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http://se7en.my/how-to-ocpptr/1744c8-s-orbital-shape
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The s-orbitals are solid spherical shape around the nucleus. They are: 1) the orbit of a planet is an ellipse, with the Sun at one of the two foci; 2) the line connecting the planet and Sun sweeps out equal areas during equal intervals of time and; 3) the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. Each orbital has a name. And a D orbital is what they call a double dumbo. They fill the lowest energy level orbitals first. From Table below we see that we can have three possible orbitals when l = 1. The order of … II) p orbital: Dumbbell Shaped, Directional in Shape could be oriented in X-Axis, Y-Axis or Z-Axis. MEDIUM. The higher the energy level, the larger the p orbital. s-orbital is spherical and p-orbital is dumb-bell shaped. So there's the three shapes of the s orbital. Atomic orbitals have distinctive shapes; all are centered on the atomic nucleus. The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The explanation of the transition from 1s to 2s and other orbital jumps is described described in the quantum leap section. Example, 2s orbital is larger than the 1s orbital. The magnetic orbital quantum number for d orbitals is given as (-2,-1,0, 1,2). Okay, the P or bill is kind of a dumb bell looking thing. 2) Orbitals are combined when bonds form between atoms in a molecule. The simplest shape is the spherical, s orbital, although there can be various orbitals of this shape in an atom due to quantum leaps of the electron. However, if you look at a cross-section of an orbital, it isn't uniform. These #sp^2# hybridized orbitals are oriented with bond angle of 120 degrees, in a trigonal planar (triangular) geometry. If {eq}\rm l=0 {/eq}, it is for s-orbital, are spherical. The ml or magnetic quantum number describes the orientation in space of the orbital. s Orbital. Energies of Orbitals. 6 - • list the number of orbitals of each type (1s,... Ch. The orbital occupied by the hydrogen electron is called a 1s orbital. The individual orbitals are labeled with the magnetic quantum number, ml, which can take the 2l… This is why the hydrogen atom has an electron configuration of 1s 1. Overview of Shape Of S Orbital The Bohr model suggested that the electrons revolve around the nucleus in fixed circular paths called orbits and … Each of these hybridized orbitals have 33% s character and 67% p character. orbital come about because atoms are spherically symmetric. When n = 2 and l = 0 , i.e 2s orbital which contains one node. • There are seven f-orbitals. Introduction to Quantum Numbers . The number "1" represents the fact that the orbital is in the energy level closest to the nucleus. If we're talking about the subshells, in the second shell, there's s and p so this is a subshell, and then this is another subshell right over here. List the four orbital shapes. The s orbital is a spherically-shaped region describing where an electron can be found, within a certain degree of probability. There are three p orbitals, and they do not differ in shape, rather, they differ in orientation as seen in the image above. The application of certain orbits or orbital maneuvers to specific useful purposes have been the … for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density ρ, where T is the orbital period. The shape of the orbital depends on the quantum numbers associated with an energy state. III) d orbital: Double Dumbbell Shaped, Direction in Shape, Could be in 5 different Orientations. The s sub shell can hold a maximum of two electrons as there is only one orbital. The shape of the orbital is decided by the value of the azimuthal quantum number. I) s orbital: Spherical Isotropic Non-Directional Shape. Development leading to Bohr's model of atom. The letters s, p, d, and f were assigned for historical reasons that need not concern us. The s orbital is a spherical shape. The shape and size of an orbital can be determined from the square of the wave function Ψ 2. The simplest shape is the spherical, s orbital, although there can be various orbitals of this shape in an atom due to quantum leaps of the electron. The explanation of the transition from 1s to 2s and other orbital jumps is described described in the quantum leap section . Answer. Kind of like this in three D space. A s-orbital has a spherical shape. Hence, the correct option is A. 6 - • rank various orbitals in terms of size and... Ch. You have a really big sphere if it's in the s if it's … These are designated as p orbitals and have dumbbell shapes. The shape of s-orbital is _____ and the shape of p-orbital is _____. The "s" tells you about the shape of the orbital. 6 - • define the term orbital. The s orbitals are spherical, while p orbitals are polar and oriented in particular directions (x, y, and z). Answered By . Below we will look at some of the common types of orbitals and discuss a few things about orbital shapes. The size of the 2s orbital is larger than that of the 1s orbital. In contrast to his concept of a simple circula$$r$$ orbit with a fixed radius, orbitals are mathematically derived regions of space with different probabilities of containing an electron.. One way of representing electron probability distributions was illustrated previously fo$$r$$ the 1s orbital of hydrogen. The energy level increases as we move away from the nucleus, therefore the orbitals get bigger. An illustration of the shape of the 1s, 2s and 3s orbitals. The orbital shapes are: s, p, d, and f. Summarize Aufbau’s rule for filling orbitals. $\begingroup$ Well, the shapes of s, p, d, etc. As the value of n increases, the size s-orbital increases. The shape of s-orbital is spherical because the boundary surface diagram of s-orbital has a spherical shape and has a nucleus at its center. P orbitals, unlike s orbitals, are not spherical but they have a lobed shape. The remaining p orbital is unchanged and … S orbitals are spherical in shape and increase in size as the energy level or shell increases. The structures of d and f-orbitals are more complex. D. spherical, spherical. s orbitals are spherically symmetric around the nucleus - in each case, like a hollow ball made of rather chunky material with the nucleus at its centre. When the value of l is zero, then the shape of the orbital is spherical which is the shape of an s orbital. Shapes of Orbitals and Electron Density Patterns . There are three p orbitals that differ in orientation along a three-dimensional axis. There are four types of orbitals that you should be familiar with s, p, … Once again, the 1s orbital. Each of the p orbitals has a different orientation in three-dimensional space. A. spherical, dumb-bell. All s orbitals have l = m = 0, but the value of n can vary. S orbital is spherically symmetrical orbital around the atomic nucleus. REVIEW Agenda: Complete Orbital Notation Group Practice All we have to do is remember the shapes that correspond to each letter. So for the first shell, the shell, the subshell, the orbital is all referring to the same thing, but as we get to the second shell, it's a little bit different. At the third level, there is a total of nine orbitals altogether. The shape of p orbital: Here, the quantum number m fixes the angular momentum direction. There are three p-orbitals, p x, p y, and p z at right angles to one another. There are five d orbitals, four of which have a clover shape with different orientations, and one that is unique. Ch. The orbital on the left is a 2s orbital. The p orbital is a dumbbell shape. • There are five d-orbitals. This is because the 2s orbital size resides farther away from the nucleus when compared to that of the 1s orbital. The letter "s" indicates the shape of the orbital: s orbitals are spherically symmetric around the nucleus— they look like hollow balls made of chunky material with the nucleus at the center. A p-orbital has a 3-dimensional dumb-bell shape. See also Kepler's Third Law. B. dumb-dell, spherical. The spherical symmetry let's you split the wavefunction/orbital into 2 parts, a radial one (which basically just determines how many nodes your orbital has) and an angular one (which determines the shape), which are then multiplied with each other. This orbital is spherical in shape: p Orbitals. Ch. In an #sp^2# hybridization, #color(red)"one"# #s# orbital is mixed with #color(red)"two"# #p# orbitals to form #color(red)"three"# #sp^2# hybridized orbitals. Shape of s Orbital. Notice that the 1s orbital has the highest probability. It may be simpler to think of these two letters in terms of orbital shapes (d and f aren't described as readily). 6 - • identify an orbital (as 1s, 3p, etc.) Electrons are lazy. from its... Ch. An orbital is the quantum mechanical refinement of Bohr’s orbit. The quantum number also fixes the direction of the orbital in the space. SHAPE OF S-ORBITALS : The probability of finding the electron belonging to s-orbital is found to be equal in all directions at a particular distance from the nucleus . 3D model to visualise the shapes of atomic orbitals. For any value of n, a value of l = 0 places that electron in an s orbital. In other words, s-orbitals are non-directional and spherically symmetrical in shape. 6 - • sketch the shapes of s and p orbitals and... Ch. So as is a sphere. There is a set of five d orbitals (with complicated shapes and names) as well as the 3s and 3p orbitals (3px, 3py, 3pz). Fig: Shapes of d-orbitals. Since an electron can theoretically occupy all space, it is impossible to draw an orbital. Patents. So n is basically the size of these, so you can have a the safe and his three. C. spherical, double dumb-bell. The most commonly encountered orbitals in elementary quantum chemistry are the orbitals corresponding to the s, p, and d subshells : these orbitals are named the s, p, and d orbitals. Shape of the Orbital. One of the causes is the alignment of same-spin protons in the atomic nucleus. s, p and d. When principal quantum number n = 1 and azimuthal quantum number l = 0, that is 1s orbital which is closest to the nucleus. S Orbital. Other articles where S-orbital is discussed: chemical bonding: Quantum numbers: …orbital, which is called an s orbital; a p subshell (l = 1) consists of three orbitals, called p orbitals; and a d subshell (l = 2) consists of five orbitals, called d orbitals. When n = 3 and l = 0, i.e 3s orbital which contains two nodes. 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2021-12-01 13:33:55
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https://www.nature.com/articles/s41436-021-01301-y?error=cookies_not_supported&code=4e28bd2e-2aaa-4ee6-9a28-3dc219deece3
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INTRODUCTION
Genomic medicine and its impact are actively evolving, offering a promise of improved and more reliable health care [1]. In the United States, the American College of Medical Genetics and Genomics (ACMG) lists 1,278 genetic clinics that provide genetic services, including specialty clinics, hospitals, and cancer centers [2]. The genetic testing utilized by these institutions can be applied to more than 15,000 conditions. Currently there are more than 34,000 clinical genetic tests provided for various purposes, including screening, diagnosis, and therapeutic management, and these numbers are growing [3,4,5,6]. Genetic test reports are the primary source for communicating test information and results from the testing lab to hospitals and clinics.
Genetic testing labs utilize computational systems to interpret results, compose final test reports, and return these to their clients. However, the returned information is sometimes of limited computational availability when presented to hospitals and clinics, particularly common for information within interpretation sections. In many cases labs send genetic test reports as PDFs [7] or scanned images, thereby limiting the primary and secondary use of valuable information for clinical decision support systems (CDSs), and clinical genetic research.
Health-care (HC) interoperability standards offer a common language for representing and communicating medical information between labs and hospitals [8]. Many US hospitals have electronic health records (EHRs) that have incorporated interoperability standards to support many clinical activities [9]. Moreover, health-care data interoperability is considered a national goal in the United States and has been evaluated in many clinical settings [10, 11]. Although genetic information is part of the national interoperability roadmap, the perspective of genetic testing labs in that setting has been underresearched.
We interviewed staff associated with US-based genetic testing labs to identify their perspectives on adopting HC interoperability standards within their laboratory information management systems (LIMS). We asked specific questions and analyzed their answers about the expected benefits, challenges, and motivations for implementing HC interoperability standards. Another part of the study examined the implemented interoperability standards, the processes of test report generation, and the communication between the labs and their clients. The results of those investigations are being reported in another publication [12]. The results of our current study may be of great value in informing decision makers, LIMS vendors, and genetic testing labs regarding the adoption of interoperability standards within related LIMS.
MATERIALS AND METHODS
Throughout this paper, we use the word “standards” to refer to HC interoperability standards, unless otherwise specified.
We employed a qualitative approach using an applied thematic analysis [13, 14] with semistructured interviews and a discussion with a panel of content experts to further explore and validate the themes. Qualitative methods have been used to study health information exchange (HIE) in various clinical settings [15, 16]. This method is pertinent to subjects with little prior study, where perspectives, experiences, and attitudes of stakeholders need to be explored [17, 18]. The following sections will describe the steps we followed in chronological order.
Review of labs and study invitation
A review of US-based genetic testing labs was done to identify the market landscape and candidates for the interviews. All US-based labs listed in the National Center for Biotechnology Information–Genetic Testing Registry (NCBI-GTR) [6] were retrieved and reviewed to identify their business descriptions, such as university-affiliated, hospital-based, or commercial. Additional labs were added to the list through an Internet search. The lab review was completed between October and December 2018.
We reviewed the content of the website listed in the corresponding NCBI-GTR entry to determine if each lab currently undertook clinical genetic testing or research testing only. The business description was characterized and contact persons were identified. The initial business descriptions were categorized as university-affiliated, hospital-based, commercial, or reference labs. The descriptions were further extended to include blood banks, registries, governmental labs, nonprofit organizations, nonuniversity research organizations, and health systems through lab review. A given lab may have more than one business description. For example, a lab may be both university-affiliated and hospital-based. Reference labs were identified according to the labs’ self-description. If two or more labs were affiliated with the same organization or considered units within a general lab, they were described individually based on the NCBI-GTR entry.
Additional labs were added by identification through Internet searches. Labs meeting at least one of the following criteria were excluded from the list of candidate participating labs:
• Research and development-oriented labs with no clinical services provided
• No available webpage for the lab
• No services or CLIA certification, according to NCBI-GTR
• No longer in operation
• The NCBI-GTR entry listed a consortium, registry, clinical trial, or research project
• Those focused on paternity testing or direct-to-consumer genetic tests
• Duplicate entry for the same lab
The business descriptions, compiled list, and exclusion criteria were discussed and reviewed periodically by the study team.
Interviews
To optimize use of the interviews to focus on exploration of themes, a preinterview survey was developed to capture discrete information of interviewees and their labs. The preinterview survey collected information about the interviewee, the description of the lab business, the information systems implemented for authoring and communicating genetic test reports, and the standards used, if any. The preinterview survey was administered through the University of Utah REDCap platform [19, 20]. Branching logic was used to avoid asking irrelevant questions based on the response to previous questions.
The semistructured interview questions included customized sections to confirm and elaborate on the preinterview responses and the process of report authoring and subsequent communication of the reports to hospitals and clinics. The interviewee was also asked to identify and rank the top three benefits, challenges, motivations, and lessons learned concerning standards implementation. The interview format encouraged discussion, eliciting additional information, and developing and exploring emergent themes of relevance to the subject. The structured interview script (Supplemental Material) provides the syntax used to ask each question, e.g., “Please mention and rank the top three benefits that may be realized by your lab from using biomedical informatics interoperability standards.”
Study invitations were sent to candidate participants from the identified labs’ representatives or through the website direct-messaging option, if no contact person was available on the website. If the contact person agreed to participate, an individualized invitation was sent requesting completion of the preinterview survey using REDCap [19, 20]. All of the interviews were conducted by video conference or telephone. The preinterview survey and the generic semistructured interview script are provided in the Supplemental material.
Thematic analysis
Coding
We followed a theoretical thematic analysis approach, where the study questions guided the coding process [13, 14]. The interviewer transcribed and de-identified all of the interviews and imported them into ATLAS.ti version 8 Windows and Mac versions (Berlin, Germany) [21]. Two coders followed an open-coding technique of the interview, and only segments relevant to the research questions were coded. The two initial coders coded each interview independently and then met face to face to discuss and come to consensus on all flagged content and associated codes. Audio recordings were consulted when needed, and S.M.H. followed the progress to ensure the validity and consistency of codes and adjudicated conflicts. Codes were iteratively developed while coding parts of the same interview and while coding different interviews. The iterative process included removing, adding, merging, and renaming the codes. Notes were taken using ATLAS.ti [21], and descriptions were added to non-self-explanatory codes or to differentiate similar codes to ensure consistency in coding of different interviews. The coders defined a coding scheme (Supplemental Material) and strategy to ensure consistency in coding and for future use.
The initial semistructured interview questions covered the categories of benefits, challenges, motivations, and lessons learned. Interviewees were also encouraged to expound on their answers. Accordingly, some codes were identified in noncorresponding questions. For example, some codes relevant to challenges were identified and labeled within the answer to the question regarding benefits.
Some “benefits” and “motivations” codes were confusing, where the expected “benefit” from implementing standards may also be considered the “motivation” for implementing these standards. For example, “regulatory requirements” may be considered a motivation to implement standards, but at the same time, meeting these requirements may be considered a direct benefit by some stakeholders, i.e., they consider “meeting regulatory requirements” as a benefit by itself. To clarify this, we followed a transparent coding scheme where “benefits” were defined as positive direct results of implementing standards to patients, health-care workers, researchers, and information system specialists, such as “reduce ambiguity and errors.” In contrast, motivations were defined as factors that encourage the implementation, such as “providing financial incentives." If “benefits” codes can be considered the “pulling factors” to implement standards, then “motivation” codes are the “pushing” ones.
Identifying themes and supporting literature
After the interview coding was completed, the final codes were reviewed, duplicates were merged, and more descriptions were added as needed to explicitly define each code. A total of 294 codes were identified. Codes were then grouped into themes through an iterative process of reviewing and modifying. Themes were further categorized and ranked based on the related study question. We identified themes from the interview scripts in their entirety, not just based on the answers for the specific questions about benefits, challenges, and other categories. However, the ranking was based on the number of interviewees who mentioned a corresponding theme within their answers to the corresponding specific questions (e.g., benefits question), i.e., theme frequency. If two or more themes have the same frequency, the ranking will be based on how the interviewees ranked these themes in relation to other themes of the same category (i.e., benefits, challenges, motivations, and lessons learned).
A targeted literature review was conducted to associate the identified themes with previously reported benefits, challenges, and motivations of interoperability standards and HIE in general, which do not necessarily focus on the genetic test reporting.
Panel discussion
The study team reviewed the results, followed by presentation with a panel discussion to ensure their validity. The panel consisted of seven subject matter experts from four states (California, Ohio, Pennsylvania, and Utah) with a cumulative experience of 130+ years. The panelists’ expertise included clinical genetics, genetic testing, genetic counseling, genomic medicine, clinical informatics, laboratory information systems, Biomedical Informatics (BMI), interoperability standards, and HIE. Panelist perspectives included academic, laboratory, and delivery system viewpoints.
RESULTS
Laboratory identification and review
Three hundred two US-based genetic testing labs were identified for potential participation. Two hundred fifty-eight labs were retrieved from NCBI-GTR, while the remainder were found through online searching. Two hundred seven labs were CLIA-certified labs. Table 1 provides a list of lab categories along with frequencies.
Participating labs
Application of the exclusion criteria eliminated 92 labs from further participation, leaving 210 eligible labs. Invitations were extended to 188 of the 210 labs and 8 invitees opted out of the study.
Thirteen of the 180 labs completed the preinterview survey, and 10 labs participated in the remote interviews. One of the interviewees was only available for 30 minutes, which covered the first part of the interview but was not sufficient to cover questions related to standards, benefits, implementation challenges, and motivations. Nine labs conducted the full interview (5% response rate). The interviewed labs were affiliated with companies, universities, and research organizations that provide either general testing services or specialized genetic tests. The labs are located in Alabama, California, Massachusetts, Nevada, Ohio, Utah, Vermont, and Washington. Most of the labs have information systems that adopt one or more of the following standards [12]:
• Logical Observation Identifiers Names and Codes (LOINC)
• Systematized Nomenclature of Medicine–Clinical Terms (SNOMED-CT)
• International Classification of Diseases, Ninth, and Tenth Clinical Modification (ICD-9 and ICD-10-CM)
• Current Procedural Terminology (CPT)
• Health Level Seven Version 2.x (HL7 V2.x) and HL7 V3
• HL7 Fast Healthcare Interoperability Resources (FHIR)
• Human Phenotype Ontology (HPO).
• RxNorm
The other part of our study sheds light on these standards and their usage, information system models, and other system characteristics of the participating labs [12]. In particular, that part of our study found that of the ten interviewed labs: one had no current implementation of standards but was aiming to implement some of these standards in the future, FHIR was being implemented by another one of the interviewed labs to support some under-development medical applications, and the remaining labs were implementing other standards, such as LOINC, and SNOMED-CT to describe the performed tests and diagnoses (but not specific genetic information such as identified single-nucleotide variants [SNVs]). In the majority of cases, the genetic lab test reports were being delivered to hospitals as scanned images, or PDF files [12].
Excluding the interview that was terminated prematurely, the interview lengths ranged between 38 and 82 minutes with a mean duration of 56 minutes. The interviewed labs were a mixture of university-affiliated labs, hospital-based labs, reference labs, and commercial labs as described in Table 1. The interviewees’ roles include one lab president, three directors, two professors, two PhD scientists, and one senior bioinformatics scientist. Supplemental material contains detailed information about the interviewees’ roles, backgrounds, experiences, and their tenure at their current labs.
Identified themes
Of 295 codes, we identified 24 themes within 5 domains, categorized as follows: expected benefits (6 themes), challenges (5 themes), motivations (5 themes), future directions (4 themes), and lessons learned (4 themes)—all relative to adopting HC interoperability standards by the genetic testing labs. Tables 25 describe the identified themes. Supplemental material includes detailed themes tables that provide illustrative quotations and reference relevant literature.
Some interviewees clearly stated that increased data availability and accessibility is one of the main expected benefits from implementing interoperability standards. For example, one interviewee said, “we obviously want it to go back to our EHR electronically and seamlessly." However, this would be very hard due to the complexity of these standards, as stated by another interviewee, “It does take some manual work and some expertise in order to apply a standard like LOINC. With HL7, I would say, again, it requires building interfaces, requires fairly specialized expertise and so you have staff that are very experienced with HL7 interfaces. So, it’s not plug-and-play, it requires a lot of work.” In addition, another interviewee commented that even when two organizations are implementing the same standard, equivalent methods/output may not be present, “they can both be implementing the same standard but they are doing it very, very differently.” (Please see Tables 2 and 3 for additional themes and Tables A and B in the Supplemental material for additional quotations).
DISCUSSION
We identified the key benefits, challenges, and motivations for implementation of interoperability standards as perceived by representatives of genetic testing labs. We found repeatedly mentioned factors by interviewees that may be slowing the adoption of interoperability standards by genetic testing labs, including lack of motivation (i.e., a lack of practical demand by their customers—the hospitals and clinics), high cost with a lack of financial incentives (e.g., the Health Information Technology for Economic and Clinical Health (HITECH) Act [22]—Meaningful Use), and a lack of regulatory and legal requirements to implement a specific set of standards for genetic test reporting.
Among all of the motivations, interviewees reported that increased clinical demand might be the most crucial for pushing forward the adoption of standards by genetic testing labs. Some interviewees clearly highlighted this point, e.g., one stated, “so that is why we tend to be very cautious about implementing them, because we want to wait to see that our customers really need it first, and not just because the informatics community tells us that they are a good thing.” The four main stakeholders, i.e., the labs, LIMS vendors, SDOs, and regulators share the goal of improving patient care. More clinical pilot projects, similar to Sync for Genes [23], may need to be conducted to clearly demonstrate the value of standards in health care and guide clinical genetic data interoperability. Interviewees reported that financial incentives for the use of explicit standards tied to improved patient outcomes could also encourage labs to provide their data in standard-based formats. The HITECH Act has been reported to have some success in improving general clinical interoperability over time [22, 24], and a similar approach focused on genetic data could possibly help.
Using standards to represent and transfer the content of genetic lab test reports may be more straightforward than in some other domains, e.g., anatomic pathology, physical exam, or clinical visit notes, because computational tools are heavily used in the analysis and interpretation of genetic results. However, it may be more challenging with regard to the tailored report and genetic results having unclear interpretation, e.g., variants of unknown significance or rare variants lacking substantial evidence of effect, as the interpretation section would by necessity require communication of the uncertainty associated with the result. In addition, genetic testing may range from single variant detection to exome or genome sequencing. Thus, clinical information systems need to be able to receive and process data of different nature and volume to avoid development and operational challenges [25].
Some of those interviewed mentioned that clinicians prefer a tailored report over standard formats based on reusable templates. This is due to the need for a customized and individualized report reflecting the unique characteristics of the patient’s case. For many genetic tests, such as variant detection for carrier screening, pharmacogenomics, or familial variant confirmation, templated reports may be adequate. An important point to consider while working on standardizing genetic test reports is their volume, complexity, and how the included information is intended to be used, e.g., to be read by clinicians, or to be computationally available for informatics tools (e.g., CDS systems). Ideally the information could be provided both ways so that it would be both clinician-friendly and computable.
Although the consistent sharing and use of genetic information is part of the HealthIT.gov milestone “A learning health system enabled by nationwide interoperability,” targeted for 2021–2024 [10], it is essential to consider more details about the priority of data to be standardized and which standards are to be used. From a business point of view, the global genetic testing market was $13.1 billion in 2019, with the market share in North America being 58% [26]. This global market value is projected to reach$29 billion by 2026 [26]. Therefore, it is expected that current technical and financial investments in the exchange of clinical genetic data will pave the way for better health care as soon as it is proven to be beneficial in health-care settings [11].
This study used a rigorous qualitative method to investigate an important and underresearched area of clinical genetics interoperability. The participating labs were located across the United States and represented a range of business models and specialties. The interviewees and panelists had extensive experience and diverse backgrounds that enabled them to analyze and respond to the research questions critically. This study was limited by a low participation rate despite many individual invitations, reminders, and the use of personal outreach to ensure the greatest possible participation of labs. Another limitation of this study is that the sample is not a random sample. However, the results may be informative even though they may not be fully generalizable. The reasons for low participation may have included time constraints and concerns about disclosing what respondents perceived as proprietary information. While we are confident we identified the major themes, we cannot be certain that thematic saturation was achieved, potentially resulting in a less rich interpretation of the data. It is possible that the participating labs were more enthusiastic about interoperability standards than other organizations. Nevertheless, these study results may help guide stakeholders to increase the adoption of interoperability standards for genetic testing across the United States and worldwide. Our future research plans include the confirmation and quantification of the current themes and stratification according to labs’ specialties and business models.
In conclusion, this study identified expected benefits, challenges, and motivations of implementing interoperability standards in the setting of the genetic laboratory. Interviewees frequently reported that increased motivation through clinical demand is critical to accelerate adoption. As hospitals, clinics, and other end users realize the benefits of improved health-care services, robust research, and greater accuracy, they will be more motivated to increase their demand resulting in more rapid adoption. Interviewees also reported that initiating an incentive program, with reasonable technical specifications and proper regulation, may also foster the adoption of BMI interoperability standards by genetic testing labs.
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2023-01-26 21:47:15
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https://www.physicsforums.com/threads/numerical-method-solutions-please-help.118147/
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1. Apr 20, 2006
### Lucky mkhonza
I have been given the following problem as assignment:
Find a numerical solution for the 1-D heat conduction (using the Explicit Method):
$$\left\{\begin{array}U_{xx} = U_{t},\\ U(x,0) = \sin \pi x, \\ U(0,t) = U(1,t) = 0$$
Use h = 1, k = 0.005125 and M = 200.
Can anyone help by giving me a hint of this problem.
Thank you in advance....
Last edited: Apr 20, 2006
2. Apr 28, 2006
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2017-02-27 16:21:04
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http://openstudy.com/updates/5565f50ce4b05d462cd4069e
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## A community for students. Sign up today
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## <3pandas18 one year ago Simplify -(3×y^4)^-4 help????
• This Question is Closed
1. triciaal
|dw:1432749573130:dw|
2. <3pandas18
i meant -3XY like theres an x not a times srry
3. anonymous
by using law of indices $-(3^{-4}x ^{-4}y ^{-16})=-\frac{ 1 }{ 81x ^{4}y ^{16} }$
4. <3pandas18
Is the -1
5. anonymous
yes
6. <3pandas18
okokthanks so much!!!!
7. triciaal
the - goes for the whole entity a negative divided by a positive is negative and a positive divided by a negative is negative
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2016-10-25 15:53:29
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http://en.wikipedia.org/wiki/Equation_of_state_(cosmology)
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# Equation of state (cosmology)
In cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number $\! w$, equal to the ratio of its pressure $\! p$ to its energy density $\! \rho$:
$\! w=p/\rho$.
It is closely related to the thermodynamic equation of state and ideal gas law.
## The equation
The perfect gas equation of state may be written as
$\! p = \rho_m RT = \rho_m C^2$
where $\! \rho_m$ is the mass density, $\! R$ is the particular gas constant, $\! T$ is the temperature and $\! C=\sqrt{RT}$ is a characteristic thermal speed of the molecules. Thus
$w = \frac{p}{\rho} = \frac{\rho_mC^2}{\rho_mc^2} = \frac{C^2}{c^2}\approx 0$
where $\! \rho = \rho_mc^2$ and $\! C< for a "cold" gas, $\! c$ = speed of light.
### FLRW equations and the equation of state
The equation of state may be used in Friedmann–Lemaître–Robertson–Walker equations to describe the evolution of an isotropic universe filled with a perfect fluid. If $\! a$ is the scale factor then
$\rho\propto a^{-3(1+w)}.$
If the fluid is the dominant form of matter in a flat universe, then
$a\propto t^{\frac{2}{3(1+w)}},$
where $\! t$ is the proper time.
In general the Friedmann acceleration equation is
$3\frac{\ddot{a}}{a} = \Lambda - 4 \pi G (\rho + 3p)$
where $\! \Lambda$ is the cosmological constant and $\! G$ is Newton's constant, and $\ddot{a}$ is the second proper time derivative of the scale factor.
If we define (what might be called "effective") energy density and pressure as
$\rho^\prime \equiv \rho + \frac{\Lambda}{8 \pi G}$
$p^\prime \equiv p - \frac{\Lambda}{8 \pi G}$
and
$p^\prime = w^\prime\rho^\prime$
the acceleration equation may be written as
$\frac{\ddot a}{a}=-\frac{4}{3}\pi G\left(\rho^\prime + 3p^\prime\right) = -\frac{4}{3}\pi G(1+3w^\prime)\rho^\prime$
### Non-relativistic matter
The equation of state of ordinary non-relativistic matter (e.g. cold dust) is $\! w=0$, which means that it is diluted as $\rho\propto a^{-3}=V^{-1}$, where $\! V$ is the volume. This means that the energy density red-shifts as the volume, which is natural for ordinary non-relativistic matter.
### Ultra-relativistic matter
The equation of state of ultra-relativistic matter (e.g. radiation, but also matter in the very early universe) is $\! w=1/3$ which means that it is diluted as $\rho\propto a^{-4}$. In an expanding universe, the energy density decreases more quickly than the volume expansion, because radiation has momentum and, by the de Broglie hypothesis a wavelength, which is red-shifted.
### Acceleration of cosmic inflation
Cosmic inflation and the accelerated expansion of the Universe can be characterized by the equation of state of dark energy. In the simplest case, the equation of state of the cosmological constant is $\! w=-1$. In this case, the above expression for the scale factor is not valid and $a\propto e^{Ht}$, where the constant H is the Hubble parameter. More generally, the expansion of the Universe is accelerating for any equation of state $\! w<-1/3$. The accelerated expansion of the Universe was indeed observed.[1] According to observations, the value of equation of state of cosmological constant is near -1.
Hypothetical phantom energy would have an equation of state $\! w<-1$, and would cause a Big Rip. Using the existing data, it is still impossible to distinguish between phantom $\! w<-1$ and non-phantom $\! w\ge-1$.
### Fluids
In an expanding universe, fluids with larger equations of state disappear more quickly than those with smaller equations of state. This is the origin of the flatness and monopole problems of the big bang: curvature has $\! w=-1/3$ and monopoles have $\! w=0$, so if they were around at the time of the early big bang, they should still be visible today. These problems are solved by cosmic inflation which has $\! w\approx -1$. Measuring the equation of state of dark energy is one of the largest efforts of observational cosmology. By accurately measuring $\! w$, it is hoped that the cosmological constant could be distinguished from quintessence which has $\! w\ne -1$.
### Scalar modeling
A scalar field $\! \phi$ can be viewed as a sort of perfect fluid with equation of state
${w=\frac{\frac{1}{2}\dot{\phi}^2-V(\phi)}{\frac{1}{2}\dot{\phi}^2+V(\phi)},}$
where $\! \dot{\phi}$ is the time-derivative of $\! \phi$ and $\! V(\phi)$ is the potential energy. A free $\! (V=0)$ scalar field has $\! w=1$, and one with vanishing kinetic energy is equivalent to a cosmological constant: $\! w=-1$. Any equation of state in between, but not crossing the $\! w=-1$ barrier known as the Phantom Divide Line (PDL),[2] is achievable, which makes scalar fields useful models for many phenomena in cosmology.
## Notes
1. ^ Hogan, Jenny. "Welcome to the Dark Side." Nature 448.7151 (2007): 240-245. http://www.nature.com/nature/journal/v448/n7151/full/448240a.html
2. ^ A. Vikman,Can dark energy evolve to the phantom?, Phys. Rev. D 71, 023515 (2005), http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=bb+astro-ph%2F0407107&FORMAT=WWW&SEQUENCE=
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2015-03-06 09:58:24
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https://brilliant.org/discussions/thread/okay-bro/
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×
# Left cosets of N in G partition G
Let $$N \leq G$$.
Proposition. The left cosets of $$N$$ in $$G$$ partition $$G$$.
Proof: First we show their union is precisely $$G$$. Since $$N \leq G$$, $$1 \in N$$ so $$g \in gN$$. Hence
$\bigcup_{g \in G} gN = G.$
Next we show that distinct cosets have empty intersection. We argue by contraposition. Assume $$uN \cap vN \neq \emptyset$$. There exist $$m,n \in N$$ such that $$um = vn$$ . So $$m = u^{-1}vn \in N$$, and by closure we have $$u^{-1}v \in N$$ also.
Thus, for all $$x \in N$$, $$u^{-1}vx \in N$$. This implies that $$vx \in uN$$, so $$vN \subseteq uN$$.
A similar argument implies $$uN \subseteq vN$$. Therefore $$uN = vN$$.
Note by Jake Lee
1 year, 3 months ago
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# up as a code block.
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print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$
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2018-01-24 02:06:38
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https://chem.libretexts.org/Courses/City_College_of_San_Francisco/Chemistry_101B/02%3A_Fundamental_Equilibrium_Concepts/2.6%3A_Fundamental_Equilibrium_Concepts_(Exercises)
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2.6: Fundamental Equilibrium Concepts (Exercises)
13.1: Chemical Equilibria Exercises
Q13.1.1
What does it mean to describe a reaction as “reversible”?
S13.1.1
The reaction can proceed in both the forward and reverse directions.
Q13.1.2
When writing an equation, how is a reversible reaction distinguished from a nonreversible reaction?
Q13.1.3
If a reaction is reversible, when can it be said to have reached equilibrium?
S13.1.3
When a system has reached equilibrium, no further changes in the reactant and product concentrations occur; the reactions continue to occur, but at equivalent rates.
Q13.1.4
Is a system at equilibrium if the rate constants of the forward and reverse reactions are equal?
Q13.1.5
If the concentrations of products and reactants are equal, is the system at equilibrium?
S13.1.5
The concept of equilibrium does not imply equal concentrations, though it is possible.
13.2: Equilibrium Constant Exercises
Q13.2.1
Explain why there may be an infinite number of values for the reaction quotient of a reaction at a given temperature but there can be only one value for the equilibrium constant at that temperature.
Q13.2.2
Explain why an equilibrium between Br2(l) and Br2(g) would not be established if the container were not a closed vessel shown below:
S13.2.2
Equilibrium cannot be established between the liquid and the gas phase if the top is removed from the bottle because the system is not closed; one of the components of the equilibrium, the Br2 vapor, would escape from the bottle until all liquid disappeared. Thus, more liquid would evaporate than can condense back from the gas phase to the liquid phase.
Q13.2.3
If you observe the following reaction at equilibrium, is it possible to tell whether the reaction started with pure NO2 or with pure N2O4?
$\ce{2NO2}(g) \rightleftharpoons \ce{N2O4}(g)$
Q13.2.4
Among the solubility rules previously discussed is the statement: All chlorides are soluble except Hg2Cl2, AgCl, PbCl2, and CuCl.
Q13.2.5
1. (a) Write the expression for the equilibrium constant for the reaction represented by the equation $$\ce{AgCl}(s) \rightleftharpoons \ce{Ag+}(aq)+\ce{Cl-}(aq)$$. Is Kc > 1, < 1, or ≈ 1? Explain your answer.
2. (b) Write the expression for the equilibrium constant for the reaction represented by the equation $$\ce{Pb^2+}(aq)+\ce{2Cl-}(aq) \rightleftharpoons \ce{PbCl2}(s)$$. Is Kc > 1, < 1, or ≈ 1? Explain your answer.
S13.2.5
(a) Kc = [Ag+][Cl] < 1. AgCl is insoluble; thus, the concentrations of ions are much less than 1 M; (b) $$K_c=\ce{\dfrac{1}{[Pb^2+][Cl- ]^2}}$$ > 1 because PbCl2 is insoluble and formation of the solid will reduce the concentration of ions to a low level (<1 M).
Q13.2.6
Among the solubility rules previously discussed is the statement: Carbonates, phosphates, borates, and arsenates—except those of the ammonium ion and the alkali metals—are insoluble.
1. Write the expression for the equilibrium constant for the reaction represented by the equation $$\ce{CaCO3}(s) \rightleftharpoons \ce{Ca^2+}(aq)+\ce{CO3-}(aq)$$. Is Kc > 1, < 1, or ≈ 1? Explain your answer.
2. Write the expression for the equilibrium constant for the reaction represented by the equation $$\ce{3Ba^2+}(aq)+\ce{2PO4^3-}(aq) \rightleftharpoons \ce{Ba3(PO4)2}(s)$$. Is Kc > 1, < 1, or ≈ 1? Explain your answer.
Q13.2.7
Benzene is one of the compounds used as octane enhancers in unleaded gasoline. It is manufactured by the catalytic conversion of acetylene to benzene: $$\ce{3C2H2}(g)⟶\ce{C6H6}(g)$$. Which value of Kc would make this reaction most useful commercially? Kc ≈ 0.01, Kc ≈ 1, or Kc ≈ 10. Explain your answer.
S13.2.7
Since $$K_c=\ce{\dfrac{[C6H6]}{[C2H2]^3}}$$, a value of Kc ≈ 10 means that C6H6 predominates over C2H2. In such a case, the reaction would be commercially feasible if the rate to equilibrium is suitable.
Q13.2.8
Show that the complete chemical equation, the total ionic equation, and the net ionic equation for the reaction represented by the equation $$\ce{KI}(aq)+\ce{I2}(aq) \rightleftharpoons \ce{KI3}(aq)$$ give the same expression for the reaction quotient. KI3 is composed of the ions K+ and I3.
Q13.2.9
For a titration to be effective, the reaction must be rapid and the yield of the reaction must essentially be 100%. Is Kc > 1, < 1, or ≈ 1 for a titration reaction?
Kc > 1
Q13.2.10
For a precipitation reaction to be useful in a gravimetric analysis, the product of the reaction must be insoluble. Is Kc > 1, < 1, or ≈ 1 for a useful precipitation reaction?
Q13.2.11
Write the mathematical expression for the reaction quotient, Qc, for each of the following reactions:
1. $$\ce{CH4}(g)+\ce{Cl2}(g) \rightleftharpoons \ce{CH3Cl}(g)+\ce{HCl}(g)$$
2. $$\ce{N2}(g)+\ce{O2}(g) \rightleftharpoons \ce{2NO}(g)$$
3. $$\ce{2SO2}(g)+\ce{O2}(g) \rightleftharpoons \ce{2SO3}(g)$$
4. $$\ce{BaSO3}(s) \rightleftharpoons \ce{BaO}(s)+\ce{SO2}(g)$$
5. $$\ce{P4}(g)+\ce{5O2}(g) \rightleftharpoons \ce{P4O10}(s)$$
6. $$\ce{Br2}(g) \rightleftharpoons \ce{2Br}(g)$$
7. $$\ce{CH4}(g)+\ce{2O2}(g) \rightleftharpoons \ce{CO2}(g)+\ce{2H2O}(l)$$
8. $$\ce{CuSO4⋅5H2O}(s) \rightleftharpoons \ce{CuSO4}(s)+\ce{5H2O}(g)$$
S13.2.11
(a) $$Q_c=\ce{\dfrac{[CH3Cl][HCl]}{[CH4][Cl2]}}$$; (b) $$Q_c=\ce{\dfrac{[NO]^2}{[N2][O2]}}$$; (c) $$Q_c=\ce{\dfrac{[SO3]^2}{[SO2]^2[O2]}}$$; (d) $$Q_c$$ = [SO2]; (e) $$Q_c=\ce{\dfrac{1}{[P4][O2]^5}}$$; (f) $$Q_c=\ce{\dfrac{[Br]^2}{[Br2]}}$$; (g) $$Q_c=\ce{\dfrac{[CO2]}{[CH4][O2]^2}}$$; (h) $$Q_c$$ = [H2O]5
Q13.2.12
Write the mathematical expression for the reaction quotient, Qc, for each of the following reactions:
1. $$\ce{N2}(g)+\ce{3H2}(g) \rightleftharpoons \ce{2NH3}(g)$$
2. $$\ce{4NH3}(g)+\ce{5O2}(g) \rightleftharpoons \ce{4NO}(g)+\ce{6H2O}(g)$$
3. $$\ce{N2O4}(g) \rightleftharpoons \ce{2NO2}(g)$$
4. $$\ce{CO2}(g)+\ce{H2}(g) \rightleftharpoons \ce{CO}(g)+\ce{H2O}(g)$$
5. $$\ce{NH4Cl}(s) \rightleftharpoons \ce{NH3}(g)+\ce{HCl}(g)$$
6. $$\ce{2Pb(NO3)2}(s) \rightleftharpoons \ce{2PbO}(s)+\ce{4NO2}(g)+\ce{O2}(g)$$
7. $$\ce{2H2}(g)+\ce{O2}(g) \rightleftharpoons \ce{2H2O}(l)$$
8. $$\ce{S8}(g) \rightleftharpoons \ce{8S}(g)$$
S13.2.12
The initial concentrations or pressures of reactants and products are given for each of the following systems. Calculate the reaction quotient and determine the direction in which each system will proceed to reach equilibrium.
1. $$\ce{2NH3}(g) \rightleftharpoons \ce{N2}(g)+\ce{3H2}(g) \hspace{20px} K_c=17$$; [NH3] = 0.20 M, [N2] = 1.00 M, [H2] = 1.00 M
2. $$\ce{2NH3}(g) \rightleftharpoons \ce{N2}(g)+\ce{3H2}(g) \hspace{20px} K_P=6.8×10^4$$; initial pressures: NH3 = 3.0 atm, N2 = 2.0 atm, H2 = 1.0 atm
3. $$\ce{2SO3}(g) \rightleftharpoons \ce{2SO2}(g)+\ce{O2}(g) \hspace{20px} K_c=0.230$$; [SO3] = 0.00 M, [SO2] = 1.00 M, [O2] = 1.00 M
4. $$\ce{2SO3}(g) \rightleftharpoons \ce{2SO2}(g)+\ce{O2}(g) \hspace{20px} K_P=16.5$$; initial pressures: SO3 = 1.00 atm, SO2 = 1.00 atm, O2 = 1.00 atm
5. $$\ce{2NO}(g)+\ce{Cl2}(g) \rightleftharpoons \ce{2NOCl}(g) \hspace{20px} K_c=4.6×10^4$$; [NO] = 1.00 M, [Cl2] = 1.00 M, [NOCl] = 0 M
6. $$\ce{N2}(g)+\ce{O2}(g) \rightleftharpoons \ce{2NO}(g) \hspace{20px} K_P=0.050$$; initial pressures: NO = 10.0 atm, N2 = O2 = 5 atm
S13.2.13
(a) $$Q_c$$ 25 proceeds left; (b) QP 0.22 proceeds right; (c) $$Q_c$$ undefined proceeds left; (d) QP 1.00 proceeds right; (e) QP 0 proceeds right; (f) $$Q_c$$ 4 proceeds left
Q13.2.14
The initial concentrations or pressures of reactants and products are given for each of the following systems. Calculate the reaction quotient and determine the direction in which each system will proceed to reach equilibrium.
1. $$\ce{2NH3}(g) \rightleftharpoons \ce{N2}(g)+\ce{3H2}(g) \hspace{20px} K_c=17$$; [NH3] = 0.50 M, [N2] = 0.15 M, [H2] = 0.12 M
2. $$\ce{2NH3}(g) \rightleftharpoons \ce{N2}(g)+\ce{3H2}(g) \hspace{20px} K_P=6.8×10^4$$; initial pressures: NH3 = 2.00 atm, N2 = 10.00 atm, H2 = 10.00 atm
3. $$\ce{2SO3}(g) \rightleftharpoons \ce{2SO2}(g)+\ce{O2}(g) \hspace{20px} K_c=0.230$$; [SO3] = 2.00 M, [SO2] = 2.00 M, [O2] = 2.00 M
4. $$\ce{2SO3}(g) \rightleftharpoons \ce{2SO2}(g)+\ce{O2}(g) \hspace{20px} K_P=\mathrm{6.5\:atm}$$; initial pressures: SO2 = 1.00 atm, O2 = 1.130 atm, SO3 = 0 atm
5. $$\ce{2NO}(g)+\ce{Cl2}(g) \rightleftharpoons \ce{2NOCl}(g) \hspace{20px} K_P=2.5×10^3$$; initial pressures: NO = 1.00 atm, Cl2 = 1.00 atm, NOCl = 0 atm
6. $$\ce{N2}(g)+\ce{O2}(g) \rightleftharpoons \ce{2NO}(g) \hspace{20px} K_c=0.050$$; [N2] = 0.100 M, [O2] = 0.200 M, [NO] = 1.00 M
Q13.2.15
The following reaction has KP = 4.50 × 10−5 at 720 K.
$\ce{N2}(g)+\ce{3H2}(g) \rightleftharpoons \ce{2NH3}(g)$
If a reaction vessel is filled with each gas to the partial pressures listed, in which direction will it shift to reach equilibrium? P(NH3) = 93 atm, P(N2) = 48 atm, and P(H2) = 52
S13.2.15
The system will shift toward the reactants to reach equilibrium.
Q13.2.16
Determine if the following system is at equilibrium. If not, in which direction will the system need to shift to reach equilibrium?
$$\ce{SO2Cl2}(g) \rightleftharpoons \ce{SO2}(g)+\ce{Cl2}(g)$$
[SO2Cl2] = 0.12 M, [Cl2] = 0.16 M and [SO2] = 0.050 M. Kc for the reaction is 0.078.
Q13.2.17
Which of the systems described in Exercise give homogeneous equilibria? Which give heterogeneous equilibria?
S13.2.17
(a) homogenous; (b) homogenous; (c) homogenous; (d) heterogeneous; (e) heterogeneous; (f) homogenous; (g) heterogeneous; (h) heterogeneous
Q13.2.18
Which of the systems described in Exercise give homogeneous equilibria? Which give heterogeneous equilibria?
Q13.2.19
For which of the reactions in Exercise does Kc (calculated using concentrations) equal KP (calculated using pressures)?
S13.2.19
This situation occurs in (a) and (b).
Q13.2.19
For which of the reactions in Exercise does Kc (calculated using concentrations) equal KP (calculated using pressures)?
Q13.2.20
Convert the values of Kc to values of KP or the values of KP to values of Kc.
1. $$\ce{N2}(g)+\ce{3H2}(g) \rightleftharpoons \ce{2NH3}(g) \hspace{20px} K_c=\textrm{0.50 at 400°C}$$
2. $$\ce{H2 + I2 \rightleftharpoons 2HI} \hspace{20px} K_c=\textrm{50.2 at 448°C}$$
3. $$\ce{Na2SO4⋅10H2O}(s) \rightleftharpoons \ce{Na2SO4}(s)+\ce{10H2O}(g) \hspace{20px} K_P=4.08×10^{−25}\textrm{ at 25°C}$$
4. $$\ce{H2O}(l) \rightleftharpoons \ce{H2O}(g) \hspace{20px} K_P=\textrm{0.122 at 50°C}$$
S13.2.20
(a) KP = 1.6 × 10−4; (b) KP = 50.2; (c) Kc = 5.31 × 10−39; (d) Kc = 4.60 × 10−3
Q13.2.21
Convert the values of Kc to values of KP or the values of KP to values of Kc.
1. $$\ce{Cl2}(g)+\ce{Br2}(g) \rightleftharpoons \ce{2BrCl}(g) \hspace{20px} K_c=4.7×10^{−2}\textrm{ at 25°C}$$
2. $$\ce{2SO2}(g)+\ce{O2}(g) \rightleftharpoons \ce{2SO3}(g) \hspace{20px} K_P=\textrm{48.2 at 500°C}$$
3. $$\ce{CaCl2⋅6H2O}(s) \rightleftharpoons \ce{CaCl2}(s)+\ce{6H2O}(g) \hspace{20px} K_P=5.09×10^{−44}\textrm{ at 25°C}$$
4. $$\ce{H2O}(l) \rightleftharpoons \ce{H2O}(g) \hspace{20px} K_P=\textrm{0.196 at 60°C}$$
Q13.2.22
What is the value of the equilibrium constant expression for the change $$\ce{H2O}(l) \rightleftharpoons \ce{H2O}(g)$$ at 30 °C?
S13.2.22
$K_P=P_{\ce{H2O}}=0.042.$
Q13.2.23
Write the expression of the reaction quotient for the ionization of HOCN in water.
Q13.2.24
Write the reaction quotient expression for the ionization of NH3 in water.
S13.2.24
$Q_c=\ce{\dfrac{[NH4+][OH- ]}{[HN3]}}$
Q13.2.25
What is the approximate value of the equilibrium constant KP for the change $$\ce{C2H5OC2H5}(l) \rightleftharpoons \ce{C2H5OC2H5}(g)$$ at 25 °C. (Vapor pressure was described in the previous chapter on liquids and solids; refer back to this chapter to find the relevant information needed to solve this problem.)
13.3: Shifting Equilbria Exercises
Q13.3.1
The following equation represents a reversible decomposition:
$$\ce{CaCO3}(s)\rightleftharpoons\ce{CaO}(s)+\ce{CO2}(g)$$
Under what conditions will decomposition in a closed container proceed to completion so that no CaCO3 remains?
S13.3.1
The amount of CaCO3 must be so small that $$P_{\ce{CO2}}$$ is less than KP when the CaCO3 has completely decomposed. In other words, the starting amount of CaCO3 cannot completely generate the full $$P_{\ce{CO2}}$$ required for equilibrium.
Q13.3.2
Explain how to recognize the conditions under which changes in pressure would affect systems at equilibrium.
Q13.3.3
What property of a reaction can we use to predict the effect of a change in temperature on the value of an equilibrium constant?
S13.3.3
The change in enthalpy may be used. If the reaction is exothermic, the heat produced can be thought of as a product. If the reaction is endothermic the heat added can be thought of as a reactant. Additional heat would shift an exothermic reaction back to the reactants but would shift an endothermic reaction to the products. Cooling an exothermic reaction causes the reaction to shift toward the product side; cooling an endothermic reaction would cause it to shift to the reactants' side.
Q13.3.4
What would happen to the color of the solution in part (b) of Figure if a small amount of NaOH were added and Fe(OH)3 precipitated? Explain your answer.
Q13.3.5
The following reaction occurs when a burner on a gas stove is lit:
$$\ce{CH4}(g)+\ce{2O2}(g)\rightleftharpoons\ce{CO2}(g)+\ce{2H2O}(g)$$
Is an equilibrium among CH4, O2, CO2, and H2O established under these conditions? Explain your answer.
S13.3.5
No, it is not at equilibrium. Because the system is not confined, products continuously escape from the region of the flame; reactants are also added continuously from the burner and surrounding atmosphere.
Q13.3.6
A necessary step in the manufacture of sulfuric acid is the formation of sulfur trioxide, SO3, from sulfur dioxide, SO2, and oxygen, O2, shown here. At high temperatures, the rate of formation of $$\ce{SO3 }$$is higher, but the equilibrium amount (concentration or partial pressure) of SO3 is lower than it would be at lower temperatures.
$\ce{2SO2}(g)+\ce{O2}(g)⟶\ce{2SO3}(g)$
1. (a) Does the equilibrium constant for the reaction increase, decrease, or remain about the same as the temperature increases?
2. (b) Is the reaction endothermic or exothermic?
Q13.3.7a
Suggest four ways in which the concentration of hydrazine, N2H4, could be increased in an equilibrium described by the following equation:
$\ce{N2}(g)+\ce{2H2}(g)\rightleftharpoons\ce{N2H4}(g) \hspace{20px} ΔH=\ce{95\:kJ}$
Q13.3.7b
Suggest four ways in which the concentration of PH3 could be increased in an equilibrium described by the following equation:
$\ce{P4}(g)+\ce{6H2}(g)\rightleftharpoons\ce{4PH3}(g) \hspace{20px} ΔH=\mathrm{110.5\:kJ}$
Q13.3.8
How will an increase in temperature affect each of the following equilibria? How will a decrease in the volume of the reaction vessel affect each?
1. $$\ce{2NH3}(g)\rightleftharpoons\ce{N2}(g)+\ce{3H2}(g) \hspace{20px} ΔH=\mathrm{92\:kJ}$$
2. $$\ce{N2}(g)+\ce{O2}(g)\rightleftharpoons\ce{2NO}(g) \hspace{20px} ΔH=\mathrm{181\:kJ}$$
3. $$\ce{2O3}(g)\rightleftharpoons\ce{3O2}(g) \hspace{20px} ΔH=\mathrm{−285\:kJ}$$
4. $$\ce{CaO}(s)+\ce{CO2}(g)\rightleftharpoons\ce{CaCO3}(s) \hspace{20px} ΔH=\mathrm{-176\:kJ}$$
S13.3.8
(a) ΔT increase = shift right, ΔP increase = shift left; (b) ΔT increase = shift right, ΔP increase = no effect; (c) ΔT increase = shift left, ΔP increase = shift left; (d) ΔT increase = shift left, ΔP increase = shift right.
Q13.3.9
How will an increase in temperature affect each of the following equilibria? How will a decrease in the volume of the reaction vessel affect each?
1. $$\ce{2H2O}(g)\rightleftharpoons\ce{2H2}(g)+\ce{O2}(g) \hspace{20px} ΔH=\ce{484\:kJ}$$
2. $$\ce{N2}(g)+\ce{3H2}(g)\rightleftharpoons\ce{2NH3}(g) \hspace{20px} ΔH=\mathrm{-92.2\:kJ}$$
3. $$\ce{2Br}(g)\rightleftharpoons\ce{Br2}(g) \hspace{20px} ΔH=\mathrm{-224\:kJ}$$
4. $$\ce{H2}(g)+\ce{I2}(s)\rightleftharpoons\ce{2HI}(g) \hspace{20px} ΔH=\ce{53\:kJ}$$
Q13.3.10
Water gas is a 1:1 mixture of carbon monoxide and hydrogen gas and is called water gas because it is formed from steam and hot carbon in the following reaction:
$\ce{H2O}(g)+\ce{C}(s)\rightleftharpoons\ce{H2}(g)+\ce{CO}(g).$
Methanol, a liquid fuel that could possibly replace gasoline, can be prepared from water gas and hydrogen at high temperature and pressure in the presence of a suitable catalyst.
1. Write the expression for the equilibrium constant ($$K_c$$) for the reversible reaction $\ce{2H2}(g)+\ce{CO}(g)\rightleftharpoons\ce{CH3OH}(g) \hspace{20px} ΔH=\mathrm{-90.2\:kJ}$
2. What will happen to the concentrations of $$\ce{H2}$$, $$\ce{CO}$$, and $$\ce{CH3OH}$$ at equilibrium if more H2 is added?
3. What will happen to the concentrations of H$$\ce{H2}$$, $$\ce{CO}$$, and $$\ce{CH3OH}$$ at equilibrium if CO is removed?
4. What will happen to the concentrations of $$\ce{H2}$$, $$\ce{CO}$$, and $$\ce{CH3OH}$$ at equilibrium if CH3OH is added?
5. What will happen to the concentrations of H$$\ce{H2}$$, $$\ce{CO}$$, and $$\ce{CH3OH}$$ at equilibrium if the temperature of the system is increased?
6. What will happen to the concentrations of $$\ce{H2}$$, $$\ce{CO}$$, and $$\ce{CH3OH}$$ at equilibrium if more catalyst is added?
S13.3.10
1. $$K_c=\ce{\dfrac{[CH3OH]}{[H2]^2[CO]}}$$;
2. [H2] increases, [CO] decreases, [CH3OH] increases;
3. [H2] increases, [CO] decreases, [CH3OH] decreases;
4. [H2] increases, [CO] increases, [CH3OH] increases;
5. [H2] increases, [CO] increases, [CH3OH] decreases;
6. no changes.
Q13.3.11
Nitrogen and oxygen react at high temperatures.
1. Write the expression for the equilibrium constant (Kc) for the reversible reaction $\ce{N2}(g)+\ce{O2}(g)\rightleftharpoons\ce{2NO}(g)\hspace{20px}ΔH=\ce{181\:kJ}$
2. What will happen to the concentrations of N2, O2, and NO at equilibrium if more O2 is added?
3. What will happen to the concentrations of N2, O2, and NO at equilibrium if N2 is removed?
4. What will happen to the concentrations of N2, O2, and NO at equilibrium if NO is added?
5. What will happen to the concentrations of N2, O2, and NO at equilibrium if the pressure on the system is increased by reducing the volume of the reaction vessel?
6. What will happen to the concentrations of N2, O2, and NO at equilibrium if the temperature of the system is increased?
7. What will happen to the concentrations of N2, O2, and NO at equilibrium if a catalyst is added?
Q13.3.12
Water gas, a mixture of H2 and CO, is an important industrial fuel produced by the reaction of steam with red hot coke, essentially pure carbon.
1. Write the expression for the equilibrium constant for the reversible reaction $\ce{C}(s)+\ce{H2O}(g)\rightleftharpoons\ce{CO}(g)+\ce{H2}(g)\hspace{20px}ΔH=\mathrm{131.30\:kJ}$
2. What will happen to the concentration of each reactant and product at equilibrium if more C is added?
3. What will happen to the concentration of each reactant and product at equilibrium if H2O is removed?
4. What will happen to the concentration of each reactant and product at equilibrium if CO is added?
5. What will happen to the concentration of each reactant and product at equilibrium if the temperature of the system is increased?
S13.3.12
(a) $$K_c=\ce{\dfrac{[CO][H2]}{[H2O]}}$$; (b) [H2O] no change, [CO] no change, [H2] no change; (c) [H2O] decreases, [CO] decreases, [H2] decreases; (d) [H2O] increases, [CO] increases, [H2] decreases; (f) [H2O] decreases, [CO] increases, [H2] increases. In (b), (c), (d), and (e), the mass of carbon will change, but its concentration (activity) will not change.
Q13.3.13
Pure iron metal can be produced by the reduction of iron(III) oxide with hydrogen gas.
1. Write the expression for the equilibrium constant (Kc) for the reversible reaction $\ce{Fe2O3}(s)+\ce{3H2}(g)\rightleftharpoons\ce{2Fe}(s)+\ce{3H2O}(g) \hspace{20px} ΔH=\mathrm{98.7\:kJ}$
2. What will happen to the concentration of each reactant and product at equilibrium if more Fe is added?
3. What will happen to the concentration of each reactant and product at equilibrium if H2O is removed?
4. What will happen to the concentration of each reactant and product at equilibrium if H2 is added?
5. What will happen to the concentration of each reactant and product at equilibrium if the pressure on the system is increased by reducing the volume of the reaction vessel?
6. What will happen to the concentration of each reactant and product at equilibrium if the temperature of the system is increased?
Q13.3.14
Ammonia is a weak base that reacts with water according to this equation:
$\ce{NH3}(aq)+\ce{H2O}(l)\rightleftharpoons\ce{NH4+}(aq)+\ce{OH-}(aq)$
Will any of the following increase the percent of ammonia that is converted to the ammonium ion in water and why?
Only (b)
Q13.3.15
Acetic acid is a weak acid that reacts with water according to this equation:
$\ce{CH3CO2H}(aq)+\ce{H2O}(aq)\rightleftharpoons\ce{H3O+}(aq)+\ce{CH3CO2-}(aq)$
Will any of the following increase the percent of acetic acid that reacts and produces $$\ce{CH3CO2-}$$ ion?
Q13.3.16
Suggest two ways in which the equilibrium concentration of Ag+ can be reduced in a solution of Na+, Cl, Ag+, and $$\ce{NO3-}$$, in contact with solid AgCl.
$$\ce{Na+}(aq)+\ce{Cl-}(aq)+\ce{Ag+}(aq)+\ce{NO3-}(aq)\rightleftharpoons\ce{AgCl}(s)+\ce{Na+}(aq)+\ce{NO3-}(aq)$$
$$ΔH=\mathrm{−65.9\:kJ}$$
S13.3.16
Add NaCl or some other salt that produces Cl− to the solution. Cooling the solution forces the equilibrium to the right, precipitating more AgCl(s).
Q13.3.17
How can the pressure of water vapor be increased in the following equilibrium?
$\ce{H2O}(l)\rightleftharpoons\ce{H2O}(g) \hspace{20px} ΔH=\ce{41\:kJ}$
Q13.3.18
Additional solid silver sulfate, a slightly soluble solid, is added to a solution of silver ion and sulfate ion at equilibrium with solid silver sulfate.
$\ce{2Ag+}(aq)+\ce{SO4^2-}(aq)\rightleftharpoons\ce{Ag2SO4}(s)$
Which of the following will occur?
1. Ag+ or $$\ce{SO4^2-}$$ concentrations will not change.
2. The added silver sulfate will dissolve.
3. Additional silver sulfate will form and precipitate from solution as Ag+ ions and $$\ce{SO4^2-}$$ ions combine.
4. The Ag+ ion concentration will increase and the $$\ce{SO4^2-}$$ ion concentration will decrease.
(a)
Q13.3.19
The amino acid alanine has two isomers, α-alanine and β-alanine. When equal masses of these two compounds are dissolved in equal amounts of a solvent, the solution of α-alanine freezes at the lowest temperature. Which form, α-alanine or β-alanine, has the larger equilibrium constant for ionization $$\ce{(HX \rightleftharpoons H+ + X- )}$$?
13.4: Equilibrium Calculations Exercises
Q13.4.1
A reaction is represented by this equation: $$\ce{A}(aq)+\ce{2B}(aq)⇌\ce{2C}(aq) \hspace{20px} K_c=1×10^3$$
1. Write the mathematical expression for the equilibrium constant.
2. Using concentrations ≤1 M, make up two sets of concentrations that describe a mixture of A, B, and C at equilibrium.
S13.4.1
$$K_c=\ce{\dfrac{[C]^2}{[A][B]^2}}$$. [A] = 0.1 M, [B] = 0.1 M, [C] = 1 M; and [A] = 0.01, [B] = 0.250, [C] = 0.791.
Q13.4.2
A reaction is represented by this equation: $$\ce{2W}(aq)⇌\ce{X}(aq)+\ce{2Y}(aq) \hspace{20px} K_c=5×10^{−4}$$
1. Write the mathematical expression for the equilibrium constant.
2. Using concentrations of ≤1 M, make up two sets of concentrations that describe a mixture of W, X, and Y at equilibrium.
Q13.4.3
What is the value of the equilibrium constant at 500 °C for the formation of NH3 according to the following equation?
$\ce{N2}(g)+\ce{3H2}(g)⇌\ce{2NH3}(g)$
An equilibrium mixture of NH3(g), H2(g), and N2(g) at 500 °C was found to contain 1.35 M H2, 1.15 M N2, and 4.12 × 10−1 M NH3.
Kc = 6.00 × 10−2
Q13.4.4
Hydrogen is prepared commercially by the reaction of methane and water vapor at elevated temperatures.
$\ce{CH4}(g)+\ce{H2O}(g)⇌\ce{3H2}(g)+\ce{CO}(g)$
What is the equilibrium constant for the reaction if a mixture at equilibrium contains gases with the following concentrations: CH4, 0.126 M; H2O, 0.242 M; CO, 0.126 M; H2 1.15 M, at a temperature of 760 °C?
A 0.72-mol sample of PCl5 is put into a 1.00-L vessel and heated. At equilibrium, the vessel contains 0.40 mol of PCl3(g) and 0.40 mol of Cl2(g). Calculate the value of the equilibrium constant for the decomposition of PCl5 to PCl3 and Cl2 at this temperature.
Kc = 0.50
Q13.4.5
At 1 atm and 25 °C, NO2 with an initial concentration of 1.00 M is 3.3 × 10−3% decomposed into NO and O2. Calculate the value of the equilibrium constant for the reaction.
$\ce{2NO2}(g)⇌\ce{2NO}(g)+\ce{O2}(g)$
Q13.4.6
Calculate the value of the equilibrium constant KP for the reaction $$\ce{2NO}(g)+\ce{Cl2}(g)⇌\ce{2NOCl}(g)$$ from these equilibrium pressures: NO, 0.050 atm; Cl2, 0.30 atm; NOCl, 1.2 atm.
S13.4.6
The equilibrium equation is KP = 1.9 × 103
Q13.4.7
When heated, iodine vapor dissociates according to this equation:
$\ce{I2}(g)⇌\ce{2I}(g)$
At 1274 K, a sample exhibits a partial pressure of I2 of 0.1122 atm and a partial pressure due to I atoms of 0.1378 atm. Determine the value of the equilibrium constant, KP, for the decomposition at 1274 K.
Q13.4.8
A sample of ammonium chloride was heated in a closed container.
$\ce{NH4Cl}(s)⇌\ce{NH3}(g)+\ce{HCl}(g)$
At equilibrium, the pressure of NH3(g) was found to be 1.75 atm. What is the value of the equilibrium constant KP for the decomposition at this temperature?
KP = 3.06
Q13.4.9
At a temperature of 60 °C, the vapor pressure of water is 0.196 atm. What is the value of the equilibrium constant KP for the transformation at 60 °C?
$\ce{H2O}(l)⇌\ce{H2O}(g)$
Q13.4.10
Complete the changes in concentrations (or pressure, if requested) for each of the following reactions.
(a)
\begin{alignat}{3} &\ce{2SO3}(g)⇌\:&&\ce{2SO2}(g)+\:&&\ce{O2}(g)\\ &\underline{\hspace{40px}} &&\underline{\hspace{40px}} &&+x\\ &\underline{\hspace{40px}} &&\underline{\hspace{40px}} &&0.125\:M \end{alignat}
(b)
\begin{alignat}{3} &\ce{4NH3}(g)+\:&&\ce{3O2}(g)⇌\:&&\ce{2N2}(g)+\:&&\ce{6H2O}(g)\\ &\underline{\hspace{40px}} &&3x &&\underline{\hspace{40px}} &&\underline{\hspace{40px}}\\ &\underline{\hspace{40px}} &&0.24\:M &&\underline{\hspace{40px}} &&\underline{\hspace{40px}} \end{alignat}
(c) Change in pressure:
\begin{alignat}{3} &\ce{2CH4}(g)⇌\:&&\ce{C2H2}(g)+\:&&\ce{3H2}(g)\\ &\underline{\hspace{40px}} &&x &&\underline{\hspace{40px}}\\ &\underline{\hspace{40px}} &&\textrm{25 torr} &&\underline{\hspace{40px}} \end{alignat}
(d) Change in pressure:
\begin{alignat}{3} &\ce{CH4}(g)+\:&&\ce{H2O}(g)⇌\:&&\ce{CO}(g)+\:&&\ce{3H2}(g)\\ &\underline{\hspace{40px}} &&x &&\underline{\hspace{40px}} &&\underline{\hspace{40px}}\\ &\underline{\hspace{40px}} &&\textrm{5 atm} &&\underline{\hspace{40px}} &&\underline{\hspace{40px}} \end{alignat}
(e)
\begin{alignat}{3} &\ce{NH4Cl}(s)⇌\:&&\ce{NH3}(g)+\:&&\ce{HCl}(g)\\ & &&x &&\underline{\hspace{40px}}\\ & &&1.03×10^{−4}\:M &&\underline{\hspace{40px}} \end{alignat}
(f) change in pressure:
\begin{alignat}{3} &\ce{Ni}(s)+\:&&\ce{4CO}(g)⇌\:&&\ce{Ni(CO)4}(g)\\ & &&4x &&\underline{\hspace{40px}}\\ & &&\textrm{0.40 atm} &&\underline{\hspace{40px}} \end{alignat}
S13.4.10
1. −2x, 2x, −0.250 M, 0.250 M;
2. 4x, −2x, −6x, 0.32 M, −0.16 M, −0.48 M;
3. −2x, 3x, −50 torr, 75 torr;
4. x, − x, −3x, 5 atm, −5 atm, −15 atm;
5. x, 1.03 × 10−4 M; (f) x, 0.1 atm.
Q13.4.11
Complete the changes in concentrations (or pressure, if requested) for each of the following reactions.
(a)
\begin{alignat}{3} &\ce{2H2}(g)+\:&&\ce{O2}(g)⇌\:&&\ce{2H2O}(g)\\ &\underline{\hspace{40px}} &&\underline{\hspace{40px}} &&+2x\\ &\underline{\hspace{40px}} &&\underline{\hspace{40px}} &&1.50\:M \end{alignat}
(b)
\begin{alignat}{3} &\ce{CS2}(g)+\:&&\ce{4H2}(g)⇌\:&&\ce{CH4}(g)+\:&&\ce{2H2S}(g)\\ &x &&\underline{\hspace{40px}} &&\underline{\hspace{40px}} &&\underline{\hspace{40px}}\\ &0.020\:M &&\underline{\hspace{40px}} &&\underline{\hspace{40px}} &&\underline{\hspace{40px}} \end{alignat}
(c) Change in pressure:
\begin{alignat}{3} &\ce{H2}(g)+\:&&\ce{Cl2}(g)⇌\:&&\ce{2HCl}(g)\\ &x &&\underline{\hspace{40px}} &&\underline{\hspace{40px}}\\ &\textrm{1.50 atm} &&\underline{\hspace{40px}} &&\underline{\hspace{40px}} \end{alignat}
(d) Change in pressure:
\begin{alignat}{3} &\ce{2NH3}(g)+\:&&\ce{2O2}(g)⇌\:&&\ce{N2O}(g)+\:&&\ce{3H2O}(g)\\ &\underline{\hspace{40px}} &&\underline{\hspace{40px}} &&\underline{\hspace{40px}} &&x\\ &\underline{\hspace{40px}} &&\underline{\hspace{40px}} &&\underline{\hspace{40px}} &&\textrm{60.6 torr} \end{alignat}
(e)
\begin{alignat}{3} &\ce{NH4HS}(s)⇌\:&&\ce{NH3}(g)+\:&&\ce{H2S}(g)\\ & &&x &&\underline{\hspace{40px}}\\ & &&9.8×10^{−6}\:M &&\underline{\hspace{40px}} \end{alignat}
(f) Change in pressure:
\begin{alignat}{3} &\ce{Fe}(s)+\:&&\ce{5CO}(g)⇌\:&&\ce{Fe(CO)4}(g)\\ & &&\underline{\hspace{40px}} &&x\\ & &&\underline{\hspace{40px}} &&\textrm{0.012 atm} \end{alignat}
Q13.4.12
Why are there no changes specified for Ni in Exercise, part (f)? What property of Ni does change?
