Ω) denote the maximum likelihood estimator and suppose we would like to compute the probabilities Pr(y = 1\|x), Pr(y = 2\|x), ..., Pr(y = J\|x), for a particular value of x. We have, Pr(y = j\|x) = Z ε:ˆ β′ jx+εj≥ˆ β′ kx+εk∀k̸=j φ(ε; 0, ˆ Ω)dε (217) = Z ε 1{ˆ β′ jx + εj ≥ˆ β′ kx + εk∀k ̸= j}φ(ε; 0, ˆ Ω)dε ≈1 R R X r=1 1{ˆ β′ jx + ˜ εrj ≥ˆ β′ kx + ˜ εrk∀k ̸= j} where ˜ εr are draws from the N(0, ˆ Ω) distribution, again relying on the law of large numbers to deliver an approximation. This approach will approximate Pr(y = j\|x), but will deliver an approximation that is not continuous in the parameters (ˆ β, ˆ Ω). This, in turn, means that the delta method cannot be used to perform inferences on the substantive eﬀects. If inferences are desired, the bootstrap can be used (discussed further in subsection 10.2). Suppose that √ N(ˆ θ −θ0) dist. − →N(0, V0) where ˆ V prob. − →V0. Consider a function C that is not necessarily continuous and diﬀerentiable. Let {˜ θ}S s=1 be a sample from N(ˆ θ, ˆ V ). We have that, Pr ˆ q(C(˜ θ), α/2) ≤C(θ0) ≤ˆ q(C(˜ θ), 1 −α/2) prob. − →α (218) as S →∞, where ˆ q(x, α) represent empirical quantiles of the data x. An alternative procedure for computing Pr(y = j\|x) is to use the GHK simulator. Since the GHK simulator is continuous and diﬀerentiable in the model parameters, the delta method can be applied to perform inferences on the substantive eﬀects. 68 6.6 Measures of Model Fit For all these models, we do not simply have a censored version of the linear regression model, so we can’t apply a McKelvey-Zavoina R-Squared. We can however consider a McFadden R-squared, by including only a choice speciﬁc constant in the multinomial probit and multi-nomial logit models. If the model we are computing the McKelvey R-Squared for does not have a choice speciﬁc intercept, there is no guarantee that the R-squared will be positive (this could be the case for the conditional logit model, or the multinomial probit model with choice-speciﬁc intercepts). 6.7 Suggested Reading 6.7.1 Background Greene (2000) Kennedy (1992) Train (1992) 6.7.2 Examples Alvarez and Nagler (1995) Burden et al. (2014) Cox and McCubbins (1993), Chapter 7 Kayser and Peress (2012) 7 Count Models 7.1 Poisson Regression The Poisson distribution is given by, pX(x; λ) = Pr(X = x; λ) = λxe−λ x! (219) for x = 0, 1, .... The Poisson regression model is used to model a dependent variable that can only take on non-negative integers as values. Note that, 69 E[X] = ∞ X x=0 xpX(x) = ∞ X x=0 xλxe−λ x! = ∞ X x=1 xλxe−λ x! = e−λλ ∞ X x=1 λx−1 (x −1)! (220) = e−λλ ∞ X y=0 λy y! = e−λλeλ = λ The Poisson regression model speciﬁes the mean to be eβ′xn—the exponential transformation here is used to force the mean to be positive. The Poisson regression model assumes that, Pr(yn = j\|xn; β) = (eβ′xn)je−eβ′xn j! (221) The log-likelihood is given by, l(β) = N X n=1 log (eβ′xn)yne−eβ′xn yn! = N X n=1 ynβ′xn −eβ′xn −log yn! (222) We can compute the marginal eﬀect as, E[yn\|xn] = eβ′xn (223) ∂ ∂xnk E[yn\|xn] = eβ′xnβk (224) We can thus interpret the coeﬃcient βk as the eﬀect of a one-unit change on the dependent variable in percentage terms. Consider the following property of the Poisson distribution, E[X(X −1)] = ∞ X x=0 x(x −1)pX(x) = ∞ X x=0 x(x −1)λxe−λ x! = ∞ X x=2 x(x −1)λxe−λ x! (225) = e−λλ2 ∞ X x=1 λx−2 (x −2)! = e−λλ2 ∞ X y=0 λy y! = e−λλ2eλ = λ2 E[X(X −1)] = E[X2] −E[X] (226) E[X2] = E[X(X −1)] + E[X] = λ2 + λ (227) 70 V ar(X) = E[X2] −E[X]2 = λ2 + λ −λ2 = λ (228) The Poisson distribution has a variance equal to its’ mean, a property that is unlikely to hold. 7.2 Negative Binomial The negative binomial model is designed to allow for over-dispersion of count data, or to allow the variance to be greater than the mean. The negative binomial distribution is given by, pX(x; λ, α) = Pr(X = x; λ, α) = Γ(α−1 + x) Γ(α−1)Γ(x + 1) α−1 α−1 + λ α−1 λ α−1 + λ x (229) for x = 0, 1, ... where λ > 0, α > 0, and Γ denotes the gamma function, Γ(x) = Z ∞ u=0 ux−1e−udu (230) The negative binomial distribution has mean E[X] = λ and variance V ar(X) = λ(1+αλ) > λ. The parameter α measures the degree of over-dispersion (note that we cannot have under-dispersion). The negative binomial regression model has, Pr(yn = j\|xn; β, α) = Γ(α−1 + j) Γ(α−1)Γ(j + 1) α−1 α−1 + eβ′xn α−1 eβ′xn α−1 + eβ′xn j (231) and the log-likelihood is given by, l(β, α) = N X n=1 log Γ(α−1 + yn) Γ(α−1)Γ(yn + 1) α−1 α−1 + eβ′xn α−1 eβ′xn α−1 + eβ′xn yn (232) Similarly to the Poisson regression model, marginal eﬀects can be interpreted in terms of percentage changes in the dependent variable. 71 7.3 Zero-Inﬂated Poisson Regression An alternative count model is the zero-inﬂated Poisson regression. The zero-inﬂated model presumes that the data-generating process for zero outcomes is diﬀerent (this can be useful if there are many zeros in the data). To derive the zero-inﬂated model, assume that there is a π probability of observing a zero and a 1 −π probability of observing a Poisson random variable. We have the following distribution for a zero-inﬂated Poisson random variable, pX(x; λ) = Pr(X = x; λ, π) = π1{x = 0} + (1 −π)λxe−λ x! (233) for x = 0, 1, .... We can specify λ as eβ′xn and π and eγ′zn 1+eγ′zn to obtain, Pr(yn = j\|xn, zn) = 1{j = 0} eγ′zn 1 + eγ′zn + 1 1 + eγ′zn (eβ′xn)je−eβ′xn j! (234) Here, the notation allows for the parameters λ and π to depend on diﬀerent covariates. We allows for xn and zn to contain some (or all) of the same covariates. We have that the mean and variance of a zero-inﬂated Poisson random variable is given by, E[X] = λ(1 −π) (235) V ar(X) = λ(1 −π)(1 + λπ) > E[X] (236) It follows that, E[yn\|xn, zn] = eβ′xn 1 + eγ′zn (237) There is no simple way to interpret marginal eﬀects in a zero-inﬂated Poisson regression model. Instead, let us write xn = (xn1, xn2) and zn = (xn1, xn3) where xn1, xn2, and xn3 are distinct covariates. We have, E[yn\|xn1, xn2, xn3] = eβ′ 1xn1+β′ 2xn2 1 + eγ′ 1xn1+γ′ 2xn3 (238) ∂ ∂xn1E[yn\|xn1, xn2, xn3] = (1 + eγ′ 1xn1+γ′ 2xn3)eβ′ 1xn1+β′ 2xn2β1 −eβ′ 1xn1+β′ 2xn2eγ′ 1xn1+γ′ 2xn3γ1 (1 + eγ′ 1xn1+γ′ 2xn3)2 (239) 72 = eβ′ 1xn1+β′ 2xn2 (1 + eγ′ 1xn1+γ′ 2xn3) β1 + (β1 −γ1)eγ′ 1xn1+γ′ 2xn3 (1 + eγ′ 1xn1+γ′ 2xn3) ∂ ∂xn2E[yn\|xn1, xn2, xn3] = β2 eβ′ 1xn1+β′ 2xn2 1 + eγ′ 1xn1+γ′ 2xn3 = β2E[yn\|xn1, xn2, xn3] (240) ∂ ∂xn3E[yn\|xn1, xn2, xn3] = − γ2eβ′ 1xn1+β′ 2xn2 (1 + eγ′ 1xn1+γ′ 2xn3)2 (241) Covariates that only appear in the speciﬁcation for λ can be interpreted as percentage changes on the dependent variable. For the other formulas, the marginal eﬀects depend on the other variables. We can interpret γ as the marginal eﬀect of the IVs on the number of “non-Poisson zeros” and we can interpret β as the marginal eﬀect of the IVs on the “Poisson expected value”, but this type of interpretation is silly. Instead, we can calculate E[yn\|xn, zn] = eβ′xn 1+eγ′zn under diﬀerence scenarios for (xn, zn). In principle, we can calculate the probability of observing any outcome. For example, Pr(yn = 0\|xn, zn) = eγ′zn 1 + eγ′zn + 1 1 + eγ′zn e−eβ′xn (242) Using a similar approach, we can derive a zero-inﬂated version of the negative binomial model. The zero-inﬂated negative binomial model has the same expected value as the zero-inﬂated Poisson, but has a diﬀerent variance. 7.4 Semi-parametric Analysis of the Count Regression Model Consider the MLE for the Poisson regression model, ˆ β = arg max β 1 N N X n=1 ψ(xn, yn; β) = arg max β 1 N N X n=1 ynβ′xn −eβ′xn −log yn! (243) Consistency of ˆ β under the assumption that Pr(yn = j\|xn; β) = (eβ′xn)je−eβ′xn j! follows from the consistency of MLEs. Here, we will establish that ˆ β is consistent under a weaker set of conditions—we assume that E[yn\|xn; β] = eβ′xn. We will treat ˆ β as an M-estimator. Recall that the identiﬁcation condition for an M-estimator is β0 = arg max β E[ψ(xn, yn; β)]. In this case, we have ψ(xn, yn; β) = yn log(eβ′xn) −eβ′xn −log yn!. We have, 73 E[ψ(xn, yn; β)] = E[ynβ′xn −eβ′xn −log yn!] (244) We use ﬁrst order conditions to β′, we obtain, 0 = E[(yn−eβ′xn)xn] = E[E[(yn−eβ′xn)xn\|xn]] = E[xn(E[yn\|xn]−eβ′xn)] = E[xn(eβ′ 0xn−eβ′xn)] (245) It is clear that β = β0 is a solution to the ﬁrst order conditions. To show that it is the only solution, consider the Taylor expansion, eβ′xn = eβ′ 0xn + e ¯ β(xn)′xnx′ n(β −β0) = eβ′ 0xn + e ¯ β(xn)′xnx′ n(β −β0) (246) We have, Exnx′ ne ¯ β(xn)′xn = 0 (247) There will be a unique solution if E[xnx′ ne ¯ β(xn)′xn] has full rank. We will demonstrate that E[xnx′ ne ¯ β(xn)′xn] is positive deﬁnite (and hence has full rank). Consider z ̸= 0. We have, z′E[xnx′ ne ¯ β(xn)′xn]z = E[z′xnx′ nze ¯ β(xn)′xn] = E[(z′xn)2e ¯ β(xn)′xn] (248) We have e ¯ β(xn)′xn > 0 and (z′xn)2 ≥0. Strict positivity of E[(z′xn)2e ¯ β(xn)′xn] will follow if (z′xn)2 > 0 with positive probability. Suppose that z′xn = 0 for z ̸= 0 with probability 1. Then one of xn is a perfect linear combination of the other covariates with probability 1. If we assume that this is not the case, then β0 is the unique minimizer of E[ynβ′xn−eβ′xn −log yn!]. Using this and certain technical conditions, we can establish that ˆ β is consistent. Applying this result, the Poisson regression estimator is consistent as long as the con-ditional mean is correctly speciﬁed and we can compute the asymptotic variance using the sandwich estimator—or robust standard errors—were the standard errors are robust to zero-inﬂation, over/under-dispersion, or any other deviation from the Poisson distribution for the dependent variable. This result allows to avoid modeling the exact distribution of the vari-able, which may make calculating marginal eﬀects much more diﬃcult. With the Poisson regression model with robust standard errors, we can compute marginal eﬀects in terms of percentage changes on the dependent variable. A similar result holds for the negative binomial regression model. The limiting objective function is given by, 74 E " log Γ(α−1 + yn) Γ(α−1)Γ(yn + 1) α−1 α−1 + eβ′xn α−1 eβ′xn α−1 + eβ′xn yn# (249) Ignoring terms that do not involve β, the limiting maximizer will maximize, E h ynβ′xn −(yn + α−1) log α−1 + eβ′xni (250) Taking ﬁrst-order conditions, we have, E ynxn −(yn + α−1) α−1 + eβ′xn eβ′xnxn = E eβ′ 0xn −eβ′ 0xn + α−1 eβ′xn + α−1 eβ′xn xn (251) = E eβ′ 0xn −eβ′xn eβ′xn + α−1 α−1xn = 0 It is clear that for all α−1, β = β0 is a solution. To demonstrate that it is the only solution, we take a Taylor expansion, eβ′xn = eβ′ 0xn + e ¯ β(xn)′xnx′ n(β −β0) (252) We have, E " e ¯ β(xn)′xnxnx′ n eβ′xn + α−1 # (β −β0) = 0 (253) Consider, z′E " e ¯ β(xn)′xnxnx′ n eβ′xn + α−1 # z = E " e ¯ β(xn)′xn(z′xn)2 eβ′xn + α−1 # ≥0 (254) with strict positivity if (z′xn)2 > 0 with positive probability. This is guaranteed to be the case, implying that β = β0 is the only solution to the ﬁrst -order conditions. Hence, the negative binomial regression estimator of β0 will be consistent as long as the conditional mean is correctly speciﬁed. 7.5 Model Fit for Parametric Models If we estimate a Poisson regression model, we can test the ﬁt of the model using the following goodness of ﬁt statistic, 75 G = N X n=1 (yn −ˆ µn)2 ˆ µn (255) where ˆ µn = e ˆ β′xn. Under the null hypothesis that yn\|xn ∼Poisson(eβ′xn), we have that G dist. − →χ2 N−K where K is the number of estimated parameters. In principle, if we fail to reject the Poisson model, we would be justiﬁed in using the Poisson model without robust standard errors. Otherwise, we would have to consider robust standard errors, the negative binomial model, the zero-inﬂated Poisson model, etc. 7.6 Suggested Reading 7.6.1 Background Cameron and Trivedi (2001) Greene (2000) Kennedy (1992) King (1989) 7.6.2 Examples Nepal, Bohara and Gawande (2011) Weghorst and Lindberg (2013) Wilson and Piazza (2013) 8 Censoring, Selection, and Truncation 8.1 The Tobit Model (Censored Regression) Consider the latent variable model given by y∗ n = β′xn + εn where εn ∼N(0, σ2). Suppose that we observe a censored version of y∗ n—we observe, yn = ( y∗ n, y∗ n ≥0 0, y∗ n < 0 (256) 76 The random variable yn is “mixed”—it is neither discrete nor continuous. This censored regression model is called the Tobit model. We can determine that the discrete part is characterized by, Pr(y∗ n < 0\|xn) = Pr(β′xn + εn < 0\|xn) = Pr(εn < −β′xn\|xn) = Φ( −β′xn σ ) (257) and the continuous part is characterized by, f(y∗ n\|xn) = 1 σ √ 2πe−1 2 (y∗ n−β′xn)2/σ2 (258) We can combine these two expressions to form the log-likelihood, l(β, σ2) = N X n=1 1{yn > 0} log 1 σ √ 2πe−1 2 (yn−β′xn)2 + 1{yn = 0} log Φ −β′xn σ (259) The above does not qualify as derivation of the Tobit model because we simply assumed we can combine the discrete part and continuous part without proof. Suppose that X ∼TN(µ, σ2). One can show that, E[X\|a < X < b] = µ −σ φ b−µ σ −φ a−µ σ Φ b−µ σ −Φ a−µ σ (260) When a = 0 and b = ∞, we obtain, E[X\|X > 0] = µ + σ φ µ σ Φ µ σ (261) Note that, E[y∗ n\|xn] = β′xn (262) E[yn\|xn] = E[yn\|xn, yn = 0] Pr(yn = 0\|xn) + E[yn\|xn, yn > 0] Pr(yn > 0\|xn) (263) 77 = E[y∗ n\|xn, y∗ n > 0] Pr(y∗ n > 0\|xn) =  β′xn + σ φ β′xn σ Φ β′xn σ  Φ β′xn σ = Φ β′xn σ β′xn + σφ β′xn σ The estimator of β0 directly reveals the marginal eﬀect of an independent variable xk on the latent (uncensored) dependent variable. The marginal eﬀect of an independent variable on the (censored) dependent variable is given by equation (263). A generalization of the Tobit model allows for censoring of the dependent variable on both sides at arbitrary points, a and b. In the more general case, we observe yn =      a, y∗ n < a y∗ n, a ≤y∗ n ≤b b, y∗ n < b (264) The random variable yn is again mixed. We have, Pr(yn = a\|xn) = Pr(y∗ n < a\|xn) = Pr(β′xn + εn < a\|xn) (265) = Pr(εn < a −β′xn\|xn) = Φ( a−β′xn σ ) Pr(yn = b\|xn) = Pr(y∗ n > b\|xn) = Pr(β′xn + εn > b\|xn) = Pr(εn > b −β′xn\|xn) (266) = 1 −Pr(εn < b −β′xn\|xn) = 1 −Φ( b−β′xn σ ) f(y∗ n\|xn) = 1 σ √ 2πe−1 2 (y∗ n−β′xn)2/σ2 (267) We can combine these expressions to form the log-likelihood, l(β, σ2) = N X n=1 1{a ≤yn ≤b} log 1 σ √ 2πe−1 2 (yn−β′xn)2 + 1{yn = a} log Φ a −β′xn σ (268) 78 + 1{yn = b} log 1 −Φ b −β′xn σ In the general case (with arbitrary censoring from above and below), we have E[y∗ n\|xn] = β′xn (269) E[yn\|xn] = a Pr(yn = a\|xn) + b Pr(yn = b\|xn) + Pr(a < yn < b\|xn)E[yn\|a < yn < b] (270) = aΦ a−β′xn σ + b 1 −Φ b−β′xn σ + Φ b−β′xn σ −Φ a−β′xn σ  β′xn −σ φ b−β′xn σ −φ a−β′xn σ Φ b−β′xn σ −Φ a−β′xn σ   Suppose that y∗ n is the number of hours people desire to work (which may depend on their wage, number of children, etc.). If people’s desired hours are less than zero, we observe them working zero hours. We let yn denote the number of hours people work. If we are interested in the eﬀect of an independent variable on the number of desired hours, we could simply look at βk. If we are interested in the eﬀect of an independent variable on the number of hours actually worked, we would have to consider E[yn\|xn] = Φ β′xn σ β′xn + σφ β′xn σ . We have, ∂ ∂xnk E[yn\|xn] = Φ β′xn σ βk (271) Now, suppose that we were to apply OLS to yn. We would obtain, ˆ βOLS prob. − →E[xnx′ n]−1E[xnyn] (272) E[xnyn] = E[xnE[yn\|xn]] (273) E[yn\|xn] = Φ β′ 0xn σ0 β′ 0xn + σ0φ β′ 0xn σ0 (274) 79 ˆ βOLS prob. − →E[xnx′ n]−1E[Φ β′ 0xn σ0 xnx′ nβ0 + xnσ0φ β′ 0xn σ0 ] ̸= β0 (275) The OLS estimator of β0 would be a biased estimator for the marginal eﬀect of the inde-pendent variables on the desired number of work hours. It could be considered a reasonable estimator for the marginal eﬀect of the independent variables on the actual number of work hours. When considering whether to apply a tobit model, it is not enough to simply examine whether there is a point mass at zero. Consider ﬁrst a case where the data collection method censors the dependent variable. Suppose, for example, that we are interested in the eﬀect of education on income, but our dataset sets all incomes above $250,000 equal to $250,000. If the data is censored in this way, we would almost certainly be interested in estimating a tobit model since we are interested in the eﬀect of education on income, not of education on censored income. Instead, consider the eﬀect of ideological location on PAC contributions to the party lead-ership Jenkins and Monroe (2012). Here, there are multiple possibilities. We could imagine the latent intentions of the party leadership involves punishing members with negative con-tributions, but this is not actually feasible. In this case, a Tobit model could be used to assess the eﬀect of ideological positions on intended PAC contributions. Alternatively, if we are not interested in intentions or if this model of intentions does not make complete sense, but an actual contribution model makes sense, a Tobit model would not be appropriate. We could also consider wages, where we observe a point mass at the minimum wage. We could attempt to explain wages by assuming that each individual has a speciﬁed labor productivity and that individuals are paid in proportion to their labor productivity. In this case, one may think that since an employer cannot pay less than the minimum wage, the process would be censored at the minimum wage. In this case, it would not make sense to apply a Tobit because according to this theory, individuals who are less productive than the minimum wage should simply not be hired (which suggested truncation rather than censoring, considered below). 8.2 Truncated Regression Consider again the model yn = β′xn + εn where εn ∼N(0, σ2), but suppose that we observe yn if yn > 0 and observe nothing otherwise. In particular, we do not observe covariates when yn < 0. This is the truncated regression model. The density of the data is given by the 80 truncated normal distribution, f(yn\|xn) = 1 σ √ 2πe−1 2 (yn−β′xn)2/σ2 Φ( β′xn σ ) (276) We have a log-likelihood of, l(β, σ) = N X n=1 log 1 σ √ 2πe−1 2 (yn−β′xn)2/σ2 Φ( β′xn σ ) (277) = N X n=1 −log σ −1 2 log(2π) −1 2(yn −β′xn)2/σ2 −log Φ( β′xn σ ) This type of model could be used if we observed a sample of hours worked (as before), but we collected the data for a sample of workers. In this case, we would not observe individuals working zero hours because they prefer to work less than zero hours. We could also let wn indicate wage, w the minimum wage and yn = wn −w is the amount above the minimum wage an individual makes. In this case, we would never observe wages for those whose productivity is below the minimum wage and hence are not employed. If we wanted to calculate the marginal eﬀect of an IV on hours of work desired, we would use, ∂ ∂xnk E[yn\|xn] = βk (278) If we wanted to calculate the marginal eﬀect of an IV on hours work desired among the employed, we would calculate, ∂ ∂xnk E[yn\|yn > 0, xn] = ∂ ∂xnk  β′xn + σ φ β′xn σ Φ β′xn σ   (279) = βk   1 + Φ β′xn σ φ′ β′xn σ −φ β′xn σ 2 Φ β′xn σ 2    8.3 The Heckman Selection Model Consider the model yn = β′xn + εn, but suppose that we only observe yn if wn = 1 where wn = 1 ⇔w∗ n = γ′zn + ηn ≥0 and where, 81 (εn, ηn) ∼N " 0 0 # , " σ2 ε σεη σεη σ2 η #! (280) For identiﬁcation purposes, we normalize σ2 η = 1. We therefore simply write σ2 ε as σ2 and we write σεη as σρ. We can observe (yn, 1) or (., 0). We have,10 f(yn, wn = 1) = Z (εn,ηn):γ′zn+ηn≥0,yn=β′xn+εn f(εn, ηn)d(εn, ηn) (281) = φ yn−β′xn σ σΦ(γ′zn) Φ ρ yn−β′xn σ + γ′zn p 1 −ρ2 ! Pr(wn = 0) = Z ηn:γ′zn+ηn<0 f(ηn)dηn = 1 −Φ(γ′zn) (282) The likelihood for the Heckman selection model is given by, l(β, γ, σ2, ρ) = N X n=1 wn log φ yn−β′xn σ σΦ(γ′zn) Φ ρ yn−β′xn σ + γ′zn p 1 −ρ2 ! +(1−wn) log(1−Φ(γ′zn)) (283) To see an example, suppose that we are interested in the eﬀect of education on wages. We will only observe wages among the employed, so we let yn be wages and wn = 1 indicate employment. We can show that,11 E[yn\|xn, zn, wn = 1] = β′xn + ρσ φ(−γ′zn) 1 −Φ(−γ′zn) = β′xn + ρσλ(−γ′zn) (284) Here, λ(u) = φ(u) 1−Φ(u) is called the Inverse Mills Ratio. Using this, we can derive the large sample bias of OLS when applied to sample selected data. We have, ˆ β prob, − →E[xnx′ n]−1E[xnyn\|wn = 1] = E[xnx′ n]−1E[xnE[yn\|xn, zn, wn = 1]] (285) = E[xnx′ n]−1E[xnβ′xn + ρσλ(−γ′zn)] = β + ρσE[xnx′ n]−1E[xnλ(−γ′zn)] The term ρσE[xnx′ n]−1E[xnλ(−γ′zn)] indicates the bias of OLS when applied to sample 10See Appendix A.3 for a derivation. 11See Appendix A.4 for a derivation. 82 selected data. It is zero when ρ = 0 and will generally be non-zero otherwise. Returning to the example, suppose that intelligence is unobserved. It is likely to enter both equations—more intelligent individuals are more likely to be employed (holding con-stant the covariates) and more likely to earn higher wages when employed (holding constant the covariates). This is likely to lead to bias in the coeﬃcients in the wage equation. There is also a variation of the Heckman selection where the DV is binary. The outcome is modeled using a latent variable, y∗ n = β′xn + εn where we observe yn = 1{y∗ n ≥0}. The selection equation is also governed by a latent variable, w∗ n = γ′zn +ηn ≥0 where we observe wn = 1{w∗ n ≥0}. We assume that (εn, ηn) are bivariate normal, (εn, ηn) ∼N " 0 0 # , " 1 ρ ρ 1 #! (286) where the variances of εn and ηn are normalized for identiﬁcation purposes. We only observe yn is wn = 1—here there are three things we can observe, (yn, wn) = (0, 1), (yn, wn) = (0, 0), and (yn, wn) = (., 0). We can easily calculate, Pr(yn = ., wn = 0\|xn, zn; β, γ, ρ) = Pr(wn = 0) = 1 −Φ(γ′zn) (287) The expressions for Pr(yn = 0, wn = 1\|xn, zn; β, γ, ρ) and Pr(yn = 1, wn = 1\|xn, zn; β, γ, ρ) are more complicated as they involve double integrals. We have, Pr(yn = 1, wn = 1\|xn, zn; β, γ, ρ) = Pr(y∗ n ≥0, w∗ n ≥0\|xn, zn; β, γ, ρ) (288) = Pr(β′xn+εn ≥0, γ′zn+ηn ≥0\|xn, zn; β, γ, ρ) = Pr(εn ≥−β′xn, ηn ≥−γ′zn\|xn, zn; β, γ, ρ) = Z ∞ ε=−β′xn Z ∞ η=−γ′zn f (ε, η); " 0 0 # , " 1 ρ ρ 1 #! d(ε, η) We can similarly demonstrate that, 83 Pr(yn = 0, wn = 1\|xn, zn; β, γ, ρ) = Z −β′xn ε=−∞ Z ∞ η=−γ′zn f (ε, η); " 0 0 # , " 1 ρ ρ 1 #! d(ε, η) (289) Using these expressions, we can form the log-likehood as, l(β, γ ρ) = N X n=1 (1 −wn)log (1 −Φ(γ′zn)) (290) + wnynlog Z ∞ ε=−β′xn Z ∞ η=−γ′zn f (ε, η); " 0 0 # , " 1 ρ ρ 1 #! d(ε, η) ! + wn(1 −yn)log Z −β′xn ε=−∞ Z ∞ η=−γ′zn f (ε, η); " 0 0 # , " 1 ρ ρ 1 #! d(ε, η) ! Unlike the expression for the Heckman selection model with a continuous dependent variable, the integrals in the likelihood for the Heckman selection model with a binary depe-dendent variable do not reduce to the simpler expression. We thus need a way of computing the integrals numerically. The GHK-Simulator (also used for estimating multinomial probit models) provides one approach. There is another variation (called a switching model) where we observe one of two equa-tions depending on the outcome of the selection process. 8.4 Applications Application 8.1 (The Eﬀect of Election Proximity on Punitive Sentences). Application 8.2 (The Eﬀect of Ideological Location on Campaign Contributions from Party Leadership). Application 8.3 (The Politics of EU Enlargement). Plumper, Schneider and Troeger (2006) describe a two-step process by which countries can join the European Union (EU). Countries must ﬁrst apply and the EU next decides whether to allow the countries to join the EU. One hypothesize that countries that are more democratic are more likely to be allowed to join on the EU. One could imagine testing this using a logit model where being admitted to the EU is the dependent variable, which is mea-sured on a sample of countries that applied to the EU. This approach could potentially yield 84 a biased and inconsistent estimate of the eﬀect of level of democracy on EU admittance—if we are interested in whether level of would increase EU acceptance in a world in which every country applied—because of selection bias (which, in this case might be more accurately called selection inconsistency). It would yield a consistent estimate of EU acceptance con-ditional on applying, but this is arguably less theoretically relevant. We could imagine an omitted variable present in both stages—for example, legal sophistication. Countries with a more sophisticated legal profession may be more inclined to apply. They may be also more eﬀective in placating the EU to the point of being admitted. If democracies have on average more legal sophistication, the eﬀect of level of democracy we estimate using a logit model might pick up the eﬀect of legal sophistication. One could consider two alternative analyses. First, one could code the dependent variable based on whether the country was admitted to the EU, coding both countries that did not apply and countries that were denied admittance as zeros. In this case, applying a logit model would yield a consistent estimate of the eﬀect of the level of democracy on admittance to the EU. This analysis would not tell us whether this eﬀect occurred because democracies were diﬀerentially likely to apply, or whether democracies would have been diﬀerentially likely to be accepted in a world in which every country applied. Second, one could estimate just the selection stage—that is, we could use a logit model to consistently estimate the eﬀect of level of democracy on apply for EU ascension. Suppose we found that (a) democratic countries were more likely to be admitted to the EU, (b) democratic countries were more likely to apply, and that (c) democratic countries were more likely to be admitted conditional on applying. It might be tempting to conclude that the mechanism by which democratic countries are disproportionably admitted to the EU is a combination of democratic countries being more likely to apply and the EU being more favorable towards democratic countries. This inference, however, might not be justiﬁed because it could be that there are unobservables correlated with level of democracy that increase the likelihood that a country applied to the EU and increases the favorability of the EU towards the country (i.e. selection inconsistency is present). Application 8.4 (The Eﬀect of Distributive Spending on Political Participation). Chen (2013) examines the eﬀects of receiving disaster aid on voter turnout. He hypothe-sizes that co-partisans of the president will increase their turnout and non-co-partisans will decrease their turnout when having their aid requests met. In a probit analysis on the sample of disaster relief applicants, he ﬁnds evidence for this claim. Chen considers the possibility that those who apply for aid are diﬀerent than those who do not apply for aid and worries 85 about “potentially limiting the external validity of the article’s ﬁndings”. Chen proposes a Heckman selection model where the ﬁrst stage is whether an individual applied for aid and the second stage captures the eﬀect of receiving aid on turnout. Chen uses a number of exclusion restrictions—median income, home value, gender, race, age, and wind speed are all included in the selection equation, but excluded from the outcome equation. Of these, only wind speed seems potentially valid. Beyond this, there is the interpretation of the selection model. It will provide the eﬀect of receiving aid on turnout if everybody applied for aid. This is probably not an interesting counterfactual. An partisan member of the government would likely be more interested in the eﬀect of receiving aid on turnout among those that applied (to determine whether politicizing aid is eﬀective is mobilizing voters). Application 8.5 (International Treaties and Current Account Restrictions). Another example of a potential selection problem that has seem some coverage in the literature is whether international treaties aﬀect the behavior of countries. Simmons (2000) argued that countries that sign Article VII are less likely to impose current account restric-tions. She determines this using a probit model where being an Article VII signatory is the main independent variable and current account restrictions is the dependent variable. Stein (2005) argues that treaties do not constrain but instead screen. In particular, Stein argues that countries who were otherwise more likely to refrain from placing current account restrictions are exactly those countries who are more likely to sign. She tests this using a switching model and ﬁnds that (i) the unobservables in the selection and outcome equations are negatively correlated and (ii) that once this is taken into account, countries that sign treaties are no less likely to impose current account restrictions. Von Stein’s model however does not justify her exclusion restrictions. It is worthwhile to comment on the diﬀerence between selection models and endogeneity. Consider the eﬀect of education on wages. More intelligent individuals are likely to spend more time in school and also likely to receive higher wages. Failing to control for intelligence when estimating the eﬀect of education on wages may lead to an upward bias on the education coeﬃcient. We may instead conceive of education as a dummy variable, indicating whether an individual chooses to attend college. We may even be tempted to say that individuals “select” into whether to attend college. The diﬀerence between this and a Heckman selection model is that while we observe wages for individuals who choose not to attend college, the Heckman model assumes that we do not observe the dependent variable for non-selected individuals. The example from Stein (2005) above somewhat confuses this distinction—in a switching model, we observe the dependent variable for selected and un-selected observations. 86 In fact, we could have applied an IV estimator to Stein’s application. In a typical IV setup, we would be assuming that an endogenous IV shifts the mean level of the DV. In a switching model, we have an entire diﬀerent equation for selected and unselected individuals. Chen’s (2013) presents an example where both selection and reverse causality could be at play— individual may select into apply for aid, for incumbent government may choose to reward individuals they believe are likely to turn out. 8.5 Suggested Reading 8.5.1 Background Greene (2000) Kennedy (1992) 8.5.2 Examples Chen (2013) Huber and Gordon (2004) Jenkins and Monroe (2012) Plumper, Schneider and Troeger (2006) Simmons (2000) Stein (2005) 9 Duration Models Duration data refers to data that records the length of time until an event happens. Examples could be the length of time before a cabinet government dissolves, the length of time between civil wars, or the length of time an individual is unemployed. Duration data can be treated as discrete or continuous. Duration data cannot take on negative values. Beyond this, duration data is often censored—some individuals may still be unemployed when our data set ends or cabinet governments may be dissolved because of a mandatory election. Duration data consists of a length of time (discrete or continuous) along with a variable indicating whether an observation is censored. 87 9.1 Survival Functions and Hazard Rates Consider a continuous (uncensored) duration T ∗ n. We can model this duration using the CDF, F(t) = Pr(T ∗ n ≤t). We can model it using the survival function S(t) = 1−F(t) = Pr(T ∗ n ≥t). We can use the density function f(t). Since S(t) = 1 −F(t), we have that S′(t) = −f(t). Finally, we can use the hazard rate, h(t) = lim δ→0 Pr(t ≤T ∗ n ≤t + δ\|yn ≥t) δ = lim δ→0 Pr(t ≤T ∗ n ≤t + δ, T ∗ n ≥t) δ Pr(T ∗ n ≥t) (291) = lim δ→0 Pr(t ≤T ∗ n ≤t + δ) δ Pr(T ∗ n ≥t) = lim δ→0 F(t + δ) −F(t) δS(t) = F ′(t) S(t) = f(t) S(t) = −S′(t) S(t) Deﬁne the integrated hazard to be H(t) = −log S(t). The implies that S(t) = e−H(t). Diﬀerentiating implies that S′(t) = −e−H(t)H′(t) or S′(t) = −S(t)H′(t) which demonstrates that H′(t) = h(t). From this, we obtain H(t) = R t u=0 h(u)du. Let Dn = 1 indicate that the duration is censored, let Tn indicate a duration, and let Xn be a vector of covariates. Let τn be the censoring point for observation n. We have T ∗ n = (1 −Dn)Tn + Dnτn. 9.2 Parametric Models If we assume a constant hazard rate, we have, h(t) = f(t) S(t) = −S′(t) S(t) = λ (292) This yields the diﬀerential equation, S(t) = −1 λS′(t) (293) which has solution S(t) = Ke−λt. The condition that S(0) = 1 implies that S(t) = e−λt which is the exponential model. It corresponds to an exponential distribution for the durations. We have that, E[T ∗ n] = λ−1. Note that F(t) = 1 −e−λt and f(t) = λe−λt. Specify λ = e−β′Xn. For any Dn = 0, we have that the duration is characterized by the density, f(Tn) = e−β′Xne−e−β′XnTn, and when Dn = 1, we have that the probability of observing a censored observation is given by S(Tn) = e−e−β′XnTn. We have the following log-likelihood for the exponential duration model, 88 l(β) = N X n=1 (1 −Dn) log e−β′Xne−e−β′XnTn + Dn log e−e−β′XnTn (294) = N X n=1 −(1 −Dn)β′Xn −e−β′XnTn Notice that E[T ∗ n\|Xn] = (e−β′Xn)−1 = eβ′Xn which allows us to interpret the model in terms of percentage changes in the expected value of the uncensored duration (an approximation which is based on small changes in the independent variable). The exponential model can be too restrictive however. More generally, we have, l(θ) = N X n=1 (1 −Dn) log f(Tn\|Xn; θ) + Dn log S(Tn\|Xn; θ) (295) We can consider the following alternative models, Weibull: f(t; λ, α) = αλαtα−1e−(λt)α, S(t; λ, α) = e−(λt)α, E[T] = λ−1Γ(α−1 + 1) (296) Log-Normal: f(t; µ, σ2) = 1 tσ √ 2πe−1 2 (log t−µ)2/σ2, S(t; µ, σ2) = 1 2 −1 2Φ log t−µ σ √ 2 , E[T] = eµ+σ2/2 (297) For the Weibull model, we have E[Tn\|Xn] = eβ′XnΓ(α−1 + 1), or, log E[Tn\|Xn] = β′Xn + log Γ(α−1 + 1) (298) so that ∂ ∂Xnk log E[Tn\|Xn] = βk. For the log-normal, we have E[Tn\|Xn] = eβ′Xn+σ2/2 so that ∂ ∂Xnk log E[Tn\|Xn] = βk and well. In either case, we can interpret a one unit change in Xnk as causing a βk percent change in the uncensored duration. 89 9.3 The Cox Proportional Hazard Model Suppose that h(Tn) = h0(Tn)eβ′Xn. The Cox-Proportional Hazard model develops an esti-mator for β0 that does not require knowledge of h0, which is called the baseline hazard. To estimate β0, we maximize the following partial log-likelihood, l(β) = N X n=1 (1 −Dn) log eβ′Xn P m:Tm≥Tn eβ′Tm ! (299) Note that log h(Tn) = log h0(Tn)+β′Xn. This implies that we can interpret βk as the increase in the log-hazard that occurs when Xnk increases by 1 unit, or as a percentage change in the hazard rate. An advantage of the Cox Proportional Hazard Model is that it generalizes the exponen-tial and Weibull duration models. To see that this is the case, note that for the exponen-tial model, we have f(Tn) = e−β′Xne−e−β′XnTn and S(Tn) = e−e−β′XnTn, which implies that h(Tn) = f(Tn) S(Tn) = e−β′Xne−e−β′Xn u e−e−β′Xn u = e−β′Xn. Using h0(Tn) = 1, we have h(Tn) = h0(Tn)e−β′Xn, which is a special case of the Cox Proportional Hazard model with the sign of β ﬂipped. For the Weibull model, we have f(Tn; β, α) = α(e−β′Xn)αT α−1 n e−(e−β′XnTn)α and S(Tn; β, α) = e−(e−β′XnTn)α, which implies that h(Tn; β, α) = α(e−β′Xn)αT α−1 n e−(e−β′Xn Tn)α e−(e−β′Xn Tn)α = α(e−αβ′Xn)T α−1 n . If we use the re-parameterization γ = −αβ, we have h(Tn; γ, α) = αT α−1 n eγ′Xn. Using h0(Tn) = αT α−1 n , we have h(Tn; γ, α) = h0(Tn)eγ′Xn, which indicates that the Weibull model is a special case of the Cox Proportional Hazards model. The log-normal model is not a special case of the Cox Proportional Hazards model. A drawback of this Cox Proportional Hazards model is that we cannot calculate substan-tive eﬀects (even in principle) since calculating substantive eﬀects requires knowledge of the baseline hazard h0. To understand why, we can start with h(Tn) = h0(Tn)eβ′Xn. The inte-grated hazard satisﬁes H(t) = R t u=0 h(u)du, which in our case implies H(t) = R t u=0 h(u)du = R t u=0 h0(Tn)eβ′Xndu = eβ′Xn R t u=0 h0(Tn)du = eβ′XnH0(t). Since S(t) = eH(t), we have S′(t) = h(t)eH(t) = eβ′Xnh0(t)eeβ′XnH0(t). Since f(t) = −S′(t), we have f(t) = −eβ′Xnh0(t)eeβ′XnH0(t). Finally, E[Tn] = R ∞ t=0 t(−eβ′Xnh0(t)eeβ′XnH0(t))dt = −eβ′Xn R ∞ t=0 th0(t)eeβ′XnH0(t)dt. We there-fore cannot calculate E[Tn] without known h0(t), which the Cox Proportional Hazards esti-mator does not reveal. We also cannot calculate diﬀerences or ratios of E[Tn] for diﬀerent values of Xn without knowing h0(t). Instead, we can interpret a positive β as indicating an increase in the hazard rate, which implies an decrease in the expected value of the uncensored duration. 90 9.4 Discrete Duration Models With discrete duration models, we model the probability of observing a positive outcome for each unit n and each time period t. We use a logit or probit model, i.e. Pr(Ynt = 1) = F(β′Xnt) where F = Λ or F = Φ. Notice that this approach naturally allows for time varying covariates. We capture duration dependence by including time-since-last-event in the covariates Xnt. For example, we could include t −arg max t∗<t 1{Ynt = 1}. Another common practice is to model duration dependence using cubic splines (Beck, Katz and Tucker, 1998). A third common practice is to include the square and cubic duration terms (Carter and Signorino, 2010). 9.5 Time Varying Covariates in Continuous Models 91 Data Type Time-Varying Censored Covariates? Durations? Discrete Duration No No Count Data (count models, OLS, logged OLS, etc.) / Logit or Probit with Duration Terms Yes Yes No Logit or Probit with Duration Terms Yes Continuous Duration No No Positive Data (OLS, logged OLS, etc.) Yes “Traditional” Duration Models (exponential, weibell, cox) Basically Tobit mixed with model for positive data Yes No Discretize: Logit or Probit with Duration Terms / Yes Don’t Discretize: “Traditional” models with data structures that indicate when IVs change Table 9: Summary of Duration Models 92 Continuous time models can handle time varying covariates. The main diﬀerence is that the time varying covariates do not truly vary continuously. Instead, the data structure only reveals when the covariates change. Based on the integrated hazard rates, we can derive the log-likelihood. 9.6 Suggested Reading 9.6.1 Background Beck, Katz and Tucker (1998) Carter and Signorino (2010) Greene (2000) 9.6.2 Examples Simmons (2000) Stein (2005) 10 Monte Carlo Simulation and the Bootstrap 10.1 Monte Carlo Simulation Consider a probit model, y∗ n = β′ 0xn+εn, yn = 1{y∗ n ≥0}, and εn ∼N(0, 1) where εn are i.i.d. The MLE for β0 should be consistent and asymptotically normal with variance-covariance matrix V0. We can examine the ﬁnite sample properties of this estimator as follows. For each r, draw vector of covariates xn from some distribution for each n and draw εn for each n. We use this to compute y∗ n, use this to further compute yn, and then estimate β0 on this simulated data set using maximum likelihood estimation. Call this estimate ˆ βr and let ˆ Vr be the associated variance covariance matrix. We can investigate the ﬁnite sample properties of the MLE estimator as follows–setting N and letting R be relatively large, [ Bias N k ≈1 R R X r=1 ˆ βr,k (300) 93 \ RMSE N k ≈ v u u t 1 R R X r=1 (ˆ βr,k −β0)2 (301) \ Overconfidence N k ≈ \ RMSE N k 1 R PR r=1 q ˆ Vr,kk (302) \ Coverage N,α k ≈1 R R X r=1 1{ˆ βr −zα/2 q ˆ Vr,kk ≤β0 ≤ˆ βr + zα/2 q ˆ Vr,kk} (303) The MLE will have good ﬁnite sample properties if [ Bias N k is small, \ RMSE N k is small, \ Overconfidence N k is close to 1, and \ Coverage N,α k is close to 1 −α. The same logic can be used when N is large to check whether an MLE is coded properly based on the theoretical large sample properties–we should have [ Bias N k →0, \ RMSE N k →0, \ Overconfidence N k →1, and \ Coverage N,α k →1 −α. 10.2 The Bootstrap Consider an estimator ˆ θ or θ0. The bootstrap provides a way to conduct inference on c(θ0) for some continuous function c using simulation methods. There are a number of diﬀerent versions of the bootstrap. The nonparametric bootstrap works as follows. Consider the data (yn, xn) where the data are i.i.d. We form R new datasets by drawing random individuals from the observed data. Denote these new datasets by (yr, xr). We then compute the same estimator ˆ θr on each of these data sets. We then perform inferences on c(θ0) using the empirical distribution of c(ˆ θr). Let ˆ cr = c(ˆ θr) and let ˆ θ(r) denote the order statistics of ˆ c. We could form a 95% conﬁdence interval for c(θ0) using [ˆ c(.05R), ˆ c(.95R)]. The parametric bootstrap works as follows. We have a parametric model for the data (xn, yn). Suppose that this data is characterized by the CDF F(xn, yn; θ). We obtain an estimate of θ0, which is ˆ θ = ˆ θ(x, y). Here, we could form a 95% conﬁdence interval for c(θ0) using [ˆ c(.05R), ˆ c(.95R)]., the function ˆ θ(x, y) denotes the estimator as a function of the data. We then take new draws (xr n, yr n) from the distribution F(xn, yn; ˆ θ). We obtain new estimates for each random sample, ˆ θr = ˆ θ(xr, yr). We deﬁne ˆ cr = c(ˆ θr). We could form a 95% conﬁdence interval for c(θ0) again using [ˆ c(.05R), ˆ c(.95R)]. An alternative version of the parametric bootstrap begins with the asymptotic approxima-tion, √ N(ˆ θ −θ0) dist. − →N(0, V0) and derives an asymptotic approximation for the distribution 94 of √ N(c(ˆ θ) −c(θ0)), where c is a continuous function. Suppose ˆ V prob. − →V0. We can draw from the asymptotic distribution of ˆ θ, ˆ θr = θ0 + ˆ V 1/2ur, where ur ∼N(0, I). We again perform inferences using the empirical distribution of c(ˆ θr). The technique is described in King, Tomz and Wittenberg (2000), though it is not described as the parametric bootstrap. The particular version of the parametric bootstrap apparently predates this article. 10.3 Recommended Reading 10.3.1 Background King, Tomz and Wittenberg (2000) 10.3.2 Advanced Horowitz (2001) 11 Nonlinear Panel Data Consider the case where the dependent variable ynt and the independent variables xnt are indexed by two things (this could be individuals and time, countries and time, individuals within states, etc.). The case where the DV and IV are indexed by two (or more) things goes by a bunch of diﬀerent names–grouped data, clustered data, longitudinal data, hierarchical models, multi-level models, and time-series cross section data. 11.1 Clustered Standard Errors Consider a panel data model with the data given by the xnt. Suppose that xnt ∼f(xnt; θ) and suppose that xnt are i.i.d. over both n and t. The MLE is then given by, ˆ θ = arg max θ 1 NT N X n=1 T X t=1 log f(xnt; θ) (304) Consistency of the MLE follows from the fact that, 1 NT N X n=1 T X t=1 log f(xnt; θ) prob. − →E[log f(xnt; θ)] (305) 95 and the fact that the information inequality implies that E[log f(xnt; θ)] is minimized at θ = θ0. With panel data model, the assumption that xnt are i.i.d. over both n and t may be implausible. Suppose instead that the marginal distribution of xnt is still f(xnt; θ), but that xnt is independent over n, but not t. In turns out that the MLE derived under the assumption that the data were independent is still consistent. The reason for this is that arg max θ 1 NT PN n=1 PT t=1 log f(xnt; θ) can be considered an M-estimator with limit E[log f(xnt; θ)] and the limiting objective function does not depend on the joint distribution of xnt, but only the marginal distribution. What remains is to characterize the asymptotic distribution. We can write, ψ(xnt; θ) = log f(xnt; θ) (306) ˆ θ = arg max θ 1 NT N X n=1 T X t=1 ψ(xnt; θ) = arg max θ 1 N N X n=1 " 1 T T X t=1 ψ(xnt; θ) # (307) = arg max θ 1 N N X n=1 φ(xn; θ) The theory of M-estimators implies that, √ N(ˆ θ −θ0) dist. − →N(0, C−1 0 B0C−1 0 ) (308) where, C0 = E[φθθ(xn; θ0)] = E[ 1 T ψθθ(xn; θ0)] = E h 1 T ∂2 ∂θ∂θ′ log f(xnt; θ0) i (309) B0 = E[φθ(xn; θ0)φθ(xn; θ0)′] (310) = E " ∂ ∂θ 1 T T X t=1 log f(xnt; θ0) ! ∂ ∂θ 1 T T X t=1 log f(xnt; θ0) !′# We can estimate these using, ˆ C = 1 NT 2 N X n=1 T X t=1 ∂2 ∂θ∂θ′ log f(xnt; ˆ θ) (311) 96 ˆ B = 1 N N X n=1 1 T T X t=1 ∂ ∂θ log f(xnt; θ0) ! 1 T T X t=1 ∂ ∂θ log f(xnt; θ0) !′ (312) This process is known as clustering the standard errors. In the derivation, we assumed that the data were independent across individuals, but not across time. In a panel data framework, this allows for individual speciﬁc unobservables that are mean zero. It also allows for time series dependence in the error terms. 11.2 Nonlinear Fixed Eﬀects Models Consider the linear ﬁxed eﬀects framework, ynt = αn0 + δt0 + β′ 0xnt + εnt (313) where E[εnt\|xnt] = 0. The OLS estimator is given by, (ˆ β, ˆ α1, ..., ˆ αN, ˆ δ1, ..., ˆ δT) = arg max (β,α1,...,αN,δ1,...,δT ) 1 NT N X n=1 T X t=1 (ynt −αn −δt −β′xnt)2 (314) We have that ˆ β prob. − →β0 if N →∞or T →∞, ˆ αn prob. − →αn0 if T →∞, and ˆ δt prob. − →δt0 if N →∞. Consider alternatively a nonlinear model with individual level ﬁxed eﬀects, (ˆ β, ˆ α1, ..., ˆ αN) = arg max (β,α1,...,αN) 1 NT N X n=1 T X t=1 ψ(xnt; β, αn) (315) We have that ˆ β prob. − →β0 and ˆ αn prob. − →αn0 if T →∞, but that ˆ β and ˆ αn are both inconsistent if T is ﬁxed and N →∞. This become relevant because we cannot apply the nonlinear ﬁxed eﬀects estimator in short panels (even though we could apply the linear ﬁxed eﬀects model is short panels). When estimating models with ﬁxed eﬀects, it is more common to apply the linear model for that reason. In fact, applications of the linear probability model is published work are often cases where panels are short and it is desired to include ﬁxed eﬀects. 97 11.3 Conditional Fixed Eﬀects Estimators The conditional ﬁxed eﬀects logit estimator is a alternative estimator for the ﬁxed eﬀects logit model that is consistent in short panels. The estimator is based on the following conditional likelihood, Pr yn1, yn2, ..., ynT\| T X t=1 ynt = τ, xnt; β, αn ! (316) = Pr yn1, yn2, ..., ynT, PT t=1 ynt = τ\|xnt; β, αn Pr PT t=1 ynt = τ\|xnt; β, αn = Pr (yn1, yn2, ..., ynT\|xnt; β, αn) Pr PT t=1 ynt = τ\|xnt; β, αn Pr (yn1, yn2, ..., ynT\|xnt; β, αn) = T Y t=1 Pr(ynt\|xnt; β, αn) = T Y t=1 (eαn+β′xnt)ynt 1 + eαn+β′xnt (317) Pr T X t=1 ynt = τ\|xnt; β, αn ! = X y:P t y=τ T Y t=1 (eαn+β′xnt)ynt 1 + eαn+β′xnt (318) Pr yn1, yn2, ..., ynT\| T X t=1 ynt = τ, xnt; β, αn ! = QT t=1 (eαn+β′xnt)ynt 1+eαn+β′xnt P y:P t y=τ QT t=1 (eαn+β′xnt)ynt 1+eαn+β′xnt (319) = QT t=1 (eαn)ynt(eβ′xnt)ynt 1+eαneβ′xnt P y:P t y=τ QT t=1 (eαn)ynt(eβ′xnt)ynt 1+eαneβ′xnt = (eαn)τ QT t=1(eβ′xnt)ynt QT t=1 1+eαneβ′xnt P y:P t y=τ(eαn)τ QT t=1(eβ′xnt)ynt QT t=1 1+eαneβ′xnt = (eαn)τ QT t=1 1+eαneβ′xnt QT t=1(eβ′xnt)ynt (eαn)τ QT t=1 1+eαneβ′xnt P y:P t y=τ QT t=1(eβ′xnt)ynt = eβ′ PT t=1 xntynt P y:P t y=τ eβ′ PT t=1 xntynt This derivation suggests that following (conditional) log-likelihood, l(β) = N X n=1 log eβ′ PT t=1 xntynt P dn:P t dnt=P t ynt eβ′ PT t=1 xntdnt ! (320) The advantage of this estimator is that it produces consistent estimates of β0 when T is small 98 and N →∞. The disadvantage is that it cannot produce a consistent estimate of αn0. This, in turn means, that we cannot compute consistent estimates of substantive eﬀects (since the substantive eﬀects would depend on αn0). There are conditional ﬁxed eﬀects estimators for the ordered logit and poison regression models (though I have never seen them used in published work). The conditional ﬁxed eﬀects estimator turns out to be equivalent to the (unconditional) ﬁxed eﬀects estimator, suggesting that for the Poisson model, the estimator of β0 is consistent when T is small and N →∞. There is believed to be no conditional ﬁxed eﬀects probit estimator. 11.4 Random Eﬀects Estimators Random eﬀects models are an alternative panel data model where β0 can be estimated consistently if T is small and N →∞. They are an alternative to the ﬁxed eﬀects model which, while less general (we typically assume that the random eﬀects are uncorrelated with the independent variables) do not require a long panel for consistency and allow for time invariant covariates to be included in the model. They are an alternative to clustered standard errors which while not valid in as general a set of circumstances, can be more eﬃcient when the assumption of the random eﬀects model are met. We can specify the random eﬀects probit as follows, y∗ nt = β′ 0xnt + νn + εnt (321) ynt = 1{y∗ nt ≥0} (322) εnt ∼N(0, 1) (323) νn ∼N(0, σ2 ν) (324) where εnt are iid, νn are iid, and εnt and νm are independent for all n, m, t. Under this framework, we have that, y∗ nt\|νn ∼N(β′xnt + νn, 1) (325) We have, 99 Pr(yn\|νn) = T Y t=1 Φ(β′xnt + νn)ynt(1 −Φ(β′xnt + νn))1−ynt (326) Pr(yn) = Z νn Pr(yn\|νn) 1 σν √ 2πe−1 2 ν2 n/2 νdν (327) We can form the likelihood for the model as, l(β, σ2 ν) = N X n=1 log Pr(yn) (328) Here, the expression for Pr(yn) involves a one-dimensional integral. We can approxi-mate this integral using simulation. Let unr ∼N(0, 1). Then σνunr ∼N(0, σ2 ν). We can approximate, l(β, σ2 ν) ≈ N X n=1 log R X r=1 T Y t=1 Φ(β′xnt + σνunr)ynt(1 −Φ(β′xnt + σνunr))1−ynt (329) Here, the draws unr should be ﬁxed ahead of time (not redrawn each time the likelihood function is evaluated). When computing marginal eﬀects or substantive eﬀects, note that y∗ nt ∼N(β′xnt, σ2 ν +1). This implies that, Pr(ynt = 1\|xnt; β, σν) = Φ β′xnt p σ2 ν + 1 ! (330) 11.5 Generalized Estimating Equations 100 Model Regular Estimator Random Eﬀects Fixed Eﬀects Conditional Fixed Eﬀects Linear-OLS ˆ β consistent as Not necessary N or T →∞; ˆ αn consistent as T →∞ Logit-MLE ˆ β consistent ˆ β consistent as as N or T →∞; ˆ β consistent N →∞, no ˆ αn Probit-MLE use clustered as T →∞ n/a Ordered Logit-MLE standard errors ˆ β consistent ˆ αn consistent Uncommon Ordered Probit-MLE to account for as N →∞ as T →∞ n/a Multinomial Logit-MLE correlated errors (for the Poisson n/a Conditional Logit-MLE within clusters model, ˆ β is n/a Multinomial Probit-MLE consistent as T →∞ n/a Poisson-MLE under assumptions Equal to ﬁxed eﬀects, necessary for consistent as N →∞, no ˆ αn Negative Binomial-MLE deriving Poisson n/a (some software CFE estimator) erroneously implements a CFE estimator) Tobit-MLE n/a Table 10: Summary of Nonlinear Panel Data Models 101 Liang and Zeger (1986) proposed the following approach for obtaining estimators that are consistent under more general conditions than random eﬀects estimators, but are potentially more eﬃcient than applying clustered standard errors to conventional estimators. Here we will derive Liang and Zeger’s estimator using an equivalent Classical Minimum Distance (CMD) estimator. Consider a dependent variable ynt and a mean function µ(xnt; β) = E[ynt\|xnt]. Suppose that the following identiﬁcation condition holds, µ(xnt; β) = µ(xnt; β0) (331) if and only if β = β0. The classical minimum distance estimator is deﬁned by, ˆ β = arg min β 1 N N X n=1 (yn −µ(xn; β))′W(yn −µ(xn; β)) (332) where, µ(xn; β) =       µ(xn1; β) µ(xn2; β) ... µ(xnT; β)       (333) and W is a positive deﬁnite weighting matrix. Liang and Zeger’s estimator works oﬀof the ﬁrst order conditions which is, of course, equivalent to optimizing the objective function above, apart from the possibility that the stationary point may not be unique. Liang and Zeger set ˆ β to be the unique solution to the nonlinear system, 1 N N X n=1 ∂µ ∂β(xn; β)′V −1(yn −µ(xn; β)) = 0 (334) To see why this is a reasonable estimator, note that, E[ ∂µ ∂β(xn; β)′V −1(yn −µ(xn; β))] = E[E[ ∂µ ∂β(xn; β)′V −1(yn −µ(xn; β))\|xn]] (335) = E[ ∂µ ∂β(xn; β)′V −1E[(yn −µ(xn; β))\|xn]] (336) = E[ ∂µ ∂β(xn; β)′V −1E[yn\|xn] −∂µ ∂β(xn; β)′V −1µ(xn; β)] 102 = E[ ∂µ ∂β(xn; β)′V −1µ(xn; β0)−∂µ ∂β(xn; β)′V −1µ(xn; β)] = E[ ∂µ ∂β(xn; β)′V −1(µ(xn; β0)−µ(xn; β))] This is equal to zero if β = β0. Demonstrating that this is the unique solution is more diﬃcult (Newey and McFadden, 1994). Provided the solution is unique and a Law or Large Numbers holds, i.e., 1 N N X n=1 ∂µ ∂β(xn; β)′V −1(yn −µ(xn; β)) prob. − →E[ ∂µ ∂β(xn; β)′V −1(yn −µ(xn; β))] (337) the estimator will be consistent. Next, consider the distribution of the estimator. We can use a Taylor expansion to obtain, µ(xn; β) ≈µ(xn; β0) + ∂µ ∂β(xn; β0)(β −β0) (338) Using the fact that, N X n=1 ∂µ ∂β(xn; ˆ β)′V −1(yn −µ(xn; ˆ β)) = 0 (339) we obtain, N X n=1 ∂µ ∂β(xn; ˆ β)′V −1(yn −µ(xn; β0)) ≈ N X n=1 ∂µ ∂β(xn; ˆ β)′V −1 ∂µ ∂β(xn; β0)(ˆ β −β0) (340) √ N(ˆ β −β0) (341) dist. − → " 1 N N X n=1 ∂µ ∂β(xn; β0)′V −1 ∂µ ∂β(xn; β0) #−1 " 1 √ N N X n=1 ∂µ ∂β(xn; β0)′V −1(yn −µ(xn; β0)) # √ N(ˆ β −β0) dist. − →N(0, C−1 0 B0C−1 0 ) (342) where, 103 C0 = E[ ∂µ ∂β(xn; β0)′V −1 ∂µ ∂β(xn; β0)] (343) B0 = V ar( ∂µ ∂β(xn; β0)′V −1(yn −µ(xn; β0))) (344) We can consider a number of alternative speciﬁcations for µ(xn; β). For the linear model, we have µ(xn; β) = β′xn, for the probit model we have µ(xn; β) = Φ(β′xn), and for the Poisson model, we have µ(xn; β) = eβ′xn. Note that, B0 = V ar( ∂µ ∂β(xn; β0)′V −1(yn −µ(xn; β0))) (345) = E[V ar( ∂µ ∂β(xn; β0)′V −1(yn −µ(xn; β0))\|xn) + V ar(E[ ∂µ ∂β(xn; β0)′V −1(yn −µ(xn; β0))\|xn]) = E[ ∂µ ∂β(xn; β0)′V −1V ar(yn\|xn)V −1 ∂µ ∂β(xn; β0)] Suppose that V ar(yn\|xn) does not depend on xn and let V0 = V ar(yn). If we select V = V0, then B0 = C0 = E[ ∂µ ∂β(xn; β0)′V −1 0 ∂µ ∂β(xn; β0)]. It can be shown that this leads to the smallest possible variance. This calculation suggests that we should select V to be close to V0. To specify V , Liang and Zeger (1986) consider four cases: 1. Independence: V = I 2. Exchangeability: V =       1 ρ ... ρ ρ 1 ... ρ ... ... ... ... ρ ρ ... 1       (346) 3. AR1 process: 104 V =         1 ρ ρ2 ... ρT−1 ρ 1 ρ ... ρT−2 ρ2 ρ 1 ... ρT−3 ... ... ... ... ... ρT−1 ρT−2 ρT−3 ... 1         (347) 4. Unstructured: V =         1 ρ12 ρ13 ... ρ1T ρ12 1 ρ23 ... ρ2T ρ13 ρ23 1 ... ρ3T ... ... ... ... ... ρ1T ρ2T ρ3T ... 1         (348) The independence case leads to an estimator that is equivalent to the MLE. For the other cases, the approach is to obtain a preliminary consistent estimator of β0 and to use this to estimate the parameters of the V matrix. For example, is the unstructured case, we would set rnt = ynt −µ(xnt; ˆ β) and estimate. ˆ ρts = 1 N PN n=1 rntrns q 1 N PN n=1 r2 nt q 1 N PN n=1 r2 ns (349) 11.6 Recommended Reading 11.6.1 Background Greene (2000) Hsiao (2003) Wooldridge (2002) 11.6.2 Examples Kayser and Peress (2012) Nepal, Bohara and Gawande (2011) Richman (2011) 105 11.6.3 Advanced Liang and Zeger (1986) Newey and McFadden (1994) 12 Time Series Dependence in Nonlinear Models 12.1 Parametric Models Consider the following model, y∗ t = β′ 0xt + εt (350) yt = 1{y∗ t ≥0} (351) εt = ρεt−1 + ut (352) where ut ∼N(0, 1) and ut are i.i.d. To form the MLE, we must consider, Pr(y\|x; β, ρ) = Pr(y∗ t ≥0 for t : yt = 1, y∗ t ≤0 for t : yt = 0) (353) This involves a T-dimensional integral of the normal distribution over a region. The integral would typically be computed using the GHK simulator. The log-likelihood is given by, l(β, ρ) = log Pr(y\|x; β, ρ) (354) A similar diﬃculty shows up in other time series models of limited dependent variables. 12.2 Semiparametric Approach Consider the same model as before, but consider Pr(yt = 1\|xt). We have that the stationary distribution for εt is given by εt ∼N(0, 1 1−ρ2). This implies that, Pr(yt = 1\|xt; β, ρ) = Φ (1 −ρ2)β′xn (355) 106 In fact, for simplicity, we can specify εt = ρεt−1 + p 1 −ρ2ut in which case the stationary distribution is εt ∼N(0, 1), so that, Pr(yt = 1\|xt; β, ρ) = Φ (β′xn) (356) Since identiﬁcation of an M-estimator only depends on the marginal distribution, this means that probit will generally be consistent even if the errors have time series dependence. 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Piazza. 2013. “Autocracies and Terrorism: Conditioning Eﬀects of Authoritarian Regime Type of Terrorist Attacks.” American Journal of Political Science 57:941–955. 110 Wolﬁnger, Raymond and Steven Rosenstone. 1980. Who Votes? New Haven, CT: Yale University Press. Wooldridge, Jeﬀrey M. 2002. Econometric Analysis of Cross Section and Panel Data. Cam-bridge University Press. A Appendix A.1 Proof that Asymptotic Normality Implies Asymptotic Unbi-asedness An estimator ˆ θ of a population parameter θ0 is said to be asymptotically unbiased if lim N→∞E[ˆ θ− θ0] = 0. Suppose that √ N(ˆ θ −θ0) dist. − →N(0, V0) where V0 is positive deﬁnite. We will show that this implies that the estimator is asymptotically unbiased. We have, V −1 0 √ NE[ˆ θ −θ0] = Z ˆ θ V −1 0 √ N(ˆ θ −θ0)dFˆ θ(ˆ θ) (357) where V −1 0 exists because V0 is assumed to be positive deﬁnite. Using the transformation x = V −1 0 √ N(ˆ θ −θ0), we have, V −1 0 √ NE[ˆ θ −θ0] = Z x xdFV −1 0 √ N(ˆ θ−θ0)(x) (358) Deﬁne the mean functional, H(G) = R x xdG(x). We have, V −1 0 √ NE[ˆ θ −θ0] = H(FV −1 0 √ N(ˆ θ−θ0)) (359) By assumption, we have that, FV −1 0 √ N(ˆ θ−θ0)(x) = ΦJ(x)∀x (360) where ΦJ is the CDF of a standard normal distribution. Continuity of the mean functional implies that, lim N→∞H(FV −1 0 √ N(ˆ θ−θ0)) = H(ΦJ) (361) Hence, we have, 111 lim N→∞V −1 0 √ NE[ˆ θ −θ0] = H(ΦJ) = 0 (362) which implies that lim N→∞E[ˆ θ −θ0] = 0. A.2 Derivation of the Expected Value of yn in the Heckman Se-lection Model We have, E[yn\|xn, zn, wn = 1] = E[β′xn + εn\|xn, zn, wn = 1] = β′xn + E[εn\|zn, wn = 1] (363) = β′xn + E[εn\|zn, ηn ≥−γ′zn] = β′xn + Z (εn,ηn):ηn≥−γ′zn εndFε,η(εn, ηn) Using the formula for the marginal and conditional normal distributions, we have, E[yn\|xn, zn, wn = 1] (364) = β′xn + Z (εn,ηn):ηn≥−γ′zn 1 √ 2πe−1 2 η2 nε 1 √ 2πσ2(1−ρ2)e − 1 2σ2(1−ρ2) (εn−ρσηn)2 d(εn, ηn) = β′xn + Z ηn:ηn≥−γ′zn 1 √ 2πe−1 2 η2 n Z εn εn 1 √ 2πσ2(1−ρ2)e − 1 2σ2(1−ρ2) (εn−ρσηn)2 dεn dηn = β′xn + Z (εn,ηn):ηn≥−γ′zn ρσηn 1 √ 2πe−1 2 η2 ndηn = ρσE[ηn\|ηn ≥−γ′zn] Finally, using the formula for the mean of a normal distribution, E[yn\|xn, zn, wn = 1] = β′xn + ρσ φ(−γ′zn) 1 −Φ(−γ′zn) (365) 112
1637	https://www.reddit.com/r/desmos/comments/1b9wpql/a_general_formula_for_the_tangents_made_from_an/	A General Formula for the tangents made from an external point away from a circle. : r/desmos Skip to main contentA General Formula for the tangents made from an external point away from a circle. : r/desmos Open menu Open navigationGo to Reddit Home r/desmos A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to desmos r/desmos r/desmos A subreddit dedicated to sharing graphs created using the Desmos graphing calculator. Feel free to post demonstrations of interesting mathematical phenomena, questions about what is happening in a graph, or just cool things you've found while playing with the calculator. Reply to people with commands! For example, "!fp" explains what floating point arithmetic is. Try out commands at 41K Members Online •2 yr. ago Just_a_spectat0r A General Formula for the tangents made from an external point away from a circle. Question i found a general equation for the tangents originated from an external point, in this you can customize all initial variables including position variables of the circle and external point coordinates. by equaling radius of the circle in terms of g, f and C to the perpendicular distance from a line to that external point as Radius = sqrt(g^2 + f^2 -C) tangent --> mx - y + c=0 Due to external point P(x_0, y_0) being on the tangent m(x_0) - y_0 + c = 0 c = y_0 - m(x_0) sqrt(g^2 + f^2 -C) = \|(m(-g) + (-1)(-f)+c)/(sqrt(m^2+1))\| sqrt(g^2 + f^2 -C) = \|(m(-g) + (-1)(-f)+y_0 - m(x_0))/(sqrt(m^2+1))\| and subjecting m getting the 2 gradients to the tangent as m_1 and m_2 substituting to: y = m_1(x-x_0) + y_0 y = m_2(x-x_0) + y_0 Link: are there any other simpler ways to do this? Read more Share Related Answers Section Related Answers Creating fractal patterns with Desmos Visualizing calculus concepts on Desmos Animating graphs using Desmos features Modeling real-world data with Desmos graphs Exploring parametric equations in Desmos New to Reddit? Create your account and connect with a world of communities. Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community 0 0 Top Posts Reddit reReddit: Top posts of March 8, 2024 Reddit reReddit: Top posts of March 2024 Reddit reReddit: Top posts of 2024 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
1638	https://www.ias.ac.in/article/fulltext/sadh/028/06/1011-1018	S¯ adhan¯ aVol. 28, Part 6, December 2003, pp. 1011–1018. © Printed in India Effect of deformation and dielectric filling on electromagnetic propagation through waveguides AJAY CHAUDHARI Department of Physics, Dr B A M University, Aurangabad 431 004, India Present address: Dept. of Chemistry, National Chung-Cheng University, Ming-Hsiung, Chia-Yi- 621, Taiwan e-mail: ajaychau5@yahoo.com MS received 19 August 2001; revised 2 November 2002 Abstract. The effect of depression and protrusion of vertical walls and dielec-tric filling on electromagnetic propagation through x-band rectangular waveguides is studied using the finite element method. The effect of these deformations and dielectric filling on TE 10 , TE 20 mode cutoff frequencies and passband is stud-ied. The results are compared with those of x-band rectangular waveguides with depression and protrusion with air medium. TE 10 , TE 20 mode cutoff frequencies for waveguides with depression and protrusion and filled with dielectric medium are less than that for air medium. However, the bandwidth in some cases of dielec-tric medium is larger than that for air medium. Keywords. Waveguides; finite element method; depression; protrusion; dielectric filling; cutoff frequencies. Introduction The study of electromagnetic propagation through various guiding structures and devices is of great significance in the development of microwave communication networks and technology. However, in the physical development of the guiding structures, there is always a possibility of the occurrence of some irregularities in these guiding structures. Many workers (Daly 1971; Ikeuchi et al 1981; Mabaya et al 1981) have used the vectorial finite element method (FEM) in terms of longitudinal electric field (E z ) and magnetic field (H z ) to compute the mode spectrum of the waveguide accurately, with arbitrary cross-section. The most serious difficulty in using FEM analysis for waveguides is the appearance of so-called spurious, non-physical modes (Daly 1971; Ikeuchi et al 1981; Mabaya et al 1981; Hayata et al 1986, 1989). The vectorial FEM in terms of all three components (H x , H y , H z ) of the magnetic field is proposed by Konard (1977) in which spurious solutions do appear (Rahman & Davies 1984; Koshiba et al 1989). Davies et al (1982) noted that the spurious solutions in the three component formulation do not satisfy the divergence relation for H, i.e. ∇.H = 0. Koshiba et al (1985) has suggested an improved FEM for analysis of dielectric waveguides in terms of all three components of H. Waveguides with arbitrary shapes are analysed by Bulley & 1011 1012 Ajay Chaudhari Davies (1969) using the Rayleigh–Ritz method. Dispersion characteristics of arbitrary-shaped waveguides with sharp metal edges are determined by Webb (1988). Srba et al (1994) have used mediam vectorial FEM for analysing rectangular waveguides half filled with dielectric media. Rectangular dielectric waveguide structures are analysed by Bierwirth et al (1986) using the finite difference method. Dielectric strip inserted waveguide is analysed by Chaud-hari et al (1998). Finite difference method has been used by Schweig & Bridges (1984) for the analysis of dielectric waveguides. Rectangular waveguides with deformations are studied by Chaudhari & Patil (1996,1997). All the above studies are on waveguides which are either partially filled by dielectric, slab loaded or of arbitrary shape, with air as a dielectric. We have studied the effect of the depression and protrusion of the vertical wall of the waveguide with air medium, on the cutoff frequencies for TE 10 and TE 20 modes in our previous work (Chaudhari & Patil 1996). The aim of this paper is to study propagation through x-band rectangular waveguides with depression and protrusion of the vertical side, as well as with dielectric filling, and to compare these results with those of waveguides with depression and protrusion, and filled with air. Finite element method with variational principle is the appropriate choice to deal with the irregular geometries of waveguides. The TE 10 and TE 20 modes are worked out by considering the propagation through waveguides as an eigenvalue problem. Consider an x-band rectangular waveguide with depression and protrusion of the vertical wall. Let the walls of the waveguide be perfectly conducting. The waveguide is filled with dielectric material of dielectric constant 2 ·5. The cross-section of the waveguide in the X-Y plane of the Cartesian coordinate system is considered a problem domain , with deformed vertical boundaries. The five cases of deformation considered are: depression on one side, depression on both sides, protrusion on one side, protrusion on both sides, depression on one side and protrusion on other side. For each case of deformation, for different deformation length L in steps of 0 ·15 cm, the eigenvalues and eigenvectors are obtained. The modes TE 10 and TE 20 are identified by using the field plots. Variational formulation The electric and magnetic fields inside the waveguide satisfy Maxwell’s equations. Thus the problem of electromagnetic propagation under consideration involves solving the Maxwell equations with suitable boundary conditions at the conducting boundaries, such that the tangential component of the electric field is zero on the conductor boundaries. To avoid spurious solutions, a magnetic field vector formulation is used for the solution of the problem. The details of the formulation can be found in the work by Hayata et al (1986). The expression for the functional 5 is 5 = 12 ∫ [(∇ × H ∗). (ε−1∇ × H ) − K2H ∗.H + (∇.H ∗).( ∇.H ) ]d. (1) The stationary character of 5 requires δ5 = 0. The first variation in 5 is given by δ ∫ [(∇ × H ∗).(ε −1∇ × H ) − K2H ∗H + (∇.H ∗)( ∇.H ) ] d  = 0. (2) Electromagnetic propagation through waveguides 1013 The domain, i.e. cross-section of the waveguide, is divided into rectangular elements with four nodes [Reddy 1996; Akin 1988). The functional over each element is, 5 e = 12 ∫ e (∇ × H e∗ ).(ε −1∇ × H e )de − ∫ e K2H e∗ .H ed e + ∫ e (∇.H e∗ ).( ∇.H e )de  , (3) where the unknown vector He has three components Hx, Hy and Hz. Each unknown component Hx, Hy and Hz is defined in terms of the nodal values He x, He y and He z respectively. A linear mapping function is used, which needs four nodes per rectangular element. Functional 5e can be written as, 5 e = (1/2)[{He}T [S e]{He} − K2{He}T [T e]{He}]. (4) The functional for the whole region is given by 5 = (1/2)[{H}T [S]{H} − K2{H}T [T ]{H}]. (5) The condition ∂5/∂ {H} = 0 leads to the following matrix equation [S]{H} − K2 [T ]{H} = 0. (6) Equation (6) is the matrix equation to be solved for eigenvalues, K2 = λ and eigenvectors, i.e. Hfield components. Numerical calculations The dimensions of the x-band waveguide considered here are a = 2·4 cm and b = 1·2 cm. For deformation, the corner points and the broad wall a are kept fixed, while the depression and protrusion of the b wall are considered. For the first case of deformation, the depression L from one side is increased in steps of 0 ·15 cm. For the second case, the depression L is increased from both sides in steps of 0 ·15 cm in opposite directions. For the third case, the protrusion L on one side is increased in steps of 0 ·15 cm. For the fourth case, the protrusion L is increased from both sides in steps of 0 ·15 cm in opposite directions. Finally, in the fifth case, the depression L from one side and protrusion L from another side are increased in steps of 0 ·15 cm. The cross-section of this waveguide is divided into 128 rectangular elements with four nodes per element. The total number of nodes in the geometry are 153, out of which 48 are the boundary nodes, on which boundary conditions are specified. For each node, there are three unknown field components. Therefore, the matrix problem to be solved for the system, consists of matrices of the order 459 × 459. As these matrices are symmetric, only half matrices are stored in the memory. Skyline storage has been used for storing these matrices. The subspace iteration method (Bathe & Wilson 1987) has been used to solve this eigenvalue problem, and consists of the following three steps. 1014 Ajay Chaudhari Table 1. Bandwidths and cutoff frequencies for TE 10 and TE 20 mode in gigahertz for waveguides with depressions on one side (ε = 2·5). L (cm) TE 10 TE 20 Bandwidth 0 6·2507 12 ·5645 6·3138 0·15 6·4744 12 ·9978 6·5234 0·30 6·7351 13 ·4718 6·7367 0·45 7·0522 13 ·9302 6·8780 0·60 7·4343 14 ·8196 7·3853 0·75 7·8889 15 ·3966 7·5077 0·90 8·4229 16 ·4282 8·0053 1·05 9·0417 17 ·7473 8·7055 1·20 9·7466 18 ·7569 9·0103 1·35 10 ·5267 19 ·4945 8·9678 (1) Establishment of q starting iteration vectors, q > p , where p is the number of eigenvalues and vectors to be calculated. (2) Use of simultaneous inverse iteration on the q vectors and Ritz analysis to extract the best eigenvalue and eigenvector approximations from the q iteration vectors. (3) Use of the strum sequence check to verify whether any eigenvalues are missed in the set that is calculated. For each case, for each L, the eigenvalues and eigenvectors are worked out. The eigenvalues obtained are the values of ω2μ0ε0 . Using these eigenvalues, corresponding cut-off frequencies f and bandwidth are calculated. For each case, for different L, the values of f for TE 10 and TE 20 mode and bandwidth are given in tables 1–5. The variation of bandwidth with L for the five cases is shown in figures 1–5. Results For waveguides with depression on one side, depressions on both side and depressions on one side and protrusion on the other, the cutoff frequencies (f ) for TE 10 , TE 20 modes as Table 2. Bandwidths and cutoff frequencies for TE 10 and TE 20 mode in gigahertz for waveguides with depressions on one sides (ε = 2·5). L (cm) TE 10 TE 20 Bandwidth 0·0 6·2507 12 ·5645 6·3138 0·15 6·7042 13 ·4549 6·7506 0·30 7·2840 14 ·1972 6·9132 0·45 8·0487 15 ·9908 7·9421 0·60 9·0512 17 ·5050 8·4538 0·75 10 ·3064 18 ·3989 8·0925 0·90 12 ·9155 20 ·2532 7·3377 1·05 14 ·1502 21 ·3168 7·1666 Electromagnetic propagation through waveguides 1015 Table 3. Bandwidths and cutoff frequencies for TE 10 and TE 20 mode in gigahertz for waveguides with protrusions on one side (ε = 2·5). L (cm) TE 10 TE 20 Bandwidth 0 6·2507 12 ·5645 6·3138 0·15 6·0814 12 ·0820 6·0006 0·30 5·9384 11 ·9004 5·9619 0·45 5·8280 11 ·6460 5·8180 0·60 5·7455 11 ·4427 5·6972 0·75 5·6849 11 ·2814 5·5965 0·90 5·6416 10 ·9745 5·3328 1·05 5·6117 11 ·0598 5·4481 1·20 5·5919 11 ·1925 5·6006 1·35 5·5805 11 ·2476 5·6671 Table 4. Bandwidths and cutoff frequencies for TE 10 and TE 20 mode in gigahertz for waveguides with protrusions on one sides (ε = 2·5). L (cm) TE 10 TE 20 Bandwidth 0 6·2507 12 ·5645 6·3138 0·15 5·9136 11 ·8668 5·9532 0·30 5·6495 11 ·2990 5·6495 0·45 5·4537 10 ·8770 5·4233 0·60 5·3114 10 ·6896 5·3781 0·75 5·2098 10 ·5734 5·3635 0·90 5·2024 10 ·5577 5·3554 1·05 5·1327 10 ·4811 5·3484 1·20 5·0551 10 ·3953 5·3403 1·35 5·0347 10 ·3798 5·3450 Table 5. Bandwidths and cutoff frequencies for TE 10 and TE 20 mode in gigahertz for waveguides with depressions on one side and protrusions on other side (ε = 2·5). L (cm) TE 10 TE 20 Bandwidth 0 6·2507 12 ·5645 6·3138 0·15 6·2845 12 ·6069 6·3224 0·30 6·3665 12 ·7008 6·3345 0·45 6·5128 12 ·8704 6·3576 0·60 6·7277 13 ·2089 6·4812 0·75 7·0115 13 ·6171 6·6056 0·90 7·3653 14 ·9462 7·5809 1·05 7·7909 15 ·1181 7·3274 1·20 8·2901 15 ·5089 7·2189 1·35 8·8631 16 ·0563 7·1932 1016 Ajay Chaudhari Figure 1. Bandwidths for air and dielectric media for waveguides with depressions on one side. Figure 2. Bandwidths for air and dielectric media for waveguides with depressions on both sides. Figure 3. Bandwidths for air and dielectric media for waveguides with protrusions on one side. Figure 4. Bandwidths for air and dielectric media for waveguides with protrusions on both sides. Electromagnetic propagation through waveguides 1017 Figure 5. Bandwidths for air and dielectric media for waveguides with depressions on one side and protrusion on the other. well as bandwidth, increase with deformation length L compared to those for normal x-band rectangular waveguides, for waveguides with air as well as dielectric media. However, for waveguides with protrusions on one side or on both sides, these modes decrease with L compared to normal x-band rectangular waveguides. In the case of waveguides with depressions on one side, f for TE 10 , TE 20 are lower for dielectric medium than for air medium. This decrease in f for TE 20 mode is very small but for TE 10 mode, the decrease increases with L. Due to this, the bandwidth for the lower value of L, for the dielectric medium, is almost the same as that for the air medium. But as decrease in f for TE 10 mode for higher value of L increases, the bandwidth in this region for the dielectric medium is larger than that for air medium. For waveguides with depressions on both sides, as the decrease in f for TE 10 , TE 20 modes for dielectric medium is less than for air medium, the bandwidth is almost same for both the media except in the middle region (for L 0·45, 0 ·6 and 0 ·75). In this region, f for TE 20 mode is almost the same for both the media but f for TE 10 mode, for dielectric medium, is less than that for air medium which causes greater increase in bandwidth for the former. In the case of waveguides with protrusion on one side, f for both, TE 10 , TE 20 mode, for dielectric medium, are much lower than that for air medium resulting in greater decrease in bandwidth for dielectric medium than air medium. For waveguides with protrusions on both side, the decrease in f for TE 10 mode for dielectric medium compared to air, is greater after L = 0·6 (i.e. L = b/ 2) but the decrease in f for TE 20 mode is not much. Due to this, the bandwidth for dielectric medium up to the length L = 0·6 cm, is less than that for air medium after which it is greater than for the air medium. In the case of waveguides with depression on one side and protrusion on the other, f for the TE 10 mode decreases more for the dielectric medium than for air but the decrease in f for TE 20 mode is not much for lower L values. However at higher L value, the decrease is more, and causes greater decrease in bandwidth for the dielectric medium in this region than for air. Conclusion x-Band rectangular waveguides with depressions and protrusions of the vertical sides have been analysed using Finite Element Method for dielectric medium. These results are compared with those for air. The cutoff frequencies for TE 10 , TE 20 modes decrease for dielectric medium compared to those for air. However this decrease is not the same for both the modes. Due to this the bandwidth is some cases for dielectric medium is larger than that for air medium. 1018 Ajay Chaudhari References Akin J E 1988 Application and implementation of finite element method (New York: Academic Press) Bathe K J, Wilson E L 1987 Numerical methods in finite element analysis (New Delhi: Prentice Hall) Bierwirth K, Schulz N, Arndt F 1986 Finite difference analysis of rectangular waveguide structure. IEEE Trans. Microwave Theory Tech. 34: 1104–1114 Bulley R M, Davies J B 1969 Comparison of approximate polynomial solutions to TE modes in an arbitrary shaped waveguide. IEEE Trans. Microwave Theory Tech. 17: 440–447 Chaudhari A, Karkare M, Patil P B 1998 Finite element analysis of dielectric strip inserted waveguide. Indian J. Pure Appl. Phys. 36: 97–100 Chaudhari A S, Patil P B 1996 Finite element analysis of rectangular waveguide with deformation as depression and protrusion of side walls. Indian J. Phys. 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Microwave Theory Tech. 29: 234–239 Konard A 1977 Higher order triangular finite elements for electromagnetic waves in anisotropic media. IEEE Trans. Microwave Theory Tech. 25: 253–259 Koshiba M, Hayata K, Suzuki M 1985 Improved finite element formulation in terms of magnetic field vector for dielectric waveguides. IEEE Trans. Microwave Theory Tech. 33: 227–233 Koshiba M, Hayata K, Suzuki M 1989 Vectorial finite element formulation without spurious solutions for dielectric waveguide. Tech. Res. Rep., Inst. Elect. Commun. Eng. Jpn. 83: 70–85 Mabaya N, Lagasse P E, Vandenbuleke P 1981 Finite element analysis of optical waveguides. IEEE Trans. Microwave Theory Tech. 20: 600–609 Rahman B M A, Davies J B 1984 Finite element analysis and optical waveguide problem. IEEE Trans. Microwave Theory Tech. 32: 20–26 Reddy J N 1986 An introduction to finite element method (New York: McGraw Hill) Schweig E, Bridges W B 1984 Computer analysis of dielectric waveguide: A finite difference method. 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1639	https://sciencenotes.org/nuclear-symbol-notation/	Home » Science Notes Posts » Chemistry » Chemistry Notes » Nuclear Symbol Notation Nuclear Symbol Notation This entry was posted on by Anne Helmenstine (updated on ) Nuclear symbol notation is a way of identifying the isotopes of elements. Here are two ways of writing nuclear symbols, along with examples. What Is Nuclear Symbol Notation? Basically, the nuclear symbol is a type of shorthand notation that identifies the element (by symbol or atomic number) and the mass number of the element. The mass number is the sum of the number of nucleons (protons and neutrons) in the atomic nucleus. Isotopes of an element have distinct mass numbers because they contain different numbers of neutrons. There are two ways of writing the symbols. The first is just the element name or symbol, followed by a dash and the mass number. For example, carbon-14 or C-14 both indicate the isotope of carbon that has 14 nucleons. The other method, sometimes called the AZE or AZX notation, lists the mass number as a superscript and the atomic number as a subscript before the element symbol. Ideally, the mass number is right on top of the atomic number, but the two numbers don’t line up in ordinary text. The general form for the notation is: AZX Where: A is the mass number or number of nucleons (protons + neutrons) Z is the atomic number or number of protons X is the element symbol For example, 146C and 126C are the nuclear symbols for the isotopes carbon-14 and carbon-12, respectively. The number of electrons is not included in the nuclear symbol because electrons are not in the nucleus. That being said, ions of isotopes may be written as nuclear symbols, followed by electric charge as a superscript. For example, 11H+ indicates a protium (hydrogen-1) ion with a +1 electrical charge. Example Problems Using Complete Nuclear Symbol Notation How to Write a Nuclear Symbol For example, with the nuclear symbol for an atom having 32 protons and 38 neutrons. First, use a periodic table and look up which element has 32 protons. This is the atomic number. The element is germanium, which has the symbol Ge. The “X” in the nuclear symbol is Ge. The atomic number Z is the number of protons. So, Z is 32. Next, find the mass number. Add up the number of protons (32) and neutrons (38). The mass number is 70. Put it all together: 7032Ge or germanium-70 Finding the Number of Neutrons Other typical problems ask you to find the number of neutrons in an atom given its nuclear symbol. For example, find the number of neutrons in an atom of 188O. This question tests your understanding of the parts of the nuclear symbol. The number 18 is A or the sum of the protons and neutrons. The number 8 is the atomic number or number of protons. A = Z + N N = A – Z N = 18 – 8 = 10 You can use the other nuclear symbol notation just as easily, although you’ll need a periodic table. For example, find the number of neutrons in an atom of uranium-238. The “238” is the mass number A (protons + neutron) and you know the element is uranium (U), but you need to look up its atomic number Z. From the periodic table, the atomic number of uranium is 92. A = Z + N N = A – Z N = 238 – 92 = 146 Note that uranium-238 is the most common isotope of uranium, even though it contains a different number of protons and neutrons. For the lighter elements, the most stable and abundant isotope has equal numbers of protons and neutrons. References Choppin, G.; Liljenzin, J. O. Rydberg, J. (1995). Radiochemistry and Nuclear Chemistry (2nd ed.). Butterworth-Heinemann. Connelly, N. G.; Damhus, T.; Hartshorn, R. M.; and Hutton, A. T. (2005). Nomenclature of Inorganic Chemistry – IUPAC Recommendations. The Royal Society of Chemistry. Scerri, Eric R. (2007). The Periodic Table. Oxford University Press. ISBN 0-19-530573-6. Related Posts What Is an Isotope? Definition and Examples Get the definition of an isotope. See examples of isotopes and learn the difference between an isotope and a nuclide of an element. Mass Number Versus Atomic Number and Atomic Mass Learn what mass number means in chemistry and how it differs from atomic number and atomic mass. What Is an Ion? Chemistry Definition Learn what an ion is in chemistry. Get the definition, examples, and the explanation for how to tell the charge of an ion.
1640	https://mathbitsnotebook.com/Geometry/TrigApps/TAlawSines.html	\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- --- --- --- --- --- \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- --- --- --- --- --- \| \| \| \| \| Law of Sines MathBitsNotebook.com Topical Outline \| Geometry Outline \| MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts \| So far, our adventure into trigonometry has been limited to working in right triangles. With a certain amount of given information about a right triangle, we have been able to find missing sides and missing angles. But can we also find missing sides and missing angles when the triangle is NOT a right triangle? What happens when the triangle is oblique? \| \| \| --- \| \| \| \| \| An oblique triangle is any triangle that is not a right triangle. \| \| The answer to the question stated above is "yes". There are methods of finding missing sides and missing angles in oblique triangles. One such method is called the Law of Sines, which relates the lengths of the sides of any triangle to the sines of its angles. \| \| \| \| --- \| Given the diagram: \| \| \| \| Law of Sines OR \| \| \| \| \| Note the lettering with side a across from ∠A, side b across from ∠B, and side c across from ∠C. \| \| \| \| Law of Sines in "words": "The ratio of the sine of an angle in a triangle to the side opposite that angle is the same for each angle in the triangle." These equal ratios are called the Law of Sines. Notice that the Law of Sines must work with at least two angles and two respective sides at a time. \| Most problems, dealing with the Law of Sines, only require that you work with one pair of the ratios at a time. Be sure to use a pair that corresponds to your given (or known) information. If you are asked to "Solve the triangle." (which means to find ALL of the remaining sides and angles), you may need to utilize all three ratios. \| \| \| \| --- \| \| \| \| Using Law of Sines \| The Law of Sines works with at least two angles and two respective sides at a time. As a result, the Law of Sines can be applied only if certain combinations are given. (1) Given two angles and the included side, find a missing side. (Given ASA) (2) Given two angles and the non-included side, find a missing side. (Given AAS) (3) Given two sides and the non-included angle, find a missing angle. (Given SSA) The third situation is called the Ambiguous Case and will be discussed on another page. Let's take a look at the first two situations. \| \| \| Since ASA and AAS are methods of proving triangles congruent, these combinations guarantee there will be only one triangle of the given size and shape (one unique solution). \| \| \| \| \| ASA - Two angles and the included side are given. \| \| \| \| --- \| \| In ΔABC, m∠C = 80º, m∠B = 34º and a = 16. Find side b to nearest integer. Solution:Since side a is given, we need the measure of ∠A. m∠A = 180 - (m∠B + m∠C) = 180 - (34 + 80) = 66º \| This answer makes sense, since a smaller side is opposite a smaller angle. ANSWER: b = 10 \| \| \| \| AAS - Two angles and the non-included side are given. \| \| \| \| --- \| \| In ΔRST, m∠R = 30º, m∠T = 95º and r = 8. Find s to nearest tenth. Solution:Since we need side s, we need the measure of ∠S. m∠S = 180 - (30 + 95) = 55º \| A larger side is opposite a larger angle. ANSWER: s = 13.1 \| \| \| \| Proving the Law of Sines: \| This proof works with an acute triangle. \| \| \| --- \| \| We start with an oblique (not-right) triangle ABC, with an altitude from C labeled h. (now cross multiply) h = b sin A and h = a sin B (since both expressions = h) b sin A = a sin B (create a proportion that yields the equation above) \| Now, draw an altitude, k, from point A. (now cross multiply) k = c sin B and k = b sin C (since both expressions = k) c sin B = b sin C (create a proportion that yields the equation above) \| \| By combining these results, we have the Law of Sines: \| \| \| \| NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use". \| Topical Outline \| Geometry Outline \| MathBitsNotebook.com \| MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts \| \|
1641	https://hinative.com/questions/1377231	Quality Point(s): 0 Answer: 10 Like: 18 What is the difference between eat dinner and have dinner ?Feel free to just provide example sentences. Do you want to eat dinner with us? Do you want to have dinner with us? ('Have' sounds more polite in this sense) Have you had dinner already? Have you eaten dinner already? (Equally polite) Was this answer helpful? Quality Point(s): 45 Answer: 42 Like: 30 "Eat dinner" focuses more on the action itself. For example, "what are you doing?" "I'm eating my dinner." If it's about who, when and where then "have dinner" is more commonly used. "When/where did you have your dinner?" Just personal experience. Don't know for sure. Was this answer helpful? Quality Point(s): 7203 Answer: 6814 Like: 4417 Please sit down and eat dinner. I am about to have dinner. I have dinner every night at 7 p.m. I eat dinner with my family. I am going to eat dinner at my friend's house tonight. I will eat dinner soon. Was this answer helpful? The Language Level symbol shows a user's proficiency in the languages they're interested in. Setting your Language Level helps other users provide you with answers that aren't too complex or too simple. Has difficulty understanding even short answers in this language. Can ask simple questions and can understand simple answers. Can ask all types of general questions and can understand longer answers. Can understand long, complex answers. Show your appreciation in a way that likes and stamps can't. By sending a gift to someone, they will be more likely to answer your questions again! If you post a question after sending a gift to someone, your question will be displayed in a special section on that person’s feed. Ask native speakers questions for free Solve your problems more easily with the app!
1642	https://www.thoughtco.com/incomplete-dominance-a-genetics-definition-373471	How is Incomplete Dominance Related to Eye Color? Skip to content Menu Home Science, Tech, Math Science Math Social Sciences Computer Science Animals & Nature Humanities History & Culture Visual Arts Literature English Geography Philosophy Issues Languages English as a Second Language Spanish French German Italian Japanese Mandarin Russian Resources For Students & Parents For Educators For Adult Learners About Us Search Close Search the site GO Science, Tech, Math Science Math Social Sciences Computer Science Animals & Nature Humanities History & Culture Visual Arts Literature English Geography Philosophy Issues Languages English as a Second Language Spanish French German Italian Japanese Mandarin Russian Resources For Students & Parents For Educators For Adult Learners About Us Contact Us Editorial Guidelines Privacy Policy Science, Tech, Math› Science› Biology› Genetics› Incomplete Dominance in Genetics Print Robert Ullmann / Getty Images Science Biology Genetics Basics Cell Biology Organisms Anatomy Physiology Botany Ecology Chemistry Physics Geology Astronomy Weather & Climate By Regina Bailey Regina Bailey Biology Expert B.A., Biology, Emory University A.S., Nursing, Chattahoochee Technical College Regina Bailey is a board-certified registered nurse, science writer and educator. Her work has been featured in "Kaplan AP Biology" and "The Internet for Cellular and Molecular Biologists." Learn about ourEditorial Process Updated on September 01, 2019 Close Incomplete dominance is a form of intermediate inheritance in which one allele for a specific trait is not completely expressed over its paired allele. This results in a third phenotype in which the expressed physical trait is a combination of the phenotypes of both alleles. Unlike complete dominance inheritance, one allele does not dominate or mask the other. Incomplete dominance occurs in the polygenic inheritance of traits such as eye color and skin color. It is a cornerstone in the study of non-Mendelian genetics. Incomplete dominance is a form of intermediate inheritance in which oneallelefor a specific trait is not completely expressed over its paired allele. Comparison With Co-Dominance Incomplete genetic dominance is similar to but different from co-dominance. Whereas incomplete dominance is a blending of traits, in co-dominance an additional phenotype is produced and both alleles are expressed completely. The best example of co-dominance is AB blood type inheritance. Blood type is determined by multiple alleles recognized as A, B, or O and in blood type AB, both phenotypes are fully expressed. Discovery Scientists have noted the blending of traits back into ancient times, although until Mendel, no one used the words "incomplete dominance." In fact, Genetics was not a scientific discipline until the 1800s when Viennese scientist and friar Gregor Mendel (1822–1884) began his studies. Bettmann Archive / Getty Images Like many others, Mendel focused on plants and, in particular, the pea plant. He helped define genetic dominance when he noticed that the plants had either purple or white flowers. No peas had lavender colors as one might suspect. Up to that time, scientists believed that physical traits in a child would always be a blend of the traits of the parents. Mendel proved that in some cases, the offspring can inherit different traits separately. In his pea plants, traits were visible only if an allele was dominant or if both alleles were recessive. Mendel described a genotype ratio of 1:2:1 and a phenotype ratio of 3:1. Both would be consequential for further research. While Mendel's work laid the foundation, it was German botanist Carl Correns (1864–1933) who is credited with the actual discovery of incomplete dominance. In the early 1900s, Correns conducted similar research on four o'clock plants. In his work, Correns observed a blend of colors in flower petals. This led him to the conclusion that the 1:2:1 genotype ratio prevailed and that each genotype had its own phenotype. In turn, this allowed the heterozygotes to display both alleles rather than a dominant one, as Mendel had found. Example: Snapdragons As an example, incomplete dominance is seen in cross-pollination experiments between red and white snapdragon plants. In this monohybrid cross, the allele that produces the red color (R) is not completely expressed over the allele that produces the white color (r). The resulting offspring are all pink. The genotypes are:Red (RR)XWhite (rr) =Pink (Rr). When the first filial (F1) generation consisting of all pink plants is allowed to cross-pollinate, the resulting plants (F2 generation) consist of all three phenotypes[1/4 Red (RR): 1/2 Pink (Rr): 1/4 White (rr)]. The phenotypic ratio is 1:2:1. When the F1generation is allowed to cross-pollinate with true breeding red plants, the resulting F2plants consist of red and pink phenotypes [1/2 Red (RR): 1/2 Pink (Rr)]. The phenotypic ratio is 1:1. When the F1generation is allowed to cross-pollinate with true breeding white plants, the resulting F2plants consist of white and pink phenotypes [1/2 White (rr): 1/2 Pink (Rr)]. The phenotypic ratio is 1:1. In incomplete dominance, the intermediate trait is the heterozygous genotype. In the case of snapdragon plants, plants with pink flowers are heterozygous with the (Rr) genotype. The red and white flowering plants are both homozygous for plant color with genotypes of (RR) red and (rr) white. Polygenic Traits Polygenic traits,such as height, weight, eye color, and skin color, are determined by more than one gene and by interactions among several alleles. The genes contributing to these traits equally influence the phenotype and the alleles for these genes are found on different chromosomes. The alleles have an additive effect on the phenotype resulting in varying degrees of phenotypic expression. Individuals may express varying degrees of a dominant phenotype, recessive phenotype, or intermediate phenotype. Those that inherit more dominant alleles will have a greater expression of the dominant phenotype. Those that inherit more recessive alleles will have a greater expression of the recessive phenotype. Those that inherit various combinations of dominant and recessive alleles will express the intermediate phenotype to varying degrees. Cite this Article Format mlaapachicago Your Citation Bailey, Regina. "Incomplete Dominance in Genetics." ThoughtCo, Jun. 25, 2024, thoughtco.com/incomplete-dominance-a-genetics-definition-373471.Bailey, Regina. (2024, June 25). Incomplete Dominance in Genetics. Retrieved from Bailey, Regina. "Incomplete Dominance in Genetics." ThoughtCo. (accessed September 24, 2025). copy citation Sponsored Stories Why You Wake Up Exhausted Even After 8 Hours of Sleep (It’s Not Stress)toptrendinggadgetreviews.com North Charleston New Policy for Senior Drivers thequotegeneral.com Heart Surgeon Begs: Throw Out Your Olive Oil Now (Here’s Why)enrichyourfood.com Ask A Pro: "I'm 70 with $1.4M in IRAs. Should I convert $120K/Year to a Roth?"SmartAsset Polygenic Inheritance of Traits Like Eye Color and Skin Color What Is Genetic Dominance and How Does It Work? What Is Mendel's Law of Segregation? How Do Alleles Determine Traits in Genetics? Mendel's Law of Independent Assortment Types of Non-Mendelian Genetics A Genetics Definition of Heterozygous Genetics Basics Sponsored Stories Lymphedema Swelling? Eat This Daily And See How It Feels nofluidretention.com Warren Buffett, Elon Musk and other moguls all use the 5-hour rule. Here's what I learned from it Blinkist Magazine Get the same tax-smart tech as the elite Betterment Dating For Serious Singles wingtalks.com Introduction to Mendel's Law of Independent Assortment Phenotype: How a Gene Is Expressed As a Physical Trait What Are Traits? Genes and Genetic Inheritance Law of Multiple Alleles What Does Homozygous Mean in Genetics? Genes, Traits and Mendel's Law of Segregation Monohybrid Cross: A Genetics Definition ThoughtCo Follow Us Science, Tech, Math Humanities Languages Resources About Us Advertise Careers Privacy Policy Editorial Guidelines Contact Terms of Service Your Privacy Choices ThoughtCo is part of the People Inc.publishing family. By clicking “Accept All Cookies”, you agree to the storing of cookies on your device to enhance site navigation, analyze site usage, and assist in our marketing efforts. Cookies Settings Accept All Cookies
1643	https://mathblog.com/reference/algebra/factoring/calculator/factors-of-4/	Factors of 4 with Prime Factorization, Factor Pairs and Factor Tree MathBlog Books Applied Math Basic Math Arithmetic Arithmetic Sequence Number Theory: Definition, Topics, Examples Whole Numbers Rounding Estimation Place Value Addition Subtraction Negative Numbers Multiplication Prime Factorization Decimals Division Fractions Find The Mean Find The Median Find The Mode Order of Operations Percentages Probability Ratios Absolute Value Number Line Equations Distance Between Two Points Logarithm Fibonacci Sequence Geometry Triangles Right Triangle SSS Triangles SSA Triangle SAS Triangle ASA Triangles Angles Area Circle Circumference Area of a Square Sphere Cone Cylinder Heptagon Congruent Shapes Hexagon Octagon The Pentagon Shape: Types, Formulas and Fun Facts Parallel Lines Parallelogram Pi Polygon Rectangle Pyramid Rectangular Pyramid Rhombus Trapezoid Shape: Definition, Area Formula and Fun Facts Tessellations Radian Measure Algebra Interval Notation How To Calculate Slope Factoring Binomial Expansion Matrix Algebra Linear Programming Quadratic Formula Variables with Exponents Synthetic Division Inverse Functions Asymptotes Hyperbolas Graphing Inequalities Trigonometry Law of Cosines Law of Sines Trig Ratios Trig Integrals Right Triangle Trig Trig Functions Calculus Minima and Maxima Improper Integrals Definite Integrals Theorems Pythagorean Theorem Bayes’ Theorem Math Education Statistics Statistics Definitions Math News History MathBlogReferenceAlgebraFactoringCalculator Factors of 4 MathBlog TeamNovember 8, 2024No Comments Factors of 4 are 1, 2, 4. Including 1 and 4 itself, there are 3 distinct factors for 4. The prime factors of 4 are 2, and its factor pairs are(1, 4), (2, 2). We've put this below in a table for easy sharing. Factors of 4 Factors of 4:1, 2, 4 Prime Factors of 4:2 Factor Pairs of 4:(1, 4), (2, 2) How to calculate factors? To be a factor of 4, a number must divide 4 exactly, leaving no remainder. In other words, when 4 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 4 and are key in understanding its mathematical properties. Below, we outline how to calculate the factorization of 4 using four methods: basic factorization, prime factorization, the division method, or using GCD and LCM. We also include a detailed analysis of factor pairs and a factor tree to illustrate the breakdown. Method 1: Basic Factorization Basic Factorization is a method to find the factors of a number by systematically testing each whole number from 2 up to the number itself to see which ones divides with zero remainder (evenly). The process is somewhat time consuming if a number is high, that's why you should master divisibility rules, to make the process faster. Here's the breakdown for 4: \| Divisor \| Is it a factor of 4? \| Verification \| --- \| 1 \| Yes, 1 is a factor of every number. \| 1 × 4 = 4 \| \| 2 \| Yes, 4 is an even number so it's divisible by 2. \| 2 × 2 = 4 \| \| 3 \| No, the sum of its digits (4) is not divisible by 3. \| 4 \| Yes, the last two digits (4) form a number divisible by 4. \| 4 × 1 = 4 \| \| ... \| continue with all the other numbers. \| \| Method 2: Prime Factorization Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. This means a prime number cannot be formed by multiplying two smaller natural numbers. For example: 2 is a prime number because its only divisors are 1 and 2. 3 is prime for the same reason—it can only be divided evenly by 1 and 3. 4 is not prime because it can be divided by 1, 2, and 4. 5, 7, 11, and 13 are also prime numbers. Prime numbers are fundamental in mathematics because they are the "building blocks" of whole numbers. Any natural number greater than 1 can be expressed as a product of prime numbers, which is known as its prime factorization. How to do prime factorization of 4 You start by dividing 4 to each prime number, multiple times, until the remainder is 0. Then you move on to the next prime number. To save time, you should test with up to 4 The prime factors of 4 are 2.\| Prime Number \| Is it a factor of 4? \| Verification \| --- \| 2 \| Yes, 4 is divisible by 2. \| 4 ÷ 2 = 2, R0 \| \| 2 \| Yes, the result 2 is divisible by 2. \| 2 ÷ 2 = 1, R0 \| Method 3: Division Method The Division Method is a systematic approach to finding all the factors of a 4 by performing successive divisions. This method involves dividing 4 by every integer from 1 up to 4 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 4, and only kept those that do: \| Divisor \| Verification \| --- \| \| 1 \| 4 ÷ 1 = 4 \| \| 2 \| 4 ÷ 2 = 2 \| \| 4 \| 4 ÷ 4 = 1 \| Using the division method, we calculated that factors of 4 are 1, 2, 4. Factor Tree of 4 The factor tree of 4 shows the step-by-step breakdown of 4 into its prime factors. Each branch of the tree represents a division of 4 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 4 and highlights the structure of its prime factorization. The factor tree for 4 4 \|\ 2 2 Factor Pairs of 4 (Visualization) Factor pairs of 4 are sets of two numbers that, when multiplied together, result in 4. Factor pairs are symmetric and mirror around the square root of 4, such as (1, 4) and (4, 1), and can be both positive and negative pairs as long as their product equals 4. Factor pairs of 4:\| Negative factor pairs \| Positive factor pairs \| --- \| \| (-1, -4) \| (1, 4) \| \| (-2, -2) \| (2, 2) \| All factor pairs of 4 are (1, 4), (2, 2), (-1, -4), (-2, -2). Why Should I Care About Factors of 4? Turns out, factors aren’t just about boring math equations—they’re like secret superpowers hiding inside numbers! Knowing them can help you split things up, share with friends, or even spot hidden patterns. Want to know how? Check out these real-life examples that show just how cool factors really are: Grouping Students: A teacher has 4 students and wants to form groups. She can create 2 groups, with 2 students in each group because 2 × 2 = 4. Filling Boxes: You have 4 toys and want to fill boxes. If each box holds 2 toys, you’ll need 2 boxes because 2 × 2 = 4. Organizing School Desks: You have 4 desks to arrange in a classroom. If each row has 1 desks, you’ll create 4 rows because 1 × 4 = 4. Arranging Books on a Shelf: You have 4 books to place on shelves. If each shelf holds 2 books, you’ll fill 2 shelves because 2 × 2 = 4. Stocking a Pantry: You have 4 cans of food to stock in the pantry. If each shelf holds 1 cans, you’ll need 4 shelves because 1 × 4 = 4. Recommended Math Books Click here to see our math book list. Search Search AboutPrivacy Policy MathBlog Copyright © 2025.
1644	https://agricarehub.com/lorentz-factor-calculator/	Skip to content Agri Care Hub Facebook Twitter Youtube Instagram Menu Lorentz Factor Calculator Calculate Lorentz Factor About the Lorentz Factor Calculator The Lorentz Factor Calculator is a scientifically precise tool designed to compute the Lorentz factor, ( \gamma ), using the formula ( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} ), from Einstein’s special relativity. This calculator provides accurate results for students, researchers, and professionals studying relativistic effects like time dilation and length contraction. By inputting an object’s velocity, users can instantly calculate the Lorentz factor, essential for understanding high-speed physics. Learn more about the Lorentz Factor or explore applications at Agri Care Hub. Importance of the Lorentz Factor Calculator The Lorentz factor, ( \gamma ), is a cornerstone of Einstein’s special relativity, describing how time, length, and mass change for objects moving at relativistic speeds. The Lorentz Factor Calculator is vital for simplifying this calculation, which is crucial in fields like particle physics, astrophysics, and advanced engineering. In educational contexts, it helps students verify calculations and grasp the implications of special relativity, such as time dilation or length contraction. In professional settings, it aids researchers in modeling high-speed particles or spacecraft, where relativistic effects are significant. For instance, in agriculture, understanding relativistic effects can inform satellite-based technologies for precision farming, as supported by Agri Care Hub. By automating the computation of ( \gamma ), the calculator eliminates errors, enhances efficiency, and supports both theoretical and applied science, making it an essential tool for exploring the physics of high velocities. User Guidelines The Lorentz Factor Calculator is designed for ease of use, ensuring accessibility for users of all levels. Follow these steps to obtain accurate results: Enter Velocity: Input the object’s velocity in meters per second (m/s), ensuring it is less than the speed of light (299,792,458 m/s). Calculate: Click the “Calculate” button to compute the Lorentz factor using ( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} ). Review Results: The output displays the Lorentz factor, along with the formula and context for transparency. Reset if Needed: Clear inputs by refreshing the page or entering a new value. Ensure the velocity is a positive number less than the speed of light, as faster-than-light speeds are physically invalid. The calculator flags invalid entries, such as negative velocities or velocities exceeding ( c ), to guide corrections. For best results, verify input accuracy and use realistic values, such as fractions of ( c ), for meaningful relativistic effects. When and Why You Should Use the Lorentz Factor Calculator The Lorentz Factor Calculator is essential in scenarios requiring precise calculations of relativistic effects: Educational Purposes: Students studying special relativity can use it to verify calculations, explore time dilation, length contraction, or relativistic mass increase. Particle Physics: Researchers calculate the Lorentz factor for high-speed particles in accelerators like the Large Hadron Collider. Astrophysics: Scientists model relativistic effects in cosmic rays, neutron stars, or spacecraft traveling at high velocities. Engineering: Engineers working on advanced technologies, such as satellite systems, use it to account for relativistic corrections in GPS or communication systems. Agricultural Technology: Relativistic corrections in satellite-based precision farming systems, as supported by Agri Care Hub. Why use it? Manual calculations of the Lorentz factor involve complex square root operations and the large constant ( c^2 ), which can lead to errors. This tool automates the process, delivering instant, reliable results, allowing users to focus on interpreting relativistic phenomena rather than performing tedious arithmetic. Purpose of the Lorentz Factor Calculator The Lorentz Factor Calculator serves multiple purposes, all aimed at making relativistic calculations accessible and accurate: Educational Support: Provides clear outputs and formula explanations, helping users understand the Lorentz factor and special relativity. Scientific Precision: Built on Einstein’s peer-reviewed formula, ensuring alignment with established physics standards. Practical Utility: Supports applications in particle physics, astrophysics, and engineering by providing accurate Lorentz factor values. Efficiency: Streamlines calculations, saving time for students, researchers, and professionals. The calculator uses the Lorentz factor formula: ( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} ), where ( v ) is the velocity in m/s and ( c ) is the speed of light (299,792,458 m/s). This formula is a fundamental component of special relativity, validated through experiments like time dilation in particle accelerators. Scientific Basis and Formulas The Lorentz Factor Calculator is grounded in Einstein’s theory of special relativity, introduced in 1905, which describes how space and time are affected by high velocities. The Lorentz factor, ( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} ), quantifies relativistic effects such as time dilation (( \Delta t' = \gamma \Delta t )), length contraction (( L' = \frac{L}{\gamma} )), and relativistic mass increase (( m' = \gamma m )). The speed of light, ( c = 299,792,458 \, \text{m/s} ), is a universal constant, and the formula has been verified through experiments in particle physics and astrophysics. The calculator uses the precise value of ( c ) and validates inputs to ensure ( v < c ), preventing non-physical results. For deeper insights, refer to the Lorentz Factor Wikipedia page. Real-World Applications The Lorentz Factor Calculator supports a wide range of applications across various fields: Particle Physics: Calculates relativistic effects for particles moving near the speed of light in accelerators. Astrophysics: Models relativistic phenomena in high-speed cosmic objects, such as pulsars or relativistic jets. Engineering: Applies relativistic corrections to GPS satellites, where time dilation affects clock accuracy. Agricultural Technology: Informs satellite-based systems for precision agriculture, accounting for relativistic effects, as supported by Agri Care Hub. Education: Helps students explore special relativity in theoretical and applied contexts. The calculator’s output can inform simulations, experiments, or educational exercises, making it versatile for both theoretical and practical applications. Advantages Over Manual Methods Manual calculations of the Lorentz factor involve complex operations, including square roots and division by large constants, increasing the risk of errors. The Lorentz Factor Calculator offers: Speed: Delivers instant results for complex relativistic calculations. Accuracy: Uses the precise speed of light and validated formula to eliminate errors. Accessibility: Intuitive interface suitable for beginners and experts. Educational Value: Displays the formula and context, aiding learning. Reliability: Adheres to peer-reviewed physics principles, ensuring trustworthy results. Its design aligns with modern demands for precision and ease of use in academic and professional settings. Potential Limitations and Tips While robust, the calculator has limitations: Velocity Constraint: Velocity must be less than the speed of light (( v < c )), as faster-than-light speeds are physically invalid. Positive Velocity: Negative velocities are not meaningful in this context, as the formula uses ( v^2 ). Real Numbers: Inputs must be real numbers; complex scenarios are not supported. Tips for optimal use: Verify velocity inputs to ensure they are less than 299,792,458 m/s. Use fractions of ( c ) (e.g., 0.9c) for intuitive relativistic effects. Combine with other relativistic calculators for time dilation or length contraction if needed. Explore the Lorentz Factor page for deeper insights. Conclusion The Lorentz Factor Calculator is a powerful, scientifically rigorous tool that simplifies the computation of the Lorentz factor while maintaining high accuracy and usability. Its adherence to Einstein’s special relativity, intuitive design, and precise outputs make it invaluable for education, particle physics, astrophysics, and engineering applications like satellite technology. Whether you’re a student exploring relativity or a professional analyzing high-speed phenomena, this calculator delivers reliable results with ease. For further reading, visit the Lorentz Factor Wikipedia page or explore related applications at Agri Care Hub. We use cookies to provide a better user experience, personalize content, and analyze traffic. By continuing to use this site, you consent to our use of cookies. You can manage your preferences or opt out anytime by clicking on "Manage Options" below. → Index
1645	https://www.chegg.com/homework-help/questions-and-answers/calculate-final-concentration-200-l-300-m-nacl-400-l-150-m-nacl-400-l-water-mixed-assume-v-q227942597	Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. Question: Calculate the final concentration if 2.00 L of 3.00 M NaCl, 4.00 L of 1.50 M NaCl and 4.00 L of water are mixed. Assume there is no volume contraction upon mixing. ... Not the question you’re looking for? Post any question and get expert help quickly. Chegg Products & Services CompanyCompany Company Chegg NetworkChegg Network Chegg Network Customer ServiceCustomer Service Customer Service EducatorsEducators Educators
1646	https://www.savemyexams.com/ap/chemistry/college-board/22/revision-notes/unit-5-kinetics/reaction-rates/concentration-time-graphs-and-rate-constants/	No subjects found Concentration-Time Graphs & Rate Constants (College Board AP® Chemistry): Study Guide Written by: Martín Reviewed by: Stewart Hird Updated on 4 June 2025 Concentration-Time Graphs & Rate Constants Line equations for the Concentration-Time Graphs Zero order concentration-time graph is a straight line The equation for this line is: 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This equation links the concentration of A at any time (), with the initial concentration of A (), the rate constant (k), and the time (t) It can be used to calculate the concentration of A at any time, just by replacing the time If the line equation is rearranged, the following equation is obtained: 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Looking carefully into the equation, if [A]t is plotted against t, an straight line going down is obtained The slope of the straight line is -k The y-intercept of the line is [A]0 An straight line can be drawn for a first order reaction by using the integrated law for a first order reaction The equation for the integrated law is shown below in two equivalent ways: ln [A]t - ln [A]0 = -kt ln [A]t = -kt +ln [A]0 An straight line can be drawn for a second order reaction by using the integrated law for a second order reaction The equation for the integrated law is shown below in two equivalent ways: 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Line equations for zero order, first order and second order reactions Comparison of the axis labels, slopes and y-intercepts between the line equations for a zero order, first order and second order reaction Worked Example A student carried the following reaction at the laboratory: A+B → products After performing the experiment, the student has plotted the following graphs, Using the graphs above, determine the order of reaction with respect to A Answer: The only graph from these three that is an straight line is Graph II The axis from this graph are ln[A] vs time By comparing the three straight line equations, the order of the reaction with respect to [A] must be 1 because it is the only one in which ln[A] is present ln [A]t = -kt + ln [A]0 Therefore, it is a first order reaction with respect to A The importance of the slope in concentration-time graphs The slope in concentration-time graphs determine the rate constant of the reaction The formula for the slope/gradient is: 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Therefore, if the overall order of reaction is determined, the rate constant can be obtained by calculating the slope of the concentration-time graphs For a zero-order reaction, plotting the concentration of reactant vs time [A]t versus t For a first-order reaction, using a plot natural logarithm of the concentration of reactant vs time ln [A]t versus t For a second-order reaction, plotting the inverse of the concentration of reactant vs time Worked Example A student carried the following reaction at the laboratory: A+B → products The student knows it is a zero order with respect to B. She has plotted the following graph: Using the graph above, determine the rate constant and write the general rate equation for the reaction Answer: Step 1: Determine the order of reaction respect to A The axis from this graph are 1/[A] vs time By comparing the three straight line equations, the order of the reaction with respect to [A] must be 2 because it is the only one in which 1/[A] is present 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Therefore, it is a second order reaction with respect to A Step 2: Determine the overall order of reaction Since the statement establishes that it is a zero order reaction respect to B, the overall order is 2 Therefore, the general rate equation should look like this: rate =k [A]2 Step 3: Choose two points and write down their coordinates Point 1 (0, 2.5) Point 2 (80, 37.5) Step 4: Calculate the slope/gradient using the coordinates 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%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7Dtext%7Bfill%3A%23000000%3B%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2225.5%22%20y%3D%2224%22%3Egradient%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19df71cc037e14b064558084cb4%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2263.5%22%20y%3D%2224%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%2273.5%22%20x2%3D%22138.5%22%20y1%3D%2219.5%22%20y2%3D%2219.5%22%2F%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2282.5%22%20y%3D%2214%22%3E37%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19df71cc037e14b064558084cb4%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2292.5%22%20y%3D%2214%22%3E.%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2299.5%22%20y%3D%2214%22%3E5%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19df71cc037e14b064558084cb4%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22110.5%22%20y%3D%2214%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22121.5%22%20y%3D%2214%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19df71cc037e14b064558084cb4%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22127.5%22%20y%3D%2214%22%3E.%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22134.5%22%20y%3D%2214%22%3E5%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2295.5%22%20y%3D%2235%22%3E80%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19df71cc037e14b064558084cb4%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22110.5%22%20y%3D%2235%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22121.5%22%20y%3D%2235%22%3E0%3C%2Ftext%3E%3C%2Fsvg%3E) %3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7Dtext%7Bfill%3A%23000000%3B%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2225.5%22%20y%3D%2224%22%3Egradient%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19df71cc037e14b064558084cb4%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2263.5%22%20y%3D%2224%22%3E%3D%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%2273.5%22%20x2%3D%22138.5%22%20y1%3D%2219.5%22%20y2%3D%2219.5%22%2F%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2282.5%22%20y%3D%2214%22%3E37%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19df71cc037e14b064558084cb4%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2292.5%22%20y%3D%2214%22%3E.%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2299.5%22%20y%3D%2214%22%3E5%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19df71cc037e14b064558084cb4%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22110.5%22%20y%3D%2214%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22121.5%22%20y%3D%2214%22%3E2%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19df71cc037e14b064558084cb4%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22127.5%22%20y%3D%2214%22%3E.%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22134.5%22%20y%3D%2214%22%3E5%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%2295.5%22%20y%3D%2235%22%3E80%3C%2Ftext%3E%3Ctext%20font-family%3D%22math19df71cc037e14b064558084cb4%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22110.5%22%20y%3D%2235%22%3E%26%23x2212%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2214%22%20text-anchor%3D%22middle%22%20x%3D%22121.5%22%20y%3D%2235%22%3E0%3C%2Ftext%3E%3C%2Fsvg%3E) gradient = 0.434 Step 5: Calculate the units for the rate constant 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units of k = M-1 s-1 Step 6: Write down the final rate equation by replacing the calculated value for k rate = k [A]2 rate = (0.434 M-1 s-1) [A]2 You've read 0 of your 5 free study guides this week Unlock more, it's free! 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Unit 1: Atomic Structure & Properties 5 Topics · 18 Study Guides Moles & Molar Mass The Mole Concept Masses & Particles Molar Mass Mass Spectra of Elements Mass Spectrum of an Element Average Atomic Mass Elements & Mixtures Elemental Composition Composition of Mixtures Elemental Analysis Atoms & Electrons Atomic Structure Using Coulomb’s Law Electron Configuration The Aufbau Principle Ionization Energy Photoelectron Spectroscopy Periodic Trends Structure of the Periodic Table Periodic Trends Predicting Properties of Elements Valence Electrons & Ionic Compounds Unit 2: Compound Structure & Properties 4 Topics · 17 Study Guides Chemical Bonding Electronegativity Values Covalent Bonds Ionic vs Covalent Valence Electrons in a Metallic Solid Intramolecular Force & Potential Energy Coulomb's Law & Attractive Forces Ionic Crystals & Metals Ionic Crystals Representing Metallic Bonding Interstitial Alloys Substitutional Alloys Lewis Diagrams Lewis Diagrams Resonance & Lewis Structures Formal Charge Limitations of Lewis Structures VSEPR & Hybridization VSEPR Theory Hybridization Sigma & Pi Bonds Unit 3: Properties of Substances & Mixtures 5 Topics · 33 Study Guides Intermolecular & Interparticle Forces London Dispersion Forces Dipole-Induced Dipole Interactions Dipole-Dipole Interactions Ion-Dipole Interactions Molecular Dipole Moment Hydrogen Bonding Interactions in Large Biomolecules Properties of Solids, Liquids & Gases Explaining the Properties of Solids & Liquids Properties of Ionic Solids Covalent Network Solids Molecular Solids Metallic Solids Noncovalent Interactions in Large Molecules The Structure of Solids The Liquid Phase The Gas Phase Gas Laws The Ideal Gas Law Partial Pressure & Total Pressure Graphical Representations of the Gas Laws Kinetic Molecular Theory The Maxwell Boltzmann Distribution Non-Ideal Behavior of Gases Solutions & Mixtures Calculations About Solutions Homogeneous & Heterogeneous Mixtures Molarity Particulate Representations of Solutions Separation By Chromatography Separation by Distillation Solubility of Ionic & Molecular Compounds Spectroscopy The Electromagnetic Spectrum Transitions Associated with Radiation Properties of Photons Beer-Lambert Law Unit 4: Chemical Reactions 4 Topics · 15 Study Guides Chemical Changes Physical & Chemical Changes Representing Chemical Changes Balancing Chemical Equations Physical & Chemical Processes Stoichiometry Conservation of Mass Mole Calculations Ideal Gas Law & Solutions Introduction to Titration Types of Chemical Reactions Identifying Acid-Base Reactions Oxidation-Reduction Reactions Assigning Oxidation Numbers Oxidation-Reduction (Redox) Reactions Precipitation Reactions Brønsted-Lowry Acids & Bases Brønsted-Lowry Acids & Bases Conjugate Acids & Bases Unit 5: Kinetics 3 Topics · 18 Study Guides Reaction Rates Measuring Reaction Rate Stoichiometry & Reaction Rate Factors Affecting Reaction Rate Introduction to Rate Law Concentration-Time Graphs Manipulating Concentration-Time Graphs Concentration-Time Graphs & Rate Constants Concentration-Time Graphs & Half-Life Catalysts & Reaction Rate Catalysts in Action Elementary Reactions & Collisions Elementary Reactions Collision Model Reaction Energy Profiles The Arrhenius Equation Reaction Mechanisms Reaction Mechanisms Reaction Mechanism & Rate Law Pre-Equilibrium Approximation Multistep Reaction Energy Profile Unit 6: Thermochemistry 3 Topics · 10 Study Guides Chemical Energy Energy of Phase Changes Exothermic & Endothermic Reactions Energy Diagrams Thermal Energy & Molecular Collisions Calorimetry Energy Transfers Calorimetry Calculations Enthalpy Changes Enthalpy of Reaction Enthalpy & Bond Energies Enthalpy of Formation Hess's Law Unit 7: Equilibrium 5 Topics · 14 Study Guides Equilibrium Reactions & Equilibrium Reversible Reactions Equilibrium Constants The Reaction Quotient Calculating the Equilibrium Constant Magnitude of the Equilibrium Constant Manipulating the Equilibrium Constant Equilibrium 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1647	https://approach0.xyz/search/?q=%24f(xy)%3Df(x)f(y)-f(x%2By)%2B1%24&p=1	@grid: 40x2 / 60%; @place-cell: center; @size: calc(100% / @size @i); transform: rotate(calc(@i 5deg)); border-radius: 30%; border: 1px solid hsla( calc(10 + 4 @i), 70%, 68%, @r.8 ); Approach Zero f ( x y ) = f ( x ) f ( y ) − f ( x + y ) + 1 f ( x y ) = f ( x ) f ( y ) − f ( x + y ) + 1 299444 136.738 Solve the functional equation and Solve the functional equation and I'm working on this problem already, the domain and co-domain are the set of Rational Numbers. we see that setting and , we obtain But I'm trying to prove that for all , the case when is obviously true, how do I approach this via induction? do you want to prove for all rational or for all positive integers? @supinf For positive integers what is ? Induction is easy - if then, ... ⋅ functional-equations 331746 136.738 f(xy) = f(x)f(y) - f(x+y) + 1 f(xy) = f(x)f(y) - f(x+y) + 1 The function is defined on the set of all rational numbers and has values in . It satisfies the conditions and for all . Determine . A short solution to a hard problem: Take y=1 and find that . Then replace y by y+1 and use again the condition given and the fact that . You will find that . Thus is aditive and then it's easy. You may study this problem in different interesting cases. Solve it for : a) b) c) Pierre. And what about if My solution: ... 81293 136.738 FE marathon!!! :P ... and . (Here represents multiplication). The solution to the equation is . Find . [hide=FE Reveal]It is division..., so if I did it right[/hide] [quote=derekzoo][hide=FE Reveal]It is division..., so if I did it right[/hide][/quote] do you have a problem? I saw that question 23 coming [hide=FE #2]If satisfies 1. 2. For all , , find .[/hide] [quote=derekzoo][hide=FE Reveal]It is division..., so if I did it right[/hide][/quote] Hm... Do your division correctly plz [quote=derekzoo][hide=FE #2]If satisfies 1. 2. For all , , find . can somebody fix the latex?[/quote] FT[size=150][b]FE[/b][/size]FY [hide = incomplete solution to FE]Because we know that , we want to use that fact. We substitute into the FE to get [im ... 109327 136.738 finding the functions satisfying [duplicate] finding the functions satisfying [duplicate] Define a function which is continuous and satisfies for all . With a supp condition . (I didn't notice that.) How to show that for all that belong to ?i got the ans from Paul that it is true for all rationals x but I still cannot show that for is correct. You probably want to be continuous. Am I right? (Or at least a way to force the real values of given the rational ones...) A duplicate of t ... ⋅ calculus ⋅ analysis ⋅ functions 269892 136.738 About finding the function such that [duplicate] About finding the function such that [duplicate] Define a function which satisfies for all . With a supp condition . (I didn't notice that.) How to show that for all that belong to ? This is homework, right? Starting point: show that . Try to continue with integers, then with rationals of the form , finish with . This is not precisely true. The function which is identically equal to satisfies your function ... ⋅ functions ⋅ functional-equations 248191 136.738 Function Equation Question Involving [duplicate] Function Equation Question Involving [duplicate] Let \mathbb{Q} denote the set of rational numbers. Find all functions from \mathbb{Q} to \mathbb{Q} which satisfy I know that f(x)=x+1 satisfies the equation but I don't know how to prove if this is the only such function. This question was in the 'Induction' section of my textbook. Any hints? Can you prove f(0)=1 ? that would be a start. You are given f(1). Now try to find f(2). As a hint: the values of f(n) and f(1) determine the value of f(n+1), by letting x=n and y=1 in the ... ⋅ functional-equations 205849 136.738 functional Equation functional Equation f: \mathbb{R} \rightarrow \mathbb{R} f(xy)=f(x)f(y)-f(x+y)+1 P(x,0) = f(x)=f(x)f(0)-f(x)-f(0)+1 \implies f(x) = \frac{1-f(0)}{2-f(0)} if f(0) \not = 2 Otherwise f(x) = 2f(x)-f(x)-f(0)+1 \implies 2 =1 which is a contradiction. So the only solution is constant. c=c^2-c-c+1 \implies c={\frac{3-\sqrt{5}}{2},\frac{3+\sqrt{5}}{2}} [quote=whiwho]P(x,0) = f(x)=f(x)f(0)-f(x)-f(0)+1 \implies f(x) = \frac{1-f(0)}{2-f(0)} if f(0) \not = 2 Otherwise f(x) = 2f(x)-f(x)-f(0)+1 \implies 2 =1 which is a contradiction. So the only solution is constant. c=c^2-c-c+1 \implies c={\frac{3-\sqrt{5}}{2},\frac{3+\sqrt{5}}{2}}[/quote ... 47520 136.738 Functional equation [duplicate] ... , we get f(0)=f(1)f(-1)-f(-1)+1 f(1)f(-1)-f(-1)=0 f(-1)(f(1)-1)=0 f(-1)=0\vee f(1)=1 So, no we have three possible cases to consider and I have no idea on how to proceed on any of them. I would appreciate any hints before giving the full solution as in the best case that's all I need right now. Putting y=1 gives you a relationship f(x + 1) = f(x) C + 1 where C = f(1) - 1. That's enough to determine f in the integers up to a constant. I'm sure you can work it out from there. Does this answer your question? About finding the function such that f(xy)=f(x)f(y)-f(x+y)+1 Also: math.stackexchange.com/q/351068, math.stackexchange.com/q/2394682, math.stackexchange.com/q/96325 – all found with SearchOnMath @MartinR Well those have the added clause that f(1)=2 I'm new to solving functional equations and found the following functional equation from a collection of functional equations. Find all functions f: \mathbb{Q}\to \mathbb{R} for f(x+y)=f(x)f(y)-f(xy)+1 for all x,y\in \mathbb{Q}. By substituting x,y\to 0, we get f(0)=f(0)^2-f(0)+1 f(0)^2-2f(0)+1=0 f(0)=1 And then doing the substitution x\to 1,\ y\to -1, we get [imath ... ⋅ functional-equations 102655 136.738 Functional equation in Q Functional equation in Q Find all functions f:\mathbb{Q}\rightarrow\mathbb{Q} such that f(1)=2 and f(xy)=f(x)f(y)-f(x+y)+1 for all x,y\in\mathbb{Q} This is [url= IMO P1[/url]. And this problem has been posted multiple times. We can solve the same equation on \mathbb{R} and also the condition f(1)=2 is unnecessary. \mathbb{Q} version: [url= [url= [url= [url= [url= ... 31282 136.738 Functional Equation in the rationals: [duplicate] Functional Equation in the rationals: f(xy)=f(x)f(y)-f(x+y)+1 [duplicate] Find all the possible functions f:\mathbb{Q}\to\mathbb{Q} such that f(1)=2 and f(xy)=f(x)f(y)-f(x+y)+1\text. I managed to find the function for the set of natural number, by putting x=1 and y=n. From it, I have that f(n+1)=f(n)+1\text, which means that f(n) is an arithmetic progression of natural numbers. I am stuck with generalizing this into the set of rational number, any hit will be appreciated. One more: math.stackexchange.com/questions/96325/finding-the-function how can I find if my question is duplicate You can check here: approach0.xyz/search/… ... ⋅ functional-equations 1 2 3 4 5 ... 10 Links About Query Logs Community AD Open Backend Developer Docs Contribute User Guide This Webpage Sponsorship Sponsor Powered by Akamai Special Thanks Current Index Math StackExchange Art of Problem Solving Help to Add More ...
1648	https://bigwww.epfl.ch/publications/nilchian1503.pdf	3826 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 24, NO. 11, NOVEMBER 2015 Optimized Kaiser–Bessel Window Functions for Computed Tomography Masih Nilchian, Student Member, IEEE, John Paul Ward, Cédric Vonesch, and Michael Unser, Fellow, IEEE Abstract—Kaiser–Bessel window functions are frequently used to discretize tomographic problems because they have two desir-able properties: 1) their short support leads to a low computa-tional cost and 2) their rotational symmetry makes their imaging transform independent of the direction. In this paper, we aim at optimizing the parameters of these basis functions. We present a formalism based on the theory of approximation and point out the importance of the partition-of-unity condition. While we prove that, for compact-support functions, this condition is incompatible with isotropy, we show that minimizing the deviation from the partition of unity condition is highly beneﬁcial. The numerical results conﬁrm that the proposed tuning of the Kaiser–Bessel window functions yields the best performance. Index Terms—Kaiser-Bessel window function, approximation theory, tomography, inverse problem, generalized sampling. I. INTRODUCTION I T IS highly desirable to reduce the radiation dose in X-ray imaging modalities. This can be achieved in two ways. The ﬁrst solution involves a reduction in the intensity of the X-ray but contaminates the data with physical noise. The second solution involves a decrease in the number of projection angles. The price to pay for this reduction is that the reconstruction problem becomes ill-posed and can no longer be solved using traditional direct methods. Instead, the deployment of more sophisticated iterative schemes is needed. In order to specify such methods, one ﬁrst discretizes the imaging operator and then selects a reconstruction scheme, which typically involves the choice of a cost functional to minimize. The second aspect is absolutely crucial when the problem is ill-posed and is typically addressed by introducing suitable regularization functionals such as sparsity or total-variation regularization. This is the aspect of the problem that is addressed primarily in the literature –. Manuscript received October 13, 2014; revised March 6, 2015 and June 2, 2015; accepted June 5, 2015. Date of publication July 1, 2015; date of current version July 30, 2015. This work was supported in part by the European Research Council through the European Union’s Seventh Framework Programme (FP7/2007-2013) under Grant 267439, in part by the Swiss National Science Foundation under Grant 200020-144355, and in part by the Center for Biomedical Imaging through the Geneva-Lausanne Univer-sities and École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. M. Nilchian, J. P. Ward, and M. Unser are with the Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland (e-mail: masih.nilchian@epﬂ.ch; jpward08@gmail.com; michael.unser@epﬂ.ch). C. Vonesch is with the Center for Biomedical Imaging–SP, École Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland (e-mail: cedric.vonesch@epﬂ.ch). Color versions of one or more of the ﬁgures in this paper are available online at Digital Object Identiﬁer 10.1109/TIP.2015.2451955 In this paper we want to concentrate on the ﬁrst aspect: the choice of a suitable reconstruction space. This space is usually determined as a set of functions of the form f (x) = k∈Zd c[k]ϕ x T −k , (1) where T is the sampling step. The reconstruction space is then speciﬁed through the choice of the generating function ϕ. In computed tomography, where the mathematical model is based on the Radon transform and its variants, it is beneﬁcial to use a generating function that has two particular properties: 1) short support for fast computation and 2) rotational symme-try for efﬁcient computation of the imaging transform. Among the functions satisﬁng these properties, Lewitt intro-duced generalized Kaiser-Bessel window functions (KBWFs) as an optimal generating function to be used in the context of computed tomography. These functions are widely used in electron microscopy – and con-ventional X-ray and differential phase-contrast computed tomography , –. KBWFs involve three parameters that need to be adjusted . These parameters are empirically tuned to improve the quality of reconstructed constant images , . In this paper, we investigate approximation-theoretic properties of the basis functions, and we show how to optimize the parameters for the best performance. We also present experimental results that corroborate our theoretical prediction. The three contributions of this paper can be brieﬂy summarized as follows: • A measure to predict the performance of the generating function used for discretizing the forward model. • A method to compute the optimal parameters for a generalized KBWF. • An experimental validation of the proposed method. The rest of this paper is organized as follows. In Section II, we describe the discretization scheme of the imaging operator and discuss the properties that should be satisﬁed by the generating function to be used for discretizing the forward model. In Section III, we study the KBWFs and propose new parameters for their use in the discretization scheme. Then, in Section IV, numerical experiments are presented to justify our choices. II. DISCRETIZATION SCHEME We ﬁrst explain how the discretization of the forward model is intimately connected with the choice of a given basis function. We then recall some fundamental results from 1057-7149 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information. NILCHIAN et al.: OPTIMIZED KAISER–BESSEL WINDOW FUNCTIONS FOR COMPUTED TOMOGRAPHY 3827 approximation theory that ensure stability and allow one to predict the expected discretization error. This will point to the importance of the partition-of-unity property which, unfortunately as we shall prove, is incompatible with the compact-support and isotropy properties. A. Matrix Formulation Reconstruction is usually formulated as a linear inverse problem. To solve it, it is convenient to introduce discrete representations of the object and the imaging operator. With-out loss of generality, we consider an object in 2D. The model of the object, from the perspective of generalized sampling theory , , is obtained by specifying a suit-able reconstruction space. Speciﬁcally, we select VT (ϕ) as the principal shift-invariant space generated by the function ϕ ∈L2(R2). This space is deﬁned by VT (ϕ) = ⎧ ⎨ ⎩ k∈Z2 c[k]ϕ x T −k : c ∈ℓ2(Z2) ⎫ ⎬ ⎭, (2) where x ∈R2. The corresponding orthogonal projection operator PT : L2(R2) →VT (ϕ) is deﬁned as PT f = argmin g∈VT (ϕ) ∥f −g∥L2. (3) In practice, however, the values of c in (1) are determined based on the solution of an inverse problem. The mathematical model of many imaging modalities in the context of X-ray computed tomography is based on the Radon transform and its derivatives. Note that the Radon transform is a linear and pseudo shift-invariant operator, R{ϕ(x −k)}(y, θ) = R{ϕ(x)}(y −⟨k, θ⟩, θ), (4) where θ = (cos θ, sin θ). Then, its application on a function f ∈VT (ϕ) is R(n){ f }(y, θ) = k∈Z2 c[k]R(n){ϕT (x −T k)}(y, θ) = k∈Z2 c[k]R(n){ϕT }(y −T ⟨k, θ⟩, θ), (5) where ϕT (x) = ϕ(x/T), and R(n) f (y, θ) = ∂nR f ∂yn (y, θ), with R : L2(R2) →L2(R×[0, π]) being the Radon-transform operator. The formulation of the reconstruction as a linear inverse problem is usually stated as the matrix equation g = Hc, (6) where g is the measurement vector, H is the system matrix, and c is the discrete representation of the object of interest. Using (5), the matrix formulation can be obtained as follows: The measurement vector g contains values of the imaging transform R(n){ f }(y, θ) at the sampled points y j = jy and θi = iθ, where i, j ∈Z. The object f is represented with its coefﬁcients c within the space VT (ϕ). The system matrix H is given by H,k = R(n){ϕT }(y j −T ⟨k, θ i⟩, θi). (7) Note that, in order to compute the imaging operator, there is no need to store the whole system matrix because it is sufﬁcient to have access to a lookup table that contains the projection of one basis function along every direction. B. Desirable Properties of the Basis Functions We require the basis function ϕ to satisfy the following four properties: 1) Riesz Basis: Every object f ∈VT (ϕ) must be uniquely speciﬁed by its coefﬁcients c. This requires the existence of a positive constant A such that ∀c ∈ℓ2, A · ∥c∥2 ℓ2 ≤ k∈Z2 c[k]ϕ x T −k L2 . (8) In addition, the representation should be stable. This requires the existence of a positive constant B such that ∀c ∈ℓ2, k∈Z2 c[k]ϕ x T −k L2 ≤B · ∥c∥2 ℓ2 . (9) Together, these two conditions are equivalent to ϕ being a Riesz basis of VT (ϕ). 2) Partition of Unity: It is constructive for such a discretiza-tion scheme that the model approximate any input function as closely as desired by choosing a sufﬁciently small sampling step. More precisely, the approximation error should vanish whenever the sampling step T tends to zero. We thus require that lim T →0 ∥f −PT f ∥L2 = 0. (10) Theorem 1 : The L2-approximation error of the oper-ator PT : L2 →VT (ϕ) can be written, for f ∈L2(R2), by ϵ f (T ) = ∥f −PT f ∥L2 = R2 Eϕ(Tω)\| f (ω)\|2 dω 2π 1/2 + ϵcorr, (11) where ϵcorr is a correction term, and Eϕ is the error kernel deﬁned in the least-squares case as Eϕ(ω) = 1 − \| ϕ(ω)\|2 k∈Z2\| ϕ(ω + 2kπ)\|2 , (12) where ϕ is the Fourier transform of ϕ. Speciﬁcally, if f ∈Wr 2 (Sobolev space of order r) with r > 1/2, then \|ϵcorr\| < γ T r∥f (r)∥L2, where γ is some constant. Under the regularity assumption that f and its ﬁrst order derivative are in L2(R2), the asymptotic convergence lim T →0 ϵ f (T) = 0 (13) is achieved if and only if the basis function ϕ satisﬁes the partition-of-unity condition [21, Appendix B] k∈Z2 ϕ(x + k) = 1, ∀x ∈R2. (14) 3828 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 24, NO. 11, NOVEMBER 2015 The equivalent formulation of the partition of unity in the frequency domain is ϕ(2πn) = δ[n], ∀n ∈Z2, (15) where δ is the 2D Kronecker delta function. 3) Compact Support: The basis function ϕ should be compactly supported in order to reduce the computational cost and also for localization in the spatial domain. 4) Isotropy: For the implementation of the imaging operator, it is required to store the values of its application on the basis function along different directions. If the basis func-tion is isotropic, its projections do not depend on the direction, which leads to simplicity and efﬁciency of implementation. C. Incompatible Properties There is a negative result that is considered in the following theorem: Theorem 2: The following properties are mutually exclusive for an isotropic basis function: 1) compact support; 2) partition of unity. Proof: The proof is given in Appendix A. Here, we provide only a sketch of the argument. The partition-of-unity condition implies the conﬁguration (15) of zeros of the Fourier transform of the basis function. At the same time, the Hankel transform of an even compactly supported function is an entire function of ﬁnite exponential type. Jensen’s theorem provides a contradiction between these two properties. D. Revisiting Optimality in the Projection Domain We now bound the error of approximation incurred by RT f = R{PT f }. It can be extended to any derivative of the Radon transform through the Fourier-slice theorem since \|\|R(n) f \|\|L2 = \|\|RF−1{\|ω\|n f (ω)}\|\|L2. To this end, we use the Sobolev norm ∥·∥2 W 1/2 2 in the projection domain. If g ∈L2(R2), then ∥g∥2 W 1/2 2 = 2π 0 ∞ 0 (1 + ω2) 1 2 \| g(ω, θ)\|dθdω, (16) where g(ω, θ) is the polar form of the Fourier transform of g. Theorem 3: Let the Sobolev approximation error of the operator RT for f ∈ L2(R2) be ϵR f (T) = ∥R f −RT f ∥W 1/2 2 . Then, there exist positive constants r1, R1 > 0 such that r1ϵ f (T ) ≤ϵR f (T) ≤R1ϵ f (T). (17) Lemma [25, Sec. II.5]: Let ⊂R2 be a compact domain. Then, there exist positive constants r2 and R2 such that, for any L2(R2) function f that is supported on , it holds that r2 ∥f ∥L2 ≤∥R f ∥W 1/2 2 ≤R2 ∥f ∥L2 . (18) Proof of Theorem 3: By letting f ←( f −PT f ) in (18), we obtain (17). □ This theorem implies that the average error over all angles is small in the transform domain when the error of approximation is small in the object domain. While the theorem is an average result that involves a continuum of angles, it is still useful practically because it gives us the approximation error in the transform domain over a family of images that would correspond to all rotated versions of a given reference image. III. OPTIMIZED KAISER-BESSEL WINDOW FUNCTION The generalized family of KBWFs is isotropic, which makes it advantageous for the representation of the imaging operator. Our goal here will be to determine the optimal set of parameters to best match the partition-of-unity condition which is so fundamental to approximation theory. A. Generalized Kaiser-Bessel Window Functions The generalized KBWF, deﬁned as ϕ(x) = ⎧ ⎨ ⎩ √ 1−(∥x∥/a)2 m Im α√ 1−(∥x∥/a)2 Im(α) 0 ≤∥x∥≤a 0 otherwise, (19) is speciﬁed by three parameters: 1) the order m of the modiﬁed Bessel function Im; 2) the window taper α; 3) the support radius a of the function. The parameter m allows us to control the smoothness of the function and the parameter α determines its shape. This function is isotropic, which makes the computation of the imaging operator signiﬁcantly faster. However, it is worth noting that this function does not satisfy the partition of unity (see Theorem 2). B. Measure of Optimality of a Basis Function If a basis function satisﬁes the partition-of-unity condition, then, as the sampling step vanishes, the error of approximation tends to zero. For those bases that do not satisfy the partition of unity, we deﬁne the residual error Aϕ = sup f ∈L2 ∥f ∥−1 L2 lim T →0 ϵ f (T) (20) for f ∈L2(Rd), which shows the deviation from the partition of unity. A basis function ϕ with lower residual error is more desirable as a generating function for the reconstruction space. Theorem 4: The residual error of a function ϕ ∈L2(Rd) is the quantity Aϕ = n̸=0\| ϕ(2πn)\|2 \| ϕ(0)\|2 . (21) Proof: We assume the regularity condition that f and its ﬁrst order derivative are in L2. From (11), we have a formula for ϵ f in terms of Eϕ as deﬁned in (12). We represent Eϕ using its Taylor series Eϕ(T ω) = N \|n\|=0 ∂nEϕ(0) n! (Tω)n + o( T N+1ω N+1 ), (22) where n = (n1, n2, . . . , nd) with nonnegative integer values, \|n\| = d i=1 ni, ω = (ω1, ω2, . . . , ωd), n! = n1!n2!...nd!, ωn = ωn1 1 ωn2 2 ...ωnd d , and ∂nEϕ(0) = ∂n1 ω1 ∂n2 ω2 · · · ∂nd ωd Eϕ(0). (23) NILCHIAN et al.: OPTIMIZED KAISER–BESSEL WINDOW FUNCTIONS FOR COMPUTED TOMOGRAPHY 3829 Therefore, we can rewrite the approximation error ϵ f as ϵ f (T) = ∥f −PT { f }∥L2 = ⎛ ⎝ Rd ⎛ ⎝ N \|n\|=0 ∂nEϕ(0) n! (T ω)n ⎞ ⎠\| f (ω)\|2 dω 2π ⎞ ⎠ 1/2 + ϵ, (24) where ϵ = o(T N+1 ∥ω∥N+1) + ϵcorr. Then, Fubini’s theorem implies that ϵ f (T) = ⎛ ⎝ N \|n\|=0 ∂nEϕ(0) n! T \|n\| Rd ωn\| f (ω)\|2 dω 2π ⎞ ⎠ 1/2 + ϵ = ⎛ ⎝ N \|n\|=0 ∂nEϕ(0) n! T \|n\| f (n/2) 2 L2 ⎞ ⎠ 1/2 + ϵ, (25) where f (n) = ∂n1 ∂x1 ∂n2 ∂x2 · · · ∂nd ∂xd f. (26) We now have that lim T →0 ϵ f (T) = Eϕ(0)1/2 ∥f ∥L2 . (27) Therefore, sup f ∈L2 ∥f ∥−2 L2 lim T →0 ϵ f (T) 2 = sup f ∈L2 Eϕ(0) = n̸=0\| ϕ(2πn)\|2 \| ϕ(0)\|2 . (28) C. Optimal Parameters for the Kaiser-Bessel Window Function There are three parameters that describe KBWFs. The radius parameter a determines its support. We set it to a = 2; this allows us to compare the optimal KBWF with the cubic B-spline. The order of the modiﬁed Bessel function is set to m = 2. In the context of 3D imaging, Matej and Lewitt empirically tune the window taper parameter α to improve the quality of reconstructed constant images. In contrast, we base our analysis on approximation-theoretic properties and determine α to minimize the residual error Aϕ. Interestingly, this leads to a condition similar to the complicated criterion of . But we go one step farther and provide a simpli-ﬁed equivalent condition in (21). The measure for different values α is depicted in Fig. 1(b). This plot indicates that values of α in the range [6, 11.2] are good choices for reconstruction, with two local optima of α = 7.05, 10.45 of comparable magnitude. The latter value is very close to 10.4, which is the value proposed in . There are modalities where the reconstruction problem is separable into a set of independent 2D problems: X-ray parallel-beam tomography, transmission electron microscopy with single-axis tilting, 2D positron emission tomography systems with septa, and single-photon emission computed Fig. 1. The Kaiser-Bessel window taper parameter is denoted by α in the Figure. Optimality measure with respect to different values of α in the (a) 2D and (b) 3D domains. tomography with parallel or fan-beam collimators. Then it is worthwhile to evaluate the optimal parameters for 2D KBWFs. We illustrate in Fig. 1(a) the residual error with respect to the parameter α in a 2D space. Again, it appears that values of α in the range [7, 11.5] are good choices for 2D reconstruction, with α = 7.91, 10.83 being the two best choices. IV. NUMERICAL EVALUATION We now present experiments where we numerically evaluate the discretization scheme based on KBWFs, with the parame-ters suggested in this paper. A. Inﬂuence of the Discretization Step By deﬁnition the optimal reconstruction in the least-square sense is the orthogonal projection of the sample on the reconstruction space, independently of the chosen algorithm. To investigate the dependence upon the grid size, we compute the optimal reconstruction with respect to different grid sizes when the generating function of the reconstruction space is a KWBF with different parameters. The reference object and signal-to-noise (SNR) computations are deﬁned with respect to the ﬁne grid. The SNR is deﬁned as the relative mean-square with respect to the reference (oracle), SNR(oracle,reconstruction) = 20log \|\|oracle\|\|L2 \|\|oracle-reconstruction\|\|L2 . (29) The grid size is progressively increased, which shows the dependences upon the sampling rate. We choose two medical samples: a coronal section of a human lung and a coronal section of a rat brain. Also, a region of interest has been chosen as shown in Figs. 2(a) and 2(b). We ﬁrst tested the KBWF with α = 2, which is well outside of the optimal interval [7, 11.5], and the results were very poor (SNR = 4 dB). We then compared the performance for the value α = 5 and α = 7.91. The former is close to, but outside of the optimal interval, while the latter is the ﬁrst of our proposed choices. Their performances are depicted in Figs. 2(c) and 2(d). It conﬁrms that using KBWFs with the proposed parameter has better optimal reconstruction compared to α = 5 for different grid sizes. This experiment shows that the “optimal choice” based on the asymptotical behavior (see (20)) is also always better for different grid sizes. The results for four times coarser grid representation 3830 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 24, NO. 11, NOVEMBER 2015 Fig. 2. (a) Coronal section of a human lung and region of interest. (b) Coronal section of a rat brain and region of interest. The performance of the optimal solution with respect to the grid size is depicted in (c) and (d). Fig. 3. The orthogonal projection of the coronal section of a human lung and region of interest using KBWFs with (a) α = 5 and (b) α = 7.91 in 4 times coarser grid. The orthogonal projection of the coronal section of a rat brain and region of interest using KBWFs with (c) α = 5 and (d) α = 7.91 in 4 times coarser grid. are depicted in Fig. 3. The artifacts are highlighted by yellow arrows. B. Reconstruction of an Analytical Phantom As data, we use the 2D synthetic phantom presented in and shown in Fig. 4. (a). The analytical formula for computing imaging transforms of the phantom is given in [7, Sec. 4.4]. Fig. 4. The Kaiser-Bessel window taper parameter is denoted by α in the Figure. (a) 2D analytical phantom with isotropic elements. (b) Zoomed version of the proposed measure. The accuracy of the reconstruction of the analytical phantom versus the window taper parameter of KBWFs is shown in (c). Its Radon- transform error in the same coarse grid is depicted in (d). 1) Conventional Tomography: The size of the phantom for the ﬁrst experiment is (2,048 × 2,048) pixels. The sinogram of the phantom is computed analytically with 1,800 viewing angles that are chosen uniformly between 0 and π; it comprises the measurements. Note that the resolution of the reconstruction in the ﬁne grid is the same as the resolution of the measurements (detector pixel size). As there is a large number of views for the reconstruction, we minimize the least-squares error J(c) = ∥Hc −g∥2 L2 , (30) where g is the measurement vector. The object is reconstructed on a grid that is (4 × 4) times coarser than the discretization grid. Then, the basis function is used to resample the object on a ﬁner grid. We use the conjugate-gradient algorithm for the minimization. As the number of directions is on the order of the size of the object, we do not use any regularization. The signal-to-noise ratio (SNR) of the reconstructions and the projection versus different values of the window taper of KBWFs are shown in Fig. 4(c) and 4(d). The best performance is obtained by using a KBWF with α = 7.75, which is very close to the ﬁrst minimum of our criterion function in Fig. 1. However, values of α in the range [7, 11.5] do also perform reasonably well, which is consistent with theoretical analysis of Section III-C. 2) Differential Phase-Contrast Tomography: We evaluate the performance of KBWFs with the proposed parameters in X-ray differential phase-contrast tomography. The mathemati-cal model of this imaging modality is based on the derivative of the Radon transform. The differentiated sinogram of the phantom with size (512 × 512) pixels is again computed analytically with 1,800 viewing angles that are chosen uniformly NILCHIAN et al.: OPTIMIZED KAISER–BESSEL WINDOW FUNCTIONS FOR COMPUTED TOMOGRAPHY 3831 Fig. 5. The Kaiser-Bessel window taper parameter is denoted by α in the Figure. Performance of the (a) reconstruction and (b) projection using KBWFs for differential phase-contrast tomography versus the window taper parameter. TABLE I COMPARISON OF THE PROJECTION AND RECONSTRUCTION ACCURACY USING CUBIC B-SPLINES AND KBWFs WITH THE PARAMETERS PROPOSED IN between 0 and π; it comprises the measurements. As there is a large number of views for the reconstruction, we minimize the least-squares error (30) for the reconstruction. This is done for different discretizations of the forward model using KBWFs with different taper parameters. Therefore, the quality of the reconstructed image depends on how well the discretization scheme represents the imaging operator, as shown in Fig. 5(a). We also compute the SNR in the transform domain (Fig. 5(b)). The results validate the importance of using KBWFs with optimized parameters in order to improve the reconstruction performance. We also repeated those experiments with measurements corrupted by additive Gaussian noise with different noise levels (10 dB, 20 dB, 30 dB). The results suggest that using KBWF with the proposed parameters results in better performance. The SNR of the reconstructions was improved by close to 3 dB with respect to α = 5. For the reconstruction of X-ray differential phase-contrast tomograms, it was shown in that using cubic B-splines results in better performance than using KBWFs, with the parameters chosen as in . Here, we compare the per-formance of three basis functions for the phantom with size (2,048 × 2,048) pixels. The projection operator is computed using KBWFs with the parameter proposed in (α = 10.4) and with the parameter suggested in Fig. 5 (α = 7.95); furthermore, we also perform the comparison with cubic B-splines. The computed SNR shown in Table I suggests that the proposed parameter provides a signiﬁcantly better perfor-mance in computing the projection operator in comparison with the others. We conclude that a KBWF with the proposed parameters improves the performance of the discretization scheme in comparison with , . In addition, its performance is as good as that of cubic B-splines in terms of quality, while its isotropy allows for a drastic reduction in its computational costs. V. CONCLUSION The projections of isotropic functions are independent of the viewing angle. Therefore, they are attractive candidates as generating functions of principal shift-invariant spaces for discretizing the imaging operators. The generalized Kaiser-Bessel window function is a member of this family that is widely used. In this paper, we proposed a measure to deter-mine the performance of a basis function for the discretization scheme. Furthermore, we suggested a method to optimize the parameters of the KBWF based on this measure. By numerical experiments, we conﬁrmed that using the proposed method improves the performance of the discretization scheme. APPENDIX PROOF OF THEOREM 2 For the proof, we recall the following theorem. Theorem 5 (J.L. Grifﬁth): Let ν > −1/2 and 1/p + 1/q = 1. Let f be an even entire function of exponential type 1. If 1 < p ≤2 and tν+1/2 f (t) ∈L p(0, ∞), then f can be represented by f (z) = 1 0 (xz)−ν Jν(xz)φ(x)dx (z ∈C), (31) with x−ν−1/2φ(x) ∈ Lq(0, 1). Conversely, if f has this representation and x−ν−1/2φ(x) ∈L p(0, 1), 1 < p ≤2, then f is an even entire function of exponential type 1 such that tν+1/2 f (t) ∈Lq(0, ∞). We prove Theorem 2 using a proof by contradiction. We suppose that there is a compactly supported isotropic function φ that satisﬁes the partition-of-unity condition. Then, using Jensen’s theorem, we obtain a contradiction. Without loss of generality, let us assume that φ(x) = 0, for ∥x∥≥1. We have the following: • The function φ is isotropic, so its Fourier transform is the Hankel transform of the function φ(x) = φ(∥x∥) with x = ∥x∥. We write F{φ}(ω) = 2π ∞ 0 xφ(x)J0(∥ω∥)dx. (32) • We deﬁne f (z) = 2π ∞ 0 xφ(x)J0(z)dx, (33) so f (∥ω∥) = F{φ}(ω). According to Theorem 5 (with ν = 0), f (z) = ∞ 0 ψ(x)J0(zx)dx, (34) where ψ(x) = 2πxφ(x). Since x−1 2 ψ(x) ∈L2(0, 1), f is an even entire function of exponential type 1. • Satisfying the partition of unity is equivalent to having the equality in the Fourier domain φ(2πn) = δ[n], (35) 3832 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 24, NO. 11, NOVEMBER 2015 where n ∈Z2 and δ is the 2D Kronecker-delta function. It means that the set of zeros of f (z) is {z = 2π ∥n∥, ∀n ∈Z2 \ {0}}. Therefore, n(R) ≥cR2, (36) where n(R) is the number of zeros in the circle with radius R and c is a positive constant. • Jensen’s theorem implies the inequality R 0 n(t) t dt ≤max \|z\|=R log\| f (z)\|. (37) This inequality restricts the number of zeros inside the disc. We have that n(R/2)log2 = R R/2 n(R/2) t dt ≤ R R/2 n(t) t dt ≤max \|z\|=R log\| f (z)\|. (38) • Since f is of exponential type 1, it implies that \| f (z)\| ≤Ae\|z\|. Therefore, max \|z\|=R log\| f (z)\| ≤C R, (39) where C is a positive constant. • Equations (36), (38), and (39) imply that c(R/2)2log2 ≤n(R/2)log2 ≤max \|z\|=R log\| f (z)\| ≤C R. (40) Taking R sufﬁciently large, we reach a contradiction. REFERENCES J. P. Oliveira, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Adap-tive total variation image deblurring: A majorization–minimization approach,” Signal Process., vol. 89, no. 9, pp. 1683–1693, 2009. M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. 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Masih Nilchian received the M.S. degree in elec-trical engineering with a minor in communication system from the Sharif University of Technology, Tehran, Iran, in 2008, and the Ph.D. degree from the Biomedical Imaging Group, Swiss Federal Insti-tute of Technology, Lausanne, Switzerland, under the direction of Prof. M. Unser, in 2015. He was involved in the applications of information and cod-ing theory in biology and bioinformatics project with the Information System and Security Laboratory, Sharif University. From 2008 to 2010, he was a Research Ofﬁcer with the Malaysian institute of microelectronic systems (MIMOS) Laboratory, University Technology Malaysia. He is currently a Post-Doctoral Researcher with the Biomedical Imaging Group. His main research area is in phase retrieving in differential phase contrast and phase contrast X-ray computed tomography. He has interest in the use of spline for image processing, information theory, coding theory, wavelet and multireso-lution algorithms, and iterative methods for reconstruction. NILCHIAN et al.: OPTIMIZED KAISER–BESSEL WINDOW FUNCTIONS FOR COMPUTED TOMOGRAPHY 3833 John Paul Ward received the B.S. degree in math-ematics from the University of Georgia, Athens, in 2005, and the Ph.D. degree in mathematics from Texas A&M University, College Station, in 2010. He is currently a Post-Doctoral Researcher with the Biomedical Imaging Group, Swiss Federal Institute of Technology, Lausanne, Switzerland. Cédric Vonesch was born in Obernai, France, in 1981. He received the M.S. degree from the Swiss Federal Institute of Technology in Lausanne (EPFL), Switzerland, in 2004, and the Ph.D. degree from the Biomedical Imaging Group, EPFL, under the direction of Prof. M. Unser, in 2009. He then joined the Program in Applied and Computational Mathe-matics with Princeton University for a post-doctoral position. Since 2012, he has been with Biomedical Imaging Group, EPFL. His primary research inter-ests are inverse problems in the imaging sciences and multiresolution (wavelet-based) methods. He has worked on applications to ﬂuorescence and electron microscopy, and seismic imaging. Michael Unser (M’89–SM’94–F’99) received the M.S. (summa cum laude) and Ph.D. degrees in electrical engineering from the École Polytech-nique Fédérale de Lausanne (EPFL), Switzerland, in 1981 and 1984, respectively. From 1985 to 1997, he was a Scientist with the National Institutes of Health, Bethesda, USA. He is currently a Full Pro-fessor and the Director of the Biomedical Imaging Group at EPFL. His main research area is biomedical image processing. He has a strong interest in sam-pling theories, multiresolution algorithms, wavelets, and the use of splines for image processing. He has authored 200 journal papers on those topics, and is one of the institute for scientiﬁc information (ISI’s) Highly Cited authors in Engineering. He is currently a member of the Editorial Boards of Foundations and Trends in Signal Processing, and Sampling Theory in Signal and Image Processing. He is a fellow of the European association for signal processing (EURASIP) and a member of the Swiss Academy of Engineering Sciences. He co-organized the ﬁrst IEEE International Symposium on Biomedical Imaging and was the Founding Chair of the Technical Committee of the IEEE-SP Society on Bio Imaging and Signal Processing. He received the 1995 and 2003 Best Paper Awards, the 2000 Magazine Award, and the two IEEE Technical Achievement Awards (2008 SPS and 2010 EMBS). He has held the position of Associate Editor-in-Chief of the IEEE TRANSACTIONS ON MEDICAL IMAGING (2003-2005), and has served as Associate Editor for the IEEE TRANSACTIONS ON MEDICAL IMAGING (1999-2002; 2006-2007), the IEEE TRANSACTIONS ON IMAGE PROCESSING (1992-1995), and the IEEE SIGNAL PROCESSING LETTERS (1994-1998).