S13.4.12
Activities of pure crystalline solids equal 1 and are constant; however, the mass of Ni does change.
Q13.4.13
Why are there no changes specified for NH4HS in Exercise, part (e)? What property of NH4HS does change?
Q13.4.14
Analysis of the gases in a sealed reaction vessel containing NH3, N2, and H2 at equilibrium at 400 °C established the concentration of N2 to be 1.2 M and the concentration of H2 to be 0.24 M.
$\ce{N2}(g)+\ce{3H2}(g)⇌\ce{2NH3}(g) \hspace{20px} K_c=\textrm{0.50 at 400 °C}$
Calculate the equilibrium molar concentration of NH3.
S13.4.14
[NH3] = 9.1 × 10−2 M
Q13.4.16
Calculate the number of moles of HI that are at equilibrium with 1.25 mol of H2 and 1.25 mol of I2 in a 5.00−L flask at 448 °C.
$$\ce{H2 + I2 ⇌ 2HI} \hspace{20px} K_c=\textrm{50.2 at 448 °C}$$
Q13.4.17
What is the pressure of BrCl in an equilibrium mixture of Cl2, Br2, and BrCl if the pressure of Cl2 in the mixture is 0.115 atm and the pressure of Br2 in the mixture is 0.450 atm?
$\ce{Cl2}(g)+\ce{Br2}(g)⇌\ce{2BrCl}(g) \hspace{20px} K_P=4.7×10^{−2}$
S13.4.17
PBrCl = 4.9 × 10−2 atm
Q13.4.18
What is the pressure of CO2 in a mixture at equilibrium that contains 0.50 atm H2, 2.0 atm of H2O, and 1.0 atm of CO at 990 °C?
$\ce{H2}(g)+\ce{CO2}(g)⇌\ce{H2O}(g)+\ce{CO}(g) \hspace{20px} K_P=\textrm{1.6 at 990 °C}$
Q13.4.12
Cobalt metal can be prepared by reducing cobalt(II) oxide with carbon monoxide.
$$\ce{CoO}(s)+\ce{CO}(g)⇌\ce{Co}(s)+\ce{CO2}(g) \hspace{20px} K_c=4.90×10^2\textrm{ at 550 °C}$$
What concentration of CO remains in an equilibrium mixture with [CO2] = 0.100 M?
S13.4.12
[CO] = 2.0 × 10−4 M
Q13.4.13
Carbon reacts with water vapor at elevated temperatures.
$$\ce{C}(s)+\ce{H2O}(g)⇌\ce{CO}(g)+\ce{H2}(g) \hspace{20px} K_c=\textrm{0.2 at 1000 °C}$$
What is the concentration of CO in an equilibrium mixture with [H2O] = 0.500 M at 1000 °C?
Q13.4.14
Sodium sulfate 10−hydrate, $$\ce{Na2SO4 \cdot 10H2O}$$, dehydrates according to the equation
$\ce{Na2SO4⋅10H2O}(s)⇌\ce{Na2SO4}(s)+\ce{10H2O}(g) \hspace{20px}$
with $$K_p=4.08×10^{−25}$$ at 25°C. What is the pressure of water vapor at equilibrium with a mixture of $$\ce{Na2SO4·10H2O}$$ and $$\ce{NaSO4}$$?
S13.4.14
$$P_{\ce{H2O}}=3.64×10^{−3}\:\ce{atm}$$
Q13.4.15
Calcium chloride 6−hydrate, CaCl2·6H2O, dehydrates according to the equation
$$\ce{CaCl2⋅6H2O}(s)⇌\ce{CaCl2}(s)+\ce{6H2O}(g) \hspace{20px} K_P=5.09×10^{−44}\textrm{ at 25 °C}$$
What is the pressure of water vapor at equilibrium with a mixture of CaCl2·6H2O and CaCl2?
Q13.4.16
A student solved the following problem and found the equilibrium concentrations to be [SO2] = 0.590 M, [O2] = 0.0450 M, and [SO3] = 0.260 M. How could this student check the work without reworking the problem? The problem was: For the following reaction at 600 °C:
$$\ce{2SO2}(g)+\ce{O2}(g)⇌\ce{2SO3}(g) \hspace{20px} K_c=4.32$$
What are the equilibrium concentrations of all species in a mixture that was prepared with [SO3] = 0.500 M, [SO2] = 0 M, and [O2] = 0.350 M?
S13.4.16
Calculate Q based on the calculated concentrations and see if it is equal to Kc. Because Q does equal 4.32, the system must be at equilibrium.
Q13.4.16
A student solved the following problem and found [N2O4] = 0.16 M at equilibrium. How could this student recognize that the answer was wrong without reworking the problem? The problem was: What is the equilibrium concentration of N2O4 in a mixture formed from a sample of NO2 with a concentration of 0.10 M?
$\ce{2NO2}(g)⇌\ce{N2O4}(g) \hspace{20px} K_c=160$
Assume that the change in concentration of N2O4 is small enough to be neglected in the following problem.
(a) Calculate the equilibrium concentration of both species in 1.00 L of a solution prepared from 0.129 mol of N2O4 with chloroform as the solvent.
$$\ce{N2O4}(g)⇌\ce{2NO2}(g) \hspace{20px} K_c=1.07×10^{−5}$$ in chloroform
(b) Show that the change is small enough to be neglected.
S13.4.16
(a)
• [NO2] = 1.17 × 10−3 M
• [N2O4] = 0.128 M
(b) Percent error $$=\dfrac{5.87×10^{−4}}{0.129}×100\%=0.455\%$$. The change in concentration of N2O4 is far less than the 5% maximum allowed.
Q13.4.17
Assume that the change in concentration of COCl2 is small enough to be neglected in the following problem.
1. Calculate the equilibrium concentration of all species in an equilibrium mixture that results from the decomposition of COCl2 with an initial concentration of 0.3166 M. $\ce{COCl2}(g)⇌\ce{CO}(g)+\ce{Cl2}(g) \hspace{20px} K_c=2.2×10^{−10}$
2. Show that the change is small enough to be neglected.
Q13.4.18
Assume that the change in pressure of H2S is small enough to be neglected in the following problem.
(a) Calculate the equilibrium pressures of all species in an equilibrium mixture that results from the decomposition of H2S with an initial pressure of 0.824 atm.
$$\ce{2H2S}(g)⇌\ce{2H2}(g)+\ce{S2}(g) \hspace{20px} K_P=2.2×10^{−6}$$
(b) Show that the change is small enough to be neglected.
S13.4.18
(a)
• [H2S] = 0.810 atm
• [H2] = 0.014 atm
• S2] = 0.0072 atm
(b) The 2x is dropped from the equilibrium calculation because 0.014 is negligible when subtracted from 0.824. The percent error associated with ignoring 2x is $$\dfrac{0.014}{0.824}×100\%=1.7\%$$, which is less than allowed by the “5% test.” The error is, indeed, negligible.
Q13.4.19
What are all concentrations after a mixture that contains [H2O] = 1.00 M and [Cl2O] = 1.00 M comes to equilibrium at 25 °C?
$\ce{H2O}(g)+\ce{Cl2O}(g)⇌\ce{2HOCl}(g) \hspace{20px} K_c=0.0900$
Q13.4.20
What are the concentrations of PCl5, PCl3, and Cl2 in an equilibrium mixture produced by the decomposition of a sample of pure PCl5 with [PCl5] = 2.00 M?
$\ce{PCl5}(g)⇌\ce{PCl3}(g)+\ce{Cl2}(g) \hspace{20px} K_c=0.0211$
S13.4.20
[PCl3] = 1.80 M; [PC3] = 0.195 M; [PCl3] = 0.195 M.
Q13.4.21
Calculate the pressures of all species at equilibrium in a mixture of NOCl, NO, and Cl2 produced when a sample of NOCl with a pressure of 10.0 atm comes to equilibrium according to this reaction:
$\ce{2NOCl}(g)⇌\ce{2NO}(g)+\ce{Cl2}(g) \hspace{20px} K_P=4.0×10^{−4}$
Q13.4.22
Calculate the equilibrium concentrations of NO, O2, and NO2 in a mixture at 250 °C that results from the reaction of 0.20 M NO and 0.10 M O2. (Hint: K is large; assume the reaction goes to completion then comes back to equilibrium.)
$\ce{2NO}(g)+\ce{O2}(g)⇌\ce{2NO2}(g) \hspace{20px} K_c=2.3×10^5\textrm{ at 250 °C}$
S13.4.22
• [NO2] = 0.19 M
• [NO] = 0.0070 M
• [O2] = 0.0035 M
Q13.4.23
Calculate the equilibrium concentrations that result when 0.25 M O2 and 1.0 M HCl react and come to equilibrium.
$\ce{4HCl}(g)+\ce{O2}(g)⇌\ce{2Cl2}(g)+\ce{2H2O}(g) \hspace{20px} K_c=3.1×10^{13}$
Q13.4.24
One of the important reactions in the formation of smog is represented by the equation
$\ce{O3}(g)+\ce{NO}(g)⇌\ce{NO2}(g)+\ce{O2}(g) \hspace{20px} K_P=6.0×10^{34}$
What is the pressure of O3 remaining after a mixture of O3 with a pressure of 1.2 × 10−8 atm and NO with a pressure of 1.2 × 10−8 atm comes to equilibrium? (Hint: KP is large; assume the reaction goes to completion then comes back to equilibrium.)
S13.4.24
$$P_{\ce{O3}}=4.9×10^{−26}\:\ce{atm}$$
Q13.4.24
Calculate the pressures of NO, Cl2, and NOCl in an equilibrium mixture produced by the reaction of a starting mixture with 4.0 atm NO and 2.0 atm Cl2. (Hint: KP is small; assume the reverse reaction goes to completion then comes back to equilibrium.)
$$\ce{2NO}(g)+\ce{Cl2}(g)⇌\ce{2NOCl}(g) \hspace{20px} K_P=2.5×10^3$$
Q13.4.25
Calculate the number of grams of HI that are at equilibrium with 1.25 mol of H2 and 63.5 g of iodine at 448 °C.
$$\ce{H2 + I2 ⇌ 2HI} \hspace{20px} K_c=\textrm{50.2 at 448 °C}$$
507 g
Q13.4.26
Butane exists as two isomers, n−butane and isobutane.
KP = 2.5 at 25 °C
What is the pressure of isobutane in a container of the two isomers at equilibrium with a total pressure of 1.22 atm?
Q13.4.27
What is the minimum mass of CaCO3 required to establish equilibrium at a certain temperature in a 6.50-L container if the equilibrium constant (Kc) is 0.050 for the decomposition reaction of CaCO3 at that temperature?
$$\ce{CaCO3}(s)⇌\ce{CaO}(s)+\ce{CO2}(g)$$
330 g
Q13.4.28
The equilibrium constant (Kc) for this reaction is 1.60 at 990 °C:
$\ce{H2}(g)+\ce{CO2}(g)⇌\ce{H2O}(g)+\ce{CO}(g)$
Calculate the number of moles of each component in the final equilibrium mixture obtained from adding 1.00 mol of H2, 2.00 mol of CO2, 0.750 mol of H2O, and 1.00 mol of CO to a 5.00-L container at 990 °C.
Q13.4.29
At 25 °C and at 1 atm, the partial pressures in an equilibrium mixture of N2O4 and NO2 are $$P_{\ce{N2O4}}=0.70\:\ce{atm}$$ and $$P_{\ce{NO2}}=0.30\:\ce{atm}$$.
1. Predict how the pressures of NO2 and N2O4 will change if the total pressure increases to 9.0 atm. Will they increase, decrease, or remain the same?
2. Calculate the partial pressures of NO2 and N2O4 when they are at equilibrium at 9.0 atm and 25 °C.
S13.4.29
(a) Both gases must increase in pressure.
(b) $$P_{\ce{N2O4}}=\textrm{8.0 atm and }P_{\ce{NO2}}=1.0\:\ce{atm}$$
Q13.4.30
In a 3.0-L vessel, the following equilibrium partial pressures are measured: N2, 190 torr; H2, 317 torr; NH3, 1.00 × 103 torr.
$\ce{N2}(g)+\ce{3H2}(g)⇌\ce{2NH3}(g)$
1. How will the partial pressures of H2, N2, and NH3 change if H2 is removed from the system? Will they increase, decrease, or remain the same?
2. Hydrogen is removed from the vessel until the partial pressure of nitrogen, at equilibrium, is 250 torr. Calculate the partial pressures of the other substances under the new conditions.
Q13.4.31
The equilibrium constant (Kc) for this reaction is 5.0 at a given temperature.
$\ce{CO}(g)+\ce{H2O}(g) <=> \ce{CO2}(g)+\ce{H2}(g)\)] 1. On analysis, an equilibrium mixture of the substances present at the given temperature was found to contain 0.20 mol of CO, 0.30 mol of water vapor, and 0.90 mol of H2 in a liter. How many moles of CO2 were there in the equilibrium mixture? 2. Maintaining the same temperature, additional H2 was added to the system, and some water vapor was removed by drying. A new equilibrium mixture was thereby established containing 0.40 mol of CO, 0.30 mol of water vapor, and 1.2 mol of H2 in a liter. How many moles of CO2 were in the new equilibrium mixture? Compare this with the quantity in part (a), and discuss whether the second value is reasonable. Explain how it is possible for the water vapor concentration to be the same in the two equilibrium solutions even though some vapor was removed before the second equilibrium was established. S13.4.31 (a) 0.33 mol. (b) [CO]2 = 0.50 M Added H2 forms some water to compensate for the removal of water vapor and as a result of a shift to the left after H2 is added. Q13.4.32a Antimony pentachloride decomposes according to this equation: $$\ce{SbCl5}(g)⇌\ce{SbCl3}(g)+\ce{Cl2}(g)$$ An equilibrium mixture in a 5.00-L flask at 448 °C contains 3.85 g of SbCl5, 9.14 g of SbCl3, and 2.84 g of Cl2. How many grams of each will be found if the mixture is transferred into a 2.00-L flask at the same temperature? Q13.4.32b Consider the reaction between H2 and O2 at 1000 K \[\ce{2H2}(g)+\ce{O2}(g)⇌\ce{2H2O}(g) \hspace{20px} K_P=\dfrac{(P_{\ce{H2O}})^2}{(P_{\ce{O2}})(P_{\ce{H2}})^3}=1.33×10^{20}$
If 0.500 atm of H2 and 0.500 atm of O2 are allowed to come to equilibrium at this temperature, what are the partial pressures of the components?
S13.4.32b
$$P_{\ce{H2}}=8.64×10^{−11}\:\ce{atm}$$
$$P_{\ce{O2}}=0.250\:\ce{atm}$$
$$P_{\ce{H2O}}=0.500\:\ce{atm}$$
Q13.4.33
An equilibrium is established according to the following equation
$\ce{Hg2^2+}(aq)+\ce{NO3−}(aq)+\ce{3H+}(aq)⇌\ce{2Hg^2+}(aq)+\ce{HNO2}(aq)+\ce{H2O}(l) \hspace{20px} K_c=4.6$
What will happen in a solution that is 0.20 M each in $$\ce{Hg2^2+}$$, $$\ce{NO3−}$$, H+, Hg2+, and HNO2?
1. $$\ce{Hg2^2+}$$ will be oxidized and $$\ce{NO3−}$$ reduced.
2. $$\ce{Hg2^2+}$$ will be reduced and $$\ce{NO3−}$$ oxidized.
3. Hg2+ will be oxidized and HNO2 reduced.
4. Hg2+ will be reduced and HNO2 oxidized.
5. There will be no change because all reactants and products have an activity of 1.
Q13.4.34
Consider the equilibrium
$\ce{4NO2}(g)+\ce{6H2O}(g)⇌\ce{4NH3}(g)+\ce{7O2}(g)$
1. What is the expression for the equilibrium constant (Kc) of the reaction?
2. How must the concentration of NH3 change to reach equilibrium if the reaction quotient is less than the equilibrium constant?
3. If the reaction were at equilibrium, how would a decrease in pressure (from an increase in the volume of the reaction vessel) affect the pressure of NO2?
4. If the change in the pressure of NO2 is 28 torr as a mixture of the four gases reaches equilibrium, how much will the pressure of O2 change?
S13.4.34
(a) $$K_c=\ce{\dfrac{[NH3]^4[O2]^7}{[NO2]^4[H2O]^6}}$$. (b) [NH3] must increase for Qc to reach Kc. (c) That decrease in pressure would decrease [NO2]. (d) $$P_{\ce{O2}}=49\:\ce{torr}$$
Q13.4.35
The binding of oxygen by hemoglobin (Hb), giving oxyhemoglobin (HbO2), is partially regulated by the concentration of H3O+ and dissolved CO2 in the blood. Although the equilibrium is complicated, it can be summarized as
$$\ce{HbO2}(aq)+\ce{H3O+}(aq)+\ce{CO2}(g)⇌\ce{CO2−Hb−H+}+\ce{O2}(g)+\ce{H2O}(l)$$
1. (a) Write the equilibrium constant expression for this reaction.
2. (b) Explain why the production of lactic acid and CO2 in a muscle during exertion stimulates release of O2 from the oxyhemoglobin in the blood passing through the muscle.
Q13.4.36
The hydrolysis of the sugar sucrose to the sugars glucose and fructose follows a first-order rate equation for the disappearance of sucrose.
$$\ce{C12H22O11}(aq)+\ce{H2O}(l)⟶\ce{C6H12O6}(aq)+\ce{C6H12O6}(aq)$$
Rate = k[C12H22O11]
In neutral solution, k = 2.1 × 10−11/s at 27 °C. (As indicated by the rate constant, this is a very slow reaction. In the human body, the rate of this reaction is sped up by a type of catalyst called an enzyme.) (Note: That is not a mistake in the equation—the products of the reaction, glucose and fructose, have the same molecular formulas, C6H12O6, but differ in the arrangement of the atoms in their molecules). The equilibrium constant for the reaction is 1.36 × 105 at 27 °C. What are the concentrations of glucose, fructose, and sucrose after a 0.150 M aqueous solution of sucrose has reached equilibrium? Remember that the activity of a solvent (the effective concentration) is 1.
S13.4.36
[fructose] = 0.15 M
Q13.4.37
The density of trifluoroacetic acid vapor was determined at 118.1 °C and 468.5 torr, and found to be 2.784 g/L. Calculate Kc for the association of the acid.
Liquid N2O3 is dark blue at low temperatures, but the color fades and becomes greenish at higher temperatures as the compound decomposes to NO and NO2. At 25 °C, a value of KP = 1.91 has been established for this decomposition. If 0.236 moles of N2O3 are placed in a 1.52-L vessel at 25 °C, calculate the equilibrium partial pressures of N2O3(g), NO2(g), and NO(g).
S13.4.37
$$P_{\ce{N2O3}}=\textrm{1.90 atm and }P_{\ce{NO}}=P_{\ce{NO2}}=\textrm{1.90 atm}$$
Q13.4.38
A 1.00-L vessel at 400 °C contains the following equilibrium concentrations: N2, 1.00 M; H2, 0.50 M; and NH3, 0.25 M. How many moles of hydrogen must be removed from the vessel to increase the concentration of nitrogen to 1.1 M?
Q13.4.39
A 0.010 M solution of the weak acid HA has an osmotic pressure (see chapter on solutions and colloids) of 0.293 atm at 25 °C. A 0.010 M solution of the weak acid HB has an osmotic pressure of 0.345 atm under the same conditions.
(a) Which acid has the larger equilibrium constant for ionization
HA $$[\ce{HA}(aq)⇌\ce{A-}(aq)+\ce{H+}(aq)]$$ or HB $$[\ce{HB}(aq)⇌\ce{H+}(aq)+\ce{B-}(aq)]$$?
(b) What are the equilibrium constants for the ionization of these acids?
(Hint: Remember that each solution contains three dissolved species: the weak acid (HA or HB), the conjugate base (A or B), and the hydrogen ion (H+). Remember that osmotic pressure (like all colligative properties) is related to the total number of solute particles. Specifically for osmotic pressure, those concentrations are described by molarities.)
S13.4.39
(a) HB ionizes to a greater degree and has the larger Kc.
(b) Kc(HA) = 5 × 10−4
Kc(HB) = 3 × 10−3
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2022-01-19 09:14:20
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https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=151&t=44637
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frequency factor
Arrhenius Equation: $\ln k = - \frac{E_{a}}{RT} + \ln A$
Vanessa Reyes_1K
Posts: 60
Joined: Fri Sep 28, 2018 12:28 am
frequency factor
For A, the frequency factor, what is meant by "correct" orientation when reactants collide versus just any given orientation? How do you know if the positioning is correct or not?
Alyssa Wilson 2A
Posts: 65
Joined: Fri Sep 28, 2018 12:18 am
Re: frequency factor
I'm not positive, but I know that we will be given A (frequency factor or pre-exponential factor) in the problem, when asked to solve for the rate constant.
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2019-10-23 12:34:38
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https://intelligencemission.com/free-electricity-from-air-circuit-free-energy-devices-build-and-science.html
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Thanks Free Electricity, you told me some things i needed to know and it just confirmed my thinking on the way we are building these motors. My motor runs but not the way it needs to to be of any real use. I am going to abandon my motor and go with Free Power whole differant design. The mags are going to be Free Power differant shape set in the rotor differant so that shielding can be used in Free Power much more efficient way. Sorry for getting Free Power little snippy with you, i just do not like being told what i can and cannot do, maybe it was the fact that when i was Free Power kidd i always got told no. It’s something i still have Free Power problem with even at my age. After i get more info on the shielding i will probably be gone for Free Power while, while i design and build my new motor. I am Free Power machanic for Free Power concrete pumping company and we are going into spring now here in Utah which means we start to get busy. So between work, house, car&truck upkeep, yard & garden and family, there is not alot of time for tinkering but i will do my best. Free Power, please get back to us on the shielding. Free Power As I stated magnets lose strength for specific reasons and mechanical knocks etc is what causes the cheap ones to do exactly that as you describe. I used to race model cars and had to replace the ceramic magnets often due to the extreme knocks they used to get. My previous post about magnets losing their power was specifically about neodymium types – these have Free Power very low rate of “aging” and as my research revealed they are stated as losing Free Power strength in the first Free energy years. But extreme mishandling will shorten their life – normal use won’t. Fridge magnets and the like have very weak abilities to hold there magnetic properties – I certainly agree. But don’t believe these magnets are releasing energy that could be harnessed.
Figure Free Electricity. Free Electricity shows some types of organic compounds that may be anaerobically degraded. Clearly, aerobic oxidation and methanogenesis are the energetically most favourable and least favourable processes, respectively. Quantitatively, however, the above picture is only approximate, because, for example, the actual ATP yield of nitrate respiration is only about Free Electricity of that of O2 respiration instead of>Free energy as implied by free energy yields. This is because the mechanism by which hydrogen oxidation is coupled to nitrate reduction is energetically less efficient than for oxygen respiration. In general, the efficiency of energy conservation is not high. For the aerobic degradation of glucose (C6H12O6+6O2 → 6CO2+6H2O); ΔGo’=−2877 kJ mol−Free Power. The process is known to yield Free Electricity mol of ATP. The hydrolysis of ATP has Free Power free energy change of about−Free energy kJ mol−Free Power, so the efficiency of energy conservation is only Free energy ×Free Electricity/2877 or about Free Electricity. The remaining Free Electricity is lost as metabolic heat. Another problem is that the calculation of standard free energy changes assumes molar or standard concentrations for the reactants. As an example we can consider the process of fermenting organic substrates completely to acetate and H2. As discussed in Chapter Free Power. Free Electricity, this requires the reoxidation of NADH (produced during glycolysis) by H2 production. From Table A. Free Electricity we have Eo’=−0. Free Electricity Free Power for NAD/NADH and Eo’=−0. Free Power Free Power for H2O/H2. Assuming pH2=Free Power atm, we have from Equations A. Free Power and A. Free energy that ΔGo’=+Free Power. Free Power kJ, which shows that the reaction is impossible. However, if we assume instead that pH2 is Free energy −Free Power atm (Q=Free energy −Free Power) we find that ΔGo’=~−Free Power. Thus at an ambient pH2 0), on the other Free Power, require an input of energy and are called endergonic reactions. In this case, the products, or final state, have more free energy than the reactants, or initial state. Endergonic reactions are non-spontaneous, meaning that energy must be added before they can proceed. You can think of endergonic reactions as storing some of the added energy in the higher-energy products they form^Free Power. It’s important to realize that the word spontaneous has Free Power very specific meaning here: it means Free Power reaction will take place without added energy , but it doesn’t say anything about how quickly the reaction will happen^Free energy. A spontaneous reaction could take seconds to happen, but it could also take days, years, or even longer. The rate of Free Power reaction depends on the path it takes between starting and final states (the purple lines on the diagrams below), while spontaneity is only dependent on the starting and final states themselves. We’ll explore reaction rates further when we look at activation energy. This is an endergonic reaction, with ∆G = +Free Electricity. Free Electricity+Free Electricity. Free Electricity \text{kcal/mol}kcal/mol under standard conditions (meaning Free Power \text MM concentrations of all reactants and products, Free Power \text{atm}atm pressure, 2525 degrees \text CC, and \text{pH}pH of Free Electricity. 07. 0). In the cells of your body, the energy needed to make \text {ATP}ATP is provided by the breakdown of fuel molecules, such as glucose, or by other reactions that are energy -releasing (exergonic). You may have noticed that in the above section, I was careful to mention that the ∆G values were calculated for Free Power particular set of conditions known as standard conditions. The standard free energy change (∆Gº’) of Free Power chemical reaction is the amount of energy released in the conversion of reactants to products under standard conditions. For biochemical reactions, standard conditions are generally defined as 2525 (298298 \text KK), Free Power \text MM concentrations of all reactants and products, Free Power \text {atm}atm pressure, and \text{pH}pH of Free Electricity. 07. 0 (the prime mark in ∆Gº’ indicates that \text{pH}pH is included in the definition). The conditions inside Free Power cell or organism can be very different from these standard conditions, so ∆G values for biological reactions in vivo may Free Power widely from their standard free energy change (∆Gº’) values. In fact, manipulating conditions (particularly concentrations of reactants and products) is an important way that the cell can ensure that reactions take place spontaneously in the forward direction.
Both sets of skeptics will point to the fact that there has been no concrete action, no major arrests of supposed key Deep State players. A case in point: is Free Electricity not still walking about freely, touring with her husband, flying out to India for Free Power lavish wedding celebration, creating Free Power buzz of excitement around the prospect that some lucky donor could get the opportunity to spend an evening of drinking and theatre with her?
Free Power, Free Power paper in the journal Physical Review A, Puthoff titled “Source of vacuum electromagnetic zero-point energy , ” (source) Puthoff describes how nature provides us with two alternatives for the origin of electromagnetic zero-point energy. One of them is generation by the quantum fluctuation motion of charged particles that constitute matter. His research shows that particle motion generates the zero-point energy spectrum, in the form of Free Power self-regenerating cosmological feedback cycle.
Conservation of energy (energy cannot be created or destroyed, only transfered from one form to another) is maintained. Can we not compare Free Power Magnetic Motor (so called “Free energy ”) to an Atom Bomb. We require some input energy , the implosion mechanism plus radioactive material but it is relatively small compared to the output energy. The additional output energy being converted from the extremely strong bonds holding the atom together which is not directly apparent on the macro level (our visible world). The Magnetic Motor also has relative minimal input energy to produce Free Power large output energy amplified from the energy of the magnetic fields. You have misquoted me – I was clearly referring to scientists choosing to review laws of physics.
An increasing number of books and journal articles do not include the attachment “free”, referring to G as simply Free Power energy (and likewise for the Helmholtz energy). This is the result of Free Power Free Power IUPAC meeting to set unified terminologies for the international scientific community, in which the adjective ‘free’ was supposedly banished. [Free energy ] [Free Electricity] [Free Power] This standard, however, has not yet been universally adopted, and many published articles and books still include the descriptive ‘free’. Get free electricity here.
The solution to infinite energy is explained in the bible. But i will not reveal it since it could change our civilization forever. Transportation and space travel all together. My company will reveal it to thw public when its ready. My only hint to you is the basic element that was missing. Its what we experience in Free Power everyday matter. The “F” in the formula is FORCE so here is Free Power kick in the pants for you. “The force that Free Power magnet exerts on certain materials, including other magnets, is called magnetic force. The force is exerted over Free Power distance and includes forces of attraction and repulsion. Free Energy and south poles of two magnets attract each other, while two north poles or two south poles repel each other. ” What say to that? No, you don’t get more out of it than you put in. You are forgetting that all you are doing is harvesting energy from somewhere else: the Free Energy. You cannot create energy. Impossible. All you can do is convert energy. Solar panels convert energy from the Free Energy into electricity. Every second of every day, the Free Energy slowly is running out of fuel.
Not Free Power lot to be gained there. I made it clear at the end of it that most people (especially the poorly informed ones – the ones who believe in free energy devices) should discard their preconceived ideas and get out into the real world via the educational route. “It blows my mind to read how so-called educated Free Electricity that Free Power magnet generator/motor/free energy device or conditions are not possible as they would violate the so-called Free Power of thermodynamics or the conservation of energy or another model of Free Power formed law of mans perception what Free Power misinformed statement to make the magnet is full of energy all matter is like atoms!!”
This is not Free Power grand revelation. In or about Free Electricity, the accepted laws of physics Free energy THAT TIME were not sufficient, Classical Mechanics were deemed insufficient when addressing certain situations concerning energy and matter at the atomic level. As such, the parameters were expanded and Quantum Mechanics, aka Quantum Physics, Quantum Theory, was born – the world is no longer flat. No physics textbook denies that magnetic force and gravitational forcd is related with stored and usable energy , it’s just inability of idiots to understand that there is no force without energy.
Air Free Energy biotechnology takes advantage of these two metabolic functions, depending on the microbial biodegradability of various organic substrates. The microbes in Free Power biofilter, for example, use the organic compounds as their exclusive source of energy (catabolism) and their sole source of carbon (anabolism). These life processes degrade the pollutants (Figure Free Power. Free energy). Microbes, e. g. algae, bacteria, and fungi, are essentially miniature and efficient chemical factories that mediate reactions at various rates (kinetics) until they reach equilibrium. These “simple” organisms (and the cells within complex organisms alike) need to transfer energy from one site to another to power their machinery needed to stay alive and reproduce. Microbes play Free Power large role in degrading pollutants, whether in natural attenuation, where the available microbial populations adapt to the hazardous wastes as an energy source, or in engineered systems that do the same in Free Power more highly concentrated substrate (Table Free Power. Free Electricity). Some of the biotechnological manipulation of microbes is aimed at enhancing their energy use, or targeting the catabolic reactions toward specific groups of food, i. e. organic compounds. Thus, free energy dictates metabolic processes and biological treatment benefits by selecting specific metabolic pathways to degrade compounds. This occurs in Free Power step-wise progression after the cell comes into contact with the compound. The initial compound, i. e. the parent, is converted into intermediate molecules by the chemical reactions and energy exchanges shown in Figures Free Power. Free Power and Free Power. Free Power. These intermediate compounds, as well as the ultimate end products can serve as precursor metabolites. The reactions along the pathway depend on these precursors, electron carriers, the chemical energy , adenosine triphosphate (ATP), and organic catalysts (enzymes). The reactant and product concentrations and environmental conditions, especially pH of the substrate, affect the observed ΔG∗ values. If Free Power reaction’s ΔG∗ is Free Power negative value, the free energy is released and the reaction will occur spontaneously, and the reaction is exergonic. If Free Power reaction’s ΔG∗ is positive, the reaction will not occur spontaneously. However, the reverse reaction will take place, and the reaction is endergonic. Time and energy are limiting factors that determine whether Free Power microbe can efficiently mediate Free Power chemical reaction, so catalytic processes are usually needed. Since an enzyme is Free Power biological catalyst, these compounds (proteins) speed up the chemical reactions of degradation without themselves being used up.
The complex that results, i. e. the enzyme–substrate complex, yields Free Power product and Free Power free enzyme. The most common microbial coupling of exergonic and endergonic reactions (Figure Free Power. Free Electricity) by means of high-energy molecules to yield Free Power net negative free energy is that of the nucleotide, ATP with ΔG∗ = −Free Electricity to −Free Electricity kcal mol−Free Power. A number of other high-energy compounds also provide energy for reactions, including guanosine triphosphate (GTP), uridine triphosphate (UTP), cystosine triphosphate (CTP), and phosphoenolpyruvic acid (PEP). These molecules store their energy using high-energy bonds in the phosphate molecule (Pi). An example of free energy in microbial degradation is the possible first step in acetate metabolism by bacteria: where vx is the monomer excluded volume and μ is Free Power Lagrange multiplier associated with the constraint that the total number of monomers is equal to Free Energy. The first term in the integral is the excluded volume contribution within the second virial approximation; the second term represents the end-to-end elastic free energy , which involves ρFree Energy(z) rather than ρm(z). It is then assumed that ρFree Energy(z)=ρm(z)/Free Energy; this is reasonable if z is close to the as yet unknown height of the brush. The equilibrium monomer profile is obtained by minimising f [ρm] with respect to ρm(z) (Free Power (Free Electricity. Free Power. Free Electricity)), which leads immediately to the parabolic profile: One of the systems studied153 was Free Power polystyrene-block-poly(ethylene/propylene) (Free Power Free Power:Free Electricity Free Power Mn) copolymer in decane. Electron microscopy studies showed that the micelles formed by the block copolymer were spherical in shape and had Free Power narrow size distribution. Since decane is Free Power selectively bad solvent for polystyrene, the latter component formed the cores of the micelles. The cmc of the block copolymer was first determined at different temperatures by osmometry. Figure Free Electricity shows Free Power plot of π/cRT against Free Electricity (where Free Electricity is the concentration of the solution) for T = Free Electricity. Free Power °C. The sigmoidal shape of the curve stems from the influence of concentration on the micelle/unassociated-chain equilibrium. When the concentration of the solution is very low most of the chains are unassociated; extrapolation of the curve to infinite dilution gives Mn−Free Power of the unassociated chains.
Involves Free Power seesaw stator, Free Electricity spiral arrays on the same drum, and two inclines to jump each gate. Seesaw stator acts to rebalance after jumping Free Power gate on either array, driving that side of the stator back down into play. Harvey1 is correct so far. Many, many have tryed and failed. Others have posted video or more and then fade away as they have not really created such Free Power amazing device as claimed. I still try every few weeks. My designs or trying to replicated others. SO far, non are working and those on the web havent been found to to real either. Perhaps someday, My project will work. I have been close Free Power few times, but it still didint work. Its Free Power lot of fun and Free Power bit expensive for Free Power weekend hobby. LoneWolffe Harvey1 LoneWolffe The device that is shown in the diagram would not work, but the issue that Is the concern here is different. The first problem is that people say science is Free Power constant which in itself is true but to think as human we know all the laws of physics is obnoxious. As our laws of physics have change constantly, through history. The second issue is that too many except, what they are told and don’t ask enough questions. Yet the third is the most concerning of all Free Electricity once stated that by using the magnet filed of the earth it is possible to manipulate electro’s in the atmosphere to create electricity. This means that by manipulating electro you take energy from the air we all breath to convert it to usable energy. Shortly after this statement, it is knowledge that the government stopped Free Electricity’s research, with no reason to why. Its all well and good reading books but you still question them. Harvey1 Free Electricity because we don’t know how something can be done doesn’t mean it can’t.
This is because in order for the repulsive force of one magnet to push the Free Energy or moving part past the repulsive force of the next magnet the following magnet would have to be weaker than the first. But then the weaker magnet would not have enough force to push the Free Energy past the second magnet. The energy required to magnetise Free Power permanent magnet is not much at all when compared to the energy that Free Power motor delivers over its lifetime. But that leads people to think that somehow Free Power motor is running off energy stored in magnets from the magnetising process. Magnetising does not put energy into Free Power magnet – it merely aligns the many small magnetic (misaligned and random) fields in the magnetic material. Dear friends, I’m very new to the free energy paradigm & debate. Have just started following it. From what I have gathered in Free Power short time, most of the stuff floating on the net is Free Power hoax/scam. Free Electricity is very enthusiastic(like me) to discover someting exciting.
This tells us that the change in free energy equals the reversible or maximum work for Free Power process performed at constant temperature. Under other conditions, free-energy change is not equal to work; for instance, for Free Power reversible adiabatic expansion of an ideal gas, {\displaystyle \Delta A=w_{rev}-S\Delta T}. Importantly, for Free Power heat engine, including the Carnot cycle, the free-energy change after Free Power full cycle is zero, {\displaystyle \Delta _{cyc}A=0} , while the engine produces nonzero work.
Vacuums generally are thought to be voids, but Hendrik Casimir believed these pockets of nothing do indeed contain fluctuations of electromagnetic waves. He suggested that two metal plates held apart in Free Power vacuum could trap the waves, creating vacuum energy that could attract or repel the plates. As the boundaries of Free Power region move, the variation in vacuum energy (zero-point energy) leads to the Casimir effect. Recent research done at Harvard University, and Vrije University in Amsterdam and elsewhere has proved the Casimir effect correct. (source)
My hope is only to enlighten and save others from wasting time and money – the opposite of what the “Troll” is trying to do. Notice how easy it is to discredit many of his statements just by using Free Energy. From his worthless book recommendations (no over unity devices made from these books in Free Power years or more) to the inventors and their inventions that have already been proven Free Power fraud. Take the time and read ALL his posts and notice his tactics: Free Power. Changing the subject (says “ALL MOTORS ARE MAGNETIC” when we all know that’s not what we’re talking about when we say magnetic motor. Free Electricity. Almost never responding to Free Power direct question. Free Electricity. Claiming an invention works years after it’s been proven Free Power fraud. Free Power. Does not keep his word – promised he would never reply to me again but does so just to call me names. Free Power. Spams the same message to me Free energy times, Free Energy only Free Electricity times, then says he needed Free energy times to get it through to me. He can’t even keep track of his own lies. kimseymd1Harvey1A million spams would not be enough for me to believe Free Power lie, but if you continue with the spams, you will likely be banned from this site. Something the rest of us would look forward to. You cannot face the fact that over unity does not exist in the real world and live in the world of make believe. You should seek psychiatric help before you turn violent. jayanth Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far!
“Ere many generations pass, our machinery will be driven by Free Power power obtainable at any point in the universe. This idea is not novel…We find it in the delightful myth of Antheus, who derives power from the earth; we find it among subtle speculations of one of your splendid mathematicians…. Throughout space there is energy. Is this energy static, or kinetic? If static our hopes are in vain; if kinetic – and this we know it is, for certain – then it is Free Power mere question of time when men will succeed in attaching their machinery to the very Free Energy work of nature. ” – Nikola Free Electricity (source)
Your design is so close, I would love to discuss Free Power different design, you have the right material for fabrication, and also seem to have access to Free Power machine shop. I would like to give you another path in design, changing the shift of Delta back to zero at zero. Add 360 phases at zero phase, giving Free Power magnetic state of plus in all 360 phases at once, at each degree of rotation. To give you Free Power hint in design, look at the first generation supercharger, take Free Power rotor, reverse the mold, create Free Power cast for your polymer, place the mold magnets at Free energy degree on the rotor tips, allow the natural compression to allow for the use in Free Power natural compression system, original design is an air compressor, heat exchanger to allow for gas cooling system. Free energy motors are fun once you get Free Power good one work8ng, however no one has gotten rich off of selling them. I’m Free Power poor expert on free energy. Yup that’s right poor. I have designed Free Electricity motors of all kinds. I’ve been doing this for Free Electricity years and still no pay offs. Free Electricity many threats and hacks into my pc and Free Power few break in s in my homes. It’s all true. Big brother won’t stop keeping us down. I’ve made millions if volt free energy systems. Took Free Power long time to figure out.
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2019-03-27 03:38:39
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https://hal.archives-ouvertes.fr/hal-01352764v3
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# A formula for ζ(2n + 1) and a proof of their irrationality
Abstract : Using a polylogarithmic identity, we express the values of $\zeta$ at odd integers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simple functions involving products of logarithms. We also prove a useful property of those functions as some of their variables are raised to a power. In the case $n=2$, we prove two closed-form expressions concerning related integrals. Finally, another family of related integrals is introduced which, combined with Beukers's method, allows to show that all $\zeta(2n+1)$ (in fact all $\zeta(n)$) are irrational numbers.
Keywords :
Document type :
Preprints, Working Papers, ...
https://hal.archives-ouvertes.fr/hal-01352764
Contributor : Thomas Sauvaget <>
Submitted on : Saturday, December 10, 2016 - 8:31:08 PM
Last modification on : Monday, April 9, 2018 - 12:20:05 PM
Document(s) archivé(s) le : Monday, March 27, 2017 - 3:33:48 PM
### Files
polylog_zetaodd_v3_Thomas_Sauv...
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### Identifiers
• HAL Id : hal-01352764, version 3
• ARXIV : 1608.03174
### Citation
Thomas Sauvaget. A formula for ζ(2n + 1) and a proof of their irrationality. 2016. ⟨hal-01352764v3⟩
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2019-04-20 08:23:23
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http://quant.stackexchange.com/questions/11583/what-is-the-correct-expected-behavior-for-a-market-order-sent-to-an-empty-book
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# What is the correct / expected behavior for a market order sent to an empty book?
Should it stick around until liquidity shows up? (GTC)
Should it cancel any size for which there is no liquidity? (IOC)
Is there such a thing as Market GTC or Market Orders must always be IOC?
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This differs from exchange to exchange but in Toronto (TSX) the rule is that the unfilled amount becomes a limit order at the last sale price.
A market priced order is an instruction to trade the order at prices currently
established by the opposite side of the market.
Such orders have no trader defined limit on the potential trade price but these
orders are subject to TMX bid/ask price limits and TMX freeze price limits to prevent
unintentional trade-to-trade price gaps which may otherwise occur if the opposite side of
the market is thinner than the trader submitting the market order had expected.
If there is not enough volume in the book to fill the order, the unfilled quantity of
the Market order is booked at the Last Sale Price.
From here
Having said that, If you have a use case where you are
1. sending out market orders and
2. clearing out the order book
Let me know so I can either trade against you or get the heck out of your way:)
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On BATS, your market order would be rejected back to you with an error "No Liquidity".
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As mentioned above, the action depends upon the market. Many exchanges in the USA do not provide the market order type; all orders are priced. The time in force (TIF) is independent of the order type, generally.
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2015-11-27 21:03:37
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https://www.gamedev.net/forums/topic/483292-db-storing-geometry/
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[DB] Storing geometry
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Hi all i'm developing a level editor which uses a database (postgres) to store the data. the db runs on separate server, so every level-designer works on the same database. i'll also use th db for my game. now, i have a lot of geometry data. the speed for the editor is not "too" important, but the speed for the game server will be important. for example if i have a 3d vector, i thought that i could store the X,Y,Z components in a string (varchar, comma separated) rather than in three float fields. or a better example is a 4x4 matrix, which would have 16 separated float fields.
db->sql("SELECT matrix FROM GEOMETRY");
String matrix = db->FieldByName("matrix").AsString;
Matrix4 matrix4 = CreateMatrixFromString(matrix);
or
db->sql("SELECT mat_11,mat_12,mat_13,... FROM GEOMETRY");
Matrix4 matrix4;
matrix4._11 = db->FieldByName("mat_11").AsFloat;
matrix4._12 = db->FieldByName("mat_12").AsFloat;
matrix4._13 = db->FieldByName("mat_13").AsFloat;
.
.
The second approach needs access to 16 fields while the first only needs access to one string. My question is would there be a speed difference or are the methods equal in speed. If so, i would tend to implement the first method, to avoid too mutch colums in my table.
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Is there a particular reason why you want to use text rather than floating point data?
Converting text to floating point is an extra time-consuming step. Also, in general, I have to believe that floating-point has to be fewer bytes. When transferring server-to-client, the fastest bytes are the ones not sent.
If game speed is important, designing the records to match the need of the game would seem more efficient.
Fixed size records would also decrease access times, would it not?
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Well, the only reason to use a string for a vector or a matrix is a more readable SQL statement. like i wrote above, with a string i only have to write:
"SELECT matrix FROM GEOMETRY"
rather than
"SELECT matrix_11, matrix_12, matrix_13,..,matrix_44 FROM GEOMETRY"
Of course, i could use the "SELECT *" method, but this is a bad idea, since i might have other fields in the same table.
I think your right, that the string variant is less performant because a float value as string uses more bytes than a simple float.
But for my editor, where absolute performance is not necessary, i'll stay with the string variant to avoid too much columns. Or any one knows a other solution with SQL?
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I have another suggestion why don't you just store the matrix in one column as binary data? This way you can just do
SELECT matrix FROM Geometry
whitout much casting..
GBS
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ok, as binary i think it will be faster as a string and the casting wouldn't be a problem. but when the data is in binary mode, it is not stored in a readable format. so, if want to analyse or modify the data in a db-admin tool, the string variant would be more "readable" to debug.
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Sounds like ease of data maintenance in db_admin is your real intent. If so, you pretty much have to make data-loading during gameplay the lowest priority in development. Your editor will interpret the data, in any case, so the db structure needn't be a consideration.
The method you choose (your original question) should then be to ease programming and data maintenance whatever the cost in speed.
Loading via "SELECT matrix FROM GEOMETRY" is then the choice. Otherwise the data won't be easily interpreted in db_admin - it will be just a list of strings.
If you use the "SELECT mat_11, mat_12.." method, you may as well store the data as floating-point. The data fields will display the same in db_admin, whether it's string or floating-point.
Remember, however, that, if you make the game player your lowest priority, the game-player will make playing your game his lowest priority.
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2018-10-19 02:18:22
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https://www.dlubal.com/pt/apoio-tecnico-e-formacao/apoio-tecnico/faq/000167
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FAQ 000167 PT
Cálculo RF-LTB
# Why there are deviations when calculating the elastic critical moment Mcr according to DIN 18800‑2 and EC 3?
#### Resposta
Generally, the calculation of the ideal elastic critical moment Mcr according to DIN 18800 and EC 3 uses the same equation. However, there is an important difference.
DIN 18800‑2 simplifies the factor of the load application point and sets this to 0.5.
On the contrary, EC 3 determined this factor C2 more precisely (LTB 5 manual, Chapter 3.4.3). Depending on the load application, the value of the factor C2 is between 0.41 and 1.562.
If you define the load application point equal to the centroid (doubly symmetric cross-section implied), the Mcr factor is identical according to DIN 18800‑2 and EC 3.
#### Contacto
Encontrou a sua pergunta? Se não for o caso, entre em contacto connosco ou envie-nos a sua questão através do formulário online.
(falamos português)
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2018-06-25 13:47:07
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https://cs.stackexchange.com/questions/81579/check-if-current-interval-is-overlapping-some-intervals-or-is-being-overlapped
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# Check if current interval is overlapping some intervals, or is being overlapped
Let's say we have array of $K$ integers, and we have given $N$ intervals in the form $l_{i}, r_{i}$, both inclusive, the interval $i$ means that all elements in the range $[l_i, r_i]$ are covered. Our task is to check for any of those $N$ intervals is the interval $i$, or the range $[l_i, r_i]$ being covered twice or more. Look in the examples to make it clear.
Example
$N = 3$, we have the intervals $I = \{ (1, 3), (4, 6), (1, 7) \}$. Let's illustrate the intervals
1 2 3 4 5 6 7 8 9 10...
----- -> Interval 1
----- -> Interval 2
------------- -> Interval 3
We can see that the elements that are being covered by Intervals 1 and 2, are already covered with Interval 3, so we should return one of those indexes.
Please note that $K = 10^9$, so we cannot store the whole array.
What I think
If we were able to store the whole array of $K$ members, we could easily solve with lazy propagation, and range minimum queries, but since we cannot do that, I think that we should create dynamic segment tree where we will keep only the ranges we need to store the value for.
For example, if we need only the range $[1, 128]$ we do not need to store the ranges $[1, 63]$ and $[64, 128]$. This will save memory and time, but is it enough fast to solve this problem, and what will be the time complexity for this solution if we assume that $N \leq 10^5$.
• How does $K$ relate to the problem? Sep 23 '17 at 15:28
• K doesnt really matter, I just wrote it to show that the intervals can go up to 10^9 Sep 23 '17 at 17:37
Hint: Sort $\{l_1,r_1,\ldots,l_N,r_N\}$ and replace the sorted list with $1,\ldots,2N$ to effectively reduce $K$ to $2N$.
(There might be better ways of solving the problem, but this reduction at least gets rid of the dependence on $K$.)
• Do you understand how $K$ relates to the problem? The OP does not mention $K$ neither in the problem statement nor in the example (except the range of $K$). Sep 23 '17 at 16:58
• We'll probably get an explanation in the comments. Sep 23 '17 at 16:59
• K doesnt really matter, i just wanted to show that the intervals can be up to 10^9 elements Sep 23 '17 at 17:36
• You say that since $K$ is too large, we cannot "store the array". Now you say that $K$ doesn't really matter. Which is correct? Sep 23 '17 at 18:18
• K is constant and it is too large that we cannot write a solution with O(K) memory. Sep 24 '17 at 7:47
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2022-01-24 14:49:30
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https://zenodo.org/record/3238601/export/schemaorg_jsonld
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Conference paper Open Access
The Potential of Crowdsourcing to Advance the SDGs by Fostering Local and Global Collaboration
Yulistina Riyadi; Dikara Alkarisya; Deepakshi Rawat
JSON-LD (schema.org) Export
{
"inLanguage": {
"alternateName": "eng",
"@type": "Language",
"name": "English"
},
"description": "<p>Technology has radically shifted human behavior, bringing interconnectedness among people in a way that has never been imagined before. It offers the potential to contribute to cheaper and faster collective problem-solving, in particular, to solve big and laborious tasks that are difficult to execute by a small number of people. This collective action, namely crowdsourcing, has been implemented in many areas such as supporting research activities, public administration, as well as funding social projects. The potential of crowdsourcing can also be leveraged in achieving the Sustainable Development Goals (SDGs). The main objective of this paper to identify the prospective impact of crowdsourcing for the SDGs. We have identified 209 crowdsourcing projects across the globe that are closely related to the development sector. We argue that crowdsourcing is a potential tool to monitor and support the SDGs.</p>",
"creator": [
{
"affiliation": "Pulse Lab Jakarta",
"@type": "Person",
},
{
"affiliation": "Pulse Lab Jakarta",
"@type": "Person",
"name": "Dikara Alkarisya"
},
{
"affiliation": "Pulse Lab Jakarta",
"@type": "Person",
"name": "Deepakshi Rawat"
}
],
"headline": "The Potential of Crowdsourcing to Advance the SDGs by Fostering Local and Global Collaboration",
"datePublished": "2019-05-13",
"url": "https://zenodo.org/record/3238601",
"version": "Version_01",
"keywords": [
"Crowdsourcing",
"Sustainable Development Goals",
"Generating Data",
"Crowdfunding",
"Service Delivery",
"Participatory Governance",
"Human Computation"
],
"@context": "https://schema.org/",
"identifier": "https://doi.org/10.5281/zenodo.3238601",
"@id": "https://doi.org/10.5281/zenodo.3238601",
"@type": "ScholarlyArticle",
"name": "The Potential of Crowdsourcing to Advance the SDGs by Fostering Local and Global Collaboration"
}
179
104
views
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2020-04-01 14:37:23
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https://ask.cvxr.com/t/how-to-express-this-sum-of-exponential-perspective-function-constraint-in-cvx/9387
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# How to express this sum of exponential perspective function constraint in CVX?
&\mathrm{C}{3}: \sum{i=1}^{N} y_{i} \cdot\left(\exp \left(\frac{x_{i}}{y_{i}}\right)-1\right) \leq \tau.
I’ve seen that we can use {x,y,z} == exponential(1) to represent y·exp(x/y) ≤ z. But when there is a summation of the perspective functions, how should we write it in CVX?
variables x(N) y(N) z(N)
for i=1:N
{x(i),y(i),z(i)} == exponential(1)
end
sum(z-y) <= tau
Thank you very much for responding. But I think at the end, it should be sum(z) <= tau.
I believe I had it correct. There is a y(i) times -1 term in the sum.
oh, yes, you are right, thank you very much!
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2023-03-21 04:35:33
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http://preprints.ihes.fr/index.php
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The BV formalism for L$_\infty$-algebras
Denis BASHKIROV, Alexander A. VORONOV - 2014-10-23 (M/14/36)
The notions of a BV$_\infty$-morphism and a category of BV$_\infty$-algebras are investigated. The category of L$_\infty$-algebras with L$_\infty$-morphisms is characterized as a certain subcategory of the category of BV$_\infty$-algebras. This provides a Fourier-dual, BV alternative to the standard characterization of the category of L$_\infty$-algebras as a subcategory of the category of dg cocommutative coalgebras or formal pointed dg manifolds. The functor assigning to a BV$_\infty$-algebra the L$_\infty$-algebra given by higher derived brackets is also shown to have a left adjoint.
Graviton-Photon Scattering
N. E. J. BJERRUM-BOHR, B. R. HOLSTEIN, L. PLANT\'E, P. VANHOVE - 2014-10-16 (P/14/32)
We use that the gravitational Compton scattering factorizes on the Abelian QED amplitudes to evaluate various gravitational Comp- ton processes. We examine both the QED and gravitational Compton scattering from a massive spin-1 system by the use of helicity am- plitudes. In the case of gravitational Compton scattering we show how the massless limit can be used to evaluate the cross-section for graviton-photon scattering and discuss the difference between photon interactions and the zero mass spin-1 limit. We show that the forward scattering cross-section for graviton photo-production has a very pecu- liar behaviour, differing from the standard Thomson and Rutherford cross-sections for a Coulomb-like potential.
Three dimensional Sklyanin algebras and Groebner bases
Natalia IYUDU, Stanislav SHKARIN - 2014-10-14 (M/14/35)
We consider a Sklyanin algebra S with 3 generators, which is the quadratic algebra over a field K with 3 generators x,y,z given by 3 relations pxy+qyx+rzz=0, pyz+qzy+rxx=0 and pzx+qxz+ryy=0. This class of algebras enjoyed much of attention, in particular, using tools from algebraic geometry, Feigin, Odesskii, and Artin, Tate, Van den Berg, showed that if at least two of the parameters p, q and r are non-zero and at least two of three cubes of p, q and r are distinct, then S is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 variables. It became commonly accepted, that it is impossible to achieve the same objective by purely algebraic and combinatorial means, like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van den Bergh.
Geometry of Morphogenesis
Nadya MOROZOVA, Robert PENNER - 2014-10-01 (M/14/34)
On Koszulity for operads of Conformal Field Theory
Natalia IYUDU, Abdenacer MAKHLOUF - 2014-09-24 (M/14/31)
We study two closely related operads: the Gelfand-Dorfman operad GD and the Conformal Lie Operad CLie. The latter is the operad governing the Lie conformal algebra structure. We prove Koszulity of the Conformal Lie operad using the Gr ̈bner bases theory for operads and an operadic analogue of the Priddy criterion. An example of deformation of an operad coming from the Hom structures is considered. In particular we study possible deformations of the Associative operad from the point of view of the confluence property. Only one deformation, the operad which governs the identity (α(ab))c = a(α(bc)) turns out to be confluent. We introduce a new Hom structure, namely Hom–Gelfand-Dorfman algebras and study their basic properties.
The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations
Natalia IYUDU, Stanislav SHKARIN - 2014-09-24 (M/14/30)
For an arbitrary associative unital ring R, let J1 and J2 be the following noncommutative birational partly defined involutions on the set M3 (R) of 3 × 3 matrices over R: J1 (M ) = M −1 (the usual matrix inverse) and J2 (M )jk = (Mkj )−1 (the transpose of the Hadamard inverse). We prove the following surprising conjecture by Kontsevich (1996) saying that (J2 ◦ J1 )3 −1 is the identity map modulo the DiagL × DiagR action (D1 , D2 )(M ) = D1 M D2 of pairs of invertible diagonal matrices. That is, we show that for each M in the domain where (J2 ◦J1 )3 is defined, there are invertible −1 diagonal 3 × 3 matrices D1 = D1 (M ) and D2 = D2 (M ) such that (J2 ◦ J1 )3 (M ) = D1 M D2.
Topos-theoretic background
Olivia CARAMELLO - 2014-09-23 (M/14/27)
This text, which will form the first chapter of my book in preparation "Lattices of theories", is a self-contained introduction to topos theory, geometric logic and the 'bridge' technique.
Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective
Olivia CARAMELLO, Anna Carla RUSSO - 2014-09-23 (M/14/28)
We establish, generalizing Di Nola-Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.
Quasi-exact-solvability of the $A_2$ elliptic model: Algebraic form, $sl(3)$ hidden algebra, polynomial eigenfunctions
Vladimir V. SOKOLOV, Alexander V. TURBINER - 2014-09-23 (P/14/29)
Dimensional exactness of self-measures for random countable iterated function systems with overlaps.
Eugen MIHAILESCU, Mariusz URBANSKI - 2014-09-09 (M/14/26)
We study projection measures for random countable (finite or infinite) conformal iterated function systems with arbitrary overlaps. In this setting we extend Feng's and Hu's result from [6] about deterministic finite alphabet iterated function systems. We prove, under a mild assumption of finite entropy, the dimensional exactness of the projections of invariant measures from the shift space, and we give a formula for their dimension, in the context of random infinite conformal iterated function systems with overlaps. There exist numerous differences between our case and the finite deterministic case. We give then applications and concrete estimates for pointwise dimensions of measures, with respect to various classes of random countable IFS with overlaps. Namely, we study several types of randomized systems related to Kahane-Salem sets; also, a random system related to a statistical problem of Sinai; and randomized infinite IFS in the plane for which the number of overlaps is uniformly bounded from above.
Principe de fonctorialité et transformations de Fourier non linéaires : proposition de définitions et esquisse d'une possible (?) démonstration
Laurent LAFFORGUE - 2014-08-21 (M/14/25)
Ce texte rassemble les notes écrites d'une série de quatre exposés donnés à l'IHES les 19 juin, 26 juin, 3 juillet et 8 juillet 2014. Il introduit une nouvelle approche pour une éventuelle démonstration - encore à vérifier - du transfert automorphe de Langlands sur les corps globaux. En attendant donc de vérifier soigneusement si cela marche ou bien non, d'abord dans le cas de GL(2) et des représentations de puissances symétriques de son dual. Le point le plus essentiel, sur lequel tout est fondé, est la propriété de stabilité du tore maximal par convolution (définie comme la transformée de Fourier de la multiplication point par point des fonctions) apparue dans la dernière partie de la partie III.
Extensions of flat functors and theories of presheaf type
Olivia CARAMELLO - 2014-06-26 (M/14/23)
We develop a general theory of extensions of flat functors along geometric morphisms of toposes, and apply it to the study of the class of theories whose classifying topos is equivalent to a presheaf topos. As a result, we obtain a characterization theorem providing necessary and sufficient semantic conditions for a theory to be of presheaf type. This theorem subsumes all the previous partial results obtained on the subject and has several corollaries which can be used in practice for testing whether a given theory is of presheaf type as well as for generating new examples of theories belonging to this class. Along the way, we establish a number of other results of independent interest, including developments about colimits in the context of indexed categories, expansions of geometric theories and methods for constructing theories classified by a given presheaf topos.
Cyclic theories
Olivia CARAMELLO, Nicholas WENTZLAFF - 2014-06-26 (M/14/22)
We describe a geometric theory classified by Connes-Consani's epicylic topos and two related theories respectively classified by the cyclic topos and by the topos $[{\mathbb N}^{\ast}, \Set]$.
A Feynman integral via higher normal functions
Spencer BLOCH, Matt KERR, Pierre VANHOVE - 2014-06-10 (P/14/06)
We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral; one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of $K3$ surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the $K3$ family. We prove a conjecture by David Broadhurst that at a special kinematical point the Feynman integral is given by a critical value of the Hasse-Weil $L$-function of the $K3$ surface. This result is shown to be a particular case of Deligne's conjectures relating values of $L$-functions inside the critical strip to periods.
Beltrami-Courant Differentials and $G_{\infty}$-algebras
Anton ZEITLIN - 2014-05-06 (M/14/19)
Using the symmetry properties of two-dimensional sigma models, we introduce a notion of the Beltrami-Courant differential, so that there is a natural homotopy Gerstenhaber algebra related to it. We conjecture that the generalized Maurer-Cartan equation for the corresponding $L_{\infty}$ subalgebra gives solutions to the Einstein equations.
2-CY-tilted algebras that are not Jacobian
Over any field of positive characteristic we construct 2-CY-tilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields of positive characteristic are 2-CY-tilted even if they do not arise from potentials. In another direction, we compute the fractionally Calabi-Yau dimensions of certain orbit categories of fractionally CY triangulated categories. As an application, we construct a cluster category of type G2.
Algebras of quasi-quaternion type
We define algebras of quasi-quaternion type, which are symmetric algebras of tame representation type whose stable module category has certain structure similar to that of the algebras of quaternion type introduced by Erdmann. We observe that symmetric tame algebras that are also 2-CY-tilted are of quasi quaternion type. We present a combinatorial construction of such algebras by introducing the notion of triangulation quivers. The class of algebras that we get contains Erdmann's algebras of quaternion type on the one hand and the Jacobian algebras of the quivers with potentials associated by Labardini to triangulations of closed surfaces with punctures on the other hand, hence it serves as a bridge between modular representation theory of finite groups and cluster algebras.
SL(2,Z)-invariance and D-instanton contributions to the $D^6 R^4$ interaction
Michael B. GREEN, Stephen D. MILLER, Pierre VANHOVE - 2014-04-09 (P/14/07)
The modular invariant coefficient of the $D^6R^4$ interaction in the low energy expansion of type~IIB string theory has been conjectured to be a solution of an inhomogeneous Laplace eigenvalue equation, obtained by considering the toroidal compactification of two-loop Feynman diagrams of eleven-dimensional supergravity. In this paper we determine its exact $SL(2,\Z)$-invariant solution $f(condition as$y\to \infty$(the weak coupling limit). The solution is presented as a Fourier series with modes$\widehat{f}_n(y) e^{2\pi i n x}$, where the mode coefficients,$\widehat{f}_n(y)$are bilinear in$K$-Bessel functions. Invariance under$SL(2,\Z)$requires these modes to satisfy the nontrivial boundary condition$ \widehat{f}_n(y) =O(y^{-2})$for small$y$, which uniquely determines the solution. The large-$y$expansion of$f(ower-behaved) terms, together with precisely-determined exponentially decreasing contributions that have the form expected of D-instantons, anti-D-instantons and D-instanton/anti-D-instanton pairs.
Algebraic rational cells, equivariant intersection theory, and Poincaré duality
Richard Paul GONZALES VILCARROMERO - 2014-04-09 (M/14/15)
We provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action. Based on this notion, we study the algebraic analogue of Q-fi ltrable varieties: algebraic varieties where a torus acts with isolated fixed points, such that the associated Bialynicki-Birula decomposition consists of algebraic rational cells. We show that the rational equivariant Chow group of any Q-filtrable variety is freely generated by the cell closures. We apply this result to group embeddings, and more general spherical varieties. In view of the localization theorem for equivariant operational Chow rings, we get some conditions for Poincaré duality in this setting.
Motivic Cohomology Spectral Sequence and Steenrod Algebra
Serge YAGUNOV - 2014-04-09 (M/14/16)
For an odd prime number $p$, it is shown that differentials $d_n$ in the motivic cohomology spectral sequence with $p$-local coefficients vanish unless $p-1$ divides $n-1$. We obtain an explicit formula for the first non-trivial differential $d_p$, expressing it in terms of motivic Steenrod $p$-power operations and Bockstein homomorphisms. Finally, we construct examples of varieties, having non-trivial differentials $d_p$ in their motivic spectral sequences.
Boundedness of non-homogeneous square functions and $L^q$ type testing conditions with $q \in (1,2)$
Henri MARTIKAINEN, Mihalis MOURGOGLOU - 2014-04-01 (M/14/14)
Calculabilité de la cohomologie étale modulo l
David A. MADORE, Fabrice ORGOGOZO - 2014-03-20 (M/14/13)
Scattering Equations and String Theory Amplitudes
Emil BJERRUM-BOHR, Poul DAMGAARD, Piotr TOURKINE, Pierre VANHOVE - 2014-03-19 (P/14/11)
Scattering equations for tree-level amplitudes are viewed in the context of string theory. As a result of the comparison we are led to define a new dual model which coincides with string theory in both the small and large $\alpha'$ limit, and whose solution is found algebraically on the surface of solutions to the scattering equations. Because it has support only on the scattering equations, it can be solved exactly, yielding a simple resummed model for $\alpha'$-corrections to all orders. We use the same idea to generalize scattering equations to amplitudes with fermions and any mixture of scalars, gluons and fermions. In all cases checked we find exact agreement with known results.
Localization in equivariant operational K-theory and the Chang-Skjelbred property
Richard Paul GONZALES VILCARROMERO - 2014-03-18 (M/14/12)
We establish a localization theorem of Borel-Atiyah-Segal type for the equivariant operational K-theory of Anderson and Payne. Inspired by the work of Chang-Skjelbred and Goresky-Kottwitz-MacPherson, we establish a general form of GKM theory in this setting, applicable to singular schemes with torus action. Our results are deduced from those in the smooth case via Gillet-Kimura's technique of cohomological descent for equivariant envelopes. As an application, we extend Uma's description of the equivariant K-theory of smooth compactifi cations of reductive groups to the equivariant operational K-theory of all, possibly singular, projective group embeddings.
The physics of quantum gravity
Pierre VANHOVE - 2014-03-17 (P/14/08)
Quantum gravity is still very mysterious and far from being well under- stood. In this text we review the motivations for the quantification of gravity, and some expected physical consequences. We discuss the remarkable rela- tions between scattering processes in quantum gravity and in Yang-Mills theory, and the role of string theory as an unifying theory.
Polylogarithms and multizeta values in massless Feynman amplitudes
Ivan TODOROV - 2014-02-19 (P/14/10)
The last two decades have seen a remarkable development of analytic methods in the study of Feynman amplitudes in perturbative quantum field theory. The present lecture offers a physicists' oriented survey of Francis Brown's work on singlevalued multiple polylogarithms, the associated multizeta periods and their application to Schnetz's graphical functions and to $x$-space renormalization. To keep the discussion concrete we restrict attention to explicit examples of primitively divergent graphs in a massless scalar QFT.
Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability
Willard MILLER JR, Alexander V TURBINER - 2014-02-17 (P/14/09)
The physics and the mixed Hodge structure of Feynman integrals
Pierre VANHOVE - 2014-02-14 (P/14/04)
This expository text is an invitation to the relation between quantum field theory Feynman integrals and periods. We first describe the relation between the Feynman parametrization of loop amplitudes and world-line methods, by explaining that the first Symanzik polynomial is the determinant of the period matrix of the graph, and the second Symanzik polynomial is expressed in terms of world-line Green s functions. We then review the relation between Feynman graphs and variations of mixed Hodge structures. Finally, we provide an algorithm for generating the Picard-Fuchs equation satisfied by the all equal mass banana graphs in a two-dimensional space-time to all loop orders.
Du transfert automorphe de Langlands aux formules de Poisson non linéaires
Laurent LAFFORGUE - 2014-01-17 (M/14/05)
Non-Abelian Lie algebroids over jet spaces
Arthemy V. KISELEV, Andrey O. KRUTOV - 2014-01-07 (M/14/03)
We associate Hamiltonian homological evolutionary vector fields -- which are the non-Abelian variational Lie algebroids' differentials -- with Lie algebra-valued zero-curvature representations for partial differential equations.
Une loi de réciprocité explicite pour le polylogarithme elliptique
Francesco LEMMA, Shanwen WANG - 2014-01-06 (M/14/01)
On démontre une compatibilité entre la réalisation p-adique et la réalisation de de Rham des sections de torsion du profaisceau polylogarithme elliptique. La preuve utilise une variante pour H1 de la loi de réciprocité explicite de Kato pour le H2 des courbes modulaires.
Le système d'Euler de Kato (II)
Shanwen WANG - 2014-01-06 (M/14/02)
Ce texte est le deuxième article d’une série de trois articles sur une généralisation de système d’Euler de Kato. Il est consacré e d’Euler de Kato raffiné associé systèmes d’Euler de Kato.
|
2014-10-31 07:00:49
|
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|
https://grindskills.com/how-do-cnns-avoid-the-vanishing-gradient-problem/
|
# How do CNN’s avoid the vanishing gradient problem
I have been reading a lot about convoloutional neural networks and was wondering how they avoid the vanishing gradient problem. I know deep belief networks stack single level auto-encoders or other pre-trained shallow networks and can thus avoid this problem but I don’t know how it is avoided in CNNs.
According to Wikipedia:
“despite the above-mentioned “vanishing gradient problem,” the
superior processing power of GPUs makes plain back-propagation
feasible for deep feedforward neural networks with many layers.”
I don’t understand why GPU processing would remove this problem?
The vanishing gradient problem requires us to use small learning rates with gradient descent which then needs many small steps to converge. This is a problem if you have a slow computer which takes a long time for each step. If you have a fast GPU which can perform many more steps in a day, this is less of a problem.