1649	https://www.khanacademy.org/math/get-ready-for-ap-calc/xa350bf684c056c5c:get-ready-for-contextual-applications-of-differentiation/xa350bf684c056c5c:volume-and-surface-area/e/solid_geometry	Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer. More information Manage Consent Preferences Strictly Necessary Cookies Always Active Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account. For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session. Functional Cookies These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences. Targeting Cookies These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission. We do not use cookies to serve third party ads on our Service. Performance Cookies These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates. For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users. We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails.
1650	https://www.geeksforgeeks.org/chemistry/stp-vs-ntp/	Difference between STP and NTP Last Updated : 23 Jul, 2025 Suggest changes 1 Like NTP and STP are two terms which are widely used in physics and chemistry to explain the various physical and chemical properties of liquids and gases. Let's learn about these and their difference in detail. Table of Content What is STP? Uses Of STP Fluid Mechanics Thermodynamics Acoustics Astrophysics What is NTP? Uses Of NTP Calibration Of Instruments Comparison of Data Prediction of Behavior Thermodynamic Calculations Difference between STP and NTP What is STP? STP, or standard temperature and pressure, is a commonly used set of conditions in chemistry to standardize measurements and compare different substances. It is defined as a temperature of 0 degrees Celsius (273.15 Kelvin) and a pressure of 1 atmosphere (101.325 kPa). STP is used in many areas of chemistry, including thermodynamics, gas laws, and phase changes. For example, the ideal gas law, PV = nRT, uses STP as a reference point for the number of moles of gas present (n) and the gas constant (R). Under STP conditions, 1 mole of any gas occupies a volume of 22.4 litres. Similarly, the density of a substance is often reported at STP, as it allows for more accurate comparisons between different materials. STP is also used in determining the standard enthalpy of formation, which is the amount of energy required to form one mole of a substance from its elements in their standard states. The standard state is defined as the substance being at a pressure of 1 atm and at its normal melting and boiling points. It is important to note that many substances do not exist at STP and must be manipulated, such as through heating or cooling, to breach these conditions. Additionally, many real-world applications occur at different temperatures and pressures than STP, so it is crucial to consider these variables when interpreting and applying data. In summary, STP is a widely used set of conditions in chemistry that serves as a reference point for measuring and comparing substances. It is defined as a temperature of 0 degrees Celsius and a pressure of 1 atmosphere. It is used in many areas of chemistry including thermodynamics, gas laws, phase changes and enthalpies. Uses Of STP STP (Standard Temperature and Pressure) is not commonly used in physics, as it is mostly used in chemistry as a reference point for measuring and comparing substances. However, it can be used in some areas of physics such as, Fluid Mechanics STP can be used as a reference point for the properties of gases and liquids, such as density, viscosity and vapour pressure. In fluid mechanics, STP (Standard Temperature and Pressure) is used as a reference point for the density and viscosity of fluids. Thermodynamics STP can be used as a reference state for thermodynamic properties such as internal energy, enthalpy and entropy. In thermodynamics, STP (Standard Temperature and Pressure) is used as a reference point for the properties of gases. For example, when a gas is at STP, its specific volume (volume per unit mass) can be calculated using the ideal gas law, PV = nRT. The value of R, the specific gas constant, is also defined at STP. Acoustics The speed of sound in gases is dependent on temperature and pressure. STP can be used as a reference point for the speed of sound in the air. In acoustics, STP (Standard Temperature and Pressure) is used as a reference point for the speed of sound in gases. The speed of sound in a gas is dependent on the temperature and pressure of the gas, and it is used to calculate the wavelength and frequency of sound waves. Astrophysics STP is used as a reference point for measuring the properties of gases in space, such as density, temperature and pressure. In astrophysics, STP (Standard Temperature and Pressure) is not a commonly used reference point, as most astrophysical phenomena occur under conditions that are vastly different from STP. Instead, other reference points such as the Cosmic Microwave Background radiation (CMB) temperature, which is about 2.725 K, or the average density of the universe, which is about 10-30 g/cm3, are used to understand the properties and behaviour of celestial objects and phenomena. What is NTP? NTP (Normal Temperature and Pressure) is a set of standard conditions used to define the physical properties of a substance, particularly in the field of thermodynamics. These standard conditions are typically set as a temperature of 20 degrees Celsius (68 degrees Fahrenheit) and a pressure of 1 atmosphere (101.325 kPa), which is roughly equivalent to the average temperature and pressure of the Earth's atmosphere at sea level. The use of NTP allows for the direct comparison of data from different experiments, as well as the prediction of the behaviour of a substance under different conditions. In thermodynamics, the properties of a substance are often described in terms of its internal energy, enthalpy, and entropy, and these properties can be measured at NTP to provide a baseline for comparison. When measuring the properties of gases, the NTP conditions are often used to measure the volume of the gas at standard temperature and pressure (STP) which is defined as 0 degree Celsius and 1 atm. It is important to note that NTP is standard, and not all experiments are conducted at NTP, but it is a way to have a common point of reference. Uses Of NTP NTP (Normal Temperature and Pressure) is used in physics for several purposes, including, Calibration Of Instruments In the calibration of instruments, NTP (Normal Temperature and Pressure) is commonly used as a reference point. NTP is defined as a temperature of 20 °C (293.15 K) and a pressure of 1 atm (101.325 kPa). Calibration is the process of adjusting an instrument to ensure that it is measuring a physical quantity accurately and consistently. When an instrument is calibrated, it is typically compared to a known standard or reference, and any discrepancies are corrected. Using NTP as a reference point allows for consistency in the calibration process. For example, a pressure gauge that is calibrated at NTP can be used to measure pressure at other temperatures and pressures, but the accuracy of the gauge will be known only at NTP. Comparison of Data In the comparison of data, NTP (Normal Temperature and Pressure) can be used as a reference point to ensure consistency and comparability of the data. NTP is defined as a temperature of 20 °C (293.15 K) and a pressure of 1 atm (101.325 kPa). When collecting data, it is important to have a consistent set of conditions to ensure that the data is comparable. For example, when measuring the pressure of a gas, it is important to ensure that the temperature and pressure of the gas are the same for all measurements. Prediction of Behavior When predicting the behaviour of a system, it is important to have a consistent set of conditions to ensure that the predictions are comparable. For example, when predicting the pressure of a gas, it is important to ensure that the temperature and pressure of the gas are the same for all predictions. Using NTP as a reference point allows for consistency in the prediction process. For example, if predictions are made at NTP, they can be compared to predictions made at other temperatures and pressures, but the accuracy of the predictions will be known only at NTP. Thermodynamic Calculations In thermodynamics, NTP (Normal Temperature and Pressure) is used as a reference point for thermodynamic calculations. NTP is defined as a temperature of 20 °C (293.15 K) and a pressure of 1 atm (101.325 kPa). Thermodynamic calculations involve the calculation of properties such as internal energy, enthalpy, entropy, and Gibbs free energy. These properties are state functions, meaning they are dependent on the initial and final state of the system and not on the path taken to reach the state. Using NTP as a reference point allows for consistency in the thermodynamic calculations. For example, if thermodynamic calculations are made at NTP, they can be compared to calculations made at other temperatures and pressures, but the accuracy of the calculations will be known only at NTP. Difference between STP and NTP STP and NTP are both acronyms used in physics that refer to specific conditions. STP stands for Standard Temperature and Pressure, which is defined as a temperature of 0 °C (273.15 K) and a pressure of 1 atm (101.325 kPa). NTP, on the other hand, stands for Normal Temperature and Pressure, which is defined as a temperature of 20 °C (293.15 K) and a pressure of 1 atm (101.325 kPa). Table given below shows the differences between STP and NTP, \| Features \| STP(Standard Temperature and Pressure) \| NTP(Normal Temperature and Pressure) \| --- \| Temperature \| The temperature in STP is 0 °C \| The temperature in NTP is 20 °C. \| \| Pressure \| The pressure in STP is 1 atm or 101.325 kPa \| The temperature in STP is also 1 atm or 101.325 kPa \| \| Area Of Usage \| STP is mostly used in chemistry and physics \| NTP is used in thermodynamics \| \| Physical/Thermodynamic Process \| STP is used for the measurement of the physical properties of gases and liquids. \| NTP is the standard condition for thermodynamic measurements. \| \| Application \| STP find its application in industrial processes. \| NTP find its application in air conditioning and refrigeration applications. \| \| Usage \| STP is used to calculate densities and volumes of gases. \| NTP is used in thermodynamics to calculate thermodynamic properties like internal energy and enthalpy. \| Read More: Ideal Gas Equation Gay Lussac’s Law Formula Charle’s Law Formula Conclusion of STP and NTP In conclusion, Standard Temperature and Pressure (STP) and Normal Temperature and Pressure (NTP) are two standardized reference conditions widely used in scientific and engineering disciplines. STP is defined as 0 degrees Celsius (273.15 Kelvin) and 1 atmosphere of pressure (101.325 kPa or 1013.25 mbar), while NTP is defined as 20 degrees Celsius (293.15 Kelvin) and the same pressure of 1 atmosphere. 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1651	https://doc.modelica.org/Modelica%203.2.3/Resources/helpWSM/Modelica/Modelica.Math.Distributions.Weibull.quantile.html	SystemModeler Modelica Math Distributions Weibull quantile quantile Quantile of Weibull distribution Information This information is part of the Modelica Standard Library maintained by the Modelica Association. Syntax Weibull.quantile(u, lambda=1, k=1); Description This function computes the inverse cumulative distribution function (= quantile) according to a Weibull distribution with scale parameter lambda and shape parameter k. Equation: y := lambda (-log( 1-u)) ^(1/k); Input argument u must be in the range: 0 ≤ u < 1 Plot of the function: For more details, see Wikipedia. Example quantile(0) // = 0 quantile(0.5,1,0.5) // = 0.41627730557884884 See also Weibull.density, Weibull.cumulative. Syntax y = quantile(u, lambda, k) Inputs (3) \| u \| Type: Real Description: Random number in the range 0 <= u <= 1 \| \| lambda \| Default Value: 1 Type: Real Description: Scale parameter of the Weibull distribution \| \| k \| Type: Real Description: Shape parameter of the Weibull distribution \| Outputs (1) \| y \| Type: Real Description: Random number u transformed according to the given distribution \|
1652	https://math.colgate.edu/~integers/y47/y47.pdf	A47 INTEGERS 24 (2024) ANOTHER TOPOLOGICAL PROOF OF THE INFINITUDE OF PRIME NUMBERS Jhixon Mac´ ıas Department of Mathematics, University of Puerto Rico at Mayaguez, Mayaguez, Puerto Rico jhixon.macias@upr.edu Received: 2/15/24, Accepted: 4/24/24, Published: 5/20/24 Abstract We present a new topological proof of the inﬁnitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the inﬁnitude of any non-empty subset of prime numbers. – I dedicate this work to my beloved wife, Alenka Calder´ on. 1. Introduction and Preliminaries In , we introduces a new topology τ on the set of positive integers N generated by the base β := {σn : n ∈N}, where σn := {m ∈N : gcd(n, m) = 1}. Indeed, below we present a proof of this fact. Theorem 1 (). The set β is a base for some topology on N. Proof. It is clear that [ n∈N σn = σ1 = N. On the other hand, note that for all positive integers x, n and m, we have gcd(x, n) = gcd(x, m) = 1 if and only if gcd(x, nm) = 1, and therefore σnm = σn ∩σm. Hence β is a base for some topology on N. Remark 1. Note that σn = [ 1≤m≤n gcd(m,n)=1 n (N ∪{0}) + m, since for every integer x we have that gcd(n, m) = gcd(n, nx+m); see [12, Theorem 1.9]. It is easily deduced from here that σn is inﬁnite for every positive integer n. Furthermore, it is deduced that the topology τ is strictly coarser than Golomb’s topology. DOI: 10.5281/zenodo.11221653 INTEGERS: 24 (2024) 2 The topological space X := (N, τ) does not satisfy the T0 axiom, is hypercon-nected, and is ultraconnected. Among other properties, one that will be useful is presented in the following lemma. Lemma 1 (). If p is a prime number, then clX({p}) = pN. Here, clX({p}) is the closure in X of the singleton set {p}. Proof. Let x ∈clX({p}). Now, if x / ∈pN, then gcd(x, p) = 1. Consequently, x ∈σp, which implies that p ∈σp, a contradiction. Therefore, x ∈pN. On the other hand, suppose x ∈pN. Then, x = np for some positive integer n. Now, take σk ∈β such that x ∈σk. This implies that gcd(np, k) = gcd(x, k) = 1, so gcd(p, k) = 1, and thus, p ∈σk. Hence, x ∈clX({p}). Remark 2. If n = 1, then clX({n}) = N. Moreover, using Lemma 1, it can be proved that for every positive integer n > 1 we have clX({n}) = T p\|n pN. The objective of this short note is to provide a new topological proof of the inﬁnitude of prime numbers, distinct from the topological proofs presented by F¨ urstenberg and Golomb , which, in fact, are similar except for the topol-ogy they use. Furthermore, we topologically characterize the inﬁnitude of any set of prime numbers; see Theorem 4. 2. The Topological Proof Let P denote the set of prime numbers. The following theorem characterizes the inﬁnitude of prime numbers in X. Theorem 2. There are inﬁnitely many prime numbers if and only if P is dense in X. Proof. Suppose there are inﬁnitely many prime numbers. Then, for any positive integer n > 1, we can choose a prime p such that p > n, and consequently, p ∈σn since gcd(n, p) = 1. Therefore, P is dense in X. On the other hand, assume that P is dense in X. Let {p1, p2, . . . , pk} be a ﬁnite collection of prime numbers and consider the non-empty basic element σx where x = p1 · p2 · · · pk. Note that none of the pi belong to σx, but since P is dense in X, there must be another prime number q, diﬀerent from each pi, such that q ∈σx. Consequently, there are inﬁnitely many prime numbers. Theorem 2 indicates that we only need to prove the density of P in X to establish the inﬁnitude of prime numbers. Precisely, that is what we will prove. To achieve our goal, consider the set N1 := N \ {1} and the subspace topology τ1 := {N1 ∩O : O ∈τ} generated by the base β1 := {N1 ∩σn : σn ∈β}. INTEGERS: 24 (2024) 3 Also, consider the topological subspace X1 := (N1, τ1) and the following lemma. Lemma 2. If P is dense in X1, then it is dense in X. Proof. It is clear that X1 is dense in X. So, by the transitive property of density, if P is dense in X1, then P is dense in X. Now, let us prove that P is dense in X1. Theorem 3. The set of prime numbers is dense in X1. Proof. In any topological space, we have that the union of closures of subsets of that space is contained in the closure of the union of those sets. Therefore, [ p∈P clX1({p}) ⊂clX1  [ p∈P {p}  = clX1(P) ⊂N1. On the other hand, from Lemma 1, it follows that [ p∈P clX1({p}) = [ p∈P (clX({p}) ∩N1) = [ p∈P (pN ∩N1) = [ p∈P pN. However, using the fundamental theorem of arithmetic, it can be easily proved that [ p∈P pN = N1, and thus, we can conclude that clX1(P) = N1, that is, P is dense in X1. From Theorem 2, Lemma 2, and Theorem 3, we can conclude that there are inﬁnitely many prime numbers. 3. Concluding Remarks There are many proofs of the inﬁnitude of prime numbers, such as Goldbach’s Proof [1, p. 3], Elsholtz’s Proof , Erdos’s Proof , Euler’s Proof , and more recent ones; see , , , and . Moreover, more than 200 proofs of the inﬁnitude of primes can be found in . However, F¨ urstenberg’s and Golomb’s proofs are the only known a priori topological proofs, which, in essence, as mentioned earlier, are based on the same idea, except for the topology used. Despite being able to present a topological proof using the same idea with the topology τ (left as an exercise to the reader), we present a completely diﬀerent proof, not only because of the topology used but also due to the underlying idea – proving that P is dense in X. Finally, we want to leave the reader with the following theorem. Theorem 4. Let A ⊂P non-empty. Then, A is dense in X, if and only if, A is inﬁnite. INTEGERS: 24 (2024) 4 Proof. Replace P with A in the proof of Theorem 2. Theorem 4 implies a new relationship between number theory and topology, at least we hope so. Indeed, to answer questions such as: are there inﬁnitely many Mersenne primes? or, are there inﬁnitely Fibonacci primes? it suﬃces to check the density of these sets on X. Certainly, it may not be easy, but it is possible. The advantage of working with X is that this space is hyperconnected, so any subset is either dense or nowhere dense. Acknowledgement. The author thanks Professor Hillel Furstenberg for valuable e-mail correspondence that greatly improved this work. References M. Aigner, and G. M. Ziegler, Proofs from the Book, Springer, Berlin, 1999. E. Aponte, J. Mac´ ıas, L. Mej´ ıas, and J. Vielma, Strongly connected topology on the positive integers, Submitted. C. Elsholtz, Prime divisors of thin sequences, Amer. Math. Monthly 119 (4) (2012), 331-333. P. Erd˝ os, ¨ Uber die Reihe P 1/p, Mathematica, Zutphen B 7 (1938), 1-2. L. Euler, Introductio in Analysin Inﬁnitorum, Apud Marcum-Michaelem Bousquet & Socios, Berlin. Germany, 1748. H. Furstenberg, On the inﬁnitude of primes, Amer. Math. Monthly 62 (5) (1955), 353. S. W. Golomb, A connected topology for the integers, Amer. Math. Monthly 66 (8) (1959), 363-665. H. G¨ oral, p-Adic metrics and the inﬁnitude of primes, Math. Mag. 93 (1) (2020), 19-22. H. G¨ oral, The Green-Tao theorem and the inﬁnitude of primes in domains, Amer. Math. Monthly 130 (2) (2023), 114-125. J. Mehta, A short generalized proof of inﬁnitude of primes, College Math. J. 53 (1) (2022), 52-53. R. Meˇ strovi´ c, Euclid’s theorem on the inﬁnitude of primes: a historical survey of its proofs (300 BC–2017) and another new proof, preprint, arXiv preprint arXiv:1202.3670. I. Niven, H. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers, John Wiley & Sons, New York, 1991. S. Northshield, Two short proofs of the inﬁnitude of primes, College Math. J. 48 (3) (2017), 21-216.
1653	https://www.yumpu.com/es/document/view/67352420/serway-10-edicion	Serway 10° Edición ES EnglishDeutschFrançaisEspañolPortuguêsItalianoRomânNederlandsLatinaDanskSvenskaNorskMagyarBahasa IndonesiaTürkçeSuomiLatvianLithuaniančeskýрусскийбългарскиالعربيةUnknown Entrar a YUMPU NewsEntrar a YUMPU Publishing TRY ADFREE productosNewsPublishing Crear ePaper Entrar a YUMPU NewsEntrar a YUMPU Publishing × Attention! Your ePaper is waiting for publication! By publishing your document, the content will be optimally indexed by Google via AI and sorted into the right category for over 500 million ePaper readers on YUMPU. This will ensure high visibility and many readers! PUBLISH DOCUMENT No, I renounce more range. Your ePaper is now published and live on YUMPU! 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Página 2 y 3: Carta pedagógica de coloresMecáni Página 4 y 5: Físicapara ciencias e ingeniería1 Página 6 y 7: Dedicamos este libro a nuestras esp Página 8 y 9: ContenidoAcerca de los autores viii Página 10 y 11: Contenido vii18.4 Expansión térmi Página 12 y 13: PrefacioAl escribir esta décima ed Página 14 y 15: Prefacio xiNuevas ampliaciones de p Página 16 y 17: Prefacio xiii1. Usted está trabaja Página 18 y 19: Prefacio xvblema cuantitativamente. Página 20 y 21: Prefacio xviiPrevenciones de riesgo Página 22 y 23: Prefacio xixProblemas de imposibili Página 24 y 25: Prefacio xxiApéndices y Notas fina Página 26 y 27: Prefacio xxiiiReconocimientosEsta D Página 28 y 29: Al estudiante xxvUsted puede compra Página 30 y 31: Físicapara ciencias e ingeniería1 Página 32 y 33: P A R T E 1MecánicaLa física, fun Página 34 y 35: 1.1 Estándares de longitud, masa y Página 36 y 37: 1.1 Estándares de longitud, masa y Página 38 y 39: 1.2 Modelado y representaciones alt Página 40 y 41: 1.2 Modelado y representaciones alt Página 42 y 43: 1.3 Análisis dimensional 11Puesto Página 44 y 45: 1. Exprese el número en notación Página 46 y 47: Resumen 15La regla de la suma y res Página 48 y 49: Problemas 17ProblemasConsulte el pr Página 50 y 51: Problemas 19iguales incrementos en Página 52 y 53: 2.1 Posición, velocidad y rapidez Página 54 y 55: 2.1 Posición, velocidad y rapidez Página 56 y 57: 2.2 Velocidad y rapidez instantáne Página 58 y 59: 2.3 Modelo de análisis: La partíc Página 60 y 61: 2.3 Modelo de análisis: La partíc Página 62 y 63: 2.4 Propuesta del modelo de anális Página 64 y 65: 2.5 Aceleración 33La pendiente de Página 66 y 67: 2.5 Aceleración 352.5 continúaxLa Página 68 y 69: 2.7 Modelo de análisis: La partíc Página 70 y 71: 2.7 Modelo de análisis: partícula Página 72 y 73: 2.8 Objetos en caída libre 412.8 c Página 74 y 75: 2.8 Objetos en caída libre 43Ejemp Página 76 y 77: 2.9 Ecuaciones cinemáticas deducid Página 78 y 79: 2.9 Piense, dialogue y comparta 47P Página 80 y 81: Problemas 4911. Una partícula part Página 82 y 83: Problemas 51res numéricos de x y a Página 84 y 85: 3.1 Sistemas coordenados 533.1 Sist Página 86 y 87: 3.3 Aritmética vectorial básica 5 Página 88 y 89: 3.3 Aritmética vectorial básica 5 Página 90 y 91: 3.4 Componentes de un vector y vect Página 92 y 93: 3.4 Componentes de un vector y vect Página 94 y 95: 3.4 Piense, dialogue y comparta 63R Página 96 y 97: 3.4 Problemas 6521. El vector S A t Página 98 y 99: 3.4 Problemas 67función de tiempo Página 100 y 101: 4.1 Vectores de posición, velocida Página 102 y 103: 4.2 Movimiento en dos dimensiones c Página 104 y 105: 4.2 Movimiento en dos dimensiones c Página 106 y 107: 4.3 Movimiento de proyectil 75La ex Página 108 y 109: 4.3 Movimiento de proyectil 77de vu Página 110 y 111: 4.3 Movimiento de proyectil 79Ejemp Página 112 y 113: 4.4 Modelo de análisis: Partícula Página 114 y 115: 4.4 Modelo de análisis: Partícula Página 116 y 117: 4.4 Velocidad y aceleración relati Página 118 y 119: 4.4 Velocidad y aceleración relati Página 120 y 121: Problemas 89› Modelo de análisis Página 122 y 123: Problemas 9114. La distancia récor Página 124 y 125: Problemas 9341. Un astronauta en la Página 126 y 127: Las leyes del movimiento5Su prima s Página 128 y 129: 5.2 Modelos de análisis utilizando Página 130 y 131: 5.4 Segunda ley de Newton 99E XAMEN Página 132 y 133: 5.4 Segunda ley de Newton 101¿Por Página 134 y 135: 5.6 Tercera ley de Newton 103portam Página 136 y 137: 5.7 Modelos de análisis utilizando Página 138 y 139: 5.7 Modelos de análisis utilizando Página 140 y 141: 5.7 Modelos de análisis utilizando Página 142 y 143: 5.7 Modelos de análisis utilizando Página 144 y 145: 5.7 Modelos de análisis utilizando Página 146 y 147: 10.8 Fuerzas de fricción 115Para p Página 148 y 149: 10.8 Fuerzas de fricción 1175.11 c Página 150 y 151: Resumen 1195.13 continuaciónComo e Página 152 y 153: Problemas 121moviéndose en la dire Página 154 y 155: Problemas 123P5.28. ¿Qué valor m Página 156 y 157: Problemas 125FSm 1 m 2 m 3Figura P5 Página 158 y 159: Movimiento circular y otrasaplicaci Página 160 y 161: 6.1 Extensión del modelo de partí Página 162 y 163: 6.1 Extensión del modelo de partí Página 164 y 165: 6.2 Movimiento circular no uniforme Página 166 y 167: 6.3 Movimiento en marcos acelerados Página 168 y 169: 6.3 Movimiento en marcos acelerados Página 170 y 171: 6.4 Movimiento en presencia de fuer Página 172 y 173: 6.4 Movimiento en presencia de fuer Página 174 y 175: Piense, dialogue y comparta 1436.10 Página 176 y 177: Problemas 145de curvatura de una ca Página 178 y 179: Problemas 147la bola en cualquier p Página 180 y 181: Problemas 149inciso (c). (d) ¿A qu Página 182 y 183: 7.2 Trabajo realizado por una fuerz Página 184 y 185: 7.2 Trabajo realizado por una fuerz Página 186 y 187: 7.3 Producto escalar de dos vectore Página 188 y 189: 7.4 Trabajo realizado por una fuerz Página 190 y 191: 7.4 Trabajo realizado por una fuerz Página 192 y 193: 7.5 Energía cinética y el teorema Página 194 y 195: 7.5 Energía cinética y el teorema Página 196 y 197: 7.6 Energía potencial de un sistem Página 198 y 199: 7.6 Energía potencial de un sistem Página 200 y 201: 7.7 Fuerzas conservativas y no cons Página 202 y 203: 7.8 Diagramas de energía y equilib Página 204 y 205: 7.9 Diagramas de energía y equilib Página 206 y 207: Piense, dialogue y comparta 175Si u Página 208 y 209: Problemas 17710. En un sistema de c Página 210 y 211: Problemas 179la figura y demuestre Página 212 y 213: Conservación de la energía8Usted Página 214 y 215: 8.1 8.1 Situaciones que incluyen fr Página 216 y 217: 8.2 8.2 Situaciones que incluyen fr Página 218 y 219: 8.2 8.2 Situaciones que incluyen fr Página 220 y 221: 8.2 8.2 Situaciones que incluyen fr Página 222 y 223: 8.3 Situaciones que incluyen fricci Página 224 y 225: 8.3 Situaciones que incluyen fricci Página 226 y 227: 8.3 Situaciones que incluyen fricci Página 228 y 229: 8.4 Cambios en energía mecánica p Página 230 y 231: 8.4 Cambios en energía mecánica p Página 232 y 233: 8.5 Potencia 201una suave elevació Página 234 y 235: Resumen 203Resumen› DefinicionesU Página 236 y 237: Problemas 205transportó lejos por Página 238 y 239: Problemas 207energía mecánica se Página 240 y 241: Problemas 20940. Un péndulo, que c Página 242 y 243: 9.1 Cantidad de movimiento lineal9. Página 244 y 245: 9.2 Modelo de análisis: Sistema ai Página 246 y 247: 9.3 Modelo de análisis: Sistema ai Página 248 y 249: 9.3 Modelo de análisis: Sistema ai Página 250 y 251: 9.4 Colisiones en una dimensión 21 Página 252 y 253: 9.4 Colisiones en una dimensión 22 Página 254 y 255: 9.4 Colisiones en una dimensión 22 Página 256 y 257: 9.4 Colisiones en una dimensión 22 Página 258 y 259: 9.5 Modelo de análisis: Sistema ai Página 260 y 261: Ejemplo 9.8 Colisión en un cruce9. Página 262 y 263: 9.6 Modelo de análisis: Sistema ai Página 264 y 265: 9.6 Modelo de análisis: Sistema ai Página 266 y 267: 9.7 Sistemas de muchas partículas Página 268 y 269: 9.8 Modelo de análisis: Sistema ai Página 270 y 271: 9.9 Modelo de análisis: Sistema ai Página 272 y 273: Resumen 2419.15 continuaciónSOLUCI Página 274 y 275: Problemas 2432. ACTIVIDAD Dibuje cu Página 276 y 277: Problemas 245rapidez de 3.00 m/s. ( Página 278 y 279: Problemas 247plazamiento de la pers Página 280 y 281: 10Rotación de un objeto rígidoen Página 282 y 283: 10.1 Posición, velocidad y acelera Página 284 y 285: 10.2 Modelo de análisis: Objeto r Página 286 y 287: 10.3 Cantidades angulares y traslac Página 288 y 289: 10.4 Momento de torsión 25710.2 co Página 290 y 291: 10.5 Modelo de análisis: Objeto r Página 292 y 293: 10.5 Modelo de análisis: Objeto r Página 294 y 295: 10.6 Cálculo de momentos de inerci Página 296 y 297: 10.6 Cálculo de momentos de inerci Página 298 y 299: 10.7 Energía cinética rotacional Página 300 y 301: 10.8 Consideraciones energéticas e Página 302 y 303: 10.8 Consideraciones energéticas e Página 304 y 305: 10.9 Movimiento de rodamiento de un Página 306 y 307: Ejemplo 10.13 Esfera que rueda haci Página 308 y 309: Resumen 27710.14 continuaciónSusti Página 310 y 311: Problemas 279ProblemasConsulte el P Página 312 y 313: Problemas 281su método de medició Página 314 y 315: Problemas 28337. Un eje gira a 65.0 Página 316 y 317: Cantidad de movimiento angular11Una Página 318 y 319: 11.1 Producto vectorial y momento d Página 320 y 321: 11.2 Modelo de análisis: sistema n Página 322 y 323: 11.2 Modelo de análisis: sistema n Página 324 y 325: 11.3 Cantidad de movimiento angular Página 326 y 327: 11.4 Modelo de análisis: sistema a Página 328 y 329: 11.4 Modelo de análisis: sistema a Página 330 y 331: 11.4 Modelo de análisis: sistema a Página 332 y 333: 11.5 El movimiento de giroscopios y Página 334 y 335: Problemas 3032. Un disco con moment Página 336 y 337: Problemas 305respectivamente. Calcu Página 338 y 339: Problemas 307su teléfono inteligen Página 340 y 341: Problemas 309Después de unos días Página 342 y 343: 12.1 Modelo de análisis: Objeto r Página 344 y 345: 12.3 Ejemplos de objetos rígidos e Página 346 y 347: 12.3 Ejemplos de objetos rígidos e Página 348 y 349: 12.3 Ejemplos de objetos rígidos e Página 350 y 351: 12.4 Propiedades elásticas de los Página 352 y 353: 12.4 Propiedades elásticas de los Página 354 y 355: 12.4 Propiedades elásticas de los Página 356 y 357: Problemas 325la tercera regla para Página 358 y 359: Problemas 327de fricción, y el pue Página 360 y 361: Problemas 329(c) ¿Qué pasaría si Página 362 y 363: Problemas 331derecha. Sugerencia: T Página 364 y 365: 13.1 Ley de Newton de gravitación Página 366 y 367: 13.2 Aceleración en caída libre y Página 368 y 369: 13.3 Modelo de análisis: Partícul Página 370 y 371: Ejemplo 13.3 El peso de la estació Página 372 y 373: 13.4 Las leyes de Kepler y el movim Página 374 y 375: 13.4 Las leyes de Kepler y el movim Página 376 y 377: 13.5 Energía potencial gravitacion Página 378 y 379: 13.6 Consideraciones energéticas e Página 380 y 381: 13.6 Consideraciones energéticas e Página 382 y 383: 13.6 Consideraciones energéticas e Página 384 y 385: Problemas 353Piense, dialogue y com Página 386 y 387: Problemas 355SECCIÓN 13.5 Energía Página 388 y 389: Problemas 357tema Tierra-Sol (b) en Página 390 y 391: 14.1 Presión 359ejercen sobre los Página 392 y 393: 14.2 Variación de la presión con Página 394 y 395: 14.2 Variación de la presión con Página 396 y 397: 14.4 Fuerzas de flotación y princi Página 398 y 399: 14.4 Fuerzas de flotación y princi Página 400 y 401: 14.5 Dinámica de fluidos 369o lami Página 402 y 403: 14.6 Ecuación de Bernoulli 37114.7 Página 404 y 405: 14.6 Ecuación de Bernoulli 373que Página 406 y 407: 14.7 Flujo de fluidos viscosos en t Página 408 y 409: 14.8 Otras aplicaciones de la diná Página 410 y 411: Problemas 379como un tubo largo. É Página 412 y 413: Problemas 381600 globos llenos con Página 414 y 415: Problemas 383Evalúe las magnitudes Página 416 y 417: P A R T E 2Oscilacionesy ondas mec Página 418 y 419: 15.1 Movimiento de un objeto unido Página 420 y 421: 15.2 Partícula en movimiento armó Página 422 y 423: 15.2 Partícula en movimiento armó Página 424 y 425: 15.2 Partícula en movimiento armó Página 426 y 427: 15.3 Energía del oscilador armóni Página 428 y 429: 15.3 Energía del oscilador armóni Página 430 y 431: 15.4 Comparación de movimiento arm Página 432 y 433: 15.5 El péndulo 401ximadamente 10 Página 434 y 435: Ya que esta ecuación es de la mism Página 436 y 437: 15.4 Oscilaciones forzadas 405compa Página 438 y 439: Resumen 407aAP ImagesbTopham/The Im Página 440 y 441: Problemas 409ProblemasConsulte el P Página 442 y 443: Problemas 411sistema, (b) la frecue Página 444 y 445: Problemas 413(e) Demuestre que el p Página 446 y 447: Movimiento ondulatorio16En la Reser Página 448 y 449: 16.1 Propagación de una perturbaci Página 450 y 451: 16.2 Modelo de análisis: onda viaj Página 452 y 453: 16.2 Modelo de análisis: onda viaj Página 454 y 455: 16.3 La rapidez de ondas en cuerdas Página 456 y 457: Ejemplo 16.3 La rapidez de un pulso Página 458 y 459: 16.4 Rapidez de transferencia de en Página 460 y 461: 16.6 Ondas sonoras 429Al combinar l Página 462 y 463: 16.7 Rapidez de ondas sonoras 431De Página 464 y 465: 16.8 Intensidad de ondas sonoras 43 Página 466 y 467: 16.8 Intensidad de ondas sonoras 43 Página 468 y 469: 16.8 Intensidad de ondas sonoras 43 Página 470 y 471: 16.9 El efecto Doppler 439Si usted Página 472 y 473: E XAMEN RÁPIDO 16.9 Considere los Página 474 y 475: Resumen 443pos posteriores, la fuen Página 476 y 477: Problemas 445agua. Sin embargo, la Página 478 y 479: Problemas 447que la tensión en las Página 480 y 481: Problemas 449por lo general es del Página 482 y 483: Sobreposicióny ondas estacionarias Página 484 y 485: 17.1 Modelo de análisis: Ondas en Página 486 y 487: 17.1 Modelo de análisis: Ondas en Página 488 y 489: 17.2 Ondas estacionarias 457La ampl Página 490 y 491: Ejemplo 17.2 Formación de una onda Página 492 y 493: 17.4 Modelo de análisis: Ondas baj Página 494 y 495: 17.4 Modelo de análisis: Ondas baj Página 496 y 497: 17.5 Resonancia 46517.4 continuaci Página 498 y 499: 17.6 Ondas estacionarias en columna Página 500 y 501: Ejemplo 17.6 Medición de la frecue Página 502 y 503: 17.7 Batimientos: Interferencia en Página 504 y 505: 17.8 Patrones de ondas no sinusoida Página 506 y 507: Problemas 475El sistema de nomencla Página 508 y 509: Problemas 47716. Problema de repaso Página 510 y 511: Problemas 479vertical de la cuerda. Página 512 y 513: P A R T E 3TermodinámicaAhora nos Página 514 y 515: 18.1 Temperatura y ley cero de la t Página 516 y 517: 18.3 Termómetro de gas a volumen c Página 518 y 519: 18.3 Termómetro de gas a volumen c Página 520 y 521: 18.4 Expansión térmica de sólido Página 522 y 523: 18.4 Expansión térmica de sólido Página 524 y 525: 18.5 Descripción macroscópica de Página 526 y 527: Piense, dialogue y comparta 495› Página 528 y 529: Problemas 497extremo para que no se Página 530 y 531: Problemas 499wT iwwFigura P18.3738. Página 532 y 533: Primera ley dela termodinámica19Se Página 534 y 535: 19.1 Calor y energía interna 503Cu Página 536 y 537: 19.2 Calor específico y calorimetr Página 538 y 539: 19.2 Calor específico y calorimetr Página 540 y 541: 19.3 Calor latente 50919.3 Calor la Página 542 y 543: 19.3 Calor latente 511Parte C. Entr Página 544 y 545: 19.4 Trabajo y calor en procesos te Página 546 y 547: 19.5 Primera ley de la termodinámi Página 548 y 549: 19.5 Primera ley de la termodinámi Página 550 y 551: 19.6 Mecanismos de transferencia de Página 552 y 553: 19.6 Mecanismos de transferencia de Página 554 y 555: 19.6 Mecanismos de transferencia de Página 556 y 557: Resumen 525La botella DewarLa botel Página 558 y 559: Problemas 527muestran en la tabla a Página 560 y 561: Problemas 529gía que se intercambi Página 562 y 563: Problemas 531bárica con volumen fi Página 564 y 565: Teoría cinéticade los gases20Un d Página 566 y 567: 20.1 Modelo molecular de un gas ide Página 568 y 569: 20.1 Modelo molecular de un gas ide Página 570 y 571: 20.2 Calor específico molar de un Página 572 y 573: 20.2 Calor específico molar de un Página 574 y 575: 20.3 Equipartición de la energía Página 576 y 577: 20.4 Procesos adiabáticos para un Página 578 y 579: 20.5 Procesos adiabáticos para un Página 580 y 581: 20.5 Procesos adiabáticos para un Página 582 y 583: Piense, dialogue y comparta 55120.5 Página 584 y 585: Problemas 5539. Calcule el cambio e Página 586 y 587: Problemas 555la expresión W 52# P Página 588 y 589: 21.1 Máquinas térmicas y segunda Página 590 y 591: 21.2 Bombas de calor y refrigerador Página 592 y 593: 21.2 Bombas de calor y refrigerador Página 594 y 595: 21.4 La máquina de Carnot 563defin Página 596 y 597: 21.4 La máquina de Carnot 565ciclo Página 598 y 599: 21.5 Motores de gasolina y diesel 5 Página 600 y 601: 21.5 Motores de gasolina y diesel 5 Página 602 y 603: 21.6 Entropía 571cero con respecto Página 604 y 605: 21.7 Entropía en sistemas termodin Página 606 y 607: Para calcular el cambio en entropí Página 608 y 609: 21.7 Entropía en sistemas termodin Página 610 y 611: 21.8 Entropía y la segunda ley 579 Página 612 y 613: Piense, dialogue y comparta 581El c Página 614 y 615: Problemas 583el inciso (c) es un pr Página 616 y 617: Problemas 585adiabáticamente a un Página 618 y 619: Apéndice A TablasTABLA A.1 Factore Página 620 y 621: Apéndice A Tablas A-3TABLA A.2Sím Página 622 y 623: B.2 Álgebra A-5Cuando se multiplic Página 624 y 625: B.2 Álgebra A-73. x 10 /x 25 5 x 1 Página 626 y 627: B.2 Álgebra A-9B.2 continuaciónSo Página 628 y 629: B.4 Trigonometría A-11La ecuación Página 630 y 631: B.6 Cálculo diferencial A-13B.3 co Página 632 y 633: B.6 Cálculo diferencial A-15Ejempl Página 634 y 635: B.7 Cálculo Integral A-17f(x)f(x i Página 636 y 637: B.7 Cálculo Integral A-19TABLA B.5 Página 638 y 639: B.8 Propagación de incertidumbre A Página 640 y 641: Apéndice C Tabla periódica de los Página 642 y 643: Respuestas a exámenes rápidos y p Página 644 y 645: Respuestas a exámenes rápidos y p Página 646 y 647: Respuestas a exámenes rápidos y p Página 648 y 649: Respuestas a exámenes rápidos y p Página 650 y 651: Respuestas a exámenes rápidos y p Página 652 y 653: Respuestas a exámenes rápidos y p Página 654 y 655: Respuestas a exámenes rápidos y p Página 656 y 657: ÍndiceNota del localizador: negrit Página 658 y 659: Índice I-3Compresión razón, para Página 660 y 661: Índice I-5Expansión térmica, com Página 662 y 663: Índice I-7datos planetarios, 343td Página 664 y 665: Índice I-9definición de, 250en eq Página 666 y 667: Índice I-11movimiento circular, 12 Página 668 y 669: Índice I-13en movimiento circular Página 670 y 671: Algunas constantes físicasCantidad Página 672 y 673: Abreviaturas estándar y símbolos Página 674: Esta edición de Física para cienc mostrar todo × Compartir or Link Short-link Embed Copiar Copiar Copiar Configuración avanzada de incrustación × Inapropiado Cargando... 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1654	https://sites.google.com/site/chiashulab/lecture_materials07	資料分析1 - 抽樣分佈與估計 Search this site Embedded Files Skip to main content Skip to navigation HOME Research NAOF Project TMDAS test The Hard Problem Studios Course MOE Students Library Short course-Thesis writing Short course-SOD/NYMU BDproject Workshop_YM_2021 TPC MSc (2022) Short course-ACTA_2023 HOME Research NAOF Project TMDAS test The Hard Problem Studios Course MOE Students Library Short course-Thesis writing Short course-SOD/NYMU BDproject Workshop_YM_2021 TPC MSc (2022) Short course-ACTA_2023 More HOME Research NAOF Project TMDAS test The Hard Problem Studios Course MOE Students Library Short course-Thesis writing Short course-SOD/NYMU BDproject Workshop_YM_2021 TPC MSc (2022) Short course-ACTA_2023 資料分析1 - 抽樣分佈與估計 [回到目錄] 抽樣分佈與估計 1. 重點就是在一個「猜」字每次聽到同學們說:「喔喔喔我要開始跑統計了...」，「電腦算出來結果是...」，「我的得到的結果不太好耶...」，我就忍不住想提醒同學 – 你剛剛講的「跑」、「算」、「得到」這些動詞，其實都應該改成猜。好啦，如果有人覺得猜這個字很不學術，那就說「估計」(estimate)好了。但講嘴巴講估計，你還是在做「猜」這件事。重點不在於到底是猜還是估計，還是亂猜或是亂估計...。重點是你到底在猜什麼? 比方某同學找了20位用過去敏感牙膏的病人，問他們用了去敏感牙膏後的經驗。他想知道的是什麼? (1) 這20位用過去敏感牙膏病人的經驗。 (2) 所有「用過去敏感牙膏病人」的經驗。如果他想問的是(1)，那麼這個研究就非常簡單了，他只要把所有20個人的滿意度或是敏感改善程度做一個表格秀出來就好了。這裡不需要任何「猜」的工作。但另一方面，如果他想問的是(2)，那麼他就必須要猜 – 他要利用現在已經有的20位病患的資料，去「猜」所有這類型病患的經驗如何。比方說，他得根據20位病患的平均使用後滿意度，去猜或估計所有這些病患的平均使用後滿意度。也就是說，用從樣本獲得的參數去猜(估計)想母群的參數。上面講的這件事(過程)我們稱為「推論」，這件事是整個統計學的核心。 p.s. 你可能會想，那假使我去調查「所有用去過敏牙膏病人」的經驗，一個都不少，把這些人通通抓來調查，那(2)不就還原到(1)的情況? 確實如此啊! 但問題是你可能沒有那麼多時間與金錢做這件事...，還有，母群的個數有沒有可能是無限的? 2. 怎麼猜? (學術語言: 怎樣從樣本去估計母群參數) 這裡有兩個非常重要的觀念要瞭解: 點估計(point estimation)與區間估計(interval estimation). (A) 點估計: 點估計基本上就是一個數值(我們稱之點估計值，estimate)，他是我們對於母群參數的猜測。回到去敏感牙膏的例子，比方現在我要猜的是母群的平均值，我就告訴你我猜母群的平均使用後滿意度是70分。這個70分就是點估計值。不過統計分析和算命不太一樣，當我說出這個估計值，我得要有所本才行。我是根據什麼說出70分這樣的猜測? 是根據我手上的樣本，還有，我所採用的估計式(estimator)，也就是我算出估計值的方法(公式)。拿平均值來說，因為我手上只有20位病患的資料，所以我採用的估計式就是「20位病患的平均滿意度」。而這個計算平均值的公式就是我用來估計母群的方法。 (B) 區間估計: 點估計雖然看起來很直觀，啊就是給你一個確切的答案(估計值)，但是它往往沒告訴我們太多東西。你說你根據樣本和估計式算出來點估計值，但是你沒告訴我這個估計的有多準確。當你講70分，啊真的就是完完全全70分? 我還想知道另外兩件事: (1)有沒有可能是69分? 有沒有可能是71分? (2)你有多大的信心可以說，母群參數是在某一個分數範圍以內? 解答上面兩個問題的方法就是區間估計。區間估計用一個區間(例如幾分到幾分)去估計母群參數，看看這個區間是否能包含母群參數。所以你看到區間估計會有兩個數字，分別是我們估計的上下區間(U/L)。區間估計一定會配合信心水準(confidence level)，例如95%的信心水準，這個信心水準表示「當我們反覆地做區間估計，做了很多很多次，有95%的個區間估計會包含母群參數」。所以區間估計比點估計提供我們更多資訊。它可以告訴我們一個猜測到底有多準(品質)，這個品質反應在兩方面: (1)區間的寬窄，如果越窄表示越準(accurate)，以及(2)估計的信心水準。這裡要釐清一下，這個信心水準(例如95%)並不是告訴你「我現在給出的區間，95%的機率是正確的」。事實上它能夠「抓到」母群參數的狀況只有兩種: 包含母群參數或是沒有包含母群參數(也就是母群參數落在區間範圍外)，所以不是正確就是不正確，沒有所謂機率的問題。再來，這個信心水準也不是告訴你「母群參數是某個數值的機率」。例如估計平均值落在[20，40]與95%的信心水準，你不能說母群平均值有95%的機率=30，你只知道是在20~40的範圍內，但你不知道是哪個確切的值。 3. 區間估計與效果量(effect size，ES) 區間估計有一個很常見的應用，就是估計效果量(ES)。記得ES談的是變項改變的程度(例如服藥前服藥後體重改變多少公斤)。記得我們在樣本上算出來的，永遠只是屬於樣本的參數，但我們真正感興趣的是根據這樣本所猜測的，母群的參數。所以以減肥藥的研究來說，你看到實驗樣本體重減少20公斤，你要做的就是去估計母群減少多少公斤。如果用區間估計的方法，你就會得到一個區段，例如介於15到25公斤之間，以及這個估計的信心水準。要做區間估計，和點估計一樣，可以從樣本和區間估計式(interval estimator)著手。這裡假定我們要估計的是平均數，區間估計式是 P(X – c ≤ μ ≤ X + c) = 0.95 (CL) 我們可以解讀成「母群平均值μ落在X - c ~ X + c這個範圍內的機率為0.95」。記得，區間估計值一定是伴隨著估計的信心水準。在樣本的分佈為常態分佈的前提下，經過推導我們可以得到估計的上下界(U,L) = X ± Z α σ/ sqrt(n) 其中α=(1-CL)/2 這裡要注意的是U和L取決於(1)信心水準，(2)母群標準差，以及(3)樣本大小。信心水準越高，注意我們估計的準確度越低(U~L越寬)，樣本越大，估計的準確度越好。而母群標準差通常是未知的，所以我們以樣本標準差(S)代替，如果樣本標準差越大，估計的準確度也越低。 4. 再談標準誤(standard error，SE) 我們先前提過標準誤(SE)這個概念，定義為樣本統計量的標準差。標準誤告訴我們「拿樣本統計量代表母群統計量」，足以代表的信心有多大。會不會有一種很熟的感覺? 怎麼感覺SE的作用和區間估計很類似? 也是去量化估計的準確度? 如果你會這樣想，那我就要恭喜你了，表示你掌握的SE的真正意涵。大家都知道SE的計算公式與標準差(standard deviation，SD)非常類似，也是用來量化資料離散程度的公式。但SE最重要的地方在於它計算的是有關「抽樣分布」的特徵。你可能要先回想一下抽樣分布(sampling distribution)是什麼東西。當你講到抽樣分布的時候，考慮的是某一個統計量(例如平均值)的分布。例如我在母群中抽樣10個樣本，算出平均值。我反覆做這件事1000次，我就會得到「1000個樣本平均值」的分佈。請注意這1000個樣本平均值肯定有高有低，可能非常離散 – 為什麼有這樣的差別? 那就適因位隨機抽樣導致的誤差。因為是「隨機」抽樣，所以也許某次抽樣，你剛好抓到了母群中最高值的那幾個樣本，但也有可能下一次抽樣，你剛好抓到了母群中最低值得那幾個樣本，而這兩次抽樣所算出的樣本平均值就可能差很多了! 而我們說SE量化離散性，它量化的就是這個抽樣分布的離散性。回到上面1000個樣本平均值的例子，我們計算的就是平均值的SE，也就是SEM(SE of mean)。SEM越小,表示我們抽樣很都次，但每次算出來的樣本平均值都差不多。 p.s. 想像一下，有沒有可能SEM=0? 什麼樣的母群，會導致SEM=0? 回到一開始講點估計的概念: 我們用樣本平均值，來(點)估計母群平均值。而SE剛好告訴我們另一見很重要的資訊: SEM越小，表示反覆抽樣獲得的樣本平均值變化不大。所以SE其實是輔助我們瞭解點估計值的重要資訊。 5. 點估計，區間估計與SE的關係我們再回想一下SEM的計算方法: SEM=σ/sqrt(n)。但是母群的標準差σ通常未知，所以我們以樣本的標準差S代替，也就是SEM=σ/sqrt(n) = S/sqrt(n)。所以SEM受到兩個因素的影響: (1)樣本標準差越大，SEM越大，(2)樣本數越大，SEM越小。關於(1)其實蠻直觀的，因為我們要問的是樣本平均數的分布，而平均數是從樣本計算而來，如果樣本本身變異都很大，那所得的平均數自然就可能被一些極端值(outlier)拉著跑。(2)樣本數越大的時候，這些極端值的效果就容易被「抵消」，所以平均值就不會那麼容易浮動。你是不是發現SEM的計算方式，其實和區間估計式很類似? 還有區間估計式中的X不是樣本平均值嗎? 那不就是我們的點估計值嗎? 所以到頭來區間估計式，其實就是點估計式加減一個誤差。如果今天我們以樣本平均值±SEM來當成區間估計，那麼根據區間估計式估計的上下界(U,L) = X ± Z α σ/ sqrt(n) = X ± Z α SEM 這裡Z α =1, 查表得CL=68% 相當於我們找的是「在樣本平均值正負1個標準差的範圍內」，也就是68%的信心水準。因此SEM本身其實就是一種區間估計的表現，它告訴我們「在68%的信心水準下，樣本平均值-SEM ~ 樣本平均值+SEM這個區間會涵蓋母群的平均值」。 Google Sites Report abuse Google Sites Report abuse
1655	https://cjhb.site/Files.php/Books/(Uncategorized)/%E5%B9%B3%E9%9D%A2%E5%87%A0%E4%BD%95%E6%95%99%E6%9C%AC_%E5%82%85%E7%A7%8D%E5%AD%99.pdf	[ G e n e r a l I n f o r ma t i o n ] 书名= 平面几何教本作者= 页数= 1 0 0 3 S S 号= 1 0 0 6 8 4 9 1 出版日期= 封面页书名页版权页前言页目录页首篇征引录第一章引言 1 . 几何论证的本源 2 . 公理、原名的选择和几何的派别 3 . 欧几里得几何第二章基本义理 4 . 点 5 . 一点对于两点间的关系及顺序公理 6 . 直线及定线公理 7 . 延长线及延长公理 8 . 线段及密布公理 9 . 共线点的顺序 1 0 . 符合及符合公理 1 1 . 半线及迁线公理 1 2 . 距离的和及加法公理 1 3 . 直线上的方向 1 4 . 形，符合形 1 5 . 直线形及其符合公理 1 6 . 不共线点及其存在公理 1 7 . 三角形及截割公理 1 8 . 平面及平面公理 1 9 . 角及迁角公理 2 0 . 垂直线及直角公理 2 1 . 符合三角形公理 2 2 . 圆及交圆公理 2 2 . 平行线及平行公理 2 4 . 共圆点的顺序及共点线的顺序 2 5 . 平面上的方向 2 6 . 角的大小和加减第三章初中平面几何摘要 2 7 . 两个三角形 2 8 . 同平面上两圆的关系 2 9 . 一直线与一圆的关系 3 0 . 等圆( 或同圆) 上的弦、弧、圆心角 3 1 . 一点到一直线的垂线与斜线 3 2 . 一线段 3 3 . 一角 3 4 . 三角形 3 5 . 两平行线与一割线 3 6 . 平行四边形 3 7 . n 边形 3 8 . 圆周角，圆内角，圆外角 3 9 . 一点与一圆的关系 4 0 . 正 n 边形 4 1 . 许多平行线 4 2 . 互等角三角形 4 3 . 由一点到一圆的切线与割线第一篇推证通法第一章顺证法及反证法 4 4 . 顺证法及反证法的意义 4 5 . 反证法的方式 4 6 . 反证法中的分别反驳第二章逆定理 4 7 . 关系语 4 8 . 逆定理及其制造法 4 9 . 逆定理的第一种证法 5 0 . 逆定理的第二种证法第三章综合法与分析法 5 1 . 综合法与分析法第四章归纳法 5 2 . 普通归纳法 5 0 . 数学归纳法第二篇证题杂术第一章相等 5 4 . 全等三角形 5 5 . 用全等三角形证相等 5 6 . 叠合法 5 7 . 证相等的其他方法第二章垂直 5 8 . 证题术 Ⅺ 5 9 . 证题术 Ⅻ 第三章平行 6 0 . 证题术 ⅩⅢ 第四章和差 6 1 . 证题术 ⅩⅣ 6 2 . 证题术 ⅩⅤ 第五章代数证法 6 3 . 证题术 ⅩⅥ 第六章共线点与共点线 6 4 . 共线点与共点线 6 5 . 证题术 ⅩⅩ—ⅩⅪ 第七章共圆点与共点圆 6 6 . 共圆点与共点圆 6 7 . 二圆合一 6 8 . 一点在一圆上 6 9 . 共圆点 7 0 . 共点圆第八章不等 7 1 . 不等的根据 7 2 . 不等腰三角形 7 3 . 证题术 ⅩⅩⅥ 7 4 . 证题术 ⅩⅩⅦ 7 5 . 有两边互等的两三角形 7 6 . 证题术 ⅩⅩⅧ 7 7 . 证题术 ⅩⅩⅨ 7 8 . 证题术 ⅩⅩⅩ 第三篇几何计算第一章线段计算 7 9 . 线段的量法 8 0 . 线段的计算 8 1 . 关于线段计算的推证法 8 2 . 线段计算的基本定理 8 3 . 证题术 ⅩⅩⅪ 8 4 . 互等角三角形法 8 5 . 多项式 8 6 . 代数证法第二章相似形 8 7 . 位似形 8 8 . 相似形 8 0 . 同序相似形第三章多边形的面积 9 0 . 多边形的面积 9 1 . 面积的比 9 2 . 相似形的加减法 9 3 . 多边形的就形变积 9 4 . 三角形的就积变形 9 5 . 多边形的就积变形第四篇作图第一章基础 9 6 . 作图题与存在定理 9 7 . 作图工具和它们的功能 9 8 . 作图的根据 9 9 . 作图的规范 1 0 0 . 推究举例第二章方法 1 0 1 . 作图题解法的寻求 1 0 2 . 拼合法 1 0 3 . 造因法 1 0 4 . 三角形奠基法 1 0 5 . 迁移法( 平移、旋转、翻折) 1 0 6 . 放大法( 位似法) 1 0 7 . 分析法和辅助线第三章代数分析法 1 0 8 . 线段作图 1 0 9 . 线段方程 1 1 0 . 代数分析法 1 1 1 . 正多边形 1 1 2 . 作图不能问题第五篇轨迹第一章类的解释 1 1 3 . 类 1 1 4 . 关于类的语言及记号第二章轨迹的意义及轨迹定理的证法 1 1 5 . 轨迹 1 1 6 . 轨迹定理的证法 1 1 7 . 轨迹定理的证法( 续) 第三章描迹 1 1 8 . 描迹 1 1 9 . 描迹中的注意事项第四章轨迹问题 1 2 0 . 怎样求轨迹 1 2 1 . 简略的轨迹定理第五章轨迹的应用 1 2 2 . 轨迹定理的用法 1 2 3 . 轨迹交点 1 2 4 . 轨迹与作图第六篇极大极小及极限第一章极大极小 1 2 5 . 定义 1 2 6 . 直接比较法 1 2 7 . 极大极小问题的发生 1 2 8 . 间接比较法 1 2 9 . 均数逐代法 1 3 0 . 反证法第二章极限 1 3 1 . 引言 1 3 2 . 序贯和它的极限 1 3 3 . 关于极限的术语和记号 1 3 4 . 升序贯及降序贯 1 3 5 . 极限原理 1 3 6 . 关于极限的重要定理 1 3 7 . 无理数 1 3 8 . 度量原理 1 3 9 . 量法 1 4 0 . 比 1 4 1 . 可通约量的公度 1 4 2 . 线段比例的基础定理 1 4 3 . 圆弧、圆心角、圆周角 1 4 4 . 面积 1 4 5 . 圆周长及圆面积 1 4 6 . 圆周率 1 4 7 . 极大极小问题的极限解法附录页
1656	https://www.hyee-current-transformer.com/info/emf-equation-of-a-transformer-and-voltage-tran-37481023.html	EMF Equation Of A Transformer And Voltage Transformation Ratio - Knowledge - Dalian Huayi Electric Power Electric Appliances Co.,Ltd Home About Us About Service About Honor Introduction Our Culture Our Customer Our History Products Current Transformer Transformador De Corriente Voltage Transformer Capacitor Voltage Transformer Transformer Surge Arrester Reactor CT PT Knowledge Dry Air Core Technical knowledge Contact Us News company news Foreign power industry news International Power Industry News PDF file Inquiry Language English Deutsch Latviešu Cymraeg Čeština Gaeilgenah Éireann hrvatski suomi O'zbek Bai Miaowen 한국어 slovenščina Navigation Menu Knowledge Home>Knowledge> Content EMF Equation Of A Transformer And Voltage Transformation Ratio Jul 29, 2019- EMF Equation Of The Transformer Let, N 1= Number of turns in primary winding N 2= Number of turns in secondary winding Φ m= Maximum flux in the core (in Wb) = (B m x A) f = frequency of the AC supply (in Hz) As, shown in the fig., the flux rises sinusoidally to its maximum value Φ m from 0. It reaches to the maximum value in one quarter of the cycle i.e in T/4 sec (where, T is time period of the sin wave of the supply = 1/f). Therefore, average rate of change of flux =Φ m/(T/4) =Φ m/(1/4f) Therefore, average rate of change of flux = 4f Φ m ....... (Wb/s). Now, Induced emf per turn = rate of change of flux per turn Therefore, average emf per turn = 4f Φ m ..........(Volts). Now, we know, Form factor = RMS value / average value Therefore, RMS value of emf per turn = Form factor X average emf per turn. As, the flux Φ varies sinusoidally, form factor of a sine wave is 1.11 Therefore, RMS value of emf per turn = 1.11 x 4f Φ m= 4.44f Φ m. RMS value of induced emf in whole primary winding (E 1) = RMS value of emf per turn X Number of turns in primary winding E 1= 4.44f N 1 Φ m ............................. eq 1 Similarly, RMS induced emf in secondary winding (E 2) can be given as E 2= 4.44f N 2 Φ m. ............................ eq 2 from the above equations 1 and 2, This is called theemf equation of transformer, which shows, emf / number of turns is same for both primary and secondary winding. For anideal transformeron no load, E 1= V 1 and E 2= V 2. where, V 1= supply voltage of primary winding V 2= terminal voltage of secondary winding Voltage Transformation Ratio (K) As derived above, Where, K = constant This constant K is known asvoltage transformation ratio. If N 2> N 1, i.e. K > 1, then the transformer is called step-up transformer. If N 2< N 1, i.e. K < 1, then the transformer is called step-down transformer. Previous: How to Calculate CT Ratio Next: CT burden, Knee point voltage, core saturation- Details of current transformer characteristics. Related Industry Knowledge Tan Delta Test \| Loss Angle Test \| Dissipation ... How to Calculate Electrical Transformer Output What is the difference between a Junction Box a... Error in Current Transformer or CT Theory of Current Transformer or CT CT Accuracy Class or Current Transformer Class How to Calculate CT Ratio Voltage Transformer Burden Calculation Voltage Transformer Theory Current transformer and voltage transformer mod... CT burden, Knee point voltage, core saturation-... What is the role of the junction box What is an electrical junction box Capacitive voltage transformer transient error Current transformer What Is a Voltage Transformer? The accuracy of a CT Phase shift of current transformer The knee-point voltage of a current transform Current transformer burden Related Products 138KV outdoor Voltage Transformer o... 230KV Voltage Transformer VT Instrument Transformer CT VT 230kv Current Transformer Live Tank 115KV Current Transformer Outdoor O... 115KV Current Transformer Electric ... TRY BEST! TO BE NO.1 With a factory area of 150,000 square meters, 500 employee, more than 60 engineering technicians and over 20 senior technicians and designers, more than 300 sets' main production equipment, our production capacity can be more than 12,000 sets per year. []( Contact Us Dalian Huayi Electric Power Electric Appliances Co.,Ltd Add: Pulandian district, Dalian city, Liaoning province, China Tel: 86-411-83199993 Cell/whatsapp: +8614741101265 Wechat: aacc20115351 E-mail: steven.yione@outlook.com Skype: steven.yi115 Fax: +86 411 83146176 Web: www.cnhuayi.com.cn Categories Current TransformerRead MoreVoltage TransformerRead MoreTransformerRead MoreCT PTRead More Home About Us Products Knowledge Contact Us News PDF file Inquiry Mobile Sitemap Copyright © Dalian Huayi Electric Power Electric Appliances Co.,Ltd All Rights Reserved. 0 正在输入中... Send a file Send a picture Your name E-mail Phone/WhatsApp Message SUBMIT upload percent: 0% Steven Steven E-Mail WeChat WeChat:aacc20115351
1657	https://en.wikipedia.org/wiki/Pie_chart	Jump to content Pie chart Afrikaans العربية Català Čeština Cymraeg Dansk Deutsch Eesti Español Euskara فارسی Français 한국어 Հայերեն हिन्दी Hrvatski Bahasa Indonesia Italiano עברית Magyar മലയാളം Bahasa Melayu Nederlands नेपाली 日本語 Norsk bokmål ਪੰਜਾਬੀ Polski Português Русский Shqip Sicilianu Simple English Srpskohrvatski / српскохрватски Suomi Svenska தமிழ் Türkçe Українська 吴语粵語中文 Betawi Edit links From Wikipedia, the free encyclopedia Circular statistical graph that illustrates numerical proportion Not to be confused with Circle graph. For the South Korean music chart, see Circle Chart. For Information on generating pie charts in Wikipedia, see Wikipedia:Graphs and charts and Template:Pie chart. A pie chart (or a circle chart) is a circular statistical graphic which is divided into slices to illustrate numerical proportion. In a pie chart, the arc length of each slice (and consequently its central angle and area) is proportional to the quantity it represents. While it is named for its resemblance to a pie which has been sliced, there are variations on the way it can be presented. The earliest known pie chart is generally credited to William Playfair's Statistical Breviary of 1801. Pie charts are very widely used in the business world and the mass media. However, they have been criticized, and many experts recommend avoiding them, as research has shown it is more difficult to make simple comparisons such as the size of different sections of a given pie chart, or to compare data across different pie charts. Some research has shown pie charts perform well for comparing complex combinations of sections (e.g., "A + B vs. C + D"). Commonly recommended alternatives to pie charts in most cases include bar charts, box plots, and dot plots. History [edit] The earliest known pie chart is generally credited to William Playfair's Statistical Breviary of 1801, in which two such graphs are used. Playfair presented an illustration, which contained a series of pie charts. One of those charts depicted the proportions of the Turkish Empire located in Asia, Europe and Africa before 1789. This invention was not widely used at first. Playfair thought that pie charts were in need of a third dimension to add additional information. Florence Nightingale may not have invented the pie chart, but she adapted it to make it more readable, which fostered its wide use, still today. Nightingale reconfigured the pie chart making the length of the wedges variable instead of their width. The graph, then, resembled a cock's comb. She was later assumed to have created it due to the obscurity and lack of practicality of Playfair's creation. Nightingale's polar area diagram,: 107 or occasionally the Nightingale rose diagram, equivalent to a modern circular histogram, to illustrate seasonal sources of patient mortality in the military field hospital she managed, was published in Notes on Matters Affecting the Health, Efficiency, and Hospital Administration of the British Army and sent to Queen Victoria in 1858. According to the historian Hugh Small, "she may have been the first to use [pie charts] for persuading people of the need for change." The French engineer Charles Joseph Minard also used pie charts, in 1858. A map of his from 1858 used pie charts to represent the cattle sent from all around France for consumption in Paris. Early types of pie charts in the 19th century Pie charts from William Playfair's "Statistical Breviary", 1801 One of the earliest pie charts, 1801 Minard's map, 1858 Polar chart by Florence Nightingale, 1858 Variants and similar charts [edit] 3D pie chart and perspective pie cake [edit] A 3D pie chart, or perspective pie chart, is used to give the chart a 3D look. Often used for aesthetic reasons, the third dimension does not improve the reading of the data; on the contrary, these plots are difficult to interpret because of the distorted effect of perspective associated with the third dimension. The use of superfluous dimensions not used to display the data of interest is discouraged for charts in general, not only for pie charts. Doughnut chart [edit] A doughnut chart (also spelled donut) is a variant of the pie chart, with a blank center allowing for additional information about the data as a whole to be included. Doughnut charts are similar to pie charts in that their aim is to illustrate proportions.[citation needed] This type of circular graph can support multiple statistics at once and it provides a better data intensity ratio to standard pie charts. It does not have to contain information in the center. Exploded pie chart [edit] A chart with one or more sectors separated from the rest of the disk is known as an exploded pie chart. This effect is used to either highlight a sector, or to highlight smaller segments of the chart with small proportions. Polar area diagram [edit] The polar area diagram is similar to a usual pie chart, except sectors have equal angles and differ rather in how far each sector extends from the center of the circle. It is used to plot cyclic phenomena (e.g., counts of deaths by month). For example, if the counts of deaths in each month for a year are to be plotted then there will be 12 sectors (one per month) all with the same angle of 30 degrees each. The radius of each sector would be proportional to the square root of the death rate for the month, so the area of a sector represents the rate of deaths in a month. If the death rate in each month is subdivided by cause of death, it is possible to make multiple comparisons on one diagram, as is seen in the polar area diagram famously developed by Florence Nightingale. The first known use of polar area diagrams was by André-Michel Guerry, which he called courbes circulaires (circular curves), in an 1829 paper showing seasonal and daily variation in wind direction over the year and births and deaths by hour of the day. Léon Lalanne later used a polar diagram to show the frequency of wind directions around compass points in 1843. The wind rose is still used by meteorologists. Nightingale published her rose diagram in 1858. Although the name "coxcomb" has come to be associated with this type of diagram, Nightingale originally used the term to refer to the publication in which this diagram first appeared—an attention-getting book of charts and tables—rather than to this specific type of diagram. "Diagram of the causes of mortality in the army in the East" by Florence Nightingale Ring chart, sunburst chart, and multilevel pie chart [edit] See also: Radial tree A ring chart, also known as a sunburst chart or a multilevel pie chart, is used to visualize hierarchical data, depicted by concentric circles. The circle in the center represents the root node, with the hierarchy moving outward from the center. A segment of the inner circle bears a hierarchical relationship to those segments of the outer circle which lie within the angular sweep of the parent segment. Spie chart [edit] A variant of the polar area chart is the spie chart, designed by Dror Feitelson. The design superimposes a normal pie chart with a modified polar area chart to permit the comparison of two sets of related data. The base pie chart represents the first data set in the usual way, with different slice sizes. The second set is represented by the superimposed polar area chart, using the same angles as the base, and adjusting the radii to fit the data. For example, the base pie chart could show the distribution of age and gender groups in a population, and the overlay their representation among road casualties. Age and gender groups that are especially susceptible to being involved in accidents then stand out as slices that extend beyond the original pie chart. Square chart / Waffle chart [edit] Square charts, also called waffle charts, are a form of pie charts that use squares instead of circles to represent percentages. Similar to basic circular pie charts, square pie charts take each percentage out of a total 100%. They are often 10 by 10 grids, where each cell represents 1%. Despite the name, circles, pictograms (such as of people), and other shapes may be used instead of squares. One major benefit to square charts is that smaller percentages, difficult to see on traditional pie charts, can be easily depicted. Example [edit] The following example chart is based on preliminary results of the election for the European Parliament in 2004. The table lists the number of seats allocated to each party group, along with the derived percentage of the total that they each make up. The values in the last column, the derived central angle of each sector, is found by taking that percentage of 360. \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- \| \| \| Group \| Seats \| Percent (%) \| Central angle (°) \| --- --- \| \| EUL \| 39 \| 5.3 \| 19.2 \| \| PES \| 200 \| 27.3 \| 98.4 \| \| EFA \| 42 \| 5.7 \| 20.7 \| \| EDD \| 15 \| 2.0 \| 7.4 \| \| ELDR \| 67 \| 9.2 \| 33.0 \| \| EPP \| 276 \| 37.7 \| 135.7 \| \| UEN \| 27 \| 3.7 \| 13.3 \| \| Other \| 66 \| 9.0 \| 32.5 \| \| Total \| 732 \| 99.9 \| 360.2 \| Because of rounding, these totals do not add up to 100 and 360. \| \| The size of each central angle is proportional to the size of the corresponding quantity, here the number of seats. Since the sum of the central angles has to be 360°, the central angle for a quantity that is a fraction Q of the total is 360Q degrees. In the example, the central angle for the largest group (European People's Party (EPP)) is 135.7° because 0.377 times 360, rounded to one decimal place, equals 135.7. Use and effectiveness [edit] A flaw exhibited by pie charts is that they cannot show more than a few values without separating the visual encoding (the “slices”) from the data they represent (typically percentages). When slices become too small, pie charts have to rely on colors, textures or arrows so the reader can understand them. This makes them unsuitable for use with larger amounts of data. Pie charts also take up a larger amount of space on the page compared to the more flexible bar charts, which do not need to have separate legends, and can display other values such as averages or targets at the same time. Statisticians generally regard pie charts as a poor method of displaying information, and they are uncommon in scientific literature. One reason is that it is more difficult for comparisons to be made between the size of items in a chart when area is used instead of length and when different items are shown as different shapes. Further, in research performed at AT&T Bell Laboratories, it was shown that comparison by angle was less accurate than comparison by length. Most subjects have difficulty ordering the slices in the pie chart by size; when an equivalent bar chart is used the comparison is much easier. Similarly, comparisons between data sets are easier using the bar chart. However, if the goal is to compare a given category (a slice of the pie) with the total (the whole pie) in a single chart and the multiple is close to 25 or 50 percent, then a pie chart can often be more effective than a bar graph. In a pie chart with many section, several values may be represented with the same or similar colors, making interpretation difficult. Several studies presented at the European Visualization Conference analyzed the relative accuracy of several pie chart formats, reaching the conclusion that pie charts and doughnut charts produce similar error levels when reading them, and square pie charts provide the most accurate reading. See also [edit] Data and information visualization References [edit] ^ Jump up to: a b c Spence (2005) ^ Jump up to: a b Tufte, p. 44 ^ Cleveland, p. 262 ^ Wilkinson, p. 23. ^ Tufte, p. 178. ^ van Belle, p. 160–162. ^ Jump up to: a b c Stephen Few. "Save the Pies for Dessert", August 2007, Retrieved 2010-02-02 ^ Steve Fenton "Pie Charts Are Bad" ^ Jump up to: a b Spence, Ian; Lewandowsky, Stephan (1 January 1991). "Displaying proportions and percentages". Applied Cognitive Psychology. 5 (1): 61–77. doi:10.1002/acp.2350050106. ^ "Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization". www.datavis.ca. ^ Palsky, p. 144–145 ^ Jump up to: a b Greenbaum, Hilary; Rubinstein, Dana (20 April 2012). "Who Made That Pie Chart?". The New York Times. ^ Dave article on this information on QI ^ Cohen, I. Bernard (March 1984). "Florence Nightingale". Scientific American. 250 (3): 128–137. Bibcode:1984SciAm.250c.128C. doi:10.1038/scientificamerican0384-128. PMID 6367033. (alternative pagination depending on country of sale: 98–107, bibliography on p. 114) online article – see documents link at left ^ Good and Hardin, chapter 8. ^ Harris, Robert L. (1999). Information graphics : a comprehensive illustrated reference ([Nachdr.] ed.). Oxford: Oxford University Press. p. 143. ISBN 9780195135329. ^ Jump up to: a b Data Design by Juergen Kai-Uwe Brock on iBooks. 21 December 2016. Retrieved 2017-06-10. {{cite book}}: \|website= ignored (help) ^ Friendly, p. 509 ^ "Florence Nightingale's Statistical Diagrams". Retrieved 2010-11-22. ^ "Multi-level Pie Charts". www.neoformix.com. ^ Webber Richard, Herbert Ric, Jiangbc Wel. "Space-filling Techniques in Visualizing Output from Computer Based Economic Models" ^ "Feitelson, Dror (2003) Comparing Partitions With Spie Charts" (PDF). 2003. Retrieved 2010-08-31. ^ Jump up to: a b Kosara, Robert; Skau, Drew (2016). "Judgment Error in Pie Chart Variations". EuroVis. ^ Krygier, John (28 August 2007). "Perceptual Scaling of Map Symbols". makingmaps.net. Retrieved 3 May 2015. ^ Cleveland, p. 86–87 ^ Simkin, D., & Hastie, R. (1987). An Information-Processing Analysis of Graph Perception. Journal of the American Statistical Association, 82(398), 454. doi:10.2307/2289447. Kosara, Robert (13 April 2011). "In Defense of Pie Charts". Retrieved April 13, 2011. ^ "An Illustrated Tour of the Pie Chart Study Results". eagereyes. 2016-06-28. Retrieved 2016-11-28. ^ Skau, Drew; Kosara, Robert (2016). "Arcs, Angles, or Areas: Individual Data Encodings in Pie and Donut Charts". EuroVis. ^ "A Reanalysis of A Study About (Square) Pie Charts from 2009". eagereyes. 2016-07-11. Retrieved 2016-11-28. Further reading [edit] Cleveland, William S. (1985). The Elements of Graphing Data. Pacific Grove, CA: Wadsworth & Advanced Book Program. ISBN 0-534-03730-5. Friendly, Michael. "The Golden Age of Statistical Graphics," Statistical Science, Volume 23, Number 4 (2008), 502–535 Good, Phillip I. and Hardin, James W. Common Errors in Statistics (and How to Avoid Them). Wiley. 2003. ISBN 0-471-46068-0. Guerry, A.-M. (1829). Tableau des variations météorologique comparées aux phénomènes physiologiques, d'aprés les observations faites à l'obervatoire royal, et les recherches statistique les plus récentes. Annales d'Hygiène Publique et de Médecine Légale, 1 :228-. Harris, Robert L. (1999). Information Graphics: A comprehensive Illustrated Reference. Oxford University Press. ISBN 0-19-513532-6. Lima, Manuel. "Why humans love pie charts: an historical and evolutionary perspective," Noteworthy, July 23, 2018 Palsky Gilles. Des chiffres et des cartes: la cartographie quantitative au XIXè siècle. Paris: Comité des travaux historiques et scientifiques, 1996. ISBN 2-7355-0336-4. Playfair, William, Commercial and Political Atlas and Statistical Breviary, Cambridge University Press (2005) ISBN 0-521-85554-3. Spence, Ian. No Humble Pie: The Origins and Usage of a statistical Chart. Journal of Educational and Behavioral Statistics. Winter 2005, 30 (4), 353–368. Tufte, Edward. The Visual Display of Quantitative Information. Graphics Press, 2001. ISBN 0-9613921-4-2. Van Belle, Gerald. Statistical Rules of Thumb. Wiley, 2002. ISBN 0-471-40227-3. Wilkinson, Leland. The Grammar of Graphics, 2nd edition. Springer, 2005. ISBN 0-387-24544-8. External links [edit] Wikimedia Commons has media related to Pie charts. \| v t e Statistics \| \| --- \| \| Outline Index \| \| \| \| Descriptive statistics \| \| --- \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- \| \| Continuous data \| \| \| \| --- \| \| Center \| Mean + Arithmetic + Arithmetic-Geometric + Contraharmonic + Cubic + Generalized/power + Geometric + Harmonic + Heronian + Heinz + Lehmer Median Mode \| \| Dispersion \| Average absolute deviation Coefficient of variation Interquartile range Percentile Range Standard deviation Variance \| \| Shape \| Central limit theorem Moments + Kurtosis + L-moments + Skewness \| \| \| Count data \| Index of dispersion \| \| Summary tables \| Contingency table Frequency distribution Grouped data \| \| Dependence \| Partial correlation Pearson product-moment correlation Rank correlation + Kendall's τ + Spearman's ρ Scatter plot \| \| Graphics \| Bar chart Biplot Box plot Control chart Correlogram Fan chart Forest plot Histogram Pie chart Q–Q plot Radar chart Run chart Scatter plot Stem-and-leaf display Violin plot \| \| \| \| \| \| \| Data collection \| \| --- \| \| \| \| \| --- \| \| Study design \| Effect size Missing data Optimal design Population Replication Sample size determination Statistic Statistical power \| \| Survey methodology \| Sampling + Cluster + Stratified Opinion poll Questionnaire Standard error \| \| Controlled experiments \| Blocking Factorial experiment Interaction Random assignment Randomized controlled trial Randomized experiment Scientific control \| \| Adaptive designs \| Adaptive clinical trial Stochastic approximation Up-and-down designs \| \| Observational studies \| Cohort study Cross-sectional study Natural experiment Quasi-experiment \| \| \| \| \| \| \| Statistical inference \| \| --- \| \| \| \| \| --- \| \| Statistical theory \| Population Statistic Probability distribution Sampling distribution + Order statistic Empirical distribution + Density estimation Statistical model + Model specification + Lp space Parameter + location + scale + shape Parametric family + Likelihood (monotone) + Location–scale family + Exponential family Completeness Sufficiency Statistical functional + Bootstrap + U + V Optimal decision + loss function Efficiency Statistical distance + divergence Asymptotics Robustness \| \| Frequentist inference \| \| \| \| --- \| \| Point estimation \| Estimating equations + Maximum likelihood + Method of moments + M-estimator + Minimum distance Unbiased estimators + Mean-unbiased minimum-variance - Rao–Blackwellization - Lehmann–Scheffé theorem + Median unbiased Plug-in \| \| Interval estimation \| Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling + Bootstrap + Jackknife \| \| Testing hypotheses \| 1- & 2-tails Power + Uniformly most powerful test Permutation test + Randomization test Multiple comparisons \| \| Parametric tests \| Likelihood-ratio Score/Lagrange multiplier Wald \| \| \| Specific tests \| \| \| \| --- \| \| Z-test (normal) Student's t-test F-test \| \| \| Goodness of fit \| Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality (Shapiro–Wilk) Likelihood-ratio test Model selection + Cross validation + AIC + BIC \| \| Rank statistics \| Sign + Sample median Signed rank (Wilcoxon) + Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova + 1-way (Kruskal–Wallis) + 2-way (Friedman) + Ordered alternative (Jonckheere–Terpstra) Van der Waerden test \| \| \| Bayesian inference \| Bayesian probability + prior + posterior Credible interval Bayes factor Bayesian estimator + Maximum posterior estimator \| \| \| \| \| \| \| Correlation Regression analysis \| \| --- \| \| \| \| \| --- \| \| Correlation \| Pearson product-moment Partial correlation Confounding variable Coefficient of determination \| \| Regression analysis \| Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS) \| \| Linear regression \| Simple linear regression Ordinary least squares General linear model Bayesian regression \| \| Non-standard predictors \| Nonlinear regression Nonparametric Semiparametric Isotonic Robust Homoscedasticity and Heteroscedasticity \| \| Generalized linear model \| Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions \| \| Partition of variance \| Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom \| \| \| \| \| \| \| Categorical / multivariate / time-series / survival analysis \| \| --- \| \| \| \| \| --- \| \| Categorical \| Cohen's kappa Contingency table Graphical model Log-linear model McNemar's test Cochran–Mantel–Haenszel statistics \| \| Multivariate \| Regression Manova Principal components Canonical correlation Discriminant analysis Cluster analysis Classification Structural equation model + Factor analysis Multivariate distributions + Elliptical distributions - Normal \| \| Time-series \| \| \| \| --- \| \| General \| Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality \| \| Specific tests \| Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey \| \| Time domain \| Autocorrelation (ACF) + partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR) \| \| Frequency domain \| Spectral density estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood \| \| \| Survival \| \| \| \| --- \| \| Survival function \| Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time \| \| Hazard function \| Nelson–Aalen estimator \| \| Test \| Log-rank test \| \| \| \| \| \| \| \| \| \| --- \| \| \| \| \| --- \| \| Biostatistics \| Bioinformatics Clinical trials / studies Epidemiology Medical statistics \| \| Engineering statistics \| Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification \| \| Social statistics \| Actuarial science Census Crime statistics Demography Econometrics Jurimetrics National accounts Official statistics Population statistics Psychometrics \| \| Spatial statistics \| Cartography Environmental statistics Geographic information system Geostatistics Kriging \| \| \| \| \| \| Category Mathematics portal Commons WikiProject \| \| Retrieved from " Category: Statistical charts and diagrams Hidden categories: CS1 errors: periodical ignored Articles with short description Short description is different from Wikidata All articles with unsourced statements Articles with unsourced statements from October 2017 Articles containing French-language text Commons category link from Wikidata
1658	https://www.stats4stem.org/geometric-distribution	STATS4STEM 1 2 3 4 5 Log in Sign Up I'm a Teacher I'm a Student Home Features Features FAQ High School/AP Statistics Resources College Statistics Resources R Help Pages Textbooks Advanced High School Statistics CK-12 Probability and Statistics OpenIntro Statistics Introductory Statistics with Randomization and Simulation Introductory Statistics Introduction to Statistics Using R for Introductory Statistics Introduction to Probability and Statistics Using R Geometric Distribution I. Introduction The geometric distribution is similar to the binomial distribution, but unlike the binomial distribution, which calculates the probability of observing a fixed number of success in n n observations, the geometric distribution allows us the probability of observing our first success on a given observation. In other words, there is no fixed n n. For the Geometric Distribution,X X is the number of trials required to obtain the first success. Problems involving the geometric distribution will ask you to flip a coin UNTIL you get the FIRST tail, or ask you for the probability of getting your FIRST tail ON the 5th flip, etc. Look for key words such as until, first, on, and after. II. Geometric Settings 1) There are two outcomes called success or failure. 2) Observations are independent. 3) The probability of success, p p, is constant. III. Calculating Geometric Probabilities (Geometric Formula) When interested in finding the probability that your first success occurs on the k t h k t h trial, one needs to use the following formula: Geometric Formula (ON):P(X=k)=q k−1 p P(X=k)=q k−1 p where q=1−p q=1−p For calculating more than problems – The probability that it takes more than n n trials to see the first success is: Geometric Formula (MORE THAN):P(X>n)=q n P(X>n)=q n IV. Calculating the Mean and Variance of a Geometric Distribution Use the following formulas to calculate the mean, variance, or standard deviation of a geometric distribution: \| Mean/Expected Value \| Variance \| Standard Deviation \| --- \| E(X)=μ x=1 p E(X)=μ x=1 p \| V A R(X)=σ 2=q p 2 V A R(X)=σ 2=q p 2 \| S D(X)=σ=q p 2−−√S D(X)=σ=q p 2 \| V. Calculator! For P(X = k) Hit 2nd Vars → Scroll to E:geometpdf→ Fill in (n, p, k) For P(X ≤ k) Hit 2nd Vars→ Scroll to F:geometcdf→ Fill in (n, p, k) IMPORTANT NOTE: Everything must be entered in the form of "less than or equal to" (≤). If the problem is asking you for "after" or "more than", draw a number line and shade in what is included. Then you can set up a"less than or equal to" (≤) problem using what is not included, as long as you remember to subtract the calculator's answer from 1. VI. Example An individual decides to roll a fair 6-sided die until he observes a 4. Calculate the following: a. Find the probability that the first time he observes a 4 is on his 3rd role of the die. Before beginning with the full solution, we must first label our outcomes. Based on the problem, rolling a 4 can be labeled as a success, and rolling any number other than a 4 can be labeled as a failure. The probability of a success in this instance is 1/6 or 0.167. P(X=4)=(1−0.167)3(0.167)P(X=4)=(1−0.167)3(0.167) P(X=4)=0.097 P(X=4)=0.097 b. What is the probability that individual must roll more than 5 times before he observes his first 4? Without a calculator,you can use the formula to solve:P(X>5)=(5 6)5=0.401 P(X>5)=(5 6)5=0.401 With a calculator,it will help to start by drawing a number line:←1 2 3 4 5 6 7 8 9 10→ As this number line shows, "more than 5" is equal to 1 - "less than or equal to 5". Using your calculator, you can solve for P(x ≤ 5), and subtract this from 1. 1 -P(x ≤ 5) = 1 - 0.599 = 0.401 c. What is the expected number of rolls before observing the first 4? Remember that "expected" is another term for "mean." Using the formula above, you know that the mean is equal to 1 p 1 p= 1 1/6=6 1 1/6=6 d. Find the standard deviation pertaining to the the number of rolls needed before observing the first 4. Using the formula above, you know that the standard deviation is equal to q p 2−−√q p 2=5/6(1/6)2−−−−√=5.477 5/6(1/6)2=5.477 STATS4STEM is supported by the National Science Foundation under NSF Award Numbers 1418163 and 0937989 Home Features FAQ Statistics Help Pages R Help Pages Contact Us About Us Privacy Policy Terms of Service Report Error Facebook Twitter Blog © 2025 STATS4STEM - RStudio® is a registered trademark of RStudio, Inc. AP® is a registered trademark of the College Board. RStudio and the College Board were not involved in the development of, and do not endorse, this product.
1659	https://teachy.ai/en/project/middle-school/7th-grade/mathematics-en/triangles-in-real-life-a-cross-disciplinary-study	Activities of Triangles in Real Life: A Cross-Disciplinary Study We use cookies Teachy uses cookies to enhance your browsing experience, analyze site traffic, and improve the overall performance of our website. You can manage your preferences or accept all cookies. Manage preferences Accept all TeachersSchoolsStudents Teaching Materials EN Log In Teachy> Projects> Mathematics> 7th grade> Area: Triangle Project: Triangles in Real Life: A Cross-Disciplinary Study Lara from Teachy Subject Mathematics Mathematics Source Teachy Original Teachy Original Topic Area: Triangle Area: Triangle Contextualization The study of geometry, more precisely about the areas of plane figures, is one of the fundamental pillars of mathematics. Area is the measure of the 'size' of the surface of a figure or flat surface and is measured in square units. In this project, we will focus on one of the most basic concepts of geometry, a triangle - a closed three-dimensional geometric figure that has three sides and three vertices. The formula for calculating the area of a triangle is one of the first ones we learn: Area = (base height) / 2. But did you know that there are other formulas to calculate the area of a triangle based on different provided information? For example, if you know the three sides of a triangle, you can use the Heron's formula. If you have a right triangle, you can use the Pythagorean theorem to find the length of the sides. Learning these important concepts will be reinforced through the creation of a project incorporating these ideas. But why is studying the calculation of triangle areas important? The ability to calculate the area of a triangle has countless applications in many fields, such as architecture, engineering, computer science, among others. For example, in architecture, it is often necessary to calculate areas to design spaces. The area of the ground of a house or building is the product of the base by the height, just like the area of a triangle! In computer science, especially in the field of computer graphics, triangles are used to model three-dimensional surfaces. The ability to calculate the area of a triangle is essential for this process. To delve deeper into how the area of triangles is calculated and its practical applications, you can check the following resources: Simple Mathematics - Triangle Area - This is an excellent resource to understand the basics of calculating the area of a triangle. Brazil School - Heron's Formula - This link has an excellent explanation of Heron's Formula to calculate the area of a triangle. Math Portal Channel - Triangle Area and Special Triangles - An informative video resource about triangle areas and special triangles. Practical Activity Activity Title: 'Triangles in Real Life: A Cross-Disciplinary Study' Project Objective This project aims to develop students' understanding of the triangle area, its different calculation formulas, and its practical applications. It also aims to develop important skills such as teamwork, time management, critical and creative thinking. Detailed Project Description Students will work in groups of 3 to 5, dedicating at least 12 hours each to carry out the project. Each group will be tasked with creating a model for a fictional city, integrating the calculation of the triangle area into various aspects of city planning and design. This project will cross mathematics and geography (urban planning) to help students understand the practical application of calculating triangle areas. Required Materials Graph paper or cardboard paper. Pencils and erasers. Ruler and calculator. Colored pencils and/or markers. Computer with Internet access for research. Detailed Step-by-Step Research and Planning: Groups will start by researching city planning and how areas (such as building areas, parks, roads, etc.) are important in this process. Additionally, they should research how the calculation of triangle areas is used in reality. City Design Sketch: Each group should draw a sketch of their city on a piece of paper, dividing the city into different areas, such as residential, commercial, industrial, among others. Each area should contain elements that are composed of triangular shapes, such as pyramids, zigzag roads, triangular soccer fields, etc. Area Calculations: Using the learned concepts of triangle area, students will calculate the area of each triangle in their city. They will use different formulas for area calculation, depending on the available information (for example, if they only know the base and height, if they know all three sides, if it is a right triangle, etc.). Geography Integration: Students will research how urban planning affects the geographical and environmental aspects of a region. They will explain the decisions they made when designing their city, such as the location of different areas and how they relate to the geography of their city. Final Model Creation: Based on the sketch and calculations, students create the final model of their city, coloring and marking the different areas. Final Report: Each group will prepare a detailed report explaining the theory behind the area calculation, the description of the process they followed to design their city, the methodology used to calculate the triangle areas, and the decisions made based on geographical aspects. The report should be divided into four main topics: Introduction, Development, Conclusions, and Bibliography used. Project Deliverables Each group must deliver the following items at the end of the project: The final model of the designed city. Detailed final report. These deliveries will serve to assess students' understanding of the triangle area, their technical and socio-emotional skills, and their ability to apply theoretical knowledge to practical situations. Need materials to present the project topic in class? On the Teachy platform, you can find a variety of ready-to-use materials on this topic! Games, slides, activities, videos, lesson plans, and much more... Explore free materials Those who viewed this project also liked... Project Exploring Counting in Our Daily Lives Lara from Teachy - Project Mapping Parabolas: An Interdisciplinary Approach Lara from Teachy - Project Conversion Game Lara from Teachy - Project Decoding Binary Language Lara from Teachy - Join a community of teachers directly on WhatsApp Connect with other teachers, receive and share materials, tips, training, and much more! 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1660	https://aast.edu/pheed/staffadminview/pdf_retreive.php?url=48_25795_ME362_2020_1__2_1_Lecture%204%20-%20Fluid%201%20-%20Forces%20on%20Submerged%20Plane%20Surfaces%20-%20Part%20I.pdf&stafftype=staffcourses	Fluid Mechanics I – ME 362 Arab Academy for Science, Technology and Maritime Transportation Dr. Ahmed Khalifa Mehanna Associate Professor a.khalifa@aast.edu ahmed_marines@yahoo.com Room No: 223 Course Assistant Lecturer: Eng. Omar Mostafa Lecture 4: Hydrostatic Forces on Submerged Plane Surfaces Fluid Mechanics I –ME 362 Hydrostatic forces on submerged plane surfaces Graphical Solution (Prism Solution) Analytical Solution Hydrostatic forces on submerged plane surfaces Analytical Solution Hydrostatic forces on submerged plane surfaces A plate is subjected to fluid pressure distributed over its surface when exposed to a liquid ; such as a gate valve in a dam, the wall of a liquid storage tank, or the hull of a ship at rest . On a plane surface, the hydrostatic forces form a system of parallel forces, and we often need to determine the magnitude of the Resultant force and its point of application , which is called the center of pressure . When analyzing hydrostatic forces on submerged surfaces, the atmospheric pressure can be subtracted for simplicity when it acts on both sides of the structure .Hydrostatic forces on submerged plane surfaces When analyzing hydrostatic forces on submerged surfaces, the atmospheric pressure can be subtracted for simplicity when it acts on both sides of the structure . Effect of atmospheric pressure on the resultant force acting on a plane vertical wall. General submerged plane General submerged plane  The total area is made up of many elemental areas .  The force on each elemental area is always normal to the surface but, in general, each force is of different magnitude as the pressure usually varies .  We can find the total or Resultant Force , FR, on the plane by summing up all of the forces on the small elements .ApApApApF nn  .... 2211R A PdA FR• For a horizontal plane submerged in a liquid, the pressure, p, will be equal at all points of the surface . • The pressure at the bottom of the container is uniform across the entire area pA F R Hydrostatic forces on Submerged Horizontal Plane Surface • For a horizontal plane submerged in a liquid, the pressure, p, will be equal at all points of the surface . • The pressure at the bottom of the tank is uniform across the entire area pA F R Hydrostatic forces on Submerged Horizontal Plane Surface h A FR  sin yzh Where: Given a plane surface AB entirely submerged in the liquid. The surface is inclined an angle θ to the liquid surface. The centroid of area is located at C. The vertical distance of C below the liquid surface is: Hydrostatic forces on Submerged Inclined or Vertical Plane Surfaces Pressure at a point at z below the liquid surface is: p(z) =  z (Gage pressure ) Or in terms of y ( distance along the plate ) pressure at point z is: p(z) =  ysin  The differential force on the differential area dA is dF(z ) Hydrostatic forces on Submerged Inclined or Vertical Plane Surfaces Pressure forces on an elemental area dA: The differential force on the differential area dA is dF(z)         dA yzdF dA zzdF dA zpzdF yzzp )(sin )sin ()(  Hydrostatic forces on Submerged Inclined or Vertical Plane Surfaces Where : Pressure forces on an elemental area dA :Hydrostatic Pressure Distribution Hydrostatic forces on Submerged Inclined or Vertical Plane Surfaces  sin ypzp The total force (F R) on the area will be obtained by integrating the differential force over the entire area :  dA ydF FR sin    and (sin ) are constants          AyAhFAhAzFAyFAyFdA yF ccRRRRR sin sin sin sin  cc hy   hyydA yAdA A Remark The total hydrostatic force on a planar inclined surface Hydrostatic forces on Submerged Inclined or Vertical Plane Surfaces The total hydrostatic force on a planar inclined surface Hydrostatic forces on Submerged Inclined or Vertical Plane Surfaces yR of the resultant force can be determined by summation of moments around the x axis . That is, the moment of the resultant force must equal the moment of the distributed pressure force, or   dA ydF FR sin   Recall that :  dA yydF yyF RR   sin So the moment is :AyF cR  sin  Recall that :The integral in the numerator is the second moment of the area (moment of inertia) , “Ix ”. Thus, we can write : Using the parallel axis theorem : where “Ixc ”. is the second moment of the area with respect to an axis through its centroid and parallel to the x axis . Thus : So, from the above Equation, it is clear that the resultant force does not pass through the centroid . but for nonhorizontal surfaces is always below it, since (Ixc / ycA) > Zero ccxc ccxc hhIyyI  AhOr Ay RRThe centroid and the centroidal moments of inertia for some common geometries The centroid and the centroidal moments of inertia for some common geometries A heavy car plunges into a lake during an accident and lands at the bottom of the lake on its wheels as shown in the Figure . The door is 1.2 m height and 1 m wide, and the top edge of the door is 8 m below the free surface of the water . Using the analytical and prism method, determine the hydrostatic force on the door and the location of the pressure center . Example m8.61 )] 21.2 ([8 1.2) 1()] 21.2 ([8 0.144 144 .0)12 1.2 1()12 (IhhIkN 101.24 kN )12.1()] 21.2 ([8 9.81 kN/m 9.81 and 433xc ccxc 3  RRRRwcwR hmba AhFFAhF   Using the Analytical Method
1661	https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/v/projection-of-a-vector	Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer. More information Manage Consent Preferences Strictly Necessary Cookies Always Active Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account. For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session. Functional Cookies These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences. Targeting Cookies These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission. We do not use cookies to serve third party ads on our Service. Performance Cookies These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates. For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users. We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails. Cookie List Consent Leg.Interest label label label
1662	https://numberline.school/about/	About \| Number Line Number Line / About Putting the Number Line Online Try numberline.school As a primary teacher, I use number lines all the time. Whether I’m teaching the four operations, place value, rounding, negative numbers, decimal numbers or fractions: number lines are the go-to tool for developing a conceptual understanding of numbers. The problem is: each teaching objective requires a different type of number line, and printed number lines are limited by their length, range, and intervals. They just aren’t flexible enough! This is why we decided to build an app: numberline.school is a zoomable, infinite number line which can be flexibly used to teach maths in any classroom. Why numberline.school How to use The Number Line Six Educational Ways to use The Number Line Consolidate Place Value Find the Number! Understand Rounding to The Nearest Power of Ten Estimating Numbers on a Number Line Ordering Numbers Visualising Addition and Subtraction Why numberline.school It's not just saving paper. Thanks to it being a digital app, you can scroll infinitely along the number line — both into negative and very large positive numbers. The zooming function allows you to easily zoom in to the decimals, and out to the millions — giving the user a flexible overall understanding of a number line. The scale is animated and shows the appropriate power of 10 as you zoom: counting first in ones, then tens, hundreds, thousands, ten thousands, hundred thousands and up to hundred millions. In decimal places, it can count in tenths, hundredths, and thousandths. How to Use The Number Line numberline.school runs on any computer. But it works best on touch screen devices such as iPads, tablets, or smartphones as the touch gestures allow for zooming and scrolling. Teachers can, for instance, project the number line to a whiteboard and use it to model to students. If enough devices are available, all students can use it for a lesson, or it may become a favourite scaffolding tool for any students with SEN. You can also mark numbers with pins. Pins are unlimited and can be placed absolutely anywhere along the number line. This way, students can pin a number and then zoom in or out to view where the number is in relation to others - perfect for rounding! Students really enjoy the number line app and even just playing around with it inspires a sense of awe about the sheer infinity of the number world. They can see just how big numbers get, but also discover more rarely used numbers such as large decimal numbers. Often kids are stumped when they see a large number with decimal places such as 30,000.25. With a Touchscreen... Swipe left and right to scroll along the infinite number line. Pinch to zoom in and out to see the number line count in tens, hundreds, thousands,… Place an unlimited amount of coloured pins by clicking, dragging, and then releasing them on the number line. If you let go over a labelled number, the pin will snap into position. Tap the reset button to centre the number line on zero and delete all pins. With a Mouse... Click and drag left and right to scroll along the infinite number line. Click the magnifying glass icons to zoom in and out between tens, hundreds, thousands,… Place an unlimited amount of coloured pins by pressing, dragging, and then releasing them on the number line. If you let go over a labelled number, the pin will snap into position. Click the reset button to centre the number line on zero and delete all pins. Six Educational Ways to use The Number Line 1. Consolidate Place Value A great way to give students a sense for place value is to let them simply play around with the number line. This could even be a warm-up starter activity for a lesson, and students often really enjoy playing with it. Some questions you could ask to lead the activity: What is the biggest number they can see? How can the zooming help them find a very big number? How can they now get back to the number 0 quickly without resetting? What is the smallest number they can find? (Introduce decimals and negative numbers here) How many numbers are in between the number 0 and 1? (Keep zooming in to get a sense of infinity!) 2. Find the Number! Whether students are learning about negative numbers, decimals, or large numbers, writing a number down is often very different from actually understanding what the number means on the big number continuum. You can make it a game to see which pair of students can find a number the fastest. Can they find a large, negative, or decimal number on the infinite number line? How quickly can they find it and place a pin on it? How many zooming steps does it take to find it quickly? E.g. to find 45,097 you would need to zoom out to the ten thousands first, then zoom in to the thousands, then hundreds, then tens, then ones… until you can place a pin on the number! Once they have found a number, what would be the most efficient way to get back to zero? (Without pressing the reset button!) 3. Understand Rounding to the Nearest Power of Ten Rounding was the main inspiration for making this number line app: every year, I find students have forgotten how to round to the nearest ten, hundred, or thousand — no matter how well they knew it the year before. To me, this was a clear sign that students are lacking the conceptual understanding of rounding. You may have heard and used the countless mnemonics which have been created to help students remember how to round: “Five or more, raise the score, four or less, let it rest” “five or more, let it soar” “five and above, give it a shove” These are just some of the most popular ones. However, none of the mnemonics help students understand why they can simply “raise the score”. In contrast, number lines are a great way to visualise how rounding works. And, thanks to its flexibility, numberline.school is an excellent tool for showing rounding to the nearest 10/100/1,000 — or even higher powers of ten. Here’s how: Ask the children to show you a number line which is counting in tens/multiples of ten (and clarify that even in 6-digit numbers you can still count up in tens!). Can they show the same for counting in hundreds, thousands, ten thousands? Ask them to find a number (e.g. 4972) and place a pin on it: Now have them zoom out until they see the tens. They should now be able to see which “ten” it is closest to, i.e. when rounding to the nearest ten, it should round to 4970… … then, by simply zooming further out, repeat for rounding to the nearest 100… … rounding to the nearest 1,000… … and so on! 4. Estimating Numbers on a Number Line One of the main stepping stones of understanding place value is being able to estimate the position of different numbers on a number line in relation to other numbers. For instance, where is 150,000 in relation to 15,000 or 50,000? Using the screenshot tools, teachers can easily create simple worksheet challenges to print out for students. Simply pin certain numbers with different colours (this can be differentiated according to students’ needs) and ask students to label the pins. As shown in the example below, students could have the task to estimate numbers shown by pins - either freely, or as multiple choice answers. Task: Label each pin with its number: 150,000 50,000 10,000 105,000 15,000 5. Ordering Numbers Whether ordering positive and negative integers or ordering decimals, having a number line to visualise the numbers is vital. With numberline.school, students can visually compare the sizes of different numbers. The zooming and scrolling give a real sense of scale. For example, ask students to order the following decimal numbers from smallest to biggest: 0.1 0.9 0.09 0.99 Students can pin each number on the number line, use the scale to check their own results, and immediately try again if they make a mistake. 6. Visualising Addition and Subtraction For simple additions and subtractions, number lines let you “count the jumps” between numbers. By practicing addition and subtraction on a number line, children will internalise it and be able to remember it when doing mental maths. Examples: 11 + 8 Ask students to place a pin on 11 and then jump 8 steps to the right (where the numbers get bigger. They can then place a pin on the answer. 23 - 15 Ask students to place a pin on 23 and to jump 6 steps to the left (to make the number smaller). They can then place a pin on the result. Alternatively, they can prove that subtractions are the same as finding the difference between two numbers: ask students to place a pin on 23 and on 15 and to count how many jumps difference it is - it should be the same answer! Number lines can be extremely helpful in teaching addition and subtraction with negative numbers! Students can scroll across zero to help them count and find the difference between -3 and 7 for example. I hope you find this number line as useful as I do — many of my students now beg me to use numberline.school! Try numberline.school numberline@blastoff.design Feedback is always welcome!
1663	https://sharmaeklavya2.github.io/theoremdep/nodes/probability/rand-vars/variance/variance-of-affine.html	Var(aX + b) = a^2 Var(X) Dependencies: Variance of a random variable Linearity of expectation Let $X$ be a random variable. Let $a \in \mathbb{R}$ and $b \in \operatorname{support}(X)$. Then $\newcommand{\E}{\operatorname{E}}\newcommand{\Var}{\operatorname{Var}}$ $\Var(aX + b) = a^2\Var(X)$. Proof [ \Var(aX + b) = \E((aX + b - \E(aX + b))^2) = \E((aX - a\E(X))^2) = a^2\E((X - \E(X))^2) = a^2\Var(X) ] Dependency for: None Info: Depth: 9 Number of transitive dependencies: 21 Transitive dependencies: /analysis/topological-space /sets-and-relations/countable-set /sets-and-relations/de-morgan-laws /measure-theory/linearity-of-lebesgue-integral /measure-theory/lebesgue-integral σ-algebra Generated σ-algebra Borel algebra Measurable function Generators of the real Borel algebra (incomplete) Measure σ-algebra is closed under countable intersections Group Ring Field Vector Space Probability Random variable Expected value of a random variable Linearity of expectation Variance of a random variable
1664	https://www.droracle.ai/articles/129985/treatment-for-lyme-in-someone-allergic-to-doxycycline-	What are alternative treatments for Lyme disease in a patient with a hypersensitivity reaction to doxycycline (Doxycycline)? Select Language▼ What are alternative treatments for Lyme disease in a patient with a hypersensitivity reaction to doxycycline (Doxycycline)? Medical Advisory BoardAll articles are reviewed for accuracy by our Medical Advisory Board Educational purpose only • Exercise caution as content is pending human review Article Review Status Submitted Under Review Approved Last updated: May 22, 2025 • View editorial policy From the Guidelines For patients allergic to doxycycline, the most effective alternative treatment for Lyme disease is amoxicillin (500 mg three times daily for 14-21 days for adults) or cefuroxime axetil (500 mg twice daily for 14-21 days for adults) 1. When considering treatment options, it's essential to weigh the potential benefits and risks of each antibiotic. Amoxicillin and cefuroxime axetil are both effective against Lyme disease and have a relatively low risk of serious adverse effects 1. For children, amoxicillin dosing is 50 mg/kg/day divided into three doses (maximum 500 mg per dose), while cefuroxime is given at 30 mg/kg/day in two divided doses (maximum 500 mg per dose) 1. In cases of neurologic Lyme disease or more severe manifestations, intravenous ceftriaxone (2 g daily for 14-28 days) may be necessary 1. For pregnant women, amoxicillin is typically the preferred alternative, as doxycycline is relatively contraindicated during pregnancy or lactation 1, 2. Azithromycin (500 mg daily for 7-10 days) can be used but is considered less effective than other options 2. The choice between these alternatives depends on the stage of Lyme disease, patient characteristics, and any other medication allergies. These antibiotics work by disrupting bacterial cell wall synthesis or protein production, effectively eliminating the Borrelia burgdorferi spirochete that causes Lyme disease 1, 2. Treatment should begin as soon as possible after diagnosis to prevent progression to later stages of the disease 1, 2. From the Research Treatment Options for Lyme Disease When a patient is allergic to doxycycline, alternative treatment options must be considered. The following alternatives are supported by existing evidence: Amoxicillin: This antibiotic has been shown to be effective in treating early Lyme disease, particularly for patients with erythema migrans 3, 4, 5, 6. Cefuroxime axetil: This antibiotic has been compared to doxycycline in several studies and has been found to be equally effective in treating early Lyme disease 6, 7. Azithromycin: This antibiotic has been found to be effective in treating Lyme disease, although it is considered a second-line agent due to its lower efficacy compared to beta-lactam antibacterials and tetracyclines 4, 5. Ceftriaxone and cefotaxime: These injectable antibiotics have been found to be effective in treating Lyme disease, particularly for patients with severe or late-stage disease 5. Considerations for Treatment When selecting an alternative treatment, the following factors should be considered: The stage and severity of the disease: Patients with severe or late-stage disease may require injectable antibiotics such as ceftriaxone or cefotaxime. The patient's age and medical history: Children and patients with certain medical conditions may require alternative treatments due to potential side effects or interactions. The potential for side effects and interactions: Each antibiotic has a unique side effect profile and potential for interactions with other medications. References 1 Guideline Guideline Directed Topic Overview Dr.Oracle Medical Advisory Board & Editors, 2025 2 Guideline Guideline Directed Topic Overview Dr.Oracle Medical Advisory Board & Editors, 2025 3 Research Diagnosis and treatment of Lyme disease. Mayo Clinic proceedings, 2008 4 Research Systematic review of the treatment of early Lyme disease. Drugs, 1999 5 Research Forty Years of Evidence on the Efficacy and Safety of Oral and Injectable Antibiotics for Treating Lyme Disease of Adults and Children: A Network Meta-Analysis. Microbiology spectrum, 2021 6 Research Comparative study of cefuroxime axetil versus amoxicillin in children with early Lyme disease. Pediatrics, 2002 7 Research Comparison of cefuroxime axetil and doxycycline in the treatment of early Lyme disease. Annals of internal medicine, 1992 Related Questions What is the treatment for suspected Lyme disease (Lyme borreliosis) in a 4-year-old pediatric patient?What are alternative treatments for Lyme disease in a patient with a hypersensitivity reaction to Doxycycline (tetracycline antibiotic)?Would a bull's eye rash from a tick bite necessitate a full 21-day course of antibiotic treatment for Lyme disease?What is the treatment for a patient who had a tick on their body for 8 hours, developed a rash that has since resolved, and vomited after one day of doxycycline (doxy) therapy?Can a subsequent Ixodes (deer tick) bite trigger a flare-up of Lyme disease in a previously infected individual?What hormone causes the glowing appearance in pregnant women?What should be checked if direct bilirubin is elevated and Gamma-Glutamyl Transferase (GGT) is normal?What are the indications for prophylaxis after a suspected tick (Ixodida) bite?Should imaging be repeated for 2 non-obstructing renal stones, each 0.5 centimeters (cm) in size, with normal renal function and no evidence of obstruction?When is a contrast-enhanced computed tomography (CT) scan indicated in gallstone-induced pancreatitis?What percentage of patients with cholecystitis (inflammation of the gallbladder) pass gallstones? Professional Medical Disclaimer This information is intended for healthcare professionals. Any medical decision-making should rely on clinical judgment and independently verified information. The content provided herein does not replace professional discretion and should be considered supplementary to established clinical guidelines. Healthcare providers should verify all information against primary literature and current practice standards before application in patient care. Dr.Oracle assumes no liability for clinical decisions based on this content. Have a follow-up question? Our Medical A.I. is used by practicing medical doctors at top research institutions around the world. Ask any follow up question and get world-class guideline-backed answers instantly. Ask Question Original text Rate this translation Your feedback will be used to help improve Google Translate
1665	https://cs.stackexchange.com/questions/79839/domino-tiling-of-a-2xn-rectangle-in-oln-n	dynamic programming - Domino tiling of a 2xN rectangle in O(ln n) - Computer Science Stack Exchange Join Computer Science By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Domino tiling of a 2xN rectangle in O(ln n) Ask Question Asked 8 years, 1 month ago Modified8 years, 1 month ago Viewed 976 times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. I solved this problem using Dynamic Programming in O(n)O(n) time. I found that is equivalent to the Fibonacci Numbers. F(0)=F(1)=1 F(0)=F(1)=1 F(n)=F(n−1)+F(n−2)F(n)=F(n−1)+F(n−2) Where the F(n−1)F(n−1) term is from fixing the left most domino vertically, and the F(n−2)F(n−2) from fixing it horizontally (which implies that the domino under it must also placed horizontally). Because the Fibonacci Numbers can be generated in O(lg n)O(lg⁡n) using F(2 n)=F(n)F(n+1)+F(n−1)F(n)F(2 n)=F(n)F(n+1)+F(n−1)F(n) F(2 n+1)=F(n)F(n)+F(n−1)F(n−1)F(2 n+1)=F(n)F(n)+F(n−1)F(n−1) Then I try to find the same expression from the domino tiling. Again, I classify all possible tilings in two set. First, all tiling like the one below Therefore, F H(n)=F(i)F(n−i−2)F H(n)=F(i)F(n−i−2). Because I want to split in the middle (or close) I consider n=2 k,i=k n=2 k,i=k and n=2 k+1,i=k n=2 k+1,i=k. Then F H(2 k)=F(k)F(2 k−k−2)=F(k)F(k−2)F H(2 k)=F(k)F(2 k−k−2)=F(k)F(k−2) and F H(2 k+1)=F(k)F(2 k+1−k−2)=F(k)F(k−1)F H(2 k+1)=F(k)F(2 k+1−k−2)=F(k)F(k−1). Then, all tiling of the form Again, F V(n)=F(i)F(n−i−1)F V(n)=F(i)F(n−i−1) and when n=2 k,i=k n=2 k,i=k and n=2 k+1,i=k n=2 k+1,i=k we have F V(2 k)=F(k)F(2 k−k−1)=F(k)F(k−1)F V(2 k)=F(k)F(2 k−k−1)=F(k)F(k−1) and F V(2 k+1)=F(k)F(2 k+1−k−1)=F(k)F(k)F V(2 k+1)=F(k)F(2 k+1−k−1)=F(k)F(k) Combining all the former expressions, F(2 k)=F H(2 k)+F V(2 k)=F(k)F(k−2)+F(k)F(k−1)F(2 k)=F H(2 k)+F V(2 k)=F(k)F(k−2)+F(k)F(k−1) F(2 k+1)=F H(2 k+1)+F H(2 k+1)=F(k)F(k−1)+F(k)F(k)F(2 k+1)=F H(2 k+1)+F H(2 k+1)=F(k)F(k−1)+F(k)F(k) However, those final expression does not produce the same numbers. Because they look really similar I belive my approach is not completly wrong but I can not find where I make my mistake. Final notes: In the first classification, I fix the top most domino horizontally and if I consider a 1-tile shifted horizontal domino (left or right) under it, then the rectangle can not be tiled. For the (correct) expression F(2 n+1)=F(n)F(n)+F(n−1)F(n−1)F(2 n+1)=F(n)F(n)+F(n−1)F(n−1), notice that F(n)F(n)F(n)F(n) can be mapped to the second case, because n+1+n=2 n+1 n+1+n=2 n+1, but the other term F(n−1)F(n−1)F(n−1)F(n−1) can not (at least in the same way), (n−1)+2+(n−1)≠2 n+1(n−1)+2+(n−1)≠2 n+1. The same can be notice in F(2 n)F(2 n) and again one term can be mapped to the second case while the other can not (to the first case). I belive my mistake should be around the first case. dynamic-programming tiling Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Improve this question Follow Follow this question to receive notifications asked Aug 7, 2017 at 22:11 Black ArrowBlack Arrow 243 1 1 silver badge 6 6 bronze badges 1 Which problem did you solve? What is the original solution to your problem? If it is the same as the Fibonacci numbers, why can't you use a fast algorithm for computing the Fibonacci numbers in order to solve your problem?Yuval Filmus –Yuval Filmus 2017-08-08 07:18:50 +00:00 Commented Aug 8, 2017 at 7:18 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 3 Save this answer. Show activity on this post. In your second attempt you tried to fixate the (k+1)(k+1)th term. But instead of the two cases 1 2 ... k k+1 k+2 ... n \| \| and 1 2 ... k k+1 k+2 ... n ----- there exist a third case: 1 2 ... k k+1 k+2 ... n ---- So if n=2 k n=2 k, then you have f(k)f(k−1)f(k)f(k−1) for the first case, f(k)f(k−2)f(k)f(k−2) for the second one, and f(k−1)f(k−1)f(k−1)f(k−1) for the third one. This gives in total: f(2 k)=f(k)f(k−1)+f(k)f(k−2)+f(k−1)f(k−1)=f(k)[f(k−1)+f(k−2)]+f(k−1)f(k−1)=f(k)2+f(k−1)2 f(2 k)=f(k)f(k−1)+f(k)f(k−2)+f(k−1)f(k−1)=f(k)[f(k−1)+f(k−2)]+f(k−1)f(k−1)=f(k)2+f(k−1)2. Now this formula looks like the formula for F(2 n+1)F(2 n+1) that you posted. Because the Fibonacci Numbers can be generated in O(lgn) using F(2n)=F(n)F(n+1)+F(n−1)F(n) F(2n+1)=F(n)F(n)+F(n−1)F(n−1) Simple reason: the formula assumes that F(1)=F(2)=1 F(1)=F(2)=1 while you have f(0)=f(1)=1 f(0)=f(1)=1. So after an index shift it should match. Identical procedure for the odd case n=2 k+1 n=2 k+1. edit: Tried to do the index shift but failed. Reason was, that you wrote down the wrong formula for F(2 n+1)F(2 n+1). It should be F(2 n+1)=F(n)2+F(n+1)2 F(2 n+1)=F(n)2+F(n+1)2. Then the index shift works: f(2 k)=F(2 k+1)=F(k)2+F(k+1)2=f(k−1)2+f(k)2 f(2 k)=F(2 k+1)=F(k)2+F(k+1)2=f(k−1)2+f(k)2. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Aug 8, 2017 at 8:34 answered Aug 8, 2017 at 8:05 JakubeJakube 1,605 10 10 silver badges 14 14 bronze badges Add a comment\| Your Answer Thanks for contributing an answer to Computer Science Stack Exchange! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. Draft saved Draft discarded Sign up or log in Sign up using Google Sign up using Email and Password Submit Post as a guest Name Email Required, but never shown Post Your Answer Discard By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. 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1666	https://mathforall.edc.org/the-problem-with-word-problems/	Log In The Problem with Word Problems March 11, 2024Language, Mathematics, Neurodevelopmental Functionsby Math for All The Problem with Word Problems By Matt McLeod Word problems can be confusing. Period. They can be set in an unfamiliar context. They can include superfluous information. They can contain ambiguous language. They can be written to trick or confuse. They can be overly complex. Here’s an example that meets a few of these criteria: The pet store sells crickets for lizards. They charge $3.65 per two-dozen crickets, but right now they are offering a 15% discount on any purchase over $40. What will be the total cost for 276 crickets if there is a 6% sales tax? How many of your students visit pet stores? How many buy crickets or even know what crickets are and that lizards eat them? If crickets are sold by the “two-dozen” why would you ask for 276 of them, especially since that’s an odd number of dozens? If you are allowed to buy 11.5 two-dozen crickets (aka 276), the total is $41.975. Is this number rounded to cents before or after you take the discount and add the sales tax? Why would you need so many crickets? How many of your students would look at this problem and say, “I don’t want to read all of that” or “Woah! There’s too much going on in that problem”? One of the constructs within the Neurodevelopmental Framework that we explore in Math for All is language. Language plays many roles in the mathematics classroom, not the least of which is reading and understanding word problems. For a student with language challenges, there are several elements of word problems that can cause frustration. Decoding the words, understanding words with several meanings, remembering vocabulary, and identifying necessary versus unnecessary information are just some of the components that can create barriers to a student understanding a problem. And all of these barriers come into play before a learner can even decide on which calculations are important and then do any of them. Word problems are a great way to get students into and better at problem solving, which should be something we strive to do in the mathematics classroom. Providing them with experience and guidance in building their skill and confidence in tackling problems is perhaps the most critical thing we do. However, any word problem that we ask students to solve must be carefully vetted. One great way to do this (that Math for All strongly encourages) is to work through the problem yourself to see where students might bump into challenges. Not all word problems have as much room for improvement as the example above, but a vast majority need at least some adaptation to be a good fit for a classroom or a student. Here are some things to ask yourself when considering a word problem for your class: What is the mathematical goal of the problem and does this align with your teaching goals? Is the context familiar to students? Is the language at students’ reading and comprehension levels? Is there too much language? Are the numbers accessible to the students? Are the numbers reasonable in the context? Is there too much going on in the problem? Another way to help students with word problems and expand their problem-solving skill is to consider when they are presented in the class. Many curricular materials use word problems as a means of applying a particular skill or algorithm after that skill has been learned and practiced ad nauseam through numeric problems. What if the order is reversed? In an issue of the Alexandria City Public Schools newsletter, Terri Mazingo presents the idea of teaching “Mathematics Through Problem Solving vs. Mathematics for Problem Solving.” This approach suggests presenting the problem as the starting point instead of the wrap-up, giving students an opportunity to engage in productive struggle, which can be effective in developing positive mindsets and confidence when approaching a new problem or type of problem. In a 2010 article, Nancy Ward from the Clairbourn School synthesizes this idea by saying that “students must change their thinking from ‘What does the teacher want me to do?’ to ‘What can I do?’” How word problems are presented also matters. Both articles offer suggestions for how to improve students’ confidence and agency in problem solving, including the idea of “headless” and “tailless” word problems. This idea is explained in more detail in a video featuring Jane Kang of Education Development Center, who describes how these types of problems are used in Transition to Algebra. Headless word problems simply ask a question—“How many steps have you taken in your life?”—while tailless problems offer a simple context—“I have 17 cents in my pocket.” Both types of problems change how students engage with the text. A traditional word problem positions students as math consumers simply using what someone else has provided. On the flip side, headless and tailless word problems allow students to be math producers, presenting them with opportunities to contribute (and subsequently answer) their own questions, drawing them into the problem in a different way. As you carefully consider how to adapt the lesson for your student, it is critical to remember to keep the mathematics rigorous! It is not helpful to students to lower the standards or expectations. Although not a comprehensive list, here are some ways that might be helpful when adapting a problem to better meet the needs of your students. Change the context of the problem. Consider using a context that students are exploring in another subject area, such as science, reading, social studies, etc. Change the language in the problem (remove ambiguities, meet students’ comprehension levels, etc.). Reduce the number of words in the problem. Add images. Offer manipulatives. Offer vocabulary reminders. Consider the option of eliminating the words completely. Here are a few parting thoughts about building your students’ skills and confidence in word problems. Be less helpful. Give students time and space for productive struggle, letting them be creative in the way they solve a problem. Avoid promoting key words for operations—total means add, less means subtract, etc.—as they are not always accurate and can lead to confusion. 3 Reads is one framework that offers students a way to deepen understanding of the problem before they begin to solve it. Playfulness is an important part of learning (for students and teachers). And finally, here are a couple more word problems. Try to answer each before you jump to the link for a discussion about it. Problem 1: There are 49 dogs signed up to compete in a dog show. There are 36 more small dogs than large dogs. How many small dogs are signed up to compete? Here’s the discussion about this problem being presented to a second-grade class for homework. Problem 2: Some birds were flying and met a bird on their way. The bird greeted them, “Hello, hundred!” They said, “We are not hundred. We need half of us plus you to make us hundred.” Here’s what others did with it. The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government. This work is licensed under CC BY-NC-SA 4.0 Math for All is a professional development program that brings general and special education teachers together to enhance their skills inplanning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics. Our Newsletter Provides Ideas for Making High-Quality Mathematics Instruction Accessible to All Students Sign Up Previous Honoring Diversity: What, why, and how? Next Parents and Teachers as Co-Constructors of Children’s Success as Mathematical Learners Comments are closed. 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1667	https://pmc.ncbi.nlm.nih.gov/articles/PMC1363913/	The first seizure and its management in adults and children - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. 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Learn more: PMC Disclaimer \| PMC Copyright Notice BMJ . 2006 Feb 11;332(7537):339–342. doi: 10.1136/bmj.332.7537.339 Search in PMC Search in PubMed View in NLM Catalog Add to search The first seizure and its management in adults and children Bernd Pohlmann-Eden Bernd Pohlmann-Eden 1 Bethel Epilepsy Centre, Bethel, Germany head and chair, professor of neurology Find articles by Bernd Pohlmann-Eden 1, Ettore Beghi Ettore Beghi 2 Epilepsy Center and Department of Neurology, University of Milano-Bicocca, Monza; Laboratory of Neurological Disorders, Instituto “Mario Negri”, Milano, Italy professor of neurology Find articles by Ettore Beghi 2, Carol Camfield Carol Camfield 3 Department of Pediatrics, Dalhousie University and the IWK Health Centre, Halifax, Nova Scotia, Canada professor of pediatrics Find articles by Carol Camfield 3, Peter Camfield Peter Camfield 3 Department of Pediatrics, Dalhousie University and the IWK Health Centre, Halifax, Nova Scotia, Canada professor of pediatrics Find articles by Peter Camfield 3 Author information Article notes Copyright and License information 1 Bethel Epilepsy Centre, Bethel, Germany 2 Epilepsy Center and Department of Neurology, University of Milano-Bicocca, Monza; Laboratory of Neurological Disorders, Instituto “Mario Negri”, Milano, Italy 3 Department of Pediatrics, Dalhousie University and the IWK Health Centre, Halifax, Nova Scotia, Canada Correspondence to: B Pohlmann-Eden pohleden@gmx.net Roles Bernd Pohlmann-Eden: head and chair, professor of neurology Ettore Beghi: professor of neurology Carol Camfield: professor of pediatrics Peter Camfield: professor of pediatrics Accepted 2005 Dec 19. Copyright © 2006, BMJ Publishing Group Ltd. PMC Copyright notice PMCID: PMC1363913 PMID: 16470055 Introduction This review presents a comprehensive approach to children and adults with a first seizure, an event that may have profound emotional, social, and vocational consequences. A first “grand mal” convulsion is frightening, yet prospective, population-based studies indicate that we all face an 8-10% lifetime risk of one seizure1 and a 3% chance of epilepsy.2 It seems likely that everyone could have a seizure if a particular set of circumstances occur—but some people have a lower seizure threshold than others. A first seizure caused by an acute disturbance of brain function (acute symptomatic or provoked) is unlikely to recur (3-10%). If a first seizure is unprovoked, however, meta-analyses suggest that 30-50% will recur; and after a second unprovoked seizure, 70-80% will recur, justifying the diagnosis of epilepsy (a tendency for recurrent seizures).3-5 When a person presents to the healthcare system with a first seizure, it is almost always a convulsive seizure, either generalised or focal. Other seizure types such as absence or complex partial seizures typically occur several times before the person or family become concerned. Methods We reviewed all literature listed in PubMed under the headings “first seizure” and “initial seizure.” Two of us have helped to develop a practice parameter on treatment of a first seizure in children, and all of us have conducted prospective studies of first seizures. We are unaware of any systematic (Cochrane) review of this topic. All references cited were judged to give strong evidence. Is it a seizure? The differential diagnosis for a first seizure is wide. Most important in our experience are syncope (including breath holding and pallid syncope), transient ischaemic attacks, metabolic encephalopathy (including hypoglycaemia or electrolyte disturbance), sleep walking, night terrors, complex migraines, cardiac arrhythmias, and pseudoseizures. “Convulsive syncope” presents a particular challenge when syncope provokes a post-anoxic convulsion. A detailed history from both patient and witness is paramount, but no single feature is diagnostic. Tongue biting is not common but is fairly specific for a convulsive seizure, while postictal confusion suggests a seizure. If the first event is ambiguous, we advocate waiting for a recurrence for clarification. In our experience, and as outlined in a thoughtful review, misdiagnosis of an “epileptic” seizure may be more stigmatising than a delayed diagnosis of epilepsy.6 Summary points The differential diagnosis of a first seizure is wide A first seizure mandates individual counselling about the risk of recurrence, the pros and cons of drug treatment, and the impact on lifestyle A first seizure provoked by an acute brain disturbance is unlikely to recur (3-10%), whereas a first unprovoked seizure has a recurrence risk of 30-50% over the next two years Many people presenting for the first time with a convulsive seizure have had prior unrecognised seizures A seizure can be diagnosed only by the history, but investigations should include prompt electroencephalography and usually magnetic resonance imaging After counselling, most patients do not choose anti-epileptic drug treatment after a first seizure Restrictions on activities after a first seizure should be individualised. Restrictions on driving vehicles vary between countries: in the United Kingdom non-commercial driving is not permitted for 12 months after an unprovoked seizure The “first” seizure may not be the first Large consecutive case series indicate that many people presenting with a dramatic first generalised tonic-clonic “grand-mal” seizure have had previous, undiagnosed simple or complex partial seizures (such as intense “deja-vu”, a sudden feeling of fear, a bad smell or taste, or brief language difficulties), absence seizures, or epileptic myoclonus.7 The first convulsive seizure may simply be the first recognised seizure pointing to the diagnosis of epilepsy. Box 1: Essential diagnostic procedures in patients with a first seizure Clinical examination Assessment of seizure semiology Routine laboratory tests (depending on clinical circumstances) Cerebrospinal fluid (if encephalitis or subarachnoid haemorrhage is suspected) Drug screening (depending on clinical circumstances) Early standard electroencephalography, if possible within 24 hours Sleep deprived electroencephalography within 1 week High resolution magnetic resonance imaging, if possible In all adults In all children except those with idiopathic (genetic) focal or generalised epilepsy syndromes What has provoked the first seizure? Population based studies indicate that 25-30% of first seizures are “acute symptomatic” or “provoked” by a brain insult or a metabolic or toxic disturbance of brain function.8-10 Provoking factors include fever, head injury, excessive alcohol intake, withdrawal from alcohol or drugs, hypoglycaemia, electrolyte disturbance, brain infection, ischaemic stroke, intracranial haemorrhage, and proconvulsive drugs (such as clozapine, maprotiline, tramadol, theophylline, baclofen). Seizures associated with reversible metabolic or toxic disturbances are associated with a minor risk of subsequent epilepsy (≤ 3% based on large case series). Those provoked by disorders that cause permanent damage to the brain, such as brain abscess, have a higher risk of recurrence (≥ 10%). Seizures that follow severe psychological stress or considerable sleep deprivation are not considered “acute symptomatic” but instead “triggered” by these factors in susceptible individuals with an underlying epilepsy disorder. Rarely, seizures are triggered by specific stimuli such as stroboscopic lights or reading. These reflex epilepsies can rarely be diagnosed with the first seizure, although identifying specific triggers may assist treatment for those with recurrences. What investigations are needed? A practice parameter noted little justification for routine investigations of blood, urine, and cerebrospinal fluid in children; however, the circumstances of a first seizure should direct investigations.11 For example, a child with insulin dependent diabetes must be assessed for hypoglycaemia, while an adult with fever and headache is a candidate for a lumbar puncture to exclude encephalitis. If a first seizure is unprovoked, large case series support the value of electroencephalography (EEG), and often magnetic resonance imaging (MRI), to identify the cause (box 1).11,12 Such images cannot be used to diagnose the event—the diagnosis can only be made from the patient's history. The value of EEG is to point to focal lesions (especially localised slow waves), predict recurrence (see below), and indicate a specific epilepsy syndrome (spike pattern). When performed within 24-48 hours of a first seizure EEG shows substantial abnormalities in about 70% of cases.7,13 The yield may be lower with longer delays after the seizure. When standard EEG is negative, systematic case series have shown that sleep deprived EEG will detect epileptiform (spike) discharges in an additional 13-31% of cases.7,13 Sleep deprived EEG may be carried out in any routine EEG laboratory. While not always available, MRI is the best method for structural imaging. Several case series comparing it with computed tomography in the same patient indicate that the latter may not detect small tumours or other subtle pathologies (figure).7 After a first seizure, abnormalities detected by MRI that lead directly to intervention are more common in adults than children.14 In a series of 166 adults with a first seizure, the most common aetiologies diagnosed with both computed tomography and MRI were cerebrovascular lesions (26%), brain tumours (12%), traumatic scar formations (5%), and other conditions (4%).15 Subcortical vascular encephalopathy itself is also associated with an increased risk for seizures.16 In elderly people, a first seizure may be caused by a silent stroke only recognisable by MRI. Figure 1. Open in a new tab A 44 year old woman with a first seizure had an apparently normal computed tomogram (left), whereas the corresponding magnetic resonance image (right) was obviously pathological, revealing a right hemispheric glioma—and showing the superiority of MRI for structural imaging If the seizure was unprovoked, does the person have an epilepsy syndrome? Once an acute provoking cause has been excluded, the next step is to decide if the first seizure indicated a focal or generalised epilepsy syndrome—a critical distinction if drug treatment is considered. An epilepsy syndrome can be diagnosed after one seizure, even though a single seizure is insufficient for the diagnosis of epilepsy.7 The diagnosis of epilepsy addresses recurrence risk, whereas epilepsy syndrome is a broader concept encompassing age of onset, aetiology, prognosis, and response to treatment. For example, a child with a first nocturnal seizure and typical EEG spikes can be diagnosed as having benign rolandic epilepsy, a disorder of genetic aetiology that constitutes 15% of childhood epilepsy and nearly always remits. In a prospective study of 300 older children and adults with a first seizure a syndrome diagnosis could be made in 80%: clinical details plus family history allowed diagnosis in 47%, EEG allowed diagnosis in an additional 30%, and plus MRI allowed diagnosis in another 4%.7 What is the recurrence risk and what sorts of activity restrictions are needed? People with a first seizure may cope more successfully once they understand the issues. Although not systematically studied, it seems intuitively correct that avoiding provoking or triggering factors should reduce recurrences. For example, a university student with a first seizure after studying all night would be best to avoid sleep deprivation. A meta-analysis concluded that the risk of recurrence after a first unprovoked seizure was 42% over the next two years.17 The significance of two definite unprovoked seizures within 24 hours is uncertain. One prospective study suggested that these two attacks should be viewed as a single, first seizure,5 whereas another concluded they should be viewed as separate events, permitting a diagnosis of epilepsy.18 Meta-analysis of case series17 shows that about 60-70% of recurrences are within six months of the first seizure, with an exponential decrease in risk thereafter. The strongest risk factors for recurrence are aetiology (pre-existing brain abnormalities indicate “remote symptomatic” epilepsy) and EEG abnormalities, especially focal spikes (box 2).3,13,17 We suggest that restrictions to recreational activity after a first untreated, unprovoked seizure should be individualised and limited to two or three months for children and adults.19 It seems likely, but unproved, that swimming, scuba diving, and climbing carry a higher risk for injury than do cross-country skiing, long distance running, or soccer. Individuals should probably be suspended from working with dangerous machines for at least six months. Laws regarding the suspension of a driving license after a first seizure vary between countries from no restriction to one year. In the United Kingdom the right to drive is granted by the Driving and Vehicle Licensing Authority. Non-commercial driving is permitted after one year's freedom from seizures after an unprovoked seizure and on a case-by-case basis for provoked seizures. Commercial driving after an unprovoked seizure is usually not permitted until 10 years' freedom from seizure with antiepileptic drug treatment. Box 2: Reported risk factors for seizure recurrence Remote symptomatic aetiology (pre-existing static brain abnormalities that are, by implication, causative) Focal neurological findings Focal seizure phenomenology (including Todd's paresis) Focal or generalised epileptiform activity on EEG Tumours or other progressive lesions as the underlying pathology Status epilepticus Family history of epilepsy Previous febrile seizures Among neurologists there is a growing consensus that non-commercial drivers with a first unprovoked seizure should stop driving only for three to six months, especially those with favourable prognostic factors. If a first seizure was acute symptomatic, then most patients should be able to drive within three months. Commercial drivers with an unprovoked seizure should be subject to a more restrictive rule (such as at least two years seizure-free without medication).20 Are antiepileptic drugs needed after a first seizure? Drug treatment after a first seizure is controversial.21-24 A practice parameter about first seizures in children concluded that antiepileptic drugs decrease but do not eliminate seizure recurrence and have no effect on long term remission.23 Two large recent randomised studies of children and adults compared antiepileptic drugs with no treatment after a first seizure and came to an identical conclusion.22,23 Any decision to start treatment must weigh the risk of another seizure against the risks of side effects from chronic drug treatment.21-23 Treatment may be justified when the risk of recurrence is high, such as with a focal structural brain deficit and corresponding EEG epileptiform activity (as after a stroke or brain abscess); when the risk of injury from a recurrent seizure is high (such as for those with a spinal cervical fracture, with severe osteoporosis, or taking anticoagulants); or when the risk of economic hardship from a recurrence is high (such as loss of employment). If drug treatment is considered, which drug is preferred? If drug treatment is considered after a first seizure, the chosen antiepileptic drug should have high efficacy, long term safety, good tolerability, and low interaction potential and allow a good quality of life, especially since half of all patients would never have another seizure without treatment. The starting dose should be in the lower range. Phenytoin and barbiturates should be avoided because of neurotoxic and cognitive side effects. If an underlying epilepsy syndrome has been established, the following antiepileptic drugs are available (listed alphabetically because there are no available comparative trials after a first seizure): For focal seizures—carbamazepine, clobazam (especially children), gabapentin, lamotrigine, oxcarbazepine, topiramate, valproate For generalised seizures—lamotrigine, topiramate, valproate. Drug choice should be individualised, and consideration given to factors such as teratogenicity, the patient's cognitive abilities, drug interactions, the doctor's familiarity with the drug, and cost. Box 3: Steps for the family doctor On the basis of the history and physical examination, be sure that the event was a first seizure Exclude acute provoking factors by history and screening laboratory tests Arrange electroencephalography and magnetic resonance imaging (if available) Review on an individual basis the risk of a recurrence and the potential social and psychological consequences of a recurrent seizure Review restrictions for the person's activities, especially for driving For unprovoked seizures, discuss but usually do not prescribe antiepileptic drug treatment Seek expert consultation for diagnosis of epilepsy syndrome and management of provoking factors How long should drug treatment be continued? In childhood epilepsy (as opposed to first seizure) drug treatment is usually continued until the child has been free of seizures for one to two years. If a child starts drug treatment after a first seizure, there is little justification for continuing treatment beyond one year free from seizures, except in the case of a few epilepsy syndromes, such as juvenile myoclonic epilepsy, that usually require long term treatment. There are no published data to guide length of treatment after a first seizure in adults. Each case must be viewed individually, including consideration of the medical and social consequences of another seizure. It is tempting to use EEG and neuroimaging to help with this decision because persistent EEG abnormalities, and a documented aetiology, are associated with a higher risk of relapse when antiepileptic drugs are withdrawn after several years of remission (affirmed by a meta-analysis).25 It would seem prudent for adult patients to decide the parameters for discontinuing before starting treatment. If drug treatment is started after a first seizure in adults, we suggest at least one year of treatment, except for those at low risk for recurrence, when six months without seizures may be sufficient. Information sources for patients Epilepsy Action (epilepsy.org.uk/)—Sponsored by the British Epilepsy Association, this site provides information about many aspects of epilepsy for patients of all ages DVLA. Medical rules. Chapter 1: neurological disorders (dvla.gov.uk/at_a_glance/ch1_neurological.htm)— Outlines the regulations for a driving permit for people in the UK with one or more seizures, provoked or unprovoked Epilepsy.com (www.epilepsy.com)—This US based website includes information written by international contributors for both healthcare providers and patients Epilepsy Foundation (epilepsyfoundation.org)—This site is sponsored by the American Epilepsy Foundation International League Against Epilepsy (ilae-epilepsy.org)— Provides some direct information, points out other educational material, and directs patients to local organisations for additional information and support Conclusions A first seizure means an uncertain future for the individual, but the consequences of a recurrence vary between individuals in different geographical areas and social situations. We agree with a practice parameter that treatment decisions must take into account medical issues and patient and family preference.23 The ultimate goal of assessment and treatment is to optimise quality of life and achieve a good balance between feeling almost healthy and yet practising some caution for at least a year. Hopefully, individualised coping strategies will be improved by careful counselling (box 3). Contributors: BP-E and PC developed the first draft of the paper. Revisions were undertaken by PC, taking into account several critical reviews by CC and EB. The final manuscript was carefully reviewed and approved by all authors. PC is guarantor for this review. Competing interests: None declared. References 1.Hauser WA, Rich SS, Annegers JF, Anderson VE. Seizure recurrence after a 1st unprovoked seizure: an extended follow-up. Neurology 1990;40: 1163-70. [DOI] [PubMed] [Google Scholar] 2.Hauser WA, Annegers JF, Kurland LT. 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[DOI] [PubMed] [Google Scholar] Articles from BMJ : British Medical Journal are provided here courtesy of BMJ Publishing Group ACTIONS View on publisher site PDF (98.7 KB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Introduction Methods Is it a seizure? The “first” seizure may not be the first What has provoked the first seizure? What investigations are needed? If the seizure was unprovoked, does the person have an epilepsy syndrome? What is the recurrence risk and what sorts of activity restrictions are needed? Are antiepileptic drugs needed after a first seizure? 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1668	https://pcbprime.com/pcb-tips/how-thick-is-1oz-copper/	Home Sign In Register Products PCB Tips About FAQ Referral How To Order Quote Contact 800.791.5370 ● 800.791.5370 Sign In Register PCBPRIME PCB-TIPS Accepted File Formats Array Design CAM Tooling Guidelines Controlled Dielectric or Controlled Impedance? Countersink vs Counterbore Default Rigid PCB Specs Dielectric Stacks Drilling and Drill File Fabrication Drawing/Fab Print Final Finish Comparison How Thick Is 1oz Copper? How to Avoid Engineering CAM Hold Minimizing Bow and Twist Solder Mask Via Tenting, Plugging, and Filling What is Copper Thieving? How Thick Is 1oz Copper? In the printed circuit board industry, the most common way to express copper thickness on a PCB is in ounces (oz). Why use a unit of weight to specify a thickness? Great question! If 1oz (28.35 grams) of copper is flattened to evenly cover 1 square foot of surface area (0.093 square meter), the resulting thickness will be 1.37mils (0.0348mm). A conversion chart for different units of measure can be found below. Copper Thickness Conversion Chart \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| oz \| 1 \| 1.5 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| \| mils \| 1.37 \| 2.06 \| 2.74 \| 4.11 \| 5.48 \| 6.85 \| 8.22 \| 9.59 \| 10.96 \| 12.33 \| \| inch \| 0.00137 \| 0.00206 \| 0.00274 \| 0.00411 \| 0.00548 \| 0.00685 \| 0.00822 \| 0.00959 \| 0.01096 \| 0.01233 \| \| mm \| 0.0348 \| 0.0522 \| 0.0696 \| 0.1044 \| 0.1392 \| 0.1740 \| 0.2088 \| 0.2436 \| 0.2784 \| 0.3132 \| \| µm \| 34.80 \| 52.20 \| 69.60 \| 104.39 \| 139.19 \| 173.99 \| 208.79 \| 243.59 \| 278.38 \| 313.18 \| How much Copper do I need? By a wide margin, most PCBs are made with 1oz copper on each layer. If your files do not include a fab print or other specifications, we'll assume 1oz finished copper weight on all copper layers. If your design requires higher voltages, resistance, or impedances, thicker copper might be necessary. There are several online tools that can help you determine what thickness, width or length your traces need to be to achieve your target results. A few such 3rd party tools are linked below. PCB Prime is not affiliated with the authors of these tools. TIP: Every copper layer is assumed to have the same finished copper weight unless otherwise specified in your fabrication notes. A 4-layer board with a defined weight of 1oz, is assumed to have a finished copper thickness of 1.37mils (or more) on every copper layer. Copper Weight Minimum Spacing Rules The thicker your copper requirements are, the more spacing is required between copper features on your PCB. \| \| \| --- \| \| CuWeight \| Min. Recommended Space between Copper Features and Min. Trace width \| \| 1oz \| 3.5 mil (0.089mm) \| \| 2oz \| 8 mil (0.203mm) \| \| 3oz \| 10 mil (0.254mm) \| \| 4oz \| 14 mil (0.355mm) \| TIP: The spacing shown in this chart is to be used as a general guide. Different factories will have slightly different capabilities. This should give you a general idea of what min spacing and min trace width to target when setting your design rules. The more spacing you can give between copper features the better. It is easier to print narrow traces than it is to etch a narrow gap between them, but for consistency, use the same dimensions for spacing and trace widths for your design rules. Copper Distribution should be as even as possible As a general rule, copper should be distributed as evenly as possible throughout your design. Not only in regard to the copper thickness on each layer, but also how it's distributed across the layer. Of course, this isn't always possible, but keep this in mind during layout. Plating and etching are organic processes in the sense that the copper clad laminate is submerged into a vat of chemicals for processing. There isn't precise control over where the copper is removed from or plated onto. During etch, the intended image is masked off to protect it from the etchant, but the chemicals in the tank dissolve the copper at slightly different rates depending on where the features are on the panel, the panel's placement within the tank itself, and how densely or sparsely the copper features are distributed. The chemical solution in the plating and etching tanks are agitated and circulated during processing to minimize these inconsistencies; however, a panel with drastically different copper densities can prove problematic. During your design phase, try to evenly distribute your copper across the entire board rather than having large open spaces with isolated features. Can I have more copper on one layer than another? Yes. It's common for there to be heavier copper on some layers than others, for example having 3oz copper on outer layers but 1oz copper on the inner layers, or vice versa. Best design rule practice calls for the same copper weight to be used on the mirror opposite layer in the stack up. Take a 4-layer board for example: layers 1 and 4 can be 2oz and layers 2 and 3 can be 1oz. This gives a balanced stack, reducing the chance the board will bow or twist when heated during use or under the high temperatures used during assembly. Outer layers should always have the same copper weight. For example, if the top copper layer is to have a finished copper weight of 2oz, then the bottom layer should also be 2oz. If a different copper weight is specified for top and bottom, say 2oz on top and 1oz on bottom, then the amount of time required to etch through the top side copper will be longer than needed for the bottom. This would result in over etching the bottom layer. Each side would have to be processed independently adding complexity and time to the process. Many factories will not attempt this so avoid this unless necessary. Not only will it add cost, and reduce the number of factories capable of manufacturing the board, but it will also increase the likelihood your board will bow or twist. Click for more information about copper distribution and thieving.