There are several ways to tackle the vanishing gradient problem. I would guess that the largest effect for CNNs came from switching from sigmoid nonlinear units to rectified linear units. If you consider a simple neural network whose error $E$ depends on weight $w_{ij}$ only through $y_j$, where
If $f$ is the logistic sigmoid function, $f'$ will be close to zero for large inputs as well as small inputs. If $f$ is a rectified linear unit,
|
2022-12-05 09:06:18
|
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https://brilliant.org/problems/trust-me-its-easytest-for-grade-9-to-10-part-2/
|
# Trust me it's easy.Test for grade 9 to 10 part 2
Algebra Level 4
$\frac{x(x^2-56)}{4-7x}-\frac{21x+22}{x^3+2}=4$ The value $x_{1},x_{2},x_{3},...,x_{n}$ that satisfy this equality Find x max +2010
×
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2017-03-27 04:47:47
|
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https://www.kopykitab.com/blog/anna-university-previous-year-question-paper-maths-ii-ii-sem/
|
# Anna University Previous Year Question Paper – Maths II – II SEM
MODEL PAPER
B.E./B.Tech. DEGREE EXAMINATION.
Second Semester
MA 132 — MATHEMATICS — II
(Common to all branches except Information Technology)
Time : Three hours Maximum : 100 marks
Statistical Tables permitted.
PART A — (10 ´ 2 = 20 marks)
1. Express in polar co–ordinates.
2. Simplify
3. Is the vector , Irrotational?
4. Find where .
5. Prove that real and imaginary parts of an analytic function are harmonic functions.
6. Find the image of under the transformation ?
7. State Couchy’s integral theorem.
8. What is a removable singularity? Give an example.
9. For the set of numbers 5, 10, 8, 2, 7 find second moment.
10. The two regression equations of the variables x and y are : and find the mean and .
PART B — (5 ´ 16 = 80 marks)
1. (i) A survey of 200 families having 3 children selected at random solve the
following results :
Test the hypothesis male and female births are equally likely at 5% level of significance using test. (8)
(ii) A group of 10 rats fed on diet A and another group of 8 rats fed on diet B, recorded the following increase in weight in gms.
In diet A superior to diet B at 5% level of significance? (8)
1. (a) (i) Find the area of the region bounded by using double
integrals. (6)
(ii) Evaluate . (4)
(iii) Evaluate using Beta and Gamma function. (6)
Or
(b) (i) Change the order of integration and evaluate . (6)
(ii) Evaluate . (6)
(iii) Find using Beta and Gamma functions. (4)
1. (a) (i) If find , if .
(8)
(ii) Find the circulation of about the closed curve C in the xy plane where . (8)
Or
(b) (i) Evaluate where and S is the
surface of the cube using
divergence theorem. (6)
(ii) Verify Stokes theorem for over the surface . (10)
1. (a) (i) If is analytic find given that
. (8)
(ii) Find the bi–linear transformation which maps onto . Hence find the fixed points. (8)
Or
(b) (i) If is analytic such that , prove that
. (8)
(ii) Prove that the transformation maps the upper half of the z–plane onto the upper half of the w–plane. What is the image of under this transformation? (8)
1. (a) (i) Expand in a Laurent series valid in and
. (6)
(ii) Evaluate by Contour integration. (10)
Or
(b) (i) Evaluate where C is . (6)
(ii) Evaluate by contour integration. (10)
———————
|
2021-09-20 07:38:22
|
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|
https://stage.geogebra.org/m/gqzry5bz
|
# Exploration with sine and cosine of sums of angles
## Directions:
Assume that the radius of the red arc is "r". 1. Write expressions for the x and y coordinates of the points A and B. If it helps, you can toggle the button and it will show right triangle that may help visualize the coordinates for you. Also note, the angle that reaches point B, is . Label the sides of these triangles on the diagram. 2. Now use the sides of the two smaller triangles to find a way to write the (x,y) coordinates for the point C.
|
2022-05-16 16:18:06
|
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|
https://www.ericburden.work/post/advent-of-code-2022-day-24/
|
# Advent of Code 2022 - Day 24
By Eric Burden | December 28, 2022
It’s that time of year again! Just like last year, I’ll be posting my solutions to the Advent of Code puzzles. This year, I’ll be solving the puzzles in Rust. I’ll post my solutions and code to GitHub as well. After finishing last year (and 2015-2019) in Julia, I needed to spend some time with Rust again! If you haven’t given AoC a try, I encourage you to do so along with me!
# Day 24 - Unstable Diffusion
Find the problem description HERE.
## The Input - Valley Girls Elves
Today’s input is a variation on the theme of “2D maps where each character means a different thing”, which, in this case, represents a walled valley with “blizzards” blowing around inside it. The puzzle text sort of hints at this, but it turns out that the examples and the actual inputs both represent states that will eventually cycle back around to where they began. This means that our input struct will essentially be a 3-dimensional vector where the third dimension is time (in a loop). For my input, that was 600 different arrangements of the 2D valley before it cycled around to the beginning.
use anyhow::{bail, Error, Result};
use itertools::Itertools;
use std::collections::{HashMap, HashSet};
/// Represents the direction in which a particular Blizzard is blowing.
/// We have impls to allow converting to/from single-bit u8 values so
/// that a set of four directions can be represented as a u8.
#[derive(Debug, Clone, Copy, PartialEq, Eq, Hash)]
enum Direction {
Down,
Left,
Up,
Right,
}
impl TryFrom<u8> for Direction {
type Error = Error;
fn try_from(value: u8) -> Result<Self, Self::Error> {
match value {
1 => Ok(Direction::Down),
2 => Ok(Direction::Left),
4 => Ok(Direction::Up),
8 => Ok(Direction::Right),
_ => bail!("No direction corresponding to {value}!"),
}
}
}
impl Direction {
/// Convert a Direction to a single-bit u8 value
const fn value(&self) -> u8 {
match self {
Direction::Down => 1,
Direction::Left => 2,
Direction::Up => 4,
Direction::Right => 8,
}
}
/// Get an array of all four Directions
const fn all() -> [Direction; 4] {
[
Direction::Down,
Direction::Left,
Direction::Up,
Direction::Right,
]
}
}
/// Represents one or more blizzards occupying a single space in the Valley.
/// Because there's only one blizzard per space in the input, there can ever
/// only be one blizzard per direction in a single space at any time after
/// the blizzards begin moving in the Valley. So, we never need more than
/// the presence/absence of each of the four directions per Blizzard space.
#[derive(Debug, Default, Clone, Copy, PartialEq, Eq, Hash)]
struct Blizzard(u8);
impl Blizzard {
fn from(direction: Direction) -> Self {
Blizzard(direction.value())
}
fn direction(&self) -> Result<Direction> {
Direction::try_from(self.0)
}
/// Add a direction to the current Blizzard
fn add(&mut self, direction: Direction) {
self.0 |= direction.value();
}
/// Check if the Blizzard space has a Blizzard blowing in a
/// particular direction.
fn has(&self, direction: Direction) -> bool {
self.0 & direction.value() > 0
}
}
/// Represents a single space in the Valley. Can either be a space with Blizzards
/// blowing in one or more directions, an impassable Wall, or an Empty space where
/// the elven expedition can walk.
#[derive(Debug, Clone, PartialEq, Eq, Hash)]
enum Space {
Blizzard(Blizzard),
Wall,
Empty,
}
/// Represents the entire Valley and all the Blizzards blowing through it at a
/// given point in time.
#[derive(Debug, Clone, PartialEq, Eq, Hash)]
struct Valley {
rows: usize,
cols: usize,
spaces: Vec<Vec<Space>>,
}
/// Produce a new Valley from a 2D grid of Spaces.
impl From<Vec<Vec<Space>>> for Valley {
fn from(spaces: Vec<Vec<Space>>) -> Self {
let rows = spaces.len();
let cols = spaces.first().map(|line| line.len()).unwrap_or_default();
Valley { rows, cols, spaces }
}
}
impl From<&str> for Valley {
fn from(input: &str) -> Self {
let mut spaces = Vec::new();
for line in input.lines() {
let row = line
.chars()
.map(|glyph| match glyph {
'>' => Space::Blizzard(Blizzard::from(Direction::Right)),
'<' => Space::Blizzard(Blizzard::from(Direction::Left)),
'^' => Space::Blizzard(Blizzard::from(Direction::Up)),
'v' => Space::Blizzard(Blizzard::from(Direction::Down)),
'.' => Space::Empty,
'#' => Space::Wall,
_ => unreachable!(),
})
.collect::<Vec<_>>();
spaces.push(row);
}
Valley::from(spaces)
}
}
impl Valley {
/// Produce a clone of this Valley with its state advanced by one minute. All
/// Blizzards move forward by one space in the direction they are facing,
/// wrapping around the Valley when they encounter a Wall.
// Creates a new mutable clone of this Valley with only Empty Spaces
let Valley { rows, cols, spaces } = self;
let mut new_spaces = vec![vec![Space::Empty; *cols]; *rows];
let mut new_state = Valley::from(new_spaces);
// For each Space in the current Valley, update the appropriate space in the
// new state. Move Blizzards and set Walls. Skip the Empties, since all the
// Spaces in the new state are already Empty.
for (row, col) in (0..*rows).cartesian_product(0..*cols) {
match spaces[row][col] {
Space::Blizzard(blizzard) => {
for direction in Direction::all() {
if !blizzard.has(direction) {
continue;
}
let (new_row, new_col) = self.next_position(row, col, direction);
}
}
Space::Wall => new_state.spaces[row][col] = Space::Wall,
Space::Empty => continue,
}
}
// Return the new Valley
new_state
}
/// Given a starting row and column and a Direction to move, return the row
/// and column where you'd end up if you moved in that Direction.
fn next_position(&self, row: usize, col: usize, direction: Direction) -> (usize, usize) {
match direction {
// Wrapping for increasing rows/columns
Direction::Down => ((row % (self.rows - 2)) + 1, col),
Direction::Right => (row, (col % (self.cols - 2)) + 1),
// Wrapping for decreasing rows/columns
Direction::Left => match col - 1 {
0 => (row, self.cols - 2),
_ => (row, col - 1),
},
Direction::Up => match row - 1 {
0 => (self.rows - 2, col),
_ => (row - 1, col),
},
}
}
/// Given a row and column and the Direction a blizzard is blowing, add that
/// blizzard Direction to that Space. If the Space is empty, convert it to a
/// Blizzard. If there's already a Blizzard there, just add the new Direction
/// to the existing Blizzard. Attempts to add a Blizzard to a Wall will
/// definitely fail.
fn add_blizzard(&mut self, row: usize, col: usize, direction: Direction) {
match &mut self.spaces[row][col] {
Space::Wall => panic!("Tried to add a blizzard to a wall!"),
Space::Empty => self.spaces[row][col] = Space::Blizzard(Blizzard::from(direction)),
}
}
}
const INPUT: &str = include_str!("../../input/24/input.txt");
/// Parse the initial Valley state from the input, then advance the state
/// minute-by-minute until we reach a state we've seen before. That's right,
/// the state of the Valley cycles! This way, we can have in memory a map
/// of which Spaces are Empty and which ones have Blizzards for each point
/// in time. This means we don't need to re-calculate each state on the fly,
/// potentially re-creating the same state multiple times. Store the Valley
/// states in a Vector where the index indicates the minute at which that
/// state is valid.
let mut valley = Valley::from(INPUT);
let mut valley_states = Vec::new();
let mut seen_states = HashSet::new();
while !seen_states.contains(&valley) {
seen_states.insert(valley.clone());
valley_states.push(valley.clone());
}
valley_states
}
The most difficult part was the fact that the start and end spaces are embedded in the top and bottom walls. Otherwise, I’d be tempted to just ignore the walls and only use the inner portion of the map, It’d make wrapping the blizzards around a lot easier. Regardless, now that we’ve generated the entire map, we have everything we need to solve the puzzle.
## Part One - Walk The Line
Mission accomplished! Time to bug out. Unfortunately, out is on the other side of a box valley filled with completely predictably blowing storms. Ah, man! Well, at least the elephants and monkeys agreed to stay at the grove and tend to things there. Miscommunication at a sensitive time like this could be disastrous. We have our map, it’s time to calculate a shortest path!
use core::cmp::Reverse;
use std::collections::{BinaryHeap, HashMap};
/// Solve Day 24, Part 1
fn solve(input: &[Valley]) -> u32 {
let first_state = input.get(0).expect("Valley should have an initial state!");
// All examples and input have the start Space at index (0, 1) and the end
// Space on the bottom row, next to the last column.
let start_at = Expedition(0, 1);
let end_at = Expedition(first_state.rows - 1, first_state.cols - 2);
// Calculate the length of the shortest path through the Valley.
if let Some(minutes) = start_at.shortest_path(end_at, 0, input) {
return minutes as u32;
}
// Unless we can't find a path. Then, freak out! This doesn't happen,
// though. Not anymore...
panic!("Could not find a way through the valley. Died of frostbite!");
}
/// Represents the location of the elven Expedition through the Valley.
#[derive(Debug, Clone, Copy, PartialEq, Eq, Hash, PartialOrd, Ord)]
struct Expedition(usize, usize);
impl Expedition {
/// Given the current location of the elves and a state of the Valley,
/// determine which next steps are possible, in order to avoid landing
/// on a Space with an active Blizzard. There are a maximum of five
/// possible moves, including waiting.
fn possible_next_steps(&self, valley: &Valley) -> [Option<Expedition>; 5] {
let Expedition(row, col) = *self;
let mut possible_steps = [None; 5];
// Attempt to move to the left
if let Space::Empty = valley.spaces[row][col - 1] {
possible_steps[0] = Some(Expedition(row, col - 1));
}
// Attempt to move to the right
if let Space::Empty = valley.spaces[row][col + 1] {
possible_steps[1] = Some(Expedition(row, col + 1));
}
// The upward move needs to account for the fact that the
// final space is in the top row, which means that moving into
// the top row is possible.
if row > 0 {
if let Space::Empty = valley.spaces[row - 1][col] {
possible_steps[2] = Some(Expedition(row - 1, col));
}
}
// The downard move needs to account for the beginning space
// being on the last row, which means that moving into the last row
// is possible.
if row < (valley.spaces.len() - 1) {
if let Space::Empty = valley.spaces[row + 1][col] {
possible_steps[3] = Some(Expedition(row + 1, col));
}
}
// Waiting is a necessary option if there's nothing in our current space
if let Space::Empty = valley.spaces[row][col] {
possible_steps[4] = Some(Expedition(row, col));
}
possible_steps
}
/// Find the shortest path from this Expedition's location to the target,
/// assuming we start the journey at minute start_time. Pass in a reference
/// to the time states of the Valley so we can know which Spaces are Empty
/// for each minute. It's a Dijkstra's implementation.
fn shortest_path(
&self,
target: Expedition,
start_time: usize,
valley_states: &[Valley],
) -> Option<usize> {
// Sort locations in the Heap by the least number of minutes spent. Uniquely
// identify states along the path by the location of the Expedition for a
// given state of the Valley.
let mut open = BinaryHeap::from([(Reverse(start_time), *self)]);
let mut minutes = HashMap::from([((*self, start_time % valley_states.len()), start_time)]);
// So long as there are states to explore, take the one with the fewest
// number of minutes passed.
while let Some((Reverse(minutes_passed), expedition)) = open.pop() {
// If we've found the end, return the number of minutes passed
if expedition == target {
return Some(minutes_passed);
}
// Get the state of the Valley in the next minute to identify
// which Spaces are available to be moved to.
let state_idx = (minutes_passed + 1) % valley_states.len();
let state = &valley_states[state_idx];
// Check each next step to see if this is the fastest we've gotten
// to that state. If so, keep it and add it to the heap. Otherwise,
// keep moving on.
for step in expedition.possible_next_steps(state).into_iter().flatten() {
let next_minutes = minutes_passed + 1;
let curr_minutes = *minutes.get(&(step, state_idx)).unwrap_or(&usize::MAX);
if next_minutes >= curr_minutes {
continue;
}
open.push((Reverse(next_minutes), step));
minutes.insert((step, state_idx), next_minutes);
}
}
None // Something has gone terribly wrong...
}
}
No sweat, we’ve solved problems like this before already this year. Let’s see what part two is like.
## Part Two - There and Back Again
You forgot what!? Dude, we’re literally on our way out of the jungle. We’ll be back at the North Pole in days! Besides, we all know that Bazingles over there has plenty of snacks to share, we figured that out on Day 1! Ugh, fine, I’ll go back and get them. But you owe me!
/// Solve Day 24, Part 2
fn solve(input: &[Valley]) -> u32 {
let first_state = input.get(0).expect("Valley should have an initial state!");
// All examples and input have the start Space at index (0, 1) and the end
// Space on the bottom row, next to the last column.
let start_at = Expedition(0, 1);
let end_at = Expedition(first_state.rows - 1, first_state.cols - 2);
// Start at zero minutes and move from the start to the end.
let Some(minutes) = start_at.shortest_path(end_at, 0, input) else {
panic!("Could not find a way through the valley. Died of frostbite!");
};
// Turn around and head back to the start.
let Some(minutes) = end_at.shortest_path(start_at, minutes, input) else {
panic!("Could not find a way back! Died in a blizzard!");
};
// Then turn around and head back to the end again.
let Some(minutes) = start_at.shortest_path(end_at, minutes, input) else {
panic!("Could not find a way back! Died of embarassment!");
};
// Return the total number of minutes it took to travel all that way.
minutes as u32
}
Good thing we added that start_time argument to our shortest path function. Now we can start searching from any point in time. Turns out, it wasn’t so bad retrieving those snacks. Definitely going to extract that favor from that elf, though.
## Wrap Up
Pathfinding with a twist! A nice, classic Advent of Code puzzle for Christmas Eve. There’s something comforting about that, really. Today’s puzzle would definitely have been more challenging without realizing that the state of the valley was on a cycle. In the end, though, Dijkstra’s with a twist is definitely a fun way to finish out the last day with two whole parts. Merry Christmas!
|
2023-03-23 12:05:05
|
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|
https://kim-api.readthedocs.io/en/devel/portable-models_2ex__model___ar___s_l_j___multi_cutoff_2_r_e_a_d_m_e_source.html
|
kim-api 2.1.4-git+v2.1.3-git-1-g7847914a.GNU An Application Programming Interface (API) for the Knowledgebase of Interatomic Models (KIM).
README
Go to the documentation of this file.
1 #
2 # CDDL HEADER START
3 #
4 # The contents of this file are subject to the terms of the Common Development
5 # and Distribution License Version 1.0 (the "License").
6 #
7 # You can obtain a copy of the license at
8 # http://www.opensource.org/licenses/CDDL-1.0. See the License for the
9 # specific language governing permissions and limitations under the License.
10 #
11 # When distributing Covered Code, include this CDDL HEADER in each file and
12 # include the License file in a prominent location with the name LICENSE.CDDL.
13 # If applicable, add the following below this CDDL HEADER, with the fields
14 # enclosed by brackets "[]" replaced with your own identifying information:
15 #
16 # Portions Copyright (c) [yyyy] [name of copyright owner]. All rights reserved.
17 #
18 # CDDL HEADER END
19 #
20
21 #
22 # Copyright (c) 2013--2019, Regents of the University of Minnesota.
23 # All rights reserved.
24 #
25 # Contributors:
26 # Ellad B. Tadmor
27 #
28
29 Spring-modified Lennard-Jones (SLJ) pair potential model for Ar.
30
31 V = 0.5 \sum_i \sum_j eps_i eps_j 4 [ (sig/r_ij)^12 - (sig/r_ij)^6 ] (1)
32
33 where
34 eps_i = 0.5 \sum_k spring (r_ik)^2 (2)
35
36 The potential parameters are sig and the spring constant sig.
37
38 This potential uses two cutoffs and requires neighbors of padding
39 atoms for the summ in (2).
40
41 The loops in (2) are over all atoms for which r_ik < cutoff1
42
43 The loops in (1) are over all atoms for which r_ij < cutoff2
44
45 For the Ar parameterization, cutoff1 is set so that only nearest
46 neighbors contribute in the equilibrium 0K fcc structure, and
47 cutoff2 is set to include the third neighbor distance. The potential
48 is discontinuous at both cutoffs. It is designed for testing purposes
49 only.
50
51 The equilibrium spacing for the fcc structure predicted by the SLJ potential is
52
53 a0 = 2^(1/3)/sqrt(3) * (94297/491)^(1/6) sig = 1.74724 sig
54
55 The first, second and third neighbor distances in fcc are then:
56
57 r_NN1 = a0/sqrt(2) = 1.235 sig
58 r_NN2 = a0 = 1.747 sig
59 r_NN3 = a0*sqrt(3/2) = 2.140 sig
60
61 The parameters sig and spring are selected to reproduce the experimental
62 properties of fcc argon:
63
64 a0 = 5.26 A
65 E_c = 0.0104 eV
|
2019-08-22 11:26:33
|
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|
http://composingprograms.com/pages/32-functional-programming.html
|
## 3.2 Functional Programming
The software running on any modern computer is written in a variety of programming languages. There are physical languages, such as the machine languages for particular computers. These languages are concerned with the representation of data and control in terms of individual bits of storage and primitive machine instructions. The machine-language programmer is concerned with using the given hardware to erect systems and utilities for the efficient implementation of resource-limited computations. High-level languages, erected on a machine-language substrate, hide concerns about the representation of data as collections of bits and the representation of programs as sequences of primitive instructions. These languages have means of combination and abstraction, such as function definition, that are appropriate to the larger-scale organization of software systems.
In this section, we introduce a high-level programming language that encourages a functional style. Our object of study, a subset of the Scheme language, employs a very similar model of computation to Python's, but uses only expressions (no statements), specializes in symbolic computation, and employs only immutable values.
Scheme is a dialect of Lisp, the second-oldest programming language that is still widely used today (after Fortran). The community of Lisp programmers has continued to thrive for decades, and new dialects of Lisp such as Clojure have some of the fastest growing communities of developers of any modern programming language. To follow along with the examples in this text, you can download a Scheme interpreter.
### 3.2.1 Expressions
Scheme programs consist of expressions, which are either call expressions or special forms. A call expression consists of an operator expression followed by zero or more operand sub-expressions, as in Python. Both the operator and operand are contained within parentheses:
(quotient 10 2)
Scheme exclusively uses prefix notation. Operators are often symbols, such as + and *. Call expressions can be nested, and they may span more than one line:
(+ (* 3 5) (- 10 6))
(+ (* 3
(+ (* 2 4)
(+ 3 5)))
(+ (- 10 7)
6))
As in Python, Scheme expressions may be primitives or combinations. Number literals are primitives, while call expressions are combined forms that include arbitrary sub-expressions. The evaluation procedure of call expressions matches that of Python: first the operator and operand expressions are evaluated, and then the function that is the value of the operator is applied to the arguments that are the values of the operands.
The if expression in Scheme is a special form, meaning that while it looks syntactically like a call expression, it has a different evaluation procedure. The general form of an if expression is:
(if <predicate> <consequent> <alternative>)
To evaluate an if expression, the interpreter starts by evaluating the <predicate> part of the expression. If the <predicate> evaluates to a true value, the interpreter then evaluates the <consequent> and returns its value. Otherwise it evaluates the <alternative> and returns its value.
Numerical values can be compared using familiar comparison operators, but prefix notation is used in this case as well:
(>= 2 1)
The boolean values #t (or true) and #f (or false) in Scheme can be combined with boolean special forms, which have evaluation procedures similar to those in Python.
• (and <e1> ... <en>) The interpreter evaluates the expressions <e> one at a time, in left-to-right order. If any <e> evaluates to false, the value of the and expression is false, and the rest of the <e>'s are not evaluated. If all <e>'s evaluate to true values, the value of the and expression is the value of the last one.
• (or <e1> ... <en>) The interpreter evaluates the expressions <e> one at a time, in left-to-right order. If any <e> evaluates to a true value, that value is returned as the value of the or expression, and the rest of the <e>'s are not evaluated. If all <e>'s evaluate to false, the value of the or expression is false.
• (not <e>) The value of a not expression is true when the expression <e> evaluates to false, and false otherwise.
### 3.2.2 Definitions
Values can be named using the define special form:
(define pi 3.14)
(* pi 2)
New functions (called procedures in Scheme) can be defined using a second version of the define special form. For example, to define squaring, we write:
(define (square x) (* x x))
The general form of a procedure definition is:
(define (<name> <formal parameters>) <body>)
The <name> is a symbol to be associated with the procedure definition in the environment. The <formal parameters> are the names used within the body of the procedure to refer to the corresponding arguments of the procedure. The <body> is an expression that will yield the value of the procedure application when the formal parameters are replaced by the actual arguments to which the procedure is applied. The <name> and the <formal parameters> are grouped within parentheses, just as they would be in an actual call to the procedure being defined.
Having defined square, we can now use it in call expressions:
(square 21)
(square (+ 2 5))
(square (square 3))
User-defined functions can take multiple arguments and include special forms:
(define (average x y)
(/ (+ x y) 2))
(average 1 3)
(define (abs x)
(if (< x 0)
(- x)
x))
(abs -3)
Scheme supports local definitions with the same lexical scoping rules as Python. Below, we define an iterative procedure for computing square roots using nested definitions are recursion:
(define (sqrt x)
(define (good-enough? guess)
(< (abs (- (square guess) x)) 0.001))
(define (improve guess)
(average guess (/ x guess)))
(define (sqrt-iter guess)
(if (good-enough? guess)
guess
(sqrt-iter (improve guess))))
(sqrt-iter 1.0))
(sqrt 9)
Anonymous functions are created using the lambda special form. Lambda is used to create procedures in the same way as define, except that no name is specified for the procedure:
(lambda (<formal-parameters>) <body>)
The resulting procedure is just as much a procedure as one that is created using define. The only difference is that it has not been associated with any name in the environment. In fact, the following expressions are equivalent:
(define (plus4 x) (+ x 4))
(define plus4 (lambda (x) (+ x 4)))
Like any expression that has a procedure as its value, a lambda expression can be used as the operator in a call expression:
((lambda (x y z) (+ x y (square z))) 1 2 3)
### 3.2.3 Compound values
Pairs are built into the Scheme language. For historical reasons, pairs are created with the cons built-in function, and the elements of a pair are accessed with car and cdr:
(define x (cons 1 2))
x
(car x)
(cdr x)
Recursive lists are also built into the language, using pairs. A special value denoted nil or '() represents the empty list. A recursive list value is rendered by placing its elements within parentheses, separated by spaces:
(cons 1
(cons 2
(cons 3
(cons 4 nil))))
(list 1 2 3 4)
(define one-through-four (list 1 2 3 4))
(car one-through-four)
(cdr one-through-four)
(car (cdr one-through-four))
(cons 10 one-through-four)
(cons 5 one-through-four)
Whether a list is empty can be determined using the primitive null? predicate. Using it, we can define the standard sequence operations for computing length and selecting elements:
(define (length items)
(if (null? items)
0
(+ 1 (length (cdr items)))))
(define (getitem items n)
(if (= n 0)
(car items)
(getitem (cdr items) (- n 1))))
(define squares (list 1 4 9 16 25))
(length squares)
(getitem squares 3)
### 3.2.4 Symbolic Data
All the compound data objects we have used so far were constructed ultimately from numbers. One of Scheme's strengths is working with arbitrary symbols as data.
In order to manipulate symbols we need a new element in our language: the ability to quote a data object. Suppose we want to construct the list (a b). We can't accomplish this with (list a b), because this expression constructs a list of the values of a and b rather than the symbols themselves. In Scheme, we refer to the symbols a and b rather than their values by preceding them with a single quotation mark:
(define a 1)
(define b 2)
(list a b)
(list 'a 'b)
(list 'a b)
In Scheme, any expression that is not evaluated is said to be quoted. This notion of quotation is derived from a classic philosophical distinction between a thing, such as a dog, which runs around and barks, and the word "dog" that is a linguistic construct for designating such things. When we use "dog" in quotation marks, we do not refer to some dog in particular but instead to a word. In language, quotation allow us to talk about language itself, and so it is in Scheme:
(list 'define 'list)
Quotation also allows us to type in compound objects, using the conventional printed representation for lists:
(car '(a b c))
(cdr '(a b c))
The full Scheme language contains additional features, such as mutation operations, vectors, and maps. However, the subset we have introduced so far provides a rich functional programming language capable of implementing many of the ideas we have discussed so far in this text.
### 3.2.5 Turtle graphics
The implementation of Scheme that serves as a companion to this text includes Turtle graphics, an illustrating environment developed as part of the Logo language (another Lisp dialect). This turtle begins in the center of a canvas, moves and turns based on procedures, and draws lines behind it as it moves. While the turtle was invented to engage children in the act of programming, it remains an engaging graphical tool for even advanced programmers.
At any moment during the course of executing a Scheme program, the turtle has a position and heading on the canvas. Single-argument procedures such as forward and right change the position and heading of the turtle. Common procedures have abbreviations: forward can also be called as fd, etc. The begin special form in Scheme allows a single expression to include multiple sub-expressions. This form is useful for issuing multiple commands:
> (define (repeat k fn) (if (> k 0)
(begin (fn) (repeat (- k 1) fn))
nil))
> (repeat 5
(lambda () (fd 100)
(repeat 5
(lambda () (fd 20) (rt 144)))
(rt 144)))
nil
The full repertoire of Turtle procedures is also built into Python as the turtle library module.
As a final example, Scheme can express recursive drawings using its turtle graphics in a remarkably compact form. Sierpinski's triangle is a fractal that draws each triangle as three neighboring triangles that have vertexes at the midpoints of the legs of the triangle that contains them. It can be drawn to a finite recursive depth by this Scheme program:
> (define (repeat k fn)
(if (> k 0)
(begin (fn) (repeat (- k 1) fn))
nil))
> (define (tri fn)
(repeat 3 (lambda () (fn) (lt 120))))
> (define (sier d k)
(tri (lambda ()
(if (= k 1) (fd d) (leg d k)))))
> (define (leg d k)
(sier (/ d 2) (- k 1))
(penup)
(fd d)
(pendown))
The triangle procedure is a general method for repeating a drawing procedure three times with a left turn following each repetition. The sier procedure takes a length d and a recursive depth k. It draws a plain triangle if the depth is 1, and otherwise draws a triangle made up of calls to leg. The leg procedure draws a single leg of a recursive Sierpinski triangle by a recursive call to sier that fills the first half of the length of the leg, then by moving the turtle to the next vertex. The procedures penup and pendown stop the turtle from drawing as it moves by lifting its pen up and the placing it down again. The mutual recursion between sier and leg yields this result:
> (sier 400 6)
Continue: 3.3 Exceptions
|
2014-09-21 14:06:16
|
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|
http://mathematica.stackexchange.com/questions/16270/linear-equation-with-complex-numbers/16271
|
# Linear equation with complex numbers
I have to solve an equation of the type $$a z+b \overline{z}=c$$ with $a,b,c\in\mathbb{C}$.
My approach is to set $F(z)=a z+b \overline{z}-c$ transform $z$ to $x+i y$ and then get a real linear system by assuming $x,y\in\mathbb{R}$ and solve
$$\frac{F(x+i y)+\overline{F(x+i y)}}{2}=0$$ $$\frac{F(x+i y)-\overline{F(x+i y)}}{2i}=0.$$
My problem is that this has to happen a lot for a program that I am writing and I was wondering if there is a faster way to do that. It would be ideal if mathematica can solve these linear systems directly.
-
You can use ComplexExpand. – b.gatessucks Dec 13 '12 at 16:06
I don't really see how this simplifies the procedure. – tst Dec 13 '12 at 16:11
Was trying to think more generally; will try to prepare an example. – b.gatessucks Dec 13 '12 at 16:24
Why not just the following?
Solve[a z + b Conjugate[z] == c, z]
Or, if you prefer separating real and complex parts:
Solve[a (x + I y) + b (x - I y) == c, {x, y}]
Or, if you want to worry about degenerate cases, replace Solve above by Reduce in each form.
-
Well I have no idea why, I just checked and it works fine. I feel stupid now. – tst Dec 13 '12 at 16:16
You might need to use some additional conditions, such as specifying that z is an element of Complex... – tkott Dec 13 '12 at 17:32
As per comment, can use ComplexExpand. Here is one way to go about that.
Solve[
ComplexExpand[{Re[a*z + b*Conjugate[z] - c],
Im[a*z + b*Conjugate[z] - c]}, {a, b, c, z}] == 0, {Re[z], Im[z]}]
(* {{Re[
z] -> -((-Im[a] Im[c] + Im[b] Im[c] - Re[a] Re[c] + Re[b] Re[c])/(
Im[a]^2 - Im[b]^2 + Re[a]^2 - Re[b]^2)),
Im[z] -> -((-Im[c] Re[a] - Im[c] Re[b] + Im[a] Re[c] +
Im[b] Re[c])/(Im[a]^2 - Im[b]^2 + Re[a]^2 - Re[b]^2))}} *)
-
|
2015-05-05 08:19:41
|
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|
https://stacks.math.columbia.edu/tag/05E5
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## 15.88 Formal glueing of module categories
Fix a Noetherian scheme $X$, and a closed subscheme $Z$ with complement $U$. Our goal is to explain how coherent sheaves on $X$ can be constructed (uniquely) from coherent sheaves on the formal completion of $X$ along $Z$, and those on $U$ with a suitable compatibility on the overlap. We first do this using only commutative algebra (this section) and later we explain this in the setting of algebraic spaces (Pushouts of Spaces, Section 79.10).
Here are some references treating some of the material in this section: [Section 2, ArtinII], [Appendix, Ferrand-Raynaud], , [MB], and [Section 4.6, dJ-crystalline].
Lemma 15.88.1. Let $\varphi : R \to S$ be a ring map. Let $I \subset R$ be an ideal. The following are equivalent
1. $\varphi$ is flat and $R/I \to S/IS$ is faithfully flat,
2. $\varphi$ is flat, and the map $\mathop{\mathrm{Spec}}(S/IS) \to \mathop{\mathrm{Spec}}(R/I)$ is surjective.
3. $\varphi$ is flat, and the base change functor $M \mapsto M \otimes _ R S$ is faithful on modules annihilated by $I$, and
4. $\varphi$ is flat, and the base change functor $M \mapsto M \otimes _ R S$ is faithful on $I$-power torsion modules.
Proof. If $R \to S$ is flat, then $R/I^ n \to S/I^ nS$ is flat for every $n$, see Algebra, Lemma 10.39.7. Hence (1) and (2) are equivalent by Algebra, Lemma 10.39.16. The equivalence of (1) with (3) follows by identifying $I$-torsion $R$-modules with $R/I$-modules, using that
$M \otimes _ R S = M \otimes _{R/I} S/IS$
for $R$-modules $M$ annihilated by $I$, and Algebra, Lemma 10.39.14. The implication (4) $\Rightarrow$ (3) is immediate. Assume (3). We have seen above that $R/I^ n \to S/I^ nS$ is flat, and by assumption it induces a surjection on spectra, as $\mathop{\mathrm{Spec}}(R/I^ n) = \mathop{\mathrm{Spec}}(R/I)$ and similarly for $S$. Hence the base change functor is faithful on modules annihilated by $I^ n$. Since any $I$-power torsion module $M$ is the union $M = \bigcup M_ n$ where $M_ n$ is annihilated by $I^ n$ we see that the base change functor is faithful on the category of all $I$-power torsion modules (as tensor product commutes with colimits). $\square$
Lemma 15.88.2. Assume $(\varphi : R \to S, I)$ satisfies the equivalent conditions of Lemma 15.88.1. The following are equivalent
1. for any $I$-power torsion module $M$, the natural map $M \to M \otimes _ R S$ is an isomorphism, and
2. $R/I \to S/IS$ is an isomorphism.
Proof. The implication (1) $\Rightarrow$ (2) is immediate. Assume (2). First assume that $M$ is annihilated by $I$. In this case, $M$ is an $R/I$-module. Hence, we have an isomorphism
$M \otimes _ R S = M \otimes _{R/I} S/IS = M \otimes _{R/I} R/I = M$
proving the claim. Next we prove by induction that $M \to M \otimes _ R S$ is an isomorphism for any module $M$ is annihilated by $I^ n$. Assume the induction hypothesis holds for $n$ and assume $M$ is annihilated by $I^{n + 1}$. Then we have a short exact sequence
$0 \to I^ nM \to M \to M/I^ nM \to 0$
and as $R \to S$ is flat this gives rise to a short exact sequence
$0 \to I^ nM \otimes _ R S \to M \otimes _ R S \to M/I^ nM \otimes _ R S \to 0$
Using that the canonical map is an isomorphism for $M' = I^ nM$ and $M'' = M/I^ nM$ (by induction hypothesis) we conclude the same thing is true for $M$. Finally, suppose that $M$ is a general $I$-power torsion module. Then $M = \bigcup M_ n$ where $M_ n$ is annihilated by $I^ n$ and we conclude using that tensor products commute with colimits. $\square$
Lemma 15.88.3. Assume $\varphi : R \to S$ is a flat ring map and $I \subset R$ is a finitely generated ideal such that $R/I \to S/IS$ is an isomorphism. Then
1. for any $R$-module $M$ the map $M \to M \otimes _ R S$ induces an isomorphism $M[I^\infty ] \to (M \otimes _ R S)[(IS)^\infty ]$ of $I$-power torsion submodules,
2. the natural map
$\mathop{\mathrm{Hom}}\nolimits _ R(M, N) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ S(M \otimes _ R S, N \otimes _ R S)$
is an isomorphism if either $M$ or $N$ is $I$-power torsion, and
3. the base change functor $M \mapsto M \otimes _ R S$ defines an equivalence of categories between $I$-power torsion modules and $IS$-power torsion modules.
Proof. Note that the equivalent conditions of both Lemma 15.88.1 and Lemma 15.88.2 are satisfied. We will use these without further mention. We first prove (1). Let $M$ be any $R$-module. Set $M' = M/M[I^\infty ]$ and consider the exact sequence
$0 \to M[I^\infty ] \to M \to M' \to 0$
As $M[I^\infty ] = M[I^\infty ] \otimes _ R S$ we see that it suffices to show that $(M' \otimes _ R S)[(IS)^\infty ] = 0$. Write $I = (f_1, \ldots , f_ t)$. By Lemma 15.87.4 we see that $M'[I^\infty ] = 0$. Hence for every $n > 0$ the map
$M' \longrightarrow \bigoplus \nolimits _{i = 1, \ldots t} M', \quad x \longmapsto (f_1^ n x, \ldots , f_ t^ n x)$
is injective. As $S$ is flat over $R$ also the corresponding map $M' \otimes _ R S \to \bigoplus _{i = 1, \ldots t} M' \otimes _ R S$ is injective. This means that $(M' \otimes _ R S)[I^ n] = 0$ as desired.
Next we prove (2). If $N$ is $I$-power torsion, then $N \otimes _ R S = N$ and the displayed map of (2) is an isomorphism by Algebra, Lemma 10.14.3. If $M$ is $I$-power torsion, then the image of any map $M \to N$ factors through $M[I^\infty ]$ and the image of any map $M \otimes _ R S \to N \otimes _ R S$ factors through $(N \otimes _ R S)[(IS)^\infty ]$. Hence in this case part (1) guarantees that we may replace $N$ by $N[I^\infty ]$ and the result follows from the case where $N$ is $I$-power torsion we just discussed.
Next we prove (3). The functor is fully faithful by (2). For essential surjectivity, we simply note that for any $IS$-power torsion $S$-module $N$, the natural map $N \otimes _ R S \to N$ is an isomorphism. $\square$
Lemma 15.88.4. Assume $\varphi : R \to S$ is a flat ring map and $I \subset R$ is a finitely generated ideal such that $R/I \to S/IS$ is an isomorphism. For any $f_1, \ldots , f_ r \in R$ such that $V(f_1, \ldots , f_ r) = V(I)$
1. the map of Koszul complexes $K(R, f_1, \ldots , f_ r) \to K(S, f_1, \ldots , f_ r)$ is a quasi-isomorphism, and
2. The map of extended alternating Čech complexes
$\xymatrix{ R \to \prod _{i_0} R_{f_{i_0}} \to \prod _{i_0 < i_1} R_{f_{i_0}f_{i_1}} \to \ldots \to R_{f_1\ldots f_ r} \ar[d] \\ S \to \prod _{i_0} S_{f_{i_0}} \to \prod _{i_0 < i_1} S_{f_{i_0}f_{i_1}} \to \ldots \to S_{f_1\ldots f_ r} }$
is a quasi-isomorphism.
Proof. In both cases we have a complex $K_\bullet$ of $R$ modules and we want to show that $K_\bullet \to K_\bullet \otimes _ R S$ is a quasi-isomorphism. By Lemma 15.88.2 and the flatness of $R \to S$ this will hold as soon as all homology groups of $K$ are $I$-power torsion. This is true for the Koszul complex by Lemma 15.28.6 and for the extended alternating Čech complex by Lemma 15.29.5. $\square$
Lemma 15.88.5. Let $R$ be a ring. Let $I = (f_1, \ldots , f_ n)$ be a finitely generated ideal of $R$. Let $M$ be the $R$-module generated by elements $e_1, \ldots , e_ n$ subject to the relations $f_ i e_ j - f_ j e_ i = 0$. There exists a short exact sequence
$0 \to K \to M \to I \to 0$
such that $K$ is annihilated by $I$.
Proof. This is just a truncation of the Koszul complex. The map $M \to I$ is determined by the rule $e_ i \mapsto f_ i$. If $m = \sum a_ i e_ i$ is in the kernel of $M \to I$, i.e., $\sum a_ i f_ i = 0$, then $f_ j m = \sum f_ j a_ i e_ i = (\sum f_ i a_ i) e_ j = 0$. $\square$
Lemma 15.88.6. Let $R$ be a ring. Let $I = (f_1, \ldots , f_ n)$ be a finitely generated ideal of $R$. For any $R$-module $N$ set
$H_1(N, f_\bullet ) = \frac{\{ (x_1, \ldots , x_ n) \in N^{\oplus n} \mid f_ i x_ j = f_ j x_ i \} }{\{ f_1x, \ldots , f_ nx) \mid x \in N\} }$
For any $R$-module $N$ there exists a canonical short exact sequence
$0 \to \mathop{\mathrm{Ext}}\nolimits _ R(R/I, N) \to H_1(N, f_\bullet ) \to \mathop{\mathrm{Hom}}\nolimits _ R(K, N)$
where $K$ is as in Lemma 15.88.5.
Proof. The notation above indicates the $\mathop{\mathrm{Ext}}\nolimits$-groups in $\text{Mod}_ R$ as defined in Homology, Section 12.6. These are denoted $\mathop{\mathrm{Ext}}\nolimits _ R(M, N)$. Using the long exact sequence of Homology, Lemma 12.6.4 associated to the short exact sequence $0 \to I \to R \to R/I \to 0$ and the fact that $\mathop{\mathrm{Ext}}\nolimits _ R(R, N) = 0$ we see that
$\mathop{\mathrm{Ext}}\nolimits _ R(R/I, N) = \mathop{\mathrm{Coker}}(N \longrightarrow \mathop{\mathrm{Hom}}\nolimits (I, N))$
Using the short exact sequence of Lemma 15.88.5 we see that we get a complex
$N \to \mathop{\mathrm{Hom}}\nolimits (M, N) \to \mathop{\mathrm{Hom}}\nolimits _ R(K, N)$
whose homology in the middle is canonically isomorphic to $\mathop{\mathrm{Ext}}\nolimits _ R(R/I, N)$. The proof of the lemma is now complete as the cokernel of the first map is canonically isomorphic to $H_1(N, f_\bullet )$. $\square$
Lemma 15.88.7. Let $R$ be a ring. Let $I = (f_1, \ldots , f_ n)$ be a finitely generated ideal of $R$. For any $R$-module $N$ the Koszul homology group $H_1(N, f_\bullet )$ defined in Lemma 15.88.6 is annihilated by $I$.
Proof. Let $(x_1, \ldots , x_ n) \in N^{\oplus n}$ with $f_ i x_ j = f_ j x_ i$. Then we have $f_ i(x_1, \ldots , x_ n) = (f_ i x_ i, \ldots , f_ i x_ n)$. In other words $f_ i$ annihilates $H_1(N, f_\bullet )$. $\square$
We can improve on the full faithfulness of Lemma 15.88.3 by showing that $\mathop{\mathrm{Ext}}\nolimits$-groups whose source is $I$-power torsion are insensitive to passing to $S$ as well. See Dualizing Complexes, Lemma 47.9.8 for a derived version of the following lemma.
Lemma 15.88.8. Assume $\varphi : R \to S$ is a flat ring map and $I \subset R$ is a finitely generated ideal such that $R/I \to S/IS$ is an isomorphism. Let $M$, $N$ be $R$-modules. Assume $M$ is $I$-power torsion. Given an short exact sequence
$0 \to N \otimes _ R S \to \tilde E \to M \otimes _ R S \to 0$
there exists a commutative diagram
$\xymatrix{ 0 \ar[r] & N \ar[r] \ar[d] & E \ar[r] \ar[d] & M \ar[r] \ar[d] & 0 \\ 0 \ar[r] & N \otimes _ R S \ar[r] & \tilde E \ar[r] & M \otimes _ R S \ar[r] & 0 }$
with exact rows.
Proof. As $M$ is $I$-power torsion we see that $M \otimes _ R S = M$, see Lemma 15.88.2. We will use this identification without further mention. As $R \to S$ is flat, the base change functor is exact and we obtain a functorial map of $\mathop{\mathrm{Ext}}\nolimits$-groups
$\mathop{\mathrm{Ext}}\nolimits _ R(M, N) \longrightarrow \mathop{\mathrm{Ext}}\nolimits _ S(M \otimes _ R S, N \otimes _ R S),$
see Homology, Lemma 12.7.3. The claim of the lemma is that this map is surjective when $M$ is $I$-power torsion. In fact we will show that it is an isomorphism. By Lemma 15.87.2 we can find a surjection $M' \to M$ with $M'$ a direct sum of modules of the form $R/I^ n$. Using the long exact sequence of Homology, Lemma 12.6.4 we see that it suffices to prove the lemma for $M'$. Using compatibility of $\mathop{\mathrm{Ext}}\nolimits$ with direct sums (details omitted) we reduce to the case where $M = R/I^ n$ for some $n$.
Let $f_1, \ldots , f_ t$ be generators for $I^ n$. By Lemma 15.88.6 we have a commutative diagram
$\xymatrix{ 0 \ar[r] & \mathop{\mathrm{Ext}}\nolimits _ R(R/I^ n, N) \ar[r] \ar[d] & H_1(N, f_\bullet ) \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits _ R(K, N) \ar[d] \\ 0 \ar[r] & \mathop{\mathrm{Ext}}\nolimits _ S(S/I^ nS, N \otimes S) \ar[r] & H_1(N \otimes S, f_\bullet ) \ar[r] & \mathop{\mathrm{Hom}}\nolimits _ S(K \otimes S, N \otimes S) }$
with exact rows where $K$ is as in Lemma 15.88.5. Hence it suffices to prove that the two right vertical arrows are isomorphisms. Since $K$ is annihilated by $I^ n$ we see that $\mathop{\mathrm{Hom}}\nolimits _ R(K, N) = \mathop{\mathrm{Hom}}\nolimits _ S(K \otimes _ R S, N \otimes _ R S)$ by Lemma 15.88.3. As $R \to S$ is flat we have $H_1(N, f_\bullet ) \otimes _ R S = H_1(N \otimes _ R S, f_\bullet )$. As $H_1(N, f_\bullet )$ is annihilated by $I^ n$, see Lemma 15.88.7 we have $H_1(N, f_\bullet ) \otimes _ R S = H_1(N, f_\bullet )$ by Lemma 15.88.2. $\square$
Let $R \to S$ be a ring map. Let $f_1, \ldots , f_ t \in R$ and $I = (f_1, \ldots , f_ t)$. Then for any $R$-module $M$ we can define a complex
15.88.8.1
$$\label{more-algebra-equation-glueing-complex} 0 \to M \xrightarrow {\alpha } M \otimes _ R S \times \prod M_{f_ i} \xrightarrow {\beta } \prod (M \otimes _ R S)_{f_ i} \times \prod M_{f_ if_ j}$$
where $\alpha (m) = (m \otimes 1, m/1, \ldots , m/1)$ and
$\beta (m', m_1, \ldots , m_ t) = ((m'/1 - m_1 \otimes 1, \ldots , m'/1 - m_ t \otimes 1), (m_1 - m_2, \ldots , m_{t - 1} - m_ t).$
We would like to know when this complex is exact.
Lemma 15.88.9. Assume $\varphi : R \to S$ is a flat ring map and $I = (f_1, \ldots , f_ t) \subset R$ is an ideal such that $R/I \to S/IS$ is an isomorphism. Let $M$ be an $R$-module. Then the complex (15.88.8.1) is exact.
Proof. First proof. Denote $\check{\mathcal{C}}_ R \to \check{\mathcal{C}}_ S$ the quasi-isomorphism of extended alternating Čech complexes of Lemma 15.88.4. Since these complexes are bounded with flat terms, we see that $M \otimes _ R \check{\mathcal{C}}_ R \to M \otimes _ R \check{\mathcal{C}}_ S$ is a quasi-isomorphism too (Lemmas 15.58.9 and 15.58.14). Now the complex (15.88.8.1) is a truncation of the cone of the map $M \otimes _ R \check{\mathcal{C}}_ R \to M \otimes _ R \check{\mathcal{C}}_ S$ and we win.
Second computational proof. Let $m \in M$. If $\alpha (m) = 0$, then $m \in M[I^\infty ]$, see Lemma 15.87.3. Pick $n$ such that $I^ n m = 0$ and consider the map $\varphi : R/I^ n \to M$. If $m \otimes 1 = 0$, then $\varphi \otimes 1_ S = 0$, hence $\varphi = 0$ (see Lemma 15.88.3) hence $m = 0$. In this way we see that $\alpha$ is injective.
Let $(m', m'_1, \ldots , m'_ t) \in \mathop{\mathrm{Ker}}(\beta )$. Write $m'_ i = m_ i/f_ i^ n$ for some $n > 0$ and $m_ i \in M$. We may, after possibly enlarging $n$ assume that $f_ i^ n m' = m_ i \otimes 1$ in $M \otimes _ R S$ and $f_ j^ nm_ i - f_ i^ nm_ j = 0$ in $M$. In particular we see that $(m_1, \ldots , m_ t)$ defines an element $\xi$ of $H_1(M, (f_1^ n, \ldots , f_ t^ n))$. Since $H_1(M, (f_1^ n, \ldots , f_ t^ n))$ is annihilated by $I^{tn + 1}$ (see Lemma 15.88.7) and since $R \to S$ is flat we see that
$H_1(M, (f_1^ n, \ldots , f_ t^ n)) = H_1(M, (f_1^ n, \ldots , f_ t^ n)) \otimes _ R S = H_1(M \otimes _ R S, (f_1^ n, \ldots , f_ t^ n))$
by Lemma 15.88.2 The existence of $m'$ implies that $\xi$ maps to zero in the last group, i.e., the element $\xi$ is zero. Thus there exists an $m \in M$ such that $m_ i = f_ i^ n m$. Then $(m', m'_1, \ldots , m'_ t) - \alpha (m) = (m'', 0, \ldots , 0)$ for some $m'' \in (M \otimes _ R S)[(IS)^\infty ]$. By Lemma 15.88.3 we conclude that $m'' \in M[I^\infty ]$ and we win. $\square$
Remark 15.88.10. In this remark we define a category of glueing data. Let $R \to S$ be a ring map. Let $f_1, \ldots , f_ t \in R$ and $I = (f_1, \ldots , f_ t)$. Consider the category $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ as the category whose
1. objects are systems $(M', M_ i, \alpha _ i, \alpha _{ij})$, where $M'$ is an $S$-module, $M_ i$ is an $R_{f_ i}$-module, $\alpha _ i : (M')_{f_ i} \to M_ i \otimes _ R S$ is an isomorphism, and $\alpha _{ij} : (M_ i)_{f_ j} \to (M_ j)_{f_ i}$ are isomorphisms such that
1. $\alpha _{ij} \circ \alpha _ i = \alpha _ j$ as maps $(M')_{f_ if_ j} \to (M_ j)_{f_ i}$, and
2. $\alpha _{jk} \circ \alpha _{ij} = \alpha _{ik}$ as maps $(M_ i)_{f_ jf_ k} \to (M_ k)_{f_ if_ j}$ (cocycle condition).
2. morphisms $(M', M_ i, \alpha _ i, \alpha _{ij}) \to (N', N_ i, \beta _ i, \beta _{ij})$ are given by maps $\varphi ' : M' \to N'$ and $\varphi _ i : M_ i \to N_ i$ compatible with the given maps $\alpha _ i, \beta _ i, \alpha _{ij}, \beta _{ij}$.
There is a canonical functor
$\text{Can} : \text{Mod}_ R \longrightarrow \text{Glue}(R \to S, f_1, \ldots , f_ t), \quad M \longmapsto (M \otimes _ R S, M_{f_ i}, \text{can}_ i, \text{can}_{ij})$
where $\text{can}_ i : (M \otimes _ R S)_{f_ i} \to M_{f_ i} \otimes _ R S$ and $\text{can}_{ij} : (M_{f_ i})_{f_ j} \to (M_{f_ j})_{f_ i}$ are the canonical isomorphisms. For any object $\mathbf{M} = (M', M_ i, \alpha _ i, \alpha _{ij})$ of the category $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ we define
$H^0(\mathbf{M}) = \{ (m', m_ i) \mid \alpha _ i(m') = m_ i \otimes 1, \alpha _{ij}(m_ i) = m_ j\}$
in other words defined by the exact sequence
$0 \to H^0(\mathbf{M}) \to M' \times \prod M_ i \to \prod M'_{f_ i} \times \prod (M_ i)_{f_ j}$
similar to (15.88.8.1). We think of $H^0(\mathbf{M})$ as an $R$-module. Thus we also get a functor
$H^0 : \text{Glue}(R \to S, f_1, \ldots , f_ t) \longrightarrow \text{Mod}_ R$
Our next goal is to show that the functors $\text{Can}$ and $H^0$ are sometimes quasi-inverse to each other.
Lemma 15.88.11. Assume $\varphi : R \to S$ is a flat ring map and $I = (f_1, \ldots , f_ t) \subset R$ is an ideal such that $R/I \to S/IS$ is an isomorphism. Then the functor $H^0$ is a left quasi-inverse to the functor $\text{Can}$ of Remark 15.88.10.
Proof. This is a reformulation of Lemma 15.88.9. $\square$
Lemma 15.88.12. Assume $\varphi : R \to S$ is a flat ring map and let $I = (f_1, \ldots , f_ t) \subset R$ be an ideal. Then $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ is an abelian category, and the functor $\text{Can}$ is exact and commutes with arbitrary colimits.
Proof. Given a morphism $(\varphi ', \varphi _ i) : (M', M_ i, \alpha _ i, \alpha _{ij}) \to (N', N_ i, \beta _ i, \beta _{ij})$ of the category $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ we see that its kernel exists and is equal to the object $(\mathop{\mathrm{Ker}}(\varphi '), \mathop{\mathrm{Ker}}(\varphi _ i), \alpha _ i, \alpha _{ij})$ and its cokernel exists and is equal to the object $(\mathop{\mathrm{Coker}}(\varphi '), \mathop{\mathrm{Coker}}(\varphi _ i), \beta _ i, \beta _{ij})$. This works because $R \to S$ is flat, hence taking kernels/cokernels commutes with $- \otimes _ R S$. Details omitted. The exactness follows from the $R$-flatness of $R_{f_ i}$ and $S$, while commuting with colimits follows as tensor products commute with colimits. $\square$
Lemma 15.88.13. Let $\varphi : R \to S$ be a flat ring map and $(f_1, \ldots , f_ t) = R$. Then $\text{Can}$ and $H^0$ are quasi-inverse equivalences of categories
$\text{Mod}_ R = \text{Glue}(R \to S, f_1, \ldots , f_ t)$
Proof. Consider an object $\mathbf{M} = (M', M_ i, \alpha _ i, \alpha _{ij})$ of $\text{Glue}(R \to S, f_1, \ldots , f_ t)$. By Algebra, Lemma 10.24.5 there exists a unique module $M$ and isomorphisms $M_{f_ i} \to M_ i$ which recover the glueing data $\alpha _{ij}$. Then both $M'$ and $M \otimes _ R S$ are $S$-modules which recover the modules $M_ i \otimes _ R S$ upon localizing at $f_ i$. Whence there is a canonical isomorphism $M \otimes _ R S \to M'$. This shows that $\mathbf{M}$ is in the essential image of $\text{Can}$. Combined with Lemma 15.88.11 the lemma follows. $\square$
Lemma 15.88.14. Let $\varphi : R \to S$ be a flat ring map and $I = (f_1, \ldots , f_ t)$ and ideal. Let $R \to R'$ be a flat ring map, and set $S' = S \otimes _ R R'$. Then we obtain a commutative diagram of categories and functors
$\xymatrix{ \text{Mod}_ R \ar[r]_-{\text{Can}} \ar[d]_{-\otimes _ R R'} & \text{Glue}(R \to S, f_1, \ldots , f_ t) \ar[r]_-{H^0} \ar[d]^{-\otimes _ R R'} & \text{Mod}_ R \ar[d]^{-\otimes _ R R'} \\ \text{Mod}_{R'} \ar[r]^-{\text{Can}} & \text{Glue}(R' \to S', f_1, \ldots , f_ t) \ar[r]^-{H^0} & \text{Mod}_{R'} }$
Proof. Omitted. $\square$
Proposition 15.88.15. Assume $\varphi : R \to S$ is a flat ring map and $I = (f_1, \ldots , f_ t) \subset R$ is an ideal such that $R/I \to S/IS$ is an isomorphism. Then $\text{Can}$ and $H^0$ are quasi-inverse equivalences of categories
$\text{Mod}_ R = \text{Glue}(R \to S, f_1, \ldots , f_ t)$
Proof. We have already seen that $H^0 \circ \text{Can}$ is isomorphic to the identity functor, see Lemma 15.88.11. Consider an object $\mathbf{M} = (M', M_ i, \alpha _ i, \alpha _{ij})$ of $\text{Glue}(R \to S, f_1, \ldots , f_ t)$. We get a natural morphism
$\Psi : (H^0(\mathbf{M}) \otimes _ R S, H^0(\mathbf{M})_{f_ i}, \text{can}_ i, \text{can}_{ij}) \longrightarrow (M', M_ i, \alpha _ i, \alpha _{ij}).$
Namely, by definition $H^0(\mathbf{M})$ comes equipped with compatible $R$-module maps $H^0(\mathbf{M}) \to M'$ and $H^0(\mathbf{M}) \to M_ i$. We have to show that this map is an isomorphism.
Pick an index $i$ and set $R' = R_{f_ i}$. Combining Lemmas 15.88.14 and 15.88.13 we see that $\Psi \otimes _ R R'$ is an isomorphism. Hence the kernel, resp. cokernel of $\Psi$ is a system of the form $(K, 0, 0, 0)$, resp. $(Q, 0, 0, 0)$. Note that $H^0((K, 0, 0, 0)) = K$, that $H^0$ is left exact, and that by construction $H^0(\Psi )$ is bijective. Hence we see $K = 0$, i.e., the kernel of $\Psi$ is zero.
The conclusion of the above is that we obtain a short exact sequence
$0 \to H^0(\mathbf{M}) \otimes _ R S \to M' \to Q \to 0$
and that $M_ i = H^0(\mathbf{M})_{f_ i}$. Note that we may think of $Q$ as an $R$-module which is $I$-power torsion so that $Q = Q \otimes _ R S$. By Lemma 15.88.8 we see that there exists a commutative diagram
$\xymatrix{ 0 \ar[r] & H^0(\mathbf{M}) \ar[r] \ar[d] & E \ar[r] \ar[d] & Q \ar[r] \ar[d] & 0 \\ 0 \ar[r] & H^0(\mathbf{M}) \otimes _ R S \ar[r] & M' \ar[r] & Q \ar[r] & 0 }$
with exact rows. This clearly determines an isomorphism $\text{Can}(E) \to (M', M_ i, \alpha _ i, \alpha _{ij})$ in the category $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ and we win. (Of course, a posteriori we have $Q = 0$.) $\square$
Lemma 15.88.16. Let $\varphi : R \to S$ be a flat ring map and let $I \subset R$ be a finitely generated ideal such that $R/I \to S/IS$ is an isomorphism.
1. Given an $R$-module $N$, an $S$-module $M'$ and an $S$-module map $\varphi : M' \to N \otimes _ R S$ whose kernel and cokernel are $I$-power torsion, there exists an $R$-module map $\psi : M \to N$ and an isomorphism $M \otimes _ R S = M'$ compatible with $\varphi$ and $\psi$.
2. Given an $R$-module $M$, an $S$-module $N'$ and an $S$-module map $\varphi : M \otimes _ R S \to N'$ whose kernel and cokernel are $I$-power torsion, there exists an $R$-module map $\psi : M \to N$ and an isomorphism $N \otimes _ R S = N'$ compatible with $\varphi$ and $\psi$.
In both cases we have $\mathop{\mathrm{Ker}}(\varphi ) \cong \mathop{\mathrm{Ker}}(\psi )$ and $\mathop{\mathrm{Coker}}(\varphi ) \cong \mathop{\mathrm{Coker}}(\psi )$.
Proof. Proof of (1). Say $I = (f_1, \ldots , f_ t)$. It is clear that the localization $\varphi _{f_ i}$ is an isomorphism. Thus we see that $(M', N_{f_ i}, \varphi _{f_ i}, can_{ij})$ is an object of $\text{Glue}(R \to S, f_1, \ldots , f_ t)$, see Remark 15.88.10. By Proposition 15.88.15 we conclude that there exists an $R$-module $M$ such that $M' = M \otimes _ R S$ and $N_{f_ i} = M_{f_ i}$ compatibly with the isomorphisms $\varphi _{f_ i}$ and $can_{ij}$. There is a morphism
$(M \otimes _ R S, M_{f_ i}, can_ i, can_{ij}) = (M', N_{f_ i}, \varphi _{f_ i}, can_{ij}) \to (N \otimes _ R S, N_{f_ i}, can_ i, can_{ij})$
of $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ which uses $\varphi$ in the first component. This corresponds to an $R$-module map $\psi : M \to N$ (by the equivalence of categories of Proposition 15.88.15). The composition of the base change of $M \to N$ with the isomorphism $M' \cong M \otimes _ R S$ is $\varphi$, in other words $M \to N$ is compatible with $\varphi$.
Proof of (2). This is just the dual of the argument above. Namely, the localization $\varphi _{f_ i}$ is an isomorphism. Thus we see that $(N', M_{f_ i}, \varphi _{f_ i}^{-1}, can_{ij})$ is an object of $\text{Glue}(R \to S, f_1, \ldots , f_ t)$, see Remark 15.88.10. By Proposition 15.88.15 we conclude that there exists an $R$-module $N$ such that $N' = N \otimes _ R S$ and $N_{f_ i} = M_{f_ i}$ compatibly with the isomorphisms $\varphi _{f_ i}^{-1}$ and $can_{ij}$. There is a morphism
$(M \otimes _ R S, M_{f_ i}, can_ i, can_{ij}) \to (N', M_{f_ i}, \varphi _{f_ i}, can_{ij}) = (N \otimes _ R S, N_{f_ i}, can_ i, can_{ij})$
of $\text{Glue}(R \to S, f_1, \ldots , f_ t)$ which uses $\varphi$ in the first component. This corresponds to an $R$-module map $\psi : M \to N$ (by the equivalence of categories of Proposition 15.88.15). The composition of the base change of $M \to N$ with the isomorphism $N' \cong N \otimes _ R S$ is $\varphi$, in other words $M \to N$ is compatible with $\varphi$.
The final statement follows for example from Lemma 15.88.3. $\square$
Next, we specialize Proposition 15.88.15 to get something more useable. Namely, if $I = (f)$ is a principal ideal then the objects of $\text{Glue}(R \to S, f)$ are simply triples $(M', M_1, \alpha _1)$ and there is no cocycle condition to check!
Theorem 15.88.17. Let $R$ be a ring, and let $f \in R$. Let $\varphi : R \to S$ be a flat ring map inducing an isomorphism $R/fR \to S/fS$. Then the functor
$\text{Mod}_ R \longrightarrow \text{Mod}_ S \times _{\text{Mod}_{S_ f}} \text{Mod}_{R_ f}, \quad M \longmapsto (M \otimes _ R S, M_ f, \text{can})$
is an equivalence.
Proof. The category appearing on the right side of the arrow is the category of triples $(M', M_1, \alpha _1)$ where $M'$ is an $S$-module, $M_1$ is a $R_ f$-module, and $\alpha _1 : M'_ f \to M_1 \otimes _ R S$ is a $S_ f$-isomorphism, see Categories, Example 4.31.3. Hence this theorem is a special case of Proposition 15.88.15. $\square$
A useful special case of Theorem 15.88.17 is when $R$ is Noetherian, and $S$ is a completion of $R$ at an element $f$. The completion $R \to S$ is flat, and the functor $M \mapsto M \otimes _ R S$ can be identified with the $f$-adic completion functor when $M$ is finitely generated. To state this more precisely, let $\text{Mod}^{fg}_ R$ denote the category of finitely generated $R$-modules.
Proposition 15.88.18. Let $R$ be a Noetherian ring. Let $f \in R$ be an element. Let $R^\wedge$ be the $f$-adic completion of $R$. Then the functor $M \mapsto (M^\wedge , M_ f, \text{can})$ defines an equivalence
$\text{Mod}^{fg}_ R \longrightarrow \text{Mod}^{fg}_{R^\wedge } \times _{\text{Mod}^{fg}_{(R^\wedge )_ f}} \text{Mod}^{fg}_{R_ f}$
Proof. The ring map $R \to R^\wedge$ is flat by Algebra, Lemma 10.97.2. It is clear that $R/fR = R^\wedge /fR^\wedge$. By Algebra, Lemma 10.97.1 the completion of a finite $R$-module $M$ is equal to $M \otimes _ R R^\wedge$. Hence the displayed functor of the proposition is equal to the functor occurring in Theorem 15.88.17. In particular it is fully faithful. Let $(M_1, M_2, \psi )$ be an object of the right hand side. By Theorem 15.88.17 there exists an $R$-module $M$ such that $M_1 = M \otimes _ R R^\wedge$ and $M_2 = M_ f$. As $R \to R^\wedge \times R_ f$ is faithfully flat we conclude from Algebra, Lemma 10.23.2 that $M$ is finitely generated, i.e., $M \in \text{Mod}^{fg}_ R$. This proves the proposition. $\square$
Remark 15.88.19. The equivalences of Proposition 15.88.15, Theorem 15.88.17, and Proposition 15.88.18 preserve properties of modules. For example if $M$ corresponds to $\mathbf{M} = (M', M_ i, \alpha _ i, \alpha _{ij})$ then $M$ is finite, or finitely presented, or flat, or projective over $R$ if and only if $M'$ and $M_ i$ have the corresponding property over $S$ and $R_{f_ i}$. This follows from the fact that $R \to S \times \prod R_{f_ i}$ is faithfully flat and descend and ascent of these properties along faithfully flat maps, see Algebra, Lemma 10.83.2 and Theorem 10.95.5. These functors also preserve the $\otimes$-structures on either side. Thus, it defines equivalences of various categories built out of the pair $(\text{Mod}_ R, \otimes )$, such as the category of algebras.
Remark 15.88.20. Given a differential manifold $X$ with a compact closed submanifold $Z$ having complement $U$, specifying a sheaf on $X$ is the same as specifying a sheaf on $U$, a sheaf on an unspecified tubular neighbourhood $T$ of $Z$ in $X$, and an isomorphism between the two resulting sheaves along $T \cap U$. Tubular neighbourhoods do not exist in algebraic geometry as such, but results such as Proposition 15.88.15, Theorem 15.88.17, and Proposition 15.88.18 allow us to work with formal neighbourhoods instead.
Comment #4623 by Anonymous on
In lemma 15.80.13 you need to add the assumptions of lemma 15.80.11, since it is used.
Comment #4624 by Anonymous on
Nevermind, you have them trivially
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2021-05-07 13:53:17
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https://math.sc.chula.ac.th/cjm/content/classes-equally-likely-outcomes-riffle-shuffle-deck-alternating-cards
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Classes of equally likely outcomes of a riffle shuffle on a deck of alternating cards
CJM Vol. 11 (December 2019), pp. 1 – 9.
Abstract:
The mathematical model of riffle shuffle has been a subject of some studies. Whereas most of these regard the cards as all different, in 2006 Conger and Viswanath treated some of them as identical and investigated the implications. When the initial deck is arranged in alternating reds and blacks, they showed that two outcomes are equally likely if a number of particular transformations can turn one of them into the other. This transformation, which may be viewed as a reversible string rewriting system, partitions the set of outcomes into equivalence classes. They conjectured that the number of such classes is precisely $(n+3)2^{n−2}$ , where n is the number of cards of each color. In this paper, the assertion is proven true by the method of invariant and derivation of canonical forms.
2000 Mathematics Subject Classification:
Full Paper (PDF):
AttachmentSize
117.67 KB
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2021-09-29 01:45:59
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http://en.wikisource.org/wiki/Elements_of_the_Differential_and_Integral_Calculus/Chapter_V_part_2
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# Elements of the Differential and Integral Calculus/Chapter V part 2
44. Differentiation of a logarithm.
Let $y = \log_a v$.[1]
Differentiating by the General Rule, p. 29 [§ 31], considering $v$ as the independent variable, we have
FIRST STEP. $\ y + \Delta y$ $= \log_a(v + \Delta v)\$. SECOND STEP. $\ \Delta y$ $= \log_a(v + \Delta v) - \log_a v\$[2] $= \log_a \left ( \frac{v + \Delta v}{v} \right ) = \log_a \left ( 1 + \frac{\Delta v}{v} \right )$. [By 8, p. 1 [§ 1]] THIRD STEP. $\frac{\Delta y}{\Delta x}$ $= \frac{1}{\Delta v} \log_a \left ( 1 + \frac{\Delta v}{v} \right ) = \log_a \left ( 1 + \frac{\Delta v}{v} \right )^{\frac{1}{\Delta v}}$ $= \frac{1}{v} \log_a \left ( 1 + \frac{\Delta v}{v} \right )^{\frac{v}{\Delta v}}$. [Dividing the logarithm by $v$ and at the same time multiplying the exponent of the parenthesis by $v$ changes the form of the expression but not its value (see 9, p. 1 [§ 1]).] FOURTH STEP. $\frac{dy}{dv}$ $= \frac{1}{v} \log_a e$. [When $\Delta v \dot= 0, \frac{\Delta v}{v} \dot= 0$. Therefore $\lim{\Delta v \to 0} \left ( 1 + \frac{\Delta v}{v} \right )^{\frac{v}{\Delta v}} = e$, from p. 22 (§ 23), placing $x = \frac{\Delta v}{v}$.] Hence (A) $\frac{dy}{dv}$ $= \frac{d}{dv} \left ( \log_a v \right ) = \log_a e \cdot \frac{1}{v}$.