1669	https://www.wyzant.com/resources/answers/496737/1_3_9_27_81_sequence	Log in Sign up Search Search Find an Online Tutor Now Ask Ask a Question For Free Login Geometery Simialr Triangle Geometer Joseph J. asked • 06/22/18 1,3,9,27,81 sequence Use inductive reasoning to find the next two terms in sequence Follow • 2 Add comment More Report 1 Expert Answer By: Mark M. answered • 06/22/18 Tutor 5.0 (278) Mathematics Teacher - NCLB Highly Qualified See tutors like this See tutors like this 1 3 = 3 3 3 = 9 9 3 = 27 27 3 = 81 Can you continue the pattern and answer? Upvote • 0 Downvote Add comment More Report Still looking for help? Get the right answer, fast. Ask a question for free Get a free answer to a quick problem.Most questions answered within 4 hours. OR Find an Online Tutor Now Choose an expert and meet online. No packages or subscriptions, pay only for the time you need. RELATED TOPICS Math Chemistry Calculus Algebra 1 Algebra Algebra 2 Geometry Precalculus Physics Other ... Biology Statistics English Trigonometry Mathematics Math Help Word Problem Math Word Problem Probability Science RELATED QUESTIONS ##### Given that the two triangles are similar, solve for x if AU = 20x + 108, UB = 273, BC = 703, UV = 444, AV = 372 and AC = 589. Answers · 1 RECOMMENDED TUTORS Andrew S. 5.0 (432) Evan W. 5.0 (2,078) JD P. 5.0 (751) See more tutors find an online tutor Geometry tutors Trigonometry tutors Abstract Algebra tutors Complex Analysis tutors Linear Algebra tutors Linear Programming tutors Discrete Math tutors Boolean Algebra tutors
1670	http://teach.files.bbci.co.uk/skillswise/ma02symb-l1-f-writing-big-numbers.pdf	Writing big numbers N1/L1.1 © BBC 2011 We come across large numbers in the news everyday and it might be helpful to be able to read them. To help with reading and writing numbers, and for calculating, we use place value. Place value is the idea that a figure has a different value when used in different places. Below is a place value table with the numbers 7,853 and 5,387. Note: each column can only contain one figure from 0 to 9. Thousands Hundreds Tens Units Th H T U 7 8 5 3 5 3 8 7 In the number 7,853 (seven thousand eight hundred and fifty three) the 7 has the value 7 thousand. This number is 7,000 + 800 + 50 + 3. In the number 5,387 (five thousand three hundred and eighty seven) the 7 has the value 7 units. This number is 5,000 + 300 + 80 + 7. In these two numbers the 7 stands for different values when it is in different places.
1671	https://math.stackexchange.com/questions/3037914/how-to-find-the-vector-equation-of-the-line-segment	calculus - How to find the vector equation of the line segment. - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more How to find the vector equation of the line segment. Ask Question Asked 6 years, 9 months ago Modified6 years, 9 months ago Viewed 1k times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. Let C C be the line segment from (0,0)(0,0) to (2,2)(2,2), and let f(x,y)=x 2+y f(x,y)=x 2+y. Write down a vector equation r(t)r(t) of the line segment, that is, find a parametrization of C C. The answer given is r(t)=⟨t,t⟩,0≤t≤2 r(t)=⟨t,t⟩,0≤t≤2. My Question: How did they get r(t)=⟨t,t⟩r(t)=⟨t,t⟩? calculus multivariable-calculus parametrization line-integrals Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications edited Dec 13, 2018 at 12:09 Rócherz 4,271 4 4 gold badges 15 15 silver badges 30 30 bronze badges asked Dec 13, 2018 at 12:03 user982787user982787 341 1 1 gold badge 4 4 silver badges 13 13 bronze badges 1 What does f f have to do with anything?amd –amd 2018-12-13 21:55:11 +00:00 Commented Dec 13, 2018 at 21:55 Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. The line going from (0,0)(0,0) to (2,2)(2,2) has equation y=x y=x, so a parametrization could be x(t)y(t)==t x(t)=t x(t)=t y(t)=x(t)=t For 0≤t≤2 0≤t≤2 a parametrization of C C is r(t)=(t,t)for 0≤t≤2 r(t)=(t,t)for 0≤t≤2 Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Dec 13, 2018 at 12:33 Rócherz 4,271 4 4 gold badges 15 15 silver badges 30 30 bronze badges answered Dec 13, 2018 at 12:16 caveraccaverac 19.8k 3 3 gold badges 22 22 silver badges 41 41 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. A (non-uniquely) parametrized straight line segment that goes from a startpoint (x 0,y 0)(x 0,y 0) to an endpoint (x 1,y 1)(x 1,y 1) is given in vector form as ⟨x,y⟩=⟨x 0,y 0⟩+λ⟨x 1−x 0,y 1−y 0⟩,0≤λ≤1.⟨x,y⟩=⟨x 0,y 0⟩+λ⟨x 1−x 0,y 1−y 0⟩,0≤λ≤1. After plugging in your numbers, you end up with ⟨x,y⟩=λ⟨2,2⟩=⟨2 λ,2 λ⟩⟨x,y⟩=λ⟨2,2⟩=⟨2 λ,2 λ⟩. Letting t:=2 λ t:=2 λ gives the desired parametrization, but that last step is more of a personal choice. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Dec 13, 2018 at 12:38 RócherzRócherz 4,271 4 4 gold badges 15 15 silver badges 30 30 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions calculus multivariable-calculus parametrization line-integrals See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Related 1Vector equation and parametric equation for a line segment 1Line integral ∫C(3 x+sin(y))d s∫C(3 x+sin⁡(y))d s where C C is line segment from (1,2)(1,2) to (5,4)(5,4) 1Evaluate a line integral using the fundamental theorem of line integrals 1Flux Through a Closed Curve - Orientation 1Integration along a line segment 4Equation of a line using parameters θ θ and ρ ρ or r r 0Line Integrals of Vector Fields, Homework Conundrum 1Vector equation of a line segment Hot Network Questions Gluteus medius inactivity while riding Origin of Australian slang exclamation "struth" meaning greatly surprised Does "An Annotated Asimov Biography" exist? 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1672	https://medium.com/the-horizon-explorer/understanding-discrete-convolution-as-polynomial-multiplication-5f94a13acd55	Sitemap Open in app Sign in Sign in ## The Horizon Explorer · The Horizon Explorer is a publication for the curious mind, delving into the latest advancements in software development, AI, machine learning, and data science. We go beyond the surface to uncover the code, the concepts, and the people shaping the future of technology. Understanding Discrete Convolution as Polynomial Multiplication Convolution is a fundamental operation in digital signal processing Shailesh Kumar 4 min readAug 24, 2021 Convolution is a fundamental operation in digital signal processing. It is usually defined by the formula: DSP books start with this definition, explain how to compute it in detail. We learn how convolution in the time domain is the same as multiplication in the frequency domain via Fourier transform. The operation of finite and infinite impulse response filters is explained in terms of convolution. This becomes the foundation for all digital filter designs. However, the definition of convolution itself remains somewhat esoteric. For me, this definition was always a bit magical until I studied polynomial algebra carefully. Then, the realization happened about how this definition arises naturally. It has remained with me since then. My sister asked me today to explain the convolution operation. Initially, I tried to explain with the typical DSP approach and it was not going anywhere. Finally, I reverted to the polynomial multiplication approach and it became obvious to her. So here it comes. Get Shailesh Kumar’s stories in your inbox Join Medium for free to get updates from this writer. Consider two simple polynomials: Compute their product: If we write h(x) as then we can match the coefficients of h(x) as: A pattern seems to be emerging. Let’s get more adventurous with second-degree polynomials: Here is their product: h(x) now has a degree of 4 with the coefficients: given by the identities: It is time to generalize. We will now look at two polynomials f(x) with degree J and g(x) with degree L. Their product is the polynomial h(x) with degree J + L: where the polynomial coefficients are given by the relationship: with the assumption that f_k and g_k are assumed to be zero for the indices for which they are not specified. i.e. both f_n and g_n are zero for negative n. f_n is 0 for n > J and g_n is zero for n > L. Reader is advised to check that this definition of h_n aligns with the previous results for multiplications for degree 1 and degree 2 polynomials. In fact, if we disregard the issue of convergence of an infinite sum, then with these assumptions of polynomial coefficients being zero outside the indices [0,J] and [0,K] for f(x) and g(x) respectively, there is nothing wrong in writing this formula as: We can now form the (finite) sequences f[n], g[n], h[n] for the coefficients of the polynomials f(x), g(x) and h(x) and define the convolution operation as: where the definition has arisen out of treating the sequences as coefficients of the corresponding polynomials. Summary: If we treat digital sequences as coefficients of polynomials, then their convolution is nothing but the product of corresponding polynomials. A note on convergence: In general, convergence is a nonissue for finite sequences. However, it becomes relevant for IIR filters as their impulse response has an infinite number of terms. From real analysis, it can be shown that the convolutional sum will converge if the sequences f_n and g_n are absolutely summable. i.e. if the sums: converge, then their convolution also converges for every n. Convolution Digital Signal Processing Image Processing Mathematics ## Published in The Horizon Explorer 1 follower ·Last published Sep 20, 2025 The Horizon Explorer is a publication for the curious mind, delving into the latest advancements in software development, AI, machine learning, and data science. We go beyond the surface to uncover the code, the concepts, and the people shaping the future of technology. ## Written by Shailesh Kumar 135 followers ·207 following Python \| JavaScript \| Web Applications \| Math \| Statistics Responses (1) Write a response What are your thoughts? Cdchavezq Feb 15, 2024 ``` Very nice article ! ``` More from Shailesh Kumar and The Horizon Explorer In The Horizon Explorer by Shailesh Kumar ## Testing in JavaScript: A Gentle, Hands-On Guide As a developer, you’re constantly asking yourself, “How do I know if my code works?” You’ve probably used console.log statements so many… Sep 18 In The Horizon Explorer by Shailesh Kumar ## Embedded Linux: A Comprehensive Overview Embedded Linux refers to a customized Linux operating system tailored for embedded devices. Unlike general-purpose Linux distributions… Sep 2 Shailesh Kumar ## Do People Hate Turning on Cameras in Virtual Meetings? I have often seen that during meetings, many people keep their cameras off. I also prefer keeping it off more often than not. I have an… Sep 1, 2024 2 In The Horizon Explorer by Shailesh Kumar ## Finding a Unique Machine ID on Macs Sometimes, during software development, I need a way to identify a machine uniquely. This may be required for purposes such as software… Sep 3 See all from Shailesh Kumar See all from The Horizon Explorer Recommended from Medium In Long. Sweet. Valuable. by Ossai Chinedum ## I’ll Instantly Know You Used Chat Gpt If I See This Trust me you’re not as slick as you think May 16 24K 1440 Tomio Kobayashi ## The Dual Nature of the Choose Function: Permutation and Convolution Perspectives Sep 19 In ImageCraft by Francisco Zavala ## Coordinate Spaces in OpenGL: Frames of Reference for Creating 3D Graphics Explore how OpenGL uses multiple coordinate spaces to transform vertices from models to pixels on the screen Sep 18 2 Abhinav ## Docker Is Dead — And It’s About Time Docker changed the game when it launched in 2013, making containers accessible and turning “Dockerize it” into a developer catchphrase. Jun 8 6.6K 182 Jordan Gibbs ## ChatGPT Is Poisoning Your Brain… Here‘s How to Stop It Before It’s Too Late. 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1673	https://www.laccd.edu/sites/laccd.edu/files/2024-05/2023-2024%20Final%20Budget%20-%20updated%20TOC%20links_May%2031%2C%202024.pdf	Final Budget 2023-2024 Office of the Chancellor September 2023 Los Angeles Community College District Los Angeles Community College District 2023-2024 Final Budget Los Angeles Community College District Board of Trustees David Vela, President Nichelle Henderson, 1st Vice President Kelsey K. Iino, Ed.D., 2nd Vice President Gabriel Buelna, Ph.D. Sara Hernandez, J.D. Andra Hoffman Steven F. Veres Alexy Cordova, Student Trustee District Administration Francisco C. Rodriguez, Ph.D., Chancellor Kathleen Burke, Ed.D., Interim Deputy Chancellor Nicole Albo-Lopez, Ed.D., Vice Chancellor Educational Programs and Institutional Effectiveness Jeanette L. Gordon, Vice Chancellor/Chief Financial Officer Carmen V. Lidz, MS, Vice Chancellor/Chief Information Officer Maribel S. Medina, J.D., General Counsel Rueben C. Smith, D.C.Sc., Vice Chancellor/Chief Facilities Executive James Lancaster, Ed.D., Vice Chancellor, Workforce Development Teyanna Williams, J.D., Vice Chancellor, Human Resources College Presidents Amanuel Gebru, Ed.D., Los Angeles City College Albert J. Román, DPA, East Los Angeles College Luis Dorado, Ed.D., Los Angeles Harbor College Armida Ornelas, Ph.D., Los Angeles Mission College Aracely Aguiar, Los Angeles Pierce College Anthony J. Culpepper, J.D., Los Angeles Southwest College Alfred McQuarters, Ed.D., Los Angeles Trade-Technical College Barry Gribbons, Ph.D., Los Angeles Valley College James M. Limbaugh, Ph.D., West Los Angeles College Interim Prepared by Office of Budget and Management Analysis City East Harbor Mission Pierce Southwest T rade-Tech Valley West Office (213) 891-2201 Fax (213) 891-2304 laccd.edu 770 Wilshire Blvd. Los Angeles, CA 90017 O f f i c e o f t h e Chancellor September 13, 2023 The Honorable Members of the Board of Trustees Los Angeles Community College District In accordance with Section 58305(c) of Title 5, California Code of Regulations, presented herein is the District’s 2023-2024 Final Budget for your consideration and approval. On July 10, 2023, Governor Gavin Newsom signed the 2023 Budget Act, totaling $310.8 billion. The Budget Act, plus associated trailer bills, provides additional resources of roughly $308.6 million to the California Community Colleges’ apportionments and categorical programs. The enacted 2023-2024 Budget Act is reflective of the state’s projected budget deficit of $31.5 billion, as State revenue projections face increased uncertainty due to the extension of the tax filing deadline to October and the increasing interest rate environment. As such, language has been included in the Budget Act that allows the Governor to delay one-time spending commitments in the event of further revenue declines. The 2023-2024 Budget Act reflects stable total funding for the California Community Colleges, and includes $678 million for a Cost of Living Allowance (COLA) of 8.22 percent for the SCFF, $97.4 million for COLA of 8.22% for certain categorical programs, $14 million in one-time funding for Workforce Training Grants, $4.2 million for Equal Opportunity Programs, all offset by a $490 million net decrease to prior year deferred maintenance funding, as well as a $5.4 million net prior year decrease in retention and enrollment funding. Of particular note for LACCD is the inclusion of $2.5 million in funding for the ELAC Entrepreneurship and Innovation Center and one-time funding of $10 million per year, for three years, for the LGBTQ+ Pilot Program, both results of strong and effective LACCD advocacy efforts. The enacted budget reaffirms the extension of the hold harmless provision in a modified form. The District’s 2024-25 funding will represent its new “floor,” below which state apportionment cannot drop. Starting in 2025-26, LACCD will be funded at the SCFF generated amount that year or our "floor” (2024-25 funding amount), whichever is higher. The District’s 2023-2024 Final Budget is based on the Student-Centered Funding Formula calculated revenue guarantee of $741.9 million, plus COLA of 8.22 percent. In 2022-2023, the Board of Trustees approved an update to the District Allocation Model that better aligns with our equity minded approach. The allocation generated by this updated model was used to develop this year’s Final Budget. 2023-2024 Final Budget September 13, 2023 Page 2 The development of the District’s 2023-2024 Final Budget has been an evolving process beginning with Governor Gavin Newsom’s proposed State Budget in January 2023 and the State Budget Development process through June 2023. The Final Budget also includes information submitted by each of the Colleges and the Educational Services Center. Each college, through its local participatory governance process, sets its own local budget priorities to meet its institutional goals and objectives, and is responsible for balancing its annual budget 2023 and the State Budget Development process through June 2023. The Final Budget also includes information submitted by each of the Colleges and the Educational Services Center. Each College, through its local participatory governance process, sets its own local budget priorities to meet its institutional goals and objectives, and is responsible for balancing its annual budget. The District’s 2023-2024 Final Budget of $10.4 billion for all funds reflects the following major budget areas: • Building Fund (Prop. A, AA, J and Measure CC) $ 8.5 billion • Unrestricted General Fund $ 1.1 billion • Student Financial Aid Fund $ 260.1 million • Restricted General Fund (categorical and specially funded) $ 434.2 million • Special Reserve Fund (State Funded Capital Outlay Projects) $ 152.4 million • Bookstore Fund $ 30.0 million • Debt Services Fund $ 7.6 million • Child Development Centers Fund $ 14.0 million • Cafeteria Fund $ 4.9 million The District continues to maintain a minimum 6.5% General Reserve and a 3.5% Contingency Reserve, and a 2.0% Deferred Maintenance Fund, as mandated by LACCD Board Policy. For 2023-2024, the District’s financial outlook remains stable under the current, state-adopted Student-Centered Funding Formula (SCFF), thanks to the current hold harmless provision. The District remains focused on increasing enrollments, retention and graduation of students, providing accessible, affordable higher education through our colleges, and maximizing the three funding streams of the SCFF that includes college access, equity and student success. In the long term, it is imperative for the District to increase enrollments, bolster access and equity, retain and graduate more students through improved course completions and the number of degrees and certificates granted, as well as generating more transfer students and achieving other progression and momentum points tied to student success. Your attention is directed to the Overview section of this document that presents a more detailed discussion of the State’s fiscal environment, and the District’s current revenue projections and planned expenditures. Respectfully, Francisco C. Rodriguez, Ph.D. Chancellor Los Angeles Community College District 2023-2024 Final Budget Table of Contents Executive Summary .............................................................................................................................. 1 Overview ............................................................................................................................................... 2 I.Summary Summary of All Funds ......................................................................................................................... 13 II. General Fund Income ................................................................................................................................................ 17 Unrestricted General Fund Appropriations Unrestricted General Fund Totals by Sub-Major Commitment Item ........................................... 23 Unrestricted General Fund Totals by Major Functional Area ...................................................... 24 Unrestricted General Fund by Sub-Major Commitment Item (repeated for each college) .......... 25 Unrestricted General Fund by Major Functional Area (repeated for each college) ..................... 26 Educational Services Center ....................................................................................................... 43 Information Technology .............................................................................................................. 45 Districtwide Accounts .................................................................................................................. 48 Unrestricted General Fund Historical Perspective ............................................................................. 48 Restricted General Fund Appropriations Restricted General Fund by Sub-Major Commitment Item ......................................................... 59 Restricted General Fund Appropriations by Program ................................................................. 60 Restricted General Fund Appropriations by Fund and Location ................................................. 61 CA Adult Education Program (CAEP) ......................................................................................... 62 CA College Promise ................................................................................................................... 62 CalWORKs (Child Care/Non-Child Care)/TANF ......................................................................... 63 Community Services ................................................................................................................... 63 Cooperative Agencies Resources for Education (CARE) ........................................................... 64 Disabled Student Programs & Services (DSPS) ......................................................................... 64 Dream Resource Liaison Support ............................................................................................... 65 Equal Employment Opportunity .................................................................................................. 65 Extended Opportunities Programs & Services (EOPS) .............................................................. 66 Federal Perkins IV (CTE) ............................................................................................................ 66 Federal Work Study .................................................................................................................... 67 Financial Aid Technology ............................................................................................................ 67 Foster & Kinship Care Education (FKCE) ................................................................................... 68 Framework for Racial Equity & Social Justice ............................................................................. 68 Health Services ........................................................................................................................... 69 Higher Education Emergency Relief Fund I (HEERF I)............................................................... 69 Higher Education Emergency Relief Fund II (HEERF II) ............................................................. 70 Higher Education Emergency Relief Fund III (HEERF III) ........................................................... 70 Los Angeles Community College District 2023-2024 Final Budget HEERF Minority Serving Institutions (HEERF MSI) Supplement ................................................ 71 HEERF Supplemental Assistance to Institutions of Higher Education – SAHIE ......................... 71 Higher Ed Emergency MSI ......................................................................................................... 72 Lottery – Prop 20 ........................................................................................................................ 72 NextUp ........................................................................................................................................ 73 One-Time Block Grants .............................................................................................................. 73 Parking ........................................................................................................................................ 74 Staff/Faculty Development .......................................................................................................... 74 Strong Workforce ........................................................................................................................ 75 Student Equity & Achievement (SEA) ......................................................................................... 75 Student Financial Aid Administration (SFAA) .............................................................................. 76 Student Retention and Enrollment .............................................................................................. 76 Unrestricted Indirects .................................................................................................................. 77 Veterans Resource Center ......................................................................................................... 77 Other Specially Funded Programs (SFP) .................................................................................... 78 Restricted General Fund Programs ............................................................................................ 79 General Fund Appropriations ............................................................................................................ 100 General Fund Summary ............................................................................................................ 100 General Fund Summary by Sub Major Commitment (repeated for each college) ..................... 101 Educational Services Center ..................................................................................................... 110 III. Other Funds Bookstore Fund ................................................................................................................................. 111 Building Fund .................................................................................................................................... 113 Cafeteria Fund .................................................................................................................................. 115 Child Development Fund .................................................................................................................. 117 Debt Service Fund ............................................................................................................................ 119 Special Reserve Fund ....................................................................................................................... 121 Student Financial Aid Fund ............................................................................................................... 142 IV. Appendices Appendix A: Definitions & Notes ....................................................................................................... 144 Appendix B: Districtwide Accounts .................................................................................................... 145 Appendix C: Budgeted Positions ....................................................................................................... 149 Appendix D: List of Active Organizational Memberships ................................................................... 165 Appendix E: Education Protection Act (EPA) Fund 10106 ................................................................ 182 Appendix F: Final Budget Allocation Mechanism .............................................................................. 192 Executive Summary Los Angeles Community College District 2023-2024 Final Budget 1 Executive Summary The following is a brief summary of the District's 2023-2024 Final Budget with reference to a more detailed discussion in the body of the report. • The 2023-2024 Final Budget for all funds is $10.4 billion distributed over eight funds - General Fund, Bookstore Fund, Cafeteria Fund, Child Development Fund, Special Reserve Fund (Capital Outlay Project Fund), Building Bond Fund, Student Financial Aid Fund, and Debt Services Fund. • The 2023-2024 General Fund, consisting of restricted and unrestricted programs, is $1.5 billion and represents 14.4% of the total budget. • The Unrestricted General Fund budget, which supports the principal operations of the District, is $1.1 billion and represents 10.2% of the Final Budget. • The 2023-2024 Education Protection Act (EPA) is $49.4 million and is included in the Unrestricted General Fund Revenue (summary; detailed expenditure plans in Appendix E). • The level of Unrestricted General Funds available for appropriation consists of beginning balances, reserve for open orders, transfers, and income. • The Beginning Balance of $187.0 million is $4.2 million greater than the 2022-2023 Beginning Balance. • State General Revenue is projected to increase from 2022-2023. The State Revenue includes the Education Protection (EPA) fund of $49.4 million. • 2023-2024 Unrestricted General Fund appropriations of $1.066 billion are $103.3 million greater than the 2022-2023 Final Budget. Appropriations for the nine colleges are $691.2 million, which is $31.0 million greater than 2022-2023 Final Budget allocations for college locations. • Appropriations in all Other Funds appear to be adequate to maintain planned levels of service. Overview Los Angeles Community College District 2023-2024 Final Budget 2 Overview The Final Budget for fiscal year 2023-2024, summarized in the following pages, has been revised from the Tentative Budget, which was adopted by the Board of Trustees on June 7, 2023, to reflect the State’s budget, the District’s 2022-2023 ending balances, and revised revenue projections generated by the colleges. The significant changes from Tentative Budget are due to 2022-2023 balances and open orders, and other allocation adjustments. The Final Budget totals $10.4 billion for the General Fund and other funds. The budget includes $8.5 billion of Proposition A, AA, and Measure J, CC and LA bond funds to finance construction, provide equipment, and improve college facilities at the various campuses of the District. The budget, as presented for final adoption by the Board of Trustees, is based on the state budget that includes the new Student Centered Funding Formula. The new funding formula includes a hold harmless provision through fiscal year 2024-25. This budget is based on the Student Centered Funding Formula calculated revenue guarantee of $741.9 million state general revenues plus COLA of $61.0 million, which includes $49.4 million for the Educational Protection Fund (Proposition 30/55). The following overview provides information on total funds available for each fund (Chart #1). Los Angeles Community College District 2023-2024 Final Budget 3 Chart 1: Summary of All Funds (In Millions) Funds & Appropriations 2021-22 Actual 2022-23 Final Budget 2022-23 Actual 2023-24 Tentative Budget 2023-24 Final Budget 2022-23 Final Bud Difference 2022-23 Final Bud Difference 2022-23 Actual Difference 2022-23 Actual Difference 2023-24 Tent Bud Difference 2023-24 Tent Bud Difference General Fund Unrestricted total 720.90 962.69 828.60 1,010.75 1,065.99 103.30 10.7% 237.39 28.6% 55.24 5.5% Less Intrafund w/in unres 5.76 0.00 5.06 0.00 0.00 0.00 0.0% -5.06 -100.0% 0.00 0.0% Unrestricted net 715.14 962.69 823.54 1,010.75 1,065.99 103.30 10.7% 242.46 29.4% 55.24 5.5% Restricted 280.19 328.83 257.41 112.89 434.80 105.97 32.2% 177.39 68.9% 321.91 285.2% Less other Intrafund 5.57 1.31 3.57 0.62 0.62 -0.69 -52.6% -2.95 -82.6% 0.00 0.0% Total General Fund 989.75 1,290.21 1,077.38 1,123.02 1,500.17 209.97 16.3% 422.79 39.2% 377.15 33.6% Bookstore Fund 15.06 28.13 17.36 18.84 29.97 1.84 6.5% 12.61 72.6% 11.13 59.1% Cafeteria Fund 0.83 4.05 1.30 1.65 4.85 0.80 19.7% 3.55 273.4% 3.20 194.0% Child Development Fund 12.62 11.48 14.92 2.13 14.01 2.54 22.1% -0.91 -6.1% 11.88 557.1% Special Reserve Fund 16.83 102.06 10.31 119.01 152.38 50.32 49.3% 142.07 1377.8% 33.37 28.0% Building Fund 218.70 3,429.60 274.02 8,511.50 8,473.68 5,044.08 147.1% 8,199.66 2992.3% -37.82 -0.4% Financial Aid Fund 264.35 351.84 229.59 235.18 260.05 -91.78 -26.1% 30.47 13.3% 24.88 10.6% Debt Service Fund 7.08 7.08 7.10 6.92 7.59 0.51 7.2% 0.49 6.9% 0.67 9.6% Total Appropriations 1,525.22 5,224.45 1,631.98 10,018.25 10,442.72 5,218.27 99.9% 8,810.73 539.9% 424.47 4.2% Less Interfund transfers 26.44 25.10 25.96 8.35 26.60 1.50 6.0% 0.64 2.4% 18.24 218.4% Total Available 1,498.78 5,199.35 1,606.02 10,009.90 10,416.12 5,216.77 100.3% 8,810.10 548.6% 406.22 4.1% Note: Interfund Transfers represent a transfer between any two of the funds listed above. In order not to account for the same funds twice, transfers are subtracted from the total. Los Angeles Community College District 2023-2024 Final Budget 4 General Fund The General Fund, which supports the basic operations of the District, totals $1.5 billion and represents 14.4% of the total Final Budget. The General Fund is further divided into 1) unrestricted programs and 2) restricted programs (i.e., federal, state, and local categorical programs). The Unrestricted General Fund, which represents those funds and expenditures over which the District retains the greatest flexibility, totals $1.1 billion or 10.2% of the total budget. This overview will focus primarily on the Unrestricted General Fund and will include a discussion of 1) State Budget Development, 2) Unrestricted General Fund revenue assumptions, 3) Appropriations, and 4) Restricted General Fund. State Budget Development On July 10, 2023, the Governor signed the $310.8 billion 2023-2024 State Budget Act plus additional trailer bills that impact the California Community Colleges. The 2023-2024 State Budget includes a total investment in Proposition 98 of $108.3 million. The 2023-24 budget provides total additional resources of roughly $308.6 million to California Community Colleges apportionments and categorical programs. Highlights of the 2023-2024 State Budget include: 1. COLA – 8.22% or $678.0 million 2. $97.4 million in COLA and adjustments for certain categorical programs 3. $50 million in one-time funding to support retention and enrollment strategies, while reducing prior year funding by $55.4 million 4. $14 million in in one-time funding for Workforce Training Grants 5. $10.0 million in one-time funding for LGBTQ+ Pilot Program 6. $4.2 million in in one-time funding for Equal Employment Opportunity Programs 7. $2.5 million in in one-time funding for the ELAC Entrepreneurship and Innovation Center Funding for existing state-funded categorical programs is currently at 80% to 95% of 2022-2023 guaranteed allocation levels based on the program. Budget adjustments will be incorporated during the year when the State releases these funds. District Budget Development The District’s 2023-2024 budget development began early in November 2022 after the Board adopted the 2023-2024 Budget Development Calendar in October 2022. In December 2022, colleges and other operating locations provided their initially-projected dedicated revenue and centralized accounts budgets. In February 2023, after the Governor’s proposed State budget was released on January 10, 2023, preliminary allocations were provided to all operating locations to begin working with their constituencies in developing their 2023-2024 Budget Operation Plans. Los Angeles Community College District 2023-2024 Final Budget 5 Year-end open order balances were added to the college budgets. As of September 5, 2023, the Final Budget was made available for public viewing. The 2023-2024 Final Budget reflects most of the projected additional funding from the State, including funding for the state-funded categorical programs. Unrestricted General Fund Chart #2, Projected Source of Funds, Unrestricted General Fund, identifies sources of revenue/income available for appropriation in the Unrestricted General Fund. Categories reflect those used in the District Budget Allocation Model provided in Appendix F. Revenues were based on the following assumptions: 1. Base revenue for State General Revenue is projected at $802.8 million, which includes $49.4 million for the Education Protection Act fund (Proposition 30/55) and $61.0 million in COLA. 2. Apprenticeship income is projected at $33,455. 3. Non-resident tuition is projected at $8.3 million based on the rate of $342 per unit. 4. Part-time faculty compensation is projected at $2.3 million. 5. Lottery revenues are projected at $17.9 million ($177/FTES) based on 2023-2024 lottery revenue and enrollment projections 6. On-Going State Mandate Block Grant is projected at $3.5 million. 7. Dedicated Revenue projections, submitted by colleges, are at $8.2 million. 8. "Other State" income is projected at $8.5 million. 9. Full Time Faculty Hire Revenue is projected at $13.4 million. 10. Interest and “Other Local” are projected at $14.0 million. These sources of income include interest earned on cash balances and other miscellaneous fees such as jury duty, royalties, handling charges, discounts, etc. 11. 2022-2023 ending balance of $187.0 million includes open orders of $23.0 million, which are allocated to colleges and district-wide accounts (Chart #3). The District ended the 2022-2023 year with an unrestricted ending balance of $187.0 million, including funded open orders, which is $4.2 million more than the 2021-2022 year ($182.9 million). Los Angeles Community College District 2023-2024 Final Budget 6 Chart 2: Projected Source of Funds - Unrestricted General Fund State General Revenues Amount Base 692,440,065 COLA (Est. @ 8.22%) 60,980,794 Growth (Est. @ 0.00%) 0 Education Protection Act (EPA) 49,418,747 Total State Apportionment 802,839,606 Total General Revenues 802,839,606 Part-Time Faculty Compensation 2,305,482 Lottery 17,892,200 Non-Resident Tuition 8,279,000 Apprenticeship 33,455 On-Going State Mandate Block Grant 3,494,286 Full Time Faculty Hiring 13,368,234 Other State 8,523,260 Interest 5,000,000 Other Local 9,000,000 Dedicated Revenue 8,210,934 Incoming Transfer 0 Less Intrafund within unrestricted 0 Total Unrestricted General Fund Income 878,946,457 Open Orders 22,994,629 Contingency Reserve 30,763,126 General Reserve 57,131,520 Other Fund Balance 76,157,677 Total Funds Available for Appropriation 1,065,993,409 Los Angeles Community College District 2023-2024 Final Budget 7 Chart 3: 2022-23 Open Orders and Ending Balances – Unrestricted General Fund Chart #3 summarizes the disposition of the 2022-2023 unrestricted carryover balances. Funded Open Orders $ 22,994,629 Balance Excluding Open Orders $ 164,052,323 Total Fund Balance $ 187,046,952 Colleges & Obligations 2022-23 Unrestricted Balance a Add’l Rev & Unrestricted Adjustments b Restricted Deficits c Budget for Open Orders d College Positive Balances e=a+b+c-d College Negative Balances f=a+b+c-d City 1,112,876 1 0 420,396 692,481 0 East 11,273,745 0 0 1,762,037 9,511,708 0 Harbor 2,093,073 2 0 292,992 1,800,083 0 Mission 758,725 0 0 184,025 574,700 0 Pierce 5,098,019 0 0 636,300 4,461,719 0 Southwest 4,552,215 0 0 364,030 4,188,185 0 Trade-Tech 15,786,535 2 0 3,541,654 12,244,883 0 Valley 6,752,661 1 0 412,584 6,340,078 0 West 1,861,240 1 0 385,356 1,475,885 0 College Total 49,289,089 7 0 7,999,374 41,289,722 0 Obligations College Positive Balances 7,999,374 41,289,722 District Office and IT balance 1,407,299 1,204,065 Van de Kamp Innovation Center 68,831 2,307,742 Districtwide 13,490,263 4,836,257 Other Districtwide 28,862 24,260,806 Contingency Reserve 27,293,160 General Reserve 50,687,298 STRS/PERS Designated Reserves 3,830,001 Restricted Program Deficits (0) Subtotal 155,709,050 Remaining Undistributed Balance to Fund Reserves 8,343,273 Total 22,994,629 164,052,323 Los Angeles Community College District 2023-2024 Final Budget 8 Appropriations Appropriations represent the planned expenditures of total funds available. The distribution of the Unrestricted General Fund budget to colleges was determined by the adopted Budget Allocation mechanism. The budget allocations were developed based on consultation with the Cabinet and the District Budget Committee. Chart #4, 2023-2024 Final Budget - Unrestricted General Fund, provides the total 2023-2024 budget allocations distributed to each operating location, including balances and open orders. The following provides a brief explanation of the changes to the appropriation categories noted in Chart #4: 1. College appropriations are at $691.2 million, an increase of $31.0 million from the prior year. 2. The Educational Services Center budget is funded at $38.8 million and Information Technology is funded at $21.5 million. 3. Total of Districtwide Accounts is $150.0 million, and includes funded open orders and carried forward balances. Major Districtwide accounts include: − $ 41.8 million for Districtwide Information Technology − $ 30.7 million for Retiree Benefits − $ 28.6 million for District Safety/Sheriff − $ 11.5 million for Liability Insurance − $ 5.1 million for Legal Expense − $ 5.0 million for Workers’ Compensation − $ 4.5 million for Board Election − $ 3.0 million for Insurance/Legal/Workers’ Compensation Reserve − $ 2.2 million for Districtwide Marketing − $ 2.0 million for Central Financial Aid Unit 4. General Reserve is funded at $57.1 million. It represents 6.5% of the Unrestricted General Fund revenue budget. 5. Contingency Reserve is funded at $30.8 million. It represents 3.5% of the Unrestricted General Fund revenue budget. Los Angeles Community College District 2023-2024 Final Budget 9 Chart 4: 2023-2024 Final Budget – Unrestricted General Fund Colleges & Obligations 2022-2023 Adopted Final budget 2023-2024 Tentative Budget 2023-2024 Final Budget City 70,203,973 73,839,649 76,844,245 East 149,292,287 143,290,001 157,801,428 Harbor 44,064,194 42,073,067 44,572,252 Mission 44,397,599 43,352,910 45,273,240 Pierce 95,708,304 89,825,305 97,325,219 Southwest 36,344,088 35,189,382 39,660,069 Trade-Tech 88,651,562 75,405,269 91,966,093 Valley 83,263,406 74,705,578 83,803,911 West 48,325,433 51,065,017 53,956,380 College Total 660,250,846 628,746,178 691,202,837 Educational Services Center 36,924,779 37,507,507 38,793,579 Information Technology 19,699,419 20,026,967 21,497,982 Districtwide 138,745,488 131,372,551 149,959,037 Contingency Reserve 27,293,160 29,505,463 30,763,126 General Reserve 50,687,298 54,795,859 57,131,520 STRS/PERS Reserve 3,830,001 0 0 Other Districtwide 1,947,141 0 1,841,622 Van de Kamp Innovation 2,943,314 1,236,396 3,612,969 Supplemental Retirement (SRP) 4,772,488 4,772,489 4,700,045 Funds for Deferred Maintenance 15,596,092 16,860,264 17,578,929 Part Time Faculty Health Benefits 0 0 2,170,443 Emergency Conditions Revenue 0 0 46,741,320 Undistributed Balance 5 85,927,869 0 Total 962,690,031 1,010,751,543 1,065,993,409 Los Angeles Community College District 2023-2024 Final Budget 10 Education Protection Act Proposition 30, the Education Protection Act (EPA), was approved by voters in November 2012. This funding was slated to end on December 31, 2018, with the sales tax portion of the funding ending on December 31, 2016. On November 8, 2016, voters extended Proposition 30 for 12 additional years to 2030-31 through the passage of Proposition 55, the California Children's Educational and Health Care Protection Act. This measure extended the increased personal income tax rates for upper income earners, but did not extend the sales tax portion of Proposition 30. The restrictions attached to revenue generated from Proposition 55 remain the same as those attached to Proposition 30, namely these funds cannot be used for administrative costs. It is estimated that the District will receive approximately $49.4 million and will use these funds for faculty salaries and benefits as reflected in Chart #5. See appendix E for detailed plans by college. Chart 5: Education Protection Act (EPA) C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 100000 Certificated Salaries 168,364,497 85.0% 29,068,107 58.8% 29,068,107 58.8% 200000 Classified, Regular 560,494 0.3% 0 0.0% 0 0.0% 300000 Employee Benefits 29,151,878 14.7% 20,350,640 41.2% 20,350,640 41.2% 400000 Books & Supplies 0 0.0% 0 0.0% 0 0.0% 500000 Operating Expenses 0 0.0% 0 0.0% 0 0.0% 600000 Capital Outlay 0 0.0% 0 0.0% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total EPA Fund 198,076,869 100.0% 49,418,747 100.0% 49,418,747 100.0% Los Angeles Community College District 2023-2024 Final Budget 11 Restricted General Fund The Restricted General Fund is the other budget category comprising the General Fund. The Restricted General Fund for the 2023-2024 fiscal year is $434.2 million or 4.2% of the total budget. The budget category of “Other SFP” is usually low at the adoption of the Final Budget due to the District’s historical practice of accepting federal program funds throughout the year. Chart #6 summarizes restricted programs in the 2023-2024 Final Budget. Other Funds A discussion of income and appropriations of the other funds may be found in the latter section of this document titled “Other Funds”. Los Angeles Community College District 2023-2024 Final Budget 12 Chart 6: Restricted General Fund (In Millions) Program 2021-22 Actual 2022-23 Final Budget 2022-23 Actual 2023-24 Tentative Budget 2023-24 Final Budget 2022-23 Final Bud Difference 2022-23 Final Bud Difference 2022-23 Actual Difference 2022-23 Actual Difference 2023-24 Tent Bud Difference 2023-24 Final Bud Difference $ % $ % $ % CA Adult Education Program (CAEP) 7.399 13.035 8.956 8.745 14.527 1.492 11.4 5.571 62.2 5.782 66.1 CA College Promise 3.780 1.801 4.915 0.000 1.751 (0.050) (2.8) (3.165) (64.4) 1.751 >100.0 CalWORKs (Child Care/non-Child Care) / TANF 5.909 11.855 6.853 6.855 12.238 0.383 3.2 5.384 78.6 5.383 78.5 Community Services 0.894 8.129 1.304 2.251 8.257 0.128 1.6 6.952 >100.0 6.006 >100.0 Cooperative Agencies Resources for Education (CARE) 0.771 1.068 0.968 0.916 1.289 0.221 20.7 0.321 33.2 0.373 40.7 Disabled Students Programs & Services (DSPS) 7.480 8.853 10.075 8.809 11.117 2.264 25.6 1.043 10.3 2.308 26.2 Dream Resource Liaison Support 0.276 1.165 0.625 0.000 0.540 (0.625) (53.7) (0.086) (13.7) 0.540 >100.0 Equal Employment Opportunity 0.009 0.184 0.009 0.000 0.314 0.129 70.3 0.304 >100.0 0.314 >100.0 Extended Opportunities Programs and Services (EOPS) 8.446 7.337 9.173 7.395 9.541 2.205 30.1 0.368 4.0 2.146 29.0 Federal Perkins IV (CTE) 4.509 4.388 5.226 0.000 4.526 0.138 3.1 (0.701) (13.4) 4.526 >100.0 Federal Work Study 1.414 2.958 2.114 3.170 3.157 0.198 6.7 1.043 49.4 (0.013) (0.4) Financial Aid Technology 0.484 0.440 0.380 0.000 0.060 (0.380) (86.3) (0.319) (84.1) 0.060 >100.0 Foster and Kinship Care Education 1.076 1.051 1.114 1.009 1.009 (0.041) (3.9) (0.104) (9.4) 0.000 0.0 Framework for Racial Equity and Social Justice 0.516 1.880 1.537 0.000 0.376 (1.504) (80.0) (1.161) (75.5) 0.376 >100.0 Health Services 2.389 8.870 2.803 4.030 8.215 (0.656) (7.4) 5.412 >100.0 4.184 >100.0 HEERF I 0.939 0.187 0.225 0.000 0.000 (0.187) (100.0) (0.225) (100.0) 0.000 0.0 HEERF II 29.189 13.001 13.115 0.000 0.000 (13.001) (100.0) (13.115) (100.0) 0.000 0.0 HEERF III 69.915 23.996 20.252 0.000 0.000 (23.996) (100.0) (20.252) (100.0) 0.000 0.0 HEERF MSI Supplement 5.644 11.554 11.472 0.000 0.082 (11.472) (99.3) (11.391) (99.3) 0.082 >100.0 HEERF SAHIE 0.000 0.917 0.917 0.000 0.000 (0.917) (100.0) (0.917) (100.0) 0.000 0.0 Higher Ed Emergency MSI 0.103 0.075 0.072 0.000 0.000 (0.075) (100.0) (0.072) (100.0) 0.000 0.0 Lottery - Prop 20 7.263 8.924 6.664 5.406 18.188 9.264 >100.0 11.524 >100.0 12.782 >100.0 NextUp 1.772 2.035 2.346 1.476 2.298 0.263 12.9 (0.048) (2.1) 0.823 55.7 One-Time Block Grants 2.349 16.199 8.282 0.000 27.463 11.264 69.5 19.181 >100.0 27.463 >100.0 Parking (0.933) 6.914 (0.116) 0.962 7.068 0.154 2.2 7.183 <100.0 6.106 >100.0 Staff/Faculty Development 0.000 0.347 0.024 0.000 0.323 (0.024) (6.8) 0.300 >100.0 0.323 >100.0 Strong Workforce 13.863 28.359 13.120 10.937 27.280 (1.079) (3.8) 14.159 >100.0 16.343 >100.0 Student Equity and Achievement (SEA) 42.699 59.895 45.561 45.629 66.728 6.833 11.4 21.167 46.5 21.099 46.2 Student Financial Aid Administration 4.492 4.038 5.386 4.586 5.223 1.185 29.4 (0.163) (3.0) 0.637 13.9 Student Retention and Enrollment 3.992 5.329 11.129 0.000 6.341 1.012 19.0 (4.788) (43.0) 6.341 >100.0 Unrestricted Indirects 0.000 1.912 (9.440) 0.000 11.352 9.440 >100.0 20.793 <100.0 11.352 >100.0 Veterans Resource Center 0.520 0.979 0.984 0.000 0.580 (0.399) (40.8) (0.404) (41.0) 0.580 >100.0 Other Specially Funded Programs 53.026 71.157 71.368 0.714 184.962 113.805 >100.0 113.594 >100.0 184.248 >100.0 Total Available 280.186 328.831 257.412 112.890 434.803 105.972 32.2 177.390 68.9 321.913 285.2 Summary Los Angeles Community College District 2023-2024 Final Budget 13 Summary of All Funds Chart #7A & B, entitled "Summary of all Funds,” describes the District's total budget by sources of funding and major objects of expenditure. These separate fund categories are established to segregate and restrict monies. While transfers between fund categories are permitted, such transfers are subject to restriction according to the source of the funds. The reader will note that transfers are deducted from both revenues and appropriations in the total columns. This is done to avoid double counting when the funds are transferred, whether between fund categories (interfund transfers), between the Unrestricted and Restricted General Fund (intrafund transfers), or between programs or locations within the Unrestricted General Fund (intrafund transfers). Income • Federal Income: Federal income represents funds projected for Student Financial Aid programs, Federal Perkins program, and other federal specially funded programs. As is customary, additional federal program award augmentations will be made as financial aid programs are noticed from the funding sources. • State Income: State income represents state general apportionment income, CalWORKs and TANF program income, Disabled Students Programs and Services (DSPS), Extended Opportunities Programs and Services (EOPS), Student Equity and Achievement (SEA), Strong Workforce, other categorical program income, Instructional Support Block Grants, child development centers income, capital outlay project income, and other specially funded state income. The categorical state funded programs such as DSPS, EOPS, CalWORKs, Strong Workforce, and Student Equity and Achievement are budgeted at 95% of the 2022-2023 budget. FKCE is budgeted at 93% of the 2022-2023 budget. Cooperative Agencies Resources for Education (CARE) and NextUp are budgeted at 90% and 70% of the 2022-2023 budget, respectively. Adjustments will be made to the budget allocation for these programs during the fiscal year when the State releases the funding for each program. Block grants for Physical Plant and Instructional Support are budgeted as allocated by the State. Income for capital outlay projects in the Special Reserve Fund is budgeted based on awarded and released construction phases. • Local Tax: This source of funds includes secured tax, unsecured tax, and property tax shift from local governments to schools. The projected income reflects the current estimated collections from the Los Angeles County Treasurer’s Office. Taxes are calculated as a portion of the state general revenue. • Other Local: Other local income includes college dedicated revenue, student fees, and other miscellaneous local income. • Intrafund Transfer: This account reflects a transfer of funds from the Unrestricted General Fund to the Restricted General Fund to comply with mandatory matching requirements of federal and state programs. • Interfund Transfer: This represents revenues received as a result of transfers between funds. The Special Reserve Fund, Cafeteria Fund, and Child Development Fund receive funds from the General Fund to support operations and projects. Los Angeles Community College District 2023-2024 Final Budget 14 • Beginning Balance and Open Orders: This reflects actual 2022-2023 ending balances and funded open orders. • Ending Balance: All unrestricted general funds available are appropriated in the Final Budget. Appropriations The reader is directed to the Appropriations section for a more complete comparison of appropriations by sub-object within each fund. Los Angeles Community College District 2023-2024 Final Budget 15 Chart 7A: Summary of All Funds Three-Year Comparison – Income Income 2021-22 Year End Actual 2022-23 Year End Actual 2023-24 Total Budget 2023-24 Unrestricted Gen Fund 2023-24 Restricted Gen Fund 2023-24 Total Gen Fund 2023-24 Bookstore 2023-24 Cafeteria 2023-24 Child Dev 2023-24 Special Reserve 2023-24 Building Fund 2023-24 Financial Aid 2023-24 Debt Services Federal 359,997,925 247,709,207 198,922,626 29,850,974 29,850,974 0 0 833,230 0 0 168,238,422 0 State 651,272,830 750,294,168 993,636,813 555,498,734 326,316,062 881,814,796 0 0 8,888,403 12,517,064 0 90,416,550 0 Local tax 248,329,453 276,695,640 272,769,790 272,769,790 272,769,790 0 0 0 0 0 0 0 Local other 95,082,822 742,575,021 103,755,512 50,677,934 25,056,298 75,734,232 18,777,194 1,403,673 105,619 1,446,958 4,887,836 1,400,000 0 Interfund Transfers 26,444,953 25,963,704 26,598,898 0 452,972 247,315 730,923 17,578,929 0 0 7,588,759 Intrafund Transfers 11,333,520 8,633,184 622,477 622,477 622,477 0 0 0 0 0 0 Total Income 1,392,461,503 2,051,870,923 1,596,306,116 878,946,458 381,845,811 1,260,792,269 19,230,166 1,650,988 10,558,175 31,542,951 4,887,836 260,054,972 7,588,759 Beginning Balance 3,929,151,555 3,742,146,432 3,608,283,646 164,052,323 45,983,904 210,036,227 11,173,061 3,203,181 3,346,420 208,324,111 3,168,792,930 3,407,716 0 Adj to Beg Balance 8,040,307 (598,706,159) 5,299,999,999 (1) 0 (1) 0 0 0 0 5,300,000,000 0 0 Reserve/Open Orders 32,789,293 83,665,670 30,078,612 22,994,629 6,974,333 29,968,962 0 0 109,650 0 0 0 0 Total Revenue 5,362,442,659 5,278,976,866 10,534,668,373 1,065,993,409 434,804,048 1,500,797,457 30,403,227 4,854,169 14,014,245 239,867,062 8,473,680,766 263,462,688 7,588,759 Less YE Open Orders 83,665,670 30,078,612 0 0 0 0 0 0 0 0 0 0 0 Less Ending Balance 3,742,222,166 3,608,283,755 91,329,593 0 1,199 1,199 432,611 0 0 87,488,067 0 3,407,716 0 Less Reserves 0 0 0 0 0 0 0 0 0 0 0 0 0 Adjusted Revenue 1,536,554,823 1,640,614,500 10,443,338,780 1,065,993,409 434,802,849 1,500,796,258 29,970,616 4,854,169 14,014,245 152,378,995 8,473,680,766 260,054,972 7,588,759 Less Intrafund within Unr 0 0 0 0 0 0 0 0 0 0 0 0 0 Adjusted Revenue 1,536,554,823 1,640,614,500 10,443,338,780 1,065,993,409 434,802,849 1,500,796,258 29,970,616 4,854,169 14,014,245 152,378,995 8,473,680,766 260,054,972 7,588,759 Less Intrafund Unr/Res 11,333,520 8,633,184 622,477 0 622,477 622,477 0 0 0 0 0 0 0 Less Interfund Transfers 26,444,953 25,963,704 26,598,898 0 0 0 0 0 0 0 0 0 0 Available for Appropriation 1,498,776,350 1,606,017,612 10,416,117,405 1,065,993,409 434,180,372 1,500,173,781 29,970,616 4,854,169 14,014,245 152,378,995 8,473,680,766 260,054,972 7,588,759 Beginning Balance includes authorized, but not yet issued bonds Measure J, CC and LA ($175,000,000 for Measure J, $2,550,000,000 for Measure CC, and $5,300,000,000 for Measure LA). Los Angeles Community College District 2023-2024 Final Budget 16 Chart 7B: Summary of All Funds Three-Year Comparison – Appropriations Appropriations 2021-22 Year End Actual 2022-23 Year End Actual 2023-24 Total Budget 2023-24 Unrestricted Gen Fund 2023-24 Restricted Gen Fund 2023-24 Total Gen Fund 2023-24 Bookstore 2023-24 Cafeteria 2023-24 Child Dev 2023-24 Special Reserve 2023-24 Building Fund 2023-24 Financial Aid 2023-24 Debt Services Certificated Salaries 363,813,712 390,093,237 427,364,466 362,128,519 62,424,052 424,552,571 0 5,000 2,806,895 0 0 0 0 Non-certificated Salaries 183,194,239 201,673,771 218,972,436 155,113,699 58,354,607 213,468,306 4,217,378 310,883 975,869 0 0 0 0 Employee Benefits 238,673,329 265,343,633 242,328,323 198,087,706 33,466,597 231,554,303 2,077,865 48,681 1,058,715 0 0 0 7,588,759 Books & Supplies 32,039,740 31,101,803 45,057,914 6,922,868 25,632,280 32,555,148 10,931,228 855,974 708,262 7,302 0 0 0 Other Expenses 196,122,084 209,480,837 6,968,440,703 202,570,444 96,897,689 299,468,133 813,482 212,418 120,787 128,454,926 6,539,370,957 0 0 Capital Outlay 206,280,762 256,510,041 1,967,635,957 7,875,027 24,658,775 32,533,802 287,704 59,605 163,747 3,656,290 1,930,934,809 0 0 Interfund Transfers 289,986,005 260,447,474 546,940,083 106,696,248 133,368,849 240,065,097 11,642,959 3,361,608 8,179,970 20,260,477 3,375,000 260,054,972 0 Other 26,444,953 25,963,704 26,598,898 26,598,898 0 26,598,898 0 0 0 0 0 0 0 Total Appropriations 1,536,554,823 1,640,614,500 10,443,338,780 1,065,993,409 434,802,849 1,500,796,258 29,970,616 4,854,169 14,014,245 152,378,995 8,473,680,766 260,054,972 7,588,759 Less Intrafund w/in Unr 0 0 0 0 0 0 0 0 0 0 0 0 0 Adjusted Appropriations 1,536,554,823 1,640,614,500 10,443,338,780 1,065,993,409 434,802,849 1,500,796,258 29,970,616 4,854,169 14,014,245 152,378,995 8,473,680,766 260,054,972 7,588,759 Less Intrafund Unr bet Loc 0 0 0 0 0 0 0 0 0 0 0 0 0 Less Intrafund Unr/Res 11,333,520 8,633,184 622,477 0 622,477 622,477 0 0 0 0 0 0 0 Less Interfund Transfers 26,444,953 25,963,704 26,598,898 0 0 0 0 0 0 0 0 0 0 Net Appropriations 1,498,776,350 1,606,017,612 10,416,117,405 1,065,993,409 434,180,372 1,500,173,781 29,970,616 4,854,169 14,014,245 152,378,995 8,473,680,766 260,054,972 7,588,759 General Fund Income Los Angeles Community College District 2023-2024 Final Budget 17 General Fund Income and Balances The District's General Fund income and balances are categorized by income sources. The following discussion summarizes the General Fund by source of funds. Total General Fund Funds of $1.5 billion available for appropriation in the General Fund (Chart #8A & B) include unrestricted and restricted income. Unrestricted funds support the general operations of the District and may be appropriated with greater discretion by the governing board. Restricted funds, whatever the source, must be used in accordance with the guidelines provided either by statute, the funding agency, or the Board of Trustees. • Beginning Balances represent unrestricted and restricted funds carried forward from the prior fiscal year. Balances are the result of income received in excess of actual expenditures. They can include funds that are obligated (contractual agreements or purchase orders) or that are committed. • Incoming Transfers: There are no interfund transfers from other funds to the General Fund. The Unrestricted General Fund contributes $7.6 million to the post-retirement health benefit trust account (Debt Services Fund), $0.7 million to support the Child Development Centers (Child Development Fund), $17.579 million for deferred maintenance (Special Reserve Fund), $0.5 million to support the Bookstore (Bookstore Fund), and $0.2 million to support the Cafeteria (Cafeteria Fund). Within the General Fund, however, funds are provided for required matching fund support from the Unrestricted General Fund for Disabled Student Programs and Services (DSPS) and the Federal Work Study (FWS) program. Colleges may also choose to provide additional subsidies to restricted programs from their unrestricted operating budgets. In addition, funds are sometimes transferred from one program or location to another within the Unrestricted General Fund. These transfers are called intrafund transfers as they occur within the General Fund. Because intrafund transfers are shown in both the originating programs or locations and the destination programs or locations, the total General Fund is overstated by this amount of the transfer. Therefore, these intrafund transfers are subtracted from the General Fund total in order to show the actual amount available to support programs. Los Angeles Community College District 2023-2024 Final Budget 18 Chart 8A: Total General Fund Income Income 2021-22 Actual 2022-23 Final Budget 2022-23 Budget 2022-23 Actual 2023-24 Budget Federal 131,953,580 79,503,518 100,159,576 77,031,596 29,850,974 General Revenue 470,036,212 521,307,832 687,478,855 687,375,276 753,420,858 Educ Protection Act (EPA) 198,102,931 198,102,931 49,418,745 49,418,747 49,418,747 Non-Resident 7,212,160 7,120,000 8,850,681 8,850,682 8,279,000 Apprenticeship 272,246 365,396 330,891 330,891 33,455 Dedicated Revenue 10,945,581 7,103,844 9,438,204 9,777,665 8,210,936 Lottery-Unrestricted 17,947,609 12,927,300 17,184,600 21,930,225 17,892,200 Lottery-Restricted (Prop 20) 8,392,626 5,094,881 5,094,881 11,079,352 7,278,181 Part-time Faculty Comp 2,102,459 2,265,548 1,775,065 1,775,067 2,305,482 Part-time Faculty Office Hours 5,860,060 4,845,498 6,464,193 6,464,192 5,252,817 Interest 1,295,226 1,000,000 1,000,000 9,724,318 5,000,000 Other State 164,030,273 188,622,850 403,490,182 184,537,601 339,170,844 Other Local 17,360,030 35,103,274 38,330,483 18,802,684 34,056,298 Incoming Transfers 11,499,115 1,312,678 8,896,266 8,896,265 622,477 Total Income 1,047,010,108 1,064,675,550 1,337,912,622 1,095,994,561 1,260,792,269 Beginning Balance 157,976,291 208,869,038 208,893,308 208,893,308 210,036,227 Open Orders 16,955,204 19,907,046 19,907,046 19,907,046 29,968,962 Adj to Beginning Balance 8,017,173 0 0 1,222,426 (1) Other Adjustments 0 0 0 0 0 Total Adj Beg Balance 182,948,668 228,776,084 228,800,354 230,022,780 240,005,188 Less Open Orders to CF 19,907,046 0 0 29,968,962 0 Less Ending Balance 208,968,321 1,930,956 24,270 210,036,336 1,199 Total General Fund Income 1,001,083,410 1,291,520,678 1,566,688,706 1,086,012,044 1,500,796,258 Less Intrafund Transfers 11,333,520 1,312,678 8,633,185 8,633,184 622,477 Net General Fund Income 989,749,890 1,290,208,000 1,558,055,521 1,077,378,860 1,500,173,781 Chart 8B: Total General Fund Appropriations Appropriations 2021-22 Actual 2022-23 Final Budget 2022-23 Budget 2022-23 Actual 2023-24 Budget Certificated Salaries 358,688,293 410,386,033 431,111,234 384,284,176 424,552,571 Non-Certificated Salaries 176,358,300 200,238,937 223,829,008 193,576,627 213,468,306 Employee Benefits 226,435,941 201,057,194 254,393,939 252,482,713 231,554,303 Books & Supplies 19,966,945 27,580,230 30,346,213 16,577,041 32,555,148 Other Operating Expenses 147,583,109 201,052,058 319,370,947 157,989,446 299,468,133 Capital Outlay 20,135,739 37,820,307 52,676,873 26,761,947 32,533,802 Interfund Transfer 26,279,358 25,099,360 25,700,623 25,700,623 26,598,898 Other 25,635,723 188,286,559 229,259,869 28,639,470 240,065,097 Total Appropriations 1,001,083,410 1,291,520,678 1,566,688,706 1,086,012,044 1,500,796,258 Less Intrafund Unr/Res 11,333,520 1,312,678 8,633,185 8,633,184 622,477 Net Appropriations 989,749,890 1,290,208,000 1,558,055,521 1,077,378,860 1,500,173,781 As of 2023 Year-End Close. Los Angeles Community College District 2023-2024 Final Budget 19 Unrestricted General Fund Chart #9 is a summary of the Unrestricted General Fund Income by source of funds. Apportionment (state revenue) constitutes the largest source of funds, followed by property tax revenue and enrollment fees. These funds, termed State General Revenues, total $802.8 million, which also include $49.4 million of the Education Protection Act (EPA) Fund, and make up 91.3% of the unrestricted fund revenue. State General Revenue General Revenues are determined by a state funding formula, which utilizes the workload measures of attendance and enrollment. The General Revenue projection is established by computation of the District's prior year base funding and adjustment for inflation and growth. The General Revenue income is made up of several sources (as reflected in Chart #2): State Apportionment, Tax Relief Subventions, Local Tax Revenue, Education Protection Act (EPA) fund, and 98% of the Enrollment Fees. Other Attendance Driven Income • Non-resident Tuition fees are paid by non-resident students whose attendance is not eligible for state support. This income is projected at $8.3 million. The non-resident rate of $342 per unit is as adopted by the Board of Trustees for fiscal year 2023-2024. • Apprenticeship funding is budgeted at the rate of $9.98 per attendance hour, pending further State information. This program is located at Los Angeles Trade-Technical College. Dedicated Revenue Dedicated Revenue, which arises from locally managed activities identified at individual colleges, includes such items as traffic citations, library fines, two percent (2%) administrative allowance for enrollment fees, etc. This income is projected by the colleges as part of the budget development process. Other Unrestricted Income • Lottery Revenue is based on a $/FTES for all FTES (as opposed to the funded FTES used in the General Revenue calculation). The current projection is at $17.9 million (or $177/FTES). • Other State includes state-mandated cost reimbursement and Part-time Office hours reimbursements. • Interest Income represents income earned from the investment by the County Treasurer of surplus District cash. • Other Local represents miscellaneous income from various sources. • Beginning Balances represent funds brought forward from the preceding year. Colleges retain their balances. Los Angeles Community College District 2023-2024 Final Budget 20 Chart 9A: Unrestricted General Fund Income Income 2021-22 Actual 2022-23 Final Budget 2022-23 Budget 2022-23 Actual 2023-24 Budget Attendance Driven General Revenue 470,036,212 521,307,832 687,478,855 687,375,276 753,420,858 Education Protection Act (EPA) 198,102,931 198,102,931 49,418,745 49,418,747 49,418,747 Non-Resident 7,212,160 7,120,000 8,850,681 8,850,682 8,279,000 Apprenticeship 272,246 365,396 330,891 330,891 33,455 Dedicated Revenue 10,945,581 7,103,844 9,438,204 9,777,665 8,210,936 Other Income Lottery-Unrestricted 17,947,609 12,927,300 17,184,600 21,930,225 17,892,200 Part-time Faculty Compensation 2,102,459 2,265,548 1,775,065 1,775,067 2,305,482 Part-time Faculty Office Hours 5,860,060 4,845,498 6,464,193 6,464,192 5,252,817 Interest 1,295,226 1,000,000 1,000,000 9,724,318 5,000,000 Other Federal 0 0 0 0 0 Other State 17,388,816 15,766,234 18,790,298 19,874,346 20,132,963 Other Local 9,162,271 9,000,000 9,000,000 10,574,428 9,000,000 Incoming Transfers 5,925,000 0 5,325,287 5,325,287 0 Total Income 746,250,572 779,804,583 815,056,819 831,421,124 878,946,458 Beginning Balance 132,935,009 164,384,701 164,408,971 164,408,971 164,052,323 Open Orders 16,439,955 18,500,747 18,500,747 18,500,747 22,994,629 Adj to Beg Bal 8,287,859 0 (38) 1,315,435 (1) Total Adj Beg Balance 157,662,823 182,885,448 182,909,680 184,225,153 187,046,951 YE Open Orders 18,500,747 0 0 22,994,629 0 Less Ending Balance 164,515,012 0 24,270 164,051,959 0 Total Unrestricted Income 720,897,636 962,690,031 997,942,229 828,599,690 1,065,993,409 Less Intrafund w/in Unrestricted 0 0 5,062,206 5,062,206 0 Net Unrestricted Income 720,897,636 962,690,031 992,880,023 823,537,484 1,065,993,409 Chart 9B: Unrestricted General Fund Appropriations Appropriations 2021-22 Actual 2022-23 Final Budget 2022-23 Budget 2022-23 Actual 2023-24 Budget Certificated Salaries 282,074,106 357,617,756 336,097,812 318,028,045 362,128,519 Non-Certificated Salaries 125,409,096 149,385,481 149,926,591 145,624,857 155,113,699 Employee Benefits 194,555,148 175,866,633 206,299,313 217,055,220 198,087,706 Books & Supplies 3,749,315 7,336,804 7,554,526 4,469,668 6,922,868 Other Operating Expenses 83,642,750 135,568,282 171,282,580 110,820,051 202,570,444 Capital Outlay 3,927,374 10,613,784 9,389,858 4,848,371 7,875,027 Interfund Transfer 22,823,741 25,099,360 23,915,635 23,915,635 26,598,898 Other 4,716,105 101,201,931 93,475,914 3,837,842 106,696,248 Total Appropriations 720,897,636 962,690,031 997,942,229 828,599,690 1,065,993,409 Less Intrafund w/in Unrestricted 0 0 5,062,206 5,062,206 0 Net Appropriations 720,897,636 962,690,031 992,880,023 823,537,484 1,065,993,409 As of 2023 Year-End Close. Los Angeles Community College District 2023-2024 Final Budget 21 Restricted General Fund Chart #10 is a summary of the Final Budget section of Restricted General Fund income budgeted to date. It is expected that the District will accept and appropriate additional federal and state categorical programs during the year. • Federal Perkins Programs comprise the largest separate category of programs from federal sources of income. "Other Federal" includes programs such as Veterans Education, Higher Education Act, FSEOG, and Federal Work Study. • State Categoricals: The State supports a number of categorical programs designed to accomplish specific objectives. Primarily, these are CalWORKs, Extended Opportunities Programs and Services (EOPS), Cooperative Agencies Resources for Education (CARE), Student Financial Aid Administration, Disabled Students Programs and Services (DSPS), Career & Technical Education, Strong Workforce, Student Equity and Achievement (SEA), and Block Grants for Physical Plant and Instructional Support. These funds are based on the projected advanced allocations by the State. • Local Restricted Programs: The primary restricted programs funded locally are Community Services, Parking, and Health Services. Community Services is restricted because a statute does not allow the District to charge more than its costs. Parking and Health Services programs are supported by a fee and this revenue is restricted by the statute establishing the fee. • Beginning Balances: Beginning balances represent unspent funds from the prior year that can be carried forward and spent within the program where they were generated. The following programs had balances: Program Balances Community Services Program $ 6,006,083 Health Services Program $ 4,184,208 Lottery $ 10,909,900 Non-Resident Capital Outlay $ 824,824 Parking Program $ 6,105,579 Unrestricted Indirects $ 11,352,370 Other Programs $ 13,575,211 Total $ 52,958,175 Los Angeles Community College District 2023-2024 Final Budget 22 Chart 10A: Restricted General Fund Income Income 2021-22 Actual 2022-23 Final Budget 2022-23 Budget 2022-23 Actual 2023-24 Final Budget Federal Federal Perkins 4,509,414 4,387,939 5,241,452 5,226,509 4,525,690 Other SFP 127,444,166 75,115,579 94,918,124 71,805,086 25,325,284 Total Federal 131,953,580 79,503,518 100,159,576 77,031,596 29,850,974 State Disabled Student Prog & Svs 7,429,154 8,070,725 11,418,557 8,201,168 10,923,237 Extended Oppor Prog & Svs 8,445,730 7,336,570 10,399,986 9,173,019 9,539,010 Instructional Equipment 16,630,635 0 0 0 0 Lottery - Restricted (Prop 20) 8,392,626 5,094,881 5,094,881 11,079,352 7,278,181 Student Equity and Achievement 42,699,430 59,895,149 66,492,061 45,560,764 66,727,950 Staff Development 0 0 0 0 0 Staff Diversity 50,000 0 138,888 0 138,888 Other 71,386,507 97,554,172 296,250,392 101,728,304 231,708,796 Total State 155,034,082 177,951,497 389,794,765 175,742,607 326,316,062 Local Community Services 1,910,350 3,118,013 3,118,013 2,297,984 2,250,632 Health Services 3,143,770 4,680,210 4,680,210 2,785,567 4,030,353 Parking 29,729 1,284,237 1,284,237 52,777 961,987 Other 3,113,910 17,020,814 20,248,023 3,091,929 17,813,326 Total Local 8,197,759 26,103,274 29,330,483 8,228,256 25,056,298 Incoming Transfers 5,574,115 1,312,678 3,570,979 3,570,978 622,477 Total Income 300,759,536 284,870,967 522,855,803 264,573,437 381,845,811 Beginning Balance 25,041,282 44,484,337 44,484,337 44,484,337 45,983,904 Open Orders 515,249 1,406,299 1,406,299 1,406,299 6,974,333 CF Balance 0 0 0 0 0 Adj to Beginning Balance (270,686) 0 38 (93,009) 0 Other Adjustments 0 0 0 0 0 Less YE Open Orders 1,406,299 0 0 6,974,333 0 Less Ending Balance 44,453,308 1,930,956 0 45,984,377 1,199 Total Restricted Income 280,185,774 328,830,647 568,746,477 257,412,354 434,802,849 Chart 10B: Restricted General Fund Appropriations Appropriations 2021-22 Actual 2022-23 Final Budget 2022-23 Budget 2022-23 Actual 2023-24 Final Budget Certificated Salaries 76,614,187 52,768,277 95,013,422 66,256,131 62,424,052 Non-Certificated Salaries 50,949,204 50,853,456 73,902,417 47,951,771 58,354,607 Employee Benefits 31,880,793 25,190,561 48,094,626 35,427,492 33,466,597 Books & Supplies 16,217,630 20,243,426 22,791,687 12,107,373 25,632,280 Other Operating Expenses 63,940,359 65,483,776 148,088,367 47,169,394 96,897,689 Capital Outlay 16,208,365 27,206,523 43,287,015 21,913,576 24,658,775 Interfund Transfer 3,455,617 0 1,784,988 1,784,988 0 Other 20,919,618 87,084,628 135,783,955 24,801,628 133,368,849 Total Appropriations 280,185,774 328,830,647 568,746,477 257,412,354 434,802,849 As of 2023 Year-End Close. Unrestricted General Fund Appropriations Los Angeles Community College District 2023-2024 Final Budget 23 Los Angeles Community College District Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 117,062,091 16.2% 126,882,462 15.3% 136,220,074 12.8% 120000 Non-Teaching, Regular 52,796,024 7.3% 60,305,294 7.3% 57,993,876 5.4% 130000 Teaching, Hourly 106,178,264 14.7% 123,538,059 14.9% 162,306,873 15.2% 140000 Non-Teaching, Hourly 6,037,727 0.8% 7,302,230 0.9% 5,307,696 0.5% 190000 Misc Certificated Salaries 0 0.0% 0 0.0% 300,000 0.0% Total Certificated Salaries 282,074,106 39.1% 318,028,045 38.4% 362,128,519 34.0% 210000 Classified, Regular 106,834,071 14.8% 121,037,072 14.6% 133,550,891 12.5% 220000 Instructional Aides, Regular 11,429,153 1.6% 13,869,809 1.7% 14,848,591 1.4% 230000 Sub/Relief, Unclassified 4,710,311 0.7% 7,250,824 0.9% 4,378,779 0.4% 240000 Instructional Aides, Non-Perm 2,435,562 0.3% 3,467,152 0.4% 2,335,438 0.2% 290000 Misc Non-Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Non-Certificated Salaries 125,409,096 17.4% 145,624,857 17.6% 155,113,699 14.6% 310000 STRS Employer Contributions 48,578,229 6.7% 58,511,855 7.1% 58,800,000 5.5% 320000 PERS Employer Contributions 39,465,243 5.5% 48,174,034 5.8% 50,700,000 4.8% 330000 OASDHI Contributions 10,662,920 1.5% 12,469,970 1.5% 12,573,308 1.2% 340000 Medical/Dental Contributions 110,747,075 15.4% 118,082,105 14.3% 123,708,465 11.6% 350000 State Unemployment Insurance 2,717,885 0.4% 3,360,018 0.4% 3,240,615 0.3% 360000 Workers Compensation Insurance 3,658,907 0.5% 3,937,678 0.5% 4,000,000 0.4% 370000 Local Retirement System 615,991 0.1% 4,424,845 0.5% 5,124,206 0.5% 390000 Misc Employee Benefits (21,891,101) -3.0% (31,905,285) -3.9% (60,058,888) -5.6% Total Benefits 194,555,148 27.0% 217,055,220 26.2% 198,087,706 18.6% 420000 Books (1,288) 0.0% 4,119 0.0% 6,708 0.0% 440000 Instructional Media Materials 69,264 0.0% 202,465 0.0% 1,510,967 0.1% 450000 Supplies 3,681,339 0.5% 4,263,084 0.5% 5,405,193 0.5% Total Printing & Supplies 3,749,315 0.5% 4,469,668 0.5% 6,922,868 0.6% 540000 Insurance 6,605,369 0.9% 8,693,083 1.0% 11,288,633 1.1% 550000 Utilities & Housekeeping Expense 18,048,683 2.5% 22,716,159 2.7% 27,231,858 2.6% 560000 Contracts & Rentals 36,977,078 5.1% 43,571,505 5.3% 68,402,408 6.4% 570000 Legal, Election, Audit 5,981,384 0.8% 15,673,753 1.9% 10,494,740 1.0% 580000 Other Expense 16,030,237 2.2% 20,114,793 2.4% 84,807,879 8.0% 590000 Misc Other Expense 0 0.0% 50,759 0.0% 344,926 0.0% Total Operating Expenses 83,642,750 11.6% 110,820,051 13.4% 202,570,444 19.0% 620000 Buildings 0 0.0% 0 0.0% 5,000 0.0% 630000 Books & Materials for Libraries 30,162 0.0% 21,179 0.0% 120,542 0.0% 640000 Equipment 3,577,256 0.5% 4,357,094 0.5% 6,277,766 0.6% 650000 Lease/Purchase 319,957 0.0% 470,098 0.1% 1,471,719 0.1% Total Capital Outlay 3,927,374 0.5% 4,848,371 0.6% 7,875,027 0.7% 730000 Interfund Transfers 22,823,741 3.2% 23,915,635 2.9% 26,598,898 2.5% 739900 Intrafund Transfer - Restr/Unrestr 5,574,115 0.8% 3,570,978 0.4% 622,477 0.1% 740000 Reallocations/Adjustments (858,010) -0.1% 257,100 0.0% 0 0.0% 750000 Loans/Grants 0 0.0% 1,186 0.0% 300 0.0% 780000 Enroll - Bad Debt Expense 0 0.0% (0) 0.0% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 8,578 0.0% 106,073,471 10.0% Total Other 27,539,846 3.8% 27,753,477 3.3% 133,295,146 12.5% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 720,897,636 100.0% 828,599,690 100.0% 1,065,993,409 100.0% Los Angeles Community College District 2023-2024 Final Budget 24 Los Angeles Community College District Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 323,231,093 44.8% 375,793,882 45.4% 422,842,792 39.7% 6000 Instructional Support 26,359,134 3.7% 31,307,629 3.8% 29,333,620 2.8% 6100 Other Instructional Services 15,980,310 2.2% 19,153,992 2.3% 21,084,307 2.0% Student Services 6200 Admissions and Records 11,721,374 1.6% 13,136,139 1.6% 13,671,198 1.3% 6300 Counseling and Guidance 15,256,084 2.1% 16,592,669 2.0% 16,268,694 1.5% 6400 Other Student Services 22,470,452 3.1% 26,867,706 3.2% 27,197,929 2.6% Total Student Services 49,447,910 6.9% 56,596,514 6.8% 57,137,821 5.4% 6500 Maintenance and Operations 76,779,415 10.7% 91,183,833 11.0% 102,384,931 9.6% Institutional Support 6600 Planning and Policymaking 24,105,048 3.3% 35,061,089 4.2% 36,577,707 3.4% 6700 General Institutional Support 167,970,804 23.3% 181,049,201 21.9% 359,532,658 33.7% Total Institutional Support 192,075,852 26.6% 216,110,290 26.1% 396,110,365 37.2% 6800 Community Service 2,147,291 0.3% 2,314,202 0.3% 1,054,394 0.1% 6900 Ancillary Services 8,187,818 1.1% 9,617,130 1.2% 9,017,699 0.8% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 26,688,812 3.7% 26,522,218 3.2% 27,027,480 2.5% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 720,897,636 100.0% 828,599,690 100.0% 1,065,993,409 100.0% Los Angeles Community College District 2023-2024 Final Budget 25 Los Angeles City College Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 13,014,517 20.0% 12,753,537 17.9% 14,383,768 18.7% 120000 Non-Teaching, Regular 6,014,341 9.2% 6,815,777 9.5% 6,617,633 8.6% 130000 Teaching, Hourly 14,351,015 22.1% 14,740,982 20.6% 15,702,637 20.4% 140000 Non-Teaching, Hourly 806,210 1.2% 810,784 1.1% 553,051 0.7% Total Certificated Salaries 34,186,082 52.6% 35,121,080 49.2% 37,257,089 48.5% 210000 Classified, Regular 8,573,433 13.2% 10,059,031 14.1% 11,427,119 14.9% 220000 Instructional Aides, Regular 1,475,880 2.3% 1,712,434 2.4% 1,973,372 2.6% 230000 Sub/Relief, Unclassified 733,446 1.1% 793,547 1.1% 475,947 0.6% 240000 Instructional Aides, Non-Perm 114,784 0.2% 127,313 0.2% 89,000 0.1% Total Non-Certificated Salaries 10,897,543 16.8% 12,692,325 17.8% 13,965,438 18.2% 390000 Misc Employee Benefits 15,203,794 23.4% 17,107,660 24.0% 15,662,515 20.4% Total Benefits 15,203,794 23.4% 17,107,660 24.0% 15,662,515 20.4% 420000 Books 0 0.0% 0 0.0% 150 0.0% 440000 Instructional Media Materials 27,237 0.0% 11 0.0% 764,551 1.0% 450000 Supplies 231,717 0.4% 348,163 0.5% 461,831 0.6% Total Printing & Supplies 258,954 0.4% 348,173 0.5% 1,226,532 1.6% 550000 Utilities & Housekeeping Expense 2,721,790 4.2% 2,985,531 4.2% 3,314,664 4.3% 560000 Contracts & Rentals 294,157 0.5% 297,230 0.4% 1,104,142 1.4% 580000 Other Expense 592,648 0.9% 1,328,806 1.9% 2,598,577 3.4% Total Operating Expenses 3,608,594 5.5% 4,611,567 6.5% 7,017,383 9.1% 630000 Books & Materials for Libraries 30,162 0.0% 21,179 0.0% 120,000 0.2% 640000 Equipment 315,602 0.5% 423,110 0.6% 735,396 1.0% 650000 Lease/Purchase 34,533 0.1% 5,400 0.0% 102,223 0.1% Total Capital Outlay 380,297 0.6% 449,689 0.6% 957,619 1.2% 730000 Interfund Transfers 241,617 0.4% 940 0.0% 0 0.0% 739900 Intrafund Transfer - Restr/Unrestr 131,436 0.2% 108,121 0.2% 22,339 0.0% 740000 Reallocations/Adjustments 131,147 0.2% 127,929 0.2% 0 0.0% 750000 Loans/Grants 0 0.0% 0 0.0% 300 0.0% 780000 Enroll - Bad Debt Expense 0 0.0% 853,974 1.2% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 735,030 1.0% Total Other 504,200 0.8% 1,090,964 1.5% 757,669 1.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 65,039,464 100.0% 71,421,457 100.0% 76,844,245 100.0% Los Angeles Community College District 2023-2024 Final Budget 26 Los Angeles City College Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 40,408,309 62.1% 42,437,642 59.4% 45,600,575 59.3% 6000 Instructional Support 3,365,762 5.2% 3,572,996 5.0% 3,990,597 5.2% 6100 Other Instructional Services 650,734 1.0% 1,316,714 1.8% 1,387,859 1.8% Student Services 6200 Admissions and Records 1,351,752 2.1% 1,641,090 2.3% 1,601,961 2.1% 6300 Counseling and Guidance 2,009,880 3.1% 1,731,402 2.4% 1,979,680 2.6% 6400 Other Student Services 3,077,837 4.7% 3,618,065 5.1% 3,255,668 4.2% Total Student Services 6,439,469 9.9% 6,990,556 9.8% 6,837,309 8.9% 6500 Maintenance and Operations 9,929,005 15.3% 11,310,266 15.8% 12,496,285 16.3% Institutional Support 6600 Planning and Policymaking 736,026 1.1% 954,468 1.3% 1,176,041 1.5% 6700 General Institutional Support 2,921,625 4.5% 4,481,317 6.3% 4,866,056 6.3% Total Institutional Support 3,657,650 5.6% 5,435,785 7.6% 6,042,097 7.9% 6800 Community Service 0 0.0% 0 0.0% 500 0.0% 6900 Ancillary Services 291,582 0.4% 338,126 0.5% 489,023 0.6% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 296,953 0.5% 19,372 0.0% 0 0.0% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 65,039,464 100.0% 71,421,457 100.0% 76,844,245 100.0% Los Angeles Community College District 2023-2024 Final Budget 27 East Los Angeles College Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 27,543,447 22.2% 30,617,251 21.3% 31,993,624 20.3% 120000 Non-Teaching, Regular 10,572,027 8.5% 12,109,831 8.4% 12,368,965 7.8% 130000 Teaching, Hourly 23,378,977 18.8% 26,975,618 18.7% 41,753,089 26.5% 140000 Non-Teaching, Hourly 1,531,427 1.2% 1,812,513 1.3% 1,803,510 1.1% Total Certificated Salaries 63,025,878 50.8% 71,515,214 49.7% 87,919,188 55.7% 210000 Classified, Regular 15,693,666 12.6% 17,277,021 12.0% 18,163,419 11.5% 220000 Instructional Aides, Regular 3,200,602 2.6% 3,526,112 2.4% 3,635,958 2.3% 230000 Sub/Relief, Unclassified 574,730 0.5% 906,326 0.6% 868,653 0.6% 240000 Instructional Aides, Non-Perm 527,130 0.4% 674,484 0.5% 600,618 0.4% Total Non-Certificated Salaries 19,996,128 16.1% 22,383,942 15.5% 23,268,648 14.7% 390000 Misc Employee Benefits 30,489,270 24.6% 34,821,237 24.2% 27,791,078 17.6% Total Benefits 30,489,270 24.6% 34,821,237 24.2% 27,791,078 17.6% 420000 Books (2,273) 0.0% 697 0.0% 500 0.0% 440000 Instructional Media Materials 0 0.0% 5,092 0.0% 5,682 0.0% 450000 Supplies 468,273 0.4% 661,649 0.5% 579,533 0.4% Total Printing & Supplies 466,000 0.4% 667,438 0.5% 585,715 0.4% 550000 Utilities & Housekeeping Expense 3,859,145 3.1% 4,666,257 3.2% 4,417,866 2.8% 560000 Contracts & Rentals 4,196,500 3.4% 5,927,202 4.1% 8,802,658 5.6% 580000 Other Expense 1,298,957 1.0% 1,263,482 0.9% 2,359,507 1.5% Total Operating Expenses 9,354,601 7.5% 11,856,941 8.2% 15,580,031 9.9% 620000 Buildings 0 0.0% 0 0.0% 0 0.0% 640000 Equipment 150,057 0.1% 504,292 0.4% 445,907 0.3% 650000 Lease/Purchase 53,558 0.0% 69,940 0.0% 328,146 0.2% Total Capital Outlay 203,614 0.2% 574,231 0.4% 774,053 0.5% 730000 Interfund Transfers 390,023 0.3% 503,408 0.3% 462,122 0.3% 739900 Intrafund Transfer - Restr/Unrestr 102,727 0.1% 56,920 0.0% 0 0.0% 740000 Reallocations/Adjustments 157,286 0.1% 30,927 0.0% 0 0.0% 780000 Enroll - Bad Debt Expense 0 0.0% 1,591,845 1.1% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 1,420,593 0.9% Total Other 650,036 0.5% 2,183,100 1.5% 1,882,715 1.2% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 124,185,527 100.0% 144,002,103 100.0% 157,801,428 100.0% Los Angeles Community College District 2023-2024 Final Budget 28 East Los Angeles College Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 79,174,123 63.8% 90,994,178 63.2% 105,660,328 67.0% 6000 Instructional Support 4,907,172 4.0% 5,792,431 4.0% 6,093,906 3.9% 6100 Other Instructional Services 3,715,373 3.0% 4,036,444 2.8% 4,020,911 2.5% Student Services 6200 Admissions and Records 2,180,984 1.8% 2,369,136 1.6% 2,402,073 1.5% 6300 Counseling and Guidance 3,852,657 3.1% 4,172,218 2.9% 4,180,057 2.6% 6400 Other Student Services 4,073,191 3.3% 4,159,771 2.9% 4,367,472 2.8% Total Student Services 10,106,832 8.1% 10,701,126 7.4% 10,949,602 6.9% 6500 Maintenance and Operations 15,260,235 12.3% 18,373,948 12.8% 18,109,357 11.5% Institutional Support 6600 Planning and Policymaking 1,866,329 1.5% 2,499,367 1.7% 2,747,060 1.7% 6700 General Institutional Support 6,577,020 5.3% 8,373,192 5.8% 7,681,885 4.9% Total Institutional Support 8,443,349 6.8% 10,872,559 7.6% 10,428,945 6.6% 6800 Community Service 391,181 0.3% 437,079 0.3% 419,453 0.3% 6900 Ancillary Services 1,792,956 1.4% 2,240,562 1.6% 1,656,804 1.0% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 394,305 0.3% 553,775 0.4% 462,122 0.3% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 124,185,527 100.0% 144,002,103 100.0% 157,801,428 100.0% Los Angeles Community College District 2023-2024 Final Budget 29 Los Angeles Harbor College Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 6,723,671 18.0% 7,382,605 17.1% 7,916,831 17.8% 120000 Non-Teaching, Regular 3,750,565 10.0% 4,652,156 10.8% 4,157,304 9.3% 130000 Teaching, Hourly 6,521,857 17.4% 7,567,631 17.5% 6,056,649 13.6% 140000 Non-Teaching, Hourly 535,542 1.4% 454,918 1.1% 535,554 1.2% Total Certificated Salaries 17,531,636 46.9% 20,057,310 46.5% 18,666,338 41.9% 210000 Classified, Regular 5,467,555 14.6% 6,306,776 14.6% 6,643,540 14.9% 220000 Instructional Aides, Regular 736,910 2.0% 756,476 1.8% 747,460 1.7% 230000 Sub/Relief, Unclassified 563,831 1.5% 448,934 1.0% 417,729 0.9% 240000 Instructional Aides, Non-Perm 204,537 0.5% 227,741 0.5% 254,316 0.6% Total Non-Certificated Salaries 6,972,832 18.7% 7,739,927 17.9% 8,063,045 18.1% 350000 State Unemployment Insurance (30,469) -0.1% 0 0.0% 0 0.0% 390000 Misc Employee Benefits 9,044,784 24.2% 10,730,969 24.9% 10,663,937 23.9% Total Benefits 9,014,315 24.1% 10,730,969 24.9% 10,663,937 23.9% 420000 Books 0 0.0% 0 0.0% 0 0.0% 440000 Instructional Media Materials 262 0.0% 120 0.0% 1,487 0.0% 450000 Supplies 381,609 1.0% 568,381 1.3% 683,511 1.5% Total Printing & Supplies 381,871 1.0% 568,502 1.3% 684,998 1.5% 540000 Insurance 0 0.0% 0 0.0% 6,000 0.0% 550000 Utilities & Housekeeping Expense 1,860,162 5.0% 1,849,573 4.3% 2,583,159 5.8% 560000 Contracts & Rentals 260,937 0.7% 391,904 0.9% 682,784 1.5% 580000 Other Expense 228,103 0.6% 616,468 1.4% 1,279,414 2.9% Total Operating Expenses 2,349,203 6.3% 2,857,946 6.6% 4,551,357 10.2% 620000 Buildings 0 0.0% 0 0.0% 5,000 0.0% 640000 Equipment 585,419 1.6% 274,366 0.6% 611,350 1.4% 650000 Lease/Purchase 9,731 0.0% 20,593 0.0% 78,170 0.2% Total Capital Outlay 595,151 1.6% 294,959 0.7% 694,520 1.6% 730000 Interfund Transfers 274,545 0.7% 175,850 0.4% 434,651 1.0% 739900 Intrafund Transfer - Restr/Unrestr 214,957 0.6% 142,593 0.3% 47,937 0.1% 740000 Reallocations/Adjustments 53,184 0.1% 88,337 0.2% 0 0.0% 780000 Enroll - Bad Debt Expense 0 0.0% 505,766 1.2% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 765,469 1.7% Total Other 542,686 1.5% 912,545 2.1% 1,248,057 2.8% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 37,387,694 100.0% 43,162,157 100.0% 44,572,252 100.0% Los Angeles Community College District 2023-2024 Final Budget 30 Los Angeles Harbor College Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 19,292,673 51.6% 21,602,638 50.0% 21,400,019 48.0% 6000 Instructional Support 1,466,283 3.9% 2,256,198 5.2% 1,963,341 4.4% 6100 Other Instructional Services 874,443 2.3% 823,972 1.9% 992,227 2.2% Student Services 6200 Admissions and Records 1,153,294 3.1% 1,396,279 3.2% 1,234,840 2.8% 6300 Counseling and Guidance 646,591 1.7% 1,548,228 3.6% 1,052,690 2.4% 6400 Other Student Services 1,421,905 3.8% 1,561,188 3.6% 1,631,202 3.7% Total Student Services 3,221,789 8.6% 4,505,695 10.4% 3,918,732 8.8% 6500 Maintenance and Operations 6,502,956 17.4% 7,184,645 16.6% 8,495,358 19.1% Institutional Support 6600 Planning and Policymaking 1,189,709 3.2% 1,419,840 3.3% 1,924,848 4.3% 6700 General Institutional Support 3,134,384 8.4% 3,932,011 9.1% 4,483,397 10.1% Total Institutional Support 4,324,093 11.6% 5,351,851 12.4% 6,408,245 14.4% 6800 Community Service 161,066 0.4% 55,823 0.1% 0 0.0% 6900 Ancillary Services 1,198,166 3.2% 1,151,166 2.7% 959,679 2.2% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 346,224 0.9% 230,169 0.5% 434,651 1.0% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 37,387,694 100.0% 43,162,157 100.0% 44,572,252 100.0% Los Angeles Community College District 2023-2024 Final Budget 31 Los Angeles Mission College Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 6,644,180 17.1% 6,697,502 15.1% 8,452,993 18.7% 120000 Non-Teaching, Regular 3,973,162 10.2% 3,675,540 8.3% 4,005,790 8.8% 130000 Teaching, Hourly 8,651,660 22.2% 10,245,688 23.2% 9,378,604 20.7% 140000 Non-Teaching, Hourly 561,592 1.4% 663,749 1.5% 186,250 0.4% 190000 Misc Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Certificated Salaries 19,830,594 50.9% 21,282,479 48.1% 22,023,637 48.6% 210000 Classified, Regular 6,137,863 15.8% 6,437,601 14.6% 6,822,726 15.1% 220000 Instructional Aides, Regular 763,214 2.0% 865,925 2.0% 940,618 2.1% 230000 Sub/Relief, Unclassified 216,990 0.6% 403,301 0.9% 200,719 0.4% 240000 Instructional Aides, Non-Perm 184,390 0.5% 229,860 0.5% 106,000 0.2% Total Non-Certificated Salaries 7,302,457 18.8% 7,936,687 17.9% 8,070,063 17.8% 390000 Misc Employee Benefits 9,529,366 24.5% 10,742,937 24.3% 11,104,505 24.5% Total Benefits 9,529,366 24.5% 10,742,937 24.3% 11,104,505 24.5% 420000 Books 0 0.0% (10) 0.0% 0 0.0% 440000 Instructional Media Materials 10,606 0.0% 0 0.0% 10,436 0.0% 450000 Supplies 50,845 0.1% 67,430 0.2% 60,583 0.1% Total Printing & Supplies 61,451 0.2% 67,420 0.2% 71,019 0.2% 550000 Utilities & Housekeeping Expense 1,457,965 3.7% 2,247,989 5.1% 2,150,409 4.7% 560000 Contracts & Rentals 147,161 0.4% 436,285 1.0% 302,746 0.7% 580000 Other Expense 460,808 1.2% 517,858 1.2% 736,062 1.6% 590000 Misc Other Expense 0 0.0% 23,340 0.1% 0 0.0% Total Operating Expenses 2,065,934 5.3% 3,225,472 7.3% 3,189,217 7.0% 640000 Equipment 37,425 0.1% 61,653 0.1% 25,778 0.1% 650000 Lease/Purchase 4,978 0.0% 14,283 0.0% 39,916 0.1% Total Capital Outlay 42,402 0.1% 75,936 0.2% 65,694 0.1% 730000 Interfund Transfers 79,041 0.2% 217,456 0.5% 287,122 0.6% 739900 Intrafund Transfer - Restr/Unrestr 25,958 0.1% 63,868 0.1% 9,229 0.0% 740000 Reallocations/Adjustments 7,444 0.0% 15,240 0.0% 0 0.0% 750000 Loans/Grants 0 0.0% 1,186 0.0% 0 0.0% 780000 Enroll - Bad Debt Expense 0 0.0% 596,223 1.3% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 452,754 1.0% Total Other 112,443 0.3% 893,974 2.0% 749,105 1.7% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 38,944,648 100.0% 44,224,904 100.0% 45,273,240 100.0% 2021-22 Expenditures include Mission and ITV. Los Angeles Community College District 2023-2024 Final Budget 32 Los Angeles Mission College Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 21,359,719 54.8% 24,001,815 54.3% 25,116,889 55.5% 6000 Instructional Support 2,445,139 6.3% 2,123,876 4.8% 1,952,133 4.3% 6100 Other Instructional Services 797,571 2.0% 872,801 2.0% 903,952 2.0% Student Services 6200 Admissions and Records 674,722 1.7% 715,282 1.6% 718,172 1.6% 6300 Counseling and Guidance 1,349,831 3.5% 1,574,368 3.6% 1,933,960 4.3% 6400 Other Student Services 1,355,305 3.5% 1,207,479 2.7% 1,492,326 3.3% Total Student Services 3,379,859 8.7% 3,497,129 7.9% 4,144,458 9.2% 6500 Maintenance and Operations 6,366,634 16.3% 7,516,533 17.0% 7,544,046 16.7% Institutional Support 6600 Planning and Policymaking 994,195 2.6% 877,701 2.0% 869,886 1.9% 6700 General Institutional Support 2,779,555 7.1% 3,898,741 8.8% 3,153,318 7.0% Total Institutional Support 3,773,750 9.7% 4,776,442 10.8% 4,023,204 8.9% 6800 Community Service 0 0.0% 0 0.0% 0 0.0% 6900 Ancillary Services 741,801 1.9% 1,173,027 2.7% 1,301,436 2.9% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 80,176 0.2% 263,280 0.6% 287,122 0.6% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 38,944,648 100.0% 44,224,904 100.0% 45,273,240 100.0% 2021-22 Expenditures include Mission and ITV. Los Angeles Community College District 2023-2024 Final Budget 33 Los Angeles Pierce College Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 19,940,917 28.4% 21,888,155 23.7% 21,747,963 22.3% 120000 Non-Teaching, Regular 5,983,270 8.5% 8,904,150 9.6% 8,417,007 8.6% 130000 Teaching, Hourly 13,216,923 18.8% 16,055,923 17.4% 23,841,823 24.5% 140000 Non-Teaching, Hourly 410,436 0.6% 539,115 0.6% 243,704 0.3% Total Certificated Salaries 39,551,545 56.3% 47,387,343 51.3% 54,250,497 55.7% 210000 Classified, Regular 8,008,291 11.4% 11,165,175 12.1% 11,668,332 12.0% 220000 Instructional Aides, Regular 1,123,481 1.6% 2,267,356 2.5% 2,567,156 2.6% 230000 Sub/Relief, Unclassified 311,218 0.4% 888,673 1.0% 50,015 0.1% 240000 Instructional Aides, Non-Perm 271,829 0.4% 492,964 0.5% 295,013 0.3% Total Non-Certificated Salaries 9,714,820 13.8% 14,814,167 16.0% 14,580,516 15.0% 310000 STRS Employer Contributions (83) 0.0% 0 0.0% 0 0.0% 390000 Misc Employee Benefits 20,329,075 28.9% 23,659,832 25.6% 22,125,182 22.7% Total Benefits 20,328,992 28.9% 23,659,832 25.6% 22,125,182 22.7% 440000 Instructional Media Materials 0 0.0% 0 0.0% 1,951 0.0% 450000 supplies 240,520 0.3% 196,943 0.2% 148,253 0.2% Total Printing & Supplies 240,520 0.3% 196,943 0.2% 150,204 0.2% 550000 Utilities & Housekeeping Expense 1,182,955 1.7% 3,100,740 3.4% 2,645,461 2.7% 560000 Contracts & Rentals 170,018 0.2% 340,147 0.4% 568,384 0.6% 580000 Other Expense 585,263 0.8% 1,327,546 1.4% 1,501,378 1.5% 590000 Misc Other Expense 0 0.0% 0 0.0% 294,926 0.3% Total Operating Expenses 1,938,236 2.8% 4,768,433 5.2% 5,010,149 5.1% 640000 Equipment 3,978 0.0% 172,654 0.2% 224,928 0.2% 650000 Lease/Purchase 0 0.0% 13,100 0.0% 32,621 0.0% Total Capital Outlay 3,978 0.0% 185,754 0.2% 257,549 0.3% 739900 Intrafund Transfer - Restr/Unrestr 98,466 0.1% 73,974 0.1% 1 0.0% 740000 Reallocations/Adjustments (1,643,633) -2.3% 42,826 0.0% 0 0.0% 780000 Enroll - Bad Debt Expense 0 0.0% 1,171,659 1.3% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 6,946 0.0% 951,121 1.0% Total Other (1,545,167) -2.2% 1,295,406 1.4% 951,122 1.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 70,232,923 100.0% 92,307,878 100.0% 97,325,219 100.0% Los Angeles Community College District 2023-2024 Final Budget 34 Los Angeles Pierce College Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 43,702,121 62.2% 56,399,788 61.1% 63,809,414 65.6% 6000 Instructional Support 2,642,666 3.8% 2,560,108 2.8% 2,370,460 2.4% 6100 Other Instructional Services 1,768,542 2.5% 1,788,094 1.9% 1,977,569 2.0% Student Services 6200 Admissions and Records 1,584,958 2.3% 1,890,968 2.0% 1,625,488 1.7% 6300 Counseling and Guidance 2,742,286 3.9% 2,793,423 3.0% 2,914,043 3.0% 6400 Other Student Services 2,219,773 3.2% 3,394,598 3.7% 3,449,345 3.5% Total Student Services 6,547,018 9.3% 8,078,989 8.8% 7,988,876 8.2% 6500 Maintenance and Operations 10,284,769 14.6% 14,664,337 15.9% 13,935,043 14.3% Institutional Support 6600 Planning and Policymaking 905,918 1.3% 940,228 1.0% 1,008,182 1.0% 6700 General Institutional Support 1,646,053 2.3% 5,104,778 5.5% 4,854,685 5.0% Total Institutional Support 2,551,971 3.6% 6,045,006 6.5% 5,862,867 6.0% 6800 Community Service 1,259,920 1.8% 1,292,625 1.4% 146,921 0.2% 6900 Ancillary Services 1,412,955 2.0% 1,472,307 1.6% 1,234,069 1.3% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 62,961 0.1% 6,624 0.0% 0 0.0% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 70,232,923 100.0% 92,307,878 100.0% 97,325,219 100.0% Los Angeles Community College District 2023-2024 Final Budget 35 Los Angeles Southwest College Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 4,485,518 14.6% 4,868,788 15.2% 5,020,213 12.7% 120000 Non-Teaching, Regular 3,990,991 13.0% 3,907,469 12.2% 4,056,892 10.2% 130000 Teaching, Hourly 5,827,048 19.0% 6,462,866 20.1% 12,261,452 30.9% 140000 Non-Teaching, Hourly 315,120 1.0% 370,424 1.2% 176,338 0.4% Total Certificated Salaries 14,618,676 47.7% 15,609,546 48.7% 21,514,895 54.2% 210000 Classified, Regular 4,747,358 15.5% 4,682,302 14.6% 5,372,296 13.5% 220000 Instructional Aides, Regular 347,975 1.1% 463,149 1.4% 481,436 1.2% 230000 Sub/Relief, Unclassified 129,111 0.4% 252,412 0.8% 101,421 0.3% 240000 Instructional Aides, Non-Perm 107,226 0.3% 145,824 0.5% 131,670 0.3% Total Non-Certificated Salaries 5,331,670 17.4% 5,543,687 17.3% 6,086,823 15.3% 390000 Misc Employee Benefits 7,085,174 23.1% 7,946,446 24.8% 6,684,518 16.9% Total Benefits 7,085,174 23.1% 7,946,446 24.8% 6,684,518 16.9% 440000 Instructional Media Materials 0 0.0% 0 0.0% 0 0.0% 450000 Supplies 55,939 0.2% 38,808 0.1% 163,988 0.4% Total Printing & Supplies 55,939 0.2% 38,808 0.1% 163,988 0.4% 550000 Utilities & Housekeeping Expense 1,943,232 6.3% 1,946,436 6.1% 2,455,998 6.2% 560000 Contracts & Rentals 536,419 1.7% 243,783 0.8% 689,572 1.7% 580000 Other Expense 805,193 2.6% 370,480 1.2% 1,273,470 3.2% Total Operating Expenses 3,284,844 10.7% 2,560,698 8.0% 4,419,040 11.1% 640000 Equipment 142,106 0.5% 8,184 0.0% 34,119 0.1% 650000 Lease/Purchase 30,681 0.1% 194,588 0.6% 402,477 1.0% Total Capital Outlay 172,787 0.6% 202,772 0.6% 436,596 1.1% 739900 Intrafund Transfer - Restr/Unrestr 114,624 0.4% 12,976 0.0% 0 0.0% 740000 Reallocations/Adjustments 15,507 0.1% (150,230) -0.5% 0 0.0% 780000 Enroll - Bad Debt Expense 0 0.0% 312,532 1.0% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 354,209 0.9% Total Other 130,131 0.4% 175,278 0.5% 354,209 0.9% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 30,679,223 100.0% 32,077,236 100.0% 39,660,069 100.0% Los Angeles Community College District 2023-2024 Final Budget 36 Los Angeles Southwest College Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 14,441,266 47.1% 15,923,271 49.6% 21,551,276 54.3% 6000 Instructional Support 2,703,436 8.8% 4,394,317 13.7% 2,073,934 5.2% 6100 Other Instructional Services 1,287,627 4.2% 1,219,674 3.8% 1,525,601 3.8% Student Services 6200 Admissions and Records 669,866 2.2% 594,627 1.9% 843,542 2.1% 6300 Counseling and Guidance 1,057,168 3.4% 1,098,521 3.4% 1,407,986 3.6% 6400 Other Student Services 1,338,012 4.4% 2,353,446 7.3% 1,454,910 3.7% Total Student Services 3,065,046 10.0% 4,046,593 12.6% 3,706,438 9.3% 6500 Maintenance and Operations 5,035,360 16.4% 4,799,667 15.0% 6,213,805 15.7% Institutional Support 6600 Planning and Policymaking 2,770,697 9.0% 1,502,434 4.7% 1,534,868 3.9% 6700 General Institutional Support 859,519 2.8% (271,421) -0.8% 2,521,853 6.4% Total Institutional Support 3,630,217 11.8% 1,231,013 3.8% 4,056,721 10.2% 6800 Community Service 195,000 0.6% 156,753 0.5% 157,546 0.4% 6900 Ancillary Services 250,627 0.8% 294,346 0.9% 374,748 0.9% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 70,644 0.2% 11,601 0.0% 0 0.0% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 30,679,223 100.0% 32,077,236 100.0% 39,660,069 100.0% Los Angeles Community College District 2023-2024 Final Budget 37 Los Angeles Trade-Technical College Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 14,891,110 24.5% 16,920,098 22.6% 18,801,297 20.4% 120000 Non-Teaching, Regular 6,014,118 9.9% 6,751,779 9.0% 4,849,218 5.3% 130000 Teaching, Hourly 10,652,119 17.5% 12,928,391 17.2% 17,970,302 19.5% 140000 Non-Teaching, Hourly 227,903 0.4% 479,214 0.6% 394,841 0.4% Total Certificated Salaries 31,785,250 52.3% 37,079,482 49.4% 42,015,658 45.7% 210000 Classified, Regular 8,944,465 14.7% 10,434,503 13.9% 12,714,441 13.8% 220000 Instructional Aides, Regular 1,143,585 1.9% 1,547,693 2.1% 1,560,183 1.7% 230000 Sub/Relief, Unclassified 464,938 0.8% 996,517 1.3% 1,326,906 1.4% 240000 Instructional Aides, Non-Perm 61,250 0.1% 67,584 0.1% 193,423 0.2% Total Non-Certificated Salaries 10,614,239 17.5% 13,046,296 17.4% 15,794,953 17.2% 390000 Misc Employee Benefits 15,163,072 24.9% 18,901,652 25.2% 16,973,921 18.5% Total Benefits 15,163,072 24.9% 18,901,652 25.2% 16,973,921 18.5% 420000 Books 0 0.0% 3,432 0.0% 4,558 0.0% 440000 Instructional Media Materials 31,159 0.1% 185,693 0.2% 713,289 0.8% 450000 Supplies 549,599 0.9% 742,675 1.0% 1,796,009 2.0% Total Printing & Supplies 580,758 1.0% 931,800 1.2% 2,513,856 2.7% 550000 Utilities & Housekeeping Expense 531,972 0.9% 913,891 1.2% 3,276,032 3.6% 560000 Contracts & Rentals 272,353 0.4% 162,142 0.2% 2,031,333 2.2% 580000 Other Expense 1,183,920 1.9% 1,838,899 2.5% 4,898,878 5.3% 590000 Misc Other Expense 0 0.0% 0 0.0% 0 0.0% Total Operating Expenses 1,988,245 3.3% 2,914,932 3.9% 10,206,243 11.1% 630000 Books & Materials for Libraries 0 0.0% 0 0.0% 542 0.0% 640000 Equipment 139,635 0.2% 369,543 0.5% 1,549,118 1.7% 650000 Lease/Purchase 105,023 0.2% 60,152 0.1% 340,285 0.4% Total Capital Outlay 244,659 0.4% 429,695 0.6% 1,889,945 2.1% 730000 Interfund Transfers 38,889 0.1% 498,365 0.7% 247,315 0.3% 739900 Intrafund Transfer - Restr/Unrestr 76,704 0.1% 338,710 0.5% 0 0.0% 740000 Reallocations/Adjustments 289,050 0.5% 183,175 0.2% 0 0.0% 780000 Enroll - Bad Debt Expense 0 0.0% 704,604 0.9% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 1,632 0.0% 2,324,202 2.5% Total Other 404,643 0.7% 1,726,486 2.3% 2,571,517 2.8% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 60,780,866 100.0% 75,030,344 100.0% 91,966,093 100.0% Los Angeles Community College District 2023-2024 Final Budget 38 Los Angeles Trade-Technical College Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 36,827,814 60.6% 44,475,732 59.3% 51,843,468 56.4% 6000 Instructional Support 2,316,464 3.8% 3,346,686 4.5% 2,653,168 2.9% 6100 Other Instructional Services 857,815 1.4% 1,081,079 1.4% 1,138,579 1.2% Student Services 6200 Admissions and Records 1,094,535 1.8% 1,296,696 1.7% 1,596,049 1.7% 6300 Counseling and Guidance 2,107,052 3.5% 2,581,662 3.4% 1,911,012 2.1% 6400 Other Student Services 2,560,236 4.2% 3,487,121 4.6% 4,395,468 4.8% Total Student Services 5,761,822 9.5% 7,365,479 9.8% 7,902,529 8.6% 6500 Maintenance and Operations 7,889,586 13.0% 9,547,563 12.7% 15,194,069 16.5% Institutional Support 6600 Planning and Policymaking 1,170,709 1.9% 872,579 1.2% 1,418,587 1.5% 6700 General Institutional Support 5,073,393 8.3% 6,419,690 8.6% 10,359,648 11.3% Total Institutional Support 6,244,102 10.3% 7,292,269 9.7% 11,778,235 12.8% 6800 Community Service 123,248 0.2% 339,701 0.5% 207,760 0.2% 6900 Ancillary Services 689,004 1.1% 791,905 1.1% 1,000,970 1.1% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 71,011 0.1% 789,930 1.1% 247,315 0.3% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 60,780,866 100.0% 75,030,343 100.0% 91,966,093 100.0% Los Angeles Community College District 2023-2024 Final Budget 39 Los Angeles Valley College Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 15,181,541 23.0% 16,257,170 20.7% 17,899,383 21.4% 120000 Non-Teaching, Regular 6,319,191 9.6% 6,293,259 8.0% 6,083,336 7.3% 130000 Teaching, Hourly 12,709,127 19.3% 17,446,273 22.3% 20,581,044 24.6% 140000 Non-Teaching, Hourly 525,469 0.8% 804,413 1.0% 773,648 0.9% Total Certificated Salaries 34,735,329 52.7% 40,801,115 52.1% 45,337,411 54.1% 210000 Classified, Regular 8,695,910 13.2% 9,951,284 12.7% 10,982,567 13.1% 220000 Instructional Aides, Regular 1,405,326 2.1% 1,457,980 1.9% 1,610,118 1.9% 230000 Sub/Relief, Unclassified 280,520 0.4% 382,633 0.5% 430,220 0.5% 240000 Instructional Aides, Non-Perm 336,425 0.5% 367,351 0.5% 402,764 0.5% Total Non-Certificated Salaries 10,718,181 16.3% 12,159,248 15.5% 13,425,669 16.0% 390000 Misc Employee Benefits 16,019,456 24.3% 19,025,670 24.3% 18,269,888 21.8% Total Benefits 16,019,456 24.3% 19,025,670 24.3% 18,269,888 21.8% 420000 Books 985 0.0% 0 0.0% 1,500 0.0% 440000 Instructional Media Materials 0 0.0% 11,999 0.0% 13,571 0.0% 450000 Supplies 597,476 0.9% 460,292 0.6% 456,022 0.5% Total Printing & Supplies 598,461 0.9% 472,292 0.6% 471,093 0.6% 540000 Insurance 1,423 0.0% 1,423 0.0% 1,423 0.0% 550000 Utilities & Housekeeping Expense 2,391,293 3.6% 2,959,759 3.8% 3,175,547 3.8% 560000 Contracts & Rentals 304,417 0.5% 223,461 0.3% 503,598 0.6% 580000 Other Expense 619,440 0.9% 933,084 1.2% 1,624,017 1.9% Total Operating Expenses 3,316,574 5.0% 4,117,727 5.3% 5,304,585 6.3% 640000 Equipment 23,714 0.0% 129,901 0.2% 109,514 0.1% 650000 Lease/Purchase 13,138 0.0% 11,216 0.0% 16,907 0.0% Total Capital Outlay 36,853 0.1% 141,117 0.2% 126,421 0.2% 730000 Interfund Transfers 0 0.0% 240 0.0% 0 0.0% 739900 Intrafund Transfer - Restr/Unrestr 440,845 0.7% 643,271 0.8% 114,389 0.1% 740000 Reallocations/Adjustments 75,315 0.1% 85,229 0.1% 0 0.0% 780000 Enroll - Bad Debt Expense 0 0.0% 919,452 1.2% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 754,455 0.9% Total Other 516,160 0.8% 1,648,192 2.1% 868,844 1.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 65,941,014 100.0% 78,365,362 100.0% 83,803,911 100.0% Los Angeles Community College District 2023-2024 Final Budget 40 Los Angeles Valley College Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 39,822,635 60.4% 48,706,321 62.2% 53,175,665 63.5% 6000 Instructional Support 2,459,549 3.7% 2,925,536 3.7% 2,939,331 3.5% 6100 Other Instructional Services 2,127,901 3.2% 2,635,355 3.4% 2,514,467 3.0% Student Services 6200 Admissions and Records 1,594,401 2.4% 1,891,166 2.4% 2,041,829 2.4% 6300 Counseling and Guidance 1,331,540 2.0% 284,350 0.4% 217,254 0.3% 6400 Other Student Services 3,443,574 5.2% 3,455,999 4.4% 3,249,290 3.9% Total Student Services 6,369,515 9.7% 5,631,515 7.2% 5,508,373 6.6% 6500 Maintenance and Operations 8,761,244 13.3% 10,297,563 13.1% 11,369,212 13.6% Institutional Support 6600 Planning and Policymaking 1,016,020 1.5% 1,265,529 1.6% 1,315,090 1.6% 6700 General Institutional Support 4,475,308 6.8% 5,944,827 7.6% 6,045,012 7.2% Total Institutional Support 5,491,328 8.3% 7,210,355 9.2% 7,360,102 8.8% 6800 Community Service 16,876 0.0% 34,356 0.0% 122,214 0.1% 6900 Ancillary Services 891,965 1.4% 924,121 1.2% 814,547 1.0% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 0 0.0% 240 0.0% 0 0.0% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 65,941,014 100.0% 78,365,362 100.0% 83,803,911 100.0% Los Angeles Community College District 2023-2024 Final Budget 41 West Los Angeles College Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 8,631,591 19.1% 9,445,312 19.5% 10,004,002 18.5% 120000 Non-Teaching, Regular 3,024,685 6.7% 3,763,574 7.8% 3,866,158 7.2% 130000 Teaching, Hourly 10,560,395 23.4% 10,740,743 22.2% 14,721,273 27.3% 140000 Non-Teaching, Hourly 637,213 1.4% 669,823 1.4% 560,800 1.0% Total Certificated Salaries 22,853,883 50.7% 24,619,452 50.9% 29,152,233 54.0% 210000 Classified, Regular 6,780,600 15.0% 7,055,134 14.6% 8,221,726 15.2% 220000 Instructional Aides, Regular 799,917 1.8% 962,310 2.0% 1,021,933 1.9% 230000 Sub/Relief, Unclassified 156,317 0.3% 212,397 0.4% 91,044 0.2% 240000 Instructional Aides, Non-Perm 229,614 0.5% 324,474 0.7% 262,634 0.5% Total Non-Certificated Salaries 7,966,448 17.7% 8,554,315 17.7% 9,597,337 17.8% 390000 Misc Employee Benefits 10,741,855 23.8% 12,143,481 25.1% 10,432,758 19.3% Total Benefits 10,741,855 23.8% 12,143,481 25.1% 10,432,758 19.3% 440000 Instructional Media Materials 0 0.0% (450) 0.0% 0 0.0% 450000 Supplies 223,712 0.5% 205,538 0.4% 218,003 0.4% Total Printing & Supplies 223,712 0.5% 205,088 0.4% 218,003 0.4% 550000 Utilities & Housekeeping Expense 1,771,625 3.9% 1,653,392 3.4% 2,185,988 4.1% 560000 Contracts & Rentals 944,261 2.1% 533,934 1.1% 1,038,591 1.9% 580000 Other Expense 196,625 0.4% 110,312 0.2% 435,484 0.8% Total Operating Expenses 2,912,511 6.5% 2,297,638 4.7% 3,660,063 6.8% 640000 Equipment 33,456 0.1% 67,618 0.1% 62,334 0.1% 650000 Lease/Purchase 4,506 0.0% 780 0.0% 6,885 0.0% Total Capital Outlay 37,962 0.1% 68,397 0.1% 69,219 0.1% 730000 Interfund Transfers 215,718 0.5% 0 0.0% 0 0.0% 739900 Intrafund Transfer - Restr/Unrestr 94,350 0.2% 2,694 0.0% 0 0.0% 740000 Reallocations/Adjustments 56,690 0.1% (166,333) -0.3% 0 0.0% 780000 Enroll - Bad Debt Expense 0 0.0% 650,577 1.3% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 826,767 1.5% Total Other 366,758 0.8% 486,938 1.0% 826,767 1.5% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 45,103,129 100.0% 48,375,309 100.0% 53,956,380 100.0% Los Angeles Community College District 2023-2024 Final Budget 42 West Los Angeles College Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 26,535,898 58.8% 28,512,725 58.9% 32,385,165 60.0% 6000 Instructional Support 1,752,398 3.9% 1,897,802 3.9% 1,915,147 3.5% 6100 Other Instructional Services 1,618,693 3.6% 1,663,406 3.4% 1,630,212 3.0% Student Services 6200 Admissions and Records 1,067,475 2.4% 952,048 2.0% 1,226,982 2.3% 6300 Counseling and Guidance 159,079 0.4% 808,498 1.7% 672,012 1.2% 6400 Other Student Services 1,770,688 3.9% 1,726,945 3.6% 1,879,767 3.5% Total Student Services 2,997,242 6.6% 3,487,491 7.2% 3,778,761 7.0% 6500 Maintenance and Operations 6,583,428 14.6% 7,285,259 15.1% 8,450,137 15.7% Institutional Support 6600 Planning and Policymaking 1,427,662 3.2% 1,565,750 3.2% 1,773,573 3.3% 6700 General Institutional Support 2,960,462 6.6% 2,733,442 5.7% 2,836,962 5.3% Total Institutional Support 4,388,124 9.7% 4,299,192 8.9% 4,610,535 8.5% 6800 Community Service 0 0.0% (2,136) 0.0% 0 0.0% 6900 Ancillary Services 918,763 2.0% 1,231,570 2.5% 1,186,423 2.2% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 308,582 0.7% 0 0.0% 0 0.0% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 45,103,129 100.0% 48,375,309 100.0% 53,956,380 100.0% Los Angeles Community College District 2023-2024 Final Budget 43 Educational Services Center Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 120000 Non-Teaching, Regular 2,239,329 7.1% 2,419,913 6.8% 2,545,207 6.6% 130000 Teaching, Hourly 12,710 0.0% 16,710 0.0% 0 0.0% 140000 Non-Teaching, Hourly 9,042 0.0% 0 0.0% 0 0.0% Total Certificated Salaries 2,261,081 7.2% 2,436,623 6.8% 2,545,207 6.6% 210000 Classified, Regular 16,231,317 51.5% 18,210,209 50.9% 19,467,689 50.2% 230000 Sub/Relief, Unclassified 546,506 1.7% 790,315 2.2% 268,275 0.7% 240000 Instructional Aides, Non-Perm 3,477 0.0% 1,397 0.0% 0 0.0% Total Non-Certificated Salaries 16,781,300 53.3% 19,001,921 53.1% 19,735,964 50.9% 390000 Misc Employee Benefits 9,875,563 31.4% 11,091,610 31.0% 11,252,895 29.0% Total Benefits 9,875,563 31.4% 11,091,610 31.0% 11,252,895 29.0% 450000 Supplies 74,909 0.2% 85,270 0.2% 169,314 0.4% Total Printing & Supplies 74,909 0.2% 85,270 0.2% 169,314 0.4% 550000 Utilities & Housekeeping Expense 46,604 0.1% 42,117 0.1% 25,000 0.1% 560000 Contracts & Rentals 350,873 1.1% 479,441 1.3% 772,251 2.0% 570000 Legal, Election, Audit 27,260 0.1% 0 0.0% 74,740 0.2% 580000 Other Expense 2,005,433 6.4% 2,541,572 7.1% 3,566,794 9.2% 590000 Misc Other Expense 0 0.0% 0 0.0% 50,000 0.1% Total Operating Expenses 2,430,170 7.7% 3,063,131 8.6% 4,488,785 11.6% 640000 Equipment 0 0.0% 19,695 0.1% 93,240 0.2% 650000 Lease/Purchase 63,808 0.2% 80,045 0.2% 124,089 0.3% Total Capital Outlay 63,808 0.2% 99,741 0.3% 217,329 0.6% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 384,085 1.0% Total Other 0 0.0% 0 0.0% 384,085 1.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 31,486,832 100.0% 35,778,297 100.0% 38,793,579 100.0% Note: Information Technology Fund Centers (D022A/B) have been excluded from this page for presentation purposes. Los Angeles Community College District 2023-2024 Final Budget 44 Educational Services Center Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 0 0.0% 0 0.0% 0 0.0% 6000 Instructional Support 978,219 3.1% 1,085,556 3.0% 1,094,296 2.8% 6100 Other Instructional Services 0 0.0% 0 0.0% 0 0.0% Student Services 6200 Admissions and Records 349,386 1.1% 388,846 1.1% 380,262 1.0% 6300 Counseling and Guidance 0 0.0% 0 0.0% 0 0.0% 6400 Other Student Services 0 0.0% 0 0.0% 0 0.0% Total Student Services 349,386 1.1% 388,846 1.1% 380,262 1.0% 6500 Maintenance and Operations 0 0.0% 0 0.0% 0 0.0% Institutional Support 6600 Planning and Policymaking 5,913,439 18.8% 6,519,336 18.2% 7,124,174 18.4% 6700 General Institutional Support 24,245,788 77.0% 27,784,558 77.7% 30,194,847 77.8% Total Institutional Support 30,159,226 95.8% 34,303,895 95.9% 37,319,021 96.2% 6800 Community Service 0 0.0% 0 0.0% 0 0.0% 6900 Ancillary Services 0 0.0% 0 0.0% 0 0.0% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 0 0.0% 0 0.0% 0 0.0% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 31,486,832 100.0% 35,778,297 100.0% 38,793,579 100.0% Note: Information Technology Fund Centers (D022A/B) have been excluded from this page for presentation purposes. Los Angeles Community College District 2023-2024 Final Budget 45 Information Technology Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 100000 Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Certificated Salaries 0 0.0% 0 0.0% 0 0.0% 210000 Classified, Regular 10,447,880 64.5% 10,971,570 60.2% 11,896,456 55.3% 220000 Instructional Aides, Regular 0 0.0% 134 0.0% 0 0.0% 230000 Sub/Relief, Unclassified 107,033 0.7% 290,258 1.6% 130,000 0.6% Total Non-Certificated Salaries 10,554,914 65.2% 11,261,963 61.8% 12,026,456 55.9% 390000 Misc Employee Benefits 4,837,158 29.9% 5,875,223 32.2% 5,946,728 27.7% Total Benefits 4,837,158 29.9% 5,875,223 32.2% 5,946,728 27.7% 450000 Supplies 122,828 0.8% 166,411 0.9% 47,931 0.2% Total Printing & Supplies 122,828 0.8% 166,411 0.9% 47,931 0.2% 550000 Utilities & Housekeeping Expense 101,639 0.6% 74,301 0.4% 222,494 1.0% 560000 Contracts & Rentals 187,011 1.2% 214,316 1.2% 499,403 2.3% 580000 Other Expense 281,159 1.7% 457,154 2.5% 410,715 1.9% Total Operating Expenses 569,808 3.5% 745,770 4.1% 1,132,612 5.3% 640000 Equipment 109,730 0.7% 185,171 1.0% 429,899 2.0% Total Capital Outlay 109,730 0.7% 185,171 1.0% 429,899 2.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 1,914,356 8.9% Total Other 0 0.0% 0 0.0% 1,914,356 8.9% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 16,194,438 100.0% 18,234,538 100.0% 21,497,982 100.0% Note: Information Technology Fund Centers (D022A/B) only. Los Angeles Community College District 2023-2024 Final Budget 46 Information Technology Unrestricted General Fund by Major Functional Area FA Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 5900 Instruction 0 0.0% 0 0.0% 0 0.0% 6000 Instructional Support 0 0.0% 0 0.0% 0 0.0% 6100 Other Instructional Services 0 0.0% 0 0.0% 0 0.0% Student Services 6200 Admissions and Records 0 0.0% 0 0.0% 0 0.0% 6300 Counseling and Guidance 0 0.0% 0 0.0% 0 0.0% 6400 Other Student Services 0 0.0% 0 0.0% 0 0.0% Total Student Services 0 0.0% 0 0.0% 0 0.0% 6500 Maintenance and Operations 0 0.0% 0 0.0% 0 0.0% Institutional Support 6600 Planning and Policymaking 0 0.0% 0 0.0% 0 0.0% 6700 General Institutional Support 16,194,438 100.0% 18,234,538 100.0% 21,497,982 100.0% Total Institutional Support 16,194,438 100.0% 18,234,538 100.0% 21,497,982 100.0% 6800 Community Service 0 0.0% 0 0.0% 0 0.0% 6900 Ancillary Services 0 0.0% 0 0.0% 0 0.0% 7000 Auxiliary Operations 0 0.0% 0 0.0% 0 0.0% 7100 Unallocated 0 0.0% 0 0.0% 0 0.0% 7300 Transfers 0 0.0% 0 0.0% 0 0.0% 7600 State Apportionment 0 0.0% 0 0.0% 0 0.0% 7900 Contingencies 0 0.0% 0 0.0% 0 0.0% 8100 Assoc. Student Organization 0 0.0% 0 0.0% 0 0.0% 9800 Prior Year Salaries & Other Adj 0 0.0% 0 0.0% 0 0.0% 9900 Restricted Prog COLA Augment 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 16,194,438 100.0% 18,234,538 100.0% 21,497,982 100.0% Note: Information Technology Fund Centers (D022A/B) only. Los Angeles Community College District 2023-2024 Final Budget 47 Information Technology Unrestricted General Fund by Sub-Major Commitment Item C/I Description 2023-24 ESC-IT Budget % of Total 2023-24 DW-IT Budget % of Total 2023-24 Total Budget % of Total 100000 Certificated Salaries 0 0.0% 27,146 0.1% 27,146 0.0% Total Certificated Salaries 0 0.0% 27,146 0.1% 27,146 0.0% 210000 Classified, Regular 11,896,456 55.3% 7,695,404 18.4% 19,591,860 31.0% 220000 Instructional Aides, Regular 0 0.0% 310,357 0.7% 310,357 0.5% 230000 Sub/Relief, Unclassified 130,000 0.6% 0 0.0% 130,000 0.2% Total Non-Certificated Salaries 12,026,456 55.9% 8,005,761 19.2% 20,032,217 31.7% 390000 Misc Employee Benefits 5,946,728 27.7% 4,346,621 10.4% 10,293,349 16.3% Total Benefits 5,946,728 27.7% 4,346,621 10.4% 10,293,349 16.3% 450000 Supplies 47,931 0.2% 354,337 0.8% 402,268 0.6% Total Printing & Supplies 47,931 0.2% 354,337 0.8% 402,268 0.6% 550000 Utilities & Housekeeping Expense 222,494 1.0% 584,275 1.4% 806,769 1.3% 560000 Contracts & Rentals 499,403 2.3% 14,852,239 35.5% 15,351,642 24.3% 580000 Other Expense 410,715 1.9% 11,741,570 28.1% 12,152,285 19.2% Total Operating Expenses 1,132,612 5.3% 27,178,084 65.0% 28,310,696 44.7% 640000 Equipment 429,899 2.0% 1,840,354 4.4% 2,270,253 3.6% Total Capital Outlay 429,899 2.0% 1,840,354 4.4% 2,270,253 3.6% 790000 Unallocated/Reserves 1,914,356 8.9% 41,800 0.1% 1,956,156 3.1% Total Other 1,914,356 8.9% 41,800 0.1% 1,956,156 3.1% Less Intrafund w/in Loc 0 0 0 Total Unrestricted 21,497,982 100.0% 41,794,103 100.0% 63,292,085 100.0% Information Technology ESC Fund Centers (D022A/B) only. Information Technology DW Fund Centers include Academic and Student Applications, College Technology Services, Cyber Security, ERP/SAP, Information Security, Network, Service Center, SIS Project Completion, Software Systems, Student Systems and Web Services, and Website Redesign. Los Angeles Community College District 2023-2024 Final Budget 48 Districtwide Accounts Unrestricted General Fund Account 2021-22 Actual Expenditure 2022-23 Actual Expenditure 2023-24 Final Budget 2023-24 % of Total A. Operating Budgets Academic Senate 1,011,533 1,240,207 1,255,372 0.84% Accreditation - 43,947 52,387 0.03% Audit Expense 620,000 522,046 700,000 0.47% Benefits - Retiree 25,842,862 28,804,156 30,680,000 20.42% Central Financial Aid Unit 1,209,930 1,859,148 1,970,094 1.31% Dolores Huerta-Intrafund 374,048 451,850 428,582 0.29% DW Mandatory Memberships 512,040 556,356 873,967 0.58% DW Marketing (Public Relations) 855,911 747,421 2,175,143 1.45% Employee Assistance Program 145,759 226,970 250,529 0.17% Environmental Health & Safety 306,956 361,515 811,730 0.54% Framework for Racial Equity & Social Justice 800,000 - - 0.00% Gold Creek 91,921 128,566 192,806 0.13% HR-Training & Development 81,790 56,016 518,432 0.35% Metro Records 100,328 113,172 108,379 0.07% Special Projects 532,736 433,480 1,180,583 0.79% Operating Budgets - Total 32,485,814 35,544,850 41,198,004 27.42% B. Operating Budgets with Var Exp Collective Bargaining 511,182 823,872 837,000 0.56% Insurance 6,842,216 8,440,880 11,503,933 7.66% Legal Expense 4,863,288 7,279,919 5,085,360 3.38% Reserve for Ins/Legal/WC - - 3,017,911 2.01% Staff Training, Legal 135,449 9,079 343,144 0.23% Workers Compensation 4,658,025 4,951,254 5,036,809 3.35% Operating Budgets with Variable Exp - Total 17,010,161 21,505,004 25,824,157 17.19% C. Other Centralized Accounts AB705 1,313,679 2,051,099 1,926,387 1.28% Board Election - 7,849,198 4,500,000 2.99% District Safety/Operations 75,368 1,375 1,376,870 0.92% District/Safety/Sheriff 22,023,615 25,113,493 28,585,207 19.02% Districtwide Benefits 172,270 162,392 150,000 0.10% Financial Services 9,210 132,431 111,000 0.07% Health Benefits Administration 458,130 570,142 549,704 0.37% LA College Promise 50,000 50,000 50,000 0.03% Project Match 25,410 125,911 117,000 0.08% Public Policy (State & Federal Advocates) 602,227 538,560 909,630 0.61% Staff Development 10,702 1,263 163,117 0.11% SW WEC Settlement - 18,287 323,877 0.22% Tuition Reimbursement 295,491 436,030 1,582,466 1.05% Vacation Balance 3,141,477 1,276,546 1,000,000 0.67% Wellness Program 13,781 4,233 98,700 0.07% Other Centralized Accounts - Total 28,191,360 38,330,961 41,443,958 23.30% D. Information Technology IT-Academic & Student Applications 2,281,609 3,716,453 4,992,930 3.32% IT-Cyber Security - - 250,000 0.17% IT-Dwide College Technology Svcs 3,476,275 4,401,839 4,482,161 2.98% IT-ERP/SAP 2,007,523 1,134,242 3,614,064 2.41% IT-Information Security 232,387 479,351 1,067,362 0.71% IT-Network 351,643 2,646,706 3,735,370 2.49% IT-Region 1 College Technology Svcs 3,911,011 4,404,075 4,670,834 3.11% IT-Region 2 College Technology Svcs 3,227,909 3,388,711 3,702,195 2.46% IT-Region 3 College Technology Svcs 2,726,672 3,555,206 3,654,452 2.43% IT-Service Center 837,266 692,660 1,196,878 0.80% IT-Systems Engineering 1,638,882 472,075 3,346,266 2.23% IT-Special Project-ERP System 458,200 272,434 92,866 0.06% IT-Special Proj-Website Redesign 19,272 211,393 2,464,464 1.64% IT-Student Systems & Web Services 1,631,804 1,990,343 4,524,261 3.01% Information Technology - Total 22,800,455 27,365,488 41,794,103 24.49% Grand Total 100,487,788 122,746,303 150,260,222 100.00% Final Budget includes $13,490,263 carryforward budget for open orders. Unrestricted General Fund Historical Perspective Los Angeles Community College District 2023-2024 Final Budget 49 Los Angeles City College Unrestricted General Fund - Historical Perspective Expenditures 2018-19 2019-20 2020-21 2021-22 2022-23 Certificated 33,098,850 32,601,054 34,043,557 34,186,082 35,121,080 Non-Certificated 12,148,512 13,223,461 12,136,353 10,897,543 12,692,325 Benefits 14,603,243 15,824,052 16,183,180 15,203,794 17,107,660 Books & Supplies 220,914 383,105 516,885 258,954 348,173 Operating Expenses 3,184,318 3,621,676 2,282,865 3,608,594 4,611,567 Capital Outlay 210,150 161,872 400,869 380,297 449,689 Other 245,698 454,305 214,525 504,200 1,090,964 Total 63,711,685 66,269,524 65,778,234 65,039,464 71,421,457 Enrollment (Fall/Spring) 29,618 30,146 28,853 25,478 25,916 FTES (Credit/Non-Credit) 10,831 11,081 10,530 8,774 8,409 Enrollment headcount is credit only. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 2018-19 2019-20 2020-21 2021-22 2022-23 Millions Expenditures by Major Object Other Capital Outlay Operating Expenses Books & Supplies Benefits Non-Certificated Certificated 2,151 2,198 2,280 2,553 2,756 1,600 1,800 2,000 2,200 2,400 2,600 2,800 3,000 3,200 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per Enrollment 5,883 5,981 6,246 7,413 8,494 4,000 4,500 5,000 5,500 6,000 6,500 7,000 7,500 8,000 8,500 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per FTES Los Angeles Community College District 2023-2024 Final Budget 50 East Los Angeles College Unrestricted General Fund - Historical Perspective Expenditures 2018-19 2019-20 2020-21 2021-22 2022-23 Certificated 61,362,151 63,789,977 63,884,007 63,025,878 71,515,214 Non-Certificated 20,287,792 21,606,598 19,039,855 19,996,128 22,383,942 Benefits 26,799,028 29,285,468 29,106,602 30,489,270 34,821,237 Books & Supplies 992,347 641,467 270,621 466,000 667,438 Operating Expenses 12,235,611 12,147,624 7,897,906 9,354,601 11,856,941 Capital Outlay 894,949 247,459 137,221 203,614 574,231 Other 567,897 541,540 341,909 650,036 2,183,100 Total 123,139,776 128,260,133 120,678,122 124,185,527 144,002,103 Enrollment (Fall/Spring) 50,058 49,024 47,554 40,927 39,154 FTES (Credit/Non-Credit) 24,818 24,981 22,671 19,029 18,806 Enrollment headcount is credit only. 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 2018-19 2019-20 2020-21 2021-22 2022-23 Millions Expenditures by Major Object Other Capital Outlay Operating Expenses Books & Supplies Benefits Non-Certificated Certificated 2,460 2,616 2,538 3,034 3,678 1,600 1,800 2,000 2,200 2,400 2,600 2,800 3,000 3,200 3,400 3,600 3,800 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per Enrollment 4,962 5,134 5,323 6,526 7,657 4,000 4,500 5,000 5,500 6,000 6,500 7,000 7,500 8,000 8,500 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per FTES Los Angeles Community College District 2023-2024 Final Budget 51 Los Angeles Harbor College Unrestricted General Fund - Historical Perspective Expenditures 2018-19 2019-20 2020-21 2021-22 2022-23 Certificated 17,110,263 18,734,214 18,110,961 17,531,636 20,057,310 Non-Certificated 7,406,042 7,626,147 6,459,143 6,972,832 7,739,927 Benefits 8,363,621 9,190,381 8,883,176 9,014,315 10,730,969 Books & Supplies 264,238 301,897 180,294 381,871 568,502 Operating Expenses 2,536,530 2,599,478 724,755 2,349,203 2,857,946 Capital Outlay 236,248 239,041 74,247 595,151 294,959 Other 386,631 1,067,973 433,445 542,686 912,545 Total 36,303,574 39,759,131 34,866,021 37,387,694 43,162,157 Enrollment (Fall/Spring) 17,635 17,600 15,936 13,449 14,980 FTES (Credit/Non-Credit) 5,688 5,578 5,318 4,441 4,549 Enrollment headcount is credit only. 0 5 10 15 20 25 30 35 40 45 2018-19 2019-20 2020-21 2021-22 2022-23 Millions Expenditures by Major Object Other Capital Outlay Operating Expenses Books & Supplies Benefits Non-Certificated Certificated 2,059 2,259 2,188 2,780 2,881 1,600 1,800 2,000 2,200 2,400 2,600 2,800 3,000 3,200 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per Enrollment 6,383 7,128 6,556 8,419 9,489 4,000 4,500 5,000 5,500 6,000 6,500 7,000 7,500 8,000 8,500 9,000 9,500 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per FTES Los Angeles Community College District 2023-2024 Final Budget 52 Los Angeles Mission College Unrestricted General Fund - Historical Perspective Expenditures 2018-19 2019-20 2020-21 2021-22 2022-23 Certificated 18,505,287 20,266,829 18,092,207 19,830,594 21,282,479 Non-Certificated 7,960,526 8,556,543 7,358,843 7,302,457 7,936,687 Benefits 8,693,387 9,811,341 9,065,134 9,529,365 10,742,937 Books & Supplies (346) 87,131 46,019 61,451 67,420 Operating Expenses 2,503,976 2,457,328 1,032,122 2,065,934 3,225,472 Capital Outlay 3,387 11,793 49,574 42,402 75,936 Other 3,654 143,917 (633,154) 112,443 893,974 Total 37,669,872 41,334,883 35,010,745 38,944,648 44,224,904 Enrollment (Fall/Spring) 19,977 20,456 18,177 16,830 17,079 FTES (Credit/Non-Credit) 6,570 6,719 5,589 5,230 5,468 Enrollment headcount is credit only. 0 5 10 15 20 25 30 35 40 45 2018-19 2019-20 2020-21 2021-22 2022-23 Millions Expenditures by Major Object Other Capital Outlay Operating Expenses Books & Supplies Benefits Non-Certificated Certificated 1,886 2,021 1,926 2,314 2,589 1,600 1,800 2,000 2,200 2,400 2,600 2,800 3,000 3,200 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per Enrollment 5,734 6,152 6,264 7,446 8,088 4,000 4,500 5,000 5,500 6,000 6,500 7,000 7,500 8,000 8,500 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per FTES Los Angeles Community College District 2023-2024 Final Budget 53 Los Angeles Pierce College Unrestricted General Fund - Historical Perspective Expenditures 2018-19 2019-20 2020-21 2021-22 2022-23 Certificated 40,783,570 42,275,837 40,401,281 39,551,545 47,387,343 Non-Certificated 15,139,704 15,395,737 13,364,722 9,714,820 14,814,167 Benefits 19,148,391 20,512,421 19,645,414 20,328,992 23,659,832 Books & Supplies 173,835 144,944 9,910 240,520 196,943 Operating Expenses 3,192,975 2,927,971 2,131,451 1,938,236 4,768,433 Capital Outlay 1,618 11,704 8,686 3,978 185,754 Other 158,666 331,579 (181,336) (1,545,167) 1,295,406 Total 78,598,759 81,600,194 75,380,129 70,232,923 92,307,878 Enrollment (Fall/Spring) 36,625 37,141 33,541 28,752 30,413 FTES (Credit/Non-Credit) 13,776 14,079 12,410 10,499 10,665 Enrollment headcount is credit only. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 2018-19 2019-20 2020-21 2021-22 2022-23 Millions Expenditures by Major Object Other Capital Outlay Operating Expenses Books & Supplies Benefits Non-Certificated Certificated 2,146 2,197 2,247 2,443 3,035 1,600 1,800 2,000 2,200 2,400 2,600 2,800 3,000 3,200 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per Enrollment 5,706 5,796 6,074 6,690 8,655 4,000 4,500 5,000 5,500 6,000 6,500 7,000 7,500 8,000 8,500 9,000 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per FTES Los Angeles Community College District 2023-2024 Final Budget 54 Los Angeles Southwest College Unrestricted General Fund - Historical Perspective Expenditures 2018-19 2019-20 2020-21 2021-22 2022-23 Certificated 15,440,285 16,706,794 14,799,482 14,618,676 15,609,546 Non-Certificated 6,931,861 7,390,279 5,803,928 5,331,670 5,543,687 Benefits 7,740,192 8,526,193 7,544,789 7,085,174 7,946,446 Books & Supplies 38,332 101,139 29,811 55,939 38,808 Operating Expenses 2,357,822 3,006,298 1,785,860 3,284,844 2,560,698 Capital Outlay 228,453 302,885 158,769 172,787 202,772 Other 0 127,192 (38,657) 130,131 175,278 Total 32,736,945 36,160,779 30,083,982 30,679,223 32,077,236 Enrollment (Fall/Spring) 11,578 12,008 10,014 9,099 9,836 FTES (Credit/Non-Credit) 4,508 4,533 3,666 3,099 3,095 Enrollment headcount is credit only. 0 5 10 15 20 25 30 35 40 2018-19 2019-20 2020-21 2021-22 2022-23 Millions Expenditures by Major Object Other Capital Outlay Operating Expenses Books & Supplies Benefits Non-Certificated Certificated 2,828 3,011 3,004 3,372 3,261 1,800 2,000 2,200 2,400 2,600 2,800 3,000 3,200 3,400 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per Enrollment 7,262 7,978 8,206 9,900 10,363 5,500 6,000 6,500 7,000 7,500 8,000 8,500 9,000 9,500 10,000 10,500 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per FTES Los Angeles Community College District 2023-2024 Final Budget 55 Los Angeles Trade-Technical College Unrestricted General Fund - Historical Perspective Expenditures 2018-19 2019-20 2020-21 2021-22 2022-23 Certificated 30,741,722 33,969,159 33,354,269 31,785,250 37,079,482 Non-Certificated 12,224,926 12,648,265 10,626,429 10,614,239 13,046,296 Benefits 14,456,949 15,879,064 15,365,603 15,163,072 18,901,652 Books & Supplies 638,187 1,125,652 315,297 580,758 931,800 Operating Expenses 3,524,987 2,455,562 1,065,250 1,988,245 2,914,932 Capital Outlay 341,439 78,253 49,109 244,659 429,695 Other 98,492 812,294 (355,396) 404,643 1,726,486 Total 62,026,702 66,968,248 60,420,561 60,780,866 75,030,344 Enrollment (Fall/Spring) 25,957 26,597 22,197 19,014 20,718 FTES (Credit/Non-Credit) 11,304 11,299 9,074 7,768 8,276 Enrollment headcount is credit only. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 2018-19 2019-20 2020-21 2021-22 2022-23 Millions Expenditures by Major Object Other Capital Outlay Operating Expenses Books & Supplies Benefits Non-Certificated Certificated 2,390 2,518 2,722 3,197 3,622 1,600 1,800 2,000 2,200 2,400 2,600 2,800 3,000 3,200 3,400 3,600 3,800 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per Enrollment 5,487 5,927 6,659 7,824 9,066 4,000 4,500 5,000 5,500 6,000 6,500 7,000 7,500 8,000 8,500 9,000 9,500 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per FTES Los Angeles Community College District 2023-2024 Final Budget 56 Los Angeles Valley College Unrestricted General Fund - Historical Perspective Expenditures 2018-19 2019-20 2020-21 2021-22 2022-23 Certificated 34,257,921 35,336,958 34,718,289 34,735,329 40,801,115 Non-Certificated 11,538,838 11,846,318 10,609,404 10,718,181 12,159,248 Benefits 14,927,321 15,987,046 15,544,191 16,019,456 19,025,670 Books & Supplies 446,843 595,855 306,302 598,461 472,292 Operating Expenses 2,814,063 2,833,671 2,826,291 3,316,574 4,117,727 Capital Outlay 31,329 21,065 75,106 36,853 141,117 Other 220,213 408,114 344,268 516,160 1,648,192 Total 64,236,529 67,029,028 64,423,851 65,941,014 78,365,362 Enrollment (Fall/Spring) 33,391 32,826 30,090 27,298 28,234 FTES (Credit/Non-Credit) 11,986 12,000 10,313 9,039 9,692 Enrollment headcount is credit only. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 2018-19 2019-20 2020-21 2021-22 2022-23 Millions Expenditures by Major Object Other Capital Outlay Operating Expenses Books & Supplies Benefits Non-Certificated Certificated 1,924 2,042 2,141 2,416 2,776 1,600 1,800 2,000 2,200 2,400 2,600 2,800 3,000 3,200 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per Enrollment 5,359 5,586 6,247 7,295 8,085 4,000 4,500 5,000 5,500 6,000 6,500 7,000 7,500 8,000 8,500 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per FTES Los Angeles Community College District 2023-2024 Final Budget 57 West Los Angeles College Unrestricted General Fund - Historical Perspective Expenditures 2018-19 2019-20 2020-21 2021-22 2022-23 Certificated 23,814,137 24,853,961 23,855,465 22,853,883 24,619,452 Non-Certificated 8,105,262 8,602,104 8,361,169 7,966,448 8,554,315 Benefits 9,963,483 10,869,348 10,776,135 10,741,855 12,143,481 Books & Supplies 249,050 172,610 52,121 223,712 205,088 Operating Expenses 1,938,701 2,262,416 787,895 2,912,511 2,297,638 Capital Outlay 80,659 170,962 10,722 37,962 68,397 Other 100,718 339,592 (66,541) 366,758 486,938 Total 44,252,010 47,270,993 43,776,967 45,103,129 48,375,309 Enrollment (Fall/Spring) 24,280 24,724 20,991 17,299 16,116 FTES (Credit/Non-Credit) 8,380 8,523 7,612 6,349 5,254 Enrollment headcount is credit only. 0 5 10 15 20 25 30 35 40 45 50 2018-19 2019-20 2020-21 2021-22 2022-23 Millions Expenditures by Major Object Other Capital Outlay Operating Expenses Books & Supplies Benefits Non-Certificated Certificated 1,823 1,912 2,086 2,607 3,002 1,600 1,800 2,000 2,200 2,400 2,600 2,800 3,000 3,200 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per Enrollment 5,280 5,546 5,751 7,104 9,207 4,000 4,500 5,000 5,500 6,000 6,500 7,000 7,500 8,000 8,500 9,000 9,500 2018-19 2019-20 2020-21 2021-22 2022-23 Expenditure per FTES Los Angeles Community College District 2023-2024 Final Budget 58 Instructional Television Unrestricted General Fund - Historical Perspective Expenditures 2018-19 2019-20 2020-21 2021-22 2022-23 Certificated 865,073 0 0 0 0 Non-Certificated 125,659 0 0 0 0 Benefits 248,844 (44) 0 0 0 Books & Supplies 3,174 0 0 0 0 Operating Expenses 72,120 0 0 0 0 Capital Outlay 0 0 0 0 0 Other 0 0 0 0 0 Total 1,314,869 (44) 0 0 0 Enrollment (Fall/Spring) 1,688 0 0 0 0 FTES (Credit/Non-Credit) 279 0 0 0 0 Enrollment headcount is credit only. 0 200 400 600 800 1,000 1,200 1,400 2018-19 2019-20 2020-21 2021-22 2022-23 Thousands Expenditures by Major Object Other Capital Outlay Operating Expenses Books & Supplies Benefits Non-Certificated Certificated 779 0 0 0 0 0 200 400 600 800 1,000 1,200 1,400 1,600 2018-19 2019-20 2019-20 2021-22 2022-23 Expenditure per Enrollment 4,711 0 0 0 0 0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000 2018-19 2019-20 2019-20 2021-22 2022-23 Expenditure per FTES Restricted General Fund Appropriations Los Angeles Community College District 2023-2024 Final Budget 59 Los Angeles Community College District Restricted General Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 110000 Teaching, Regular 1,143,924 0.4% 3,316,160 1.3% 690,230 0.2% 120000 Non-Teaching, Regular 31,612,340 11.3% 32,883,692 12.8% 34,051,971 7.8% 130000 Teaching, Hourly 23,293,069 8.3% 9,969,906 3.9% 3,080,775 0.7% 140000 Non-Teaching, Hourly 20,564,855 7.3% 20,086,374 7.8% 24,601,076 5.7% 190000 Misc Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Certificated Salaries 76,614,187 27.3% 66,256,131 25.7% 62,424,052 14.4% 210000 Classified, Regular 26,760,464 9.6% 24,116,576 9.4% 28,605,046 6.6% 220000 Instructional Aides, Regular 3,950,226 1.4% 2,797,307 1.1% 2,563,789 0.6% 230000 Sub/Relief, Unclassified 18,651,334 6.7% 19,815,296 7.7% 22,224,712 5.1% 240000 Instructional Aides, Non-Perm 1,587,180 0.6% 1,222,592 0.5% 4,911,060 1.1% 290000 Misc Non-Certificated Salaries 0 0.0% 0 0.0% 50,000 0.0% Total Non-Certificated Salaries 50,949,204 18.2% 47,951,771 18.6% 58,354,607 13.4% 390000 Misc Employee Benefits 31,880,793 11.4% 35,427,492 13.8% 33,466,597 7.7% Total Benefits 31,880,793 11.4% 35,427,492 13.8% 33,466,597 7.7% 420000 Books 679,027 0.2% 1,175,735 0.5% 753,895 0.2% 440000 Instructional Media Materials 3,739,489 1.3% 4,555,344 1.8% 19,080,619 4.4% 450000 Supplies 11,790,848 4.2% 6,367,111 2.5% 5,789,266 1.3% 470000 Materials Fees 8,267 0.0% 9,183 0.0% 8,500 0.0% Total Printing & Supplies 16,217,630 5.8% 12,107,373 4.7% 25,632,280 5.9% 540000 Insurance 175,376 0.1% 0 0.0% 0 0.0% 550000 Utilities & Housekeeping Expense 5,679,844 2.0% 1,391,834 0.5% 182,574 0.0% 560000 Contracts & Rentals 18,474,947 6.6% 28,357,471 11.0% 54,846,106 12.6% 580000 Other Expense 39,044,221 13.9% 17,187,119 6.7% 41,559,142 9.6% 590000 Misc Other Expense 565,971 0.2% 232,971 0.1% 309,867 0.1% Total Operating Expenses 63,940,359 22.8% 47,169,394 18.3% 96,897,689 22.3% 610000 Sites 0 0.0% 0 0.0% 0 0.0% 620000 Buildings 0 0.0% 0 0.0% 0 0.0% 630000 Books and Materials for Libraries 241,164 0.1% 315,566 0.1% 75,951 0.0% 640000 Equipment 15,965,020 5.7% 21,593,760 8.4% 24,545,814 5.6% 650000 Lease/Purchase 2,181 0.0% 4,250 0.0% 37,010 0.0% Total Capital Outlay 16,208,365 5.8% 21,913,576 8.5% 24,658,775 5.7% 720000 Tuition Transfers 2,434,004 0.9% 3,842,445 1.5% 135,167 0.0% 730000 Interfund Transfers 3,455,617 1.2% 1,784,988 0.7% 0 0.0% 739900 Intrafund Transfers 5,759,405 2.1% 5,062,206 2.0% 0 0.0% 750000 Loans/Grants 12,714,956 4.5% 15,881,626 6.2% 4,323,047 1.0% 760000 Other Payments 0 0.0% 0 0.0% 5,250 0.0% 790000 Unallocated/Reserves 11,253 0.0% 15,351 0.0% 128,905,385 29.6% Total Other 24,375,235 8.7% 26,586,616 10.3% 133,368,849 30.7% Less Intrafund w/in Loc 0 0 0 Total Restricted General Fund 280,185,774 100.0% 257,412,354 100.0% 434,802,849 100.0% Los Angeles Community College District 2023-2024 Final Budget 60 Restricted General Fund Appropriations by Program Restricted Program 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total CA Adult Education Program (CAEP) 7,398,850 2.64 8,955,887 3.48 14,526,593 3.34 CA College Promise 3,780,230 1.35 4,915,404 1.91 1,750,500 0.40 CalWORKs (Child Care/Non-Child Care) / TANF 5,908,532 2.11 6,853,346 2.66 12,237,788 2.81 Community Services 894,354 0.32 1,304,279 0.51 8,256,714 1.90 Cooperative Agencies Resources for Education (CARE) 770,589 0.28 967,790 0.38 1,289,203 0.30 Disabled Student Programs & Services (DSPS) 7,479,746 2.67 10,074,565 3.91 11,117,132 2.56 Dream Resource Liaison Support 276,442 0.10 625,102 0.24 539,533 0.12 Equal Employment Opportunity 9,320 0.00 9,486 0.00 313,567 0.07 Extended Opportunities Programs & Services (EOPS) 8,446,487 3.01 9,173,019 3.56 9,541,485 2.19 Federal Perkins IV (CTE) 4,508,823 1.61 5,226,306 2.03 4,525,690 1.04 Federal Work Study 1,413,751 0.50 2,113,542 0.82 3,156,693 0.73 Financial Aid Technology 484,440 0.17 379,542 0.15 60,454 0.01 Foster and Kinship Care Education 1,075,533 0.38 1,113,804 0.43 1,009,309 0.23 Framework for Racial Equity and Social Justice 516,299 0.18 1,536,788 0.60 376,243 0.09 Health Services 2,388,987 0.85 2,802,977 1.09 8,214,560 1.89 HEERF I 938,673 0.34 225,382 0.09 0 0.00 HEERF II 29,189,490 10.42 13,114,617 5.09 0 0.00 HEERF III 69,915,175 24.95 20,251,599 7.87 0 0.00 HEERF MSI Supplement 5,644,049 2.01 11,472,283 4.46 81,685 0.02 HEERF SAHIE 0 0.00 917,445 0.36 0 0.00 Higher Ed Emergency MSI 103,222 0.04 71,692 0.03 0 0.00 Lottery - Prop 20 7,263,091 2.59 6,664,119 2.59 18,188,081 4.18 NextUp 1,772,049 0.63 2,346,499 0.91 2,298,265 0.53 One-Time Block Grants 2,348,967 0.84 8,281,665 3.22 27,462,929 6.32 Parking -932,817 -0.33 -115,774 -0.04 7,067,565 1.63 Staff/Faculty Development 22 0.00 23,596 0.01 323,177 0.07 Strong Workforce 13,862,599 4.95 13,120,107 5.10 27,279,539 6.27 Student Equity and Achievement (SEA) 42,699,430 15.24 45,560,764 17.70 66,727,950 15.35 Student Financial Aid Administration 4,491,822 1.60 5,385,696 2.09 5,223,081 1.20 Student Retention and Enrollment 3,992,054 1.42 11,129,085 4.32 6,340,638 1.46 Unrestricted Indirects 0 0.00 -9,440,265 -3.67 11,352,370 2.61 Veterans Resource Center 519,769 0.19 983,826 0.38 580,204 0.13 Other Specially Funded Programs 53,025,797 18.93 71,368,179 27.73 184,961,901 42.54 Total Restricted General Fund 280,185,774 100.00 257,412,354 100.00 434,802,849 100.00 Los Angeles Community College District 2023-2024 Final Budget 61 Restricted General Fund Appropriations by Fund and Location Restricted Program City East Harbor Mission Pierce Southwest Trade-tech Valley West ESC/DW Total Final CA Adult Education Program (CAEP) 1,351,105 1,145,212 691,459 695,343 619,321 1,470,051 1,112,418 1,319,943 644,146 5,477,595 14,526,593 CA College Promise 0 0 0 0 0 0 0 0 0 1,750,500 1,750,500 CalWORKs (Child Care/Non-Child Care) / TANF 1,167,864 1,643,527 561,241 572,699 633,555 2,741,539 2,241,906 1,744,124 931,333 0 12,237,788 Community Services 1,202,745 1,628,592 32,792 1,573 2,073,574 0 0 2,927,600 389,838 0 8,256,714 Cooperative Agencies Resources for Education (CARE) 139,725 216,738 242,236 134,878 21,915 191,465 134,375 103,224 104,647 0 1,289,203 Disabled Student Programs & Services (DSPS) 1,115,489 2,336,035 1,128,869 683,943 1,663,250 452,692 1,248,558 1,862,275 626,021 0 11,117,132 Dream Resource Liaison Support 67,191 119,167 86,814 34 115,468 30,476 87,156 1,140 32,087 0 539,533 Equal Employment Opportunity 0 0 0 0 0 0 0 0 0 313,567 313,567 Extended Opportunities Programs & Services (EOPS) 1,732,819 1,701,600 952,615 879,702 1,171,988 304,259 980,252 1,257,647 560,603 0 9,541,485 Federal Perkins IV (CTE) 561,192 886,872 251,635 380,617 509,599 235,512 509,599 499,925 464,455 226,284 4,525,690 Federal Work Study 478,547 674,690 178,222 233,840 452,675 137,735 329,873 380,380 267,068 23,663 3,156,693 Financial Aid Technology 27,356 0 3,333 0 1,854 13,291 12,876 1,729 0 15 60,454 Foster and Kinship Care Education 135,534 96,738 135,661 198,751 79,992 95,176 148,005 0 119,452 0 1,009,309 Framework for Racial Equity and Social Justice 0 0 0 0 0 0 0 0 0 376,243 376,243 Health Services 1,279,258 1,832,633 368,645 767,745 1,012,541 121,072 942,396 1,342,533 547,737 0 8,214,560 HEERF I 0 0 0 0 0 0 0 0 0 0 0 HEERF II 0 0 0 0 0 0 0 0 0 0 0 HEERF III 0 0 0 0 0 0 0 0 0 0 0 HEERF MSI Supplement 1,714 0 0 0 4,361 0 75,610 0 0 0 81,685 HEERF Supplemental Assistance to Institutions of Higher Education 0 0 0 0 0 0 0 0 0 0 0 Higher Ed Emergency MSI 0 0 0 0 0 0 0 0 0 0 0 Lottery - Prop 20 1,531,036 3,553,908 756,380 947,612 2,493,839 596,029 1,510,419 1,683,225 1,136,549 3,979,084 18,188,081 NextUp 119,451 189,935 333,172 243,531 321,567 237,926 321,961 236,600 294,122 0 2,298,265 One-Time Block Grants 6,443,698 5,271,378 831,243 1,326,163 4,330,265 2,527,623 3,082,201 645,466 3,004,892 0 27,462,929 Parking 63,938 1,646,067 134,532 0 1,643,166 574,578 810,330 2,124,498 70,456 0 7,067,565 Staff/Faculty Development 23,106 34,978 11,599 13,425 13,611 0 19,303 18,550 14,795 173,810 323,177 Strong Workforce 3,291,905 7,551,813 822,020 1,233,532 2,633,417 657,388 6,120,142 2,336,667 2,632,655 0 27,279,539 Student Equity and Achievement (SEA) 6,519,210 15,643,211 4,480,900 3,917,657 6,667,870 4,955,046 11,917,950 5,737,297 6,888,809 0 66,727,950 Student Financial Aid Administration 691,593 1,110,044 389,849 462,372 599,496 264,294 478,924 575,841 493,740 156,928 5,223,081 Student Retention and Enrollment 120,213 929,811 420,778 679,346 594,454 978,484 1,196,363 867,322 465,477 88,390 6,340,638 Unrestricted Indirects 275,989 1,308,918 646,920 842,910 2,566,836 784,991 2,196,352 2,353,974 375,480 0 11,352,370 Veterans Resource Center 24,960 134,400 0 67,292 42,603 34,631 117,553 71,490 87,275 0 580,204 Other Specially Funded Programs 15,117,104 30,960,340 8,145,728 11,191,945 15,803,834 11,360,498 27,337,615 14,424,716 21,456,767 29,163,354 184,961,901 Total Restricted General Fund 43,482,742 80,616,607 21,606,643 25,474,910 46,071,051 28,764,756 62,932,137 42,516,166 41,608,404 41,729,433 434,802,849 Los Angeles Community College District 2023-2024 Final Budget 62 CA Adult Education Program (CAEP) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 1,000,871 13.53 1,181,113 13.19 1,351,105 9.30 East 947,477 12.81 956,148 10.68 1,145,212 7.88 Harbor 642,019 8.68 511,280 5.71 691,459 4.76 Mission 687,439 9.29 1,413,352 15.78 695,343 4.79 Pierce 521,963 7.05 698,149 7.80 619,321 4.26 Southwest 848,475 11.47 789,420 8.81 1,470,051 10.12 Trade-Tech 147,348 1.99 844,607 9.43 1,112,418 7.66 Valley 649,518 8.78 864,830 9.66 1,319,943 9.09 West 1,108,385 14.98 1,027,793 11.48 644,146 4.43 ESC 845,356 11.43 669,196 7.47 5,477,595 37.71 Total CA Adult Education Program 7,398,850 100.00 8,955,887 100.00 14,526,593 100.00 Includes funds 10460-10464. CA College Promise Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 0 0.00 0 0.00 East 0 0.00 0 0.00 0 0.00 Harbor 0 0.00 0 0.00 0 0.00 Mission 0 0.00 0 0.00 0 0.00 Pierce 0 0.00 0 0.00 0 0.00 Southwest 0 0.00 0 0.00 0 0.00 Trade-Tech 0 0.00 0 0.00 0 0.00 Valley 0 0.00 0 0.00 0 0.00 West 0 0.00 0 0.00 0 0.00 ESC 3,780,230 100.00 4,915,404 100.00 1,750,500 100.00 Total CA College Promise 3,780,230 100.00 4,915,404 100.00 1,750,500 100.00 Includes funds 10407-10409, and 10459. Los Angeles Community College District 2023-2024 Final Budget 63 CalWORKs (Child Care/Non-Child Care)/TANF Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 792,321 13.41 975,377 14.23 1,167,864 9.54 East 846,342 14.32 842,761 12.30 1,643,527 13.43 Harbor 505,610 8.56 586,968 8.56 561,241 4.59 Mission 623,040 10.54 536,493 7.83 572,699 4.68 Pierce 485,415 8.22 528,873 7.72 633,555 5.18 Southwest 420,790 7.12 490,049 7.15 2,741,539 22.40 Trade-Tech 721,973 12.22 1,050,570 15.33 2,241,906 18.32 Valley 1,040,222 17.61 1,325,255 19.34 1,744,124 14.25 West 472,818 8.00 516,999 7.54 931,333 7.61 ESC 0 0.00 0 0.00 0 0.00 Total CalWORKs 5,908,532 100.00 6,853,346 100.00 12,237,788 100.00 Includes funds 10440-10444, 10445-10447, 10448-10451. Community Services Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 423,327 47.33 956,568 73.34 1,202,745 14.57 East 43,139 4.82 (227,236) -17.42 1,628,592 19.72 Harbor 625,867 69.98 0 0.00 32,792 0.40 Mission 0 0.00 0 0.00 1,573 0.02 Pierce (387,794) -43.36 99,170 7.60 2,073,574 25.11 Southwest 0 0.00 0 0.00 0 0.00 Trade-Tech 0 0.00 0 0.00 0 0.00 Valley (150,709) -16.85 486,064 37.27 2,927,600 35.46 West 340,524 38.07 (10,287) -0.79 389,838 4.72 ESC 0 0.00 0 0.00 0 0.00 Total Community Services 894,354 100.00 1,304,279 100.00 8,256,714 100.00 Includes funds 10010. Los Angeles Community College District 2023-2024 Final Budget 64 Cooperative Agencies Resources for Education (CARE) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 106,429 13.81 97,671 10.09 139,725 10.84 East 149,442 19.39 193,793 20.02 216,738 16.81 Harbor 137,870 17.89 212,996 22.01 242,236 18.79 Mission 41,718 5.41 70,434 7.28 134,878 10.46 Pierce 36,475 4.73 21,423 2.21 21,915 1.70 Southwest 98,315 12.76 104,695 10.82 191,465 14.85 Trade-tech 46,010 5.97 106,536 11.01 134,375 10.42 Valley 82,646 10.73 95,875 9.91 103,224 8.01 West 71,684 9.30 64,366 6.65 104,647 8.12 ESC 0 0.00 0 0.00 0 0.00 Total CARE 770,589 100.00 967,790 100.00 1,289,203 100.00 Includes only funds in General Fund portion of the program (funds 10867-10869 and 11405). Disabled Student Programs & Services (DSPS) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 988,655 13.22 1,488,017 14.77 1,115,489 10.03 East 1,198,669 16.03 1,742,253 17.29 2,336,035 21.01 Harbor 883,619 11.81 962,010 9.55 1,128,869 10.15 Mission 464,532 6.21 700,806 6.96 683,943 6.15 Pierce 1,078,972 14.43 1,145,490 11.37 1,663,250 14.96 Southwest 254,186 3.40 574,346 5.70 452,692 4.07 Trade-Tech 794,129 10.62 1,460,807 14.50 1,248,558 11.23 Valley 1,300,286 17.38 1,321,858 13.12 1,862,275 16.75 West 516,698 6.91 678,979 6.74 626,021 5.63 ESC 0 0.00 0 0.00 0 0.00 Total DSPS 7,479,746 100.00 10,074,565 100.00 11,117,132 100.00 Includes funds 10404-10406 and 10420. Los Angeles Community College District 2023-2024 Final Budget 65 Dream Resource Liaison Support Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 93,600 14.97 67,191 12.45 East 49,488 17.90 101,804 16.29 119,167 22.09 Harbor 4,301 1.56 28,624 4.58 86,814 16.09 Mission 31,936 11.55 91,089 14.57 34 0.01 Pierce 8,942 3.23 59,471 9.51 115,468 21.40 Southwest 37,289 13.49 43,461 6.95 30,476 5.65 Trade-Tech 55,150 19.95 22,192 3.55 87,156 16.15 Valley 70,022 25.33 98,687 15.79 1,140 0.21 West 19,314 6.99 86,175 13.79 32,087 5.95 ESC 0 0.00 0 0.00 0 0.00 Total Dream Resource Liaison Support 276,442 100.00 625,102 100.00 539,533 100.00 Includes funds 18207 and 18209. Equal Employment Opportunity Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 0 0.00 0 0.00 East 0 0.00 0 0.00 0 0.00 Harbor 0 0.00 0 0.00 0 0.00 Mission 0 0.00 0 0.00 0 0.00 Pierce 0 0.00 0 0.00 0 0.00 Southwest 0 0.00 0 0.00 0 0.00 Trade-tech 0 0.00 0 0.00 0 0.00 Valley 0 0.00 0 0.00 0 0.00 West 0 0.00 0 0.00 0 0.00 ESC 9,320 100.00 9,486 100.00 313,567 100.00 Total Equal Employment Opportunity 9,320 100.00 9,486 100.00 313,567 100.00 Includes fund 10436. Los Angeles Community College District 2023-2024 Final Budget 66 Extended Opportunities Programs & Services (EOPS) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 1,453,401 17.21 1,579,831 17.22 1,732,819 18.16 East 1,340,702 15.87 1,321,306 14.40 1,701,600 17.83 Harbor 977,656 11.57 1,027,683 11.20 952,615 9.98 Mission 684,548 8.10 902,894 9.84 879,702 9.22 Pierce 732,144 8.67 893,168 9.74 1,171,988 12.28 Southwest 698,384 8.27 700,268 7.63 304,259 3.19 Trade-tech 1,204,721 14.26 1,155,845 12.60 980,252 10.27 Valley 977,989 11.58 1,241,512 13.53 1,257,647 13.18 West 376,941 4.46 350,511 3.82 560,603 5.88 ESC 0 0.00 0 0.00 0 0.00 Total EOPS 8,446,487 100.00 9,173,019 100.00 9,541,485 100.00 Includes only funds in General Fund portion of the program (funds 10486-10490). Federal Perkins IV (CTE) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 506,464 11.23 661,805 12.66 561,192 12.40 East 839,177 18.61 1,059,741 20.28 886,872 19.60 Harbor 257,652 5.71 278,007 5.32 251,635 5.56 Mission 361,806 8.02 389,875 7.46 380,617 8.41 Pierce 505,597 11.21 564,265 10.80 509,599 11.26 Southwest 266,523 5.91 275,795 5.28 235,512 5.20 Trade-tech 551,733 12.24 593,381 11.35 509,599 11.26 Valley 472,866 10.49 561,489 10.74 499,925 11.05 West 531,782 11.79 580,782 11.11 464,455 10.26 ESC 215,223 4.77 261,165 5.00 226,284 5.00 Total Federal Perkins IV (CTE) 4,508,823 100.00 5,226,306 100.00 4,525,690 100.00 Includes funds 10580-10585. Los Angeles Community College District 2023-2024 Final Budget 67 Federal Work Study Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 377,086 26.67 421,046 19.92 478,547 15.16 East 444,908 31.47 607,228 28.73 674,690 21.37 Harbor 49,340 3.49 110,526 5.23 178,222 5.65 Mission 163,624 11.57 202,960 9.60 233,840 7.41 Pierce 51,355 3.63 105,656 5.00 452,675 14.34 Southwest 57,252 4.05 181,561 8.59 137,735 4.36 Trade-tech 71,498 5.06 166,825 7.89 329,873 10.45 Valley 132,582 9.38 242,059 11.45 380,380 12.05 West 66,107 4.68 75,681 3.58 267,068 8.46 ESC 0 0.00 0 0.00 23,663 0.75 Total Federal Work Study 1,413,751 100.00 2,113,542 100.00 3,156,693 100.00 Includes funds 10453-10458. Financial Aid Technology Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 1,150 0.24 0 0.00 27,356 45.25 East 9,714 2.01 0 0.00 0 0.00 Harbor 881 0.18 0 0.00 3,333 5.51 Mission 0 0.00 0 0.00 0 0.00 Pierce 12,166 2.51 16,491 4.35 1,854 3.07 Southwest 0 0.00 5,809 1.53 13,291 21.99 Trade-tech 0 0.00 5,755 1.52 12,876 21.30 Valley 10,500 2.17 19,455 5.13 1,729 2.86 West 862 0.18 11,875 3.13 0 0.00 ESC 449,167 92.72 320,158 84.35 15 0.02 Total Financial Aid Technology 484,440 100.00 379,542 100.00 60,454 100.00 Includes funds 10484-10485 and 10492-10494. Los Angeles Community College District 2023-2024 Final Budget 68 Foster & Kinship Care Education (FKCE) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 137,155 226.87 145,735 13.08 135,534 13.43 East 116,129 192.10 104,019 9.34 96,738 9.58 Harbor 152,179 251.73 160,220 14.38 135,661 13.44 Mission 222,399 367.88 207,689 18.65 198,751 19.69 Pierce 71,718 118.63 106,212 9.54 79,992 7.93 Southwest 110,043 182.03 102,340 9.19 95,176 9.43 Trade-tech 144,846 239.60 159,145 14.29 148,005 14.66 Valley 0 0.00 0 0.00 0 0.00 West 121,064 200.26 128,443 11.53 119,452 11.84 ESC 0 0.00 0 0.00 0 0.00 Total FKCE 1,075,533 100.00 1,113,804 100.00 1,009,309 100.00 Includes funds 10422-10425. Framework for Racial Equity & Social Justice Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 0 0.00 0 0.00 East 0 0.00 0 0.00 0 0.00 Harbor 0 0.00 0 0.00 0 0.00 Mission 0 0.00 0 0.00 0 0.00 Pierce 0 0.00 0 0.00 0 0.00 Southwest 0 0.00 0 0.00 0 0.00 Trade-tech 0 0.00 0 0.00 0 0.00 Valley 0 0.00 0 0.00 0 0.00 West 0 0.00 0 0.00 0 0.00 ESC 516,299 100.00 1,536,788 100.00 376,243 100.00 Total Framework for Racial Equity & Social Justice 516,299 100.00 1,536,788 100.00 376,243 100.00 Includes fund 19660. Los Angeles Community College District 2023-2024 Final Budget 69 Health Services Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 98,359 4.12 516,154 18.41 1,279,258 15.57 East 502,823 21.05 500,583 17.86 1,832,633 22.31 Harbor 123,150 5.15 161,750 5.77 368,645 4.49 Mission 164,000 6.86 121,375 4.33 767,745 9.35 Pierce 498,166 20.85 565,707 20.18 1,012,541 12.33 Southwest 204,238 8.55 167,016 5.96 121,072 1.47 Trade-tech 213,009 8.92 253,497 9.04 942,396 11.47 Valley 376,069 15.74 400,371 14.28 1,342,533 16.34 West 209,174 8.76 116,524 4.16 547,737 6.67 ESC 0 0.00 0 0.00 0 0.00 Total Health Services 2,388,987 100.00 2,802,977 100.00 8,214,560 100.00 Includes fund 10135. Higher Education Emergency Relief Fund I (HEERF I) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 98,593 43.74 0 0.00 East 237,881 25.34 1,456 0.65 0 0.00 Harbor (37,822) -4.03 43,714 19.40 0 0.00 Mission 214,599 22.86 11,696 5.19 0 0.00 Pierce 498,852 53.14 65,330 28.99 0 0.00 Southwest 8,182 0.87 3,562 1.58 0 0.00 Trade-tech 0 0.00 0 0.00 0 0.00 Valley (1,021) -0.11 1,030 0.46 0 0.00 West 18,002 1.92 0 0.00 0 0.00 ESC 0 0.00 0 0.00 0 0.00 Total HEERF I 938,673 100.00 225,382 100.00 0 0.00 Includes fund 17637. Los Angeles Community College District 2023-2024 Final Budget 70 Higher Education Emergency Relief Fund II (HEERF II) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 3,049,732 10.45 897,614 6.84 0 0.00 East 3,654,649 12.52 5,044,144 38.46 0 0.00 Harbor 88,993 0.30 238,430 1.82 0 0.00 Mission 2,588,256 8.87 302,852 2.31 0 0.00 Pierce 6,920,413 23.71 1,643,789 12.53 0 0.00 Southwest 1,376,746 4.72 302,277 2.30 0 0.00 Trade-tech 2,987,667 10.24 1,995,978 15.22 0 0.00 Valley 2,494,752 8.55 612,370 4.67 0 0.00 West 836,896 2.87 508,307 3.88 0 0.00 ESC 5,191,386 17.79 1,568,856 11.96 0 0.00 Total HEERF II 29,189,490 100.00 13,114,617 100.00 0 0.00 Includes fund 17650. Higher Education Emergency Relief Fund III (HEERF III) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 10,268,907 14.69 556,280 2.75 0 0.00 East 18,246,673 26.10 5,139,392 25.38 0 0.00 Harbor 2,907,182 4.16 35,563 0.18 0 0.00 Mission 2,709,661 3.88 2,342,330 11.57 0 0.00 Pierce 11,404,118 16.31 2,926,533 14.45 0 0.00 Southwest 2,294,491 3.28 2,009,407 9.92 0 0.00 Trade-tech 7,574,790 10.83 3,512,801 17.35 0 0.00 Valley 9,634,949 13.78 2,119,377 10.47 0 0.00 West 3,585,634 5.13 794,705 3.92 0 0.00 ESC 1,288,771 1.84 815,211 4.03 0 0.00 Total HEERF III 69,915,175 100.00 20,251,599 100.00 0 0.00 Includes fund 17651. Los Angeles Community College District 2023-2024 Final Budget 71 HEERF Minority Serving Institutions (HEERF MSI) Supplement Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 1,987,687 17.33 1,714 2.10 East 937,918 16.62 3,319,635 28.94 0 0.00 Harbor 927,854 16.44 40,225 0.35 0 0.00 Mission 17,781 0.32 1,090,588 9.51 0 0.00 Pierce 1,372,410 24.32 1,200,843 10.47 4,361 5.34 Southwest 0 0.00 1,098,416 9.57 0 0.00 Trade-tech 0 0.00 1,989,980 17.35 75,610 92.56 Valley 1,714,054 30.37 470,771 4.10 0 0.00 West 674,031 11.94 274,138 2.39 0 0.00 ESC 0 0.00 0 0.00 0 0.00 Total HEERF MSI Supplement 5,644,049 100.00 11,472,283 100.00 81,685 100.00 Includes funds 17652. HEERF Supplemental Assistance to Institutions of Higher Education – SAHIE Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 0 0.00 0 0.00 East 0 0.00 0 0.00 0 0.00 Harbor 0 0.00 0 0.00 0 0.00 Mission 0 0.00 0 0.00 0 0.00 Pierce 0 0.00 0 0.00 0 0.00 Southwest 0 0.00 917,445 100.00 0 0.00 Trade-tech 0 0.00 0 0.00 0 0.00 Valley 0 0.00 0 0.00 0 0.00 West 0 0.00 0 0.00 0 0.00 ESC 0 0.00 0 0.00 0 0.00 Total HEERF SAHIE 0 0.00 917,445 100.00 0 0.00 Includes funds 17662. Los Angeles Community College District 2023-2024 Final Budget 72 Higher Ed Emergency MSI Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 0 0.00 0 0.00 East 0 0.00 0 0.00 0 0.00 Harbor 44,735 43.34 64,816 90.41 0 0.00 Mission 19,659 19.05 12 0.02 0 0.00 Pierce 38,356 37.16 6,865 9.58 0 0.00 Southwest 0 0.00 0 0.00 0 0.00 Trade-tech 0 0.00 0 0.00 0 0.00 Valley 472 0.46 0 0.00 0 0.00 West 0 0.00 0 0.00 0 0.00 ESC 0 0.00 0 0.00 0 0.00 Total Higher Ed Emergency MSI 103,222 100.00 71,692 100.00 0 0.00 Includes fund 17638. Lottery – Prop 20 Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 1,289,891 17.76 702,248 10.54 1,531,036 8.42 East 1,488,088 20.49 1,588,566 23.84 3,553,908 19.54 Harbor 236,271 3.25 714,613 10.72 756,380 4.16 Mission 293,436 4.04 755,302 11.33 947,612 5.21 Pierce 424,057 5.84 819,348 12.29 2,493,839 13.71 Southwest 161,268 2.22 117,597 1.76 596,029 3.28 Trade-tech 655,767 9.03 620,739 9.31 1,510,419 8.30 Valley 257,268 3.54 580,619 8.71 1,683,225 9.25 West 522,138 7.19 708,457 10.63 1,136,549 6.25 ESC 1,934,908 26.64 56,630 0.85 3,979,084 21.88 Total Lottery – Prop 20 7,263,091 100.00 6,664,119 100.00 18,188,081 100.00 Includes fund 10421. Los Angeles Community College District 2023-2024 Final Budget 73 NextUp Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 152,432 8.60 226,508 9.65 119,451 5.20 East 243,604 13.75 178,309 7.60 189,935 8.26 Harbor 238,027 13.43 278,631 11.87 333,172 14.50 Mission 233,315 13.17 305,742 13.03 243,531 10.60 Pierce 182,860 10.32 312,361 13.31 321,567 13.99 Southwest 266,077 15.02 407,948 17.39 237,926 10.35 Trade-tech 143,125 8.08 198,259 8.45 321,961 14.01 Valley 174,986 9.87 205,523 8.76 236,600 10.29 West 137,623 7.77 233,218 9.94 294,122 12.80 ESC 0 0.00 0 0.00 0 0.00 Total NextUp 1,772,049 100.00 2,346,499 100.00 2,298,265 100.00 Includes only funds in General Fund portion of the program (funds 10400-10403). One-Time Block Grants Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 36,508 1.55 1,345,827 16.25 6,443,698 23.46 East 935,935 39.84 2,273,129 27.45 5,271,378 19.19 Harbor 72,542 3.09 575,912 6.95 831,243 3.03 Mission 141,848 6.04 972,930 11.75 1,326,163 4.83 Pierce 228,329 9.72 153,180 1.85 4,330,265 15.77 Southwest 50,129 2.13 13,252 0.16 2,527,623 9.20 Trade-tech 499,572 21.27 1,416,791 17.11 3,082,201 11.22 Valley 233,256 9.93 1,203,474 14.53 645,466 2.35 West 150,847 6.42 327,169 3.95 3,004,892 10.94 ESC 0 0.00 0 0.00 0 0.00 Total One-Time Block Grants 2,348,967 100.00 8,281,665 100.00 27,462,929 100.00 Includes One-Time Block Grants (funds 10116, 10125-10128, 10131-10134, 10136-10138). Los Angeles Community College District 2023-2024 Final Budget 74 Parking Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 24,337 -2.61 28,910 -24.97 63,938 0.90 East (296,432) 31.78 (315,313) 272.35 1,646,067 23.29 Harbor 6,955 -0.75 0 0.00 134,532 1.90 Mission 4,765 -0.51 66,813 -57.71 0 0.00 Pierce (250,933) 26.90 (3,976) 3.43 1,643,166 23.25 Southwest (151,331) 16.22 (101,573) 87.73 574,578 8.13 Trade-tech (100,274) 10.75 (54,572) 47.14 810,330 11.47 Valley (168,245) 18.04 332,796 -287.45 2,124,498 30.06 West (1,660) 0.18 (68,858) 59.48 70,456 1.00 ESC 0 0.00 0 0.00 0 0.00 Total Parking (932,817) 100.00 (115,774) 100.00 7,067,565 100.00 Includes fund 10145. Staff/Faculty Development Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 0 0.00 23,106 7.15 East 0 0.00 0 0.00 34,978 10.82 Harbor 0 0.00 0 0.00 11,599 3.59 Mission 0 0.00 0 0.00 13,425 4.15 Pierce 22 100.00 10,090 42.76 13,611 4.21 Southwest 0 0.00 10,137 42.96 0 0.00 Trade-tech 0 0.00 0 0.00 19,303 5.97 Valley 0 0.00 3,369 14.28 18,550 5.74 West 0 0.00 0 0.00 14,795 4.58 ESC 0 0.00 0 0.00 173,810 53.78 Total Staff/Faculty Development 22 100.00 23,596 10.00 323,177 100.00 Includes fund 10435. Los Angeles Community College District 2023-2024 Final Budget 75 Strong Workforce Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 1,341,565 9.68 1,173,322 8.94 3,291,905 12.07 East 3,386,786 24.43 3,776,178 28.78 7,551,813 27.68 Harbor 355,264 2.56 547,637 4.17 822,020 3.01 Mission 904,681 6.53 744,344 5.67 1,233,532 4.52 Pierce 1,566,445 11.30 1,803,749 13.75 2,633,417 9.65 Southwest 363,051 2.62 219,228 1.67 657,388 2.41 Trade-tech 3,156,742 22.77 2,791,974 21.28 6,120,142 22.43 Valley 1,209,771 8.73 979,057 7.46 2,336,667 8.57 West 1,126,437 8.13 1,084,618 8.27 2,632,655 9.65 ESC 451,858 3.26 0 0.00 0 0.00 Total Strong Workforce 13,862,599 100.00 13,120,107 100.00 27,279,539 100.00 Includes funds 10496-10500. Student Equity & Achievement (SEA) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 5,143,847 12.05 6,072,559 13.33 6,519,210 9.77 East 10,264,521 24.04 10,453,698 22.94 15,643,211 23.44 Harbor 2,870,668 6.72 2,492,487 5.47 4,480,900 6.72 Mission 3,340,759 7.82 3,503,568 7.69 3,917,657 5.87 Pierce 4,511,106 10.56 6,979,854 15.32 6,667,870 9.99 Southwest 3,016,245 7.06 2,696,212 5.92 4,955,046 7.43 Trade-tech 4,596,835 10.77 3,791,851 8.32 11,917,950 17.86 Valley 5,190,062 12.15 5,897,293 12.94 5,737,297 8.60 West 3,665,387 8.58 3,673,243 8.06 6,888,809 10.32 ESC 100,000 0.23 0 0.00 0 0.00 Total SEA 42,699,430 100.00 45,560,764 100.00 66,727,950 100.00 Includes funds 11400-11404. Los Angeles Community College District 2023-2024 Final Budget 76 Student Financial Aid Administration (SFAA) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 529,747 11.79 678,895 12.61 691,593 13.24 East 955,261 21.27 1,079,222 20.04 1,110,044 21.25 Harbor 91,333 2.03 331,203 6.15 389,849 7.46 Mission 320,672 7.14 359,406 6.67 462,372 8.85 Pierce 551,741 12.28 679,860 12.62 599,496 11.48 Southwest 355,404 7.91 309,412 5.75 264,294 5.06 Trade-tech 460,822 10.26 550,783 10.23 478,924 9.17 Valley 577,669 12.86 672,745 12.49 575,841 11.02 West 414,136 9.22 507,273 9.42 493,740 9.45 ESC 235,035 5.23 216,897 4.03 156,928 3.00 Total SFAA 4,491,822 100.00 5,385,696 100.00 5,223,081 100.00 Includes funds 10415-10419. Student Retention and Enrollment Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 1,670,910 15.01 120,213 1.90 East 1,142,562 28.62 2,130,744 19.15 929,811 14.66 Harbor 148,453 3.72 320,174 2.88 420,778 6.64 Mission 153,753 3.85 447,218 4.02 679,346 10.71 Pierce 325,385 8.15 1,594,664 14.33 594,454 9.38 Southwest 8,000 0.20 12,230 0.11 978,484 15.43 Trade-tech 211,559 5.30 940,228 8.45 1,196,363 18.87 Valley 480,155 12.03 761,018 6.84 867,322 13.68 West 497,879 12.47 614,595 5.52 465,477 7.34 ESC 1,024,308 25.66 2,637,304 23.70 88,390 1.39 Total Student Retention and Enrollment 3,992,054 100.00 11,129,085 100.00 6,340,638 100.00 Includes funds 18187, 18192, and 18198. Los Angeles Community College District 2023-2024 Final Budget 77 Unrestricted Indirects Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 0 0.00 (163,632) 1.73 275,989 2.43 East 0 0.00 (1,161,789) 12.31 1,308,918 11.53 Harbor 0 0.00 (199,737) 2.12 646,920 5.70 Mission 0 0.00 (750,904) 7.95 842,910 7.42 Pierce 0 0.00 (2,459,201) 26.05 2,566,836 22.61 Southwest 0 0.00 (506,501) 5.37 784,991 6.91 Trade-tech 0 0.00 (1,940,950) 20.56 2,196,352 19.35 Valley 0 0.00 (1,982,302) 21.00 2,353,974 20.74 West 0 0.00 (275,249) 2.92 375,480 3.31 ESC 0 0.00 0 0.00 0 0.00 Total Unrestricted Indirects 0 0.00 (9,440,265) 100.00 11,352,370 100.00 Includes fund 10022. Veterans Resource Center Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 64,195 12.35 127,266 12.94 24,960 4.30 East 17,561 3.38 101,351 10.30 134,400 23.16 Harbor 51,612 9.93 123,757 12.58 0 0.00 Mission 48,924 9.41 23,845 2.42 67,292 11.60 Pierce 84,146 16.19 185,683 18.87 42,603 7.34 Southwest 37,407 7.20 77,283 7.86 34,631 5.97 Trade-tech 85,887 16.52 132,554 13.47 117,553 20.26 Valley 57,267 11.02 102,290 10.40 71,490 12.32 West 72,772 14.00 109,797 11.16 87,275 15.04 ESC 0 0.00 0 0.00 0 0.00 Total Veterans Resource Center 519,769 100.00 983,826 100.00 580,204 100.00 Includes funds 10471-10474. Los Angeles Community College District 2023-2024 Final Budget 78 Other Specially Funded Programs (SFP) Location 2021-22 Actual Expenditure % of total 2022-23 Actual Expenditure % of total 2023-24 Final Budget % of total City 7,665,984 14.46 13,171,054 18.46 15,117,104 8.17 East 6,786,404 12.80 7,196,327 10.08 30,960,340 16.74 Harbor 4,048,468 7.63 3,996,817 5.60 8,145,728 4.40 Mission 3,577,147 6.75 6,540,035 9.16 11,191,945 6.05 Pierce 4,037,718 7.61 5,304,461 7.43 15,803,834 8.54 Southwest 4,422,188 8.34 4,703,421 6.59 11,360,498 6.14 Trade-tech 4,720,358 8.90 4,812,370 6.74 27,337,615 14.78 Valley 7,399,741 13.95 11,807,561 16.54 14,424,716 7.80 West 8,843,607 16.68 11,134,110 15.60 21,456,767 11.60 ESC 1,524,181 2.87 2,702,024 3.79 29,163,354 15.77 Total Other SFP 53,025,797 100.00 71,368,179 100.00 184,961,901 100.00 Includes Customized Workshop Program, Community Partnership Training & Education, Customized Training Program, Community Services-Other, Technical & Career Ed, Non-Resident Capital Outlay, Veteran's Reporting Fees, On-Going Block Grants, Telecommunication & Technology Program, Federal PELL Grant (Funds 10465-10470), FSEOG (Funds 10475-10479), and funds above 10700 (if not previously listed). Los Angeles Community College District 2023-2024 Final Budget 79 Restricted General Fund Programs Locations & Programs Fund # Final Budget Los Angeles City College 2022-23 Block Grant-Instructional Support 10131 5,337,105 AANHPI Student Achievement 18049 150,697 AB 131 Stipend CSPP 17541 19,800 Adult Education Block Grant 2021-22 10460 77,815 Adult Education Block Grant 2022 -23 10461 773,117 Adult Education Block Grant FY 2023-24 10462 500,173 Advancing Scholars Successful Undergraduates 17186 112,395 Basic Needs Centers and Staffing 2021-22 11406 66,083 Basic Needs Centers and Staffing 2022-23 11407 281,596 CA Stat Presch Pr Quality Start LA CCALA 19694 890 CAI Academy Innovative Tech Professional 18205 281,968 CAI Cal Apprentic Init Nursing Assistant 18217 497,490 CalFresh Outreach 18186 2,684 CalWORKs 10450 61,519 CalWORKs 10451 10,883 CalWORKs 2023-24 10449 604,297 CalWORKs 22-23 10448 257,579 CalWORKs Child Care 2023-24 10442 126,012 CCAP Instructional Materials Dual Enroll 18197 39,630 Classified Staff Development 10435 23,106 Community Services Program 10010 1,202,745 Cooperative Agencies Resources for Education (CARE) 2023-24 11405 116,481 Cooperative Agencies Resources for Education (CARE) 2022-23 10869 23,244 COVID-19 Recovery Block Grant 18216 1,250,585 Culturally Competent Professional Deve 18601 149,625 Disabled Students Program & Serv (DSPS) 10405 15,912 Disabled Students Program & Serv 2023-24 10406 1,099,577 DPSS-CalWORKs 23-24 19266 155,083 Dream Resource Liaison Support 18207 3,314 Dream Resource Liaison Support 18209 63,877 Education Fund SEIU-UHW-West & Joint Emp 19693 525,000 Extended Opportunity Prog & Svcs 2023-24 10487 1,701,310 Extended Opportunity Prog & Svcs 22-23 10486 31,509 Federal Work Study (FWS) 23-24 10453 478,547 Financial Aid Technology 2020-21 10494 27,356 Focus on Creative Technology 17647 1,358,600 Foster and Kinship Care Education 23-24 10422 135,534 FSEOG Program 23-24 ACA 10477 24,883 Gear Up 4 LA 17703 30,804 Los Angeles Community College District 2023-2024 Final Budget 80 Locations & Programs Fund # Final Budget Gear Up 4 LA 17704 70,296 Gear Up 4 LA 17705 85,320 Guided Pathways 2022-23 18589 130,567 HACU Grow Google HSI Career Readiness 19692 48,384 Health Services 10135 1,279,258 Higher Ed Emergency Relief MSI Supplemen 17652 1,714 LACC Basic Needs 17696 902,541 Learning-Aligned Employment Program LAEP 18048 3,028,416 LGBTQ+ Students 18194 35,000 Lottery-Prop 20 - Restricted 10421 1,531,036 LSP Allocation for Operational Ser 21-22 11414 11,656 Math, Engineering, & Science Achievement 18598 1,400,000 Mental Health Services Support 2022-23 11411 353,009 NextUp (CAFYES) 2023-24 10400 117,609 NextUp (CAFYES) 22-23 10402 1,842 Non-Resident Capital Outlay 10020 2,016 Nursing Ed Support 18213 118,355 One Time Block Grant 21-22 10128 1,063,940 One-Time Block Grant 10127 42,653 Parking Fees 10145 63,938 Pathway to Law School (Cal Law) Initiati 18219 97,920 Perkins V 2023-24 10580 561,192 Puente Project 19678 30,000 Regional Equity and Recovery Partnership 18710 143,261 Resource Family Approval Training Servic 18575 346 Resource Family Approval Training Servic 18597 13,600 Resource for Success 17661 593,807 Rising Scholars Network 18585 42,658 Seamless Transfer of Ethnic Studies 18225 48,695 SFRF Emergency Financial Assistance 18045 22,966 State Administrativ Match Grant for Snap 17517 138,682 Strong Workforce Program - Local 10497 1,179,805 Strong Workforce Program - Local 2021-22 10496 976,660 Strong Workforce Program - Local 2023-24 10498 1,135,440 Strong Workforce Program -Regional 21-22 18899 265,422 Strong Workforce Program -Regional 22-23 18700 1,095,810 Student Equity and Achievement 2022-23 11403 1,282,223 Student Equity and Achievement 2023-24 11404 5,236,987 Student Financial Aid Adm 22-23 10419 122,505 Student Financial Aid Adm 23-24 10415 569,088 Student Food and Housing Support 18193 51,799 Student Food and Housing Support 2022-23 18199 311,363 Student Retention and Enrollment 2022-23 18198 120,213 Los Angeles Community College District 2023-2024 Final Budget 81 Locations & Programs Fund # Final Budget Student Support Services TRIO 17641 79,196 TANF Funding 2023-24 10445 107,574 Telecommunication & Technology Program 10437 270 UMOJA Community Foundation 19667 4 Undocumented Dream Resource Liaisons 18212 106,884 Unrestricted Indirect Fund 10022 275,989 Upward Bound Fairfax & Manual Arts H. S. 17663 441,632 Upward Bound Hollywood & Belmont H. S. 17664 309,505 Veterans Resource Center 2021-22 10471 3,709 Veterans Resource Center 2022-23 10472 21,251 Veterans Reporting Fees 10021 11,260 Zero Textbook Costs 2021-22 18044 178,651 Los Angeles City College Total $43,482,742 East Los Angeles College Community Learning Partnership CYLC 19683 8,679 2022-23 Block Grant-Instructional Support 10131 3,058,698 AB 131 Stipend CSPP 17541 33,600 Adult Education Block Grant 2021-22 10460 268 Adult Education Block Grant 2022 -23 10461 644,944 Adult Education Block Grant FY 2023-24 10462 500,000 Basic Needs Centers and Staffing 2021-22 11406 410,138 Basic Needs Centers and Staffing 2022-23 11407 602,850 CA Apprenticeship Initiative Med. Assit. 18204 180,525 CalFresh Outreach 18186 31,708 CalWORKs 2023-24 10449 642,992 CalWORKs 22-23 10448 580,665 CalWORKs Child Care 2023-24 10442 145,359 CalWORKs-Child Care 2022-23 10441 147,460 Career to Bridge Program 19670 7,363 CCAP Instructional Materials Dual Enroll 18197 115,468 Classified Staff Development 10435 34,978 Collaborative Research: Eager NSF 17518 99,554 College and Career 19695 31,756 Community Services Program 10010 1,628,592 Competency Based Education Collaborative 18580 460,123 Cooperative Agencies Resources for Education (CARE) 2023-24 11405 215,904 Cooperative Agencies Resources for Education (CARE) 22-23 10869 834 COVID-19 Recovery Block Grant 18216 11,566,892 Culturally Competent Professional Deve 18601 149,169 Disabled Students Program & Serv (DSPS) 10405 585,701 Disabled Students Program & Serv 2023-24 10406 1,750,334 Los Angeles Community College District 2023-2024 Final Budget 82 Locations & Programs Fund # Final Budget DPSS-CalWORKs 23-24 19266 168,415 Dream Resource Liaison Support 18207 92 Dream Resource Liaison Support 18209 119,075 Extended Opportunity Prog & Svcs 2023-24 10487 1,333,180 Extended Opportunity Prog & Svcs 22-23 10486 368,416 Extended Opportunity Prog & Svcs. (EOPS) 10490 4 Federal Pell Grant 19-20 ACA 10469 3,232 Federal Pell Grant 22-23 ACA 10467 29,351 Federal Work Study (FWS) 23-24 10453 674,690 Foster and Kinship Care Education 23-24 10422 96,738 FSEOG Program 22-23 ACA 10476 1,000 FSEOG Program 23-24 ACA 10477 31,432 Guided Pathways 2022-23 18589 453,911 Health Services 10135 1,832,633 Increasing Retention of Veterans in Engr 17138 968,046 Invention & Inclusive Innovation Initiat 18588 54,330 Learning-Aligned Employment Program LAEP 18048 6,022,941 LGBTQ+ Students 18194 56,727 Lottery-Prop 20 - Restricted 10421 3,553,908 LSP Allocation for Operational Ser 21-22 11414 26,278 Math, Engineering, & Science Achievement 18896 425,299 Mental Health Services Support 2021-22 11410 529,608 Mental Health Services Support 2022-23 11411 597,871 NextUp (CAFYES) 10401 11,684 NextUp (CAFYES) 2023-24 10400 163,116 NextUp (CAFYES) 22-23 10402 15,135 Non-Resident Capital Outlay 10020 728,332 Nursing Ed Support 18213 64,394 One Time Block Grant 21-22 10128 2,212,680 Parking Fees 10145 1,646,067 Perkins V 2023-24 10580 886,872 Puente Project 19675 35,000 Regional Equity and Recovery Partnership 18712 143,261 Rising Scholars Network 18585 119,647 Seamless Transfer of Ethnic Studies 18225 48,695 SFRF Emergency Financial Assistance 18045 56,425 Song-Brown Health Care Workforce Trainin 18596 625,662 Southeast Training Hub in Healthcare Car 17546 925,000 Strong Workforce Program - Local 10497 3,358,655 Strong Workforce Program - Local 2021-22 10496 994,638 Strong Workforce Program - Local 2023-24 10498 3,198,520 Strong Workforce Program -Regional 21-22 18899 779,930 Strong Workforce Program -Regional 22-23 18701 1,182,909 Los Angeles Community College District 2023-2024 Final Budget 83 Locations & Programs Fund # Final Budget Student Equity and Achievement 2022-23 11403 4,858,221 Student Equity and Achievement 2023-24 11404 10,784,990 Student Financial Aid Adm 22-23 10419 101,183 Student Financial Aid Adm 23-24 10415 1,008,861 Student Food and Housing Support 18193 57,578 Student Food and Housing Support 2022-23 18199 441,557 Student Retention and Enrollment 18187 517 Student Retention and Enrollment 2021-22 18192 221,297 Student Retention and Enrollment 2022-23 18198 707,997 Substance Use Disorder Earn and Learn 18223 2,043,813 TANF Funding 2023-24 10445 127,051 Teaching & Learning Community of Practic 18581 49,805 Tutoring/Mentoring Program 19705 180,000 Undocumented Dream Resource Liaisons 18212 179,781 Unrestricted Indirect Fund 10022 1,308,918 Veterans Resource Center 2021-22 10471 52,219 Veterans Resource Center 2022-23 10472 82,181 Veterans Program One Time 18185 40,814 Veterans Reporting Fees 10021 11,576 Zero Textbook Costs 2021-22 18044 179,895 East Los Angeles College Total $80,616,607 Los Angeles Harbor College 2022-23 Block Grant-Instructional support 10131 338,631 AANHPI Student Achievement 18049 150,697 AB 131 Stipend 17531 1,060 AB 131 Stipend CSPP 17541 20,400 Adult Education Block Grant 2021-22 10460 324 Adult Education Block Grant 2022 -23 10461 191,135 Adult Education Block Grant FY 2023-24 10462 500,000 Basic Needs Centers and Staffing 2021-22 11406 41,128 Basic Needs Centers and Staffing 2022-23 11407 158,809 CA Stat Presch Pr Quality Start LA CCALA 19694 5,416 CalFresh Outreach 18186 8,004 CalWORKs 2023-24 10449 298,830 CalWORKs 22-23 10448 89,700 CalWORKs Child Care 2023-24 10442 95,675 CCAP Instructional Materials Dual Enroll 18197 16,975 Classified Staff Development 10435 11,599 College Action for Student Achievement 17659 248,989 Community Services Program 10010 32,792 Cooperative Agencies Resources for Education (CARE) 2023-24 11405 210,275 Los Angeles Community College District 2023-2024 Final Budget 84 Locations & Programs Fund # Final Budget Cooperative Agencies Resources for Education (CARE) 22-23 10869 31,961 COVID-19 Recovery Block Grant 18216 2,954,696 CSPP Quality Improvement Block Grant 19294 998 Disabled Students Program & Serv (DSPS) 10405 151,782 Disabled Students Program & Serv 2023-24 10406 977,087 DPSS-CalWORKs 23-24 19266 112,277 Dream Resource Liaison Support 18207 35,927 Dream Resource Liaison Support 18209 50,887 Extended Opportunity Prog & Svcs 2023-24 10487 846,136 Extended Opportunity Prog & Svcs 22-23 10486 106,479 Federal Pell Grant 20-21 ACA 10470 129 Federal Pell Grant 22-23 ACA 10467 2,885 Federal Work Study (FWS) 23-24 10453 178,222 Financial Aid Technology 2020-21 10494 3,333 Foster and Kinship Care Education 23-24 10422 135,661 FSEOG Program 23-24 ACA 10477 7,908 Guided Pathways 2022-23 18589 256,946 Health Services 10135 368,645 Learning-Aligned Employment Program LAEP 18048 1,625,918 LGBTQ+ Students 18194 28,831 Lottery-Prop 20 - Restricted 10421 756,380 LSP Allocation for Operational Ser 21-22 11414 5,867 Mental Health Services Support 2021-22 11410 61,026 Mental Health Services Support 2022-23 11411 193,981 Middle College High School 18191 260,430 NextUp (CAFYES) 2023-24 10400 257,744 NextUp (CAFYES) 22-23 10402 75,428 Non-Resident Capital Outlay 10020 27 One Time Block Grant 21-22 10128 492,612 Parking Fees 10145 134,532 Perkins V 2023-24 10580 251,635 Puente Project 19689 32,694 Regional Equity and Recovery Partnership 18714 71,631 RUPE Foundation CNA Program 19662 8,329 Seamless Transfer of Ethnic Studies 18225 48,695 SFRF Emergency Financial Assistance 18045 7,623 Strong Workforce Program - Local 10497 334,956 Strong Workforce Program - Local 2021-22 10496 137,160 Strong Workforce Program - Local 2023-24 10498 349,904 Strong Workforce Program -Regional 21-22 18898 288,010 Strong Workforce Program -Regional 22-23 18702 658,689 Student Equity and Achievement 2022-23 11403 1,435,180 Student Equity and Achievement 2023-24 11404 3,045,720 Los Angeles Community College District 2023-2024 Final Budget 85 Locations & Programs Fund # Final Budget Student Financial Aid Adm 22-23 10419 84,209 Student Financial Aid Adm 23-24 10415 305,640 Student Food and Housing Support 18193 74,486 Student Food and Housing Support 2022-23 18199 220,728 Student Retention and Enrollment 18187 885 Student Retention and Enrollment 2021-22 18192 85,852 Student Retention and Enrollment 2022-23 18198 334,041 Student Support Services - TRIO 17655 98,123 TANF Funding 2023-24 10445 77,036 Teaching & Learning Community of Practic 18581 11,404 Telecommunication & Technology Program 10437 733 Tutoring/Mentoring Program 19705 180,000 Undocumented Dream Resource Liaisons 18212 80,082 Unrestricted Indirect Fund 10022 646,920 Veterans Reporting Fees 10021 1,104 Zero Textbook Costs 2021-22 18044 200,000 Los Angeles Harbor College Total $21,606,643 Los Angeles Mission College 17-18 Block Grant-One Time 10125 7,728 18-19 Block Grant-Instructional Support 10126 5,743 1-Train South Bay Workforce Invest Board 17989 43,336 2022-23 Block Grant-Instructional Support 10131 680,004 AB 131 Stipend CSPP 17541 26,400 Adult Education Block Grant 2021-22 10460 9,142 Adult Education Block Grant 2022 -23 10461 186,201 Adult Education Block Grant FY 2023-24 10462 500,000 Arthur N. RUPE Foundation CNA Program 19674 4,974 Basic Needs Centers and Staffing 2022-23 11407 75,805 Biology Major in Mathematics, NSF 17136 282,909 Biotechnology Program NSF 17513 110,283 Birmingham Pathways Development 18584 50,000 Block Grant-Instr Material/Equip; Lib Mat 10116 3,251 CA Stat Presch Pr Quality Start LA CCALA 19694 44 CalFresh Outreach 18186 9,992 California Wellness COVID 19 19671 9,100 CalWORKs 2023-24 10449 373,481 CalWORKs 22-23 10448 41,115 CalWORKs Child Care 2023-24 10442 86,344 CCAP Instructional Materials Dual Enroll 18197 60,659 Classified Staff Development 10435 13,425 Closing Equity Gaps in Allied Health DOL 17545 1,482,991 Los Angeles Community College District 2023-2024 Final Budget 86 Locations & Programs Fund # Final Budget Community Services Program 10010 1,573 Cooperative Agencies Resources for Education (CARE) 2023-24 11405 88,370 Cooperative Agencies Resources for Education (CARE) 22-23 10869 46,508 COVID-19 Recovery Block Grant 18216 3,569,022 Disabled Students Program & Serv (DSPS) 10405 101,174 Disabled Students Program & Serv 2023-24 10406 582,769 DPSS-CalWORKs 23-24 19266 102,453 Dream Resource Liaison Support 18207 32 Dream Resource Liaison Support 18209 2 Equitable Calculus for Life Sciences 18188 172,937 Extended Opportunity Prog & Svcs 2023-24 10487 822,431 Extended Opportunity Prog & Svcs 22-23 10486 57,271 Federal Pell Grant 18-19 ACA 10468 4,875 Federal Pell Grant 19-20 ACA 10469 2,071 Federal Pell Grant 20-21 ACA 10470 13,429 Federal Pell Grant 21-22 ACA 10465 9,889 Federal Pell Grant 22-23 ACA 10467 10,942 Federal Work Study (FWS) 23-24 10453 233,840 Foster and Kinship Care Education 23-24 10422 198,751 FSEOG Program 23-24 ACA 10477 10,619 Gear Up 4 LA 17702 82,874 Guided Pathways 2022-23 18589 121,604 HACU Grow Google HSI Career Readiness 19688 50,000 Health Services 10135 767,745 Increasing Short Term Vocational Trainin 17698 162,391 Learning-Aligned Employment Program LAEP 18048 1,738,883 LGBTQ+ Students 18194 63,892 Lottery-Prop 20 - Restricted 10421 947,612 LSP Allocation for Operational Ser 21-22 11414 7,067 Math, Engineering, & Science Achievement 18599 432,039 Mental Health Services Support 2021-22 11410 24,608 Mental Health Services Support 2022-23 11411 193,093 NextUp (CAFYES) 2023-24 10400 208,824 NextUp (CAFYES) 22-23 10402 34,707 One Time Block Grant 21-22 10128 592,120 One-Time Block Grant 10127 37,317 Pathway to Success Title V 17646 125,939 Perkins V 2023-24 10580 380,617 Regional Equity and Recovery Partnership 18713 167,138 Rising Scholars Network 18585 62,002 RUPE Foundation CNA Program 19658 561 Seamless Transfer of Ethnic Studies 18225 48,695 SFRF Emergency Financial Assistance 18045 9,478 Los Angeles Community College District 2023-2024 Final Budget 87 Locations & Programs Fund # Final Budget STEM METAS 17656 423,431 Strong Workforce Program - Local 10497 511,941 Strong Workforce Program - Local 2021-22 10496 132,206 Strong Workforce Program - Local 2023-24 10498 589,385 Strong Workforce Program -Regional 21-22 18897 233,543 Strong Workforce Program -Regional 22-23 18703 690,112 Student Equity and Achievement 2022-23 11403 647,380 Student Equity and Achievement 2023-24 11404 3,270,277 Student Financial Aid Adm 22-23 10419 107,440 Student Financial Aid Adm 23-24 10415 354,932 Student Food and Housing Support 18193 5,104 Student Food and Housing Support 2022-23 18199 34,552 Student Retention and Enrollment 18187 1,496 Student Retention and Enrollment 2021-22 18192 36,877 Student Retention and Enrollment 2022-23 18198 640,973 Student Support Services TRIO 17642 179,426 TANF Funding 2023-24 10445 71,759 Telecommunication & Technology Program 10437 54 Undocumented Dream Resource Liaisons 18212 86,132 Unrestricted Indirect Fund 10022 842,910 Veterans Resource Center 2021-22 10471 23,622 Veterans Resource Center 2022-23 10472 43,670 Veterans Reporting Fees 10021 5,035 Zero Textbook Costs 2021-22 18044 191,562 Los Angeles Mission College Total $25,474,910 Los Angeles Pierce College 18-19 Block Grant-Instructional Support 10126 91,125 2022-23 Block Grant-Instructional Support 10131 4,175,551 AB 131 Stipend 17531 43 AB 131 Stipend CSPP 17541 25,800 Adult Education Block Grant 2022 -23 10461 119,321 Adult Education Block Grant FY 2023-24 10462 500,000 Alcanzando Las Estrellas HE 17669 384,586 Basic Needs Centers and Staffing 2022-23 11407 456,556 CA Stat Presch Pr Quality Start LA CCALA 19694 1,407 CalFresh Outreach 18186 9,575 CalWORKs 10451 409 CalWORKs 2023-24 10449 330,796 CalWORKs 22-23 10448 153,358 CalWORKs Child Care 2023-24 10442 83,855 CCAP Instructional Materials Dual Enroll 18197 18,648 Los Angeles Community College District 2023-2024 Final Budget 88 Locations & Programs Fund # Final Budget Classified Staff Development 10435 13,611 Community Services Program 10010 2,073,574 Cooperative Agencies Resources for Education (CARE) 10868 2 Cooperative Agencies Resources for Education (CARE) 2023-24 11405 19,457 Cooperative Agencies Resources for Education (CARE) 22-23 10869 2,456 COVID-19 Recovery Block Grant 18216 6,388,226 Disabled Students Program & Serv (DSPS) 10405 464,213 Disabled Students Program & Serv 2023-24 10406 1,199,037 DPSS-CalWORKs 23-24 19266 100,348 Dream Resource Liaison Support 18209 115,468 Extended Opportunity Prog & Svcs 2023-24 10487 903,790 Extended Opportunity Prog & Svcs 22-23 10486 268,198 Federal Pell Grant 16-17 ACA 10466 11 Federal Pell Grant 18-19 ACA 10468 241 Federal Pell Grant 19-20 ACA 10469 685 Federal Pell Grant 20-21 ACA 10470 522 Federal Pell Grant 21-22 ACA 10465 1,120 Federal Pell Grant 22-23 ACA 10467 11,462 Federal Work Study (FWS) 23-24 10453 452,675 Financial Aid Technology 2020-21 10494 1,854 Foster and Kinship Care Education 23-24 10422 79,992 FSEOG Program 22-23 ACA 10476 287 FSEOG Program 23-24 ACA 10477 20,833 Guided Pathways 2022-23 18589 515,521 Health Services 10135 1,012,541 Higher Ed Emergency Relief MSI Supplemen 17652 4,361 Improving Student Career Readiness Exper 17185 109,956 Innovation in Higher Education 18837 13,018 Learning-Aligned Employment Program LAEP 18048 3,820,445 LGBTQ+ Students 18194 31,693 Lottery-Prop 20 - Restricted 10421 2,493,839 LSP Allocation for Operational Ser 21-22 11414 14,809 Math, Engineering, & Science Achievement 18599 1,650,137 Mental Health Services Support 2021-22 11410 52,750 Mental Health Services Support 2022-23 11411 240,120 NextUp (CAFYES) 2023-24 10400 145,096 NextUp (CAFYES) 22-23 10402 176,471 Non-Resident Capital Outlay 10020 3,241 Nursing Ed Support 18213 169,583 One Time Block Grant 21-22 10128 17,478 One-Time Block Grant 10127 46,111 Parking Fees 10145 1,643,166 Pathways and Career Explorations in Stem 17521 51,982 Los Angeles Community College District 2023-2024 Final Budget 89 Locations & Programs Fund # Final Budget Perkins V 2023-24 10580 509,599 Pritzker Grant-SFV Guardian Scholars Net 19572 3,972 Rising Scholars Network 18585 85,527 Seamless Transfer of Ethnic Studies 18225 48,695 SFP Tech ASO 19639 20,190 SFRF Emergency Financial Assistance 18045 13,524 Strong Workforce Program - Local 10497 1,145,934 Strong Workforce Program - Local 2021-22 10496 154,018 Strong Workforce Program - Local 2023-24 10498 1,333,465 Strong Workforce Program -Regional 21-22 18898 278,889 Student Equity and Achievement 2022-23 11403 1,380,490 Student Equity and Achievement 2023-24 11404 5,287,380 Student Financial Aid Adm 22-23 10419 15,354 Student Financial Aid Adm 23-24 10415 584,142 Student Food and Housing Support 18193 26,105 Student Food and Housing Support 2022-23 18199 301,672 Student Retention and Enrollment 18187 558 Student Retention and Enrollment 2021-22 18192 40,567 Student Retention and Enrollment 2022-23 18198 553,329 TANF Funding 2023-24 10445 65,137 Telecommunication & Technology Program 10437 4,926 Training Skilled Biomanufa Workforce NSF 17516 77,845 Undocumented Dream Resource Liaisons 18212 122,234 Unrestricted Indirect Fund 10022 2,566,836 Veterans Resource Center 2022-23 10472 42,603 Veterans Reporting Fees 10021 11,654 Wellness Vending Machines Pilot 18224 15,000 Zero Emission, Clean Energy, Electr Trn 18586 500,000 Zero Textbook Costs 2021-22 18044 199,996 Los Angeles Pierce College Total $46,071,051 Los Angeles Southwest College 17-18 Block Grant-One Time 10125 142,337 18-19 Block Grant-Instructional Support 10126 25,237 2022-23 Block Grant-Instructional Support 10131 1,230,447 AB 131 Stipend 17531 3,670 AB 131 Stipend CSPP 17541 31,800 Adult Education Block Grant 2021-22 10460 67,605 Adult Education Block Grant 2022 -23 10461 902,446 Adult Education Block Grant FY 2023-24 10462 500,000 Basic Needs Centers and Staffing 2021-22 11406 76,568 Los Angeles Community College District 2023-2024 Final Budget 90 Locations & Programs Fund # Final Budget Basic Needs Centers and Staffing 2022-23 11407 245,869 CA Stat Presch Pr Quality Start LA CCALA 19694 270 CalFresh Outreach 18186 7,107 CalWORKs 10450 241,530 CalWORKs 10451 726,588 CalWORKs 2023-24 10449 594,179 CalWORKs 22-23 10448 712,468 CalWORKs Child Care 10444 175,281 CalWORKs Child Care 2021-22 10440 20,000 CalWORKs Child Care 2023-24 10442 124,969 CalWORKs-Child Care 2022-23 10441 40,000 CCAP Instructional Materials Dual Enroll 18197 42,986 Cooperative Agencies Resources for Education (CARE) 2023-24 11405 118,066 Cooperative Agencies Resources for Education (CARE) 22-23 10869 73,399 COVID-19 Recovery Block Grant 18216 2,286,817 CSPP Quality Improvement Block Grant 19294 377 Disabled Students Program & Serv (DSPS) 10405 28,254 Disabled Students Program & Serv 2023-24 10406 424,438 DPSS-CalWORKs 23-24 19266 115,084 Dream Resource Liaison Support 18209 30,476 Extended Opportunity Prog & Svcs 2023-24 10487 209,794 Extended Opportunity Prog & Svcs 22-23 10486 94,465 Federal Pell Grant 16-17 ACA 10466 90 Federal Pell Grant 18-19 ACA 10468 1,433 Federal Pell Grant 20-21 ACA 10470 75 Federal Pell Grant 21-22 ACA 10465 1,592 Federal Pell Grant 22-23 ACA 10467 6,440 Federal Work Study (FWS) 23-24 10453 137,735 Financial Aid Technology 2020-21 10494 13,291 Foster and Kinship Care Education 23-24 10422 95,176 Foster Care Counts-Guardian Scholars Pro 19582 11,811 FSEOG Program 23-24 ACA 10477 8,146 Guardian Scholars Program 19581 88,111 Guided Pathways 2022-23 18589 259,428 Health Services 10135 121,072 Learning-Aligned Employment Program LAEP 18048 1,352,157 LGBTQ+ Students 18194 35,000 Lottery-Prop 20 - Restricted 10421 596,029 LSP Allocation for Operational Ser 21-22 11414 4,768 Mental Health Services Support 2021-22 11410 276,130 Mental Health Services Support 2022-23 11411 122,312 NextUp (CAFYES) 10403 31,286 NextUp (CAFYES) 2023-24 10400 173,073 Los Angeles Community College District 2023-2024 Final Budget 91 Locations & Programs Fund # Final Budget NextUp (CAFYES) 22-23 10402 33,567 Non-Resident Capital Outlay 10020 58,443 Nursing Ed Support 18213 132,952 One Time Block Grant 21-22 10128 1,077,830 One-Time Block Grant 10127 51,772 Parking Fees 10145 574,578 Perkins V 2023-24 10580 235,512 Predominantly Black Institutions-Formula 17657 523,297 Puente Project 19675 40,500 Rapid Rehousing 2020-21 18892 136,659 Rapid Rehousing 2021-22 18895 648,873 Rapid Rehousing 2022-23 18590 933,333 Regional Equity and Recovery Partnership 18713 119,384 Resource Family Approval Training Servic 18575 8,538 RISE Center & Family Resource Center 17699 1,500,000 Seamless Transfer of Ethnic Studies 18225 48,695 SFRF Emergency Financial Assistance 18045 6,563 Strong Workforce Program - Local 10497 133,931 Strong Workforce Program - Local 2021-22 10496 271,395 Strong Workforce Program - Local 2023-24 10498 252,062 Strong Workforce Program -Regional 21-22 18897 432,224 Strong Workforce Program -Regional 22-23 18705 795,856 Student Equity and Achievement 2022-23 11403 1,757,279 Student Equity and Achievement 2023-24 11404 3,197,767 Student Financial Aid Adm 22-23 10419 2,736 Student Financial Aid Adm 23-24 10415 261,558 Student Food and Housing Support 18193 108,654 Student Food and Housing Support 2022-23 18199 208,972 Student Retention and Enrollment 18187 143,413 Student Retention and Enrollment 2021-22 18192 283,368 Student Retention and Enrollment 2022-23 18198 551,703 Student Support Services Stem TRIO 17645 60,389 Student Support Services TRIO 17648 346,015 TANF Funding 2023-24 10445 106,524 Undocumented Dream Resource Liaisons 18212 73,935 Unrestricted Indirect Fund 10022 784,991 Veterans Resource Center 2022-23 10472 34,631 Veterans Reporting Fees 10021 3,025 Zero Textbook Costs 2021-22 18044 196,150 Los Angeles Southwest College Total $28,764,756 Los Angeles Trade-Tech College Los Angeles Community College District 2023-2024 Final Budget 92 Locations & Programs Fund # Final Budget 2022-23 Block Grant-Instructional Support 10131 1,460,081 AB 131 Stipend 17531 2,149 AB 131 Stipend CSPP 17541 21,600 Adult Education Block Grant 2022 -23 10461 612,418 Adult Education Block Grant FY 2023-24 10462 500,000 Basic Needs Centers and Staffing 2021-22 11406 243,463 Basic Needs Centers and Staffing 2022-23 11407 361,296 Block Grant-Instr Material/Equip; Lib Mat 10116 53,617 CalFresh Outreach 18186 15,908 CalWORKs 10450 1,049 CalWORKs 10451 86,333 CalWORKs 2023-24 10449 797,234 CalWORKs 22-23 10448 1,040,898 CalWORKs Child Care 10444 3,236 CalWORKs Child Care 2023-24 10442 165,665 CCAP Instructional Materials Dual Enroll 18197 59,173 Chemical Technical Program 19231 1,201 Chemical Technology-NSF (Matchg-10762) 10918 9,186 Chemical/Process Technology Prog-Bp 19367 17,474 Classified Staff Development 10435 19,303 Cooperative Agencies Resources for Education (CARE) 10868 2 Cooperative Agencies Resources for Education (CARE) 2023-24 11405 92,000 Cooperative Agencies Resources for Education (CARE) 22-23 10869 42,373 COVID-19 Recovery Block Grant 18216 4,731,528 Disabled Students Program & Serv (DSPS) 10405 247,747 Disabled Students Program & Serv 2023-24 10406 1,000,811 DPSS-CalWORKs 23-24 19266 207,712 Dream Resource Liaison Support 18209 87,156 ECMC Foundation 2023 19709 12,881 Extended Opportunity Prog & Svcs 2023-24 10487 973,728 Extended Opportunity Prog & Svcs 22-23 10486 6,521 Extended Opportunity Prog & Svcs. (EOPS) 10490 3 Family Support Program 19339 8,716 Federal Pell Grant 16-17 ACA 10466 19,262 Federal Pell Grant 18-19 ACA 10468 8,447 Federal Pell Grant 19-20 ACA 10469 5,565 Federal Pell Grant 20-21 ACA 10470 32,105 Federal Pell Grant 21-22 ACA 10465 14,618 Federal Pell Grant 22-23 ACA 10467 41,878 Federal Work Study (FWS) 23-24 10453 329,873 Financial Aid Technology 2020-21 10494 12,876 Formerly Incarcerated 19706 35,000 Foster and Kinship Care Education 23-24 10422 148,005 Los Angeles Community College District 2023-2024 Final Budget 93 Locations & Programs Fund # Final Budget Foster Care Counts-Guardian Scholars Pro 19559 3,330 FSEOG Program 23-24 ACA 10477 18,151 Gear Up 4 LA 17700 51,261 Gear Up 4 LA 17701 50,109 Guardian Scholars Jewish Commu Foundatio 19608 3,717 Guardian Scholars Program 19556 15,601 Guardian Scholars Support Donation 19607 15,065 Guided Pathways 2022-23 18589 477,238 Health Services 10135 942,396 Higher Ed Emergency Relief MSI Supplemen 17652 75,610 IEPI Innovation and Effectiveness 18894 99,881 Innovation in Higher Education 18838 199,830 LA DWP Training 19624 13,381,442 Learning-Aligned Employment Program LAEP 18048 2,931,176 LGBTQ+ Students 18194 74,902 Lottery-Prop 20 - Restricted 10421 1,510,419 LSP Allocation for Operational Ser 21-22 11414 11,886 Mental Health Services Support 2021-22 11410 224,006 Mental Health Services Support 2022-23 11411 275,167 New Car Dealers Association 19704 10,000 NextUp (CAFYES) 2023-24 10400 247,000 NextUp (CAFYES) 22-23 10402 74,961 Non-Resident Capital Outlay 10020 30,551 Nursing Ed Support 18213 89,492 One Time Block Grant 21-22 10128 1,568,503 Outreach Support Foster Youth Services 19707 58,314 Parking Fees 10145 810,330 Perkins V 2023-24 10580 509,599 Puente Project 19675 40,500 Regional Equity and Recovery Partnership 18220 143,261 Resource Family Approval Training Servic 18575 56,214 Rising Scholars Network 18585 424,343 Seamless Transfer of Ethnic Studies 18225 48,695 SFRF Emergency Financial Assistance 18045 15,541 South Bay-Chem Tech/ Process Tech Major 19347 7,983 Strong Workforce Program - Local 10497 2,386,371 Strong Workforce Program - Local 2021-22 10496 1,456,024 Strong Workforce Program - Local 2023-24 10498 2,277,747 Strong Workforce Program -Regional 21-22 18897 386,142 Strong Workforce Program -Regional 22-23 18706 838,729 Student Equity and Achievement 2022-23 11403 6,125,287 Student Equity and Achievement 2023-24 11404 5,792,663 Student Financial Aid Adm 22-23 10419 7,412 Los Angeles Community College District 2023-2024 Final Budget 94 Locations & Programs Fund # Final Budget Student Financial Aid Adm 23-24 10415 471,512 Student Food and Housing Support 18193 81,376 Student Food and Housing Support 2022-23 18199 252,720 Student Retention and Enrollment 18187 66,211 Student Retention and Enrollment 2021-22 18192 259,654 Student Retention and Enrollment 2022-23 18198 870,498 TANF Funding 2023-24 10445 147,491 Technical & Career Ed 10017 950,410 Undocumented Dream Resource Liaisons 18212 21,420 Unrestricted Indirect Fund 10022 2,196,352 Veterans Resource Center 2021-22 10471 34,143 Veterans Resource Center 2022-23 10472 83,410 Zero Textbook Costs 2021-22 18044 200,000 Los Angeles Trade-Tech College Total $62,932,137 Los Angeles Valley College 18-19 Block Grant-Instructional Support 10126 7,245 2022-23 Block Grant-Instructional Support 10131 622,216 AB 131 Stipend 17531 1,709 AB 131 Stipend CSPP 17541 158 Adult Education Block Grant 2021-22 10460 70,360 Adult Education Block Grant 2022 -23 10461 748,900 Adult Education Block Grant FY 2023-24 10462 500,683 Basic Needs Centers and Staffing 2021-22 11406 83,592 Basic Needs Centers and Staffing 2022-23 11407 435,706 CA Stat Presch Pr Quality Start LA CCALA 19694 853 CalFresh Outreach 18186 1,323 CalWORKs 2023-24 10449 846,750 CalWORKs 22-23 10448 560,622 CalWORKs Child Care 2023-24 10442 164,123 CalWORKs-Child Care 2022-23 10441 26,690 CCAP Instructional Materials Dual Enroll 18197 13,058 Classified Staff Development 10435 18,550 Community Services Program 10010 2,927,600 Cooperative Agencies Resources for Education (CARE) 10868 2 Cooperative Agencies Resources for Education (CARE) 2023-24 11405 94,679 Cooperative Agencies Resources for Education (CARE) 22-23 10869 8,543 COVID-19 Recovery Block Grant 18216 2,377,618 CSPP Quality Improvement Block Grant 19294 175 Disabled Students Program & Serv (DSPS) 10405 709,426 Disabled Students Program & Serv 2023-24 10406 1,152,849 DPSS-CalWORKs 23-24 19266 218,238 Los Angeles Community College District 2023-2024 Final Budget 95 Locations & Programs Fund # Final Budget Dream Resource Liaison Support 18209 1,140 Dual Enrollment to Career Title V 17643 758,954 Equitable Calculus tor Life Sciences 18210 190,740 Extended Opportunity Prog & Svcs 2023-24 10487 1,184,852 Extended Opportunity Prog & Svcs 22-23 10486 72,795 Family Resource Center 18866 514,588 Federal Pell Grant 16-17 ACA 10466 1,322 Federal Pell Grant 18-19 ACA 10468 6,410 Federal Pell Grant 19-20 ACA 10469 12,626 Federal Pell Grant 20-21 ACA 10470 19,457 Federal Pell Grant 21-22 ACA 10465 20,322 Federal Pell Grant 22-23 ACA 10467 24,490 Federal Work Study (FWS) 23-24 10453 380,380 Financial Aid Technology 2020-21 10494 1,729 FSEOG Program 23-24 ACA 10477 17,120 Guided Pathways 2022-23 18589 427,008 Health Services 10135 1,342,533 Increasing Short Term Vocational Trainin 17698 115,094 Innovation in Higher Education 18837 69,696 JTPA City of Inglewood Voucher 10712 279,758 Learning-Aligned Employment Program LAEP 18048 3,600,761 LGBTQ+ Students 18194 54,505 Lottery-Prop 20 - Restricted 10421 1,683,225 LSP Allocation for Operational Ser 21-22 11414 12,623 Math, Engineering, & Science Achievement 18599 453,262 Mental Health Services Support 2022-23 11411 335,488 NextUp (CAFYES) 2023-24 10400 216,414 NextUp (CAFYES) 22-23 10402 20,186 Non-Resident Capital Outlay 10020 1,722 Nursing Ed Support 18213 19,088 One Time Block Grant 21-22 10128 9,402 One-Time Block Grant 10127 6,603 Parking Fees 10145 2,124,498 Pathways and Career Explorations in STEM 17521 227,244 Perkins V 2023-24 10580 499,925 Puente Project 19675 45,001 Regional Equity and Recovery Partnership 18712 77,600 Seamless Transfer of Ethnic Studies 18225 48,695 SFRF Emergency Financial Assistance 18045 15,846 Strong Workforce Program - Local 10497 989,209 Strong Workforce Program - Local 2021-22 10496 407,704 Strong Workforce Program - Local 2023-24 10498 939,754 Strong Workforce Program -Regional 21-22 18899 489,644 Los Angeles Community College District 2023-2024 Final Budget 96 Locations & Programs Fund # Final Budget Strong Workforce Program -Regional 22-23 18707 921,353 Student Equity and Achievement 2022-23 11403 710,344 Student Equity and Achievement 2023-24 11404 5,026,953 Student Financial Aid Adm 22-23 10419 47 Student Financial Aid Adm 23-24 10415 575,794 Student Food and Housing Support 18193 115,127 Student Food and Housing Support 2022-23 18199 305,591 Student Parent Support Program 17658 118,658 Student Retention and Enrollment 2022-23 18198 867,322 Student Support Services TRIO 17649 64,666 TANF Funding 2023-24 10445 145,939 Teaching & Learning Community of Practic 18581 522 Telecommunication & Technology Program 10437 54,042 Title V Proyecto Adelante (PAC3) 17630 112,499 TRIO - Upward Bound 17625 153,322 TRIO Upward Bound 17622 25,256 TRIO Upward Bound 17667 110,204 TRIO Upward Bound Math & Science 17666 120,917 Tutoring/Mentoring Program 19705 83,102 Undocumented Dream Resource Liaisons 18212 82,530 Unrestricted Indirect Fund 10022 2,353,974 Valley Academic and Cultural Center 18047 999,620 Veterans Resource Center 2021-22 10471 1,089 Veterans Resource Center 2022-23 10472 70,401 Veterans Reporting Fees 10021 5,626 Zero Textbook Costs 2021-22 18044 180,187 Los Angeles Valley College Total $42,516,166 West Los Angeles College 2022-23 Block Grant-Instructional Support 10131 1,709,677 AB 131 Stipend CSPP 17541 7,590 Adult Education Block Grant 2022 -23 10461 144,146 Adult Education Block Grant FY 2023-24 10462 500,000 Arts, Media and Entertainment Flex Appre 18218 477,196 Basic Needs Centers and Staffing 2021-22 11406 81,005 Basic Needs Centers and Staffing 2022-23 11407 334,261 Bioflex CAI Apprenticeship 18996 38,295 CA Center for Climate Change Education 18214 4,935,875 CalFresh Outreach 18186 10,781 CalWORKs 2023-24 10449 365,147 CalWORKs 22-23 10448 405,636 CalWORKs Child Care 2023-24 10442 89,614 Los Angeles Community College District 2023-2024 Final Budget 97 Locations & Programs Fund # Final Budget CCAP Instructional Materials Dual Enroll 18197 18,955 Classified Staff Development 10435 14,795 Community Connect 18600 3,050 Community Partnership Training & Ed 10013 369,403 Community Services Program 10010 389,838 Cooperative Agencies Resources for Education (CARE) 2023-24 11405 53,039 Cooperative Agencies Resources for Education (CARE) 22-23 10869 51,608 COVID-19 Recovery Block Grant 18216 4,041,767 CSPP Quality Improvement Block Grant 19294 3 Culturally Competent Professional Deve 18601 300,000 Customized Trng Program 10014 7,220 Disabled Students Program & Serv (DSPS) 10405 4,157 Disabled Students Program & Serv 2023-24 10406 621,864 DPSS-CalWORKs 23-24 19266 74,383 Dream Resource Liaison Support 18209 32,087 Educational Opportunity Centers TRIO 17660 32,847 Extended Opportunity Prog & Svcs 2023-24 10487 339,927 Extended Opportunity Prog & Svcs 22-23 10486 220,671 Extended Opportunity Prog & Svcs. (EOPS) 10490 5 Federal Pell Grant 22-23 ACA 10467 3,786 Federal Work Study (FWS) 23-24 10453 267,068 Foster and Kinship Care Education 23-24 10422 119,452 Fresh Success 17520 215,639 FSEOG Program 23-24 ACA 10477 10,891 Guided Pathways 2022-23 18589 162,768 H-1B Job Training Grant 17181 3,176,630 Health Flex Ca Apprenticeship Initiative 18211 429,919 Health Services 10135 547,737 Hispanic Serving Institution Precise NSF 17539 92,578 Institutional Effect Partnership Initiat 18709 200,000 Invention & Inclusive Innovation Initiat 18588 9,746 LACOE Early Childhood Educator Apprentic 19697 266,409 Learning Lab 18591 62,066 Learning-Aligned Employment Program LAEP 18048 1,774,138 LGBTQ+ Students 18194 35,000 Lottery-Prop 20 - Restricted 10421 1,136,549 LSP Allocation for Operational Ser 21-22 11414 1,912 Mental Health Services Support 2021-22 11410 188,248 Mental Health Services Support 2022-23 11411 272,359 NextUp (CAFYES) 2023-24 10400 193,740 NextUp (CAFYES) 22-23 10402 100,382 Non-Resident Capital Outlay 10020 492 One Time Block Grant 21-22 10128 1,295,215 Los Angeles Community College District 2023-2024 Final Budget 98 Locations & Programs Fund # Final Budget Parking Fees 10145 70,456 Perkins V 2023-24 10580 464,455 Puente Project 19675 20,000 Regional Equity and Recovery Partnership 18711 202,953 Resource Family Approval Training Servic 18575 37,200 Rising Scholars Network 18585 68,346 Seamless Transfer of Ethnic Studies 18225 48,695 SFRF Emergency Financial Assistance 18045 35,847 Stem Teacher Success He 17629 878,086 Strong Workforce Program - Local 10497 1,054,733 Strong Workforce Program - Local 2021-22 10496 534,268 Strong Workforce Program - Local 2023-24 10498 1,043,654 Strong Workforce Program -Regional 21-22 18898 93,697 Strong Workforce Program -Regional 22-23 18708 753,820 Student Equity and Achievement 2022-23 11403 2,902,390 Student Equity and Achievement 2023-24 11404 3,986,419 Student Financial Aid Adm 22-23 10419 35,879 Student Financial Aid Adm 23-24 10415 457,861 Student Food and Housing Support 18193 55,740 Student Food and Housing Support 2022-23 18199 254,513 Student Retention and Enrollment 2022-23 18198 465,477 Student Support Services TRIO 17644 280,562 TANF Funding 2023-24 10445 70,936 Teaching & Learning Community of Practic 18581 4,367 TRIO - Talent Search 17653 59,137 TRIO Upward Bound 17623 14,938 TRIO Upward Bound at Dorsey High WLAC 17665 89,537 TRIO Upward Bound at Hamilton & La High 17668 98,887 TRIO-Upward Bound M&S Dorsey 17671 199,184 TRIO-Upward Bound M&S La High School 17670 206,304 Undocumented Dream Resource Liaisons 18212 94,115 Unrestricted Indirect Fund 10022 375,480 Veterans Resource Center 2021-22 10471 20,045 Veterans Resource Center 2022-23 10472 67,230 Veterans Reporting Fees 10021 8,148 Workforce Training Program 10012 138,669 Zero Textbook Costs 2021-22 18044 178,810 West Los Angeles College Total $41,608,404 Educational Services Center Adult Education Block Grant 2022 -23 10461 403,566 Adult Education Block Grant FY 2023-24 10462 5,074,029 Los Angeles Community College District 2023-2024 Final Budget 99 Locations & Programs Fund # Final Budget Apple Community Education Initiatives 19691 14,984 Automobile Emissions Research & Technolo 19701 719,568 California College Promise 22-23 10408 1,750,500 Campus Safety and Sexual Assault 18366 104,930 Center for International Business Educat 17992 19,000 Classified Staff Development 10435 173,810 COVID-19 Recovery Block Grant 18216 6,238,547 Culturally Competent Faculty Pd 18196 453,915 Diversity, Equity & Inclusion Deities 17510 275,926 Dolores Huerta Labor Institute 19362 428,582 DPSS-CalWORKs 23-24 19266 62,250 EEO Best Practices 18195 20,044 Equal Employment Opportunity 10436 313,567 Federal Pell Grant 16-17 ACA 10466 85,872 Federal Pell Grant 18-19 ACA 10468 50,974 Federal Pell Grant 19-20 ACA 10469 42,313 Federal Pell Grant 20-21 ACA 10470 100,750 Federal Pell Grant 21-22 ACA 10465 93,288 Federal Pell Grant 22-23 ACA 10467 94,979 Federal Work Study (FWS) 23-24 10453 23,663 Financial Aid Technology 2020-21 10494 15 Financial Aid Technology 2022-23 10485 317,472 Framework for Racial Equity and Social 19660 376,243 FSEOG Program 23-24 ACA 10477 26,470 Institutional Effect Partnership Initiat 18602 189,000 Learning-Aligned Employment Program LAEP 18048 1,364,465 LGBTQ+ Students 18194 23,814 Local & Systemwide Technology & Data Sec 18221 540,000 Lottery-Prop 20 - Restricted 10421 3,979,084 Perkins V 2023-24 10580 226,284 Regional K-16 Education Collaboratives 18046 16,952,023 Student Financial Aid Adm 22-23 10419 156,928 Student Retention and Enrollment 2021-22 18192 88,390 Telecommunication & Technology Program 10437 19,534 Transportation Assistance to Students 17697 836,648 WestEd Reading Apprenticeship 19710 88,006 Educational Services Center Total $41,729,433 Restricted General Fund Programs Total $434,802,849 General Fund Appropriations Los Angeles Community College District 2023-2024 Final Budget 100 General Fund Summary C/I Description Unrestricted General Fund % of Total Restricted General Fund % of Total Total General Fund % of Total 110000 Teaching, Regular 136,220,074 12.8% 690,230 0.2% 136,910,304 9.1% 120000 Non-Teaching, Regular 57,993,876 5.4% 34,051,971 7.8% 92,045,847 6.1% 130000 Teaching, Hourly 162,306,873 15.2% 3,080,775 0.7% 165,387,648 11.0% 140000 Non-Teaching, Hourly 5,307,696 0.5% 24,601,076 5.7% 29,908,772 2.0% 190000 Misc Certificated Salaries 300,000 0.0% 0 0.0% 300,000 0.0% Total Certificated Salaries 362,128,519 34.0% 62,424,052 14.4% 424,552,571 28.3% 210000 Classified, Regular 133,550,891 12.5% 28,605,046 6.6% 162,155,937 10.8% 220000 Instructional Aides, Regular 14,848,591 1.4% 2,563,789 0.6% 17,412,380 1.2% 230000 Sub/Relief, Unclassified 4,378,779 0.4% 22,224,712 5.1% 26,603,491 1.8% 240000 Instructional Aides, Non-Perm 2,335,438 0.2% 4,911,060 1.1% 7,246,498 0.5% 290000 Misc Non-Certificated Salaries 0 0.0% 50,000 0.0% 50,000 0.0% Total Non-Certificated Salaries 155,113,699 14.6% 58,354,607 13.4% 213,468,306 14.2% 310000 STRS Employer Contributions 58,800,000 5.5% 0 0.0% 58,800,000 3.9% 320000 PERS Employer Contributions 50,700,000 4.8% 0 0.0% 50,700,000 3.4% 330000 OASDHI Contributions 12,573,308 1.2% 0 0.0% 12,573,308 0.8% 340000 Medical/Dental Contributions 123,708,465 11.6% 0 0.0% 123,708,465 8.2% 350000 State Unemployment Insurance 3,240,615 0.3% 0 0.0% 3,240,615 0.2% 360000 Workers Compensation Insurance 4,000,000 0.4% 0 0.0% 4,000,000 0.3% 370000 Local Retirement System 5,124,206 0.5% 0 0.0% 5,124,206 0.3% 390000 Misc Employee Benefits (60,058,888) -5.6% 33,466,597 7.7% (26,592,291) -1.8% Total Benefits 198,087,706 18.6% 33,466,597 7.7% 231,554,303 15.4% 420000 Books 6,708 0.0% 753,895 0.2% 760,603 0.1% 440000 Instructional Media Materials 1,510,967 0.1% 19,080,619 4.4% 20,591,586 1.4% 450000 Supplies 5,405,193 0.5% 5,789,266 1.3% 11,194,459 0.7% 470000 Materials Fees 0 0.0% 8,500 0.0% 8,500 0.0% Total Printing & Supplies 6,922,868 0.6% 25,632,280 5.9% 32,555,148 2.2% 540000 Insurance 11,288,633 1.1% 0 0.0% 11,288,633 0.8% 550000 Utilities & Housekeeping Expense 27,231,858 2.6% 182,574 0.0% 27,414,432 1.8% 560000 Contracts & Rentals 68,402,408 6.4% 54,846,106 12.6% 123,248,514 8.2% 570000 Legal, Election, Audit 10,494,740 1.0% 0 0.0% 10,494,740 0.7% 580000 Other Expense 84,807,879 8.0% 41,559,142 9.6% 126,367,021 8.4% 590000 Misc Other Expense 344,926 0.0% 309,867 0.1% 654,793 0.0% Total Operating Expenses 202,570,444 19.0% 96,897,689 22.3% 299,468,133 20.0% 620000 Buildings 5,000 0.0% 0 0.0% 5,000 0.0% 630000 Books & Materials for Libraries 120,542 0.0% 75,951 0.0% 196,493 0.0% 640000 Equipment 6,277,766 0.6% 24,545,814 5.7% 30,823,580 2.1% 650000 Lease/Purchase 1,471,719 0.1% 37,010 0.0% 1,508,729 0.1% Total Capital Outlay 7,875,027 0.7% 24,658,775 5.7% 32,533,802 2.2% 720000 Tuition Transfers 0 0.0% 135,167 0.0% 135,167 0.0% 730000 Interfund Transfers 26,598,898 2.5% 0 0.0% 26,598,898 1.8% 739900 Intrafund Transfer - Restr/Unrestr 622,477 0.1% 0 0.0% 622,477 0.0% 750000 Loans/Grants 300 0.0% 4,323,047 1.0% 4,323,347 0.3% 760000 Other Payments 0 0.0% 5,250 0.0% 5,250 0.0% 790000 Unallocated/Reserves 106,073,471 10.0% 128,905,385 29.7% 234,978,856 15.7% Total Other 133,295,146 12.5% 133,368,849 30.7% 266,663,995 17.8% Less Intrafund w/in Loc 0 0 Less Total Intrafund Transfers 622,477 622,477 Total General Fund 1,065,993,409 100.0% 434,180,372 100.0% 1,500,173,781 100.0% Los Angeles Community College District 2023-2024 Final Budget 101 Los Angeles City College General Fund C/I Description Unrestricted General Fund % of Totl Restricted General Fund % of Total Total General Fund % of Total 110000 Teaching, Regular 14,383,768 18.7% 42,693 0.1% 14,426,461 12.0% 120000 Non-Teaching, Regular 6,617,633 8.6% 3,206,512 7.4% 9,824,145 8.2% 130000 Teaching, Hourly 15,702,637 20.4% 793,072 1.8% 16,495,709 13.7% 140000 Non-Teaching, Hourly 553,051 0.7% 2,115,522 4.9% 2,668,573 2.2% Total Certificated Salaries 37,257,089 48.5% 6,157,799 14.2% 43,414,888 36.1% 210000 Classified, Regular 11,427,119 14.9% 3,811,434 8.8% 15,238,553 12.7% 220000 Instructional Aides, Regular 1,973,372 2.6% 303,612 0.7% 2,276,984 1.9% 230000 Sub/Relief, Unclassified 475,947 0.6% 2,927,836 6.7% 3,403,783 2.8% 240000 Instructional Aides, Non-Perm 89,000 0.1% 544,454 1.3% 633,454 0.5% Total Non-Certificated Salaries 13,965,438 18.2% 7,587,336 17.5% 21,552,774 17.9% 390000 Misc Employee Benefits 15,662,515 20.4% 3,907,653 9.0% 19,570,168 16.3% Total Benefits 15,662,515 20.4% 3,907,653 9.0% 19,570,168 16.3% 420000 Books 150 0.0% 100,868 0.2% 101,018 0.1% 440000 Instructional Media Materials 764,551 1.0% 1,776,483 4.1% 2,541,034 2.1% 450000 Supplies 461,831 0.6% 802,953 1.8% 1,264,784 1.1% Total Printing & Supplies 1,226,532 1.6% 2,680,304 6.2% 3,906,836 3.2% 550000 Utilities & Housekeeping Expense 3,314,664 4.3% 58,960 0.1% 3,373,624 2.8% 560000 Contracts & Rentals 1,104,142 1.4% 4,228,675 9.7% 5,332,817 4.4% 580000 Other Expense 2,598,577 3.4% 6,957,025 16.0% 9,555,602 7.9% Total Operating Expenses 7,017,383 9.1% 11,244,660 25.9% 18,262,043 15.2% 630000 Books & Materials for Libraries 120,000 0.2% 0 0.0% 120,000 0.1% 640000 Equipment 735,396 1.0% 4,956,668 11.4% 5,692,064 4.7% 650000 Lease/Purchase 102,223 0.1% 0 0.0% 102,223 0.1% Total Capital Outlay 957,619 1.2% 4,956,668 11.4% 5,914,287 4.9% 739900 Intrafund Transfer - Restr/Unrestr 22,339 0.0% 0 0.0% 22,339 0.0% 750000 Loans/Grants 300 0.0% 1,476,773 3.4% 1,477,073 1.2% 790000 Unallocated/Reserves 735,030 1.0% 5,471,549 12.6% 6,206,579 5.2% Total Other 757,669 1.0% 6,948,322 16.0% 7,705,991 6.4% Less Intrafund w/in Loc 0 0 0 Less Total Intrafund Transfers 22,339 22,339 Total General Fund 76,844,245 100.0% 43,460,403 100.0% 120,304,648 100.0% Los Angeles Community College District 2023-2024 Final Budget 102 East Los Angeles College General Fund C/I Description Unrestricted General Fund % of Total Restricted General Fund % of Total Total General Fund % of Total 110000 Teaching, Regular 31,993,624 20.3% 50,000 0.1% 32,043,624 13.4% 120000 Non-Teaching, Regular 12,368,965 7.8% 6,014,163 7.5% 18,383,128 7.7% 130000 Teaching, Hourly 41,753,089 26.5% 913,201 1.1% 42,666,290 17.9% 140000 Non-Teaching, Hourly 1,803,510 1.1% 4,802,841 6.0% 6,606,351 2.8% Total Certificated Salaries 87,919,188 55.7% 11,780,205 14.6% 99,699,393 41.8% 210000 Classified, Regular 18,163,419 11.5% 6,298,264 7.8% 24,461,683 10.3% 220000 Instructional Aides, Regular 3,635,958 2.3% 226,630 0.3% 3,862,588 1.6% 230000 Sub/Relief, Unclassified 868,653 0.6% 2,910,146 3.6% 3,778,799 1.6% 240000 Instructional Aides, Non-Perm 600,618 0.4% 1,791,556 2.2% 2,392,174 1.0% Total Non-Certificated Salaries 23,268,648 14.7% 11,226,596 13.9% 34,495,244 14.5% 390000 Misc Employee Benefits 27,791,078 17.6% 5,919,015 7.3% 33,710,093 14.1% Total Benefits 27,791,078 17.6% 5,919,015 7.3% 33,710,093 14.1% 420000 Books 500 0.0% 191,243 0.2% 191,743 0.1% 440000 Instructional Media Materials 5,682 0.0% 3,498,550 4.3% 3,504,232 1.5% 450000 Supplies 579,533 0.4% 2,104,080 2.6% 2,683,613 1.1% 470000 Materials Fees 0 0.0% 2,500 0.0% 2,500 0.0% Total Printing & Supplies 585,715 0.4% 5,796,373 7.2% 6,382,088 2.7% 550000 Utilities & Housekeeping Expense 4,417,866 2.8% 0 0.0% 4,417,866 1.9% 560000 Contracts & Rentals 8,802,658 5.6% 6,136,758 7.6% 14,939,416 6.3% 580000 Other Expense 2,359,507 1.5% 6,741,555 8.4% 9,101,062 3.8% Total Operating Expenses 15,580,031 9.9% 12,878,313 16.0% 28,458,344 11.9% 640000 Equipment 445,907 0.3% 5,862,623 7.3% 6,308,530 2.6% 650000 Lease/Purchase 328,146 0.2% 2,000 0.0% 330,146 0.1% Total Capital Outlay 774,053 0.5% 5,864,623 7.3% 6,638,676 2.8% 730000 Interfund Transfers 462,122 0.3% 0 0.0% 462,122 0.2% 750000 Loans/Grants 0 0.0% 1,487,300 1.8% 1,487,300 0.6% 790000 Unallocated/Reserves 1,420,593 0.9% 25,664,182 31.8% 27,084,775 11.4% Total Other 1,882,715 1.2% 27,151,482 33.7% 29,034,197 12.2% Less Intrafund w/in Loc 0 0 0 Less Total Intrafund Transfers 0 0 Total General Fund 157,801,428 100.0% 80,616,607 100.0% 238,418,035 100.0% Los Angeles Community College District 2023-2024 Final Budget 103 Los Angeles Harbor College General Fund C/I Description Unrestricted General Fund % of Total Restricted General Fund % of Total Total General Fund % of Total 110000 Teaching, Regular 7,916,831 17.8% 668 0.0% 7,917,499 12.0% 120000 Non-Teaching, Regular 4,157,304 9.3% 2,459,233 11.4% 6,616,537 10.0% 130000 Teaching, Hourly 6,056,649 13.6% 156,975 0.7% 6,213,624 9.4% 140000 Non-Teaching, Hourly 535,554 1.2% 2,516,027 11.7% 3,051,581 4.6% Total Certificated Salaries 18,666,338 41.9% 5,132,903 23.8% 23,799,241 36.0% 210000 Classified, Regular 6,643,540 14.9% 1,139,201 5.3% 7,782,741 11.8% 220000 Instructional Aides, Regular 747,460 1.7% 101,580 0.5% 849,040 1.3% 230000 Sub/Relief, Unclassified 417,729 0.9% 1,325,121 6.1% 1,742,850 2.6% 240000 Instructional Aides, Non-Perm 254,316 0.6% 239,727 1.1% 494,043 0.7% Total Non-Certificated Salaries 8,063,045 18.1% 2,805,629 13.0% 10,868,674 16.4% 390000 Misc Employee Benefits 10,663,937 23.9% 1,850,504 8.6% 12,514,441 18.9% Total Benefits 10,663,937 23.9% 1,850,504 8.6% 12,514,441 18.9% 420000 Books 0 0.0% 45,427 0.2% 45,427 0.1% 440000 Instructional Media Materials 1,487 0.0% 781,079 3.6% 782,566 1.2% 450000 Supplies 683,511 1.5% 273,093 1.3% 956,604 1.4% Total Printing & Supplies 684,998 1.5% 1,099,599 5.1% 1,784,597 2.7% 540000 Insurance 6,000 0.0% 0 0.0% 6,000 0.0% 550000 Utilities & Housekeeping Expense 2,583,159 5.8% 31,518 0.1% 2,614,677 4.0% 560000 Contracts & Rentals 682,784 1.5% 1,063,359 4.9% 1,746,143 2.6% 580000 Other Expense 1,279,414 2.9% 2,390,719 11.1% 3,670,133 5.5% Total Operating Expenses 4,551,357 10.2% 3,485,596 16.2% 8,036,953 12.2% 620000 Buildings 5,000 0.0% 0 0.0% 5,000 0.0% 640000 Equipment 611,350 1.4% 1,085,525 5.0% 1,696,875 2.6% 650000 Lease/Purchase 78,170 0.2% 2,384 0.0% 80,554 0.1% Total Capital Outlay 694,520 1.6% 1,087,909 5.0% 1,782,429 2.7% 730000 Interfund Transfers 434,651 1.0% 0 0.0% 434,651 0.7% 739900 Intrafund Transfer - Restr/Unrestr 47,937 0.1% 0 0.0% 47,937 0.1% 750000 Loans/Grants 0 0.0% 232,125 1.1% 232,125 0.4% 790000 Unallocated/Reserves 765,469 1.7% 5,912,378 27.4% 6,677,847 10.1% Total Other 1,248,057 2.8% 6,144,503 28.5% 7,392,560 11.2% Less Intrafund w/in Loc 0 0 0 Less Total Intrafund Transfers 47,937 47,937 Total General Fund 44,572,252 100.0% 21,558,706 100.0% 66,130,958 100.0% Los Angeles Community College District 2023-2024 Final Budget 104 Los Angeles Mission College General Fund C/I Description Unrestricted General Fund % of Total Restricted General Fund % of Total Total General Fund % of Total 110000 Teaching, Regular 8,452,993 18.7% 169,716 0.7% 8,622,709 12.2% 120000 Non-Teaching, Regular 4,005,790 8.8% 2,797,535 11.0% 6,803,325 9.6% 130000 Teaching, Hourly 9,378,604 20.7% 89,075 0.3% 9,467,679 13.4% 140000 Non-Teaching, Hourly 186,250 0.4% 347,659 1.4% 533,909 0.8% Total Certificated Salaries 22,023,637 48.6% 3,403,985 13.4% 25,427,622 35.9% 210000 Classified, Regular 6,822,726 15.1% 1,677,432 6.6% 8,500,158 12.0% 220000 Instructional Aides, Regular 940,618 2.1% 162,670 0.6% 1,103,288 1.6% 230000 Sub/Relief, Unclassified 200,719 0.4% 1,358,813 5.3% 1,559,532 2.2% 240000 Instructional Aides, Non-Perm 106,000 0.2% 420,442 1.7% 526,442 0.7% 290000 Misc Non-Certificated Salaries 0 0.0% 50,000 0.2% 50,000 0.1% Total Non-Certificated Salaries 8,070,063 17.8% 3,669,357 14.4% 11,739,420 16.6% 390000 Misc Employee Benefits 11,104,505 24.5% 2,423,064 9.5% 13,527,569 19.1% Total Benefits 11,104,505 24.5% 2,423,064 9.5% 13,527,569 19.1% 420000 Books 0 0.0% 34,596 0.1% 34,596 0.0% 440000 Instructional Media Materials 10,436 0.0% 1,009,297 4.0% 1,019,733 1.4% 450000 Supplies 60,583 0.1% 334,871 1.3% 395,454 0.6% Total Printing & Supplies 71,019 0.2% 1,378,764 5.4% 1,449,783 2.0% 550000 Utilities & Housekeeping Expense 2,150,409 4.7% 41,451 0.2% 2,191,860 3.1% 560000 Contracts & Rentals 302,746 0.7% 1,392,087 5.5% 1,694,833 2.4% 580000 Other Expense 736,062 1.6% 3,454,183 13.6% 4,190,245 5.9% 590000 Misc Other Expense 0 0.0% 154,463 0.6% 154,463 0.2% Total Operating Expenses 3,189,217 7.0% 5,042,184 19.8% 8,231,401 11.6% 640000 Equipment 25,778 0.1% 1,220,533 4.8% 1,246,311 1.8% 650000 Lease/Purchase 39,916 0.1% 0 0.0% 39,916 0.1% Total Capital Outlay 65,694 0.1% 1,220,533 4.8% 1,286,227 1.8% 730000 Interfund Transfers 287,122 0.6% 0 0.0% 287,122 0.4% 739900 Intrafund Transfer - Restr/Unrestr 9,229 0.0% 0 0.0% 9,229 0.0% 750000 Loans/Grants 0 0.0% 292,286 1.1% 292,286 0.4% 790000 Unallocated/Reserves 452,754 1.0% 8,044,737 31.6% 8,497,491 12.0% Total Other 749,105 1.7% 8,337,023 32.7% 9,086,128 12.8% Less Intrafund w/in Loc 0 0 0 Less Total Intrafund Transfers 9,229 9,229 Total General Fund 45,273,240 100.0% 25,465,681 100.0% 70,738,921 100.0% Los Angeles Community College District 2023-2024 Final Budget 105 Los Angeles Pierce College General Fund C/I Description Unrestricted General Fund % of Total Restricted General Fund % of Total Total General Fund % of Total 110000 Teaching, Regular 21,747,963 22.3% 120,945 0.3% 21,868,908 15.3% 120000 Non-Teaching, Regular 8,417,007 8.6% 3,628,356 7.9% 12,045,363 8.4% 130000 Teaching, Hourly 23,841,823 24.5% 285,047 0.6% 24,126,870 16.8% 140000 Non-Teaching, Hourly 243,704 0.3% 1,544,677 3.4% 1,788,381 1.2% Total Certificated Salaries 54,250,497 55.7% 5,579,025 12.1% 59,829,522 41.7% 210000 Classified, Regular 11,668,332 12.0% 3,209,468 7.0% 14,877,800 10.4% 220000 Instructional Aides, Regular 2,567,156 2.6% 630,975 1.4% 3,198,131 2.2% 230000 Sub/Relief, Unclassified 50,015 0.1% 2,317,125 5.0% 2,367,140 1.7% 240000 Instructional Aides, Non-Perm 295,013 0.3% 842,996 1.8% 1,138,009 0.8% Total Non-Certificated Salaries 14,580,516 15.0% 7,000,564 15.2% 21,581,080 15.0% 390000 Misc Employee Benefits 22,125,182 22.7% 3,696,488 8.0% 25,821,670 18.0% Total Benefits 22,125,182 22.7% 3,696,488 8.0% 25,821,670 18.0% 420000 Books 0 0.0% 44,591 0.1% 44,591 0.0% 440000 Instructional Media Materials 1,951 0.0% 1,855,966 4.0% 1,857,917 1.3% 450000 Supplies 148,253 0.2% 528,648 1.1% 676,901 0.5% Total Printing & Supplies 150,204 0.2% 2,429,205 5.3% 2,579,409 1.8% 550000 Utilities & Housekeeping Expense 2,645,461 2.7% 8,250 0.0% 2,653,711 1.9% 560000 Contracts & Rentals 568,384 0.6% 2,847,862 6.2% 3,416,246 2.4% 580000 Other Expense 1,501,378 1.5% 3,996,451 8.7% 5,497,829 3.8% 590000 Misc Other Expense 294,926 0.3% 0 0.0% 294,926 0.2% Total Operating Expenses 5,010,149 5.1% 6,852,563 14.9% 11,862,712 8.3% 630000 Books & Materials for Libraries 0 0.0% 75,951 0.2% 75,951 0.1% 640000 Equipment 224,928 0.2% 5,204,077 11.3% 5,429,005 3.8% 650000 Lease/Purchase 32,621 0.0% 0 0.0% 32,621 0.0% Total Capital Outlay 257,549 0.3% 5,280,028 11.5% 5,537,577 3.9% 739900 Intrafund Transfer - Restr/Unrestr 1 0.0% 0 0.0% 1 0.0% 750000 Loans/Grants 0 0.0% 80,198 0.2% 80,198 0.1% 760000 Other Payments 0 0.0% 5,250 0.0% 5,250 0.0% 790000 Unallocated/Reserves 951,121 1.0% 15,147,730 32.9% 16,098,851 11.2% Total Other 951,122 1.0% 15,233,178 33.1% 16,184,300 11.3% Less Intrafund w/in Loc 0 0 0 Less Total Intrafund Transfers 1 1 Total General Fund 97,325,219 100.0% 46,071,050 100.0% 143,396,269 100.0% Los Angeles Community College District 2023-2024 Final Budget 106 Los Angeles Southwest College General Fund C/I Description Unrestricted General Fund % of Total Restricted General Fund % of Total Total General Fund % of Total 110000 Teaching, Regular 5,020,213 12.7% 0 0.0% 5,020,213 7.3% 120000 Non-Teaching, Regular 4,056,892 10.2% 1,990,050 6.9% 6,046,942 8.8% 130000 Teaching, Hourly 12,261,452 30.9% 0 0.0% 12,261,452 17.9% 140000 Non-Teaching, Hourly 176,338 0.4% 2,660,586 9.2% 2,836,924 4.1% Total Certificated Salaries 21,514,895 54.2% 4,650,636 16.2% 26,165,531 38.2% 210000 Classified, Regular 5,372,296 13.5% 1,367,121 4.8% 6,739,417 9.8% 220000 Instructional Aides, Regular 481,436 1.2% 0 0.0% 481,436 0.7% 230000 Sub/Relief, Unclassified 101,421 0.3% 1,816,377 6.3% 1,917,798 2.8% 240000 Instructional Aides, Non-Perm 131,670 0.3% 623,505 2.2% 755,175 1.1% Total Non-Certificated Salaries 6,086,823 15.3% 3,807,003 13.2% 9,893,826 14.5% 390000 Misc Employee Benefits 6,684,518 16.9% 1,572,957 5.5% 8,257,475 12.1% Total Benefits 6,684,518 16.9% 1,572,957 5.5% 8,257,475 12.1% 420000 Books 0 0.0% 91,800 0.3% 91,800 0.1% 440000 Instructional Media Materials 0 0.0% 666,023 2.3% 666,023 1.0% 450000 Supplies 163,988 0.4% 381,128 1.3% 545,116 0.8% Total Printing & Supplies 163,988 0.4% 1,138,951 4.0% 1,302,939 1.9% 550000 Utilities & Housekeeping Expense 2,455,998 6.2% 2,587 0.0% 2,458,585 3.6% 560000 Contracts & Rentals 689,572 1.7% 2,298,036 8.0% 2,987,608 4.4% 580000 Other Expense 1,273,470 3.2% 2,854,206 9.9% 4,127,676 6.0% Total Operating Expenses 4,419,040 11.1% 5,154,829 17.9% 9,573,869 14.0% 640000 Equipment 34,119 0.1% 753,826 2.6% 787,945 1.2% 650000 Lease/Purchase 402,477 1.0% 0 0.0% 402,477 0.6% Total Capital Outlay 436,596 1.1% 753,826 2.6% 1,190,422 1.7% 750000 Loans/Grants 0 0.0% 356,856 1.2% 356,856 0.5% 790000 Unallocated/Reserves 354,209 0.9% 11,329,698 39.4% 11,683,907 17.1% Total Other 354,209 0.9% 11,686,554 40.6% 12,040,763 17.6% Less Intrafund w/in Loc 0 0 0 Less Total Intrafund Transfers 0 0 Total General Fund 39,660,069 100.0% 28,764,756 100.0% 68,424,825 100.0% Los Angeles Community College District 2023-2024 Final Budget 107 Los Angeles Trade-Technical College General Fund C/I Description Unrestricted General Fund % of Total Restricted General Fund % of Total Total General Fund % of Total 110000 Teaching, Regular 18,801,297 20.4% 79,292 0.1% 18,880,589 12.2% 120000 Non-Teaching, Regular 4,849,218 5.3% 3,259,102 5.2% 8,108,320 5.2% 130000 Teaching, Hourly 17,970,302 19.5% 458,674 0.7% 18,428,976 11.9% 140000 Non-Teaching, Hourly 394,841 0.4% 7,339,416 11.7% 7,734,257 5.0% Total Certificated Salaries 42,015,658 45.7% 11,136,484 17.7% 53,152,142 34.3% 210000 Classified, Regular 12,714,441 13.8% 3,755,870 6.0% 16,470,311 10.6% 220000 Instructional Aides, Regular 1,560,183 1.7% 665,858 1.1% 2,226,041 1.4% 230000 Sub/Relief, Unclassified 1,326,906 1.4% 4,487,163 7.1% 5,814,069 3.8% 240000 Instructional Aides, Non-Perm 193,423 0.2% 165,062 0.3% 358,485 0.2% Total Non-Certificated Salaries 15,794,953 17.2% 9,073,953 14.4% 24,868,906 16.1% 390000 Misc Employee Benefits 16,973,921 18.5% 5,478,381 8.7% 22,452,302 14.5% Total Benefits 16,973,921 18.5% 5,478,381 8.7% 22,452,302 14.5% 420000 Books 4,558 0.0% 38,958 0.1% 43,516 0.0% 440000 Instructional Media Materials 713,289 0.8% 2,646,091 4.2% 3,359,380 2.2% 450000 Supplies 1,796,009 2.0% 681,673 1.1% 2,477,682 1.6% Total Printing & Supplies 2,513,856 2.7% 3,366,722 5.3% 5,880,578 3.8% 550000 Utilities & Housekeeping Expense 3,276,032 3.6% 2,000 0.0% 3,278,032 2.1% 560000 Contracts & Rentals 2,031,333 2.2% 8,620,102 13.7% 10,651,435 6.9% 580000 Other Expense 4,898,878 5.3% 6,528,540 10.4% 11,427,418 7.4% 590000 Misc Other Expense 0 0.0% 55,404 0.1% 55,404 0.0% Total Operating Expenses 10,206,243 11.1% 15,206,046 24.2% 25,412,289 16.4% 630000 Books & Materials for Libraries 542 0.0% 0 0.0% 542 0.0% 640000 Equipment 1,549,118 1.7% 3,204,136 5.1% 4,753,254 3.1% 650000 Lease/Purchase 340,285 0.4% 0 0.0% 340,285 0.2% Total Capital Outlay 1,889,945 2.1% 3,204,136 5.1% 5,094,081 3.3% 730000 Interfund Transfers 247,315 0.3% 0 0.0% 247,315 0.2% 750000 Loans/Grants 0 0.0% 58,541 0.1% 58,541 0.0% 790000 Unallocated/Reserves 2,324,202 2.5% 15,407,874 24.5% 17,732,076 11.4% Total Other 2,571,517 2.8% 15,466,415 24.6% 18,037,932 11.6% Less Intrafund w/in Loc 0 0 0 Less Total Intrafund Transfers 0 0 Total General Fund 91,966,093 100.0% 62,932,137 100.0% 154,898,230 100.0% Los Angeles Community College District 2023-2024 Final Budget 108 Los Angeles Valley College General Fund C/I Description Unrestricted General Fund % of Total Restricted General Fund % of Total Total General Fund % of Total 110000 Teaching, Regular 17,899,383 21.4% 110,792 0.3% 18,010,175 14.3% 120000 Non-Teaching, Regular 6,083,336 7.3% 4,689,822 11.1% 10,773,158 8.5% 130000 Teaching, Hourly 20,581,044 24.6% 254,747 0.6% 20,835,791 16.5% 140000 Non-Teaching, Hourly 773,648 0.9% 1,391,897 3.3% 2,165,545 1.7% Total Certificated Salaries 45,337,411 54.1% 6,447,258 15.2% 51,784,669 41.0% 210000 Classified, Regular 10,982,567 13.1% 3,405,446 8.0% 14,388,013 11.4% 220000 Instructional Aides, Regular 1,610,118 1.9% 195,272 0.5% 1,805,390 1.4% 230000 Sub/Relief, Unclassified 430,220 0.5% 2,862,706 6.8% 3,292,926 2.6% 240000 Instructional Aides, Non-Perm 402,764 0.5% 260,111 0.6% 662,875 0.5% Total Non-Certificated Salaries 13,425,669 16.0% 6,723,535 15.9% 20,149,204 16.0% 390000 Misc Employee Benefits 18,269,888 21.8% 3,950,102 9.3% 22,219,990 17.6% Total Benefits 18,269,888 21.8% 3,950,102 9.3% 22,219,990 17.6% 420000 Books 1,500 0.0% 91,450 0.2% 92,950 0.1% 440000 Instructional Media Materials 13,571 0.0% 1,666,163 3.9% 1,679,734 1.3% 450000 Supplies 456,022 0.5% 490,478 1.2% 946,500 0.7% 470000 Materials Fees 0 0.0% 6,000 0.0% 6,000 0.0% Total Printing & Supplies 471,093 0.6% 2,254,091 5.3% 2,725,184 2.2% 540000 Insurance 1,423 0.0% 0 0.0% 1,423 0.0% 550000 Utilities & Housekeeping Expense 3,175,547 3.8% 30,339 0.1% 3,205,886 2.5% 560000 Contracts & Rentals 503,598 0.6% 1,778,534 4.2% 2,282,132 1.8% 580000 Other Expense 1,624,017 1.9% 5,996,946 14.1% 7,620,963 6.0% Total Operating Expenses 5,304,585 6.3% 7,805,819 18.4% 13,110,404 10.4% 640000 Equipment 109,514 0.1% 1,039,024 2.5% 1,148,538 0.9% 650000 Lease/Purchase 16,907 0.0% 2,626 0.0% 19,533 0.0% Total Capital Outlay 126,421 0.2% 1,041,650 2.5% 1,168,071 0.9% 739900 Intrafund Transfer - Restr/Unrestr 114,389 0.1% 0 0.0% 114,389 0.1% 750000 Loans/Grants 0 0.0% 114,160 0.3% 114,160 0.1% 790000 Unallocated/Reserves 754,455 0.9% 14,179,551 33.4% 14,934,006 11.8% Total Other 868,844 1.0% 14,293,711 33.7% 15,162,555 12.0% Less Intrafund w/in Loc 0 0 0 Less Total Intrafund Transfers 114,389 114,389 Total General Fund 83,803,911 100.0% 42,401,777 100.0% 126,205,688 100.0% Los Angeles Community College District 2023-2024 Final Budget 109 West Los Angeles College General Fund C/I Description Unrestricted General Fund % of Total Restricted General Fund % of Total Total General Fund % of Total 110000 Teaching, Regular 10,004,002 18.5% 116,124 0.3% 10,120,126 10.6% 120000 Non-Teaching, Regular 3,866,158 7.2% 5,366,498 12.9% 9,232,656 9.7% 130000 Teaching, Hourly 14,721,273 27.3% 120,827 0.3% 14,842,100 15.5% 140000 Non-Teaching, Hourly 560,800 1.0% 1,882,451 4.5% 2,443,251 2.6% Total Certificated Salaries 29,152,233 54.0% 7,485,900 18.0% 36,638,133 38.3% 210000 Classified, Regular 8,221,726 15.2% 3,239,786 7.8% 11,461,512 12.0% 220000 Instructional Aides, Regular 1,021,933 1.9% 277,192 0.7% 1,299,125 1.4% 230000 Sub/Relief, Unclassified 91,044 0.2% 2,198,818 5.3% 2,289,862 2.4% 240000 Instructional Aides, Non-Perm 262,634 0.5% 23,207 0.1% 285,841 0.3% Total Non-Certificated Salaries 9,597,337 17.8% 5,739,003 13.8% 15,336,340 16.0% 390000 Misc Employee Benefits 10,432,758 19.3% 3,978,301 9.6% 14,411,059 15.1% Total Benefits 10,432,758 19.3% 3,978,301 9.6% 14,411,059 15.1% 420000 Books 0 0.0% 114,962 0.3% 114,962 0.1% 440000 Instructional Media Materials 0 0.0% 1,201,883 2.9% 1,201,883 1.3% 450000 Supplies 218,003 0.4% 185,926 0.4% 403,929 0.4% Total Printing & Supplies 218,003 0.4% 1,502,771 3.6% 1,720,774 1.8% 550000 Utilities & Housekeeping Expense 2,185,988 4.1% 7,469 0.0% 2,193,457 2.3% 560000 Contracts & Rentals 1,038,591 1.9% 5,297,106 12.7% 6,335,697 6.6% 580000 Other Expense 435,484 0.8% 2,259,844 5.4% 2,695,328 2.8% Total Operating Expenses 3,660,063 6.8% 7,564,419 18.2% 11,224,482 11.7% 640000 Equipment 62,334 0.1% 1,219,381 2.9% 1,281,715 1.3% 650000 Lease/Purchase 6,885 0.0% 30,000 0.1% 36,885 0.0% Total Capital Outlay 69,219 0.1% 1,249,381 3.0% 1,318,600 1.4% 750000 Loans/Grants 0 0.0% 224,808 0.5% 224,808 0.2% 790000 Unallocated/Reserves 826,767 1.5% 13,863,821 33.3% 14,690,588 15.4% Total Other 826,767 1.5% 14,088,629 33.9% 14,915,396 15.6% Less Intrafund w/in Loc 0 0 0 Less Total Intrafund Transfers 0 0 Total General Fund 53,956,380 100.0% 41,608,404 100.0% 95,564,784 100.0% Los Angeles Community College District 2023-2024 Final Budget 110 Educational Services Center General Fund C/I Description Unrestricted General Fund % of Total Restricted General Fund % of Total Total General Fund % of Total 120000 Non-Teaching, Regular 2,545,207 4.2% 640,700 1.5% 3,185,907 3.1% 130000 Teaching, Hourly 0 0.0% 9,157 0.0% 9,157 0.0% Total Certificated Salaries 2,545,207 4.2% 649,857 1.6% 3,195,064 3.1% 210000 Classified, Regular 31,364,145 52.0% 701,024 1.7% 32,065,169 31.4% 230000 Sub/Relief, Unclassified 398,275 0.7% 20,607 0.0% 418,882 0.4% Total Non-Certificated Salaries 31,762,420 52.7% 721,631 1.7% 32,484,051 31.8% 390000 Misc Employee Benefits 17,199,623 28.5% 690,132 1.7% 17,889,755 17.5% Total Benefits 17,199,623 28.5% 690,132 1.7% 17,889,755 17.5% 440000 Instructional Media Materials 0 0.0% 3,979,084 9.5% 3,979,084 3.9% 450000 Supplies 217,245 0.4% 6,416 0.0% 223,661 0.2% Total Printing & Supplies 217,245 0.4% 3,985,500 9.6% 4,202,745 4.1% 550000 Utilities & Housekeeping Expense 247,494 0.4% 0 0.0% 247,494 0.2% 560000 Contracts & Rentals 1,271,654 2.1% 21,183,587 50.8% 22,455,241 22.0% 570000 Legal, Election, Audit 74,740 0.1% 0 0.0% 74,740 0.1% 580000 Other Expense 3,977,509 6.6% 379,673 0.9% 4,357,182 4.3% 590000 Misc Other Expense 50,000 0.1% 100,000 0.2% 150,000 0.1% Total Operating Expenses 5,621,397 9.3% 21,663,260 51.9% 27,284,657 26.7% 640000 Equipment 523,139 0.9% 21 0.0% 523,160 0.5% 650000 Lease/Purchase 124,089 0.2% 0 0.0% 124,089 0.1% Total Capital Outlay 647,228 1.1% 21 0.0% 647,249 0.6% 720000 Tuition Transfers 0 0.0% 135,167 0.3% 135,167 0.1% 790000 Unallocated/Reserves 2,298,441 3.8% 13,883,865 33.3% 16,182,306 15.9% Total Other 2,298,441 3.8% 14,019,032 33.6% 16,317,473 16.0% Less Intrafund w/in Loc 0 0 0 Less Total Intrafund Transfers 0 0 Total General Fund 60,291,561 100.0% 41,729,433 100.0% 102,020,994 100.0% Includes Information Technology Fund Centers (D022A/B). Other Funds Los Angeles Community College District 2023-2024 Final Budget 111 Bookstore Fund Income 2021-22 Year-End Actual 2022-23 Year-End Actual 2023-24 Final Budget State 0 0 0 Other - Local 12,954,853 15,704,010 18,777,194 Net Income 12,954,853 15,704,010 18,777,194 Plus: Incoming Transfers 3,527,848 1,962,876 452,972 Total Income 16,482,701 17,666,886 19,230,166 Beginning Balance 10,051,176 10,790,304 11,173,061 Adjustment to Beg. Balance (685,942) 77,219 0 Reserve/Open Orders 0 0 0 Less: Ending Balance 10,790,305 11,173,061 432,611 Amount Available 15,057,630 17,361,348 29,970,616 Comments: The Bookstore Fund generates income through sales. The Fund comprises the bookstore operations of the nine colleges. The beginning balance includes reserves for inventory, improvement reserves, and individual college balances, which are required for the operation of the bookstores. For 2023-24, the requirement for colleges to reserve 3% of projected annual sales for the Campus Improvement and Inventory Reserves continues to be suspended. Los Angeles Community College District 2023-2024 Final Budget 112 Bookstore Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 100000 Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Certificated Salaries 0 0.0% 0 0.0% 0 0.0% 210000 Classified, Regular 2,990,822 19.9% 3,188,606 18.4% 3,634,866 12.1% 230000 Sub/Relief, Unclassified 474,282 3.1% 741,707 4.3% 582,512 1.9% 240000 Instructional Aides, Non-Perm 0 0.0% 0 0.0% 0 0.0% Total Non-Certificated Salaries 3,465,104 23.0% 3,930,312 22.6% 4,217,378 14.1% 390000 Misc Employee Benefits 1,865,970 12.4% 2,029,847 11.7% 2,077,865 6.9% Total Benefits 1,865,970 12.4% 2,029,847 11.7% 2,077,865 6.9% 450000 Supplies 15,135 0.1% 7,685 0.0% 60,350 0.2% 460000 Bookstore 9,328,048 61.9% 10,887,055 62.7% 10,870,878 36.3% Total Printing & Supplies 9,343,183 62.0% 10,894,740 62.8% 10,931,228 36.5% 540000 Insurance 0 0.0% 0 0.0% 0 0.0% 550000 Utilities & Housekeeping Expense 115,279 0.8% 157,544 0.9% 159,661 0.5% 560000 Contracts & Rentals 8,204 0.1% 137,356 0.8% 117,822 0.4% 580000 Other Expense 256,977 1.7% 211,452 1.2% 535,999 1.8% 590000 Misc Other Expense 0 0.0% 0 0.0% 0 0.0% Total Operating Expenses 380,460 2.5% 506,351 2.9% 813,482 2.7% 620000 Buildings 0 0.0% 0 0.0% 0 0.0% 640000 Equipment 2,913 0.0% 97 0.0% 287,704 1.0% 650000 Lease/Purchase 0 0.0% 0 0.0% 0 0.0% Total Capital Outlay 2,913 0.0% 97 0.0% 287,704 1.0% 730000 Interfund Transfers 0 0.0% 0 0.0% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 11,642,959 38.8% Total Other 0 0.0% 0 0.0% 11,642,959 38.8% Less Intrafund w/in Loc 0 0 0 Total Bookstore 15,057,630 100.0% 17,361,348 100.0% 29,970,616 100.0% Los Angeles Community College District 2023-2024 Final Budget 113 Building Fund Income 2021-22 Year-End Actual 2022-23 Year-End Actual 2023-24 Final Budget New GO Bond Proceeds 0 0 0 Other - Local 1,166,224 614,279,453 4,887,836 Net Income 1,166,224 614,279,453 4,887,836 Plus: Incoming Transfers 0 0 0 Total Income 1,166,224 614,279,453 4,887,836 Beginning Balance 3,646,073,294 3,382,327,000 443,792,930 Recognition of Remaining Issues 0 0 2,725,000,000 Adjustment to Beg. Balance 0 (600,000,000) 5,300,000,000 Reserve/Open Orders 0 46,209,667 0 Less: YE Open Orders 46,209,667 0 0 Less: Ending Balance 3,382,327,000 3,168,792,930 0 Amount Available 218,702,851 274,023,190 8,473,680,766 For presentation purposes, the remaining GO Bonds is $175,000,000 for Measure J, $2,550,000,000 for Measure CC, and $5,300,000,000 for Measure LA. Comments: On April 10, 2001, the District passed a $1.245 billion General Obligation bond (Proposition A) to finance the construction, equipping and improvement of college and support facilities at the nine campuses of the District. On May 20, 2003, the District passed another General Obligation bond (Proposition AA) for $980 million. These funds were for District and college debt, the Educational Services Center building, satellite locations, and capital outlay at the colleges. All authorized funds for both bonds have been issued as of 2008. On November 4, 2008, the District passed a third General Obligation bond (Measure J) for $3.5 billion and on November 8, 2016 the District passed a fourth General Obligation bond (Measure CC) for $3.3 billion, for the construction, acquisition, furnishing, and equipping of District facilities. On May 3, 2023 the District passed a fifth General Obligation Bond (Measure LA) for $5.3 billion to finance infrastructure and technology upgrades, as well as increase Los Angeles Community College District’s investment in sustainability, athletics facilities, and other improvements throughout the system. Los Angeles Community College District 2023-2024 Final Budget 114 Building Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 100000 Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Certificated Salaries 0 0.0% 0 0.0% 0 0.0% 210000 Non-Certificated Salaries 367,004 0.2% 382,647 0.1% 0 0.0% Total Non-Certif Salaries 367,004 0.2% 382,647 0.1% 0 0.0% 390000 Employee Benefits 137,623 0.1% 139,959 0.1% 0 0.0% Total Benefits 137,623 0.1% 139,959 0.1% 0 0.0% 420000 Books 0 0.0% 0 0.0% 0 0.0% 450000 Supplies 1,694,554 0.8% 2,039,151 0.7% 0 0.0% Total Printing & Supplies 1,694,554 0.8% 2,039,151 0.7% 0 0.0% 540000 Insurance 0 0.0% 0 0.0% 0 0.0% 560000 Contracts & Rentals 22,661,860 10.4% 21,575,000 7.9% 42,350,000 0.5% 570000 Legal, Election, Audit 3,706,591 1.7% 13,513,670 4.9% 16,675,000 0.2% 580000 Other Expense 3,582,755 1.6% 6,692,524 2.4% 6,480,311,473 76.5% 590000 Misc Other Expense 452,072 0.2% (373,188) -0.1% 34,484 0.0% Total Operating Expenses 30,403,277 13.9% 41,408,006 15.1% 6,539,370,957 77.2% 610000 Sites 0 0.0% 0 0.0% 0 0.0% 620000 Buildings 179,945,371 82.3% 216,717,724 79.1% 1,717,212,945 20.3% 640000 Equipment 6,155,022 2.8% 11,113,193 4.1% 213,721,864 2.5% Total Capital Outlay 186,100,393 85.1% 227,830,917 83.1% 1,930,934,809 22.8% 710000 Debt Service 0 0.0% 2,222,509 0.8% 3,375,000 Total Other 0 0.0% 2,222,509 0.8% 3,375,000 0.0% Less Intrafund w/in Loc 0 0 0 Total Building Fund 218,702,851 100.0% 274,023,190 100.0% 8,473,680,766 100.0% Los Angeles Community College District 2023-2024 Final Budget 115 Cafeteria Fund Income 2021-22 Year-End Actual 2022-23 Year-End Actual 2023-24 Final Budget Federal 0 0 0 State 0 0 0 Other - Local 593,937 1,537,247 1,403,673 Net Income 593,937 1,537,247 1,403,673 Plus: Incoming Transfers 311,250 388,510 247,315 Total Income 905,187 1,925,757 1,650,988 Beginning Balance 1,782,765 2,583,362 3,203,181 Adjustment to Beg. Balance 726,664 (5,804) 0 Reserve/Open Orders 0 0 0 Less: Ending Balance 2,583,362 3,203,181 0 Amount Available 831,253 1,300,134 4,854,169 Comments: Projected income from food and beverage sales and vending machines commission is budgeted at a level necessary to support projected costs. Historically, cafeteria operations have not produced sufficient sales to cover its costs, requiring support from the General Fund. Los Angeles Community College District 2023-2024 Final Budget 116 Cafeteria Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 120000 Non-Teaching, Regular 0 0.0% 0 0.0% 0 0.0% 140000 Non-Teaching, Hourly 656 0.1% 1,056 0.1% 5,000 0.1% Total Certificated Salaries 656 0.1% 1,056 0.1% 5,000 0.1% 210000 Classified, Regular 77,656 9.3% 82,801 6.4% 111,383 2.3% 230000 Sub/Relief, Unclassified 101,944 12.3% 175,763 13.5% 180,500 3.7% 240000 Instructional Aides, Non-Perm 0 0.0% 38,768 3.0% 19,000 0.4% Total Non-Certificated Salaries 179,600 21.6% 297,332 22.9% 310,883 6.4% 390000 Misc Employee Benefits 39,812 4.8% 32,814 2.5% 48,681 1.0% Total Benefits 39,812 4.8% 32,814 2.5% 48,681 1.0% 450000 Supplies 577,262 69.4% 902,610 69.4% 855,974 17.6% Total Printing & Supplies 577,262 69.4% 902,610 69.4% 855,974 17.6% 550000 Utilities & Housekeeping Expense 0 0.0% 93 0.0% 33,270 0.7% 560000 Contracts & Rentals 3,560 0.4% 6,980 0.5% 17,000 0.4% 580000 Other Expense 27,190 3.3% 53,341 4.1% 162,148 3.3% 590000 Misc Other Expense 0 0.0% 0 0.0% 0 0.0% Total Operating Expenses 30,750 3.7% 60,414 4.6% 212,418 4.4% 640000 Equipment 3,172 0.4% 5,907 0.5% 59,605 1.2% Total Capital Outlay 3,172 0.4% 5,907 0.5% 59,605 1.2% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 3,361,608 69.3% Total Other 0 0.0% 0 0.0% 3,361,608 69.3% Less Intrafund w/in Loc 0 0 0 Total Cafeteria 831,253 100.0% 1,300,134 100.0% 4,854,169 100.0% Los Angeles Community College District 2023-2024 Final Budget 117 Child Development Fund Income 2021-22 Year-End Actual 2022-23 Year-End Actual 2023-24 Final Budget Federal 1,701,917 2,152,860 833,230 State 10,437,563 12,675,524 8,888,403 Other - Local 101,525 108,279 105,619 Net Income 12,241,006 14,936,663 9,827,252 Plus: Incoming Transfers 856,352 829,861 730,923 Total Income 13,097,358 15,766,524 10,558,175 Beginning Balance 2,109,774 2,529,207 3,346,420 Adjustment to Beg. Balance (17,587) (1) 0 Reserve/Open Orders 41,899 80,538 109,650 Less: YE Open Orders 80,538 109,650 0 Less: Ending Balance 2,529,927 3,346,419 0 Amount Available 12,620,980 14,920,199 14,014,245 Comments: Since 1980-81, the State Department of Education has provided funding for all community college child development centers. This method of funding is expected to continue indefinitely. While no specific rate of funding (i.e., per student allowances for child-hour rate) was established, a funding level was determined based upon the provisions for inflation. The amount of state funds shown represents the funding level established by the State Department of Education. Projected parent fees total $105,619. The program is augmented by college support through interfund transfers of $730,923 from the General Fund. Los Angeles Community College District 2023-2024 Final Budget 118 Child Development Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 120000 Non-Teaching, Regular 3,059,824 24.2% 3,668,086 24.6% 1,977,804 14.1% 140000 Non-Teaching, Hourly 2,064,939 16.4% 2,139,918 14.3% 829,091 5.9% Total Certificated Salaries 5,124,763 40.6% 5,808,004 38.9% 2,806,895 20.0% 210000 Classified, Regular 2,504,081 19.8% 2,717,334 18.2% 788,216 5.6% 230000 Sub/Relief, Unclassified 320,149 2.5% 769,518 5.2% 187,653 1.3% Total Non-Certificated Salaries 2,824,230 22.4% 3,486,852 23.4% 975,869 7.0% 390000 Misc Employee Benefits 3,116,081 24.7% 3,557,544 23.8% 1,058,715 7.6% Total Benefits 3,116,081 24.7% 3,557,544 23.8% 1,058,715 7.6% 420000 Books 2,577 0.0% 0 0.0% 0 0.0% 440000 Instructional Media Materials 17,867 0.1% 34,466 0.2% 500 0.0% 450000 Supplies 437,351 3.5% 653,793 4.4% 707,762 5.1% Total Printing & Supplies 455,218 3.6% 688,260 4.6% 708,262 5.1% 550000 Utilities & Housekeeping Expense 976 0.0% 0 0.0% 0 0.0% 560000 Contracts & Rentals 714,262 5.7% 737,952 4.9% 129,479 0.9% 580000 Other Expense 38,635 0.3% (36,983) -0.2% (64,097) -0.5% 590000 Other Expense 16,000 0.1% 55,405 0.4% 55,405 0.4% Total Operating Expenses 769,873 6.1% 756,374 5.1% 120,787 0.9% 620000 Buildings 0 0.0% 0 0.0% (2,048) 0.0% 640000 Equipment 161,553 1.3% 357,768 2.4% 162,195 1.2% 650000 Lease/Purchase 1,090 0.0% 2,315 0.0% 3,600 0.0% Total Capital Outlay 162,643 1.3% 360,083 2.4% 163,747 1.2% 730000 Interfund Transfers 165,595 1.3% 263,081 1.8% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 8,179,970 58.4% Total Other 0 0.0% 0 0.0% 8,179,970 58.4% Less Intrafund w/in Loc 0 0 0 Total Child Development 12,620,980 100.0% 14,920,199 100.0% 14,014,245 100.0% Los Angeles Community College District 2023-2024 Final Budget 119 Debt Service Fund Income 2021-22 Year-End Actual 2022-23 Year-End Actual 2023-24 Final Budget Federal 0 0 0 Other - Local 5,953 177,472 0 Net Income 5,953 177,472 0 Plus: Incoming Transfers 7,071,948 6,923,284 7,588,759 Total Income 7,077,901 7,100,756 7,588,759 Beginning Balance 0 0 0 Adjustment to Beg. Balance 0 0 0 Reserve/Open Orders 0 0 0 Less: Ending Balance 0 0 0 Amount Available 7,077,901 7,100,756 7,588,759 Comments: For fiscal year 2023-24, incoming transfer of $7,588,759 is estimated for post-retirement health insurance contribution (GASB 45/75). Los Angeles Community College District 2023-2024 Final Budget 120 Debt Service Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 100000 Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Certificated Salaries 0 0.0% 0 0.0% 0 0.0% 200000 Non-Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Non-Certif Salaries 0 0.0% 0 0.0% 0 0.0% 390000 Misc Employee Benefits 7,077,901 100.0% 7,100,756 100.0% 7,588,759 100.0% Total Benefits 7,077,901 100.0% 7,100,756 100.0% 7,588,759 100.0% 400000 Book & Supplies 0 0.0% 0 0.0% 0 0.0% Total Printing & Supplies 0 0.0% 0 0.0% 0 0.0% 580000 Other Expense 0 0.0% 0 0.0% 0 0.0% Total Operating Expenses 0 0.0% 0 0.0% 0 0.0% 600000 Capital Outlay 0 0.0% 0 0.0% 0 0.0% Total Capital Outlay 0 0.0% 0 0.0% 0 0.0% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 0 0.0% Total Other 0 0.0% 0 0.0% 0 0.0% Less Intrafund w/in Loc 0 0 0 Total Debt Service 7,077,901 100.0% 7,100,756 100.0% 7,588,759 100.0% Los Angeles Community College District 2023-2024 Final Budget 121 Special Reserve Fund Income 2021-22 Year-End Actual 2022-23 Year-End Actual 2023-24 Final Budget Federal 0 0 0 State 26,528,492 37,396,970 12,517,064 Other - Local 1,321,424 16,548,693 1,446,958 Net Income 27,849,916 53,945,663 13,964,022 Plus: Incoming Transfers 14,511,960 15,596,092 17,578,929 Total Income 42,361,876 69,541,755 31,542,951 Beginning Balance 107,773,887 131,625,272 208,324,111 Adjustment to Beg. Balance 0 0 0 Reserve/Open Orders 15,788,445 17,468,419 0 Less: Year-End Open Orders 17,468,419 0 0 Less: Ending Balance 131,625,272 208,324,111 87,488,067 Amount Available 16,830,517 10,311,335 152,378,995 Comments: Projected income for fiscal year 2023-24 includes $1,446,958 from Other-Local Income, which is interest income restricted for Capital Outlay Programs and utility incentive for Prop 39 (California Clean Energy Jobs Act) projects. Beginning Balances include projected funds carried forward for various on-going projects continuing from previous fiscal years. Los Angeles Community College District 2023-2024 Final Budget 122 Special Reserve Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 100000 Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Certificated Salaries 0 0.0% 0 0.0% 0 0.0% 200000 Non-Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Non-Certificated Salaries 0 0.0% 0 0.0% 0 0.0% 300000 Employee Benefits 0 0.0% 0 0.0% 0 0.0% Total Benefits 0 0.0% 0 0.0% 0 0.0% 450000 Supplies 0 0.0% 0 0.0% 7,302 0.0% Total Printing & Supplies 0 0.0% 0 0.0% 7,302 0.0% 560000 Contracts & Rentals 2,397 0.0% 3,335 0.0% 4,268 0.0% 580000 Other Expense 155,415 0.9% 211,517 2.1% 4,359,550 2.9% 590000 Misc Other Expense 16,796,803 99.8% 8,545,393 82.9% 124,091,108 81.4% Total Operating Expenses 16,954,615 100.7% 8,760,245 85.0% 128,454,926 84.3% 620000 Buildings (256,074) -1.5% 1,390,052 13.5% 3,114,806 2.0% 640000 Equipment 46,976 0.3% 65,904 0.6% 541,484 0.4% 650000 Lease/Purchase 85,000 0.5% 95,134 0.9% 0 0.0% Total Capital Outlay (124,098) -0.7% 1,551,090 15.0% 3,656,290 2.4% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 20,260,477 13.3% Total Other 0 0.0% 0 0.0% 20,260,477 13.3% Less Intrafund w/in Loc 0 0 0 Total Special Reserve 16,830,517 100.0% 10,311,335 100.0% 152,378,995 100.0% Los Angeles Community College District 2023-2024 Final Budget 123 Scheduled Maintenance, Hazmat & Other Projects FY 2023-2024 Location and Projects Final Budget Los Angeles City College Replace/Upgrade HVAC Kinesiology South 2,000 Central Plant Chiller Repair 10,440 Theatre Drapes Replacement 3,407 Flooring & Base Replacement Communication Building 3,049 Replace Roof Kinesiology South 2,215 Replace Air Handlers Kinesiology South 3,948 Replace Air Handlers Communications Building 7,539 Replace Flooring FacStaffCtr(Caf)&Rstrm 518 Central Plant Boiler Replacement Phase II 1,764 Replace Roof North Building 4,150 Cafeteria Air Handler Replacement 17,600 Morgan Freeman Theater Seat/Lighting 18,035 Replace Kinesiology South Elevator and Controls 266,511 Repair/replace Facilities Deck 3,802 Replace South Kinesiology Roof Vent Hoods and Exhaust Fans 2,309 Replace Communications Elevator Controls and Cab 227,127 Replace Kinesiology South Fire Alarm System 44,990 Replace VFDs Central Plant 1,852 Doors & Hardware Replacement Ph 2 -Communications & Rad Tech 194,055 Fire Alarm System 197,879 Replace Elevator Controls & Cab - Franklin Hall and Jefferson Hall 8,649 Repair/Replace Basement Sewage Ejectors 14,220 Repair/Replace Ceiling & Acoustic Tile 105,500 Repair Restrooms Rad Tech 192,204 Replace Communications Elevator Controls 276,640 Replace Kinesiology South Elevator and C 206,765 Fire Alarm System 100,000 Replace Kinesiology South Fire Alarm Sys 100,000 Replace Switchgear - FH, JH, HAM 1,748 Repair Emergency Lighting System - Student Union 45,000 Paint Interior Communications 10,000 Paint Exterior - South Kinesiology & Rad 115,000 Replace Card Access Control Systems 313,490 ReP FI-Chem Base Lev 651 Campus wide Duct Cleaning 55 Majestic Bldg Roof & Ceiling Repair 2,785 Franklin Hall Roof Repair 10,960 Replace Window Tint SSB 927 Los Angeles Community College District 2023-2024 Final Budget 124 Location and Projects Final Budget Repair Fire Alarm System 12,979 Repair Roof Deck Science Technology 1,878 Replace Sump Pump Franklin Hall 107,000 Repair Deck Majestic Building 3rd Floor 80,000 Replace Fan Coils Kinesiology South 131,000 Repair Campus Central Gates Park L & Camp 175,000 Repair Artificial Turf Athletic Field 17,536 Replace Irrigation System Park & Student Services 70,000 Replace Irrigation System Northwest Campus 200,000 Repair HVAC Controls - Air Balancing 400,000 Kinesiology North Secondary Containment Chem 50,000 Campus-wide Duct Cleaning 359,583 Repair Roof NEB - VDK 150,000 Com Theater Fire Alarm 335,075 Flooring Chemistry Building 206 Repair Sidewalk Heliotrope C Chavez 281,150 Repair Roof/Coating Jefferson Hall 182,900 Paint Exterior - Facilities Management 104,600 Replace Drinking Fountains Cwide 235,570 Replace Emergency Lighting Inverter System 212,000 Repair Exterior Lighting Cwide 976,000 Repair Roof/Coating RadTech 69,459 Replace Central Plant Rooftop Boilers 589,469 Cchavez - Roof Replacement 182,900 Replace MFreeman Theater Classroom Ph2 154,967 Replace Ball Retention Netting-Athletic 2,984 Replace Artificial Turf-Athletic Field 1,200,782 Swimming Pool Kinesiology 333,179 Playground Rubberized Surface CDC 80,000 Stage Fly Systems Repair - Theater Arts 79,000 Los Angeles City College Total $9,293,001 East Los Angeles College Corporate Center Replace Elevator Controls 104,094 P1 Automotive Lifts Replacement 106,325 G3 Drinking Water Fountain Replacement 14,112 Corporate Center Repair Parking Structure Entry Gate 5,598 B5 Plumbing Repair 181,860 D5 Pool Repair/Replacement Tile & Plumbing Ph II 34,910 Corporate Center Air Conditioning Phase 32,098 Replace/Repair Cooling Tower E-7 39,106 Repair/Replace 5 KV G3, E9 P1 255,669 Los Angeles Community College District 2023-2024 Final Budget 125 Location and Projects Final Budget Repair Roof A4 & A5 15,742 Repair 5 KV Electrical System Campus-wide 271,915 Repair Roof 1700 Building 123,770 Repair/Replace D5 Access Road 192,328 Repair/Replace Asphalt on Avalanche Way 290,000 Repair Pedestrian Access E9-F9-B5 250,000 Repair/Replace Elevator G1 502,187 Replace Waterless Urinal Corroded Pipes 38,975 Repair/Replace Roof C1 8,931 D1 Repair Damaged Elevator Doors 7,157 ReP A/C Pkage-D7 18,032 ReP Volt TF-G3E9P1 309,998 Replace Failing & Obsolete Condenser Units 30,483 Chiller Overhaul 14,699 Replace ADA Lifts & Blocks D5 21,583 Replace Cooling Tower F3 16,831 E7 Chillers Overhaul 17,416 D5 Pump Room Surge Pit 236 D5 Swimming Pool Mechanical System Phase II 215 Rf Area AutoTech P1 400,696 Boiler Replacement (G9) 29,143 Corporate Center Exterior Envelope Waterproofing 85,860 Fire Alarm Repair 462 Corporate Center Roof Repair 9,002 D5 Swimming Pool Roof Replacement Ph I 11,801 Corporate Center Air Conditioning Ph II 1,075 Corporate Center Exterior Envelope Waterproofing 2,542 Repair/Replace Flooring G1 6,695 Paint C2 Building 58,380 Replace Water Intensice Landscaping D1 and E1 9,601 Replace UPS System for Emergency Lighting 50 Central Plant Microturbine Replacement 7,299 Repair/Replace Fire Access Road Near D1 60,106 Repair Elevator Cab Lights S1 1,875 Repair HVAC Chill Water Pipe 612 Replace Carpet E7 6,523 C1 Fire Sprinkler Head Replacement 89,165 D5 Roll Up Door Replacement 280,240 G3 Fire Sprinkler Head Replacement 119,008 Replace Corroded Electrical Panels D5 Pool 144,647 Replace Damaged Pipe and Urinals Campus wide 15,051 Seal & Paint D5 Building 54,825 Repair Student Services E1 Shade Structure 82,533 Los Angeles Community College District 2023-2024 Final Budget 126 Location and Projects Final Budget Repair HVAC Women’s Gymnasium E9 62,524 Replace AC Unit C1 49,186 Slurry/Seal Existing Asphalt C1-C2 4,780 Repair/Replace Playground Surface Equipm 462,961 Repair Fire Alarm System Corporate Cente 250,000 Repair Entry Doors S1 46,819 Replace Facilities Radio System 41,055 Rep Road/Sideways 1,155,000 A/C PackUnits CDC 245,000 Rep SurgePumpTnk 250,000 Pump JeffersonWell 8,469 Dom&Irrigation PmpS 200,000 PmpStation Bladders 150,000 Repl A/C System E7 140,000 Repl A/C System K5 113,100 Repl ExhFan Motors 86,000 Repl AlarmSys CW 700,000 Repl Wayfinding CW 237,000 Repl SwimPool TBrd 64,424 ReplDomesticPiping 280,000 Ceramics RmRep S2 200,000 Paint/Seal E1 Tower 300,000 Paint Ext D7/D7A 160,000 Repl Flooring G1 175,000 PaintSeal PipingSys 200,000 Corroded LightPoles 300,000 Lighting CntlSys G1 282,000 K5 Bldg P3&P4 Prkng 192,471 B5 ShopLeadAbate 200,000 Rep CracksCbles-S4 260,000 Rep FieldSafetyNet 293,051 Rep LightSystem-B5 700,000 Potable Water East Restroom B5 330,000 Water Intrusion and Paint Walls S4 750,000 Lower Slab S4 Parking Structure 362,100 Pedestrian Doors S4 Parking Structure 132,000 D1 Parking Structure Elevators 232,500 EMS - Tech Center E7 1,416,451 Replace Way finding Campus-wide 36,797 HVAC Automation System 554,200 Replace Damaged Seat - Swimming Pool 362,100 Grout Slab Joints Repair 132,000 G3 Auditorium Repairs Walls and Paint 884,293 Los Angeles Community College District 2023-2024 Final Budget 127 Location and Projects Final Budget Repair Doors G3 Auditorium 11,579 VFD Tech Center E7 360,436 Repair Vandalized Area Campus-wide 124,034 Repair Slab Walkways - Corporate Center 112,000 Repair Gates B1 & C2 Modulars 154,000 Refurbish Water Heating 75,952 Replace Roof - Corporate Center 39,691 Repair Chilled Water Tie 500,000 Replace E3 Elevators 2,350,500 Repair Blue Phones Campus-wide 389,542 Repair Gas Seismic Shut Valve 72,084 Replace Landscape Drought K5 650,000 Replace Landscape Drought C1 300,438 Replace Landscape Drought E9 650,000 East Los Angeles College Total $23,172,998 Los Angeles Harbor College Replace Campus Fire Alarm and Phone System 17,879 Repair Urinal Drain Lines Campus-wide 4,991 ReP SWEjector&Alarm 3,771 ReP Irrigation Cont 8,202 Tech Bldg -Repair Hydronic Leak 60 Replace Cooling Tower Panels & Central Plant System 21,789 Solar Panel Repair 3,000 Refurbish Boilers 49,213 Replace PE Exhaust Fans 99,589 Repair Central Plant Small Chiller 714,757 Replace Wallboard General classroom 1,927 Replace Football Field 761 Remove Underground Storage Tanks 1,439 Replace Campus Trash Compactors 1,274 Replace Irrigation Controllers and Flow Switches 93 Emergency Lighting Inverter Repairs 775 Repair Hallways and Common Area Walls 4,000 Replace HVAC Valves & Gaskets-N,S,T,P,C 4,570 Replace Flooring - Tech, Science & FMO 247,915 Central Plant Cooling Towers Repair/Repl 4,901 Campus wide video and security systems 253,047 Repair Roof Ducting Seal & Insulation 42,100 HVAC Air Separator Replacement 60,016 Repl Fence&Gates 782,726 Repl CeilingTile CW 148,594 Los Angeles Community College District 2023-2024 Final Budget 128 Location and Projects Final Budget ECommunicationSys 7,302 ECommunicationSys 2,545 ECommunicationSys 177,638 CarpetMusicBldg 245,000 Rep ElectrDisplay 15,721 Repair HVAC CoolSupp 250,438 Rep Backflow&ShutOff 87,502 Refurbish Boilers 88,966 Emergency FenceRep 5,183 Repl DamagedMirrors 53,055 HVAC Cooling MDF/IDF 332,500 Cntrl Plant Repairs 70,000 Reg 4 Repairs 40,000 Emergency Pipe Repairs Hydronic Heating & Cooling 269 Hydronic Heating & Pipe Repairs 159,500 Water Proof Leaking Wast & South Walls 600,000 Reair HVAC Mechanic System 740,118 Repair Access Control Systems 1,500,000 Paint Rooms NEA/SSA 200,000 Rentry Door Repairs 250,000 Water Damage Restrooms 213,000 Los Angeles Harbor College Total $7,516,126 Los Angeles Mission College Replace IA Atrium Skylights 3,000 Repair Waste Line & Replace Waterless 9,880 ReP Lib Eltor Ctrl P 163,685 Upgrd Facilities Emgncy Pwr&Lighting Sys 117,644 Replce HVCA Unit IT 321,751 Replace Baseball Field Lighting 37,023 Chiller Repair East Campus 22,160 Repair Elevator Collaborative Studies 60,000 Replace Boiler IA 281,600 Repair Chiller Center for Math & Science 268 Replace EMS System Campus-wide 4,355 Campus-wide Duct Cleaning 25,001 LRC Water Damage Cleanup 121,122 Replace Boiler CDC 14,700 Replace Gym Lights 1,639 Repair/Replace Electrical and Plumbing C 97 Air Conditioner Replacement 11,209 Replace Air Compressor in Facilities Shops 1,479 Los Angeles Community College District 2023-2024 Final Budget 129 Location and Projects Final Budget Replace LRC Sump Pumps 789 Replace Non-operational Water Isolation valves 103,679 Repair Fire Damage Math Science 5,620 Replace Safety Switch for AHU #1 CSB 55 Replace HFAC Booster Water Pump 1,866 Paint Exterior of the Campus Services Building 8,134 Paint Exterior of the Instructional Admin Building 2,083 Repair Duplex Sewer Pump System 4,573 Entrance Monument Repair 9,226 Replace Fountain Equipment by LRC 440 Replace HVAC Unit IT 39,797 Replace Playground Equipment 232,520 Replace All Old Irrigation Controllers 5,807 Replace Flooring Entrance Foyer & Lobby 33,178 Renovate IT Spaces 23,413 Repair UPS System - Health Fitness Bldg 10,904 Emergency Power Inverter Repair 637 Repl CoolingCoil 200,000 HVAC MathScience 150,000 HVAC VFD CDC 35,351 Ceiling Tile&Pnt 50,000 EMS Controllers CW 56,950 Repl Carpet Lib 69,526 HVAC VFD CollabSt 39,009 EMS ControlSys 300,000 Repair Roofs - IA/CS etc 100,000 E Campus CoolingTwr 100,000 Repl Roof-CDC 305,000 Reg 4 Repairs 12,302 Rep MainBreaker LRC 150,000 Roof replace & Water Intrusion - LRC 900,000 Replace Exterior Lights 600,000 Floor Repairs - Math & Science 350,000 Repair Roll Down Fire Door IA 360,000 Repair Lab - IA 300,000 Repair Gym Floor - Health & Fitness 200,000 Privacy Screen - CDC 50,000 Scale Central Plant Chiller 30,000 Repair Fume Hood Control CMS 100,000 Replace Campus-wide Wayfinding 450,000 Los Angeles Mission College Total $6,587,472 Los Angeles Community College District 2023-2024 Final Budget 130 Location and Projects Final Budget Los Angeles Pierce College Repair Pool Deck 131 Drinking Fountain Repair/Replacement 10,484 Central Plant Systems Replace/Repair 596,885 Replace Roof North & South Gym 294,706 Replace Trash System 203,251 Repair/Replace Football Scoreboard 3,771 Ind Repairs Electrical Tech Bldg Ph II 931 ReP Main Ent Rway 19,800 Mold Abatement-Tech 2,041 Replace Heat Actuators 371 Fine Arts - Repl Booster Pump 26,750 Emergency Rep Campus Electrical Feeder 23,295 Replace Boiler Art FH Music 341,313 Replace Booster Pump College Services 200,000 Replace Evaporative Industrial Tech 150,000 Replace HVAC Units Faculty Office Building 26,657 Campus-wide Duct Cleaning 6,901 Emergency Replace LLRC Inverter Batteries 10,309 AC IT Rm-Ginger 1600 175,000 Replace Electrical Infrastructure Campus Wide 25,702 Electrical Infrastructure Campus Wide P2 6,335 PAB Woodslat Ceiling - Theater 57,540 Fire Alarm NFPA 72Code Compliance Project 34,448 Pepper Tree Roadway 8,667 Faculty Office Roadway 717 Electrical Repairs Industrial Technology 10,001 Replace Variable Speed Drive Ph III 482 Replace Water Intensive Landscaping 200,000 Repair Pool Decking 42,571 Replace Old Irrigation Rocky Young Park 305,000 Campus Wide Fencing 320,472 Replace Old HVAC Controls 281,750 Nonslip Floor North & South Gym Locker/Shower Rooms 4,943 Remove Hazardous Underground Propane Tan 398,723 Replace Boiler 900 Building 15,430 HVAC System 1500 & Chilled Water Pump CP 278,732 Repair/Replace Crosswalk Signage & Light 258,760 Repair/Replace Irrigation Control System 185,318 Repl Roof CollServ 1,272,539 Campus IrrigatnSys 692,780 Repl Field PhyEdu 2,831,750 Los Angeles Community College District 2023-2024 Final Budget 131 Location and Projects Final Budget Refurb CrossCountry 1,407,901 CW Fencing Ph2 309,712 2ndCont Generator 175,000 PoolDeck SwimPool 234,747 HVAC Contrl P2 CW 499,400 Rep Hazard S'fields 15,975 LED Lights Stadium & Controls Tennis Court 600,000 Replace Roof - 3200 Building 1,000,000 Main Fire Alarm Display Panels 100,000 Three Side - Great Hall 2,094,555 Repair Electric UV System 58,001 Replace Wayfinding - MainMall 100,000 Repair Fire Alarm System 40,060 Replace Fountain Control 200,000 Los Angeles Pierce College Total $16,160,607 Los Angeles Southwest College Central Plant System Maintenance 1,063 Repair Emergency Lighting Invertors Campus Wide 2,799 Repair Elevator Field House & Parking 10,946 Central Plant Repair 2,169 ReP Football Field 49,455 Ctl P-Cond Pump Motor Replacement 614 Repair Main Electrical Lakin Center 3,328 Repair Hot Water Piping 419 Stadium Scoreboard Restoration 3,547 Repair Fire Alarm Communications 11,466 Repair Roof Leaks Cox & Little Theater 24,365 Replace Fuel Storage & Dispensary M&O 18,857 Repair Swimming Pool Boiler 600 Replace Carpet Student Services 1,811 Replace Ice Cream maker Sensor 24,508 Electrical Switchgear Repair 5,586 Repair Swimming Pool 22,784 Central Plant Repairs Phase 3 38,571 Campus-wide Duct Cleaning 417 Repair Irrigation Water Leak 5,410 Replace M&O HVAC 241,576 R'diateMld-Ltheater 50,500 Replace Doors at PE 1,414 Repair Automatic Entry Doors 1,125 TEC ED Replumb Chilled & Hot Water Lines 2,184 Los Angeles Community College District 2023-2024 Final Budget 132 Location and Projects Final Budget Deferred Maintenance 3,787 Repair Student Services HVAC 23,088 Repair Main Domestic Waterline PE Building 902 Roof Repair West Campus 2,573 SBSS Elevator Upgrad 9,564 Repaint Exterior Tech Ed Center 3,453 Gym Bleachers Lakin 1,080,876 Repl GymFlr-LakinFt 1,155,000 Hydronic Pipe Rep 29,238 Chillers Boilers Control - CDC 150,000 Emergency HVAC Repairs 27,985 Replace Site Generator Filter 50,000 Repair Damaged Fire Sprinkler 70,719 Repair Lifted Concrete 158,411 Repair Library Skylight 119,700 Repair Elevator - Tech Ed 402,828 Repair Radio Communicator 75,000 Repair Elevator Lakin Hall 353,789 SSEC - Repair Field Damage 91,700 Los Angeles Southwest College Total $4,334,127 Los Angeles Trade-Technical College CDC Chiller Repair 75,000 Cypress Hall Refurbish/Replace Freight Elevator Controller and Door Controls 25,021 Cedar Hall Install New Sewer Piping 4,822 Sage Hall Install New Sewer Piping 1,183 Cypress Hall Install New Steam and Condensate Piping 6,589 Cypress Hall Install New Sewer Piping 16,906 Sequoia Hall Refurbish and Replace Freight Elevator Controller and Door Controls (29,026) Rpr/Rplc Rdwd Hall A/C Rooftop Units 12,316 Rpr/Rpl Laurel Hall Air Handler 252,637 Cypress Hall Broiler Replacement 228,658 Replace Roof Redwood Hall 20,738 Replace Switchgear Redwood Hall 11,357 Replace & Maintenance Switchgear Campus- 466,444 Replace Roof Laurel Gym 183,000 Replace Lighting Controls Cypress Hall 161,250 Repair/Replace Cypress Hall South End 2nd floor deck 5,869 Replace Sewage Lift Stations 142,780 ReP CH Cooling Tower 400,001 ReP CH Chem Ctr Top 49,052 Los Angeles Community College District 2023-2024 Final Budget 133 Location and Projects Final Budget Aspen Hall - Repl seats in 2 Two Clsrms 150,979 Repair Chiller Leak Detectors 270,000 Seal Windows Aspen & Juniper 170,000 Repair Marble Surfaces Aspen & Juniper 25,000 Repair A/C Wall Units Tom Bradley Ctr 13,203 Replace Sewage Line Oak Hall 650,000 Replace Library Security Door 60,000 Repair Fire Pump System 75,000 Campus-wide Duct Cleaning 123,718 Fencing 22nd & Flower 110,000 AirComp AdvTransp 100,000 K Building Freight Elevator 17,728 Replace Waterproof Membrane-Cedar Building 3,582 Chiller Tie In at Cedar Hall 2,800 Oak Hall Roof Repair (F Building) 5,157 Oak Hall Air Handler Repair/Replacement (F Building) 3,875 Repaint Exterior of Cedar Hall (K Building) 5,104 Repaint Exterior of Sequoia Hall (B Building) 13,183 Replace/Upgrade Cypress Hall Passenger Elevator Controls 20,121 Swimming Pool Deck and Tile Repair 115 Elevator Doors & Fixt Sequoia/Cypress 156,032 Oak Hall Replace Exhaust Ventilation 119,280 Oak Hall Replace Fiber Optic Cables 122,025 Replace and Upgrade Security Camera System Campus Wide 239,126 Upgrade Core Network Switches 5,042 Upgrade Security Retrofit Access Controls 203,000 Repair Campus Uninterruptible Power Supp 92,159 Replace Wooden Floor 164 Replace Roof G2 Gymnasium 149,237 Replace Carpeting Liberal Arts (F5) 6,046 SteamPress VacTnk 16,104 CondensatePump D4 13,542 ExtLight DesignMed 189,832 GlassDoors AdminSrv 100,650 HotWater StorageTnk 83,570 ExtLight AthBldg F2 26,474 WaterValves SurgePt 34,526 ExtLight AdvTrnsprt 168,360 PntExt Cosmotology 246,928 AirComp AdvTrans B1 71,563 PntExt AdvTransp B1 368,928 EmerGenerator C4 369,050 PntExt Tom Bradley 140,300 Los Angeles Community College District 2023-2024 Final Budget 134 Location and Projects Final Budget EntryDoors TBradley 37,262 ExtLighting AS C4 38,042 PntSpray Booth B1 447,740 DrainsPool Ath F2 18,423 VehicleExhaust B1 189,344 ReplFnce-CnstrcYrd 224,000 PoolChem FiltSys-F2 1,159,000 Fire Damage Repairs Building B4 1,299,239 HVAC System Cosmetology 2,123,748 Repair Gym Roof G2 Ph2 750,000 Repair Elevators Campus-wide 525,000 Storm Drain - T Bradley C2 45,000 Replace EMS - Magnolia 247,500 Repair Roof Joints B1 250,000 Replace Roof - Athletics 707,383 Repair Restrooms - Design 366,000 Repair Restrooms - T Bradley 305,000 Repair Locker Training F2 1,000,000 Gas Seismic Shut Valve 129,360 Repair D4 Passenger Elevator 55,000 Los Angeles Trade-Technical College Total $16,693,141 Los Angeles Valley College Repair South Gym Roof 6,000 Replace Art Roof 1,779 Repair Campus Roadway-Coldwater Extension Ph II 35,619 Replace Football Field 15,054 Repair Roadway College Road North 163 Campus Center Repair Windows & Wall 128,088 Replace Oil Pump Chiller 631 ReP AHU-N Gym 477,215 ReP Fan Coil-Art 408,700 Repair GoldCreek Eco Field Station Rdway 1,210 Campus Ctr - Repl Exterior Doors 219,600 Repair Overhead Doors M&O Shops 295 Repair Fire Alarm System Music & Art 8,443 Remove Trees 63,000 Replace Filtration System Central Plant 1,526,881 Repair EMS Data Points 50,000 Repair Hallway Lights Student Services 104,607 Repair Marquees & Scoreboard 367,360 Repair Pavillion Gold Creek 24,000 Los Angeles Community College District 2023-2024 Final Budget 135 Location and Projects Final Budget Replace Playround Surfacing CDC 3,953 Repair Gym Flooring 17,000 Campus-wide Duct Cleaning 369,970 Campus Center Water Damage Repairs 3,929 Batt&Comp SouthGym 62,000 Basketball Wenches North & South Gym 6,508 Lighting Controls LARC & SSC 52,000 Rep O2 Sensors - LARC 5,311 Repair Arcade Main Entrance Roof 651 Repair Motion Picture Roof 4,815 Replace Training Pool Heater 1,000 Flood Remediation 1,403,112 Repair Irrigation Controllers Campus Wide 1,646 repair Central Plant Tower Fan & Board Chill 6,896 Repair Campus Roadway-Coldwater Extension 2,398 Rep Allied Health&Sci Fume Hoods 4,854 Repair Walls & Lighting Student Services 69,126 Repair Emergency Lighting - LARC 4,707 South Gym Basketball Court 12,649 Rep Walk Pads&Roofs 12,613 Acoustical Panels 75,000 HiighBay Lighting 50,000 Mold&Asbestos 367,192 HVACSys MotionPic 150,000 Window Cover AHS149 11,000 Plumbing ArtBldg 65,000 SouthGym AirHandler 185,000 Paint Ext&Interior 2,127,567 Greenhouse/Aviary 27,000 Repl Wayfinding CW 663,370 SolarPanels/Collect 50,000 HVAC Music IDF 56,750 Ventilation ArtRms 62,000 Rep StudioRm 112B 16,250 LightInvert/BkupBat 175,000 Heating Water Pumps - VFD 115,630 EST Fan Coil Units 243,750 Roof Admin1&3 40,881 Repl SGypBleachers 140,000 Lighting Controls Parking Structure 57,000 Replace Wayfinding Ph2 115,630 Repair Gas Line - Behavioral Science 185,311 Irrigation Lines Valve 577,337 Los Angeles Community College District 2023-2024 Final Budget 136 Location and Projects Final Budget ARR Water Piping 1,811,016 Replace Swimming Pool Equipment 140,011 Repair South Gym Air Handlers 1,156,363 Replace M&O Restroom Fixtures 60,646 Replace Emergency Lights & Exit Signs 158,297 Replace Internal Lights - South Gym 572,519 Los Angeles Valley College Total $14,939,303 West Los Angeles College CDC Storefront Window Replacement 96,392 Replace PE Complex South Roof 168 Replace PEC Complex Roof 487 Replace HVAC System Physical Education C 396,776 Repair/Replace HVAC PE Complex Center 514,115 Replace Flooring & Ceiling Fans PEC 4,546 ReP Aviation Comx Rf 240,976 Rep DomesticWater&Fire mn valves&piping 85,358 Repave Access Roadways 579,500 Replace Waterless Urinals 378,130 Repair Solar Panels South Parking Structure 326,723 Replace Baseball Scoreboard 46,490 Replace Sound System PE 18,583 Replace Fine Arts A & B Roof 5,250 Campus-wide Duct Cleaning 11,714 Replace Hydraulic Elevator 12,556 PE Complex Transformer/Feeder Replacement 25 Removal of Asbestos Flooring PE Complex 85,060 Repair Water Sealing Building Envelope Fine Arts/Aviation Complex 194,915 Replace Flooring 8-7 33,225 Replace Rain Gutters 149 Repair Restrooms & Shower Rooms PE 213,852 Replace Flooring PE North 19 Repair Restrooms & Shower Rooms Phase 2 99,397 Repair Roof A9 & B5 448 Rep Bungalow Rf 167,569 Repl Playground CDC 453,750 VCT Flooring ATC 951,000 Siemens EMS Repair 292,400 CntrlPlnt ChckVlve 16,375 Reg 4 Repairs 150,000 Light Inverter 184,000 Remove Mold & Mildew - Campus-wide 33,159 Los Angeles Community College District 2023-2024 Final Budget 137 Location and Projects Final Budget Refurbish Pump House A17 500,000 Rooftop HVAC Equipment Fine Arts B 819,000 Paint Exterior - Fine Arts 309,951 Replace Landscape East FAB 555,283 Patch & Slurry C St. 118,237 Patch & Slurry AlbVera 36,162 Paint Exterior - ATC Buildings 369,552 Curb & Bollard Paint 346,767 Replace FA-A Lighting System 346,767 West Los Angeles College Total $8,994,826 Educational Service Center Repair ESC Roof 225,000 Replace ESC Electrical System 785,476 ESC Alleyway Repairs 22,051 Paint Parking Garage 7,317 Deferred Maintenance - Holding Account 8,181 Deferred Maintenance 197,254 ESC Retrocommisioning 32,878 Window Washing ESC 110,000 Campus-wide Duct Cleaning 7,250 Evaluation Pathogen Killing Techniques 23,620 District M&DR Ph1 5,216 Deferred Maintenance 197,244 Utility CostMngt-Utility BillReview 335,000 ESC Duct Cleaning 1,300 Infrastructure Final Project Proposals 245,399 Secondary Chilled Water Pump-ESC 100,000 Development Initial Project Proposals 117,679 Junior Achievement Finance Park Alternations & Improvement 127,174 Central Plant Analysis 37,222 EV Charging Station 1,194,203 District-wide support Safe Clean Water 380,000 Hydration Station 232,330 Deferred Maintenance - Holding Account 2,681,548 Parking Structure-Water Intrusion Repair 17,603,929 Educational Service Center Total $24,677,271 Grand Total $132,368,872 Los Angeles Community College District 2023-2024 Final Budget 138 Capital Outlay Projects FY 2023-2024 Location and Projects Final Budget Los Angeles City College Cellphone Tower 1,656 Theater Arts Replacement project 3,221 Theater Arts Replacement project 86,781 AB183 Student Housing 110,000 Los Angeles City College Total $201,658 East Los Angeles College Ernest H Moreno Language Arts Humanity 190,819 Cellphone Tower 207 Cellphone Tower 3 3,903 South Gate Vandalism Repair 267,759 Safe Clean Water 60,525 AB183 Student Housing 110,000 East Los Angeles College Total $633,213 Los Angeles Harbor College AB183 Student Housing 110,000 Los Angeles Harbor College Total $110,000 Los Angeles Mission College Cellphone Tower 2 6,641 Plant Facilities Warehouse & Shop Replacement 85,855 AB183 Student Housing 110,000 San Fernando Valley STEM Hub 10,000,000 Los Angeles Mission College Total $10,202,496 Los Angeles Pierce College T Mobile/Sprint Wireless 15,000 Industrial Technology Replacement project 591,374 Cellular Tower Project 6,448 Safe Clean Water 476,698 AB183 Student Housing 110,000 Los Angeles Pierce College $1,199,520 Los Angeles Southwest College AB183 Student Housing 110,000 Los Angeles Southwest College Total $110,000 Los Angeles Community College District 2023-2024 Final Budget 139 Location and Projects Final Budget Los Angeles Trade-Tech College Design and Media Arts 248,143 AB183 Student Housing 110,000 Los Angeles Trade-Tech College Total $358,143 Los Angeles Valley College Repair Irrigation Ph2 (14,337) Cellphone Tower 1,656 Cell Tower - Sequoia 10,000 Academic Building 2 project 1,400,524 AB183 Student Housing 110,000 Los Angeles Valley College Total $1,507,843 West Los Angeles College Plant Facilities/Shops Replacement project 312,001 AB183 Student Housing 110,000 West Los Angeles College Total $422,001 Education Service Center Gold Creek Ecological Reserve 4,781,892 General Administrative Services 4,268 Green Revolving Project 318,276 Education Service Center Total $5,104,436 Grand Total $19,849,310 Los Angeles Community College District 2023-2024 Final Budget 140 Proposition 39 Projects FY 2023-2024 Location and Projects Final Budget Los Angeles City College Los Angeles City College Total $0 East Los Angeles College Parking Structure 3 & 4 Lighting Retrofit $35,570 East Los Angeles College Total $35,570 Los Angeles Harbor College Boiler $1,240 Baseball Field Lighting and Controls $3,075 Interior Lighting Retrofit $5,122 Los Angeles Harbor College Total $9,437 Los Angeles Mission College Campus Center, Fitness Center, Parking 2,725 Los Angeles Mission College Total $2,725 Los Angeles Pierce College Tennis Court Lighting & North and South 38,152 Exterior Lighting Campuswide 29,812 Los Angeles Pierce College $67,964 Los Angeles Southwest College SC Building Retrocommissioning (RCx) 4,602 Los Angeles Southwest College Total $4,602 Los Angeles Trade-Tech College Pool Cover $2,013 Los Angeles Trade-Tech College Total $2,013 Los Angeles Valley College Football Field and Swimming Pool Lighting $5,321 LARC, Art, Campus Exterior Lighting Retr $76 Los Angeles Valley College Total $5,397 West Los Angeles College AHU Replacement $23,165 Fine Arts Exterior Lighting $9,270 West Los Angeles College Total $32,435 Los Angeles Community College District 2023-2024 Final Budget 141 Location and Projects Final Budget Districtwide Lighting Retrofit $670 Districtwide Total $670 Grand Total $160,813 Los Angeles Community College District 2023-2024 Final Budget 142 Student Financial Aid Fund Income 2021-22 Year-End Actual 2022-23 Year-End Actual 2023-24 Final Budget Federal 226,261,891 168,545,007 168,238,422 State 37,558,399 60,573,689 90,416,550 Other - Local 539,859 476,535 1,400,000 Net Income 264,360,149 229,595,231 260,054,972 Plus: Incoming Transfers 0 0 0 Total Income 264,360,149 229,595,231 260,054,972 Beginning Balance 3,384,368 3,397,980 3,407,716 Adjustment to Beg. Balance 0 0 0 Reserve/Open Orders 3,745 0 0 Less: Year-End Open Orders 0 0 0 Less: Ending Balance 3,397,980 3,407,716 3,407,716 Amount Available 264,350,282 229,585,495 260,054,972 Comments: The nine campuses of the Los Angeles Community College District have participated in the following student financial aid grant/loan programs in previous years, and the