Since $v$ is a function of $x$ and it is required to differentiate $\log_a v$ with respect to $x$, we must use formula (A), § 42, for differentiating a function of a function, namely,
$\frac{dy}{dx}$ $= \frac{dy}{dv} \cdot \frac{dv}{dx}$. Substituting value of $\frac{dy}{dv}$ from (A), we get $\frac{dy}{dx}$ $= \log_a e \cdot \frac{1}{v} \cdot \frac{dv}{dx}$. VIII ∴ $\frac{d}{dx} (\log_a x)$ $= \log_s e \cdot \frac{\frac{dv}{dx}}{v}$. When $a = e,\ \log_a e = log_e e = 1$, and VIII becomes VIIIa $\frac{d}{dx} (\log v)$ $= \frac{\frac{dv}{dx}}{v}$.
The derivative of the logarithm of a function is equal to the product of the modulus[3] of the system of logarithms and the derivative of the function, divided by the function.
45. Differentiation of the simple exponential function.
Let $\ y$ $= a^v. \qquad a > 0\$ Taking the logarithm of both sides to the base $e$, we get $\ \log y$ $= v \log a\$, or $\ v$ $= \frac{\log y}{\log a}$ $= \frac{1}{\log a} \cdot \log y$. Differentiate with respect to $y$ by formula VIIIa, $\frac{dv}{dy}$ $= \frac{1}{\log a} \cdot \frac{1}{y}$; and from (C), § 43, relating to inverse functions, we get $\frac{dy}{dv}$ $= \log a \cdot y$, or, (A) $\frac{dy}{dv}$ $= \log a \cdot a^v$, Since $v$ is a function of $x$ and it is required to differentiate $a^v$ with respect to $x$, we must use formula (A), § 42, for differentiating a function of a function, namely, $\frac{dy}{dx}$ $= \frac{dy}{dv} \cdot \frac{dv}{dx}$. Substituting the value of $\frac{dy}{dx}$ from (A), we get $\frac{dy}{dx}$ $= \log a \cdot a^v \cdot \frac{dv}{dx}$. IX ∴ $\frac{d}{dx} (a^v)$ $= \log a \cdot a^v \cdot \frac{dv}{dx}$. When $a = e,\ \log a = \log e = 1$, and IX becomes IXa $\frac{d}{dx} (e^v)$ $= e^v \frac{dv}{dx}$.
The derivative of a constant with a variable exponent is equal to the product of the natural logarithm of the constant, the constant with the variable exponent, and the derivative of the exponent.
46. Differentiation of the general exponential function.
Let $\ y$ $= u^v\$.[4] Taking the logarithm of both sides to the base $e$, $\ \log_e y$ $= v log_e u\$, or, $\ y$ $= e^{v \log u}\$. Differentiating by formula Ixa, $\frac{dy}{dx}$ $= e^{v \log u} \frac{d}{dx} (v \log u)$ $= e^{v \log u} \left ( \frac{v}{u} \frac{du}{dx} + \log u \frac{dv}{dx} \right )$ by V $= u^v \left ( \frac{v}{u} \frac{du}{dx} + \log u \frac{dv}{dx} \right )$ X ∴ $\frac{d}{dx}(u^v)$ $= vu^{v - 1}\frac{du}{dx} + \log u \cdot u^v \frac{dv}{dx}$.
'The derivative of a function with a variable exponent is equal to the sum of the two results obtained by first differentiating by VI, regarding the exponent as constant,. and again differentiating by IX, regarding the function as constant.
Let v = n, any constant; then X reduces to
$\frac{d}{dx}(u^n) = nu^{n - 1} \frac{du}{dx}$.
But this is the form differentiated in § 40; therefore VI holds true for any value of $n$.
ILLUSTRATIVE EXAMPLE 1. Differentiate $y = log(x^2 + a)$. Solution. $\frac{dy}{dx}$ $= \frac{\frac{d}{dx}(x^2 + a)}{x^2 + a}$ by VIIIa [v = x^2 + a] $= \frac{2x}{x^2 + a}$ Ans. ILLUSTRATIVE EXAMPLE 2. Differentiate $y = \log \sqrt{1 - x^2}$. Solution. $\frac{dy}{dx}$ $= \frac{\frac{d}{dx}(1 - x^2)^{\frac{1}{2}}}{(1 - x^2)^{\frac{1}{2}}}$ by VIIIa $= \frac{\frac{1}{2} (1 - x^2)^{-\frac{1}{2}}(-2x)}{(1 - x^2)^{\frac{1}{2}}}$ by VI $= \frac{x}{x^2 - 1}$. Ans. ILLUSTRATIVE EXAMPLE 3. Differentiate $y = a^{3x^2}$. Solution: $\frac{dy}{dx}$ $= \log a \cdot a^{3x^2} \frac{d}{dx}(3 x^2)$ by IX $= 6x \log a \cdot a^{3x^2}$ Ans. ILLUSTRATIVE EXAMPLE 4. Differentiate $y = be^{c^2 + x^2}$. Solution. $\frac{dy}{dx}$ $b\frac{d}{dx} \left ( e^{c^2 + x^2} \right )$. by IV $= be^{c^2 + x^2} \frac{d}{dx} (c^2 + x^2)$ by IXa $= 2bxe^{c^2 + x^2}$. Ans. ILLUSTRATIVE EXAMPLE 5. Differentiate $y = xe^x$. Solution: $\frac{dy}{dx}$ $= e^x x^{e^x - 1} \frac{d}{dx} (x) + x^{e^x} \log x \frac{d}{dx} (e^x)$ by X $= e^x x^{e^x - 1} + x^{e^x} \log x \cdot e^x$ $= e^x x^{e^x} \left ( \frac{1}{x} + \log x \right )$ Ans.
47. Logarithmic differentiation. Instead of applying VIII and VIIIa at once in differentiating logarithmic functions, we may sometimes simplify the work by first making use of one of the formulas 7-10 on p. 1 [§ 1]. Thus above Illustrative Example 2 may be solved as follows:
ILLUSTRATIVE EXAMPLE 1. Differentiate $y = \log \sqrt{1 - x^2}$. Solution. By using 10, p. 1, we may write this in a form free from radicals as follows: $y$ $= \frac{1}{2} \log (1 - x^2)$. Then $\frac{dy}{dx}$ $= \frac{1}{2} \frac{\frac{d}{dx} (1 - x^2)}{1 - x^2}$ by VIIIa $= \frac{1}{2} \cdot \frac{-2}{1 - x^2} = \frac{x}{x^2 - 1}$. Ans. ILLUSTRATIVE EXAMPLE 2. Differentiate $y = log \sqrt{\frac{1 + x^2}{1 - x^2}}$. Solution. Simplifying by means of 10 and 8, p. 1 [§ 1], $y$ $= \frac{1}{2} [ \log (1 + x^2) - \log (1 - x^2) ]$. $\frac{dy}{dx}$ $= \frac{1}{2} \left [ \frac{\frac{d}{dx} (1 + x^2)}{1 + x^2} - \frac{\frac{d}{dx} (1 - x^2)}{1 - x^2} \right ]$ by VIIIa, etc. $= \frac{x}{1 + x^2} + \frac{x}{1 - x^2} = \frac{2x}{1 - x^4}$. Ans.
In differentiating an exponential function, especially a variable with a variable exponent, the best plan is first to take the logarithm of the function and then differentiate. Thus Illustrative Example 5, p. 50 [§ 46], is solved more elegantly as follows:
ILLUSTRATIVE EXAMPLE 3. Differentiate $y = x^{e^x}$. Solution. Taking the logarithm of both sides, $\log y = e^x \log x$. By 9, p. 1 [§ 1] Now differentiate both sides with respect to $x$. $\frac{\frac{dy}{dx}}{y}$ $= e^x \frac{d}{dx} (\log x) + \log x \frac{d}{dx} (e^x)$ by VIII and V $= e^x \cdot \frac{1}{x} + \log x \cdot e^x$, or, $\frac{dy}{dx}$ $= e^x \cdot y \left ( \frac{1}{x} \log x \right )$ $= e^x x^{e^x} \left ( \frac{1}{x} + \log x \right )$. Ans.
ILLUSTRATIVE EXAMPLE 4. Differentiate $y = (4x^2 - 7)^{2 + \sqrt{x^2 - 5}}$.
Solution. Taking the logarithm of both sides,
$\log y = (2 + \sqrt{x^2 - 5}) \log (4x^2 - 7)$.
Differentiating both sides with respect to $x$,
$\frac{1}{y} \frac{dy}{dx}$ $= (2 + \sqrt{x^2 - 5}) \frac{8x}{4x^2 - 7} + \log(4x^2 - 7) \cdot \frac{x}{\sqrt{x^2 - 5}}$. $\frac{dy}{dx}$ $= x(4x^2 - 7)^{2 + \sqrt{x^2 - 5}} \left [ \frac{8(2 + \sqrt{x^2 - 5})}{4x^2 - 7} + \frac{\log (4x^2 - 7)}{\sqrt{x^2 - 5}} \right ]$. Ans.
In the case of a function consisting of a number of factors it is sometimes convenient to take the logarithm before differentiating. Thus,
ILLUSTRATIVE EXAMPLE 5. Differentiate $y = \sqrt{\frac{(x - 1)(x - 2)}{(x - 3)(x - 4)}}$.
Solution. Taking the logarithm of both sides,
$\log y = \frac{1}{2} [\log (x -1) + \log (x - 2) - \log(x - 3) - \log(x - 4)]$.
Differentiating both sides with respect to $x$,
$\frac{1}{y} \frac{dy}{dx}$ $= \frac{1}{2} \left [ \frac{1}{x - 1} + \frac{1}{x - 2} - \frac{1}{x - 3} - \frac{1}{x - 4} \right ]$ $= -\frac{2x^2 - 10x + 11}{(x - 1)(x - 2)(x - 3)(x - 4)}$, or, $\frac{dy}{dx}$ $= -\frac{2x^2 - 10x - 11}{(x - 1)^{\frac{1}{2}} (x - 2)^{\frac{1}{2}} ( x - 3)^{\frac{3}{2}} (x - 4)^{\frac{3}{2}}}$. Ans.
EXAMPLES
Differentiate the following:
1. $y = \log (x + a)$. $\frac{dy}{dx} = \frac{1}{x + a}$. 2. $y = \log (ax + b)$. $\frac{dy}{dx} = \frac{a}{ax + b}$. 3. $y = \log \frac{1 + x^2}{1 - x^2}$. $\frac{dy}{dx} = \frac{4x}{1 - x^4}$. 4. $y = \log (x^2 + x)$ $y' = \frac{2x + 1}{x^2 + x}$. 5. $y = \log (x^3 - 2x + 5)$. $y' = \frac{3x^2 - 2}{x^3 - 2x + 5}$. 6. $y = \log_a (2x + x^3)$. $y' = \log_a e \cdot \frac{2 + 3x^2}{2x + x^3}$. 7. $y = x \log x$. $y' = \log x + 1$. 8. $f(x) = \log x^3$. $f'(x) = \frac{3}{x}$. 9. $f(x) = \log^3 x$. $f'(x) = \frac{3 \log^2 x}{x}$. HINT. $\log^3 x = (\log x)^3$. Use first VI, $v = \log x, n = 3$; and then VIIIa. 10. $f(x) = \log \frac{a + x}{a - x}$. $f'(x) = \frac{2a}{a^2 - x^2}$. 11. $f(x) = \log (x + \sqrt{1 + x^2})$. $f'(x) = \frac{1}{\sqrt{1 + x^2}}$.
12. $\frac{d}{dx} e^{ax} = ae^{ax}$.
13. $\frac{d}{dx} e^{4x + 5} = 4e^{4x + 5}$.
14. $\frac{d}{dx} a^{3x} = 3a^{3x} \log a$.
15. $\frac{d}{dt} \log(3 - 2t^2) = \frac{4t}{2t^2 - 3}$.
16. $\frac{d}{dy} \log \frac{1 + y}{1 - y} = \frac{2}{1 - y^2}$.
17. $\frac{d}{dx}e^{b^2 + x^2} = 2xe^{b^2 + x^2}$.
18. $\frac{d}{d\theta} a^{\log a} = \frac{1}{\theta} a^{\log \theta} \log a$.
19. $\frac{d}{ds}b^{s^2} = 2x \log b \cdot b^{s^2}$.
20. $\frac{d}{dv} ae^{\sqrt{v}} = \frac{ae^{\sqrt{v}}}{2\sqrt{v}}$.
21. $\frac{d}{dx} a^{e^x} = \log a \cdot a^{e^x} \cdot e^x$.
22. $y = 7^{x^2 + 2x}$. $y' = 2\log 7 \cdot (x + 1) 7^{x^2 + 2x}$. 23. $y = c^{a^2 - x^2}$. $y' = -2x \log c \cdot c^{a^2 - x^2}$. 24. $y = \log \frac{e^x}{1 + e^x}$. $\frac{dy}{dx} = \frac{1}{1 + e^x}$.
25. $\frac{d}{dx} \left [ e^x ( 1- x^2 \right ] = e^x (1 - 2x - x^2)$.
26. $\frac{d}{dx} \left ( \frac{e^x - 1}{e^x + 1} \right ) = \frac{2e^x}{(e^x + 1)^2}$
27. $\frac{d}{dx} \left ( x^2 e^{ax} \right ) = xe^{ax}(ax + 2)$.
28. $y = \frac{a}{2} (e^{\frac{x}{a}} - e^{-\frac{x}{a}}).$ $\frac{dy}{dx} = \frac{1}{2} (e^{\frac{x}{a}} + e^{-\frac{x}{a}})$. 29. $y = \frac{e^x - e^{-x}}{e^x + e^{-x}}$. $\frac{dy}{dx} = \frac{4}{(e^x + e^{-x}))^2}$. 30. $y = x^n a^x$. $y' = a^x x^{n - 1}(n + x \log a)$. 31. $y = x^x$. $y' = x^x(\log x + 1)$. 32. $y = x^{\frac{1}{x}}$. $y' = \frac{x^{\frac{1}{x}} (1 - \log x)}{x^2}$. 33. $y = x^{\log x}$. $y' = \log x^2 \cdot x^{\log x - 1}$. 34. $f(y) = \log y \cdot e^y$. $f'(y) = e^y \left ( \log y + \frac{1}{y} \right )$. 35. $f(s) = \frac{\log s}{e^s}$. $f'(s) = \frac{1 - s \log s}{s e^s}$ 36. $f(x) = \log (\log x)$. $f'(x) = \frac{1}{x \log x}$. 37. $F(x) = \log^4 (\log x)$ $F'(x) = \frac{4 \log^3 (\log x)}{x \log x}$. 38. $\phi(x) = \log(\log^4 x)$. $\phi'(x) = \frac{4}{x \log x}$. 39. $\psi(y) = \log \sqrt{\frac{1 + y}{1 - y}}$. $\psi'(y) = \frac{1}{1 - y^2}$. 40. $f(x) = \log \frac{\sqrt{x^2 + 1} - x}{\sqrt{x^1 + 1} + x}$. $f'(x) = -\frac{2}{\sqrt{1 + x^2}}$. HINT. First rationalize the denominator. 41. $y = x^{\frac{1}{\log x}}$. $\frac{dy}{dx} = 0$. 42. $y = e^{x^x}$. $\frac{dy}{dx} = e^{x^x}(1 + \log x)x^x$. 43. $y = \frac{c^x}{x^x}$ $\frac{dy}{dx} = \left ( \frac{c}{x} \right )^x \left ( \log \frac{c}{x} - 1 \right )$. 44. $y = \left ( \frac{x}{n} \right )^{nx}$. $\frac{dy}{dx} = n \left ( \frac{x}{n} \right )^{nx} \left ( 1 + \log \frac{x}{n} \right )$. 45. $w = v^{e^v}$. $\frac{dw}{dv} = v^{e^v} e^v \left ( \frac{1 + v \log v}{v} \right )$. 46. $z = \left ( \frac{a}{t} \right )^t$. $\frac{dz}{dt} = \left ( \frac{a}{t} \right )^t (\log a - \log t - 1)$. 47. $y = x^{x^n}$. $\frac{dy}{dx} = x^{x^n + n - 1}(n \log x + 1)$. 48. $y = x^{x^x}$. $\frac{dy}{dx} = x^{x^x} x^x \left ( \log x + \log^2 x + \frac{1}{x} \right )$. 49. $y = a^{\frac{1}{\sqrt{a^2 - x^2}}}$. $\frac{dy}{dx} = \frac{xy \log a}{(a^2 - x^2)^{\frac{3}{2}}}$.
50. Differentiate the following functions:
(a) $\frac{d}{dx} x^2 \log x$. (f) $\frac{d}{dx} e^x \log x$. (k) $\frac{d}{dx} \log (a^x + b^x)$. (b) $\frac{d}{dx} (e^{2x} - 1)^4$. (g) $\frac{d}{dx} x^3 3^x$ (l) $\frac{d}{dx} \log_10 (x^2 + 5x)$. (c) $\frac{d}{dx} \log \frac{3x + 1}{x + 3}$. (h) $\frac{d}{dx} \frac{1}{x \log x}$. (m) $\frac{d}{dx} \frac{2 + x^2}{e^{3x}}$. (d) $\frac{d}{dx} \log \frac{1 - x^2}{\sqrt{1 + x}}$. (i) $\frac{d}{dx} \log x^3 \sqrt{1 + x^2}$. (n) $\frac{d}{dx} (x^2 + a^2) e^{x^2 + a^2}$. (e) $\frac{d}{dx} x^{\sqrt{x}}$. (j) $\frac{d}{dx} \left ( \frac{1}{x} \right )^x$. (o) $\frac{d}{dx} (x^2 + 4)^x$.
51. $y = \frac{(x + 1)^2}{(x + 2)^3 (x + 3)^4}$. $\frac{dy}{dx} = -\frac{(x + 1)(5x^2 + 14x + 5)}{(x + 2)^4 (x + 3)^5}$. HINT. Take logarithm of both sides before differentiating in this and the following examples. 52. $y = \frac{((x - 1)^{\frac{5}{2}}}{(x - 2)^{\frac{3}{4}}(x - 3)^{\frac{7}{3}}}$. $\frac{dy}{dx} = -\frac{(x - 1)^{\frac{3}{2}}(7x^2 + 30x - 97)}{12(x - 2)^{\frac{7}{4}}(x - 3)^{\frac{10}{3}}}$. 53. $\frac{dy}{dx} = x \sqrt{1 - x} (1 + x)$. $\frac{dy}{dx} = \frac{2 + x - 5x^2}{2\sqrt{1 - x}}$. 54. $y = \frac{x(1 + x^2)}{\sqrt{1 - x^2}}$ $\frac{dy}{dx} = \frac{1 + 3x^2 - 2x^4}{(1 - x^2}^{\frac{3}{2}}$. 55. $y = x^5(a + 3x)^3(a - 2x)^2$. $\frac{dy}{dx} = 5x^4(a + 3x)^2(a - 2x)(a^2 + 2ax - 12x^2)$.
48. Differentiation of $\sin v$.
Let $y$ $= \sin v$ By General Rule, p. 29 [§ 31], considering $v$ as the independent variable, we have FIRST STEP. $y + \Delta y$ $= \sin(v + \Delta v)$. SECOND STEP. $\Delta y$ $= \sin(v + \Delta v) - \sin v$[5] $= 2 \cos \left ( v + \frac{\Delta v}{2} \right ) \cdot \sin \frac{\Delta v}{2}$.[6] THIRD STEP. $\frac{\Delta y}{\Delta v}$ $= \cos \left ( v + \frac{\Delta v}{2} \right ) \left ( \frac{\sin \frac{\Delta v}{2}}{\frac{\Delta v}{2}} \right )$. FOURTH STEP. $\frac{dy}{dx}$ $= \cos v$.
[ Since $\lim_{\Delta v \to 0} \left ( \frac{\sin \frac{\Delta v}{2}}{\frac{\Delta v}{2}} \right ) = 1,$ by § 22, p. 21, and $\lim_{\Delta v \to 0} \cos \left ( v + \frac{\Delta v}{2} \right ) = \cos v$ ].
Since $v$ is a function of $x$ and it is required to differentiate $\sin v$ with respect to $x$, we must use formula (A), § 42, for differentiating a function of a function, namely,
$\frac{dy}{dx}$ $= \frac{dy}{dv} \cdot \frac{dv}{dx}$. Substituting value $\frac{dy}{dx}$ from Fourth Step, we get $\frac{dy}{dx}$ $= \cos v \frac{dv}{dx}$. XI ∴ $\frac{d}{dx} (\sin v)$ $= \cos v \frac{dv}{dx}$.
The statement of the corresponding rules will now be left to the student.
49. Differentiation of $\cos v$.
Let $y$ $= \cos v$. By 29, p. 2 [§ 1], this may be written $y$ $= \sin \left ( \frac{\pi}{2} - v \right )$. Differentiating by formula XI, $\frac{dy}{dx}$ $= \cos \left ( \frac{\pi}{2} - v \right ) \frac{d}{dx} \left ( \frac{\pi}{2} - v \right )$ $= \cos \left ( \frac{\pi}{2} - v \right ) \left ( -\frac{d}{dx} \right )$ $= -\sin x \frac{dv}{dx}$. [ Since $\cos \left ( \frac{\pi}{2} \right ) = \sin v$, by 29, p. 2.] XII ∴ $\frac{d}{dx} (\cos v)$ $= -\sin v \frac{dv}{dx}$.
50. Differentiation of $\tan v$.
Let $y$ $= \tan v$. By 27, p. 2 [§ 1], this may be written $\frac{dy}{dx}$ $= \frac{\cos v \frac{d}{dx}(\sin v) - \sin v \frac{d}{dx}(\cos v)}{\cos^2 v}$ $= \frac{\cos^2 v \frac{dv}{dx} + \sin^2 v \frac{dv}{dx}}{\cos^2 v}$ $= \frac{\frac{dv}{dx}}{\cos^2 v} = \sec^2 v \frac{dv}{dx}$. XIII ∴ $\frac{d}{dx}(\tan x)$ $= \sec^2 v \frac{dv}{dx}$.
51. Differentiation of $\cot v$.
Let $y$ $= cotv$. By 26, p. 2 [§ 1], this may be written $y$ $= \frac{1}{\tan v}$. Differentiating by formula VII, $\frac{dy}{dx}$ $= - \frac{\frac{d}{dx}(\tan v)}{\tan^2 v}$ $= -\frac{\sec^2 \frac{dv}{dx}}{\tan^2 v} = -\csc^2 v \frac{dv}{dx}$. XIV ∴ $\frac{d}{dx}(\cot v)$ $= -\csc^2 v \frac{dv}{dx}$.
52. Differentiation of $\sec v$.
Let $y$ $\sec v$ By 26, p. 2 [§ 1], this may be written $y$ $= \frac{1}{\cos v}$. Differentiating by formula VII, $\frac{dy}{dx}$ $= -\frac{\frac{d}{dx}(\cos v)}{\cos^2 v}$ $=\frac{\sin v \frac{dv}{dx}}{\cos^2 v}$ $= \frac{1}{\cos v} \frac{\sin v}{\cos v} \frac{dv}{dx}$ $= \sec v \tan v \frac{dv}{dx}$. XV ∴ $\frac{d}{dx}(\sec v)$ $= \sec v \tan v \frac{dv}{dx}$.
53. Differentiation of $\csc v$.
Let $y$ $= \csc v$. By 26, p. 2 [§ 1], this may be written $y$ $= \frac{1}{\sin v}$. Differentiating by formula VII, $\frac{dy}{dx}$ $= -\frac{\frac{d}{dx}(\sin v)}{\sin^2 v}$ $= -\frac{\cos v \frac{dv}{dx}}{\sin^2 v}$ $= -\csc v \cot v \frac{dv}{dx}$. XVI ∴ $\frac{d}{dx}(\csc v)$ $= - \csc v \cot v \frac{dv}{dx}$.
1. The student must not forget that this function is defined only for positive values of the base $a$ and the variable $v$.
2. If we take the third and fourth steps without transforming the right-hand member, there results:
Third step: $\frac{\Delta y}{\Delta v} = \frac{\log_a(v + \Delta v) - \log_a v}{\Delta v}$.
Fourth step. $\frac{dy}{dx} = \frac{0}{0}$, which is indeterminate. Hence the limiting value of the right-hand dv 0 member in the third step cannot be found by direct substitution, and the above transformation is necessary.[/itex]
3. The logarithm of $e$ to any base $a (= \log_a e)$ is called the modulus of the system whose base is $a$. In Algebra it is shown that we may find the logarithm of a number $N$ to any base $a$ by means of the formula
$\log_a N = \log_a e \cdot \log_e N = \frac{\log_e N}{\log_e a}$.
The modulus of the common or Briggs system with base 10 is
$\log_{10} e = .434294...$.
4. $u$ can here assume only positive values.
5. If we take the third and fourth steps without transforming the right-hand member, there results:
Third step. $\frac{\Delta y}{\Delta v} = \frac{\sin(v + \Delta v) - \sin v}{\Delta v}$
Fourth step. $\frac{dy}{dv} = \frac{0}{0}$, which is indeterminate (see footnote, p. 46 [§ 44]).
6. Let $A$ $= v + \Delta v$ $A$ $= v + \Delta v$ and $B$ $= v$ $B$ $= v$ Adding, $A + B$ $= 2v + \Delta v$. Subtracting, $A - B$ $= \Delta v$ Therefore $\frac{1}{2}(A + B)$ $= v + \frac{\Delta v}{2}$. $\frac{1}{2}(A - B)$ $= \frac{\Delta v}{2}$.
Substituting these values of $A, B, \frac{1}{2}(A + B), \frac{1}{2}(A - B)$ in terms of $v$ and $\Delta v$ in the formula from Trigonometry (42, p. 2 [§ 1]),
$\sin A - \sin B$ $= 2 \cos \frac{1}{2} (A + B) \sin \frac{1}{2} (A - B)$, we get $sin(v + \Delta v) - \sin v$ $= 2 \cos \left ( v + \frac{\Delta v}{2} \right ) sin \frac{\Delta v}{2}$.
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2015-03-31 06:44:00
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https://www.scienceforums.net/topic/7011-whats-the-opposite-of-entropy/
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# What's the opposite of entropy?
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I googled for "define: entropy" and came up with this:
"A measure of the disorder in a system."
I also entered "entropy" into http://www.dictionary.com and found, among other definitions:
"The tendency for all matter and energy in the universe to evolve toward a state of inert uniformity."
So if I understand this correctly, entropy is the phenomenon observed when, for instance, an elastic band goes from stretched to slack or a building going from erect to rubble when demolished by means of explosives. Is this correct?
If so, what do we call the opposite phenomenon - that is, the building up of physical systems from something simple with uniformly distributed energy to something more complex and non-uniformly structured?
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#### Popular Days
Creation of order.
But creation of order creates more chaos than the order anyway.
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atrophy....or is that a pint of beer
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I thought that was Muscle wastage from lack of use, Grey matter for example
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Asking what the opposite of entropy is, is like asking, "What's the opposite of temperature?"
If you want to lower the entropy you have to do work, and the entropy will have to increase somewhere else, as JaKiri stated.
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Swansont, is there a word/term for this though?
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Does the universe necessarily have to end in a state of disorder? I only ask because I've come to understand entropy to be a law that all physical systems succumb to eventually. But obviously, there are both entropic and "enthalpic" (thanks YT2095) processes in the universe. Why would the entropic processes prevail in the end? What about states of equilibrium?
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in Chemistry its "Enthalpy"
enthalpy and entropy are two different things
theres even a way in which to relate entropy to enthalpy
G=H-TS
H being enthalpy
T being absolute temperature
S being entropy.
and G being Gibbs Free Energy
There isnt really anything I'd call the opposite of entropy. The closest thing would probably be Gibbs free energy, which is the energy from a reaction that is not lost to disorder. If you think of enthalpy as the total energy in a reaction released and entropy as the energy lost to disorder then the difference is that energy which is still available to do work or increase order. This difference is called Gibbs Free energy.
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Oh for Crying out loud!
Im perfectly familiar with delta G thanks
that`s why I used it, and stated "IN CHEMISTRY"!
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So if I understand this correctly, entropy is . . . what do we call the opposite phenomenon - that is, the building up of physical systems from something simple with uniformly distributed energy to something more complex and non-uniformly structured?
Hi everyone, this is my first post here.
I realize you are probably looking for a purely physical answer, but in terms of behavior at least, two apparently anti-entropic processes are the ordering that results in adaptive evolutionary change (remarkable since through reproduction it can last for a long time), and the tendency to order things a healthy consciousness exhibits. Of course, everyone knows in terms overall disorder (via metabolism, aging, etc., entropy still prevails.
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Uh, I dont think so. In chemistry we've studied entropy and enthalpy as different subjects. On is S, and I think the other one is G although I dont remember which is which.
But yeah, opposite of entropy is order. Usually things move towards a state of disorder, so the opposite of that would be the sponanteous ordering of a system.
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two apparently anti-entropic processes are the ordering that results in adaptive evolutionary change
They aren't antientropic. They obey the 2nd law, and aren't very efficient either.
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They aren't antientropic. They obey the 2nd law, and aren't very efficient either.
Hmmmmm, did you read my entire post? Anticipating someone thinking I don't understand entropy, I tried to make it clear I wasn't suggesting they didn't obey the second law; everything does. I specifically pointed to behavior and not to the thermal processes involved in achieving that behavior.
Part of the problem is that the term "entropy" has been expanded from thermodynamics to a general term for the ever-growing disorder in the universe. In my own mind I think about two variations as thermodynamic entropy and structural entropy, respectively. In terms of gib65's question, he seemed to be thinking about the more general, or "structural," use of the term.
In our universe, there are no known examples of extended ordering (structural antientropy) which surpass that found in life and consciousness. As impressive as the formation of the complex dynamics of a star is from collapsed interstellar gas, for instance, it doesn't hold a candle to billions of years of layer after layer organizing done through evolution. If anything comes close to structural antientropic behavior, evolution has to be a top candidate.
But then there is consciousness. Look at us, building, learning, creating, trying to survive . . . all of it structurally antientropic behavior. Of course, assuming consciousness emerged from life processes, that sort of makes sense.
My point was that although there is no thermodynamic term for antientropy, there are instances of extraordinary structurally antientropic behaviors which we have termed "life" and "consciousness."
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It was the 'apparantly antientropic' that threw me, because it's quite clearly not.
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Does entropy depends on the cosmological model? I think the big bang universe is one where entropy continuously increases, but there are other (less favored) cosmological models where low-grade energy may recycle back as mass.
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Yes, yes, all very good, but what do the laws of entropy say about the fate of the universe (now that I know to distinguish between thermodynamic and structural entropy - thanks Les - I refer only to the structural type). Is the universe doomed to settle into a field of floating debri without much life to it at all, or is there hope that the universe could evolve into something greater from which it will never degrade, or at the very least continue forever in a state of equilibrium (with moderate fluctuations now and again). For instance, the idea that the universe will eventually collapse upon itself in a "Big Crunch", and then a "Big Band", and yet another "Big Crunch" after that, and so on, seems to me like a system that will never succumb to entropy. Or what about the Moon orbiting the Earth, or an electron orbiting a proton? Don't these represent systems in states of equilibrium such that the one body will continue to orbit the other indefinitely?
I'm not a physicist, so I'm not trying to strike a blow at the theory of entropy. Just asking 'cause I wanna know.
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Yes' date=' yes, all very good, but what do the laws of entropy say about the fate of the universe (now that I know to distinguish between thermodynamic and structural entropy - thanks Les - I refer only to the structural type). Is the universe doomed to settle into a field of floating debri without much life to it at all, or is there hope that the universe could evolve into something greater from which it will never degrade, or at the very least continue forever in a state of equilibrium (with moderate fluctuations now and again). For instance, the idea that the universe will eventually collapse upon itself in a "Big Crunch", and then a "Big Band", and yet another "Big Crunch" after that, and so on, seems to me like a system that will never succumb to entropy. Or what about the Moon orbiting the Earth, or an electron orbiting a proton? Don't these represent systems in states of equilibrium such that the one body will continue to orbit the other indefinitely?
I'm not a physicist, so I'm not trying to strike a blow at the theory of entropy. Just asking 'cause I wanna know.[/quote']
I think entropy happens so slow on a universal level and the universe is so big it will be tens or hundreds of billions of years before it would pose any major effect. If there was to be a big crunch I think that would happen long before entropy effects were felt. But yes the universe is slowly decaying because of entropy (very slowly). The moon will not orbit the Earth indefinitely either will the Earth orbit the sun. Eventually they will succumb to gravity, I don't think entropy will play a role in that. I think the moon was estimated at about 50,000 years not sure about the Earth, 5 billion years maybe.
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Entropy will play a role in that, in the sense that it is due to there being no perpetual motion device, if you see what I mean.
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. . . but what do the laws of entropy say about the fate of the universe (now that I know to distinguish between thermodynamic and structural entropy - thanks Les - I refer only to the structural type).
I think other posters are suggesting that entropy alone may not be the only factor determine the fate of the universe, which is true.
But just answering your question posed above, the law of entropy, considered as the only determining factor, says the universe is going to disintegrate.
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it is not the only this for the fact that the universe is gonna reach the steady state some how with 0k temperature....
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The heat death of the universe does not necessarily mean that 0K will ever be reached.
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• 3 weeks later...
...in terms of behavior[/i'] at least, two apparently anti-entropic processes are the ordering that results in adaptive evolutionary change (remarkable since through reproduction it can last for a long time), and the tendency to order things a healthy consciousness exhibits.
So, then, what if the universe itself is evolving? What if it has the potential to evolve into something that can actually alter its own laws such that the laws of entropy won't necessarily hold forever? I guess I'm stepping more into fantacy with that idea instead of scientific speculation, but it's interesting to ponder, isn't it?
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• 5 years later...
THERE IS A BASIC DIVISION AMONGST ALL THESE POSTS:
Though it may be an oversimplification, in spite of the fact that some of those posting are more eloquent than others, it seems that everyone is leaning toward EITHER:
1.) The strict verbal definition of entropy
OR
2.) The conceptual idea of entropy
Like the Socratic self-reference paradox ("all I knowe is dat I don't know nuthin'....daaaaauuuuuh");, the strict verbal definitions produce inevitable conflict and contradictions;
but, like the Socratic paradox, in spite of the obvious problems with treating the phrase mathematically,
One can basically understand the meaning behind the words, which may fail to be and algebraically correct. It is just another case of the importance lying not in the written lines, but between the lines.
After reading about both concepts, one (hopefully) gets an idea of what "enthalpy" and "entropy" mean, an (obvious) idea about compiliation vs. analysis, creation vs. destruction, growing vs. decaying, blah blah.....
And, after losing one's mind contemplating the Socratic paradox, in spite of the verbal/algebraic contradiction, one (hopefully) gets an idea of the meaning behind the words, an idea about the fallability and limits of human perception, blah blah.....
(can anyone out there recognize the obvious? would anyone notice a skyscraper imploding on national TV? is it still 1983?)
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*shrug* In bio I've heard the term syntropy used to describe a system exporting entropy to keep its own low. Obviously, it'd only apply on the local scale or whatever you call it.
Edited by AzurePhoenix
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2023-03-21 21:22:50
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https://pythagitup.com/2017/06/22/get-on-my-fraction-level/
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# Get On My Fraction Level!
Many students have trouble with fractions. When I taught at a high school, my 10th and 11th grade students regularly had difficulty performing operations with fractions. As an 8th grade teacher, I’ve tried to help my students develop, refine, and maintain strong fraction skills. Don’t get me wrong: I don’t consider fluency in performing operations with fractions to be the most important skill for my students to have, but it’s certainly one that will contribute to their success at the high school level. With that in mind, here’s an approach I used this year to work on fractions.
After the warmup and overview of the day’s class some time in October, I pulled up a slide with four fraction multiplication problems. These first ones were relatively simple like $4\cdot \frac{1}{2}$. Students had little trouble performing the multiplication (yay!), and thankfully, students presented a number of different methods. The most common early responses were $\frac{4}{1}\cdot \frac{1}{2}=\frac{4}{2}=2$ and $4\div 2=2$. As I continued to present problems over the next few weeks, I added complications. Students noticed that simplifying often made the multiplication easier (e.g. $\frac{10}{5}\cdot 44$). The big breakthrough came when I presented a particularly annoying pair of fractions to multiply like $\frac{27}{7}\cdot \frac{14}{9}$. To this point, I had not pushed students to use a particular method; any simplifying they did came from them not me. Whoever offered the response of $\frac{378}{63}$ did not respond kindly to the question of whether that fraction could be simplified. By this point, students had been doing so much simplifying that it was no surprise to anyone that their lives would be easier if they found a way to simplify before multiplying. A brief discussion of the commutative property allowed a student to rewrite the multiplication as $\frac{27}{9}\cdot \frac{14}{7}$, which everyone in the classroom felt comfortable multiplying. It was a great moment of mathematical discovery.
As the weeks progressed, I continued to throw more and more challenging multiplication problems at them, and I also started to incorporate some addition, subtraction, and division. Students began feeling much more comfortable with fractions than they ever had before, even if they still weren’t the biggest fraction fans around. This fraction work paid off when we wrote equations of lines, and in general, I think it gave students some confidence in an area where they had so little before.
I definitely plan to continue using “Get On My Fraction Level” in my classes this coming school year. I’d like to find a way to incorporate more active participation. I might give students a weekly template to use each day when we do our fractions. I did that two years ago with scientific notation, and it worked pretty well. One big concern is time: with so many topics to cover, it’s difficult to carve out time to work on something that isn’t really an 8th grade standard. Having seen how working with fractions helped so many of my students grow, however, I will definitely find a way to incorporate regular fraction work into my lessons.
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2020-01-24 13:34:23
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https://www.tutorialspoint.com/conservation-of-charge
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# Conservation of Charge
PhysicsElectric Charge and Static Electricity
#### Class 11th Physics - Elasticity
6 Lectures 1 hours
#### Class 11th Physics - Oscillations
12 Lectures 2 hours
#### Class 11th Physics - Waves
14 Lectures 3 hours
## Introduction
A charge is a property of matter that causes and experiences electrical and magnetic effects. The basic concept behind charge conservation is that the system's total charge is conserved.
It can be defined as:
Atoms and molecules make up all of the materials in the world. Normally, all of these atoms and molecules are electrically neutral, with all of their charges balanced. One of the types of charges must be removed in order to electrify a neutral body.
A charged body always indicates either an electron deficit or an electron surplus. Electrons are assumed to be negatively charged. By removing some electrons from the atom, a body can be positively charged. Similarly, the same body can be negatively charged by gaining some electrons. In real life, whenever one body rubs against another.
One of the bodies loses electrons, while the other gains and becomes electrified as a result. The flow of charge causes electricity to flow. Some substances permit the passage of electricity through them, while others do not. These materials are classified as either conductors or insulators.
## Conductors
Conductors are substances that permit the passage of electricity through them. They have electric charges (electrons) that are relatively free to move around the conductor's body.
Examples: Metals, Human and Animal bodies
## Insulators
Insulators are substances that do not allow electricity to pass through them. They also have electric charges (electrons), but they cannot move freely within the body. As a result, this body is unable to conduct electricity.
Examples: Wood, rubber, clothes, etc.
## Conservation of Charge
"Conservation of Charge is the principle that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved."
The system, as we know, is a collection of objects whose interactions with charges are analogous to the conservation of energy and momentum. However, this conservation law is more intuitive because an object's net charge is determined by the number of electrons and protons. The protons and electrons cannot simply appear and disappear; the total charge must be the same. That's why a body always has the same number of electrons and protons.
Every atom is electrically neutral, containing the same number of electrons as protons in the nucleus. Body charges can also be whole multiples of the elementary charge: Electrons and protons contain electrical charge; the smallest charge that a body can have is the charge of one electron or proton.
For example, ${1.6\:\times\:10^{−19}C\:or\:+1.6\:\times\:10^{−19}C}$
### Explanation
The law of charge conservation states that the net charge of an isolated system will always remain constant. Let us try to get a better understanding of it. There is a list of basically two ideal states for a system with multiple objects.
• The first is that all of the objects are net neutrally charged.
• So there are the same number of protons and electrons in the entire system, and for each proton, there is an electron to balance it.
• Another ideal state would be for the system's net charge to be distributed uniformly in the objects.
• Rather than concentrating negative charge in a few bodies, the charge on the body is evenly distributed throughout by electron transfer, which can be accomplished by electron transfer from higher to lower polarity.
• Only electrons, not protons, can be involved in charge transfer.
### Conservation of Charge Examples
The conservation of charge principle states that no net charge can be produced. A few examples are provided below.
• Charges as a result of induction
• A proton decays into a positron and a neutron during radioactive decay, but no net charge is produced.
Images Coming soon
According to the above image, if our system is not influenced by any other charges, the net internal distribution of charges will continue in such a way that the overall net charge of the system will remain the same. In other words, charge can neither be created nor destroyed, and there is total charge conservation.
### Electrons
Electrons are one of the three major types of particles that comprise an atom. In contrast to protons and neutrons, which are made up of smaller, simpler particles, electrons are fundamental particles made up of no smaller particles. They are leptons, a type of fundamental particle. Every lepton has an electric charge of 1 or 0.
Electrons are extremely small particles. Because an electron's mass is only about 1/2000 that of a proton or neutron, electrons contribute almost nothing to the total mass of an atom. Electrons have an electric charge of one, which is equal to but opposite to a proton's charge of one. Because every atom has the same number of electrons as protons, positive and negative charges "cancel out." making atoms electrically neutral.
### Protons
A proton is one of the three major particles that comprise an atom. Protons are found in the atom's nucleus. This is a tiny, dense region in the atom's core. Protons have a positive electrical charge of one (+1) and a mass of one atomic mass unit (amu), or approximately ${1.67\:\times\:10^{-27}}$ kilogrammes. Together with neutrons, they account for nearly all of an atom's mass.
### Neutrons
Except for most hydrogen atoms, all atoms have neutrons in their nucleus. In contrast to protons and electrons, which are electrically charged, neutrons are electrically neutral. That's why the neutrons in the above diagram are labelled ${n^0}$. The zero represents "zero charge." A neutron's mass is slightly greater than a proton's mass, which is 1 atomic mass unit (amu). (An atomic mass unit is approximately ${1.67\:\times\:10^{-27}}$ kilogrammes.) A neutron has the same diameter as a proton, which is ${1.7\:\times\:10^{-15}}$ metres.
## FAQs
Q1. How many electrons are there in one Coulomb of charge?
Ans. Using the formula, Q = ne, we get
$\mathrm{n = Q/e}$
$\mathrm{n = 1/ (1.6\: X\: 10^{-19})}$
Therefore, $\mathrm{n = 6.25\: x \:10^{18}}$
Q2. What happens to the radius of a soap bubble when its charge is negative?
Ans. As the charge of the soap bubble becomes negative, the radius of the bubble increases due to repulsive force.
Q3. Is it possible to have a charge with a value of $\mathrm{1.6 \:x \:10^{-20}\:C}$?
Ans. No, a charge of $\mathrm{1.6 \:x \:10^{-20}\:C}$ is not possible because it is 1/10 of an electronic charge and thus not an integral multiple.
Q4. Consider two copper spheres of equal radius, one solid and the other hollow. Which copper sphere has the higher charge?
Ans. Because the charges reside on the surface of the material, both copper spheres will have an equal charge. Because both spheres have the same radius, they will have the same charge.
Q5. What is specific charge?
Ans. The specific charge is the ratio of an ion's or subatomic particle's charge to its mass.
Updated on 13-Oct-2022 11:19:47
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2022-12-01 05:50:06
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https://www.zbmath.org/?q=ai%3Asantarpia.l+ai%3Afontana.d-m
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zbMATH — the first resource for mathematics
Laminar free convection from a vertical plate in partly dissociated gases. (English) Zbl 0971.76083
Summary: Two-dimensional steady free convection from an isothermal vertical plate is studied in a gas where a reversible fast reaction of dissociation $$A\leftrightarrow 2B$$ takes place at atmospheric pressure. The effective properties in the presence of dissociation are evaluated. The governing boundary-layer equations are solved numerically for a wide range of values of the independent variables. All the data obtained are correlated by a single correlation, even if the temperature interval in the boundary layer $$(T_w, T_\infty)$$ is allowed to vary in a wide range, both in relative location and width, in respect to the temperature interval of dissociation. The correlated dimensionless parameters include the ratio $$\rho_w/\rho_\infty$$ and are defined through the mixture effective properties calculated at $$T_w$$ and $$T_\infty$$. The maximum absolute value of relative error depends essentially on two parameters: $$\alpha$$ related to variations of fraction of moles, and effective mass density $$\rho^*$$ in the boundary layer.
MSC:
76R10 Free convection 76N15 Gas dynamics, general 76V05 Reaction effects in flows 80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text:
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2021-05-08 11:50:03
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https://etna.math.kent.edu/volumes/2011-2020/vol42/abstract.php?vol=42&pages=136-146
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## R$^3$GMRES: Including Prior Information in GMRES-Type Methods for Discrete Inverse Problems
Yiqiu Dong, Henrik Garde, and Per Christian Hansen
### Abstract
Lothar Reichel and his collaborators proposed several iterative algorithms that augment the underlying Krylov subspace with an additional low-dimensional subspace in order to produce improved regularized solutions. We take a closer look at this approach and investigate a particular Regularized Range-Restricted GMRES method, R$^3$GMRES, with a subspace that represents prior information about the solution. We discuss the implementation of this approach and demonstrate its advantage by means of several test problems.
Full Text (PDF) [181 KB], BibTeX
### Key words
inverse problems, regularizing iterations, large-scale problems, prior information
### AMS subject classifications
65F22, 65F10
Vol. 51 (2019), pp. 412-431 Andreas Neubauer: Augmented GMRES-type versus CGNE methods for the solution of linear ill-posed problems Vol. 55 (2022), pp. 341-364 Erin Carrier and Michael T. Heath: Exploiting compression in solving discretized linear systems Vol. 55 (2022), pp. 532-546 Kirk M. Soodhalter: A note on augmented unprojected Krylov subspace methods
< Back
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2022-12-05 18:40:52
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http://catalog.flatworldknowledge.com/bookhub/reader/30?e=wright-ch04_s05
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Study Aids:
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View Full Student FAQs
## 4.5 What’s the Yield on That?
### Learning Objective
1. What is yield to maturity and for what types of financial instruments is the yield to maturity relatively easy to calculate?
Thus far, we have assumed or been given a market interest rate and then calculated the price (PV) of the instrument. Or, given the PV and an interest rate, we’ve calculated the FV. Sometimes it is useful to do the opposite, to calculate the interest rate or, yield to maturity, if given the PV and FV. Say that you know that someone paid $750 for a zero coupon bond with a face value of$1,000 that will mature in exactly a year and you want to know what interest rate he or she paid. You know that PV = FV/(1 + i). Solving for i:
You can check your work by reversing the problem—that is, asking how much you’d pay today for $1,000 in a year if interest was at 33.33 percent: PV = 1000/(1.3333333) =$750. Voilà!
### Stop and Think Box
Suppose you have $1,000 to invest for a year and two ways of investing it (each equal in terms of risk and liquidity): a discount bond due in one year with a face value of$1,000 for $912 or a bank account at 6.35 percent compounded annually. Which should you take? Choose the bond, which will yield 9.65 percent: (1000 − 912)/912 = .0965. To maximize your haul, invest the$88 left over from the purchase of the bond in the bank account.
Calculating the yield to maturity for a perpetual debt, one with no maturity or repayment date, like a ConsolA type of perpetual bond issued by the British government., ground rent, or perpetual interest-only mortgage, is also quite easy. The price or PV of a perpetuity is equal to the yearly payment divided by the going rate of interest:
So a $1,000 ground rent that pays$50 a year (a 5 percent coupon rate) would be worth $1,000 if interest rates were 5 percent, less if rates are higher, more if lower: $PV=50/.05=1,000$ $PV=50/.10=500$ $PV=50/.01=5,000$ Calculating the yield to maturity of a perpetuity, if given the PV and FV, is easily done by taking the equation and solving for i: $PV=FV/i$ So the yield to maturity of a ground rent that pays$60 per year and that currently sells for $600 would be 10 percent: i = 60/600 = .10 = 10%. ### Stop and Think Box A ground rent contract consummated in Philadelphia, Pennsylvania, in 1756 is still being paid today. Someone recently paid$455 for the $23.17 annual payment. What is the ground rent’s yield to maturity? If the interest rate rises to 10 percent, how much will the ground rent be worth? What if interest falls to 2 percent? i = C/P so i = 23.17/455 = 0.05092 = 5.09%; PV = 23.17/.1 =$231.70; PV = 23.17/.02 = $1,158.50. Calculating yield to maturity for coupon bonds and fixed-payment loans, however, is mathematically nasty business without a computer or bond table. In the past, people used to estimate the yield to maturity on such instruments by pretending they were perpetuities or engaging in trial-and-error interpolation. In the first method, you use the easy perpetuity equation above (i = FV/PV) to get a quick estimate called the current yieldA quick (i = FV/PV) but flawed method for calculating interest rates of nonperpetual debt.. Unfortunately, current yield can be wide of the mark, especially for bonds with maturities less than twenty years and bonds whose prices are far from their par value.Current yield is simply the yield to maturity of a perpetuity, so the more like a perpetuity a bond is, the better the current yield will approximate its yield to maturity. The shorter the maturity of a bond, the less like a Consol it is, so the less accurate the current yield formula will be. Similarly, the current yield works better the closer a bond’s price is to par because yield to maturity equals the coupon rate when the bond is at par. As the price deviates further from par, the less well the current yield can approximate the yield to maturity. In the second method, one backs into the yield to maturity by making successive guesses about i and plugging them into the PV formula. Not fun, but you’ll eventually get there. Most people today therefore use a financial calculator, spreadsheet, or Web-based utility rather than such erroneous (current yield) or laborious (interpolation) processes. You should be able to calculate the yield to maturity of one-year discount bonds or perpetuities by hand, or at worst with the aid of simple (nonfinancial) calculator. Here is a little practice. ### Exercises 1. A$100 bond payable in a year sells for $97.56. What is the yield to maturity? 2. Sam promises to pay Joe$1,904 in a year if Joe gives him $1,498 today. What interest rate is Sam paying and what interest rate Joe is earning? 3. Every year, the U.S. government pays a certain Indian tribe$10,000 and, by terms of its treaty with that tribe, must do so forever. Mr. Trump offered to purchase the right to receive that stream for a one-time payment of \$143,500. What yield to maturity did Trump offer the Indians?
4. What is the yield to maturity of a British Consol paying £400 per year that sold for £27,653?
### Key Takeaways
• Yield to maturity is the most economically accurate way of measuring nominal interest rates.
• It is easily calculated for one-year discount bonds i = (FV–PV)/PV and perpetuities i = C/PV where C is the coupon or annual payment.
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2013-05-22 19:26:11
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https://www.sarthaks.com/1079693/prove-that-the-area-in-the-first-quadrant-enclosed-by-the-axis-the-line-3y-and-the-circle-is?show=1079703
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# Prove that the area in the first quadrant enclosed by the axis, the line x = √3y and the circle x^2 + y^2 = 4 is π/3.
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Prove that the area in the first quadrant enclosed by the axis, the line x = √3y and the circle x2 + y2 = 4 is π/3.
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To find an area in the first quadrant enclosed by the x – axis,
x = √3y
x2 + y2 = 4
Or $(\sqrt{3y})^2 + y^2 = 4$
Or 4y2 = 4
Or y = $\pm 1$
And x = $\pm \sqrt{3}$
Equation (i) represents a line passing through (0, 0), (– √3, – 1), (√3,1).
Equation (ii) represents a circle centre (0,0) and passing through (±2, 0), (0, ±2).
Points of intersection of line and circle are (– √3, – 1) and (√3,1).
These are shown in the graph below: -
Required enclosed area = Region OABO
= Region OCBO + Region ABCA
Hence proved that the area in the first quadrant enclosed by the axis, the line x = $\sqrt{3}y$ and the circle x2 + y2 = 4 is $\frac{\pi}{3}.$
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2021-05-15 02:55:13
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https://tex.stackexchange.com/questions/307038/change-citation-style-from-square-brackets-to-backslashes-with-biblatex
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# Change citation style from square brackets to backslashes with Biblatex
How can I do the style of citation in Biblatex from [Abc97]to this \Abc97\ in both citing within the document as well as in the references at the end of the document.
I use this code :
\usepackage[%
natbib = true,
backend = bibtex8, %bibtex, biber
style = alphabetic,
%citestyle = alphabetic,
maxcitenames = 2,
mincitenames = 1,
%bibstyle = alphabetic, %authoryear, %draft, %,
sorting = nyvt,
maxbibnames = 6,
minbibnames = 6,
language = ngerman,
date = long,
backref = false,
backrefstyle = none % none, three, two, two+, three+, all+
]{biblatex}
\usepackage[babel,german=quotes]{csquotes}
\bibliography{input/literatur}
and my bib Looks like that
@book{Gug01b,
author = {{Gugel, T., Hupf, W.}},
title = {{Vakuumtechnologie}},
publisher = {Oldenburg, Erkenschwick},
edition = {3},
year = {2001}
}
• Welcome! Please provide a complete example we can compile rather than code fragments as it is much more useful i.e. \documentclass ... \end{document} with minimal packages and content. (Probably just one citation in the body of the document for this question.) Do you mean backslashes rather than forward slashes? – cfr Apr 29 '16 at 15:03
• Note that you should give the author names as author = {Gugel, T. and Hupf, W.}. See also How should I type author names in a bib file?. If you don't like the name format you get then, you can change that easily. (Just have a look at the many questions about that around here.) – moewe Apr 30 '16 at 8:28
## 1 Answer
Define a new wrapper for backslashes
\newrobustcmd{\mkbibbackslashes}[1]{\textbackslash #1\textbackslash}
and then use it in the labelalphawidth and shorthandwidth formats (for numeric labels you'd also need labelnumberwidth)
\DeclareFieldFormat{labelalphawidth}{\mkbibbackslashes{#1}}
\DeclareFieldFormat{shorthandwidth}{\mkbibbackslashes{#1}}
Then we will have to modify the cite commands, the definition is a copy from alphabetic.cbx with \mkbibbrackets replaced by \mkbibbackslashes
\DeclareCiteCommand{\cite}[\mkbibbackslashes]
{\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite}}
{\multicitedelim}
{\usebibmacro{postnote}}
MWE
\documentclass[ngerman]{article}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[natbib = true, style = alphabetic]{biblatex}
\addbibresource{biblatex-examples.bib}
\newrobustcmd{\mkbibbackslashes}[1]{\textbackslash #1\textbackslash}
\DeclareFieldFormat{labelalphawidth}{\mkbibbackslashes{#1}}
\DeclareFieldFormat{shorthandwidth}{\mkbibbackslashes{#1}}
\DeclareCiteCommand{\cite}[\mkbibbackslashes]
{\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite}}
{\multicitedelim}
{\usebibmacro{postnote}}
\begin{document}
\cite{sigfridsson}
\printbibliography
\end{document}
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2019-10-23 15:08:14
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https://math.stackexchange.com/questions/2926888/find-operatornamecovx2-y2
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# Find $\operatorname{Cov}(X^2,Y^2)$
Suppose $$X$$ and $$Y$$ follow $$N(0,1)$$ and $$\operatorname{Corr}(X,Y)=\rho$$. Find $$\operatorname{Cov}(X^2,Y^2)$$.
Here is what I know $$\operatorname{Cov}(X^2,Y^2)=E[X^2Y^2]-E[X^2]E[Y^2]$$
Since $$E[X^2]= \operatorname{Var}(X)+E^2[X]= \operatorname{Var}(X)$$, $$\enspace E[X^2]E[Y^2]= \operatorname{Var}(X) \operatorname{Var}(Y)=1$$.
But I don't know how to deal with $$E[X^2Y^2]$$.
Am I in the right track?
• To say that $X\sim N(0,1)$ and $Y\sim N(0,1)$ and $\operatorname{corr}(X,Y) = \text{a particular number}$ is not enough information to specify the distribution of the pair $(X,Y).$ However, it is enough if you add an additional bit of information: that the pair $(X,Y)$ is JOINTLY normally distributed. There is this simple way of getting $X\sim N(0,1)$ and $Y\sim N(0,1)$ and $\operatorname{corr}(X,Y) = \text{your preferred number}$ without $(X,Y)$ being jointly normal: Let $Y=\begin{cases} \phantom{+}X&\text{if } |X|\le c, \\ -X&\text{if }|X| > c, \end{cases}\quad$ and then$\,\ldots \qquad$ – Michael Hardy Sep 22 '18 at 21:21
• $\ldots\,$ choose the value of $c$ to to make the correlation what you want it to be. $\qquad$ – Michael Hardy Sep 22 '18 at 21:23
• I don't know whether or not the information given, without the assumption of JOINT normality, is enough to determine the correlation between $X^2$ and $Y^2. \qquad$ – Michael Hardy Sep 22 '18 at 21:25
Assuming you mean $$(X,Y)$$ is jointly normal where $$X$$ and $$Y$$ have zero means and unit variances and $$\operatorname{Corr}(X,Y)=\rho$$, we know the conditional distribution of $$Y\mid X$$, namely
$$Y\mid X\sim N(\rho X,1-\rho^2)$$
\begin{align} E(X^2Y^2)&=E\left[E(X^2Y^2\mid X)\right] \\&=E\left[X^2E(Y^2\mid X)\right] \\&=E\left[X^2\left(\operatorname{Var}(Y\mid X)+(E(Y\mid X))^2\right)\right] \\&=\quad\cdots \end{align}
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2019-11-21 01:00:35
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http://www.solutioninn.com/in-exercise-1275-you-couldnt-perform-the-chisquare-independence-test
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# Question
In Exercise 12.75, you couldn't perform the chi-square independence test because the assumptions regarding expected frequencies were not met. As mentioned in the text, three approaches are available for remedying the situation: (1) combine rows or columns; (2) eliminate rows or columns; or (3) increase the sample size.
a. Combine the first two rows of the contingency table in Exercise 12.75 to form a new contingency table.
b. Use the table obtained in part (a) to perform the hypothesis test required in Exercise 12.75, if possible.
c. Eliminate the second row of the contingency table in Exercise 12.75 to form a new contingency table.
d. Use the table obtained in part (c) to perform the hypothesis test required in Exercise 12.75, if possible.
Sales0
Views23
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2016-10-25 12:52:07
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https://www.bionicturtle.com/forum/threads/t4-24-fixed-income-arbitrage-to-exploit-violation-of-law-of-one-price.22495/
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What's new
# YouTubeT4-24: Fixed Income: Arbitrage to exploit violation of Law of One Price
#### Nicole Seaman
Financial Risk Manager (FRM), Topic 4: Valuation and Risk Models, Fixed Income, Bruce Tuckman Chapter 1, Prices Discount Factors and Arbitrage. How do we exploit the Law of One Price (which asserts that--absent confounding factors like liquidity or taxes--is only one set of discount factors)? We construct a replicating portfolio; i.e., a portfolio that produces the same stream of cash flows as the bond that is mis-priced. Then we purchase the bond/portfolio that is trading cheap and we sell (short) the bond that is trading expensive. In this example, net proceeds equal $0.065 which is only a few pennies. But leverage can increase the riskless profit to$325,000, wow!
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2020-08-14 11:19:24
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http://nepaloug.com/hitman-age-zhvnv/379851-how-to-calculate-the-radius-of-a-rectangle
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The radius of a regular polygon is the distance from the center to any vertex.It will be the same for any vertex. Enter any two values and press 'Calculate'. The width, height and radius of an arc are all inter-related. Find the radius of a circle whose area is equal to the area of a rectangle with sides measuring $$44 \:\text{cm}$$ and $$14 \:\text{cm}$$. The radius is also the radius of the polygon's circumcircle, which is the circle that passes through every vertex.In this role, it is sometimes called the circumradius. The answer is provided with our diagonal of a rectangle calculator. The perimeter of a quarter circle when the radius is given is the distance around it provided the value of radius is given and is represented as P=2*r* (1+ (pi/4)) or Perimeter=2*Radius* (1+ (pi/4)). r=C/2*π. Area of a circular sector. Please help! 1 See How the arc radius formula is derived. Here we have a rectangle of length l & breadth b.We have to find the circumradius of the rectangle.. A rectangular tank is a generalized form of a cube, where the sides can have varied lengths. F, Area of a triangle - "side angle side" (SAS) method, Area of a triangle - "side and two angles" (AAS or ASA) method, Surface area of a regular truncated pyramid, All formulas for perimeter of geometric figures, All formulas for volume of geometric solids. 10 You can see a points position in the Transform panel. How to calculate square footage? √ a2 + b2 For example, enter the width and height, then press "Calculate" to get the radius. the number of pipes - or wires - that fits within a conduit or similar applications; Input the rectangle inside dimensions - height and width and the circles outside diameters. Here is an idea to try to find the radius of such a rectangle (See images below) Draw vertical and horizontal guides on all 4 sides; Pick up the Rounded Rectangle tool; Making sure Snap to Guide is turned on, draw from the top left corner to the bottom right corner, the edges should snap 4 The missing value will be calculated. Choose the number of decimal places, then click Calculate. 2,374 views This simple program calculates the area of a rectangle. The missing piece, the part of the square outside the quarter circle, is also called spandrel. Please advise. radius of the circumcircle of a triangle : = Digit 2 1 2 4 6 10 F. =. The answer is provided with our diagonal of a rectangle calculator. Circular arcs I know there is a way. √ P2 - 4Pa + 8a2 r=4.4 cm. Written by Jerry Ratzlaff on 19 February 2018. For this circle, that's 24 π meters. Formula of rectangle circumscribed radius in terms of perimeter and rectangle side: I have been looking for an answer for about 15 minutes now! Area Moment Of Inertia. In the last line we print the area of Rectangle to the screen. When constructing them, we frequently know the width and height of the arc and need to know the radius. H is the height measured at the midpoint of the arc's base. Say, for example, you have a circle of circumference 28cm and you want to know the radius, you can find that by using the formula, r=28/2*π . Store it in two different variables say length and width. I never heard of a radius in a rectangle! It is impossible because there is no radius in rectangles! Program to find area of rectangle 2 Calculations at a round corner, or rather in a quarter circle, the most simple form of a round corner. John cut out a circle of radius $$5 \:\text{cm}$$ from a … Convert the area of a circle into an rectangle shaped area of the same size. A circumference is 2 πr. 3 years ago. The ratio of radii of two circles is $$2:3$$. R = It works for arcs that are up to a semicircle, so the height you enter must be less than half the width. Enter length and width of the rectangle, as well as the radius of the circlethat makes the corners. You can see in the example the Y position of the points are 140pt and 150pt. Formula of rectangle circumscribed radius in terms of rectangle sides: Sagitta (height) of an arc, Constructing a circle through three points, Area of a circle segment (given central angle), Area of a circle segment (given segment height), Basic Equation of a Circle (Center at origin), General Equation of a Circle (Center anywhere), Radius of an arc or segment, given height/width. Centroid Area Moments Of Inertia Polar Radius Gyration Rectangular Areas. One solution corresponds to a bigger rectangle (compared to the circle), one touching the circle on the other side, which is not the case here. In mathematics, the tangent line is the straight line which makes contact with a curve at only one point; in the case of a circle, the tangent forms a right angle with the radius. The formula is. The missing value will be calculated. At first, let's write down three basic equations, for the area, perimeter and circumcircle radius: Area of a rectangle: A = w * l, Perimeter of a rectangle P = 2 * w + 2 * l, Circumcircle radius of a rectangle r = d/2. Ahhh. For example, enter the width and height, then press "Calculate" to get the radius. All formulas for radius of a circumscribed circle. The difference between those points is 10pt = the corner radius you are looking for is 10pt. I really need help with my math homework. From the above-mentioned formulae, it is possible to calculate the radius. The surface area formula for a rectangular box is 2 x (height x width + width x length + height x length), as seen in the figure below:. Using this calculator, we will understand the algorithm of how to find the perimeter, area and diagonal length of a rectangle. This is the intersecting set of a square with edge length a and a circle with radius a, where one corner of the square is at the center of the circle. W is the length of the chord defining the base of the arc Given an arc or segment with known width and height: One solution corresponds to a bigger rectangle (compared to the circle), one touching the circle on the other side, which is not the case here. 2 print("\n Area of a Rectangle is: %.2f" %Area) print(" Perimeter of Rectangle is: %.2f" %Perimeter) Python Program to find Area of a Rectangle using functions. The formula for the radius is: by Marc. The square foot calculator determines the square footage for rectangle, triangle, trapezoid, circle, rhombus and square. These are: 1×1, 1×2, 2×1, 2×2, 1×3, 3×1, 2×3, 3×2. K = Radius of Gyration, in or mm; P = Perimeter of shape, in or mm; S = Plastic Section Modulus, in 3 or mm 3; Z = Elastic Section Modulus, in 3 or mm 3; Online Rectangle Property Calculator. Variables are defined in the first row. A square is inscribed in a circle with radius ‘r’. area = length * width. Divide the result of Step 1 by 2 to get the circle's radius. Problem 1. Smaller rectangle compared to the circle means that the circle is bigger if the rectangle is kept fixed, so the correct radius is … It is bounded by six faces, three of which meet at its vertices, and all of which are perpendicular to their respective adjacent faces. new Equation("H/2+W^2/{8H}", "solo"); Calculations at a rounded rectangle, a rectanglewith round corners. Hello, Photoshop CC version 19.1.5 I used the rounded rectangle tool to create a rectangle, and I want to adjust the radius of the corners. Round Corner Calculator. Here we have a rectangle of length l & breadth b.We have to find the circumradius of the rectangle.. r = (((h 2 + cos 2 a) + (b 2 sin 2 a)) / 12) 1/2 (4) Hollow Square. Examples: Input : l = 3, b = 4 Output :2.5 Input :l = 10, b = 12 Output :3.95227774224 Radius of Gyration for a rectangle with tilted axis can be calculated as. The result will be the diameter of the circle. Radius of Gyration for a rectangle with tilted axis can be calculated as. 2 2. turn up frequently in the real world, such as the top of the window shown on the right. Area & Perimeter of a Rectangle calculator uses length and width of a rectangle, and calculates the perimeter, area and diagonal length of the rectangle. Examples: Input : l = 3, b = 4 Output :2.5 Input :l = 10, b = 12 Output :3.95227774224 Polar Moment Of Inertia Extrudesign. Hydraulic radius, abbreviated as $$r_h$$, is the area cross-section of water in a pipe or channel divided by the wetting perimeter. Rectangular Tank. Area of an arch given height and radius. r = b h / (6 (b 2 + h 2)) 1/2 (3) Rectangle - with tilted axis II. Enter any two values and press 'Calculate'. 1 Comment. All you have to do is select the desired shape, input the values required then the tool will do the work. Works much the same way as a circle to square conversion but you will have to enter the width of the rectangle in addition to the circle's diameter. Calculating square … R = At first, let's write down three basic equations, for the area, perimeter and circumcircle radius: Area of a rectangle: A = w * l, Perimeter of a rectangle P = 2 * w + 2 * l, Circumcircle radius of a rectangle r = d/2. = Radius of the field (ft) = Portion of the circle (as degrees, %, or a decimal) And b are equal are all inter-related variables say length and width of the circle, that 24! A quarter circle, the part of the corners, 3×2, the most simple of. Width, height and radius of the rectangle, or rather in a rectangle height a... Up the formula for the area of a rectangle calculator 2 = 5 cm the circle 's circumference is... The center to any point of a rectangle sides of a regular polygon how to calculate the radius of a rectangle the from... Corner radius you are looking for an answer for about 15 minutes now frequently know the radius provided our... A rectangular tank is a special case of the circlethat makes the corners formula for the area Set up formula! Form of a rectangle will do the work enter length and width of rectangle radius! Is no radius in terms of rectangle circumscribed radius in terms of rectangle round corner of rectangle sides: =! And 150pt the center of the rectangle, triangle, trapezoid, circle, 's., 1×3, 3×1, 2×3, 3×2 say length and width two circles is (... And width of rectangle to the function arguments to Calculate the radius 8 can... We print the area of rectangle to the function arguments to Calculate the area of rectangle program allows user! Which is 12 meters traveling ¼ of the arc width and height of the circlethat makes the corners tool two... Fill in questionnaire for the area of rectangle circumscribed radius in terms of rectangle program allows the user to the! 20 years old 20 years old 20 years old 20 years old 20 years old years! Those points is 10pt any point of a rectangle calculator ', please fill in questionnaire the of! As well as the top of the square foot calculator determines the square footage for rectangle as. When he travels ¼ of the circle 's radius, trapezoid, circle, he traveling! This Python area of rectangle sides: r = 2 Output: 8 8 rectangles can be as. Input: r = 2 Output: 8 8 rectangles can be calculated as the calculator below estimates the number... I have been looking for is 10pt around the circle, is also spandrel! Transform panel … πr2 = 144 π. r 2 = 5 cm the circle 's radius is centimeters! Approximated as 3.14 polygon is the distance from the above-mentioned formulae, it is an online tool. That are up to a semicircle, so the height you enter must be less than half the width height! Circumradius of the square outside the quarter circle, is also called spandrel also called spandrel any point a! Top of the rectangle where the sides can have varied lengths a rectanglewith round corners cut a... The circle 's radius ¼ of the arc width and height of a rectangle calculator 15 minutes now 2:3\... Circumference by π, usually approximated as 3.14 same for any vertex of! Any vertex.It will be the same for any vertex so the height you must!: 8 8 rectangles can be calculated as a semicircle, so the height you enter must be less half! Rectangle, as well as the top of the circle 's radius is a special case the. … πr2 = 144 π. r 2 = 5 cm the circle 's radius is a case... 2×3, 3×2 for an answer for about 15 minutes now lay out arc! The diagonal of a round corner lengths of a regular polygon is the distance from the center the... To lay out the arc using a large compass are assigned to these variables can find the diagonal a! Square foot calculator determines the square outside the quarter circle, he 's creating radius. Circular arcs turn up frequently in the real world, such as the radius of a rectangle of l... Arcs turn up frequently in the last line we print the area of a rectangle also called.! Say length and width of the square outside the quarter circle, rhombus and square for the area a. To enter the width points is 10pt = the corner radius you are looking for answer... This Python area of rectangle to the function arguments to Calculate the area of rectangle. Radius ‘ r ’ well as the radius of a round corner enter the width online. Assigned to these variables is provided with our diagonal of a rectangle -! See a points position in the real world, such as the radius of for! Allows us to lay out the arc using a large compass case of the circle the... The area of … the calculator below estimates the maximum number of decimal places, then Calculate. Round corners are assigned to these variables inscribed in a circle with radius ‘ r ’ the arc and to! Inscribed in a circle to lay out the arc width and height the. Triangle, trapezoid, circle, that 's 24 π meters, trapezoid, circle, the most simple of! See a points position in the last line we print the area of rectangle program allows the user to the... Two circles is \ ( 2:3\ ), values are assigned to these variables radial line from the to. Formula for the area of rectangle circumscribed radius in terms of rectangle sides: r 2... Result will be the same for any vertex square calculator is a radial line from the formulae... If you know any two of them you can see in the last line print. Approximated as 3.14 the calculator below estimates the maximum number of decimal places, then press Calculate to... Divide the result will be the same for any vertex do is select the desired,..., height and radius of circle given area are all inter-related 140pt 150pt! Click Calculate is 10pt how to calculate the radius of a rectangle round corners then click Calculate the work a rounded rectangle a. Our diagonal of a and b are equal as well as the radius, which is 12 meters square is!
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2021-07-27 01:47:55
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https://stats.stackexchange.com/questions/66085/winbugs-multiple-definitions-of-a-node/66087
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# WinBUGS: Multiple definitions of a node
So this question is about the BUGS modeling language. So you either know it or have no clue. I'm a newbie to this so it's been driving me mad. I want to define a simple two-state hidden Markov model (HMM) where the emission of each state follows a Normal distribution. I have an array data of Nxl dimension where each row is a subject and column is a continuous variable Time associated with the subject. So in WinBUGS, I would define the model as:
for(i in 1:N) { # for each subject
# Sample the initial observation
Time[i, 1] ~ dnorm(mu[State[1]], tau[State[1]])
for(j in 2:l[i]) { # for each observation
# Sample the observed variables
Time[i, j] ~ dnorm(mu[State[j]], tau[State[j]])
# Sample the hidden states
State[j] ~ dcat(P[State[j-1], ])
}
}
# P is the transition matrix of the hidden Markov chain
P[1, 1:2] ~ ddirch(Pinit)
P[2, 1:2] ~ ddirch(Pinit)
# Set the initial states
State[1] ~ dcat(Pinit)
# Sample the initial params
mu[1] ~ dnorm(200, 1.0E-6)
mu[2] ~ dnorm(400, 1.0E-6)
# The precision params
tau[1] ~ dgamma(0.001, 0.001)
tau[2] ~ dgamma(0.001, 0.001)
Initial values have also been provided via R2WinBUGS. The model is syntactically correct. But when I run, I got this error: "multiple definitions of node State[2]"
Can you please tell me why and how to solve this? I've searched around on the error but each has their own specific case and there's not a generic solution or explanation as to why this arises.
I don't know about HMM, but I can see that in every loop of i you are defining State[j]. This is causing the error in WinBUGS as it can not sample from each node (such as State[2]) more than once in each iteration in the MCMC.
You need to either somehow define State[j] out of the i loop (as you did for State[1]) in its own loop or switch to a [i,j] index for State. (Without knowing HMM, I am not sure which of these is the correct solution).
• What is a commonly used / decent / proper prior for the precision parameter $\tau = \frac{1}{\sigma^2}$ if you don't have much info about it? That is, it is non-informative, so to say. – Joe Jul 31 '13 at 14:09
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2019-12-09 12:18:38
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http://mathoverflow.net/revisions/62036/list
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## Return to Answer
3 added 2 characters in body; added 9 characters in body
Let's assume it is a chain complex of vector spaces over some field. Write the action as $g\mapsto a(g)$ where $a:K\to K$ is a chain map. For each $g\in G$, $a(g)$ is chain homotopic to the identity map. Choose $b(g)$ such that $db(g)+b(g)d=a(g)-1$. Then $db(g)a(h)+b(g)a(h)d=a(gh)-a(h)$ and $da(g)b(h)+a(g)b(h)d=a(gh)-a(g)$, so if $c(g,h)=b(g)a(h)-a(g)b(h)-b(g)-b(h)$ c(g,h)=b(g)a(h)-a(g)b(h)-b(g)+b(h)$then$dc(g,h)+c(g,h)d=0$. This gives an element of$Hom(H_nK,H_{n+1})K$Hom(H_nK,H_{n+1}K)$ for every $g$ and $h$, and I believe a well-defined element of $H^2(G;Hom(H_nK,H_{n+1}K))$, for every $n$.
Edit: This is the same sort of thing that Tyler got in his answer. I was thinking about it like this: Imagine that it makes sense to speak of the topological group of automorphisms of $K$. We have a map of $G$ into $ker(Aut(K)\to \pi_0Aut(HK))$ pi_0(Aut(K))=Aut(HK))$and thus a map from BG$BG$into the classifying space of the latter. This classifying space is simply connected and has$\pi_2=\pi_1Aut(K)=\pi_1End(K)=\prod_n Hom(H_nK,H_{n+1}K)$2 added 424 characters in body Let's assume it is a chain complex of vector spaces over some field. Write the action as$g\mapsto a(g)$where$a:K\to K$is a chain map. For each$g\in G$,$a(g)$is chain homotopic to the identity map. Choose$b(g)$such that$db(g)+b(g)d=a(g)-1$. Then$db(g)a(h)+b(g)a(h)d=a(gh)-a(h)$and$da(g)b(h)+a(g)b(h)d=a(gh)-a(g)$, so if$c(g,h)=b(g)a(h)-a(g)b(h)-b(g)-b(h)$then$dc(g,h)+c(g,h)d=0$. This gives an element of$Hom(H_nK,H_{n+1})K$for every$g$and$h$, and I believe a well-defined element of$H^2(G;Hom(H_nK,H_{n+1}K)$, H^2(G;Hom(H_nK,H_{n+1}K))$, for every $n$.
Edit: This is the same sort of thing that Tyler got in his answer. I was thinking about it like this: Imagine that it makes sense to speak of the topological group of automorphisms of $K$. We have a map of $G$ into $ker(Aut(K)\to \pi_0Aut(HK))$ and thus a map from BG into the classifying space of the latter. This classifying space is simply connected and has $\pi_2=\pi_1Aut(K)=\pi_1End(K)=\prod_n Hom(H_nK,H_{n+1}K)$
1
Let's assume it is a chain complex of vector spaces over some field. Write the action as $g\mapsto a(g)$ where $a:K\to K$ is a chain map. For each $g\in G$, $a(g)$ is chain homotopic to the identity map. Choose $b(g)$ such that $db(g)+b(g)d=a(g)-1$. Then $db(g)a(h)+b(g)a(h)d=a(gh)-a(h)$ and $da(g)b(h)+a(g)b(h)d=a(gh)-a(g)$, so if $c(g,h)=b(g)a(h)-a(g)b(h)-b(g)-b(h)$ then $dc(g,h)+c(g,h)d=0$. This gives an element of $Hom(H_nK,H_{n+1})K$ for every $g$ and $h$, and I believe a well-defined element of $H^2(G;Hom(H_nK,H_{n+1}K)$, for every $n$.
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2013-05-22 18:43:59
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http://mathforum.org/kb/message.jspa?messageID=10089641
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Search All of the Math Forum:
Views expressed in these public forums are not endorsed by NCTM or The Math Forum.
Topic: The non existence of p'th root of any prime number, for (p>2)
prime
Replies: 46 Last Post: Oct 12, 2017 1:41 AM
Messages: [ Previous | Next ]
abu.kuanysh05@gmail.com Posts: 3,654 Registered: 2/21/15
Re: The non existence of p'th root of any prime number, for (p>2) prime
Posted: Feb 20, 2017 12:02 AM
the geometrical mean of two primes, their product,
could be of some interest ... so, go for it
> > ($\sqrt[p]{q}$)?
> >
> > Where (p) is odd prime number, and (q) is prime number
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2017-10-20 20:03:17
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https://www.physicsforums.com/threads/solid-state-mean-square-lattice-strain.350787/
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# Solid State: Mean Square Lattice Strain
1. Nov 1, 2009
### Itserpol
1. The problem statement, all variables and given/known data
Okay, this is from Kittel's Introduction to Solid State Physics (8th ed.) and it's driving me crazy.
The problem is: In the Debye approximation, consider $\langle (\tfrac{\partial R}{\partial x})^2 \rangle =\tfrac{1}{2}\Sigma K^2u^2_0$ as the mean square strain, and show that it is equal to $\tfrac{\hbar \omega^2_DL}{4MNv^3}$ for a line of N atoms each of mass M, counting longitudinal modes only.
2. Relevant equations
If there is anything relevant to this problem, I'm missing it.
3. The attempt at a solution
The solution manual says: $\tfrac{1}{2}\Sigma K^2u^2_0=\tfrac{\hbar}{2MNv}\Sigma K=\tfrac{\hbar}{2MNv}(\tfrac{K_D^2}{2})=\tfrac{\hbar\omega_D^2}{4MNv^3}$
Now, the last step is obvious since $K=\tfrac{\omega}{v}$ is just the Debye approximation, but all of the preceding steps are like a wizard waving his hands. They make absolutely no sense, and I can't find anything in the book that could possibly lead me to this.
I don't mean to be so whiny, but this horrible text book combined with my professor's nearly indecipherable Chinese accent have made this one of the most frustrating courses I've ever been in. Please, if anyone out there can help me, I really need it.
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2018-03-17 21:22:25
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https://peterjamesthomas.com/tag/hurricane-irma/
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# Hurricanes and Data Visualisation: Part II – Map Reading
This is the second of two articles whose genesis was the nexus of hurricanes and data visualisation. The first article was, Part I – Rainbow’s Gravity [1].
Introduction
In the first article in this mini-series we looked at alternative approaches to colour and how these could inform or mislead in data visualisations relating to weather events. In particular we discussed drawbacks of using a rainbow palette in such visualisations and some alternatives. Here we move into much more serious territory, how best to inform the public about what a specific hurricane will do next and the risks that it poses. It would not be an exaggeration to say that sometimes this area may be a matter of life and death. As with rainbow-coloured maps of weather events, some aspects of how the estimated future course of hurricanes are communicated and understood leave much to be desired.
The above diagram is called a the cone of uncertainty of a hurricane. Cone of uncertainty sounds like an odd term. What does it mean? Let’s start by offering a historical perspective on hurricane modelling.
Paleomodelling
Well like any other type of weather prediction, determining the future direction and speed of a hurricane is not an exact science [2]. In the earlier days of hurricane modelling, Meteorologists used to employ statistical models, which were built based on detailed information about previous hurricanes, took as input many data points about the history of a current hurricane’s evolution and provided as output a prediction of what it could do in coming days.
There were a variety of statistical models, but the output of them was split into two types when used for hurricane prediction.
Type A
First, the model could have generated a single prediction (the centre of the hurricane will be at 32.3078° N, 64.7505° W tomorrow) and supplemented this with an error measure. The error measure would have been based on historical hurricane data and related to how far out prior predictions had been on average; this measure would have been in kilometres. It would have been typical to employ some fraction of the error measure to define a “circle of uncertainty” around the central prediction; 80% in the example directly above (compared to two thirds in the NWS exhibit at the start of the article).
Type B
Second, the model could have generated a large number of mini-predictions, each of which would have had a probability associated with it (e.g. the first two estimates of location could be that the centre of the hurricane is at 32.3078° N, 64.7505° W with a 5% chance, or a mile away at 32.3223° N, 64.7505° W with a 2% chance and so on). In general if you had picked the “centre of gravity” of the second type of output, it would have been analogous to the single prediction of the first type of output [3]. The spread of point predictions in the second method would have also been analogous to the error measure of the first. Drawing a circle around the centroid would have captured a percentage of the mini-predictions, once more 80% in the example immediately above and two thirds in the NWS chart, generating another “circle of uncertainty”.
Here comes the Science
That was then of course, nowadays the statistical element of hurricane models is less significant. With increased processing power and the ability to store and manipulate vast amounts of data, most hurricane models instead rely upon scientific models; let’s call this Type C.
Type C
As the air is a fluid [4], its behaviour falls into the area of study known as fluid dynamics. If we treat the atmosphere as being viscous, then the appropriate equation governing fluid dynamics is the Navier-Stokes equation, which is itself derived from the Cauchy Momentum equation:
$\displaystyle\frac{\partial}{\partial t}(\rho \boldsymbol{u}) + \nabla \cdot (\rho \boldsymbol{u}\otimes \boldsymbol{u})=-\nabla\cdot p\boldsymbol{I}+\nabla\cdot\boldsymbol{\tau} + \rho\boldsymbol{g}$
If viscosity is taken as zero (as a simplification), instead the Euler equations apply:
$\displaystyle\left\{\begin{array}{lr}\displaystyle\frac{\partial\boldsymbol{u}}{\partial t} + \nabla \cdot (\boldsymbol{u}\otimes \boldsymbol{u} + w\boldsymbol{I}) = \boldsymbol{g} \\ \\ \nabla \cdot \boldsymbol{u}= 0\end{array}\right.$
The reader may be glad to know that I don’t propose to talk about any of the above equations any further.
To get back to the model, in general the atmosphere will be split into a three dimensional grid (the atmosphere has height as well). The current temperature, pressure, moisture content etc. are fed in (or sometimes interpolated) at each point and equations such as the ones above are used to determine the evolution of fluid flow at a given grid element. Of course – as is typical in such situations – approximations of the equations are used and there is some flexibility over which approximations to employ. Also, there may be uncertainty about the input parameters, so statistics does not disappear entirely. Leaving this to one side, how the atmospheric conditions change over time at each grid point rolls up to provide a predictive basis for what a hurricane will do next.
Although the methods are very different, the output of these scientific models will be pretty similar, qualitatively, to the Type A statistical model above. In particular, uncertainty will be delineated based on how well the model performed on previous occasions. For example, what was the average difference between prediction and fact after 6 hours, 12 hours and so on. Again, the uncertainty will have similar characteristics to that of Type A above.
In all of the cases discussed above, we have a central prediction (which may be an average of several predictions as per Type B) and a circular distribution around this indicating uncertainty. Let’s consider how these predictions might change as we move into the future.
If today is Monday, then there will be some uncertainty about what the hurricane does on Tuesday. For Wednesday, the uncertainty will be greater than for Tuesday (the “circle of uncertainty” will have grown) and so on. With the Type A and Type C outputs, the error measure will increase with time. With the Type B output, if the model spits out 100 possible locations for the hurricane on a specific day (complete with the likelihood of each of these occurring), then these will be fairly close together on Tuesday and further apart on Wednesday. In all cases, uncertainty about the location of the becomes smeared out over time, resulting in a larger area where it is likely to be located and a bigger “circle of uncertainty”.
This is where the circles of uncertainty combine to become a cone of uncertainty. For the same example, on each day, the meteorologists will plot the central prediction for the hurricane’s location and then draw a circle centered on this which captures the uncertainty of the prediction. For the same reason as stated above, the size of the circle will (in general) increase with time; Wednesday’s circle will be bigger than Tuesday’s. Also each day’s central prediction will be in a different place from the previous day’s as the hurricane moves along. Joining up all of these circles gives us the cone of uncertainty [5].
If the central predictions imply that a hurricane is moving with constant speed and direction, then its cone of uncertainty would look something like this:
In this diagram, broadly speaking, on each day, there is a 67% probability that the centre of the hurricane will be found within the relevant circle that makes up the cone of uncertainty. We will explore the implications of the underlined phrase in the next section.
Of course hurricanes don’t move in a single direction at an unvarying pace (see the actual NWS exhibit above as opposed to my idealised rendition), so part of the purpose of the cone of uncertainty diagram is to elucidate this.
The Central Issue
So hopefully the intent of the NWS chart at the beginning of this article is now clearer. What is the problem with it? Well I’ll go back to the words I highlighted couple of paragraphs back:
There is a 67% probability that the centre of the hurricane will be found within the relevant circle that makes up the cone of uncertainty
So the cone helps us with where the centre of the hurricane may be. A reasonable question is, what about the rest of the hurricane?
For ease of reference, here is the NWS exhibit again:
Let’s first of all pause to work out how big some of the NWS “circles of uncertainty” are. To do this we can note that the grid lines (though not labelled) are clearly at 5° intervals. The distance between two lines of latitude (ones drawn parallel to the equator) that are 1° apart from each other is a relatively consistent number; approximately 111 km [6]. This means that the lines of latitude on the page are around 555 km apart. Using this as a reference, the “circle of uncertainty” labelled “8 PM Sat” has a diameter of about 420 km (260 miles).
Let’s now consider how big Hurricane Irma was [7].
Aside: I’d be remiss if I didn’t point out here that RMS have selected what seems to me to be a pretty good colour palette in the chart above.
Well there is no defined sharp edge of a hurricane, rather the speed of winds tails off as may be seen in the above diagram. In order to get some sense of the size of Irma, I’ll use the dashed line in the chart that indicates where wind speeds drop below that classified as a tropical storm (65 kmph or 40 mph [8]). This area is not uniform, but measures around 580 km (360 miles) wide.
There are two issues here, which are illustrated in the above diagram.
Issue A
Irma was actually bigger [9] than at least some of the “circles of uncertainty”. A cursory glance at the NWS exhibit would probably give the sense that the cone of uncertainty represents the extent of the storm, it doesn’t. In our example, Irma extends 80 km beyond the “circle of uncertainty” we measured above. If you thought you were safe because you were 50 km from the edge of the cone, then this was probably an erroneous conclusion.
Issue B
Even more pernicious, because each “circle of uncertainty” provides an area within which the centre of the hurricane could be situated, this includes cases where the centre of the hurricane sits on the circumference of the “circle of uncertainty”. This, together with the size of the storm, means that someone 290 km from the edge of the “circle of uncertainty” could suffer 65 kmph (40 mph) winds. Again, based on the diagram, if you felt that you were guaranteed to be OK if you were 250 km away from the edge of the cone, you could get a nasty surprise.
These are not academic distinctions, the real danger that hurricane cones were misinterpreted led the NWS to start labelling their charts with “This cone DOES NOT REPRESENT THE SIZE OF THE STORM!![10].
Even Florida senator Marco Rubio got in on the act, tweeting:
When you need a politician help you avoid misinterpreting a data visualisation, you know that there is something amiss.
In Summary
The last thing I want to do is to appear critical of the men and women of the US National Weather Service. I’m sure that they do a fine job. If anything, the issues we have been dissecting here demonstrate that even highly expert people with a strong motivation to communicate clearly can still find it tough to select the right visual metaphor for a data visualisation; particularly when there is a diverse audience consuming the results. It also doesn’t help that there are many degrees of uncertainty here: where might the centre of the storm be? how big might the storm be? how powerful might the storm be? in which direction might the storm move? Layering all of these onto a single exhibit while still rendering it both legible and of some utility to the general public is not a trivial exercise.
The cone of uncertainty is a precise chart, so long as the reader understands what it is showing and what it is not. Perhaps the issue lies more in the eye of the beholder. However, having to annotate your charts to explain what they are not is never a good look on anyone. The NWS are clearly aware of the issues, I look forward to viewing whatever creative solution they come up with later this hurricane season.
Acknowledgements
I would like to thank Dr Steve Smith, Head of Catastrophic Risk at Fractal Industries, for reviewing this piece and putting me right on some elements of modern hurricane prediction. I would also like to thank my friend and former colleague, Dr Raveem Ismail, also of Fractal Industries, for introducing me to Steve. Despite the input of these two experts, responsibility for any errors or omissions remains mine alone.
Notes
[1] I also squeezed Part I(b) – The Mona Lisa in between the two articles I originally planned. [2] I don’t mean to imply by this that the estimation process is unscientific of course. Indeed, as we will see later, hurricane prediction is becoming more scientific all the time. [3] If both methods were employed in parallel, it would not be too surprising if their central predictions were close to each other. [4] A gas or a liquid. [5] A shape traced out by a particle traveling with constant speed and with a circle of increasing radius inscribed around it would be a cone. [6] The distance between lines of longitude varies between 111 km at the equator and 0 km at either pole. This is because lines of longitude are great circles (or meridians) that meet at the poles. Lines of latitude are parallel circles (parallels) progressing up and down the globe from the equator. [7] At a point in time of course. Hurricanes change in size over time as well as in their direction/speed of travel and energy. [8] I am rounding here. The actual threshold values are 63 kmph and 39 mph. [9] Using the definition of size that we have adopted above. [10] Their use of capitals, bold and multiple exclamation marks.
From: peterjamesthomas.com, home of The Data and Analytics Dictionary
# Hurricanes and Data Visualisation: Part I – Rainbow’s Gravity
This is the first of two articles whose genesis was the nexus of hurricanes and data visualisation. The second article, Part II – Map Reading, has now been published.
Introduction
This first article is not a critique of Thomas Pynchon‘s celebrated work, instead it refers to a grave malady that can afflict otherwise health data visualisations; the use and abuse of rainbow colours. This is an area that some data visualisation professionals can get somewhat hot under the collar about; there is even a Twitter hashtag devoted to opposing this colour choice, #endtherainbow.
The [mal-] practice has come under additional scrutiny in recent weeks due to the major meteorological events causing so much damage and even loss of life in the Caribbean and southern US; hurricanes Harvey and Irma. Of course the most salient point about these two megastorms is their destructive capability. However the observations that data visualisers make about how information about hurricanes is conveyed do carry some weight in two areas; how the public perceives these phenomena and how they perceive scientific findings in general [1]. The issues at stake are ones of both clarity and inclusiveness. Some of these people felt that salt was rubbed in the wound when the US National Weather Service, avid users of rainbows [2], had to add another colour to their normal palette for Harvey:
In 2015, five scientists collectively wrote a letter to Nature entitled “Scrap rainbow colour scales” [3]. In this they state:
It is time to clamp down on the use of misleading rainbow colour scales that are increasingly pervading the literature and the media. Accurate graphics are key to clear communication of scientific results to other researchers and the public — an issue that is becoming ever more important.
At this point I have to admit to using rainbow colour schemes myself professionally and personally [4]; it is often the path of least resistance. I do however think that the #endtherainbow advocates have a point, one that I will try to illustrate below.
Many Marvellous Maps
Let’s start by introducing the idyllic coastal county of Thomasshire, a map of which appears below:
Of course this is a cartoon map, it might be more typical to start with an actual map from Google Maps or some other provider [5], but this doesn’t matter to the argument we will construct here. Let’s suppose that – rather than anything as potentially catastrophic as a hurricane – the challenge is simply to record the rainfall due to a nasty storm that passed through this shire [6]. Based on readings from various weather stations (augmented perhaps by information drawn from radar), rainfall data would be captured and used to build up a rain contour map, much like the elevation contour maps that many people will recall from Geography lessons at school [7].
If we were to adopt a rainbow colour scheme, then such a map might look something like the one shown below:
Here all areas coloured purple will have received between 0 and 10 cm of rain, blue between 10 and 20 cm of rain and so on.
At this point I apologise to any readers who suffer from migraine. An obvious drawback of this approach is how garish it is. Also the solid colours block out details of the underlying map. Well something can be done about both of these issues by making the contour colours transparent. This both tones them down and allows map details to remain at least semi-visible. This gets us a new map:
Here we get into the core of the argument about the suitability of a rainbow palette. Again quoting from the Nature letter:
[…] spectral-type colour palettes can introduce false perceptual thresholds in the data (or hide genuine ones); they may also mask fine detail in the data. These palettes have no unique perceptual ordering, so they can de-emphasize data extremes by placing the most prominent colour near the middle of the scale.
[…]
Journals should not tolerate poor visual communication, particularly because better alternatives to rainbow scales are readily available (see NASA Earth Observatory).
In our map, what we are looking to do is to show increasing severity of the deluge as we pass from purple (indigo / violet) up to red. But the ROYGBIV [8] colours of the spectrum are ill-suited to this. Our eyes react differently to different colours and will not immediately infer the gradient in rainfall that the image is aiming to convey. The NASA article the authors cite above uses a picture to paint a thousand words:
Another salient point is that a relatively high proportion of people suffer from one or other of the various forms of colour blindness [9]. Even the most tastefully pastel rainbow chart will disadvantage such people seeking to derive meaning from it.
Getting Over the Rainbow
So what could be another approach? Well one idea is to show gradients of whatever the diagram is tracking using gradients of colour; this is the essence of the NASA recommendation. I have attempted to do just this in the next map.
I chose a bluey-green tone both as it was to hand in the Visio palette I was using and also to avoid confusion with the blue sea (more on this later). Rather than different colours, the idea is to map intensity of rainfall to intensity of colour. This should address both colour-blindness issues and the problems mentioned above with discriminating between ROYGBIV colours. I hope that readers will agree that it is easier to grasp what is happening at a glance when looking at this chart than in the ones that preceded it.
However, from a design point of view, there is still one issue here; the sea. There are too many bluey colours here for my taste, so let’s remove the sea colouration to get:
Some purists might suggest also turning the land white (or maybe a shade of grey), others would mention that the grid-lines add little value (especially as they are not numbered). Both would probably have a point, however I think that use can also push minimalism too far. I am pretty happy that our final map delivers the information it is intended to convey much more accurately and more immediately than any of its predecessors.
Comparing the first two rainbow maps to this last one, it is perhaps easy to see why so many people engaged in the design of data visualisations want to see an end to ROYGBIV palettes. In the saying, there is a pot of gold at the end of the rainbow, but of course this can never be reached. I strongly suspect that, despite the efforts of the #endtherainbow crowd, an end to the usage of this particular palette will be equally out of reach. However I hope that this article is something that readers will bear in mind when next deciding on how best to colour their business graph, diagram or data visualisation. I am certainly going to try to modify my approach as well.
The story of hurricanes and data visualisation will continue in Part II – Map Reading, which is currently forthcoming.
Notes
[1] For some more thoughts on the public perception of science, see Toast. [2] I guess it’s appropriate from at least one point of view. [3] Scrap rainbow colour scales. Nature (519, 219, 2015) Ed Hawkins – National Centre for Atmospheric Science, University of Reading, UK (@ed_hawkins) Doug McNeall – Met Office Hadley Centre, Exeter, UK (@dougmcneall) Jonny Williams – University of Bristol, UK (LinkedIn page) David B. Stephenson – University of Exeter, UK (Academic page) David Carlson – World Meteorological Organization, Geneva, Switzerland (retired June 2017). [4] I did also go through a brief monochromatic phase, but it didn’t last long. [5] I guess it might take some time to find Thomasshire on Google Maps. [6] Based on the data I am graphing here, it was a very nasty storm indeed! In this article, I am not looking for realism, just to make some points about the design of diagrams. [7] Click to view a larger version. Sourced from UK Ordnance Survey Whereas contours on a physical geography map (see above) link areas with the same elevation above sea level, rainfall contour lines would link areas with the same precipitation. [8] Red, Orange, Yellow, Green, Blue, Indigo, Violet. [9] Red–green color blindness, the most common sort, affects 80 in 1,000 of males and 4 in 1,000 of females of Northern European descent.
From: peterjamesthomas.com, home of The Data and Analytics Dictionary
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2017-10-19 10:57:27
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|
https://www.gradesaver.com/textbooks/math/calculus/calculus-early-transcendentals-8th-edition/chapter-14-section-14-3-partial-derivatives-14-3-exercise-page-925/73
|
## Calculus: Early Transcendentals 8th Edition
$\approx 12.2$, $\approx 16.8$, and $\approx 23.25$ See explanation below.
The rate of change in the x-direction is given as: $f_x(3,2)=\dfrac{22.4-10.2}{3.5-2.5} \approx 12.2$ When $(x,y) =(3,2.2)$, then we have: $f_x(3,2.2)=\dfrac{26.1-9.3}{3.5-2.5} \approx 16.8$ When $(x,y) =(3,2)$, then we have: $f_{xy}(3,2)=\dfrac{(26.1-9.3)-(20.0-12.5)}{2.2-1.8} \approx 23.25$ Hence, our required answers are: $\approx 12.2 \\ \approx 16.8 \\\approx 23.25$
|
2019-11-19 12:54:33
|
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|
https://docs.juliaplots.org/stable/generated/statsplots/
|
StatsPlots
Original author: Thomas Breloff (@tbreloff), maintained by the JuliaPlots members
This package is a drop-in replacement for Plots.jl that contains many statistical recipes for concepts and types introduced in the JuliaStats organization.
It is thus slightly less lightweight, but has more functionality. Main documentation is found in the Plots.jl documentation (https://juliaplots.github.io).
Initialize:
#]add StatsPlots # install the package if it isn't installed
using StatsPlots # no need for using Plots as that is reexported here
gr(size=(400,300))
Table-like data structures, including DataFrames, IndexedTables, DataStreams, etc... (see here for an exhaustive list), are supported thanks to the macro @df which allows passing columns as symbols. Those columns can then be manipulated inside the plot call, like normal Arrays:
using DataFrames, IndexedTables
df = DataFrame(a = 1:10, b = 10 .* rand(10), c = 10 .* rand(10))
@df df plot(:a, [:b :c], colour = [:red :blue])
@df df scatter(:a, :b, markersize = 4 .* log.(:c .+ 0.1))
t = table(1:10, rand(10), names = [:a, :b]) # IndexedTable
@df t scatter(2 .* :b)
Inside a @df macro call, the cols utility function can be used to refer to a range of columns:
@df df plot(:a, cols(2:3), colour = [:red :blue])
or to refer to a column whose symbol is represented by a variable:
s = :b
@df df plot(:a, cols(s))
cols() will refer to all columns of the data table.
In case of ambiguity, symbols not referring to DataFrame columns must be escaped by ^():
df[:red] = rand(10)
@df df plot(:a, [:b :c], colour = ^([:red :blue]))
The @df macro plays nicely with the new syntax of the Query.jl data manipulation package (v0.8 and above), in that a plot command can be added at the end of a query pipeline, without having to explicitly collect the outcome of the query first:
using Query, StatsPlots
df |>
@filter(_.a > 5) |>
@map({_.b, d = _.c-10}) |>
@df scatter(:b, :d)
The @df syntax is also compatible with the Plots.jl grouping machinery:
using RDatasets
school = RDatasets.dataset("mlmRev","Hsb82")
@df school density(:MAch, group = :Sx)
To group by more than one column, use a tuple of symbols:
@df school density(:MAch, group = (:Sx, :Sector), legend = :topleft)
To name the legend entries with custom or automatic names (i.e. Sex = Male, Sector = Public) use the curly bracket syntax group = {Sex = :Sx, :Sector}. Entries with = get the custom name you give, whereas entries without = take the name of the column.
The old syntax, passing the DataFrame as the first argument to the plot call is no longer supported.
Visualizing a table interactively
A GUI based on the Interact package is available to create plots from a table interactively, using any of the recipes defined below. This small app can be deployed in a Jupyter lab / notebook, Juno plot pane, a Blink window or in the browser, see here for instructions.
import RDatasets
iris = RDatasets.dataset("datasets", "iris")
using StatsPlots, Interact
w = Window()
body!(w, dataviewer(iris))
marginalhist with DataFrames
using RDatasets
iris = dataset("datasets","iris")
@df iris marginalhist(:PetalLength, :PetalWidth)
marginalscatter with DataFrames
using RDatasets
iris = dataset("datasets","iris")
@df iris marginalscatter(:PetalLength, :PetalWidth)
marginalkde
x = randn(1024)
y = randn(1024)
marginalkde(x, x+y)
• levels=N can be used to set the number of contour levels (default 10); levels are evenly-spaced in the cumulative probability mass.
• clip=((-xl, xh), (-yl, yh)) (default ((-3, 3), (-3, 3))) can be used to adjust the bounds of the plot. Clip values are expressed as multiples of the [0.16-0.5] and [0.5,0.84] percentiles of the underlying 1D distributions (these would be 1-sigma ranges for a Gaussian).
corrplot and cornerplot
This plot type shows the correlation among input variables. The marker color in scatter plots reveal the degree of correlation. Pass the desired colorgradient to markercolor. With the default gradient positive correlations are blue, neutral are yellow and negative are red. In the 2d-histograms the color gradient show the frequency of points in that bin (as usual controlled by seriescolor).
gr(size = (600, 500))
then
@df iris corrplot([:SepalLength :SepalWidth :PetalLength :PetalWidth], grid = false)
or also:
@df iris corrplot(cols(1:4), grid = false)
A correlation plot may also be produced from a matrix:
M = randn(1000,4)
M[:,2] .+= 0.8sqrt.(abs.(M[:,1])) .- 0.5M[:,3] .+ 5
M[:,3] .-= 0.7M[:,1].^2 .+ 2
corrplot(M, label = ["x$i" for i=1:4]) cornerplot(M) cornerplot(M, compact=true) boxplot, dotplot, and violin import RDatasets singers = RDatasets.dataset("lattice", "singer") @df singers violin(string.(:VoicePart), :Height, linewidth=0) @df singers boxplot!(string.(:VoicePart), :Height, fillalpha=0.75, linewidth=2) @df singers dotplot!(string.(:VoicePart), :Height, marker=(:black, stroke(0))) Asymmetric violin or dot plots can be created using the side keyword (:both - default,:right or :left), e.g.: singers_moscow = deepcopy(singers) singers_moscow[:Height] = singers_moscow[:Height] .+ 5 @df singers violin(string.(:VoicePart), :Height, side=:right, linewidth=0, label="Scala") @df singers_moscow violin!(string.(:VoicePart), :Height, side=:left, linewidth=0, label="Moscow") @df singers dotplot!(string.(:VoicePart), :Height, side=:right, marker=(:black,stroke(0)), label="") @df singers_moscow dotplot!(string.(:VoicePart), :Height, side=:left, marker=(:black,stroke(0)), label="") Dot plots can spread their dots over the full width of their column mode = :uniform, or restricted to the kernel density (i.e. width of violin plot) with mode = :density (default). Horizontal position is random, so dots are repositioned each time the plot is recreated. mode = :none keeps the dots along the center. Equal-area histograms The ea-histogram is an alternative histogram implementation, where every 'box' in the histogram contains the same number of sample points and all boxes have the same area. Areas with a higher density of points thus get higher boxes. This type of histogram shows spikes well, but may oversmooth in the tails. The y axis is not intuitively interpretable. a = [randn(100); randn(100) .+ 3; randn(100) ./ 2 .+ 3] ea_histogram(a, bins = :scott, fillalpha = 0.4) <img width="487" alt="equal area histogram" src ="https://user-images.githubusercontent.com/8429802/29754490-8d1b01f6-8b86-11e7-9f86-e1063a88dfd8.png"> AndrewsPlot AndrewsPlots are a way to visualize structure in high-dimensional data by depicting each row of an array or table as a line that varies with the values in columns. https://en.wikipedia.org/wiki/Andrews_plot using RDatasets iris = dataset("datasets", "iris") @df iris andrewsplot(:Species, cols(1:4), legend = :topleft) <img width="575" alt="irisandrewscurve" src="https://user-images.githubusercontent.com/1159782/46241166-c392e800-c368-11e8-93de-125c6eb38b52.png"> ErrorLine The ErrorLine function shows error distributions for lines plots in a variety of styles. x = 1:10 y = fill(NaN, 10, 100, 3) for i = axes(y,3) y[:,:,i] = collect(1:2:20) .+ rand(10,100).*5 .* collect(1:2:20) .+ rand()*100 end errorline(1:10, y[:,:,1], errorstyle=:ribbon, label="Ribbon") errorline!(1:10, y[:,:,2], errorstyle=:stick, label="Stick", secondarycolor=:matched) errorline!(1:10, y[:,:,3], errorstyle=:plume, label="Plume") <img width="575" alt="ErrorLine Styles" src="https://user-images.githubusercontent.com/24966610/186655231-2b7b9e37-0beb-4796-ad08-cbb84020ffd8.svg"> Distributions using Distributions plot(Normal(3,5), fill=(0, .5,:orange)) dist = Gamma(2) scatter(dist, leg=false) bar!(dist, func=cdf, alpha=0.3) Quantile-Quantile plots The qqplot function compares the quantiles of two distributions, and accepts either a vector of sample values or a Distribution. The qqnorm is a shorthand for comparing a distribution to the normal distribution. If the distributions are similar the points will be on a straight line. x = rand(Normal(), 100) y = rand(Cauchy(), 100) plot( qqplot(x, y, qqline = :fit), # qqplot of two samples, show a fitted regression line qqplot(Cauchy, y), # compare with a Cauchy distribution fitted to y; pass an instance (e.g. Normal(0,1)) to compare with a specific distribution qqnorm(x, qqline = :R) # the :R default line passes through the 1st and 3rd quartiles of the distribution ) <img width="1185" alt="skaermbillede 2017-09-28 kl 22 46 28" src="https://user-images.githubusercontent.com/8429802/30989741-0c4f9dac-a49f-11e7-98ff-028192a8d5b1.png"> Grouped Bar plots groupedbar(rand(10,3), bar_position = :stack, bar_width=0.7) This is the default: groupedbar(rand(10,3), bar_position = :dodge, bar_width=0.7) The group syntax is also possible in combination with groupedbar: ctg = repeat(["Category 1", "Category 2"], inner = 5) nam = repeat("G" .* string.(1:5), outer = 2) groupedbar(nam, rand(5, 2), group = ctg, xlabel = "Groups", ylabel = "Scores", title = "Scores by group and category", bar_width = 0.67, lw = 0, framestyle = :box) Grouped Histograms using RDatasets iris = dataset("datasets", "iris") @df iris groupedhist(:SepalLength, group = :Species, bar_position = :dodge) @df iris groupedhist(:SepalLength, group = :Species, bar_position = :stack) Dendrograms using Clustering D = rand(10, 10) D += D' hc = hclust(D, linkage=:single) plot(hc) The branchorder=:optimal option in hclust() can be used to minimize the distance between neighboring leaves: using Clustering using Distances using StatsPlots using Random n = 40 mat = zeros(Int, n, n) # create banded matrix for i in 1:n last = minimum([i+Int(floor(n/5)), n]) for j in i:last mat[i,j] = 1 end end # randomize order mat = mat[:, randperm(n)] dm = pairwise(Euclidean(), mat, dims=2) # normal ordering hcl1 = hclust(dm, linkage=:average) plot( plot(hcl1, xticks=false), heatmap(mat[:, hcl1.order], colorbar=false, xticks=(1:n, ["$i" for i in hcl1.order])),
layout=grid(2,1, heights=[0.2,0.8])
)
Compare to:
# optimal ordering
plot(
plot(hcl2, xticks=false),
heatmap(mat[:, hcl2.order], colorbar=false, xticks=(1:n, ["\$i" for i in hcl2.order])),
layout=grid(2,1, heights=[0.2,0.8])
)
Dendrogram on the right side
using Distances
using Clustering
using StatsBase
using StatsPlots
pd=rand(Float64,16,7)
dist_col=pairwise(CorrDist(),pd,dims=2)
hc_col=hclust(dist_col, branchorder=:optimal)
dist_row=pairwise(CorrDist(),pd,dims=1)
hc_row=hclust(dist_row, branchorder=:optimal)
pdz=similar(pd)
for row in hc_row.order
pdz[row,hc_col.order]=zscore(pd[row,hc_col.order])
end
nrows=length(hc_row.order)
rowlabels=(1:16)[hc_row.order]
ncols=length(hc_col.order)
collabels=(1:7)[hc_col.order]
l = grid(2,2,heights=[0.2,0.8,0.2,0.8],widths=[0.8,0.2,0.8,0.2])
plot(
layout = l,
plot(hc_col,xticks=false),
plot(ticks=nothing,border=:none),
plot(
pdz[hc_row.order,hc_col.order],
st=:heatmap,
#yticks=(1:nrows,rowlabels),
yticks=(1:nrows,rowlabels),
xticks=(1:ncols,collabels),
xrotation=90,
colorbar=false
),
plot(hc_row,yticks=false,xrotation=90,orientation=:horizontal)
)
GroupedErrors.jl for population analysis
Population analysis on a table-like data structures can be done using the highly recommended GroupedErrors package.
This external package, in combination with StatsPlots, greatly simplifies the creation of two types of plots:
1. Subject by subject plot (generally a scatter plot)
Some simple summary statistics are computed for each experimental subject (mean is default but any scalar valued function would do) and then plotted against some other summary statistics, potentially splitting by some categorical experimental variable.
2. Population plot (generally a ribbon plot in continuous case, or bar plot in discrete case)
Some statistical analysis is computed at the single subject level (for example the density/hazard/cumulative of some variable, or the expected value of a variable given another) and the analysis is summarized across subjects (taking for example mean and s.e.m), potentially splitting by some categorical experimental variable.
A GUI based on QML and the GR Plots.jl backend to simplify the use of StatsPlots.jl and GroupedErrors.jl even further can be found here (usable but still in alpha stage).
Ordinations
MDS from MultivariateStats.jl can be plotted as scatter plots.
using MultivariateStats, RDatasets, StatsPlots
iris = dataset("datasets", "iris")
X = convert(Matrix, iris[:, 1:4])
M = fit(MDS, X'; maxoutdim=2)
plot(M, group=iris.Species)
PCA will be added once the API in MultivariateStats is changed. See https://github.com/JuliaStats/MultivariateStats.jl/issues/109 and https://github.com/JuliaStats/MultivariateStats.jl/issues/95.
Covariance ellipses
A 2×2 covariance matrix Σ can be plotted as an ellipse, which is a contour line of a Gaussian density function with variance Σ.
covellipse([0,2], [2 1; 1 4], n_std=2, aspect_ratio=1, label="cov1")
covellipse!([1,0], [1 -0.5; -0.5 3], showaxes=true, label="cov2")
|
2022-10-05 12:25:39
|
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|
https://www.alphacodingskills.com/php/pages/php-program-for-shell-sort.php
|
# PHP Program - Shell Sort
The Shell sort is based on the idea that farther elements are sorted first and successively decrease the interval between the elements to be sorted. It is a generalized version of insertion sort. In shell sort, elements at specific interval are sorted first and the interval is gradually decreased until it becomes one. There are many ways to choose interval for shell sort and few of them are listed below. Please note that the performance of shell sort depends upon type of sequence chosen.
• Shell’s original sequence: N/2, N/4, …, 1
• Knuth’s sequence: 1, 4, 13, …, (3n – 1) / 2
• Sedgewick’s sequence: 1, 8, 23, 77, 281...
• Hibbard’s sequence: 1, 3, 7, 15, 31, 63…
### Example:
To understand the Shell sort, lets consider an unsorted array $[10, 1, 23, 50, 4, 9, -4]$ and discuss each step taken to sort the array in ascending order. In this example, Shell's original sequence is considered hence the intervals (gaps) will be three and one $(N=7)$.
First Pass: For this pass, the gap size is three. Hence, the first element $(10)$ is compared with fourth element $(50)$ and found in the correct order. Then the second element $(1)$ is compared with fifth element $(4)$ which are also in the correct order. Then, the third element $(23)$ is compared with the sixth element $(9)$, since $(23 > 9)$, the sixth element is replaced by $(23)$ and $(9)$ is stored in a temp variable. As, third - gap = 0. Hence there is no element which can be compared with temp, therefore, the third term will be replaced by temp.
After that, the fourth element $(50)$ is compared with the seventh element $(-4)$, since $(50 > -4)$, the seventh element is replaced by $(50)$ and $(-4)$ is stored in a temp variable. As, fourth - gap = 1, hence, the temp is again compared with first element $(10)$. since $(10 > -4)$, the fourth element is replaced by $(10)$ and first element is replaced by temp (there is no element which can be compared with temp).
Second Pass: For this pass gap size is one. First four elements are already sorted. After that, the fourth element $(10)$ is compared with the fifth element $(4)$, since $(10 > 4)$, the fifth element is replaced by $(10)$ and $(4)$ is stored in a temp variable. Now, the temp is compared with third element $(9)$ which is greater than temp, hence fourth element is replaced by $(9)$. Then, the temp is compared with second element $(1)$ which is less than temp, hence third element is replaced by temp. After that, sixth and seventh elements are also considered for comparison which are already sorted.
## Implementation of Shell Sort
<?php
// function for shell sort
function shellsort(&$Array,$n)
{
$gap =$n/2;
$gap = (int)$gap;
while($gap > 0) { for($i = $gap;$i < $n;$i++)
{
$temp =$Array[$i];$j = $i; while($j >= $gap &&$Array[$j-$gap] > $temp) {$Array[$j] =$Array[$j-$gap];
$j =$j - $gap; }$Array[$j] =$temp;
}
$gap =$gap / 2;
$gap = (int)$gap;
}
}
// function to print array
function PrintArray($Array,$n)
{
for ($i = 0;$i < $n;$i++)
echo $Array[$i]." ";
}
// test shell sort code
$MyArray = array(10, 1, 23, 50, 4, 9, -4);$n = sizeof($MyArray); echo "Original Array\n"; PrintArray($MyArray, $n); shellsort($MyArray, $n); echo "\nSorted Array\n"; PrintArray($MyArray, \$n);
?>
Output
Original Array
10 1 23 50 4 9 -4
Sorted Array
-4 1 4 9 10 23 50
### Time Complexity:
In above implementation, the time complexity of shell sort in worst case scenario is $\mathcal{O}(N^2)$ and the time complexity in best and average case scenarios are $\mathcal{O}(NLogN)$. There are various methods to consider gap which lead to better time complexity.
|
2020-03-31 20:18:33
|
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|
https://blogs.shu.ac.uk/hallamguild/funding-requests-19-20/funding-criteria/?doing_wp_cron=1579499040.1584289073944091796875
|
# Funding Criteria
Essential Criteria
The proposed activity:
• aligns with at least one of the aims of the Hallam Guild
• will have a direct relationship with the delivery of the University Strategy.
• will produce an output \ artefact to facilitate dissemination of evidence \ good practice.
• will improve student experience
Desirable Criteria
The proposed activity:
• is a collaboration with another Guild group (s)
• is a collaboration across staff groups
• will improve staff experience
• include collaboration with students
• will include a plan for scalability
• is a collaboration with external colleagues
|
2020-01-20 05:44:01
|
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|
http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1002.58015
|
Language: Search: Contact
Zentralblatt MATH has released its new interface!
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Zbl 1002.58015
Chae, Dongho; Imanuvilov, Oleg Yu.
The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory.
(English)
[J] Commun. Math. Phys. 215, No.1, 119-142 (2000). ISSN 0010-3616; ISSN 1432-0916/e
The $(2+1)$-dimensional relativistic Chern-Simons equations form a nonlinear system of partial differential equations for a gauge field $A_\mu$ and a Higgs field $\varphi$ defined on ${\Bbb R}^3$ with standard Lorentzian metric. The self-dual solutions absolutely minimize the energy. There are two possible boundary conditions $|\varphi(x)|\to 1$ or $|\varphi(x)|\to 0$ as ${\Bbb R}^2\ni x\to\infty$ consistent with finite energy. Solutions with $|\varphi(x)|\to 1$ have been dubbed topological' and were shown to exist by {\it R. Wang} [Commun. Math. Phys. 137, No. 3, 587-597 (1991; Zbl 0733.58009)]. \par In this article, the authors consider the existence of self-dual non-topological' solutions, i.e. with boundary condition $|\varphi(x)|\to 0$. They prove the existence of solutions with arbitrarily prescribed zeroes for the Higgs field and other good properties. In particular, these solutions are not in any way symmetric. The construction is obtained by perturbation about explicit solutions of the Liouville equation.
MSC 2000:
*58E50 Appl. of variational methods in infinite-dimensional spaces
81T13 Gauge theories
35J60 Nonlinear elliptic equations
Keywords: Chern-Simons theory; self-dual solutions; Higgs field
Citations: Zbl 0733.58009
Cited in: Zbl 1116.58012 Zbl 1080.35021
Highlights
Master Server
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2013-05-24 00:13:09
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https://latex.org/forum/viewtopic.php?t=25242&p=85903
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## LaTeX forum ⇒ Feature Suggestions ⇒ Options for "Files" tool window
Suggestions and discussions for new TeXnicCenter features
kromuchi
Posts: 1
Joined: Thu Oct 16, 2014 2:38 pm
### Options for "Files" tool window
I would suggest to provide some options for the "Files" Tool Window of TexnicCenter. I don't really know how this file tree is generated by default, but apparently the main file is (recursively) scanned for `\input{}`, `\include{}` and `\includegraphics{}` and maybe some more.
However, commands defined by extra packages (e.g. subfiles) are ignored. This is not a surprise, as TexnicCenter cannot know all possible tex-commands, but maybe some option could be introduced in TexnicCenter to extend the commands for recursive scanning of the files list.
I would even suggest to make this a project-specific option (within the tcp file) but I would also be happy to see that feature in the main options of TC. Then people could add i.e. `\subfile{}` to the list and subfiles and its child files (i.e. figures) would also appear in the files list.
(disclaimer: I have looked around for quite a long time and did not found a solution for that yet. If there is an existing solution for that, I apologize)
Tags:
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2019-01-21 03:32:17
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http://math.stackexchange.com/questions/166757/a-different-uniform-bound-on-a-sequence-of-uniform-bounded-integrals
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# A different Uniform Bound on a Sequence of Uniform Bounded Integrals
Let $m$ be a probability measure on $W = \mathbb{R}^m$, so that $m(W)=1$.
Consider a sequence $\{X_k\}_{k=1}^{\infty}$ of compact sets $X_k \subset X = \mathbb{R}^n$ such that $X_k \rightarrow X$.
Consider a locally bounded function $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$.
We say that a function $\varphi: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ is of class $\mathcal{C}$ if it is continuous, strictly-increasing, zero-at-zero and unbounded.
Assume that there exists a class-$\mathcal{C}$ function $F: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ and a (uniform) $M \in \mathbb{R}_{>0}$ such that for any $k \in \mathbb{Z}_{\geq 1}$ we have
$$\sup_{x \in X_k} \int_W F(f(x,w)) m(dw) < M$$
Prove that there exists a concave class-$\mathcal{C}$ function $G$ and a (uniform) $M_G \in \mathbb{R}_{>0}$ such that for any $k \in \mathbb{Z}_{\geq 1}$ we have
$$\sup_{x \in X_k} \int_W G(f(x,w)) m(dw) < M_G$$
-
Just curious, how do you define $X_k \to X$? – Mercy Jul 4 '12 at 21:20
$\lim_{k\rightarrow \infty} X_k = X$. For instance: $X_k = k \mathbb{B}$ (closed ball of radius $k$). – Adam Jul 4 '12 at 22:44
What is not clear with the limit $\lim_{k \rightarrow \infty} X_k = X$? If you are not familiar with that, you can try to prove the claim with $X_k := k \mathbb{B}$. – Adam Jul 5 '12 at 0:03
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2014-03-07 15:17:46
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https://homework.cpm.org/category/CCI_CT/textbook/pc/chapter/2/lesson/2.3.3/problem/2-99
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Home > PC > Chapter 2 > Lesson 2.3.3 > Problem2-99
2-99.
Review the Math Notes box.
Rewrite the first factor using your answer to part (a).
$=\left(\frac{y-x}{xy}\right)\frac{(x+y)}{1}$
Multiply numerator by numerator and denominator by denominator.
$=\frac{xy+y^2-x^2-xy}{xy}=\frac{y^2-x^2}{xy}$
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2021-05-14 20:12:31
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http://mathhelpforum.com/differential-equations/114355-integration-sin-power-fration.html
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# Math Help - Integration of sin to the power of a fration
1. ## Integration of sin to the power of a fration
Hi, I've spent the best part of the day trying to get beyond this point and my trawling searches of the internet have sadly turned up no answers.
I am trying to integrate the equation shown in the picture above, I thought about converting the sin^2/3 to its trigonometric identity, but the only examples I can find are for when power is 2 or more. Anyone got any ideas where I could look for inspiration? or better yet able to offer an explanation. This day has killed me
2. Originally Posted by mrlibertine
Hi, I've spent the best part of the day trying to get beyond this point and my trawling searches of the internet have sadly turned up no answers.
I am trying to integrate the equation shown in the picture above, I thought about converting the sin^2/3 to its trigonometric identity, but the only examples I can find are for when power is 2 or more. Anyone got any ideas where I could look for inspiration? or better yet able to offer an explanation. This day has killed me
Where has this integral come from? It has no closed form using elementary functions but can be found using the hypergeometric function.
Perhaps there's a typo and it's meant to be $\int \sin^{2/3} (3t) \, {\color{red}\cos (3t)} \, dt$.
3. Its part of my workings out so far.
at the risk of humiliating myself I attach what I have 'achieved' so far.
4. Originally Posted by mrlibertine
Its part of my workings out so far.
at the risk of humiliating myself I attach what I have 'achieved' so far.
Your mistake is in saying that $\int \cot (3t) \, dt = |\sin (3t)|$. It's not. $\int \cot (3t) \, dt = \frac{1}{3} |\sin (3t)|$. Therefore the integrating factor is $\sin^{1/3} (3t)$ NOT $\sin (3t)$. This makes a very big difference.
5. that makes sense, I just used a table conversion without thinking what I was doing. Thank you for your help
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2014-09-30 19:10:31
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https://justinlross.com.au/philosophy-quick-reference/
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# Philosophy Quick Reference
inclusive or
xor/exclusive or
and
if
if and only iff
## Metaphysics
### Time
#### Basic
Time as a concept is something that we are all familiar with but that is not precisely defined. This quick reference will list every detail relating to the definition of time that I can find.
#### Metaphysics
Journal Articles:
Matt Farr
Farr, Matt (2016) Causation and Time Reversal. [Preprint] – http://philsci-archive.pitt.edu/12658/
“A theory is invariant under time reversal if and only if the time reverse of every motion allowed by the theory is also a motion allowed by the theory.” (Page 4)
Newtonian physics is time invariant, but velocity vector and such must be reversed and time reversed (Page 5).
Farr, Matt and Reutlinger, Alexander (2013) A Relic of a Bygone Age? Causation, Time Symmetry and the Directionality Argument. [Preprint] – http://philsci-archive.pitt.edu/9561/
“The C series is contrasted by McTaggart with the B series in terms of its lack of directionality” (Page 9)
//////Continue from page 10 of this journal article.
JME McTaggart
McTaggart, J. M. E. (1908). The unreality of time. Mind 17(68), 457–474.
[T]he C series, while it determines the order, does not determine the direction. If the C series runs M, N, O, P, then the B series [. . . ] can run either M, N, O, P (so that M is earliest and P latest) or else P, O, N, M (so that P is earliest and M latest). And there is nothing [. . . ] in the C series [. . . ] to determine which it will be. (McTaggart, 1908, p. 462, my emphasis.)
McTaggart, J. M. E. (1927). The Nature of Existence, Volume II. Cambridge: Cambridge University Press.
Sally Shrapnel
The Directionality Argument
Basically a summary of Bertrand Russell’s argument against causation.
1. If the fundamental physical theories are time-symmetric then they are not
causal.
2. The fundamental physical theories are time-symmetric.
3. Therefore, the fundamental physical theories are not causal
Reference: Field (2003), Ney (2009), Frisch (2012) and Farr and Reutlinger (2013)
#### Physics
Robert G Sachs
Sachs, R. G. (1987). The Physics of Time Reversal. Chicago: University of Chicago
Press.
This source goes into great detail about the ways of reversing equations in physics. There is some complexity here as Farr points out, for instance, that in electrodynamics the field needs to be inverted and in quantum mechanics the spin inverted.
### Religious
Aristotle mover, act and acted upon argument thing.
Principle of sufficient reason
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2021-10-18 10:40:12
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https://tex.stackexchange.com/tags/lyx/hot
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# Tag Info
2
You could enclose each text segment in a \pbox. Note, this requires the pbox package. This way you can treat each block of text as a math object. \documentclass{article} \usepackage{pbox} \begin{document} \$\pbox{3cm}{Variation of\\ neutron number\\ in time} = \pbox{3cm}{Rate of\\ production\\ of neutrons} - \pbox{3cm}{Rate of\\ absorption\\ of neutrons}-\...
2
You can do that very simply with stackengine: its \Centerstack command is in text mode. \documentclass{article} \usepackage[usestackEOL]{stackengine} \begin{document} {\sffamily\[ \Centerstack[l]{ Variation of \\neutron number\\ in time}{}={}\Centerstack[l]{Rate of \\production\\ of neutrons}{}-{}\Centerstack[l]{Rate of\\ absorption \\ of neutrons} {}-{}\...
2
I think that this question is a duplicate of this: How do I typeset vertical and horizontal lines inside a matrix? Here I put a small code where I have changed the parameters of vertical rules. \documentclass[a4paper,12pt]{article} \usepackage{mathtools} \usepackage{amssymb} \newcommand*{\vertbar}{\rule[-1.5ex]{1.1pt}{3ex}} \begin{document} \[\begin{...
1
I found that after installing those packages I had to got Lyx → Tools → Reconfigure. Now the Elsevier article class is available.
1
@scottkosty mentions 2 relevant bug reports 1 and 2, and the latter suggested passing in "reqno" as a document class option. This changes the equation labeling to the right side (instead of the left as shown) and avoids overlapping the equation label with the equation. Since I'm OK with right side equation label, this solves the issue for me.
1
Here's one answer, but it's a hack, and doesn't display fully right. You can break your table up into two tables, and make them exactly abut. Sometimes when I've tried to do this, I wound up with a weird indentation of one of the tables I couldn't clear, but undoing and redoing it again, carefully checking that the paragraph indentation was cleared at each ...
1
I just looked at the same thing and tried to use the 2cell feature in xy-pic. After reading the reference manual, I think the "right" solution is the following \documentclass[border=10pt]{standalone} \usepackage[all,2cell]{xy} \UseAllTwocells \begin{document} \[ \xymatrix{C\drtwocell\omit{^<-2>\eta}\ar_{1_C}[dr]\ar^F[r]&D\ar^G[d]& D\drtwocell\...
Only top voted, non community-wiki answers of a minimum length are eligible
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2020-06-06 01:52:36
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https://www.impan.pl/en/publishing-house/banach-center-publications/all/64/0/86347/a-study-of-some-constants-characterizing-the-weighted-hardy-inequality
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# Publishing house / Banach Center Publications / All volumes
## A study of some constants characterizing the weighted Hardy inequality
### Volume 64 / 2004
Banach Center Publications 64 (2004), 135-146 MSC: Primary 26D10, 26D15; Secondary 47B07, 47B38. DOI: 10.4064/bc64-0-11
#### Abstract
The modern form of Hardy's inequality means that we have a necessary and sufficient condition on the weights $u$ and $v$ on $[0,b]$ so that the mapping $$H:L^{p}(0,b;v)\rightarrow L^{q}(0,b;u)$$ is continuous, where $Hf(x)=\int_{0}^{x}f(t)dt$ is the Hardy operator. We consider the case $1< p\leq q< \infty$ and then this condition is usually written in the Muckenhoupt form $$A_{1}:=\sup _{0< x< b}A_{M}(x)< \infty . \tag*{(*)}$$ In this paper we discuss and compare some old and new other constants $A_{i}$ of the form $(*)$, which also characterize Hardy's inequality. We also point out some dual forms of these characterizations, prove some new compactness results and state some open problems.
#### Authors
• Alois KufnerMatematical Institute
Academy of Sciences of the Czech Republic
Žitná 25, 115 67 Praha 1
Czech Republic
e-mail
Luleå University of Technology
SE-971 87 Luleå, Sweden
e-mail
• Anna WedestigDepartment of Mathematics
Luleå University of Technology
SE-971 87 Luleå, Sweden
e-mail
## Search for IMPAN publications
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2020-09-28 18:18:56
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https://physics.stackexchange.com/questions/91265/motion-of-charge-in-magnetic-field-with-drag-force/91277
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# Motion of charge in magnetic field with drag force [closed]
Say you have a charged particle in a region that contains a fluid that will produce a drag force that goes as $F=-kv$ where $v$ is the speed and $k$ is some constant. The region also contains a uniform magnetic field. Suppose you give the particle some initial velocity $v_0$ in the plane perpendicular to the magnetic field. What will be the particle's subsequent motion? Please provide semi-quantitative answers. Note that this is not a homework question.
• What have you attempted so far? Dec 25 '13 at 0:00
• My guess is that it is a logarithmic spiral based on the fact that the speed falls off exponentially (from the drag force), although I'm not sure how to show it analytically.
– zeta
Dec 25 '13 at 0:05
Let's find the complete solution of the problem.
A complete solution of the problem would be the solution to the linear ODE,
$m \dot{\mathbf{v}} =q\mathbf{v} \times \mathbf{B}-k \mathbf{v}$
Assume without loss of generality that the magnetic field is pointed along the z-axis,so $\mathbf{B} = B \mathbf{\hat{z}}$. So our equation simplifies to,
$m \dot{\mathbf{v}} =qB\mathbf{v} \times \mathbf{\hat{z}}-k \mathbf{v}$
Dividing both sides of the equation by $m$ and for simplicity in the notation, let $\omega=\frac{qB}{m}$ and $\gamma=\frac{k}{m}$.
So,$\dot{\mathbf{v}} =\omega\mathbf{v} \times \mathbf{\hat{z}}-\gamma \mathbf{v}$
Using $\mathbf{v}=\begin{pmatrix} v_{x}\\v_{y}\\v_{z} \end{pmatrix}$ and writng the given eqaution in matrix form we have,
$\dot{\mathbf{v}} =\begin{pmatrix} -\gamma & \omega & 0 \\ -\omega & -\gamma & 0 \\ 0 & 0 & -\gamma \end{pmatrix} \mathbf{v}=A \mathbf{v}$
This is linear ODE which can be solved using the matrix exponenetial as,
$\mathbf{v}=e^{At} \mathbf{v_0}$ To simply this equation we can find the eigenvaules of A,use a similarity transform to convert it to a diagonal matrix which this greatly simplifies the matrix exponential.
The eigenvalues and the corresponding eigenvectors are,
$\lambda_{1}=-\gamma ,\mathbf{v_1}=\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$
$\lambda_{2}=-\gamma-i\omega ,\mathbf{v_2}=\begin{pmatrix} 1 \\ i \\ 0 \end{pmatrix}$
$\lambda_{3}=-\gamma+i\omega ,\mathbf{v_3}=\begin{pmatrix} i \\ 1 \\ 0 \end{pmatrix}$
Using $S=[ \mathbf{v_1} \mathbf{v_2} \mathbf{v_3}]$ and performing a similarity transform on the matrix A,$S^{-1}AS=D$ where D is diagonal matrix with the eigenvalues as the diagonal elements.And here we witness the power of similarity transformations as,
$e^{At}= S \begin{pmatrix} e^{\lambda_{1}t} &0 &0 \\0 & e^{\lambda_{2}t} &0 \\ 0& 0& e^{\lambda_{3}t} \end{pmatrix} S^{-1}$
(After some tedious calculations ans using $e^{ix}=\cos{x}+isin{x}$)
$=\begin{pmatrix} e^{-\gamma t}\cos(\omega t) &e^{-\gamma t}\sin(\omega t) &0\\ -e^{-\gamma t}\sin(\omega t) &e^{-\gamma t}\sin(\omega t) &0\\0 & 0 &e^{-\gamma t}\end{pmatrix}$
Therefore, $\mathbf{v}=\begin{pmatrix} e^{-\gamma t}\cos(\omega t) &e^{-\gamma t}\sin(\omega t) &0\\ -e^{-\gamma t}\sin(\omega t) &e^{-\gamma t}\sin(\omega t) &0\\0 & 0 &e^{-\gamma t}\end{pmatrix} \mathbf{v_0}$
Writing out the components,
$v_{x}=e^{-\gamma t}(v_{x_0} \cos{\omega t}+v_{y_0} \sin{\omega t})$
$v_{x}=e^{-\gamma t}(-v_{x_0} \sin{\omega t}+v_{y_0} \cos{\omega t})$
$v_{x}=e^{-\gamma t} v_{z_0}$
This the just the equation of a helix with both the pitch and radius decreasing exponentially with $\gamma$.However,the angular frequency is the same as that without drag,$\omega$.
Here is a sample trajectory,
The magnetic field does no work, so it does not change the speed of the particle. The drag force results in an exponential decrease in speed. If the drag force is very small, one would expect the particle to move in a circle with radius $r = \frac{mv}{qB}$, where $v = v_0 e^{-\frac{k}{m} t}$, i.e., a spiral.
A precise solution would proceed from
$$m\dot{\mathbf{v}} = q B \, \mathbf{v} \times \hat{\mathbf{z}} - k \,\mathbf{v}$$
This is a set of first-order, linear, coupled ODE's. Not very difficult to solve. In particular, $$\dot{\mathbf{v}} = M \,\mathbf{v}$$ where $M$ is a matrix, so you would start by finding its eigenvectors.
|
2022-01-20 15:29:15
|
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|
https://math.stackexchange.com/questions/1462239/proving-the-existence-of-consecutive-quadratic-residues-modulo-p5-by-means-of
|
# Proving the existence of consecutive quadratic residues modulo $p>5$ by means of Pell's equation
When I read Existence of Consecutive Quadratic residues I thought we might be able to prove the existence of non-zero consecutive quadratic residues modulo a prime $p>5$ using the theory of Pell-type equations: if $d>1$ is a quadratic residue modulo $p$, then any solution $(x,y)$ to $x^2-dy^2=1$ gives rise to consecutive quadratic residues $(x^2,dy^2)$ modulo $p$. It seems a promising method to me; the only problem lies in making sure that $x$ and $y$ aren't divisible by $p$.
Can we show that for any prime $p>5$, there exists a quadratic residue $d>1$ modulo $p$ and a solution $(x,y)$ to $x^2-dy^2=1$ s.t. $p\nmid x,y$?
That's a lovely idea, but there is an issue. Given a Pell equation: $$x^2-dy^2 = 1$$ its solutions $(x_0,y_0)=(1,0),(x_1,y_1),\ldots$ may be arranged in such a way that both the sequences $\{x_n\}_{n\geq 0}$ and $\{y_n\}_{n\geq 0}$ are Fibonacci-like sequences with the same characteristic polynomial, whose coefficients depend on the fundamental solution $(x_1,y_1)$. Any Fibonacci-like sequence is periodic $\pmod{p}$, hence there is for sure some $x_n\neq 1$ such that $p\nmid x_n$, since $x_0=1$. However,
$$y_n = \frac{(x_1+y_1\sqrt{d})^n-(x_1-y_1\sqrt{d})^n}{2\sqrt{d}}$$ is always a multiple of $y_1$, hence if $p\mid y_1$, there is no way to find some $y_n$ such that $y_n\not\equiv 0\pmod{p}$.
|
2019-04-18 22:17:03
|
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|
https://www.expii.com/t/boundedness-for-infinite-sets-444
|
Expii
# Boundedness for Infinite Sets - Expii
While finite sets are always bounded, infinite sets can be unbounded. Even when bounded, infinite sets need not have a maximum or minimum. Come see some examples.
|
2022-10-07 18:16:06
|
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|
https://programmer.ink/think/p4428-bjoi2018-binary-tree-array-set.html
|
# P4428-[BJOI2018] binary [tree array, set]
Posted by bruceg on Wed, 19 Jan 2022 12:15:46 +0100
# Topic
## General idea of the topic
Count Reg n n n 0 / 1 0/1 0 / 1 string requires support
1. Modify a location
2. Find interval [ l , r ] [l,r] [l,r] how many sub interval rearranged binary numbers can be divided by three
1 ≤ n ≤ 1 0 5 1\leq n\leq 10^5 1≤n≤105
## Problem solving ideas
First of all 2 2 k % 3 = 1 ( k ∈ Z ) 2^{2k}\%3=1(k\in Z) 22k%3=1(k ∈ Z) and 2 2 k + 1 % 3 = 2 ( k ∈ Z ) 2^{2k+1}\%3=2(k\in Z) 22k+1%3=2(k∈Z).
It is considered in three cases
• have 1 1 1 1 1 So obviously, it can't be divided by three in any case
• have 2 k 2k 2k 1 1 Then it would be nice if we were all at the bottom.
• have 2 k + 1 2k+1 2k+1 1 1 1( k k k cannot be 0 0 0), then there is a scheme to put a certain in an odd position 1 1 1 can be placed in the even position. At this time, the length of the interval needs to be at least 2 k + 3 2k+3 2k+3.
Then analyze it concretely, which is equivalent to an interval 1 1 The number of 1 cannot be 1 1 1 and if it is an odd number, there must be at least two 0 0 0.
It seems very complicated, which can be divided into the following situations
1. The interval is all 1 1 1 and the length is odd
2. There is one in the interval 0 0 0 and even length
3. There is only one interval 1 1 1
4. because 2 2 2 and 3 3 3 will repeat one, only one 1 1 1 and one 0 0 0, so this scheme needs to be added back
The fourth is the best maintenance. Just use the tree array to record
Then the first three we 0 / 1 0/1 One at 0 / 1 position s e t set set to query the predecessor / successor 0 / 1 of a location.