budgets are established in anticipation of awards from the granting agencies based upon prior year receipts and projected increases in student need. Budgets for student financial aid programs will be augmented as additional grants and loans are received throughout the year. Program Budget Americorps Program $399,000 Cal Grant A $1,889,020 Cal Grant B $27,625,796 Cal Grant C $1,162,170 CalKIDS $1,000,000 Chafee Grant $900,000 Direct and Private Loan $20,567,750 EOPS Cash Grant $13,656,379 Care Cash Grant $2,609,929 NextUp Cash Grant $1,991,923 Federal Pell Grant $144,493,731 Federal FSEOG Grant $3,477,941 Student Success Completion Grant $39,081,333 Osher Scholarships $1,200,000 Total $260,054,972 Los Angeles Community College District 2023-2024 Final Budget 143 Student Financial Aid Fund by Sub-Major Commitment Item C/I Description 2021-22 Expenditure % of Total 2022-23 Expenditure % of Total 2023-24 Final Budget % of Total 100000 Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Certificated Salaries 0 0.0% 0 0.0% 0 0.0% 200000 Non-Certificated Salaries 0 0.0% 0 0.0% 0 0.0% Total Non-Certificated Salaries 0 0.0% 0 0.0% 0 0.0% 300000 Employee Benefits 0 0.0% 0 0.0% 0 0.0% Total Benefits 0 0.0% 0 0.0% 0 0.0% 450000 Supplies 0 0.0% 0 0.0% 0 0.0% Total Printing & Supplies 0 0.0% 0 0.0% 0 0.0% 550000 Utilities & Housekeeping Expense 0 0.0% 0 0.0% 0 0.0% 580000 Other Expense 0 0.0% 0 0.0% 0 0.0% Total Operating Expenses 0 0.0% 0 0.0% 0 0.0% 600000 Capital Outlay 0 0.0% 0 0.0% 0 0.0% Total Capital Outlay 0 0.0% 0 0.0% 0 0.0% 750000 Loans/Grants 264,350,282 100.0% 229,585,495 100.0% 257,189,896 98.9% 790000 Unallocated/Reserves 0 0.0% 0 0.0% 2,865,076 1.1% Total Other 264,350,282 100.0% 229,585,495 100.0% 260,054,972 100.0% Less Intrafund w/in Loc 0 0 0 Total Student Financial Aid 264,350,282 100.0% 229,585,495 100.0% 260,054,972 100.0% Appendices Los Angeles Community College District 2023-2024 Final Budget 144 Appendix A: Definitions & Notes Appropriation: An allocation of funds for a specified time and purpose; used synonymously with budget. Budget: A plan of financial operation for a given period for specified purposes consisting of income, revenues and expenditures. Debt Service: The Debt Service fund consists of both Debt Service funds and the State revenue bond repayment. Income: Funds upon which appropriations are based. Revenue. Interfund Transfer: This account reflects a transfer of funds from the Unrestricted General Fund to the Restricted General Fund to comply with mandatory matching requirements of federal and state programs. Intrafund Transfer: This refers to the transfer of funds between Unrestricted and Restricted programs of the General Fund, as opposed to interfund transfers, which are transfers between the General Fund and other funds. Intrafund transfers most often occur when additional support from Unrestricted programs is needed in Restricted programs -- i.e., in cases where District matching is required, or when a location wishes to expand a Restricted program beyond its funding. Additionally, income generated from specific dedicated revenue sources (e.g., Swap Meet) that is transferred to another unrestricted program requires an intrafund transfer. Transferring of funds between locations is also established via an intrafund transfer. Restricted General Fund: The restricted portion of the General Fund (Fund Application 1) is used to account for resources available for the operation and support of the educational programs that are specifically restricted by laws, regulations, donors, or other outside agencies as to their expenditure. Funds are restricted based on the following funding sources or types: Federal, State, State Categorical, Local, and Board-mandated. Unallocated Funds: The Unallocated category in this document serves two purposes -- to establish a budget in specific programs for colleges which have not yet submitted an approved budget; and to indicate an estimate of new year income and appropriations in Restricted programs based on prior year data. Unallocated funds may not yet be reflected in the operating budget. Unrestricted General Fund: The unrestricted portion of the General Fund (Fund Application 1) consists of Worker's Compensation (fund 10009), plus funds 10020 through 10299, excluding the programs listed under the definition of Restricted General Fund. The General Purpose budget is synonymous with the Unrestricted General Fund, except that Worker's Compensation is omitted from the General Purpose budget. Los Angeles Community College District 2023-2024 Final Budget 145 Appendix B: Districtwide Accounts A. Operating Budgets Academic Senate – funding for District Academic Senate Operations and release time. Accreditation – funding for assignments, contracts, travel expense, and other logistical support pertaining to accreditation efforts for the nine colleges. African American Outreach Initiative – funds for promoting student success and retention among African American students. Audit Expense – cost of annual and special audits. Benefits-Retiree – cost of retirees’ medical/dental benefits. Central Financial Aid Unit (CFAU) – the Central Financial Aid Unit operates at the Educational Services Center and is associated with loan collection and districtwide financial aid administration. Compliance Officers – Regional Compliance Officers – no longer used Dolores Huerta Center – funding for the Dolores Huerta Labor Institute is used to educate community college students about labor history, the current labor movement, the impact of unions, and workers’ issues to promote critical thinking, enhance career prospects, and encourage civic participation among students. The Dolores Huerta Labor Institute is an educational partnership between the Los Angeles Community College District and Los Angeles unions. Districtwide Mandatory Memberships – funds for mandatory institutional memberships for the colleges. Mandatory memberships budgeted in Districtwide Accounts include the Accrediting Commission for Community and Junior Colleges (ACCJC), American Association of Community Colleges (AACC), and Community College League of California (CCLC). Districtwide Marketing (Public Relations) – funds for districtwide recruitment of prospective students and public relations. Employee Assistance Program – funds for this program are based on contractual agreements and used to cover costs for service fees and supplies supporting the coordination of professional counseling, work/life programs, employee development workshops, and other employee support services. Environmental Health & Safety – districtwide costs of safety and emergency supplies, equipment, tuberculosis testing of employees, and renewal of existing contract in compliance with the Division of Occupational Safety and Health (DOSH) asbestos screenings, respirator physicals, blood chemistry panels, and blood-borne pathogens standard for employees exposed to regulated hazardous substances and “select carcinogens.” Los Angeles Community College District 2023-2024 Final Budget 146 Framework for Racial Equality & Social Justice – funds to support the identification of structural and systemic barriers to the recruitment, hiring, onboarding, supervision, and promotion of historically underrepresented and marginalized communities; to construct and redesign curriculum to support and build upon equitable, anti-racist classroom environments; to establish mandated cultural proficiency, anti-bias, and cultural responsiveness training germane to community policing and de-escalation techniques; and to engage and invest in Districtwide advocacy efforts aimed at introducing and supporting state and national legislation focused on racial equity, inclusion, and diversity. Gold Creek – funds for the maintenance of the District’s instructional laboratory in the San Gabriel Mountains. Human Resources Training & Development – funding for contracts for professional development. Metro Records – funding to cover the costs of record keeping and transcripts for the District’s defunct Metropolitan College. Special Projects – funding to cover expenses for special projects. Current special projects include Client Advantage Group consulting services for the District’s purchase of a new fleet of multifunction devices (MFDs) and their associated software and print services, as well as a Title IX workgroup tasked with ensuring District compliance with new Title IX regulations. B. Operating Budgets with Variables Collective Bargaining – funds for Labor Union representatives’ release time, faculty travel, Local 99 equipment, and negotiation expenses. Liability Insurance – funds for insurance premiums for athletics, property, and excess worker’s compensation liability and costs of claims, litigation, and settlements related to District property. Legal Expense – funds for districtwide legal expenses including outside counsel and case settlement. Reserve for Insurance/Legal/Worker’s Compensation – funds set aside as Reserve for any claim associated with Collective Bargaining, Liability, Legal Expense, and Worker’s Compensation which is based on 20% increase of the 3-year average expenditures. Staff Training, Legal – funds for diversity training. Worker’s Compensation – payments of worker’s compensation claims and administration. C. Other Centralized Accounts AB705 – funds to support imbedded face-to-face student tutoring in entry-level courses in math and English. Board Election Expense – funds to cover costs incurred in the election of the District’s Board member(s) that are conducted every other year. Los Angeles Community College District 2023-2024 Final Budget 147 District Safety/Operations – funds to cover costs for conducting emergency exercises and drills, update all college emergency plans, creating online floor warden training and certification r Educational Services Center employees, developing a standard for Safety and Security Technologies to be deployed throughout the District. District Safety/Sheriff– funds for District’s security contract. Districtwide Benefits – funds to cover the annual OPEB contribution of District employees charged to Districtwide Accounts. Financial Services– funds to cover the actuarial services needed to implement GASB Statement No. 75, Accounting and Financial for Postemployment Benefits Other Than Pensions and to provide reporting information to CALPERS. Health Benefits Administration – funds cover contracts pertaining to health benefits administration. Los Angeles College Promise – funds provide admin support to the Los Angeles College Promise program. Project Match – funds for an instructional development program designed to promote quality instruction and diversity in community college teaching. Public Policy – funds for services provided by lobbyists who advocate and communicate legislation, policy, and regulatory developments and activities to the state and federal legislatures that may impact the District operations, priorities, and goals. Staff Development – funds for the enhancement and developmental activities of staff based on contractual agreements. Tuition Reimbursement – funds for tuition reimbursement of District employees as specified in the collective bargaining contract and Board authorization. Vacation Balance – funds for accrual lump sum vacation payments for employees who leave the Los Angeles Community College District. Wellness Program – funds to provide health and wellness awareness and intervention programs for Los Angeles Community College District employees and their families through districtwide health promotions that support initiatives identified by the Joint Labor-Management Benefits Committee (JLMBC) and the Board of Trustees. D. Districtwide Information Technology Academic and Student Applications – cost of various academic software support applications, including Mathematica, VoteNet, and CurriQnet. College Technology Services – funds for Information Technology personnel, supplies, and equipment that directly support operations within the three college regions. Los Angeles Community College District 2023-2024 Final Budget 148 Cyber Security – funds to recover from Information Technology security compromises and to protect against unauthorized access. ERP/SAP – funds set aside for support and maintenance of SAP enterprise resource planning (ERP) software. Information Security – funds for anti-phishing software and security consulting services pertaining to technology. Network – funds for the support and maintenance of the District’s data transmission and network resources. Service Center – funds for the support and maintenance of various districtwide information systems, including email servers and cloud services, licenses for Adobe and other electronic signature software, remote desktop access and support, and other management software. Software Systems – funds for support and maintenance of server hardware and related software at Educational Services Center and regional data centers. Special Project-Website Redesign – funds to support redesign of district and campus websites Student Systems and Web Services – funds for support and maintenance of various districtwide information systems, including cloud hosting for college websites, licenses for Zoom, and PeopleSoft support. Los Angeles Community College District 2023-2024 Final Budget 149 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Appendix C: 2023-2024 Budgeted Positions Fund Application: 1 Program: Unrestricted General Fund Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Academic Senate Officer A0756 1.45 2.20 2.00 1.20 2.00 0.50 1.00 1.00 2.60 13.95 Associate Dean A0650 2.50 1.00 3.50 Associate Vice Chanc, Institu Effectiv A0095 1.00 1.00 Athletic Director A0750 0.20 1.00 0.80 0.60 1.00 0.80 0.80 5.20 Bargaining Unit Representative A0755 1.00 1.00 1.00 1.00 2.00 1.00 0.60 7.60 Chancellor A0023 1.00 1.00 Child Development Center Teacher A0553 2.00 1.00 3.00 Consulting Instruc (Learning Skills Ctr) A0401 0.80 0.80 Consulting Instructor A0403 1.00 2.00 2.40 2.00 0.40 1.00 8.80 Consulting Instructor (SFP) A0407 1.00 1.00 Counselor A0706 8.00 17.90 4.10 8.55 14.00 7.09 6.00 0.10 1.40 67.14 Dean A0640 4.40 12.00 5.25 1.70 7.50 4.00 8.00 6.30 4.80 5.00 58.95 Dean (Acting) A0638 1.00 1.00 Department Chair A0711 9.70 16.70 5.40 5.30 11.60 2.62 5.20 13.70 6.60 76.82 Department Chair, Counseling A0712 1.00 1.00 1.00 1.00 0.90 1.00 0.90 6.80 Department Chair, Library A0713 0.80 0.20 0.50 0.20 0.50 0.50 2.70 Department Chair, Library A0795 0.20 0.80 0.50 0.50 0.80 0.50 0.50 3.80 Department Chair, Teaching A0798 9.25 11.00 0.40 3.80 11.00 1.00 1.80 7.50 4.40 50.15 Deputy Chancellor A0025 1.00 1.00 Director of Diversity Programs A0136 1.00 1.00 Handicap Specialist A0734 0.50 0.50 Instr (Special Assign) (Learning Skills) A0751 1.20 1.30 2.50 Instr (Special Assignment) A0753 3.60 11.27 2.00 3.30 2.40 4.30 1.00 3.60 1.45 2.70 35.62 Instr (Special Assignment) (SFP) A0759 0.11 0.40 0.51 Instructor A0741 101.45 225.97 63.77 61.90 164.80 38.30 148.50 134.20 73.15 1,012.03 Instructor, Coach A0743 1.20 1.20 Librarian A0730 3.00 8.72 1.00 1.00 4.00 1.80 2.00 3.60 2.00 27.12 Nurse A0467 0.50 0.50 President A0602 1.00 1.00 1.00 1.00 1.00 1.00 1.00 7.00 President (Interim) A0604 1.00 1.00 Senior Lead SIS Administrator A0091 1.00 1.00 Vice Chancellor A0038 2.00 2.00 Vice Department Chair A0721 0.20 0.80 0.35 0.30 1.20 0.20 0.20 0.20 0.50 3.95 Vice President of Academic Affairs A0630 1.00 0.75 1.00 0.25 1.00 2.00 1.00 1.00 8.00 Vice President of Student Services A0632 1.00 1.00 1.00 1.00 1.00 1.00 6.00 VP Of Academic Affairs (Acting) A0627 1.00 1.00 VP Of Academic Affairs (Interim) A0628 1.00 1.00 VP of Student Services (Acting) A0629 1.00 1.00 Total Certificated Assignments 151.75 315.30 92.17 91.40 231.90 62.80 180.70 179.20 102.60 19.30 1,427.13 Los Angeles Community College District 2023-2024 Final Budget 150 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Non-Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Academic Scheduling Specialist C2442 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 9.50 Accountant C1163 2.50 3.00 1.00 2.00 1.00 1.00 14.00 24.50 Accounting Assistant C1348 2.00 3.00 1.00 3.00 7.00 0.50 2.00 18.50 Accounting Manager C1123 2.00 2.00 Accounting Systems Analyst C1129 1.00 1.00 Accounting Technician C1328 5.50 10.50 1.00 3.50 3.00 2.00 3.00 4.00 3.50 8.00 44.00 ADA Compliance Officer C2207 1.00 1.00 2.00 Admin Assistant to the Board of Trustees C2452 1.00 1.00 Admin Assistant to the Chancellor C2415 1.00 1.00 Administrative Analyst C5075 1.00 1.00 1.00 2.00 1.30 3.00 2.00 1.50 5.00 17.80 Administrative Assistant C2478 3.00 11.00 2.00 1.00 4.00 5.00 6.00 5.00 3.00 40.00 Administrative Assistant (Confidential) C2475 1.00 1.00 Administrative Assistant, Admin Services C2440 1.00 1.00 Administrative Intern C5090 2.00 2.00 Administrative Operations Technician C2460 3.00 1.00 1.00 3.00 3.00 1.00 3.00 2.00 2.00 19.00 Admissions & Records Assistant C2598 5.00 14.00 3.00 3.00 7.00 4.00 7.00 8.00 5.00 56.00 Admissions & Records Evaluation Tech C2596 4.00 5.00 3.00 2.00 3.00 1.00 4.00 3.85 2.00 27.85 Admissions & Records Office Supervisor C2560 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 8.00 Agricultural Technician C4505 5.00 5.00 Applications and Programming Manager C1036 1.00 1.00 Applications Developer/Programmer C1093 3.00 3.00 Art Gallery and Museum Director C5153 1.00 1.00 Art Gallery Preparator C5253 2.00 0.20 2.20 Assignment Auditor C1209 1.00 1.00 Assistant Accounting Systems Analyst C1311 1.00 1.00 Assistant Administrative Analyst C5084 1.00 1.00 9.00 11.00 Assistant Director of Accounting C1029 1.00 1.00 Assistant Personnel Director C5008 1.00 1.00 Assistant Research Analyst C2081 1.50 1.00 1.00 3.50 Assistant Technology Services Specialist C1102 1.00 14.00 15.00 Assoc Vice President, Admin Services C1054 1.00 1.00 1.00 1.00 4.00 Associate General Counsel C1023 3.00 3.00 Asst. Dir. of Employee & Labor Relations C5005 1.00 1.00 Asst. Financial Aid Systems Specialist C2575 2.00 2.00 Athletic Trainer C5310 1.00 2.00 1.00 1.00 2.00 1.00 1.00 2.00 2.00 13.00 Auditor C1216 4.00 4.00 Automotive Mechanic C5770 1.00 1.00 1.00 1.00 1.00 5.00 Carpenter C3433 2.00 2.00 1.00 2.00 2.00 1.00 10.00 Cashier C5166 0.50 1.00 2.00 1.00 4.50 Central Plant/Util. Infras. Project Mgr. C1442 1.00 1.00 Chemistry Lab Technician C5254 1.00 2.00 1.00 1.50 2.00 1.00 2.00 1.00 11.50 Chief Advancement Officer C1017 1.00 1.00 Chief Information Security Officer C1061 1.00 1.00 Chief IT Mgr, Engr & Tech Svc Delivery C1040 1.00 1.00 Los Angeles Community College District 2023-2024 Final Budget 151 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Class Description Job Code C E H M P S T V W ESC/DW Total FTES College Event and Venue Coordinator C5304 1.00 1.00 1.00 1.00 1.00 1.00 6.00 College Event and Venue Technician C5334 2.00 1.00 1.00 4.00 College Financial Administrator C1121 1.00 1.00 1.00 1.00 1.00 1.00 1.00 7.00 College Public Relations Manager C2109 1.00 1.00 1.00 1.00 1.00 1.00 6.00 Community Services Aide C5064 1.00 1.00 Compliance Investigator C2204 2.00 2.00 Compliance Officer C5011 1.00 1.00 Computer & Network Support Specialist C1144 1.00 1.00 Computer Laboratory Assistant C4595 2.00 1.00 2.00 5.00 Computer Operations Supervisor C1151 1.00 1.00 Computer Systems Operator C1149 2.00 2.00 Construction Inspector C1596 1.00 1.00 Cosmetology Lab Technician C5257 1.00 1.00 Costume Maker C5393 1.00 1.00 1.00 0.50 3.50 Courier C5864 2.00 2.00 Custodial Supervisor C4053 3.00 4.00 1.00 3.00 3.00 1.00 3.50 3.00 3.00 24.50 Custodian C4076 31.00 57.00 19.00 21.00 40.00 18.00 39.00 27.00 25.00 3.00 280.00 Data Management Support Assistant C1158 1.00 1.00 Database Administrator C1041 2.00 2.00 Deputy CIO, College Technology Services C1067 1.00 1.00 Deputy CIO, IT Infra. & Soft. Sys. Supp. C1068 1.00 1.00 Director of Budget and Managemt Analysis C1011 1.00 1.00 Director of Business Services C1003 1.00 1.00 Director of College Facilities C3158 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 9.00 Director of Communications & Marketing C2105 1.00 1.00 Director of Employee and Labor Relations C1004 1.00 1.00 Director of Facilities Planning & Devel C1012 1.00 1.00 Director of Foundation C2106 1.00 1.00 1.00 3.00 Director of Human Resources C5004 1.00 1.00 Director of Internal Audit C1203 1.00 1.00 Division Operations Specialist C2083 1.00 1.00 Division Operations Specialist (Conf.) C2084 1.00 1.00 Electrician C3322 1.00 4.00 1.00 2.00 2.00 1.00 2.00 1.00 2.00 16.00 Electronics Laboratory Technician C4558 1.00 1.00 2.00 Electronics Technician C3547 1.00 1.00 2.00 Employee and Labor Relations Specialist C5016 2.00 2.00 Employee Benefits Specialist C5068 1.00 1.00 Engineering Lab Technician C5261 1.00 1.00 Environ. & Occupa. Health & Safety Spec C4266 1.00 1.00 2.00 ERP Functional Business Analyst (SI) C5444 7.00 7.00 ERP Team Leader (Student Systems) C5424 1.00 1.00 Event Assistant C5389 0.28 1.00 0.45 1.73 Exec Assistant to the Board of Trustees C2448 1.00 1.00 Executive Assistant C2431 3.00 3.00 Executive Assistant (Confidential) C2430 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 4.00 13.00 Executive Assistant to the Chancellor C2405 1.00 1.00 Los Angeles Community College District 2023-2024 Final Budget 152 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Class Description Job Code C E H M P S T V W ESC/DW Total FTES Executive Legal Assistant C2437 1.00 1.00 Facilities Operations Technician C2445 1.00 1.00 Facilities Programs Specialist C5065 1.00 1.00 Facilities Project Manager C1441 5.00 5.00 Farm Manager C4503 1.00 1.00 Finance Proj Mgr-Bond & Special Funding C1119 1.00 1.00 Financial Aid Assistant C2584 3.20 3.00 1.00 2.00 1.00 10.20 Financial Aid Manager C1125 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 10.00 Financial Aid Supervisor C2580 1.00 1.00 1.00 1.00 1.00 1.00 6.00 Financial Aid Technician C2582 6.49 9.00 2.00 3.00 6.00 3.00 9.00 8.00 4.70 7.00 58.19 Financial Analyst C5073 3.00 3.00 Fitness Center Coordinator C5305 1.00 1.00 2.00 Foundation Development Assistant C5098 1.00 1.00 Foundation Development Officer C2206 0.75 1.00 1.75 Gardener C4183 3.00 9.00 3.00 1.00 12.00 2.00 2.00 6.00 3.00 41.00 Gardening Supervisor C4157 1.00 1.00 1.00 1.00 4.00 General Counsel C1016 1.00 1.00 General Foreman C3301 1.00 3.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 12.00 General Services Supervisor C4710 1.00 1.00 Grants Coordinator C2209 1.00 1.00 Graphic Designer C4613 1.00 1.00 1.00 3.00 Groundskeeper C4187 3.00 2.00 4.00 1.00 1.00 11.00 Heating & Air Conditioning Technician C4036 2.00 3.00 2.00 1.00 3.00 4.00 2.00 2.00 1.00 20.00 Human Resources Assistant C2278 2.00 1.00 2.00 1.00 7.00 13.00 Information Security Analyst C1078 2.00 2.00 Instructional Aide, Vocational Arts C5283 1.00 2.00 3.00 Instructional Assistant - Admin of Justi C4587 2.00 2.00 Instructional Assistant - Architecture C5259 1.00 1.00 Instructional Assistant – Art C5252 2.00 1.00 1.00 4.00 Instructional Assistant - Automotive Tec C4577 2.00 1.50 3.50 Instructional Assistant - CAOT C4582 1.00 1.00 1.00 1.00 1.00 5.00 Instructional Assistant - Child Develop C4583 1.00 1.00 2.00 Instructional Assistant - Culinary Arts C4578 1.00 3.00 2.00 6.00 Instructional Assistant - Horticulture C4153 1.00 1.00 Instructional Assistant - Information Te C4569 4.00 7.00 1.00 3.00 1.00 2.00 3.00 21.00 Instructional Assistant - Language Arts C4560 1.00 3.00 1.00 1.00 1.00 7.00 Instructional Assistant - Mathematics C4579 3.00 1.00 1.00 5.00 Instructional Assistant - Music C5268 2.00 1.00 1.00 1.00 5.00 Instructional Assistant - Nursing C4580 1.00 1.00 1.00 1.00 1.00 1.00 6.00 Instructional Assistant - Photography C5273 1.50 1.00 1.00 1.00 4.50 Instructional Assistant, Dental Hygiene C5266 1.00 1.00 2.00 Instructional Assistant, Industrial Tech C5275 1.00 2.40 1.60 5.00 Instructional Asst, Assistive Technology C4584 1.00 1.00 Instructional Asst, Registrd Vet Technol C4586 1.00 1.00 Instructional Media Specialist C4623 1.00 1.00 Instructional Media Technician C4571 1.00 1.00 2.00 2.00 1.00 1.00 8.00 Los Angeles Community College District 2023-2024 Final Budget 153 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Class Description Job Code C E H M P S T V W ESC/DW Total FTES Insurance Claims Specialist C5066 1.00 1.00 Investigator C4264 1.00 1.00 Lead Carpenter C3432 1.00 1.00 Lead Electrician C3321 1.00 1.00 1.00 1.00 4.00 Lead Gardener C4174 1.00 2.00 1.00 4.00 Lead Heating & Air Conditioning Technici C4035 1.00 1.00 1.00 3.00 Lead Painter C3471 1.00 1.00 Lead Plumber C3342 1.00 1.00 Lead Support Services Assistant C4765 1.00 1.00 1.00 1.00 4.00 Legal Secretary C2462 2.00 2.00 Legislative & Governmental Rel. Officer C2104 1.00 1.00 Library Technician C2618 5.00 7.00 2.00 2.00 4.00 4.00 4.00 5.00 3.00 36.00 Life Sciences Lab Technician C5263 2.00 3.00 2.00 3.00 3.00 1.63 1.00 3.00 1.60 20.23 Locksmith C3445 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 9.00 Machinist C3522 1.00 1.00 2.00 Maintenance & Ops Standard Coordinator C3170 1.00 1.00 Maintenance Assistant C3768 3.00 6.00 1.00 1.00 3.00 2.00 8.00 3.00 3.00 1.00 31.00 Network Architect C1082 1.00 1.00 Network Engineer C1096 7.00 7.00 Office Assistant C2694 3.96 10.00 2.00 3.90 2.00 2.00 1.50 5.00 30.36 Online Multimedia Specialist C4620 1.00 1.00 1.00 1.00 4.00 8.00 Online Technical Support Assistant C4622 1.00 1.00 1.00 1.00 1.00 5.00 Operations Manager C4023 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 10.00 Painter C3473 2.00 3.00 1.00 1.00 1.00 4.00 2.00 1.00 15.00 Paralegal C2301 1.00 1.00 Paralegal (Litigation) C2303 1.00 1.00 Patient Care Simulation Technician C5258 1.00 1.00 Payroll Assistant C1347 2.00 4.00 1.00 1.00 2.00 1.00 1.00 2.00 1.00 15.00 Payroll Systems Coordinator C1127 1.00 1.00 Payroll Systems Manager C1118 1.00 1.00 Payroll Systems Technician C1338 9.00 9.00 Performing Arts Technician C5256 4.00 2.00 2.00 1.00 1.00 10.00 Personnel Analyst C5017 4.00 4.00 Personnel Director C5003 1.00 1.00 Phys Educa/Athletics Facil Asst(F) C5978 1.00 2.00 2.00 1.00 1.00 1.00 1.00 1.00 10.00 Phys Educa/Athletics Facil Asst(M) C5973 1.00 1.00 1.00 1.00 1.00 1.00 6.00 Physical Sciences Lab Technician C5274 1.00 2.00 2.00 1.00 1.00 7.00 Piano Accompanist/Coach C5378 1.23 2.00 1.00 1.58 5.81 Plasterer C3330 1.00 1.00 Plumber C3343 2.00 4.00 1.00 2.00 2.00 1.00 1.00 2.00 1.00 16.00 Pool Lifeguard C5383 0.50 0.50 3.71 0.50 0.80 3.00 9.01 Pool Operations Technician C4056 1.00 1.00 1.00 1.00 4.00 Power Equipment Mechanic C5775 1.00 1.00 2.00 Procurement Manager C2060 1.00 1.00 Procurement Specialist C5123 8.00 8.00 Procurement Technician C5140 1.00 1.00 1.00 1.00 1.00 1.00 6.00 Los Angeles Community College District 2023-2024 Final Budget 154 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Class Description Job Code C E H M P S T V W ESC/DW Total FTES Professional Development Coordinator C5043 1.00 1.00 2.00 Projectionist C4609 0.50 0.50 Public Information Officer C2112 1.00 1.00 2.00 Recruitment and Assessment Manager C1087 1.00 1.00 Regional Mgr., College Tech. Services C1070 3.00 3.00 Registrar C2510 1.00 0.75 1.00 1.00 1.00 1.00 5.75 Reprographic Equipment Operator C4770 2.00 1.00 2.00 2.00 1.00 2.00 10.00 Research Analyst C2079 0.50 2.00 2.00 1.00 2.00 2.00 1.00 2.00 12.50 Risk Manager C2062 2.00 2.00 Safety & Emergency Preparedness Manager C4265 1.00 1.00 SAP ABAP Programmer C5418 3.00 3.00 SAP Basis Administrator C5409 1.00 1.00 SAP Business Data Warehouse Developer C5431 1.00 1.00 SAP Functional Business Analyst C5441 5.00 5.00 SAP Functional Team Leader C5425 1.00 1.00 SAP Quality Assurance Analyst C5417 1.00 1.00 SAP/ERP Manager C5405 1.00 1.00 Secretary C2480 6.00 9.00 4.00 6.00 3.00 3.00 7.00 2.00 4.00 44.00 Senior Accountant C1161 1.00 1.00 1.00 1.00 5.50 9.50 Senior Accounting Technician C1325 1.00 1.00 1.00 2.00 1.00 1.00 7.00 Senior Administrative Analyst C5023 1.00 2.00 3.00 Senior Administrative Assistant C2468 4.00 4.00 2.00 1.00 2.00 2.00 2.00 3.00 3.00 4.00 27.00 Senior Administrative Assistant (Conf.) C2465 1.00 1.00 Senior Administrative Asst. (Steno/Conf) C2461 1.00 1.00 Senior Applications Developer/Programmer C1092 3.00 3.00 Senior Assessment and Selection Analyst C5033 1.00 1.00 Senior Compliance Investigator C2203 1.00 1.00 Senior Custodial Supervisor C4048 1.00 1.00 1.00 1.00 1.00 5.00 Senior Financial Analyst C5071 1.00 1.00 Senior Human Resources Assistant C2270 2.00 1.00 1.00 1.00 6.00 11.00 Senior Human Resources Technician C2249 1.00 1.00 1.00 5.00 8.00 Senior Network Engineer C1079 2.00 2.00 Senior Office Assistant C2425 5.00 7.00 4.00 3.00 6.00 2.00 4.50 3.00 3.00 3.00 40.50 Senior Payroll Systems Technician C1317 5.00 5.00 Senior Procurement Specialist C5116 2.00 2.00 Senior SAP ABAP Programmer C5415 3.00 3.00 Senior SAP Functional Business Analyst C5439 2.00 2.00 SFP-Program Technician C5998 1.00 1.00 Sign Language Interpreter Specialist Ii C4556 0.50 0.50 Software Systems Engineer C1045 6.00 6.00 Sound Engineer C4607 1.00 1.00 2.00 Special Services Assistant C5038 1.00 1.00 2.00 Sports Event Technician C5388 0.75 0.75 Sports Information Specialist C2115 1.00 1.00 2.00 Sr Admissions & Records Office Spvr C2554 1.00 1.00 2.00 Stock Control Aide C5292 1.00 1.00 2.00 Los Angeles Community College District 2023-2024 Final Budget 155 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Class Description Job Code C E H M P S T V W ESC/DW Total FTES Stock Control Assistant C5248 2.00 2.00 1.00 1.00 1.00 2.00 2.00 1.00 1.00 13.00 Stock Control Supervisor C5203 1.00 1.00 1.00 3.00 Student Programs Specialist C5049 1.00 1.00 Student Recruiter C5042 1.00 2.00 3.00 Student Recruitment Coordinator C5040 1.00 1.00 2.00 Student Services Aide C5048 3.00 1.00 1.00 2.00 1.00 1.00 9.00 Student Services Assistant C5046 1.50 7.60 1.00 1.00 4.00 5.00 2.00 22.10 Student Services Specialist C5044 2.00 1.00 3.00 Student Support Services Representative C5051 2.00 2.00 Supervising Accounting Technician C1320 1.00 1.00 1.00 1.00 1.00 1.00 6.00 Supervising Auditor C1206 1.00 1.00 Supervising Construction Inspector C1595 1.00 1.00 Supervising Instructional Media Tech. C4553 1.00 1.00 Supervising Payroll Systems Technician C1301 3.00 3.00 Supervising Technology Svcs. Specialist C1100 2.00 2.00 10.00 14.00 Supervising Television/Cinema Engineer C3536 1.00 1.00 Sustainability & Utility Program Manager C1435 1.00 1.00 Swimming Pool Supervisor C5358 1.00 1.00 2.00 Team Leader, App. Develop. & Programming C1090 3.00 3.00 Team Leader, SAP ABAP Programming C5407 1.00 1.00 Technology Project Manager C1081 3.00 3.00 Technology Service Desk Manager C1084 1.00 1.00 Technology Services Specialist C1101 0.40 8.00 5.00 22.00 35.40 Television/Cinema Engineer C4605 1.00 1.00 Theater Management Assistant C4540 1.00 1.00 2.00 V.C./Chief Facilities Executive C1002 1.00 1.00 Vice Chancellor of Human Resources C5000 1.00 1.00 Vice Chancellor/Chief Financial Officer C1010 1.00 1.00 Vice Chancellor/Chief Info. Officer C1005 1.00 1.00 Vice President, Administrative Services C1009 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 9.00 Web Architect C1134 5.00 5.00 Web Designer C1141 5.00 5.00 Workers' Compensation Claims Specialist C5067 1.00 1.00 Total Non-Certificated Assignments 179.77 295.88 97.00 103.00 213.91 79.13 202.65 178.38 127.40 364.50 1,841.61 Total Unrestricted General Fund 331.52 611.18 189.17 194.40 445.81 141.93 383.35 357.58 230.00 383.80 3,268.74 Los Angeles Community College District 2023-2024 Final Budget 156 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Program: Community Services (10010) Class Description Job Code C E H M P S T V W ESC/DW Total FTES Community Services Aide C5064 1.00 0.50 1.00 2.50 Community Services Assistant C5062 1.00 1.00 Community Services Manager C5058 1.00 1.00 1.00 3.00 Office Assistant C2694 0.04 1.00 1.00 2.04 Total Non-Certificated Assignments 3.04 2.00 0.00 0.00 1.50 0.00 0.00 2.00 0.00 0.00 8.54 Total Community Services (10010) 3.04 2.00 0.00 0.00 1.50 0.00 0.00 2.00 0.00 0.00 8.54 Los Angeles Community College District 2023-2024 Final Budget 157 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Program: Health Services (10135) Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Dean A0640 0.25 0.25 Nurse A0467 0.50 0.50 Total Certificated Assignments 0.00 0.00 0.00 0.00 0.50 0.00 0.00 0.25 0.00 0.00 0.75 Non-Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Student Health Center Assistant C2600 2.00 2.00 Student Services Assistant C5046 1.00 1.00 2.00 Total Non-Certificated Assignments 0.00 1.00 0.00 0.00 2.00 0.00 0.00 1.00 0.00 0.00 4.00 Total Health Services (10135) 0.00 1.00 0.00 0.00 2.50 0.00 0.00 1.25 0.00 0.00 4.75 Los Angeles Community College District 2023-2024 Final Budget 158 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Program: Parking Services (10145) Class Description Job Code C E H M P S T V W ESC/DW Total FTES Administrative Analyst C5075 0.50 0.50 Senior Office Assistant C2425 1.00 1.00 0.50 2.50 Total Non-Certificated Assignments 0.00 1.00 0.00 0.00 1.50 0.00 0.50 0.00 0.00 0.00 3.00 Total Parking Services (10145) 0.00 1.00 0.00 0.00 1.50 0.00 0.50 0.00 0.00 0.00 3.00 Los Angeles Community College District 2023-2024 Final Budget 159 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Program: Disabled Student Programs & Services (10404-10406, 10420) Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Associate Dean A0650 1.00 1.00 Consulting Instructor A0403 1.00 1.00 Counselor A0706 2.00 1.00 2.00 2.00 7.00 Counselor (SFP) A0715 0.50 0.50 Dean A0640 0.20 0.35 0.55 Handicap Specialist A0734 1.00 1.00 1.00 0.90 1.00 4.90 Handicap Specialist (SFP) A0735 1.00 1.00 Vice President of Student Services A0632 0.10 0.10 Total Certificated Assignments 3.00 1.00 2.50 1.30 1.90 0.00 0.00 4.00 2.35 0.00 16.05 Non-Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Event Assistant C5389 0.90 0.90 Exam Proctor C2293 1.45 1.45 Handicap Specialist A0734 1.00 1.00 Instructional Asst, Assistive Technology C4584 1.00 1.00 1.00 1.00 1.00 1.00 6.00 Sign Language Interpreter Specialist II C4556 3.63 3.13 3.00 0.50 10.26 Special Services Assistant C5038 1.00 2.00 1.00 1.00 5.00 Sr Sign Language Interpreter Specialist C4551 1.00 1.00 1.00 1.00 4.00 Student Services Assistant C5046 1.00 1.00 0.80 2.80 Total Non-Certificated Assignments 6.63 5.00 2.00 2.00 6.13 0.00 4.00 4.15 1.50 0.00 31.41 Total Disabled Students Prog & Svs (10420) 9.63 6.00 4.50 3.30 8.03 0.00 4.00 8.15 3.85 0.00 47.46 Los Angeles Community College District 2023-2024 Final Budget 160 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Program: Extended Opportunities Programs & Services (10486-10490) Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Counselor A0706 4.10 2.65 2.00 1.65 3.00 0.62 4.05 0.86 18.93 Total Certificated Assignments 4.10 2.65 2.00 1.65 3.00 0.62 0.00 4.05 0.86 0.00 18.93 Non-Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Accountant C1163 0.50 0.50 Data Management Support Assistant C1158 1.00 1.00 Instructional Assistant - Information Te C4569 0.50 0.50 Office Assistant 1.00 1.00 Senior Office Assistant C2425 0.90 0.75 1.00 1.00 3.65 Student Recruitment Coordinator C5040 0.35 0.35 Student Services Assistant C5046 0.50 2.50 0.25 0.50 1.25 0.40 5.40 Student Services Specialist C5044 0.75 0.75 Technology Services Specialist C1101 0.60 0.60 Total Non-Certificated Assignments 2.50 2.50 1.00 2.60 1.00 0.00 0.50 2.25 1.40 0.00 13.75 Total Extended Opp Prog & Svs (10486-10490) 6.60 5.15 3.00 4.25 4.00 0.62 0.50 6.30 2.26 0.00 32.68 Los Angeles Community College District 2023-2024 Final Budget 161 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Program: Other Specially Funded Programs Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Consulting Instructor A0403 1.00 2.00 3.00 Counselor A0706 1.65 0.95 1.00 1.85 1.00 1.50 1.00 0.80 1.24 10.99 Dean A0640 0.60 3.00 0.25 0.70 1.50 1.25 1.68 1.00 9.98 Dean (SFP) A0642 1.00 1.00 Department Chair A0711 0.60 0.60 Department Chair - Varied Cap Utilizatio A0790 0.10 0.10 Department Chair, Teaching A0798 0.90 0.90 Instr (Special Assignment) A0753 1.00 1.30 2.30 Instructor A0741 1.00 1.00 0.80 2.80 Vice President of Academic Affairs A0630 0.25 0.50 0.75 Total Certificated Assignments 2.25 5.20 2.25 4.05 3.50 2.10 1.00 4.05 5.02 3.00 32.42 Non-Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Academic Scheduling Specialist C2442 0.50 0.50 Accountant C1163 1.00 1.00 2.00 Administrative Assistant C2478 1.00 1.00 Admissions & Records Evaluation Tech C2596 0.15 0.15 Community Services Aide C5064 0.50 0.50 Community Services Specialist C5059 1.00 1.00 Financial Aid Assistant C2584 0.20 4.00 2.00 1.00 7.20 Financial Aid Supervisor C2580 2.00 1.00 2.00 1.00 6.00 Financial Aid Technician C2582 2.00 3.00 2.00 2.00 3.00 2.00 2.00 2.00 2.00 2.00 22.00 Grants Coordinator C2209 1.00 1.00 2.00 Graphic Designer C4613 1.00 1.00 Office Assistant C2694 1.00 1.00 Research Analyst C2079 1.00 1.00 Senior Office Assistant C2425 0.10 0.25 1.00 1.35 SFP-Program Director C5996 1.00 1.00 1.00 1.00 4.00 SFP-Program Office Assistant C5999 2.00 1.00 3.00 SFP-Program Specialist C5997 1.00 1.00 1.00 0.50 1.00 4.50 SFP-Program Technician C5998 3.43 1.00 1.00 5.43 Student Recruitment Coordinator C5040 0.15 0.15 Student Services Assistant C5046 1.50 1.00 0.75 1.60 4.85 Student Services Specialist C5044 0.90 0.25 1.00 1.80 3.95 Student Support Services Representative C5051 5.00 5.00 Total Non-Certificated Assignments 7.20 21.43 6.00 4.65 9.50 3.00 4.50 8.90 7.40 5.00 77.58 Total Specially Funded Programs 9.45 26.63 8.25 8.70 13.00 5.10 5.50 12.95 12.42 8.00 110.00 Los Angeles Community College District 2023-2024 Final Budget 162 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Fund Application: 6 Program: Cafeteria Class Description Job Code C E H M P S T V W ESC/DW Total FTES Administrative Analyst C5075 0.20 0.20 Cashier C5166 2.00 2.00 Total Non-Certificated Assignments 0.00 0.00 0.00 0.00 0.20 0.00 2.00 0.00 0.00 0.00 2.20 Total Cafeteria 0.00 0.00 0.00 0.00 0.20 0.00 2.00 0.00 0.00 0.00 2.20 Los Angeles Community College District 2023-2024 Final Budget 163 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Fund Application: 7 Program: Child Development Center Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Child Development Center Teacher A0553 2.00 1.00 1.00 4.00 Director, Child Development Center A0551 1.00 1.00 1.00 0.15 1.00 1.00 1.00 1.00 0.50 7.65 Vice Director, Child Development Center A0552 1.00 1.00 Total Certificated Assignments 1.00 4.00 2.00 0.15 2.00 1.00 1.00 1.00 0.50 0.00 12.65 Non-Certificated Assignments Class Description Job Code C E H M P S T V W ESC/DW Total FTES Child Develop. Center Food Services Aide C4524 0.88 1.00 1.88 Child Development Center Assistant C4529 3.88 4.00 7.88 Senior Office Assistant C2425 1.00 1.00 2.00 Total Non-Certificated Assignments 0.00 0.88 0.00 0.00 4.88 0.00 6.00 0.00 0.00 0.00 11.75 Total Child Development Center 1.00 4.88 2.00 0.15 6.88 1.00 7.00 1.00 0.50 0.00 24.40 Los Angeles Community College District 2023-2024 Final Budget 164 Source: March 2023 PBF submitted positions May not reflect positions at Final Budget Fund Application: 8 Program: Bookstore Class Description Job Code C E H M P S T V W ESC/DW Total FTES Accountant C1163 1.00 1.00 Accounting Assistant C1348 1.00 0.50 1.50 Accounting Technician C1328 0.50 0.50 0.50 0.50 2.00 Cashier C5166 4.00 3.00 1.00 1.00 2.00 3.00 1.00 2.00 17.00 College Store Buyer C5162 1.00 2.00 1.00 2.00 1.00 1.00 8.00 College Store Manager C2140 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 8.00 College Store Supervisor C2144 1.00 1.00 1.00 1.00 4.00 Custodian C4076 0.00 0.00 Office Assistant C2694 1.00 1.00 Senior Accountant C1161 0.50 0.50 Senior Cashier C2136 1.00 1.00 Stock Control Aide C5292 1.00 1.00 2.00 Stock Control Assistant C5248 1.00 1.00 1.00 1.00 1.00 5.00 Total Non-Certificated Assignments 8.50 6.50 3.00 4.50 10.00 1.00 8.00 4.50 3.50 1.50 51.00 Total Bookstore 8.50 6.50 3.00 4.50 10.00 1.00 8.00 4.50 3.50 1.50 51.00 Los Angeles Community College District 2023-2024 Final Budget 165 Appendix D: List of Active Organizational Memberships According to Education Code Section 72014, the Board of Trustees may authorize participation in any organization which has for its purpose the promotion and advancement of education. Membership is not allowed in organizations whose membership practices are discriminatory on the basis of the characteristics listed in Education Code section 66270. Listed below are organizational memberships that are considered active and which have been previously approved by the Board of Trustees, including new memberships requested by locations. These consist of any memberships that have been paid by a location over the past three years and memberships submitted by the colleges in their Operational Plans. Any memberships that have not been used for the last three years will be moved to Inactive Status and will be removed from the list of Active Memberships. The Budget Office will maintain a complete list of all Active and Inactive Memberships and will submit to the Board any requests by colleges to reactivate an Inactive Membership. Please note that inclusion on the list does not indicate that funds have been allocated to pay for the annual dues. Board approval of the Final Budget will constitute approval of this membership list. Following each title are abbreviations for the locations that have requested membership in the organization (D = District Offices) through their submitted Operational Plans. Brief descriptions are provided for each membership. New memberships are indicated with an asterisk (). Active Organizational Memberships Academic Senate of the California Community Colleges (ASCCC) – D This organization assists in promoting the interests of Higher Education in the State of California and represents the faculty of all the community colleges at the state level. Accreditation Commission for Education in Nursing (ACEN) – V This organization is responsible for the specialized accreditation of nursing education programs, both postsecondary and higher degree, which offer a certificate, a diploma, or a recognized professional degree. Accrediting Commission for Community and Junior Colleges (ACCJC) – EMPSV This organization is a part of the Western Association of Schools and Colleges, which accredits institutions of higher education by making periodic site visits and evaluations. Advancing Professional Construction and Program Management Worldwide (CMAA) – D This organization ensures that staff utilize best practices to complete projects on-time and on budget. Alhambra Chamber of Commerce (ACC) – E This organization promotes community participation and provides colleges with ties to the private sector. Los Angeles Community College District 2023-2024 Final Budget 166 Alpha Gamma Sigma, Inc. – E Alpha Gamma Sigma, Inc. is a National Honors Society for California’s community colleges. The function of the society is to encourage local chapters to offer cultural, social, or enrichment experiences as part of the total experience of community college students. Students benefit from opportunities to learn, grow, and serve from the society’s available workshops, resources, networking, and scholarships. America’s SAP User’s Group (ASUG) – D This organization supports licensed SAP customers actively involved in installing and operating SAP software in their business or industry. American Association for Paralegal Education (AAPE) – W This national organization is dedicated to improving the quality of higher education by working on a broad range of issues in order to create effective changes at the local, state and national levels. American Association of Community Colleges (AACC) – EMPST This organization is concerned with all issues affecting two-year colleges. American Bar Association – C This organization provides benefits, programs, and services that promote members’ professional growth and quality of life. American Choral Directors – M This organization assists all music professors, particularly those who teach choir classes, with achieving a deeper understanding of choral education through networking opportunities with other choral directors, including those at university level. American College Health Association (ACHA) – P This organization provides continual update of health-related information appropriate to College Health Services. It also provides in-service to medical and related professionals engaged in serving health needs of college community. American Council on Education (ACE) – MTV This organization focuses on research concerning specific educational problems, and provides liaison with agencies of the Federal Government. American Culinary Federation Educational Institute (ACFEI) – T This is the primary accreditation organization in the culinary arts; its purpose is to promote high quality programs in the field of Culinary Arts, Restaurant and Institutional management. American Dental Association Council on Education (ADACE) – C This organization provides guidance for students enrolled in the Dental Program. Los Angeles Community College District 2023-2024 Final Budget 167 American Dental Education Association (ADEA) – C This organization provides excellent professional development opportunities focusing on enhancing teaching, management, and leadership skills. Conferences and workshops also provide fundamental, hands-on experience with other educators on competencies, legislation, and minority recruitment and retention. American Health Information Management Association (AHIMA) – E This organization promotes the art and science of medical record administration. It is responsible for accrediting the Medical Record Technician program. American Institute of Architects, Los Angeles (AIA) – E This organization connects a global community of over 90,000 professionals who share a passion for architecture, design, and the built environment. Members are afforded opportunities to mold the architecture profession, public policy, and practice. American Veterinary Medical Association (AVMA) – P This organization is the accrediting agency for the Animal Health Technology program. American Volleyball Coaches Association (AVCA) – M This organization provides education to volleyball coaches, recognition of elite players and coaches, promotion of volleyball competitions throughout the world, and networking opportunities for volleyball products and services providers. Associate Degree Nursing Directors of Southern California (ADNDSC) – V This membership provides support and resources for nursing programs. Association for Career and Technical Education (ACTE) – C This organization informs members of the latest trends and issues affecting career and technical education. Association for Community and Continuing Education (ACCE) – HPW This organization provides leadership in the development of Community Services and Continuing Education practitioners. It provides special assistance in professional growth and development opportunities. Association for Nutrition and Food Service Professionals (ANFP) – C This organization allows college to maintain the Pathway I certifying program. Successful completion of Dietetic Service Supervisor (DSS) certificate qualifies students to take the Dietary Manager Certifying Exam through Pathway I. Association for the Advancement of Sustainability in Higher Education (AASHE) – W This organization’s mission is to promote sustainability in all sectors of higher education. It Los Angeles Community College District 2023-2024 Final Budget 168 provides access to curriculum and operational best practices that have been developed by other higher education institutions. Association for the Study of Higher Education (ASHE) – W This organization provides a forum for the discussion of issues effecting higher education. It includes a journal, a newsletter and discounts on conferences. Association of Chief Human Resources Officers and Equal Employment Officers (ACHRO/EEO) – D This organization shares information involving key issues relating to Affirmative Action in the State of California. Association of Collegiate Educators in Radiological Technology (ACERT) – C This organization enables the Radiological Technology program to be informed of new trends and changes in the field. Association of International Educators – MP This organization provides assistance in developing the knowledge and competence of people concerned with international education. It also provides professional training and information through national and regional conferences, workshops and publications. The organization helps advisors gain valuable skills in aiding foreign students. (Formerly known as National Association for Foreign Student Affairs - NAFSA) Aviation Technician Education Council – W This organization will provide West Los Angeles College access to materials and conferences that will help promote and advance current and future programs pertaining to its Aviation Technician Certificate. California Association of College Stores (CACS) – MPTV This organization provides an exchange of trade information among college stores located in California. It acts as a liaison between college stores, publishers, manufacturers, and distributors. California Association of Community College Registrars and Admissions Officers (CACCRAO) – CMW This organization provides professional development opportunities for members, including a day-long regional workshop, a four-day annual conference, and at least one full-day training session for specialized staff in the Offices of Admissions and Records. California Association of Latino Community College Trustees and Administrators (CALCCTA) – M This organization gives the District an opportunity to assist the Latino community in Latino leadership development, mentoring, succession planning, and strategies essential for innovative and ethical management. Los Angeles Community College District 2023-2024 Final Budget 169 California Campus Compact (CCC) – W This organization is a coalition of college and university leaders that seeks to encourage student involvement in community and public service. It provides a forum through which presidents, chancellors, faculty, and students can share information and address issues related to collegiate service. The project is designed to recruit, train, and support students to work as mentors with at-risk sixth grade youths, helping colleges to participate in the welfare of the community at large. California Child Development Administrators Association (CCDAA) – E This organization promotes the advocacy of children services and development. It offers seminars, conferences, and workshops. California Community College Athletic Association (CCCAA) – CEHMPSTVW This organization is a portion of the Community College League of California. (Formerly known as Commission on Athletics - COA) California Community College Athletic Director Association (CCCADA) – MW This organization provides the colleges with information on current team regulations that are essential in the support of a successful Athletic program. It serves as a voice for Athletic Directors on matters of regulations and legislation regarding State Athletics. California Community College Chief Instructional Officers (CCCCIO) – CHPSTVW This organization provides information and advocacy on instructional issues, and general suggestions to the CCCIO Executive Board and all CIO’s in general. California Community College Cross Country and Track & Field Coaches (5CTCA) – HMW This organization provides students-athletes and coaches with staff opportunities for professional growth in cross country and track programs. California Community College Fastpitch Coaches Association (3CFCA) – HM This organization provides members with a NCAA Rule Book, policy updates for the 3CFCA Handbook, access to the 3CFCA statistics website, voting rights, and free admission to Regional and State tournaments. California Community College Football Coaches Association (CCCFCA) – MW This organization enables all member football players to be eligible for all-State selection. California Community College Men’s Basketball Coaches Association (CCCMBCA) – SW Coaches at participating colleges must be members of this organization to be able to nominate for academic or athletic awards at the end of the season. California Community College Mental Health and Wellness Association (CCCMHWA) – P The purpose of this Association is to enhance student success, wellness, and retention by the Los Angeles Community College District 2023-2024 Final Budget 170 support and promotion of quality mental health services programs throughout the California Community College System. California Community College Soccer Coaches Association (CCCSCA) – CHM This organization is required to allow coaches access to student athlete’s statistics, records, and player transfers. California Community College Student Affairs Association (CCCSAA) – EMSV This professional organization provides training and support for student government advisors. The Association meets two to three times annually, conducts workshops and presentations for members, and holds business meetings. The Association also presents an annual Leadership conference for student government officers from community colleges throughout the state. California Community College Student Financial Aid Administrators (CCCSFAA) – T Participation in the organization provides members the opportunity to meet with colleagues and shares methods for administering financial aid programs. Association also provides training workshops and newsletters that are of great use. California Community College Women’s Basketball Coaches Association (CCCWBCA) – HSTW Coaches at participating colleges need to be members of this organization to be able to nominate for academic or athletic awards at the end of the season. California Community College Women’s Volleyball Coaches Association (CCCWVCA) – HM This organization is a part of the California Community College Athletic Association (CCCAA), which is the administrative governing entity responsible for statewide rules and policies for intercollegiate athletic programs. It maintains a membership representative of all the community colleges in the state of California and provides information and resources to their members. California Community Colleges Baseball Coaches Association (CCCBCA) – W Coaches at participating colleges need to be members of this organization to vote for players in post conference play in intercollegiate sports. California Community Colleges Women's Caucus – D This organization aims to unify and support women from across California Community Colleges while identifying unique opportunities to strengthen student, faculty, and employee success in our system. California Consortium of Addiction Programs and Professions (CCAPP) – C This organization allows the college to provide career ladder certification for the drug/alcohol careers. It certifies Human Services curriculum as approved and allows students to take the certifying exams. Approved colleges are listed on the CCAPP webpage. Los Angeles Community College District 2023-2024 Final Budget 171 California Department of Public Health - Radiologic Health Branch (CDPH - RHB) – C All colleges teaching radiological technology are required to belong to this organization. California Fashion Association (CFA) – T This is the premier organization in the Los Angeles apparel industry dedicated to the promotion of local business, expanding contacts and sponsoring educational seminars. California Law Inc. – E This organization will allow for participation in the California Pathways Law Consortia and will allow students from high schools, community colleges, four-year institutions, and law schools to explore career pathways in law. California Organization of Associate Degree Nursing Program Directors (COADNPD) – CEPV Nursing directors from all nursing programs in Southern California meet monthly to collaborate and discuss issues relating to the profession of nursing and nursing programs in California. Directors also have an opportunity to discuss issues related to their individual programs with a Board of Nursing representative who is always present at these meetings. Many ideas and pertinent information are received from these meetings and used in strengthening our Registered Nursing Program. California Placement Association (CPA) – HMS This is a professional organization for Community College Job Placement centers. California School Personnel Commissioners Association (CSPCA) – D This organization focuses on ways to improve school classified personnel management. California Swap Meet Association (CSMA) – H This organization provides publicity and directories of swap meets all around the world. California Veterinary Medical Association - P This membership helps provide opportunities to research scholarships in the California Veterinary Medical Association for the Registered Veterinary Technology Program. Central City Association (CCA) – D This neighborhood network of business and community groups has the common purpose of addressing issues regarding career preparation and other local issues. Chief Student Services Officers Association (CSSO) – CEHSTVW This organization promotes statewide collaboration on new and changing initiatives that affect statewide student services programs to ensure that student voices and needs are being kept at the forefront. Members serve on committees, task groups, ad advisories representing Student Services and Equity perspectives. Los Angeles Community College District 2023-2024 Final Budget 172 Clery Center – D Clery Center is a non-profit organization committed to helping college and universities meet the standards of the Jeanne Clery Act, a consumer protection law aimed to provide transparency around campus crime policy and statistics. Under the membership, Los Angeles Community College District will receive technical assistance on annual security reporting and compliance trainings, which will improve annual crime reporting and maintain safer campuses for the entire district. Coalition of Community College Architecture Programs – HP This organization provides access to regional and national data, including curriculum, articulation agreements, and surveys of students. It also provides access to educational resources, including sample course curricula and articulation agreements and strategies. Collaborative Online International Learning (COIL) – E This organization connects students and professors in different countries for collaborative projects and discussions as part of their coursework, providing global experiences built into programs of study. COIL enhances intercultural student interaction through proven approaches to meaningful online engagement. Memberships includes access to members-only resources, special interest groups, webinars, meetings, and events Commission on Accreditation for Health Informatics and Information Management Education (CAHIIM) Organization – E This is the fee for Continuous Accreditation of the Health Information Technology Program so students will be able to take their certification exam (RHIT). Commission on Accreditation for Respiratory Care (CARC) – E This organization is the accrediting body that recognizes students who are eligible to take the national examination. It is a member of the Council on Medical Education. Commission on Accreditation of Allied Health Education Programs (CAAHEP) – EV This organization is the accrediting body for Health Education programs. Community College Association of Math Engineering and Science Achievement (MESA) Directors (CAMD) – EV This organization entitles the director to attend meetings and students of the program to attend sponsored events at no charge to the individual MESA Program. These events include activities such as student retreats and symposia. Community College Baccalaureate Association (CCBA) – W This organization provides the college with relevant and up to date information on various models for promoting access to baccalaureate degree programs, changes to legislation dealing with the community college baccalaureate degree, and other resources. The organization hosts an annual conference and provides opportunities for networking and camaraderie. Los Angeles Community College District 2023-2024 Final Budget 173 Community College Counselors/Advisors Academic Association for Athletics (3C4A) – HM This organization is an advocate of student athlete academic success. It benefits the athletic department. Community College Executive Forum - Education Advisory Board (EAB) – MS This organization provides best practice research and practical advice to leaders of academic affairs, business affairs, student affairs, advancement, continuing, online, and professional education, and community colleges across North America. Some of the services provided are best practice briefs, access to experts, performance audit tools, and online research database. Community College Facility Coalition (CCFC) – D This organization of community college facility planners, industry and financial personnel provides a forum for improving delivery systems of facilities by education, training, and interchange of ideas. Community College League of California (CCLC) – EMPSTV This organization promotes inter-college relations and the representation of junior colleges to other organizations. Council for Advancement and Support of Education (CASE) – W This organization will allow the college to receive support and gain access to various resources toward enhancing academic, administrative, and student support services at the college through access to comprehensive data, analytics, and research; professional development opportunities; advocacy and public policy at the national level; and global networking. Council for Higher Education Accreditation (CHEA) – T This is a national coordinating organization for accreditation. Council for Opportunity in Education (COE) – HSW This organization provides support for recipients of Federal Trio Grants by providing training and discounts on all activities. Council of Chief Librarians, California Community Colleges (CCLCCC) – CMPSTW This organization represents, promotes, and advances libraries in public California community college education and provides a vehicle for communication among chief librarians, other community college personnel, and state agencies. Culver City Chamber of Commerce (CCCC) – W This organization gives colleges visibility in area businesses and in the community at large. It provides contacts with people who serve on advisory committees and offers off-campus locations for both Community Services and Outreach classes. Los Angeles Community College District 2023-2024 Final Budget 174 Economic Alliance of the San Fernando Valley (EASFV) – MP This organization developed a new contract education training partnership with several District colleges. It is the marketing arm of the workplace training partnership and the colleges will provide the employment training. AKA Valley Economic Alliance EDUCAUSE – D The mission of this organization is to advance higher education by promoting the intelligent use of information technology. It helps those who lead, manage, and use information resources to shape strategic decisions at every level. Gartner – D This organization provides access to leading research and advisory services, consulting, conferences, business and IT insight, advice, and tools necessary for IT leaders and their teams. Greater San Fernando Valley Chamber of Commerce – MPV This organization aids colleges in establishing ties with the business community. Health Services Association of California Community Colleges (HSACCC) – P This organization provides services useful to the new Student Health Center at member colleges. HealthImpact – CPTV Membership in organization is required to enable students to be placed at Providence clinical facilities. Formerly the California Institute for Nursing and Healthcare (CINHC). Hispanic Association of Colleges and Universities (HACU) – CHMPV This organization assists its member institutions with procurement of funds that will assist in improving needed educational services for Hispanic students, for the expansion of instructional facilities, for upgrading the affirmative action programs regarding Hispanic faculty, and for providing a national network of resources, contacts, and legislative impetus where needed. Hispanic Educational Technology Services (HETS) – D This organization provides, promotes, and supports the capabilities of member institutions to enhance Hispanic/ Latino success and opportunities in higher education. Membership provides access to online trainings, innovative IT strategies in higher education, and representation on the HETS Board of Directors. Hollywood Chamber of Commerce (HCC) – C This organization promotes and fosters college/community relations and develops a support system for contacts with local businessmen and agencies. Honors Transfer Council of California (HTCC) – CHVW This organization is a consortium of Southern California community college honors transfer and Los Angeles Community College District 2023-2024 Final Budget 175 scholars program directors and coordinators. Independent College Bookstore Association (ICBA) – HPTV This organization is a co-op buying group formed to assist institutionally related stores with an aggregated buying service. Institute of Internal Auditors – D This organization provides access to local chapter activities such as seminars and training for the internal auditor’s required professional development. It provides professional networking and certification in particular areas of the internal audit profession and numerous tools for the internal auditor’s professional development. Intercollegiate Tennis Coaches Association (ITCA) – M This organization allows for tennis coaches and their coaching staff to have access to the Coaches' resources portal and have voting rights within the association which allows for student-athletes to be eligible to receive academic and athletic awards. International Facility Management Association (IFMA) – D IFMA provides its members with a wealth of educational career enhancement and personal development resources; the bi-monthly, award-winning Facility management Journal; the Association Newsletters, IFMA news, featuring updates on Association activities, research projects, news and events; IFMAnet, the members-only area of ifma.org. International Public Management Association for Human Resources (IPMA-HR) – D The primary purposes of this organization are to advance merit principles of employment and to develop sound policies and practices in the public personnel field. Joint Review Committee on Education in Radiologic Technology (JRCERT) – C This organization is the accrediting body that recognizes and approves training programs in radiologic technology in hospitals and institutions of higher learning. Graduates of JRS examination receive the title of Radiologic Technologist. Journalism Association of Community Colleges (JACC) – V This organization focuses on improvement of journalism in education. Lambda Beta Society (LBS) – V This organization provides for the National Honor Society for the Profession of Respiratory Care. Only graduates of the Respiratory Therapy Programs from member institutions may be nominated and inducted into the Respiratory Therapy Honor Society. Leadership in Educational Facilities – D This organization promotes the development and maintenance of high standards in the Los Angeles Community College District 2023-2024 Final Budget 176 administration, planning, and operation of the physical plant of its member institutions. (Formerly known as Association of Higher Education Facilities Officers - APPA) League for Innovation in the Community Colleges – PW This organization provides website resources, conferences, seminars, and speakers which effectively serve educators in their professional development. Learning Resources Network (LERN) – C This organization provides ongoing analysis and help in Community Services program development and marketing. Liebert Cassidy Whitmore’s Employment Relations Consortium (LCWERC) – D This organization joins agencies and school/community college districts in a geographic area for the purpose of securing quality employment relations trainings. Member institutions may attend educational lectures, workshops, and seminars. The District receives five-full days of training, which include reference materials, workbooks, case studies, and pretests for all attendees. In addition, the District receives a subscription to the firm’s monthly newsletter. Los Angeles Area Chamber of Commerce (LAACC) – T This organization has had a long relationship with industrial and business community which surrounds it. The college president relates to the education and industry committees of the chamber as an observer. Membership allows the president to continue to serve as a full member of these important committees. Los Angeles Coalition for the Economy & Jobs – D This organization brings together leaders from the region's business, labor, academic, and nonprofit communities to advance sound policy initiatives that will help to responsibly grow the economy and create quality jobs throughout the Los Angeles region. Los Angeles County Bar Association (LACBA) – D This organization provides General Counsel with several useful services, including LEXIS/NEXIS computer search system discounts, discounts on legal education programs and video tapes, issues of Los Angeles Lawyer and County Bar Update publications, attorney/messenger service discounts, section mailings, advance announcements of programs and member discounts at educational events, lawyer referral, and information services. Los Angeles County Business Federation (BizFed) – V This organization allows the District up to five representatives to participate in the advocacy committee; participate in BizFed events, meetings, working groups; and receive BizFed intel, action alerts, and notices. Los Angeles County School Trustees Association Council (LACSTAC) – D This organization provides a forum for issues and discussion for governing boards members Los Angeles Community College District 2023-2024 Final Budget 177 within Los Angeles County. Los Angeles Economic Development Corporation (LAEDC) – C This organization assists the District and the nine colleges in facilitating efforts to partner with private and public entities in efforts to enhance and broaden participation in community development activities and programs. Los Angeles Paralegal Association (LAPA) – M This organization provides advertising, networking, and exhibition opportunities for the Paralegal Program at West Los Angeles College. Mountain Measurement, Inc. – V This organization assists colleges in analyzing examination results for candidates applying for Registered Nursing Licensure in California and is used in the Education Master Plan for accreditation. National Asian Pacific Islander Council (NAPIC) – P This organization advances career and leadership development of Asian Pacific Islander (API) professionals and serves as a resource to community colleges on matters related to Asian Pacific Islanders. National Association of College and University Attorneys (NACUA) – D This organization of colleges and universities provides mutual assistance in resolving legal problems. It operates an Exchange of Legal Information program to which member institutions contribute legal memoranda, pleadings, model statues, or regulations and other significant legal materials. National Association of College Auxiliary Services (NACAS) – P This organization provides members with the latest information regarding college auxiliary services. National Association of College Stores (NACS) – MPTV This organization provides members with the following benefits: textbook, tradebook and publishers' information; new products information; seminars, conventions and a training school for bookstore managers and staff. National Association of Collegiate Directors of Athletics (NACDA) – M This organization serves as the professional association for those in the field of athletics administration, providing educational opportunities, and serves as a vehicle for networking and the exchange of information to others in the profession National Association of Dental Laboratories (NADL) – C This organization offers a subscription to the Journal of Dental Technology; opportunities to Los Angeles Community College District 2023-2024 Final Budget 178 administer RG and CDT exams at the college; confidential reports on the college’s and student’s test results; discounted member rates for RG and CDT study materials, NADL products, videos, and manuals; etc. National Association of Student Financial Aid Administrators (NASFAA) – T This organization promotes the effective administration of student financial aid in the United States. It provides training, conferences, and published material for members. Also provides up-to-date information on pending legislation as well as action of state organizations. National Association of Veterans Program Administrators (NAVPA) – C This organization helps to provide community-oriented services for veterans in education, employment, legal assistance and psychological readjustment. National Athletic Trainers Association (NATA) – MV This organization benefits college coaches and trainers by keeping them abreast of the latest research and innovations in sports technology. National Collegiate Honors Council (NCHC) – W This organization provides valuable input on honors education, curricular development, selection of students, etc. National Community College Hispanic Council (NCCHC) – E The NCCHC is a non-profit, charitable and educational affiliate of the American Association of Community Colleges (AACC) organization that addresses the special needs of Hispanic students in the nation's learning institutions. National Council for Marketing and Public Relations (NCMPR) – T This organization offers a broad range of support services in marketing, public and media relations, community and alumni relations, publications, sports marketing, legislative and governmental relations, and special events coordination. National Institute for Staff and Organizational Development (NISOD) – P This organization is a non-profit consortium of colleges who share a philosophical commitment to support excellence in teaching and learning. National League for Nursing Accrediting Commission (NLNAC) – V This organization is an obligation for recognition of accreditation status for the Nursing Program. National League of Nursing (NLN) – V This organization promotes improvements of nursing training programs and provides liaison between the academic institution and its professional counterpart. Los Angeles Community College District 2023-2024 Final Budget 179 National Student Clearinghouse – SW This organization maintains a database of more than 3,600 colleges and universities that report enrollment and degree information on a yearly basis. It is used to verify student enrollments, certificates, and degrees. With this type of data, more informed decisions can be made that may assist with student outcomes. Network California Community College Foundation (NCCCF) – D Membership in the Network will help to professionalize the development office of member colleges through exchange of information, training sessions, and fellowship with other development officers throughout California. Membership in the Network includes the newsletter, information about workshop sessions and seminars, participation in the fall symposium and access to a network of professional expertise in all aspects of resource development. President's Round Table (PRT) – T The President's Round Table is an organization consisting of Presidents and Chancellors of community colleges throughout the country. The Presidents' Round Table is affiliated with AACJC and the National Council on Black American Affairs. This organization provides CEOs of community colleges a national forum to express and share issues of interest and concern pertaining to education and specifically community colleges. Printing United Alliance – T Membership in this organization provides the latest materials, information on technology and annual conferences for instructors in the Design and Media Arts (DMA) pathway to acquire materials and industry knowledge for use in instruction. Public Agency Risk Managers Association (PARMA) – D This organization provides training covering issues in liability workers’ compensation, property, employee benefits, loss prevention, and a newsletter on risk management and legislative issues. It also allows members to post job vacancies on its website. Public Risk Insurance Management Association (PRIMA) – D This organization includes annual education programs, newsletters, publications, and the latest revisions and training regarding public sector risk management and legislation and regulations. Research and Planning Group for California Community Colleges (RPGCCC) – CEHPVW This organization acts as the cohesive voice for researchers in the community colleges. Services include workshops, newsletters and bulletins on recent and specialized research issues. Risk and Insurance Management Society (RIMS) This organization helps professionals in the field of risk management to expand their knowledge through workshops, on-line courses, and cost-effective interactive forums for networking. It also promotes the growth and development of educational programs for risk management. Los Angeles Community College District 2023-2024 Final Budget 180 School Employers Association of California (SEAC) – D This organization provides resources to maintain programs, policies, and procedures necessary to comply with the provisions of Educational Employers Relations Act, CA Gov. Code Se. 3540, et seq. South Coast Conference (SCC) – HT Intercollegiate athletic teams must join this conference to officially compete with member colleges. Southern California Football Association (SCFA) – SVW This organization is the new football conference for all Southern California Community Colleges. Southern California Intersegmental Articulation Council (SCIAC) – ECPTW This organization promotes the continuing improvement of articulation among and between the segments of post-secondary education in California. It provides channels of communications among the post-secondary segments and strengthens the role, functions, and support of articulation. Southern California Marine Institute (SCMI) – DSVW This organization provides access to the following resources at the Southern California Marine Institute (SCMI): the research vessel, Yellowfin, and several small vessels for coastal research (for an annual cruise and education for students and faculty professional development). Members are given a seat at the table as plans are being drafted for the facility. Southern California Rules Committee Association (SCRCA) – H This organization administers track and field and cross-country regional fees at member colleges. Southern California Wrestling Alliance – E This organization promotes sustainable partnerships for the improvement, preservation, and promotion of the wrestling program. Study California Inc. – C A non-profit consortium of California educational institutions that offer educational services to international students holding an applicable visa from the U.S. State Department. SurveyMonkey – CP This organization enables members to create professional online surveys quickly and easily. Sylmar Chamber of Commerce (SCC) – M Membership in the Sylmar Chamber of Commerce will enable the college to develop industrial and professional ties with the local business community. Los Angeles Community College District 2023-2024 Final Budget 181 United Soccer Coaches Association of America – H This organization provides information to aid the Soccer coaches and players at schools that care to join. (Formerly known as the National Soccer Coaches Association of America - NSCAA) University Risk Management & Insurance Association (URMIA) – D The organization allows Risk Management access to over 2,200 experienced risk managers and insurance professionals when we need assistance with an issue or need feedback on an issue that is unique to educational institutions. Some of the areas addressed are business continuity and emergency response, safety, enterprise risk management, information technology, insurance, international programs, loss prevention, regulatory compliance, and student activity risks. Valley Industry and Commerce Association (VICA) – MPV This organization is dedicated to the development and maintenance of cooperative efforts between business, labor, and government groups that serve the community and its economic wellbeing. Membership in the Association is comprised of key leaders and corporate chief executive officers throughout the Valley community who directly influence decisions that impact the economic, social, and educational conditions in the community. Western Association of Education Opportunity Personnel Southern California (WESTOP SoCal) Chapter – H This organization provides the TRIO program with information and services designed to establish a community of communication, coordination, and professional development among education equity personnel; to advocate for continued financial and legislative support at the federal, state, and local levels for educational programs working with economically and educationally disadvantaged persons and persons with disabilities; to promote and engage in research and evaluation which will enhance the effectiveness of programs and personnel; and to provide technical assistance, resources, and expertise for educational equity programs. Western Association of Student Financial Aid Administrators (WASFAA) – T This professional organization for financial aid practitioners from the Western state supports the professional preparation of student financial aid administrators and facilitates communication among institutions and private agencies that deal with financial aid programs. Western Association of Veteran Education Specialists (WAVES) – EPW This organization promotes high professional standards, policies, and ethical practices among members; and serves the needs and interests of veterans, faculty members, and administrators in the member institutions. Western State Conference (WSC) – MVW Intercollegiate athletic teams must join this conference to officially compete with member colleges. Los Angeles Community College District 2023-2024 Final Budget 182 Appendix E: Education Protection Act (EPA) Fund 10106 Proposed Spending Plan College Summer 2023 Fall 2023 Winter 2024 Spring 2024 Summer 2024 # of Planned Classes Final Budget City $2,460,068 $1,259,078 $419,693 $1,259,078 $419,692 $3,188 $5,817,609 East $0 $5,960,810 $0 $5,960,810 $4,202 $11,921,620 Harbor $0 $1,621,580 $0 $1,621,579 $0 $205 $3,243,159 Mission $0 $1,894,397 $0 $1,894,397 $0 $420 $3,788,794 Pierce $0 $3,308,663 $0 $3,308,663 $0 $914 $6,617,326 Southwest $0 $1,417,132 $0 $1,081,532 $0 $212 $2,498,664 Trade-Tech $0 $2,661,356 $0 $2,661,357 $0 $309 $5,322,713 Valley $0 $3,187,628 $0 $3,164,218 $0 $390 $6,351,846 West $0 $1,928,508 $0 $1,928,508 $0 $205 $3,857,016 Total $2,460,068 $23,239,152 $419,693 $22,880,142 $419,692 $10,045 $49,418,747 On November 6, 2012, voters passed Proposition 30, The Schools and Local Public Safety Protection Act of 2012 (EPA) to provide funding for K-12, community colleges, and public safety. In 2016, voters extended the provisions of Proposition 30 through the passage of Proposition 55. It is prohibited to use EPA funds for salaries and benefits of administrators or any administrative costs. Please note that the EPA Fund cannot be used to support administrative salaries and benefits or other administrative costs consistent with the State Chancellor’s Office Accounting Advisory FS 13-03, April 17, 2013. Los Angeles Community College District 2023-2024 Final Budget 183 Los Angeles City College 2023-2024 Proposed Prop 30 EPA Plan Semester 2021-2022 Actual 2022-2023 Actual 2023-2024 Budget Summer Session (Fund 10221): Number of Class Offerings 158 137 144 Number of Students Served 3,242 3,317 3,729 Instructional Budget/Actual ($) $2,407,716 $2,312,636 $2,460,068 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Fall Semester: Number of Class Offerings 1,355 1,347 1,304 Number of Students Served 14,301 15,541 16,318 Instructional Budget/Actual ($) $11,967,248 $13,202,859 $1,259,078 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Winter Intersession (Fund 10099): Number of Class Offerings 217 212 210 Number of Students Served 4,781 4,845 5,087 Instructional Budget/Actual ($) $1,599,537 $1,893,414 $419,693 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Spring Semester: Number of Class Offerings 1,265 1,377 1,300 Number of Students Served 13,963 15,896 16,691 Instructional Budget/Actual ($) $11,967,247 $13,202,859 $1,259,078 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Summer Session (Fund 10222): Number of Class Offerings 235 232 230 Number of Students Served 5,116 5,510 5,730 Instructional Budget/Actual ($) $777,982 $859,677 $419,692 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Total: Number of Class Offerings 3,230 3,305 3,188 Number of Students Served 41,403 45,109 47,555 Instructional Budget/Actual ($) $28,719,730 $31,471,445 $5,817,609 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Beginning July 1, 2023. Beginning prior to July 1, 2024. Los Angeles Community College District 2023-2024 Final Budget 184 East Los Angeles College 2023-2024 Proposed Prop 30 EPA Plan Semester 2021-2022 Actual 2022-2023 Actual 2023-2024 Budget Summer Session (Fund 10221): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Fall Semester: Number of Class Offerings 1,984 2,007 2,109 Number of Students Served 19,402 21,648 21,648 Instructional Budget/Actual ($) $16,360,899 $18,645,476 $5,960,810 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Winter Intersession (Fund 10099): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Spring Semester: Number of Class Offerings 1,046 2,093 2,093 Number of Students Served 16,588 21,648 21,648 Instructional Budget/Actual ($) $16,446,075 $13,699,870 $5,960,810 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Summer Session (Fund 10222): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Total: Number of Class Offerings 3,030 4,100 4,202 Number of Students Served 35,990 43,296 43,296 Instructional Budget/Actual ($) $32,806,974 $32,345,346 $11,921,620 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Beginning July 1, 2023. Beginning prior to July 1, 2024. Los Angeles Community College District 2023-2024 Final Budget 185 Los Angeles Harbor College 2023-2024 Proposed Prop 30 EPA Plan Semester 2021-2022 Actual 2022-2023 Actual 2023-2024 Budget Summer Session (Fund 10221): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Fall Semester: Number of Class Offerings 298 265 105 Number of Students Served 3,650 3,394 1,310 Instructional Budget/Actual ($) $4,157,385 $2,580,517 $1,621,580 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Winter Intersession (Fund 10099): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Spring Semester: Number of Class Offerings 280 257 100 Number of Students Served 3,508 3,438 1,294 Instructional Budget/Actual ($) $3,538,814 $4,933,712 $1,621,579 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Summer Session (Fund 10222): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Total: Number of Class Offerings 578 522 205 Number of Students Served 7,158 6,832 2,604 Instructional Budget/Actual ($) $7,696,199 $7,514,229 $3,243,159 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Beginning July 1, 2023. Beginning prior to July 1, 2024. Los Angeles Community College District 2023-2024 Final Budget 186 Los Angeles Mission College 2023-2024 Proposed Prop 30 EPA Plan Semester 2021-2022 Actual 2022-2023 Actual 2023-2024 Budget Summer Session (Fund 10221): Number of Class Offerings 436 219 0 Number of Students Served 4,223 4,106 0 Instructional Budget/Actual ($) $2,629,217 $2,515,254 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Fall Semester: Number of Class Offerings 1,276 1,111 457 Number of Students Served 12,352 10,754 4,426 Instructional Budget/Actual ($) $7,689,764 $7,428,848 $3,308,663 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Winter Intersession (Fund 10099): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Spring Semester: Number of Class Offerings 1,268 1,533 457 Number of Students Served 12,270 14,834 4,424 Instructional Budget/Actual ($) $7,639,260 $7,190,109 $3,308,663 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Summer Session (Fund 10222): Number of Class Offerings 0 110 0 Number of Students Served 0 2,242 0 Instructional Budget/Actual ($) $0 $860,199 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Total: Number of Class Offerings 2,980 2,973 914 Number of Students Served 28,845 31,937 8,850 Instructional Budget/Actual ($) $17,958,241 $17,994,409 $6,617,326 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Beginning July 1, 2023. Beginning prior to July 1, 2024. Los Angeles Community College District 2023-2024 Final Budget 187 Los Angeles Pierce College 2023-2024 Proposed Prop 30 EPA Plan Semester 2021-2022 Actual 2022-2023 Actual 2023-2024 Budget Summer Session (Fund 10221): Number of Class Offerings 436 219 0 Number of Students Served 4,223 4,106 0 Instructional Budget/Actual ($) $2,629,217 $2,515,254 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Fall Semester: Number of Class Offerings 1,030 1,347 1,375 Number of Students Served 39,537 42,772 42,772 Instructional Budget/Actual ($) $7,203,133 $6,777,866 $6,079,094 Non-Instructional and Others Budget/Actual ($) $23,720 $0 $0 Winter Intersession (Fund 10099): Number of Class Offerings 0 138 146 Number of Students Served 0 4,631 4,911 Instructional Budget/Actual ($) $0 $1,275,057 $2,372,096 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Spring Semester: Number of Class Offerings 948 1,281 1,281 Number of Students Served 34,811 37,737 37,737 Instructional Budget/Actual ($) $6,656,728 $6,708,929 $13,097,480 Non-Instructional and Others Budget/Actual ($) $9,891 $0 $0 Summer Session (Fund 10222): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Total: Number of Class Offerings 1,978 2,766 2,802 Number of Students Served 74,348 85,140 85,420 Instructional Budget/Actual ($) $13,859,861 $14,761,852 $21,548,670 Non-Instructional and Others Budget/Actual ($) $33,611 $0 $0 Beginning July 1, 2023. Beginning prior to July 1, 2024. Los Angeles Community College District 2023-2024 Final Budget 188 Los Angeles Southwest College 2023-2024 Proposed Prop 30 EPA Plan Semester 2021-2022 Actual 2022-2023 Actual 2023-2024 Budget Summer Session (Fund 10221): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Fall Semester: Number of Class Offerings 298 306 109 Number of Students Served 7,107 7,163 2,628 Instructional Budget/Actual ($) $2,733,201 $3,015,428 $1,417,132 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Winter Intersession (Fund 10099): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Spring Semester: Number of Class Offerings 273 285 103 Number of Students Served 6,120 6,693 2,456 Instructional Budget/Actual ($) $2,571,801 $2,250,898 $1,081,532 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Summer Session (Fund 10222): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Total: Number of Class Offerings 571 591 212 Number of Students Served 13,227 13,856 5,084 Instructional Budget/Actual ($) $5,305,002 $5,266,326 $2,498,664 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Beginning July 1, 2023. Beginning prior to July 1, 2024. Los Angeles Community College District 2023-2024 Final Budget 189 Los Angeles Trade-Technical College 2023-2024 Proposed Prop 30 EPA Plan Semester 2021-2022 Actual 2022-2023 Actual 2023-2024 Budget Summer Session (Fund 10221): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Fall Semester: Number of Class Offerings 433 379 153 Number of Students Served 7,664 8,387 4,233 Instructional Budget/Actual ($) $7,222,012 $6,604,990 $2,661,357 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Winter Intersession (Fund 10099): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Spring Semester: Number of Class Offerings 355 388 156 Number of Students Served 5,787 7,368 3,704 Instructional Budget/Actual ($) $5,908,919 $6,604,990 $2,661,357 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Summer Session (Fund 10222): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Total: Number of Class Offerings 788 767 309 Number of Students Served 13,451 15,755 7,937 Instructional Budget/Actual ($) $13,130,931 $13,209,980 $5,322,714 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Beginning July 1, 2023. Beginning prior to July 1, 2024. Los Angeles Community College District 2023-2024 Final Budget 190 Los Angeles Valley College 2023-2024 Proposed Prop 30 EPA Plan Semester 2021-2022 Actual 2022-2023 Actual 2023-2024 Budget Summer Session (Fund 10221): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Fall Semester: Number of Class Offerings 530 806 196 Number of Students Served 13,250 20,125 4,900 Instructional Budget/Actual ($) $7,594,576 $12,023,143 $3,187,628 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Winter Intersession (Fund 10099): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Spring Semester: Number of Class Offerings 512 800 194 Number of Students Served 12,800 22,400 4,850 Instructional Budget/Actual ($) $7,328,682 $11,934,847 $3,164,218 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Summer Session (Fund 10222): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Total: Number of Class Offerings 1,042 1,606 390 Number of Students Served 26,050 42,525 9,750 Instructional Budget/Actual ($) $14,923,258 $23,957,990 $6,351,846 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Beginning July 1, 2023. Beginning prior to July 1, 2024. Los Angeles Community College District 2023-2024 Final Budget 191 West Los Angeles College 2023-2024 Proposed Prop 30 EPA Plan Semester 2021-2022 Actual 2022-2023 Actual 2023-2024 Budget Summer Session (Fund 10221): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Fall Semester: Number of Class Offerings 294 269 97 Number of Students Served 5,276 6,901 2,484 Instructional Budget/Actual ($) $5,507,585 $5,398,310 $1,928,508 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Winter Intersession (Fund 10099): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Spring Semester: Number of Class Offerings 299 301 108 Number of Students Served 4,970 6,849 2,466 Instructional Budget/Actual ($) $5,507,585 $5,398,309 $1,928,508 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Summer Session (Fund 10222): Number of Class Offerings 0 0 0 Number of Students Served 0 0 0 Instructional Budget/Actual ($) $0 $0 $0 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Total: Number of Class Offerings 593 570 205 Number of Students Served 10,246 13,750 4,950 Instructional Budget/Actual ($) $11,015,170 $10,796,619 $3,857,016 Non-Instructional and Others Budget/Actual ($) $0 $0 $0 Beginning July 1, 2023. Beginning prior to July 1, 2024. Los Angeles Community College District 2023-2024 Final Budget 192 Appendix F: 2023-2024 Final Budget Allocation Mechanism In 2019-20, the Board approved a new District Allocation Model that better aligns with the new Student Centered Funding Formula. In 2022-23, this District Allocation Model was reviewed and updated with an equity minded approach and approved by the Board in July 2023. This updated District Budget Allocation Model has been used for the Final Budget Allocation. Funding Principles • Aligns with the State’s Student Centered Funding Formula (SCFF) in support of student access, equity and success. • Allocation Model should be easily understood, fair and predictable. • Recognizes there are core services and unique characteristics associated with a College regardless of size. • Recognizes that there are Districtwide costs and Educational Service Center operations that must be funded. • Balances will be retained by Colleges, Educational Service Center and Information Technology locations. • Colleges are encouraged to collaborate and promote innovation with each other that will maximize student access and success. • Apply an equity minded approach, as in the SCFF, recognizing college resources and student needs vary across the District. I. Parameters Used to Determine State Apportionment Revenue 1. Base Allocation The Base Allocation is the enrollment-based component of the State Student Centered Funding Formula (SCFF) and is the sum of the Basic Allocation funding (which is based on the number of colleges and centers in a district and its size) and the funding for enrollment in credit (utilizing a three-year average), noncredit, and career development and college preparation (CDCP) noncredit courses, as well as enrollment of special admit students and inmates in correctional facilities. For fiscal year 2023-24, the basic allocation base rate is estimated to be: • FTES >= 20,000 $8,586,065 large college • 10,000 <= FTES < 20,000 $7,512,806 medium college • FTES < 10,000 $6,439,546 small college • State Approved Center $2,146,516 center For fiscal year 2023-24, the FTES allocation rates are estimated to be: Los Angeles Community College District 2023-2024 Final Budget 193 • Credit $5,238 • Special Admit Credit $7,346 • Incarcerated Credit $7,346 • Non-Credit $4,417 • Non-Credit Enhanced (CDCP) $7,346 2. Supplemental Allocation The Supplemental Allocation of the SCFF recognizes that districts must provide additional support to remove barriers to access and success for certain groups of students. It is determined based on the number of low-income students in a district. For fiscal year 2023-24, the Supplemental Allocation rates are estimated to be: • Pell Grant Recipients $1,239 • College Promise Grant Recipients $1,239 • AB 540 students $1,239 3. Student Success Allocation The Student Success Allocation encourages progress on outcomes linked to the goals included in the State Chancellors Office Vision for Success. This allocation assigns funding rates for eight outcomes with additional funding for outcomes attained by students who received Pell Grants and College Promise Grants (Equity). For fiscal year 2023-24, the Student Success Allocation rates are estimated to be: • Associate degree for transfer (ADT) $2,922 • Associate degree granted $2,191 • Baccalaureate degree granted $2,191 • Credit certificate granted $1,461 • Transfer-level Math or English course $1,461 • Transfer to four-year university $1,096 • Completion of nine or more CTE units $730 • Attainment of regional living wage $730 For fiscal year 2023-24, the Equity Allocation rates for Pell Students are estimated to be: • Associate degree for transfer (ADT) $1,105 • Associate degree granted $829 • Baccalaureate degree granted $829 • Credit certificate granted $553 Los Angeles Community College District 2023-2024 Final Budget 194 • Transfer-level Math or English course $553 • Transfer to four-year university $415 • Completion of nine or more CTE units $276 • Attainment of regional living wage $276 For fiscal year 2022-23, the Equity Allocation rates for CA Promise Grant Students are estimated to be: • Associate degree for transfer (ADT) $737 • Associate degree granted $553 • Baccalaureate degree granted $553 • Credit certificate granted $368 • Transfer-level Math or English course $368 • Transfer to four-year university $276 • Completion of nine or more CTE units $184 • Attainment of regional living wage $184 4. COLA COLA (cost of living adjustment) will be distributed as specified in the State Apportionment notice. 5. College Growth • Growth will not be budgeted until earned • Earned College Growth is defined as the amount of SCFF apportionment calculated (adjusted for the minimum base allocation) in excess of the College hold harmless amount • College Growth not resulting in additional revenue from the State will be paid out of the contingency reserve II. Parameters to Allocate State Apportionment Revenue 1. Educational Services Center (ESC) The District recognizes that there are certain services that are provided more efficiently through a central operation. Examples of these services include Human Resources, Payroll, Accounts Payable and Purchasing and Information Technology. Funding for the ESC will be determined by a percentage of LACCD Base Allocation determined by the state Student Centered Funding Formula (SCFF). During the hold harmless period of the SCFF, the allocation will be determined by the formula: Prior Year Allocation + Current Year COLA + Board Approved Adjustments +/- cost transfers from/to other locations. At the end of the hold harmless period, (currently 2024-25) a percentage will be established equal to the 2024-25 allocation amount (minus ending balance) divided by the 2024- 25 General Fund Unrestricted Revenue Final Budget (less Los Angeles Community College District 2023-2024 Final Budget 195 dedicated revenue). This percentage will be adjusted in subsequent years by any Board Approved Adjustments +/- cost transfers from/to other locations. Funding for the ESC will come off the top of the Base Allocation, the remaining Base Allocation will be proportionately reduced across all locations and shall be distributed to colleges based on their proportion of the Districts base allocation plus hold harmless amount. The percentage and methodology will be reviewed a few years after the SCFF funding floor is fully implemented. 2. Districtwide (Centralized) Accounts There are annual expenditures which support the District as a whole or that cannot be easily broken out by college. Examples of these expenditures include Property & Liability Insurance, Legal, Audit, etc. Budgets in these accounts do not carryover but are replenished each year. Funding for the Districtwide Accounts is based on need, the Presidents will make budget recommendations on Districtwide Accounts to the District Budget Committee. Funding for the Districtwide Accounts will come off the top of the Base Allocation, the remaining Base Allocation will be proportionately reduced across all locations and shall be distributed to colleges based on their proportion of the District’s funded FTES. 3. Other Districtwide Accounts There are Districtwide projects and expenditures that are one time in nature that tend to take multiple years to complete. Budgets in these accounts carryover until project completion or are self-supporting operations. Examples of these expenditures include the President and Dean Academy, DAS professional college, DAS sustainability and Van de Kamp. Funding for these other Districtwide accounts come from one-time budget requests or from unique funding streams and does not come from the Base Allocation. 4. Reserves The District shall maintain a District General Reserve of six and a half percent (6.5%) and a Contingency Reserve of three and a half percent (3.5%) of total unrestricted general fund revenue at the districtwide account level. Such reserves shall be established to ensure the District’s financial stability, to meet emergency situations or budget adjustments due to any revenue projection shortfalls during the fiscal year. Use of the reserve must be approved by the Board prior to any expenditure. State Apportionment Base Allocation Revenue will be utilized to maintain the General Reserve (6.5%) and replenish the Contingency Reserve (3.5%). 5. College Set Asides One percent (1.0%) of total college unrestricted allocation is to be set aside in the college budget to ensure College financial stability, to meet emergency situations or budget adjustments due to any revenue projection shortfalls during the fiscal year. 6. Other Set Asides The District shall maintain a Deferred Maintenance fund, setting aside two percent (2.0%) of total unrestricted general fund revenue at the districtwide account level. State Apportionment Base Allocation Revenue will be utilized to establish the Deferred Maintenance fund each budget year. Los Angeles Community College District 2023-2024 Final Budget 196 7. College Allocation a. College Minimum Base To recognize that there are fixed expenses and core services associated with a College regardless of size, each College will receive an annual minimum base allocation determined by the following parameters: • Minimum Administrative Staffing: 1. (1) President; 2. (3) Vice Presidents; 3. (1) Institutional Research Dean; 4. (1) Facilities Manager; 5. Deans a. (4) Deans => small colleges (FTES<10,000); b. (8) Deans => medium colleges (FTES>=10,000 and <20,000); c. (12) Deans => large colleges (FTES>=20,000). • Maintenance and Operations costs based on average cost per gross square footage. b. Remaining State Apportionment Allocation The colleges shall receive 100 % of their earned Supplemental Allocation and 100% of their earned Student Success Allocation, as well as their proportional share of their earned amount of the remaining Base Allocation (after ESC/IT, Districtwide and Reserves). c. Assessment Calculation The proportionate share of the total allocated base plus hold harmless amount will be used to determine the college assessment. III. Parameters to Allocate Other Revenue 1. Non-Resident Tuition/Enrollment Fees Revenue shall be distributed to colleges based on college projections of tuition earnings. 2. Local Revenue and Other Federal and State Revenue (Dedicated Revenue) Revenue that is directly generated by colleges shall be distributed to colleges based on college projections and adjusted for actual. 3. Lottery Revenue Revenue shall be distributed to colleges based on the proportion of a college’s prior year FTES over the total District FTES and adjusted for actual. Los Angeles Community College District 2023-2024 Final Budget 197 4. Interest and Other Federal, State, and Local Income Not Directly Generated by the Colleges. Interest and other federal, state, and local income that is not directly generated by colleges shall be utilized to fund the District’s reserves. IV. Parameters for Allocations 1. A College total budget shall be the sum of the adjusted base allocation, 100% of the calculated supplemental allocation, 100% of the calculated student success allocation, plus other revenue; minus college deficit payments; plus, balances. 2. Additional funding received by the District after Final Budget, not directly attributable to an individual college, shall be distributed through the new allocation model as delineated in the Revenue Parameters above. 3. In the event that actual revenues are less than the amounts projected and allocated to colleges for the fiscal year, the college budgets will be recalculated and adjusted accordingly. 4. As the District is being ‘held harmless’ by the State, and will be held to a ‘funding floor’ in the future; Colleges will be ‘held harmless’ to the total of the prior year allocated State Apportionment Revenue. 5. The College ‘hold harmless’ amount will increase by State COLA if the District ‘hold harmless’ revenue also increases by the same. 6. The College ‘funding floor’ amount, currently scheduled to be implemented in 2025-26, will not increase by COLA. 7. Colleges shall keep their ending balances through fiscal year 2024-25. Beginning in 2025-26, colleges shall keep their year-end balance up to five (5%) of their prior year’s Unrestricted General Fund budget, excluding prior year balances. Colleges are allowed to carry over their accumulated balances from fiscal year 2025-26 and subsequent fiscal years up to ten (10%) of their prior year Unrestricted General Fund budget. 8. Colleges with balances in the General Reserve will be allowed to use up to $5 million or twenty five percent (25%) of that balance annually, whichever is less. Additional access is allowed with the Chancellor’s approval. 9. The Educational Services Center (ESC) and Information Technology (IT) shall retain its prior year ending balance including open orders. Open orders for Educational Services Center/IT and Districtwide Accounts shall be funded up to the available balances from these locations. Any uncommitted balances in Districtwide Accounts shall be redistributed to colleges at the end of the fiscal year. Los Angeles Community College District 2023-2024 Final Budget 198 10. The college president is the authority for college matters within the parameters of law and Board operating policy. The college president shall be responsible for the successful operation and performance of the college. 11. During Budget Preparation, the Presidents will make a recommendation on Districtwide (Centralized) Accounts allocation to the District Budget Committee. 12. Prior to Budget Preparation, the Presidents will meet to forecast FTES and other metrics and set goals to maximize revenues to be generated by the colleges. 13. Each operating location shall prepare a quarterly report to include annual projected expenditures and identify steps necessary to maintain a balanced budget. 14. The budget allocation will be recalculated using this mechanism at Final Budget, First Principal Apportionment (February) and at year-end. Los Angeles Community College District 2023-2024 Final Budget 199 Funds Available for 2023-2024 Unrestricted General Fund Income & Balances 2022-2023 Final Budget (COLA@6.56%, GR@0.00%) 2023-2024 Final Budget (COLA@8.22%, GR@0.00%) Difference Base (excluding EPA Funds) 477,019,782 692,440,065 215,420,283 EPA Funds 198,102,933 49,418,747 (148,684,186) COLA 44,288,051 60,980,794 16,692,743 Growth 0 0 0 Lottery 12,927,300 17,892,200 4,964,900 Non-Resident 7,120,000 8,279,000 1,159,000 Apprenticeship 365,396 33,455 (331,941) Part-Time Faculty Compensation 2,265,548 2,305,482 39,934 On-Going State Mandate Block Grant 2,398,000 3,494,286 1,096,286 Full-Time Faculty Hiring 13,368,234 13,368,234 0 Part-Time Office Hours 4,845,499 5,252,817 407,318 Part-Time Faculty Health Benefits 0 2,170,443 2,170,443 BOG Fee Waiver Administration 0 1,100,000 1,100,000 Local Interest and RDA Pass through 10,000,000 14,000,000 4,000,000 Dedicated Revenue 7,103,840 8,210,934 1,107,094 Total Income 779,804,583 878,946,457 99,141,874 Fund Balances Open Orders 18,500,747 22,994,629 4,493,882 Contingency Reserve 27,293,160 30,763,126 3,469,966 General Reserve 50,687,298 57,131,520 6,444,222 Other Fund Balance 86,404,243 76,157,677 (10,246,566) Total Fund Balance 182,885,448 187,046,952 4,161,504 Total Projected Funds Available 962,690,031 1,065,993,409 103,303,378 Los Angeles Community College District 2023-2024 Final Budget 200 Unrestricted General Fund Colleges & Obligations 2022-2023 Final Budget w/ distributed balances 2022-2023 Final Budget w/o distributed balances 2023-2024 Final Budget City 70,203,973 68,894,646 76,844,245 East 149,292,287 132,737,664 157,801,428 Harbor 44,064,194 39,345,407 44,572,252 Mission 44,397,599 41,111,541 45,273,240 Pierce 95,708,304 83,432,451 97,325,219 Southwest 36,344,088 33,819,419 39,660,069 Trade-Tech 88,651,562 71,120,518 91,966,093 Valley 83,263,406 71,268,198 83,803,911 West 48,325,433 46,371,025 53,956,380 College Total 660,250,846 588,100,870 691,202,837 Educational Services Center 36,924,779 34,536,380 38,793,579 Information Technology 19,699,419 18,605,657 21,497,982 Districtwide Services 138,745,488 122,502,004 149,959,037 Contingency Reserve 27,293,160 27,293,160 30,763,126 General Reserve 50,687,298 50,687,298 57,131,520 STRS/PERS Reserve 3,830,001 3,830,001 0 Other District-wide 1,947,141 0 1,841,622 Van de Kamp Innovation 2,943,314 1,018,604 3,612,969 Supplemental Retirement (SRP) 4,772,488 4,772,488 4,700,045 Funds for Deferred Maintenance 15,596,092 15,596,092 17,578,929 Part Time Faculty Health Benefits 0 0 2,170,443 Emergency Conditions Revenue 0 0 46,741,320 Undistributed Balance 5 95,747,476 0 Total 962,690,031 962,690,031 1,065,993,409 Los Angeles Community College District 2023-2024 Final Budget 201 Revenue Allocation Detail Colleges & Obligations Minimum Base Rev Base Funds Remaining EPA Funds Supplemental Student Success COLA SCFF Hold Harmless Total SCFF Apportionment Allocated Full Time Faculty Hiring Other State/ Local Apprentice State Mandate Revenue Lottery Non-Resident Dedicated Revenue Total Revenues City 16,127,369 31,804,678 5,817,609 15,099,827 8,960,229 6,956,319 6,817,038 91,583,069 1,722,084 1,059,703 0 391,926 2,037,529 2,500,000 339,356 99,633,667 East 19,714,760 65,175,106 11,921,620 23,691,345 17,207,021 13,665,949 28,542,574 179,918,375 1,603,410 1,840,770 0 883,591 4,547,965 1,627,000 445,282 190,866,393 Harbor 9,323,098 17,730,243 3,243,159 6,105,403 4,861,645 3,955,224 6,853,533 52,072,305 901,752 549,941 0 197,277 1,003,304 278,000 1,404,618 56,407,197 Mission 9,706,165 20,713,216 3,788,794 7,988,303 5,277,126 4,141,381 2,908,155 54,523,140 1,840,756 762,767 0 237,639 1,202,404 311,000 390,776 59,268,482 Pierce 12,934,351 36,176,707 6,617,326 15,954,858 12,576,638 8,333,266 17,118,048 109,711,194 1,310,188 1,087,988 0 497,961 2,552,291 1,558,000 1,169,792 117,887,414 Southwest 10,988,428 13,660,115 2,498,664 4,350,701 3,054,809 3,388,152 6,665,680 44,606,549 920,378 384,233 0 160,315 812,475 150,000 615,371 47,649,321 Trade-Tech 15,355,834 29,099,098 5,322,713 11,316,858 8,093,783 7,008,754 16,076,364 92,273,404 1,369,524 967,220 33,455 399,662 2,039,191 481,000 924,987 98,488,443 Valley 15,203,561 34,725,336 6,351,846 15,730,513 9,787,711 7,167,131 5,392,408 94,358,506 2,074,642 1,217,064 0 424,452 2,151,896 650,000 452,249 101,328,809 West 10,412,592 21,086,181 3,857,016 7,357,617 6,027,952 4,709,649 8,553,637 62,004,644 1,625,495 788,613 0 301,463 1,545,145 724,000 1,232,107 68,221,467 College Total 119,766,158 270,170,680 49,418,747 107,595,425 75,846,914 59,325,825 98,927,437 781,051,186 13,368,229 8,658,299 33,455 3,494,286 17,892,200 8,279,000 6,974,538 839,751,193 Educational Services Ctr Information Technology Districtwide Services Contingency Reserve General Reserve STRS/PERS Reserve Other District-wide Van de Kamp Innovation 1,236,396 1,236,396 SRP- Early Retirement Funds for Def Maintenance PT Fac Health Ben 2,170,443 2,170,443 Emerg. Conditions Rev 1,845,229 22,448,046 24,293,275 24,293,275 Undistributed (Projected Bal) (190,260) (2,314,595) (2,504,855) 5 14,000,000 11,495,150 Total 119,766,158 270,170,680 49,418,747 107,595,425 75,846,914 60,980,794 119,060,888 802,839,606 13,368,234 24,828,742 33,455 3,494,286 17,892,200 8,279,000 8,210,934 878,946,457 Los Angeles Community College District 2023-2024 Final Budget 202 Assessment and Adjustment Detail Colleges & Obligations Total Revenues Assessment SRP Faculty Overbase Centralized at Colleges PERS/STRS Contingency Bud Alloc w/o Balances Balances Balances Held Back Budget for Open Orders Bud Alloc before Debt Pymt Assessment Adjustment Budget Allocation City 99,633,667 (22,341,334) (712,174) 22,995 0 427,994 77,031,148 692,481 0 420,396 78,144,025 (1,299,780) 76,844,245 East 190,866,393 (46,239,554) (539,448) 75,100 0 706,738 144,869,229 9,511,708 0 1,762,037 156,142,974 1,658,454 157,801,428 Harbor 56,407,197 (13,703,593) (471,955) 11,550 0 228,151 42,471,350 1,800,083 0 292,992 44,564,425 7,827 44,572,252 Mission 59,268,482 (13,691,160) (287,613) 27,878 108,379 267,542 45,693,508 574,700 0 184,025 46,452,233 (1,178,993) 45,273,240 Pierce 117,887,414 (26,870,981) (400,796) 0 0 487,740 91,103,377 4,461,719 0 636,300 96,201,396 1,123,823 97,325,219 Southwest 47,649,321 (12,472,614) (406,324) 11,550 0 189,910 34,971,843 4,188,185 0 364,030 39,524,058 136,011 39,660,069 Trade-Tech 98,488,443 (24,291,674) (504,038) 891,843 0 397,825 74,982,399 12,244,883 0 3,541,654 90,768,936 1,197,157 91,966,093 Valley 101,328,809 (22,749,475) (407,721) 40,425 192,806 436,671 78,841,515 6,340,078 0 412,584 85,594,177 (1,790,266) 83,803,911 West 68,221,467 (16,196,941) (399,617) 46,047 0 278,416 51,949,372 1,475,885 0 385,356 53,810,613 145,767 53,956,380 College Total 839,751,193 (198,557,326) (4,129,686) 1,127,388 301,185 3,420,987 641,913,741 41,289,722 0 7,999,374 691,202,837 0 691,202,837 Educational Services Ctr 0 37,758,876 (361,228) 249,449 37,647,097 244,495 901,987 38,793,579 38,793,579 Information Technology 0 20,041,838 (88,797) 80,060 20,033,101 959,569 505,312 21,497,982 21,497,982 Districtwide Services 0 131,933,702 (301,185) 131,632,517 4,836,257 13,490,263 149,959,037 149,959,037 Contingency Reserve 0 (3,705,090) (120,334) (1,127,388) 79,505 (4,873,307) 35,636,433 30,763,126 30,763,126 General Reserve 0 6,444,222 6,444,222 50,687,298 57,131,520 57,131,520 STRS/PERS Reserve 0 0 0 0 0 Other District-wide 0 0 0 1,812,760 28,862 1,841,622 1,841,622 Van de Kamp Innovation 1,236,396 1,236,396 2,307,742 68,831 3,612,969 3,612,969 SRP- Early Retirement 0 4,700,045 4,700,045 4,700,045 4,700,045 Funds for Def Maintenance 0 17,578,929 17,578,929 0 17,578,929 17,578,929 PT Fac Health Ben 2,170,443 2,170,443 2,170,443 2,170,443 Emergency Cond Rev 24,293,275 (1) 24,293,274 22,448,046 46,741,320 46,741,320 Undistributed (Projected Bal) 11,495,150 (11,495,150) 0 0 0 0 0 0 Total 878,946,457 0 0 0 0 3,830,001 882,776,458 160,222,322 0 22,994,629 1,065,993,409 0 1,065,993,409 Los Angeles Community College District 2023-2024 Final Budget 203 Total Unrestricted General Fund Revenues Location Base Allocation (Less EPA Funds) EPA Funds Supplemental Student Success Hold Harmless COLA Total SCFF Apportionment Generated Funds for FT Faculty Hiring Apprenticeship Non- Resident Dedicated Lottery Interest/ Other State On-Going State Mandate Block Grant Total Revenue City 45,903,633 5,817,609 15,099,827 8,960,229 6,817,038 6,956,319 89,554,655 1,722,084 0 2,500,000 339,356 2,037,529 1,059,703 391,926 97,605,253 East 94,067,108 11,921,620 23,691,345 17,207,021 28,542,574 13,665,949 189,095,617 1,603,410 0 1,627,000 445,282 4,547,965 1,840,770 883,591 200,043,635 Harbor 25,590,026 3,243,159 6,105,403 4,861,645 6,853,533 3,955,224 50,608,990 901,752 0 278,000 1,404,618 1,003,304 549,941 197,277 54,943,882 Mission 29,895,346 3,788,794 7,988,303 5,277,126 2,908,155 4,141,381 53,999,105 1,840,756 0 311,000 390,776 1,202,404 762,767 237,639 58,744,447 Pierce 52,213,774 6,617,326 15,954,858 12,576,638 17,118,048 8,333,266 112,813,910 1,310,188 0 1,558,000 1,169,792 2,552,291 1,087,988 497,961 120,990,130 Southwest 19,715,619 2,498,664 4,350,701 3,054,809 6,665,680 3,388,152 39,673,625 920,378 0 150,000 615,371 812,475 384,233 160,315 42,716,397 Trade-Tech 41,998,674 5,322,713 11,316,858 8,093,783 16,076,364 7,008,754 89,817,146 1,369,524 33,455 481,000 924,987 2,039,191 967,220 399,662 96,032,185 Valley 50,119,012 6,351,846 15,730,513 9,787,711 5,392,408 7,167,131 94,548,621 2,074,642 0 650,000 452,249 2,151,896 1,217,064 424,452 101,518,924 West 30,433,646 3,857,016 7,357,617 6,027,952 8,553,637 4,709,649 60,939,517 1,625,495 0 724,000 1,232,107 1,545,145 788,613 301,463 67,156,340 PT Fac Health Ben 0 0 0 0 0 0 0 0 0 0 0 0 2,170,443 0 2,170,443 Emergency Cond Rev 0 0 0 0 22,448,046 1,845,229 24,293,275 0 0 0 0 0 0 0 24,293,275 Undistributed/Other DW 0 0 0 0 (2,314,595) (190,260) (2,504,855) 5 0 0 0 0 14,000,000 0 11,495,150 ESC/Info Tech/VDK 0 0 0 0 0 0 0 0 0 0 1,236,396 0 0 0 1,236,396 Total 389,936,838 49,418,747 107,595,425 75,846,914 119,060,888 60,980,794 802,839,606 13,368,234 33,455 8,279,000 8,210,934 17,892,200 24,828,742 3,494,286 878,946,457 Los Angeles Community College District 2023-2024 Final Budget 204 2023-2024 Education Protection Act (EPA) College Total Calculated Base % of Total Total EPA Funds City 51,721,242 11.8% $5,817,609 East 105,988,728 24.1% $11,921,620 Harbor 28,833,185 6.6% $3,243,159 Mission 33,684,140 7.7% $3,788,794 Pierce 58,831,100 13.4% $6,617,326 Southwest 22,214,283 5.1% $2,498,664 Trade-Tech 47,321,387 10.8% $5,322,713 Valley 56,470,858 12.9% $6,351,846 West 34,290,662 7.8% $3,857,016 Total 439,355,585 100.0% $49,418,747 Funds to be restricted in the Education Protection Act (EPA) and cannot be used for salaries and benefits of administrators or any administrative costs. Los Angeles Community College District 2023-2024 Final Budget 205 Minimum Base Funding Revised M&O Cost based on FY 2021-22 Obligation City East Harbor Mission Pierce S-west Trade Valley West Total Annual Salary President $292,886 $292,886 $292,886 $292,886 $292,886 $292,886 $292,886 $292,886 $292,886 $2,635,975 Academic Affairs VP $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $2,008,127 Student Services VP $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $2,008,127 Administrative Services VP $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $223,125 $2,008,127 Director of College Facilities $170,868 $170,868 $170,868 $170,868 $170,868 $170,868 $170,868 $170,868 $170,868 $1,537,810 Institutional Research Dean $177,635 $177,635 $177,635 $177,635 $177,635 $177,635 $177,635 $177,635 $177,635 $1,598,718 Total Funding for Presidents and VPs $1,310,765 $1,310,765 $1,310,765 $1,310,765 $1,310,765 $1,310,765 $1,310,765 $1,310,765 $1,310,765 $11,796,884 Estimated Benefits for Presidents/VPs/DCF/Dean $588,744 $588,744 $588,744 $588,744 $588,744 $588,744 $588,744 $588,744 $588,744 $5,298,694 Deans Current Number of Deans funded from 10100 5.2 13.0 4.8 0.5 8.5 4.0 7.6 6.3 4.8 54.7 FTE Faculty (Credit Instruction) 274 504 139 181 353 101 269 324 183 2,328 FTES (Students) 8,409 18,806 4,549 5,468 10,665 3,095 8,276 9,692 5,254 74,214 Number of Faculty per Dean 53 39 29 362 41 25 35 51 38 43 Number of FTES per Dean 1,617 1,447 948 10,936 1,255 774 1,089 1,538 1,095 1,357 Proposed Number of Deans (per Total # of FTES) 6 14 3 4 8 2 6 7 4 55 Proposed Number of Deans (per Total # of FTEF) 6 12 3 4 8 2 6 8 4 55 Proposed Number of Deans 8 12 4 4 8 4 8 8 4 60 Dean Salary $177,635 $177,635 $177,635 $177,635 $177,635 $177,635 $177,635 $177,635 $177,635 $177,635 Total Funding for Deans Position $1,421,082 $2,131,623 $710,541 $710,541 $1,421,082 $710,541 $1,421,082 $1,421,082 $710,541 10,658,117 Estimated Benefits for Deans $549,959 $824,938 $274,979 $274,979 $549,959 $274,979 $549,959 $549,959 $274,979 $4,124,691 M&O Costs by Square Footage Gross Square Footage 1,046,319 1,268,431 549,594 582,295 773,743 691,757 980,456 967,457 642,600 7,502,652 Average Cost per sq.ft. $11.71 $11.71 $11.71 $11.71 $11.71 $11.71 $11.71 $11.71 $11.71 $11.71 Total Funding for M&O Costs $12,256,819 $14,858,689 $6,438,069 $6,821,136 $9,063,802 $8,103,399 $11,485,285 $11,333,011 $7,527,563 $87,887,773 Total Proposed Minimum Base Funding $16,127,369 $19,714,760 $9,323,098 $9,706,165 $12,934,351 $10,988,428 $15,355,834 $15,203,561 $10,412,592 $119,766,158 Source: Salary schedule (top step) - for Presidents ($23,907) plus auto allowance ($500) totals to $24,407 per month; for Academic Affairs and Student Services VPs ($18,594); Administrative Services VP ($18,594); Director of College Facilities ($14,239); Dean ($14,803). Average Cost per square feet is based on the average cost for all colleges, and not by individual college. Benefits are estimated based on FY 2022-23 rates - 59.38% for classified (Administrative Services VP and Director of College Facilities); and 38.70% for certificated (Presidents, other VPs and Deans). Current Number of Deans is based on the result of a college survey conducted in July 2023. FTE Faculty (Credit Instruction) is based on the Report WSCH Trends and Staffing Patterns by College in the Fall 2022 (P) Data book as reported by the Office of Attendance Accounting. FTES (Students) is based on the 2022-23 Annual FTES report, including Credit, Non-Credit and Enhanced Non-Credit FTES, as reported by the Office of Attendance Accounting. Proposed Number of Deans is 4 for small colleges (FTES < 10,000 - H,M,S,W); 8 for medium (FTES < 20,000 - C,P,T,V); and 12 for large (FTES > 20,000 - E). Source: Data for M&O Costs and Gross Square Footage for FY 2021-22 is based on data from the Fusion Space Inventory Report. Los Angeles Community College District 2023-2024 Final Budget 206 Student Centered Funding Formula Calculated Revenue Location Base Supplemental Student Success Total Calculated SCFF Revenue City 51,721,242 15,099,827 8,960,229 75,781,298 East 105,988,728 23,691,345 17,207,021 146,887,094 Harbor 28,833,185 6,105,403 4,861,645 39,800,233 Mission 33,684,140 7,988,303 5,277,126 46,949,569 Pierce 58,831,100 15,954,858 12,576,638 87,362,596 Southwest 22,214,283 4,350,701 3,054,809 29,619,793 Trade-Tech 47,321,387 11,316,858 8,093,783 66,732,028 Valley 56,470,858 15,730,513 9,787,711 81,989,082 West 34,290,662 7,357,617 6,027,952 47,676,231 Adjustment for hold harmless 0 0 0 0 Total 439,355,585 107,595,425 75,846,914 622,797,924 Los Angeles Community College District 2023-2024 Final Budget 207 Base Allocation Revenue (FTES + Basic Allocation) Location Basic Allocation 3-Year Average Credit Special Admit Credit Incarcerated CDCP Noncredit Total Calculated Base % of Base Allocation City 6,942,161 33,783,585 3,957,382 - 6,831,631 206,483 51,721,242 11.8% East 9,917,373 80,056,095 4,980,555 2,715 10,418,903 613,087 105,988,728 24.1% Harbor 5,950,421 19,316,542 2,651,595 - 830,288 84,339 28,833,185 6.6% Mission 5,950,421 22,037,315 2,407,696 292,573 2,718,669 277,466 33,684,140 7.7% Pierce 6,942,161 45,198,567 3,680,696 - 434,442 2,575,234 58,831,100 13.4% Southwest 5,950,421 11,686,360 2,067,483 - 2,462,259 47,760 22,214,283 5.1% Trade-Tech 6,942,161 36,228,381 1,824,318 - 2,065,172 261,355 47,321,387 10.8% Valley 6,942,161 38,715,533 3,783,107 - 6,898,552 131,505 56,470,858 12.9% West 5,950,421 23,894,263 2,696,116 - 1,651,973 97,889 34,290,662 7.8% Total 61,487,701 310,916,641 28,048,948 295,288 34,311,889 4,295,118 439,355,585 100% Includes South Gate Center Los Angeles Community College District 2023-2024 Final Budget 208 Paid FTES Workload Measures Location 3-Year Average Credit Special Admit Credit Incarcerated CDCP Noncredit City 6,979 583 - 1,006 51 East 16,539 734 0 1,535 150 Harbor 3,991 391 - 122 21 Mission 4,553 355 43.10 401 68 Pierce 9,338 542 - 64 631 Southwest 2,414 305 - 363 12 Trade-Tech 7,484 269 - 304 64 Valley 7,998 557 - 1,016 32 West 4,936 397 - 243 24 Total 64,232 4,132 43.50 5,055 1,052 FTES Funding Rates $4,840.49 $6,787.96 $6,787.96 $6,787.96 $4,081.79 Multi District Basic Allocation Rates Size FTES Allocation Small <10,000 $5,950,421 Medium 10,000 - 19,999 $6,942,161 Large >=20,000 $7,933,899 State Approved Center Allocation Rates >=1,000 $1,983,474 Los Angeles Community College District 2023-2024 Final Budget 209 Base Funds Remaining Adjustment to FTES Base Description Amount Minimum Base 119,766,158 EPA 49,418,747 Base Distributed to Colleges 1 169,184,905 1 Distributed using different methodology Calculation of Base Funds Remaining Description Amount Total Base Allocation 439,355,585 Less: Base Revenue to Colleges 1 (169,184,905) FTES Base Funds Remaining 270,170,680 1 Distributed using different methodology Distribution of Base Funds Remaining Location % of Base Allocation Funds Remaining City 11.8% 31,804,678 East 24.1% 65,175,106 Harbor 6.6% 17,730,243 Mission 7.7% 20,713,216 Pierce 13.4% 36,176,707 Southwest 5.1% 13,660,115 Trade-Tech 10.8% 29,099,098 Valley 12.9% 34,725,336 West 7.8% 21,086,181 Total 100.0% 270,170,680 Los Angeles Community College District 2023-2024 Final Budget 210 2023-2024 FTES Workload Measures Location Credit w/o Special Admit Special Admit Incarcerated Total Credit CDCP Noncredit Total FTES City 6,769 583 - 7,352 1,006 51 8,409 East 16,386 734 0.40 17,120 1,535 150 18,806 Harbor 4,015 391 - 4,406 122 21 4,549 Mission 4,601 355 43.10 4,999 401 68 5,468 Pierce 9,428 542 - 9,971 64 631 10,665 Southwest 2,416 305 - 2,721 363 12 3,095 Trade-Tech 7,639 269 - 7,908 304 64 8,276 Valley 8,086 557 - 8,644 1,016 32 9,692 West 4,590 397 - 4,987 243 24 5,254 Total 63,931 4,132 44 68,107 5,055 1,052 74,214 projected Calculation of 3-Year Average Location Total Credit 2021-22 Total Credit 2022-23 Total Credit 2023-24 Special Admit & Incarcerated Credit 2021-22 Special Admit & Incarcerated Credit 2022-23 Special Admit & Incarcerated Credit 2023-24 Credit w/o Special Admit or Incarcerated 2021-22 Credit w/o Special Admit or Incarcerated 2022-23 Credit w/o Special Admit or Incarcerated 2023-24 3-year average City 7,939 7,352 7,352 538 583 583 7,401 6,769 6,769 6,979 East 17,553 17,120 17,120 709 734 734 16,844 16,386 16,386 16,539 Harbor 4,325 4,406 4,406 383 391 391 3,942 4,015 4,015 3,991 Mission 4,972 4,999 4,999 517 398 398 4,455 4,601 4,601 4,553 Pierce 9,652 9,971 9,971 496 542 542 9,156 9,428 9,428 9,338 Southwest 2,686 2,721 2,721 276 305 305 2,410 2,416 2,416 2,414 Trade-Tech 7,364 7,908 7,908 189 269 269 7,175 7,639 7,639 7,484 Valley 8,428 8,644 8,644 606 557 557 7,822 8,086 8,086 7,998 West 5,952 4,987 4,987 322 397 397 5,630 4,590 4,590 4,936 Total 68,870 68,107 68,107 4,035 4,176 4,176 64,835 63,931 63,931 64,232 Projected using 2022-23 Annual data. Los Angeles Community College District 2023-2024 Final Budget 211 Supplemental Allocation Revenue Location AB 540 Totals Pell Grant Totals CA Promise Grant Students Totals Subtotal % of Total Unallocated Adj Total Supplemental Rates $1,144.62 $1,144.62 $1,144.62 City 725,689 5,351,099 9,023,039 15,099,827 14% - 15,099,827 East 966,059 8,031,799 14,693,487 23,691,345 22% - 23,691,345 Harbor 225,490 2,162,187 3,717,726 6,105,403 6% - 6,105,403 Mission 407,485 2,589,130 4,991,688 7,988,303 7% - 7,988,303 Pierce 745,148 5,428,933 9,780,778 15,954,858 15% - 15,954,858 Southwest 136,210 1,471,981 2,742,510 4,350,701 4% - 4,350,701 Trade-Tech 684,483 3,639,892 6,992,484 11,316,858 11% - 11,316,858 Valley 858,465 5,240,070 9,631,977 15,730,513 15% - 15,730,513 West 289,589 2,335,025 4,733,004 7,357,617 7% - 7,357,617 Total District 5,038,617 36,250,115 66,306,692 107,595,425 - 107,595,425 Total State 5,038,617 36,250,115 66,306,692 107,595,425 2022-23 data and revenue. Los Angeles Community College District 2023-2024 Final Budget 212 Supplemental Workload Measures Location AB 540 Totals Pell Grant Totals Promise Grant Students Totals City 634 4,675 7,883 East 844 7,017 12,837 Harbor 197 1,889 3,248 Mission 356 2,262 4,361 Pierce 651 4,743 8,545 Southwest 119 1,286 2,396 Trade-Tech 598 3,180 6,109 Valley 750 4,578 8,415 West 253 2,040 4,135 Unallocated - 10 28 Total District 4,402 31,680 57,957 Total State 4,402 31,680 57,957 2021-22 data Los Angeles Community College District 2023-2024 Final Budget 213 Student Success Allocation – Total Revenue Location All Students Pell CA Promise Grant Total Success City 6,383,020 1,342,634 1,234,575 8,960,229 East 12,169,956 2,662,454 2,374,611 17,207,021 Harbor 3,516,887 707,281 637,477 4,861,645 Mission 3,718,244 801,640 757,242 5,277,126 Pierce 9,221,594 1,731,514 1,623,530 12,576,638 Southwest 2,110,763 510,862 433,184 3,054,809 Trade-Tech 5,774,338 1,197,974 1,121,471 8,093,783 Valley 6,893,050 1,515,008 1,379,653 9,787,711 West 4,424,907 801,598 801,447 6,027,952 Total 54,212,759 11,270,965 10,363,190 75,846,914 Los Angeles Community College District 2023-2024 Final Budget 214 Student Success Allocation – All Student Revenue Location Associate Degree for Transfer Associate Degree Baccalaureate Degree Credit Certificates Transfer-level Math and English Transfers to 4-yr 9 or more CTE Units Regional Living Wage Subtotal % of Total Revenue Adjustment Total rates $2,699.76 $2,024.82 $2,024.82 $1,349.88 $1,349.88 $1,012.41 $674.94 $674.94 City 1,351,680 1,004,986 - 802,279 394,615 621,282 1,198,693 1,009,485 6,383,020 12% - 6,383,020 East 3,806,662 1,721,097 - 411,263 556,601 1,238,852 2,419,435 2,016,046 12,169,956 22% - 12,169,956 Harbor 895,420 1,044,132 - 17,998 297,874 415,763 365,593 480,107 3,516,887 6% - 3,516,887 Mission 1,036,708 698,563 - 140,837 227,680 374,592 670,215 569,649 3,718,244 7% - 3,718,244 Pierce 2,700,660 1,557,087 - 83,693 1,049,307 1,413,662 1,293,635 1,123,550 9,221,594 17% - 9,221,594 Southwest 473,358 601,372 - 20,248 121,489 225,430 248,828 420,038 2,110,763 4% - 2,110,763 Trade-Tech 487,757 950,316 - 625,894 83,693 269,976 2,094,789 1,261,913 5,774,338 11% - 5,774,338 Valley 1,981,624 1,191,944 - 203,382 438,711 904,757 1,150,323 1,022,309 6,893,050 13% - 6,893,050 West 1,001,611 548,051 105,966 213,281 159,286 402,264 1,037,608 956,840 4,424,907 8% - 4,424,907 Total District 13,735,480 9,317,548 105,966 2,518,875 3,329,256 5,866,578 10,479,119 8,859,937 54,212,759 - 54,212,759 Total State - Proj 13,735,480 9,317,548 105,966 2,518,875 3,329,256 5,866,578 10,479,119 8,859,937 54,212,759 Student Success Data – 3-Year Average - All Student Data Location Associate Degree for Transfer Associate Degree Baccalaureate Degree Credit Certificates Transfer level Math and English Transfers to 4-yr 9 or more CTE Units Regional Living Wage City 501 496 - 594 292 614 1,776 1,496 East 1,410 850 - 305 412 1,224 3,585 2,987 Harbor 332 516 - 13 221 411 542 711 Mission 384 345 - 104 169 370 993 844 Pierce 1,000 769 - 62 777 1,396 1,917 1,665 Southwest 175 297 - 15 90 223 369 622 Trade-Tech 181 469 - 464 62 267 3,104 1,870 Valley 734 589 - 151 325 894 1,704 1,515 West 371 271 52 158 118 397 1,537 1,418 Unallocated 6 4 - 25 3 33 8 33 Total 5,094 4,606 52 1,891 2,469 5,827 15,534 13,160 Los Angeles Community College District 2023-2024 Final Budget 215 Associate Degree for Transfer (ADT) All Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 562 470 470 501 East 1,594 1,318 1,318 1,410 Harbor 345 325 325 332 Mission 428 362 362 384 Pierce 1,075 963 963 1,000 Southwest 214 156 156 175 Trade-Tech 190 176 176 181 Valley 772 715 715 734 West 403 355 355 371 Unallocated 9 5 5 6 Total 5,592 4,845 4,845 5,094 Associate Degrees (AA/AS) All Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 481 504 504 496 East 890 830 830 850 Harbor 603 472 472 516 Mission 333 351 351 345 Pierce 821 743 743 769 Southwest 307 292 292 297 Trade-Tech 438 485 485 469 Valley 624 571 571 589 West 280 266 266 271 Unallocated 9 2 2 4 Total 4,786 4,516 4,516 4,606 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 216 Baccalaureate Degrees All Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 0 0 0 0 East 0 0 0 0 Harbor 0 0 0 0 Mission 0 0 0 0 Pierce 0 0 0 0 Southwest 0 0 0 0 Trade-Tech 0 0 0 0 Valley 0 0 0 0 West 43 57 57 52 Unallocated 0 0 0 0 Total 43 57 57 52 Credit Certificates All Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 619 582 582 594 East 276 319 319 305 Harbor 18 11 11 13 Mission 67 123 123 104 Pierce 90 48 48 62 Southwest 15 15 15 15 Trade-Tech 393 499 499 464 Valley 144 154 154 151 West 226 124 124 158 Unallocated 4 36 36 25 Total 1,852 1,911 1,911 1,891 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 217 Transfer Level Math & English All Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 319 279 279 292 East 425 406 406 412 Harbor 240 211 211 221 Mission 146 180 180 169 Pierce 858 737 737 777 Southwest 80 95 95 90 Trade-Tech 54 66 66 62 Valley 223 376 376 325 West 118 118 118 118 Unallocated 2 3 3 3 Total 2,465 2,471 2,471 2,469 Transfer to a 4-Year University All Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 565 638 638 614 East 1,089 1,291 1,291 1,224 Harbor 364 434 434 411 Mission 328 391 391 370 Pierce 1,297 1,446 1,446 1,396 Southwest 204 232 232 223 Trade-Tech 276 262 262 267 Valley 887 897 897 894 West 364 414 414 397 Unallocated 20 39 39 33 Total 5,394 6,044 6,044 5,827 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 218 Nine or More CTE Units All Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 2,052 1,638 1,638 1,776 East 3,866 3,444 3,444 3,585 Harbor 623 501 501 542 Mission 1,055 962 962 993 Pierce 2,178 1,786 1,786 1,917 Southwest 424 341 341 369 Trade-Tech 3,173 3,069 3,069 3,104 Valley 1,939 1,587 1,587 1,704 West 1,774 1,419 1,419 1,537 Unallocated 3 11 11 8 Total 17,087 14,758 14,758 15,534 Regional Living Wage All Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 1,239 1,624 1,624 1,496 East 3,163 2,899 2,899 2,987 Harbor 606 764 764 711 Mission 796 868 868 844 Pierce 1,648 1,673 1,673 1,665 Southwest 639 614 614 622 Trade-Tech 1,985 1,812 1,812 1,870 Valley 1,382 1,581 1,581 1,515 West 1,279 1,487 1,487 1,418 Unallocated 27 36 36 33 Total 12,764 13,358 13,358 13,160 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 219 Student Success Allocation – Pell Student Revenue Location Associate Degree for Transfer Associate Degree Baccalaureate Degree Credit Certificates Transfer level Math and English Transfers to 4-yr 9 or more CTE Units Regional Living Wage Subtotal % of Total Revenue Adjustment Total rates $1,021.46 $766.10 $766.10 $510.73 $510.73 $383.05 $255.37 $255.37 City 348,658 230,341 - 169,903 85,122 141,473 251,965 115,172 1,342,634 12% - 1,342,634 East 1,023,843 440,508 - 48,519 111,850 315,378 454,473 267,883 2,662,454 24% - 2,662,454 Harbor 198,504 219,615 - 3,916 54,308 88,357 73,291 69,290 707,281 6% - 707,281 Mission 255,025 162,413 - 33,538 37,794 81,334 145,561 85,975 801,640 7% - 801,640 Pierce 577,465 323,805 - 16,173 160,710 268,518 234,515 150,328 1,731,514 15% - 1,731,514 Southwest 138,919 158,327 - 4,086 20,429 55,287 60,863 72,951 510,862 5% - 510,862 Trade-Tech 141,302 233,405 - 142,664 17,195 68,566 426,213 168,629 1,197,974 11% - 1,197,974 Valley 506,985 291,118 - 36,262 78,142 201,995 257,924 142,582 1,515,008 13% - 1,515,008 West 235,957 120,278 23,494 37,283 23,494 78,908 170,417 111,767 801,598 7% - 801,598 Total District 3,426,658 2,179,810 23,494 492,344 589,044 1,299,816 2,075,222 1,184,577 11,270,965 - 11,270,965 Total State - Proj 3,426,658 2,179,810 23,494 492,344 589,044 1,299,816 2,075,222 1,184,577 11,270,965 Student Success Data – 3-Year Average - Pell Student Data Location Associate Degree for Transfer Associate Degree Baccalaureate Degree Credit Certificates Transfer level Math and English Transfers to 4-yr 9 or more CTE Units Regional Living Wage City 341 301 - 333 167 369 987 451 East 1,002 575 - 95 219 823 1,780 1,049 Harbor 194 287 - 8 106 231 287 271 Mission 250 212 - 66 74 212 570 337 Pierce 565 423 - 32 315 701 918 589 Southwest 136 207 - 8 40 144 238 286 Trade-Tech 138 305 - 279 34 179 1,669 660 Valley 496 380 - 71 153 527 1,010 558 West 231 157 31 73 46 206 667 438 Unallocated 3 2 - 16 - 26 3 16 Total 3,358 2,847 31 980 1,153 3,419 8,129 4,655 Los Angeles Community College District 2023-2024 Final Budget 220 Associate Degree for Transfer (ADT) Pell Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 400 312 312 341 East 1,133 937 937 1,002 Harbor 207 188 188 194 Mission 281 234 234 250 Pierce 600 548 548 565 Southwest 170 119 119 136 Trade-Tech 145 135 135 138 Valley 515 487 487 496 West 249 222 222 231 Unallocated 4 3 3 3 Total 3,704 3,185 3,185 3,358 Associate Degrees (AA/AS) Pell Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 308 297 297 301 East 587 569 569 575 Harbor 350 255 255 287 Mission 212 212 212 212 Pierce 486 391 391 423 Southwest 216 202 202 207 Trade-Tech 288 313 313 305 Valley 418 361 361 380 West 161 155 155 157 Unallocated 6 - - 2 Total 3,032 2,755 2,755 2,847 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 221 Baccalaureate Degrees Pell Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City - - - East - - - Harbor - - - Mission - - - Pierce - - - Southwest - - - Trade-Tech - - - Valley - - - West 26 33 33 31 Unallocated - - - Total 26 33 33 31 Credit Certificates Pell Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 346 326 326 333 East 93 96 96 95 Harbor 11 6 6 8 Mission 43 77 77 66 Pierce 45 25 25 32 Southwest 12 6 6 8 Trade-Tech 240 299 299 279 Valley 83 65 65 71 West 91 64 64 73 Unallocated 3 23 23 16 Total 967 987 987 980 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 222 Transfer Level Math & English Pell Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 170 165 165 167 East 243 207 207 219 Harbor 95 112 112 106 Mission 66 78 78 74 Pierce 326 309 309 315 Southwest 40 40 40 40 Trade-Tech 27 37 37 34 Valley 101 179 179 153 West 50 44 44 46 Unallocated - - - - Total 1,118 1,171 1,171 1,153 Transfer to a 4-Year University Pell Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 354 377 377 369 East 754 858 858 823 Harbor 214 239 239 231 Mission 199 219 219 212 Pierce 683 710 710 701 Southwest 131 151 151 144 Trade-Tech 191 173 173 179 Valley 554 514 514 527 West 220 199 199 206 Unallocated 17 30 30 26 Total 3,317 3,470 3,470 3,419 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 223 Nine Or More CTE Units Pell Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 1,086 937 937 987 East 1,931 1,704 1,704 1,780 Harbor 309 276 276 287 Mission 596 557 557 570 Pierce 1,011 872 872 918 Southwest 275 220 220 238 Trade-Tech 1,657 1,675 1,675 1,669 Valley 1,112 959 959 1,010 West 734 634 634 667 Unallocated 1 4 4 3 Total 8,712 7,838 7,838 8,129 Regional Living Wage Pell Student Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 337 508 508 451 East 939 1,104 1,104 1,049 Harbor 210 302 302 271 Mission 288 361 361 337 Pierce 514 626 626 589 Southwest 273 292 292 286 Trade-Tech 645 668 668 660 Valley 457 609 609 558 West 367 473 473 438 Unallocated 11 19 19 16 Total 4,041 4,962 4,962 4,655 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 224 Student Success Allocation – CA Promise Grant Revenue Location Associate Degree for Transfer Associate Degree Baccalaureate Degree Credit Certificates Transfer level Math and English Transfers to 4-yr 9 or more CTE Units Regional Living Wage Subtotal % of Total Revenue Adjustment Total rates $680.98 $510.73 $510.73 $340.49 $340.49 $255.37 $170.24 $170.24 City 293,502 207,867 - 164,116 74,340 121,811 231,697 141,242 1,234,575 12% - 1,234,575 East 860,078 378,791 - 45,512 103,282 272,054 408,122 306,772 2,374,611 23% - 2,374,611 Harbor 179,552 189,651 - 3,518 46,080 76,526 67,188 74,962 637,477 6% - 637,477 Mission 228,809 149,644 - 30,644 40,291 73,547 135,454 98,853 757,242 7% - 757,242 Pierce 519,815 291,627 - 14,641 153,561 247,794 223,355 172,737 1,623,530 16% - 1,623,530 Southwest 110,319 124,618 - 3,859 21,791 45,030 53,115 74,452 433,184 4% - 433,184 Trade-Tech 119,172 209,570 - 132,110 15,095 56,607 407,555 181,362 1,121,471 11% - 1,121,471 Valley 435,146 254,003 - 38,475 79,788 176,546 234,364 161,331 1,379,653 13% - 1,379,653 West 215,190 113,042 21,621 40,972 25,083 78,313 173,304 133,922 801,447 8% - 801,447 Total District 2,961,583 1,918,813 21,621 473,847 559,311 1,148,228 1,934,154 1,345,633 10,363,190 - 10,363,190 Total State - Proj 2,961,583 1,918,813 21,621 473,847 559,311 1,148,228 1,934,154 1,345,633 10,363,190 Student Success Data – 3-Year Average – Promise Grant Recipient Data Location Associate Degree for Transfer Associate Degree Baccalaureate Degree Credit Certificates Transfer level Math and English Transfers to 4-yr 9 or more CTE Units Regional Living Wage City 431 407 - 482 218 477 1,361 830 East 1,263 742 - 134 303 1,065 2,397 1,802 Harbor 264 371 - 10 135 300 395 440 Mission 336 293 - 90 118 288 796 581 Pierce 763 571 - 43 451 970 1,312 1,015 Southwest 162 244 - 11 64 176 312 437 Trade-Tech 175 410 - 388 44 222 2,394 1,065 Valley 639 497 - 113 234 691 1,377 948 West 316 221 42 120 74 307 1,018 787 Unallocated 5 3 - 23 1 28 6 27 Total 4,354 3,760 42 1,415 1,643 4,525 11,368 7,931 Los Angeles Community College District 2023-2024 Final Budget 225 Associate Degree for Transfer (ADT) Promise Grant Recipient Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 499 397 397 431 East 1,429 1,180 1,180 1,263 Harbor 275 258 258 264 Mission 382 313 313 336 Pierce 834 728 728 763 Southwest 198 144 144 162 Trade-Tech 185 170 170 175 Valley 661 628 628 639 West 340 304 304 316 Unallocated 7 4 4 5 Total 4,810 4,126 4,126 4,354 Associate Degrees (AA/AS) Promise Grant Recipient Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 409 406 406 407 East 765 730 730 742 Harbor 456 329 329 371 Mission 273 303 303 293 Pierce 639 537 537 571 Southwest 262 235 235 244 Trade-Tech 385 423 423 410 Valley 526 483 483 497 West 232 216 216 221 Unallocated 8 - - 3 Total 3,955 3,662 3,662 3,760 Projected using 2021-22 annual-data. Los Angeles Community College District 2023-2024 Final Budget 226 Baccalaureate Degrees Promise Grant Recipient Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 0 0 0 0 East 0 0 0 0 Harbor 0 0 0 0 Mission 0 0 0 0 Pierce 0 0 0 0 Southwest 0 0 0 0 Trade-Tech 0 0 0 0 Valley 0 0 0 0 West 31 48 48 42 Unallocated 0 0 0 0 Total 31 48 48 42 Credit Certificates Promise Grant Recipient Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 504 471 471 482 East 135 133 133 134 Harbor 13 9 9 10 Mission 58 106 106 90 Pierce 69 30 30 43 Southwest 14 10 10 11 Trade-Tech 328 418 418 388 Valley 121 109 109 113 West 151 105 105 120 Unallocated 3 33 33 23 Total 1,396 1,424 1,424 1,415 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 227 Transfer Level Math & English Promise Grant Recipient Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 225 215 215 218 East 326 292 292 303 Harbor 138 134 134 135 Mission 105 125 125 118 Pierce 487 433 433 451 Southwest 62 65 65 64 Trade-Tech 41 46 46 44 Valley 161 271 271 234 West 73 74 74 74 Unallocated - 1 1 1 Total 1,618 1,656 1,656 1,643 Transfer to a 4-Year University Promise Grant Recipient Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 453 489 489 477 East 970 1,113 1,113 1,065 Harbor 269 315 315 300 Mission 264 300 300 288 Pierce 919 996 996 970 Southwest 165 182 182 176 Trade-Tech 233 216 216 222 Valley 702 686 686 691 West 280 320 320 307 Unallocated 19 33 33 28 Total 4,274 4,650 4,650 4,525 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 228 Nine or More CTE Units Promise Grant Recipient Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 1,535 1,274 1,274 1,361 East 2,652 2,270 2,270 2,397 Harbor 442 371 371 395 Mission 837 775 775 796 Pierce 1,496 1,220 1,220 1,312 Southwest 362 287 287 312 Trade-Tech 2,404 2,389 2,389 2,394 Valley 1,550 1,290 1,290 1,377 West 1,148 953 953 1,018 Unallocated 1 9 9 6 Total 12,427 10,838 10,838 11,368 Regional Living Wage Promise Grant Recipient Data Location 2020-21 2021-22 2022-23 3-Yr Avg City 643 923 923 830 East 1,662 1,872 1,872 1,802 Harbor 349 486 486 440 Mission 514 614 614 581 Pierce 934 1,055 1,055 1,015 Southwest 436 438 438 437 Trade-Tech 1,070 1,063 1,063 1,065 Valley 817 1,013 1,013 948 West 704 828 828 787 Unallocated 20 30 30 27 Total 7,149 8,322 8,322 7,931 Projected using 2021-22 annual data. Los Angeles Community College District 2023-2024 Final Budget 229 College Hold Harmless Calculation Location 2022-23 FY22 TCR +FY23 COLA 2023-24 Min Base 2023-24 Base Funds Remaining 2023-24 EPA 2023-24 Total Allocated Base 2023-24 Supplemental 2023-24 Student Success 2023-24 Total TCR 2023-24 Hold Harmless Amount 2023-24 COLA 2023-24 FY23 TCR + FY24 COLA City 84,626,750 16,127,369 31,804,678 5,817,609 53,749,656 15,099,827 8,960,229 77,809,712 6,817,038 6,956,319 91,583,069 East 166,252,426 19,714,760 65,175,106 11,921,620 96,811,486 23,691,345 17,207,021 137,709,852 28,542,574 13,665,949 179,918,375 Harbor 48,117,081 9,323,098 17,730,243 3,243,159 30,296,500 6,105,403 4,861,645 41,263,548 6,853,533 3,955,224 52,072,305 Mission 50,381,759 9,706,165 20,713,216 3,788,794 34,208,175 7,988,303 5,277,126 47,473,604 2,908,155 4,141,381 54,523,140 Pierce 101,377,928 12,934,351 36,176,707 6,617,326 55,728,384 15,954,858 12,576,638 84,259,880 17,118,048 8,333,266 109,711,194 Southwest 41,218,397 10,988,428 13,660,115 2,498,664 27,147,207 4,350,701 3,054,809 34,552,717 6,665,680 3,388,152 44,606,549 Trade-Tech 85,264,650 15,355,834 29,099,098 5,322,713 49,777,645 11,316,858 8,093,783 69,188,286 16,076,364 7,008,754 92,273,404 Valley 87,191,375 15,203,561 34,725,336 6,351,846 56,280,743 15,730,513 9,787,711 81,798,967 5,392,408 7,167,131 94,358,506 West 57,294,995 10,412,592 21,086,181 3,857,016 35,355,789 7,357,617 6,027,952 48,741,358 8,553,637 4,709,649 62,004,644 Adjustment (2,314,595) - - (2,314,595) (190,260) (2,504,855) Emergency Cond Rev 22,448,046 - - 22,448,046 1,845,229 24,293,275 Total 741,858,812 119,766,158 270,170,680 49,418,747 439,355,585 107,595,425 75,846,914 622,797,924 119,060,888 60,980,794 802,839,606 Includes growth paid to West & South Gate Center paid to East Includes revenues earned through Emergency Conditionals Allowance (ECA) adjustment Los Angeles Community College District 2023-2024 Final Budget 230 Assessment Calculation Total Assessment 198,557,326 FY24 Assessment Methodology Location Total Allocated Base Hold Harmless Total Assessment Base Base % Assessment City 53,749,656 6,817,038 60,566,694 11.3% 22,341,334 East 96,811,486 28,542,574 125,354,060 23.3% 46,239,554 Harbor 30,296,500 6,853,533 37,150,033 6.9% 13,703,593 Mission 34,208,175 2,908,155 37,116,330 6.9% 13,691,160 Pierce 55,728,384 17,118,048 72,846,432 13.5% 26,870,981 Southwest 27,147,207 6,665,680 33,812,887 6.3% 12,472,614 Trade-Tech 49,777,645 16,076,364 65,854,009 12.2% 24,291,674 Valley 56,280,743 5,392,408 61,673,151 11.5% 22,749,475 West 35,355,789 8,553,637 43,909,426 8.2% 16,196,941 Total 439,355,585 98,927,437 538,283,022 198,557,326 Prior Assessment Methodology Location Total Allocated Base Hold Harmless Total Assessment Base Base % Assessment City 53,749,656 - 53,749,656 12.2% 24,291,004 East 96,811,486 - 96,811,486 22.0% 43,751,873 Harbor 30,296,500 - 30,296,500 6.9% 13,691,853 Mission 34,208,175 - 34,208,175 7.8% 15,459,650 Pierce 55,728,384 - 55,728,384 12.7% 25,185,247 Southwest 27,147,207 - 27,147,207 6.2% 12,268,597 Trade-Tech 49,777,645 - 49,777,645 11.3% 22,495,938 Valley 56,280,743 - 56,280,743 12.8% 25,434,874 West 35,355,789 - 35,355,789 8.0% 15,978,290 Total 439,355,585 - 439,355,585 198,557,326 Change due to the FY24 Budget Allocation Model Assessment Base to be implemented over 3 years. Location Total Change FY24 Adjustment City 1,949,670 (1,299,780) East (2,487,681) 1,658,454 Harbor (11,740) 7,827 Mission 1,768,490 (1,178,993) Pierce (1,685,734) 1,123,823 Southwest (204,017) 136,011 Trade-Tech (1,795,736) 1,197,157 Valley 2,685,399 (1,790,266) West (218,651) 145,767 Total - - Los Angeles Community College District 2023-2024 Final Budget 231 Dedicated Revenue Projections/Distribution Income Type City East Harbor Mission Pierce Southwest Trade Valley West ESC Total Salvage Sales 200 8,000 5,000 45,000 0 0 2,750 1,000 4,500 66,450 Admin Allowance 50,976 78,132 28,618 32,826 82,792 13,921 34,727 55,699 34,307 411,998 Class Audit Fees 4,000 12,000 500 1,500 6,000 0 2,200 3,500 3,500 33,200 SEVIS Fees 12,000 15,000 500 750 7,000 250 2,750 1,000 2,300 41,550 Library Fines 10 0 0 0 0 0 0 0 0 10 Forgn St Appl Fee 7,000 4,500 1,000 450 6,000 0 550 50 2,500 22,050 Transcripts 25,000 20,000 20,000 28,000 60,000 1,200 22,000 50,000 35,000 261,200 Emerg Transcr Fees 0 0 0 0 0 0 2,200 0 0 2,200 Facility Rental 200,000 0 200,000 90,000 1,000,000 600,000 575,000 100,000 950,000 3,715,000 Civic Center Rental 0 300,000 0 0 0 0 0 150,000 0 450,000 Baccalaureate fees 0 0 0 0 0 0 0 0 200,000 200,000 Gym Membership Fees 0 0 0 90,000 0 0 0 0 0 90,000 Program Development 1,000 0 0 0 0 0 0 1,000 0 2,000 Traffic Citations 500 0 0 200 1,000 0 5,500 0 0 7,200 Donations 0 0 0 0 0 0 0 10,000 0 10,000 Dup Reg Receipt 0 0 0 900 1,000 0 0 0 0 1,900 Dup Diploma/Certif 150 0 0 0 0 0 110 0 0 260 Verification Fees 1,500 650 3,000 800 0 0 550 0 0 6,500 Copy Machine 0 0 0 2,200 0 0 1,100 0 0 3,300 Returned Checks 20 0 0 150 0 0 0 0 0 170 Other: Income 0 0 0 80,000 0 0 0 60,000 0 140,000 Other: Local 0 7,000 0 0 0 0 550 20,000 0 27,550 Subtotal Non-Specific 302,356 445,282 258,618 372,776 1,163,792 615,371 649,987 452,249 1,232,107 0 5,492,538 Farm Sales 5,000 5,000 Swap Meet 800,000 800,000 Golf Driving Range 110,000 110,000 Contract Educ 35,000 0 236,000 18,000 0 0 275,000 0 0 564,000 Journalism 2,000 0 0 0 1,000 0 0 0 0 3,000 Van de Kamp 1,236,396 1,236,396 Subtotal Specific 37,000 0 1,146,000 18,000 6,000 0 275,000 0 0 1,236,396 2,718,396 Location Total 339,356 445,282 1,404,618 390,776 1,169,792 615,371 924,987 452,249 1,232,107 1,236,396 8,210,934 Dedicated revenues are those arising from locally managed activities, which can be associated with individual locations. Colleges are now responsible for their own projections of dedicated revenues. Administrative Allowance. (2% of enrollment revenue) provided by Budget & Management Analysis. Los Angeles Community College District 2023-2024 Final Budget 232 Districtwide Services Appropriations Program C E H M P S T V W D Total A. Operating Budgets Academic Senate 0 0 0 0 0 0 0 0 0 1,230,425 1,230,425 Accreditation 0 0 0 0 0 0 0 0 0 25,000 25,000 Audit Expense 0 0 0 0 0 0 0 0 0 700,000 700,000 Benefits-Retiree 0 0 0 0 0 0 0 0 0 30,680,000 30,680,000 Central Financial Aid Unit (CFAU) 0 0 0 0 0 0 0 0 0 1,908,034 1,908,034 Dolores Huerta Center 0 0 0 0 0 0 0 0 0 428,582 428,582 DW Mandatory Memberships 0 0 0 0 0 0 0 0 0 600,000 600,000 DW Marketing (Public Relations) 0 0 0 0 0 0 0 0 0 2,000,000 2,000,000 Employee Assistance Program 0 0 0 0 0 0 0 0 0 220,000 220,000 Environmental Health and Safety 0 0 0 0 0 0 0 0 0 645,000 645,000 Gold Creek 0 0 0 0 0 0 0 192,806 0 0 192,806 HR-Training & Development 0 0 0 0 0 0 0 0 0 285,000 285,000 Metro Records 0 0 0 108,379 0 0 0 0 0 0 108,379 Special Projects 0 0 0 0 0 0 0 0 0 1,028,296 1,028,296 Total Operating Budgets 40,051,522 B. Operating Budget w/ Variable Expenses Collective Bargaining 0 0 0 0 0 0 0 0 0 837,000 837,000 Insurance 0 0 0 0 0 0 0 0 0 11,223,872 11,223,872 Legal Expense 0 0 0 0 0 0 0 0 0 5,085,000 5,085,000 Reserve for Insur/Legal/WC 0 0 0 0 0 0 0 0 0 3,017,911 3,017,911 Staff Training-Legal 0 0 0 0 0 0 0 0 0 165,000 165,000 Worker's Compensation 0 0 0 0 0 0 0 0 0 5,036,809 5,036,809 Total Op Budgets w/ Variable Expenses 25,365,592 C. Other Districtwide Accounts Board Election Expense 0 0 0 0 0 0 0 0 0 4,500,000 4,500,000 District Safety/Operations 0 0 0 0 0 0 0 0 0 1,376,870 1,376,870 District Safety/Sheriff 0 0 0 0 0 0 0 0 0 26,038,988 26,038,988 Districtwide Benefits 0 0 0 0 0 0 0 0 0 150,000 150,000 Financial Services 0 0 0 0 0 0 0 0 0 90,000 90,000 Health Benefits Administration 0 0 0 0 0 0 0 0 0 475,000 475,000 LA College Promise 0 0 0 0 0 0 0 0 0 50,000 50,000 Project Match 0 0 0 0 0 0 0 0 0 117,000 117,000 Public Policy (State & Federal Advocates) 0 0 0 0 0 0 0 0 0 545,700 545,700 Staff Development 0 0 0 0 0 0 0 0 0 35,000 35,000 SW WEC Settlement 0 0 0 0 0 0 0 0 0 323,877 323,877 Tuition Reimbursement 0 0 0 0 0 0 0 0 0 593,000 593,000 Vacation Balance 0 0 0 0 0 0 0 0 0 1,000,000 1,000,000 Wellness Program 0 0 0 0 0 0 0 0 0 75,000 75,000 Total Other Districtwide Accounts 35,370,435 D. Districtwide Information Technology IT- Academic & Student Applications 0 0 0 0 0 0 0 0 0 4,228,675 4,228,675 IT- Cyber Security 0 0 0 0 0 0 0 0 0 250,000 250,000 IT- Dwide College Technology Svcs 0 0 0 0 0 0 0 0 0 3,816,079 3,816,079 IT- ERP/SAP 0 0 0 0 0 0 0 0 0 2,051,893 2,051,893 IT- Information Security 0 0 0 0 0 0 0 0 0 740,500 740,500 IT- Network 0 0 0 0 0 0 0 0 0 3,191,522 3,191,522 IT- Region 1 College Technology Svcs 0 0 0 0 0 0 0 0 0 4,670,834 4,670,834 IT- Region 2 College Technology Svcs 0 0 0 0 0 0 0 0 0 3,702,195 3,702,195 IT- Region 3 College Technology Svcs 0 0 0 0 0 0 0 0 0 3,654,452 3,654,452 IT- Service Center 0 0 0 0 0 0 0 0 0 776,000 776,000 IT- Systems Engineering 0 0 0 0 0 0 0 0 0 1,697,694 1,697,694 IT- Student Systems & Web Services 0 0 0 0 0 0 0 0 0 2,366,309 2,366,309 Total Op Budgets w/ Variable Expenses 31,146,153 Total Districtwide Services 0 0 0 108,379 0 0 0 192,806 0 131,632,517 131,933,702 Indicates items funded separately from college/office allocations but not budgeted in Business Area D000. Other Designated Program C E H M P S T V W ESC DW Total Campus Safety Blue Ribbon 0 0 0 0 0 0 0 0 0 0 0 1,769,850 DAS Sustainability 0 0 0 0 0 0 0 0 0 0 0 3,823 Dean's Academy 0 0 0 0 0 0 0 0 0 0 0 16,330 President's Academy 0 0 0 0 0 0 0 0 0 0 0 22,757 Total Other Districtwide 0 0 0 0 0 0 0 0 0 0 0 1,812,760
1674	https://artofproblemsolving.com/wiki/index.php/Proofs_of_AM-GM?srsltid=AfmBOooyT5LrOihuETAvn0NRkykXQRAwoMFC78hJFa34z-UfnNSE1S3c	Art of Problem Solving Proofs of AM-GM - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Proofs of AM-GM Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Proofs of AM-GM This page lists some proofs of the weighted AM-GM Inequality. The inequality's statement is as follows: for all nonnegative reals and nonnegative reals such that , then with equality if and only if for all such that . We first note that we may disregard any for which , as they contribute to neither side of the desired inequality. We also note that if and , for some , then the right-hand side of the inequality is zero and the left hand of the inequality is greater or equal to zero, with equality if and only if whenever . Thus we may henceforth assume that all and are strictly positive. Contents [hide] 1 Proofs of Unweighted AM-GM 1.1 Proof by Cauchy Induction 1.2 Proof by Rearrangement 1.3 Proof by Calculus 1.4 Proof by An Easier Induction 2 Proof of Weighted AM-GM 2.1 Proof by Convexity 2.2 Alternate Proof by Convexity Proofs of Unweighted AM-GM These proofs use the assumption that , for all integers . Proof by Cauchy Induction We use Cauchy Induction, a variant of induction in which one proves a result for , all powers of , and then that implies . Base Case: The smallest nontrivial case of AM-GM is in two variables. By the properties of perfect squares (or by the Trivial Inequality), with equality if and only if , or . Then because and are nonnegative, we can perform the following manipulations: with equality if and only if , just as before. This completes the proof of the base case. Powers of Two: We use induction. Suppose that AM-GM is true for variables; we will then prove that the inequality is true for . Let be any list of nonnegative reals. Then, because the two lists and , each have variables, Adding these two inequalities together and dividing by yields From here, we perform AM-GM in two variables on and to get Combining this inequality with the previous one yields AM-GM in variables, with one exception — equality. For equality, note that every AM-GM application mentioned must have equality as well; thus, inequality holds if and only if all the numbers in are the same, all the numbers in are the same, and . From here, it is trivial to show that this implies , which is the equality condition for AM-GM in variables. This completes the induction and proves that the inequality holds for all powers of . Backward Step: Assume that AM-GM holds for variables. We will then use a substitution to derive AM-GM for variables. Letting , we have that Because we assumed AM-GM in variables, equality holds if and only if . However, note that the last equality is implied if all the numbers of are the same; thus, equality holds if and only if . We first simplify the lefthand side. Multiplying both sides of the fraction by and combining like terms, we get that Plugging this into the earlier inequality yields Raising both sides to the th power yields From here, we divide by and take the root to get that This is the inequality in variables. Note that every step taken also preserves equality, which completes the backward step. Then by Cauchy Induction, the AM-GM inequality holds. Proof by Rearrangement Define the sequence as , for all integers . Evidently, these sequences are similarly sorted. Then by the Rearrangement Inequality, where we take our indices modulo , with equality exactly when all the , and therefore all the , are equal. Dividing both sides by gives the desired inequality. Proof by Calculus We will start the proof by considering the function . We will now find the maximum of this function. We can do this simply using calculus. We need to find the critical points of , we can do that by finding and setting it equal to . Using the linearity of the derivative . We need Note that this is the only critical point of . We can confirm it is the maximum by finding its second derivative and making sure it is negative. letting we get . Since the second derivative , is a maximum. . Now that we have that is a maximum of , we can safely say that or in other words . We will now define a few more things and do some manipulations with them. Letting , notice that . This fact will come into play later. Now we can do the following. Letting and plugging this into , we get Adding all these results together we get Now exponentiating both sides we get This proves the AM-GM inequality. Proof by An Easier Induction We can always rearrange the order of without changing its geometric mean and arithmetic mean. Hence, WLOG, let . This follows by For the case of , , so its true. For , since , Hence GM-AM Inequality is true for . If we assume it is true for , then for : By our inductive hypothesis, Now we need to prove a lemma to finish our induction process. Recall that , this implies that Hence, Thus, substitute this part in, By mathematical induction, the statement is true for as it's true for , and since it's true for and , we can conclude that the statement is true for all . Proof of Weighted AM-GM Proof by Convexity We note that the function is strictly concave. Then by Jensen's Inequality, with equality if and only if all the are equal. Since is a strictly increasing function, it then follows that with equality if and only if all the are equal, as desired. Alternate Proof by Convexity This proof is due to G. Pólya. Note that the function is strictly convex. Let be the line tangent to at ; then . Since is also a continuous, differentiable function, it follows that for all , with equality exactly when , i.e., with equality exactly when . Now, set for all integers . Our earlier bound tells us that so Multiplying such inequalities gives us Evaluating the left hand side: for Evaluating the right hand side: Substituting the results for the left and right sides: as desired. 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1675	https://www.youtube.com/watch?v=Ohv9VCdTm4s	Multiplying Integers: Multiplying a Negative by a Negative \| Negative x Negative \| Math with Mr. J Math with Mr. J 1690000 subscribers 2478 likes Description 247612 views Posted: 18 Jul 2022 Welcome to Multiplying a Negative by a Negative with Mr. J! Need help with how to multiply negative integers? You're in the right place! Whether you're just starting out, or need a quick refresher, this is the video for you if you're looking for help with multiplying integers. Mr. J will go through examples of multiplying two negative integers and explain the steps of how to multiply integers. More Videos and Examples: ✅ Adding Integers -A Quick Review of Adding Integers = -Adding a Positive and Negative Integer \| Positive + Negative = -Adding a Negative and Positive Integer \| Negative + Positive = -Adding Two Negative Integers \| Negative + Negative = ✅ Subtracting Integers -A Quick Review of Subtracting Integers = -Subtracting a Positive Integer from a Negative Integer \| Negative - Positive = -Subtracting a Negative Integer from a Positive Integer \| Positive - Negative = -Subtracting a Negative Integer from a Negative Integer \| Negative - Negative = ✅ Multiplying Integers -A Quick Review of Multiplying Integers = -Multiplying a Positive by a Negative \| Positive x Negative = -Multiplying a Negative by a Positive \| Negative x Positive = -Multiplying a Negative Integer by a Negative Integer \| Negative x Negative = -Multiplying Three Integers = -Why Does a Negative Times a Negative Equal a Positive? = ✅ Dividing Integers -A Quick Review of Dividing Integers = -Dividing a Positive by a Negative \| Positive ÷ Negative = -Dividing a Negative by a Positive \| Negative ÷ Positive = -Dividing a Negative Integer by a Negative Integer \| Negative ÷ Negative = ✅ Integers How to Add and Subtract Integers = How to Multiply and Dividing Integers = How to Add Subtract Multiply and Divide Integers = About Math with Mr. J: This channel offers instructional videos that are directly aligned with math standards. Teachers, parents/guardians, and students from around the world have used this channel to help with math content in many different ways. All material is absolutely free. Click Here to Subscribe to the Greatest Math Channel On Earth: Follow Mr. J on Twitter: @MrJMath5 Email: math5.mrj@gmail.com Music: Hopefully this video is what you're looking for when it comes to multiplying a negative integer by a negative integer. Have a great rest of your day and thanks again for watching! ✌️✌️✌️ ✅ Thanks to Aloud, this video has been dubbed into Spanish and Portuguese. #DubbedWithAloud English This video has been dubbed into Spanish (United States) and Portuguese (Brazil) using an artificial voice via to increase accessibility. You can change the audio track language in the Settings menu. Spanish Este video ha sido doblado al español con voz artificial con para aumentar la accesibilidad. Puede cambiar el idioma de la pista de audio en el menú Configuración. Portuguese Este vídeo foi dublado para o português usando uma voz artificial via para melhorar sua acessibilidade. Você pode alterar o idioma do áudio no menu Configurações. 256 comments Transcript: [Music] welcome to math with mr j [Music] in this video i'm going to cover how to multiply a negative integer by a negative integer so a negative times a negative let's jump into number one where we have negative eight times negative three with this being a negative times a negative we have the same signs this means our product the answer to a multiplication problem will be positive same signs equal a positive answer we can think of this as 8 times 3 which is 24 and again this is positive because we had a negative times a negative so the same signs final answer positive 24. now let's think a little further about this problem and why we end up with a positive 24 when we multiplied two negatives when we multiply a negative by a negative we are taking the inverse of the inverse meaning the opposite of the opposite the inverse or opposite of positive eight is negative eight and then the inverse or opposite of positive three is negative three we have two inverses or two opposites here so the inverse of the inverse gives us a positive answer in other words the opposite of the opposite here now instead of thinking about the inverse of the inverse in terms of a whole multiplication problem a negative times a negative let's just think of this in terms of one number it will be a little easier to picture it that way let's work with a positive eight so we start with positive eight and we're going to take the inverse or opposite that's going to give us negative eight then if we take the inverse or opposite of that negative eight we get back to a positive eight so the inverse of the inverse or opposite of the opposite so as a brief explanation as to why a negative times a negative equals a positive it's because we have the inverse of the inverse or opposite of the opposite and that gives us that positive product now there are other ways of explaining and thinking through why a negative times a negative equals a positive i'll go through a couple more in another video that link is down in the description let's move on to number two where we have negative eleven times negative five so a negative times a negative same signs we will have a positive product let's think of this as eleven times five which is 55 and again it's going to be positive our final answer positive 55. so there you have it there's how you multiply a negative times a negative i hope that helped thanks so much for watching until next time peace
1676	https://www.youtube.com/watch?v=ktckOMr6T4s	Simplifying a radical expression to rational exponents Brian McLogan 1600000 subscribers 49 likes Description 6700 views Posted: 13 Mar 2013 👉 Learn how to convert a radical to rational power. A radical is an expression having the root/radical symbol. The number outside the radical symbol (nth root) is called the index, the number/expression inside the radical symbol is called the radical. To convert a radical to a rational power, the base of the radical becomes the base of the rational power, the exponent of the radical becomes the numerator of the exponent of the rational power, while the index (nth root) of the radical expression becomes the denominator of the rational expression. 👏SUBSCRIBE to my channel here: ❤️Support my channel by becoming a member: 🙋‍♂️Have questions? Ask here: 🎉Follow the Community: Organized Videos: ✅Fractional Exponents and Radicals ✅Numbers Raised to Fractional Exponents ✅Convert Radicals to Fractional Exponents ✅Convert Fractional Exponents to Radicals ✅Converting Racials and Exponents \| Learn About ✅Rationalize the Denominator with Fractional Exponent ✅Simplify Fractional Exponents using Power to Quotient ✅Raise an Exponent to a fraction ✅Simplify Fractional Exponents using Power to Product ✅Multiply Fractional Exponents ✅Divide Rational Exponents ✅Solve Equations with Fractional Exponents 🗂️ Organized playlists by classes here: 🌐 My Website - 🎯Survive Math Class Checklist: Ten Steps to a Better Year: Connect with me: ⚡️Facebook - ⚡️Instagram - ⚡️Twitter - ⚡️Linkedin - 👨‍🏫 Current Courses on Udemy: 👨‍👩‍👧‍👧 About Me: I make short, to-the-point online math tutorials. I struggled with math growing up and have been able to use those experiences to help students improve in math through practical applications and tips. Find more here: Fraction #BrianMclogan 3 comments Transcript: i'm going to simplify this problem the main important thing we need to do is let's reduce this or let's rewrite this as a rational exp as a a number with a rational exponent right so therefore we're going to have 5 to the 10th power so 8 to the 5 to the 10 or 8 to the 5 over 10 power is exactly the same as 8 or as the 10th root of 8. now we can reduce 5 over 10 right we don't need to use 5 or 10 we can just simplify that to one half good now can we rewrite 8 as a a number raised to another power 2 to the third power and then what do we do when we have a power of a power multiply across there you go simplified evaluated or not evaluated but simplified form oh
1677	https://www.cuemath.com/calculus/modulus-function/	Modulus Function A modulus function gives the magnitude of a number irrespective of its sign. It is also called the absolute value function. In mathematics, the modulus of a real number x is given by the modulus function, denoted by \|x\|. It gives the non-negative value of x. The modulus or absolute value of a number is also considered as the distance of the number from the origin or zero. In this article, we will learn about the modulus function definition and its properties, its domain and range, and how to apply this function. Also, we will see how to graph modulus function. \| \| \| --- \| \| 1. \| What is Modulus Function? \| \| 2. \| Modulus Function Formula \| \| 3. \| Domain and Range of Modulus Function \| \| 4. \| Application of Modulus Function \| \| 5. \| Modulus Function Graph \| \| 6. \| Properties of Modulus Function \| \| 7. \| Derivative And Integral of Modulus Function \| \| 8. \| FAQs on Modulus Function \| What is Modulus Function? The modulus function, which is also called the absolute value function gives the magnitude or absolute value of a number irrespective of the number being positive or negative. It always gives a non-negative value of any number or variable. The parent modulus function is denoted as y = \|x\| or f(x) = \|x\|, where f: R → [0,∞) and x ∈ R. \|x\| is the modulus of x, where x is a real number. If x is non-negative then f(x) will be of the same value x. If x is negative, then f(x) will be the magnitude of x, that is, f(x) = -x if x is negative. Let us sum up the modulus function formula below. Modulus Function Formula The value of the modulus function is always non-negative. If f(x) is a modulus function, then we have: This means if the value of x is greater than or equal to 0, then the modulus function takes the actual value, but if x is less than 0 then the function takes minus of the actual value 'x'. Domain and Range of Modulus Function We can apply the modulus function f(x) = \|x\| to any real number. i.e., its input can be any real number and hence its domain is the set of all real numbers (ℝ). The output of the modulus function is always a of non-negative real number and hence its range is [0,∞). Thus, Note that the domain of a modulus function f(x) = a\|x - h\| + k is still ℝ but its range varies on the values of 'a' and 'k'. Its range is Application of Modulus Function Now, that we know the modulus function formula, let us consider a few examples to understand its application. The steps to calculate the modulus function are given below: Example: Consider the modulus function f(x) = \|x\|. Note that f(-3) = f(3) here. In other words, \|3\| = \|-3\| = 3. Modulus Function Graph Now let us see how to plot the graph for a modulus function. Let us consider x to be a variable, taking values from -5 to 5. Calculating modulus for the positive values of 'x', the line plotted in the graph is 'y = x' and for the negative values of 'x', the line plotted in the graph is 'y = -x'. \| x \| f(x) = \|x\| \| --- \| \| -5 \| 5 \| \| -4 \| 4 \| \| -3 \| 3 \| \| -2 \| 2 \| \| -1 \| 1 \| \| 0 \| 0 \| \| 1 \| 1 \| \| 2 \| 2 \| \| 3 \| 3 \| \| 4 \| 4 \| \| 5 \| 5 \| ☛ Note: The modulus function is NOT one-one function as it fails the horizontal line test. Properties of Modulus Function Now, that we have the formula for the modulus function and the graph of the modulus function, let us now explore the properties of the modulus function: \|x\| = a; a > 0 ⇒ x = ± a ; \|x\| = a; a = 0 ⇒ x = 0 ; \|x\| = a; a < 0 ⇒ There doesn't exist such x ☛ Note: Here, the property 2 helps in solving the absolute value inequalities. Derivative And Integral of Modulus Function Since we know that a modulus function f(x) = \|x\| is equal to x if x > 0 and -x if x < 0, therefore the derivative of modulus function is 1 if x > 0 and -1 if x < 0. The derivative of the modulus function is NOT defined for x = 0. Hence the derivative of modulus function can be written as d(\|x\|)/dx = x/\|x\|, for all values of x and x ≠ 0. Using the formula of the modulus function and integration formulas, the integral of the modulus function is (1/2)x2 + C if x ≥ 0, and its integral is -(1/2)x2 + C if x < 0. Hence the integration of the modulus function can be clubbed as: Important Notes on Modulus Function ☛ Related Topics: Modulus Function Examples Example 1: Find the value of the modulus function \|x\| for x = -5 and x = 10. Solution: If x = -5, then \|x\| = \|-5\| = 5 If x = 10, then \|x\| = \|10\| = 10 Answer: \|x\| = 5 for x = -5 and \|x\| = 10 for x = 10 Example 2: Solve \|x + 3\| = 8 using modulus function definition. Solution: We know that the modulus function value is always non-negative, therefore we have two cases: If x + 3 > 0, then \|x + 3\| = x + 3 and if x + 3 < 0, then \|x + 3\| = -(x + 3). Case 1: If x + 3 > 0, we have \|x + 3\| = x + 3 ⇒ x + 3 = 8 ⇒ x = 8 - 3 = 5 Case 2: If x + 3 < 0, we have \|x + 3\| = -(x + 3) ⇒ -(x + 3) = 8 ⇒ -x - 3 = 8 ⇒ x = -3 - 8 = -11 Answer: Hence, the solution is x = 5, -11. Example 3: Solve the inequality \|x - 1\| < 3. Solution: Here, 3 > 0. So by the properties of the modulus function, -3 < x - 1 < 3 Adding 1 on all sides, -3 + 1 < x < 3 + 1 -2 < x < 4 Answer: The solution of the given inequality is -2 < x < 4. go to slidego to slidego to slide Book a Free Trial Class Practice Questions on Modulus Function go to slidego to slide FAQs on Modulus Function What is the Definition of Modulus Function? The modulus function gives the magnitude or absolute value of a number irrespective of the number is positive or negative. The modulus function is denoted as y = \|x\| or f(x) = \|x\|, where f: ℝ → [0,∞) and x ∈ ℝ. This is also called the absolute value function. What is the Domain and Range of the Modulus Function? The domain of the modulus function is ℝ (where ℝ refers to the set of all real numbers) and the range of the modulus function is the set of non-negative real numbers which is denoted as [0,∞). What is the Integration of Modulus Function? The integration of the modulus function depends on the value of x. It is: How to Differentiate Modulus Function? We have f(x) = \|x\| is equal to x if x > 0 and -x if x < 0, hence, the derivative of modulus function is 1 if x > 0 and -1 if x < 0. The derivative of the modulus function is not defined for x = 0. To summarize, the derivative of modulus function \|x\| is x/\|x\|, where x not equal to 0. Is Modulus Function Always Positive? The modulus of a positive number is positive. The modulus of a negative number is obtained by ignoring the minus sign. Thus, the modulus function is always positive. What is the Equation of Modulus Function? The general form of the parent function of the modulus function is f(x) = \|x\|. Its vertex is at (0, 0). After transformations, it may look like g(x) = a \|x - h\| + k, whose vertex is (h, k). Why Modulus Function is Not Differentiable? Modulus function \|x\| is not differentiable at x = 0 as the graph of Mod(x) has a sharp point at x = 0. Also, the left-hand limit and the right-hand limit of the derivative are not equal at x = 0. How do you Graph Modulus Function? Take some positive and negative values of x. Also, take x = 0. Frame a table with two columns x and y with all the random x-values that we have chosen. Calculate the modulus of every number and fill in the column of y. Then just plot all the ordered pairs (x, y) and join them by lines. We will get a 'V' - shaped graph and it is the graph of the modulus function.
1678	https://www.quora.com/How-do-I-convert-gauge-pressure-to-absolute-pressure	How to convert gauge pressure to absolute pressure - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Physics Pressure Measurement Inst... Conversion Process Units of Measure Absolute Pressure Engineering Gauge Pressure Pressure Testing Unit Conversions 5 How do I convert gauge pressure to absolute pressure? All related (36) Sort Recommended M. Sharma Lives in New Delhi ·Updated 7y Absolute pressure = Guage pressure + Atmospheric pressure. So for converting gauge pressure into absolute pressure, just add atmospheric pressure in it. Upvote · 9 8 Sponsored by Grammarly 92% of professionals who use Grammarly say it has saved them time Work faster with AI, while ensuring your writing always makes the right impression. Download 999 209 Igor Francois Engineer (2015–present) · Author has 252 answers and 270.3K answer views ·7y P _abs-P _amb = P _g Where: P_abs is the asolute pressure, P_amb is the ambient pressure (or atmospheric pressure) and P_g is the gauge pressure. If you want to get absolute pressure from gauge pressure, you simply add the ambient pressure value to the gauge pressure. Upvote · 9 5 Alan Grimes Author has 124 answers and 136.7K answer views ·7y As always, add then multiply... A gauge will have a BASE and a SCALE. A standard gauge will be based on one atmosphere of pressure, something like 14.3 psi So add that to the reading. The gauge may also need to be scaled, depending on what unit of measure it uses. This is done by multiplying depending on what the conversion constant is. Upvote · 9 1 Related questions More answers below How do you convert absolute pressure to gauge? A pressure gauge attached in a pipeline reads 30 psi. What is the absolute pressure in absolute pressure? What is the need of gauge pressure with respect to absolute pressure? Does the equation P= F/A give gauge or absolute pressure or neither? How does one calculate the gauge pressure from an absolute pressure reading on a transducer correctly (mechanics engineering)? James Rickman Former Environmental Engineer at USDA Natural Resources Conservation Service (1972–1998) · Author has 9.3K answers and 6.6M answer views ·7y You add the atmospheric pressure to the gauge pressure. On earth the atmospheric pressure due to the weight of air is about 14.7 psi so if your tire pressure was 30 psi from your gauge the absolute pressure would be 44.7 psi. Jim Upvote · 9 1 Assistant Bot · 1y To convert gauge pressure to absolute pressure, you can use the following formula: P absolute=P gauge+P atmospheric P absolute=P gauge+P atmospheric Where: P absolute P absolute is the absolute pressure. P gauge P gauge is the gauge pressure (the pressure relative to atmospheric pressure). P atmospheric P atmospheric is the atmospheric pressure, which is approximately 101.3 kPa (or 1 atm) at sea level. Steps to Convert: Identify Gauge Pressure: Determine the gauge pressure you have. Find Atmospheric Pressure: Use the standard atmospheric pressure value (101.3 kPa or 14.7 psi) unless the conditions sp Continue Reading To convert gauge pressure to absolute pressure, you can use the following formula: P absolute=P gauge+P atmospheric P absolute=P gauge+P atmospheric Where: P absolute P absolute is the absolute pressure. P gauge P gauge is the gauge pressure (the pressure relative to atmospheric pressure). P atmospheric P atmospheric is the atmospheric pressure, which is approximately 101.3 kPa (or 1 atm) at sea level. Steps to Convert: Identify Gauge Pressure: Determine the gauge pressure you have. Find Atmospheric Pressure: Use the standard atmospheric pressure value (101.3 kPa or 14.7 psi) unless the conditions specify otherwise. Apply the Formula: Add the gauge pressure to the atmospheric pressure to find the absolute pressure. Example: If you have a gauge pressure of 200 kPa, the absolute pressure would be: P absolute=200 kPa+101.3 kPa=301.3 kPa P absolute=200 kPa+101.3 kPa=301.3 kPa This method will give you the absolute pressure in the same units as the gauge pressure. Upvote · Related questions More answers below What is the equivalent of 250 lb pressure using a pressure gauge? What would be the absolute pressure if the gauge pressure is 6.2 bars? What is the gauge pressure if the absolute pressure is 400 kPa? If the absolute pressure is 300 kPa, what would a pressure gauge read? How do you find gauge pressure from absolute pressure? Mukesh Kumar Maintenance Planning Engineer. ·7y By adding atmospheric pressure of 14.7 Upvote · 9 1 Promoted by HP HP Tech Takes Tech Enthusiast \| Insights, Tips & Guidance ·Updated Sep 18 What are the pros and cons of getting an expensive laser printer like HP versus a cheap but good quality inkjet/multi-function machine like Canon Pixma MX490? Choosing between an expensive laser printer and a more affordable inkjet or multi-function machine ultimately depends on your printing habits, the type of documents you produce, and your long-term budget. Laser printers and inkjet printers serve different purposes, and understanding their strengths and limitations can help you make a more informed decision. Laser printers are typically designed for speed and efficiency, making them ideal for office environments or users who print frequently. Inkjet and ink tank printers are slower but they offer excellent colour reproduction and are often more Continue Reading Choosing between an expensive laser printer and a more affordable inkjet or multi-function machine ultimately depends on your printing habits, the type of documents you produce, and your long-term budget. Laser printers and inkjet printers serve different purposes, and understanding their strengths and limitations can help you make a more informed decision. Laser printers are typically designed for speed and efficiency, making them ideal for office environments or users who print frequently. Inkjet and ink tank printers are slower but they offer excellent colour reproduction and are often more compact and affordable upfront. If your printing needs are frequent and primarily text-based, a laser printer such as the HP Color Laser 179fnw or the HP LaserJet M234sdw would be a strong choice. These models deliver fast print speeds and sharp text output. Toner cartridges used in these printers last significantly longer than ink cartridges, resulting in a lower cost per page over time. Although the initial investment is higher, the long-term savings and reliability could make them well-suited for your business use or heavy personal workloads. LaserJet Printers - Black & White or Color Document Printers Alternatively, if your printing volume is low to moderate and you value colour accuracy for photos or creative projects, an ink-based solution like the HP Smart Tank 7605 or HP Smart Tank 5105 may be more convenient. These printers offer refillable ink tanks that reduce running costs compared to traditional cartridge-based inkjets. They are also compact and versatile, supporting scanning and copying functions in addition to printing. However, they require occasional maintenance to prevent ink from drying out or clogging, and their print speed is generally slower than laser models. HP Smart Tank Printers – Refillable Ink Tank Printers So to break it down, if you prioritise speed, durability, and cost-efficiency for high-volume printing, a laser printer from HP’s 200 or 3000 series is the better investment. On the other hand, if your needs include occasional printing with a focus on colour and photo output, HP’s Smart Tank series provides a more economical and flexible alternative. The right choice will depend on how often you print and what kind of documents you’re producing. Check out the blog linked below to learn more about the different models and which is best for your printing needs, hope this helps! Inkjet vs LaserJet vs OfficeJet: HP Printer Buying Guide By Lizzie - HP Tech Expert Upvote · 999 112 99 33 9 4 Sudersanan Professor at Dr T Thimmaiah Institute of Technology (1989–present) · Author has 65 answers and 448.2K answer views ·6y Related What is the difference between gauge pressure and absolute pressure? Gauge pressure is the pressure measured relative to the atmospheric pressure. Gauge pressure measured can be above or below atmospheric pressure. The pressure above atmospheric pressure is positive gauge pressure. If the pressure measured is below atmospheric pressure, it indicates a negative gauge pressure or vacuum pressure. Absolute pressure is the pressure measured with respect to the absolute zero pressure. It is always a positive quantity. Absolute pressure can be obtained by adding atmospheric pressure to the gauge pressure. Points to note: Gauge pressure can be positive or negative. Gauge Continue Reading Gauge pressure is the pressure measured relative to the atmospheric pressure. Gauge pressure measured can be above or below atmospheric pressure. The pressure above atmospheric pressure is positive gauge pressure. If the pressure measured is below atmospheric pressure, it indicates a negative gauge pressure or vacuum pressure. Absolute pressure is the pressure measured with respect to the absolute zero pressure. It is always a positive quantity. Absolute pressure can be obtained by adding atmospheric pressure to the gauge pressure. Points to note: Gauge pressure can be positive or negative. Gauge pressure is zero-referenced against atmospheric pressure. Gauge pressure is absolute pressure minus atmospheric pressure Minimum gauge pressure is -Patm (-1.013bar). That is absolute zero pressure or perfect vacuum. Absolute pressure is always positive. Absolute pressure is zero-referenced against perfect vacuum. Absolute pressure is gauge pressure plus atmospheric pressure. Minimum absolute pressure is zero (perfect vacuum) Upvote · 99 17 9 1 9 1 Paul Montgomery Retired Engineer (Metrology & Control Systems) · Author has 6.2K answers and 39.4M answer views ·Updated 3y Related Do pressure gauges measure absolute pressure? Most pressure gauges do not measure absolute pressure. For a pressure gauge to measure absolute pressure, it must have a reference pressure that is near absolute zero pressure. Usually, this will be a sealed chamber where a vacuum pump has removed the air. This adds a lot of expense to the gauge. An absolute pressure gauge will be priced at ten to a hundred times as expensive as a regular “gauge” pressure gauge. An absolute gauge will always have some marking that indicates it measures absolute pressure, even if this is only by the scale indicator of “PSIA” instead of “PSI” P.S. A gauge that reads Continue Reading Most pressure gauges do not measure absolute pressure. For a pressure gauge to measure absolute pressure, it must have a reference pressure that is near absolute zero pressure. Usually, this will be a sealed chamber where a vacuum pump has removed the air. This adds a lot of expense to the gauge. An absolute pressure gauge will be priced at ten to a hundred times as expensive as a regular “gauge” pressure gauge. An absolute gauge will always have some marking that indicates it measures absolute pressure, even if this is only by the scale indicator of “PSIA” instead of “PSI” P.S. A gauge that reads vacuum, including compound gauges that read both vacuum and pressure are not absolute pressure gauges. These gauges use ambient pressure as a reference zero. They only work accurately when used inside the atmosphere of the earth at sea level. If taken into space or used inside a vacuum chamber, they would read zero. Upvote · 99 93 99 15 9 2 Sponsored by Avnet Silica We're at the Pulse of the Market. Access deep technical insight and real-time market intelligence to take your business to the next level. Learn More 9 2 Arpit Gupta Mechanical Engineer in the making ·9y Related How do we measure absolute gauge pressure at any point in a fluid mechanics setup? Pressure is either Gauge or Absolute. It cannot be both. The difference between the two is the atmospheric pressure. Let me elaborate. Let us take an example of a manometer. There is a pressurized fluid (say gas) in the circular chamber. The manometer fluid (dark colored 'B') is in the U-tube. If the circular chamber had not been there, the level of B would have equal in both the arms of the U-tube because then, the pressure on both sides would have been equal to the atmospheric pressure (Patm). Since, there is a gas A with a pressure P1, the levels are not same. The difference in the levels of Continue Reading Pressure is either Gauge or Absolute. It cannot be both. The difference between the two is the atmospheric pressure. Let me elaborate. Let us take an example of a manometer. There is a pressurized fluid (say gas) in the circular chamber. The manometer fluid (dark colored 'B') is in the U-tube. If the circular chamber had not been there, the level of B would have equal in both the arms of the U-tube because then, the pressure on both sides would have been equal to the atmospheric pressure (Patm). Since, there is a gas A with a pressure P1, the levels are not same. The difference in the levels of the fluid B is used to measure the Gauge Pressure of the gas A. The diagram shows that the difference is x. Consider the line PQ. The forces should balance on this line. Force on P = Force on Q Force on P = Pressure on P = P1 Force on Q = Pressure on Q = Patm + weight of liquid above Q Equating, P1 = Patm + weight of liquid P1 = Patm + Densitygx Now, P1 can either be reported as (Patm + Densitygx) or as simply (Densitygx). When it includes Patm, its called Absolute Pressure and when Patm is excluded, its called Gauge pressure. Hope it helps. Upvote · 9 3 Partha Raman B.E in Mechanical Engineering&Watcing Movies, Arignar Anna Institute of Science and Technology (Graduated 2020) ·4y Related How do you convert absolute pressure to gauge? Absolute pressure=atmospheric pressure+ gauge pressure Now, gauge pressure= absolute pressure - atmospheric pressure Upvote · 9 1 Sponsored by Visit Dubai Plan your dream Dubai getaway. Explore traditional souks, luxury malls and desert adventures in one epic trip! Learn More 99 41 Roland Yonaba Engineer in Hydraulics · Author has 79 answers and 664K answer views ·9y Related What is Absolute Pressure? Pressure at a given point is due to the weight of the fluid above that point. Two types of pressure are commonly used in hydraulics : gauge pressure and absolute pressure. Absolute pressure is the pressure measured with absolute zero taken as a reference. Hence, a perfect vacuum, called absolute zero, is the datum. Gauge pressure is the pressure measured with atmospheric pressure taken as its datum. For example, a pressure gauge located on earth's surface and open to the atmosphere will dial zero (gauge pressure). Those two pressure are related in the following manner : P_{abs}=P_{gauge}+P_{a P_{abs}=P_{gauge}+P_{a Continue Reading Pressure at a given point is due to the weight of the fluid above that point. Two types of pressure are commonly used in hydraulics : gauge pressure and absolute pressure. Absolute pressure is the pressure measured with absolute zero taken as a reference. Hence, a perfect vacuum, called absolute zero, is the datum. Gauge pressure is the pressure measured with atmospheric pressure taken as its datum. For example, a pressure gauge located on earth's surface and open to the atmosphere will dial zero (gauge pressure). Those two pressure are related in the following manner : P a b s=P g a u g e+P a t m P a b s=P g a u g e+P a t m Upvote · 99 19 9 1 Will Caruthers Chemical Engineer, Trader, Renaissance Man · Author has 1.4K answers and 6.5M answer views ·7y Related What should I assume if it is not specified whether the given pressure is gauge or absolute? Q: What should I assume if it is not specified whether the given pressure is gauge or absolute? Assumptions are a necessary evil. It is not generally possible to find a practical solution to an engineering or other physical problem without making them, yet they are to be avoided if it is both possible and reasonable to do so (i.e. if a variable fluctuates somewhat yet incorporating a function to model this would cause unwieldy calculations that were more precise than is needed, you might instead use an assumed variable value that is constant or varied in a more straightforward manner). Here, you Continue Reading Q: What should I assume if it is not specified whether the given pressure is gauge or absolute? Assumptions are a necessary evil. It is not generally possible to find a practical solution to an engineering or other physical problem without making them, yet they are to be avoided if it is both possible and reasonable to do so (i.e. if a variable fluctuates somewhat yet incorporating a function to model this would cause unwieldy calculations that were more precise than is needed, you might instead use an assumed variable value that is constant or varied in a more straightforward manner). Here, you are asking for a blanket assumption that you might make, a rule of thumb. Normally, if provided a pressure in a textbook problem and it is not given that it is gauge or was not worded in a manner like, “the pressure gauge reads…” then I’d think it was absolute. HOWEVER. In the real world, you’re going to be getting your pressure readings from an instrument. If you are talking about a practical engineering question, then you wouldn’t blanket assume. You can either deduce or look up which it would be based on the conditions of the mounting of the pressure plate or whatever. If it is hooked to a PLC so you can model its data there should be an “EngUnits” or similar tag in your data processing software. If it’s a manual gauge in the field it should be showing the differential between the atmospheric pressure surrounding it and the internal pressure on the other side of the diaphragm within the dial, thus this is (logically) gauge pressure. Therefore, the best answer is DON’T blanket assume. Either find your answer through research or inspection or, at the least, use logic to determine which of the two it either must be or most likely is. Just putting a little thought into how the pressure is being taken should reveal which of the two it must be. I hope that helps. Upvote · 9 4 Haruhit Singh Kunwar there is always a pressure on me known as atm pressure! · Author has 58 answers and 310.7K answer views ·8y Related What is the relation between Gauge pressure absolute pressure atmospheric pressure and vacuum pressure? Absolute Pressure - It is the zero referenced against a perfect vacuum. Atmospheric Pressure - It is the pressure exerted by the weight of the atmosphere. Basically it is the ambiant pressure i.e the pressure around you. Gauge Pressure - It is the zero reference against the ambient pressure. Vacuum Pressure - It is the pressure below atmospheric pressure. So the releation between gauge pressure, absolute pressure, atmospheric pressure and vacuum pressure is :- P(abs) = P(gauge) + P(atm) P(vac) = P(atm) - P(abs) Or P(vac) = -P(gauge) . Continue Reading Absolute Pressure - It is the zero referenced against a perfect vacuum. Atmospheric Pressure - It is the pressure exerted by the weight of the atmosphere. Basically it is the ambiant pressure i.e the pressure around you. Gauge Pressure - It is the zero reference against the ambient pressure. Vacuum Pressure - It is the pressure below atmospheric pressure. So the releation between gauge pressure, absolute pressure, atmospheric pressure and vacuum pressure is :- P(abs) = P(gauge) + P(atm) P(vac) = P(atm) - P(abs) Or P(vac) = -P(gauge) . Upvote · 99 24 9 6 9 2 Related questions How do you convert absolute pressure to gauge? A pressure gauge attached in a pipeline reads 30 psi. What is the absolute pressure in absolute pressure? What is the need of gauge pressure with respect to absolute pressure? Does the equation P= F/A give gauge or absolute pressure or neither? How does one calculate the gauge pressure from an absolute pressure reading on a transducer correctly (mechanics engineering)? What is the equivalent of 250 lb pressure using a pressure gauge? What would be the absolute pressure if the gauge pressure is 6.2 bars? What is the gauge pressure if the absolute pressure is 400 kPa? If the absolute pressure is 300 kPa, what would a pressure gauge read? How do you find gauge pressure from absolute pressure? When we say 30 psi is the pressure, then what pressure are we talking about, the gauge pressure or the absolute pressure? What is an absolute pressure gauge? What is its use in measuring pressures? If the gauge pressure is reported as 6 ATM, what is the absolute pressure? How is gauge pressure, absolute pressure, and differential pressure measured in a smart transmitter? Are the pressure gauges generally used measure pressure at absolute pressure or gauge pressure? Related questions How do you convert absolute pressure to gauge? A pressure gauge attached in a pipeline reads 30 psi. What is the absolute pressure in absolute pressure? What is the need of gauge pressure with respect to absolute pressure? Does the equation P= F/A give gauge or absolute pressure or neither? How does one calculate the gauge pressure from an absolute pressure reading on a transducer correctly (mechanics engineering)? What is the equivalent of 250 lb pressure using a pressure gauge? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025 Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer. 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1679	https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Seeburger)/03%3A_Derivatives/3.04%3A_Derivatives_of_Trigonometric_Functions	3.4: Derivatives of Trigonometric Functions - Mathematics LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 3: Derivatives MTH 210 Calculus I (Seeburger) { "3.4E:_Exercises_for_Section_3.4" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } { "3.00:_Prelude_to_Derivatives" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "3.01:_Defining_the_Derivative" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "3.02:_The_Derivative_as_a_Function" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "3.03:_Differentiation_Rules" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "3.04:_Derivatives_of_Trigonometric_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "3.05:_Derivatives_as_Rates_of_Change" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "3.06:_The_Chain_Rule" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "3.07:_Implicit_Differentiation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "3.08:_Derivatives_of_Exponential_and_Logarithmic_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "3.09:_Derivatives_of_the_Inverse_Trigonometric_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "3R:_Chapter_3_Review_Exercises" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } { "00:_Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_Functions_and_Graphs" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Limits" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Derivatives" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Applications_of_Derivatives" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Integration" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", Appendices : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } Fri, 17 Jul 2020 16:56:24 GMT 3.4: Derivatives of Trigonometric Functions 43637 43637 Demo Instructor { } Anonymous Anonymous 2 false false [ "article:topic", "Derivative of sine function", "Derivative of cosine function", "Derivative of tangent function", "Derivative of cotangent function", "Derivative of secant function", "Derivative of cosecant function", "authorname:openstax", " "license:ccbyncsa", "showtoc:yes", "transcluded:yes", "source-math-2494", "licenseversion:40", "source@ ] [ "article:topic", "Derivative of sine function", "Derivative of cosine function", "Derivative of tangent function", "Derivative of cotangent function", "Derivative of secant function", "Derivative of cosecant function", "authorname:openstax", " "license:ccbyncsa", "showtoc:yes", "transcluded:yes", "source-math-2494", "licenseversion:40", "source@ ] Search site Search Search Go back to previous article Sign in Username Password Sign in Sign in Sign in Forgot password Expand/collapse global hierarchy 1. 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MTH 210 Calculus I (Seeburger) 5. 3: Derivatives 6. 3.4: Derivatives of Trigonometric Functions Expand/collapse global location 3.4: Derivatives of Trigonometric Functions Last updated Jul 17, 2020 Save as PDF 3.3E: Exercises for Section 3.3 3.4E: Exercises for Section 3.4 Page ID 43637 OpenStax OpenStax ( \newcommand{\vecs}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}) ( \newcommand{\vecd}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash{#1}}} ) ( \newcommand{\id}{\mathrm{id}}) ( \newcommand{\Span}{\mathrm{span}}) ( \newcommand{\kernel}{\mathrm{null}\,}) ( \newcommand{\range}{\mathrm{range}\,}) ( \newcommand{\RealPart}{\mathrm{Re}}) ( \newcommand{\ImaginaryPart}{\mathrm{Im}}) ( \newcommand{\Argument}{\mathrm{Arg}}) ( \newcommand{\norm}{\\| #1 \\|}) ( \newcommand{\inner}{\langle #1, #2 \rangle}) ( \newcommand{\Span}{\mathrm{span}}) ( \newcommand{\id}{\mathrm{id}}) ( \newcommand{\Span}{\mathrm{span}}) ( \newcommand{\kernel}{\mathrm{null}\,}) ( \newcommand{\range}{\mathrm{range}\,}) ( \newcommand{\RealPart}{\mathrm{Re}}) ( \newcommand{\ImaginaryPart}{\mathrm{Im}}) ( \newcommand{\Argument}{\mathrm{Arg}}) ( \newcommand{\norm}{\\| #1 \\|}) ( \newcommand{\inner}{\langle #1, #2 \rangle}) ( \newcommand{\Span}{\mathrm{span}}) ( \newcommand{\AA}{\unicode[.8,0]{x212B}}) ( \newcommand{\vectorA}{\vec{#1}} % arrow) ( \newcommand{\vectorAt}{\vec{\text{#1}}} % arrow) ( \newcommand{\vectorB}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}) ( \newcommand{\vectorC}{\textbf{#1}}) ( \newcommand{\vectorD}{\overrightarrow{#1}}) ( \newcommand{\vectorDt}{\overrightarrow{\text{#1}}}) ( \newcommand{\vectE}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} ) ( \newcommand{\vecs}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}) ( \newcommand{\vecd}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash{#1}}} ) (\newcommand{\avec}{\mathbf a}) (\newcommand{\bvec}{\mathbf b}) (\newcommand{\cvec}{\mathbf c}) (\newcommand{\dvec}{\mathbf d}) (\newcommand{\dtil}{\widetilde{\mathbf d}}) (\newcommand{\evec}{\mathbf e}) (\newcommand{\fvec}{\mathbf f}) (\newcommand{\nvec}{\mathbf n}) (\newcommand{\pvec}{\mathbf p}) (\newcommand{\qvec}{\mathbf q}) (\newcommand{\svec}{\mathbf s}) (\newcommand{\tvec}{\mathbf t}) (\newcommand{\uvec}{\mathbf u}) (\newcommand{\vvec}{\mathbf v}) (\newcommand{\wvec}{\mathbf w}) (\newcommand{\xvec}{\mathbf x}) (\newcommand{\yvec}{\mathbf y}) (\newcommand{\zvec}{\mathbf z}) (\newcommand{\rvec}{\mathbf r}) (\newcommand{\mvec}{\mathbf m}) (\newcommand{\zerovec}{\mathbf 0}) (\newcommand{\onevec}{\mathbf 1}) (\newcommand{\real}{\mathbb R}) (\newcommand{\twovec}{\left[\begin{array}{r}#1 \ #2 \end{array}\right]}) (\newcommand{\ctwovec}{\left[\begin{array}{c}#1 \ #2 \end{array}\right]}) (\newcommand{\threevec}{\left[\begin{array}{r}#1 \ #2 \ #3 \end{array}\right]}) (\newcommand{\cthreevec}{\left[\begin{array}{c}#1 \ #2 \ #3 \end{array}\right]}) (\newcommand{\fourvec}{\left[\begin{array}{r}#1 \ #2 \ #3 \ #4 \end{array}\right]}) (\newcommand{\cfourvec}{\left[\begin{array}{c}#1 \ #2 \ #3 \ #4 \end{array}\right]}) (\newcommand{\fivevec}{\left[\begin{array}{r}#1 \ #2 \ #3 \ #4 \ #5 \ \end{array}\right]}) (\newcommand{\cfivevec}{\left[\begin{array}{c}#1 \ #2 \ #3 \ #4 \ #5 \ \end{array}\right]}) (\newcommand{\mattwo}{\left[\begin{array}{rr}#1 \amp #2 \ #3 \amp #4 \ \end{array}\right]}) (\newcommand{\laspan}{\text{Span}{#1}}) (\newcommand{\bcal}{\cal B}) (\newcommand{\ccal}{\cal C}) (\newcommand{\scal}{\cal S}) (\newcommand{\wcal}{\cal W}) (\newcommand{\ecal}{\cal E}) (\newcommand{\coords}{\left{#1\right}_{#2}}) (\newcommand{\gray}{\color{gray}{#1}}) (\newcommand{\lgray}{\color{lightgray}{#1}}) (\newcommand{\rank}{\operatorname{rank}}) (\newcommand{\row}{\text{Row}}) (\newcommand{\col}{\text{Col}}) (\renewcommand{\row}{\text{Row}}) (\newcommand{\nul}{\text{Nul}}) (\newcommand{\var}{\text{Var}}) (\newcommand{\corr}{\text{corr}}) (\newcommand{\len}{\left\|#1\right\|}) (\newcommand{\bbar}{\overline{\bvec}}) (\newcommand{\bhat}{\widehat{\bvec}}) (\newcommand{\bperp}{\bvec^\perp}) (\newcommand{\xhat}{\widehat{\xvec}}) (\newcommand{\vhat}{\widehat{\vvec}}) (\newcommand{\uhat}{\widehat{\uvec}}) (\newcommand{\what}{\widehat{\wvec}}) (\newcommand{\Sighat}{\widehat{\Sigma}}) (\newcommand{\lt}{<}) (\newcommand{\gt}{>}) (\newcommand{\amp}{&}) (\definecolor{fillinmathshade}{gray}{0.9}) Table of contents 1. Learning Objectives 2. Derivatives of the Sine and Cosine Functions 1. The Derivatives of (\sin x) and (\cos x) 2. Proof 3. Example (\PageIndex{1}): Differentiating a Function Containing (\sin x) 1. Solution 4. Exercise $\PageIndex{1}$/03:_Derivatives/3.04:_Derivatives_of_Trigonometric_Functions#Exercise_.5C(.5CPageIndex.7B1.7D.5C)) 5. Example $\PageIndex{2}$: Finding the Derivative of a Function Containing cos x/03:_Derivatives/3.04:_Derivatives_of_Trigonometric_Functions#Example_.5C(.5CPageIndex.7B2.7D.5C):_Finding_the_Derivative_of_a_Function_Containing_cos_x) 1. Solution/03:_Derivatives/3.04:_Derivatives_of_Trigonometric_Functions#Solution_2) 6. Exercise $\PageIndex{2}$/03:_Derivatives/3.04:_Derivatives_of_Trigonometric_Functions#Exercise_.5C(.5CPageIndex.7B2.7D.5C)) 7. Example $\PageIndex{3}$: An Application to Velocity/03:_Derivatives/3.04:_Derivatives_of_Trigonometric_Functions#Example_.5C(.5CPageIndex.7B3.7D.5C):_An_Application_to_Velocity) 1. Solution/03:_Derivatives/3.04:_Derivatives_of_Trigonometric_Functions#Solution_3) 8. Exercise $\PageIndex{3}$/03:_Derivatives/3.04:_Derivatives_of_Trigonometric_Functions#Exercise_.5C(.5CPageIndex.7B3.7D.5C)) Derivatives of Other Trigonometric Functions Example (\PageIndex{4}): The Derivative of the Tangent Function Solution Exercise (\PageIndex{4}) Derivatives of (\tan x), (\cot x), (\sec x), and (\csc x) Example (\PageIndex{5}): Finding the Equation of a Tangent Line Solution Example (\PageIndex{6}): Finding the Derivative of Trigonometric Functions Solution Exercise (\PageIndex{5}) Exercise (\PageIndex{6}) Higher-Order Derivatives Example (\PageIndex{7}): Finding Higher-Order Derivatives of (y=\sin x) Solution Analysis Exercise (\PageIndex{7}) Example (\PageIndex{8}): Using the Pattern for Higher-Order Derivatives of (y=\sin x) Solution Exercise (\PageIndex{8}) Example (\PageIndex{9}): An Application to Acceleration Solution Exercise (\PageIndex{9}) Key Concepts Key Equations Learning Objectives Find the derivatives of the sine and cosine function. Find the derivatives of the standard trigonometric functions. Calculate the higher-order derivatives of the sine and cosine. One of the most important types of motion in physics is simple harmonic motion, which is associated with such systems as an object with mass oscillating on a spring. Simple harmonic motion can be described by using either sine or cosine functions. In this section we expand our knowledge of derivative formulas to include derivatives of these and other trigonometric functions. We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion. Derivatives of the Sine and Cosine Functions We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Recall that for a function (f(x),) [f′(x)=\lim_{h→0}\dfrac{f(x+h)−f(x)}{h}. \nonumber ] Consequently, for values of (h) very close to (0), [f′(x)≈\dfrac{f(x+h)−f(x)}{h}. \nonumber ] We see that by using (h=0.01), [\dfrac{d}{dx}(\sin x)≈\dfrac{\sin (x+0.01)−\sin x}{0.01} \nonumber ] By setting [D(x)=\dfrac{\sin (x+0.01)−\sin x}{0.01} \nonumber ] and using a graphing utility, we can get a graph of an approximation to the derivative of (\sin x) (Figure (\PageIndex{1})). Figure (\PageIndex{1}): The graph of the function (D(x)) looks a lot like a cosine curve. Upon inspection, the graph of (D(x)) appears to be very close to the graph of the cosine function. Indeed, we will show that [\dfrac{d}{dx}(\sin x)=\cos x. \nonumber ] If we were to follow the same steps to approximate the derivative of the cosine function, we would find that [\dfrac{d}{dx}(\cos x)=−\sin x. \nonumber ] The Derivatives of (\sin x) and (\cos x) The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine. [\dfrac{d}{dx}(\sin x)=\cos x \nonumber ] [\dfrac{d}{dx}(\cos x)=−\sin x \nonumber ] Proof Because the proofs for (\dfrac{d}{dx}(\sin x)=\cos x) and (\dfrac{d}{dx}(\cos x)=−\sin x) use similar techniques, we provide only the proof for (\dfrac{d}{dx}(\sin x)=\cos x). Before beginning, recall two important trigonometric limits: (\displaystyle \lim_{h→0}\dfrac{\sin h}{h}=1) and (\displaystyle \lim_{h→0}\dfrac{\cos h−1}{h}=0). The graphs of (y=\dfrac{\sin h}{h}) and (y=\dfrac{\cos h−1}{h}) are shown in Figure (\PageIndex{2}). Figure (\PageIndex{2}): These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions. We also recall the following trigonometric identity for the sine of the sum of two angles: [\sin (x+h)=\sin x\cos h+\cos x\sin h. \nonumber ] Now that we have gathered all the necessary equations and identities, we proceed with the proof. [\begin{align} \dfrac{d}{dx}(\sin x) &=\lim_{h→0}\dfrac{\sin(x+h)−\sin x}{h} && \text{Apply the definition of the derivative.}\[4pt] &=\lim_{h→0}\dfrac{\sin x\cos h+\cos x\sin h−\sin x}{h} && \text{Use trig identity for the sine of the sum of two angles.}\[4pt] &=\lim_{h→0}\left(\dfrac{\sin x\cos h−\sin x}{h}+\dfrac{\cos x\sin h}{h}\right) && \text{Regroup.}\[4pt] &=\lim_{h→0}\left(\sin x\left(\dfrac{\cos h−1}{h}\right)+(\cos x)\left(\dfrac{\sin h}{h}\right)\right) && \text{Factor out }\sin x\text{ and }\cos x \[4pt] &=(\sin x)\lim_{h→0}\left(\dfrac{\cos h−1}{h}\right)+(\cos x)\lim_{h→0}\left(\dfrac{\sin h}{h}\right) && \text{Factor }\sin x\text{ and }\cos x \text{ out of limits.} \[4pt] &=(\sin x)(0)+(\cos x)(1) && \text{Apply trig limit formulas.}\[4pt] &=\cos x && \text{Simplify.} \end{align} \nonumber ] □ Figure (\PageIndex{3}) shows the relationship between the graph of (f(x)=\sin x) and its derivative (f′(x)=\cos x). Notice that at the points where (f(x)=\sin x) has a horizontal tangent, its derivative (f′(x)=\cos x) takes on the value zero. We also see that where f((x)=\sin x) is increasing, (f′(x)=\cos x>0) and where (f(x)=\sin x) is decreasing, (f′(x)=\cos x<0.) Figure (\PageIndex{3}): Where (f(x)) has a maximum or a minimum, (f'(x)=0) that is, (f'(x)=0) where (f(x)) has a horizontal tangent. These points are noted with dots on the graphs Example (\PageIndex{1}): Differentiating a Function Containing (\sin x) Find the derivative of (f(x)=5x^3\sin x). Solution Using the product rule, we have [ \begin{align} f'(x) &=\dfrac{d}{dx}(5x^3)⋅\sin x+\dfrac{d}{dx}(\sin x)⋅5x^3 \[4pt] &=15x^2⋅\sin x+\cos x⋅5x^3. \end{align}] After simplifying, we obtain [f′(x)=15x^2\sin x+5x^3\cos x. \nonumber ] Exercise (\PageIndex{1}) Find the derivative of (f(x)=\sin x\cos x.) Hint Don’t forget to use the product rule. Answer [f′(x)=\cos^2x−\sin^2x \nonumber ] Example (\PageIndex{2}): Finding the Derivative of a Function Containing cos x Find the derivative of (g(x)=\dfrac{\cos x}{4x^2}). Solution By applying the quotient rule, we have [g′(x)=\dfrac{(−\sin x)4x^2−8x(\cos x)}{(4x^2)^2}. \nonumber ] Simplifying, we obtain [g′(x)=\dfrac{−4x^2\sin x−8x\cos x}{16x^4}=\dfrac{−x\sin x−2\cos x}{4x^3}. \nonumber ] Exercise (\PageIndex{2}) Find the derivative of (f(x)=\dfrac{x}{\cos x}). Hint Use the quotient rule. Answer (f'(x) = \dfrac{\cos x+x\sin x}{\cos^2x}) Example (\PageIndex{3}): An Application to Velocity A particle moves along a coordinate axis in such a way that its position at time (t) is given by (s(t)=2\sin t−t) for (0≤t≤2π.) At what times is the particle at rest? Solution To determine when the particle is at rest, set (s′(t)=v(t)=0.) Begin by finding (s′(t).) We obtain [s′(t)=2 \cos t−1, \nonumber ] so we must solve [2 \cos t−1=0\text{ for }0≤t≤2π. \nonumber ] The solutions to this equation are (t=\dfrac{π}{3}) and (t=\dfrac{5π}{3}). Thus the particle is at rest at times (t=\dfrac{π}{3}) and (t=\dfrac{5π}{3}). Exercise (\PageIndex{3}) A particle moves along a coordinate axis. Its position at time (t) is given by (s(t)=\sqrt{3}t+2\cos t) for (0≤t≤2π.) At what times is the particle at rest? Hint Use the previous example as a guide. Answer (t=\dfrac{π}{3},\quad t=\dfrac{2π}{3}) Derivatives of Other Trigonometric Functions Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives. Example (\PageIndex{4}): The Derivative of the Tangent Function Find the derivative of (f(x)=\tan x.) Solution Start by expressing (\tan x ) as the quotient of (\sin x) and (\cos x): (f(x)=\tan x =\dfrac{\sin x}{\cos x}). Now apply the quotient rule to obtain (f′(x)=\dfrac{\cos x\cos x−(−\sin x)\sin x}{(\cos x)^2}). Simplifying, we obtain [f′(x)=\dfrac{\cos^2x+\sin^2 x}{\cos^2x}. \nonumber ] Recognizing that (\cos^2x+\sin^2x=1,) by the Pythagorean theorem, we now have [f′(x)=\dfrac{1}{\cos^2x} \nonumber ] Finally, use the identity (\sec x=\dfrac{1}{\cos x}) to obtain (f′(x)=\text{sec}^2 x). Exercise (\PageIndex{4}) Find the derivative of (f(x)=\cot x .) Hint Rewrite (\cot x ) as (\dfrac{\cos x}{\sin x}) and use the quotient rule. Answer (f′(x)=−\csc^2 x) The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem. Derivatives of (\tan x), (\cot x), (\sec x), and (\csc x) The derivatives of the remaining trigonometric functions are as follows: [\begin{align} \dfrac{d}{dx}(\tan x )&=\sec^2x\[4pt] \dfrac{d}{dx}(\cot x )&=−\csc^2x\[4pt] \dfrac{d}{dx}(\sec x)&=\sec x \tan x\[4pt] \dfrac{d}{dx}(\csc x)&=−\csc x \cot x. \end{align} \nonumber ] Example (\PageIndex{5}): Finding the Equation of a Tangent Line Find the equation of a line tangent to the graph of (f(x)=\cot x ) at (x=\frac{π}{4}). Solution To find the equation of the tangent line, we need a point and a slope at that point. To find the point, compute (f\left(\frac{π}{4}\right)=\cot\frac{π}{4}=1). Thus the tangent line passes through the point (\left(\frac{π}{4},1\right)). Next, find the slope by finding the derivative of (f(x)=\cot x ) and evaluating it at (\frac{π}{4}): (f′(x)=−\csc^2 x) and (f′\left(\frac{π}{4}\right)=−\csc^2\left(\frac{π}{4}\right)=−2). Using the point-slope equation of the line, we obtain (y−1=−2\left(x−\frac{π}{4}\right)) or equivalently, (y=−2x+1+\frac{π}{2}). Example (\PageIndex{6}): Finding the Derivative of Trigonometric Functions Find the derivative of (f(x)=\csc x+x\tan x .) Solution To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find (f′(x)=\dfrac{d}{dx}(\csc x)+\dfrac{d}{dx}(x\tan x )). In the first term, (\dfrac{d}{dx}(\csc x)=−\csc x\cot x ,) and by applying the product rule to the second term we obtain (\dfrac{d}{dx}(x\tan x )=(1)(\tan x )+(\sec^2 x)(x)). Therefore, we have (f′(x)=−\csc x\cot x +\tan x +x\sec^2 x). Exercise (\PageIndex{5}) Find the derivative of (f(x)=2\tan x −3\cot x .) Hint Use the rule for differentiating a constant multiple and the rule for differentiating a difference of two functions. Answer (f′(x)=2\sec^2 x+3\csc^2 x) Exercise (\PageIndex{6}) Find the slope of the line tangent to the graph of (f(x)=\tan x ) at (x=\dfrac{π}{6}). Hint Evaluate the derivative at (x=\dfrac{π}{6}). Answer (\dfrac{4}{3}) Higher-Order Derivatives The higher-order derivatives of (\sin x) and (\cos x) follow a repeating pattern. By following the pattern, we can find any higher-order derivative of (\sin x) and (\cos x.) Example (\PageIndex{7}): Finding Higher-Order Derivatives of (y=\sin x) Find the first four derivatives of (y=\sin x.) Solution Each step in the chain is straightforward: [\begin{align} y&=\sin x \[4pt] \dfrac{dy}{dx}&=\cos x \[4pt] \dfrac{d^2y}{dx^2}&=−\sin x \[4pt] \dfrac{d^3y}{dx^3}&=−\cos x \[4pt] \dfrac{d^4y}{dx^4}&=\sin x \end{align}] Analysis Once we recognize the pattern of derivatives, we can find any higher-order derivative by determining the step in the pattern to which it corresponds. For example, every fourth derivative of (\sin x) equals (\sin x), so [\dfrac{d^4}{dx^4}(\sin x)=\dfrac{d^8}{dx^8}(\sin x)=\dfrac{d^{12}}{dx^{12}}(\sin x)=…=\dfrac{d^{4n}}{dx^{4n}}(\sin x)=\sin x \nonumber ] [\dfrac{d^5}{dx^5}(\sin x)=\dfrac{d^9}{dx^9}(\sin x)=\dfrac{d^{13}}{dx^{13}}(\sin x)=…=\dfrac{d^{4n+1}}{dx^{4n+1}}(\sin x)=\cos x. \nonumber ] Exercise (\PageIndex{7}) For (y=\cos x), find (\dfrac{d^4y}{dx^4}). Hint See the previous example. Answer (\cos x) Example (\PageIndex{8}): Using the Pattern for Higher-Order Derivatives of (y=\sin x) Find (\dfrac{d^{74}}{dx^{74}}(\sin x)). Solution We can see right away that for the 74th derivative of (\sin x), (74=4(18)+2), so [\dfrac{d^{74}}{dx^{74}}(\sin x)=\dfrac{d^{72+2}}{dx^{72+2}}(\sin x)=\dfrac{d^2}{dx^2}(\sin x)=−\sin x. \nonumber ] Exercise (\PageIndex{8}) For (y=\sin x), find (\dfrac{d^{59}}{dx^{59}}(\sin x).) Hint (\dfrac{d^{59}}{dx^{59}}(\sin x)=\dfrac{d^{4⋅14+3}}{dx^{4⋅14+3}}(\sin x)) Answer (−\cos x) Example (\PageIndex{9}): An Application to Acceleration A particle moves along a coordinate axis in such a way that its position at time (t) is given by (s(t)=2−\sin t). Find (v(π/4)) and (a(π/4)). Compare these values and decide whether the particle is speeding up or slowing down. Solution First find (v(t)=s′(t)) [v(t)=s′(t)=−\cos t . \nonumber ] Thus, (v\left(\frac{π}{4}\right)=−\dfrac{1}{\sqrt{2}}=-\dfrac{\sqrt{2}}{2}). Next, find (a(t)=v′(t)). Thus, (a(t)=v′(t)=\sin t) and we have (a\left(\frac{π}{4}\right)=\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}). Since (v\left(\frac{π}{4}\right)=−\dfrac{\sqrt{2}}{2}<0) and (a\left(\frac{π}{4}\right)=\dfrac{\sqrt{2}}{2}>0), we see that velocity and acceleration are acting in opposite directions; that is, the object is being accelerated in the direction opposite to the direction in which it is traveling. Consequently, the particle is slowing down. Exercise (\PageIndex{9}) A block attached to a spring is moving vertically. Its position at time t is given by (s(t)=2\sin t). Find (v\left(\frac{5π}{6}\right)) and (a\left(\frac{5π}{6}\right)). Compare these values and decide whether the block is speeding up or slowing down. Hint Use Example (\PageIndex{9}) as a guide. Answer (v\left(\frac{5π}{6}\right)=−\sqrt{3}<0) and (a\left(\frac{5π}{6}\right)=−1<0). The block is speeding up. Key Concepts We can find the derivatives of (\sin x) and (\cos x) by using the definition of derivative and the limit formulas found earlier. The results are (\dfrac{d}{dx}\big(\sin x\big)=\cos x\quad\text{and}\quad\dfrac{d}{dx}\big(\cos x\big)=−\sin x). With these two formulas, we can determine the derivatives of all six basic trigonometric functions. Key Equations Derivative of sine function (\dfrac{d}{dx}(\sin x)=\cos x) Derivative of cosine function (\dfrac{d}{dx}(\cos x)=−\sin x) Derivative of tangent function (\dfrac{d}{dx}(\tan x )=\sec^2x) Derivative of cotangent function (\dfrac{d}{dx}(\cot x )=−\csc^2x) Derivative of secant function (\dfrac{d}{dx}(\sec x)=\sec x\tan x ) Derivative of cosecant function (\dfrac{d}{dx}(\csc x)=−\csc x\cot x ) This page titled 3.4: Derivatives of Trigonometric Functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform. Back to top 3.3E: Exercises for Section 3.3 3.4E: Exercises for Section 3.4 Was this article helpful? Yes No Recommended articles 3.5: Derivatives of Trigonometric FunctionsWe can find the derivatives of sin x and cos x by using the definition of derivative and the limit formulas found earlier. With these two formulas, we... 3.6: Derivatives of Trigonometric FunctionsWe can find the derivatives of sin x and cos x by using the definition of derivative and the limit formulas found earlier. With these two formulas, we... 3.6: Derivatives of Trigonometric FunctionsWe can find the derivatives of sin x and cos x by using the definition of derivative and the limit formulas found earlier. With these two formulas, we... 3.5: Derivatives of Trigonometric FunctionsWe can find the derivatives of sin x and cos x by using the definition of derivative and the limit formulas found earlier. With these two formulas, we... 2.3: Derivatives of Trigonometric FunctionsThis section explains how to differentiate the six basic trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. It covers th... Article typeSection or PageAuthorOpenStaxLicenseCC BY-NC-SALicense Version4.0Show Page TOCyesTranscludedyes Tags Derivative of cosecant function Derivative of cosine function Derivative of cotangent function Derivative of secant function Derivative of sine function Derivative of tangent function source-math-2494 source@ © Copyright 2025 Mathematics LibreTexts Powered by CXone Expert ® ? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org. Support Center How can we help? Contact Support Search the Insight Knowledge Base Check System Status×
1680	https://www.chemguide.co.uk/organicprops/alkanes/oxygen.html	\| \| \| THE COMBUSTION OF ALKANES AND CYCLOALKANES This page deals briefly with the combustion of alkanes and cycloalkanes. In fact, there is very little difference between the two. Complete combustion Complete combustion (given sufficient oxygen) of any hydrocarbon produces carbon dioxide and water. It is quite important that you can write properly balanced equations for these reactions, because they often come up as a part of thermochemistry calculations. Don't try to learn the equations - there are far too many possibilities. Work them out as you need them. Some are easier than others. For example, with alkanes, the ones with an even number of carbon atoms are marginally harder than those with an odd number! For example, with propane (C3H8), you can balance the carbons and hydrogens as you write the equation down. Your first draft would be: Counting the oxygens leads directly to the final version: With butane (C4H10), you can again balance the carbons and hydrogens as you write the equation down. Counting the oxygens leads to a slight problem - with 13 on the right-hand side. The simple trick is to allow yourself to have "six-and-a-half" O2 molecules on the left. If that offends you, double everything: \| \| Note:You might well come across either version of these equations. The ones with the halves left in are often used in calculation work. Forgive me if you find this last bit on equations unbearably trivial - not everybody does! Just be grateful that you have been well taught. \| \| Trends The hydrocarbons become harder to ignite as the molecules get bigger. This is because the bigger molecules don't vaporise so easily - the reaction is much better if the oxygen and the hydrocarbon are well mixed as gases. If the liquid isn't very volatile, only those molecules on the surface can react with the oxygen. Bigger molecules have greater Van der Waals attractions which makes it more difficult for them to break away from their neighbours and turn to a gas. \| \| Note:If you aren't sure about Van der Waals forces, then you should follow this link before you go on. Use the BACK button on your browser to return to this page. \| \| Provided the combustion is complete, all the hydrocarbons will burn with a blue flame. However, combustion tends to be less complete as the number of carbon atoms in the molecules rises. That means that the bigger the hydrocarbon, the more likely you are to get a yellow, smoky flame. Incomplete combustion Incomplete combustion (where there isn't enough oxygen present) can lead to the formation of carbon or carbon monoxide. As a simple way of thinking about it, the hydrogen in the hydrocarbon gets the first chance at the oxygen, and the carbon gets whatever is left over! The presence of glowing carbon particles in a flame turns it yellow, and black carbon is often visible in the smoke. Carbon monoxide is produced as a colourless poisonous gas. Why carbon monoxide is poisonous Oxygen is carried around the blood by haemoglobin (US: hemoglobin). Unfortunately carbon monoxide binds to exactly the same site on the haemoglobin that oxygen does. The difference is that carbon monoxide binds irreversibly - making that particular molecule of haemoglobin useless for carrying oxygen. If you breath in enough carbon monoxide you will die from a sort of internal suffocation. \| \| Note:There is more about haemoglobin towards the bottom of the page about complex ions. If you want some description of catalytic converters which help to remove carbon monoxide and some other pollutants, see the introductory page on catalysis. If you want full details about any of the environmental problems associated with burning hydrocarbons, you can't do better than explore the excellent US Environmental Protection Agency site. If you want details about the role of hydrocarbons in the formation of photochemical smog, a Google search on photochemical smog will quickly lead you to some detailed chemistry. Use the BACK button (or the HISTORY or GO menus) on your browser if you want to return to this page later. \| \| \| \| \| Questions to test your understanding If this is the first set of questions you have done, please read the introductory page before you start. You will need to use the BACK BUTTON on your browser to come back here afterwards. questions on the combustion of alkanes and cycloalkanes answers \| Where would you like to go now? To the alkanes menu . . . To the menu of other organic compounds . . . To Main Menu . . . © Jim Clark 2003 (modified August 2015) \|
1681	https://www.teacherspayteachers.com/browse?search=rational%20and%20irrational%20numbers%20lesson%20plan	Log InSign Up Cart is empty Total: View Wish ListView Cart Rational and Irrational Numbers Lesson Plan 100+results Relevance Rating Rating Count Price (Ascending) Price (Descending) Most Recent Relevance Rating Rating Count Price (Ascending) Price (Descending) Most Recent Filters Grade Elementary 4th grade 5th grade Middle school 6th grade 7th grade 8th grade High school 9th grade 10th grade 11th grade 12th grade Higher education Adult education Subject Art Coloring pages English language arts Writing Math Algebra Algebra 2 Applied math Arithmetic Basic operations Decimals Fractions Geometry Graphing Math test prep Mental math Numbers Order of operations Place value Statistics Other (math) Price Format Easel Easel Activities Google Apps Image Interactive whiteboards SMART Notebook Microsoft Microsoft Excel Microsoft OneDrive Microsoft PowerPoint Microsoft Word PDF Video Resource type Classroom decor Posters Word walls Forms Professional documents Teacher tools Lessons Homeschool curricula Lectures Outlines Teacher manuals Thematic unit plans Unit plans Yearlong curriculum Printables Hands-on activities Activities Centers Internet activities Games Project-based learning Instruction Handouts Interactive notebooks Scaffolded notes Student assessment Assessment Critical thinking and problem solving Test preparation Student practice Independent work packet Worksheets Graphic organizers Homework Workbooks Standard Theme Seasonal Back to school Summer Holiday Halloween Audience Homeschool Staff & administrators Rational vs Irrational Numbers: Lesson Plan, Activities & Visuals for 8th Grade Created by Breaking Printers Struggling to make the real number system real for your 8th-grade students? This resource is designed to take your students on a mathematical discovery, helping them master the crucial difference between rational and irrational numbers in a way that’s engaging, visual, and deeply conceptual. ✨ What's Inside?Comprehensive Lesson Plan: A detailed, step-by-step 60-75 minute lesson plan that covers everything from an engaging hook to a formative assessment. Aligned with Common Core Standard 8.NS.A. 5th - 9th Basic Operations, Decimals, Numbers CCSS 8.NS.A.1 , 8.NS.A.2 $2.95 Original Price $2.95 A BUNDLE: Rational and Irrational Numbers–Two Lesson Plans and Assessments Created by The Institute for Standards-Based Learning This 23-page bundle addresses grade 8 rational and irrational Common Core State Standards (CCSS) 8.NS.A.1 and 8. NS.A.2. The Bundle includes:1. Two Understanding by Design(UbD) lesson plans2. Worksheets, formative and summative assessments, applications, and critical problem multiple choice assessments with rubrics and solutions.Discover engaging educational resources that spark curiosity and inspire learning! Follow our store on TPT to stay updated with our latest products and offerings. If yo 8th Arithmetic, Math, Math Test Prep CCSS 8.NS.A.1 , 8.NS.A.2 $3.00 Original Price $3.00 Integers and Rational and irrational numbers - 2 PPT + lesson plan Created by Ilene's Room this is a full lesson to introduce integers and rational and irrational numbers. can be split into two lessons or one. includes two separate PPTs...one on integers, one on rational and irrational numbers. 6th - 8th Decimals, Numbers, Other (Math) CCSS 7.NS.A.1 , 7.NS.A.2d $2.00 Original Price $2.00 Rated 4.5 out of 5, based on 2 reviews 4.5 (2) Lesson Plan - Rational and Irrational Number Created by RESUL GEYIK Lesson plan template for rational and Irrational Numbers 8th Math FREE Rated 4.33 out of 5, based on 3 reviews 4.3 (3) Number Systems Unit (Rational/Irrational Numbers, Square Roots) 8th Grade Math Created by Teacher Twins Teaching square roots, cube roots, estimating square roots, irrational numbers, comparing and ordering real numbers? Make lesson planning easy with this no prep Number Systems - Rational and Irrational Numbers - Square and Cube Roots - Unit! Included are 10 ready-made lessons to teach Number Systems, Rational and Irrational Numbers, and Square and Cube Roots to your students. This unit is very easy to use and will save you a lot of time! Teach your students about Number Systems by using our 7th - 8th Algebra, Math, Numbers CCSS 8.NS.A.1 , 8.NS.A.2 Also included in: 8th Grade Math Curriculum for Full Year - 13 Units Plus Over 60 Extra Activities $14.50 Original Price $14.50 Rated 4.95 out of 5, based on 52 reviews 5.0 (52) Rational and Irrational Numbers Unit Bundle for 8th Grade Math Created by Rise over Run This unit bundle is full of engaging activities to help you teach your students all about identifying and estimating irrational numbers. These resources are included:Rational & Irrational Numbers VocabularyIdentifying Rational & Irrational Numbers PuzzleIdentifying Rational & Irrational Numbers Group ChallengeEstimating Roots PowerPointLocating Rational & Irrational Number on a Number LineRational & Irrational Numbers on a Number Line Group ChallengePlus in the Bonus File, you will receive a P 8th - 9th Math, Math Test Prep, Numbers CCSS 8.NS.A.1 , 8.NS.A.2 $13.00Original Price $13.00 $9.00 Price $9.00 Rated 5 out of 5, based on 6 reviews 5.0 (6) Rational and Irrational Numbers Guided Notes & Pixel Art \| Digital Print Bundle Created by Congruent Math Teach your 8th grade math students to classify rational and irrational numbers, approximate irrational square roots, convert between rational numbers (decimals, fractions, and percents), and convert repeating decimals to fractions with this Rational & Irrational Numbers Print & Digital Bundle. Products 1-4 are guided notes (sketch notes & doodles) covering concepts, vocabularies, and steps. Products 5-7 are sets of eight self-checking digital Google Sheets pixel art activities. This bundl 7th - 8th Math CCSS 8.NS.A.1 , 8.NS.A.2 $29.97Original Price $29.97 $19.99 Price $19.99 Rated 5 out of 5, based on 1 reviews 5.0 (1) Identifying & Classifying Rational and Irrational Number Activity Puzzles 8.NS.1 Created by Ready Lessons Looking for a fun interactive teaching idea for classifying rational and irrational numbers? Well bingo! Look no further as Identifying Rational and Irrational Numbers Activity Puzzles, for CCSS 8.NS.1, will serve as an exciting lesson plan for 8th grade middle school classrooms. This is a great resource to incorporate into your unit as a guided math center rotation, review game exercise, small group work, morning work, remediation, intervention, or rti. It can also be used as a quiz, drill or a 7th - 9th Math, Numbers CCSS 8.NS.A.1 $4.90 Original Price $4.90 Rated 4.83 out of 5, based on 12 reviews 4.8 (12) Classify Rational & Irrational Numbers Digital Pixel Art \| Real Numbers Integers Created by Congruent Math Students practice classifying numbers as real, rational, irrational, whole, or integer with this Classify Rational & Irrational Numbers Digital Pixel Art. Contains 2 pixel art Google Sheets with 24 questions. Level 1 involves identifying whether a number is rational or irrational. Level 2 involves classifying a number as a whole number, integer, rational number, irrational number, real number, or multiple. Images are autumn themed, perfect for fall, back to school, Halloween, and Thanksgiving 7th - 8th Math CCSS 8.NS.A.1 Also included in: Rational and Irrational Numbers Guided Notes & Pixel Art \| Digital Print Bundle $3.99 Original Price $3.99 Rated 5 out of 5, based on 1 reviews 5.0 (1) Rational and Irrational Numbers Lesson {Repeating and Terminating Decimals} Created by Taylor J's Math Materials Are you looking for a COMPLETE lesson to practice identifying terminating and repeating decimals? This product is complete with video lesson, guided notes, classroom activity & more! Great for use as a flipped lesson or mastery learning station!SAVE 20% WHEN YOU PURCHASE THE ENTIRE UNIT!INCLUDEDlesson plan & implementation guidecode for access to video lesson (must have access to YouTube)guided notesthis cut and paste in-class activity to practice identification of rational (terminating an 7th - 9th Algebra, Math, Math Test Prep CCSS 8.NS.A.1 $3.00 Original Price $3.00 Rated 4.62 out of 5, based on 8 reviews 4.6 (8) Algebra 1 Scope and Sequence Lesson Planning Guide Created by Rise over Run Plan your year of Algebra I with this FREE time-saving planning guide. If you love hands-on activities, discovery lessons, and tasks that get students thinking, this is for YOU!The entire year is broken down into units. Each unit has an information sheet with CCSS standards, daily objectives ("I can" statements), and resources to make planning simple. Included Units: Expressions & EquationsEquations with Variables on Both Sides & Literal EquationsInterpreting GraphsGraphing Linear Relationship 8th - 9th Algebra, Math Also included in: Algebra 1 and 8th Grade Math Curriculum Bundle for Entire Year FREE Rated 4.64 out of 5, based on 25 reviews 4.6 (25) Pre-Algebra Curriculum - Unit 1: Rational and Irrational Numbers Created by Activities by Jill Pre-Algebra Unit 1 Course Materials Unit 1: Rational & Irrational Numbers (This unit also includes Bell work for Pre-Algebra and Two Interactive Notebook Covers) - $109.00 value for $55.00 All of my Pre-Algebra course materials are now available bundled at a discounted price! Bell work, notes, instructional activities, study guides, tests, and more are included. This is a versatile collection of materials that may be used for pre-algebra students of varying abilities at multiple grade levels. 6th - 10th Algebra, Math $109.00Original Price $109.00 $55.00 Price $55.00 Rated 4.5 out of 5, based on 2 reviews 4.5 (2) The Ultimate Rational & Irrational Numbers Activity Pack \| 8th Grade Math Lesson Created by Breaking Printers Stop the blank stares and endless questions! Transform your number sense unit with this all-in-one lesson pack that makes the abstract world of rational and irrational numbers concrete, engaging, and actually fun. Watch your 8th graders go from confused to confident as they classify, sort, and connect numbers to the real world. This isn't just a worksheet; it's a complete, print-and-go teaching toolkit designed to save you precious prep time and keep your students hooked from start to finish. 7th - 9th Basic Operations, Numbers, Order of Operations CCSS 8.NS.A.1 , 8.NS.A.2 $3.95 Original Price $3.95 High School Math: Rational & Irrational Numbers \| Real World Algebra, Geometry Created by Breaking Printers Bring math to life with "The Real World is Irrational," a complete, project-based learning unit for Algebra 1, Geometry, and Algebra 2. This isn't just another worksheet; it's an engaging, multi-day lesson plan that shows students how numbers shape everything around us—from the music they hear to the buildings they admire. What's Inside? Complete 3-Day Lesson Plan: A step-by-step guide to take your students from introductory concepts to deep, project-based exploration.Engaging Student Workshe 9th - 12th Algebra 2, Geometry, Numbers CCSS HSN-RN.B.3 $2.00 Original Price $2.00 Ordering Real Numbers Lesson (Rational and Irrational) Created by Taylor J's Math Materials Are you looking for a COMPLETE lesson on ordering rational and irrational numbers? Students will learn about and practice ordering rational, irrational, square roots, and cube roots on the number line. This product is complete with video lesson, guided notes, riddle activity, and more! Great for use as a flipped lesson or mastery learning station!SAVE 20% WHEN YOU PURCHASE THE ENTIRE UNIT!Students will learn how to locate, compare and order real numbers using the number line. The main activ 7th - 9th Algebra, Math, Math Test Prep CCSS 8.NS.A.2 $4.00 Original Price $4.00 Rated 4 out of 5, based on 2 reviews 4.0 (2) Rational and Irrational Numbers Algebra Unit with Activities Bundle Created by Organized in Algebra Creating Algebra materials to meet the needs of all your students, especially those who struggle, takes TIME! Get back hours in your day with this full, differentiated mini unit on Rational and Irrational Numbers!This mini-unit includes learning targets, warm ups (bell ringers), notes, practice worksheets, activities, quizzes, and more. Download the preview to see examples of each! This curriculum was designed to support students on or below grade level in Algebra 1 that have struggled in mat 9th - 10th Algebra, Math CCSS HSN-RN.B.3 $22.00Original Price $22.00 $12.99 Price $12.99 Rated 5 out of 5, based on 2 reviews 5.0 (2) Rational and Irrational Numbers Unit 8th Grade Math Guided Notes Activity Quiz Created by Robin Cornecki - Round Robin Math Are you looking for a complete lesson plan unit on Rational and irrational numbers? Look no further than this complete unit that will teach your students everything they need to know about Real Numbers including Rational and Irrational Numbers through this investigation, notes, practice, homework, task card activity (digital & printable), digital drag, and drop activity, and an EDITABLE quiz! ✅ 16 pages plus answer keys of investigation, notes, and practice problems include: Keywords ( 7th - 9th Algebra, Applied Math, Math CCSS 8.NS.A.1 , 8.NS.A.2 , 8.EE.A.2 +1 $15.00Original Price $15.00 $12.00 Price $12.00 8th Grade Math Lesson Plans - Florida Standards Created by Math Radio Are you looking for a comprehensive resource to support your 8th-grade math instruction? Look no further! Our 8th Grade Math Lesson Plan Bundle is designed to provide you with everything you need to engage your students, differentiate instruction, and promote deep understanding of key mathematical concepts. This bundle includes a collection of 40 meticulously crafted lesson plans, aligned with the Florida BEST standards for 8th-grade math. Each lesson plan follows a consistent format, making it 8th Algebra, Math, Math Test Prep $5.00 Original Price $5.00 Algebra 1 Curriculum - Unit 1: Rational and Irrational Numbers Created by Activities by Jill Algebra 1 Curriculum - Unit 1: Rational and Irrational Numbers Unit 1: Rational & Irrational Numbers (This unit also includes Bell Work for Algebra, Two Interactive Notebook Covers, & Problem Trail Back To School Algebra 1) – $53.50 value for $27.00 ALL of my Algebra course materials are now available bundled at a discounted price as a Dropbox file! (Individual units are NOT Dropbox files, but are downloads from the TPT site.) Bell work, notes, instructional activities, study guides, tests 7th - 10th Algebra, Math $53.50Original Price $53.50 $27.00 Price $27.00 Rated 5 out of 5, based on 1 reviews 5.0 (1) 8th Grade Common Core Math Unit: The Number System Rational & Irrational Numbers Created by Past The Potholes This middle school math unit for The Number System includes detailed 3-part Google lesson plans and slides, activities, and assessments that correspond to the Common Core Math curriculum for 8th grade . This unit is part of our ENTIRE YEAR OF COMMON CORE MATH PACKAGE which includes all of our math resources for Grade 8th Grade. "This unit is excellent! It provides simple to use lessons that cover Statistics and Probability and align to the common core standards for grade 8. I love that each le 7th - 9th Geometry, Math, Numbers CCSS 8.NS.A.1 , 8.NS.A.2 Also included in: 8th Grade Common Core Math: Entire Year of Lesson Plans, Activities, Assessments $15.00Original Price $15.00 Price $10.00 Rated 5 out of 5, based on 2 reviews 5.0 (2) The Real Number System: Lesson Plan Unit Created by Adele Levin This detailed mini lesson unit will help guide your teaching of The Real Number System from start to finish. Through engaging hands-on activities, students will... Know - and be able to classify - natural, whole, integer, rational, irrational and real numbers.Be able to identify rational numbers by writing them as a fraction.Be able to convert terminating and recurring decimals to fractions.Know what perfect squares are, and be able to identify them.Be able to place the square root of imperfec 6th - 8th Math, Numbers CCSS , 8.NS.A.1 , 8.NS.A.2 +1 Also included in: The Real Number System: UNIT BUNDLE Original Price $6.99 THE REAL NUMBER SYSTEM PDF & Powerpoint Algebra I Lesson Plan Created by Scholastic Champions Unlock the vast landscape of numbers with this engaging lesson plan bundle on The Real Number System (aligned with CCSS 6.NS.C.6, 7.NS.A.1, 8.NS.A.1)! Designed for middle school Algebra I students (available in both PDF and editable PowerPoint formats), this resource explores the intricate relationships within the real number system. Through interactive activities, clear explanations, and real-world examples, students will confidently classify numbers as natural, whole, integers, rational, and i 6th - 8th Algebra, Applied Math, Math CCSS , 7.NS.A.1 , 8.NS.A.1 $3.00 Original Price $3.00 Rational v. Irrational Numbers Lessons Plans for MN Math Standard 7.1.1.1 Created by Making Our Math Time Count Title: Rational v. Irrational Numbers Lessons Plans for MN Math Standard 7.1.1.1 This is a digital resource using Google Slides, Google Sheets, and Google Docs. MN Math Standard 7.1.1.1 Know that every rational number can be written as the ratio of two integers or as a terminating or repeating decimal. Recognize that π is not rational, but that it can be approximated by rational numbers such as 22/7 and 3.14 .Everything you need to teach and assess the MN Math Standard 7.1.1.1 the way it is mean 7th Math, Math Test Prep, Other (Math) Original Price $14.99 Rational and Irrational Numbers Unit Algebra Differentiated Created by Organized in Algebra Creating Algebra materials to meet the needs of all your students, especially those who struggle, takes TIME! Get back hours in your day with this full, differentiated mini unit on Rational and Irrational Numbers!This mini-unit includes learning targets, warm ups (bell ringers), notes, practice worksheets, quizzes, and more. Download the preview to see examples of each! This curriculum was designed to support students on or below grade level in Algebra 1 that have struggled in math. It includ 9th - 10th Algebra, Math CCSS HSN-RN.B.3 Also included in: Rational and Irrational Numbers Algebra Unit with Activities Bundle $15.00Original Price $15.00 Price $9.99 Showing 1-24of 100+ results TPT is the largest marketplace for PreK-12 resources, powered by a community of educators. Who we are We're hiring Press Blog Gift Cards Help & FAQ Security Privacy policy Student privacy Terms of service Tell us what you think Get our weekly newsletter with free resources, updates, and special offers. 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1682	https://www.youtube.com/watch?v=sIhE9pztPJA	The unit cost method for comparing cost Joel Speranza Math 26100 subscribers 26 likes Description 2362 views Posted: 8 Jan 2021 Find 100's more videos linked to the Australia Senior Maths Curriculum at There are videos for: Queensland: General Mathematics Queensland: Mathematical Methods Queensland: Mathematics Specialist Western Australia: Mathematics Applications Western Australia: Mathematics Methods Western Australia: Mathematics Specialist 1 comments Transcript: it is true that some things at school you won't really use much once you leave but the unit cost method you'll use all the time you'll go to the supermarket you'll see a whole isle of toilet paper and you'll wonder hmm should i buy the packet of 12 rolls for 14.88 or should i buy the packet of 25 rolls for 31.50 which of these is the better deal because i'm going to use a lot of toilet paper anyway right over the next three four five months i may as well have 25 rolls if it's cheaper to buy it like this than it is to buy it like this and you can use the unit cost method to figure it out now it's called the unit cost method because it figures out what one unit of stuff is worth okay in other words what one roll of toilet paper is worth and then if you know what one roll of this toilet paper is and you know what one roll of this toilet paper is you can then directly compare them we can't directly compare these because this is 12 rolls and this is 25 rows all right so let's figure out what one roll of toilet paper costs so one roll or one unit of the item is equal to the total cost over the number of items now in this case 14.88 divided by 12 because that's the number of items now if you type that into your calculator you're going to get one dollar and 24 cents each roll of toilet paper is worth a dollar twenty four now what about this one one roll will be equal to the total cost over the number of items now in this case that's 31.50 over 25 and when you type that into your calculator you're going to get 1.26 now what does this mean well if you buy in packs of 12 you're paying a dollar 24 per roll of toilet paper if you're buying in packs of 25 you're paying a dollar 26 per roll of toilet paper you're much or you're a little bit better off buying this packet then you are buying this packet you should always buy in rolls of 12 in packets of 12 not in packets of 25. now you might be saying well you didn't save much yeah but you might save a little bit more depending on how they've set it up but across the entirety of your shop you buy some toilet paper here you buy some detergent here you buy something else here you can save a lot of money and it's nice to save money because you can spend that money on other stuff so we'll do it with dishwashing liquid as well here so you can see that doesn't need to be nice little packets like the toilet paper one roll one roll we can do 400 mils of dishwashing liquid at a dollar twelve nine hundred mils at two dollars twenty five now we can set this up we can figure out what one unit is and we might call it one unit one mil of dishwashing liquid now one mil of dishwashing liquid is going to be equal to the total cost over the not number of items but in this case the amount so the total cost is dollar 12 and we divide that by the amount 400 and we'll get an answer in this case .0028 that's a very small number here it's it's less than a cent all right now what about this one here well one mil is going to be equal to the total cost over the total amount now that's going to be 2.25 divided by 900 and that's going to be 0.0025 all right 0.0025 is smaller than 0.0028 therefore i should buy this packet here it's cheaper to buy it in the 900ml bottle than it is to buy it in the 400ml bottle now in this case and in this case i've done some comparing but you can also do something else with the unit cost method and that's to figure out how much a certain amount of dishwashing liquid or toilet paper might cost so someone else might ask you how much is 350 rolls of toilet paper now it depends on how this question is asked but if it's asked just like this how much for 350 rolls of toilet paper the answer is total cost equals 350 times the unit cost 1.24 and that is going to be 434 dollars now i said it depends on has questions asked right because if you needed to buy 350 rolls of toilet paper you couldn't do it because 12 doesn't go into 350 neatly right if you do 350 divided by 12 you don't get a nice neat number of these packets you would actually need to buy more than 350 rolls i'll show you what i mean if i do 350 divided by 12 my answer is 29.17 now that means that if i want exactly 350 rolls of toilet paper i'd have to buy 29.1 packets of toilet paper but you can't buy 29.17 packets of toilet paper so you have to round up to 30 packets of toilet paper and if you have to round up to 30 packets of toilet paper it means that the total cost is going to be 30 packets of toilet paper times not a dollar 24 but the total cost of each packet 14.88 which is 446.40 so two answers here 446.40 434 dollars they're different things this one is how much for 350 rolls of toilet paper this question is more complicated it's if you have to buy 350 rolls of toilet paper in these packets how much will it cost you'll have to round up to 30 packets and therefore it's going to be 446.40 all right that is the unit cost method you divide down so you find the cost of one item and then you can multiply up to find out whatever you want to find out
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If contacting us regarding your consent, please state your consent ID and date from that page. skip to main content Advanced Search Proceedings of the ACM on Programming Languages research-article Open access Share on Non-linear reasoning for invariant synthesis Authors: Zachary Kincaid Zachary Kincaid Princeton University, USA View Profile , John Cyphert John Cyphert University of Wisconsin-Madison, USA View Profile , Jason Breck Jason Breck University of Wisconsin-Madison, USA View Profile , Thomas Reps Thomas Reps University of Wisconsin-Madison, USA / GrammaTech, USA View Profile Authors Info & Claims Proceedings of the ACM on Programming Languages, Volume 2, Issue POPL Article No.: 54, Pages 1 - 33 Published: 27 December 2017 Publication History Metrics Total Citations56Total Downloads1,169 Last 12 Months217 Last 6 weeks34 PDFeReader Abstract Automatic generation of non-linear loop invariants is a long-standing challenge in program analysis, with many applications. For instance, reasoning about exponentials provides a way to find invariants of digital-filter programs, and reasoning about polynomials and/or logarithms is needed for establishing invariants that describe the time or memory usage of many well-known algorithms. An appealing approach to this challenge is to exploit the powerful recurrence-solving techniques that have been developed in the field of computer algebra, which can compute exact characterizations of non-linear repetitive behavior. However, there is a gap between the capabilities of recurrence solvers and the needs of program analysis: (1) loop bodies are not merely systems of recurrence relations---they may contain conditional branches, nested loops, non-deterministic assignments, etc., and (2) a client program analyzer must be able to reason about the closed-form solutions produced by a recurrence solver (e.g., to prove assertions). This paper presents a method for generating non-linear invariants of general loops based on analyzing recurrence relations. The key components are an abstract domain for reasoning about non-linear arithmetic, a semantics-based method for extracting recurrence relations from loop bodies, and a recurrence solver that avoids closed forms that involve complex or irrational numbers. Our technique has been implemented in a program analyzer that can analyze general loops and mutually recursive procedures. Our experiments show that our technique shows promise for non-linear assertion-checking and resource-bound generation. Formats available You can view the full content in the following formats: PDF Supplementary Material Auxiliary Archive (popl18-p201-aux.zip) The artifact for this publication is a zip file containing a virtual machine in OVA (Open Virtualization Archive) format. The virtual machine contains an installation of ICRA, which is the program analysis tool that implements the ideas described in the associated publication, "Non-Linear Reasoning for Invariant Synthesis." For more information about the virtual machine, and for detailed information about how to use the virtual machine to run ICRA, see the "/home/icrauser/Code/README.txt" file inside the virtual machine. Download 3039.60 MB WEBM File (invariantsynthesis.webm) Download 88.96 MB References E. Albert, P. Arenas, S. Genaim, and G. Puebla. 2008. Automatic Inference of Upper Bounds for Recurrence Relations in Cost Analysis. In SAS. 221–237. Digital Library Google Scholar E. Albert, S. Genaim, and A.N. Masud. 2013. On the inference of resource usage upper, lower bounds. Trans. on Computational Logic 14, 3 (2013). Digital Library Google Scholar Z. Ammarguellat and W. L. Harrison, III. 1990. Automatic recognition of induction variables and recurrence relations by abstract interpretation. 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Google Scholar Cited By View all Cao WWu GXu TYao YWei HChen TMa X(2025)Clause2Inv: A Generate-Combine-Check Framework for Loop Invariant InferenceProceedings of the ACM on Software Engineering10.1145/37289202:ISSTA(1009-1030)Online publication date: 22-Jun-2025 Wu HWang QXue BZhan NZhi LYang Z(2025)Synthesizing Invariants for Polynomial Programs by Semidefinite ProgrammingACM Transactions on Programming Languages and Systems10.1145/370855947:1(1-35)Online publication date: 17-Feb-2025 Ke JFu HLiu HSun ZChen LLi G(2025)Affine Disjunctive Invariant Generation with Farkas’ LemmaVerification, Model Checking, and Abstract Interpretation10.1007/978-3-031-82700-6_9(187-213)Online publication date: 20-Jan-2025 Show More Cited By Index Terms Non-linear reasoning for invariant synthesis Software and its engineering Software organization and properties Software functional properties Formal methods Automated static analysis Theory of computation Semantics and reasoning Program reasoning Program analysis Recommendations ### A note on a family of two-variable polynomials Abstract The main object of this paper is to construct a two-variable analogue of Jacobi polynomials and to give some properties of these polynomials. We show that these polynomials are orthogonal, then we obtain various recurrence formulas for them. ... Read More ### Laguerre–Angelesco multiple orthogonal polynomials on an r-star Abstract We investigate the type I and type II multiple orthogonal polynomials on an r-star with weight function \| x \| β e − x r, with β > − 1. Each measure μ j, for 1 ≤ j ≤ r, is supported on the semi-infinite interval [ 0 , ω j − 1 ∞ ) with ω ... Read More ### Derivatives of a finite class of orthogonal polynomials defined on the positive real line related to F-distribution Among the six classes of classical orthogonal polynomials, three of them are infinite, namely Jacobi, Hermite and Laguerre and the remaining three are finite and characterized by Masjed Jamei (2002) . In this work, we consider derivatives of one such ... 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Want a personalized experience? Download our free app Follow Us Baby & Toddler Breastfeeding Breastfeeding Let-Down 101: Everything You Should Know During a breastfeeding let-down, your body’s releasing milk for baby. Here are all the details—including troubleshooting tips. save article Save this article to view it later on your Bump dashboard . It’s free! ByNatalie Gontcharova,Senior Editor Updated February 26, 2024 Medically Reviewed byKendra Segura, MD \| Fact Checked by G. O’Hara Image:Rawpixel.com \| Shutterstock If you’re breastfeeding or pumping, chances are you’ve felt the pins-and-needles sensation of the let-down reflex. But what exactly is a breastfeeding let-down and why does it happen? And what should you do if you have a painful let-down—or if it just isn’t working? Read on for expert advice. In this article: What is the breastfeeding let-down reflex? What does let-down feel like? How to stimulate the let-down reflex What to do about painful let-down What to do about an overactive let-down What to do about a slow let-down Frequently asked questions What Is the Breastfeeding Let-Down Reflex? The breastfeeding let-down reflex, aka oxytocin reflex or milk-ejection reflex, occurs when a surge of the hormone oxytocin triggers the milk ducts to contract and release milk for baby, says Jacque Ordner, IBCLC, BSN, RN, a lactation consultant, registered nurse and medical advisor at Motif Medical. This typically happens because of “nipple stimulation or thinking about or being close to baby,” she says. Oxytocin is also known as the “love hormone,” notes Nicole Peluso, IBCLC, manager of lactation services and perinatal education at Aeroflow Breastpumps. “It’s released in many situations, including lovemaking, nurturing, childbirth and lactation,” she says. What Does Let-Down Feel Like? There’s a wide range of “normal” when it comes to what breastfeeding let-down feels like, experts say. “Some women feel the let-down as a tingly, warm sensation, while others may feel sharp pain, and some feel nothing at all,” says Jaimie Zaki, IBCLC, MCPCD, a lactation consultant, doula and author. Ordner agrees: “Many women describe the let-down sensation as feeling a gentle tightening or squeezing within their breasts. Others feel a pins-and-needles type of sensation.” Moreover, it’s also completely fine and normal not to feel let-down at all. The release of oxytocin can have a calming effect, says Peluso. “Women often feel a sense of calm or relaxation as the release happens,” she says. “Some may suddenly feel very sleepy due to the relaxing effects of oxytocin.” It’s important to note that while the effects of oxytocin are typically positive, a 2019 study suggests that as many as 9 percent of breastfeeding people experience D-MER, an abrupt, negative emotional response to let-down. If this sounds like you, know you’re not alone; be sure to reach out to your provider. Related Video How to Stimulate the Let-Down Reflex “Nipple stimulation is the most reliable way to trigger let-down,” says Ordner. “This can be from baby’s suckling, your hands or the use of a breast pump.” You can also encourage oxytocin release simply by thinking about baby, says Peluso. “Looking at baby, smelling an article of baby’s clothing, or listening to baby’s coos or noises can help raise oxytocin levels,” she says. “These activities can also help make pumping easier when mothers are at work or away from their babies.” Ordner suggests looking at pictures or videos of baby—you likely have no shortage of those! To get oxytocin flowing, it’s also important to relax, notes Cleveland Clinic. (Easier said than done, we know.) Try stress-managing techniques like deep breathing or listening to calming music. Another popular tip is to get warm, since being cold can make your body tense up: Throw on a cozy blanket, or use a warm washcloth on your breasts to encourage milk to flow. What to Do About Painful Let-Down There are several reasons painful let-down can happen. Ordner says it could be due to clogged milk ducts—which can lead to breast inflammation, or mastitis—oversupply, nipple damage or an infection. Zaki says that painful let-down could also be related to Raynaud’s Phenomenon of the Nipple, a treatable condition that gets worse when you’re exposed to cold. “For others yet, past trauma can amplify the intensity of painful let-downs,” she adds. No matter what the reason for your painful let-down, make sure to reach out to an International Board Certified Lactation Consultant (IBCLC) if you need help. Applying heat can help deal with painful let-down, says Peluso. Zaki adds: “Some women may find that warmth can help ease physical discomfort, while others may find cold compresses helpful.” What to Do About an Overactive Let-Down “If baby tends to cough, gulp, sputter or choke with let-down, you may have what’s often called an overactive let-down or forceful milk ejection reflex,” explains Ordner. Other signs include baby arching or stiffening—or milk sprays when baby comes off the breast, she adds. You may also notice excessive leaking from the side where baby’s not feeding, says La Leche League Canada. Here are a few tips that could help with an overactive let-down: Change positions. A laid-back or side-lying breastfeeding position can help baby better manage the flow of milk, says Ordner. Peluso also recommends an upright position. “[It can] make the milk flow against gravity,” she says. Hand-express. Sometimes, hand-expressing before latching baby can make it easier for baby to handle the milk flow, says Ordner. Take frequent breaks for burping. This can help slow your flow, says Peluso. Apply (a little) pressure. “Using your hand to gently press your breast toward your chest wall for some counter pressure can help slow the flow,” says Ordner. Ordner says that overactive let-down gets better for many moms when their milk supply regulates, which often happens between six and 12 weeks postpartum. “If you think an oversupply is what’s causing your overactive let-down, an IBCLC can help you create a safe plan to manage it,” she adds. What to Do About a Slow Let-Down? The best way to encourage let-down is to spend as much time as possible skin-to-skin with baby before breastfeeding or pumping sessions, says Ordner. “Babywearing is a great way to keep baby close, whether skin-to-skin or not, and can also help boost oxytocin levels to aid let-down,” she says. Rest assured that a slow let-down doesn’t necessarily mean you have low milk supply, according to experts. The important thing is that baby is gaining weight properly, as well as pooping and peeing enough, says Peluso. “If that’s all within an expected range, Mom can be assured she has enough milk,” she says. Check in with a lactation consultant if you’re unsure. Frequently Asked Questions What is a “phantom let-down”? “Phantom let-down is the sense that you have a let-down of milk even after you’ve stopped lactating,” says Peluso. “It’s often triggered by the same thing that triggers an actual let-down—hearing baby cry or thinking about a person you love.” Should you worry about a phantom let-down? Typically no, says Ordner: “As long as there’s no nipple discharge or breast abnormalities associated with phantom let-down, there’s nothing to worry about.” Why do I get a let-down headache and what can I do about it? People can get headaches related to breastfeeding let-down because of postpartum hormonal fluctuations, says Ordner. She recommends eating a nutrient-dense diet, avoiding processed foods and staying hydrated. Making sure you have micronutrients such as magnesium and zinc can also be important, says Zaki. Can you stop your let-down reflex? If you feel a let-down coming on at an inconvenient time, experts recommend using your wrist or forearm to apply firm pressure to your breast for a few seconds to slow the response. When does let-down stop? The breastfeeding let-down reflex starts to fade as you wean baby, and stops altogether once milk production tapers down, says Ordner. “I don’t think there’s good science on when it stops,” adds Peluso. “Let-down will happen for the whole time that you breastfeed, but your sensations of let-down will usually diminish the longer you feed.” When your period returns, the let-down sensation tends to diminish the most, she says. Knowing the facts about your let-down reflex can help you figure out what to do if something goes wrong, such as if your let-down is painful. In the meantime, let the milk flow and enjoy all the benefits oxytocin can provide! Please note: The Bump and the materials and information it contains are not intended to, and do not constitute, medical or other health advice or diagnosis and should not be used as such. You should always consult with a qualified physician or health professional about your specific circumstances. Plus, more from The Bump: Is D-MER the Reason You Feel Sad While Breastfeeding? Extended Breastfeeding 101: Everything You Need to Know 30 Breastfeeding Tips Every Nursing Mom Should Know Sources Jacque Ordner, IBCLC, BSN, RN, RLC, is a lactation consultant, registered nurse and medical advisor at Motif Medical. Nicole Peluso, IBCLC, CD, CAHPE, is the manager of lactation services and perinatal education at Aeroflow Breastpumps. She is also a parenting educator certified by Attachment Parenting International, a La Leche League Leader, and a birth and postpartum doula. Jaimie Zaki, IBCLC, MCPCD, is a lactation consultant, doula and author in Wichita Falls, Texas. World Health Organization, Infant and Young Child Feeding: Model Chapter for Textbooks for Medical Students and Allied Health Professionals, 2009 Breastfeeding Medicine, Dysphoric Milk Ejection Reflex: A Descriptive Study, November 2019 Cleveland Clinic, What to Know About the Breastfeeding Let-Down, September 2023 Pediatrics (American Academy of Pediatrics), Raynaud’s Phenomenon of the Nipple: A Treatable Cause of Painful Breastfeeding, April 2004 La Leche League Canada, Oversupply and Forceful Letdown (Milk Ejection Reflex), November 2022 Learn how we ensure the accuracy of our content through our editorial and medical review process. save article Was this article helpful? Yes No Amazon Baby Registry Free Welcome Box 15% Registry Discount Free 1-Year Returns Quick Free Shipping Group Gifting Thank-You List Collect Amazon Freebies View all registry retailers Subject to availability and Retailer's terms. We earn commissions from these links. Want a personalized experience? Download The Bump App for daily pregnancy and newborn updates with our free app Want a personalized experience? Download The Bump App for daily pregnancy and newborn updates with our free app Your Child’s Age Select your child's age in months or weeks to begin tracking their development. First Year Second Year Third Year Baby Month by Month Baby Week by Week NB123456789101111314151617181920212223225262728293031323334353 Advertisement Most popular on The Bump Strategies for Changing a Squirmy Toddler's Diaper Baby Poop 101: A Comprehensive Guide to Newborn and Infant Poop What to Do When Baby Refuses the Bottle Advertisement Advertisement Most popular on The Bump When Can Newborns Go Outside? Diaper Bag Checklist: What to Pack in a Diaper Bag Your Ultimate Guide to Baby Rashes Advertisement ADVERTISEMENT Next on Your Reading List One-Third of Working Parents Still Lack a Place to Pump By Wyndi Kappes Half of Moms Say They Stopped Breastfeeding Earlier Than They Wanted By Wyndi Kappes Medela Shares Why It Phased Out Bottles in Commitment to Breastfeeding By Wyndi Kappes ADVERTISEMENT Feeling Guilty About Not Breastfeeding? Read This. Medically Reviewed by Dina DiMaggio Walters, MD When Does Breastfeeding Get Easier? 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1685	https://math.stackexchange.com/questions/770248/counting-sets-addition-principle	discrete mathematics - Counting sets - Addition principle - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Counting sets - Addition principle Ask Question Asked 11 years, 5 months ago Modified11 years, 5 months ago Viewed 4k times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. Theorem: If A and B are non-empty sets, and A and B are disjoint, then \|A⋃B\|=\|A\|+\|B\|\|A⋃B\|=\|A\|+\|B\| If I have n sets and all of them are disjoint, then \|A 1⋃A 2⋃...⋃A n\|=\|A 1\|+\|A 2\|+...+\|A n\|\|A 1⋃A 2⋃...⋃A n\|=\|A 1\|+\|A 2\|+...+\|A n\| If I want to prove that, can I do like this? 1) By the theorem we know that is it true fpr n=2 2) we suppose that it is A 1,A 2,...,A k A 1,A 2,...,A k is disjoint then \|A 1⋃A 2⋃...⋃A k\|=\|A 1\|+\|A 2\|+...+\|A k\|\|A 1⋃A 2⋃...⋃A k\|=\|A 1\|+\|A 2\|+...+\|A k\| 3) we suppose that A 1,A 2,...,A k,A k+1 A 1,A 2,...,A k,A k+1 is disjoint sets, then it follows that A 1⋃A 2⋃...⋃A k A 1⋃A 2⋃...⋃A k and A k+1 A k+1 are disjoint sets. The theorem above then applies that \|A 1⋃A 2⋃...⋃A k⋃A k+1\|=\|A 1\|+\|A 2\|+...+\|A k\|+\|A k+1\|\|A 1⋃A 2⋃...⋃A k⋃A k+1\|=\|A 1\|+\|A 2\|+...+\|A k\|+\|A k+1\| and by 2) \|A 1⋃A 2⋃...⋃A k\|=\|A 1\|+\|A 2\|+...+\|A k\|\|A 1⋃A 2⋃...⋃A k\|=\|A 1\|+\|A 2\|+...+\|A k\| so \|A 1⋃A 2⋃...⋃A k⋃A k+1\|=\|A 1\|+\|A 2\|+...+\|A k\|+\|A k+1\|\|A 1⋃A 2⋃...⋃A k⋃A k+1\|=\|A 1\|+\|A 2\|+...+\|A k\|+\|A k+1\| The result follows now from induction principle. discrete-mathematics Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Apr 26, 2014 at 17:36 Unwisdom 4,600 20 20 silver badges 23 23 bronze badges asked Apr 26, 2014 at 16:51 KimKim 35 1 1 silver badge 7 7 bronze badges 0 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 0 Save this answer. Show activity on this post. What you have is fine. I've tidied it up a little, exploiting the ⋃⋃ notation a little more here: Theorem: If A A and B B are non-empty sets, and A A and B B are disjoint, then \|A∪B\|=\|A\|+\|B\|\|A∪B\|=\|A\|+\|B\| Proposition: If we have n n pairwise disjoint sets, then ∣∣⋃n j=1 A j∣∣=∑n j=1\|A j\|\|⋃j=1 n A j\|=∑j=1 n\|A j\| Proof of Proposition: The case of n=1 n=1 is trivial. Base Case: By the theorem we know that the proposition holds for n=2 n=2 Induction hypothesis: Assume that for some k k, whenever A 1,A 2,…,A k A 1,A 2,…,A k are disjoint then ∣∣∣∣⋃j=1 k A j∣∣∣∣=∑j=1 k\|A j\|\|⋃j=1 k A j\|=∑j=1 k\|A j\| We now suppose that A 1,A 2,…,A k,A k+1 A 1,A 2,…,A k,A k+1 are pairwise disjoint. It follows that ⋃k j=1 A j⋃j=1 k A j and A k+1 A k+1 are disjoint sets. The Theorem tells us that ∣∣∣∣⋃j=1 k+1 A j∣∣∣∣=∣∣∣∣(⋃j=1 k A j)∪A k+1∣∣∣∣=∣∣∣∣⋃j=1 k A j∣∣∣∣+\|A k+1\|.\|⋃j=1 k+1 A j\|=\|(⋃j=1 k A j)∪A k+1\|=\|⋃j=1 k A j\|+\|A k+1\|. Now applying the induction hypothesis to this, we get ∣∣∣∣⋃j=1 k+1 A j∣∣∣∣=(∑j=1 k\|A j\|)+\|A k+1\|=∑j=1 k+1\|A j\|\|⋃j=1 k+1 A j\|=(∑j=1 k\|A j\|)+\|A k+1\|=∑j=1 k+1\|A j\| concluding our proof. I've tried to follow your reasoning as closely as I could here. It is actually possible to be slightly more efficient. The base case really should be n=1 n=1. The proof for n=2 n=2 follows from this base case and the argument for the inductive step (which is really just an invocation of the Theorem). Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Apr 26, 2014 at 18:34 answered Apr 26, 2014 at 17:31 UnwisdomUnwisdom 4,600 20 20 silver badges 23 23 bronze badges 3 I originally included this version as an edit, but it is more appropriate here. Sorry about any confusion.Unwisdom –Unwisdom 2014-04-26 17:37:17 +00:00 Commented Apr 26, 2014 at 17:37 1 The very first indexed union symbol has a typo in the lower index, it says n=j n=j instead of j=1 j=1.Asaf Karagila –Asaf Karagila♦ 2014-04-26 17:58:12 +00:00 Commented Apr 26, 2014 at 17:58 @AsafKaragila - Thanks - I have made the correction.Unwisdom –Unwisdom 2014-04-26 18:13:25 +00:00 Commented Apr 26, 2014 at 18:13 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions discrete-mathematics See similar questions with these tags. 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1686	https://www2.lbl.gov/Science-Articles/Archive/Kamen-Fermi-Award.html	Martin Kamen, discoverer of carbon-14, wins Enrico Fermi Award Martin Kamen, Who Discovered Carbon-14 Here, Wins Fermi Award December 15, 1995 By Lynn Yarris, LCYarris@LBL.gov M artin Kamen, the chemist who used the 60-Inch Cyclotron in the days of the Radiation Laboratory to create carbon-14 and change biochemistry research forever, has been named one of two winners of this year's Enrico Fermi Award. The 82-year old Kamen is joined by 83-year-old physicist Ugo Fano, who won for his pioneering contributions to the theory of atomic and radiation physics. The Fermi award, which was announced by President Clinton on Dec. 12, is the nation's oldest prize for achievements in science and technology. It is granted for lifetime achievements in the field of nuclear energy. Winners receive a gold medal and a $100,000 honorarium. In winning the award, Kamen joins a list that includes two of his colleagues from the Rad Lab--Glenn Seaborg and Luis Alvarez. Kamen is currently professor emeritus at both UC San Diego and the University of Southern California. During the 1950s and '60s, he cemented his claims to scientific greatness with his ground-breaking research in the field of bacterial cytochromes and photosynthesis. However, it was his discovery of carbon-14, the radioisotope of carbon used to trace biochemical pathways and mechanisms and to date archeological and anthropological objects, that proved to be the source of his brightest and darkest moments. Born in Toronto, Canada, in 1913, Kamen earned his B.S. and Ph.D. degrees in chemistry and physical chemistry from the University of Chicago. After completing his doctoral research on neutron scattering, he began his career in 1937 as a radiochemist with the legendary group under Ernest O. Lawrence at the Rad Lab. Kamen was interested in studying plant photosynthesis and the related problem of carbon dioxide assimilation. In 1938, working with the late UC chemist Samuel Ruben, Kamen demonstrated that water is the source of molecular oxygen in photosynthesis and not carbon dioxide. This work had been accomplished through the use of carbon-11, which was the only radiocarbon known at that time. Kamen, however, was frustrated by that tracer's short half-life (21 minutes) and set out to find a radioisotope of carbon that would be better suited for biochemistry. In 1940, again working with Ruben, Kamen used the 60-Inch Cyclotron to bombard a graphite target with a beam of deuterium nuclei. The result was carbon-14, which, with a half-life of 5,000 years, allowed biochemists to trace carbon's movement through photosynthesis, metabolism, and a host of other biochemical processes. Kamen's resulting scientific prominence, however, coupled with his liberal viewpoints and association with leftist intellectuals, brought him under the scrutiny of government agencies, including the FBI. In July 1944 he was declared a security risk and Lawrence was forced to dismiss him. A few years later, Kamen was brought before the House Un-American Activities Committee and his passport was revoked. He spent 10 years in the courts before finally clearing his name and winning back his passport. He also won settlements against the Washington Times-Herald and Chicago Tribune newspapers for publishing libelous articles about him.