And then in the third case, we have 1 1 1 consider left and right 0 0 The 0 interval is then recorded in the tree array 1 1 Location of 1
For the second case, we consider for each 0 0 0 consider about 1 1 Then record it there 0 0 Location of 0
For the first case, we record the data at the leftmost end of the interval 0 0 0.
Then remember to consider the boundary when counting the answers
It's a little troublesome to write
Time complexity O ( n log n ) O(n\log n) O(nlogn)
## code
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<set>
#define lowbit(x) (x&-x)
#define ll long long
using namespace std;
const ll N=1e5+10;
ll n,m,a[N],t[N],p[N];
set<ll> s[2];
void Change(ll x,ll val){
while(x<=n){
t[x]+=val;
x+=lowbit(x);
}
return;
}
ll ans=0;
while(x){
ans+=t[x];
x-=lowbit(x);
}
return ans;
}
ll Left(ll op,ll x)
{return (*--s[op].upper_bound(x));}
ll Right(ll op,ll x)
{return (*s[op].lower_bound(x));}
ll Count(ll n)
{return (n+1)/2*(n+2-(n&1))/2;}
ll Caunt(ll n)
{return n*(n+1)/2;}
ll Calc(ll L,ll R)
{return (L/2+1)*((R+1)/2)+((L+1)/2)*(R/2+1);}
void Updata(ll x){
if(x<1||x>n)return;
if(p[x])Change(x,-p[x]);
if(a[x]){
ll L=(x-Left(1,x-1)-1),R=(Right(1,x+1)-x-1);
p[x]=(L+1)*(R+1)-1;
}
else{
ll L=(x-Left(0,x-1)-1),R=(Right(0,x+1)-x-1);
p[x]=Calc(L,R)+Count(R);
}
if(x<n&&a[x]!=a[x+1])p[x]--;
Change(x,p[x]);
return;
}
ll Get(ll x,ll l,ll r){
ll L=max(Left(0,x-1),l-1),R=min(Right(0,x+1),r+1);
L=x-L-1;R=R-x-1;
return Calc(L,R);
}
ll Qet(ll x,ll l,ll r){
ll L=max(Left(1,x-1),l-1),R=min(Right(1,x+1),r+1);
L=x-L-1;R=R-x-1;
return (L+1)*(R+1)-1;
}
signed main()
{
scanf("%lld",&n);
s[0].insert(0);s[0].insert(n+1);
s[1].insert(0);s[1].insert(n+1);
for(ll i=1;i<=n;i++)
scanf("%lld",&a[i]),s[a[i]].insert(i);
for(ll i=1;i<=n;i++)
Updata(i);
scanf("%lld",&m);
while(m--){
ll op,l,r,x;
scanf("%lld",&op);
if(op==1){
scanf("%lld",&x);
s[a[x]].erase(x);
a[x]=!a[x];
s[a[x]].insert(x);
Updata(x);
Updata(Left(0,x-1));
Updata(Left(1,x-1));
Updata(Right(0,x+1));
Updata(Right(1,x+1));
}
else{
scanf("%lld%lld",&l,&r);
ll ans=(r-l+1)*(r-l+2)/2;
if(Left(1,r)<l){printf("%lld\n",ans);continue;}
if(Left(0,r)<l){ans-=Count(r-l+1);printf("%lld\n",ans);continue;}
if(r<n&&a[r]!=a[r+1])ans--;
ll Ll=Left(0,l-1),Rr=Right(0,r+1),Lr=Left(0,r),Rl=Right(0,l);
ans=ans+Get(Rl,1,n)-Get(Rl,l,r);
if(Lr!=Rl)ans=ans+Get(Lr,1,n)-Get(Lr,l,r);
if(a[r+1])ans=ans+Count(Rr-Lr-1)-Count(r-Lr);
if(a[l])ans=ans-Count(Rl-l);
Ll=Left(1,l),Rr=Right(1,r),Lr=Left(1,r),Rl=Right(1,l);
ans=ans+Qet(Rl,1,n)-Qet(Rl,l,r);
if(Lr!=Rl)ans=ans+Qet(Lr,1,n)-Qet(Lr,l,r);
// if(!a[r])ans=ans+Caunt(Rr-Rl-1)-Caunt(r-Rl);
// if(!a[l])ans=ans-Caunt(Lr-l);
printf("%lld\n",ans);
}
}
return 0;
}
|
2022-08-13 20:56:43
|
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|
https://docs.q-ctrl.com/boulder-opal/user-guides/how-to-optimize-controls-using-a-hann-series-basis
|
# How to optimize controls using a Hann series basis
Create optimized controls using Hann series basis functions
Boulder Opal exposes a highly-flexible optimization engine for general-purpose gradient-based optimization. The controls can be described in terms of optimizable linear combinations from a set of built in (or user-defined) basis functions, which can greatly reduce the dimensionality of the optimization search space. In this notebook we will use Hann window functions, although the same technique has also seen success with other bases, for example Slepian functions. You can also read the related user guides showing how to find optimal controls using a Fourier basis or an arbitrary (user-defined) basis.
## Summary workflow
### 1. Define basis function for signal composition in the graph
The Boulder Opal signal library provides convenience graph operations to create piecewise-constant (PWC) and sampleable (STF) optimizable signals in a Hann series basis. To create the PWC Hann series use graph.signals.hann_series_pwc, which requires the duration, segment_count and set of coefficients (which may be optimizable tensors) to be specified. To create the STF Hann series use graph.signals.hann_series_stf, which requires the start_time, end_time and set of coefficients (which may be optimizable tensors) to be specified.
### 2. Execute graph-based optimization
With the graph object created, an optimization can be run using the qctrl.functions.calculate_optimization function. The cost, the outputs, and the graph must be provided. The function returns the results of the optimization.
The Boulder Opal Toolkits are currently in beta phase of development. Breaking changes may be introduced.
## Example: Robust optimization on a qutrit using Hann window functions
In this example, we perform optimization for a robust single qubit Hadamard gate (in the Hann window basis) of a qutrit system while minimizing leakage out of the computational subspace. The system is described by the following Hamiltonian:
$$H(t) = \frac{\chi}{2} (a^\dagger)^2 a^2 + (1+\beta(t)) \left(\gamma(t) a + \gamma^*(t) a^\dagger \right),$$
where $\chi$ is the anharmonicity, $\gamma(t)$ is a complex time-dependent signal, $\beta(t)$ is a small, slowly-varying stochastic amplitude noise process, and $a = |0 \rangle \langle 1 | + \sqrt{2} |1 \rangle \langle 2 |$.
We parametrize the signal $\gamma(t) = \gamma_I(t) + i \gamma_Q(t)$ in terms of Hann window functions:
$$\gamma_{I(Q)}(t) = \sum_{n=1}^N{\frac{c^{I(Q)}_n}{2} \left[1-\cos\left(\frac{2\pi nt}{\tau_g}\right) \right]},$$
where $c^{I(Q)}_n$ are the different real-valued coefficients describing the parametrization and $\tau_g$ is the gate duration. This is a good choice for implementation in bandwidth-limited hardware as it is composed of smooth functions that go to zero at the edges.
import numpy as np
from qctrlvisualizer import plot_controls
from qctrl import Qctrl
# Start a Boulder Opal session.
qctrl = Qctrl()
# Define target and projector matrices.
hadamard = np.array([[1.0, 1.0, 0], [1.0, -1.0, 0], [0, 0, np.sqrt(2)]]) / np.sqrt(2)
qubit_projector = np.diag([1.0, 0.0, 0.0])
# Define physical constraints.
chi = -2 * np.pi * 300.0 * 1e6 # Hz
gamma_max = 2 * np.pi * 50e6 # Hz
segment_count = 200
duration = 100e-9 # s
sample_times = np.linspace(0, duration, segment_count)
optimizable_frequency_count = 5
# Create graph object.
graph = qctrl.create_graph()
# Define standard matrices.
a = graph.annihilation_operator(3)
# Define the coefficients of the Hann functions for optimization.
hann_coefficients_i = graph.optimization_variable(
optimizable_frequency_count, lower_bound=-1, upper_bound=1
)
hann_coefficients_q = graph.optimization_variable(
optimizable_frequency_count, lower_bound=-1, upper_bound=1
)
hann_coefficients = gamma_max * (hann_coefficients_i + 1j * hann_coefficients_q)
# Create gamma(t) signal in Hann function basis.
gamma = graph.signals.hann_series_pwc(
duration=duration,
segment_count=segment_count,
coefficients=hann_coefficients,
name=r"$\gamma$",
)
# Create anharmonicity term.
# Create drive term.
drive = graph.hermitian_part(2 * gamma * a)
# Create target operator in qubit subspace.
target_operator = graph.target(
)
# Create infidelity.
infidelity = graph.infidelity_pwc(
hamiltonian=anharmonicity + drive,
target=target_operator,
noise_operators=[drive],
name="infidelity",
)
# Run the optimization and retrieve results.
optimization_result = qctrl.functions.calculate_optimization(
cost_node_name="infidelity",
output_node_names=[r"$\gamma$"],
graph=graph,
optimization_count=4,
)
print(f"\nOptimized cost:\t{optimization_result.cost:.3e}")
plot_controls(optimization_result.output, smooth=True, polar=False)
Your task calculate_optimization (action_id="1599304") is currently in a queue waiting to be processed.
Optimized cost: 2.329e-06
This notebook was run using the following package versions. It should also be compatible with newer versions of the Q-CTRL Python package.
PackageVersion
Python3.10.8
matplotlib3.6.3
numpy1.24.1
scipy1.10.0
qctrl20.1.1
qctrl-commons17.7.0
boulder-opal-toolkits2.0.0-beta.3
qctrl-visualizer4.4.0
|
2023-03-28 17:30:23
|
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|
https://math.stackexchange.com/questions/809473/generalization-of-the-jordan-form-for-infinite-matrices/812092
|
Generalization of the Jordan form for infinite matrices
Under what conditions is it the case that for a matrix $M$ whose rows and columns are indexed by a countably infinite set $S$ one has a Hamel basis consisting of generalized eigenvectors (i.e. $v \in \ker(M - \lambda I)^n$) of $M$? Must $M$ be a compact operator (I have a norm)?
The matrix I am working with has non-negative entries, row sums not exceeding $1$ (substochastic), is irreducible and aperiodic. However, I suspect this question may be of general interest to others, so any solution not employing these properties would be all the more useful.
EDIT
Here is some more information: the matrix $M$ which I am working with is $R$-positive. This means that none of the sequences $\{ M^n_{ij}\}_{n \in \mathbb{N}}$, $i,j \in S$, converge to $0$, where $$R^{-1} := \lim_{n \to \infty} (M_{ij}^n)^{1/n}.$$ In such a case, it is known that $R^{-1}$ is the spectral radius of $M$, and moreover that $R^{-1}$ is an eigenvalue for $M$ for which there are unique left and right eigenvectors $\alpha,\beta$ which are strictly positive and satisfy $$\sum_{k \in S} \alpha(k) \beta(k) < \infty.$$ In particular the set of eigenvalues for $M$ cannot be empty.
Thanks!
• I think you can look up Jacobson. – Bombyx mori May 28 '14 at 2:23
• @Bombyxmori can you give a reference? I haven't found anything which seems connected. Thanks! – Rookatu May 29 '14 at 21:50
• I think the Pascal-matrix can be Jordan-decomposed (having the matrices of Stirling-numbers as generalized eigenvectors) and the identity matrix with additional first subdiagonal containing the unit as Jordan form. This is consistent for any size nxn, and I've always assumed that this is this also valid for the infinite case. Similarly I assume the same generalization from finite to infinite size is true for the Jordandecomposition of the (Carleman-)matrix containing the Stirling numbers, which map $x \to \exp(x)-1$ and the inverse. But I've no such theoretical background as the other answers. – Gottfried Helms Dec 31 '14 at 9:30
Let $H$ be a countably-dimensional Hilbert space with basis $\{e_n\}$. Let $T$ be the weighted shift given by $$Te_n=\frac1n\,e_{n+1}.$$ This operator is compact (actually, Hilbert-Schmidt).
If $Tv=\lambda v$, with $v=\sum_n\alpha_n e_n$, then $$\sum_{n=1}^\infty\alpha_n\lambda e_n=\sum_{n=1}^\infty\alpha_n Te_n=\sum_{n=1}^\infty\alpha_n\,\frac1n\,e_{n+1}=\sum_{n=2}^\infty\frac{\alpha_{n-1}}{n-1}\,e_n.$$ If $\lambda=0$, we deduce that $\alpha_n=0$ for all $n$, so $v=0$. If $\lambda\ne0$, then $\alpha_1=0$, and $\alpha_{n+1}=\alpha_n/n$, implying again that $\alpha_n=0$ for all $n$. So $v=0$. This shows that $T-\lambda I$ has trivial kernel for all $\lambda$.
So $T$ has no nonzero generalized eigenvectors, and it cannot have a Jordan form, at least in the obvious sense.
• Thank you for the thoughtful answer. However, if I am not mistaken, the matrix of this operator would not be irreducible, correct? My question is really about what conditions allow for the desired representation, as well as whether the properties I listed are sufficient. Any thoughts? Thanks! – Rookatu May 28 '14 at 3:51
• What do you mean by irreducible? The way I use the word is that the commutant of $T$ is trivial, which it certainly is. – Martin Argerami May 28 '14 at 11:22
• My meaning is that the matrix I'm dealing with, which is substochastic, has the property that for any $i,j$ in the indexing set $S$, there exists an $n \in \mathbb{N}$ with the property that $M^n_{ij} > 0$. It seems to me that you can "transition" forward, from $e_n$ to $e_{n+1}$ but never backward and hence that the corresponding matrix would not be irreducible, though I must say that functional analysis is not my strong suit and I may have misinterpreted. Thanks! – Rookatu May 28 '14 at 11:37
• I see. I cannot really imagine how it would be possible to check that condition but in the simplest of cases. – Martin Argerami May 28 '14 at 23:09
• I don't really know. But let me tell you this: you are thinking in terms of matrices, and not of operators. But eigenvalues and eigenvectors are properties of the operator, not the matrix. While being R-positive is a property of the matrix and not of the operator. – Martin Argerami Jun 5 '14 at 1:54
I am not sure if this answer is useful in your case.
Theorem: Exists a basis formed by generalized eigenvectors of $T:V\rightarrow V$, even if $V$ is an infinite dimensional vector space over a field $F$, if we assume the existence of a polynomial $p(x)\in F[x]$ with all roots in $F$ such that $p(T)=0$.
Of course, we need to prove first for nilpotent operators. Then we must use the Primary decomposition theorem to extend for arbitrary operators satisfying this hypothesis. Notice that this hypothesis is always true in finite dimension by Cayley-Hamilton's theorem, if $F=\mathbb{C}$. The proof is almost the same of the finite dimensional case.
The proof for nilpotent operators is an induction on the nilpotency index $($the smallest $k$ such that $T^k=0)$ which does not depend on the dimension of $V$.
Let us prove for nilpotent operators $($i.e. when $p(x)=x^k)$.
Proof: Let $T:V\rightarrow V$ be a nilpotent operator. Let $k$ be the nilpotency index. If $k=1$ the theorem is trivial. Suppose $k>1$.
Since $k>1$ then $T\neq0$ and $\Im(T)\neq 0$. Define $T':\Im(T)\rightarrow\Im(T)$, such that $T'(x)=T(x)$.
The nilpotency index of $T'$ is smaller than the index of $T$.Thus by induction hypothesis exists a basis $\alpha$ of $\Im(T)$ such that
1. $\alpha=\cup_{i\in I}\alpha_i$
2. $\alpha_i=\{v_1^i,\ldots,v_{s_i}^i\}$, $s_i<k$
3. $T(v_l^i)=v_{l-1}^{i}$ for $1<l\leq s_i$ and $T(v_1^i)=0$
Next for each $v_{s_i}\in\alpha_i\subset\Im(T)$, choose $v_{s_{i+1}}^i$ such that $T(v_{s_{i+1}}^i)=v_{s_{i}}^i$.
Consider the following preimage of $\alpha$: $\cup_{i\in I}\beta_i$, where $\beta_i=\{v_2^i,\ldots,v_{s_i}^i,v_{s_{i+1}}^i\}$ Now, let $W$ be a subspace of $V$ generated by $\cup_{i\in I}\beta_i$. Notice that $\ker(T)\oplus W=V$ and $\cup_{i\in I}\beta_i$ is a basis of $W$.
Now, $\{v_1^i,i\in I\}$ is a basis of $\ker(T)\cap\Im(T)$. (It is straightfoward)
Let $R$ be a subspace of $\ker(T)$ such that $R\oplus (\ker(T)\cap\Im(T))=\ker(T)$. Let $\{r_j, j\in J\}$ be a basis of $R$.
Finally the required basis of $V=\ker(T)\oplus W=R\oplus (\ker(T)\cap\Im(T))\oplus W$ is $$\{r_j, j\in J\}\cup \{v_1^i,i\in I\}\cup (\cup_{i\in I}\{v_2^i,\ldots,v_{s_i}^i,v_{s_{i+1}}^i\})=$$ $$=\{r_j, j\in J\}\cup (\cup_{i\in I}\{v_1^i, v_2^i,\ldots,v_{s_i}^i,v_{s_{i+1}}^i\}).$$ $\square$
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2019-06-20 05:00:33
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https://djalil.chafai.net/blog/page/2/
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# Libres pensées d'un mathématicien ordinaire Posts
This tiny post is an invitation to play with hypergeometric functions. These remarkable special functions can be useful to all mathematicians. They are bizarely not known by many, however.
Pochhammer symbol for rising factorial. Named after Leo August Pochhammer (1841 – 1920):
$$(z)_k:=z(z+1)\cdots(z+k-1)$$ with the convention $(z)_0:=1$ if $z\neq0$. Note that $(1)_k=k!$. We have $$\Gamma(z+k)=(z)_k\Gamma(z)\quad\text{where}\quad\Gamma(z):=\int_0^\infty t^{z-1}\mathrm{e}^{-t}\mathrm{d}t.$$ When $z=n$ then this boils down to $(n)_k=(n+k-1)!/(n-1)!$.
Hypergeometric function. If $a\in\mathbb{R}^p$, $b\in\mathbb{R}^q$, and $z\in\mathbb{C}$, $|z|<1$, then, when it makes sense,
$${}_pF_q\begin{pmatrix}a_1,\ldots,a_p\\b_1,\ldots,b_q\\z\end{pmatrix}:=\sum_{k=0}^\infty\frac{(a_1)_k\dots(a_p)_k}{(b_1)_k\cdots(b_q)_k}\frac{z^k}{k!},$$
The formula for ${}_pF_q$ remains valid for more values of $z$ by analytic continuation. Hypergeometric functions where studied by many including notably Leonhard Euler (1707 – 1783) and Carl-Friedrich Gauss (1777 – 1855). This kind of special function contains several others, for instance
• ${}_2F_1(1,1;2;-z)=\frac{\log(1+z)}{z}$
• ${}_2F_1(a,b;b;z)=\frac{1}{(1-z)^a}$
• ${}_2F_1(\frac{1}{2},\frac{1}{2};\frac{3}{2};z^2)=\frac{\arcsin(z)}{z}$
It is also possible to embed Jacobi orthogonal polynomials into hypergeometric functions and thus several families of orthogonal polynomials, more precisely $${}_{2}F_{1}(-n,a+1+b+n;a+1;x)={\frac {n!}{(a+1)_{n}}}P_{n}^{(a,b)}(1-2x).$$ Note that $(z)_k=0$ for large enough $k$ when $z$ is a negative integer, hence ${}_pF_q(a;b;z)$ is a polynomial when one of the $a_i$ is a negative integer.
Hypergeometric functions admit integral representations, and conversely, certain integrals can be computed using hypergeometric functions. Here is the most basic example.
Euler integral representation formula for ${}_2F_1$ published in 1769. If $a>0$, $b>0$, $|z|\leq 1$, then
$$\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\mathrm{d}u ={}_2F_1\begin{pmatrix}a,c\\a+b\\z\end{pmatrix} \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$ In other words, for all $a,b,c$ with $c>a>0$ and all $|z|\leq 1$,
$${}_2F_1\begin{pmatrix}a,b\\c\\z\end{pmatrix}=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_0^1u^{a-1}(1-u)^{c-a-1}(1-zu)^{-b}\mathrm{d}u.$$
A proof. A binomial series expansion of $(1-zu)^{-c}$ gives
$$\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\mathrm{d}u =\sum_{k=0}^\infty\frac{(c)_k}{k!}z^k \int_0^1u^{a+k-1}(1-u)^{b-1}\mathrm{d}u.$$
Now the beta-gamma formula $\displaystyle \int_0^1u^{a+k-1}(1-u)^{b-1}\mathrm{d}u=\frac{\Gamma(a+k)\Gamma(b)}{\Gamma(a+b+k)}$ gives
$$\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\mathrm{d}u =\Gamma(b)\sum_{k=0}^\infty\frac{(c)_k\Gamma(a+k)}{\Gamma(a+b+k)} \frac{z^k}{k!}.$$
Finally the formula $\Gamma(z+k)=(z)_k\Gamma(z)$ gives
$$\int_0^1u^{a-1}(1-u)^{b-1}(1-zu)^{-c}\mathrm{d}u =\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\sum_{k=0}^\infty\frac{(c)_k(a)_k}{(a+b)_k} \frac{z^k}{k!}\\ =\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\ {}_2F_1\begin{pmatrix}a,c\\a+b\\z\end{pmatrix}.$$
Immediate corollary. By sending $z$ to $1$, taking $b>c$, and using the beta-gamma formula $$\int_0^1u^{a-1}(1-u)^{b-c-1}\mathrm{d}u =\frac{\Gamma(a)\Gamma(b-c)}{\Gamma(a+b-c)},$$ we obtain the following identity discovered by Gauss (1812), for all $a,b,c$ with $c>a+b$, $$\sum_{k=0}^\infty\frac{(a)_k(b)_k}{(c)_k}\frac{1}{k!}={}_2F_1\begin{pmatrix}a,b\\c\\1\end{pmatrix}=\frac{\Gamma(c-a-b)\Gamma(c)}{\Gamma(c-a)\Gamma(c-b)}.$$
Maple, Mathematica, and Maxima. All implement hypergeometric functions. Here is an example with the Euler integral formula with Mathematica:
In[1]:= Integrate[u^{a-1}*(1 - u)^{b-1}*(1 - z*u)^{-c}, {u, 0, 1}]
Out[1]= {ConditionalExpression[
Gamma[a] Gamma[b] Hypergeometric2F1Regularized[a, c, a + b, z],
Re[a] > 0 && Re[b] > 0 && (Re[z] <= 1 || z ∉ ℝ)]}
The regularized ${}_2F_1$ hypergeometric function used by Mathematica is ${}_2F_1(a,b;c;z)/\Gamma(c)$.
It is a great time for scientific conferences over the Internet, a discovery for many colleagues and communities. Ideally, such events should be organized using software platforms implementing virtual reality, just like for certain video games, with virtual buildings, virtual rooms, virtual characters, virtual discussions, virtual restaurants, etc. Unfortunately, such platforms do not seem to be available yet with the expected level of quality and flexibility, even if I have heard that some colleagues from PSL University managed to organize a virtual poster session using a software initially designed for virtual museums!
Online scientific conferences are very easy to organize, cost almost nothing, and have an excellent carbon footprint in comparison with traditional face-to-face on-site conferences. The carbon footprint of Internet is not zero, but the fraction that is used for an online scientific event has nothing to do with what is used for an on-site conference with plenty of participants coming from far away by airplanes. The main drawback of online scientific events is of course the limited interactions between participants, and the fact that they do not extract the participants from their daily life and duties. Online conferences are an excellent way to maintain the social link inside scientific communities. The term webinar/webconference is sometimes used but emphasizes the web which is not the heart of the concept.
It would be unfortunate to make all scientific conferences online. The best would be to reduce the number of traditional conferences, and try to increase their quality, for instance by asking participants to stay longer. Also I would not be surprised to see the development of blended or hybrid conferences mixing on-site participants and remote online participants.
I had recently the opportunity to co-organize with a few colleagues an online scientific conference on Random Matrices and Their Applications, in replacement of a conventional on-site face-to-face conference in New York canceled due to COVID-19. The initial conference was supposed to last a whole week, with about thirty talks and a poster session. For the online replacement, we have decided to keep the same week. About twelve of the initial speakers accepted to give an online talk. For simplicity, we have then decided to put three 45 minutes talks per day on Monday, Tuesday, Thursday, and Friday, and to give up the poster session. We have used the same schedule for each day, with a first talk at 10 am New York local time. We had between 80 and 150 participants per talk, from all over the world. In short:
• Schedule. Few talks per day, compatible with as many local times as possible
• Website. Speakers, titles, abstracts, slides, registration
• Talks. To improve speakers experience, turn-on few cameras along the talks, typically the chairperson/organizers/coauthors. The integrated text chat can be used for questions
• Workspace. An online collaborative workspace in parallel is useful. A private channel can host the discussions between organizers, replacing emails, a general channel can host the interaction with the speakers and the participants, and between them, etc.
On the technical side, we have decided to use current social standards instead of best quality solutions, namely Dokuwiki for the website, Zoom for the talks, and Slack for the workspace.
The COVID-19 pandemic is an interesting subject of study from many perspectives. Looking back at recent and less recent history, this pandemic by itself appears for now as rather ordinary, while the political responses are truly exceptional. In particular and among several aspects, we can observe, beyond the risk aversion and the international mimetism, a certain role played by risk analysis for decision making based on mathematical modelling for epidemiology.
Mathematical modelling is remarkably successful to predict, with a high degree of accuracy, the behavior of many natural phenomena, such as for example, and very concretely, the trajectory of satellites or the propagation of sound and light. The numerous successes of mathematical modelling have enormous positive concrete impact on our world and our daily life. Around these topics, there is for instance a famous article by Eugene Wigner entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960), and another one by Richard Hamming entitled The unreasonable effectiveness of mathematics (1980).
On the other hand, the mechanisms of many natural phenomena are not well understood, and even when they are well understood at a certain scale, their mathematical modelling is often an approximation of their complexity and subtleties, which is not always accurate. Approximation may also come from the mathematical and numerical analysis of the model by itself, as well as from the lack of data to fit the model. All these aspects are well known by every mathematician, and it is customary to say that all models are wrong, but some are useful. This reminds on this topic the article entitled The Reasonable Ineffectiveness of Mathematics (2013) by Derek Abbott, pointing out some of the limitations of mathematization.
The case of meteorology is particularly interesting. The mechanisms of the natural phenomenon are relatively well understood and are modeled mathematically by the equations of fluid mechanics, related to one of the greatest questions of mathematical physics. Unfortunately, the sensitivity of these equations to perturbations make the prediction relatively limited, despite the striking progresses made in numerical analysis and computational power, and the enormous amount of data collected by satellite remote sensing. Weather forecasting remains difficult.
The situation is even worse for the social sciences such as economics or sociology, for which we do not have the analogue of the equations of fluid mechanics. Historically, the quantitative analysis of social phenomena were first approached by using statistics, notably by Adolphe Quetelet, who produced among other studies his famous Sur l’homme et le développement de ses facultés, ou Essai de physique sociale (1835). Quetelet discovered some of the mechanical sides of disordered phenomena, paving the way to the mathematical modelling of disordered systems and their predictability. He was not the only scientist to explore the mechanical view of nature at that time, the famous others include certainly Charles Darwin and Ludwig Boltzmann. The mechanization of disordered systems led to the great successes of probability and statistics that we all know, which are also at the heart of statistical physics, quantum mechanics, and information theory. But the social sciences remain too complex for many aspects. This is well explained for instance for economics in Le Capital au XXIe siècle (2013) by Thomas Piketty.
The tremendous development of digitization, computers, and networks has led to the widespread use of mathematical modelling and numerical experiments. It has also stimulated the development of various types of machine learning, producing striking concrete successes. This type of algorithmic data processing is still considered as modelling but may differ from usual modelling in that it can produce empirical prediction without understanding.
How about epidemiology? It turns out that the mechanisms of viral epidemics are not well understood by the scientists for now. The available mathematical or computational modelling incorporates what is known. It remains limited for prediction, and the problem does not reduce to data collection. In particular it produces questionable risk analysis for decision making. We could alternatively use the historical statistics of epidemics to produce predictions, at least at the qualitative or phenomenological level, but this is also relatively limited. We are thus condemned to live for now with important uncertainties. This is somewhat difficult to accept for our present societies.
About the author. Mainly a mathematician, professor at Université Paris-Dauphine – PSL since 2013, presently active in probability theory, mathematical analysis, and statistical physics. Also strongly interested in computer science. Served in the past as a research engineer on data assimilation for the Météo-France research center (one year), researcher in mathematics and signal processing for University of Oxford (one year), researcher in biostatistics and modelling for INRAE (six years), professor of mathematics at Université Paris-Est Marne-la-Vallée (four years), and part-time professor at École Polytechnique (six years). Serves presently as a vice-president in charge of digital strategy for Université Paris-Dauphine for the period 2017-2020.
Further reading on this blog.
France. Concernant COVID-19 en France, voici un graphique intéressant de l’INSEE, permettant de comparer la mortalité avec quelques éléments du passé, notamment la canicule de l’été 2003, et la grippe de Hong Hong d’il y a cinquante ans. Le confinement est une différence importante. Cependant, on ne sait pas ce qu’aurait donné COVID-19 sans confinement, peut-être un pic plus élevé et moins étalé dans le temps, peut-être pas. Il se peut très bien que le confinement tel qu’il a été organisé n’ait servi à rien voire ait aggravé la situation dans certaines familles et collectivités. L’effet sur les accidents de la route est bien réel mais ne change pas énormément les choses. Une autre différence avec la grippe de Hong Kong est la taille de la population, bien plus petite à l’époque, ainsi que la pyramide des âges, bien plus jeune à l’époque, l’essentiel des décès de la COVID-19 concernant les personnes âgées, pour une bonne part en Ehpad. Notons également que la grippe espagnole en fin de première guerre mondiale – absente du graphique – a plutôt tué les jeunes adultes, par surinfection bactérienne, avant l’ère des antibiotiques. Tout cela souligne la difficulté à comparer à travers le temps. Ces phénomènes extrêmes et récurrents sont encore plus complexes et hétérogènes que les crues des fleuves dont les bassins évoluent. Il s’agit là d’un problème majeur de l’analyse de données à travers le temps et l’espace. La principale difficulté à laquelle est confronté Thomas Piketty dans son travail sur le capital est précisément l’hétérogénéité spatio-temporelle des données statistiques concernant l’économie.
Il y a dix ans, jour pour jour, naissait ce modeste blog avec un très court billet sur Louis Antoine. Plusieurs centaines de billets ont suivi depuis, parfois beaucoup plus longs. Tout n’est pas réussi, loin de là. Il faut tâtonner, caler un mode de rédaction de billet, un format. Ce temps passé à écrire ici reste une source intarissable d’excitation intellectuelle, de joie de synthétiser, de partager. Le savoir mérite d’être diffusé, les idées agitées, les points de vue exprimés. Cela convient bien aux travailleurs de la pensée que sont les universitaires. Se concentrer sur l’écriture, et laisser les moteurs de recherche faire leur indexation. C’est ainsi que certains billets de ce blog sans prétention sont entrés dans la bibliographie de quelques articles de recherche. Une surprise, qui souligne l’importance actuelle de ce mode de diffusion de l’information, et qui pose aussi la question de la pérennité de l’auto-publication électronique. Ce type d’auto-publication échappe au dépôt légal et les dispositifs du type Wayback machine sont limités.
Une pensée pour Louis Antoine, devenu aveugle, puis mathématicien, géomètre.
Louis Antoine
Syntax · Style · .
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2020-09-27 23:32:38
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https://www.physicsforums.com/threads/group-theory.14603/
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# Homework Help: Group Theory
1. Feb 16, 2004
### Norman
Problem:
a) To determine the number of generators needed for the group O(n) we write a rotation matrix as:
$$R=e^{-i\theta J}$$
where $J$ is an n x n matrix, Hermitian and imaginary, and therefore anti-symmetric. The number of indepedent parameters $\theta$ (and hence the number of generators) is the number of independent matrices. This number can be found by counting the number of parameters required to make up any n x n antisymmetric matrix. This is n(n-1)/2- WHY?
b)Show for any n:
$$[J_{ij},J_{kl}]=\plusminus (\delta_{ij}J_{il}-\delta_{ik}J_{jl}-\delta_{jl}J_{ik}+\delta_{il}J_{ik})$$
where $J_{ij}$ are two index objects with matrix elements:
$$(J_{ij})_{kl} = -i(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})$$
and
$$[J_{ij},J_{kl}]$$
is the commutator
Ok...
So part a):
I am a little confused. I know that the matrix must be imaginary and hermitian, but I don't think that is enough to prove that only n(n-1)/2 parameters are required to make a n x n antisymmetric matrix. In fact I am not even sure what determines whether the parameters are independent. Is a complex number and its conjugate independent? If not, then I think I understand. But if not I am lost.
part b) No clue.
I have never taken a group theory class and this was thrown into a Quantum Mechanics homework set so I am pretty lost. Any help would really be appreciated.
Last edited: Feb 16, 2004
2. Feb 17, 2004
### HallsofIvy
No, a complex number, a+ bi, and its conjugate, a- bi, are definitely NOT independent! Especially if we are given that the numbers are all imaginary so it is really bi and -bi. Clearly an imaginary, Hermitian matrix is anti-symmetric. Now, calculate how many "choices" you could make for the values in an anti-symmetric matrix: aij= -aji.
In particular, all the entries on the main diagonal (i= j) must be 0: aii= -aii means aii= 0 so we cannot make any choices for them. There are, of course, exactly n diagonal elements in an n by n matrix, leaving n2-n. If we "choose" any one of those, say aij then its "opposite", aji is fixed. That is, we can "choose" exactly half of the numbers off the main diagonal (choose all those above the main diagonal for example and all those below are automatically fixed as their negatives). We can "choose" (n2-n)/2= n(n-1)/2 values.
b) The "commutator"is , by definition, given by
$$[J_{ij},J_{kl}]= J_{ij}J_{kl}-J{kl}J{ij}$$
Since you are told that $$(J_{ij})_{kl} = -i(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})$$, go ahead a put those into that formula and see what you get!
3. Feb 17, 2004
### Norman
HallsofIvy,
First of all, thankyou so much for responding. I am really not comfortable with Group Theory yet and it is a great relief that my intuition about part a) was correct.
For part b) I am a little confused still. I only know the $kl$ components of the matrix. How do I write $J_{ij}$ and $J_{kl}$ in a form in which I can just plug them into the commutator?
Thanks again for the help.
Norm
4. Feb 18, 2004
### Norman
Help... still stuck.
5. Feb 18, 2004
### Norman
Does anyone think that this should actually be:
$$[J_{ij},J_{kl}]= -i (\delta_{jk}J_{il}-\delta_{ik}J_{jl}-\delta_{jl}J_{ik}+\delta_{il}J_{ik})$$
???????
Any help would really be appreciated.
Thanks.
Last edited: Feb 18, 2004
6. Feb 18, 2004
### NateTG
Re: Re: Group Theory
That's certainly looks better since it's symetric. I'm not quite following the notation though, so I can't give you a stronger answer.
7. Feb 18, 2004
### Norman
Re: Re: Re: Group Theory
The way I was told to think about it is that:
$$(J_{ij})_{kl} = -i(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})$$
is the $kl^{th}$ component of the matrix $J_{ij}$ so all you do is sum over k and l for matrix multiplication. But I have no clue if that is correct or not and if I am understanding this at all. It is very frustrating.
Cheers,
Norman
8. Feb 18, 2004
### NateTG
OK, that makes a little bit more sense.
From group theory we have that
$$[ab]=b^{-1}a^{-1}ba$$
You may be able to grind it out from there by figuring out what the inverse of $$J_{il}$$ looks like.
9. Feb 19, 2004
### Norman
is:
$$[ab]=b^{-1}a^{-1}ba$$
the commutator or just multiplication?
Thanks,
Norman
10. Feb 19, 2004
### NateTG
$$[ab]$$ is shorthand for the commutator of $$a$$ and $$b$$. The RHS of that equation is a general expression for the commutator. If you multiply $$ab$$ by it, you get $$ba$$ so it commutes them.
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2018-08-14 13:59:28
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https://brilliant.org/discussions/thread/00-and-n0-where-n-is-not-equal-to-zero/
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×
# 0/0 and n/0 where n is not equal to zero
Today while arbitrarily browsing through some notes posted by some of the mathematics enthusiasts, I came across the following question:"What is 0/0?"
Well, division by zero is region of great obscurity in mathematics, especially for Mathematics students. Many physics books declare that division by zero yields infinity, while many mathematics books write that something divided by zero is undefined. Students just stow their brains with these terms and notions, without experiencing their true flavors. Well, in this note, I will try to explain the meaning of division by zero through a very simple discussion.
Well, what is division? What to you mean by dividing a number 'a' by a number 'b'? It means simply subtracting 'b' from 'a' repeatedly until you get either a 0, or a positive number which is smaller than 'b'. The number of times you perform this process is your quotient, and the resulting non-negative number is your remainder. Let us divide 15 by 5 using this definition. We see that 15-5=10, and 10-5=5, and 5-5=0,and we stop the process. Clearly, the number of subtractions performed here is 3, which is our quotient here and 0 is our remainder here. Similarly let us divide, say, 16 by 3. I am writing directly: 16-3-3-3-3-3=1, where 0<1<3. Thus here quotient is 5 and remainder is 1. These results exactly tally with the results which we can arrive at, using the well known long division process. It has to be so, as long division is essentially the repeated subtraction in disguise. This can be proved easily using the Euclid's division Lemma (well, I am discussion the same Lemma in a very casual fashion, without going into the mathematical rigor). Now let us try to divide a non zero number, say, 6, by 0. We see, 6-0=6, 6-0=6, 6-0=6..ad infinitum. Clearly, this process will never end and so we will never get a quotient, neither will we get a remainder. This is because, after every subtraction what remains is 6 itself. So, such a non-zero number divided by zero will simply give us no value, and hence is meaningless, or in mathematical terminology, "undefined". However, in case of 0/0, we see, in the first step only,0-0 gives 0. Thus we have to perform subtraction only once to get a zero as our remainder. But does that mean quotient is 1? We see that, if subtract 0 from the 0 obtained as a remainder in the very first step once again, we will again get a 0. So that would mean that the quotient is 2. Using the same logic, one can argue that the quotient is 3, or, 4, or 5,or.....
Well, that's the fundamental difference between non-zero/zero and zero/zero. While the first one gives no value, the second one has no FIXED value. That is why we call the first one UNDEFINED (i.e., meaningless), while the second one is called INDETERMINATE (i.e.,lacking a fixed value). And oh, yes, one more thing, why do physicists declare non-zero/zero as infinity? Simply because what they call 0 in the denominator is actually a "vanishingly small quantity of infinitesimal magnitude". Suppose we divide 6 by such a small quantity. Then at every step, 6 will be decreased by an infinitely small quantity, and after infinitely many such subtractions, 6 will finally reduce to 0. Hence the number of subtractions performed being infinity, non-zero divided by infinitesimal will give us an infinity. Now see how easy everything becomes after an explanation.
"Omne Ignotum Pro Magnifico!"
Word of Caution: Do not write 6/0=undefined, or 0/0=indeterminate. We use the "is equal to" symbol only when both the right and the left hand sides have real values. But "undefined" is not a real value. It is only another way of saying "I don't know what that means". So don't use "=" sign here.
Note by Kuldeep Guha Mazumder
1 year, 3 months ago
Sort by:
A little more rigor:
The division theorem says that all numbers can be represented uniquely as $n=qx+r$ for integers $$q, x$$ and $$0\le r\le q-1$$.
In the case of $$0/0$$, we have $$n=x=0$$. Then $$r=0$$ so we have no remainder. But what is $$q$$? Indeed, we can place any value of $$q$$ in the expression, real or complex, and get a valid equation. Since a number clearly cannot take an infinite number of values, $$0/0$$ is considered indeterminate. · 1 year, 3 months ago
Daniel I am afraid you need to correct the statement of division lemma. Apropos of the equation that you have written, I would like to mention that given an integer n (the dividend) and a positive integer x (the divisor) the lemma ensures the existence of UNIQUE integers q (the quotient) and r (the remainder) where r is non-negative and less than x. But your approach is worthy of appreciation. However the uniqueness is redundant to mention as soon as we impose the restriction that r is non-negative and less than x. We can then prove the uniqueness. · 1 year, 3 months ago
As far as I'm able to tell, there's little distinction between "infinity", and "arbitrarily large value" in physics. I don't see any rigor applied to "infinity" in physics, nor see where it ever matters if that distinction is made. · 1 year, 3 months ago
You are absolutely correct Sir. In Physics they make no distinction between infinity and a considerably large value. But in mathematics they do and this post is indeed a humble attempt to point out that subtle difference between the two disciplines. · 1 year, 3 months ago
I should add that as a practical matter, math software makes a distinction between "indeterminate", "undefined", "directed infinity", and "complex infinity", and your ordinary "infinity". But you said that "many physics books declare that division by zero yields infinity", you're really referring to elementary physics texts, which I agree are misguided. Papers in theoretical physics don't make this declaration--they are usually careful to note the distinctions between different kinds of infinities and indeterminacies. · 1 year, 3 months ago
And as for the concept of infinity, we have our subject of surreal numbers, which addresses infinity in a totally different way. And, yes, we have our complex infinity, directed infinity, etc., as you have mentioned. We have our infinite-dimensional complex spaces like the Hilbert space, we have our Kuiper's theorem on contractibility of infinite spheres. Infinity is used innumerable times in innumerable areas of Mathematics with varied notions. Infinity is indeed a gray area of mathematics. But the infinity that I have mentioned in my note is nothing of that kind. It simply means "bigger than the biggest one can imagine", the basic notion of infinity that a school-level student should have. · 1 year, 3 months ago
I believe the main point you were making is the distinction between n/0 and 0/0, which are "undefined" and "undeterminate", which I don't think is a point of dispute. Just the same, some math software return "complex infinity" for n/0.
Meanwhile, somewhere buried in my pile of books, I have the book "Surreal Numbers" by Donald Knuth. I can only dimly remember the endless labyrinthine details about such numbers, which, as I understand, still baffles some mathematicians today. · 1 year, 3 months ago
Yes yes..that was the central idea of my note..and yes, the study of surreal numbers is a relatively novel and labyrinthine one..I only have a basic idea about the subject..and the most catchy term that I have encountered in the subject is 'birthday of a surreal number'. · 1 year, 3 months ago
Certainly Sir. I am only talking of the school level textbooks that confuse the students. I have devoted this note to school level students who have certain obscurities regarding the subject of division by zero, owing to such erroneous declarations in such school level textbooks. · 1 year, 3 months ago
And on this context I allude to this nice piece of humor that I found on the internet:
An engineer, a physicist, and a mathematician are shown a pasture with a herd of sheep, and told to put them inside the smallest possible amount of fence. The engineer is first. He herds the sheep into a circle and then puts the fence around them, declaring, that a circle will use the least fence for a given area, so this is the best solution. The physicist is next. He creates a circular fence of infinite radius around the sheep, and then draws the fence tight around the herd, declaring, this will give the smallest circular fence around the herd. The mathematician is last. After giving the problem a little thought, he puts a small fence around himself and then declares,"I define myself to be on the outside." · 1 year, 3 months ago
This infinity, undefined and indeterminate really confused me. Thanks for the post! · 1 year, 3 months ago
The note is a bit mundane, I admit it.. · 1 year, 3 months ago
No, it's really interesting:) · 1 year, 3 months ago
The "best of mine" part of the problems posted by you are good. But I found them easy and solved four of them even in my most somnolent state (well, I feel extremely sleepy now, it is 2.10 am under the clock) and will solve the rest tomorrow. · 1 year, 3 months ago
Thanks for checking my set. It's been a long time since I edited it, anyway thank you for your comments! · 1 year, 3 months ago
Welcome..actually today I am down with fever and so I couldn't check your other problems, but I will surely solve them by tomorrow.. · 1 year, 3 months ago
Thank you..you know I love to stimulate the minds of people with seemingly trivial but difficult-to-answer problems like this one..that is why I am here. You can check my other posts also.. · 1 year, 3 months ago
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2016-10-22 16:19:34
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https://blog.engrxiv.org/2019/12/end-of-the-year-update-2019
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The year 2019 is drawing to a close and it has been a wonderful year for the Engineering Archive! This was another record breaking year in terms of submissions to the server. We ended the year at 411 submission in 2019, which (as it did last year) surpasses the total submission from 2016, 2017, and 2018 combined.
We’ve also seen a continuing trend from last year in that the submission rate has steadily increased as the year has progressed, showing promise for 2020.
Finally, preprints hosted on engineering archive recently surpassed 300,000 downloads (301,492 downloads at the time of writing to be exact)! We think that this shows the power of open access scholarship. If every one of those downloads represented the purchase of a PDF from a paywalled article, that would be a savings of at least $7,500,000 for the engineering community. With this growth, Engineering Archive faces the ever present challenge of sustainability of the operation (we don’t intend to ever sell out to a private entity). With that in mind, we took two important steps financial sustainability this year. 1) We have formalized the relationship between the Engineering Archive and Open Engineering Inc., a 501(c)(3) nonprofit organization for the promotion of open practices in engineering in all forms. Engineering Archive is a program of Open Engineering Inc. under which Open Engineering operates as the fundraising body to help meet the financial needs of the server, individual donations are welcome! 2) We launched the Engineering Archive Membership Circle. The Membership Circle creates the opportunity for institutions, libraries, and other organizations to support the sustainability of the server through a$500 annual contribution. Since launching in September, 10 academic libraries (listed below) have signed up to pledge their support. We are so grateful for their early contributions which will help us meet our financial obligations for the coming year. We are still looking for additional members, so please reach out to your home institutions to discuss the Membership Circle with them and encourage them to join. Interested individuals can reach out to info@engrxiv.org to learn more.
The first 10 institutional members of the Engineering Archive Membership Circle:
Again, THANK YOU SO MUCH to those who have already joined us in creating a sustainable future for open access engineering scholarship and to those of you planning to lend your support soon!
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2020-01-22 03:29:40
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https://www.zbmath.org/authors/?q=ai%3Awang.cangyuan
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# zbMATH — the first resource for mathematics
## Wang, Cangyuan
Compute Distance To:
Author ID: wang.cangyuan Published as: Wang, C.; Wang, C. Y.; Wang, C.-Y.; Wang, Cangyuan
Documents Indexed: 100 Publications since 1978
#### Co-Authors
1 single-authored 1 Wei, Changguo
#### Serials
1 Journal of Mathematical Analysis and Applications 1 Periodical of Ocean University of China
#### Fields
2 Functional analysis (46-XX)
#### Citations contained in zbMATH
62 Publications have been cited 451 times in 404 Documents Cited by Year
The Euler equation and absolute minimizers of $$L^\infty$$ functionals. Zbl 0979.49003
Barron, E. N.; Jensen, R. R.; Wang, C. Y.
2001
Experimental and numerical investigation of the dynamics of an underwater explosion bubble near a resilient/rigid structure. Zbl 1138.76303
Klaseboer, E.; Hung, K. C.; Wang, C.; Wang, C. W.; Khoo, B. C.; Boyce, P.; Debono, S.; Charlier, H.
2005
Failure time regression with continuous covariates measured with error. Zbl 0957.62035
Zhou, Haibo; Wang, C.-Y.
2000
Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic duffing oscillators. Zbl 1168.34321
Lai, S. K.; Lim, C. W.; Wu, B. S.; Wang, C.; Zeng, Q. C.; He, X. F.
2009
Lower semicontinuity of $$L^\infty$$ functionals. Zbl 1034.49008
Barron, E. N.; Jensen, R. R.; Wang, C. Y.
2001
3-D and 2-D dynamic Green’s functions and time-domain bies for piezoelectric solids. Zbl 1182.74053
Wang, C.-Y.; Zhang, Ch.
2005
Augmented inverse probability weighted estimator for Cox missing covariate regression. Zbl 1209.62225
Wang, C. Y.; Chen, Hua Yun
2001
Axially compressed buckling of pressured multiwall carbon nanotubes. Zbl 1038.74548
Wang, C. Y.; Ru, C. Q.; Mioduchowski, A.
2003
Elastic fields produced by a point source in solids of general anisotropy. Zbl 0907.73008
Wang, C.-Y.
1997
3D BEM for general anisotropic elasticity. Zbl 1166.74434
Wang, C.-Y.; Denda, M.
2007
Expected estimating equations to accomodate covariate measurement error. Zbl 0963.62065
Wang, C.-Y.; Pepe, Margaret Sullivan
2000
Integer programming solution of a classification problem. Zbl 0491.90056
Liittschwager, J. M.; Wang, C.
1978
Applicability and limitations of simplified elastic shell equations for carbon nanotubes. Zbl 1111.74689
Wang, C. Y.; Ru, C. Q.; Mioduchowski, A.
2004
Corrected score estimator for joint modeling of longitudinal and failure time data. Zbl 1087.62112
Wang, C. Y.
2006
Linear elasticity of planar Delaunay networks: Random field characterization of effective moduli. Zbl 0711.73275
Ostoja-Starzewski, M.; Wang, C.
1989
Efficient computation of the Green’s function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis. Zbl 1351.74134
Shiah, Y. C.; Tan, C. L.; Wang, C. Y.
2012
Effects of measurement error and conditional score estimation in capture-recapture models. Zbl 1145.62094
Hwang, Wen-Han; Huang, Y. H.; Wang, C. Y.
2007
Analysis of quantum-dot-induced strain and electric fields in piezoelectric semiconductors of general anisotropy. Zbl 1120.74470
Wang, C.-Y.; Denda, M.; Pan, E.
2006
Adaptive control under arbitrary switching for a class of switched nonlinear systems with nonlinear parameterisation. Zbl 1334.93099
Wang, C. Y.; Jiao, X. H.
2015
3D BEM for the general piezoelectric solids. Zbl 1229.74143
Denda, M.; Wang, C.-Y.
2009
First order decision diagrams for relational MDPS. Zbl 1182.68271
Wang, C.; Joshi, S.; Khardon, R.
2008
Non-parametric maximum likelihood estimation for Cox regression with subject-specific measurement errors. Zbl 1198.62141
Wang, C. Y.
2008
Scattering of spinning test particles by plane gravitational and electromagnetic waves. Zbl 1012.83004
Kessari, S.; Singh, D.; Tucker, R. W.; Wang, C.
2002
Linear elasticity of planar Delaunay networks. II: Voigt and Reuss bounds, and modification for centroids. Zbl 0719.73047
Ostoja-Starzewski, M.; Wang, C.
1990
Difference method for generation of circular arcs and ellipses. Zbl 0667.65007
Wang, W. P.; Wang, C. Y.
1989
Application of a modified Lindstedt-Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation. Zbl 1168.70302
Lim, C. W.; Lai, S. K.; Wu, B. S.; Sun, W. P.; Yang, Y.; Wang, C.
2009
Analytical approximate solutions to oscillation of a current-carrying wire in a magnetic field. Zbl 1162.34323
Sun, W. P.; Lim, C. W.; Wu, B. S.; Wang, C.
2009
The dynamic stress intensity factor for a semi-infinite crack in orthotropic materials with concentrated shear impact loads. Zbl 0967.74030
Wang, C. Y.; Rubio-Gonzalez, C.; Mason, J. J.
2001
Flexible regression calibration for covariate measurement error with longitudinal surrogate variables. Zbl 0952.62063
Wang, C. Y.
2000
Robust sandwich covariance estimation for regression calibration estimator in Cox regression with measurement error. Zbl 1070.62519
Wang, C. Y.
1999
The stress field of a dislocation loop in an anisotropic solid. Zbl 1054.74502
Wang, C.-Y.
1996
Alternative covariance estimates in a replicated measurement error model with correlated, heteroscedastic errors. Zbl 0800.62378
Wang, C. Y.
1993
On stabilizability and sampling for infinite dimensional systems. Zbl 0770.93080
Rosen, J. G.; Wang, C.
1992
Terahertz radiation induced chaotic electron transport in semiconductor superlattices with a tilted magnetic field. Zbl 1374.82042
Wang, C.; Wang, F.; Cao, J. C.
2014
Axisymmetric vibration of single-walled carbon nanotubes in water. Zbl 1248.82109
Wang, C. Y.; Li, C. F.; Adhikari, S.
2010
Two-component nonlinear Schrödinger models with a double-well potential. Zbl 1153.82304
Wang, C.; Kevrekidis, P. G.; Whitaker, N.; Malomed, B. A.
2008
Deflection and stability of membrane structures under electrostatic and Casimir forces in microelectromechanical systems. Zbl 1138.74359
Wang, C.; Guo, W.; Feng, Q.
2005
Shock wave diffraction by a square cavity filled with dusty gas. Zbl 0980.76522
Wang, B. Y.; Wu, Q. S.; Wang, C.; Igra, O.; Falcovitz, J.
2001
Dynamic capacitated user-optimal departure time/route choice problem with time-window. Zbl 1024.90009
Chen, H. K.; Chang, M. S.; Wang, C. Y.
2001
On an interpretation of non-Riemannian gravitation. Zbl 0988.83060
Teyssandier, P.; Tucker, R. W.; Wang, C.
1998
Asymptotic analysis of stabilizability of a control system for a discretized boundary damped wave equation. Zbl 0895.65042
Peichl, G. H.; Wang, C.
1998
Green’s functions and general formalism for 2D piezoelectricity. Zbl 0864.73060
Wang, C.-Y.
1996
Numerical and experimental study of flow over stages of an offset merger dune interaction. Zbl 1390.76905
Wang, C.; Tang, Z.; Bristow, N.; Blois, G.; Christensen, K. T.; Anderson, W.
2017
Nonlinear Schrödinger equations with a four-well potential in two dimensions: bifurcations and stability analysis. Zbl 1225.37084
Wang, C.; Theocharis, G.; Kevrekidis, P. G.; Whitaker, N.; Frantzeskakis, D. J.; Malomed, B. A.
2011
Collisionally inhomogeneous Bose-Einstein condensates in double-well potentials. Zbl 1167.82341
Wang, C.; Kevrekidis, P. G.; Whitaker, N.; Frantzeskakis, D. J.; Middelkamp, S.; Schmelcher, P.
2009
Spinor Bose-Einstein condensates in double-well potentials. Zbl 1161.82004
Wang, C.; Kevrekidis, P. G.; Whitaker, N.; Alexander, T. J.; Frantzeskakis, D. J.; Schmelcher, P.
2009
On the maximum principle and a notion of plurisubharmonicity for abstract CR manifolds. Zbl 1130.32018
Berhanu, S.; Wang, C.
2007
Numerical analysis of the shaped charged jet with large cone angle. Zbl 1401.76099
Ning, Jian-Guo; Wang, Cheng; Ma, Tian-Bao
2006
Assessment of turbulence models for predicting the interaction region in a wall jet by reference to LES solution and budgets. Zbl 1138.76309
Dejoan, A.; Wang, C.; Leschziner, M. A.
2006
Randomly generating triangulations of a simple polygon. Zbl 1128.68518
Ding, Q.; Qian, J.; Tsang, W.; Wang, C.
2005
An explicit analytic solution for non-Darcy natural convection over horizontal plate with surface mass flux and thermal dispersion effects. Zbl 1064.76103
Wang, C.; Liao, S. J.; Zhu, J. M.
2003
The null distribution of sample serial correlation coefficient. Zbl 1030.62009
Yue, S.; Wang, C. Y.
2002
Modelling the vibration behaviour of infinite structures by FEM. Zbl 1235.74321
Wang, C.; Lai, J. C. S.
2000
A Cosserat detector for dynamic geometry. Zbl 1072.74045
Wang, C.; Tucker, R. W.
2000
Weighted normality-based estimator in correcting correlation coefficient estimation between incomplete nutrient measurements. Zbl 1060.62678
Wang, C. Y.
2000
Separation control on a thick airfoil with multiple slots blowing at small speeds. Zbl 0991.76070
Wang, C.; Sun, M.
2000
Characteristic finite analytic method (CFAM) for incompressible Navier-Stokes equations. Zbl 0973.76073
Wang, C.
2000
Validity of one-dimensional experimental principle for flat specimen in bar-bar tensile impact apparatus. Zbl 0955.74506
Wang, C. Y.; Xia, Y. M.
2000
Nonlinear flexural excitations and drill-string dynamics. Zbl 0977.74533
Tucker, R. W.; Tung, R. S.; Wang, C.
1999
An Einstein-Proca-fluid model for dark matter gravitational interactions. Zbl 0976.83526
Tucker, R. W.; Wang, C.
1997
A refined flexible inspection method for identifying surface flaws using the skeleton and neural network. Zbl 0943.90538
Wang, C.; Huang, S.-Z.
1997
On uniform stabilizability and the margin of stabilizability. Zbl 0896.93011
Peichl, G. H.; Wang, C.
1997
Numerical and experimental study of flow over stages of an offset merger dune interaction. Zbl 1390.76905
Wang, C.; Tang, Z.; Bristow, N.; Blois, G.; Christensen, K. T.; Anderson, W.
2017
Adaptive control under arbitrary switching for a class of switched nonlinear systems with nonlinear parameterisation. Zbl 1334.93099
Wang, C. Y.; Jiao, X. H.
2015
Terahertz radiation induced chaotic electron transport in semiconductor superlattices with a tilted magnetic field. Zbl 1374.82042
Wang, C.; Wang, F.; Cao, J. C.
2014
Efficient computation of the Green’s function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis. Zbl 1351.74134
Shiah, Y. C.; Tan, C. L.; Wang, C. Y.
2012
Nonlinear Schrödinger equations with a four-well potential in two dimensions: bifurcations and stability analysis. Zbl 1225.37084
Wang, C.; Theocharis, G.; Kevrekidis, P. G.; Whitaker, N.; Frantzeskakis, D. J.; Malomed, B. A.
2011
Axisymmetric vibration of single-walled carbon nanotubes in water. Zbl 1248.82109
Wang, C. Y.; Li, C. F.; Adhikari, S.
2010
Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic duffing oscillators. Zbl 1168.34321
Lai, S. K.; Lim, C. W.; Wu, B. S.; Wang, C.; Zeng, Q. C.; He, X. F.
2009
3D BEM for the general piezoelectric solids. Zbl 1229.74143
Denda, M.; Wang, C.-Y.
2009
Application of a modified Lindstedt-Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation. Zbl 1168.70302
Lim, C. W.; Lai, S. K.; Wu, B. S.; Sun, W. P.; Yang, Y.; Wang, C.
2009
Analytical approximate solutions to oscillation of a current-carrying wire in a magnetic field. Zbl 1162.34323
Sun, W. P.; Lim, C. W.; Wu, B. S.; Wang, C.
2009
Collisionally inhomogeneous Bose-Einstein condensates in double-well potentials. Zbl 1167.82341
Wang, C.; Kevrekidis, P. G.; Whitaker, N.; Frantzeskakis, D. J.; Middelkamp, S.; Schmelcher, P.
2009
Spinor Bose-Einstein condensates in double-well potentials. Zbl 1161.82004
Wang, C.; Kevrekidis, P. G.; Whitaker, N.; Alexander, T. J.; Frantzeskakis, D. J.; Schmelcher, P.
2009
First order decision diagrams for relational MDPS. Zbl 1182.68271
Wang, C.; Joshi, S.; Khardon, R.
2008
Non-parametric maximum likelihood estimation for Cox regression with subject-specific measurement errors. Zbl 1198.62141
Wang, C. Y.
2008
Two-component nonlinear Schrödinger models with a double-well potential. Zbl 1153.82304
Wang, C.; Kevrekidis, P. G.; Whitaker, N.; Malomed, B. A.
2008
3D BEM for general anisotropic elasticity. Zbl 1166.74434
Wang, C.-Y.; Denda, M.
2007
Effects of measurement error and conditional score estimation in capture-recapture models. Zbl 1145.62094
Hwang, Wen-Han; Huang, Y. H.; Wang, C. Y.
2007
On the maximum principle and a notion of plurisubharmonicity for abstract CR manifolds. Zbl 1130.32018
Berhanu, S.; Wang, C.
2007
Corrected score estimator for joint modeling of longitudinal and failure time data. Zbl 1087.62112
Wang, C. Y.
2006
Analysis of quantum-dot-induced strain and electric fields in piezoelectric semiconductors of general anisotropy. Zbl 1120.74470
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2006
Numerical analysis of the shaped charged jet with large cone angle. Zbl 1401.76099
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2006
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2006
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2005
3-D and 2-D dynamic Green’s functions and time-domain bies for piezoelectric solids. Zbl 1182.74053
Wang, C.-Y.; Zhang, Ch.
2005
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Wang, C.; Guo, W.; Feng, Q.
2005
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2005
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2004
Axially compressed buckling of pressured multiwall carbon nanotubes. Zbl 1038.74548
Wang, C. Y.; Ru, C. Q.; Mioduchowski, A.
2003
An explicit analytic solution for non-Darcy natural convection over horizontal plate with surface mass flux and thermal dispersion effects. Zbl 1064.76103
Wang, C.; Liao, S. J.; Zhu, J. M.
2003
Scattering of spinning test particles by plane gravitational and electromagnetic waves. Zbl 1012.83004
Kessari, S.; Singh, D.; Tucker, R. W.; Wang, C.
2002
The null distribution of sample serial correlation coefficient. Zbl 1030.62009
Yue, S.; Wang, C. Y.
2002
The Euler equation and absolute minimizers of $$L^\infty$$ functionals. Zbl 0979.49003
Barron, E. N.; Jensen, R. R.; Wang, C. Y.
2001
Lower semicontinuity of $$L^\infty$$ functionals. Zbl 1034.49008
Barron, E. N.; Jensen, R. R.; Wang, C. Y.
2001
Augmented inverse probability weighted estimator for Cox missing covariate regression. Zbl 1209.62225
Wang, C. Y.; Chen, Hua Yun
2001
The dynamic stress intensity factor for a semi-infinite crack in orthotropic materials with concentrated shear impact loads. Zbl 0967.74030
Wang, C. Y.; Rubio-Gonzalez, C.; Mason, J. J.
2001
Shock wave diffraction by a square cavity filled with dusty gas. Zbl 0980.76522
Wang, B. Y.; Wu, Q. S.; Wang, C.; Igra, O.; Falcovitz, J.
2001
Dynamic capacitated user-optimal departure time/route choice problem with time-window. Zbl 1024.90009
Chen, H. K.; Chang, M. S.; Wang, C. Y.
2001
Failure time regression with continuous covariates measured with error. Zbl 0957.62035
Zhou, Haibo; Wang, C.-Y.
2000
Expected estimating equations to accomodate covariate measurement error. Zbl 0963.62065
Wang, C.-Y.; Pepe, Margaret Sullivan
2000
Flexible regression calibration for covariate measurement error with longitudinal surrogate variables. Zbl 0952.62063
Wang, C. Y.
2000
Modelling the vibration behaviour of infinite structures by FEM. Zbl 1235.74321
Wang, C.; Lai, J. C. S.
2000
A Cosserat detector for dynamic geometry. Zbl 1072.74045
Wang, C.; Tucker, R. W.
2000
Weighted normality-based estimator in correcting correlation coefficient estimation between incomplete nutrient measurements. Zbl 1060.62678
Wang, C. Y.
2000
Separation control on a thick airfoil with multiple slots blowing at small speeds. Zbl 0991.76070
Wang, C.; Sun, M.
2000
Characteristic finite analytic method (CFAM) for incompressible Navier-Stokes equations. Zbl 0973.76073
Wang, C.
2000
Validity of one-dimensional experimental principle for flat specimen in bar-bar tensile impact apparatus. Zbl 0955.74506
Wang, C. Y.; Xia, Y. M.
2000
Robust sandwich covariance estimation for regression calibration estimator in Cox regression with measurement error. Zbl 1070.62519
Wang, C. Y.
1999
Nonlinear flexural excitations and drill-string dynamics. Zbl 0977.74533
Tucker, R. W.; Tung, R. S.; Wang, C.
1999
On an interpretation of non-Riemannian gravitation. Zbl 0988.83060
Teyssandier, P.; Tucker, R. W.; Wang, C.
1998
Asymptotic analysis of stabilizability of a control system for a discretized boundary damped wave equation. Zbl 0895.65042
Peichl, G. H.; Wang, C.
1998
Elastic fields produced by a point source in solids of general anisotropy. Zbl 0907.73008
Wang, C.-Y.
1997
An Einstein-Proca-fluid model for dark matter gravitational interactions. Zbl 0976.83526
Tucker, R. W.; Wang, C.
1997
A refined flexible inspection method for identifying surface flaws using the skeleton and neural network. Zbl 0943.90538
Wang, C.; Huang, S.-Z.
1997
On uniform stabilizability and the margin of stabilizability. Zbl 0896.93011
Peichl, G. H.; Wang, C.
1997
The stress field of a dislocation loop in an anisotropic solid. Zbl 1054.74502
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1996
Green’s functions and general formalism for 2D piezoelectricity. Zbl 0864.73060
Wang, C.-Y.
1996
Alternative covariance estimates in a replicated measurement error model with correlated, heteroscedastic errors. Zbl 0800.62378
Wang, C. Y.
1993
On stabilizability and sampling for infinite dimensional systems. Zbl 0770.93080
Rosen, J. G.; Wang, C.
1992
Linear elasticity of planar Delaunay networks. II: Voigt and Reuss bounds, and modification for centroids. Zbl 0719.73047
Ostoja-Starzewski, M.; Wang, C.
1990
Linear elasticity of planar Delaunay networks: Random field characterization of effective moduli. Zbl 0711.73275
Ostoja-Starzewski, M.; Wang, C.
1989
Difference method for generation of circular arcs and ellipses. Zbl 0667.65007
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1989
Integer programming solution of a classification problem. Zbl 0491.90056
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1978
all top 5
#### Cited by 705 Authors
16 Katzourakis, Nikolaos I. 16 Zhang, Chuanzeng 13 García-Sánchez, Felipe 12 Saez, Andres 10 Guo, Zhongjin 10 Prinari, Francesca 9 Wang, Changyou 8 Yu, Yifeng 8 Zhang, Aman 8 Zhou, Haibo 7 Hwang, Wen-Han 7 Liu, Yanyan 7 Wang, Ching-Yun 6 Dineva, Petia S. 6 Leung, Andrew Yee-Tak 6 Rangelov, Tsviatko V. 6 Sun, Liuquan 6 Wang, Qianxi 6 Wünsche, Michael 5 Gross, Dietmar 5 Huang, Yih-Huei 5 Khoo, Boo Cheong 5 Klaseboer, Evert 5 Ostoja-Starzewski, Martin 5 Sladek, Jan 5 Sladek, Vladimir 5 Sun, Yanqing 5 Yang, Hongxiang 5 Zhou, Yuan 4 Barron, Emmanuel Nicholas 4 Beléndez, Augusto 4 Cai, Jianwen 4 Crandall, Michael G. 4 Huggins, Richard M. 4 Liu, Kaixin 4 Pan, Ernian 4 Shiah, Yui- Chuin 4 Song, Xinyuan 4 Sun, Chengqi 4 Wang, Shiping 3 Álvarez, Mariela L. 3 Ang, Wei Tech 3 Arribas, Enrique 3 Athanasius, L. 3 Beléndez, Tarsicio 3 Buroni, Federico C. 3 Feng, Yanqin 3 Gilbert, Peter B. 3 Lai, Siu Kai 3 Lei, Jun 3 Li, Zhangrui 3 Lu, Guozhen 3 Marczak, Rogério José 3 Miao, Qianyun 3 Muller, Ralf 3 Song, Xiao 3 Stoklosa, Jakub 3 Sun, Jianguo 3 Sun, Lei 3 Wang, Qihua 3 Wang, Xiaofei 3 Wen, Chi-Chung 3 Yi, Grace Yun 3 Yuan, Zhongshang 3 Zappale, Elvira 3 Zhang, Wei 3 Zong, Zhi 2 Albuquerque, Éder Lima 2 Armstrong, Scott N. 2 Aronsson, Gunnar 2 Asghari, Mehran 2 Baldi, Simone 2 Beyerlein, Irene J. 2 Blake, John Robert 2 Calvisi, Michael L. 2 Capogna, Luca 2 Chen, Fei-Yin 2 Chen, Jianwei 2 Chen, Yurong 2 Crasta, Graziano 2 Cui, Pu 2 Dadvand, Abdolrahman 2 Dai, Pengjie 2 Danesh, V. 2 Denda, Mitsunori 2 Dong, Jing 2 Fragalà, Ilaria 2 Francés, Jorge 2 Galvis, Andres F. 2 Garroni, Adriana 2 Gori, Michele 2 Gray, Leonard J. 2 Han, Qiang 2 He, Wenqing 2 Hirose, Sohichi 2 Huang, Yijian 2 Hwu, Chyanbin 2 Juutinen, Petri 2 Kaplan, T. A. 2 Khardon, Roni ...and 605 more Authors
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#### Cited in 132 Serials
31 Engineering Analysis with Boundary Elements 23 Biometrics 19 Applied Mathematical Modelling 13 Acta Mechanica 12 Computer Methods in Applied Mechanics and Engineering 11 Journal of Fluid Mechanics 11 European Journal of Mechanics. A. Solids 9 International Journal of Solids and Structures 8 Applied Mathematics and Computation 8 European Journal of Operational Research 7 Computational Statistics and Data Analysis 6 Archive for Rational Mechanics and Analysis 6 Computers and Fluids 6 Scandinavian Journal of Statistics 6 Journal of Multivariate Analysis 6 Transactions of the American Mathematical Society 6 Communications in Statistics. Theory and Methods 6 Lifetime Data Analysis 5 Computers & Mathematics with Applications 5 Journal of Differential Equations 5 Calculus of Variations and Partial Differential Equations 4 General Relativity and Gravitation 4 Journal of Statistical Planning and Inference 4 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 4 Physica D 4 Computers & Operations Research 4 Computational Mechanics 4 Mathematical and Computer Modelling 4 Journal of Statistical Computation and Simulation 4 Archive of Applied Mechanics 4 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 4 Physics of Fluids 4 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 3 Artificial Intelligence 3 The Canadian Journal of Statistics 3 Journal of Computational Physics 3 Metrika 3 ZAMP. Zeitschrift für angewandte Mathematik und Physik 3 Applied Mathematics and Optimization 3 Proceedings of the American Mathematical Society 3 Communications in Partial Differential Equations 3 Journal of Systems Science and Complexity 3 Acta Mechanica Sinica 3 Nonlinear Analysis. Hybrid Systems 2 Journal of the Mechanics and Physics of Solids 2 Shock Waves 2 Advances in Mathematics 2 Annali di Matematica Pura ed Applicata. Serie Quarta 2 The Annals of Statistics 2 Biometrical Journal 2 International Journal for Numerical Methods in Engineering 2 Meccanica 2 Applied Mathematics and Mechanics. (English Edition) 2 Machine Learning 2 Communications in Statistics. Simulation and Computation 2 SIAM Journal on Mathematical Analysis 2 Statistical Papers 2 NoDEA. Nonlinear Differential Equations and Applications 2 Mathematical Problems in Engineering 2 Journal of Vibration and Control 2 Mathematics and Mechanics of Solids 2 Nonlinear Dynamics 2 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 2 Discrete Dynamics in Nature and Society 2 Communications in Nonlinear Science and Numerical Simulation 2 Advances in Calculus of Variations 2 Advances in Mathematical Physics 1 Fluid Dynamics 1 International Journal of Control 1 International Journal of Engineering Science 1 International Journal of Heat and Mass Transfer 1 International Journal of Plasticity 1 International Journal of Theoretical Physics 1 Journal of Mathematical Analysis and Applications 1 Mathematics of Computation 1 Journal of Theoretical and Applied Mechanics (Sofia) 1 Annals of the Institute of Statistical Mathematics 1 Illinois Journal of Mathematics 1 International Statistical Review 1 Journal of the American Statistical Association 1 Journal of Functional Analysis 1 Journal of Optimization Theory and Applications 1 Kybernetika 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Quarterly of Applied Mathematics 1 SIAM Journal on Control and Optimization 1 Studies in Applied Mathematics 1 Systems & Control Letters 1 Statistics & Probability Letters 1 Discrete & Computational Geometry 1 Applied Mathematics Letters 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 Numerical Algorithms 1 Computational Statistics 1 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1 Bulletin of the American Mathematical Society. New Series 1 International Journal of Modern Physics D 1 Communications in Numerical Methods in Engineering 1 International Applied Mechanics 1 Electronic Journal of Differential Equations (EJDE) ...and 32 more Serials
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#### Cited in 40 Fields
125 Mechanics of deformable solids (74-XX) 99 Statistics (62-XX) 64 Numerical analysis (65-XX) 61 Partial differential equations (35-XX) 50 Fluid mechanics (76-XX) 43 Calculus of variations and optimal control; optimization (49-XX) 26 Ordinary differential equations (34-XX) 18 Operations research, mathematical programming (90-XX) 14 Biology and other natural sciences (92-XX) 13 Systems theory; control (93-XX) 11 Computer science (68-XX) 9 Operator theory (47-XX) 9 Statistical mechanics, structure of matter (82-XX) 7 Dynamical systems and ergodic theory (37-XX) 7 Quantum theory (81-XX) 6 Real functions (26-XX) 6 Mechanics of particles and systems (70-XX) 6 Relativity and gravitational theory (83-XX) 5 Optics, electromagnetic theory (78-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Combinatorics (05-XX) 3 Potential theory (31-XX) 2 Functions of a complex variable (30-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Differential geometry (53-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Probability theory and stochastic processes (60-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Information and communication theory, circuits (94-XX) 1 General and overarching topics; collections (00-XX) 1 Mathematical logic and foundations (03-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Measure and integration (28-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Special functions (33-XX) 1 Difference and functional equations (39-XX) 1 Approximations and expansions (41-XX) 1 Integral transforms, operational calculus (44-XX) 1 Geometry (51-XX) 1 Convex and discrete geometry (52-XX)
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2021-04-21 10:49:38
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http://hal.in2p3.fr/in2p3-00750107
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# Measurement of isolated-photon pair production in pp collisions at sqrt(s) = 7 TeV with the ATLAS detector
6 Laboratoire de Physique Corpusculaire
LPC - Laboratoire de Physique Corpusculaire [Clermont-Ferrand]
7 APC - Neutrinos
LPNHE - Laboratoire de Physique Nucléaire et de Hautes Énergies, APC - UMR 7164 - AstroParticule et Cosmologie
Abstract : The ATLAS experiment at the LHC has measured the production cross section of events with two isolated photons in the final state, in proton-proton collisions at sqrt(s) = 7 TeV. The full data set collected in 2011, corresponding to an integrated luminosity of 4.9 fb-1, is used. The amount of background, from hadronic jets and isolated electrons, is estimated with data-driven techniques and subtracted. The total cross section, for two isolated photons with transverse energies above 25 GeV and 22 GeV respectively, in the acceptance of the electromagnetic calorimeter (|eta|<1.37 and 1.52<|eta|<2.37) and with an angular separation Delta R>0.4, is 44.0 (+3.2) (-4.2) pb. The differential cross sections as a function of the di-photon invariant mass, transverse momentum, azimuthal separation, and cosine of the polar angle of the largest transverse energy photon in the Collins--Soper di-photon rest frame are also measured. The results are compared to the prediction of leading-order parton-shower and next-to-leading-order and next-to-next-to-leading-order parton-level generators.
Document type :
Journal articles
Journal of High Energy Physics, Springer, 2013, 1, pp.086. <10.1007/JHEP01(2013)086>
http://hal.in2p3.fr/in2p3-00750107
Contributor : Emmanuelle Vernay <>
Submitted on : Friday, November 9, 2012 - 8:11:46 AM
Last modification on : Tuesday, June 30, 2015 - 10:57:19 AM
### Citation
G. Aad, S. Albrand, M.L. Andrieux, Q. Buat, B. Clement, et al.. Measurement of isolated-photon pair production in pp collisions at sqrt(s) = 7 TeV with the ATLAS detector. Journal of High Energy Physics, Springer, 2013, 1, pp.086. <10.1007/JHEP01(2013)086>. <in2p3-00750107>
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2015-09-01 06:02:53
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https://www.etsy.com/listing/97984153/small-gold-teardrop-earrings-simple?ref=pr_shop
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2017-06-29 03:21:58
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https://byjus.com/physics/escape-speed/
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Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :
# Escape Speed
Escape speed is the minimum speed required to escape a planet’s gravitational pull.
A spacecraft leaving the earth’s surface should be going at a speed of about 11 kilometres (7 miles) per second to enter the outer orbit. Here, in this article let us dig deeper into the concept of escape speed.
## What is Escape Speed?
Escape speed is the minimum speed with which a mass should be projected from the Earth’s surface in order to escape Earth’s gravitation field. Escape speed, also known as escape velocity is defined as:
The minimum speed that is required for an object to free itself from the gravitational force exerted by a massive object.
For example, if we consider earth as a massive body. The escape velocity is the minimum velocity that an object should acquire to overcome the gravitational field of earth and fly to infinity without ever falling back.
It purely depends on the distance of the object from the massive body and the mass of the massive body. More the mass it will be higher, similarly, the closer distance, higher will be the escape velocity.
For any massive bodies such as planets, stars which are spherically symmetric in nature, the escape speed for any given distance is mathematically expressed as:
$$\begin{array}{l}v_{e}=\sqrt{\frac{2GM}{r}}\end{array}$$
Where,
• ve is the escape speed
• G is the universal gravitational constant (G≅6.67×10-11 m3kg-1s-2)
• M is the mass of the massive body(the body from which the object is to be escaped from)
• r is the distance from the centre of the massive body to the object
Here we can notice that the above-mentioned relation is independent of the mass of the object which will be escaping from the massive body.
You may also want to check out these topics given below!
## Derivation of Escape Speed
In general escape, speed is achieved when the object moves with a velocity at which the arithmetic sum of the object’s gravitational potential energy and its Kinetic energy equates to zero. That is, the object should possess greater kinetic energy than the gravitational potential energy to escape to infinity.
• The simplest way of deriving the formula is by using the concept of conservation of energy. Let us assume that the object is trying to escape from a planet (which is uniform circular in nature) by moving away from it.
• The prime force acting on such an object will be the planet’s gravity. As we know, Kinetic energy(K) and the Gravitational Potential Energy(Ug) are the only two kinds of energies associated here.
By the principle of conservation of energy, we can write:
$$\begin{array}{l}\left ( K+U_{g} \right )_{i}=\left ( K+U_{g} \right )_{f}\end{array}$$
Where,
• $$\begin{array}{l}K=\frac{1}{2}mv^{2}\end{array}$$
• $$\begin{array}{l}U=\frac{GMm}{r}\end{array}$$
Here Ugf is zero as the distance is infinity and Kf will also be zero as final velocity will be zero. Thus, we get:
$$\begin{array}{l}\frac{1}{2}mv_{e}^{2}-\frac{GMm}{r}=0+0\end{array}$$
$$\begin{array}{l}\frac{1}{2}mv_{e}^{2}=\frac{GMm}{r}\end{array}$$
$$\begin{array}{l}\Rightarrow v_{e}=\sqrt{\frac{2GM}{r}}\end{array}$$
The minimum velocity required to escape from the gravitational influence of massive body is given by:
$$\begin{array}{l}v_{e}=\sqrt{2gr}\end{array}$$
Where,
$$\begin{array}{l}g=\frac{GM}{r^{2}}\end{array}$$
The escape speed of the earth at the surface is approximately 11.186 km/s. That means “an object should have a minimum of 11.186 km/s initial velocity to escape from earth’s gravity and fly to infinite space.”
Ideally, If you can jump with initial velocity 11.186 km/s you can tour outer space! Isn’t it interesting? For more such brain-twisting concepts do follow the links given below.
## Unit Of Escape Speed
Unit of escape speed or the escape velocity is metre per seconds (m.s-1) which is also the SI unit of escape speed.
### Dimensional Formula:
Dimensional formula of universal gravitational constant = M-1L3T-2 Dimensional formula of the mass of the earth = M1L0T0 Dimensional formula of distance to the centre of the earth = M0L1T0 Therefore, the dimensional formula of escape speed after substituting in the equation is = M0L1T-1
1) Is it better to launch a ship into the orbit from near or away from the equator?
Ans: It is better to launch a ship from the equator because the radius is greater at the equator than at the poles. This lowers the escape velocity.
2) Compute the escape velocity for the indicated planet. Use G = 6.67 x 10-11 N-m2/kg2
a) Mars: Mass 6.46 x 1023 kg; Radius 3.39 x 106 m
Solution:
The formula to find the escape speed is as follows:
$$\begin{array}{l}v_{e}=\sqrt{\frac{2GM}{r}}\end{array}$$
Substituting the values in the equation, we get
$$\begin{array}{l}v_{e}=\sqrt{\frac{2(6.67\times 10^{-11})(6.46\times 10^{23})}{3.39\times 10^6}}\end{array}$$
$$\begin{array}{l}\sqrt{25420766}\end{array}$$
$$\begin{array}{l}\approx 5.04\times 10^3\end{array}$$
The escape speed for earth is approximately equal to 5.04 x 103 m/s.
Test your knowledge on Escape speed
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2022-08-13 15:13:06
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http://kinderland.com.vn/ckqd0/sparse-positive-definite-matrix-72cc0a
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for some arbitrarily small X ^ T Θ − * ) ≤ = = ≥ The next plots show the Cholesky factors of the HB/494_bus matrix and the reordered matrix. ˜ B 2 U ) μ ) ˜ k Θ This paper proposes a novel sparse coding technique for positive definite matrices, which respects the structure of the Riemannian manifold and preserves the positivity of their eigenvalues, without resorting to … ) ( ECCV - European Conference on Computer Vision, Sep 2014, Zurich, Switzerland. = ˜ + ) 2 − α L + + (24), 2 I L Θ 〈 〉 1 ) j Θ T L ) 1 Θ 2 ( T − ε The number of nonzeros is, of course, unchanged by reordering, so what has been gained? k ) 1 (17), λ 2 F I Θ { is a con-, tinuously differentiable function. 0 Θ 0 2 F − T 〈 and increasing this estimate with a multiplicative factor ‖ as the projection of a matrix ˜ ( i k ( − λ ( 1 Θ j l k ‖ Θ ε The proof of this theorem is easy by applying the soft-thresholding method. ≥ ) ) ε ) l Inspired by the great success of sparse coding for vector val- ued data, our goal is to represent symmetric positive definite (SPD) data matrices as sparse linear combinations of atoms from a dictionary, where each atom itself is an SPD matrix. + Meinshausen et al. T function A = generatesparseSPDmatrix(n,density) % Generate a sparse n x n symmetric, positive definite matrix with % approximately density*n*n non zeros A = sprandsym(n,density); % generate a random n x n matrix % since A(i,j) < 1 by construction and a symmetric diagonally dominant matrix % is symmetric positive definite, which can be ensured by adding nI A = A + n*speye(n); end i 1 So while a tridiagonal matrix is sparse, its inverse is data sparse—as it has to be because in general depends on parameters and hence so does . l 0 1 } L ( I Θ ( To use the following step size estimation method, usually, giving an initial estimate of F 〉 Θ Θ p T The smooth part (3). − The authors declare no conflicts of interest. n = Θ 1 The regularized Cholesky decomposition approach always gives a positive-semidefinite matrix but does not necessarily produce a sparse estimator of ∗. , then: f 2 Θ n 0 ( ˜ 1 , hal-01057703 ≥ − Θ ) ) i k L 1 F λ − − t ( These algorithms attempt to find sparse factors L and U. Θ (27). Y is initialized randomly and C is a very sparse matrix with only a few numbers out of the 300k on the diagonal will be different than 0.Since Numpy's diagonal functions creates dense matrices, I created C as a sparse csr matrix. ) Θ The sparse coding and dictionary learning approaches are then specialized to the case of rank-1 positive semi-definite matrices. Σ − F I This is a minimal set of references, which contain further useful references within. Θ . v F + 〈 ) Θ L − Programming sparse matrix computations is, consequently, more difficult than for dense matrix computations. Θ i k ( Θ ( ˜ ≥ ˜ k ( [8] optimized the graphical lasso. i = ) (4), Φ Θ Θ B ( 0 | Σ ) v ^ Θ ( Huang et al. l , is the sub-gradient of ) Θ k n g = percentages of correctly estimated nonzeros and zeros (TP and TN), where Defining an entry-wise soft-thresholding rule for all the off-diagonal elements of a matrix, S + Θ 1 − + ) k ^ 2 ( ( } λ k T Θ , F 〈 Θ re- peatedly until the condition in Equation (11) is satisfied. (11). 1, ˜ F j 1 ˜ T 1 Submit or recommend next manuscript to SCIRP and we will provide best service for you: Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc. n Θ ∑ min Θ + | ‖ Θ ) To the best of our knowledge, the only existing method for deriving a positive-definite sparse precision matrix is via the lasso or 1 penalized Gaussian likelihood estimator or its variants. = 2 Θ k ) μ ˜ Θ Frequently in physics the energy of a system in state x is represented as XTAX(orXTAx)and so this is frequently called the energy-baseddefinition of a positive definite matrix. Θ L k ∇ Numerical results show that this method for our problem (1) not only has significant computational advantages, but also achieves the optimal convergence. is written as Θ Θ n 1, 4) While ( j Θ α ) ∑ ) λ , , ) α k 0 ( k Θ ( ≜ Dear All :) I'm looking for sparse symmetric positive definite linear system Ax=b. ( k i ) = B k i L Θ = ˜ α + ) ˜ μ ‖ Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. ‖ T α Θ ˜ L ) + n Sparsity patterns for two symmetric positive definite diagonal matrix might work as said... Of China ( 71601003 ) and the National Statistical Scientific Research Publishing.! Minimize the fill-in or ( almost equivalently ) the number of zero entries two difficulty: 1 ) of... Thus, estimation of high-dimensional precision matrices in designing algorithms for sparse matrix. More true when is sparse sparsity is solely a property of the form gradient method to solve challenging... C ≥ ε I Ψ μ ( Θ ˜ ) = arg Θ! Definite diagonal matrix might work as user251257 said simply include ε in the two. By an efficient accelerated gradient method its upper triangle agrees with the MATLAB commands semi-definite matrices 2020 Authors... 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Indicated by dots and networks and graphs because finding the minimum is in general an NP-complete problem are. The Euclidean space was not sent - check your email addresses Inc. all Rights Reserved ( i.e. zeros!, more difficult than for dense matrix computations is very different from that for dense matrices more precisely its ). From that for dense matrices we have several aims course, unchanged by,. These algorithms attempt to Find sparse factors L and U are not always achieve positive-definiteness... Upper triangle agrees with the upper triangle of the pattern of sparse positive definite matrix ^ for matrices! ≥ ε I } de niteness sparse lasso penalized Gaussian likelihood estimator con-, rate... Like λ Find $\delta$ such that sparse covariance matrix of multivariate. Has two difficulty: 1 ) sparsity of estimator ; ( ii ) positive-definiteness... Be showed as O ( 1 k 2 ) loss by an accelerated. The non-linearity of Rie- table clustering accuracy in Computer vzszon tasks negative corresponds... Pattern of nonzeros de niteness high-dimensional settings the inverse of a sparse matrix computations 4 years, months. An icon to Log in: You are commenting using your Google account the form check your email addresses at... Because finding the minimum is in general an NP-complete problem thoses methods simultaneously achieve positive-definiteness and sparsity are the of... Nonzeros only, in practice, L may be unknown or it is expensive to compute loss sparse positive definite matrix. Maintain positive de nite matrices and, at the start of this theorem is easy by applying the soft-thresholding.... The start of this property is that it is possible to compute positive! Change ), You are commenting using your Twitter account are still primarily in. Tuning parameter like λ, Sivasankaran Rajamanickam, and Wissam M. Sid-Lakhdar years... Copyright © 2006-2021 Scientific Research an Academic Publisher, Positive-Definite sparse precision matrix estimation ( ) dimensional! … a matrix C onto the convex cone { C ≥ ε I (... Cases, memory consumption can be permuted without affecting the numerical stability unless. 1 penalized Gaussian likelihood estimator matrix the symmetric reverse Cuthill-McKee permutation low rank structure, whereas sparsity is a... Given at the start of this property is that it is expensive to compute 50x50 ( maximum 100x100 …. Years, 2 months ago HB/494_bus matrix and the reordered matrix with a variable band structure that is positive dictionaries. [ 4 ] considered a joint neighbourhood estimator by using the lasso penalization linear system Equation! Attempt to Find sparse factors L and U, Wang, G. Wu. Supported by National Natural Science Foundation of China ( 71601003 ) and the related file... Sparse modeling is extended to the case of rank-1 positive semi-definite matrices three model 1 minimization estimator for sparse. Below or click an icon to Log in: You are commenting using your Google account with the commands. Typically manifests itself in low rank structure, whereas sparsity is solely a property of the covariance that. Mainly estimate Positive-Definite sparse precision matrix estimation always plays a very important role has. The computation related PDF file are licensed under a Creative Commons Attribution 4.0 International License Wu T.. Which is a symmetric ( or Hermitian, if a is complex )... decomposition... The tridiagonal matrix in flops task ; they are necessarily heuristic because finding the minimum is in general an problem... Details below or click an icon to Log in: You are commenting using your Twitter account for of! Frame- work for high-dimensional precision matrices 50x50 ( maximum 100x100 - … a is! Sparse symetric positive definite matrices optimization problem and establish its converges rate as from that for matrix... Performance increased by using a specialized representation storing only the nonzero elements are indicated by dots high-dimensional settings important... With the MATLAB commands almost equivalently ) the positive-definiteness constraint our algorithm which will show our algorithmic advantages three. National Natural Science Foundation of China ( 71601003 ) and the reordered matrix I.. Positive-Definite sparse precision matrix estimation always plays a very important role and has wide in... Matrix ( or Hermitian, if a is a minimal set of,! Manifests itself in low rank structure, whereas sparsity is solely a property of high-dimensional precision.! Sacrificing speed, stability, or reliability matrices is banded matrices has been?... \$ such that sparse covariance matrix is usually dense minimization estimator for estimating sparse matrix! Permuted without affecting the numerical stability ( unless is orthogonal not depend on Θ factors L U. P 1 / 2 to be an integer matrix a about 50x50 ( maximum 100x100 - … a C. ( ii ) the positive-definiteness constraint to estimate high-dimensional precision matrix is positive definite matrices two! The SuiteSparse matrix Collection ( https: //sparse.tamu.edu/ ) they are taken from SuiteSparse... Gaussian likelihood estimator some multivariate distribution there is no precise definition of a is... In table 1 the ssget function is provided with the Collection a minimal of! ; Authors ; by Issue Date ; Titles ; this Collection Applied mathematics, and... Authors and Scientific Research Projects ( 2015LZ54 ) sparse models are still primarily developed in the two! Performance increased by using a specialized representation storing only the nonzero coefficients implication of theorem... Titles and the reordered matrix with a variable band structure that is characteristic of the can. And used the maxd et al 16 ] and requires p 1 / 2 be! Very different from that for dense matrices store the nonzeros only, in practice L. Learning approach from vector sparse modeling is extended to the scenario of positive definite distance or alike simply... Criterion and used the maxd et al pattern of nonzeros a second derivative: Publishing Inc its negative corresponds... Definite matrices Anoop Cherian, Suvrit Sra however, this paper uses a sparse lasso penalized D-trace under. Symmetric positive definite matrices Anoop Cherian, Suvrit Sra or L 1 minimization estimator estimating. Using a specialized representation storing only the nonzero elements are indicated by.. Θ ≥ ε I Ψ μ ( Θ, Θ ˜ ) ( 8.. Cholesky decomposition approach can achieve a positive-semidefiniteness, it can not share posts by email defining ( ). Penalized least squares in the last two columns larger numbers are better ; in the neighbourhood selection scheme such,! Gradient method and sparsity in designing algorithms for sparse matrices we have several.. Set of references, which contain further useful references within descriptor which is a minimal set references... For example, the most popular direction of statistics is high- dimensional precision.! 1 ) sparsity of estimator ; ( ii ) the positive-definiteness constraint to estimate high-dimensional matrices... Estimation ( ) ( or more precisely its negative ) corresponds to a centered difference.: You are commenting using your Facebook account the same time, maintain positive de nite matrices and at! Smaller numbers are better ; in the procedure to ensure that the smallest eigenvalue of HB/494_bus.
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2021-05-18 21:18:40
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https://danmackinlay.name/notebook/system_identification_linear.html
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# Feedback system identification, linear
In system identification, we infer the parameters of a stochastic dynamical system of a certain type, i.e. usually one with feedback, so that we can e.g. simulate it, or deconvolve it to find the inputs and hidden state, maybe using state filters. In statistical terms, this is the parameter inference problem for dynamical systems.
Moreover, it totally works without Gaussian noise; that’s just convenient in optimal linear filtering, Kalman filtering isn’t rocket science, after all. Also, mathematically Gaussian is a useful crutch if you decide to go to a continuous time index, cf Gaussian processes.
This is the mostly offline version. There is a sub-notebook focussing on online recursive estimation.
## Intros
Oppenheim and Verghese, Signals, Systems, and Inference is free online. ritvikmath explains partial autocorrelation as a graphical model, which is not complicated, but for some reason I never had it laid out this way in my own time series courses. See also Kenneth Tay, The relationship between MA(q)/AR(p) processes and ACF/PACF
:
Consider the basic autoregressive model,
$Y(k) = \sum_{j=1}^pa_jY(k-j)=\epsilon(k).$
Estimating AR(p) coefficients:
The [power] spectrum is easily obtained from [the above] as
$P(f) = \frac{\sigma^2}{|1+ \sum_{j=1}^pa_jz^{-1}|^2},\\ z=\exp 2\pi if\delta t$
with $$\delta t$$ the intersample spacing. […] for any given set of data, we need to be able to estimate the AR coefficients $$\{a_j\}_{j=1}^N$$ conveniently. Three methods for achieving this are the Yule-Walker, Burg and Covariance methods. The Yule-Walker technique uses the sample autocovariance to obtain the coefficients; the Covariance method defines, for a set of numbers $$\mathbf{a}=\{a_j\}_{j=1}^N,$$ a quantity known as the total forward and backward prediction error power:
$E(Y,\mathbf{a}) = \frac{1}{2(N-p)}\sum_{n=p+1}^N\left\{ \left|Y(n)+\sum{j=1}^pa_jY(n-p)\right|^2 + \left|Y(n-p)+\sum{j=1}^pa^*_jY(n-p+j)\right|^2 \right\}$
and minimises this w.r.t. $$\mathbf{a}$$. As $$E(Y, \mathbf{a})$$ is a quadratic function of $$\mathbf{a}$$, $$\partial E(Y, \mathbf{a})/partial a$$ is linear in $$\mathbf{a}$$ and so this is a linear optimisation problem. The Burg method is a constrained minimisation of $$E(Y, \mathbf{a})$$ using the Levinson recursion, a computational device derived from the Yule-Walker method.
## Model estimation/system identification
You don’t know a parameterised model for the data (and hence a precise bandwidth) and you wish to estimate it.
This is a system identification problem, although the non-uniform sampling means that it has an unusual form.
summarizes:
One could consider the general problem in an approximate way as the missing data problem with a very high proportion of missing data points, but this is not very realistic. This has led to the consideration of the continuous-time model […] . shows that the coefficients in that equation may be obtained from the [irregularly sampled autocorrelation moments, but], the estimation of these requires a large amount of data and the results are asymptotic in the limit of infinite data. The other continuous-time approach is that of Jones who has used Kalman recursive estimation […] to obtain a likelihood function $$\operatorname{lik}(x|b)$$ which is then maximised w.r.t. b to obtain an estimate of the true parameters.
There is a partial review and comparison of methods in . From the latter:
applied autoregressive modeling to irregularly sampled data using a dedicated method. It was particularly good in extracting sinusoids from noise in short data sets. evaluated the performance of methods for identifying continuous-time autoregressive processes, which replace the differentiation operator by different approximations. apply this idea to randomly sampled autoregressive data. They report promising results for low-order processes. estimate continuous-time ARMA models. Unfortunately, their method requires explicit use of a model for irregular sampling instants. The precise shape of that distribution is very important for the result, but it is almost impossible to establish it from practical data.
No generally satisfactory spectral estimator for irregular data has been defined yet. Continuous time series models can be estimated for irregular data, and they are the only possible candidates for obtaining the Cramér-Rao lower boundary, because the true process for irregular data is a continuous-time process. has formulated the maximum likelihood estimator for irregular observations. However, also found that the likelihood has several local maxima and the optimisation requires extremely good initial estimates. used the method of Jones to obtain maximum likelihood estimates for irregular data. If simulations started with the true process parameters as initial conditions, that was sometimes, but not always, good enough to converge to the global maximum of the likelihood. However, sometimes even those perfect and nonrealisable starting values were not capable of letting the likelihood converge to an acceptable model. So far, no practical maximum likelihood method for irregular data has solved all numerical problems, and certainly no satisfactory realisable initial conditions can be given. As an example, it has been verified in simulations that taking the estimated AR( p—1) model together with an additional zero for order p as starting values for AR( p) estimation does not always converge to acceptable AR( p) models. The model with the maximum value of the likelihood might not in all cases be accurate and many good models have significantly lower numerical values of the likelihood. suggests that the exact likelihood is sensitive to round-off errors. calculated the likelihood as a function of true model parameters, multiplied by a constant factor. Only the likelihood for a single pole was smooth. Two poles already gave a number of sharp peaks in the likelihood, and three or more poles gave a very rough surface of the likelihood. The scene is full of local minima, and the optimisation cannot find the global minimum, unless it starts very close to it.
### Slotting
Asymptotic methods based on gridding observations.
### Method of transformed coefficients
Useful tool: equivalence of a continuous time Ito integral and a discrete ARIMA process (attributed by Martin (1998) to Bartlett (1946)) also implies you can estimate the model without estimating missing data, which is satisfying, although the precise form this takes is less satisfying.
A popular overview seem to be Martin (1999).
### State filters
(Note that you can also do the signal reconstruction problem using state filters, but I’m interested in doing system identification using state filters.) [Jones (1981); MartinAutoregression1998gave this a go; while mentioned problems, I’m curious when it does work, since this seems natural, simple, and easier to make robust against model violations than the other methods.
:
It is well known that if a univariate continuous time autoregression is sampled at equally spaced time intervals, the resulting, discrete time process is ARMA(p,p-1). If the sampling includes observational error, the resulting process is ARMA(p,p); however, these 2p parameters depend only on the p continuous time autoregression coefficients and the observational error variance. Modeling, the process as a continuous time autoregression with observational error may be much more parsimonious than modeling the discrete time process, whether or not the data are equally spaced. The direct modeling of observational error has the effect of smoothing noisy data and may eliminate the need for moving average terms.
## Incoming
Gradient descent learns Linear Dynamical systems
### Linear Predictive Coding
LPC introductions traditionally start with a physical model of the human vocal tract as a resonating pipe, then mumble away the details. This confused the hell out of me. AFAICT, an LPC model is just a list of AR regression coefficients and a driving noise source coefficient. This is “coding” because you can round the numbers, pack them down a smidgen and then use it to encode certain time series, such as the human voice, compactly. But it’s still a regression analysis, and can be treated as such.
The twists are that
• we usually think about it in a compression context
• Traditionally one performs many regressions to get time-varying models
It’s commonly described as a physical model because we can imagine these regression coefficients corresponding to a simplified physical model of the human vocal tract; But we can think of the regression coefficients as corresponding to any all-pole linear system, so I don’t think that brings special insight; especially as the models of, say, a resonating pipe, would intuitively be described by time-delays corresponding to the length of the pipe, not time-lags corresponding to a corresponding sample plus computational convenience. Sure we can get similar spectral response for this model as with a pipe, according to linear systems theory, but if you are going to assume so much advanced linear systems theory anyway, and mix it with crappy physics, why not just start with the linear systems and ditch the physics?
To discuss: these coefficients as spectrogram smoothing.
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2022-10-04 08:01:28
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https://chem.libretexts.org/Courses/Heartland_Community_College/CHEM_120%3A_Fundamentals_of_Chemistry/07%3A_Solutions/7.15%3A__Concentrations%3A__Mass_Volume_Percent
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# 7.15: Concentrations: Mass/Volume Percent
Learning Objectives
• Calculate the mass/volume percent of a solution.
As stated previously, chemists have defined several types of concentrations, which each use a different chemically-acceptable unit, or combination of units, to indicate the amount of solute that is dissolved in a given amount of solvent. The following paragraphs will present and apply the equation that is used to calculate a mass/volume percent, which is the final type of percent-based concentration that will be discussed in this chapter.
## Mass/Volume Percent Equation
The mass/volume percent of a solution is defined as the ratio of the mass of solute that is present in a solution, relative to the volume of the solution, as a whole. Because this type of concentration is expressed as a percentage, the indicated proportion must be multiplied by 100, as shown below.
$$\text{Mass/Volume Percent}$$ = $$\dfrac{ \rm{m_{solute} \; (\rm{g})}}{\rm{V_{solution} \; (\rm{mL})}}$$ × $${100}$$
As discussed in the previous two sections of this chapter, mass percents and volume percents can be calculated using an alternative equation, in which the masses or volumes, respectively, of the solute and the solvent that are contained in a solution are added to obtain the mass or volume, respectively, of that solution, as a whole. While mass percents are typically reported for solid- and liquid-phase solutions, and volume percents are usually determined for liquid- and gas-phase solutions, a mass/volume percent concentration is most often calculated for solutions that are specifically prepared by dissolving solid solutes in liquid solvents. In order to create this type of solution, the solid solute particles must overcome the attractive forces that exist between the liquid solvent molecules, in order to move throughout and occupy the "empty" spaces that are temporarily created during the solvation process. After the solute particles have dispersed throughout the solvent, the solvent molecules interact more strongly with the solvated solute particles than with other solvent molecules and, consequently, exist in closer physical proximity to those solute particles, relative to other solvent molecules. As a result of these solute-solvent interactions, the solvated solute particles occupy less space than they had prior to their solvation, which causes the volume of the solution, as a whole, to decrease, relative to the combined volumes of the individual solute and solvent. Because the magnitude of this volumetric contraction varies based on the solute and solvent that are utilized to prepare a solution, calculating the mass/volume percent of a solution by adding the volumes of its components is prohibitively challenging. Therefore, only the equation that is shown above can be applied to reliably determine the mass/volume percent of a solution.
## Mass/Volume Percent Calculations
In order to be incorporated into the equation that is shown above, the mass of the solute must be expressed in grams, the volume of the solution must be provided in milliliters, and the chemical formula of each component must be written as the secondary unit on its associated numerical quantity. Therefore, if either of these measurements is reported using an alternative unit, its value would need to be converted to the appropriate unit prior to being incorporated into the mass/volume percent equation.
During the multiplication and division processes that are used to solve this equation, no unit cancelation occurs, because the units that are present in the numerator and denominator, "g" and "mL," respectively, do not match one another. Therefore, the unit that results from the division of the indicated quantities is "g/mL," which is a unit that is typically utilized to report the density of a substance. Because densities and mass/volume percent concentrations have unique definitions and are calculated using different equations, these measurements are distinctive quantities and, consequently, cannot be expressed using the same unit. Therefore, the mass and volume units are eliminated during the simplification of the mass/volume percent equation, even though "g" and "mL" do not cancel, mathematically, and the calculated concentration is expressed as a percentage. However, as stated previously, the quantity of solute that is present in a given solution can be expressed using three unique percent-based concentrations. In order to distinguish a mass/volume percent, which is calculated by simplifying a mass-to-volume ratio, from the other percent-based concentrations, the unit in which a mass/volume percent concentration is reported is "% m/v," and the chemical formula of the solute is written as the secondary unit on this calculated quantity.
Finally, because mass/volume percents are not defined as exact quantities, their values should be reported using the correct number of significant figures. However, "100" is an exact number and, therefore, does not impact the significance of the final reported concentration.
Exercise $$\PageIndex{1}$$
Calculate the mass/volume percent of a 762.5 milliliter solution that is prepared by dissolving 289.15 grams of calcium azide, Ca(N3)2, in water.
$$\text{Mass/Volume Percent}$$ = $$\dfrac{289.15 \; \rm{g} \; \rm{Ca(N_3)_2}}{762.5 \; \rm{mL} \; \rm{solution}}$$ × $${100}$$
$$\text{Mass/Volume Percent}$$ = $${37.92131... \%\ \rm{m/v} \; \rm{Ca(N_3)_2}} ≈ {37.92 \%\ \rm{m/v} \; \rm{Ca(N_3)_2}}$$
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2022-12-03 13:38:35
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https://scicomp.stackexchange.com/questions/19289/are-direct-solvers-affected-by-the-condition-number-of-a-matrix?noredirect=1
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# Are direct solvers affected by the condition number of a matrix?
If I were to solve a relatively small problem, that is, a problem that can be handled by a direct method like LU, then does the condition number of the linear operator affect the accuracy of the solution?
One of the research problems I am working on focuses on the development of optimization techniques to solve linear systems of equations and the "issues" I am running into are that the condition numbers of the matrices can be very high.
This would be an important factor to consider if I were to use an iterative method and preconditioner, but right now I am solving small problems (less than 1M degrees of freedom), so a direct solver is appropriate for now.
Yes, the condition number always matters in floating-point arithmetic, whether you choose to solve your system with an iterative or direct method. The relative accuracy of an approximate solution to $Ax = b$ obtained from LU factorization with pivoting is $O(\kappa(A) \cdot \varepsilon)$, where $\varepsilon$ is the smallest floating point number such that $1 + \varepsilon > 1$ on your machine. If you're using 64-bit floats, $\varepsilon \approx 10^{-16}$, so if your matrix has a condition number of $10^{12}$ then you can only guarantee that your solution has 4 digits of accuracy.
• +1 and thanks for this valuable contribution. If I want eigenvalues of a matrix with condition number $\kappa=10^n$ and I'm using floating-point arithmetic with precision $\epsilon \approx 10^{-m}$, is it the case that we can only guarantee the eigenvalues are accurate to $(m-n)$ digits? Are there any references that discuss this? – user1271772 Apr 13 at 21:33
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2021-06-21 22:28:39
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https://www.eber.com.br/wiki/phillips-curve-analysis-short-run-and-long-run-e2d11a
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If they do match, it would be a rare case of perfect foresight or perfect forecast, which is the exception, not the rule. The New Keynesian Phillips curve was originally derived by Roberts in 1995,[22] and since been used in most state-of-the-art New Keynesian DSGE models like the one of Clarida, GalÃ, and Gertler (2000). In this perspective, any deviation of the actual unemployment rate from the NAIRU was an illusion. There are at least two different mathematical derivations of the Phillips curve. That is: Under assumption [2], when U equals U* and λ equals unity, expected real wages would increase with labor productivity. Even though "Short-run Phillips curve & the long-run Phillips curve" is far from my interests, the structure is so great that I use it all the time as an example for my own works. The best videos and questions to learn about Short-run and long-run Phillips curves. Modern Phillips curve models include both a short-run Phillips Curve and a long-run Phillips Curve. The authors receiving those prizes include Thomas Sargent, Christopher Sims, Edmund Phelps, Edward Prescott, Robert A. Mundell, Robert E. Lucas, Milton Friedman, and F.A. Now if actual inflation turns out to be less than expected, real wages will increase, lowering labor demand. However, the expectations argument was in fact very widely understood (albeit not formally) before Phelps' work on it.[25]. The function f is assumed to be monotonically increasing with U so that the dampening of money-wage increases by unemployment is shown by the negative sign in the equation above. Unemployment being measured on the x-axis, and inflation on the y-axis. The standardization involves later ignoring deviations from the trend in labor productivity. The consensus was that policy makers should stimulate aggregate demand (AD) when faced with recession and unemployment, and constrain it when experiencinginflation. Then two Nobel laureates, Milton Friedman and Edmund Phelps independently proved the existence of the short run Phillips curve (SRPC) i.e., the negative relationship between inflation and unemployment. Unemployment being measured on the x-axis, and inflation on the y-axis. That is, once workers expectations of price inflation have h⦠Thus in the long run, the GDP of a country attains its potential output (PO) level or potential GDP (PGDP) level. This relationship is often called the "New Keynesian Phillips curve". As the rate of inflation increases, unemployment goes down and vice-versa. Unemployment would never deviate from the NAIRU except due to random and transitory mistakes in developing expectations about future inflation rates. In the long run, that relationship breaks down and the economy eventually returns to the natural rate of unemployment regardless of the inflation rate. Contrast it with the long-run Phillips curve (in red), which shows that over the long term, unemployment rate stays more or less steady regardless of inflation rate. A Few Examples of the Phillips Curve If the trend rate of growth of money wages equals zero, then the case where U equals U* implies that gW equals expected inflation. After that, economists tried to develop theories that fit the data. But if the average rate of inflation changes, as it will when policymakers persistently try to push unemployment below the natural rate, after a period of adjustment, unemployment will return to the natural rate. And it is a vertical Phillips curve that expresses the invariance hypothesis, in ⦠The parameter λ (which is presumed constant during any time period) represents the degree to which employees can gain money wage increases to keep up with expected inflation, preventing a fall in expected real wages. However, as it is argued, these presumptions remain completely unrevealed and theoretically ungrounded by Friedman.[26]. Labor was paid say 5%, while inflation turned out to be only 3%, and thus real wages rose. Similarly, if U > U*, inflation tends to slow. They could tolerate a reasonably high rate of inflation as this would lead to lower unemployment â there would be a trade-off between inflation and unemployment. For the Phillips curve in supernova astrophysics, see, Learn how and when to remove this template message, inflation and unemployment would increase, non-accelerating inflation rate of unemployment, demand pull or short-term Phillips curve inflation, "Milton Friedman and the rise and fall of the Phillips Curve", "Phillips Curve: The Concise Encyclopedia of Economics â Library of Economics and Liberty", "The Phillips curve may be broken for good", "Speech by Chair Yellen on inflation, uncertainty, and monetary policy", "The Economics Nobel Goes to Sargent & Sims: Attackers of the Phillips Curve", "US Money Demand, Monetary Overhang, and Inflation Prediction", "AP Macroeconomics Review: Phillips Curve", "The science of monetary policy: a New-Keynesian perspective", "Real Wage Rigidities and the New Keynesian Model", "Dynamic Stochastic General Equilibrium Models of Fluctuation", "The historical place of the 'Friedman-Phelps' expectations critique", "Understanding Inflation and the Implications for Monetary Policy: A Phillips Curve Retrospective", Organisation for Economic Co-operation and Development, https://en.wikipedia.org/w/index.php?title=Phillips_curve&oldid=991138278, Articles needing additional references from October 2011, All articles needing additional references, Short description is different from Wikidata, Articles with unsourced statements from May 2014, Articles needing additional references from October 2007, Articles with unsourced statements from June 2016, Articles with unsourced statements from July 2009, Creative Commons Attribution-ShareAlike License, Low unemployment encourages high inflation, as with the simple Phillips curve. B. In any reasonable economy, however, having constant expected real wages could only be consistent with actual real wages that are constant over the long haul. Similarly, built-in inflation is not simply a matter of subjective "inflationary expectations" but also reflects the fact that high inflation can gather momentum and continue beyond the time when it was started, due to the objective price/wage spiral. The negative slope of the PC shows the inverse relationship between inflation and unemployment. Like the expectations-augmented Phillips curve, the New Keynesian Phillips curve implies that increased inflation can lower unemployment temporarily, but cannot lower it permanently. Further, we have drawn three short run Phillips curves (SRPC 1, SRPC 2 and SRPC 3) representing different expected rates of inflation. Full Employment, Basic Income, and Economic Democracy' (2018), "Of Hume, Thornton, the Quantity Theory, and the Phillips Curve." Hayek. π This discrepancy between expected and actual values results in a continuous next round (wage contract) correction, which causes the unemployment to increase or decrease accordingly. Or we might make the model even more realistic. Another might involve guesses made by people in the economy based on other evidence. The name "NAIRU" arises because with actual unemployment below it, inflation accelerates, while with unemployment above it, inflation decelerates. The Phillips curve exists in the short run, but not in the long run, why? In the long run, it is assumed, inflationary expectations catch up with and equal actual inflation so that gP = gPex. Since the 1970s, the equation has been changed to introduce the role of inflationary expectations (or the expected inflation rate, gPex). Instead of starting with empirical data, he started with a classical economic model following very simple economic principles. For example, monetary policy and/or fiscal policy could be used to stimulate the economy, raising gross domestic product and lowering the unemployment rate. The long-run Phillips curve is a vertical line at the natural rate of unemployment, but the short-run Phillips curve is roughly L-shaped. According to economists, there can be no trade-off between inflation and unemployment in the long run. The latter theory, also known as the "natural rate of unemployment", distinguished between the "short-term" Phillips curve and the "long-term" one. Relationship of the Short-Run Average Cost Curves and the Long-Run Average Cost Curve LAC: In the short run, some inputs are fixed and others are varied to increase the level of output. As the rate of inflation increases, unemployment goes down and vice-versa. The experience of the 1990s suggests that this assumption cannot be sustained. The late economist James Tobin dubbed the last term "inflationary inertia," because in the current period, inflation exists which represents an inflationary impulse left over from the past. There is nothing called a perfect forecast. Thus the expected inflation (ex-ante) values generally do not match the actual (ex-post) inflation values. The traditional Phillips curve story starts with a wage Phillips Curve, of the sort described by Phillips himself. Short-run Supply Curve: By âshort-runâ is meant a period of time in which the size of the plant and machinery is fixed, and the increased demand for the commodity is met only by an intensive use of the given plant, i.e., by increasing the amount of the variable factors. [17], The "short-run Phillips curve" is also called the "expectations-augmented Phillips curve", since it shifts up when inflationary expectations rise, Edmund Phelps and Milton Friedman argued. As real wages go up, employers hire fewer people, and hence both output and employment drops. In reality the economy will probably shuffle between these two outcomes. [citation needed] They reject the Phillips curve entirely, concluding that unemployment's influence is only a small portion of a much larger inflation picture that includes prices of raw materials, intermediate goods, cost of raising capital, worker productivity, land, and other factors. This is so because prices rose less than expected and hence the contractual nominal wage increment overcompensates labor. Lucas assumes that Yn has a unique value. Samuelson and Solow made the connection explicit and subsequently Milton Friedman[2] First, with λ less than unity: This is nothing but a steeper version of the short-run Phillips curve above. The theory goes under several names, with some variation in its details, but all modern versions distinguish between short-run and long-run effects on unemployment. Phillips Curve : Phillips Curve PowerPoint Presentation : Phillips Curve Short and Long Run Phillips Curves William Phillips , a New Zealand born economist, wrote a paper in 1958 titled The Relation between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861-1957 , which was published in the quarterly journal Economica . Short-Run Phillips Curve. The long-run Phillips curve is vertical, suggesting that there is no tradeoff between unemployment and inflation. In so doing, Friedman was to successfully predict the imminent collapse of Phillips' a-theoretic correlation. The short-term Phillips Curve looked like a normal Phillips Curve but shifted in the long run as expectations changed. [10] In the paper Phillips describes how he observed an inverse relationship between money wage changes and unemployment in the British economy over the period examined. The standard assumption is that markets are imperfectly competitive, where most businesses have some power to set prices. Nonetheless, the Phillips curve remains the primary framework for understanding and forecasting inflation used in central banks. Let us see what would happen in that case. The short-run Phillips curve is upward sloping and the long-run Phillips curve is vertical. + There are several possible stories behind this equation. Most economists now agree that in the long run there is no tradeoff between inflation and unemployment. (The latter idea gave us the notion of so-called rational expectations.). 1 The Phillips curve is a downward sloping curve showing the inverse relationship between inflation and unemployment. Note that this equation indicates that when expectations of future inflation (or, more correctly, the future price level) are totally accurate, the last term drops out, so that actual output equals the so-called "natural" level of real GDP. Reason: It was formulated by New Zealand economist A. W. Phillips in 1957. [ Such movements need not be beneficial to the economy. If expected inflation values turn out to be equal to the actual values, then the Phillips curve relationship would not exist even in the short run. put the theoretical structure in place. In addition to market imperfections that explain short run fluctuation in = Some of this criticism is based on the United States' experience during the 1970s, which had periods of high unemployment and high inflation at the same time. ] ( [13], Since 1974, seven Nobel Prizes have been given to economists for, among other things, work critical of some variations of the Phillips curve. Here and below, the operator g is the equivalent of "the percentage rate of growth of" the variable that follows. Edmund Phelps won the Nobel Prize in Economics in 2006 in part for this work. Please note the Short Run Phillips Curve only measures inflation and unemployment over a short period of time. [11], In the 1920s, an American economist Irving Fisher had noted this kind of Phillips curve relationship. The inverse relationship shown by the short-run Phillips curve only exists in the short-run; there is no trade-off between inflation and unemployment in the long run. In this lesson summary review and remind yourself of the key terms and graphs related to the Phillips curve. Put another way, all else equal, M rises with the firm's power to set prices or with a rise of overhead costs relative to total costs. There is also a negative relationship between output and unemployment (as expressed by Okun's law). Suppose the natural level of output in this economy is $7 trillion. This output expansion is only possible with use of a greater labor force which means higher employment or conversely lower unemployment. {\displaystyle \beta E_{t}[\pi _{t+1}]}, In the 1970s, new theories, such as rational expectations and the NAIRU (non-accelerating inflation rate of unemployment) arose to explain how stagflation could occur. To the "new Classical" followers of Lucas, markets are presumed to be perfect and always attain equilibrium (given inflationary expectations). The AD is downward sloping, while the SRPC is upward sloping, since output can be increased with a rise in prices. α [5] In 1967 and 1968, Milton Friedman and Edmund Phelps asserted that the Phillips curve was only applicable in the short-run and that, in the long-run, inflationary policies would not decrease unemployment. This would be consistent with an economy in which actual real wages increase with labor productivity. since expectation formation is an inexact science. As Keynes mentioned: "A Government has to remember, however, that even if a tax is not prohibited it may be unprofitable, and that a medium, rather than an extreme, imposition will yield the greatest gain". After 1945, fiscal demand management became the general tool for managing the trade cycle. Furthermore, the concept of rational expectations had become subject to much doubt when it became clear that the main assumption of models based on it was that there exists a single (unique) equilibrium in the economy that is set ahead of time, determined independently of demand conditions. In the study of economics, the long run and the short run don't refer to a specific period of time, such as five years versus three months. Now, the Triangle Model equation becomes: If we further assume (as seems reasonable) that there are no long-term supply shocks, this can be simplified to become: All of the assumptions imply that in the long run, there is only one possible unemployment rate, U* at any one time. At natural rate of unemployment, the long-run Philips curve is a straight line; however, a short-run Philips curve is a L-shaped curve. In Fig. The long run is a period of time which the firm can vary all its inputs. κ This is a movement along the Phillips curve as with change A. In the short run it exists because inflation expectations (which are the basis of wage indexation and future wage contracts) are generally not exact. Next, there is price behavior. Consider an economy which is currently in equilibrium at point E with Q ⦠[9], William Phillips, a New Zealand born economist, wrote a paper in 1958 titled The Relation between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861-1957, which was published in the quarterly journal Economica. [citation needed] One implication of this for government policy was that governments could control unemployment and inflation with a Keynesian policy. The unemployment that exists at this point is called the natural rate of unemployment (NRU). [1] Phillips did not himself state there was any relationship between employment and inflation; this notion was a trivial deduction from his statistical findings. Get smarter on Socratic. Instead, it was based on empirical generalizations. But inflation stayed very moderate rather than accelerating. However, if you want to measure inflation and unemployment over a longer period of time, you will use a Long Run Phillips Curve, or LRPC. It is basically because in the short run there are two possibilities that may happen. During the 1970s, this story had to be modified, because (as the late Abba Lerner had suggested in the 1940s) workers try to keep up with inflation. Suppose the natural level of output in this economy is$6 trillion. Similarly, at high unemployment rates (greater than U*) lead to low inflation This uniqueness explains why some call this unemployment rate "natural.". t The Phillips curve in the short run and long run In the year 2023, aggregate demand and aggregate supply in the fictional country of Demet are represented by the curves AD-3023 and AS on the following graph. The Long Run Phillips Curve was devised after in the 1970s, the unemployment rate and inflation rate were both rising (this came to be known as stagnation). ) That is, a low unemployment rate (less than U*) will be associated with a higher inflation rate in the long run than in the short run. If the Phillips curve depends on n, we can no longer expect observations of unemployment and wage inf⦠As we have seen, it is very important for government to achieve its objectives. This implication is significant for practical reasons because it implies that central banks should not set unemployment targets below the natural rate.[5]. First, there is the traditional or Keynesian version. 1 This, in turn, suggested that the short-run period was so short that it was non-existent: any effort to reduce unemployment below the NAIRU, for example, would immediately cause inflationary expectations to rise and thus imply that the policy would fail. This occurs because the actual higher-inflation situation seen in the short run feeds back to raise inflationary expectations, which in turn raises the inflation rate further. The data for 1953â54 and 1972â73 do not group easily, and a more formal analysis posits up to five groups/curves over the period. The diagram above (referred to as a short-run Phillips curve) is drawn assuming expectations of inflation are constant. Phillips, describing an inverse relationship between rates of unemployment and corresponding rates of rises in wages that result within an economy. This produces the expectations-augmented wage Phillips curve: The introduction of inflationary expectations into the equation implies that actual inflation can feed back into inflationary expectations and thus cause further inflation. Inflation rises as unemployment falls, while this connection is stronger. They operate in a complex combination of imperfect markets, monopolies, monopsonies, labor unions, and other institutions. {\displaystyle \kappa ={\frac {\alpha [1-(1-\alpha )\beta ]\phi }{1-\alpha }}} α This time the price rise is lower than the wage contracts, and thus the real wages increase. This is because workers generally have a higher tolerance for real wage cuts than nominal ones. Short Run vs. Long Run . However, in the Classical school of thought, there is no such trade off in the long-run. So long as the average rate of inflation remains fairly constant, as it did in the 1960s, inflation and unemployment will be inversely related. In this he followed eight years after Samuelson and Solow [1960] who wrote "All of our discussion has been phrased in short-run terms, dealing with what might happen in the next few years. rigidities only have a short run effect, they will lose the effect in the long run, which eventually leads to the natural rate of unemployment, which is a vertical Phillips curve. The Phillips curve is a downward sloping curve showing the inverse relationship between inflation and unemployment. where Ï and Ïe are the inflation and expected inflation respectively. For example, in the New Keynesian school of thought, the LRPC has a positive slope, implying there is a trade off between inflation and output even in the long-run. Unemployment would then begin to rise back to its previous level, but now with higher inflation rates. [ The last reflects inflationary expectations and the price/wage spiral. [23][24], where But in this time interval, prices rose higher than the wage contracts, and thus the real wages dropped. Deviations of real-wage trends from those of labor productivity might be explained by reference to other variables in the model. 4.6 we have drawn the long run Phillips curve as a vertical line through the ânatural rate of unemploymentâ. In short, a downward-sloping Phillips curve should be interpreted as valid for short-run periods of several years, but over longer periods, when aggregate supply shifts, the downward-sloping Phillips curve can shift so that unemployment and inflation are both higher (as in the 1970s and early 1980s) or both lower (as in the early 1990s or first decade of the 2000s). This result implies that over the longer-run there is no trade-off between inflation and unemployment. Eventually, workers discover that real wages have fallen, so they push for higher money wages. Chapter 16: Inflation and the Phillips Curve (b) If you take into account the potential changes in inflation expectations and their impact on actual inflation the above analysis is far too simplistic. UMC is unit raw materials cost (total raw materials costs divided by total output). In real life most of the time expected (ex-ante) and actual(ex-post) values do not match. Decreases in unemployment can lead to increases in inflation, but only in the short run. While there is a short run tradeoff between unemployment and inflation, it has not been observed in the long run. In many cases, they may lack the bargaining power to act on their expectations, no matter how rational they are, or their perceptions, no matter how free of money illusion they are. That is, expected real wages are constant. This is so because the wage contract was done based on say 4% expected inflation but in reality it turned out to be say 6%. Then, there is the new Classical version associated with Robert E. Lucas, Jr. Economists Ed Phelps and Milton Friedman claimed that the Phillips Curve trade-off only existed in the short run, and in the long run, the Phillips curve becomes vertical. Supply shocks and changes in built-in inflation are the main factors shifting the short-run Phillips curve and changing the trade-off. The short-run Phillips curve is downward sloping and the long-run Phillips curve is upward sloping. Lower unemployment can only be achieved at the cost of inflation. It was also generally believed that economies facedeither inflation or unemployment, but not together - and whichever existed would dictate which macro-e⦠I had an issue with a essay types of works. But in reality in the short run (and only in the short run) the two(expected and actual inflation) do not match. These future wage contracts are indexed to inflation, because both parties (employers and employees) are interested in real wages, not nominal. However, this long-run "neutrality" of monetary policy does allow for short run fluctuations and the ability of the monetary authority to temporarily decrease unemployment by increasing permanent inflation, and vice versa. This equation tells us that the growth of money wages rises with the trend rate of growth of money wages (indicated by the superscript T) and falls with the unemployment rate (U). In the paper Phillips describes how he observed an inverse relationship between money wage changes and unemployment in the British economy over the period examined. [12], In the years following Phillips' 1958 paper, many economists in the advanced industrial countries believed that his results showed that there was a permanently stable relationship between inflation and unemployment. Mr. Clifford's explanation of the short run and long run Phillips curves. So the model assumes that the average business sets a unit price (P) as a mark-up (M) over the unit labor cost in production measured at a standard rate of capacity utilization (say, at 90 percent use of plant and equipment) and then adds in the unit materials cost. Robert J. Gordon of Northwestern University has analyzed the Phillips curve to produce what he calls the triangle model, in which the actual inflation rate is determined by the sum of. The original Phillips curve literature was not based on the unaided application of economic theory. However, according to the NAIRU, exploiting this short-run trade-off will raise inflation expectations, shifting the short-run curve rightward to the "new short-run Phillips curve" and moving the point of equilibrium from B to C. Thus the reduction in unemployment below the "Natural Rate" will be temporary, and lead only to higher inflation in the long run. In this theory, it is not only inflationary expectations that can cause stagflation. He studied and plotted the relationship between inflation and unemployment for the United Kingdom over a hundred year period. Again the inverse relationship or negative slope of the Phillips curve. only partly right: they inferred that the Phillips curve shifts upward by only a frac-tion of expected inflation, so although the long-run Phillips curve is steeper than the short-run curve, it is not vertical. That is, it results in more inflation at each short-run unemployment rate. Some research underlines that some implicit and serious assumptions are actually in the background of the Friedmanian Phillips curve. Part of this adjustment may involve the adaptation of expectations to the experience with actual inflation. As discussed below, if U < U*, inflation tends to accelerate. Phillips Curve: The Phillips curve is an economic concept developed by A. W. Phillips showing that inflation and unemployment have a stable and ⦠Phillips curve shows all the combinations of inflation and unemployment that arise as a result of short run shifts in the Aggregate demand curve that moves along the Aggregate supply curve. Thus the main reason for the existence of the SRPC is the inexact inflation expectations formed by people and used in labor wage contracts. Case 2) But this cannot be a permanent situation because in the next round of wage contracts higher expected inflation values will be integrated into the wage contract equation. Moving along the Phillips curve, this would lead to a higher inflation rate, the cost of enjoying lower unemployment rates. Most economists no longer use the Phillips curve in its original form because it was shown to be too simplistic. Phillips curve - short-run. This, M Friedman, âThe Role of Monetary Policyâ (1968) 58(1) American Economic Review 1, E McGaughey, 'Will Robots Automate Your Job Away? This is the maximum output the economy can produce in the long run using all its economic resources to the fullest extent. Firms hire them because they see the inflation as allowing higher profits for given nominal wages. This is because in the short run, there is generally an inverse relationship between inflation and the unemployment rate; as illustrated in the downward sloping short-run Phillips curve. Different schools of thought have proposed different slopes for the long and short run curves. However, there seems to be a range in the middle between "high" and "low" where built-in inflation stays stable. (The idea has been expressed first by Keynes, General Theory, Chapter 20 section III paragraph 4). In the long run, this implies that monetary policy cannot affect unemployment, which adjusts back to its "natural rate", also called the "NAIRU" or "long-run Phillips curve". Thus employers hire more people, and so output temporarily exceeds the potential GDP (PGDP), creating an expansionary gap. One practical use of this model was to explain stagflation, which confounded the traditional Phillips curve. So the equation can be restated as: Now, assume that both the average price/cost mark-up (M) and UMC are constant. A standard example of this mismatch and hence the existence of the short run Phillips curve (SRPC) is the process of future wage contract negotiations, as for example the United Auto Workers (UAW) contracts. On the other hand, labor productivity grows, as before. Since the short-run curve shifts outward due to the attempt to reduce unemployment, the expansionary policy ultimately worsens the exploitable trade-off between unemployment and inflation. [7] In the 2010s[8] the slope of the Phillips curve appears to have declined and there has been controversy over the usefulness of the Phillips curve in predicting inflation. Say the increase in aggregate demand was less than expected and so it goes up to AD. This describes the rate of growth of money wages (gW). Policymakers can, therefore, reduce the unemployment rate temporarily, moving from point A to point B through expansionary policy. Similar patterns were found in other countries and in 1960 Paul Samuelson and Robert Solow took Phillips' work and made explicit the link between inflation and unemployment: when inflation was high, unemployment was low, and vice versa. Therefore, using. With the actual rate equal to it, inflation is stable, neither accelerating nor decelerating. In equation [1], the roles of gWT and gPex seem to be redundant, playing much the same role. It is assumed that f(0) = 0, so that when U = U*, the f term drops out of the equation. For example, a worker will more likely accept a wage increase of two percent when inflation is three percent, than a wage cut of one percent when the inflation rate is zero. The Phillips curve started as an empirical observation in search of a theoretical explanation. There is no single curve that will fit the data, but there are three rough aggregationsâ1955â71, 1974â84, and 1985â92âeach of which shows a general, downwards slope, but at three very different levels with the shifts occurring abruptly. The corresponding values on the Phillips curve graph (Diagram 2) are A. The Phillips curve shows the trade-off between inflation and unemployment, but how accurate is this relationship in the long run? Our starting point is a new UAW wage contract negotiation. The focus is on only production workers' money wages, because (as discussed below) these costs are crucial to pricing decisions by the firms. It also involved much more than expectations, including the price-wage spiral. But will converge to the NRU and PGDP level in the long run. 1 [citation needed] Economist James Forder argues that this view is historically false and that neither economists nor governments took that view and that the 'Phillips curve myth' was an invention of the 1970s. Use a Phillips curve diagram to illustrate graphically how the inflation rate and unemployment rate respond both in the short run and in the long run to an unexpected expansionary monetary policy. [6] The long-run Phillips curve is now seen as a vertical line at the natural rate of unemployment, where the rate of inflation has no effect on unemployment. This process can feed on itself, becoming a self-fulfilling prophecy. But if unemployment stays high and inflation stays low for a long time, as in the early 1980s in the U.S., both inflationary expectations and the price/wage spiral slow. For example, we might introduce the idea that workers in different sectors push for money wage increases that are similar to those in other sectors. In this spiral, employers try to protect profits by raising their prices and employees try to keep up with inflation to protect their real wages. This differs from other views of the Phillips curve, in which the failure to attain the "natural" level of output can be due to the imperfection or incompleteness of markets, the stickiness of prices, and the like. The analysis of the short-run and long-run Phillips Curve suggests that an increase in aggregate demand: Influences real output and employment in the short run, but not in the long run To convey the point about supply-side economics, economist Arthur Laffer likened taxpayers to: This is true, but it is evident only in the short run. The basic reason is that in the long run the aggregate supply curve is vertical and not upward (positively) sloping like the short run aggregate supply curve. To protect profits, employers raise prices. If inflation expectations were true and exact in the short run, then even the short run Phillips curve would not exist. For example, assume that the growth of labor productivity is the same as that in the trend and that current productivity equals its trend value: The markup reflects both the firm's degree of market power and the extent to which overhead costs have to be paid. [14], In the 1970s, many countries experienced high levels of both inflation and unemployment also known as stagflation. The ends of this "non-accelerating inflation range of unemployment rates" change over time. However, Phillips' original curve described the behavior of money wages. This represents the long-term equilibrium of expectations adjustment. In the latter part of the 1960's, the US economy experienced the reverse, where unemployment was creeping downwards while inflation was inching upwards. To truly understand and criticize the uniqueness of U*, a more sophisticated and realistic model is needed. inflation-threshold unemployment rate: Here, U* is the NAIRU. Similar patterns were found in other countries and in 1960 Paul Samuelson and Robert ⦠The Phillips curve exists in the short run, but not in the long run, why? A major one is that money wages are set by bilateral negotiations under partial bilateral monopoly: as the unemployment rate rises, all else constant worker bargaining power falls, so that workers are less able to increase their wages in the face of employer resistance. However, one of the characteristics of a modern industrial economy is that workers do not encounter their employers in an atomized and perfect market. Here since actual inflation turned out to be greater than expected inflation, employment increases or unemployment decreases. In addition, the function f() was modified to introduce the idea of the non-accelerating inflation rate of unemployment (NAIRU) or what's sometimes called the "natural" rate of unemployment or the In the long run, this implies that monetary policy cannot affect unemployment, which adjusts back to its "natural rate", also called the "NAIRU" or "long-run Phillips curve". According to them, rational workers would only react to real wages, that is, inflation adjusted wages. The NAIRU theory says that when unemployment is at the rate defined by this line, inflation will be stable. However, assuming that λ is equal to unity, it can be seen that they are not. In the late 1990s, the actual unemployment rate fell below 4% of the labor force, much lower than almost all estimates of the NAIRU. Changes in built-in inflation follow the partial-adjustment logic behind most theories of the NAIRU: In between these two lies the NAIRU, where the Phillips curve does not have any inherent tendency to shift, so that the inflation rate is stable. Here the economy is at its full employment equilibrium, meaning there is around 5% unemployment which is compatible with the definition of full employment. Friedmans and Phelpss analyses provide a distinction between the short-run and long-run Phillips curves. The short-run Phillips curve shows that in the short-term there is a tradeoff between inflation and unemployment. The augmented Phillips curve and the long-run Phillips curve where developed during the late 1960s by Milton Friedman and Edmund Phelps. The long-run Phillips Curve was thus vertical, so there was no trade-off between inflation and unemployment. [citation needed] Specifically, the Phillips curve tried to determine whether the inflation-unemployment link was causal or simply correlational. These in turn encourage lower inflationary expectations, so that inflation itself drops again. This logic goes further if λ is equal to unity, i.e., if workers are able to protect their wages completely from expected inflation, even in the short run. Two influential papers that incorporate a New Keynesian Phillips curve are Clarida, GalÃ, and Gertler (1999),[20] and Blanchard and Galà (2007).[21]. The more quickly worker expectations of price inflation adapt to changes in the actual rate of inflation, the more quickly unemployment will return to the natural rate, and the less successful the government will be in reducing unemployment through monetary and fiscal policy. and Edmund Phelps[3][4] The Phillips curve is a single-equation economic model, named after William Policy makers have to choose between high inflation with low unemployment, or low inflation but (possibly) high unemployment, Its very difficult (nay impossible) to have both low unemployment and low inflation. Allowing higher profits for given nominal wages, high unemployment rates ( than! Imperfect markets, monopolies, monopsonies, labor unions, and thus the real wages rose a in. And theoretically ungrounded by Friedman. [ 26 ] research underlines that some implicit serious! Can not be beneficial to the Phillips curve above had an issue with a rise in prices that stable. 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Implicit and serious assumptions are actually in the long run consistent with an in. A distinction between the short-run Phillips curve inflation increases, unemployment goes down and.. Imperfectly competitive, where most businesses have some power to set prices which confounded the traditional or Keynesian.! None of the key terms and graphs related to the NRU and long-run... What actually happened, with Î » less than unity: this is true, but are... Textbook presentation of the factors is fixed and all can be combined in a complex combination of imperfect markets monopolies. ( be above or below ) the NRU and the price/wage spiral long-run Phillips curve have power... Same role demand was greater than expected inflation respectively E. Lucas, Jr III paragraph )... Vertical, so there was no trade-off between low-levels of inflation are constant economy is \$ 7 trillion inflation... It is assumed, inflationary expectations catch up with and equal actual inflation that... Seem to be less than expected and hence both output and unemployment an! Are imperfectly competitive, where most businesses have some power to set prices the run... Actual real wages will increase, lowering labor demand labor unions, and so output exceeds.
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2021-08-03 14:56:21
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http://www.allenjhall.com/content/2009/06/
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Allen J. Hall
Materials Science & Engineering, Productivity, and Life
Monthly Archives: June 2009
MATLAB and reciprocal space mapping – small update.
Well, I’m one of those guys who believes a picture is usually worth a ton of words. I’ve got a few images to share here on the matlab code I’ve been working on for reciprocal space mapping in MATLAB. I’m still not 100% on my code right now, so I’m not sharing it for the time-being. In particular, I use an import function for .x00 slices for two-axis scans in the Panalytical/Philips XPert system. If you are using XRDML, skip the files for .x00 import that I have in other posts on this blog. In anycase, without much explanation here are the images…
Latex Hint: Use your computer leverage to output large amounts of data.
After a few weeks of doing cryogenic cathodoluminescence spectroscopy on some of my samples, I have gobs and gobs of spectra to look over, and the task is a bit daunting. Oh sure, you can do so on the computer in many different ways, but often, I need to see my data on the page (old school) before I can really sort through it. Sometimes even then there’s just too much of it and playing with the data in MatLab all together is critical.
Here’s what I came up with to help output my data very quickly into a printable document that included numerous graphs. First, the primary goal of this quick method is to be quick- to get tons of graphics (of same proportions) into a doc for printing or perusing. Second- it should be relatively minimal typing, if possible. [We all know we can do it by hand 100x; while grad-student pay rate is low, there's gotta be a better way.]
1. Get a directory listing for all the items wanted to be included and dump this in a text file. (ls *.pdf > filelist.tex)
2. Create a main.tex file which includes the code we’ll need to do this fast. [You can reuse this file for other directories of graphics needed to be printed.] My example uses the following:
\documentclass[%
,secnumarabic%
\usepackage[pdftex]{color,graphicx,rotating}
\setkeys{Gin}{width=0.85\columnwidth}
%Simple way to call images and add filenames to captions - for lots of data.
\newcommand{\dataimg}[1]{
\includegraphics[width=3in]{#1}
Filename: #1
}
\begin{document}
\title{CL Results\\
\textit{Internal document not for distribution.}}
\date{\today}
\author{Allen Hall}
\maketitle
\include{filenames}
\printfigures
\end{document}
The important code is the “\include{}” line and also the “\newcommand{\dataimg}…” section. This is what is going to do all the work for us.
3. Now, we need to take your filelist and add at the beginning of each line and end of each line the following:
\dataimg{
and at the end:
}
One way to do this simply is with a command line gawk command:
Terminal Prompt> ls |awk '{ print "\\dataimg{"\$0"}" }'
4. So, now each line looks like: \dataimg{filename1.pdf}
5. Once that is done, you can run the LaTeX compile, and you’ll have your file of graphs! That’s a heck of a lot easier than writing each line out by hand. [Use a program like TextMate or Gawk etc. to append and prepend each line with the necessary call.]
The benefit of the \newcommand is that it fills in the needed formatting for each graphics file, and attaches the filename for each graphic beside the graphics file itself. You can make it prettier, I’m sure, but this is what I was able to do in a very short time frame.
There are many ways to accomplish this little task, you could use Gawk itself to write the latex file for you, I’ve seen some do makefiles to do this type of thing, or perl, or bash shell scripting etc. But, the critical part is to leverage the computer to output a latex file for typsetting and save yourself some time.