1687	https://www.gov.uk/guidance/marburg-virus-disease-origins-reservoirs-transmission-and-guidelines	Skip to main content Emergency Alerts Test on Sunday 7 September, 3pm Guidance Marburg virus disease: origins, reservoirs, transmission and guidelines Marburg virus is a filovirus which, along with Ebola virus, can cause a severe and often fatal viral haemorrhagic fever (VHF). From: : UK Health Security Agency Published : 5 September 2014 Last updated : 29 July 2025 — See all updates Marburg virus belongs to the Filovirus family, along with Ebola. It can cause a severe and often fatal haemorrhagic fever called Marburg disease (MARD) which is clinically almost indistinguishable from Ebola virus disease. Marburg virus is known to affect both humans and non-human primates. MARD was first recognised in 1967 when outbreaks of haemorrhagic fever occurred simultaneously in laboratories in both Marburg and Frankfurt in Germany, and Belgrade in Yugoslavia (now Serbia). A total of 31 people became ill, including 25 laboratory workers, medical personnel and a family member who had cared for them. The laboratory workers all had contact with the blood, organs or cell-cultures from a batch of imported African green monkeys from north-western Uganda. It is generally accepted that Marburg virus is a zoonotic (animal borne) virus. Fruit bats (Rousettus aegyptii) are considered to be the natural host of the virus. Monkeys are susceptible to Marburg virus infection but are not considered the reservoir hosts as they usually die rapidly once infected. Experimental infections have shown how pigs are susceptible to filovirus infection and can shed the virus. Epidemiology Recorded cases of MARD are rare. Outbreaks and sporadic cases have been reported in Angola, the Democratic Republic of the Congo (DRC), Equatorial Guinea, Kenya, Ghana, Guinea, Rwanda, Uganda, South Africa (in a person who had recently travelled to Zimbabwe) and Tanzania. The first recorded large outbreak occurred in the DRC between 1998 and 2000 and involved 154 cases, including 128 deaths. The largest outbreak on record occurred in 2005 in Angola and involved 252 cases, including 227 deaths. While reports of MARD outbreaks in West Africa are rare, sporadic cases have been identified, including 1 fatal case reported from Guinea in August 2021, and 2 fatal cases reported from Ghana in July 2022. Cases detected outside the African Region are extremely rare. Two unrelated sporadic cases in travellers occurred during 2008 following visits to the ‘python cave’ in the Maramagambo Forest in western Uganda; this cave is home to a large colony of Egyptian fruit bats. Both people became ill upon return to their home country; one in the Netherlands and one in the United States. A single case was detected in Russia in 1990 and occurred through laboratory exposure. Map of MARD in Africa See the HCID: country specific risk webpage for further information on incidents, outbreaks and travel associated MARD cases outside of Africa. Symptoms The incubation period of MARD is typically 3 to 10 days, but can take up to 21 days from the date of exposure to the virus for symptoms to appear. There have been rare reports of longer incubation periods up to 28 days (although the precise mechanism of transmission in these cases was not well documented). The onset of illness is sudden, with: severe headache malaise high fever progressive and rapid debilitation By about the third day symptoms include: watery diarrhoea abdominal pain abdominal cramping nausea vomiting Symptoms can become increasingly severe, and many patients develop a maculopapular rash after 5 to 7 days. Severe cases usually exhibit some form of bleeding, including bleeding under the skin, bleeding from mucous membranes and from venepuncture sites. Many of the early symptoms of MARD are similar to those of other infectious diseases, such as malaria or typhoid. Confirmation of the disease requires laboratory testing. Transmission Evidence gathered to date suggests that natural infection is most likely associated with contact with Rousettus bat colonies. Subsequent transmission of virus from person-to-person requires close contact with blood or bodily fluids from an infected patient. Faeces, vomit, urine, saliva and respiratory secretions contain a high concentration of virus, particularly when these fluids contain the patient’s blood. Sexual transmission of the virus can occur, and the virus may remain in semen for several weeks after clinical recovery, with some reports of virus present up to 203 days after disease onset. Transmission of the virus via contaminated injection equipment or needle-stick injuries is associated with more severe disease. Close contact with the body or body fluids of people who have died of MARD during preparation for burial is a recognised source of infection. Diagnosis In the UK, clinicians who suspect that a patient may have MARD should seek urgent advice from the UK Health Security Agency (UKHSA)’s Imported Fever Service (IFS) on 0844 778 8990. The IFS operates 24/7 and provides advice on risk assessment, immediate management and infection control. The IFS will also coordinate urgent testing at UKHSA’s Rare and Imported Pathogens Laboratory (RIPL), Porton Down. RIPL provides polymerase chain reaction (PCR) testing for MARD including out of hours if indicated. See VHF sample testing advice. Treatment There are currently no licensed vaccines or specific antivirals to treat MARD. Treatment is therefore mainly supportive and includes: replacing blood components balancing fluids and electrolytes maintaining oxygen status and blood pressure organ support as needed UK guidelines The UK has specialist guidance on the management (including infection control) of patients with viral haemorrhagic fevers (VHFs) including MARD. The guidelines provide advice on risk assessment, testing and management of suspected MARD cases presenting to healthcare services within the UK. Read the Marburg contact tracing guidance for public health recommendations for asymptomatic contacts of MARD in UK settings. Prevention and control MARD is a VHF: measures for prevention of secondary transmission of Marburg virus are similar to those used for other haemorrhagic fever viruses, and focus on avoiding contact with infected bodily fluids. Suspected cases must be immediately isolated and assessed using appropriate VHF assessment personal protective equipment. See the ACDP algorithm and guidance on management of patients and the NIPCM for further information. In the UK, confirmed cases will be notified immediately to the High Consequence Infectious Diseases Network to arrange urgent transport to a High Level Isolation Unit. Updates to this page Published 5 September 2014 Last updated 29 July 2025 + show all updates Updated current accepted acronym from MVD (Marburg virus disease) to MARD (Marburg disease) Added a link to Marburg contact tracing guidance. Updated the epidemiology section and map. Added map and guidance updates. Updated to reflect the end of 2 MVD outbreaks. Update to reflect Marburg virus disease outbreak in Tanzania. Update to reflect Marburg virus disease outbreak in Equatorial Guinea. Updated epidemiological information relating to a Marburg virus outbreak. Updated with latest MVD information. Updated text to reflect 2021 case of MVD in Guinea. Updated as the Marburg outbreak in Uganda was declared over. Added new outbreak in Uganda. First published. Sign up for emails or print this page Contents Related content
1688	https://study.com/academy/lesson/copper-ii-oxide-formula-properties-structure.html	Copper II Oxide \| Formula, Properties & Structure - Lesson \| Study.com Log In Sign Up Menu Plans Courses By Subject College Courses High School Courses Middle School Courses Elementary School Courses By Subject Arts Business Computer Science Education & Teaching English (ELA) Foreign Language Health & Medicine History Humanities Math Psychology Science Social Science Subjects Art Business Computer Science Education & Teaching English Health & Medicine History Humanities Math Psychology Science Social Science Art Architecture Art History Design Performing Arts Visual Arts Business Accounting Business Administration Business Communication Business Ethics Business Intelligence Business Law Economics Finance Healthcare Administration Human Resources Information Technology International Business Operations Management Real Estate Sales & Marketing Computer Science Computer Engineering Computer Programming Cybersecurity Data Science Software Education & Teaching Education Law & Policy Pedagogy & Teaching Strategies Special & Specialized Education Student Support in Education Teaching English Language Learners English Grammar Literature Public Speaking Reading Vocabulary Writing & Composition Health & Medicine Counseling & Therapy Health Medicine Nursing Nutrition History US History World History Humanities Communication Ethics Foreign Languages Philosophy Religious Studies Math Algebra Basic Math Calculus Geometry Statistics Trigonometry Psychology Clinical & Abnormal Psychology Cognitive Science Developmental Psychology Educational Psychology Organizational Psychology Social Psychology Science Anatomy & Physiology Astronomy Biology Chemistry Earth Science Engineering Environmental Science Physics Scientific Research Social Science Anthropology Criminal Justice Geography Law Linguistics Political Science Sociology Teachers Teacher Certification Teaching Resources and Curriculum Skills Practice Lesson Plans Teacher Professional Development For schools & districts Certifications Teacher Certification Exams Nursing Exams Real Estate Exams Military Exams Finance Exams Human Resources Exams Counseling & Social Work Exams Allied Health & Medicine Exams All Test Prep Teacher Certification Exams Praxis Test Prep FTCE Test Prep TExES Test Prep CSET & CBEST Test Prep All Teacher Certification Test Prep Nursing Exams NCLEX Test Prep TEAS Test Prep HESI Test Prep All Nursing Test Prep Real Estate Exams Real Estate Sales Real Estate Brokers Real Estate Appraisals All Real Estate Test Prep Military Exams ASVAB Test Prep AFOQT Test Prep All Military Test Prep Finance Exams SIE Test Prep Series 6 Test Prep Series 65 Test Prep Series 66 Test Prep Series 7 Test Prep CPP Test Prep CMA Test Prep All Finance Test Prep Human Resources Exams SHRM Test Prep PHR Test Prep aPHR Test Prep PHRi Test Prep SPHR Test Prep All HR Test Prep Counseling & Social Work Exams NCE Test Prep NCMHCE Test Prep CPCE Test Prep ASWB Test Prep CRC Test Prep All Counseling & Social Work Test Prep Allied Health & Medicine Exams ASCP Test Prep CNA Test Prep CNS Test Prep All Medical Test Prep College Degrees College Credit Courses Partner Schools Success Stories Earn credit Sign Up Science Courses / Organic & Inorganic Compounds Study Guide Course Copper II Oxide \| Formula, Properties & Structure Lesson Transcript Maram Ghadban, Matthew Bergstresser, Christianlly Cena Author Maram Ghadban A freelance tutor currently pursuing a master's of science in chemical engineering. Graduated from the American University of the Middle East with a GPA of 3.87, performed a number of scientific primary and secondary research. Tutored university level students in various courses in chemical engineering, math, and art. Has experience tutoring middle school and high school level students in science courses. View bio Instructor Matthew Bergstresser Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years. View bio Expert Contributor Christianlly Cena Christianlly has taught college Physics, Natural science, Earth science, and facilitated laboratory courses. He has a master's degree in Physics and is currently pursuing his doctorate degree. View bio Learn to identify the copper oxide formula, copper(II) oxide formula and copper ion charge. See the preparation, properties, structure, and uses of copper oxides. Updated: 11/21/2023 Table of Contents Copper Oxide Formula Preparation of Copper(II) Oxide Properties of Copper Oxide Structure of Copper Oxide Uses of Copper Oxide Lesson Summary Show FAQs Activities Copper II Oxide: True or False Activity This activity will help you assess your knowledge of the chemical formula and physical properties of copper(II) oxide. Directions Determine whether the following statements are true or false. To do this, print or copy this page on a blank paper and underline or circle the answer. True \| False 1. The chemical element copper is a reddish metal that constitutes 46 percent of the Earth's crust. True \| False 2. In ionic bonds, a metal loses electrons while the nonmetal accepts those electrons. True \| False 3. As the copper atom loses electrons, it then becomes negatively charged. True \| False 4. The total charge within an ionic compound must be +2. True \| False 5. The chemical reaction between oxygen and copper forms copper(II) oxide. True \| False 6. Oxidation states are typically represented by integers, which can either be positive or negative. True \| False 7. Copper(II) oxide is made up of atoms arranged in a lattice, held together via electrostatic attraction. True \| False 8. CuO boils over 2000 degrees Celsius; it is highly-flammable and dissolves well in water. Answer Key False, because the correct statement is: The chemical element copper is a reddish metal that constitutes 0.007 percent of the Earth's crust. True False, because the correct statement is: As the copper atom loses electrons, it becomes positivelycharged. False, because the correct statement is: The total charge within an ionic compound must be 0 or neutral. True True True False, because the correct statement is: CuO boils over 2000 degrees Celsius; it is non-flammable and insoluble in water. How do you name Cu2O? This atom is composed of two copper ions and one oxygen ion. The oxygen's formal charge is -2, and copper's formal charge is +1. The copper ion with a +1 formal charge is called cuprous ion Cu+1. Cu2O's chemical name is cupric oxide or copper(I) oxide. What are the elements in copper(II) oxide? Copper(II) oxide is composed of two elements; oxygen (nonmetal) and copper (metal). This compound is formed when cupric ions Cu+2 ionically bond with oxygen ions O-2 to form the neutral CuO. How do you write copper(II) oxide? Copper (II) oxide is made of two ions; the cupric ion, Cu+2, and the oxygen ion, O-2. One cupric ion transfers its two electrons to the oxygen ion. Cu+2 + O-2 → CuO One cupric ion is required to neutralize oxygen's -2. The chemical formula for this compound is CuO. How do you make Cu2O? Cu2O is copper(I) oxide, it can be produced through the reduction of copper(II) oxide CuO. The reduction of copper(II) oxide solution involves its treatment with sulfuric dioxide. Create an account Table of Contents Copper Oxide Formula Preparation of Copper(II) Oxide Properties of Copper Oxide Structure of Copper Oxide Uses of Copper Oxide Lesson Summary Show Copper Oxide Formula -------------------- Copper oxide is an inorganic chemical compound that exist as a pitch black solid at room temperature. It is a naturally occurring mineral that is composed of the nonmetal, oxygen, and the transition metal, copper. It is found naturally in a mineral ore by the name tenorite. The copper oxide formula can be derived by determining the ions that make this compound, which are the oxygen ions O−2 and copper ions C u+x. What is the charge of copper? The reason why x was used in place of copper ion's formal charge is because copper has multiple oxidative states. The two possible oxidation numbers for copper are +1 and +2, meaning there are two possible copper ions—C u+1 and C u+2. The copper(I) ion, cuprous, C u+1 is formed when elemental copper Cu gives up one of its valence electrons. The copper(II) ion, cupric ion, C u+2 is formed when elemental copper Cu gives up its two valence electrons. There are two possible copper oxide compounds—copper(I) oxide and copper(II) oxide. Both of these compounds are electrically neutral. For the case of copper(I) oxide, the chemical formula is C u 2 O; two cuprous ions C u+1 are needed to neutralize the negative charge on the oxygen ions O−2. 2 C u+1+O−2→C u 2 O As for the case of copper(II) oxide, the chemical formula is C u O; only one cupric ion C u+2 is needed to neutralize O−2 charge. C u+2+O−2→C u O In both cases, the copper ions give a total of two electrons to the oxygen atom. The difference that in copper(I) oxide, two cuprous C u+1 ions give their electrons to O−2. In copper(II) oxide, only one cupric C u+2 ion gives its two valence electrons to O−2. The copper oxide that has been introduced on the start of the lesson was made of cupric ions C u+2; it was cupric oxide C u O. C u 2 O Compound Name The copper ion charge is +2 in the cupric ion case and +1 in the cuprous ion case. It has been revealed that the copper(I) oxide formula is C u 2 O; it is composed of the metallic cuprous ions C u+1 and the nonmetallic oxygen ions O−2. The C u 2 O compound name is cuprous oxide; it contains cuprous ions. It is also called dicopper oxide; the di- prefix signifies the presence of two copper atoms. The systematic name, copper(I) oxide, and its common name, cuprous oxide, are preferred. As revealed earlier, the charge on the cuprous ion is +1. In order to neutralize the oxygen ion's -2 charge, two cuprous ions are needed. The combination of two cuprous ions and one oxygen ion leads to the formation of the neutrally charged C u 2 O. To unlock this lesson you must be a Study.com Member. 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Teacher Try it now Back Coming up next: Allotropes of Silicon: Definition, Appearance & Differences You're on a roll. Keep up the good work! Take QuizWatch Next Lesson Replay Just checking in. Are you still watching? Yes! Keep playing. Your next lesson will play in 10 seconds 0:04 A Natural Compound 0:27 Copper II Oxide Formula 1:33 Copper II Oxide Properties 2:41 Copper II Oxide Structure 3:06 Lesson Summary View Video OnlySaveTimeline 33K views Video Quiz Course Video Only 33K views Preparation of Copper(II) Oxide ------------------------------- Copper(II) oxide is composed one cupric ion C u+2 and one oxygen ion O−2. The large-scale production of this specific copper oxide is conducted through the thermal treatment of ores and minerals in a process called pyrometallurgy. This process involves the following: Ores that are rich with copper are treated with an aqueous solution. The aqueous solution is a mixture of the following compounds: ammonia N H 3, oxygen O 2 and ammonium carbonate (N H 4)2 C O 3. Copper(I) and copper(II) ammine complexes are produced upon treating the ore with the solution. The copper ammine complexes are extracted then decomposed into copper(II) oxide C u O. Copper(II) oxide can be prepared from the following substances through thermal treatment: From copper(II) nitrate: 2 C u(N O 3)2(s)→2 C u O(s)+4 N O 2(g)+O 2(g) From copper(II) hydroxide: C u(O H)2(s)→C u O(s)+H 2 O(g) From copper(II) carbonate: C u C O 3(s)→C u O(s)+C O 2(g) Copper(II) oxide can be reduced into copper(I) oxide when using sulfur dioxide as a reagent. To unlock this lesson you must be a Study.com Member. Create your account Properties of Copper Oxide -------------------------- Both copper oxides may be made from the exact same elements but, they have completely different properties. For instance, copper(I) oxide is a red-to-brown solid while copper(II) oxide is a pitch black solid. They do, however, share a few properties. They are both amphoteric chemicals; they are capable of acting as acids and as bases. The table serves as a comparison between both types of copper oxides. Figure 1: Copper(I) oxide is a red leaning to brown solid Figure 2: Copper(II) oxide is a black solid \| Parameter \| Copper(I) Oxide \| Copper(II) Oxide \| --- \| Ions \| Cu+1, O-2 \| Cu+2, O-2 \| \| Chemical Formula \| Cu2O \| CuO \| \| Molar Mass \| 143 g/mol \| 79.545 g/mol \| \| Odor \| No odor \| No odor \| \| Appearance \| Red-to-brown solid \| Black solid \| \| Boiling Point \| 1800C \| 2000C \| \| Melting Point \| 1232C \| 1326C \| \| Density \| 6.0 g/mL \| 6.315 g/mL \| \| Flammability \| Nonflammable \| Nonflammable \| \| Solubility \| Insoluble in water \| Insoluble in water \| \| Solubility in Other Solvents \| Soluble in acids \| Soluble in potassium cyanide and ammonium chloride \| \| Band Gap \| 2.137 eV \| 1.2 eV \| The following are some of copper(II) oxide's chemical properties; its reactions with different reagents: Reaction with nitric acid H N O 3:C u O+2 H N O 3→C u(N O 3)2+H 2 O Reaction with hydrochloric acid H C l:C u O+2 H C l→C u C l 2+H 2 O Reaction with sulfuric acid H 2 S O 3:C u O+H 2 S O 4→C u S O 4+H 2 O Reaction with hydrogen H 2:C u O+H 2→C u+H 2 O Reaction with carbon monoxide C O:C u O+C O→C u+C O 2 Reaction with carbon C:2 C u O+C→2 C u+C O 2 The complex [C u(N H 3)2]+ is formed when copper(I) oxide C u 2 O is dissolved in ammonia N H 3. Exposing the complex to air oxidizes it into [C u(N H 3)4(H 2 O)2]+2. Dissolving the mentioned complex in hydrochloric acid H C l leads to the formation of a solution concentrated with copper(II) chloride C u C l 2. To unlock this lesson you must be a Study.com Member. Create your account Structure of Copper Oxide ------------------------- The crystal system in copper(I) oxide is cubic; the copper and the oxygen ions are arranged in a rigid cubic crystal structure, as shown in figure 3. The crystal system in copper(II) oxide is monoclinic. In both types of copper oxides, the copper and the oxygen ions are attracted to one another through an electrostatic attraction that is born from their different charges. The type of bond between the ions are ionic bonds. The crystal structures of copper(I) oxide and copper(II) oxide are depicted in figures 3 and 4 respectively. Figure 3: Copper(I) oxide has a cubic crystal structure Figure 4: Copper(II) oxide has a monoclinic crystal structure To unlock this lesson you must be a Study.com Member. Create your account Uses of Copper Oxide -------------------- There are various applications in which copper(I) oxide and copper(II) oxide are used. The following list summarizes the uses of copper(II) oxide: Copper(II) oxide is used as pigment mainly producing colored ceramic glazes. It is used as an additive in livestock feed. It is used as a blue coloring agent in blue flames. It is used in batteries; in the electrode, specifically. The following list summarizes some of copper(I) oxide uses: Copper(I) oxide's semi-conducting properties were utilized in manufacturing semi-conductive materials. It is used as an antifouling agent for the paint in marine ships. It is also used in minimizing corrosion. It is used in the production of fungicides and paints for glass. To unlock this lesson you must be a Study.com Member. Create your account Lesson Summary -------------- Copper oxide is an inorganic chemical compound that is made of copper and oxygen ions bonding together through ionic bonds. The copper ion has two possible oxidative states; +1 and +2. There are two possible copper oxide ions: Copper(I) oxide, which is a brownish red solid whose chemical formula is C u 2 O. It is made of two cuprous ions C u+1 giving each of their electrons to oxygen O−2. Copper(II) oxide, which is a black or dark brown solid whose chemical formula is C u O. It is made of one cupric ion C u+2 giving its two valence electrons to oxygen O−2. The general properties of these two compounds are different, but there are some shared properties; they are both amphoteric, they can act as acids and as bases depending on the reaction. The large-scale production of copper(II) oxide is done through a process called pyrometallurgy, which involves the thermal treatment of copper ores. Copper(II) oxide is used as a nutritional supplement in livestock feed, in battery electrodes and in making colored ceramic glazes. Copper(I) oxide is used in making semiconductive materials and in producing fungicides. To unlock this lesson you must be a Study.com Member. Create your account Video Transcript A Natural Compound Just under 0.007% of Earth's crust is comprised of copper. Around 46% of Earth's crust is oxygen, and just over 20% of Earth's atmosphere is oxygen. These two elements can come together chemically to form copper(II) oxide. Let's figure out how these two elements combine and discuss the properties and structure of the compound. Copper(II) Oxide Formula The chemical combination of a metal and a non-metal generates an ionic compound. We can determine the formula for an ionic compound based on how many electrons the metal atom loses and how many electrons the non-metal atom gains. The Roman numeral II tells us the electric charge, or oxidation state, of the copper ion, which is +2. This means each copper atom loses two electrons to form the ion Cu+2. Oxygen is a non-metal and will always gain two electrons, giving it the oxidation state -2. Since each oxygen atom has two extra electrons, the oxide ion is formed, which is O-2. All ionic compounds have to be electrically neutral, which means the ratio of each element in the compound must contribute enough charge to cancel the charge of the other ion. In the case of the copper(II) ion and the oxygen ion, we can see both ions have equal and opposite charges. This means we only need one of each ion to form the neutral compound copper(II) oxide, which is CuO. Let's discuss the properties of copper(II) oxide. Copper(II) Oxide Properties When you watch police television dramas you might hear them describe the characteristics of the person they're looking for. They might note their physical appearance, including height, weight, eye color, and so on. They also might also note their personality. Are they friendly, reserved, or generally cranky? We can use this analogy to understand the types of properties of a compound, including its physical characteristics and its chemical reactivity. Let's pretend we're a private detective searching for the copper(II) oxide compound like we're searching for a person. We're on the lookout for an ionic compound with the following physical characteristics: Molar mass equal to 78.92 g/mole Black or dark brown solid, possibly in powder form Density of around 6.31 g/cm 3 Melts at just over 1200°C Boils at around 2000°C Copper(II) oxide is an amphoteric substance, which means it can act as an acid or a base. It's also non-flammable and insoluble in water. Now that we know what to look for in terms of copper(II) oxide's properties, let's talk about its structure. Copper(II) Oxide Structure Since copper(II) oxide is an ionic compound, the Cu+2 and the O-2 stick together due to electrostatic attraction. This type of bond is very similar to how two opposite ends of a magnet stick together, except this is on the atomic scale. These ions stick together to form a lattice structure, which is a specific arrangement of the ions, which you can see playing out here: Three-dimensional lattice structure of copper(II) oxide Lesson Summary Let's review. Ionic compounds are compounds where a metal and a non-metal stick together due to electrostatic attraction. This is how copper(II) oxide comes to be, since it's a compound in which oxygen and copper stick together. It's the quantity of electrons the metal loses and the non-metal gains determines the ions' oxidation state. Copper(II) has the Roman numeral II because it loses two electrons forming Cu+2. Oxygen gains two electrons forming O-2. Since these ions are plus and minus 2, they cancel each other out, which is a requirement for an ionic compound. The formula for copper(II) oxide is CuO. Ionic compounds like copper(II) oxide are formed because the oppositely charged ions stick together like opposite ends of two magnets. The ions stick together in a three-dimensional lattice structure. The properties of copper(II) oxide are: Molar mass equal to 78.92 g/mole Black or dark brown solid possibly in powder form Density of around 6.31 g/cm 3 Melts at just over 1200°C Boils at around 2000°C Finally, we learned that copper(II) oxide is also amphoteric, meaning it can act as an acid or a base. Register to view this lesson Are you a student or a teacher? I am a student I am a teacher Unlock Your Education See for yourself why 30 million people use Study.com Become a Study.com member and start learning now. Become a Member Already a member? 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blogs Support Recommended product Popular links Popular links Euclidean Geometry in Mathematical Olympiads All titles Euclidean Geometry in Mathematical Olympiads Series: MAA Problem Book Series Author: Evan Chen, Massachusetts Institute of Technology Published: October 2016 Availability: This item is not supplied by Cambridge University Press in your region. Please contact Mathematical Association of America for availability. Format: Paperback ISBN: 9780883858394 Looking for an examination copy? If you are interested in the title for your course we can consider offering an examination copy. To register your interest please contact collegesales@cambridge.org providing details of the course you are teaching. $69.95 (P) USD Paperback Description Contents About the authors Description This challenging problem-solving book on Euclidean geometry requires nothing of the reader other than courage. Readers will encounter cyclic quadrilaterals, power of a point, homothety, and triangle centers, as well as such classical gems as the nine-point circle, the Simson line, and the symmedian. Both a traditional and a computational view of the use of complex numbers and barycentric coordinates is offered, while more advanced topics are covered in the final part. These include inversion in the plane, the cross ratio and projective transformations, and the theory of the complete quadrilateral. The emphasis of this book is placed squarely on the problems. Each chapter contains carefully chosen worked examples, which explain not only the solutions to the problems but also how one would invent the solution to begin with. Providing over 300 practice problems of varying difficulty from contests around the world, with extensive hints and selected solutions, this book is especially suitable for students preparing for national or international mathematical Olympiads. Includes over 300 practice problems, of varying difficulty, taken from contests around the world Worked examples explain not only the solutions to problems, but also how to invent solutions to begin with Requires no prerequisites Product details October 2016 Paperback 9780883858394 326 pages 254 × 177 × 17 mm 0.58kg This item is not supplied by Cambridge University Press in your region. Please contact Mathematical Association of America for availability. Often bought together Euclidean and Non-Euclidean Geometry : Paperback An Analytic Approach Euclidean and Non-Euclidean Geometry : Paperback An Analytic Approach Hyperbolic Geometry from a Local Viewpoint : Paperback Lectures on Kähler Geometry : Paperback Often bought together Related Journals Journal of the Institute of Mathematics of Jussieu : Journal Journal of the Institute of Mathematics of Jussieu covers all domains in pure mathematics. Compositio Mathematica : Journal Compositio Mathematica is a prestigious, well-established journal publishing first-class research papers that traditionally focus on the mainstream of pure mathematics. Compositio Mathematica has a broad scope which includes the fields of algebra, number theory, topology, algebraic and differential geometry and (geometric) analysis. Papers on other topics are welcome if they are of broad interest. All contributions are required to meet high standards of quality and originality. The Journal has an international editorial board reflected in the journal content. Online access is free for all papers published 5 years ago or more. Papers published before 1997 are available from the NUMDAM website. Compositio is owned and published by non-profit organisations (the Foundation Compositio Mathematica and the London Mathematical Society) that use any surplus income from journal subscriptions to sponsor mathematics and mathematical research. Ergodic Theory and Dynamical Systems : Journal Ergodic Theory and Dynamical Systems focuses on a rich variety of research areas which, although diverse, employ as common themes global dynamical methods. The journal provides a focus for this important and flourishing area of mathematics and brings together many major contributions in the field. The journal acts as a forum for central problems of dynamical systems and of interactions of dynamical systems with areas such as differential geometry, number theory, operator algebras, celestial and statistical mechanics, and biology. Mathematical Proceedings of the Cambridge Philosophical Society : Journal Mathematical Proceedings is one of the few high-quality journals publishing original research papers that cover the whole range of pure and applied mathematics, theoretical physics and statistics. All branches of pure mathematics are covered, in particular logic and foundations, number theory, algebra, geometry, algebraic and geometric topology, classical and functional analysis, dynamical systems, probability and statistics. On the applied side, mechanics, mathematical physics, relativity and cosmology are included. Related Journals Also by this Author Contents Table of Contents Preface Preliminaries Part I. Fundamentals: Angle chasing Circles Lengths and rules Assorted configurations Part II. Analytic Techniques: Computational geometry Complex numbers Barycentric coordinates Part III. Farther from Kansas: Inversion Projective geometry Complete quadrilaterals Personal favorites Part IV. Appendices: Appendix A. An ounce of linear algebra Appendix B. Hints Appendix C. Selected solutions Appendix D. List of contests and abbreviations Bibliography Index About the author. Show more About the authors Author Evan Chen , Massachusetts Institute of Technology Browse by related subject Abstract analysis Algebra Computational science Differential and integral equations, dynamical systems and control Discrete mathematics, information theory and coding Fluid dynamics and solid mechanics Geometry and topology Historical mathematical texts History of mathematics Logic, categories and sets Mathematical biology Mathematical finance Mathematical modelling and methods Mathematical physics Mathematical tables and handbooks Mathematics (general) Number theory Numerical analysis Numerical recipes Optimization, OR and risk analysis Real and complex analysis ✕ Sorry, this item cannot be purchased in the same transaction as the existing items in your cart. 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You are now able to request access to teacher restricted resources. If you are a teacher, simply complete the teacher resource request form, and wait for your request to be validated. Request resources Email verified ✕ Your email has been verified. Continue This challenging problem-solving book on Euclidean geometry requires nothing of the reader other than courage. Readers will encounter cyclic quadrilaterals, power of a point, homothety, and triangle centers, as well as such classical gems as the nine-point circle, the Simson line, and the symmedian. Both a traditional and a computational view of the use of complex numbers and barycentric coordinates is offered, while more advanced topics are covered in the final part. These include inversion in the plane, the cross ratio and projective transformations, and the theory of the complete quadrilateral. The emphasis of this book is placed squarely on the problems. Each chapter contains carefully chosen worked examples, which explain not only the solutions to the problems but also how one would invent the solution to begin with. Providing over 300 practice problems of varying difficulty from contests around the world, with extensive hints and selected solutions, this book is especially suitable for students preparing for national or international mathematical Olympiads. Includes over 300 practice problems, of varying difficulty, taken from contests around the world Worked examples explain not only the solutions to problems, but also how to invent solutions to begin with Requires no prerequisites This website uses cookies By clicking “Accept all cookies”, you agree to the storing of cookies on your device to enhance site navigation, analyse site usage, and assist in our marketing efforts which includes personalised advertising on certain pages.Cookie notice Use of Cookies by Cambridge When you visit our website, Cambridge may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalised web experience including personalised advertising. 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1690	https://www.wordsclarity.com/dictionary/vilify	vilify \| Meaning, Synonyms, Keydifference & Examples \| Wo... ☰ Home About Contact Words Have Power, Let’s Unlock It! Every word tells a story, sparks an idea, and shapes the way we communicate. Dive into a world where meanings come alive, synonyms open new possibilities, and subtle distinctions make all the difference! Supercharge your vocabulary, sharpen your writing, and fall in love with the art of words. Ready to explore? Let’s begin! Word of the Day Fight A violent confrontation or struggle between people or groups, which can be physical or verbal, and ranges from small-scale personal disputes to large-scale conflicts like wars. Know its synonyms with key difference Find the Right Word with Clarity Search🎤 Voice Search vilify🔊 Meaning: To speak or write about someone in a way that harshly criticizes and damages their reputation. Key Difference: Unlike general criticism, vilification involves malicious intent to harm someone's character publicly. Examples: The politician was vilified in the media after the scandal broke. Some historical figures were vilified by their opponents but later recognized for their contributions. Synonyms of vilify: Similar words with related meanings. defame🔊 Meaning: To damage someone's good reputation through false statements. Key Difference: Defamation often involves false accusations, while vilification can include harsh truths. Examples: The celebrity sued the tabloid for attempting to defame her with fabricated stories. In medieval times, rivals would defame each other to gain political advantage. slander🔊 Meaning: To make false spoken statements damaging to a person's reputation. Key Difference: Slander is strictly oral defamation, whereas vilification can be written or spoken. Examples: He was accused of slander after spreading lies about his coworker. Slander was a serious offense in ancient legal systems, punishable by law. malign🔊 Meaning: To speak harmful untruths about someone with ill intent. Key Difference: Maligning is broader and can include subtle attacks, while vilification is more direct and severe. Examples: The author felt maligned by critics who misunderstood her work. Leaders often malign their opponents during election campaigns. denigrate🔊 Meaning: To unfairly criticize someone, often to belittle them. Key Difference: Denigration can be subtle or indirect, while vilification is openly hostile. Examples: Some historians denigrate certain rulers without considering their achievements. Social media can be used to denigrate people anonymously. disparage🔊 Meaning: To express a negative opinion about someone, often unfairly. Key Difference: Disparagement may lack the extreme hostility of vilification. Examples: She disparaged his efforts, calling them insignificant. Artists often face disparagement before gaining recognition. smear🔊 Meaning: To damage someone's reputation by spreading false or misleading accusations. Key Difference: Smearing is often part of a deliberate campaign, while vilification can be a one-time act. Examples: The smear campaign against the scientist backfired when the truth emerged. Political smears are common during heated elections. revile🔊 Meaning: To criticize someone with abusive language. Key Difference: Reviling is more about verbal abuse, whereas vilification can be written or systemic. Examples: Protesters reviled the corrupt official during the trial. Historical rebels were often reviled before being celebrated. berate🔊 Meaning: To scold or criticize someone angrily. Key Difference: Berating is direct verbal reprimand, while vilification aims to publicly shame. Examples: The coach berated the team for their poor performance. Parents should avoid berating children in public. traduce🔊 Meaning: To speak badly of someone with the intent to misrepresent. Key Difference: Traducing involves misrepresentation, while vilification can be based on truth or lies. Examples: The journalist was accused of traducing the activist's motives. Historical figures were often traduced by biased chroniclers. Conclusion: ✅ Vilify is best used when describing deliberate, public attempts to ruin someone's reputation. ✅ Defame is appropriate when false statements are involved, especially in legal contexts. ✅ Slander should be used when referring specifically to spoken falsehoods. ✅ Malign works well for describing indirect or subtle character attacks. ✅ Denigrate fits when the criticism is belittling but not necessarily vicious. ✅ Disparage is suitable for milder, often unfair criticism. ✅ Smear is ideal for describing orchestrated reputation-damaging campaigns. ✅ Revile is best for situations involving abusive verbal attacks. ✅ Berate applies to direct, angry scolding rather than public shaming. ✅ Traduce should be used when misrepresentation is a key factor. 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1691	https://www.sciencedirect.com/science/article/pii/S0006349523001054	The effects of RNA.DNA-DNA triple helices on nucleosome structures and dynamics - ScienceDirect Typesetting math: 100% Skip to main contentSkip to article Journals & Books ViewPDF Download full issue View Open Manuscript Other access options Search ScienceDirect Outline Abstract Significance Introduction Materials and methods Results Discussion Author contributions Acknowledgments Supporting material References Show full outline Cited by (5) Figures (11) Show 5 more figures Tables (1) Table 1 Extras (2) Download all Document S1. Tables S1–S14 and Figures S1–S18 Document S2. Article plus supporting material Biophysical Journal ------------------- Volume 122, Issue 7, 4 April 2023, Pages 1229-1239 Article The effects of RNA.DNA-DNA triple helices on nucleosome structures and dynamics Author links open overlay panel Havva Kohestani 1, Jeff Wereszczynski 2 Show more Outline Add to Mendeley Share Cite rights and content Under an Elsevier user license Open archive Abstract Noncoding RNAs (ncRNAs) are an emerging epigenetic factor and have been recognized as playing a key role in many gene expression pathways. Structurally, binding of ncRNAs to isolated DNA is strongly dependent on sequence complementary and results in the formation of an RNA.DNA-DNA (RDD) triple helix. However, in vivo DNA is not isolated but is rather packed in chromatin fibers, the fundamental unit of which is the nucleosome. Biochemical experiments have shown that ncRNA binding to nucleosomal DNA is elevated at DNA entry and exit sites and is dependent on the presence of the H3 N-terminal tails. However, the structural and dynamical bases for these mechanisms remain unknown. Here, we have examined the mechanisms and effects of RDD formation in the context of the nucleosome using a series of all-atom molecular dynamics simulations. Results highlight the importance of DNA sequence on complex stability, elucidate the effects of the H3 tails on RDD structures, show how RDD formation impacts the structure and dynamics of the H3 tails, and show how RNA alters the local and global DNA double-helical structure. Together, our results suggest ncRNAs can modify nucleosome, and potentially higher-order chromatin, structures and dynamics as a means of exerting epigenetic control. Previous article in issue Next article in issue Significance Noncoding RNAs play an essential role in gene regulation by binding to DNA and forming RNA.DNA-DNA (RDD) triple helices. In the cell, this occurs in the context where DNA is not a free double helix but is instead condensed into chromatin fibers. At the fundamental level, this compaction involves wrapping approximately 147 DNA basepairs around eight histone proteins to form the nucleosome. Here, we have used molecular dynamics simulations to understand the interplay between the structure and dynamics of RDD triple helices with the nucleosome. Results highlight the importance of RNA sequence on RDD stability regardless of its environment and suggest potential mechanisms for cross talk between epigenetic factors and the effects of noncoding RNA binding on local and higher-order chromatin structures. Introduction Epigenetic regulation mechanisms are critical in inducing heritable phenotype changes in biological systems without altering their core geneticDNA sequences. In vivo, reversible epigenetic mechanisms engage various molecular structures from small posttranslational modifications (PTMs) to RNAs to large protein complexes. For example, chromosome X-inactivation, histonedeacetylation, histone modification, DNA methylation, and noncoding RNA (ncRNA)-mediated processes are a few examples of epigenetic control mechanisms (1,2,3,4,5). In particular, experiments have shown the involvement of different ncRNAs like Piwi-interacting RNAs, microRNAs, and small interfering RNAs in embryonic development and diseases such as cancer. For example, there is an association between active mRNA transcription sites and trimethylation of lysine 4 and lysine 36 of histone H3. In contrast, monomethylation of lysine 4 and acetylation at lysine 27 in H3 is a sign of transcription upregulation from enhancer RNA-activated promoters (6,7). These factors suggest that RNA-based molecular markers may be useful for the detection and targeting of cancer cells and that ncRNAs may have therapeutic applications in eukaryotic systems by inducing viral resistance (5,6,7,8,9,10). Structurally, ncRNAs fit into two categories based on nucleotide size. Long ncRNAs are composed of a minimum of 200 nucleotides, while RNAs with 20–30 nucleotides are considered small ncRNAs (8,11). The activity and influence of small or long ncRNAs depends on several factors such as their localization and the type of RNA-DNA or RNA-protein interactions they make (5,9,11,12). Various elements play a role in initiating a successful interaction between ncRNAs and genomic DNA to create functional RNA.DNA-DNA (RDD) triple helices. Experimental evidence has shown that the formation of proper H-bonding between triplex-forming oligonucleotides (TFOs) and a DNA double helix is highly sequence dependent, where these factors have a preference to bind to DNA at specific triplex targeting sites (TTSs) that have a higher presence at an enhancer or promoter locus (5,13,14,15,16,17,18). Besides the significant sequence specificity, efficient binding of RNA to a DNA counterpart in solution is reliant on factors such as the RNA length, C-G content of the TTS, the parallel or antiparallel alignment of nucleotide strands, and the presence or absence of point mutations (16,17,18,19,20). In vivo, ncRNAs exert their influence by interacting with DNA that is packaged into chromatin fibers. The fundamental building block of chromatin is the nucleosome, a complex consisting of two copies of the H4, H3, H2A, and H2B histones, along with approximately 145–147 DNA basepairs (21). Structurally, each core histone has an unstructured N-terminal tail of varying lengths and three α-helices that are connected by two loops, and H2B also has a short and unstructured C-terminal tail (22). The histones assemble into a disc-like structure around which the DNA wraps (Fig.1) (23,24). The nucleosome itself is a highly dynamic entity and adopts an ensemble of structures characterized by core histone, histone tail, and DNA breathing motions (23,24,25,26,27). Other epigentic factors, such as reader domain binding, linker histone interactions, and the replacement of canonical histones with variants, have been shown to influence these dynamics as a way to influence gene expression and higher-order chromatin structures (27,28,29,30,31). 1. Download: Download high-res image (756KB) 2. Download: Download full-size image Figure 1. (Top) A general view of the nucleosome and its structure. The H3 (blue), H4 (yellow), H2A (red), and H2B (green) histones form the protein core, around which approximately 147 basepairs of DNA (silver) wrap. Here, there are an additional 20 DNA basepairs at the entry DNA site, which we term the DNA arm. The reference dyad axis is shown by a red arrow. By convention, DNA locations are denoted by their superhelical locations (SHLs) relative to the dyad, with DNA closer to the entry site having SHLs less than 0 and closer to the exit site greater than 0. (Bottom left) Pyrimidine RNA triplex forming oligonucleotides have a preference for A-T-rich TTS. An RNA-TFO is represented in red against a blue TTS DNA sequence (16). (Bottom right) Hoogsteen basepairs help to stabilize RNA.DNA-DNA triple helices, and we have selected a parallel (Y) RNA-TFO sequence to maximize the complementary binding between the RNA and DNA double helix. To see this figure in color, go online. To understand how ncRNAs interact with DNA in chromatin, Maldonado et al. recently examined the stability of RDD triple helices in nucleosome structures. In particular, electrophoretic mobility shift assays showed that formation of a stable RDD triple helix occurs primarily at the entry/exit sites of the nucleosomal DNA. In addition, the H3 tails were shown to play a key role in increasing RDD stability compared with the truncated nucleosome, which did not include these long, disordered regions (16,32). However, the molecular processes underlying these mechanisms remain elusive, as there are no published crystal structures available for RDD helices in chromatin structures, and there is little information about how RNA binding affects nucleosome dynamics. Here, we have sought to examine the molecular mechanisms of RDD triple helices when bound to nucleosome systems. To do so, we generated models of nucleosome structures with an extended DNA arm at the entry DNA site to which a TFO RNA could bind. We performed a series of molecular dynamics (MD) simulations of isolated RDD triplexes and full-length and H3-tailless nucleosomes containing RDD triplexes with either TTS or noTTS DNA sequences in the DNA arm. Results are in agreement with experiments that have shown that the stable formation of an RDD triplex is driven by the DNA sequence, as in all cases of noTTS-containing models, there was unstable RNA binding regardless of its environment. Furthermore, the H3 tails are shown to affect RNA binding and adopt distinct conformations in the presence of the RDD triplexes. Finally, analyses of the DNA nucleic acid parameters show that RNA binding induces a transition to an intermediate structure between B and A forms. Overall, these results demonstrate that RDD triplex formation is largely dependent on the correct positioning of the TTS sequence and suggest that RDD binding may influence other epigentic factors, particularly those involving the H3 tails. Materials and methods System setup We designed five nucleosome systems based on the protein core of the PDB: 1KX5 and DNA of the PDB: 6VYP crystal structures (Table S1) (22,33). PDB: 6VYP was stripped of all protein segments. We kept only one DNA arm to avoid potential interactions of the two extended DNA arms in the simulation box. For the TTS systems, the sequence of the entry DNA arm was mutated in the w3DNA webserver to match the En3-TTS sequence for the samples with the TTS sequences (16,34). For the noTTS DNA sequence, we utilized the extranucleosomal DNA sequence present in the PDB: 6VYP structure. Experimental studies have suggested that an RNA-TFO should be a minimum of 19 nucleotides in length to generate an RDD triplex, while increases in the RNA length beyond 27 bases do not enhance its DNA affinity (17). As a middle ground, we constructed a 24 basepair RNA-TFO in the 3D-NuS webserver for the triplex targeting and nontargeting sequences (35). RDD triplexes were replaced and adjusted at the DNA arm via Chimera’s modeller function (36). The canonical protein core was added to the final mutated edited nucleic acid structure and assembled in Visual Molecular Dynamics and Chimera (36,37). The model pool consisted of seven systems, including isolated TTS and noTTS triplexes, TTS nucleosomes with and without H3 tails, noTTS nucleosomes with and without H3 tails, and a nucleosome model without RNA. For clarity, we have used superscript t H 3 for nucleosomes with truncated H3 tails and labeled DNA locations by their superhelical locations (SHLs), which range from −7 to 7 with a value of SHL= 0 corresponding to the nucleosome dyad (Fig.1) (24,38,39,40). All systems were built in the tleap tool of the Amber MD package (41,42). The Amber ff14SB, OL15, OL3, and OPC force-fields were used for the protein, DNA, RNA, and water force-fields, while ion parameters were based on the Joung and Cheatham model (43,44,45,46,47,48). All systems were neutralized and solvated in a water box with a 12 Å solvent buffer and an additional 0.15 M NaCl concentration. The ParmEd (parameter file editor) program was used to repartition the hydrogen atom masses to reduce high-frequency motions, which allowed the use of a 4 ps MD timestep instead of 2 ps (49). All systems were minimized twice for 10,000 steps, with and without heavy atom restraints. Systems were then gradually heated in the canonical ensemble from 0 to 300 K over 5000 steps with restraints on the solute heavy atoms. Restraints were released by relaxing the systems in the isothermal-isobaric ensemble over 300 ps. The Langevin piston was used for a barostat with Langevin dynamics for temperature control with a collision frequency of 2 ps−1 (50). All simulations were performed with the Amber 20 GPU accelerated version of PMEMD for three replicas of each system (51). Table S1 represents a summary of the simulations performed. Simulation analyses CPPTRAJ was used to measure root-mean-square deviations (RMSDs), RMS fluctuations, inter- and intramolecular distances, and various nucleic acid parameters (52). For all analyses, 100 ns simulation time was taken for equilibration. Visual Molecular Dynamics was used to calculate the number of H-bonds and also produce figures (37), and python was used for data processing and generating plots (53). The molecular mechanics with generalized Born and surface areasolvation (MM/GBSA) approach was used to estimate the interaction energy between the H3 histones and nucleic acid helices (54,55). The GB model with igb= 5 implicit solvent models was utilized for the polar term along with a salt concentration of 0.15 M and the mbondi2 radii set (56,57). For interpretation, the energy terms were grouped into two terms: Δ E v d W and Δ E e l e c. Δ E v d W is the sum of the MM van der Waals energy and the solvation apolar energy differences, and E e l e c is the sum of the MM electrostatics and polar solvation energy differences. A residue decomposition of the MM/GBSA analysis was performed to estimate the energy values for each protein residue interacting with the RDD. Results TTSs in nucleosomes form stable sequence-dependent RDD helices Seven sets of simulations were performed, which included isolated TTS DNA and noTTS DNA RDD systems, nucleosome/RNA systems with and without the TTS DNA and with and without the H3 tails, and a canonical nucleosome without RNA (Table S1). For each system, three 1.1 μ s simulations were performed. Simulations showed that a TTS DNA sequence is required for maintaining the stability of an RDD triplex, regardless of its environment. In simulations of the isolated triplex with the TTS sequence, the nucleic acid RMSD remained relatively low and constant, with values of ∼2–4 Å (Fig.2a). In contrast, for isolated triplexes with a noTTS DNA sequence the system quickly destabilized, typically in the first 200 ns, and reached RMSD values in the range of 40 Å (Fig.2b). To ensure that this rapid destabilization was not an artifact of inadequate system equilibration or solvent placement, an additional set of three 240 ns simulations of the isolated noTTS system were performed. In those simulations, the solute heavy atoms were restrained for 100 ns. However, even after this extended equilibration period, the RNA quickly separated from the DNA double helix, as shown by an increased RMSD of the RDD to values between 5 and 18 Å (Fig.S2). 1. Download: Download high-res image (822KB) 2. Download: Download full-size image Figure 2. RMSDs of RDD triplexes in isolated forms and in assembly with nucleosomes. The triplex dynamics are influenced by the DNA sequence at the triplex site. Models with the triplex targeting site sequence had the lowest RMSD values, while in samples with a noTTS sequence, the local and global stability of the RDD triplex was volatile and transient regardless of whether it was isolated or nucleosome bound. Note that the y axis of (b), (d), and (f) are five times larger than the y axis of (a), (c), and (e). To see this figure in color, go online. In the context of the full-length nucleosome, the TTS triplex stayed well formed with RMSD values of 2–6 Å, whereas some destabilization of the triplex was noted in the noTTS triplex, with one simulation sampling RMSD values of ∼20 Å (Fig.2d). Removal of the H3 tails resulted in similar sampling of the TTS triplex (Fig.2e); however, there was a marked increase in the RMSD of the noTTS sequence to values as large as 40 Å (Fig.2f). In each of these systems, the RNA component of the RDD triplex was the most dynamic, with trends in RNA RMSD values largely mirroring the overall trends in the RDD RMSD values (Fig.S1). These large RMSD values corresponded to the RNA detaching and unzipping from the DNA helix, as observed in snapshots of the final frames of the simulation trajectories (Fig.3d). A similar trend is noted by monitoring the RNA/DNA H-bond occupancies and the RNA/DNA distances (Figs. 4 and 5). In all systems in which the DNA has a TTS sequence, the number of RNA/DNA H-bonds was well maintained beyond the terminal two residues, whereas the number of H-bonds dropped dramatically in noTTS systems. For noTTS triplexes in complex with full-length nucleosomes, there was additional H-bond occupancy for the central basepairs 7–11 and 17–21, albeit not to the level as observed in TTS-containing sequences. This partial localized stability was due to interactions with the H3 tail, as these additional H-bonds were lost by removal of the H3 tail. The differences between the TTS and noTTS systems are highly statistically significant, with p values well below 0.001 (Table S2). 1. Download: Download high-res image (730KB) 2. Download: Download full-size image Figure 3. Final, representative snapshots from simulation trajectories. (a) Isolated RDD sequence with a triplex targeting site (TTS; in cyan) DNA sequence in cyan maintained its structural integrity. The RDD helix structure was maintained in nucleosome systems with (b) and without (c) the histone tails. For noTTS systems, the RNA strand detached from the isolated DNA double helix (d), as well as the DNA in the full-length nucleosome (e) and the nucleosome lacking H3 tails (f). To see this figure in color, go online. 1. Download: Download high-res image (366KB) 2. Download: Download full-size image Figure 4. Occupancy of H-bonds in RDD triplex between RNA and the adjacent DNA for systems containing a TTS DNA sequence (top) and lacking the TTS DNA sequence (bottom). The TFO sequence dictates the presence of two H-bonds formed between U-A pairs in the RNA-DNA double helix. Table S2 shows the statistical differences between various models. To see this figure in color, go online. 1. Download: Download high-res image (1MB) 2. Download: Download full-size image Figure 5. DNA and RNA distance in isolated and nucleosome systems. In the absence of the TTS sequence, RNA is released from its adjacent DNA double helix. and the sequence-dependent event is observed in isolated noTTS triplex and noTTS nucleosomes. Note that the y axis of (b), (d), and (f) are eight times larger than the y axis of (a), (c), and (e). To see this figure in color, go online. H3 tails form stronger interactions with RDD triplexes In all simulations with the nucleosome, the protein core maintained its overall structure regardless of whether RNA was bound or not (Fig.S3). The protein regions with the most variability between structures were the H3 N-terminal tails. These regions are long, unstructured extensions that are key sites of multiple epigenetic markers. They are heavily charged, with four arginines and six lysines in the first 35 residues, and these extensions have been shown to influence nucleosome dynamics, DNA breathing, PTM accessibility, and effector protein binding (27,40,58,59,60,61,62,63,64,65). Visual inspection of the simulation trajectories showed a dramatic difference in the H3 tail and entry DNA dynamics based on the presence or absence of RNA (Figs. 3 and S4). This is in agreement with experiments in which the N-terminal tails of all histones, including H3, were truncated, which showed a measurable reduction in the formation of RDD triplexes in the nucleosome (32). This reduced RDD stability in the absence of tails is likely a result of the lack of positively charged H3 residues in the tails at the DNA entry and exit sites of H3-truncated nucelosomes. Given these key roles of the H3 tails in epigenetic regulation, we sought to identify how the H3 tails helped stabilize RNA binding and to what degree RNA binding influenced the structure and dynamics of the H3 tails. In systems without RNA, the H3 tails at both the DNA entry and exit sites tended to form contacts with the DNA proximal to their location, which was primarily SHLs 0–1 and −7 and the DNA arm for the entry DNA and SHLs 6–7 for the exit DNA (Figs. 1 and 6). In RNA- and TTS-containing systems, the entry DNA was more elongated along the DNA arm, with contacts primarily on the DNA arm and SHLs −6.5 and −7. In addition, the exit DNA had increased contacts with the DNA near the dyad, with significantly more contacts formed to SHLs −1.5 to 0.5. The noTTS case presented an intermediate of the two cases, with the entry H3 tail primarily having contacts with the DNA arm and SHL −7, whereas the exit H3 tail made some contacts with the dyad DNA, albeit fewer than in the case of the TTS-containing sequence (Fig.6). This shifting of H3 tail contacts with the DNA by RNA is highlighted in a clustering analysis, which shows that in the TTS-containing systems, the dominant clusters have increased binding of the entry DNA along the DNA arm (Figs. 7 and S5), whereas without RNA, the H3 tail binding along the DNA arm is not the dominant structural cluster (Figs. 7 and S6). 1. Download: Download high-res image (858KB) 2. Download: Download full-size image Figure 6. SHLs of interactions between the H3 tails and nucleosomal DNA. The additional negative charge of RNA in the TTS and noTTS systems is attractive for both H3 tails at the entry and exit sites, resulting in a shift of interaction locations between the H3 tails and DNA. To see this figure in color, go online. 1. Download: Download high-res image (480KB) 2. Download: Download full-size image Figure 7. Top five dominant structural clusters of the H3 tails in the noRNA and TTS nucleosomes. The introduction of a negatively charged RNA strand to nucleosomal DNA encourages the alignment of the H3 tail along the entry DNA arm in TTS nucleosomes. To see this figure in color, go online. An analysis of the H-bond occupancy between the H3 tails and the nucleic acids shows a dramatic increase in H-bonds upon RNA binding (Fig.8). In particular, for systems with RNA, H-bond occupancies were significantly increased for Arg2, Arg8, Lys9, Arg17, and Arg26. This is likely due to the increased polyanionic properties of the RDD triplex, as its increased density of negative charges relative to the standard DNA double helix increased the attraction of and formation of H-bonds to the H3 tails. These differences were found to be highly statistically significant (Table S3). In addition, while the RNA presence helps to stabilize H3/nucleic acid interactions, the H3 tail presence also helps to stabilize RNA/DNA interactions. For systems that lacked the TTS sequence, the absence of the H3 tail exacerbated disengagement of the RNA strand from the entry DNA arm (Fig.3). We also found that RNA binding affects the H3 tail flexibility and binding energy to the nucleic acids. The RMS fluctuations of the tails showed that for the entry H3 tail, systems with a TTS sequence had lower fluctuations in the first 10 residues, suggesting that tail binding was stabilized by interactions with the RDD triplex (Figs. 9 and S7). The interaction energies between the entry and exit H3 histones and the nucleic acids were estimated through an MM/GBSA analysis, which primarily accounts for enthalpic terms in the interaction energies (Tables 1 and S4–S6). Results show a significantly stronger binding of the entry H3 tails in systems with RNA, with E total for systems with RNA and with and without the TTS sequence of −80.23 ± 1.65 and −77.74± 1.5 kcal/mol compared with −52.15± 0.99 kcal/mol in systems without RNA (Table 1). The energetics of the positively charged residues along the H3 tail interacting with the negatively charged nucleic acids varied significantly between systems (Tables S7–S12). The most significant differences are associated with Arg2, Arg8, Arg17, Arg26, and Lys27, which had approximately doubled the affinity in TTS nucleosomes compared with noRNA nucleosomes. In particular, this interaction energy decreased by 3.98 ± 1.01 and 2.38±1.12 kcal/mol, respectively, in TTS and noTTS systems compared with the RNA-free system in the case of Arg2, and for Arg8, there was a decrease of 6.27 ± 0.83 and 5.07± 0.9 kcal/mol for TTS and noTTS nucleosomes compared with the noRNA system (Tables S10–S12). Notably, these residues are associated with multiple PTMs that have been implicated in chromatin regulation (27,62,65,66). A similar trend is observable for Lys4, Lys9, Arg17, Lys18, Lys23, Arg26, and Lys27, highlighting the influence of RNA on H3-DNA binding. 1. Download: Download high-res image (173KB) 2. Download: Download full-size image Figure 8. Average H-bond occupancy formed between RDD and the adjacent H3 tails in each model. Arg2, Arg8, Lys9, Lys14, Arg17, and Arg26 of H3 tail are among the most favorable residues to form H-bonds with the nucleic acid triplex. To see this figure in color, go online. 1. Download: Download high-res image (467KB) 2. Download: Download full-size image Figure 9. Average RMS fluctuation of the H3 tails at the entry (top) and exit (bottom) DNA sites show that RNA binding reduces tail flexibility in the H3 entry tail, particularly for the first 10 residues. Error bars represent standard error of mean. To see this figure in color, go online. Table 1. MM/GBSA interaction energy interaction (kcal/mol) between entry and exit H3 histone tails and nucleic acid (RNA-DNA) in nucleosome systems \| H3 tail \| Nucleosome system \| E vaW (kcal/mol) \| E elec (kcal/mol) \| E total (kcal/mol) \| --- --- \| Entry \| noRNA \| −33.27 ± 0.53 \| −18.88 ± 0.57 \| −52.15 ± 0.99 \| \| noTTS \| −59.41 ± 0.70 \| −18.33 ± 0.87 \| −77.74 ± 1.50 \| \| TTS \| −52.47 ± 0.65 \| −27.75 ± 1.12 \| −80.23 ± 1.65 \| \| Exit \| noRNA \| −36.93 ± 0.55 \| −22.11 ± 0.67 \| −59.04 ± 1.17 \| \| noTTS \| −21.71 ± 0.35 \| −12.99 ± 0.48 \| −34.70 ± 0.79 \| \| TTS \| −35.88 ± 0.50 \| −14.22 ± 0.69 \| −50.1 ± 1.10 \| Errors are reported as standard error of mean. RNA influences the nucleosomal DNA structure In addition to different H3 tail dynamics, RNA binding influenced the global and local structure of nucleosomal DNA. We did not observe any large-scale DNA unwrapping during our simulations; however, introduction of RNA to the nucleosome did increases DNA breathing. Here, we have focused on only the TTS/nucleosome systems, as the noTTS systems have a heterogeneous mixture of RDD triplex and nontriplex states. More frequent DNA breathing is reflected in elevated entry-exit DNA end-to-end distances in the TTS RNA systems (Figs. 10a and S8). The DNA entry-exit distances reached up to 120 Å in TTS RNA and noRNA nucleosomes, although there was a higher frequency for distance above 100 Å in the TTS RNA systems. We also observed an increase in the minimum entry-exit DNA distance in nucleosomes with RNA from values as low as 60 Å to a minimum separation of ∼80 Å. 1. Download: Download high-res image (186KB) 2. Download: Download full-size image Figure 10. Histograms of DNA end to end distances for systems with H3 tails (a) and without H3 tails (b). For reference, the H3 tail containing noRNA system is shown in the bottom. The inclusion of RNA increased the distance distributions for systems with H3 tails, suggesting the propensity for increased DNA breathing. To see this figure in color, go online. Previous studies have suggested that the proper interaction of single-stranded RNA with a DNA double helix can result in the formation of a DNA triple helix with an intermediate structure between A- and B-DNA forms (67). To examine how the DNA structures change upon RNA binding in the context of the nucleosome, interbasepair nucleic acid parameters were computed for the DNA in the DNA arm for the TTS RNA and noRNA simulation sets. The averages of these values are summarized in Table S13 along with values for ideal A- and B-DNA structures (67). From these six metrics, the slide parameter had the most deviation from ideal B-DNA toward ideal A-DNA in TTS systems (Figs. 11 and S9–S14). In fact, in models with a TTS sequence, the slide values were close to the A-DNA state, with twist and roll values similar to the B-DNA state. Although the range of change for the basepair shift is narrow (the shift for ideal B-DNA is 0.0 Å and for A-DNA is 0.01 Å), all models showed an increasing trend for this parameter. The basepair step inclination and X displacement are the major geometrical parameters that track the global structural changes in the DNA double helix (68). These parameters are zero for an ideal B-DNA duplex, while for the case of ideal A-DNA, they are 20° and −5 Å, respectively (Table S14). Slightly elevated values of the helix X displacement and inclination were observed in all TTS systems, indicating DNA structural variations from linear B-DNA structures toward structures with global shapes resembling the A form (Figs. S11, S15, and S16). We also observed a strong correlation between variations of local basepair and and helix parameters in all TTS systems (Figs. S17 and S18). These correlations indicate that angular and linear variations in basepair coordinates have a cumulative influence on the DNA helix structure, which is invariant under sequence type or the presence or absence of RNA and H3 tails. 1. Download: Download high-res image (406KB) 2. Download: Download full-size image Figure 11. Deviations from ideal A- and B-DNA values for interbasepair parameters of roll, slide, and shift and basepair step helix parameters of X displacement and inclination. Basepair step helix inclination, X-displacement, and interbasepair shift values are zero for ideal B-DNA. To see this figure in color, go online. Discussion The binding of small ncRNAs to nucleosomal DNA triggers a sequence of events that modifies the structure and energetics of both the DNA and the critical N-terminal H3 tails. Simulations show that the stability of an RDD triple helix is highly sequence dependent, as even in the context of a nucleosome with full-length H3 tails, RNA separation was observed with a noncomplimentary DNA sequence on the hundreds of nanoseconds timescale. Stable RNA binding increases the attraction of the H3 tails with the nucleic acids, as shown by both the increased hydrogen binding and decreased MM/GBSA-derived energies computed here. This effect appears to be largely due to the increased acidic nature of the triple helix, which creates a strong electrostatic attraction for the multiple arginines and lysines in the tails. RDD triplex formation also has the effect of altering the DNA structure to a shape that is intermediate of ideal B and A forms. The effects of RNA binding on the H3 tails likely have significant effects on gene regulation. Over a third of the residues in these tails have been identified as sites for PTMs that are recognized by various reader proteins (66). Simulations and NMR experiments have shown that the energetics and dynamics of these tails on nucleosomal DNA can be tuned by reducing the charge states of specific residues, for example through acetylation or phosphorylation, and that by doing so, other epigentic markers may become more accessible to reader proteins (27). This provides one example of cross talk in which one epigenetic factor influences another at a distant location (62). Our results suggest that RNA binding may have the opposite effect. By forming a stronger H3/nucleic acid interface, PTM locations used for various signaling pathways are likely more occluded than in systems without RNA, making the addition and reading of some PTMs more difficult. Given that RNA binding and TTSs are enriched at regulatory elements that correlate with active chromatin locations in vivo (32), the interplay between RNA and other epigenetic marks, particularly those that utilize the H3 tails, is likely to have implications for highly expressed genes. Our results also show how RNA binding influences global and local DNA structures. Although these simulations were performed on only single-nucleosome systems, we can infer that these changes may also influence higher-order chromatinstructures. Indeed, cryoelectron microscopy and crystal structures have shown that heterogeneous poly-nucleosomal arrays can form regular structures in solution (69,70,71), while simulations have shown that these structures are dynamic in nature and can be modulated by factors such as linker histone binding and linker DNA length (72,73,74,75). The introduction of heterogeneity to these structures, such as through subtle changes to the nucleosomal DNA structure or by changing the linker DNA length, can disrupt these higher-order structures (76,77). As shown here, ncRNA binding influences local and global DNA structures. These changes likely create another source of conformational heterogeneity in chromatin fibers, thus disrupting the repeating structures observed in some experiments. Additional experiments and simulations with ncRNA binding will, therefore, likely show a loss of ordering from regularly repeating poly-nucleosomal arrays upon the introduction of ncRNAs. Author contributions H.K. performed the simulations. H.K. and J.W. designed the experiments, analyzed data, and wrote the manuscript. Acknowledgments The authors thank members of the Wereszczynski group for valuable discussions concerning this work. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) (78). This research was supported by the National Institute of General Medical Sciences of the National Institutes of Health grant R35GM119647. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Declaration of interests The authors declare no competing interests. Supporting material Download all supplementary files included with this articleWhat’s this? Download: Download Acrobat PDF file (17MB) Document S1. Tables S1–S14 and Figures S1–S18. Download: Download Acrobat PDF file (23MB) Document S2. Article plus supporting material. Recommended articles References 1F. Perri, F. Longo, et al., S. Pisconti Epigenetic control of gene expression: potential implications for cancer treatment Crit. Rev. Oncol. Hematol., 111 (2017), pp. 166-172 View PDFView articleView in ScopusGoogle Scholar 2C.D. Allis, T. Jenuwein The molecular hallmarks of epigenetic control Nat. Rev. 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1692	https://mathoverflow.net/questions/380593/confusing-notation-for-sets-of-unordered-vs-ordered-pairs	Skip to main content Confusing notation for sets of unordered vs ordered pairs Ask Question Asked Modified 4 years, 7 months ago Viewed 1k times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. Given two finite sets X and Y, one may consider the ordered pairs (x,y) with x∈X and y∈Y. Then, (x,y)≠(y,x), and (x,x) exists if x∈X and x∈Y. One may also consider unordered pairs xy with x∈X and y∈Y, with x≠y. Then, {x,y}={y,x}, and {x,x} does not exist. The set of all ordered pairs is denoted by X×Y, and the set of unordered pairs is often denoted by X×Y too, which is confusing. For instance, in graph theory ordered vs unordered pairs make the distinction between directed and undirected graphs. Therefore, most graph papers dealing with undirected graphs start by saying that they use the ordered pair notations, but consider unordered pairs. In most cases, this is sufficient. In some situations, though, one deals with both kinds of pairs jointly and then needs to make a clear distinction. It was for instance the case in our paper Stream Graphs and Link Streams for the Modeling of Interactions over Time, where we used the X⊗Y notation for unordered pairs. My questions are: is there a standard notation for sets of unordered pairs? Why not ? And is the ⊗ notation appropriate, although it is already used in various contexts? A typical example where the distinction is important, although trivial: \|X×Y\|=\|X\|⋅\|Y\|, and so \|X×X\|=n2 if \|X\|=n. This is different from \|X⊗Y\|=\|(X∖Y)×Y\|+\|(Y∖X)×X\|−\|(X∖Y)×(Y∖X)\|+\|X∩Y\|2−\|X∩Y\|2 leading to \|X⊗X\|=n⋅(n−1)2 if \|X\|=n. reference-request graph-theory soft-question notation products Share CC BY-SA 4.0 Improve this question Follow this question to receive notifications edited Jan 7, 2021 at 10:39 Matthieu Latapy asked Jan 7, 2021 at 9:31 Matthieu LatapyMatthieu Latapy 1,53411 gold badge1212 silver badges3232 bronze badges 14 7 I would not consider the set of unordered pairs from two sets X,Y meaningful unless X=Y, in which case I would write it as (X2) (assuming we want distinct pairs). Qiaochu Yuan – Qiaochu Yuan 01/07/2021 10:34:03 Commented Jan 7, 2021 at 10:34 2 In fact the usual convention is that {x}={x}, so that one does not have to bother about repetitions. As to the set of unordered pairs, X×X is of course a totally wrong notation; X⊗X is at least different from X×X, but confusing in that it has nothing to do with any other use of ⊕. (X2) follows the standard convention about cardinality, it is self-explanatory and natural. Also used X(n) for unordered n-ples Pietro Majer – Pietro Majer 01/07/2021 11:30:56 Commented Jan 7, 2021 at 11:30 2 @QiaochuYuan The notation "[X]2" for the set of two-element subsets of X is standard at least in logic. Noah Schweber – Noah Schweber 01/07/2021 13:07:22 Commented Jan 7, 2021 at 13:07 1 If X≠Y, I am not sure that the notion of "unordered pair" really makes sense, because one does not have the Z2-action anymore. Francesco Polizzi – Francesco Polizzi 01/07/2021 14:23:29 Commented Jan 7, 2021 at 14:23 1 Thanks for your interesting comments! It seems that there is no consensus, rather a nice bestiary of relevant notations, each with its own strengths and weaknesses. Matthieu Latapy – Matthieu Latapy 01/07/2021 16:48:05 Commented Jan 7, 2021 at 16:48 \| Show 9 more comments 0 Reset to default You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions reference-request graph-theory soft-question notation products See similar questions with these tags. Linked 3 Notation for a graph without any edges? 3 Temporal generalization of graphs: density vs n and m? Related 2 Intersection graphs for "conflicting" directed paths in trees 4 Probability of two vertices being connected in a random graph 3 Temporal generalization of graphs: density vs n and m? 2 Best notation for tensor product with associativity 19 Is the powerset graph construction injective? Question feed By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
1693	https://staff.tiiame.uz/storage/users/43/books/rTQgmAwTIE1kaXiFZy9RntjqQR0qTvEM9OATuAk4.pdf	SCHAUM’S Easy OUTLINES PRINCIPLES OF ECONOMICS Other Books in Schaum’s Easy Outlines Series Include: Schaum’s Easy Outline: Calculus Schaum’s Easy Outline: College Algebra Schaum’s Easy Outline: College Mathematics Schaum’s Easy Outline: Discrete Mathematics Schaum’s Easy Outline: Differential Equations Schaum’s Easy Outline: Elementary Algebra Schaum’s Easy Outline: Geometry Schaum’s Easy Outline: Linear Algebra Schaum’s Easy Outline: Mathematical Handbook of Formulas and Tables Schaum’s Easy Outline: Precalculus Schaum’s Easy Outline: Probability and Statistics Schaum’s Easy Outline: Statistics Schaum’s Easy Outline: Trigonometry Schaum’s Easy Outline: Business Statistics Schaum’s Easy Outline: Principles of Accounting Schaum’s Easy Outline: Applied Physics Schaum’s Easy Outline: Biology Schaum’s Easy Outline: Biochemistry Schaum’s Easy Outline: Molecular and Cell Biology Schaum’s Easy Outline: College Chemistry Schaum’s Easy Outline: Genetics Schaum’s Easy Outline: Human Anatomy and Physiology Schaum’s Easy Outline: Organic Chemistry Schaum’s Easy Outline: Physics Schaum’s Easy Outline: Programming with C++ Schaum’s Easy Outline: Programming with Java Schaum’s Easy Outline: Basic Electricity Schaum’s Easy Outline: Electromagnetics Schaum’s Easy Outline: Introduction to Psychology Schaum’s Easy Outline: French Schaum’s Easy Outline: German Schaum’s Easy Outline: Spanish Schaum’s Easy Outline: Writing and Grammar SCHAUM’S Easy OUTLINES PRINCIPLES OF ECONOMICS Ba s e d o n S c h a u m ’ s Outline of Theory and Problems of Principles of Economics (Second Edition) b y D o m i n i c k S a lvat o r e , Ph.D. and E u g e n e A . D i u l i o , Ph.D. A b r i d g e m e n t E d i t o r W m. A l a n Ba r t l e y, Ph.D. SCHAUM’S OUTLINE SERIES M c G R AW- H I L L New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be repro-duced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior writ-ten permission of the publisher. 0-07-142583-7 The material in this eBook also appears in the print version of this title: 0-07-139873-2 All trademarks are trademarks of their respective owners. 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DOI: 10.1036/007145837 Contents v Chapter 1 Introduction to Economics 1 Chapter 2 Demand, Supply, and Equilibrium 13 Chapter 3 Unemployment, Inﬂation, and National Income 25 Chapter 4 Consumption, Investment, Net Exports, and Government Expenditures 37 Chapter 5 Traditional Keynesian Approach to Equilibrium Output 46 Chapter 6 Fiscal Policy 56 Chapter 7 The Federal Reserve and Monetary Policy 64 Chapter 8 Monetary Policy and Fiscal Policy 74 Chapter 9 Economic Growth and Productivity 81 Chapter 10 International Trade and Finance 88 Chapter 11 Theory of Consumer Demand and Utility 96 Chapter 12 Production Costs 104 Chapter 13 Perfect Competition 111 Chapter 14 Monopoly 118 Chapter 15 Monopolistic Competition and Oligopoly 125 Chapter 16 Demand for Economic Resources 132 Chapter 17 Pricing of Wages, Rent, Interest, and Proﬁts 139 Index 149 For more information about this title, click here. Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. This page intentionally left blank. Chapter 1 Introduction to Economics In the chapter: ✔ Methodology of Economics ✔ Problem of Scarcity ✔ Production-Possibility Frontier ✔ Principle of Increasing Costs ✔ Scarcity and the Market System ✔ True or False Questions ✔ Solved Problems Methodology of Economics Economics is a social science that studies individu-als and organizations engaged in the production, dis-tribution, and consumption of goods and services. The goal is to predict economic occurrences and to develop policies that might prevent or correct such problems as unemployment, inﬂation, or waste in the economy. Economics is subdivided into macroeconomics and microeconomics. Macroeconomics studies ag-gregate output, employment, and the general price level. Microeconom-1 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. ics studies the economic behavior of individual decision makers such as consumers, resource owners, and business ﬁrms. The discipline of economics has developed principles, theories, and models that isolate the most important determinants of economic events. In constructing a model, economists make assumptions to eliminate un-necessary detail to reduce the complexity of economic behavior. Once modeled, economic behavior may be presented as a relationship between dependent and independent variables. The behavior being explained is the dependent variable; the economic events explaining that behavior are the independent variables. The dependent variable may be presented as depending upon one independent variable, with the inﬂuence of the oth-er independent variables held constant (the ceteris paribus assumption). An economic model will also specify whether the dependent and inde-pendent variables are positively or negatively related, i.e., moving in the same or opposite directions. Note! Ceteris paribus is Latin for “other things being equal.” This phrase is used often by economists in modeling to isolate the relationship between spe-ciﬁc dependent and independent variables. Example 1.1 We shall assume that the amount a consumer spends (C) is positively re-lated to her disposable income (Yd), i.e., C = f(Yd). Table 1.1 presents data on consumer spending for ﬁve individuals with different levels of in-come. As seen in the table, consumption and disposable income display a positive relationship. The data from Table 1.1 are plotted in Figure 1-1 and labeled C1. The dependent variable, consumer spending, is plotted on the vertical axis and the independent variable, disposable income, is plotted on the horizontal axis. Graphs are used to present data and the positive or negative rela-tionship of the dependent and independent variables visually. 2 PRINCIPLES OF ECONOMICS Problem of Scarcity Economics is the study of scarcity—the study of the allocation of scarce resources to satisfy human wants. People’s material wants, for the most part, are unlimited. Output, on the other hand, is limited by the state of CHAPTER 1: Introduction to Economics 3 Table 1.1 (in $) Figure 1-1 technology and the quantity and quality of the economy’s resources. Thus, the production of each good and service involves a cost. A good is usually deﬁned as a physical item such as a car or a hamburger, and a ser-vice is something provided to you such as insurance or a haircut. Scarcity is a fundamental problem for every society. Decisions must be made regarding what to produce, how to produce it, and for whom to produce. What to produce involves decisions about the kinds and quanti-ties of goods and services to produce. How to produce requires decisions about what techniques to use and how economic resources (or factors of production) are to be combined in producing output. The economic re-sources used to produce goods and services include: • Land. The economy’s natural resources—such as land, trees, and minerals. • Labor. The mental and physical skills of individuals in a soci-ety. • Capital. Goods—such as tools, machines, and factories—used in production or to facilitate production. The for whom to produce involves decisions on the distribution of output among members of a society. Remember Economics helps to solve the three important questions of what to pro-duce, how to produce it, and for whom to produce. These decisions involve opportunity costs. An opportunity cost is what is sacriﬁced to implement an alternative action, i.e., what is given up to produce or obtain a particular good or service. For example, the op-portunity cost of expanding a country’s military arsenal is the decreased production of nonmilitary goods and services. Opportunity costs are found in every situation in which scarcity necessitates decision making. Opportunity cost is the value—monetary or otherwise—of the next 4 PRINCIPLES OF ECONOMICS best alternative, or that which is given up. This concept is used in both macroeconomics and microeconomics. Production-Possibility Frontier A production-possibility frontier shows the maximum number of alter-native combinations of goods and services that a society can produce at a given time when there is full utilization of economic resources and tech-nology. Table 1.2 presents alternative combinations of guns and butter output for a hypothetical economy (guns represent the output of military goods, while butter represents nonmilitary goods and services). In choos-ing what to produce, decision makers have a choice of producing, for ex-ample, alternative C—5,000 guns and 14 million units of butter—or any other alternative presented. This production-possibility schedule is plotted in Figure 1-2. The curve, labeled PP, is called the production-possibility frontier. Point C plots the combination of 5,000 guns and 14 million units of butter, as-suming full employment of the economy’s resources and full use of its technology, as do all of the alternatives presented in Table 1.2. The production-possibility frontier depicts not only limited produc-tive capability and therefore the problem of scarcity, but also the concept of opportunity cost. When an economy is situated on the production-possibility frontier, such as at point C, gun production can be increased only by decreasing butter output. Thus, to move from alternative C (5,000 guns and 14 million units of butter) to alternative D (9,000 guns and 6 million units of butter), the opportunity cost of the additional 4,000 units of gun production is the 8 million less units of butter that are produced. CHAPTER 1: Introduction to Economics 5 Table 1.2 The production-possibility frontier shifts outward over time as more resources become available and/or technology is improved. Growth in an economy’s productive capability is depicted in Figure 1-2 by the outward shift of the production-possibility frontier from PP to P′P′. Suppose a so-ciety chooses to be at point C. When the production-possibility frontier shifts outward, 4,000 additional guns can be produced without sacriﬁc-ing any butter production, as seen at C′. This example should not be con-strued as a refutation of the law of opportunity cost just because fewer sacriﬁces may be made when growth occurs. When there is full utiliza-tion of resources and an absence of growth, additional gun production is possible only when the output of butter is decreased. Points on a production-possibility frontier are considered to be efﬁ-cient. Points within the frontier are inefﬁcient, and points outside the frontier are unattainable. Points C and D are efﬁcient because all avail-able resources are utilized and there is full use of existing technology. Po-sitions outside the production-possibility frontier are unattainable since the frontier deﬁnes the maximum amount that can be produced at a giv-en time. Positions within the frontier are inefﬁcient because some re-sources are either unemployed or underemployed. 6 PRINCIPLES OF ECONOMICS Figure 1-2 Principle of Increasing Costs Resources are not equally efﬁcient in the production of all goods and ser-vices, i.e., they are not equally productive when used to produce an al-ternative good. This imperfect substitutability of resources is due to dif-ferences in the skills of labor and to the specialized function of most machinery and many buildings. Thus, when the decision is made to pro-duce more guns and less butter, the new resources allocated to the pro-duction of guns are usually less productive. It therefore follows that as larger amounts of resources are transferred from the production of butter to the production of guns, increasing units of butter are given up for few-er incremental units of guns. This increasing opportunity cost of gun pro-duction illustrates the principle of increasing costs. Note! The principle of increasing opportunity cost is the reason why the production-possibility frontier is bowed outward from the origin of the graph, and not a straight line. Scarcity and the Market System As we have seen, two of the most important economic decisions faced by a society are deciding what goods and services to produce and how to al-locate resources among their competing uses. The combination of goods and services produced can be resolved by government command or through a market system. In a command economy, a central planning board determines the mix of output. The experience with this system, however, has not been very successful, as evidenced by the changing eco-nomic and political events in the 1990s in the command economies of Eastern Europe and the former USSR. In a market economy, economic decisions are decentralized and are made by the collective wisdom of the marketplace, i.e., prices resolve the three fundamental economic questions of what, how, and for whom. The CHAPTER 1: Introduction to Economics 7 only goods and services produced are those that individuals are willing to purchase at a price sufﬁcient to cover the cost of producing them. Be-cause resources are scarce, goods and services are produced using the technique and resource combination that minimizes the cost of produc-tion. And the goods and services produced are sold (distributed) to those who are willing and have the money to pay the prices. Most countries have a mixed economy, a mixture of both command and market economies. For example, the United States has primarily a market economy, although the government produces some goods, such as roads, and ﬁnances these expenditures by taxing the income of indi-viduals and businesses. The government may also regulate how the mar-ket operates, such as with minimum wage laws. True or False Questions 1. Economic models and theories are accurate statements of reality. 2. In the statement “consumption is a function of disposable in-come,” consumption is the dependent variable. 3. Graphs provide a visual representation of the relationship between two variables. 4. A production-possibility frontier depicts the unlimited wants of a society. 5. When there is full employment, the decision to produce more of one good necessitates decreased production of another good. 6. There are increasing costs of production because economic re-sources are not equally efﬁcient in the production of all goods and ser-vices. Answers: 1. False; 2. True; 3. True; 4. False; 5. True; 6. True Solved Problems Solved Problem 1.1 What are some of the problems associated with the study of economics? Solution: Multiple difﬁculties may arise with the study of economics. a. Generalizing from individual experiences often leads to wrong conclusions (this is called the fallacy of composition). For example, an 8 PRINCIPLES OF ECONOMICS individual becomes richer by increasing his savings, but society as a whole may become poorer if everyone saves because people may be put out of work. b. The fact that one economic event precedes another does not nec-essarily imply cause and effect, which is what economists want to study. For example, the U.S. stock market collapse in 1929 did not cause the worldwide Great Depression of the 1930s. c. Since economics is a social science and laboratory experiments cannot be conducted, economic theories can only describe expected be-havior. Thus, these theories are not as precise or reliable as the natural laws established in the pure sciences. Solved Problem 1.2 a. The demand for purchasing videos might be presented as a func-tion of the video’s purchase price. Does this mean that income and the cost of renting a video are unimportant? b. What is the meaning of and the economist’s use of the term ceteris paribus? Solution: a. To further simplify the demand-for-videos function—Qvideos = f(Yd, Pvideos, Prental)—we could assume that the individual’s disposable income and the cost of renting videos are unchanged. Thus, while income and rental cost do inﬂuence the demand for videos, here more videos are purchased only because of a lower price. b. The phrase ceteris paribus means that other independent variables affecting the dependent variable are held constant, or are unchanged. When other independent variables that inﬂuence the quantity of videos purchased are held constant, the demand for videos can be presented as Qvideos = f(Pvideos), ceteris paribus. Solved Problem 1.3 Suppose an economy has the production-possibili-ty frontier depicted in Figure 1-3? a. What implication does the selection of point A or C have regard-ing the economy’s current and future production of consumer goods and services? b. What linkage is there between saving and economic growth? CHAPTER 1: Introduction to Economics 9 Solution: a. At point A, society has more consumer goods and services in the current period. Point C, however, provides the possibility of a larger quantity of consumer goods and services in the future because of addi-tions to the economy’s stock of capital resources. Here the economy’s productive capabilities and thus production-possibilities frontier will ex-pand (perhaps through the addition of a new factory) and thereby provide an increased output of consumer goods and services in a future period. b. As discussed in a., society must forgo purchases of consumer goods and services now if it is to increase its capital and thereby expand production capabilities. Thus, people must be willing to save, and have 10 PRINCIPLES OF ECONOMICS Figure 1-3 fewer goods and services now, so that resources can be used in the cur-rent period to produce capital goods. Solved Problem 1.4 Figure 1-4 presents a production-possibility fron-tier for food and clothing. a. What is the opportunity cost of increasing food production from 0 to 2 million units, from 2 million to 4 million units, and from 4 million to 6 million units? b. What is happening to the opportunity cost of increasing food pro-duction from 0 to 6 million units? c. Explain how the shape of the production-possibility frontier im-plies increasing costs for the production of clothing. CHAPTER 1: Introduction to Economics 11 Figure 1-4 Solution: a. In increasing food production from 0 to 2 million units, produc-tion of clothing decreases from 16,000 to 15,000 units. Thus, the oppor-tunity cost of producing the ﬁrst 2 million units of food is 1 thousand units of clothing. The opportunity cost of a second and third additional 2 mil-lion units is 2,000 and 3,000 units of clothing, respectively. b. The opportunity cost of increasing food production is increasing from 1,000 units of clothing to 2,000 to 3,000 units of clothing. c. Increasing clothing and food costs are reﬂected in a concave (out-ward-sloping) production-possibility frontier. Moving down the frontier from point A to points B, C, D, E, and F shows that to produce 2 million incremental units of food (the 2-million-unit-length horizontal dashed lines in Figure 1-4), we must give up more and more units of clothing (the vertical dashed lines of increasing length). Solved Problem 1.5 Explain how division of labor and specialization enhance production in an advanced society. Solution: Through the division of labor and specialization, the population within a given geographic region, instead of being self-sufﬁcient and producing the full range of goods and services wanted, can concentrate its energies and time in the production of only a few goods and services in which its efﬁciency is greatest. Thus, specialization and division of labor allow greater output. By then exchanging some of the goods and services so produced for different goods and services produced similarly within a dif-ferent geographic region, the regions’populations as a whole end up con-suming a larger number and greater diversity of goods and services than would otherwise be the case. 12 PRINCIPLES OF ECONOMICS Chapter 2 Demand, Supply, and Equilibrium In This Chapter: ✔ Demand ✔ Supply ✔ Equilibrium Price and Quantity ✔ Government and Price Determination ✔ Elasticity ✔ True or False Questions ✔ Solved Problems Demand The demand schedule for an individual speciﬁes the units of a good or service that the individual is willing and able to purchase at alternative prices during a given period of time. The relationship be-tween price and quantity demanded is inverse: more units are purchased at lower prices because of a substitution effect and an income effect. As a commodity’s price falls, an individual normally purchases more of this good since he or she is like-13 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. ly to substitute it for other goods whose price has remained unchanged. Also, when a commodity’s price falls, the purchasing power of an indi-vidual with a given income increases, allowing for greater purchases of the commodity. When graphed, the inverse relationship between price and quantity demanded appears as a negatively sloped demand curve. A market demand schedule speciﬁes the units of a good or service all indi-viduals in the market are willing and able to purchase at alternative prices, i.e., Qd = f (P). Example 2.1 Table 2.1 gives an individual’s demand and the market demand for a com-modity. Column 2 shows one individual’s demand for corn—the bushels of corn that one individual is willing and able to buy per month at alter-native prices. We ﬁnd, for example, that the individual buys 3.5 bushels of corn each month when the price is $5 per bushel. If there are 1,000 in-dividuals in the market, the market demand for corn is the sum of the quantities the 1,000 individuals will buy at each price. So for example, 1,000 individuals collectively are willing to purchase 3,500 bushels of corn each month at $5 per bushel. The market demand is shown in the last column, which shows the typical relationship between quantity de-manded and price, i.e., more units of a commodity are demanded at low-er prices. The market demand for corn is plotted in Figure 2-1 and the curve is labeled D. Note that the demand curve is negatively sloped. The market demand for a good or service is inﬂuenced not only by the commodity’s price, but also by the price of other goods and services, 14 PRINCIPLES OF ECONOMICS Table 2.1 disposable income, wealth, tastes, and the size of the market. In present-ing the market demand for corn of Table 2.1 and Figure 2-1, variables oth-er than the commodity’s price are held constant. This relationship is pre-sented as Qd = f (Pcorn), ceteris paribus, where ceteris paribus indicates that variables other than the price of corn are unchanged. When one or more of these variables change, there is a change in demand and there-fore a shift of the demand curve. For example, the market demand curve shifts up and to the right when there is an increased preference for the commodity, when income increases, and when the price of a substitute commodity rises and/or the price of a complementary good declines. A substitute good can be used instead of the good considered (wheat for corn), and a complementary good is used together with the good consid-ered (butter with corn). A common error made by the beginning economics student is failure to differentiate between a change in demand and a change in quantity de-manded. A change in demand refers to a shift of the demand curve be-cause a variable other than price has changed. A change in quantity de-manded occurs when there is a change in the commodity’s price, resulting in a movement along an existing demand curve. CHAPTER 2: Demand, Supply, and Equilibrium 15 Figure 2-1 Remember There is a distinct difference be-tween demand and quantity de-manded, and the two must not be confused. Example 2.2 The market demand for corn from Table 2.1 was plotted in Figure 2-1 and labeled D. The market demand shifts up and to the right from D to D1 when the market size increases—for example, when the number of indi-viduals in this market increases from 1,000 to 1,200. Should the price of wheat then increase—and individuals substitute corn for wheat in their diets—the market demand curve for corn again shifts up and to the right, this time from D1 to D2. Supply A supply schedule speciﬁes the units of a good or service that a produc-er is willing to supply (Qs) at alternative prices over a given period of time, i.e, Qs = f (P). The supply curve normally has a positive (upward) slope, indicating that the producer must receive a higher price for in-creased output due to the principle of increasing costs. (Review Chapter 1). A market supply curve is derived by summing the units each individ-ual producer is willing to supply at alternative prices. A typical market supply curve (labeled S) is plotted in Figure 2-2. The market supply curve shifts when the number and/or size of pro-ducers changes, factor prices (wages, interest, and/or rent paid to eco-nomic resources) change, the cost of materials changes, technological progress occurs, and/or the government subsidizes or taxes output. The market supply curve shifts down and to the right with more pro-ducers entering the market, decreases in factor or materials prices, im-provement in technology, and government subsidization. A change in supply thereby denotes a shift of the supply curve. A change in quantity 16 PRINCIPLES OF ECONOMICS supplied indicates a change in the commodity’s price and therefore a movement along an existing supply curve. In Figure 2-2, if the number of producers increases, the market supply curve shifts down and to the right from S to S1. If a technological improvement in corn production also develops, the market supply curve shifts further downward from S1 to S2. Equilibrium Price and Quantity Equilibrium occurs at the intersection of the market supply and market demand curves. At this intersection, quantity demanded equals quantity supplied, i.e., the quantity that individuals are willing to purchase exact-ly equals the quantity producers are willing to supply. A surplus exists at prices higher than the equilibrium price since the quantity demanded falls short of the quantity supplied. At prices lower than the equilibrium price, there is a shortage of output since quantity demanded exceeds quantity supplied. Once achieved, the equilibrium price and quantity persist until there is a change in demand and/or supply. CHAPTER 2: Demand, Supply, and Equilibrium 17 Figure 2-2 You Need to Know Economists spend much time and effort in analyz-ing where and how market equilibrium is achieved. Its importance cannot be overstated. Equilibrium price and/or equilibrium quantity change when the mar-ket demand and/or market supply curves shift. Equilibrium price and equilibrium quantity both rise when there is an increase in market demand with no change in the market supply curve. Equilibrium price falls while equilibrium quantity increases when market supply increases and de-mand is unchanged. Government and Price Determination The government may intervene in the market and mandate a maximum price (price ceiling) or minimum price (price ﬂoor) for a good or service. For example, some city governments in the U.S. legis-late the maximum price that a landlord can charge a ten-ant for rent. Such rent-control policies, though well-intentioned, result in a disequilibrium in the housing market since, at the government-mandated price ceiling, the quantity of housing supplied falls short of the quan-tity of housing demanded. An example of minimum prices (price ﬂoors) in the U.S. is the minimum wage. Price ﬂoors result in market disequilibrium in that quantity supplied at the mandated price exceeds quantity demanded. The government can alter an equilibrium price by changing market demand and/or market supply. The government can restrict demand by rationing a good, as occurred for many items during World War II. Equi-librium price can be altered by shifting the market supply curve. A tax on a good raises its supply price—shifts the market supply curve up and to the left—and causes the equilibrium price to increase and the equilibri-um quantity to fall. A subsidy to the producer will do the opposite and lower equilibrium price and raise equilibrium quantity. 18 PRINCIPLES OF ECONOMICS Elasticity Market prices will change whenever shifts in supply or demand occur. Example 2.3 Table 2.2 gives a hypothetical market demand and supply schedule for wheat; it shows whether a surplus or shortage occurs at each price and in-dicates the pressure on price toward equilibrium. Thus, the equilibrium price is $2 because the quantity demanded, 4,500 bushels of wheat per month, equals the quantity supplied. The elasticity of demand (ED) measures the percentage change in the quantity demanded of a commodity as a result of a given percentage in its price. The formula is ED can be calculated in terms of the new quantity and price, or with the original quantity and price. However, different results would then be ob-tained. To avoid this problem, economists generally measure ED in terms of the average quantity and the average price, as follows: ED is a pure number. Thus, it is a better measurement tool than the slope, which is expressed in terms of the units of measurement. ED is always ED = ÷ change in quantity demanded (sum of quantities demanded) / 2 change in price (sum of prices) / 2 ED = percentage change in the quantity demanded percentage change in price CHAPTER 2: Demand, Supply, and Equilibrium 19 Table 2.2 expressed as a positive number, even though price and quantity demand-ed move in opposite directions. The demand curve is said to be elastic if ED > 1, unitary elastic if ED = 1, and inelastic if ED < 1. Don’t Forget! Different formulas are used to compute elasticity and slope. Asimple glance at a graph is not enough to determine whether a curve has a high or low elasticity. Example 2.4 The elasticity between the quantities demanded at $4 and $3 of Table 2.2 is calculated below using the average quantities and prices. Thus, we say that this demand curve is elastic (on the average) be-tween these two points. The elasticity of demand is greater (1) the greater the number of good substitutes available, (2) the greater the proportion of income spent on the commodity, and (3) the longer the time period con-sidered. When the price of a commodity falls, the total revenue of producers (price times quantity) increases if ED > 1, remains unchanged if ED = 1, and decreases if ED < 1. This occurs because when ED > 1, the percent-age increase in quantity exceeds the percentage decline in price and so total revenue (TR) increases. When ED = 1, the percentage increase in quantity equals the percentage decline in price and so TR remains un-changed. Finally, when ED < 1, the percentage increase in quantity is less than the percentage decline in price, and so TR falls. The elasticity of supply (ES) measures the percentage change in the quantity supplied of a commodity as a result of a given percentage change in its price. We again use the average quantity and price as follows: ED = + ÷ + = ÷ = = 1 2 3 2 1 4 3 2 1 2 5 1 3 5 3 5 2 5 1 4 ( ) / ( ) / . . . . . 20 PRINCIPLES OF ECONOMICS ES is a pure number and is positive because price and quantity move in the same direction. Supply is said to be elastic if ES > 1, unitary elas-tic if ES = 1, and inelastic if ES < 1. Example 2.5 The (average) elasticity between the quantities supplied at $1 and $2 of the supply schedule of Table 2.2 is True or False Questions 1. There is a decrease in the demand for a commodity when the price of a substitute commodity increases. 2. When the supply curve is positively sloped, an increase in de-mand will result in a larger quantity supplied. 3. A surplus exists when the market price is above the equilibrium price. 4. Government subsidization of ﬁrms producing Good A results in an increase in the demand for Good A. 5. Demand is inelastic if the percentage increase in quantity exceeds the percentage decrease in price. 6. A decline in price leaves total revenue unchanged when ED = 1. Answers: 1. False; 2. True; 3. True; 4. False; 5. False; 6. True Solved Problems Solved Problem 2.1 Explain what happens to the demand curve for air transportation between New York City and Washington, D.C., as a result of the following events: a. The income of households in metropolitan New York and Wash-ington, D.C., increases 20%. ES = + ÷ + = ÷ ≅ 2 2 5 4 5 2 1 1 2 2 1 3 5 1 1 5 0 43 ( . . ) / ( ) / . . . . ES = ÷ change in quantity supplied (sum of quantities supplied) / 2 change in price (sum of prices) / 2 CHAPTER 2: Demand, Supply, and Equilibrium 21 b. The cost of a train ticket between New York City and Washington, D.C., is reduced 50%. c. The price of an airline ticket decreases 20%. Solution: a. Individuals will travel more since they have more disposable in-come. The demand for air transportation between NYC and Washington increases; the demand curve shifts up and to the right. b. The cost of an alternative mode of transportation between NYC and Washington has decreased; thus, more individuals will travel by train between NYC and Washington. The demand for air transportation de-creases; the demand curve shifts down and to the left. c. There is no shift, but there is a movement down the existing de-mand curve; the lower price for an airline ticket results in an increase in the number of people traveling (quantity demanded) by air between NYC and Washington. Solved Problem 2.2 Suppose the market supply and demand curves for Good A are initially S and D, respectively, in Figure 2-3; equilibrium price is $3 and equilibrium quantity is 280 units. 22 PRINCIPLES OF ECONOMICS Figure 2-3 a. Suppose improved technology in the production of Good A shifts the market supply curve from S to S, ceteris paribus. After the initial supply shift, what is the relationship between quantity demanded and quantity supplied at the initial $3 equilibrium price? b. What is the new equilibrium price and quantity after the techno-logical advance has increased the supply of Good A? Solution: a. Quantity demanded for market schedule D is 280 units when the price is $3, while market supply is 330 units. There is a surplus of Good Aat the initial $3 equilibrium price which puts downward pressure on the price of Good A. b. Equilibrium price falls from $3 to $2 as a result of the increase in market supply; equilibrium quantity increases from 280 to 320 units. Solved Problem 2.3 Why has the federal government placed price ﬂoors on some agricultural goods? Solution: Aprice ﬂoor is a government-mandated price that exists above the market’s equilibrium price; price ﬂoors result in a surplus of produc-tion. While market demand for most agricultural commodities is rela-tively stable over time, market supply is very much inﬂuenced by the weather. A drought, for example, decreases supply and pushes up prices while a bumper crop can severely depress agricultural prices. The prof-itability of farming becomes uncertain, as does the price of food products and the income needed to feed a household. Thus, the reasons for agri-cultural price supports (price ﬂoors) are: (1) to stabilize farmer incomes and encourage farmers to continue farming whether there are bumper crops or droughts; (2) to provide a steadier ﬂow of agricultural products at relatively stable prices; and (3) to stabilize the amount of income that households need to spend on food. Solved Problem 2.4 a. Is the demand for table salt elastic or inelastic? Why? b. Is the demand for stereos elastic or inelastic? Why? Solution: a. The demand for salt is inelastic because there are no good substi-tutes for salt and households spend a very small portion of their total in-CHAPTER 2: Demand, Supply, and Equilibrium 23 come on this commodity. Even if the price of salt were to rise substan-tially, households would reduce their purchases of salt little. b. The demand for stereos is elastic because stereos are expensive and, as a luxury rather than a necessity, their purchase can be postponed or avoided when their price rises. One could also use the radio as a par-tial substitute for a stereo. Solved Problem 2.5 a. Should the price of a subway ride or bus ride be increased or de-creased if total revenue needs to be increased? b. What about the price of a taxi ride? Solution: a. To the extent that there are no inexpensive good substitutes for public transportation in metropolitan areas, the demand for subway and bus rides is inelastic. Their prices should, therefore, be increased to in-crease total revenue. However, this can be self-defeating. Sharply in-creasing the price of public transportation will encourage people to use their cars and increase congestion and pollution. b. For taxi rides, the case is likely to be different. Taxi rides are rel-atively expensive; an increase in their price may encourage people to rely much more on their cars and public transportation. To the extent that this makes the demand for taxi rides elastic, total revenue will fall when the price of taxi rides is increased. 24 PRINCIPLES OF ECONOMICS Chapter 3 Unemployment, Inﬂation, and National Income In This Chapter: ✔ Gross Domestic Output ✔ Aggregate Demand, Aggregate Supply, and Equilibrium Output ✔ Changes in Aggregate Output ✔ Business Cycles ✔ Unemployment and the Labor Force ✔ Inﬂation ✔ True or False Questions ✔ Solved Problems Gross Domestic Output Gross domestic product (GDP) measures total output in the domestic economy. Nominal GDP, real GDP, and potential GDP are three different measures of aggregate output. Nominal GDP is the market value of all ﬁnal goods and ser-vices produced in the domestic economy in a one-year pe-25 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. riod at current prices. By this deﬁnition, (1) only output exchanged in a market is included (do-it-yourself services such as cleaning your own house are not included); (2) output is valued in its ﬁnal form (output is in its ﬁnal form when no further alteration is made to the good which would change its market value); and (3) output is measured using current-year prices. Because nominal GDPvalues are inﬂated by prices that increase over time, aggregate output is also measured holding the prices of all goods and services constant over time. This valuation of GDP at constant prices is called real GDP. The third measure of aggregate output is potential GDP, the maxi-mum production that can take place in the domestic economy without putting upward pressure on the general level of prices. Conceptually, po-tential GDP represents a point on a given production-possibility frontier. The U.S. economy’s potential output increases at a fairly steady rate each year while actual real GDP ﬂuctuates around potential GDP. These ﬂuctuations of real GDP are identiﬁed as business cycles. The GDP gap is the difference between potential GDP and real GDP; it is positive when potential GDP exceeds real GDP and negative when real GDP exceeds potential GDP. A positive gap indicates that there are unemployed re-sources and the economy is operating inefﬁciently within its production-possibility frontier. It therefore follows that an economy’s rate of unem-ployment rises as its GDP gap increases, and falls when the gap declines. An economy is operating above its normal productive capacity when there is a negative gap. Aggregate Demand, Aggregate Supply, and Equilibrium Output The economy’s equilibrium level of output occurs at the point of inter-section of aggregate supply and aggregate demand. In microeconomics, equilibrium price exists where quantity demanded equals quantity sup-plied. The supply and demand schedules in macroeconomics differ in that they relate the aggregate quantity supplied and the aggregate quantity de-manded to the price level. 26 PRINCIPLES OF ECONOMICS Important Things to Remember Supply and demand curves may appear similar to aggregate supply and aggregate demand curves in graphs, but they are substantially different. An aggregate demand curve represents the collective spending of consumers, businesses, and government, as well as net foreign purchas-es of goods and services, at different price levels. An aggregate demand curve, like the demand curve in microeconomics, is negatively related to price, holding constant other factors that inﬂuence aggregate spending decisions. Price, presented as price level in macroeconomics, affects aggregate spending because of an interest rate effect, a wealth effect, and an inter-national purchasing power effect. The interest rate effect traces the effect that interest rate levels have upon aggregate spending. The nominal rate of in-terest is directly related to the price level, ceteris paribus. Increases in the price level push up interest rates, which usually will depress interest-sensitive spending. The wealth effect relates changes in wealth to changes in aggregate spending. The mar-ket value of many ﬁnancial assets falls as price lev-el and interest rates increase. Ahigher price level will decrease the house-hold sector’s net wealth, lower consumer spending, and cause lower aggregate spending. A country’s imports and exports are also affected by a changing price level, i.e., by an international purchasing power effect. When the price level increases in the home country and is unchanged in foreign countries, foreign-made commodities become relatively less ex-pensive, the home country’s exports fall, its imports increase, and there is less aggregate spending on the home country’s output. An aggregate demand curve shifts when there is a change in a vari-able (other than price level) that affects aggregate spending decisions. Outward shifts (to the right) occur when consumers become more will-ing to spend or there are increases in investment spending, government expenditures, and net exports. Determinants of these factors will be tak-en up in the next chapter. CHAPTER 3: Unemployment, Inﬂation, and Income 27 An aggregate supply schedule depicts the relationship of aggregate output and price level, holding constant other variables that could affect supply. There is some disagreement among economists on the shape of the aggregate supply curve. Three distinct curves can characterize this disagreement. The Keynesian aggregate supply curve is horizontal until it reaches the economy’s full-employment level of output, at which point it becomes positively sloped. Others view the aggregate supply curve as always being positively sloped. The classical aggregate supply curve is vertical at the full-employment level, indicating there is no relationship between aggregate output and the price level. Changes in economy-wide resource availability, resource cost, and technology shift the aggregate supply curve. The aggregate supply curve shifts rightward when (1) improved technology increases the potential output of a given quantity of resources; (2) the quantity of economic re-sources increases; or (3) the cost of resources declines. Changes in Aggregate Output The effect of changes in aggregate demand and/or aggregate supply upon equilibrium output and the price level depends upon the shape of the ag-gregate supply curve. With a Keynesian aggregate supply curve, an in-crease in aggregate demand affects only output as long as the economy is below full-employment output, whereas an increase in aggregate sup-ply has no effect upon either the price level or output. Increases in ag-gregate demand and/or aggregate supply affect both the price level and real output when aggregate supply is positively sloped, as can be seen in Figure 3-1. For a classical aggregate supply curve, increases in aggregate demand result in only a higher price level, whereas increases in aggre-gate supply result in a higher level of output and a lower price level. Example 3.1 Equilibrium real output is y1 and the price level is p1 for aggregate sup-ply and aggregate demand curves AS and AD in Figure 3-1. Increased government spending shifts the aggregate demand curve outward to AD, and the point of equilibrium changes from E1 to E2. Equilibrium output increases from y1 to y2 as the price level rises from p1 to p2. When ag-gregate supply increases to AS and aggregate demand remains at AD, 28 PRINCIPLES OF ECONOMICS CHAPTER 3: Unemployment, Inﬂation, and Income 29 the equilibrium point changes from point E1 to E3. Equilibrium output in-creases from y1 to y2 and the price level falls from p1 to p0. There are two approaches to measuring aggregate output: an expen-diture approach, which measures the value of ﬁnal sales, and a cost ap-proach, which measures the value added in producing ﬁnal output. The expenditure or ﬁnal sales approach consists of summing the consumption spending of individuals (C), investment spending by businesses (I), gov-ernment expenditures (G), and net exports (Xn). [GDP = C + I + G + Xn]. The cost approach consists of summing the value added to ﬁnal output at each stage of production. Gross domestic product consists of all output produced within the country’s boundaries. Business Cycles A business cycle is a cumulative ﬂuctuation in aggregate output that lasts for some time. Although recurrent, the duration and intensity of each ﬂuc-tuation varies. Points at which aggregate output changes direction are marked by peaks and troughs. A peak is a point which marks the end of economic expansion (rising aggregate output) and the beginning of a re-cession (decline in economic activity). A trough marks the end of a re-Figure 3-1 cession and the beginning of economic recovery. The time span between troughs and peaks is classi-ﬁed as an expansionary period (trough to peak) or a contractionary period (peak to trough). There are a number of explanations for the cyclical behavior of ag-gregate output. The central focus of many of these theories is investment spending and consumer purchases of durable goods. These expenditures consist of large-ticketed items whose purchase, in most cases, can be postponed. For example, an individual can repair an existing car rather than purchase a new one. Thus, purchases of such items occur when cred-it (borrowing) is more readily available or less costly, individuals are more optimistic about the future, and/or cash ﬂows are more certain. However no one theory is able to explain why some business cycles are more severe than others. This suggests that there are numerous causes and that the importance of each cause varies. Unemployment and the Labor Force The U.S. labor force does not include the entire population but only those who are at least 16 years old, employed, or unemployed and looking for work. A working-age person who is not looking for work is considered voluntarily unemployed and is not included in the labor force. Thus, the size of the labor force and the number of people unemployed can be un-derstated when a signiﬁcant number of workers, after some searching, be-come discouraged and stop looking for work. The unemployment rate is the percent of the total labor force that is unemployed. Unemployment arises for frictional, structural, and cyclical reasons. Frictional unemployment is temporary and occurs when a per-son (1) quits a current job before securing a new one, (2) is not immedi-ately hired when entering the labor force, or (3) is let go by a dissatisﬁed employer. Workers who lose their jobs due to a change in the demand for a particular commodity or because of technological advance are struc-turally unemployed; their unemployment normally lasts for a longer period since they usually possess specialized skills which are not de-manded by other employers. Cyclical unemployment is the result of in-sufﬁcient aggregate demand. Workers have the necessary skills and are available to work, but there are insufﬁcient jobs because of inadequate aggregate spending. Cyclical unemployment occurs when real GDP falls below potential GDP. 30 PRINCIPLES OF ECONOMICS Note! In the economist’s deﬁnition of unemployment, not everyone that is without a job is unemployed. Full employment exists when there is no cyclical unemployment but normal amounts of frictional and structural unemployment; thus, full em-ployment exists at an unemployment rate greater than zero. This is re-ferred to as the natural rate of unemployment. It may change when there is a change in the normal amount of frictional and structural unemploy-ment. The cyclical unemployment rate can be negative when real GDP exceeds potential GDP and the economy is producing beyond its normal full-employment level. This negative cyclical unemployment rate indi-cates that the normal job search period for the frictionally and structural-ly unemployed is shortened because of an abnormally large number of job openings. Cyclical unemployment imposes costs upon both society and the person unemployed. Society’s opportunity cost is the amount of output which is not produced and therefore is lost forever. The personal costs that occur during an economic downturn are unevenly distributed between different types of workers. Example 3.2 Table 3.1 presents the unemployment rate by sex, age, and race in 1992, when U.S. real GDP was considerably below potential output, and in 1987, when U.S. real GDP equaled potential GDP. Note that the unem-ployment rate is always higher for teenagers than for those older, and higher for blacks and others than for whites. This difference worsens when the economy is below its potential GDP. Inﬂation A price index relates prices in a speciﬁc year, month, or quarter to prices during a reference period. For example, the consumer price index (CPI), the most frequently quoted price index, relates the prices that urban con-sumers paid for a ﬁxed basket of approximately 400 goods and services CHAPTER 3: Unemployment, Inﬂation, and Income 31 in a given month to the prices that existed during a reference period. The producer price index (PPI) and GDP deﬂator are the other two major price indexes. The PPI measures the prices for ﬁnished goods, in-termediate materials, and crude materials at the wholesale level. Because wholesale prices are even-tually translated into retail prices, changes in the PPI are usually a good predictor of changes in the CPI. The GDP deﬂator is the most compre-hensive measure of the price level since it measures prices for net exports, investment, and government expenditures, as well as for consumer spending. Inﬂation is the annual rate of increase in the price level. Disinﬂation is a term used to denote a slowdown in the rate of inﬂation; deﬂation ex-ists when there is an annual rate of decrease in the price level. While there have been some monthly decreases in the price level, the U.S. economy has not experienced deﬂation since the 1930s. You Need to Know Inﬂation refers to an increase in the general price level, not the price of a speciﬁc good or service. 32 PRINCIPLES OF ECONOMICS Table 3.1 Economists identify two distinct causes of inﬂation. Demand-pull in-ﬂation is inﬂation that occurs when aggregate spending exceeds the econ-omy’s normal full-employment level of output, i.e., when aggregate de-mand is pushed too far to the right along a given aggregate supply curve. Demand-pull inﬂation is normally characterized by both a rising price and output level. It often results in an unemployment rate lower than the natural rate. Cost-push inﬂation originates from increases in the cost of producing goods and services, such as wages or the prices of raw mate-rials. Aggregate supply is pushed to the left, which is referred to as stagﬂation. It is associated with increases in the price level, decreases in aggregate output, and an increase in the unemployment rate above the natural rate. Inﬂation can slow economic growth, redistribute income and wealth, and cause economic activity to contract. Inﬂation impairs decision mak-ing since it creates uncertainty about future prices and/or costs and distorts economic values. For example, a business may postpone the pur-chase of equipment because of increasing uncertainty about the purchas-ing power of future money streams. Such postponed capital outlays slow capital formation and economic growth. True or False Questions 1. Increases in nominal GDP always result in increases in real GDP. 2. Increases in a positive GDP gap are associated with increases in the unemployment rate. 3. All economists agree that an increase in aggregate demand will re-sult in an increase in both the price level and real output. 4. A business cycle occurs every two years. 5. Unemployment only imposes a cost upon those who are unem-ployed. 6. Cyclical unemployment is unevenly distributed among members of the labor force. Answers: 1. False; 2. True; 3. False; 4. False; 5. False; 6. True Solved Problems Solved Problem 3.1 a. Distinguish between a ﬁnal good and an intermediate good. b. Is a loaf of bread a ﬁnal or an intermediate good? CHAPTER 3: Unemployment, Inﬂation, and Income 33 Solution: a. A ﬁnal good is one that involves no further processing and is pur-chased for ﬁnal use. An intermediate good is one that: (1) involves fur-ther processing; (2) is being purchased, modiﬁed, and then resold by the purchaser; or (3) is resold during the year for a proﬁt. b. A loaf of bread could be either a ﬁnal or intermediate good, de-pending upon the purchaser’s use of the good. It is a ﬁnal good when pur-chased by a household for consumption; it is an intermediate good when purchased by a deli which resells the bread as part of a sandwich. Solved Problem 3.2 An economy’s potential output is depicted by the production-possibility frontier in Figure 3-2. a. Explain the relationship between potential GDP and real GDP when output is at point A. b. What is a GDP gap? c. Is there a GDP gap for the situation described in part a? d. Can a GDP gap be negative? Solution: a. Point A is within the economy’s production-possibility frontier. Thus, actual output is less than the economy’s ability to produce, i.e., real GDP is less than potential GDP. b. A GDP gap exists when real GDP does not equal potential GDP. It is measured by subtracting real GDP from potential GDP. c. There is a positive GDP gap at point A since the economy’s pro-duction of goods and services is below its ability to produce. d. The production-possibility frontier measures the economy’s abil-ity to produce goods and services without putting upward pressure on out-put prices. The production-possibility frontier can thus be exceeded, but in doing so there are increases in both output and the price level. Thus, a negative GDP gap can exist—real GDP can exceed potential GDP— when real GDP is, for example, at point B in Figure 3-2 and the econo-my is producing beyond its full-employment level of output. Solved Problem 3.3 Use aggregate demand and aggregate supply curves AD and AS in Figure 3-3 to answer the following questions: a. Is the aggregate supply curve Keynesian or classical? b. Find the economy’s equilibrium level of output and price level. c. Does an increase in government spending, ceteris paribus, shift 34 PRINCIPLES OF ECONOMICS aggregate demand or aggregate supply? What happens to equilibrium output and the price level? d. Suppose there is a technological advance rather than an increase in government spending. What happens to aggregate demand? Aggregate supply? Equilibrium output? The price level? CHAPTER 3: Unemployment, Inﬂation, and Income 35 Figure 3-2 Figure 3-3 Solution: a. Figure 3-3 depicts a classical aggregate supply curve since it shows no relationship between aggregate output and the price level. b. Equilibrium exists where the aggregate demand curve intersects the aggregate supply curve. Equilibrium for curves AD and AS exists at point A; the price level is p0 and output is y1. c. Increased government spending results in an outward shift of ag-gregate demand. There is no change in aggregate supply since there has been no change in the economy’s ability to produce goods and services. If aggregate demand shifts from AD to AD, then equilibrium now exists at point B. Equilibrium output remains at y1 and equilibrium prices in-creases from p0 to p2. d. The technological advance has no effect on aggregate demand, but it shifts aggregate supply rightward from AS to AS. Equilibrium changes from point A to point C. Equilibrium output has increased from y1 to y2, while the price level has decreased from p0 to p1. Solved Problem 3.4 a. What effect does unanticipated inﬂation have upon: (1) individu-als who are retired and living on a ﬁxed income; (2) debtors, and (3) cred-itors? b. How does indexation protect one from the redistribution effect of inﬂation? Solution: a. (1) Unanticipated inﬂation lowers the real income of those on a ﬁxed income. An increase in the price level reduces the purchasing pow-er of a ﬁxed nominal income; the result is the purchase of fewer goods and services. (2) Debtors beneﬁt from unanticipated inﬂation since the dollars they pay back have less purchasing power. (3) Creditors (lenders), on the other hand, lose from unanticipated inﬂation since the dollars they are repaid purchase fewer goods and services. b. Indexation ties money payments to a price level so that the sum of money payments rises proportionately with the price level. For example, a $20,000 salary would increase to $22,000 when the monetary payments of $20,000 are indexed and there is a 10 percent increase in the price level. 36 PRINCIPLES OF ECONOMICS Chapter 4 Consumption, Investment, Net Exports, and Government Expenditures In This Chapter: ✔ Consumption ✔ Investment ✔ Net Exports ✔ Government Taxes and Expenditures ✔ True or False Questions ✔ Solved Problems Consumption Because consumption represents two-thirds of total aggregate spending in the U.S., understanding the determinants of consumer spending is cen-tral to any analysis of the economy’s level of output. Consumer spending 37 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. is largely determined by personal income, income taxes, consumer expectations, consumer indebted-ness, wealth, and price level. Since consumption is impossible for most individuals without income from employment or through transfers from busi-ness or government, personal income is the most important of these variables. Personal income taxes are also central in that one’s ability to spend de-pends not upon the income received but on the in-come available for spending. A consumption function is the relationship of consumption to dis-posable income, holding nonincome determinants of consumption con-stant. Figure 4-1 plots the consumption function for a hypothetical econ-omy, labeled C. A change in a nonincome determinant of consumption alters the relationship of consumption to disposable income. Such changes are depicted graphically by upward or downward shifts of the consumption function. Shifts of the consumption function affect the lev-el of consumption and saving. The 45 line in Figure 4-1 is equidistant from both the consumption and disposable income axes. As drawn, C = Yd at each point on this 45 line. For linear consumption function C, there is only one level of dis-posable income at which consumer spending equals disposable income, and that is the point of intersection of the consumption line and the 45 line. Since the consumption line is below the 45 line at disposable in-come levels above $500 billion, it follows that consumers are not con-suming their entire income and therefore are saving. Thus, consumer sav-ing is the distance between the consumption line and the 45 line at each level of disposable income. Example 4.1 Should consumers expect an increase in the price level, they are likely to spend more in the current period before prices rise. An upward shift of consumption function C to C in Figure 4-1 results. We now ﬁnd that at disposable income of $500 billion, consumption exceeds disposable in-come, i.e., consumers are dissaving. (Consumers can dissave by borrow-ing or by spending accumulated savings). Consumption now equals dis-posable income when Yd is $600 billion; for consumption function C there is less saving at each level of disposable income than there is for consumption function C. 38 PRINCIPLES OF ECONOMICS The marginal propensity to consume is the ratio of the change in con-sumption relative to the change in disposable income between two levels of disposable income (MPC = ∆C/∆Yd), while the marginal propensity to save is the ratio of the change in saving relative to the change in dispos-able income (MPS = ∆S/∆Yd). Also MPS = 1 −MPC. Example 4.2 From Figure 4-1, consumption (under C) increases from $500 billion to $540 billion when disposable income increases from $500 billion to $550 billion; the MPC is therefore 0.80 since ∆C of $40 billion divided by ∆Yd CHAPTER 4: Consumption, Investment, Exports and Government 39 ED: Please shorten RH. Figure 4-1 of $50 billion equals 0.80. For consumption function C, the MPC = 0.80 for each change in disposable income. It is constant for any linear con-sumption function since the MPC is the consumption function’s slope, and all straight lines have a constant slope. Note! Consumers comprise the largest percentage of ag-gregate spending, so their actions are very impor-tant to the strength of the economy. Investment Gross investment is the least stable component of aggregate spending and a principal cause of the business cycle. In calculating GDP, investment consists of residential construction; nonresidential construction (ofﬁces, hotels, and other commercial real estate); producers’ durable equipment; and changes in inventories. While the rate of interest is only one of many variables that inﬂuence investment decisions, it is customary to present investment demand as a negative function of the interest rate, holding constant the other variables which inﬂuence the decision to invest. Thus, a lower interest rate is associated with a higher level of investment, and vice versa. Holding other variables constant, we expect that at a lower rate of in-terest (1) more households are ﬁnancially able to carry a mortgage, and a greater number of housing units will be demanded; (2) businesses are more willing and able to purchase durable equipment and to carry larger inventories; and (3) real estate developers ﬁnd that there are a larger num-ber of purchasers for newly constructed commercial real estate. Net Exports Gross exports are the value of goods and services produced in a home country and sold abroad, i.e., they are the value of foreign spending on U.S.-produced goods and services. Gross imports are the value of U.S. 40 PRINCIPLES OF ECONOMICS purchases of goods and services produced in other countries. When com-modities are imported, some of the consumption and gross investment spending is for foreign-produced rather than U.S.-produced goods. Im-ports thereby lower aggregate spending on domestically produced goods. Net exports are the value of gross exports less gross imports, i.e., the net addition to domestic aggregate spending that results from importing and exporting goods and services. Net exports are positive when the home country exports more than it imports, and negative when the home country imports more than it exports. Numerous variables affect a country’s imports and exports. A coun-try’s imports are related to its level of income, foreign exchange rate, do-mestic prices relative to prices in foreign countries, import tariffs, and restrictions on imported goods. Exports are inﬂuenced by the same vari-ables, except that the income levels of foreign countries rather than that of the home country affect the amount exported. Because these variables change with time, it is reasonable to expect a country’s net export balance to change over time. You Need to Know In the United States net exports are often referred to as the trade deﬁcit because U.S. imports have been greater than exports for some time. Government Taxes and Expenditures Government spending increases when Congress passes legislation au-thorizing new spending. Tax revenues ﬁnance this government spending and government transfer payments to the private sector. Government transfer payments (in the form of unemployment insurance, social secu-rity payments, and various government assistance programs) can be viewed as a negative tax. Net tax revenues consist of income taxes plus lump-sum taxes less transfer payments. Net tax revenues fall when trans-fer payments increase, and rise when greater per capita taxes are imposed. CHAPTER 4: Consumption, Investment, Exports and Government 41 With respect to income tax receipts, net tax revenues increase when out-put increases and more taxes are collected or when government imposes a higher income tax rate. Don’t Forget! Income tax receipts may decrease if the economy slows, not just because Congress cuts tax rates. True or False Questions 1. A change in disposable income causes an equal change in con-sumption. 2. Investment spending is the most unstable component of aggregate spending. 3. Consumption and investment spending in the national income ac-counts is solely for domestically produced goods and services. 4. Imports by a country are unrelated to its level of GDP. Answers: 1. False; 2. True; 3. False; 4. False Solved Problems Solved Problem 4.1 Suppose the economy’s consumption function is speciﬁed by the equation C = $50 + 0.80Yd. a. Find consumption when disposable income (Yd) is $400, $500, and $600. b. Plot this consumption equation and label it C. c. Use the plotted consumption function to ﬁnd saving when dispos-able income is $400, $500, and $600. d. What amount of consumption for consumption function C is au-tonomous, and what amount is induced when disposable income is $400? $500? $600? Solution: a. Consumption for each level of disposable income is found by sub-stituting the speciﬁed disposable income level into the consumption 42 PRINCIPLES OF ECONOMICS equation. Thus, for Yd = $400, C = $50 + 0.80($400) = $50 + $320 = $370. So, C is $450 when Yd = $500, and $530 when Yd = $600. b. The linear consumption function C = $50 + 0.80Yd is plotted in Figure 4-2. c. Saving is the difference between disposable income and con-sumption. Using the calculation from part a., we ﬁnd that saving is $30 when Yd is $400 (Yd −C = S = $400 −$370 = $30), $50 when Yd is $500, and $70 when Yd is $600. Saving is the difference between the con-sumption line and the 45 line at each level of disposable income in Fig-ure 4-2. Thus, reading up from the $400 income level, we ﬁnd that C is $370; the distance from consumption function C to the 45 line at the $400 income level is $30—the amount of saving. d. Autonomous consumption is the amount consumed when dispos-able income is 0. In Figure 4-2, autonomous consumption is $50, the amount consumed when the consumption line C intersects the vertical axis and disposable income is 0. Since autonomous consumption is un-related to income, autonomous consumption is $50 for all levels of in-come. Induced consumption is the amount of consumption that depends upon the receipt of income. Consumption is $370 when disposable in-CHAPTER 4: Consumption, Investment, Exports and Government 43 Figure 4-2 come is $400. Since $50 is consumed regardless of the income level, $320 of the $370 level of consumption is induced by disposable income. In-duced consumption is $400 when disposable income is $500, and $480 when disposable income is $600. Solved Problem 4.2 What will happen to consumption function C in Figure 4-2 when: a. Consumers consider their job secure and therefore become more conﬁdent about the future level of disposable income? b. Credit card issuers implement tighter credit standards and con-sumers are less able to buy goods and services on credit? c. Consumers expect the price level to increase 10 percent by year end? Solution: a. Consumers become more willing to consume their current dispos-able income. Consumption function C in Figure 4-2 shifts upward to C, and consumption is greater for each level of disposable income. b. Some consumers are no longer able to borrow to purchase goods and services in the current period. The consumption function will shift downward from C to C. Consumption is lower for each level of dis-posable income. c. Consumers reschedule future purchases to the current period be-cause of the expected rise in prices for goods and services. Consumption function C shifts upward to C. Solved Problem 4.3 Variables other than the rate of interest affect gross investment. Changes in these other variables cause investment demand to shift downward or upward. What should happen to the economy’s in-vestment demand when there is a change in the following variables? a. There is an increase in consumer conﬁdence. b. Manufacturers’ utilization of existing capacity declines. c. There is an increase in vacancy rates in commercial buildings. Solution: a. Investment demand should shift upward. Housing sales should in-crease as consumers become more conﬁdent; builders would construct more new housing to meet this increased demand. b. Investment demand should shift downward. Businesses’ purchas-es of durable equipment should fall since such purchases would expand 44 PRINCIPLES OF ECONOMICS productive capacity and there is no need to expand productive capacity when utilization of existing capacity is declining. c. Investment demand should shift downward. Increased vacancy rates for existing commercial buildings indicate that there will be difﬁ-culty selling newly constructed commercial real estate. Thus, commer-cial real estate construction will decline. Solved Problem 4.4 What is the difference between a lump-sum tax and an income tax? Solution: A lump-sum tax is a ﬁxed-sum tax that is unrelated to income. An income tax is related directly to earned income. In the case of a pro-portional income tax the government collects a ﬁxed percent of income earned, while for a progressive income tax the rate of taxation increases with the level of income. Lump-sum taxes and proportional and progres-sive income taxes are illustrated in Figure 4-3. Note that lump-sum tax-es remain at $1,000 as income increases from $10,000 to $11,000. When there is a 10 percent proportional income tax rate, tax payments increase from $1,000 to $1,100 as income increases from $10,000 to $11,000. When the tax rate is 10 percent on the ﬁrst $10,000 earned and 20 per-cent on income greater than $10,000, tax payments increase from $1,000 to $1,200 when income increases from $10,000 to $11,000. CHAPTER 4: Consumption, Investment, Exports and Government 45 Figure 4-3 Chapter 5 Traditional Keynesian Approach to Equilibrium Output In This Chapter: ✔ Keynesian Model of Equilibrium Output ✔ Income-Expenditure Model of Equilibrium Output ✔ Leakage-Injection Model of Equilibrium Output ✔ The Multiplier ✔ Changes in Equilibrium Output When Aggregate Supply Is Positively Sloped 46 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. ✔ True or False Questions ✔ Solved Problems Keynesian Model of Equilibrium Output John Maynard Keynes developed the framework for modern-day macro-economics in the 1930s. Because there was considerable unemployment at that time, he assumed that changes in aggregate demand have no effect upon the price level as long as output is below the full-employment lev-el, i.e., as long as aggregate supply is horizontal. A positive GDP gap— where real GDP is below potential GDP—is identiﬁed as a recessionary gap and is the distance between equilibrium output and full-employment output on an AD-AS graph. An inﬂationary gap exists when there is ex-cessive aggregate spending, such that aggregate demand intersects the Keynesian aggregate supply curve in the positively sloped region beyond full-employment output. This results in an increase in the price level. Income-Expenditure Model of Equilibrium Output The Keynesian model of output can be expressed as a circular ﬂow of income and output between businesses and individuals. In a capitalist, free-market economy, individuals own, directly or indi-rectly, the economy’s economic resources (land, la-bor, and capital). Businesses hire resources to produce output and pay individuals a money income for the services of these resources in the form of wages, rent, interest, and proﬁts. Individuals in turn spend their mon-ey income and purchase output. Assuming no supply constraints (when aggregate supply is horizontal), we can expect businesses to supply out-put as long as the receipts from selling output equal the payments made by businesses to the owners of economic resources and the owners of the business ﬁrms. CHAPTER 5: Keynesian Approach to Equilibrium Output 47 Note! The circular ﬂow of income and output helps to ex-plain why we should all study economics because one person’s actions have repercussions for oth-ers. The circular ﬂow of income and expenditure can be used to ﬁnd the economy’s equilibrium level of output. The market value of ﬁnal output for a hypothetical economy appears in column 1 of Table 5.1. Assuming a capitalist system with no government spending or tax-es, the value of output in column 1 is also the disposable income of indi-viduals since individuals receive all the payments made to the factors of production. Aggregate spending in column 5 is the sum of consumer spending (column 2), investment spending (column 3), and net exports (column 4). Note that consumer spending increases with the level of out-put and thus the level of personal disposable income, as discussed in Chapter 4. Investment and net exports are assumed here to be unrelated to the output level and remain constant. The equilibrium level of output is $800 billion since this is the only level of production at which output equals aggregate spending. This equilibrium condition is depicted in col-48 PRINCIPLES OF ECONOMICS Table 5.1 (in Billions of $) umn 6 by the absence of production shortages or surpluses. At output lev-els below $800 billion there is a shortage of output (aggregate spending is greater than production), while at output levels greater than $800 bil-lion there is surplus production. The income-expenditure approach to output can be presented graph-ically beginning with the linear consumption function and the 45 line en-countered in Chapter 4. Adding investment spending and net exports shifts the linear consumption function upward to the aggregate spending line (C + I + Xn). Aggregate spending equals output at only one level of output, determined by the intersection of the 45 line and the aggregate spending line. It can be shown that increases (upward shifts) or decreases (downward shifts) of aggregate spending increase or decrease the econ-omy’s equilibrium level of output. Example 5.1 The equilibrium level of output can be found algebraically by equating output and aggregate spending. Suppose the equation for the linear con-sumption function is C = $70 + 0.8Y; I is $120 and Xn = 0. Equilibrium output is found by solving Y = C + I + Xn for Y. Y = C + I + Xn Y = $70 + 0.8Y + $120 + 0 Y −0.8Y = $70 + $120 + 0 0.2Y = $190 Y = $190.2 = $950 Leakage-Injection Model of Equilibrium Output The leakage-injection model of equilibrium output fo-cuses on saving and gross imports as leakages from the circular ﬂow and on investment and gross exports as spending injections. Leakages depress aggregate spend-ing, while injections increase aggregate spending. For example, spending on domestic output declines when in-dividuals buy more imported rather than domestically produced com-modities. An equilibrium level of output exists in the leakage-injection model when the sum of leakages equals the sum of injections. CHAPTER 5: Keynesian Approach to Equilibrium Output 49 The paradox of thrift demonstrates that increases or decreases in con-sumers’ desire to save, ceteris paribus, affect the economy’s output lev-el but not its saving level. Suppose that there are no exports or imports and that the government neither taxes nor spends. Individuals begin to save more of their income, which is a leakage. If investment does not in-crease, leakages will not equal injections. However, output will be de-creased because individuals are not spending as much with more saving. Since output becomes income to individuals, income will decrease until saving equals investment again, i.e., leakages equal injections. If invest-ment does not change, saving must end up at its initial value. Individuals may be saving a greater percentage of their income, but they have less in-come. So saving stays the same while output decreases. Saving has hurt the economy, thus the paradox. Remember The paradox of thrift explains why it is good for an individual to save, but not necessarily good for the econo-my in general. The Multiplier Shifts of the aggregate spending curve result in a change in the equilibri-um level of output that is several times larger than the initial shift of the curve. This multiplied effect upon output arises from consumption’s pos-itive relationship to income. For example, an increase in investment spending of $10 billion will raise consumers’ income by $10 billion, which results in numerous rounds of induced consumer spending. The new recipients of the $10 billion will consume 80 percent or $8 billion (if the MPC = 0.80). This new $8 billion in spending will become new in-come to individuals who will also spend 80 percent or $6.40 billion, and so on. 50 PRINCIPLES OF ECONOMICS Important! The multiplier is extremely important in explaining how strong an impact one person’s actions can have upon the economy. The value of the multiplier (k) is found by relating the change in out-put (∆Y) to the initial change in aggregate spending. The value of the mul-tiplier can also be found from the equation k = 1/(1 −MPC). Thus, the multiplier is 5 if the MPC = 0.80. Changes in Equilibrium Output When Aggregate Supply Is Positively Sloped When the aggregate supply curve is positively sloped, increases in ag-gregate demand raise both equilibrium output and the price level, even though output may be below its full-employment level. The $50 billion outward shift of aggregate demand from AD to AD in Figure 5-1 rais-es equilibrium output $40 billion rather than $50 billion because of a pos-itively sloped aggregate supply curve. If the aggregate supply curve had been horizontal in this range and the price level had remained at p1, equilibrium output would have in-creased $50 billion, an amount equal to the shift in the aggregate demand curve. It therefore follows that an increase in aggregate spending, when aggregate supply is positively sloped, has a smaller multiplier effect since the increase in the price level decreases wealth (which limits the expan-sion of induced consumer spending), causes higher interest rates (which slow investment spending), and reduces the purchasing power of the home currency (which increases imports and decreases exports). True or False Questions 1. An inﬂationary gap exists when output is above the economy’s equilibrium level of output. CHAPTER 5: Keynesian Approach to Equilibrium Output 51 2. A production shortage exists when the output level is to the right of the point of intersection of the aggregate spending line and the 45 line. 3. A decrease in saving, ceteris paribus, results in a decrease in the equilibrium level of output. 4. A $5 billion increase in investment spending results in a $50 bil-lion increase in the equilibrium level of output when the MPC is 0.90 and aggregate supply is horizontal. 5. An equilibrium level of output exists when output is $550 billion, investment spending is $70 billion, net exports equal $30 billion, and the consumption function is C = $10 billion + 0.80Y. Answers: 1. False; 2. False; 3. False; 4. True; 5. True Solved Problems Solved Problem 5.1 Suppose individuals own all businesses and eco-nomic resources, government does not tax or spend, and the business sec-tor produces 500 units at an average price of $1.50 per unit. a. What is the money value of output? 52 PRINCIPLES OF ECONOMICS Figure 5-1 b. What is the money income of individuals? c. Find consumer spending when individuals spend 90 percent of their income. d. What money revenues are received by the business sector from consumer spending? e. What is the relationship of the cost of producing output and the money receipts of businesses when there are only consumer expendi-tures? What should happen to the level of output? Solution: a. The money value of output equals the output times the average price per unit. The money value of output is $750 (500 × $1.50). b. Since individuals receive an amount equal to the money value of output, their money income is $750. c. Individuals are spending $675—0.90 times their $750 money in-come. There is a $75 saving leakage from the circular ﬂow. d. Business revenues equal the sum of aggregate spending. Since in-dividuals are the only source of spending, revenues equal $675. e. Businesses are making payments of $750 to produce output, while individuals are purchasing only $675 of what is produced. Because busi-ness ﬁrms are left holding unsold output valued at $75, they can be ex-pected to decrease output. Solved Problem 5.2 Suppose there are no gross imports, gross exports, government taxes, or government expenditures for Figure 5-2. a. What is the equilibrium level of output? b. What is the level of saving and investment when output is $400, $500, and $600? c. There being no gross imports or exports, what is the relationship of saving leakages and investment injections when output is above and below the equilibrium output? Solution: a. The equilibrium level of output is $500, determined by the point of intersection of the aggregate spending line (C + I) and the 45 line. b. Saving at each level of output is the difference between the income received and the amount consumed; in the graph, it is the difference be-tween the consumption line C and the 45 line. Consumption is $350 and saving is $50 when Y = $400. Saving is $100 when Y = $500, and $150 CHAPTER 5: Keynesian Approach to Equilibrium Output 53 when Y = $600. Investment spending is $100, the distance between the consumption line and the aggregate spending line. c. When output is below the equilibrium level of output, investment injections are greater than saving leakages, e.g., saving is $50 when Y = $400 while I = $100. When output is above the equilibrium level of out-put, investment injections are smaller than saving leakages, e.g., saving is $150 when Y = $600 while I = $100. At equilibrium, saving leakages equal investment injections. Solved Problem 5.3 a. Find the value of the multiplier when MPC = 0.50, 0.75, and 0.80. b. Find the relationship between the multiplier and the MPC. c. Find the change in the equilibrium level of output when there is a $10 increase in net export spending and the MPC = 0.50, 0.75, and 0.80. 54 PRINCIPLES OF ECONOMICS Figure 5-2 Solution: a. When the MPC = 0.50, the value of the multiplier is 2 [k = 1/(1 − 0.50) = 2]. The multiplier is 4 when the MPC is 0.75 and 5 when it is 0.80. b. The value of the multiplier is directly related to the magnitude of MPC, i.e., the greater the MPC, the larger the value of the multiplier. c. The change in the equilibrium level of output is found by solving the equation ∆Y = k(∆Xn) for ∆Y. When MPC = 0.50, the change in the equilibrium level of output is +$20 [∆Y = 2($10) = $20]. The change in equilibrium level of output is +$40 when the MPC = 0.75, and +$50 when the MPC = 0.80. CHAPTER 5: Keynesian Approach to Equilibrium Output 55 Chapter 6 Fiscal Policy In This Chapter: ✔ Level of Output with Government Expenditures or Taxes ✔ Discretionary Fiscal Policy ✔ Built-In Stabilizers ✔ Government Deﬁcit and Debt ✔ Implementing Fiscal Policy ✔ True or False Questions ✔ Solved Problems Level of Output with Government Expenditures or Taxes Taxes reduce personal disposable income and therefore consumption and aggregate spending, whereas government expenditures increase aggre-gate spending. The inﬂuence of government expenditures and of taxes upon aggregate spending is shown in Figure 6-1 in the shift of aggregate spending line (C + I + Xn + G). An increase in net lump-sum tax revenues, ceteris paribus, shifts the aggregate spending line downward to (C + I + Xn + G), since higher taxes reduce consumer disposable income and therefore consumer spending at each level of output. An increase in gov-56 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. ernment spending, ceteris paribus, shifts the aggregate spending line up-ward to (C + I + Xn + G). It therefore follows that the government can alter the economy’s equilibrium level of output by changing its expendi-tures or net tax revenues. Such government actions are classiﬁed as dis-cretionary ﬁscal policy. Discretionary Fiscal Policy Discretionary ﬁscal policy involves intentional changes in government spending and/or net tax revenues in order to alter the level of aggregate spending. We have already found that an increase in government spend-ing and/or a decrease in lump-sum taxes shifts the aggregate spending line upward and raises the equilibrium level of output, while a decrease in government spending and/or an increase in lump-sum taxes shifts the aggregate spending line downward and lowers the equilibrium level of output. The government can use discretionary ﬁscal actions (changing government spending and/or lump-sum taxes) to eliminate an inﬂation-ary or recessionary gap. CHAPTER 6: Fiscal Policy 57 Figure 6-1 Important Things to Remember In the real world, the government may change its spending and taxing policies for economic reasons or for purely political reasons. A discretionary ﬁscal action has a multiplier effect upon the equilib-rium level of output. The size of the multiplier depends upon whether there is a change in government spending, or in net lump-sum tax rev-enues, and there is an income tax. The value for the multiplier for the change in government spending is ∆Y/∆G, while the value of the multi-plier for the change in net lump-sum tax revenues is ∆Y/∆T. When there is no income tax, a change in government spending has the same multi-plier effect [k = 1/(1 −MPC)] as does a similar change in investment spending or net exports. The multiplier is smaller for changes in net lump-sum tax revenues; the tax multiplier kt = −MPC(k) or −MPC/(1 −MPC) is for an economy with no income tax. An income tax reduces the value of both the ex-penditure and the lump-sum tax revenue multiplier since the amount of taxes paid to government is di-rectly related to income earned. For example, when the income tax rate is 20 percent and personal in-come increases $10, tax payments to the govern-ment rise $2 and personal disposable income in-creases $8 rather than $10. Thus, an increase in personal income results in smaller increments in induced consumption, and therefore results in a smaller multiplied effect. When there is an in-come tax, the equation for the expenditure multiplier is k = 1/[1 −MPC + MPC(t)], where t is the income tax rate. The equation for the lump-sum tax multiplier is kt = −MPC(k) or −MPC/[1 −MPC + MPC(t)]. Built-In Stabilizers Personal income taxes and various government transfers automatically change the level of net tax revenues when the economy moves away (or 58 PRINCIPLES OF ECONOMICS toward) the full-employment level of output. For example, government collects smaller revenues from income taxes when output decreases; lump-sum tax revenues also fall when output decreases because of in-creased government transfer payments to individuals in the form of un-employment insurance beneﬁts, food stamps, and other government as-sistance programs. Because of such automatic changes in net tax revenues, consumer disposable income is not completely dependent on the level of output, consumer spending is more stable over the business cycle, and the amplitude of economic ﬂuctuations is lessened. Note! Built-in stabilizers only help to mitigate economic ﬂuctuations, not to correct them. Government Deﬁcit and Debt A federal deﬁcit exists when government outlays exceed revenues. The structural deﬁcit is the deﬁcit that exists when output is at its full-em-ployment level; a cyclical deﬁcit is the amount of the deﬁcit that is at-tributable to output being below its full-employment level. In Figure 6-2, yf represents full-employment output. Here the economy’s structur-al deﬁcit is $200 ($500 in government spending less $300 in net tax re-ceipts). Note that the deﬁcit increases to $300 when output declines to y1, which is not surprising since there are smaller tax receipts and larger gov-ernment transfers at output levels below yf. Thus, at output y1, the $300 deﬁcit consists of a $200 structural deﬁcit and a $100 cyclical deﬁcit. The public debt is the amount of interest-bearing debt issued by the federal government at a given point in time and arises from previous year-ly deﬁcits. Some argue that a large public debt will result in default and federal bankruptcy. The federal government will not default, however, since it has the power to print money and the power to tax. The govern-ment can also repay a maturing debt obligation by issuing a new debt ob-ligation. However, due to possible redistribution effects, there is concern about the large increases in the U.S. federal debt. A rapidly rising debt level necessitates larger interest payments. If the government increases CHAPTER 6: Fiscal Policy 59 taxes to pay its higher interest expense, it could cause a redistribution of income from those who pay taxes to those who have substantial wealth. Important! There is a difference between the deﬁcit and the debt. The former occurs yearly and the latter is an accumulation of the former deﬁcits over time. Implementing Fiscal Policy Since discretionary changes in tax revenues and government spending have a multiplier effect upon equilibrium output, it would appear that government has the ability to maintain full-employment output by ma-nipulating its net tax revenues and/or spending. Fiscal policy, however, is not as easily implemented or as successful as ﬁrst suggested. Suppose a recessionary gap exists. Will Congress and the administration agree on 60 PRINCIPLES OF ECONOMICS Figure 6-2 an immediate course of action? In reality, an action lag is likely to occur because of conﬂicting priorities. For example, some individuals may advocate in-creased government expenditures on public goods, such as the rebuilding of roads, while others may pre-fer government expenditures on services such as public education. An-other group may advocate expanded welfare services or reduced tax rates for middle-income workers. And once a ﬁscal plan of action is reached and implemented, will Congress and the administration be prepared to scale down or eliminate any of these measures should the ﬁscal stimulus eventually become excessive? Besides political priorities, we must also recognize that economic ac-tivity exists in a dynamic, changing environment, where other variables may change. Thus, while a ﬁscal stimulus may close a recessionary gap and bring the economy to full employment, ceteris paribus, it is possible that investment and/or net export spending may increase after the ﬁscal stimulus is implemented, which would result in an inﬂationary gap. In ad-dition, economists are uncertain about the output level at which full em-ployment exists and have been unable to establish precise values for mul-tipliers for the U.S. economy. True or False Questions 1. Fiscal policy refers to any change in government tax revenue and/or in government spending. 2. With no income tax and the MPC equal to 0.80, a $10 increase in transfer payments shifts the aggregate spending line upward by $8. 3. With no income tax and the MPC equal to 0.75, a $10 decrease in net tax revenues results in a $30 increase in the equilibrium level of out-put. 4. When the MPC is 0.75 and the income tax rate is 0.20, the lump-sum multiplier is −3. 5. The availability of food stamps is an example of discretionary ﬁs-cal policy. Answers: 1. True; 2. True; 3. True; 4. False; 5. False CHAPTER 6: Fiscal Policy 61 Solved Problems Solved Problem 6.1 How do the following events affect an aggregate spending line? a. A $15 increase in government spending. b. A $10 decrease in investment spending. c. A $15 decrease in net tax revenues when the MPC is 0.80. Solution: a. The aggregate spending line shifts upward by ∆G, the amount of the change in government spending. In this case, there is a correspond-ing $15 upward shift of the aggregate spending line. b. Changes in investment shift the aggregate spending line by ∆I. Here, there is a $10 downward shift of the aggregate spending line. c. Changes in lump-sum taxes shift the aggregate spending line by − MPC(∆T). Since net tax revenues decrease $15, there is a $12 upward shift of the aggregate spending line [$12 = −0.80(−$15)]. Solved Problem 6.2 Suppose there is full employment at the $600 lev-el of output and the MPC is 0.80 in Figure 6-3. a. Does the aggregate spending line (C + I + Xn + G) depict the ex-istence of an inﬂationary or recessionary gap? b. What discretionary ﬁscal action can government implement to close this gap? c. What discretionary ﬁscal action is needed when investment spend-ing decreases $5? Solution: a. There is a $60 inﬂationary gap since the equilibrium level of out-put is $660 and full-employment output is $600. b. Government spending should be decreased $12 since the neces-sary decrease in aggregate spending is $60 and the multiplier is 5 [∆Y = k(∆G); −$60 = 5(∆G); ∆G = −$12]. An alternative ﬁscal action is a $15 increase in lump-sum taxes since the tax multiplier is −4 [∆Y = kt(∆T); − $60 = −4(∆T); ∆T = +$15]. c. The inﬂationary gap is $35 rather than $60 since the $5 decrease in investment spending lowers aggregate spending $25. To close the smaller inﬂationary gap, lump-sum taxes need to be increased $8.75, or government expenditures need to be reduced $7. 62 PRINCIPLES OF ECONOMICS CHAPTER 6: Fiscal Policy 63 Figure 6-3 64 Chapter 7 The Federal Reserve and Monetary Policy In This Chapter: ✔ Functions of Money ✔ Financial Instruments and Markets ✔ Creation of M1 Money Supply ✔ Federal Reserve System ✔ Monetary Tools ✔ Open-Market Operations ✔ True or False Questions ✔ Solved Problems Functions of Money Money serves as a medium of exchange, a measure of value, and a store of value. As a medium of exchange, money is the payment made to eco-nomic resources for their services, which the owners of these resources use to purchase goods and services. For example, labor is paid a money Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. wage; individuals use this money to purchase food and clothing. Paper currency and checking ac-counts comprise the medium of exchange in most countries. Money serves as a measure of value in that it is the common denominator for measuring prices and income. For example, a newspaper costs $0.50 and workers may earn $9.85 per hour. Mon-ey functions as a store of value in that the money re-ceived today can be saved and held for expenditures at some future date. Financial Instruments and Markets Savings can be held in ﬁnancial assets other than money. Since currency and checking accounts offer savers little or no interest, many savers are willing to transfer money balances they do not intend to spend for a pe-riod of time into a higher-yielding ﬁnancial instrument. A credit or debt ﬁnancial instrument is one which requires that a borrower make period-ic interest payments and repay the amount loaned at the end of a contract period. An equity ﬁnancial instrument gives the saver partial ownership of a ﬁrm and a share of its proﬁts. Many ﬁnancial instruments are marketable and can be sold to another party in a secondary ﬁnancial market. A ﬁnancial instrument is liquid when the current owner can quickly convert it into a money balance with a minimal loss of nominal capital value. A saver therefore has a choice of holding a liquid ﬁnancial instrument or money as a store of value. The portfolio decision of holding money, liquid ﬁnancial instruments, or illiq-uid ﬁnancial instruments depends upon the time horizon of the saver, the return on these alternative instruments, and the willingness of the saver to assume risk. You Need to Know Savers have a host of mediums in which to store extra money. We mention some major categories primarily for information purposes. CHAPTER 7: The Federal Reserve and Monetary Policy 65 66 PRINCIPLES OF ECONOMICS Depository institutions (commercial banks, savings and loan associ-ations, and credit unions) borrow savers’ money balances and lend them to individuals, businesses, or government. By pooling the funds of many small savers and investing in a diversiﬁed portfolio of ﬁnancial instru-ments, these institutions reduce the transaction costs and risks associated with lending to a borrower. In the U.S., the Federal Deposit Insurance Corporation (FDIC) insures the liabilities of deposit intermediaries. Savers therefore readily hold these liquid liabilities because they nor-mally offer a higher interest return than money. Because the liabilities are liquid and therefore good stores of value, the Federal Reserve presents an M1, M2, and M3 deﬁnition of money. The M1 deﬁnition is a transaction deﬁnition and consists of currency and checking accounts, while M2 and M3 add other liquid ﬁnancial instruments to the M1 deﬁnition. Current measures of the money supply appear in Table 7.1. Small time deposits are certiﬁcates of deposit (CDs) issued by these same ﬁ-nancial intermediaries in amounts less than $100,000, and large time de-posits exceed this dollar amount. CDs are classiﬁed as time deposits since the depositor agrees to keep these funds on deposit for a speciﬁed period of time or incur an interest penalty. Repurchase agreements (RPs) are large (at least $1 million) overnight, collateralized loans. Table 7.1 CHAPTER 7: The Federal Reserve and Monetary Policy 67 Creation of M1 Money Supply When a bank lends, it gives the borrower a check drawn upon itself. The Federal Reserve controls the banking system’s ability to issue check-writing deposits by imposing a reserve requirement on checking deposits. U.S. bank reserves consist of curren-cy held by banks and deposits that banks have at the Federal Reserve. The reserve requirement (r) on check-writing deposits is currently 10 percent (r = 0.10); it requires that a bank hold $1 in reserves for each $10 in checking account liability it has. Since U.S. banks are managed to maximize proﬁts, they usually expand loans and issue check-writing deposits when they have more reserves than they are required to hold. Even when a bank has no excess reserves, it can lend and create new checking deposits if it can borrow the excess reserves of other banks. With banks behaving in this manner, there is a tendency for the ex-cess reserves of the banking system to approximate zero and for the com-bined sum of check-writing deposits to be a multiple of the amount of re-serves held by all banks. When excess reserves for the combined banking system equal zero, the relationship of check-writing deposits (D) and re-serves (R) can be presented as Dmax = dR, where d, the check-writing de-posit multiplier, equals 1/r. It therefore follows that the Federal Reserve can control the maximum amount of check-writing deposits by control-ling the amount of reserves held by banks and by setting the reserve re-quirement on checking deposits. Example 7.1 Suppose the reserve requirement on check-writing deposits is 0.10 and reserves held by all banks total $500,000. The maximum amount of check-writing deposits for the banking system is $5,000,000. Dmax = dR; d = 1/r; d = 1/0.10 = 10; since R = $500,000, Dmax = (10)$500,000 = $5,000,000. Note! Banks are essential to the U.S. monetary system as the Federal Reserve System can only work through these businesses. 68 PRINCIPLES OF ECONOMICS Federal Reserve System The Federal Reserve System manages the U.S. money supply in order to minimize inﬂationary pressures and promote economic stability. The Federal Reserve System, frequently referred to as the Fed, consists of twelve Federal Reserve Banks, a Board of Governors, and a Federal Open Market Committee. The Federal Reserve System is considered indepen-dent in that its policy directives are not directly inﬂuenced by the con-gressional or executive branches of the federal government. Federal Reserve Banks. Each of the twelve Federal Banks has its own president, services banks in a speciﬁc geographical area, and acts as a central bank for that region. A Federal Reserve Bank clears checks be-tween banks, supervises and regulates banks in its region, performs bank examinations, provides currency to banks, and holds bank reserves. Pri-vate individuals and corporations do not deal directly with a Federal Re-serve Bank. Board of Governors. The seven-member Board of Governors is the policy-making body of the Fed. Each member of the Board is nominated by the President of the United States for a fourteen-year, nonrenewable term. Because appointments to the Board are terminal and last for many years, members of the Board of Governors are free of political consider-ations in the formulation of monetary policy. Federal Open Market Committee (FOMC). The twelve-member FOMC is responsible for implementing U.S. monetary policy. It estab-lishes directives for open-market operations which determine the M1 money supply. The seven-member Board of Governors and ﬁve Federal Reserve Bank presidents comprise the FOMC. Monetary Tools The Fed supplies the private sector with whatever amount of currency it wants to hold—thus, the currency component of the M1 money supply is determined by the private sector. Monetary tools available to the Fed include changes in the reserve requirement, open-market operations that control the amount of reserves held by banks, and adjusting the discount rate, which may inﬂuence the amount of reserves banks borrow from Federal Reserve Banks. Reserve-Requirement Variation. A decrease in the reserve require-CHAPTER 7: The Federal Reserve and Monetary Policy 69 ment on check-writing deposits (monetary ease) creates excess reserves and increases the amount of check-writing deposits issued by banks. Sim-ilarly, an increase in the reserve requirement (monetary tightness) de-creases the check-writing component of M1. While reserve-requirement variation is a powerful means of changing the M1 money supply, it is used infrequently. Monetary ease or tightness is usually done incrementally. Weekly or monthly changes in the reserve requirement are abrupt and would create management problems for the large number of banks that exist in the U.S. Open-Market Operations. Open-market operations consist of the purchase and sale of government securities (debt obligations of the U.S. Treasury) by the Fed and are directed by the FOMC. When the Fed pur-chases government securities, it pays for these bonds by crediting the de-posit account banks have at a Federal Reserve Bank. Since reserves in-clude these deposits, such security purchases increase bank reserves and eventually the amount of check-writing deposits issued by banks. Feder-al Reserve sales of government bonds reduce bank reserves and check-writing deposits, and thereby the M1 money supply. Discount Rate. A bank may borrow reserves (discount) from a Fed-eral Reserve Bank when it has a reserve deﬁciency; the rate of interest it pays the Fed is the discount rate. Banks are encouraged to remedy a de-ﬁciency by borrowing the excess reserves of other banks in the Fed funds market rather than borrow at the Fed. So, the Fed frequently changes the discount rate after an increase or decrease in the Fed funds rate to en-courage this practice. Adiscount rate change is newsworthy in that it con-ﬁrms the direction of the movement in the Fed funds rate of interest and interest rates in general. Important! Open-market operations are the primary tool of the Fed and are used practically daily to accomplish the Fed’s goals. Open-Market Operations A change in the M1 money supply affects the short-term rate of interest when there is no change in the private sector’s demand for money. There 70 PRINCIPLES OF ECONOMICS are numerous reasons why the private sector holds money balances. These reasons can be categorized into types of demand, as follows: (1) a transaction demand, since money is needed to purchase goods and ser-vices, to pay employees, etc.; (2) a precautionary demand, since money may be held to meet emergency and unforeseen needs that may arise; (3) a portfolio (asset) demand, since some money balances are held in the ex-pectation of opportunities in the ﬁnancial markets. When there is a ﬁxed demand for money, an increase in the M1 money supply lowers the short-term nominal rate of interest, ceteris paribus. Example 7.2 L in Figure 7-1 depicts the demand for money. The amount of money de-manded is inversely related to the rate of interest since the holder of mon-ey forgoes a higher interest return from an alternative ﬁnancial asset. When the Fed purchases government securities in the open market, bank reserves increase as does the M1 money supply. Thus, money supply curve S in Figure 7-1 shifts rightward to S as the M1 money supply in-creases and the short-term rate of interest falls from i0 to i1. Figure 7-1 CHAPTER 7: The Federal Reserve and Monetary Policy 71 The downward pressure on short-term interest rates due to an in-crease in the money supply is also evident when we consider the effect that an increase in bank reserves has upon bank lending. In purchasing government securities and supplying more reserves to the banking sys-tem, the Fed increases the supply of excess reserves. Banks can encour-age more borrowers to apply for bank loans by lowering their lending rates. Consumer spending on large-ticketed items such as houses and cars is interest-sensitive since individuals are likely to take out loans to pay for major purchases. Business investment—purchases of new buildings and equipment—is also interest sensitive. Thus, as depicted below, a Fed increase in the money supply should lower the rate of interest, increase interest-sensitive spending, and result in a higher level of spending and gross domestic output. ↑M →↓i →↑total spending →↑gross domestic product True or False Questions 1. A savings deposit is a medium of exchange. 2. A marketable ﬁnancial instrument can be traded on a secondary market. 3. The banking system’s ability to issue check-writing deposits is limited by the reserve requirement on checking deposits and the amount of reserves held by the bank. 4. The maximum increase in check-writing deposits is $100,000, ce-teris paribus, when the Fed purchases government securities valued at $10,000 and the reserve requirement is 10 percent. 5. When there is a stable demand-for-money curve, a decrease in the M1 money supply lowers the short-term nominal rate of interest. 6. Bank reserves increase and the fed funds rate decreases when the Fed purchases government securities, ceteris paribus. Answers: 1. False; 2. True; 3. True; 4. True; 5. False; 6. True Solved Problems Solved Problem 7.1. a. Why are check-writing deposits included in the deﬁnition of mon-ey? 72 PRINCIPLES OF ECONOMICS b. Is there backing for coins, paper currency, and check-writing de-posits? c. How can money have value without commodity backing? Solution: a. In most cases, one can pay for the purchase of a good or service with cash or by writing a check. Since checks are accepted as payment, they are classiﬁed as money along with coins and paper currency. b. In the U.S., coins, paper currency, and checking accounts have no intrinsic value. While coins have a metallic content, the market value of the coined material is considerably less than the face (monetary) value of the coin. Paper currency is issued by the Fed and has no commodity back-ing, while check-writing deposits are noncollateralized liabilities of de-posit institutions. c. Anything has value when its supply is limited and demand is vir-tually unlimited. The basis for value for an inconvertible paper standard (coins, paper currency, and checking accounts) is that government can and is willing to limit its supply, economic units are willing to receive it in payment for services, and spending units can use it to obtain goods and services. Solved Problem 7.2 Why are ﬁnancial intermediaries essential to the ef-ﬁcient operation of the economy? Solution: An economic system is judged efﬁcient when it achieves max-imum use of economic resources and maximum satisfaction of consumer wants. Financial instruments and institutions generate efﬁciency in the following ways: a. The ﬁnancial system increases consumer satisfaction by facilitat-ing the allocation of spending over time. It allows some units to spend more than their current income (dissave) and allows other spending units to increase their future spending level by earning interest on the money they have saved. b. The creation of safe and liquid ﬁnancial claims by ﬁnancial inter-mediaries reduces the likelihood that some savers will hold money bal-ances idle. By rechanneling savings into the circular ﬂow, spending ﬂows are stabilized. This in turn stabilizes employment and economic activity. c. Financial instruments encourage savers to lend their savings to those who want to spend more than their current money inﬂow. A large CHAPTER 7: The Federal Reserve and Monetary Policy 73 portion of the funds borrowed from savers is used by business ﬁrms to add to the economy’s capital stock. This increases productive capacity. d. Since the proﬁt motive guides the operation of ﬁnancial institu-tions, money saving is distributed to those capital uses that have the great-est productivity. Solved Problem 7.3 Suppose the banking system holds no excess re-serves. a. What is the maximum amount of check-writing deposits issued by the banking system when reserves total $1,000 and the reserve require-ment is (1) 0.20, (2) 0.16, and (3) 0.10? b. Find the maximum amount of check-writing deposits when the re-serve requirement is 0.20 and reserves total (1) $1,000, (2) $1,250, and (3) $2,000. c. Compare the quantity of check-writing deposits when reserves are held constant and the reserve requirement is lowered in (a) with the quan-tity of deposits when the amount of reserves held by banks is increased and the reserve requirement remains constant in b. Solution: a. The maximum amount of check-writing deposits is found by solv-ing Dmax = R/r. (1) Dmax is $5,000 (Dmax = $1,000/0.20); (2) $6,250; and (3) $10,000. b. When the reserve requirement remains at 0.20 and bank reserves increase from $1,000 to $1,250 to $2,000, check-writing deposits in-crease from (1) $5,000 to (2) $6,250 to (3) $10,000. c. The situations in a. and b. show that the Fed has two alternative ways of bringing about similar increases in the amount of check-writing deposits; by lowering the reserve requirement or by increasing the amount of reserves held by the banking system. Chapter 8 Monetary and Fiscal Policy In This Chapter: ✔ Using Monetary and Fiscal Policy ✔ Problems with Fiscal and Monetary Policy ✔ Price Level Changes ✔ Choosing Fiscal or Monetary Policy ✔ True or False Questions ✔ Solved Problems Using Monetary and Fiscal Policy Previous chapters have shown that monetary and ﬁscal policies are alter-native ways of changing aggregate spending to close GDP gaps. For ex-ample, if output is below its full-employment level, an increase in the money supply, an increase in government spending, or a decrease in tax-es raises aggregate spending and increases equilibrium output. Example 8.1 Suppose the expenditure multiplier ke is 5, the tax multiplier kt is −4, and full-employment output exists at $900. If equilibrium output is $800, shifting the aggregate spending line upward can close the $100 recession-74 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. ary gap and bring the economy to full-employment output. This could be accomplished by a $20 increase in government spending [∆Y = ke∆G; $100 = 5($20)], a $25 decrease in lump-sum taxes, or an increase in the money supply which lowers interest rates and in-creases investment spending $20. Problems with Fiscal and Monetary Policy An expansionary ﬁscal policy might not result in an increase in output exactly equal to ke∆G because of the crowding-out effect. Government spending increases, which raise the level of output, will usually push the rate of interest higher. Private-sector interest-sensitive spending will thereby fall and be crowded out by the ﬁscal action. Thus, the net increase in equilibrium output due to increased government spending is usually less than ke∆G. How much less depends upon the interest sensitivity of the demand for money and the interest sensitivity of investment spend-ing. Example 8.2 Suppose ke is 5, full-employment output exists when output is $900, and equilibrium output is initially $800. A$20 increase in government spend-ing, ceteris paribus, should increase spending $100 and bring output to its full-employment level. But suppose the rate of interest increases as a result of the $20 increase in government spending and investment spend-ing declines $5. The net effect of the government’s ﬁscal action is then $75 rather than $100, and full-employment output is not reached. The net effect equals ∆G(ke) + ∆I(ke) = $20(5) −$5(5) = $75. Since policymak-ers do not know in advance the extent to which there will be crowding out, the effect of a stimulative ﬁscal policy upon output is uncertain. You Need to Know Crowding out may partially negate ﬁscal policies and so must be considered carefully when imple-menting any action. CHAPTER 8: Monetary and Fiscal Policy 75 Normally, the rate of interest falls and interest-sensitive spending and equilibrium output increase when the Fed increases the money sup-ply. While most economists agree that changes in the money supply im-pact interest-sensitive spending, there is substantial disagreement about the predictability of the effect. Keynesians have traditionally argued that there is considerable uncertainty about the effect a money supply change has upon the rate of interest and the level of investment. Monetarists con-tend that a change in the money supply has a highly predictable effect upon nominal GDP. The disagreement about the predictability of a mon-ey supply change has centered around the velocity of money (average cir-culation of a unit of money in the economy) and its variability. When the demand for money and/or the investment demand are subject to unpre-dictable shifts, the effect of a money supply change upon equilibrium out-put is uncertain. Price Level Changes In an aggregate demand and aggregate supply framework, an economic stimulus is constrained by a possible increase in the price level. (Note: previ-ous chapters assumed there would be no increase in the price level until the economy reached its full-employment level of output. Such a scenario is un-likely to exist in the real world.) It therefore follows that the effect on output of a monetary or ﬁscal stim-ulus depends upon the slope of the aggregate supply curve. A steeply sloped aggregate supply curve has a smaller effect upon output than a rel-atively ﬂat one. While the actual steepness of the aggregate supply curve is unknown, it is generally believed that aggregate supply is more steeply sloped the closer output is to its full-employment level. Important! Price level changes constrain any monetary or ﬁs-cal policy, but are difﬁcult to counteract because the slopes of the aggregate demand and aggre-gate supply curves can only be estimated. 76 PRINCIPLES OF ECONOMICS Example 8.3 Suppose ke is 5, there is no crowding out, and full-employment output is $600. Equilibrium output is initially $500 in Figure 8-1 for aggregate de-mand and aggregate supply curves AD1 and AS1. The recessionary gap is $100 since full-employment output is $600; the price level is initially p0. Since the expenditure multiplier is 5, a $20 increase in government spending should increase output $100 and bring output to its full-em-ployment level when the price level remains at p0. This $100 increase in spending is presented in Figure 8-1 by the shift of aggregate demand from AD1 to AD2. Since aggregate supply AS1 is positively sloped, the price level rises from p0 to p1. This increase in the price level decreases pri-vate-sector spending; equilibrium output thus increases to $580 rather than to $600. Suppose the aggregate supply curve is AS2 rather than AS1. Figure 8-1 shows that the increase in aggregate demand from AD1 to AD2 raises the price level to p2 rather than p1, and equilibrium output is $540. CHAPTER 8: Monetary and Fiscal Policy 77 Figure 8-1 Note that there is a smaller increase in equilibrium output but a larger in-crease in the price level because aggregate supply curve AS2 is more steeply sloped than AS1. Choosing Fiscal or Monetary Policy Other factors also inﬂuence the choice of a monetary or ﬁscal stimulus: how quickly the economic stimulus impacts economic activity and how the economic stimulus affects the economy’s struc-ture of output. Achange in government spending nor-mally has the most immediate impact on economic activity since a change immediately affects spending levels. The response to money supply changes is more likely to lag. While money supply changes immediately impact the rate of interest, the response of interest-sensitive spending to an interest rate change may not be as immediate since many investment projects are not ready to be started when funding costs decrease. A money supply change, however, has a short action lag since the Fed, unlike Congress, can respond quickly to changing economic conditions. Thus, in spite of its longer impact lag, monetary policy is the principal economic stabi-lization measure used in the US because of its short action lag. Those who advocate minimal interference with the market prefer monetary policy to ﬁscal policy. Monetary policy, through its interest rate effect, works through the ﬁnancial markets and impacts private-sector spending; a ﬁscal action may redistribute income within the private sec-tor or expand public rather than private-sector goods and services. For ex-ample, a change in the personal income tax rate does not equally impact each income class. Note! Monetary policy is the primary type of policy used to affect the U.S. economy. True or False Questions 1. An increase in government spending always crowds out an equal amount of private-sector interest-sensitive spending. 78 PRINCIPLES OF ECONOMICS 2. The net increase in equilibrium output is $10 when ke is 5, gov-ernment spending increases $10, and higher interest rates crowd out $8 of investment spending. 3. An increase in aggregate demand has no effect upon real output when aggregate supply is vertical. 4. A $10 increase in the money supply increases equilbrium output $50 when ke is 5, there is no crowding out, and aggregate supply is pos-itively sloped. 5. Monetary policy is more frequently used than ﬁscal policy since it more quickly impacts aggregate spending. Answers: 1. False; 2. True; 3. True; 4. False; 5. False Solved Problems Solved Problem 8.1 What happens to equilibrium output and the price level in Figure 8-2 when an increase in the money supply shifts aggre-gate demand from AD1 to AD2? CHAPTER 8: Monetary and Fiscal Policy 79 Figure 8-2 Solution: The rightward shift of aggregate demand, caused by an in-crease in the money supply, has no effect upon equilibrium output but in-creases the price level from p2 to p1. Aggregate demand shifts have no ef-fect upon output whenever the aggregate supply curve is vertical; demand shifts in such an economic situation only affect the price level. Solved Problem 8.2 a. A stimulative monetary or ﬁscal action should increase aggregate demand. What factors may limit the actual increase in aggregate demand? b. An increase in aggregate demand should raise equilibrium output. What is responsible for the size of the increase in equilibrium output? Solution: a. Factors that constrain the aggregate demand shift when there is a ﬁscal or monetary stimulus are crowding out and the interest sensitivity of the demand for money and investment spending. An increase in gov-ernment spending and/or a decrease in taxes raises output, usually re-sulting in an increase in the rate of interest. Higher interest rates can crowd out private-sector interest-sensitive investment spending. So, the actual increase in aggregate demand due to a ﬁscal stimulus depends upon the magnitude of the crowding-out effect. An increase in money supply raises private-sector spending by lowering the rate of interest. The actu-al decrease in the interest rate depends upon the interest sensitivity of the demand for money. The effect that a decrease in the interest rate has upon spending in turn depends upon the interest sensitivity of investment spending. Thus, a money supply increase can cause a large or small shift of the aggregate demand. b. An increase in aggregate demand should raise equilibrium output; the actual increase in output depends upon the slope of the aggregate sup-ply curve. When aggregate supply is steeply sloped, demand increases have a smaller effect upon output than when aggregate supply is less steeply sloped. 80 PRINCIPLES OF ECONOMICS 81 Chapter 9 Economic Growth and Productivity In This Chapter: ✔ Concept of Economic Growth ✔ Growth through Population and Capital Accumulation ✔ Supply-Side Economics ✔ True or False Questions ✔ Solved Problems Concept of Economic Growth Economic growth is concerned with the expansion of an economy’s abil-ity to produce (potential GDP) over time. Expansion of potential output occurs when there is an increase in natural re-sources, human resources, or capital, or when there is a technological advance. The two most common measures of economic growth are an increase in real GDP and an increase in output per capita. Of these two measures, an increase in output per capi-ta is more meaningful since it indicates there are more goods and services available per person and hence a rise in the economy’s standard of living. An increase in potential Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. output can be conceptualized by an outward shift of an economy’s pro-duction-possibility frontier. In our discussion of economic growth, we as-sume that increases in potential output are matched by equal increases in aggregate spending so that full-employment growth is assured. Growth through Population and Capital Accumulation An increase in the labor supply, ceteris paribus, expands potential out-put. The law of diminishing returns shows that the incremental output from an additional labor input decreases when other economic resources and technology are unchanged. Thus, the possibility exists that aggregate output could increase while output per capita decreases. Expecting rapid population growth, economists in the early nineteenth century predicted such growth would result in declining output per capita. Thomas Malthus, in particular, held that the population would increase at such a rapid rate that the economy would increasingly be unable to grow enough food to feed its population; eventually output per capita would fall to a subsis-tence level. While technology has allowed highly industrialized countries to avoid these gloomy projections, rapid population growth is a problem for many developing countries. Note! Such theories as that of Thomas Malthus have helped to label economics as the dismal science. The neoclassical model of economic growth maintains that, in the absence of technological change, an economy reaches a steady state— where there are no further increases in output per capita. In the steady state, capital deepening ceases, although capital widening can occur be-cause of growth in the labor supply. Capital widening exists when capi-tal is added to keep the ratio of capital per worker constant due to in-creases in the supply of labor. Capital deepening occurs when there is an increase in the ratio of capital to labor. With no technological advance, capital additions that are capital 82 PRINCIPLES OF ECONOMICS widening do not change output per worker; however, capital additions that are capital deepening increase output per worker. When there is no change in the labor force, capital additions result in diminishing returns and have decreased rates of re-turn. Capital additions cease—and the economy reaches a steady state—when the rate of return from capital additions equals the economy’s real rate of interest. Since there is a limit to capital deepening when there is no change in technology, there must also be a limit to output per worker and therefore to the economy’s standard of living. An economy’s steady-state position can be pushed to a higher level of output per worker by an in-crease in its rate of saving, by improved technology, and/or by better ed-ucation of its population. Productivity is measured by dividing real GDP by the total number of hours worked by labor. Over time, the growth of labor productivity in the U.S. has slowed. Economists have been unable to empirically estab-lish the cause of this productivity growth slowdown, but some potential factors may be: (1) an increase in environmental regulations; (2) high en-ergy costs, resulting in the substitution of more labor and capital for en-ergy; and (3) an increase in the number of less skilled workers in the la-bor force. A productivity growth slowdown has implications for a country’s standard of living. Standard of living is measured by an economy’s real GDP per capita (total output divided by population), whereas productiv-ity is measured as real GDP per hour of labor input (output divided by the number of hours worked to produce this output). Suppose an economy’s labor force is always 50 percent of its population. Increases in labor’s out-put per hour will result in higher GDP per capita and therefore raise the economy’s standard of living. When output per hour is unchanged, there will be no increase in output per capita and therefore no improvement in the economy’s standard of living. Important! Increased productivity has been the major source of growth for many countries during certain peri-ods, including the U.S. CHAPTER 9: Economic Growth and Productivity 83 84 PRINCIPLES OF ECONOMICS Supply-Side Economics Concern about the slowdown in U.S. productivity growth during the 1970s helped popularize the theory of supply-side economics. Supply-siders stressed that U.S. productivity would be enhanced by actions that promoted incentives to produce. A decrease in private-sector taxes was proposed. Proponents of this theory called for a decrease in corporate in-come tax rates, which would increase corporate proﬁts and therefore busi-ness saving. This in turn would encourage business investment and cap-ital accumulation. A reduction in the personal income tax rate would increase the reward from working, which might increase labor produc-tivity. Decreased tax rates on interest income and corporate dividends would increase household saving, which would result in capital deepen-ing. While not identiﬁed as supply-side economics, various measures were promoted in the 1990s that would also increase the economy’s abil-ity to produce. Improvement of the U.S. educational system would en-hance labor skills, increase labor productivity, and thereby promote eco-nomic growth. You Need to Know Supply-siders can come from any political party, but different groups tend to favor speciﬁc policies (i.e., tax cuts or increased education). Example 9.1 In Figure 9-1, assume that the labor supply and population are unchanged and that a combination of tax incentives and a better-educated population shift the aggregate supply curve AS1 to AS2. An increase in the money supply shifts aggregate demand from AD1 to AD2; the price level remains constant and output increases from y1 to y2. Since there is no change in population, output per capita has increased with an attending rise in the economy’s standard of living. CHAPTER 9: Economic Growth and Productivity 85 True or False Questions 1. An economy’s standard of living is rising when real GDP is in-creasing 10 percent while population is increasing 5 percent. 2. Assuming full employment of resources, an increase in the labor force, ceteris paribus, always increases output per capita. 3. Additions to the economy’s stock of capital always result in cap-ital deepening and capital widening. 4. Supply-side policies intend to increase the economy’s ability to produce. 5. The slowdown in U.S. productivity growth was caused by capital widening. Answers: 1. True; 2. False; 3. False; 4. True; 5. False Solved Problems Solved Problem 9.1 Table 9.1 presents growth in real GDP for Country A and Country B. For each country ﬁnd: a. Relative increase in output between 1984 and 1994. b. Output per capita for 1984 and 1994. Figure 9-1 86 PRINCIPLES OF ECONOMICS c. Relative increase in output per capita between 1984 and 1994. d. Which measure of economic growth, as calculated in a. or c., is more useful? Solution: a. The relative increase in output is found by dividing 1994 GDP by that for 1984. The relative increase in Country A’s real GDP is 2.0; thus, Country A’s real GDP doubled between 1984 and 1994. The relative in-crease for Country B is 2.45 for the same period. b. An economy’s per capita GDP is found by dividing real GDP by the economy’s population. In Country A, per capita output is $3,915.66 in 1984 and $5,803.57 in 1994. Country B’s per capita output increased from $3,915.66 in 1984 to $4,796.67 in 1994. c. The relative increase in per capita output between 1984 and 1994 is found by dividing per capita output in 1994 by that in 1984. The rela-tive increase in per capita output is 1.48 for Country Aand 1.22 for Coun-try B. d. Economic growth is frequently presented as the increase in real GDP. While useful for some analysis, increases in real GDP do not mea-sure the economic well-being of individuals in an economy, which is best measured by increases in per capita output. The calculations in parts a. and c. show how one might reach different conclusions about an econo-my’s economic growth. Country B’s real GDP more than doubled be-tween 1984 and 1994, while A’s only doubled; B obviously has increased its output at a faster rate than A. On a per capita basis, however, output per capita increased more rapidly in Country A than B. Which measure is more useful depends upon one’s intent in measuring growth. When one is only interested in the increase in aggregate output, growth in real GDP Table 9.1 is the relevant measure. However, when the focus is upon the standard of living in the economy, output per capita is the better measure of economic growth. Solved Problem 9.2 What objections, if any, are there to economic growth? Solution: Some economists object to maximizing economic growth be-cause in doing so it may possibly affect the quality of life, in such ways as pollution of the environment or waste of natural resources. Maximized economic growth may also fail to resolve socioeconomic problems or may exacerbate them. Rapid economic growth through technological change in many instances increases worker obsolescence (workers no longer have skills needed in the labor market), brings about new anxieties and insecurities, and undermines family relationships as the workplace takes on greater importance than human relationships. Although attempts are being made to curb pollution, industrial waste is a by-product of in-creased output. It therefore can be expected that water, land, and air pol-lution will increase with time. Waste of economic resources may also re-sult when least-cost methods dictate current resource use with little attention paid to the possible effect that current use may have upon fu-ture generations. And there is no guarantee that growth resolves socio-economic problems such as poverty. Poverty in an economy is relative to the economy’s standard of living. Thus, growth does not resolve the prob-lem of relative poverty, which is only resolved by a redistribution of cur-rent income. CHAPTER 9: Economic Growth and Productivity 87 Chapter 10 International Trade and Finance In This Chapter: ✔ Basis of and Gains from Trade ✔ Obstacles to Trade ✔ Balance of Payments ✔ Exchange Rates ✔ True or False Questions ✔ Solved Problems Basis of and Gains from Trade Thus far, we have assumed a relatively closed econ-omy, or an economy isolated from the rest of the world. In reality, most nations are open economies. That is, they are connected to other nations through a network of trade and ﬁnancial relationships. These relationships have great advantages but they may also result in problems. Even though trade is gener-ally more important to small than to large developed nations, the welfare of the latter is also greatly dependent on trade. Since the availability of resources differs among nations, the oppor-88 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. tunity cost of producing more of a commodity (in terms of the amount of another commodity that would not be produced) also usually differs among nations. In a two-nation, two-commodity world, each nation should specialize in the production of the commodity with the lower op-portunity cost; this is the commodity in which the nation has a compara-tive advantage. The nation should trade part of its output with the other nation for the commodity with the higher opportunity cost (the one in which the nation has a comparative disadvantage). This leads to a larger combined output of both commodities than would occur in the absence of specialization and trade. Note! Comparative advantage is the key to trade among countries. Example 10.1 Figure 10-1 shows a hypothetical production-possibilities frontier for cloth (C) and food (F) for the U.S. and U.K. under constant costs (the solid lines). CHAPTER 10: International Trade and Finance 89 Figure 10-1 It shows that the U.S. could produce alternatives including 40C and 0F, or 0C and 80F. For each unit of cloth the U.S. gives up, it releases re-sources to produce two additional units of food. The domestic cost ratio is 1C = 2F, or 1/2C = 1F, and is constant in the U.S. In the U.K., 2C = 1F. Since the opportunity cost of F is 1/2C in the U.S. and 2C in the U.K., the U.S. has a comparative advantage in F. Similarly, the U.K. has a com-parative advantage in C. Suppose that in the absence of trade, the U.S. and U.K. produced and consumed at points A (20C and 40F) and A (20C and 20F), respectively. With trade, the U.S. should specialize in the pro-duction of F and produce at B (80F and 0C) and the U.K. should spe-cialize in C and produce at B (60C and 0F). By then exchanging, say, 30F for 30C with the U.K., the U.S. would end up consuming at E (30C and 50F) and the U.K. would consume at E (30C and 30F). Thus, both the U.S. and the U.K. end up consuming 10C and 10F more than without specialization and trade (compare E with A and E with A). With in-creasing opportunity costs, the production-possibilities frontiers are con-cave or bulge outward, and there would be incomplete specialization in production. Obstacles to Trade Even though trade can be the source of major gains, most nations restrict the free ﬂow of trade by imposing tariffs, quotas, and other obstructions. An import tariff is a tax on the imported commodity. An import quota is a quantitative restriction on the amount of a good that may be imported during a year. Other restric-tions include health regulations and safety and pol-lution standards. Trade restrictions are advocated by labor and ﬁrms in some industries as a protec-tion against foreign competition. These restric-tions, however, generally impose a burden on society as a whole because they reduce the availability of goods and increase their prices. Some of the speciﬁc arguments advanced for trade restrictions are: (1) to protect domestic labor against cheap foreign labor; (2) to reduce domestic un-employment; (3) to protect young or “infant” industries; and (4) to pro-tect industries important for national defense. Most of the arguments are invalid and are based on misconceptions. 90 PRINCIPLES OF ECONOMICS You Need to Know Trade restrictions are often supported by groups that will speciﬁcally beneﬁt from the restrictions, of-tentimes at the expense of other groups. Balance of Payments The balance of payments is a yearly summary statement of a nation’s transactions with the rest of the world. The balance of payments is di-vided into three major sections: (1) current account, which shows ﬂows of good and services and government grants; (2) capital account, which shows ﬂows of investments and loans; and (3) ofﬁcial reserve account, which shows the change in the nation’s ofﬁcial government reserves and liabilities to balance the current and capital accounts. The nation gains foreign currencies by exporting goods and services and receiving capital inﬂows (i.e., investments and loans) from abroad; all of these are credits. The nation spends these foreign currencies to im-port goods and services and to invest and lend abroad; these are debits. When the sum of all these debits exceeds the sum of the credits in the cur-rent and capital accounts, the nation has a deﬁcit in its balance of pay-ments equal to the difference. The deﬁcit is settled by a reduction in the nation’s reserves of foreign currency or by an increase in the foreign country’s holdings of the deﬁcit nation’s currency. The opposite is true for a balance-of-payments surplus. Exchange Rates Anation generates a supply of foreign currencies or monies in the process of exporting goods and services and receiving grants, investments, and loans from abroad. On the other hand, the nation uses foreign currencies to import goods and services and to make grants, investments, and loans abroad. When foreign currencies can be freely bought and sold, the rate of exchange between the domestic and a foreign currency is determined by the market demand for and the supply of the foreign currency. If the CHAPTER 10: International Trade and Finance 91 demand for the foreign currency increases, the rate of exchange rises. That is, more domestic currency is required to purchase one unit of the foreign currency (so that the domestic currency depreciates). Example 10.2 In Figure 10-2, D is the U.S. demand and S is the U.S. supply curve for pounds (£, the currency of the U.K.). 92 PRINCIPLES OF ECONOMICS Figure 10-2 D is downward sloping because at lower dollar prices for pounds it is cheaper for the U.S. to import from, invest in, and extend loans to the U.K. S is upward sloping because at higher dollar prices for pounds, it is cheaper for the U.K. to import from, invest in, and extend loans to the U.S. D and S intersect at the equilibrium rate of exchange of $2 = £1 and the equilibrium quantity of £300 million. If D shifts up to D, the rate of exchange rises to $3 = £1. If, on the other hand, the rate of exchange is not allowed to rise (as under the ﬁxed-exchange system), the U.S. would have a deﬁcit with the U.K. of EF = £200 = $400 million in its balance of payments. This deﬁcit could only be corrected by reducing the level of national income, by allowing domestic prices to rise less than abroad, or by government control of trade and payments. Note! Foreign currencies have prices (i.e., exchange rates), just like any other good or service. From the end of World War II until 1971, the world operated under a ﬁxed-exchange-rate system known as the Bretton Woods System. Un-der this system, the U.S. faced large and chronic deﬁcits, which it was justiﬁably unwilling to correct by domestic deﬂation or direct controls on trade and payments. The resulting lack of adjustment forced the aban-donment of the ﬁxed-exchange-rate system and the establishment of a ﬂexible-exchange-rate system. However, the system that is in operation today is not freely ﬂexible or completely ﬂoating because national mon-etary authorities intervene in foreign exchange markets to prevent errat-ic and unwanted ﬂuctuations in exchange rates. True or False Questions 1. Large countries are generally more open than small countries. 2. When each nation specializes in the production of the commodi-ty of its comparative advantage, the combined output of both commodi-ties increases. 3. Import restrictions are required to protect the nation’s labor against foreign competition. 4. A nation has a surplus in its balance of payments if its total cred-its exceed its total debits in its current and capital accounts. 5. A deﬁcit in a nation’s balance of payments is corrected by a de-preciation of its currency under a ﬁxed-exchange-rate system. Answers: 1. True; 2. True; 3. False; 4. True; 5. False Solved Problems Solved Problem 10.1 a. How can we measure a nation’s degree of economic interdepen-dence with the rest of the world? b. Why does the U.S. rely less on trade than most other developed nations? c. What would happen to its standard of living if the U.S. withdrew completely from international trade? Solution: a. Arough measure of the degree of interdependence of a nation with the rest of the world is given by the value of its exports as a percentage of its GDP. CHAPTER 10: International Trade and Finance 93 b. The U.S. is a nation of continental size with immense natural and human resources. As such, it can produce with relative efﬁciency most of the products it needs. In contrast, a small nation can only specialize in the production and export of a small range of commodities and must import all the others. In general, the larger the nation, the smaller its economic interdependence with the rest of the world. c. Even though the U.S. relies only to a relatively small extent on for-eign trade, a signiﬁcant part of its high standard of living depends on it. For one thing, the U.S. is incapable of producing such commodities as coffee, tea, and Scotch whiskey. In addition, the U.S. has no known de-posits of such minerals as tin and tungsten, which are important for in-dustrial production. It also needs to import huge quantities of petroleum. In addition, there are many commodities that the U.S. could produce do-mestically but only at a relatively higher cost than the costs of some for-eign countries. Thus, trade is very important to the welfare of the U.S. Solved Problem 10.2 a. Cite some of the speciﬁc arguments advanced in favor of trade pro-tection. b. Evaluate these arguments. Solution: a. Protection is often advocated to protect domestic labor against cheap foreign labor. That is, since wages are generally higher in the U.S. than in other nations, without protection foreign nations can undersell the U.S. because of the lower wages. Another argument for protection is that it reduces domestic unemployment. By restricting imports, domestic pro-duction is stimulated and unemployment is reduced. A third argument in favor of protection is the “infant industry” argument. This states that a newly established industry requires protection until it can grow in size and efﬁciency so as to be able to face foreign competition. Finally, pro-tection is advocated in order to protect such industries as shipyards that are important for national defense. b. The argument for protection against cheap foreign labor is gener-ally invalid because it incorrectly implies that higher wages necessarily mean higher labor costs. This is not true if the higher U.S. wages are based on even higher labor productivity. Restrictions on U.S. imports to reduce U.S. unemployment is a beggar-thy-neighbor policy because it leads to higher unemployment in those nations whose exports to the U.S. have 94 PRINCIPLES OF ECONOMICS been restricted. As a result, these other nations can retaliate and also re-duce imports from the U.S., and all nations lose in the end. The infant-in-dustry argument is generally invalid for the U.S. and other industrial na-tions but may be valid for poor developing nations. However, the same degree of protection can generally be better achieved by subsidies to the infant industry rather than by tariffs and quotas. Subsidies are also gen-erally preferable to tariffs and quotas as protection to industries impor-tant for national defense. Solved Problem 10.3 a. What happens to the equilibrium rate of exchange and to the equi-librium quantity of foreign exchange if the nation’s demand for the for-eign currency decreases? Why? b. How is a deﬁcit or a surplus in a nation’s balance of payments cor-rected under a ﬂexible-exchange-rate system? Solution: a. Given the nation’s supply curve of the foreign currency, a down-ward shift in the nation’s demand curve for the foreign currency will de-termine a new and lower equilibrium exchange rate and equilibrium quantity. A decrease in the nation’s demand for a foreign currency may result from a change in tastes for less imported goods and services. It may also occur if the nation decreases its investments and loans abroad in the expectation of decreased returns. b. A deﬁcit in a nation’s balance of payments means that at a given rate of exchange, there is a shortage (an excess of quantity demanded over quantity supplied) of the foreign currency. If the exchange rate is freely ﬂexible or ﬂoating, the exchange rate will rise until the quantity de-manded of the foreign currency equals the quantity supplied and the deﬁcit is completely eliminated. This rise in the exchange rate means that the relative value of the domestic currency is falling or depreciating. The exact opposite occurs when there is a surplus and the nation’s currency appreciates (or increases) in relative value. CHAPTER 10: International Trade and Finance 95 Chapter 11 Theory of Consumer Demand and Utility In This Chapter: ✔ Law of Diminishing Marginal Utility ✔ Utility Maximization ✔ Derivation of Individual Demand Curve ✔ True or False Questions ✔ Solved Problems Law of Diminishing Marginal Utility In previous chapters, we saw that the market demand curve for a com-modity is derived by adding the individual’s demand curves for the com-modity. We also saw that each individual’s (and thus the market) demand curve for a commodity is downward-sloping because of the substitution and income effects. However, an individual demands a particular com-modity because of the satisfaction, or utility, he or she re-96 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. ceives from consuming it. The more units of a commodity the individual consumes per unit of time, the greater is the total utility he receives. Al-though total utility increases, the extra, or marginal, utility received from consuming each additional unit of the commodity decreases. This is re-ferred to as the law of diminishing marginal utility. Example 11.1 For purposes of illustration, we assume in Table 11.1 that satisfaction can actually be measured in terms of units of utility called utils. The ﬁrst two columns of Table 11.1 give an individual’s hypothetical total utility (TU) schedule from consuming various quantities of commodity X (say or-anges) per unit of time. Note that as the individual consumes more units of X, TUx increases. Columns 1 and 3 of the table give this individual’s marginal utility (MU) schedule for commodity X. Each value of column 3 is obtained by sub-tracting two successive values of column 2. For example, if the individ-ual’s consumption of X rises from 1 unit to 2 units, TUx rises from 10 to 18, and the MU of the second unit of X is 8. Remember Utility models can be used to predict how consumers will act even if the consumers do not speciﬁcally think in terms of units of utility. CHAPTER 11: Theory of Consumer Demand and Utility 97 Table 11.1 Utility Maximization A consumer maximizes the total utility or satisfaction obtained from spending his income and is in equilibrium when the marginal utility of the last dollar spent on each commodity is the same. This equilibrium condition for utility maximization can be restated as follows: Example 11.2 Table 11.2 shows the marginal utility that an individual receives from consuming various units of X and Y per unit of time. Suppose that the con-sumer has $7 to spend on X and Y, and that Px (the price of X) = $2 and Py = $1. This consumer maximizes total utility and is in equilibrium by spending $4 of his $7 to buy 2X and the remaining $3 to purchase 3Y. At this point, MUx / Px = 8 utils / $2 = MUy / Py = 4 utils / $1 = MU of 4 utils from the last $ spent on X and Y. By purchasing 2X and 3Y, TUx = 18 (from 10 + 8), TUy = 15 (from 6 + 5 + 4), and TU from both is 33 utils. MU MU common MU of last $ spent on each commodity x x y y P P = = = ... 98 PRINCIPLES OF ECONOMICS Table 11.2 Derivation of Individual Demand Curve Starting with an equilibrium, we get one point on a consumer’s demand curve. At a lower commodity price, the consumer must purchase more of the commodity to be in equilibrium, and so we can get another point on that demand curve. From these and other points of consumer equilibri-Note! Utility maximization theory gives a formula to help explain the shape of the individual demand curve. um, we can derive a downward sloping demand curve because of dimin-ishing MU. Because MU declines, the price must fall to induce the indi-vidual consumer to buy more of the commodity. Consumer’s surplus refers to the difference between what the con-sumer would be willing to pay to purchase a given number of units of a commodity and what he actually pays for them. It arises because the con-sumer pays for all units of the commodity the price he is just willing to pay for the last unit purchased, even though the MU on earlier units is greater. Consumer surplus can be measured by the area under the con-sumer’s demand curve and above the commodity price. You Need to Know Consumer surplus arises from the different prices consumers would be willing to pay for different quantities of the same good. Imagine how much you would be willing to pay for your ﬁrst phone call each month, as opposed to your ﬁftieth call. Example 11.3 In Figure 11-1, the consumer purchases AF units of the commodity at price AB and spends AB times AF (the area of the rectangle ABCF) on this commodity. However, this consumer would have been willing to pay a higher price for all but the last unit of this commodity purchased (as in-dicated by the height of her demand curve) because these previous units give her a greater MU than the last unit purchased. The difference be-CHAPTER 11: Theory of Consumer Demand and Utility 99 tween what she would be willing to pay for AF units of the commodity (the area of AGCF) and what she actually pays for them (the area of ABCF) is an estimate of this consumer’s surplus (the area of triangle BGC). True or False Questions 1. The demand curve is downward sloping because of the substitu-tion and income effects. 2. The more of a commodity is consumed, the higher is the total util-ity derived. 3. The law of diminishing marginal utility states that each successive unit of the commodity consumed leads to a larger addition to total utili-ty. 4. Consumer utility maximization is satisﬁed by the condition that MUx = MUy = MUz. 5. Consumer’s surplus can be measured by the area under the de-mand curve and below the commodity price. Answers: 1. True; 2. True; 3. False; 4. False; 5. False Solved Problems Solved Problem 11.1 Suppose that a consumer has the MUx and MUy of Table 11.3, money income of $10, Px = $2, and Py = $1. a. Describe how this consumer should spend each dollar of her $10 100 PRINCIPLES OF ECONOMICS Figure 11-1 to purchase each unit of X and Y so as to maximize her total utility or sat-isfaction. b. Show that her TU would be less if she bought one more unit of ei-ther X or Y. c. Show that the equilibrium condition for utility maximization is satisﬁed when the consumer purchases 2X and 6Y. Solution: a. Because Px = $2, if this consumer spent her ﬁrst $2 to buy the ﬁrst unit of X, she would receive a MUx of only 14, or 7 utils per dollar spent on X. On the other hand, if this consumer spent her ﬁrst dollar to purchase the ﬁrst unit of Y, she would receive a MUy of 13, or 13 utils per dollar. Thus, she should spend her ﬁrst dollar to purchase the ﬁrst unit of Y and receive 13 utils of satisfaction. Similarly, this consumer should spend her second, third, and fourth dollars to purchase the second, third, and fourth units of Y and receive 11, 10, and 8 utils, respectively. This consumer is indifferent between purchasing the ﬁfth unit of Y or the ﬁrst unit of X be-cause she receives 7 utils per dollar spent on each. She would purchase both and spend her ﬁfth, sixth, and seventh dollars to purchase the ﬁfth Y and the ﬁrst X (remember, Px = $2). Similarly, the consumer should spend her eighth, ninth, and tenth (or last) dollar to purchase the sixth Y (and re-ceive 6 utils) and the second X (and receive 12 utils, or 6 utils per dollar). By purchasing 2X and 6Y, this consumer is receiving 81 utils (14 + 12 from X and 13 + 11 + 10 + 8 + 7 + 6 from Y). This is the maximum TU she can receive by spending her total income of $10 on X and Y when Px = $2 and Py = $1. Thus, the consumer is in equilibrium by purchasing 2X and 6Y. b. To buy the third unit of X (at Px = $2), this consumer would have to give up the ﬁfth and sixth units of Y (at Py = $1). She would gain 11 utils by purchasing the third unit of X but lose 13 utils (7 + 6) by giving up her ﬁfth and sixth Y, with a net loss of 2 utils. The consumer’s TU would be only 79 utils if she purchased 3X and 4Y (compared with a TU CHAPTER 11: Theory of Consumer Demand and Utility 101 Table 11.3 of 81 utils with 2X and 6Y) and she would not be maximizing the TU from spending her $10 of income. On the other hand, by giving up her second X (thus losing 12 utils), this consumer could purchase her seventh and eighth Y (gaining only a total of 5 utils), with a net loss of 7 utils. Pur-chasing 1X and 8Y, this consumer would receive a total of 74 utils (81 −7) and would not be in equilibrium. c. With 2X and 6Y, the consumer is in equilibrium because MUx / Px = 12 utils / $2 = MUy / Py = 6 utils / $1 = MU of 6 utils from the last dollar spent on X and Y. Plus, the consumer’s income is ex-hausted. Solved Problem 11.2 Why is water, which is essential to life, so cheap, while diamonds, which are not essential to life, so expensive? Solution: Because water is essential to life, the TU received from water exceeds the TU received from diamonds. However, the price we are will-ing to pay for each unit of a commodity depends not on the TU but on the MU. We consume so much water that the MU of the last unit of water consumed is very low. Therefore, we are willing to pay only a very low price for the last unit of water consumed. Since all units of water con-sumed are identical we pay the same low price on all other units of wa-ter consumed. On the other hand, we purchase so few diamonds that the MU of the last diamond purchased is very high. Therefore, we are willing to pay a high price for this last diamond and for all the other diamonds purchased. Solved Problem 11.3 With MUx and MUy of Table 11.3, income of $10, Px = $2, and Py = $1, the consumer is in equilibrium by purchasing 2X and 6Y (see Solved Problem 11.1) a. Find the point of consumer equilbrium with Px = $1. b. How is this consumer’s demand schedule for commodity X de-rived? Solution: a. If Px fall to $1, the consumer will no longer be in equilibrium by continuing to purchase 2X and 6Y because MU of 12 utils of $1 MU of 6 utils of $1 x x y y P P > 102 PRINCIPLES OF ECONOMICS and she is spending only $8 of her $10 income. Since the second dollar spent to purchase the second unit of X (at Px = $1) gives this individual more (marginal) utility than the sixth dollar spent to purchase the sixth unit of Y, the individual should spend more on X and less on Y. As she buys more X, the consumer moves down her diminishing MUx schedule. As she buys less of Y, she moves up her diminishing MUy. The consumer will be in equilibrium when the MU of the last dollar spent on X equals the MU of the last dollar spent on Y. This occurs when this consumer spends her $10 to purchase 6X and 4Y because b. When Px = $2, this consumer purchases 2X in order to be in equi-librium. This gives one point of this demand schedule for commodity X. Other points on the consumer’s demand schedule for X can be similarly obtained by allowing Px to change again and recording qx at equilibrium (as done in part a.). MU of 8 utils of $1 MU of 8 utils of $1 x x y y P P > CHAPTER 11: Theory of Consumer Demand and Utility 103 Chapter 12 Production Costs In This Chapter: ✔ Explicit and Implicit Costs ✔ Short-Run Costs ✔ Long-Run Costs ✔ True or False Questions ✔ Solved Problems Explicit and Implicit Costs In this chapter we concentrate on the ﬁrm’s produc-tion costs—or what lies behind its supply curve. Ex-plicit costs are the actual, out-of-pocket expendi-tures of the ﬁrm to purchase the services of the factors of production it needs. Implicit costs are the costs of the factors owned by the ﬁrm and used in its own production processes. These costs should be estimated from what these factors could earn in their best alternative use or employment. In economics, costs include both ex-plicit and implicit costs. Proﬁt is the excess of revenues over these costs. Example 12.1 The explicit costs of a ﬁrm are the wages it must pay to hire labor, the in-terest to borrow money capital, and the rent on land and buildings used 104 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. in the production process. To these, the ﬁrm must add such implicit costs as the wage that the entrepreneur would earn working as a manager for somebody else, the interest he would get by supplying his money capital (if any) to someone else in a similarly risky business, and the rent on his own land and buildings, if he were not using them himself. Only if the to-tal revenue received from selling the output exceeds both its explicit and implicit costs is the ﬁrm making an economic or pure proﬁt. Don’t Forget Both explicit and implicit costs must be considered any time we are con-sidering the true economic costs of a project. The law of diminishing returns is one of the most important and un-challenged laws of production. This law states that as we get more and more units of a factor of production to work with one or more ﬁxed fac-tors, after a point we get less and less extra or marginal output from each additional unit of the variable factor used. The time period when at least one factor of production is ﬁxed in quantity (i.e., cannot be varied) is re-ferred to as the short run. Thus, the law of diminishing returns is a short-run law. In the long run, all factors are variable. Short-Run Costs In the short run, there are total ﬁxed costs, total variable costs, and total costs. Total ﬁxed costs (TFC) are the costs that the ﬁrm incurs in the short run for its ﬁxed in-puts; these are constant regardless of the level of output and of whether it produces or not. An example of TFC is the rent that a producer must pay for the factory build-ing over the life of a lease. Total variable costs (TVC) are costs incurred by the ﬁrm for the variable inputs it uses. These vary directly with the lev-el of output produced and are zero when output is zero. Examples of TVC are raw material costs and some labor costs. Total costs (TC) are equal to the sum of total ﬁxed costs and total variable costs. CHAPTER 12: Production Costs 105 Though total costs are very important, per-unit or average costs are even more important in the short-run analysis of the ﬁrm. The short-run per-unit costs that we consider are the average ﬁxed cost, the average variable cost, the average cost, and the marginal cost. Average ﬁxed cost (AFC) equals total ﬁxed costs divided by output. Average variable cost (AVC) equals total variable costs divided by output. Average cost (AC) equals total costs divided by output; AC also equals AFC plus AVC. Mar-ginal cost (MC) equals the change in TC or the change in TVC per unit change in output. Example 12.2 Table 12.1 presents the AFC, AVC, AC, and MC schedules derived from the TFC, TVC, and TC schedules. The AFC schedule (column 5) is ob-tained by dividing TFC (column 2) by the corresponding quantities of output produced (Q in column 1). The AVC schedule (column 6) is ob-tained by dividing TVC (column 3) by Q. The AC schedule (column 7) is obtained by dividing TC (column 4) by Q. The MC schedule (column 8) is obtained by subtracting successive values of TC (column 4) or TVC (column 3). Thus, MC does not depend on the level of TFC. The AFC, AVC, AC, and MC schedules of Table 12.1 are graphed in Figure 12-1. Note that the values of the MC schedule (from column 8) are plotted halfway between successive levels of output. Also note that while the AFC curve falls continuously as output is expanded, the AVC, AC, and MC curves are U-shaped. The MC curve reaches its lowest point at a low-er level of output than either the AVC curve or the AC curve. Also, the rising portion of the MC curve intersects the AVC and AC curves at their lowest points. This will always be the case. 106 PRINCIPLES OF ECONOMICS Table 12.1 Note! The law of diminishing returns is the reason that the marginal cost curve is U-shaped. Long-Run Costs In the long run, there are no ﬁxed factors, and a ﬁrm can build a plant of any size. Once a ﬁrm has constructed a particular plant, it operates in the short run. A plant size can be represented by its short-run average cost (SAC) curve. Larger plants can be represented by SAC curves, which lie further to the right. The long-run average cost (LAC) curve shows the minimum per-unit costs of producing each level of output when any de-sired size of plant can be built. The LAC curve is thus formed from the relevant segment of the SAC curves. Example 12.3 Figure 12-2 shows four hypothetical plant sizes that a ﬁrm could build in the long run. Each plant is shown by a SAC curve. To produce up to 300 CHAPTER 12: Production Costs 107 Figure 12-1 units of output, the ﬁrm should build and utilize plant 1 (given by SAC1). From 300 to 550 units of output, it should build the larger plant given by SAC2, etc. Note that the ﬁrm could produce an output of 400 with plant 1, but only at a higher cost than with plant 2. The irrelevant portions of the SAC curves are dashed. The remaining (undashed) portions form the LAC curve. By drawing many more SAC curves, we would get a smoother LAC curve. If in the long run we increase all factors used in production by a giv-en proportion, there are three possible outcomes: (1) output increases in the same proportion, so that there are constant returns to scale or constant costs; (2) output increases by a greater proportion, giving increasing re-turns to scale or decreasing costs; and (3) output increases in a smaller proportion, giving decreasing returns to scale or increasing costs. In-creasing returns to scale or economies of mass production may result be-cause of division of labor and specialization in production. Beyond a cer-tain size, however, management problems resulting in decreasing returns to scale may arise. Remember The LAC curve derives its shape from the possible SAC curves that a ﬁrm has over ranges of outputs. 108 PRINCIPLES OF ECONOMICS Figure 12-2 Example 12.4 The LAC curve of Figure 12-2 at ﬁrst shows increasing returns to scale. Then for a small range of outputs (around 800 units), it shows constant returns to scale. For larger outputs, we have decreasing returns to scale. Whether and when this occurs in the real world depends on the ﬁrm and industry under consideration. True or False Questions 1. Implicit costs are the costs of factors of production owned by the ﬁrm. 2. The law of diminishing returns holds in both the short-run and long-run periods. 3. TFC is constant regardless of the level of ﬁrm output. 4. TC is zero when the ﬁrm does not produce any output. 5. Decreasing costs refers to the situation wherein output increases proportionately more than inputs. Answers: 1. True; 2. False; 3. True; 4. False; 5. True Solved Problems Solved Problem 12.1 A ﬁrm pays $200,000 in wages, $50,000 in inter-est on borrowed money capital, and $70,000 for the yearly rental of its factory building. If the entrepreneur worked for somebody else as a man-ager she would earn at most $40,000 per year, and if she lent out her mon-ey capital to somebody else in a similarly risky business, she would at most receive $10,000 per year. She owns no land or building. a. Calculate the entrepreneur’s economic proﬁt if she received $400,000 from selling her year’s output. b. How much proﬁt is the entrepreneur earning from the point of view of the person on the street? To what is the difference in the results due? c. What would happen if the entrepreneur’s total revenue were $360,000 instead? Solution: a. The explicit costs of this entrepreneur are $320,000 ($200,000 in wages plus $50,000 in interest plus $70,000 in rents). Her implicit costs CHAPTER 12: Production Costs 109 are $50,000 ($40,000 in wages in her best alternative employment plus $10,000 interest on her money capital). Thus, her total costs are $370,000. Since the total revenue from selling the year’s output is $400,000, she earns an economic proﬁt of $30,000 for the year. b. The person on the street would instead say that this entrepreneur’s proﬁt is $80,000 (the total revenue of $400,000 minus the out-of-pocket expenditures, or explicit costs, of $320,000). However, $50,000 of this $80,000 represents the normal return on the entrepreneur’s owned factors and is appropriately considered a cost by the economist. c. If the entrepreneur’s total revenue were $360,000, she would earn less than a normal return on her owned factors (her wage and interest in alternative employment) and it would be best to eventually go out of busi-ness and work as a manager for and lend her money to someone else. This shows that implicit costs are part of the costs of production because they must be covered in order for the ﬁrm to remain in business and continue to supply the goods and services it produces. Solved Problem 12.2 a. Why are the MC, AVC, and AC curves U-shaped? b. Why does the MC curve intersect the AVC and AC curves at their respective lowest points? Solution: a. As we start using variable factors with some ﬁxed factors, we may ﬁrst obtain increasing returns, but eventually diminishing returns will set in. As a result, the MC, AVC, and AC curves ﬁrst fall but eventually rise, giving them their U shapes. b. The MC curve always intersects the AVC and AC curves at their respective lowest points because as long as MC is below AC, it pulls the average down. When the MC is above AC, it pulls the average up. Only when MC equals AC is AC neither falling nor rising (i.e., AC is at its low-est point). This is logical. For example, if your grade on a quiz is lower than your previous average, your average will fall and vice versa. 110 PRINCIPLES OF ECONOMICS 111 Chapter 13 Perfect Competition In This Chapter: ✔ Perfect Competition Deﬁned ✔ Proﬁt Maximization in the Short Run ✔ Short-Run Proﬁt or Loss ✔ Long-Run Equilibrium ✔ True or False Questions ✔ Solved Problems Perfect Competition Deﬁned An industry is said to be perfectly competitive if: (1) it is composed of a large number of independent sellers of a commodity, each too small to affect the commodity price; (2) all ﬁrms in the industry sell homogenous (identical) products; and (3) there is perfect mobility of resources, so ﬁrms can enter or leave the industry in the long run without much dif-ﬁculty. As a result, the perfectly competitive ﬁrm is a “price taker” and can sell any amount of the com-modity at the prevailing market price. Perhaps the closest we come to perfect competition is in the market for such agricultural commodities as wheat and cotton. There, we may have a large number of producers each too small to affect commodity Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. 112 PRINCIPLES OF ECONOMICS price. The output of each farmer (say wheat of a given grade) is identi-cal, and it is rather easy to enter or leave this industry. The perfectly com-petitive model is used to analyze markets, such as these, that approximate perfect competition. It is also used to evaluate the efﬁciency of the other forms of market organization to be covered in subsequent chapters. Note! Perfect competition is rarely seen in the real world, but is an economic benchmark for all types of busi-nesses. Proﬁt Maximization in the Short Run A ﬁrm maximizes total proﬁts in the short run when the (positive) differ-ence between total revenue (TR) and total costs (TC) is greatest. TR equals price times quantity. In general, it is more useful to analyze the short-run behavior of the ﬁrm by using the marginal-revenue–marginal-cost approach. Marginal revenue (MR) is the change in TR per unit change in the quantity sold. Since the perfectly competitive ﬁrm can sell any quantity of the com-modity at the prevailing price, its MR = P, and the demand curve it faces is horizontal at that price. The perfectly competitive ﬁrm maximizes its short-run total proﬁts at the output at which MR or P equals MC (and MC is rising). Example 13.1 In Table 13.1, MR (column 4) is the change in TR and is recorded be-tween the various quantities sold. MC (column 7) is the change in TC and is also entered between the various levels of output. Proﬁt per unit (col-umn 10) equals P −AC. Total proﬁts (column 11) equal proﬁts per unit times the quantities sold. Note that total proﬁts are maximized at $16.90 when the ﬁrm produces and sells 6.5 units of output where MR = MC. CHAPTER 13: Perfect Competition 113 Example 13.2 The proﬁt-maximizing (or best) level of output of this ﬁrm can also be viewed in Figure 13-1. The MC and AC values are from Table 13.1. The demand curve facing the ﬁrm is horizontal at P = $8 = MR. As long as MR exceeds MC, it pays for the ﬁrm to expand output. Thus, the ﬁrm maximizes its total proﬁts at the output level of 6.5 units (given by point C where MR = MC). The proﬁt per unit at this level of output is CF, or $2.60, and total proﬁt is given by the area of rectangle CFGH, which equals $16.90. Figure 13-1 Table 13.1 114 PRINCIPLES OF ECONOMICS Remember MR = MC is the key to proﬁt maxi-mization. Short-Run Proﬁt or Loss If, at the point where P = MR = MC, P exceeds AC, the ﬁrm is maxi-mizing its total proﬁts. If P = AC, the ﬁrm is breaking even. If P is larg-er than AVC but smaller than AC, the ﬁrm minimizes total losses. If P is smaller than AVC, the ﬁrm minimizes total losses by shutting down. Thus, P = AVC is the shutdown point for the ﬁrm. Note! A business ﬁrst determines the proﬁt-maximizing quantity and then determines whether it will have a proﬁt or loss. Since the perfectly competitive ﬁrm always produces where P = MR = MC (as long as P exceeds AVC), the ﬁrm’s short-run supply curve is given by the rising portion of its MC curve over and above its AVC, or shutdown point. Long-Run Equilibrium If the ﬁrms in a perfectly competitive industry are making short-run prof-its, more ﬁrms will enter the industry in the long run. This increases mar-ket supply of the commodity and reduces the market price until all prof-its are competed away and all ﬁrms just break even. The exact opposite occurs if we start with ﬁrms with short-run losses. As a result, all ﬁrms in a perfectly competitive industry with long-run equilibrium produce where P = lowest LAC and resources are utilized in the most efﬁcient way. CHAPTER 13: Perfect Competition 115 True or False Questions 1. In a perfectly competitive industry, each ﬁrm can affect the com-modity price. 2. The marginal revenue of a ﬁrm in perfect competition is equal to the commodity price. 3. The perfectly competitive ﬁrm maximizes proﬁts at the quantity where its MR curve intersects the rising portion of its MC curve. 4. A ﬁrm breaks even when price equals its average variable cost. 5. All ﬁrms in perfect competition break even in the long run. Answers: 1. False; 2. True; 3. True; 4. False; 5. True Solved Problems Solved Problem 13.1 a. Deﬁne marginal revenue. How is it calculated? Why is marginal revenue constant and equal to price under perfect competition? b. What is the shape and elasticity of the demand curve facing a per-fectly competitive ﬁrm? Why? c. How does the ﬁrm determine how much to produce in the short run? Solution: a. MR is deﬁned as the change in TR for a one-unit change in the quantity sold. Since the perfectly competitive ﬁrm can sell any amount of the commodity at the prevailing market price, its MR is constant. For example, if P = $4, TR = $4 when the ﬁrms sells one unit and TR = $8 for two units. Thus, MR = change in TR = $4 = P. b. Since the perfectly competitive ﬁrm can sell any amount at the market price, the demand curve it faces is horizontal or inﬁnitely elastic at this price. With a horizontal demand curve, an inﬁnitely small fall in price causes an inﬁnitely large increase in sales because all consumers will go to the seller with the lowest price. As the denominator of the elas-ticity formula (the percentage change in price) approaches zero and the numerator (the percentage change in quantity) becomes very large, the value of the fraction and elasticity (ED) approaches inﬁnity. c. We can determine how much a ﬁrm produces in the short run by 116 PRINCIPLES OF ECONOMICS making the reasonable assumption that the ﬁrm wants to maximize its to-tal proﬁts or minimize its total losses. The general rule is that the ﬁrm should expand its output until MR = MC (as long as P exceeds AVC). A ﬁrm should expand its output as long as the addition to TR from an ad-ditional unit sold (its MR) exceeds the addition to TC to produce this ex-tra unit (its MC). As long as MR > MC, the ﬁrm can increase its total prof-its by expanding output. The ﬁrm should not produce any unit for which MR < MC. If it did, it would be adding more to its TC than to its TR and its total proﬁts would fall. Solved Problem 13.2 From Figure 13-2, set up a table indicating for each alternative demand curve, the best level of output, AC, proﬁt per unit, total proﬁts, whether the ﬁrm produces or not, and whether it makes proﬁts or losses (if TFC = $65). Figure 13-2 Solution: Table 13.2 shows that with d5, the ﬁrm maximizes total profits. Table 13.2 With d4, P =AC so that the ﬁrm breaks even. With d3, the ﬁrm minimizes total losses at $33.80 by producing 65 units of output. If the ﬁrm stopped producing, it would incur losses equal to its TFC of $65. Thus, by pro-ducing, the ﬁrm recovers all of its TVC plus part of TFC. With d2, the ﬁrm’s total losses equal $65 (by rounding) whether it produces or not. This is the shutdown point for the ﬁrm. With d1, the best level of output is 55 units where MR = MC). At this output, the ﬁrm’s total losses would equal $92.40. But by stopping production altogether and going out of business, the ﬁrm would lose only $65 (its TFC). Thus, the ﬁrm would not produce at P = $1.50. Solved Problem 13.3 Discuss the advantages of perfect competition. Solution: The most important advantages of the perfectly competitive form of market organization are that resources are utilized in the most ef-ﬁcient way to produce the goods and services most wanted by society and that consumers pay the lowest possible prices. In long-run equilibrium, each perfectly competitive ﬁrm operates the optimum scale of plant at the optimum level of output. This is given by the lowest point of the SAC curve, which generates the lowest point of the LAC curve. Resources could not possibly be arranged more efﬁciently. Furthermore, since the forces of competition eliminate all proﬁts in the long run, consumers get the good or service at P = lowest LAC. Finally, since the price of the com-modity measures the utility of the last unit of the commodity consumed, and this is equated to the MC of producing this unit, there is no better use of these resources. That is, the same resources could not be used to pro-duce goods and services that give greater utility to consumers. Thus, per-fect competition is used as the standard against which the efﬁciency of other market forms is compared. CHAPTER 13: Perfect Competition 117 118 Chapter 14 Monopoly In This Chapter: ✔ Monopoly Deﬁned ✔ Proﬁt Maximization ✔ Price Discrimination ✔ Regulation of Monopoly ✔ True or False Questions ✔ Solved Problems Monopoly Deﬁned Pure monopoly is the form of market organization in which there is a sin-gle seller of a commodity for which there are no close substitutes. Thus, it is at the opposite extreme from perfect competition. Monopoly may be the result of: (1) increasing returns to scale; (2) control over the supply of raw materials; (3) patents; or (4) government franchise. For example, electrical companies, telephone companies, and other “public utilities” usually have increasing returns to scale (i.e., falling long-run av-erage costs) over a sufﬁcient range of outputs as to enable a single ﬁrm to satisfy the entire market at a lower per-unit cost than two or more ﬁrms could. These natural monopolies usually operate under a Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. CHAPTER 14: Monopoly 119 government franchise and are subject to government regulation. A mo-nopoly may also arise because a ﬁrm may own a patent which precludes other ﬁrms from producing the same commodity. Under pure monopoly, the ﬁrm is the industry and faces the nega-tively sloped industry demand curve for the commodity. As a result, if the monopolist wants to sell more of the commodity, it must lower its price. Thus, for a monopolist, MR is less than P, and its MR curve lies below its demand curve. Important! A monopoly is opposite of perfect competition in every facet of its organization. Proﬁt Maximization The proﬁt-maximizing or best level of output for the monopolist is the output at which MR = MC. Price is then read off the demand curve. De-pending on the level of AC at this output, the monopolist can have prof-its, break even, or minimize the short-run total losses. Example 14.1 From Table 14.1, the monopolist maximizes total proﬁts at $3.75 when it produces and sells 2.5 units of output at the price of $5.50. At this output, MR = MC = $3. As long as MR > MC, the monopolist will expand out-put and sales because doing so adds more to TR than to TC (and proﬁts Table 14.1 120 PRINCIPLES OF ECONOMICS rise). The opposite is true when MR < MC. Thus total proﬁts are maxi-mized where MR = MC. The proﬁt-maximizing or best level of output for this monopolist can also be seen in Figure 14-1 (obtained by plotting the value of columns 1, 2, 4, 6, and 7 of Table 14.1). In this ﬁgure, the best level of output is at the point where MR = MC. At this best output level of 2.5 units, the mo-nopolist makes a proﬁt of $1.50 per unit (the vertical distance between D and AC at 2.5 units of output) and $3.75 in total (2.5 units times the $1.50 proﬁt per unit). Note that since P > MR where MR = MC, the rising por-tion of the MC curve above the AVC does not represent the monopolist supply curve. In the long run, the monopolist can adjust the scale of plant, and proﬁts may persist because of blocked or restricted entry. Note! Even though organized completely differently from a perfect competitor, a monopolist still maximizes proﬁt where MR = MC. Figure 14-1 CHAPTER 14: Monopoly 121 Price Discrimination A monopolist can increase TR and proﬁts at a given level of output and TC by practicing price discrimination. This involves charging different prices for the commodity for different quantities purchased, to different classes of consumers, or in different markets. For example, a telephone company may charge individuals 15 cents for each of the ﬁrst 50 telephone calls made during each month, 10 cents for each of the next 100 calls, and so on. Electrical companies usually charge less per kilowatt-hour to industrial users than to households be-cause industrial users have more substitutes available (such as generat-ing their own electricity) and thus have a more elastic demand curve than households. Regulation of Monopoly Since a monopoly produces output where MR = MC and P > MR, the mo-nopolist produces less and charges a higher price than a perfect competi-tor with the same cost curves. For example, if Figure 14-1 was for a per-fectly competitive industry, output would be 3 units and price $5 (given where P = MC), rather than Q = 2.5 and P = $5.50 for the monopolist. Thus, monopoly leads to a misallocation of resources. For efﬁciency considerations, government (federal, state, or local) often allows natural monopolies (such as public utilities) to operate but subjects them to regulation. This usually takes the form of setting a price that allows the monopolist to earn the “normal or fair” return of about 8–10 percent on its investment. However, such regulation only partly cor-rects the more serious problem of misallocation of resources. Remember Monopolies rarely exist in the real world except when regulated by a government body. 122 PRINCIPLES OF ECONOMICS True or False Questions 1. Pure monopoly is the opposite of perfect competition. 2. The monopoly maximizes proﬁt at the output level where P = MC. 3. The monopolist always earns proﬁts in the short run. 4. A monopoly leads to a higher commodity price and less output than perfect competition. 5. All monopoly proﬁts disappear in the long run. Answers: 1. True; 2. False; 3. False; 4. True; 5. False Solved Problems Solved Problem 14.1 a. Draw a ﬁgure showing, for a monopolist, the best level of output. Include three alternative AC curves, showing that the ﬁrm (1) makes a proﬁt, (2) breaks even, and (3) incurs a loss. b. What would happen to this monopolist in the long run if it incurs short-run losses? Short-run proﬁts? Solution: a. In Figure 14-2, the best level of output for the monopolist is OB, given by point C where MR = MC. With AC1, the monopolist makes a per-unit proﬁt of GF and a total proﬁt of GF times OB. With AC2, P =AC and the monopolist breaks even. With AC3, the monopolist incurs a per-unit loss of HG and a total loss of HG times OB. Only if P > AVC (so that TR > TVC) will the monopolist stay in business and minimize short-run total losses by producing OB. b. If the monopolist has short-run losses, it could, in the long run, build a more appropriate scale of plant to produce the best long-run lev-el of output. The monopolist might also advertise in an attempt to cause an upward shift in the demand curve it faces. (This, however, will also shift cost curves up.) If this monopolist still incurs a loss after having con-sidered all of these possibilities, it will stop producing the commodity in the long run. If the monopolist was already making short-run proﬁts, it will still build the most appropriate plant in the long run and increase to-tal proﬁts. CHAPTER 14: Monopoly 123 Solved Problem 14.2 Refer to Figure 14-3, which contains the market demand curve facing a monopolist. a. What price should the monopolist charge without price discrimi-nation if its best level of output (given by the point where MR = MC) is OB? What would the TR be? How much is the consumers’ surplus? b. Suppose the monopolist sold OA units at price OF. In order to in-duce consumers to buy AB additional units, it lowers its price to OC only on AB units. How much would TR be now? How much of the consumers’ surplus remains? Solution: a. The highest price the monopolist can charge (without price dis-crimination) to sell OB units is OC. The TR would then equal the area of rectangle OCKB. Consumers’ surplus is CGK. b. TR is OFHA (for OA units) plus AJKB (for AB units). Note that price discrimination has increased TR by CFHJ (and this is the amount by which the consumers’ surplus declined). Consumers’ surplus is now only FGH plus HKJ. Figure 14-2 124 PRINCIPLES OF ECONOMICS Solved Problem 14.3 a. Should the government break up a monopoly into a large number of perfectly competitive ﬁrms? Why? b. Does monopoly lead to more technological progress than perfect competition? Why? Solution: a. In industries operating under cost conditions (such as constant re-turns to scale) that make the existence of perfect competition feasible, the dissolution of a monopoly (by government antitrust action) into a large number of perfectly competitive ﬁrms will result in a greater long-run equilibrium output for the industry, a lower commodity price, and usual-ly a lower LAC than under monopoly. However, because of cost and tech-nological conditions, it is not desirable to break up a natural monopoly into a large number of perfectly competitive ﬁrms. In dealing with nat-ural monopolies, the government usually chooses to regulate them rather than break them up. b. There is a great deal of disagreement on whether monopoly or per-fect competition leads to more technological progress. Since a monopo-list usually makes long-run proﬁts while perfect competitors do not, the monopolist has more resources to devote to research and development (R & D). The monopolist is also more likely to retain the beneﬁts of the tech-nological advance it introduces. A technological advance introduced by a perfect competitor which leads to lower costs and short-run proﬁts is easily and quickly copied by other ﬁrms, and this eliminates the proﬁts of the ﬁrm that introduced it. On the other hand, a monopolist may feel very secure in its position and have no incentive to engage in research and development and to innovate. Figure 14-3 125 Chapter 15 Monopolistic Competition and Oligopoly In This Chapter: ✔ Monopolistic Competition Deﬁned ✔ Proﬁt Maximization ✔ Oligopoly Deﬁned ✔ Collusion ✔ Long-Run Efﬁciency Implications ✔ True or False Questions ✔ Solved Problems Monopolistic Competition Deﬁned In monopolistic competition there are many ﬁrms sell-ing a differentiated product or service. It is a blend of competition and monopoly. The competitive elements result from the large number of ﬁrms and the easy en-try. The monopoly element results from differentiated (i.e., similar but not identical) products or services. Product differentiation may be real or imaginary and can be created through advertising. However, the availability of close substitutes se-verely limits the “monopoly” power of each ﬁrm. Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. 126 PRINCIPLES OF ECONOMICS Monopolistic competition is the most prevalent form of market or-ganization in retailing. The numerous grocery stores, gasoline stations, dry cleaners, etc. within close proximity of each other are good examples. Examples of differentiated products include the numerous brands of headache remedies (e.g., aspirin, Bufferin, Excedrin, etc.), soaps, deter-gents, breakfast cereals, and cigarettes. Even if the differences are imag-inary (as in the case of various brands of aspirin), they are economically important if the consumer is willing to pay a little more or travel a little further for a preferred brand. You Need to Know Most businesses ﬁt under the category of monop-olistic competition in terms of market organization. Proﬁt Maximization The monopolistic competitor faces a demand curve which is negatively sloped (because of product differentiation) but highly elastic (because of the availability of close substitutes). The monopolistic competitor’s prof-it-maximizing or best level of output is the output at which MR = MC, provided P > AVC. At that output, the ﬁrm can make a proﬁt, break even, or minimize losses in the short run. In the long run, ﬁrms are either at-tracted into an industry by short-run proﬁts or leave it if faced with loss-es until the demand curve (d) facing remaining ﬁrms is tangent to its AC curve, and the ﬁrm breaks even (P = AC). Example 15.1 Panel A of Figure 15-1 shows a monopolistic competitor producing 550 units of output (where MR = MC), selling it at $10.50 (on d), and mak-ing a proﬁt of $3.50 per unit and $1925 in total. These proﬁts attract more ﬁrms into the industry. This causes a downward (leftward) shift in this ﬁrm’s demand curve to d (in Panel B), at which the ﬁrm sells 400 units at $8 and breaks even. Since P > MR where MR = MC, the MC curve above AVC does not represent the ﬁrm’s supply curve. CHAPTER 15: Monopolistic Competition and Oligopoly 127 Oligopoly Deﬁned Oligopoly is the form of market organization in which there are few sell-ers of a product. If the product is homogenous, there is a pure oligopoly. If the product is differentiated, there is a differentiated oligopoly. Since there are only a few sellers of a product, the actions of each seller affect others. That is, the ﬁrms are mutually interdependent. Figure 15-1 128 PRINCIPLES OF ECONOMICS Note! It is difﬁcult to graphically analyze an oligopoly (as we do with the other types of market organizations) due to the mutual interdependence among the ﬁrms. Pure oligopoly is found in the production of cement, aluminum, and many other industrial products which are are virtually standardized. Ex-amples of differentiated oligopolies are industries producing automo-biles, cigarettes, PCs, and most electrical appliances, where three or four large ﬁrms dominate the market. Because of mutual interdependence, if one ﬁrm lowered its price, it could take most of the sales away from the other ﬁrms. Other ﬁrms are then likely to retaliate and possibly start a price war. As a result, there is a strong compulsion for oligopolists not to change prices but, rather, to compete on the basis of quality, product de-sign, customer service, and advertising. Collusion An orderly price change (i.e., one that does not start a price war) is usually accomplished by collusion that can be overt or tacit. The most extreme form of overt collusion is the centralized cartel, in which the oligopolists produce the monopoly output, charge the monopoly price, and somehow allocate produc-tion and proﬁts among the cartel members. Antitrust laws make overt collusion illegal in the U.S. In tacit collusion, the oligopolists informally follow a recognized price leader in their pricing policies or agree on how to share the market. Until the 1980s, U.S. Steel (now called USX) was a recognized price leader. When rising costs required it, U.S. Steel raised the price on some of its products on the tacit understanding that other domestic steel pro-ducers would match the price within a few days. An orderly price increase was thus achieved without exposing producers to government antitrust action or the danger of a price war. CHAPTER 15: Monopolistic Competition and Oligopoly 129 Long-Run Efﬁciency Implications The monopolistically competitive ﬁrm misallocates resources because it produces where P > MC (see Figure 15-1). In addition, it does not pro-duce at the lowest point on its LAC curve as a perfect competitor does. However, these inefﬁciencies are usually not great because of the highly elastic demand faced by monopolistic competitors. In contrast to the perfect competitor, the monopolistic competitor en-gages in nonprice competition, which takes the form of advertising and product differentiation. Such tactics are intended to increase the ﬁrm’s share of the market and shift its demand curve upward (to the right). However, they also increase the ﬁrm’s costs and shift the ﬁrm’s cost curves upward. While some advertising informs the consumer and prod-uct differentiation satisﬁes the consumers’desire for variety, both may be excessive and wasteful. While the oligopolist can make proﬁts, break even, or incur losses in the short run, in the long run the ﬁrm will leave the industry rather than incur losses. Oligopolists underallocate resources and can earn long-run proﬁts because of restricted entry. Usually they also engage in excessive advertising and product differentiation. However, efﬁciency considera-tions may allow only a few ﬁrms in the industry, and oligopolists may use their proﬁts for research and development. Don’t Forget! Monopolistic competitors and oligopolists are like monopolists in that they do not allocate resources as efﬁciently as perfect competitors, as far as so-ciety is concerned. True or False Questions 1. The monopoly power of a monopolistic competitor is limited by the availability of close substitutes. 2. A monopolistic competitor produces at the lowest point on its LAC curve. 130 PRINCIPLES OF ECONOMICS 3. Restricted entry is a characteristic of monopolistic competition. 4. In tacit collusion, oligopolists meet and decide on a price leader to follow in their pricing policies. 5. In the long run oligopolists can earn proﬁts. Answers: 1. True; 2. False; 3. False; 4. False; 5. True Solved Problems Solved Problem 15.1 a. Why does a prospective monopolistic competitor ﬁnd it relatively easy to start production in the long run? b. Why does the demand curve of a monopolistic competitor shift down when more ﬁrms start production? c. Why is it difﬁcult or impossible to deﬁne the industry under mo-nopolistic competition? d. Why is there a cluster of prices rather than a single equilibrium price in this kind of industry? Solution: a. A prospective monopolistic competitor usually ﬁnds it relatively easy to start production because very little capital and no great technical know-how are required to open a small gasoline station, grocery store, barber shop, etc. b. When more ﬁrms start producing a differentiated product, the de-mand curve of previously existing monopolistic competitors shifts down because each ﬁrm now has a smaller share of the market. c. Technically speaking, we cannot deﬁne the monopolistically com-petitive industry because each ﬁrm produces a somewhat different prod-uct. We simply cannot add together aspirins, Bufferins, Excedrins, etc. to get the market demand and supply curve because they are similar, but not identical, products. Thus, our graphical analysis must be conﬁned to the “typical” or “representative” ﬁrm. d. Slightly differentiated products also permit and cause slightly dif-ferent prices. That is, even in long-run equilibrium, there will be a clus-ter of equilibrium prices, one for each differentiated product, rather than a single, industry-wide equilibrium price. CHAPTER 15: Monopolistic Competition and Oligopoly 131 Solved Problem 15.2 a. What are some of the natural and artiﬁcial barriers to entry into oligopolistic industries? b. What are the possible harmful effects of oligopoly? c. What are the possible beneﬁcial effects of oligopoly? Solution: a. The natural barriers to entry into oligopolistic industries like the automobile, aluminum, and steel industries are the smallness of the mar-kets in relation to efﬁcient operation and the huge amounts of capital and specialized inputs required to start efﬁcient operation. Some artiﬁcial bar-riers to entry are control over raw materials, patents, and government franchise. When entry is blocked or at least restricted, the ﬁrms in an oli-gopolistic industry can earn long-run proﬁts. b. In the long run, oligopoly may lead to the following harmful ef-fects: (1) P > MC and so there is an underallocation of the economy’s re-sources to the ﬁrms in the oligopolistic industry; (2) the oligopolist usu-ally does not produce at the lowest point on its LAC curve; and (3) when oligopolists produce a differentiated product, too much may be spent on advertising and model changes. c. For technological reasons, many products (such as automobiles, steel, etc.) cannot be produced under conditions of perfect competition (because their cost of production would be prohibitively high). In addi-tion, oligopolists spend a great deal of their proﬁts on research and de-velopment, and this may lead to faster technological advance and a high-er standard of living than if the industry were organized along more competitive lines. Finally, some advertising is useful since it informs cus-tomers, and some product differentiation has the economic value of sat-isfying the different tastes of different consumers. Chapter 16 Demand for Economic Resources In This Chapter: ✔ Resource Pricing ✔ Resource Demand ✔ Changes in Resource Demand ✔ True or False Questions ✔ Solved Problems Resource Pricing We now examine how the prices of productive resources such as wages, rents, interest, and proﬁts are determined in a mixed economy. Resource prices are a major determinant of money incomes and of the allocation of resources to various uses and ﬁrms. Broadly speaking, the price of a resource is de-termined by its market demand and supply. Firms demand resources in order to produce commodities. The demand for resources is a derived demand—de-rived from the demand for the commodities that re-quire the resources in production. The greater the demand for the commodity and the more productive the resource, the greater the price that ﬁrms are will-ing to pay for the resource. 132 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. For example, as a result of consumers’ demand for a ﬁnal commod-ity, say, shoes, ﬁrms hire labor and other resources in order to produce shoes. The greater the demand for shoes, the greater the ﬁrms’ demands for labor. In the absence of market imperfections (minimum wage laws, union power, etc.), the wage rate of labor is determined exclusively by the market demand and supply of labor. To derive a ﬁrm’s demand for a resource, we must ﬁrst deﬁne the marginal revenue product (MRP). MRP measures the increase in the ﬁrm’s total revenue from selling the extra product that results from em-ploying one additional unit of the resource. If the ﬁrm is a perfect com-petitor in the commodity market, it can sell this extra output at the given market price for the commodity. However, as additional units of the vari-able resource are used together with ﬁxed resources, after a point the ex-tra output or marginal physical product (MPP) declines because of the op-eration of the law of diminishing returns. Because of the declining MPP, MRP also declines. Important! Resources are priced just as goods and services are—by the strength of the demand and supply for them—but resource demand is a derived demand. Resource Demand In order to maximize total proﬁts, a ﬁrm should hire additional units of a resource as long as each adds more to the ﬁrm’s total revenue than to its total costs. The increase in total revenue is the MRP. The increase in total cost gives the marginal resource cost (MRC). If the ﬁrm is a perfect competitor in the resource market, it can hire any quantity of the variable resource at the given resource price, so MRC equals the resource price. Thus to maximize total proﬁts, the ﬁrm should hire the resource until MRP equals the resource price. The de-clining MRP schedule then represents the ﬁrm’s demand schedule for the resource. CHAPTER 16: Demand for Economic Resources 133 If the ﬁrm is an imperfect competitor in the commodity market, the MRP declines both because the MPP declines and because the ﬁrm must lower the commodity price in order to sell more units. If the ﬁrm remains a perfect competitor in the resource market, the ﬁrm again maximizes to-tal proﬁts when it hires the resource until MRP equals the resource price. The declining MRP schedule then represents the ﬁrm’s demand schedule for the variable resource. Example 16.1 In Table 16.1, column 1 refers to units of a variable resource, say, labor, employed in a given plant. Column 2 gives the total product produced. Column 3 gives the marginal physical product or the change in total prod-uct per unit change in the use of the resource. Commodity price (column 4) declines because of imperfect competition in the commodity market. TR (column 5) is obtained by multiplying commodity price by total prod-uct. Column 6 gives the MRP, measured as the change in total revenue. MRP declines both because MPP declines and because the product price declines. A ﬁrm that is a perfect competitor in the resource market would maximize its total proﬁts by employing the resource until the MRPequals the resource price. 134 PRINCIPLES OF ECONOMICS Table 16.1 The declining MRPschedule of columns 6 and 1 in Table 16.1 is the ﬁrm’s demand schedule for the resource and is graphed as d in Figure 16-1. At the resource price of $50, the ﬁrm will hire one unit of the resource. At the resource price of $31, the ﬁrm will hire two units of the resource, and so on. Changes in Resource Demand A ﬁrm’s demand for a productive resource will increase (i.e., shift up) if: (1) the product demand increases; (2) the productivity of the resource ris-es; (3) the prices of substitute resources rise; or (4) the prices of comple-mentary resources fall. Remember A ﬁrm’s demand for a resource (say, labor) depends in large part upon cir-cumstances beyond the ﬁrm’s con-trol. For example, if the market demand for shoes rises and if the ﬁrm pro-vides each worker with better but more expensive equipment, the ﬁrm’s demand for labor will also rise. That is, to produce more shoes requires more labor; better equipment makes labor more productive so the demand for labor increases; an increase in the price of equipment encourages the substitution of labor for capital in production. If a ﬁrm uses more than one variable resource, say labor (L) and cap-ital (K), the ﬁrm will maximize total proﬁts when it uses labor and capi-tal until the marginal revenue product of each resource equals the re-CHAPTER 16: Demand for Economic Resources 135 Figure 16-1 source price (if the ﬁrm is a perfect competitor in the resource markets). That is, the ﬁrm will maximize total proﬁts when MRPL = PL or wage rate, and MRPK = PK or the rate of interest. This can be rewritten as MRPL/PL = MRPK/PK = 1 and can be generalized to any number of re-sources. If the ﬁrm is an imperfect competitor in the resource markets, the proﬁt maximization condition is generalized to MPPL = MRCL and MPPK = MRCK or MPPL/MRCL = MPPK/MRCK = 1 (where MRC refers to the marginal resource cost). True or False Questions 1. The price of a resource is determined by the demand for the re-source. 2. If the ﬁrm is a perfect competitor in the product market, its MRP curve is downward-sloping only because the marginal physical product curve of the resource is downward sloping. 3. Marginal resource cost refers to the increase in the ﬁrm’s total costs in hiring each additional unit of the resource. 4. To maximize proﬁts, a ﬁrm should hire resources as long as each additional unit of the resource adds more to the ﬁrm’s total costs than to its total revenue. 5. A ﬁrm’s demand for a resource shifts up if the productivity of the resource increases. Answers: 1. False; 2. True; 3. True; 4. False; 5. True Solved Problems Solved Problem 16.1 a. Why do ﬁrms demand resources? In what way is a ﬁrm’s demand for a resource a derived demand? How does this differ from consumers’ demand for ﬁnal commodities? b. What determines the strength of a ﬁrm’s demand for a productive resource? Solution: a. Firms demand resources in order to produce ﬁnal commodities. It is the consumers’demand for ﬁnal commodities that ultimately gives rise 136 PRINCIPLES OF ECONOMICS to the ﬁrm’s demand for productive resources. Because of this, the de-mand for a resource is referred to as a derived demand. It is derived from the demand for the ﬁnal commodities that require the resource in pro-duction. While consumers demand ﬁnal commodities because of the di-rect utility that they get from consuming commodities, producers demand resources only because the resource can be used to produce the com-modities that consumers demand. b. The strength of a ﬁrm’s demand for a resource depends on: (1) the strength of the demand for the commodity that the resource is used to pro-duce; (2) the productivity of the resource in producing the ﬁnal com-modity; and (3) the prices of other related (i.e., substitute and comple-mentary) resources. The higher the demand for the ﬁnal commodity, the more productive is the resource. The higher the price of substitute re-sources and the lower the price of complementary resources, the greater the ﬁrm’s demand for the resource. Solved Problem 16.2 From Table 16.2, a. Find the marginal physical product (MPP), total revenue, and the marginal revenue product (MRP) schedules. b. Why does the MPP decline? Why does MRP decline? How do we know this ﬁrm is a perfect competitor in the product market? CHAPTER 16: Demand for Economic Resources 137 Table 16.2 Solution: a. Column 3 in Table 16.3 gives the MPP. It is obtained from the change in total product per unit change in the use of the variable resource. Column 5 gives the total revenue of the ﬁrm. It is obtained by multiply-ing the product price (column 4) by the total product (column 2). Column 6 gives the marginal revenue product. It is obtained from the increase in the total revenue in column 5. b. The MPP that results from employing each additional unit of the variable resource (together with ﬁxed amounts of other resources) de-clines because of the law of diminishing returns. The MRP declines be-cause MPP declines. We know that this ﬁrm is a perfect competitor in the product market because product price remains constant at $1 per unit re-gardless of the quantity of the product sold by the ﬁrm. Solved Problem 16.3 Explain how much of each variable resource a ﬁrm should hire in order to maximize total proﬁts, if the ﬁrm is an im-perfect competitor in the resource markets. Solution: When an imperfect competitor in the resource markets wants to hire more of a resource, it will have to pay a higher price, not only on the additional units of the resource but also on all previous units of the resource hired. Thus, the increase in the total costs of hiring an addition-al unit of the resource or marginal resource cost (MRC) exceeds the re-source price. The ﬁrm will maximize total proﬁts when it hires variable resources as long as each resource MRP exceeds its MRC and until they are equal. With variable resources labor (L) and capital (K), the ﬁrm max-imizes total proﬁts when MRPL = MRCL and MRPK = MRCK. Another way of stating the proﬁt-maximizing condition is to say that a ﬁrm should hire resources until the MRPper dollar spent on each resource is the same. Once again, this rule can be extended to any number of variable re-sources. 138 PRINCIPLES OF ECONOMICS Table 16.3 Chapter 17 Pricing of Wages, Rent, Interest, and Proﬁts In This Chapter: ✔ Wage Determination ✔ Unions and Wage Differentials ✔ Rent ✔ Interest ✔ Proﬁts ✔ Epilogue on Commodity and Resource Pricing ✔ True or False Questions ✔ Solved Problems Wage Determination The wage rate refers to the earnings per hour of labor. The wage rate di-vided by the price index gives the real wage rate or “purchasing power” of wages. We are primarily concerned with real wages. 139 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. The level of real wages depends on the produc-tivity of labor. Real wages are higher the greater the amount of capital available per worker, the more ad-vanced the technology, and the greater the availabili-ty of natural resources (fertile land, mineral deposits, etc.). In preceding chapters, we saw that ﬁrms demand labor (and other resources) in order to produce the products demanded by consumers. By adding each ﬁrm’s demand for la-bor, we get the market demand for labor. The market supply of labor de-pends on the population size, the proportion of the population in the la-bor force, the state of the economy (such as boom or recession), and the level of real wages. The competitive equilibrium real-wage rate is determined at the in-tersection of the market demand and supply of labor curves. The ﬁrm then hires labor until the marginal revenue product of labor (MRPL) or its de-mand for labor (dL) equals the wage rate. Example 17.1 In Panel B of Figure 17-1, the competitive equilibrium real-wage rate of $6 per hour is determined at the intersection of the market demand and supply of labor. The supply of labor to the competitive ﬁrm of Panel A (sL) is horizontal at the wage rate of $6. This means that the ﬁrm is so small (say, one of 1,000 identical ﬁrms in the market) that it can hire any quantity of labor at the equilibrium market wage rate without affecting that wage rate. To maximize total proﬁts the ﬁrm hires 30 units of labor because MRPL = W = $6 at 30 units of labor. 140 PRINCIPLES OF ECONOMICS Figure 17-1 Note! Wages are determined by the intersection of the demand and supply of labor, not just by ﬁrms or just by workers. Workers may not be hired competitively. A dominant employer has monopoly power in the local labor market and is referred to as a monop-sonist. It faces the rising market supply curve of labor which indicates that it must pay higher wages to hire more workers. Thus, the change in the total cost of hiring an additional worker or marginal resource cost of labor (MRCL) exceeds the wage rate. To maximize proﬁts, the ﬁrm hires labor until MRPL = MRCL and pays the wage indicated on the supply curve of labor for that quantity of labor. Unions and Wage Differentials Labor unions attempt to increase wages in three ways. First, unions at-tempt to increase the demand for labor by increasing productivity, by ad-vertising union-made products, and by lobbying to restrict imports. These are the most desirable but also the least effective methods. Second, unions attempt to raise wages by restricting the supply of labor through the imposition of high initiation fees and long apprenticeships and requirements that employ-ers hire only union members. This is done primarily by craft unions (i.e., unions of such skilled workers as electricians). Third, unions attempt to raise wage rates directly by bargaining with employers, under the threat of a strike. This is the most common method and is used primarily by in-dustrial unions (i.e., unions of all the workers of a particular industry, such as automobile workers). Empirical studies seem to indicate that, in general, unions in the U.S. have raised real wages for their members by only about 10 to 15 percent. CHAPTER 17: Pricing of Wages, Rent, Interest, and Proﬁts 141 Example 17.2 In Panel A of Figure 17-2, the equilibrium real wage rate is $4 and em-ployment is 3000 workers (at point E, where DL intersects SL). If the union can increase DL to DL, W = $6 and employment rises to 4000. Starting from the same original equilibrium point E in Panel B, a craft union could instead attempt to reduce SL to SL so that W = $6 but only 2000 are employed. In Panel C, an industrial union could attempt to negotiate W = $6 at which 2000 workers are employed and another 2000 workers (EA) are unable to ﬁnd jobs. If all jobs and individuals were exactly alike and all markets perfectly competitive, there would be a single wage for all jobs and all workers. However, jobs requiring equal qualiﬁcations may differ in attractiveness, and higher wages must be paid to attract and retain workers in more un-pleasant jobs. Such wage differentials are known as equalizing differ-ences. Even if all jobs were equally attractive, wage differences would persist because individuals such as doctors and clerks differ widely in ca-142 PRINCIPLES OF ECONOMICS Figure 17-2 pacities, skills, training, and education. Thus, labor falls into many non-competing groups, each requiring different training and receiving differ-ent wages. Finally, some wage differences are the result of imperfect mar-kets. Market imperfections include lack of information, unwillingness to move, union power, minimum-wage laws, and monopsony power. The wide wage differences actually observed in the real world among differ-ent categories of people and jobs are in general the result of a combina-tion of these three factors. Rent Rent is the price for the use of land and other natural resources that are ﬁxed in total supply. If all land is alike and has one competitive use (say, the growing of wheat), then rent is determined at the intersection of the market demand curve and the vertical market supply curve of land. Re-gardless of the market demand and the rent paid, the same amount of land remains available. You Need to Know Pricing rent for land is different from pricing other resources in that land is relatively ﬁxed in supply. Example 17.3 With the supply of land ﬁxed (S) in Figure 17-3, rent is equal to r when the market demand curve for land is D, and r when it is D. If from the equilibrium rent of r, the government imposed a tax of rr on rental in-comes, land users would continue to pay r but landowners would retain only r. The quantity of land supplied, however, would remain unchanged. In the real world, we have different types and uses of land with different rental values. The supply of land can also be increased somewhat (say, by drainage) or reduced (by improper use). CHAPTER 17: Pricing of Wages, Rent, Interest, and Proﬁts 143 Interest Interest is the price paid for using money or loanable funds, expressed as a percentage of the amount borrowed. If the rate of interest is 8 percent per year, this means that for $100 borrowed today, $108 will have to be repaid a year from today. For simplicity, we discuss the pure rate of in-terest or the interest on a riskless loan (as on a U.S. government bond). Other interest rates are higher depending on the risk, maturity, adminis-trative cost, and competitiveness of the loanable-funds market. The equilibrium interest rate is determined at the intersection of the market demand and supply curves of loanable funds. The demand for loanable funds comes from the borrowing of ﬁrms, consumers, and gov-ernment, and is negatively sloped. To maximize proﬁts, a ﬁrm will bor-row in order to invest in machinery, inventory, etc., as long as the return, or marginal productivity, of the investment exceeds the rate of interest on borrowed funds. Thus, interest rates allocate the scarce loanable funds to the most productive uses. The supply of loanable funds stems from the past and current savings of individuals and ﬁrms. It is upward sloped and is greatly affected by monetary policy. Proﬁts Economic proﬁts are the excess of total revenue over total costs, includ-ing both explicit and implicit costs. Proﬁts stem from the introduction of a successful innovation, a reward for uninsurable risk-bearing or uncer-tainty, and monopoly power. They serve as incentives for innovation, to 144 PRINCIPLES OF ECONOMICS Figure 17-3 shift resources to the production of those commodities that society wants most, and as a reward for efﬁciency. Firms introduce new products and new production methods in the expectation of proﬁts. If successful, other ﬁrms may imitate the success-ful innovator and compete these proﬁts away. Similarly, more risky ven-tures (such as petroleum exploration) require the expectation of a higher proﬁt to induce investments. Finally, monopoly power allows a ﬁrm to restrict output artiﬁcially, keep competitors out, and charge a price that allows proﬁts to persist. Remember The search for proﬁts is what brings most new goods and services to the market. In 1990, the breakdown of U.S. national income was as follows: wages and salaries, 74 percent; proprietors’ income, 7 percent, corporate proﬁts, 8 percent; interest, 10 percent; and rents, 1 percent. Since 1990, wages and salaries have increased relatively and proprietors’ incomes have fallen relatively. This is due to the increase in the importance of cor-porations relative to individual-owned businesses. Epilogue on Commodity and Resource Pricing In a free-enterprise economy, commodity and factor prices are deter-mined by their respective demands and supplies. Firms demand resources owned by households in order to produce the goods and services de-manded by households. Households then use the income they receive to purchase the goods and services produced by ﬁrms. This circular ﬂow of economic activity determines what, how, and for whom to produce. It is a general equilibrium system because a change in any part of the econo-my affects every other part of the economy. When markets are perfectly competitive and are in long-run equilibrium, resources are allocated most efﬁciently and the economy’s output of goods and services is maximized. CHAPTER 17: Pricing of Wages, Rent, Interest, and Proﬁts 145 In the real world, however, this most efﬁcient resource allocation is dif-ﬁcult to achieve. True or False Questions 1. The competitive equilibrium real-wage rate is determined at the intersection of the market demand and supply curves of labor. 2. The wage differentials observed in the real world are generally due to market imperfections. 3. The supply curve of land and natural resources is upward-sloping. 4. The factors affecting interest rates are risk, maturity, administra-tive costs, and competitiveness of the loanable funds market. 5. In order for ﬁrms to invest in higher risk ventures the expected proﬁt must be higher. Answers: 1. True; 2. False; 3. False; 4. True; 5. True Solved Problems Solved Problem 17.1 a. On what does the market supply of labor depend? b. How does the state of the economy affect the market supply of la-bor? c. What is the effect of the real-wage rate level on the quantity of la-bor supplied in the market? Solution: a. The market supply of labor depends on the population size, the proportion of the population in the labor force, and the state of the econ-omy. In general, the larger the population and the greater the participa-tion rate of the population in the labor force, the greater the market sup-ply of labor. b. The state of the economy (boom or recession) affects the market supply of labor. When the economy is booming, many people not previ-ously employed may, attracted by the availability of high-paying jobs, de-cide to enter the labor force. On the other hand, someone who felt the need to look for a job under less prosperous conditions may leave the la-bor force when a spouse or parent gets a high-paying job in a booming 146 PRINCIPLES OF ECONOMICS economy. Thus, the supply of labor may increase, decrease, or remain un-changed depending on the net effect of these two opposing forces. The opposite is true in a recession. c. The level of real wages also gives rise to two opposing forces af-fecting the quantity of labor supplied. On the one hand, a high level of real wages induces workers to substitute work for leisure and work more hours per week to take advantage of the high real wages. On the other hand, a high real wage (and income) results in workers demanding more of every normal commodity, including leisure, and working fewer hours per week. Once again, the quantity of labor supplied may increase, de-crease, or remain unchanged, depending on the net effect of these two op-posing forces. Solved Problem 17.2 Getting an education and training is sometimes re-ferred to as an “investment in human capital.” a. In what ways is this similar to any other investment? b. Why is treating education and training as investment in human capital useful? c. What are its shortcomings? Solution: a. Getting an education and training can be considered an investment in human capital because, as with any other investment, it involves a cost and entails a return. The cost of getting an education and training involves such explicit expenses as tuition, books, etc. and such implicit cost as the forgone wages while in school or the lower wages received while in train-ing. The return on education and training takes the form of the higher wages received over the individual’s working life. By discounting all costs and extra income to the present and comparing returns to costs, we can calculate the rate of return on the investment in human capital and compare it to the returns from other investments. b. Viewing education and training as investment in human capital can explain real-world occurrences such as why we educate and train the young more than the old, why young people migrate more readily than old, etc. The answer is that young people have a longer working time over which to receive the beneﬁts of education, training, and migration. c. Some shortcomings of this line of thinking are as follows: (1) Not all expenses for education and training represent costs. Some should be regarded as consumption since they do not contribute to subsequent high-CHAPTER 17: Pricing of Wages, Rent, Interest, and Proﬁts 147 er earnings (for example, when an engineering student takes a course in poetry). (2) Higher subsequent earnings may be the result of innate abil-ity and greater intelligence and effort rather than training. Solved Problem 17.3 a. What are the functions of proﬁts? b. What are some objections to proﬁts? Solution: a. Proﬁts serve as incentives for innovators to shift resources to the production of those commodities that society wants most, and as a reward for efﬁciency. The introduction of an innovation involves uncertainty and may result in ﬁnancial loss if it is not successful. The expectation of a ﬁ-nancial reward in the form of proﬁts is required to induce innovations. Similarly, proﬁt in some industries and losses in others is the indication that society wants more commodities from the former and less from the latter. Related to this is the fact that more efﬁcient ﬁrms in a given in-dustry are rewarded with proﬁts which they can then use to expand, while less efﬁcient ﬁrms incur losses and have to contract operations or go out of business. b. Among the objections to proﬁts are the following: (1) Proﬁts aris-ing from monopoly serve no socially useful purpose (except when they lead to more innovations). Therefore, such proﬁts should be taxed away or the monopoly should be regulated (if it is not feasible to break it up). (2) Proﬁts may lead to an excessively unequal distribution of income. This, too, can be corrected by progressive taxation. However, a general attack on all proﬁts is not justiﬁed, because proﬁts, as we have seen be-fore, do perform socially useful functions. 148 PRINCIPLES OF ECONOMICS Index AC (average costs), 106–107 AFC (average ﬁxed costs), 106– 107 Aggregate demand, 26–28 Aggregate output, 28–29 Aggregate supply, 27–28, 51 AVC (average variable costs), 106–107 Average costs (AC), 106–107 Average ﬁxed costs (AFC), 106– 107 Average variable costs (AVC), 106–107 Balance of payments, 91 Board of Governors (Federal Re-serve), 68 Bretton Woods System, 93 Built-in stabilizers, 58–59 Business cycles, 29–30 Capital, 4 Capital accumulation, 82–83 CD (certiﬁcates of deposit), 66 Certiﬁcates of deposit (CD), 66 Ceteris paribus, 2 Circular ﬂow of income and out-put, 48–49 Collusion, 128 Competition, 111–114 Competitive equilibrium real-wage rate, 140–141 Consumer demand, 96–100 Consumer price index (CPI), 31– 32 Consumer surplus, 99–100 Consumption, 37–40 Cost-push inﬂation, 33 Costs, 4, 5, 104–109 CPI (consumer price index), 31– 32 Credit ﬁnancial instruments, 65 Crowding-out effect, 75–76 Cyclical unemployment, 30 Debit ﬁnancial instruments, 65 Debt, 59–60 Deﬁcits, 59–60 Demand, 13–16 aggregate, 26–28 resource, 133–136 Demand-pull inﬂation, 33 Depository institutions, 66 Discount rate, 69 Discretionary ﬁscal policy, 57–58 Disinﬂation, 32 Economic growth, 81–85 Economic resources, 132–136 Economics, 1–3 Elasticity, 19–21 Equilibrium, 17–18 long-run, 114 Equilibrium output, 47–51 149 Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. Equity ﬁnancial instruments, 65 Eurodollars, 66 Exchange rates, 91–93 Expenditures: government, 41–42, 56–57 Explicit costs, 104–105 Exports, 40–41 FDIC (Federal Deposit Insurance Corporation), 66 Fed. See Federal Reserve System Federal Deposit Insurance Corpo-ration (FDIC), 66 Federal Open Market Committee (FOMC), 68 Federal Reserve Bank, 68 Federal Reserve System, 64–71 Financial instruments, 65–66 Fiscal policy, 56–61, 74–78 FMOC (Federal Open Market Committee), 68 Full employment, 31 GDP (Gross Domestic Product): deﬂator, 32 gap, 47 nominal, 25–26 potential, 26 real, 26 Goods, 4 Government: debt, 59–60 deﬁcit, 59–60 expenditures, 41–42, 56–57 price determination, 18 taxes, 41–42, 56–57 Gross Domestic Product. See GDP Gross exports, 40 Gross imports, 40 Guns and butter, 5–6 Illiquid ﬁnancial instruments, 65 Implicit costs, 104–105 Import quotas, 90 Imports, 40 Import tariffs, 90 Income-expenditure model, 47– 49 Individual demand curve, 98– 100 Inﬂation, 31–33 Interest, 144 Interest rate effect, 27 International purchasing power effect, 27 International trade, 88–93 Investment, 40 Keynes, John Maynard, 47 Keynesian model, 47–51 Labor, 4 LAC (long-run average costs), 107–109 Land, 4, 143 Law of diminishing marginal util-ity, 96–97 Law of diminishing returns, 105, 107 Leakage-injection model, 49–50 Liquid ﬁnancial instruments, 65 Long-run average costs (LAC), 107–109 Long-run costs, 107–109 Long-run equilibrium, 114 Loss, 114 150 PRINCIPLES OF ECONOMICS M1, 66–71 M2, 66 M3, 66 Malthus, Thomas, 82 Marginal costs (MC), 106–107 Marginal propensity to consume, 39 Marginal revenue (MR), 112–113 Marginal utility, 96–97 Market equilibrium, 18 Markets, 65–66 Market system, 7–8 MC (marginal costs), 106–107 Methodology of economics, 1–3 Models: income-expenditure, 47–49 Keynesian, 47–51 leakage-injection, 49–50 Monetary policy, 64–71, 74–78 Monetary tools, 68–69 Money, 64–71 Monopolistic competition, 125– 127 Monopoly, 118–121 MR (marginal revenue), 112– 113 Multiplier, 50–51 Net exports, 40–41 Nominal GDP, 25–26 Oligopoly, 127–128 Open-market operations, 69–71 Opportunity cost, 4–5 Output aggregate, 28–29 equilibrium, 47–51 with government expenditure or taxes, 56–57 Paradox of thrift, 50 Peaks, 29 Perfect competition, 111–114 Population, 82–83 Potential GDP, 26 PPI (producer price index), 32 Price, 17–18 discrimination, 121 elasticity, 19–21 government determination, 18 level, 27 level changes, 76–78 resources, 132–133 wages, 139–143 Principle of increasing cost, 7 Producer price index (PPI), 32 Production costs, 104–109 Production-possibility frontier, 5–6 Productivity, 83 Proﬁt, 113–114, 144–145 Proﬁt maximization, 112–113, 119–120, 126–127 Quantity, 17–18 Quotas, 90 Real GDP, 26 Real wage rate, 139–140 Regulation, 121 Rent, 143–144 Repurchase agreements (RP), 66 Reserve requirement, 67, 68–69 Revenue, 112–113 RP (repurchase agreements), 66 SAC (short-run average costs), 107–109 INDEX 151 Savings, 65 Scarcity, 3–5 market system, 7–8 Services, 4 Short run, 105 Short-run average costs (SAC), 107–109 Short-run costs, 105–107 Short-run loss, 114 Short-run proﬁt, 114 Stabilizers, 58–59 Stagﬂation, 33 Supply, 16–17 aggregate, 27–28 elasticity, 20–21 Supply and demand, 27 Supply-side economics, 84–85 Tariffs, 90 Taxes, 41–42, 56–57 TC (total costs), 105–106 TFC (total ﬁxed costs), 105–106 Total costs (TC), 105–106 Total ﬁxed costs (TFC), 105–106 Total variable costs (TVC), 105– 106 Trade, 88–93 Trade deﬁcits, 41, 91 Trade restrictions, 90–91 Troughs, 29–30 TVC (total variable costs), 105– 106 Unemployment, 30–31 Unions, 141–143 United States Steel, 128 USX, 128 Utility, 96–100 Utility maximization theory, 98– 99 Wage differentials, 141–143 Wage rate, 139–140 Wealth effect, 27 152 PRINCIPLES OF ECONOMICS
1694	https://bio.libretexts.org/Courses/City_College_of_San_Francisco/Introduction_to_Genetics/11%3A_Pedigrees_and_Populations/11.06%3A_Population_Genetics	Skip to main content 11.6: Population Genetics Last updated : Sep 5, 2025 Save as PDF 11.5: Calculating Probabilities 11.E: Pedigrees and Populations (Exercises) Page ID : 27259 Ying Liu City College of San Francisco ( \newcommand{\kernel}{\mathrm{null}\,}) A population is a large group of individuals of the same species, who are capable of mating with each other. It is useful to know the frequency of particular alleles within a population, since this information can be used to calculate disease risks. Population genetics is also important in ecology and evolution, since changes in allele frequencies may be associated with migration or natural selection. Allele frequencies may also be studied at the population level The frequency of different alleles in a population can be determined from the frequency of the various phenotypes in the population. In the simplest system, with two alleles of the same locus (e.g. A,a), we use the symbol p to represent the frequency of the dominant allele within the population, and q for the frequency of the recessive allele. Because there are only two possible alleles, we can say that the frequency of p and q together represent 100% of the alleles in the population (p+q=1). We can calculate the values of p and q, in a representative sample of individuals from a population, by simply counting the alleles and dividing by the total number of alleles examined. For a given allele, homozygotes will count for twice as much as heterozygotes. For example: \| \| \| --- \| \| genotype \| number of individuals \| \| AA Aa aa \| 320 160 20 \| \| \| \| \| --- \| \| A (p) \| a (q) \| \| A (p) \| p2 \| pq \| \| a (q) \| pq \| q2 \| Hardy-Weinberg formula With the allele frequencies of a population we can use an extension of the Punnett Square, and the product rule, to calculate the expected frequency of each genotype following random matings within the entire population. This is the basis of the Hardy-Weinberg formula: Here is the frequency of homozygotes AA, is the frequency of the heterozygotes, and is the frequency of homozygotes aa. Notice that if we substitute the allele frequencies we calculated above (p=0.8, q=0.2) into the Equation , we obtain expected probabilities for each of the genotypes that exactly match our original observations: This is a demonstration of the Hardy-Weinberg Equilibrium, where both the genotype frequencies and allele frequencies in a population remain unchanged following successive matings within a population, if certain conditions are met. These conditions are listed in Table . Few natural populations actually satisfy all of these conditions. Nevertheless, large populations of many species, including humans, appear to approach Hardy-Weinberg equilibrium for many loci. In these situations, deviations of a particular gene from Hardy-Weinberg equilibirum can be an indication that one of the alleles affects the reproductive success of organism, for example through natural selection or assortative mating. Conditions for the Hardy-Weinberg equilibrium Random mating: Individuals of all genotypes mate together with equal frequency. Assortative mating, in which certain genotypes preferentially mate together, is a type of non-random mating. No natural selection: All genotypes have equal fitness. No migration: Individuals do not leave or enter the population. No mutation: The allele frequencies do not change due to mutation. Large population: Random sampling effects in mating (i.e. genetic drift) are insignificant in large populations. The Hardy-Weinberg formula can also be used to estimate allele frequencies, when only the frequency of one of the genotypic classes is known. For example, if 0.04% of the population is affected by a particular genetic condition, and all of the affected individuals have the genotype aa, then we assume that q2 = 0.0004 and we can calculate p, q, and 2pq as follows: Thus, approximately 4% of the population is expected to be heterozygous (i.e. a carrier) of this genetic condition. Note that while we recognize that the population is probably not exactly in Hardy-Weinberg equilibrium for this locus, application of the Hardy-Weinberg formula nevertheless can give a reasonable estimate of allele frequencies, in the absence of any other information. 11.5: Calculating Probabilities 11.E: Pedigrees and Populations (Exercises)
1695	https://stackoverflow.com/questions/1838401/general-formula-to-calculate-polyhedron-volume	math - General formula to calculate Polyhedron volume - Stack Overflow Join Stack Overflow By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google Sign up with GitHub OR Email Password Sign up Already have an account? Log in Skip to main content Stack Overflow 1. About 2. Products 3. 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Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more General formula to calculate Polyhedron volume Ask Question Asked 15 years, 10 months ago Modified2 years, 4 months ago Viewed 17k times This question shows research effort; it is useful and clear 7 Save this question. Show activity on this post. Given a list of vertices (v), and a list of edges connecting the vertices (e), and a list of surfaces that connect the edges (s), how to calculate the volume of the Polyhedron? math Share Share a link to this question Copy linkCC BY-SA 4.0 Improve this question Follow Follow this question to receive notifications edited Feb 17, 2021 at 6:35 GravitonGraviton asked Dec 3, 2009 at 8:12 GravitonGraviton 83.2k 149 149 gold badges 442 442 silver badges 614 614 bronze badges Add a comment\| 6 Answers 6 Sorted by: Reset to default This answer is useful 11 Save this answer. Show activity on this post. Take the polygons and break them into triangles. Consider the tetrahedron formed by each triangle and an arbitrary point (the origin). Sum the signed volumes of these tetrahedra. Notes: This will only work if you can keep a consistent CW or CCW order to the triangles as viewed from the outside. The signed volume of the tetrahedron is equal to 1/6 the determinant of the following matrix: [ x1 x2 x3 x4 ] [ y1 y2 y3 y4 ] [ z1 z2 z3 z4 ] [ 1 1 1 1 ] where the columns are the homogeneous coordinates of the verticies (x,y,z,1). It works even if the shape does not enclose the origin by subracting off that volume as well as adding it in, but that depends on having a consistent ordering. If you can't preserve the order you can still find some way to break it into tetrahedrons and sum 1/6 absolute value of the determinant of each one. Edit: I'd like to add that for triangle mesh where one vertex (say V4) of the tetrahedron is (0,0,0) the determinante of the 4x4 matrix can be simplified to the upper left 3x3 (expansion along the 0,0,0,1 column) and that can be simplified to Vol = V1xV2.V3 where "x" is cross product and "." is dot product. So compute that expression for every triangle, sum those volumes and divide by 6. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications edited Nov 29, 2022 at 14:14 StayOnTarget 13.3k 11 11 gold badges 65 65 silver badges 115 115 bronze badges answered Dec 4, 2009 at 21:28 phkahlerphkahler 5,777 1 1 gold badge 25 25 silver badges 31 31 bronze badges 3 Comments Add a comment Fady Mohamed Othman Fady Mohamed OthmanOver a year ago What do you mean by "It works even if the shape does not enclose the origin by subracting off that volume as well as adding it in, but that depends on having a consistent ordering." ?? How can this by done mathematically?? 2014-01-19T13:23:52.33Z+00:00 0 Reply Copy link phkahler phkahlerOver a year ago It sums the volume of tetrahedra formed by connecting each triangle to the origin. If the origin in outside the shape, the triangles facing away from the origin will produce a volume with sign opposite those facing the origin. This causes the volume between the origin and the shape to be included in the total, but then subtracted back out, resulting in a correct volume. 2014-07-22T14:10:06.21Z+00:00 0 Reply Copy link Jan JanOver a year ago Does this work for non convex polyhedron conceptdraw.com/a1942c3/p12/preview/640/… ? And your algorithm also works for an irregular polyhedron, since the polygons are breaked up into triangles right? 2018-11-01T09:53:08.517Z+00:00 0 Reply Copy link Add a comment This answer is useful 2 Save this answer. Show activity on this post. Similarly with a polygon where we can split it into triangles and sum the areas, you could split a polyhedron into pyramids and sum their volumes. But I'm not sure how hard is to implement an algorithm for that. (I believe there is a mathematical way/formula, like using vectors and matrices. I suggest to post your question also on Share Share a link to this answer Copy linkCC BY-SA 2.5 Improve this answer Follow Follow this answer to receive notifications edited Apr 13, 2017 at 12:57 CommunityBot 1 1 1 silver badge answered Dec 3, 2009 at 8:46 Nick DandoulakisNick Dandoulakis 43.3k 17 17 gold badges 106 106 silver badges 139 139 bronze badges Comments Add a comment This answer is useful 2 Save this answer. Show activity on this post. Actually it's not necessary to divide the polyhedron's polygons into triangles. All that is needed is to sum the volumes of the pyramids formed by each polygon with an apex at the origin using the one-third base area times height rule. The base area can be computed using the shoelace method and the height can be accounted for by forming the dot product of that (directed) area with any one vertex of the given polygon. It is apparent that the volume of each pyramid is identical to the sum of the volumes formed from a set of triangles that make up its base polygon, so it must generate the same answer. Potentially this is a much faster method if the polyhedron has faces that are intricate, perhaps reentrant, polygons when dividing them into triangles is not a simple task. After some thought I worked out there is a simple formula for the volume contribution of an n-sided polygon: formula for volume contribution of n-sided polygon in terms of the co-ordinates of its vertices xi, yi, zi Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications edited May 3, 2023 at 15:35 CommunityBot 1 1 1 silver badge answered May 2, 2023 at 19:39 JohnJohn 21 3 3 bronze badges 1 Comment Add a comment JKreft JKreftOver a year ago There's a hidden rotational directionality requirement here that's not stated. If I take the cube with side length two with one vertex at (0.5, 0.5, 0.5) and the opposite at (2.5, 2.5, 2.5), I get very different answers if my second polygon is listed in the order (.5,.5.5)->(0.5,2.5,0.5)->(0.5,2.5,2.5)->(0.5,0.5,2.5) vs swapping the second and fourth vertices. It flips the sign of the individual contribution of the face and each face is sensitive to this, so they all have to share the same rotational orientation to get the correct answer in the absolute value of the sum. 2024-08-31T20:18:58.417Z+00:00 1 Reply Copy link This answer is useful 1 Save this answer. Show activity on this post. I have done this before, but the surface mesh I used always had triangular facets. If your mesh has non triangular facets, you can easily break them up into triangular facets first. Then I fed it to TetGen to obtain a tetrahedralization of the interior. Finally, I added up all the volumes of the tetrahedra. TetGen is reasonably easy to use, and is the only library other than CGAL I know of that can handle complicated meshes. CGAL is pretty easy to use if you don't mind installing a gigantic library and use templates like crazy. Share Share a link to this answer Copy linkCC BY-SA 2.5 Improve this answer Follow Follow this answer to receive notifications answered Dec 3, 2009 at 20:16 Victor LiuVictor Liu 3,663 2 2 gold badges 27 27 silver badges 38 38 bronze badges Comments Add a comment This answer is useful 1 Save this answer. Show activity on this post. First, break every face into triangles by drawing in new edges. Now look at one triangle, and suppose it's on the "upper" surface (some of these details will turn out to be unimportant later). Look at the volume below the triangle, down to some horizontal plane below the polyhedron. If {h1, h2, h3} are the heights of the three points, and A is the area of the base, then the volume of the solid will be A(h1+h2+h3)/3. Now we have to add up the volumes of these solids for the upper faces, and subtract them for the lower faces to get the volume of the polyhedron. Play with the algebra and you'll see that the height of the polyhedron above the horizontal plane doesn't matter. The plane can be above the polyhedron, or pass through it, and the result will still be correct. So what we need is (1) a way to calculate the area of the base, and (2) a way to tell an "upper" face from a "lower" one. The first is easy if you have the Cartesian coordinates of the points, the second is easy if the points are ordered, and you can combine them and kill two birds with one stone. Suppose for each face you have a list of its corners, in counter-clockwise order. Then the projection of those points on the x-y plane will be counterclockwise for an upper face and clockwise for a lower one. If you use this method to calculate the area of the base, it will come up positive for an upper face and negative for a lower one, so you can add them all together and have the answer. So how do you get the ordered lists of corners? Start with one triangle, pick an ordering, and for each edge the neighbor that shares that edge should list those two points in the opposite order. Move from neighbor to neighbor until you have a list for every triangle. If the volume of the polyhedron comes up negative, just multiply by -1 (it means you chose the wrong ordering for that first triangle, and the polyhedron was inside-out). EDIT: I forgot the best part! If you check the algebra for adding up these volumes, you'll see that a lot of terms cancel out, especially when combining triangles back into the original faces. I haven't worked this out in detail, but it looks as if the final result could be a surprisingly simple function. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications edited Nov 1, 2018 at 11:58 Jan 2,293 1 1 gold badge 19 19 silver badges 13 13 bronze badges answered Dec 3, 2009 at 15:49 BetaBeta 99.4k 18 18 gold badges 159 159 silver badges 156 156 bronze badges Comments Add a comment This answer is useful 0 Save this answer. Show activity on this post. Here's a potential implementation for that in Python. Can anyone please check if it's correct? I believe that I am missing permutations of the points because my second test (cube) gives 0.666 and not 1. Ideas anyone? Cheers EL ``` class Simplex(object): ''' Simplex ''' def __init__(self,coordinates): ''' Constructor ''' if not len(coordinates) == 4: raise RuntimeError('You must provide only 4 coordinates!') self.coordinates = coordinates def volume(self): ''' volume: Return volume of simplex. Formula from ''' import numpy vA = numpy.array(self.coordinates) - numpy.array(self.coordinates) vB = numpy.array(self.coordinates) - numpy.array(self.coordinates) vC = numpy.array(self.coordinates) - numpy.array(self.coordinates) return numpy.abs(numpy.dot(numpy.cross(vA,vB),vC)) / 6.0 class Polyeder(object): def __init__(self,coordinates): ''' Constructor ''' if len(coordinates) < 4: raise RuntimeError('You must provide at least 4 coordinates!') self.coordinates = coordinates def volume(self): pivotCoordinate = self.coordinates volumeSum = 0 for i in xrange(1,len(self.coordinates)-3): newCoordinates = [pivotCoordinate] for j in xrange(i,i+3): newCoordinates.append(self.coordinates[j]) simplex = Simplex(newCoordinates) volumeSum += simplex.volume() return volumeSum coords = [] coords.append([0,0,0]) coords.append([1,0,0]) coords.append([0,1,0]) coords.append([0,0,1]) s = Simplex(coords) print s.volume() coords.append([0,1,1]) coords.append([1,0,1]) coords.append([1,1,0]) coords.append([1,1,1]) p = Polyeder(coords) print p.volume() ``` Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications edited Oct 28, 2013 at 19:56 answered Oct 28, 2013 at 19:38 El DudeEl Dude 5,678 11 11 gold badges 65 65 silver badges 114 114 bronze badges Comments Add a comment Your Answer Thanks for contributing an answer to Stack Overflow! 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1696	https://chem.libretexts.org/Courses/Oregon_Institute_of_Technology/OIT%3A_CHE_201_-_General_Chemistry_I_(Anthony_and_Clark)/Unit_4%3A_Quantifying_Chemicals/4.3%3A_Empirical_and_Molecular_Formulas_(Problems)	Skip to main content 4.3: Empirical and Molecular Formulas (Problems) Last updated : Sep 18, 2020 Save as PDF 4.3: Empirical and Molecular Formulas Unit 5: Transformations of Matter Page ID : 98703 ( \newcommand{\kernel}{\mathrm{null}\,}) PROBLEM 4.3.1 Determine the empirical formulas for compounds with the following percent compositions: (a) 15.8% carbon and 84.2% sulfur (b) 40.0% carbon, 6.7% hydrogen, and 53.3% oxygen Answer a : CS2 Answer b : CH2O Click here to see a video of the solution PROBLEM 4.3.2 Determine the empirical and molecular formula for chrysotile asbestos. Chrysotile has the following percent composition: 28.03% Mg, 21.60% Si, 1.16% H, and 49.21% O. The molar mass for chrysotile is 520.8 g/mol. Answer : Mg3Si2H3O8 (empirical formula), Mg6Si4H6O16 (molecular formula) PROBLEM 4.3.3 Polymers are large molecules composed of simple units repeated many times. Thus, they often have relatively simple empirical formulas. Calculate the empirical formulas of the following polymers: (a) Lucite (Plexiglas); 59.9% C, 8.06% H, 32.0% O (b) Saran; 24.8% C, 2.0% H, 73.1% Cl (c) polyethylene; 86% C, 14% H (d) polystyrene; 92.3% C, 7.7% H (e) Orlon; 67.9% C, 5.70% H, 26.4% N Answer a : C5H8O2 Answer b : CHCl Answer c : CH2 Answer d : CH Answer e : C3H3N Click here to see a video of the solution PROBLEM 4.3.4 A major textile dye manufacturer developed a new yellow dye. The dye has a percent composition of 75.95% C, 17.72% N, and 6.33% H by mass with a molar mass of about 240 g/mol. Determine the molecular formula of the dye. Answer : C15H15N3 Contributors Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at Adelaide Clark, Oregon Institute of Technology 4.3: Empirical and Molecular Formulas Unit 5: Transformations of Matter
1697	https://www.mathworksheets4kids.com/ratio.php	Child Login Main Menu Math Language Arts Science Social Studies Interactive Worksheets Browse By Grade Become a Member Become a Member Math Interactive Worksheets Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade Worksheets by Grade Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade Number Sense and Operations Number Recognition Counting Skip Counting Place Value Number Lines Addition Subtraction Multiplication Division Word Problems Comparing Numbers Ordering Numbers Odd and Even Prime and Composite Roman Numerals Ordinal Numbers Properties Patterns Rounding Estimation In and Out Boxes Number System Conversions More Number Sense Worksheets Measurement Size Comparison Time Calendar Money Measuring Length Weight Capacity Metric Unit Conversion Customary Unit Conversion Temperature More Measurement Worksheets Financial Literacy Writing Checks Profit and Loss Discount Sales Tax Simple Interest Compound Interest Statistics and Data Analysis Tally Marks Pictograph Line Plot Bar Graph Line Graph Pie Graph Scatter Plot Mean, Median, Mode, Range Mean Absolute Deviation Stem-and-leaf Plot Box-and-whisker Plot Factorial Permutation and Combination Probability Venn Diagram More Statistics Worksheets Geometry Positions Shapes - 2D Shapes - 3D Lines, Rays and Line Segments Points, Lines and Planes Angles Symmetry Transformation Area Perimeter Rectangle Triangle Circle Quadrilateral Polygon Ordered Pairs Midpoint Formula Distance Formula Slope Parallel, Perpendicular and Intersecting Lines Scale Factor Surface Area Volume Pythagorean Theorem More Geometry Worksheets Pre-Algebra Factoring GCF LCM Fractions Decimals Converting between Fractions and Decimals Integers Significant Figures Percents Convert between Fractions, Decimals, and Percents Ratio Proportions Direct and Inverse Variation Order of Operations Exponents Radicals Squaring Numbers Square Roots Scientific Notations Logarithms Speed, Distance, and Time Absolute Value More Pre-Algebra Worksheets Algebra Translating Algebraic Phrases Evaluating Algebraic Expressions Simplifying Algebraic Expressions Algebraic Identities Equations Quadratic Equations Systems of Equations Functions Polynomials Inequalities Sequence and Series Matrices Complex Numbers Vectors More Algebra Worksheets Trigonometry Calculus Math Workbooks English Language Arts Summer Review Packets Science Social Studies Holidays and Events Support Worksheets> Math> Pre-Algebra> Ratio Ratio Worksheets Immerse yourself in practice with our printable ratio worksheets. Whether it is part-to-part ratios, part-to-whole ratios, identifying parts from the whole, or finding the whole from the parts, dividing quantities, generating equivalent ratios, or expressing the ratio in three different ways, that you are looking for, these pdfs have them all covered for your grade 5 through grade 8 learners. Instantly evaluate your ratio problems with the answer keys provided. Our free ratio worksheets are a must-have to get the ball rolling! Select the Measurement Units Part-to-Part Ratio \| Level-1 Perk up with these part-to-part ratio worksheets and get children in grade 5 and grade 6 to count how many of one thing there is as compared to another and express it as a ratio. Part-to-Part Ratio \| Level-2 Fuel learning with these ratio pdfs where the two sets of objects are mixed up. Count the number of each kind and frame a ratio comparing the size of one number to the size of the other. Part-to-Whole Ratio \| Level-1 Comparing the number of butter cookies to the assorted cookies in a jar is an example of part-to-whole ratios. Measure the quantity of one item against the total and write the ratio. Part-to-Whole Ratio \| Level-2 Upscale with these printable part-to-whole ratio worksheets. Direct children to count the number of specified objects, add the three terms to find the whole and express them as a ratio. Expressing Ratio in Three Ways \| Pictures Get acquainted with the different representations of a ratio. Count the two different sets of pictures and express the ratio by using "to", the colon symbol ":", and the fraction notation. Representing Ratio in Three Ways \| Standard Doing away with pictures and presenting ratio in word format, these ratio worksheets are a sure-fire way to help learners transit easily as they read phrases and express the ratio in three ways. Drawing Shapes to Represent the Ratio Break away from the humdrum of regular printable ratio worksheets and get children to stay in the groove as they follow the instructions and sketch shapes to represent the ratio. Coloring Objects to Represent the Ratio Add a splash of color with these pdf worksheets where 5th grade, and 6th grade children are expected to color the specified number of objects as instructed and figure out the ratio. Reducing Ratios to the Lowest Terms \| Easy Reducing is nothing but converting the ratio to its simplest form. Divide the antecedent and the consequent terms of the ratio, which are numbers within 50, by the GCF to reduce the ratio. Reducing Ratios to the Lowest Terms \| Moderate Raise the bar for grade 6 and grade 7 learners with these reducing the ratios to the lowest terms worksheets where the number range gradually increases to 100. Simplifying Ratios Involving Unit Conversion \| Easy Take your simplifying ratio skills to new heights with these pdfs. Using appropriate conversion formulas, make the units uniform and reduce the ratio to the simplest terms. Simplifying Ratios Involving Unit Conversion \| Moderate Bring uniformity in the units before reducing the ratio to lowest terms as you make headway with these printable simplifying ratio worksheets involving numbers within 100. Finding the Parts from the Whole Add up the terms of the ratio. Divide the whole quantity by the sum of the parts to find the unit rate. Multiply each term with the unit rate to find the two parts. Not as complicated as it sounds! Finding the Whole from Parts The ratio of two numbers and the quantity of one part is provided. Scale up the other term by multiplying it and figure out the other quantity, add the two quantities to find the whole. Dividing Quantities into 3-Part Ratios Add the three terms to find the total number of parts the quantity should be divided into. Multiply each term in the ratio with the unit rate and share the quantity in the given ratio. Writing the Equivalent Ratios Scale up each ratio by multiplying the antecedent and the consequent terms by the same number and write an equivalent ratio and complete the table with the missing equivalent ratios. Finding Unknown Quantities from Equivalent Ratios Instruct 7th grade and 8th grade learners to simplify ratios and check for equivalence, work out the unknown quantity in an equivalent equation, and solve word problems too. Ratio Word Problems Go beyond books and practice the real-time application of ratio concepts as you solve word problems involving part-to-part ratios, part-to-whole ratios, and much more. (27 Worksheets) Proportion Worksheets Comprehend the distinction between ratio and proportion with this batch of proportion worksheets offering exercises like finding proportions, identifying them from graphs, and more! 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1698	https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Structure_and_Properties_(Tro)/18%3A_Aqueous_Ionic_Equilibrium/18.07%3A_Complex_Ion_Equilibria	Skip to main content 18.7: Complex Ion Equilibria Last updated : Jul 16, 2022 Save as PDF 18.6: Precipitation 19: Free Energy and Thermodynamics Page ID : 392360 ( \newcommand{\kernel}{\mathrm{null}\,}) Learning Objectives To be introduced to complex ions, including ligands. Previously, you learned that metal ions in aqueous solution are hydrated—that is, surrounded by a shell of usually four or six water molecules. A hydrated ion is one kind of a complex ion (or, simply, complex), a species formed between a central metal ion and one or more surrounding ligands, molecules or ions that contain at least one lone pair of electrons, such as the [Al(H2O)6]3+ ion. A complex ion forms from a metal ion and a ligand because of a Lewis acid–base interaction. The positively charged metal ion acts as a Lewis acid, and the ligand, with one or more lone pairs of electrons, acts as a Lewis base. Small, highly charged metal ions, such as Cu2+ or Ru3+, have the greatest tendency to act as Lewis acids, and consequently, they have the greatest tendency to form complex ions. As an example of the formation of complex ions, consider the addition of ammonia to an aqueous solution of the hydrated Cu2+ ion {[Cu(H2O)6]2+}. Because it is a stronger base than H2O, ammonia replaces the water molecules in the hydrated ion to form the [Cu(NH3)4(H2O)2]2+ ion. Formation of the [Cu(NH3)4(H2O)2]2+ complex is accompanied by a dramatic color change, as shown in Figure . The solution changes from the light blue of [Cu(H2O)6]2+ to the blue-violet characteristic of the [Cu(NH3)4(H2O)2]2+ ion. Figure : The Formation of Complex Ions. An aqueous solution of consists of hydrated Cu2+ ions in the form of pale blue [Cu(H2O)6]2+ (left). The addition of aqueous ammonia to the solution results in the formation of the intensely blue-violet [Cu(NH3)4(H2O)2]2+ ions, usually written as [Cu(NH3)4]2+ ion (right) because ammonia, a stronger base than H2O, replaces water molecules from the hydrated Cu2+ ion. For a more complete description, see www.youtube.com/watch?v=IQNcLH6OZK0. The Formation Constant The replacement of water molecules from [Cu(H2O)6]2+ by ammonia occurs in sequential steps. Omitting the water molecules bound to Cu2+ for simplicity, we can write the equilibrium reactions as follows: The sum of the stepwise reactions is the overall equation for the formation of the complex ion: The hydrated Cu2+ ion contains six H2O ligands, but the complex ion that is produced contains only four ligands, not six. The equilibrium constant for the formation of the complex ion from the hydrated ion is called the formation constant (Kf). The equilibrium constant expression for Kf has the same general form as any other equilibrium constant expression. In this case, the expression is as follows: The formation constant (Kf) has the same general form as any other equilibrium constant expression. Water, a pure liquid, does not appear explicitly in the equilibrium constant expression, and the hydrated Cu2+(aq) ion is represented as Cu2+ for simplicity. As for any equilibrium, the larger the value of the equilibrium constant (in this case, Kf), the more stable the product. With Kf = 2.1 × 1013, the [Cu(NH3)4(H2O)2]2+ complex ion is very stable. The formation constants for some common complex ions are listed in Table . Table : Common Complex Ions \| \| Complex Ion \| Equilibrium Equation \| Kf \| \| Reported values are overall formation constants. Source: Data from Lange’s Handbook of Chemistry, 15th ed. (1999). \| \| \| \| \| Ammonia Complexes \| [Ag(NH3)2]+ \| Ag+ + 2NH3 ⇌ [Ag(NH3)2]+ \| 1.1 × 107 \| \| [Cu(NH3)4]2+ \| Cu2+ + 4NH3 ⇌ [Cu(NH3)4]2+ \| 2.1 × 1013 \| \| [Ni(NH3)6]2+ \| Ni2+ + 6NH3 ⇌ [Ni(NH3)6]2+ \| 5.5 × 108 \| \| Cyanide Complexes \| [Ag(CN)2]− \| Ag+ + 2CN− ⇌ [Ag(CN)2]− \| 1.1 × 1018 \| \| [Ni(CN)4]2− \| Ni2+ + 4CN− ⇌ [Ni(CN)4]2− \| 2.2 × 1031 \| \| [Fe(CN)6]3− \| Fe3+ + 6CN− ⇌ [Fe(CN)6]3− \| 1 × 1042 \| \| Hydroxide Complexes \| [Zn(OH)4]2− \| Zn2+ + 4OH− ⇌ [Zn(OH)4]2− \| 4.6 × 1017 \| \| [Cr(OH)4]− \| Cr3+ + 4OH− ⇌ [Cr(OH)4]− \| 8.0 × 1029 \| \| Halide Complexes \| [HgCl4]2− \| Hg2+ + 4Cl− ⇌ [HgCl4]2− \| 1.2 × 1015 \| \| [CdI4]2− \| Cd2+ + 4I ⇌ [CdI4]2− \| 2.6 × 105 \| \| [AlF6]3− \| Al3+ + 6F− ⇌ [AlF6]3− \| 6.9 × 1019 \| \| Other Complexes \| [Ag(S2O3)2]3− \| Ag+ + 2S2O32− ⇌ [Ag(S2O3)2]3− \| 2.9 × 1013 \| \| [Fe(C2O4)3]3− \| Fe3+ + 3C2O42− ⇌ [Fe(C2O4)3]3− \| 2.0 × 1020 \| Example If 12.5 g of is added to 500 mL of 1.00 M aqueous ammonia, what is the equilibrium concentration of Cu2+(aq)? Given: mass of Cu2+ salt and volume and concentration of ammonia solution Asked for: equilibrium concentration of Cu2+(aq) Strategy: Calculate the initial concentration of Cu2+ due to the addition of copper(II) nitrate hexahydrate. Use the stoichiometry of the reaction shown in Equation to construct a table showing the initial concentrations, the changes in concentrations, and the final concentrations of all species in solution. Substitute the final concentrations into the expression for the formation constant (Equation ) to calculate the equilibrium concentration of Cu2+(aq). Solution Adding an ionic compound that contains Cu2+ to an aqueous ammonia solution will result in the formation of [Cu(NH3)4]2+(aq), as shown in Equation . We assume that the volume change caused by adding solid copper(II) nitrate to aqueous ammonia is negligible. A The initial concentration of Cu2+ from the amount of added copper nitrate prior to any reaction is as follows: Because the stoichiometry of the reaction is four NH3 to one Cu2+, the amount of NH3 required to react completely with the Cu2+ is 4(0.0846) = 0.338 M. The concentration of ammonia after complete reaction is 1.00 M − 0.338 M = 0.66 M. These results are summarized in the first two lines of the following table. Because the equilibrium constant for the reaction is large (2.1 × 1013), the equilibrium will lie far to the right. Thus we will assume that the formation of [Cu(NH3)4]2+ in the first step is complete and allow some of it to dissociate into Cu2+ and NH3 until equilibrium has been reached. If we define x as the amount of Cu2+ produced by the dissociation reaction, then the stoichiometry of the reaction tells us that the change in the concentration of [Cu(NH3)4]2+ is −x, and the change in the concentration of ammonia is +4x, as indicated in the table. The final concentrations of all species (in the bottom row of the table) are the sums of the concentrations after complete reaction and the changes in concentrations. \| \| [Cu2+] \| [NH3] \| [[Cu(NH3)4]2+] \| --- --- \| \| initial \| 0.0846 \| 1.00 \| 0 \| \| after complete reaction \| 0 \| 0.66 \| 0.0846 \| \| change \| +x \| +4x \| −x \| \| final \| x \| 0.66 + 4x \| 0.0846 − x \| B Substituting the final concentrations into the expression for the formation constant (Equation ) and assuming that x << 0.0846, which allows us to remove x from the sum and difference, The value of x indicates that our assumption was justified. The equilibrium concentration of Cu2+(aq) in a 1.00 M ammonia solution is therefore 2.1 × 10−14 M. Exercise The ferrocyanide ion {[Fe(CN)6]4−} is very stable, with a Kf of 1 × 1035. Calculate the concentration of cyanide ion in equilibrium with a 0.65 M solution of K4[Fe(CN)6]. Answer : 2 × 10−6 M The Effect of the Formation of Complex Ions on Solubility What happens to the solubility of a sparingly soluble salt if a ligand that forms a stable complex ion is added to the solution? One such example occurs in conventional black-and-white photography. Recall that black-and-white photographic film contains light-sensitive microcrystals of AgBr, or mixtures of AgBr and other silver halides. AgBr is a sparingly soluble salt, with a Ksp of 5.35 × 10−13 at 25°C. When the shutter of the camera opens, the light from the object being photographed strikes some of the crystals on the film and initiates a photochemical reaction that converts AgBr to black Ag metal. Well-formed, stable negative images appear in tones of gray, corresponding to the number of grains of AgBr converted, with the areas exposed to the most light being darkest. To fix the image and prevent more AgBr crystals from being converted to Ag metal during processing of the film, the unreacted AgBr on the film is removed using a complexation reaction to dissolve the sparingly soluble salt. The reaction for the dissolution of silver bromide is as follows: with The equilibrium lies far to the left, and the equilibrium concentrations of Ag+ and Br− ions are very low (7.31 × 10−7 M). As a result, removing unreacted AgBr from even a single roll of film using pure water would require tens of thousands of liters of water and a great deal of time. Le Chatelier’s principle tells us, however, that we can drive the reaction to the right by removing one of the products, which will cause more AgBr to dissolve. Bromide ion is difficult to remove chemically, but silver ion forms a variety of stable two-coordinate complexes with neutral ligands, such as ammonia, or with anionic ligands, such as cyanide or thiosulfate (S2O32−). In photographic processing, excess AgBr is dissolved using a concentrated solution of sodium thiosulfate. The reaction of Ag+ with thiosulfate is as follows: with The magnitude of the equilibrium constant indicates that almost all Ag+ ions in solution will be immediately complexed by thiosulfate to form [Ag(S2O3)2]3−. We can see the effect of thiosulfate on the solubility of AgBr by writing the appropriate reactions and adding them together: Comparing K with Ksp shows that the formation of the complex ion increases the solubility of AgBr by approximately 3 × 1013. The dramatic increase in solubility combined with the low cost and the low toxicity explains why sodium thiosulfate is almost universally used for developing black-and-white film. If desired, the silver can be recovered from the thiosulfate solution using any of several methods and recycled. If a complex ion has a large Kf, the formation of a complex ion can dramatically increase the solubility of sparingly soluble salts. Example Due to the common ion effect, we might expect a salt such as AgCl to be much less soluble in a concentrated solution of KCl than in water. Such an assumption would be incorrect, however, because it ignores the fact that silver ion tends to form a two-coordinate complex with chloride ions (AgCl2−). Calculate the solubility of AgCl in each situation: in pure water in 1.0 M KCl solution, ignoring the formation of any complex ions the same solution as in part (b) except taking the formation of complex ions into account, assuming that AgCl2− is the only Ag+ complex that forms in significant concentrations At 25°C, Ksp = 1.77 × 10−10 for AgCl and Kf = 1.1 × 105 for AgCl2−. Given: Ksp of AgCl, Kf of AgCl2−, and KCl concentration Asked for: solubility of AgCl in water and in KCl solution with and without the formation of complex ions Strategy: Write the solubility product expression for AgCl and calculate the concentration of Ag+ and Cl− in water. Calculate the concentration of Ag+ in the KCl solution. Write balanced chemical equations for the dissolution of AgCl and for the formation of the AgCl2− complex. Add the two equations and calculate the equilibrium constant for the overall equilibrium. Write the equilibrium constant expression for the overall reaction. Solve for the concentration of the complex ion. Solution A If we let x equal the solubility of AgCl, then at equilibrium [Ag+] = [Cl−] = x M. Substituting this value into the solubility product expression, Ksp = [Ag+][Cl−] = (x)(x) = x2 =1.77×10−10 x = 1.33×10−5 Thus the solubility of AgCl in pure water at 25°C is 1.33 × 10−5 M. B If x equals the solubility of AgCl in the KCl solution, then at equilibrium [Ag+] = x M and [Cl−] = (1.0 + x) M. Substituting these values into the solubility product expression and assuming that x << 1.0, Ksp = [Ag+][Cl−] = (x)(1.0 + x) ≈ x(1.0) = 1.77×10−10 = x If the common ion effect were the only important factor, we would predict that AgCl is approximately five orders of magnitude less soluble in a 1.0 M KCl solution than in water. C To account for the effects of the formation of complex ions, we must first write the equilibrium equations for both the dissolution and the formation of complex ions. Adding the equations corresponding to Ksp and Kf gives us an equation that describes the dissolution of AgCl in a KCl solution. The equilibrium constant for the reaction is therefore the product of Ksp and Kf: D If we let x equal the solubility of AgCl in the KCl solution, then at equilibrium [AgCl2−] = x and [Cl−] = 1.0 − x. Substituting these quantities into the equilibrium constant expression for the net reaction and assuming that x << 1.0, That is, AgCl dissolves in 1.0 M KCl to produce a 1.9 × 10−5 M solution of the AgCl2− complex ion. Thus we predict that AgCl has approximately the same solubility in a 1.0 M KCl solution as it does in pure water, which is 105 times greater than that predicted based on the common ion effect. (In fact, the measured solubility of AgCl in 1.0 M KCl is almost a factor of 10 greater than that in pure water, largely due to the formation of other chloride-containing complexes.) Exercise Calculate the solubility of mercury(II) iodide (HgI2) in each situation: pure water a 3.0 M solution of NaI, assuming [HgI4]2− is the only Hg-containing species present in significant amounts Ksp = 2.9 × 10−29 for HgI2 and Kf = 6.8 × 1029 for [HgI4]2−. Answer a : 1.9 × 10−10 M Answer a : 1.4 M Complexing agents, molecules or ions that increase the solubility of metal salts by forming soluble metal complexes, are common components of laundry detergents. Long-chain carboxylic acids, the major components of soaps, form insoluble salts with Ca2+ and Mg2+, which are present in high concentrations in “hard” water. The precipitation of these salts produces a bathtub ring and gives a gray tinge to clothing. Adding a complexing agent such as pyrophosphate (O3POPO34−, or P2O74−) or triphosphate (P3O105−) to detergents prevents the magnesium and calcium salts from precipitating because the equilibrium constant for complex-ion formation is large: with However, phosphates can cause environmental damage by promoting eutrophication, the growth of excessive amounts of algae in a body of water, which can eventually lead to large decreases in levels of dissolved oxygen that kill fish and other aquatic organisms. Consequently, many states in the United States have banned the use of phosphate-containing detergents, and France has banned their use beginning in 2007. “Phosphate-free” detergents contain different kinds of complexing agents, such as derivatives of acetic acid or other carboxylic acids. The development of phosphate substitutes is an area of intense research. Commercial water softeners also use a complexing agent to treat hard water by passing the water over ion-exchange resins, which are complex sodium salts. When water flows over the resin, sodium ion is dissolved, and insoluble salts precipitate onto the resin surface. Water treated in this way has a saltier taste due to the presence of Na+, but it contains fewer dissolved minerals. Figure : An MRI Image of the Heart, Arteries, and Veins. When a patient is injected with a paramagnetic metal cation in the form of a stable complex known as an MRI contrast agent, the magnetic properties of water in cells are altered. Because the different environments in different types of cells respond differently, a physician can obtain detailed images of soft tissues. Another application of complexing agents is found in medicine. Unlike x-rays, magnetic resonance imaging (MRI) can give relatively good images of soft tissues such as internal organs. MRI is based on the magnetic properties of the 1H nucleus of hydrogen atoms in water, which is a major component of soft tissues. Because the properties of water do not depend very much on whether it is inside a cell or in the blood, it is hard to get detailed images of these tissues that have good contrast. To solve this problem, scientists have developed a class of metal complexes known as “MRI contrast agents.” Injecting an MRI contrast agent into a patient selectively affects the magnetic properties of water in cells of normal tissues, in tumors, or in blood vessels and allows doctors to “see” each of these separately (Figure ). One of the most important metal ions for this application is Gd3+, which with seven unpaired electrons is highly paramagnetic. Because Gd3+(aq) is quite toxic, it must be administered as a very stable complex that does not dissociate in the body and can be excreted intact by the kidneys. The complexing agents used for gadolinium are ligands such as DTPA5− (diethylene triamine pentaacetic acid), whose fully protonated form is shown here. Summary The formation of complex ions can substantially increase the solubility of sparingly soluble salts if the complex ion has a large Kf. A complex ion is a species formed between a central metal ion and one or more surrounding ligands, molecules or ions that contain at least one lone pair of electrons. Small, highly charged metal ions have the greatest tendency to act as Lewis acids and form complex ions. The equilibrium constant for the formation of the complex ion is the formation constant (Kf). The formation of a complex ion by adding a complexing agent increases the solubility of a compound. 18.6: Precipitation 19: Free Energy and Thermodynamics
1699	https://data.worldbank.org/indicator/SP.POP.TOTL	THEMES DATA & RESOURCES ABOUT # Population, total World Population Prospects, United Nations ( UN ), uri: population.un.org/wpp, publisher: UN Population Division; Statistical databases and publications from national statistical offices, National Statistical Offices, uri: unstats.un.org/home/nso_sites, publisher: National Statistical Offices; Eurostat: Demographic Statistics, Eurostat ( ESTAT ), uri: ec.europa.eu/eurostat/data/database?node_code=earn_ses_monthly, publisher: Eurostat; Population and Vital Statistics Report ( various years ), United Nations ( UN ), uri: unstats.un.org, publisher: UN Statistics Division License : CC BY-4.0 LineBarMap None Population & Labor All Countries and Economies Country Most Recent Year Most Recent Value(Thousands) Help us improve this site IBRD IDA IFC MIGA ICSID Legal Privacy Notice Access to Information Jobs Contact © 2025The World Bank Group, All Rights Reserved. REPORT FRAUD OR CORRUPTION This site uses cookies to optimize functionality and give you the best possible experience. If you continue to navigate this website beyond this page, cookies will be placed on your browser. To learn more about cookies, click here.

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