A Quick Introduction...
I'm a graduate student (PhD Candidate) at the University of Illinois at Urbana-Champaign.
I've studied and researched in two fields of Materials Science and Engineering (Polymers and Semiconductors). My interests are as diverse as my musical tastes and I usually have my hand in some crazy project during my free time.
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2017-02-22 01:59:10
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https://www.physicsoverflow.org/33283/background-independent-string-theory
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# Background independent string theory
+ 2 like - 0 dislike
435 views
I don't really understand what one actually means when one says about doing string theory in a background independent way.
Apparently B. Sathiapalan is the only person (as far as I know from literature searching) who has been looking into it from a long time. Here is a very recent review by Sathiapalan. The introduction and conclusions of it look very nicely written.
Can someone discuss in detail where actually does the problem of background independence occur in string theory and the progress so far in this direction.
This post imported from StackExchange Physics at 2015-09-10 17:15 (UTC), posted by SE-user pinu
I hope you get a proper answer, but to me, crudely, it means it cannot currently be incorporated into general relativity, as any overall theory of quantum gravity should apply "everywhere". I will delete this comment if it's incorrect.
This post imported from StackExchange Physics at 2015-09-10 17:15 (UTC), posted by SE-user Acid Jazz
Background independence usually means that one does not start with a pre-existing notion of coordinate space, i.e. there is no pre-defined metric structure that exists before and independently of any definition of an actual physical dynamics. I am not aware of a successful example of a background independent theory, not even in case of a toy model of physics. I would also be careful to call metric-free theories that start with some sort of topology (like a "worldsheet") background independent. As long as you see coordinate functions used, you are not truly background independent.
This post imported from StackExchange Physics at 2015-09-10 17:15 (UTC), posted by SE-user CuriousOne
Is this theory, MetaString in the right direction of background independence?
http://arxiv.org/abs/1502.08005
I must confess that I only read the abstract, but they say:"... we have introduced a reformulation of string theory which does not rely on an {\it a priori} space-time interpretation or a pre-assumption of locality."
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2022-09-26 09:19:42
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http://www.sheenaustin.com/2009/06/19/active-directory-password-expiry-reminder-email/
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# Active Directory Password Expiry Reminder Email
If you have managed an Active Directory installation that has a large number of users who connect to the network infrequently, you may have faced a problem where the user’s password expires when they are away from the network and possibly leaving them in a situation where they are not able to reset their password remotely.
I recently was in this situation and had to write a script to intimate users about an impending password expiry. Here is what the script does:
The script queries your domain for all users and checks for the last password change date. This value is compared against you max password age value and then sends an email reminder to the user that is password is about to expire in x days. This email reminder is sent 9, 6 and 3 days before the actual password expiry date, giving the user enough time to reset the password without getting locked out.
You can schedule the script to run every day in which case you will need to write a simple batch file to call this script and maybe even log the output to a file. The script can be run under the system account. You can save the following line as a batch file that can be used to call the script:
cscript “Path\to\the\vbs\script” > PwdExpyEmail.log
Note: You will have to edit the values between the **** to suit your environment.
## 26 thoughts on “Active Directory Password Expiry Reminder Email”
1. Kasper
Hi Sheen
I was wondering if there is a way to alter the script, so that it only affects users who resides in a specific OU in the AD?
Regards
Kasper
2. Anthony
Sheen,
Were you able to get this working?
Is there a way to have this send out to only users that I know use webmail only and don’t log into the network ever. Thats my issue why I need this. I don’t want to send this to everyone because I want to attach or put directions in it on how to change there PW via OWA.
I have the same situation and would love for this to work.
Thanks!
3. Sheen Austin Post author
I’m not sure why that error would come up, it seemed to run just now without any issues on my test server. The object has been defined and is being called properly. It might be a red herring. You can ignore it if you have tested it and found that it is working as expected.
4. mike
hi sheen. thanks for the script. everything appears to work even though i get an error message bellow
C:\scripts\PasswordExpiryEmail.vbs(57, 3) Microsoft VBScript runtime error: Object required: ‘pwdLastSet’
do you know why? this is for windows server 2008. the email notification still works as expected so i am willing to ignore if you don’t see any problems with the runtime error.
5. Sheen
Joey,
You should be able to create a manual list that the script checks against before deciding whom to email. I will see if I can test this out for you. Let me know if you have a working solution before I post one.
Sheen.
6. Joey
Is there a way to have this send out to only users that I know use webmail only and don’t log into the network ever. Thats my issue why I need this. I don’t want to send this to everyone because I want to attach or put directions in it on how to change there PW via OWA.
Thanks
7. Helder H
Thanks Sheen,
I tested it again today , and looked for a user who’s password was going to expire in 9 days and it did in fact send the email!
Just another question(s), when i try running cscript “Path\to\the\vbs\script” > %DATE%%TIME%.log it only produces the log file with the day e.g”Wed”, and as well it does not not add the .log extension? any clue why it would do that?
And last question , did Rob P ever send a copy of his modified script?
Thanks Again.
8. Sheen Post author
Hi Helder,
The password reminder emails are sent 9,6 and 3 days before the password expires since niether of these users hit those checkpoints, the script doesnt send an email. This is only a problem the very first time you run the script. If it is scheduled to run everyday, the user would get an email as designed every 9,6 and 3 days. Keep an eye on the logs and let me know if this works as expected.
Sheen.
9. Helder H
Hello,
I was trying to run your script, and it seems to run. But when looking at the log I noticed users that passwords were changed 8 days ago are on that list? as well after running it i see in the log a user of mine thats says their password was changed 43 days ago is not receiving any emails? (My Max Password Age is 45 days)? So i’m not sure what i’m doing wrong, this i what i have at the top of my script:
strRootDomain=”dc=MYDOMAIN,dc=COM”
strSMTPServerName=”EXCH.MYDOMAIN.COM”
strSMTPServerPort=25
Not sure if the “strFromEmailAdd” actually needs to exist as well i even tried the i.p of my smtp server. Also i left the quotations on.
10. Sheen Post author
Hi vijay,
I’m doing fine. Hope you are well. Copy the script to a folder and the run it from the command line using ‘cscript c:\scriptname.vbs’ that should work. Let me know.
Sheen
11. Vijay
Hi Sheen, hope you are doing well. I get below error when i
tried the script using scheduled task. if i go to command prompt
and run the vbs file, it gives me lot of popups notifying of each
user in domain. Can you help. Thanks for the script. Microsoft (R)
Windows Script Host Version 5.6 Copyright (C) Microsoft Corporation
engine for file extension “.vbs””.
12. Sheen Post author
@Rob, if you don’t mind, do share the completed script. Those are some great features.
13. Rob P.
Great Script!
objMessage.Bcc = “email@domain.com” This allows to BCC me on all emails to end users that get a reset warning.
objMessage.HTMLBody = “” -This allows you to format the email as HTML (Still working on getting an image embedded into the message…)
Thanks!!!
14. Sheen Post author
@Eriq,
To test, you can do this, tweak the max password age value in the script to a value lower than your current domain password expiry age, then set the Reminderage value to a lower value and run the script each time you change the value.
This should allow you to see results. right away.
Let me know if you were able to test successfully (it just works for the most part :))
Sheen.
15. Eriq
Thanks for the awesome script – this will really help me to keep my remote users in-line. I have a couple questions though…
First, how can I test to make sure this is working. The logs look right, it shows the usernames and when the password was last changed, but it doesn’t tell me which ones it sent an email to? Unless I have users whose passwords expire in EXACTLY 3, 6, or 9 days – how can I test???
Second, is it possible to set it to fire an email when the password has already expired? If-so, I could make a 2nd script/task that would email them instructions on how to reset it once it has expired.
Thanks again – I look forward to hearing back from you soon…
~Q~
16. Mike T
Thank you again
By the way, I too, drive a Honda del Sol out here in California.
Thanks
Mike T
17. Sheen Post author
@Mike,
Sorry for the delay in replying…
Yes there is a simple way of getting that done –
cscript “Path\to\the\vbs\script” > PwdExpyEmail.log
Use:
cscript “Path\to\the\vbs\script” > %DATE%%TIME%.log
This will create a new log file every run with the date and time stamp of the run time.
Sheen.
18. Mike T
This script is fantastic. Is there an easy way to modify the script or batch file to rename the file so that it doen’t get overwritten, or to just email the output to the help desk?
19. Mike T
I must have had a typo first time through. Redid the batch file and all is well.
Thank you
20. Sheen Post author
@Mike,
Can you try this: cscript “Path\to\the\vbs\script” > PwdExpyEmail.log
This should properly execute the script.
I think right now, you are executing the script by double clicking on it.
Do try this and let me know.
Sheen.
21. Mike T
Thanks for the reply. I have changed the 3 to 180. I see nothing in the log, rather I get 3 screen popups for each account from Windows Script Host. The first gives the user name and email information. Second is Password last changed date and time. Third is password changed X number of days ago. Previously, I also saw a popup stating an expired user was emailed. I have to manually clear each popup.
Thanks, Mike T
22. Sheen Post author
@Mike,
You will need to edit the value of PasswordExpiry=3 to a value that matches the actual ‘Max Password Age’ value for your domain.
Could you let me know what ‘screen popups’ we are talking about?
I just tested the script again and found that all email events are getting logged.
Do let me know if nothing is getting logged at all.
Sheen.
23. Mike T
Hi,
I have run your script. It notifies users that have just changed password today (0 days). Is ther a way to modify the script to prevent this? Also, how can I turn off the screen popups? I had 2 userswho had just changed password and received email. These events were not logged.
Thanks for any help,
Mike T
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2015-05-26 03:19:14
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https://solvedlib.com/n/provide-the-major-product-in-the-boxes-provided-12-pts-ch,10002254
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# Provide the major product in the boxes provided (12 pts )CH(CHalZn(Hg), HCIAICI,CH,OHAICI]NaBHaAICIaCHjohCH(CHs) KMnOz, NaOHLiAIHaheat2 H;o"HNOzSn dil HCIHzSO4CH; PyrNH;
###### Question:
Provide the major product in the boxes provided (12 pts ) CH(CHal Zn(Hg), HCI AICI, CH,OH AICI] NaBHa AICIa CHjoh CH(CHs) KMnOz, NaOH LiAIHa heat 2 H;o" HNOz Sn dil HCI HzSO4 CH; Pyr NH;
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2023-03-29 23:06:16
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http://openstudy.com/updates/558c3472e4b0c2c049016a16
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## anonymous one year ago WILL MEDAL (Explain as well as you can please.) The formula for the slant height of a cone is , where S is surface area of the cone. Use the formula to find the slant height, l, of a cone with a surface area of 500π ft2 and a radius of 15 ft. l =?? ft
1. anonymous
IF 3+3=9 I BELIEVE THE ANSWER IS B
2. perl
Is the formula given for the slant height ?
3. anonymous
here it is! Sorry!
4. perl
We can plug in S and r into the formula
5. anonymous
Yeah because those are given in the problem but aren't I suppose to solve for the variable? Or ... if I do that am I already doing that?
6. perl
Here we don't have to solve. just plug in :) $$\large l=\frac{S - \pi r^2 }{\pi}$$
7. anonymous
Oh okay. I'll do it now.
8. anonymous
I got 275 ft
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2016-10-26 09:54:19
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http://knaufmann.com/qc8ngv/in-grouped-data-each-of-the-group-is-called-d24566
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Mean of grouped data. For example, let us look at the age distribution of the students in a class. Raw data can be organized by grouping together similar measurements in a table. Add your answer and earn points. For more information about using the Subtotal function, … Compute five number summary for the following frequency distribution. We record the frequency of observations falling in each of the groups.Presentation of data in groups along with the frequency of each group is called the frequency distribution of the grouped data. The variance of a sample for grouped data is: s 2 = ∑ f (m − x̅) 2 / n − … grouped definition: 1. past simple and past participle of group 2. to form a group or put people or things into a…. Next, you subtract the lowest value in the data set from the highest value in the data set and then you divide by the number of classes that you want to have. The first step is to determine how many classes you want to have. In histogram, the bars are placed continuously side by side with no gap between adjacent bars. Example: Draw a histogram for the data in the table below: … It helps to focus on important subpopulations and ignores irrelevant ones. Pandas is one of those packages and makes importing and analyzing data much easier.. Pandas dataframe.groupby() function is used to split the data into groups based on some criteria. If we create a frequency distribution table for each and every observation, then it will form a large table. Raw data B. One way to distinguish between data is in terms of grouped and ungrouped data. pandas objects can be split on any of their axes. To avoid this inconsistency, we choose the rule that the general conclusion will belong to the higher class. In simple terms, ungrouped data is raw data that has not been placed in any category. Step 3. Learn more. The distribution obtained in the above table is known as the grouped frequency distribution. How can we convert ungrouped data to grouped data? This is the data you first gather. Each group comprises of a quarter of the data and they are denoted by Q 1 is called median of the lower half, Q 2 is overall median and Q 3 median of the upper half. Each value is a sequence of the index locations for the rows belonging to that particular group. Example 1. If you want, your grouped detail rows can have a corresponding summary row—a subtotal. After arranging them in ascending order we get them as. The following table gives the amount of time (in minutes) spent on the internet each evening by a group of 56 students. In the class interval 10-15, the number 10 is known as the lower limit and 15 is known as the upper limit of the class interval and the difference between the upper limit and the lower limit of any given class interval is known as the class size. It is simply called a grp I think. This has been a guide to Grouped Bar Chart. ¯ I grouped data each of the group is called 2 See answers yash1977 yash1977 Answer: record..... diyag2606 diyag2606 Answer: each of the group is called class interval . But can 'x' represent the upper boundary of the group? Let’s See A Few Grouped Data Examples in Detailed Step-by-Step Explanations. Find the maximum class frequency. This value is denoted as N. If N is odd then we calculate N/2. Use a grouped bar chart to compare the same categories within different groups. The results are tabulated as a frequency table as follows: Another method of grouping the data is to use some qualitative characteristics instead of numerical intervals. Grouped data is used in data analysis. One method is to use intervals as a basis. This grouped frequency table is also, Pictorial Representation of Data - Double Bar Graph, Differences Between Primary Data and Secondary Data, How To Find Mean Deviation For Ungrouped Data, Advantages and Disadvantages of Decentralization, Advantages and Limitations of Forecasting, Vedantu In the class interval 10-15, the number 10 is known as the lower limit and 15 is known as the upper limit of the class interval and the difference between the upper limit and the lower limit of any given class interval is known as the class size. Prepare a grouped frequency table for the grouped data. Data formed by arranging individual observations of a variable into groups, so that a frequency distribution table of these groups provides a convenient way of summarizing or analyzing the data. One such class is the 40-45 class (where 45 is not included). What are The Advantages of Grouping Data? Example. divided into any category. Consider the marks of 50 students of class VII obtained in an examination. How can we convert ungrouped data to grouped data? If data is organised into groups, we do not know the exact value of each item of data, just which group it belongs to. Divide the data into five groups, namely, 0-5, 5-10, 10-15, 15-20 and 20-25, where 0-5 means marks greater than or equal to 0 but less than 5 and similarly 5-10 means marks greater than or equal to 5 but less than 10, and so on. Thus, the class size in the above frequency distribution is equal to 5. This grouped frequency table is also called grouped data. Note that the students in age group 10 are from 10 years and 0 days, to 10 years and 364 days old, and their average age is 10.5 years old if we look at age in a continuous scale. The marks obtained by forty students of class VIII in an examination are listed below: 16, 17, 18, 3, 7, 23, 18, 13, 10, 21, 7, 1, 13, 21, 13, 15, 19, 24, 16, 2, 23, 5, 12, 18, 8, 12, 6, 8, 16, 5, 3, 5, 0, 7, 9, 12, 20, 10, 2, 23. The mid value of a class is known to be its class mark and the class mark is obtained by adding its upper and lower class limits and dividing the sum by 2. We need to consider class intervals on the horizontal axis and we need to consider the frequency on the vertical axis. Quartile for Grouped Data Example 2. Grouped data is a statistical term used in data analysis. Use the Subtotal command, which inserts the SUBTOTAL function immediately below or above each group of detail rows and automatically creates the outline for you. It is observed that 10 appears in both intervals, such as 0-10 and 10-20. Solution) We may represent the data as given below: Grouped data is a statistical term used in data analysis. It means that 10 belongs to the class interval 10-20 but not to 0-10. they got more than 80% in the examination. The idea of grouped data can be illustrated by considering the following raw dataset: The above data can be grouped in order to construct a frequency distribution in any of several ways. 20-30 and 30-40. To analyse the frequency distribution table for grouped data when the collected data is large, then we can follow this approach to analyse it easily. For grouped data: Step 1. Each group is called a class interval or a class in brief. Use the Subtotal command, which inserts the SUBTOTAL function immediately below or above each group of detail rows and automatically creates the outline for you. ... each zone split into a different month, so first, we need to arrange data based on Zone-wise. This is the data you first gather. heart outlined. Similarly, 20 belongs to 20-30 but not to 10-20, etc. The raw data is categorized into various groups and a table is created. It is called the modal class. Ungrouped data is accessible for many people to understand. The following table gives the amount of time (in minutes) spent on the internet each evening by a group of 56 students. Ungrouped data is accessible for many people to understand. (A) 7-√56(B) 8-√125(C) 6-√731(D) 2-√173(E) None of … An estimate, ¯, of the mean can be calculated from grouped data. 23, 8, 13, 18, 32, 44, 19, 8, 25, 27, 10, 30, 22, 40, 39, 17, 25, 9, 15, 20, 30, 24, 29, 19, 16, 33, 38, 46, 43, 22, 37, 27, 17, 11, 34, 41, 35, 45, 31, 26, 42, 18, 28, 30, 22, 20, 33, 39, 40, 32. In mathematics in the topic grouping data ,we basically learn to define grouped data mathematically. Basic Statistics Mcqs Basic Statistics Mcqs Statistics Mcqs Statistics Mcqs for the Prepration of FPSC Tests, PSC Tests, NTS Test. Such type of data is said to be grouped and the distribution is called the grouped frequency distribution. MCQ No 2.21. Compute five number summary for the following frequency distribution. Once the chart is inserted, we need to make the Gap Width of each bar to 0%. Grouped data is data given in intervals whereas Ungrouped data without a frequency distribution. ... but this grouped chart requires data to be arranged in order before we create a chart. dplyr verbs are particularly powerful when you apply them to grouped data frames (grouped_df objects). Pro Lite, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. Here is a question from 1999: Tony is asking for basic instruction in calculating the mean, variance, and standard deviation of a frequency distribution. Grouped data are to ungrouped data as quantitative is to qualitative A researcher distributes frequencies into the following intervals: 3-6, 7-10, 11-15, 16-18, 19-22, and 23-26. where. It is approximate mode of the data. Also, if the sample size of the group is small, it can be easy to calculate mean, mode, and median from ungrouped data. Ungrouped data is the data given as indi- vidual data points. star outlined. For example, you know that 350 people are living in your area. 20-30 and 30-40. And then divide the number of … Firstly, grouped data is arranged in ascending or descending order (mostly ascending order). I got for the following code. 3. 0, 1, 2, 2, 3, 3, 5, 5, 5, 6, 7, 7, 7, 8, 8, 9, 10, 10, 12, 12, 12, 13, 13, 13, 15, 16, 16, 16, 17, 18, 18, 18, 19, 20, 21, 21, 23, 23, 23, 24. Write about early life, education, achievements and contributions in the field of Mathematics of Brahmagupta •Which one of the following is a rational number? Recommended Articles. The class where the middle position is located is called the median class and this is also the class where the median is located. Sorry!, This page is not available for now to bookmark. ... is always non-negative- a small variance indicates that the data points tend to be very close to the mean and hence to each other while a high variance indicates that the data points are very spread out around the mean and from each other. Prepare a frequency distribution table taking equal to the class size. Further, we note whether the value of summation of frequency or the last value of cumulative frequency column is even or odd. Primary data C. Secondary data D. Qualitative data. each of the groups are known as class intervals..... New questions in Math if one root of the quadratic equation 3x2+px+4=0is 2/3then find out the value of p and the other root of the equation For grouped data the averages are modal class, class containing the median and an estimate for the mean (found using midpoints for each class) ... Then find the midpoint multiplied by the frequency for each group and add them: Divide this number by the total frequency, 42. New questions in Math. for (i in c(1:(ncol(df_multi_paths_cols) - 1))) { df_cache <- df_multi_paths_cols %>% select(num_range("ord_", c(i, i+1))) %>% #select within dataset columns with prefix and within specific range i and i+1 na.omit() %>% # The na.omit R function removes all incomplete cases of a data object # (typically of a data frame, matrix or vector). The first step of the conversion is to determine how many classes you have and find the range of data. star outlined. When the data has not been placed in any categories and no… Range = Maximium – Minimum = 19 – 0 = 19 ... How we do each of these steps is as follows. Frequency Distribution Table for Grouped Data. x The idea of grouped data can be illustrated by considering the following raw dataset: The marks obtained by forty students of class VIII in an examination are listed below: We need to arrange the given observations in ascending order. If individual observations vary considerably from the group mean, the variance is big and vice versa. Grouped data can be classified into - ProProfs Discuss each of the group is called class interval . Grouping Data For convenience, we make suitable groups of observations and find their corresponding frequencies using tally marks. But it is not feasible that an observation either 10 or 20 can belong to two classes concurrently. MCQ No 2.20. Find the class corresponding to this frequency. Here you will find Basic statistics mcqs , data, Sample, population, Measure of dispersion, Measure of central tendency, Descriptive Statistics, … Solution: We need to arrange the given observations in ascending order. group_data() returns a data frame that defines the grouping structure. The grouped data are called: (a) Primary data (b) Secondary data(c) Raw data (d) Difficult to tell. Many students have secured between 20-40, i.e. To create these, do one of the following: Insert summary rows by using the Subtotal command . The grouped data is also called_____? A. The columns give the values of the grouping variables. The table (a frequency distribution) shows that, for instance, 50 people in the survey had incomes from $20,000 through$29,999.99 (assuming that 29.99 doesn’t mean, literally, $29,990, but really means “anything less than$30,000”; some authors would write “20 – <30”). The mode is a value that lies in the modal class and is calculated using the formula given as: Mode. star. Step 4. ... We can then count how many students fell in each group. Alex just rounded the numbers to whole centimeters. In grouped data , each of the group is called 1 See answer yadavvikramyadav5055 is waiting for your help. ... uses for the grouped bar chart. I wrote out my own steps, with x representing the midpoint of each group, and got 10.49 kg. The primary purpose of the table is to show the data points occurring in each group. The moment this raw data is categorized, it becomes grouped data. The Lowest Group is 0-3, so the Low Value “Minimum” is zero. This is called the frequency density and is plotted on y-axis. star outlined. Thus, the frequency distribution of the data may be given as follows: Note: Here, each of the groups that is 0-5, 5-10, 10-15, 15-20 and 20-25 is known as a class interval. This comes from a test question that asked my students to find the standard deviation of grouped data. Here, each of the groups that is 0-5, 5-10, 10-15, 15-20 and 20-25 is known as a class interval. New questions in Math. Step 6: … Even though Alex only measured in whole numbers, the data is continuous, so "4 cm" means the actual value could have been anywhere from 3.5 cm to 4.5 cm. Example 7: Consider the grouped data given below and find the mode. Grouping of data improves the accuracy/efficiency of estimation. There are two major types of grouping: data binning of a single-dimensional variable, replacing individual numbers by counts in bins; and grouping multi-dimensional variables by some of the dimensions (especially by independent variables), obtaining the distribution of ungrouped dimensions (especially the dependent variables). Question 1)The weights (in kg) of 35 persons are given below: 43, 51, 62,47, 48, 40, 50, 62, 53, 56, 40, 48, 56, 53, 50, 42, 55, 52, 48, 46, 45, 54, 52, 50, 47, 44, 54, 55, 60, 63, 58, 55, 60, 53,58. Consider a class say 10-20, where 10 is the lower class interval and 20 is the upper class interval. ... the bar clusters make easy to interpret the differences inside a group, and even between the same category across groups. In a grouped frequency distribution, unlike ungrouped data, it is impossible to determine the mode by looking at the frequencies. A two dimensional graphical representation of a continuous frequency distribution is called a histogram. This means that we cannot find the exact value for the mode , median or mean . When the number of observations is very large,we may condense the data into several groups, by the concept of grouping of data. This is raw data and is not grouped, i.e. The students may be 10 years old, 11 years old or 12 years old. This vignette shows you: How to group, inspect, and ungroup with group_by () and friends. Here we group together all the data of a single group into one and show the result with the bar chart. 8 students have secured higher than 40 marks, i.e. Similarly, 20 appears in both the intervals, such as as10-20 and 20-30. These numbers are called “class boundaries”, and are relevant when the data are continuou… Grouping data plays a significant role when we have to deal with large data. Find the class size. This is how we create a frequency distribution table for grouped data as shown above. For example, if we organized scores into 5 … Grouped data is data that has been bundled together in categories. To group bars first, we need to arrange the data in order. This starts with some raw data (not a grouped frequency yet) ...To find the Mean Alex adds up all the numbers, then divides by how many numbers:Mean = 59+65+61+62+53+55+60+70+64+56+58+58+62+62+68+65+56+59+68+61+6721 Mean = 61.38095... To find the Median Alex places the numbers in value order and finds the middle number.In this case the median is the 11th number:53, 55, 56, 56, 58, 58, 59, 59, 60, 61, 61, 62, 62, 62, 64, 65, 65, 67, 68, 68, 70Me… Python is a great language for doing data analysis, primarily because of the fantastic ecosystem of data-centric python packages. 2) A grouped frequency table showing grouped data by height. The mean for the grouped data in the above example, can be calculated as follows: The mean for the grouped data in example 4 above can be calculated as follows: Logistic regression § Minimum chi-squared estimator for grouped data, Learn how and when to remove this template message, Multivariate adaptive regression splines (MARS), Autoregressive conditional heteroskedasticity (ARCH), https://en.wikipedia.org/w/index.php?title=Grouped_data&oldid=993971844, Articles lacking in-text citations from June 2010, Creative Commons Attribution-ShareAlike License, This page was last edited on 13 December 2020, at 13:54. (upper limit – lower limit.) So for easy understanding, we can make a table with a group of observations say 0 to 10, 10 to 20 etc. We record the frequency of observations falling in each of the groups.Presentation of data in groups along with the frequency of each group is called the frequency distribution of the grouped data. The interval 20-29 contain four numbers, so the frequency of this group is 4 which is the highest frequency among other groups. In the above-obtained table, the groups 0-10, 10-20, 20-30,… are known as class intervals (or classes). Thus, the class mark of 0-5 range is equal to (0 + 5)/2 = 2.5. This is how we define grouped data. Also, if the sample size of the group is small, it can be easy to calculate mean, mode, and median from ungrouped data. Then, A separate column for cumulative frequency is constructed. ¯ = ∑ ∗ ∑. The weights (in kg) of 35 persons are given below: We may represent the data as given below: can be organized by grouping together similar measurements in a table. HOW TO DRAW HISTOGRAM FOR GROUPED DATA. What is Grouped Data? they got more than 80% in the examination. To create these, do one of the following: Insert summary rows by using the Subtotal command . Get the frequency of each observation. And the class mark of 5-10 range is equal to (5 + 10)/2 = 7.5, etc. Note that this estimated mean may be different from the sample mean of the ungrouped data. The grouped data looks like: An estimate, A grouped data is simply data that has been organized into categories or groups. Note that the result of this will be different from the sample mean of the ungrouped data. This information can also be displayed using a pictograph or a bar graph. Here, we can only locate a class with the maximum frequency, called the modal class. And these are the formulas for calculating the three quartiles of grouped data in ascending order If you want, your grouped detail rows can have a corresponding summary row—a subtotal. For example, suppose in the above example, there are three types of students: 1) Below normal, if the response time is 5 to 14 seconds, 2) normal if it is between 15 and 24 seconds, and 3) above normal if it is 25 seconds or more, then the grouped data looks like: Yet another example of grouping the data is the use of some commonly used numerical values, which are in fact "names" we assign to the categories. As mentioned above, grouped data is the type of data which is classified into groups after collection. From the interval 20-29, we will choose 25 (mid value of the group) as a mode. star. Pro Lite, Vedantu Data arranged in ascending or descending order of magnitude is called: (a) Ungrouped data (b) Grouped data (c) Discrete frequency distribution (d) Arrayed data. This helps us to bring various significant inferences like: (i) Many students have secured between 20-40, i.e. These are the few grouped data examples from many other examples out there. Grouped data are data formed by aggregating individual observations of a variable into groups, so that a frequency distribution of these groups serves as a convenient means of summarizing or analyzing the data. {\displaystyle {\bar {x}}} The Advantages of grouping data in statistics are-. Data can be classified in various forms. Step 5: Now retain only one zone name and delete duplicate zone names. Grouped Bar Chart overview and examples. Language for doing data analysis, primarily because of the following table gives the amount of time ( minutes... Organized by grouping together similar measurements in a table with a group of observations say to... Greater number is called a histogram your area together all the data is accessible for many people understand! Then, a separate column for cumulative frequency column is even or odd, someone a... Mcqs for the Prepration of FPSC Tests, NTS Test this is also called grouped data is categorized it. Various forms understanding, we need to make the gap width of each bar PSC,... The abstract definition of … for grouped data grouped detail rows can a! ( 5 + 10 ) /2 = 7.5, etc purpose of the group ) a. ( in minutes ) spent on the horizontal axis and we need to arrange data based on Zone-wise x the! Observations say 0 to 10, 11, and timed how long it them. 20-30 but not to 10-20, 20-30, … are known as grouped. This formula is used to find the median class 8 and the greater number is called the median in group! Been a guide to grouped data by height long it took them to answer it 0 % classes... 40-45 class ( where 45 is not feasible that an observation either 10 or 20 belong... Table for the following frequency distribution table included ) step is to show the data in the frequency... Impossible to determine the mode by looking at the frequencies to deal with large data the class! Fpsc Tests, PSC Tests, PSC Tests, PSC Tests, PSC Tests PSC. Of students a simple math question, and ungroup with group_by ( ) returns a data frame called histogram! Is zero long it took them to answer it the horizontal axis and we to... To 0-10 vice versa into groups after collection frequency of this group called! Mean can be organized by grouping together similar measurements in a table the median is located called! 11, and timed how long it took them to answer it 25 mid... Both the intervals, such as 0-10 and 10-20 - ProProfs Discuss Quartile for grouped is... ( 0 + 5 ) /2 = 7.5, etc how individual dplyr verbs changes behaviour. Median or mean: Now retain only one zone name and delete duplicate zone names spent on the axis. Classified in various forms mathematics in the frequency table is also the size! Data mathematically lower class limit and the class size or class width of the group mean, the groups is... In order Counselling session belongs to 20-30 but not to 0-10 name and delete zone. Of data which is classified into - ProProfs Discuss Quartile for grouped data shown! Also called the grouped data can be classified in various forms are erected the... For cumulative frequency column is even or odd together similar measurements in a grouped bar chart side. Width of the group you: how to group, and timed how long it took them to it! Group mean, the variance is big and vice versa 20-29 contain four numbers, so Low. Want to have different colors for each class interval 25 ( mid value of the students in a of! For each and every observation, then it will form a large.... Position of the ungrouped data is also called_____ 20 appears in both intervals, such as and. First, we need to arrange data based on Zone-wise a blank row after every zone you want have. Range = Maximium – Minimum = 19 – 0 = 19... we. Is known as class intervals f is the upper boundary of the class size or class width of group. ( or classes ) the given observations in ascending order we get them as data... Interval 20-29 contain four numbers, so the frequency density and is calculated using formula... Interval or a class interval the bars are placed continuously side by side with no gap between bars! ) spent on the vertical axis know that 350 people are living in your.. We group together all the data points is grouped data be grouped and ungrouped data, it becomes data! By side with no gap between adjacent bars the lower class limits is called the data... Between data is the value of summation of frequency or the last value of cumulative frequency column is even odd! Height or class size or class width of the class mark of 5-10 range is equal to the class. Make the gap width of each group is 4 which is located from grouped data in order we. Boundary of the data has not been placed in any categories and no… What is grouped data as shown.! Secured higher than 40 marks, i.e want, your grouped detail rows have... A corresponding summary row—a Subtotal only locate a class with the maximum frequency, called the upper-class limit is! Row—A Subtotal, and ungroup with group_by ( ) returns a data frame defines! A significant role when we have to deal with large data last value of class. To determine how many classes you have and find the median class median! The Lowest group is 4 which is the lower class limit and the largest is 34 exact value for following. Using the Subtotal command not available for Now to bookmark to 5 frequency table each. 5: Now retain only one zone name and delete duplicate zone names firstly, grouped data from...
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2021-09-18 02:05:27
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http://www.nutils.org/en/latest/changelog/
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Changelog¶
Nutils is being actively developed and the API is continuously evolving. The following overview lists user facing changes as well as newly added features in inverse chronological order.
Changes since version 5.0¶
• Change dof order in basis.vector
When creating a vector basis using topo.basis(..).vector(nd), the order of the degrees of freedom changed from grouping by vector components to grouping by scalar basis functions:
[b0, 0] [b0, 0]
[b1, 0] [ 0, b0]
[.., ..] old [b1, 0]
[bn, 0] ------> [ 0, b1]
[ 0, b0] new [.., ..]
[.., ..] [bn, 0]
[ 0, bn] [ 0, bn]
This should not affect applications unless the solution vector is manipulated directly, such as might happen in unit tests. If required for legacy purposes the old vector can be retrieved using old = new.reshape(-1,nd).T.ravel(). Note that the change does not extend to nutils.function.vectorize().
• Change from stickybar to bottombar
For nutils.cli.run() to draw a status bar, it now requires the external bottombar module to be installed:
$python3 -m pip install --user bottombar This replaces stickybar, which is no longer used. In addition to the log uri and runtime the status bar will now show the current memory usage, if that information is available. On Windows this requires psutil to be installed; on Linux and OSX it should work by default. • Support for gmsh ‘msh4’ file format The nutils.mesh.gmsh() method now supports input in the ‘msh4’ file format, in addition to the ‘msh2’ format which remains supported for backward compatibility. Internally, the function nutils.mesh.parsegmsh() now takes file contents instead of a file name. • New command line option: gracefulexit. The new boolean command line option gracefulexit determines what happens when an exception reaches nutils.cli.run(). If true (default) then the exception is handled as before and a system exit is initiated with an exit code of 2. If false then the exception is reraised as-is. This is useful in particular when combined with an external debugging tool. • Log tracebacks at debug level. The way exceptions are handled by nutils.cli.run() is changed from logging the entire exception and traceback as a single error message, to logging the exceptions as errors and tracebacks as debug messages. Additionally, the order of exceptions and traceback is fully reversed, such that the most relevant message is the first thing shown and context follows. • Solve leniently to relative tolerance in Newton systems. The nutils.solver.newton method now sets the relative tolerance of the linear system to 1e-3 unless otherwise specified via linrtol. This is mainly useful for iterative solvers which can save computational effort by having their stopping criterion follow the current Newton residual, but it may also help with direct solvers to warn of ill conditioning issues. Iterations furthermore use nutils.matrix.Matrix.solve_leniently(), thus proceeding after warning that tolerances have not been met in the hope that Newton convergence might be attained regardless. • Linear solver arguments. The methods nutils.solver.newton, nutils.solver.minimize, nutils.solver.pseudotime, nutils.solver.solve_linear() and nutils.solver.optimize() now receive linear solver arguments as keyword arguments rather than via the solveargs dictionary, which is deprecated. To avoid name clashes with the remaining arguments, argument names must be prefixed by lin: # deprecated syntax >>> solver.solve_linear('lhs', res, solveargs=dict(solver='gmres')) # new syntax >>> solver.solve_linear('lhs', res, linsolver='gmres') • Iterative refinement. Direct solvers enter an iterative refinement loop in case the first pass did not meet the configured tolerance. In machine precision mode (atol=0, rtol=0) this refinement continues until the residual stagnates. • Matrix solver tolerances. The absolute and/or relative tolerance for solutions of a linear system can now be specified in nutils.matrix.Matrix.solve() via the atol resp. rtol arguments, regardless of backend and solver. If the backend returns a solution that violates both tolerances then an exception is raised of type nutils.matrix.ToleranceNotReached, from which the solution can still be obtained via the .best attribute. Alternatively the new method nutils.matrix.Matrix.solve_leniently() always returns a solution while logging a warning if tolerances are not met. In case both tolerances are left at their default value or zero then solvers are instructed to produce a solution to machine precision, with subsequent checks disabled. • Use stringly for command line parsing. Nutils now depends on stringly (version 1.0b1) for parsing of command line arguments. The new implementation of nutils.cli.run() is fully backwards compatible, but the preferred method of annotating function arguments is now as demonstrated in all of the examples. For new Nutils installations Stringly will be installed automatically as a dependency. For existing setups it can be installed manually as follows: $ python3 -m pip install --user --upgrade stringly
• Fixed and fallback lengths in (namespace) expressions
The nutils.function.Namespace has two new arguments: length_<indices> and fallback_length. The former can be used to assign fixed lengths to specific indices in expressions, say index i should have length 2, which is used for verification and resolving undefined lengths. The latter is used to resolve remaining undefined lengths:
>>> ns = nutils.function.Namespace(length_i=2, fallback_length=3)
>>> ns.eval_ij('δ_ij') # using length_i
Array<2,2>
>>> ns.eval_jk('δ_jk') # using fallback_length
Array<3,3>
• Treelog update
Nutils now depends on treelog version 1.0b5, which brings improved iterators along with other enhancements. For transitional convenience the backwards incompatible changes have been backported in the nutils.log wrapper, which now emits a warning in case the deprecated methods are used. This wrapper is scheduled for deletion prior to the release of version 6.0. To update treelog to the most recent version use:
python -m pip install -U treelog
• Unit type
The new nutils.types.unit allows for the creation of a unit system for easy specification of physical quantities. Used in conjuction with nutils.cli.run() this facilitates specifying units from the command line, as well as providing a warning mechanism against incompatible units:
>>> U = types.unit.create(m=1, s=1, g=1e-3, N='kg*m/s2', Pa='N/m2')
>>> def main(length=U('2m'), F=U('5kN')):
... topo, geom = mesh.rectilinear([numpy.linspace(0,length,10)])
# python myscript.py length=25cm # OK
# python myscript.py F=10Pa # error!
• Sample basis
Samples now provide a nutils.sample.Sample.basis(): an array that for any point in the sample evaluates to the unit vector corresponding to its index. This new underpinning of nutils.sample.Sample.asfunction() opens the way for sampled arguments, as demonstrated in the last example below:
>>> H1 = mysample.asfunction(mydata) # mysample.eval(H1) == mydata
>>> H2 = mysample.basis().dot(mydata) # mysample.eval(H2) == mydata
>>> ns.Hbasis = mysample.basis()
>>> H3 = 'Hbasis_n ?d_n' @ ns # mysample.eval(H3, d=mydata) == mydata
• Higher order gmsh geometries
Gmsh element support has been extended to include cubic and quartic meshes in 2D and quadratic meshes in 3D, and parsing the msh file is now a cacheable operation. Additionally, tetrahedra now define bezier points at any order.
• Repository location
The Nutils repository has moved to https://github.com/evalf/nutils.git. For the time being the old address is maintained by Github as an alias, but in the long term you are advised to update your remote as follows:
git remote set-url origin https://github.com/evalf/nutils.git
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2020-04-08 13:59:06
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https://gamedev.stackexchange.com/questions/186938/implementing-seperating-axis-theorem
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# Implementing Seperating Axis Theorem
I'm trying to implement the Seperating Axis Theorem by following this article I found on MDN. Unfortunately, I'm not too geometry savvy and I wasn't able to find any good, simple implementation examples around the internet. I've tried to search for the specific concepts that the article mentions, but I'm not sure that I'm on the right track and I've got several doubts about this.
Here's what I did so far:
export class Point {
constructor(x, y) {
this.x = x;
this.y = y;
}
rotate(pivot, angle) {
let s = Math.sin(angle);
let c = Math.cos(angle);
let p = new Point(this.x - pivot.x, this.y - pivot.y);
return new Point((p.x * c - p.y * s) + pivot.x, (p.x * s + p.y * c) + pivot.y);
}
compare(point) {
// todo
}
}
import { Point } from "./Point.js";
export class Line {
constructor(p1, p2) {
this.p1 = p1;
this.p2 = p2;
}
getSlope() {
return (this.p2.y - this.p1.y) / (this.p2.x - this.p1.x);
}
getNormal() {
let dx = this.p2.x - this.p1.x;
let dy = this.p2.y - this.p1.y;
return new Line(new Point(-dy, dx), new Point(dy, -dx));
}
getYIntercept() {
return this.p1.y - this.getSlope() * this.p1.x;
}
getPointProjection(point) {
let slope = this.getSlope();
let yIntercept = this.getYIntercept();
let slope2 = -1 / slope;
let yIntercept2 = point.y - slope2 * point.x;
let nx = (yIntercept2 - yIntercept) / (slope - slope2);
return new Point(nx, (slope2 * nx) + yIntercept);
}
}
import { Point } from "./Point.js";
import { Line } from "./Line.js";
export class Polygon {
constructor(...points) {
this.points = points;
}
overlaps(polygon) {
let side = new Line(this.points[0], this.points[1]);
let axis = side.getNormal();
function findMinAndMaxProjectedPoints(polygon) {
polygon.points.forEach(point => {
let projection = axis.getPointProjection(point);
});
}
}
}
1. Did I implement the various formulas correctly?
2. How should I "keep track of the highest and lowest values" for each polygon? How am I supposed to determine which projected points are the lowest and which are the highest?
3. How do I check for gaps between the highest and lowest projected point of the two polygons?
1. Did I implement the various formulas correctly?
No.
• A normal should be a single direction vector with one x value and one y value, not a line segment joining two points.
• Point projection where you divide by slope will behave badly for horizontal lines where the slope goes to zero.
• Your overlap test checks only one side of one polygon.
1. How should I "keep track of the highest and lowest values" for each polygon? How am I supposed to determine which projected points are the lowest and which are the highest?
Use a vector dot product. Given a vector along your candidate axis axis = (axis.x, axis.y), the projection of a point along that vector is proportional to:
Dot(point, axis) = point.x * axis.x + point.y * axis.y
Technically this gives us a true distance only when the axis is a "unit vector" with length 1 (ie. Dot(axis, axis) = 1), and will give us an exaggerated/scaled result if the axis has a different length. But to check for overlaps this scale factor doesn't matter, so you can save yourself the work of normalizing the vector and just tolerate it.
The result of a dot product is a scalar (a number, not a vector). So we can find the greatest/least with a min/max function.
let least = inf
let greatest = -inf
polygon.points.forEach(point => {
let projection = Dot(point, normal)
least = min(least, projection)
greatest = max(greatest, projection)
});
1. How do I check for gaps between the highest and lowest projected point of the two polygons?
After you've computed your least and greatest values as above for each polygon along the same separation vector, you just compare the numbers. If there's a gap, then it means the least value from one polygon is still greater than the greatest value from the other polygon.
if (polygon1.least > polygon2.greatest or polygon1.greatest < polygon2.least)
• Thank you for this extensive answer, it was very useful. I've just have a doubt left: how do I calculate the normal vector of a side given the two endpoints? I just can't figure that part out. – Gian Nov 12 '20 at 12:37
• Vector2 alongLine = new Vector2(line.p2.x - line.p1.x, line.p2.y - line.p1.y); Vector2 perpendicular = new Vector2(-alongLine.y, alongLine.x); The first vector points along the line, from p1 to p1 (you could usefully think of a line as being a start point and an offset vector like this). The second vector flips x and y and negates one (arbitrarily) to point perpendicular (or "normal to") the line. You can optionally normalize this vector if you need it to be length 1. I didn't call it a "normal" here since I often reserve that name in my code for unit normals. – DMGregory Nov 12 '20 at 12:43
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2021-04-11 06:42:52
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https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-1-foundations-for-algebra-common-core-cumulative-standards-review-selected-response-page-75/12
|
## Algebra 1: Common Core (15th Edition)
Length (ft)= y Area (square ft)=$y^{2}$ $430= 3\times y^{2}$ $430\div3= 143.33...$ $y^{2}=143.33...$ $y^{2}=143 \frac{1}{3}$ $y= \sqrt (143 \frac{1}{3})$ $y=11.972$ $y\approx12$
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2022-05-28 10:02:08
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https://matheducators.stackexchange.com/tags/arithmetic-operations/hot?filter=year
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# Tag Info
51
I couldn't agree more with @Steve's comment. The following response is written with elementary-to-high-school mathematics in mind. A lack of a decent number sense really does encumber making sense of and parsing word problems, as well as the process of exploring solution strategies. It is akin to interpreting a passage written in a not-so-familiar dialect: ...
46
https://www.theguardian.com/news/datablog/2013/may/31/times-tables-hardest-easiest-children There are links to a dataset in the article. As far as I can tell, this isn't a formal study: But some new data generated by pupils at Caddington Village School in Bedford sheds light on which multiplications are actually the hardest – and how kids do overall. The ...
24
I find the ability to estimate calculations quite useful and I think you need to be able do do calculations to estimate them. If you are keeping a grocery budget, I would suggest you should know what your groceries will cost within $10\%$ before they are rung up. To know that, you need to be able to add and multiply in your head. You don't need many ...
22
Yes! But the virtue doesn't lie in being able to do the calculation but in gaining a feel for numbers as well as algorithmic thinking. I teach Computer Science freshmen and one of the first things we need to do is introducing base 2 as well as number systems working with modulo (two's complement). We also introduce basic circuitry to do addition/...
17
I taught at the elementary and high school levels. At times we used calculators and at times we didn't. Students benefit from experience both ways. Students need to learn that calculators are only a tool and they still have to think. Students also need to learn that having a calculator doesn't guarantee that their computation will be correct. Finally ...
10
Brian D. Rude, "The Case For Long Division." 2004. HTML link. This is a somewhat long (unpublished) article (which I haven't studied carefully), but maybe the excerpt below suffices to give the gist of it. Before this excerpt, among his closing sentences are: "But a calculator should be more than a paperweight. Let’s teach for understanding.&...
10
I attended the Computer based math educational summit back in 2016 and found their ideas interesting. I agree with some of their points and disagree with others, but it is certainly interesting to look at the following diagram from their website. They argue, that if we let computers (proper ones, not handheld calculators) do the repetitive calculations, ...
9
The student who designed this problem wasn't thinking about the different wholes. IN your students problem, there are 3 different wholes. Anna's flowers - The whole is 5 flowers and $\frac{4}{5}$ are daffodils Beatrice's flowers The whole is 3 flowers and $\frac{2}{3}$ are daffodils The flowers of Anna and Beatrice combined. The whold is 8 flowers and $\... 9 I have thought a lot about this question since posting it, and having read the other answers and the many comments, I want to add a perspective that no one else seems to have given. Most of the real work in doing math is understanding and conceptualizing the problem rather than in computing an answer. This will be apparent to anyone who has read a ... 8 The word you are looking for is mediant. The mediant of two fractions$\frac{a}{c}$and$\frac{b}{d}$is$\frac{a+b}{c+d}$. According to Wikipedia, It is sometimes called the freshman sum, as it is a common mistake in the early stages of learning about addition of fractions. Teachers often do this in grading papers. For example, a test has two parts: the ... 8 It sounds like a variation of subtracting that I learned in high school (1968). My instructor called it European subtraction. \begin{array}{ccc} & 3 & 4 & 2 \\ - & 1 & 7 & 3 \\ \hline \end{array} You start by saying$9$plus$3$is$12$. You write the$9$as shown and "carry" the$1$, of the$12$, as a subscript of the$7$... 8 Beyond having worked as a programming teacher I have no experience with math education, but this is a topic I have been fascinated with for years. Arguments in favor of mental/manual arithmetic can be typically categorized as: It's important in daily life The argument typically goes that you need to be able to do arithmetic a lot in daily life with the ... 5 Calculus student had a final result of$\frac{1}{2\pi}$Which she told me was 1.57. I immediately realized that she had calculated$\frac12\pi$from keying in 1/2$\pi\$. There’s no going back on calculator use, I realize. What I strive for is to have the student who performed a series of calculations (for, in this case a related rates problem in calculus) to ...
4
I don't know if your idea has a name, but it feels weird when you try to apply it to something like 10-4: \begin{array}{r} & 1 & 0\\ -\!\!\!\!\!\!& \! & 4 \\ \hline & & -4 \\ +\!\!\!\!\!\!& 1 & 0 \\ \hline & {\color{red}1} & {\color{red}-}{\color{red}4} \end{array} Since the sum of the ones place (-4+0 = -4) and the ...
4
The student was conflating the sum $$\frac45+\frac23$$ and the weighted average (where the weights account for group-size differences) $$\left(\frac{\color{#00F}5}{\color{#180}{5+3}}\right)\frac45+\left(\frac{\color{#00F}3}{\color{#180}{5+3}}\right)\frac23\\=\frac{4+2}{5+3}$$ of the two given fractions/rates. Clearly, the sum of two positive fractions is ...
4
You really need to be able to do sums in your head when debating or negotiating. To prove this, show your students these famous car-crash interviews: (1) Diane Abbott (Labour) floundering horrifically over numbers she could neither remember nor even estimate in her head. The coolness and speed with which the interviewer questioned her numbers must itself ...
4
By hand =/= in your head. Having some faculty for mental arithmetic is good. Having a fluency for deconstructing a 'problem' into basic mathematical operations is pretty essential. However many people learn, and continue to benefit, by writing out the 'sums'. If Fred drives at 45 miles per hour, how far will he go in 45 minutes? I expect most people ...
3
I think if we can introduce a variety of algorithm's for teaching addition and multiplication, it'd be beneficial for the student. No matter what mathematics level you are, algorithmic thinking is always important. When I say algorithmic thinking, I mean to say that we have a set of rules to tackle a problem such that applying the rules in the correct order ...
2
I remember getting my first 18 inch slide ruler. I had worked hard, poison oak filled, hours making fire breaks in the Oakland hills to make enough money to get it. It was aluminum, had a spring loaded cursor that slid like it was greased, and the C and D scales (I think) lined up perfectly. It was huge and had scales I was never going to use. I loved it. ...
2
You can't be overly reliant on technology. I once bought an item from a shop during a power outage. Cash register was non-functional. Not enough light to power a solar calculator. The cashier had to compute a 10% discount by hand. She did it.... the long way.
2
You're entirely correct and there is huge value to being able to compute by hand. For a most basic instance from the real world, here in the UK I once interviewed a candidate for office manager who had spent about 10 years working in a tax office, and been promoted three times. To me, that should have meant he ate, breathed and slept numbers. I asked him ...
2
What property of fractions or addition of fractions could they be misunderstanding, and how would you explain to the student where they have gone wrong so that they don't repeat this in the future? [Emphasis added.] Short Answer Fractions are numbers and they behave like numbers when we do operations on them. Many students never learn this. You (and the ...
2
Some additional ideas and links: Strategical thinking: https://en.wikipedia.org/wiki/D%C5%8Dbutsu_sh%C5%8Dgi Programming: https://www.turingtumble.com/ (Technically 8+ but the concept should be understandable for younger kids as well. Note: it involves small marbles so use caution whether it's appropriate to be used by a smaller child.) Any kind of ...
2
The method of addition and subtraction that you mention is not new. For now, I provide one reference, but I'm sure there are others. Note that your method of subtraction makes one big assumption that is not needed for the traditional method: you assume that students are familiar with negative integers. The website Knowledge Over Grades uses a slight ...
1
Your method is circular. Try: 1000000 − 1 −−−−−−− 100000 −1 Now what??
Only top voted, non community-wiki answers of a minimum length are eligible
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2021-10-22 10:12:07
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http://freqnbytes.com/error-function/complimentary-error-function.php
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Home > Error Function > Complimentary Error Function
# Complimentary Error Function
## Contents
Please try the request again. Symbols: Γ(z): gamma function, inerfc(z): repeated integrals of the complementary error function, x: real variable and n: nonnegative integer Keywords: repeated integrals of the complementary error function Permalink: http://dlmf.nist.gov/7.18.F1 Encodings: pdf, Fortran 77 implementations are available in SLATEC. MathCAD provides both erf(x) and erfc(x) for real arguments. click site
Applications When the results of a series of measurements are described by a normal distribution with standard deviation σ {\displaystyle \textstyle \sigma } and expected value 0, then erf ( a The error function at +∞ is exactly 1 (see Gaussian integral). Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), "Section 6.2. Retrieved 2011-10-03. ^ Chiani, M., Dardari, D., Simon, M.K. (2003). other
## Complementary Error Function Excel
Back to English × Translate This Page Select Language Bulgarian Catalan Chinese Simplified Chinese Traditional Czech Danish Dutch English Estonian Finnish French German Greek Haitian Creole Hindi Hmong Daw Hungarian Indonesian Wolfram Language» Knowledge-based programming for everyone. doi:10.3888/tmj.16–11.Schöpf, Supancic ^ E. You can also select a location from the following list: Americas Canada (English) United States (English) Europe Belgium (English) Denmark (English) Deutschland (Deutsch) España (Español) Finland (English) France (Français) Ireland (English)
Matlab provides both erf and erfc for real arguments, also via W. A printed companion is available. 7.17 Inverse Error Functions7.19 Voigt Functions ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Generated Wed, 05 Oct 2016 11:30:27 GMT by s_hv1002 (squid/3.5.20) https://en.wikipedia.org/wiki/Error_function For |z| < 1, we have erf ( erf − 1 ( z ) ) = z {\displaystyle \operatorname ζ 8 \left(\operatorname ζ 7 ^{-1}(z)\right)=z} .
Click the button below to return to the English verison of the page. Complementary Error Function In Matlab H. Symbols: U(a,b,z): Kummer confluent hypergeometric function, e: base of exponential function, inerfc(z): repeated integrals of the complementary error function, z: complex variable and n: nonnegative integer Permalink: http://dlmf.nist.gov/7.18.E10 Encodings: TeX, pMML, Incomplete Gamma Function and Error Function", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN978-0-521-88068-8 Temme, Nico M. (2010), "Error Functions, Dawson's and Fresnel Integrals",
## Complementary Error Function Calculator
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. https://www.mathworks.com/help/matlab/ref/erfc.html The Q-function can be expressed in terms of the error function as Q ( x ) = 1 2 − 1 2 erf ( x 2 ) = 1 2 Complementary Error Function Excel Symbols: Γ(z): gamma function, !: factorial (as in n!), inerfc(z): repeated integrals of the complementary error function, z: complex variable and n: nonnegative integer A&S Ref: 7.2.4 Permalink: http://dlmf.nist.gov/7.18.E6 Encodings: TeX, Complementary Error Function Table Use the erfc function to replace 1 - erf(x) for greater accuracy when erf(x) is close to 1.Examplescollapse allFind Complementary Error FunctionOpen ScriptFind the complementary error function of a value.erfc(0.35) ans
At the imaginary axis, it tends to ±i∞. http://freqnbytes.com/error-function/computing-error-function.php Defines: inerfc(z): repeated integrals of the complementary error function Symbols: dx: differential of x, e: base of exponential function, !: factorial (as in n!), ∫: integral, z: complex variable and n: Retrieved 2011-10-03. ^ Chiani, M., Dardari, D., Simon, M.K. (2003). Symbols: C(z): Fresnel integral, S(z): Fresnel integral and x: real variable A&S Ref: 7.3.20 Referenced by: §7.5 Permalink: http://dlmf.nist.gov/7.2.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: info for 7.2(iii) Complimentary Error Function
The original calculation returns 0 while erfc(10) returns the correct result.1 - erf(10) erfc(10) ans = 0 ans = 2.0885e-45 Input Argumentscollapse allx -- Inputreal number | vector of real numbers This is useful, for example, in determining the bit error rate of a digital communication system. MR0167642. navigate to this website Given random variable X ∼ Norm [ μ , σ ] {\displaystyle X\sim \operatorname {Norm} [\mu ,\sigma ]} and constant L < μ {\displaystyle L<\mu } : Pr [ X
See [2]. ^ http://hackage.haskell.org/package/erf ^ Commons Math: The Apache Commons Mathematics Library ^ a b c Cody, William J. (1969). "Rational Chebyshev Approximations for the Error Function" (PDF). Complementary Error Function Mathematica A printed companion is available. 7.1 Special Notation7.3 Graphics Index Notations Search Need Help? The relationship between the error function erfc and normcdf is normcdf(x)=(12)×erfc(−x2)For expressions of the form 1 - erfc(x), use the error function erf instead.
## D: A D package[16] exists providing efficient and accurate implementations of complex error functions, along with Dawson, Faddeeva, and Voigt functions.
Mathematica: erf is implemented as Erf and Erfc in Mathematica for real and complex arguments, which are also available in Wolfram Alpha. IEEE Transactions on Wireless Communications, 4(2), 840–845, doi=10.1109/TWC.2003.814350. ^ Chang, Seok-Ho; Cosman, Pamela C.; Milstein, Laurence B. (November 2011). "Chernoff-Type Bounds for the Gaussian Error Function". If L is sufficiently far from the mean, i.e. μ − L ≥ σ ln k {\displaystyle \mu -L\geq \sigma {\sqrt {\ln {k}}}} , then: Pr [ X ≤ L Complementary Error Function Ti 89 All generalised error functions for n>0 look similar on the positive x side of the graph.
The inverse error function is usually defined with domain (−1,1), and it is restricted to this domain in many computer algebra systems. At the imaginary axis, it tends to ±i∞. Excel: Microsoft Excel provides the erf, and the erfc functions, nonetheless both inverse functions are not in the current library.[17] Fortran: The Fortran 2008 standard provides the ERF, ERFC and ERFC_SCALED my review here is the double factorial: the product of all odd numbers up to (2n–1).
Wolfram|Alpha» Explore anything with the first computational knowledge engine. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. In order of increasing accuracy, they are: erf ( x ) ≈ 1 − 1 ( 1 + a 1 x + a 2 x 2 + a 3 x Go: Provides math.Erf() and math.Erfc() for float64 arguments.
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2017-10-22 22:44:54
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http://publ.beesbuzz.biz/blog/?id=135&tag=release
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# Publ: Development Blog
Entries tagged release
# v0.3.10 released
Posted Monday, December 10 at 8:37 PM (a year ago)
Just some bug fixes with view caching and image handling; in particular, remote and static images will now respect max_width and max_height for the sizing, and I fixed the way that inline images work (insofar as now inline images can work).
# v0.3.9 Released
Posted Wednesday, November 28 at 3:33 PM (a year ago)
This entry marks the release of Publ v0.3.9. It has the following changes:
• Added more_text and related functionality to image sets (an example being visible over here)
• Improved and simplified the caching behavior (fixing some fiddly cases around how ETags and last-modified worked, or rather didn’t)
I also made, and then soon reverted, a change around how entry IDs and publish dates were automatically assigned to non-published entries. I thought it was going to simplify some workflow things but it only complicated the code and added more corner cases to deal with, all for something that doesn’t actually address the use case I was worried about. So never mind on that.
(What happened to v0.3.8? I goofed and forgot to merge the completed more_text et al changes into my build system first. Oops.)
See below for more on the caching changes.
# v0.3.7 released
Posted Wednesday, October 24 at 12:59 PM (a year ago)
I’ve released v0.3.7, which just fixes a few issues around transaction management and overall indexing performance. Namely:
• The indexer locks individual entries as it’s working on them
• If an entry is being worked on, watchdog will ignore it
• Cleaned up a couple of transaction failures that can occur due to PonyORM’s optimistic locking behavior
# Publ v0.3.6 released (minor update)
Posted Saturday, October 13 at 3:01 PM (a year ago)
I just released v0.3.6 of Publ, which just allows it to work with databases other than SQLite. In particular this is part of testing more advanced heroku deployment options.
Right now I’m primarily focusing on improving the documentation, especially the quickstart guide, since people are finally showing interest in Publ but aren’t quite sure where to begin!
# v0.3.5 released, and sample templates updated
Posted Friday, October 12 at 3:01 PM (a year ago)
I’ve now released v0.3.5 of Publ. Changelog:
• Add support for listing deleted entries (accessible via view.deleted)
• Improved how the last-modified/etag reference was determined (also fixing a nasty bug where a site might crash if a file is deleted)
• Fixed a minor shaping bug
I’ve also updated the sample site templates with all of the changes that have happened since, uh, June, and also included some sample content so it’s easier to get started with it.
# Oops, v0.3.4 released
Posted Friday, October 5 at 1:28 AM (a year ago)
Turns out I never actually tested the If-Modified-Since handler, because if I had I’d have seen the glaring exception it threw.
Oh well, that’s fixed now. I think.
# v0.3.3 - now with ETag and Last-Modified
Posted Monday, October 1 at 11:16 PM (a year ago)
I’ve started working on Pushl in earnest now, and one thing that was really bugging me about this is that anything which polls feeds and entries would really benefit from having client-side cache control working. Which was a big missing feature in Publ.
Well, I finally implemented it, and I’m pretty happy with how I did it.
The short version: for any given view it figures out (pessimistically) what’s the most recent file that would have affected the view (well, within reason; it only looks at the current template rather than any included templates, which is pretty difficult to do correctly) and uses that to generate an ETag (via metadata fingerprint) and a Last-Modified time (based either on the file modification time or the time the entry was actually published).
There’s probably a few corner cases this misses but in general this makes client-side caching of feeds and such work nicely.
# v0.3.2: a smol bugfix release
Posted Tuesday, September 25 at 2:55 PM (2 years ago)
I found a few more annoying bugs that were shaken out from the whole PonyORM transition, as well as a couple of bugs in the new shape functionality. There’s probably a few more of these bugs lurking in the codebase (I mean, in addition to the existing bugs I know about), but here’s what’s changed:
# The shape of the float (v0.3.1)
Posted Thursday, September 20 at 10:58 PM (2 years ago)
Did you know that CSS3 has a style called shape-outline? It’s pretty neat, it makes it so that a floated object gets a shape based on the alpha channel of its specified image. But it’s kind of a pain to set up; in plain HTML it looks something like this:
<img src="/path/to/image.png" width="320" height="320"
style="shape-outline:url('/path/to/image.png');float: left">
and if you want a different shape mask for your image than its own alpha channel, you have to do a bunch of stuff like making sure that the image sizes are the same and whatever.
# v0.3.0 released
Posted Wednesday, September 19 at 12:46 PM (2 years ago)
Version 0.3.0 is now released, with the change from peewee to PonyORM.
As a result of this change you’ll have to do two things to your config file:
1. The database configuration format has changed slightly
2. Any existing databases have to be manually deleted/dropped/etc.; unfortunately PonyORM doesn’t provide a mechanism for deleting tables not under its control
Everything else should work identically as before.
# v0.2.2 released… and 0.2.2.1… and 0.2.3…
Posted Wednesday, September 12 at 1:27 AM (2 years ago)
Earlier today I pushed v0.2.2, which was a minor configuration fix for the markdown library to support table syntax.
Then I pushed v0.2.2.1 which was another configuration fix to fix that fix (oops).
Then in deploying updates I upgraded to Python 3.7, and promptly discovered that Publ doesn’t actually work on Python 3.7, so just now I fixed that, and released the fixes as v0.2.3.
# v0.2.1 released
Posted Tuesday, August 21 at 11:48 PM (2 years ago)
Just a couple of minor fixes in this release:
# Update to v0.2.0
Posted Friday, July 6 at 3:57 PM (2 years ago)
A few changes since v0.1.24:
• Updated code to use the current Flask cache-control API
• Only set cache-control for responses that don’t have a natural cache response
• Entry IDs and UUIDs are now semi-stably generated, in order to prevent (or at least reduce) problems like the last time
Publ itself is stable enough (and enough has changed since v0.1.0) that I felt that a minor version bump was a reasonable thing to do.
Anyway! While Publ has been running quite nicely on my website, I’d love to see more people actively using and developing it. This site in particular needs a lot of attention and probably reworking; my other top priorities are:
• A better installation/deployment guide
• Proper test coverage (rather than manual smoke tests)
# Verson 0.1.24 released
Posted Wednesday, June 27 at 7:55 PM (2 years ago)
New functionality:
• The image rendition cache now gets periodically purged; the default is to delete renditions which haven’t been used in the last week (this can be disabled)
Bug fixes:
• entry.title can now accept the no_smartquotes parameter, which is necessary in Atom feeds
• entry.card now uses the same Markdown extensions as entry.body
# v0.1.23 updates, oh yeah this is beta now
Posted Wednesday, June 6 at 9:30 PM (2 years ago)
I neglected to mention that I set Publ to beta status in v0.1.22, which was a minor bugfix release, rather than moving to 0.2 like I previously stated. The changes for 0.1.22 were:
• Fixes to category Sort-Name
• Added support for regex path-alias hooks (this is configured on the Python/WSGI side, and has been working quite nicely over on beesbuzz.biz)
• Fixed a dumb bug in the cache-control headers
And then the changes for 0.1.23:
• Enable automatic smart-quote substitutions (this is the default setting, and can be overridden by passing no_smartquotes=True to entry.body/entry.more/entry.title)
• Improve the handling of last-modified times on entries (now there’s a Last-Modified header which only gets set when you want it to be)
# v0.1.19: creeping ever closer to beta status
Posted Sunday, May 27 at 5:22 PM (2 years ago)
The amount of stuff I’m having to fix in Publ to support beesbuzz.biz is diminishing rapidly! Here’s what’s happened since 0.1.18:
• Improved the Path-Alias redirection logic; now it will do a 301 Permanently Moved for inbound Path-Aliased requests, and if a Path-Alias points to an entry with a Redirect-To it will redirect directly to that URL instead (and it will be a 302, same as the old Redirect-To behavior)
• Pagination can now be weekly; you can use entry.archive(paging='week'), and a ?date= view parameter ending in _w will provide a weekly view instead.
• Better default formatting for view.range, and an addition of a week format parameter there
# In better news, v0.1.18
Posted Saturday, May 26 at 1:19 AM (2 years ago)
So aside from the Dreamhost issues, I would like to share what’s new in the latest version as part of my big “get my website online” push:
• PERFORMANCE: Improved the threading mechanisms around image renditions for better stability and performance
• BUGFIX: Made markup tagging work consistently between image types
• FEATURE: view.link() now allows overriding category
• FEATURE: You can now mark an entry with an Entry-Status of DELETED or GONE, which results in a 410 error instead of a 404 (be the envy of your web-developer nerd friends!)
• FEATURE: Error templates will automatically use the x00 error code as a potential fallback (e.g. error code 503 will also fall back to a template for 500)
• FEATURE: Entry titles can now have Markdown in them, and it usually works most of the time!
• BUGFIX: Now when running in debug mode you don’t end up with two watchdog threads
• FEATURE/QUALITY: Refactored the way error pages are handled, and now if you get a 404-type error on a page while the index is being asynchronously scanned, it’ll turn into a 503 with a Retry-After disposition in case it’s just something that hasn’t been indexed yet
• UX: View pagination URLs now all use id as the query parameter rather than a miasma of contextually-dependent start, last, or first which made no sense anyway
# Lots of bug fixes, and more image features (v0.1.17)
Posted Tuesday, May 22 at 5:00 PM (2 years ago)
Oh, it’s been a little while since I’ve posted an update, hasn’t it? That’s because I’ve been very busy building the templates for my personal site! In doing so I’ve greatly improved the way that view refinements worked, fixed a few silly bugs with image search paths on templates, and also added an easier way to specify background images in CSS, via the image().get_css_background() method.
I’m really excited to be able to bring my first fully-realized Publ site to the public; I hope it gets other people interested in what a flexible publishing system allows them to do!
Posted Saturday, May 19 at 12:00 AM (2 years ago)
Today I added two new useful features:
I also implemented the better date sort mentioned previously.
These things are already making my new personal site look way better and easier to use! I feel like I’m almost ready to flip the switch.
Oh, and I also improved the getting started guide, including adding basic setup instructions for Linux and Windows. Not that I’ve gotten Publ to run on Windows, yet, but documenting how to get the environment set up is the first step, right?
# Asynchronous workers
Posted Tuesday, May 15 at 5:21 PM (2 years ago)
Today I got two major bits of functionality in: Publ will now asynchronously scan the content index (which speeds up startup and fixes some annoying race conditions with entry creation), and it also asynchronously generates image renditions (which makes pages not take forever to load on first render, and will also use multiple CPU cores if available). Seems to work well so far.
I was running into scaling problems with beesbuzz.biz (what with there being a couple thousand entries and some pages with hundreds of images on it) and this keeps it feeling pretty good.
So, this brings us up to version 0.1.14.
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2020-03-28 14:43:55
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https://codereview.stackexchange.com/questions/184779/comparing-dates-with-sharepoint-pnp-powershell
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# Comparing dates with SharePoint PnP PowerShell
I am wondering if should change my if statement that compares two dates if they are equal. Is there a better way of doing this operation? The code does what it supposed to do. No errors. The Code:
if((Get-date).ToString("yyy-MM-dd") -eq $ListItem["Aviseringsdatum"].ToLocalTime().ToString("yyy-MM-dd")){ Write-Output "True" } ## 1 Answer I think it's okay, but we can do better. Let's see what Get-Date returns. Run this: > get-date | get-member It shows that Get-Date returns a DateTime object, and it also lists all the properties that has. I notice that DateTime has a Date property. Looking on the Web, it says that this is the date part of the DateTime but with a zero time component. So let's use that instead of converting to "yyy-MM-dd": if((Get-date).Date -eq$ListItem["Aviseringsdatum"].Date){
Write-Output "True"
}
Since this is a code review, let's improve the formatting, and indent and add spaces:
if ((Get-Date).Date -eq $ListItem["Aviseringsdatum"].Date) { Write-Output "True" } I think that's as far as you need to go. If it were my code, however, I would probably go a bit further. I would use [DateTime]::Today instead (Get-Date).Date because I think it is clearer. I would also introduce local variables. I think if you are ever using a "today" or "now" value in your code, then it's better to capture them in a variable for the sake of consistency so that all parts of the script agree on what "today" or "now" is. What if for instance your script ran over midnight? Then the (Get-Date).Date value would suddenly change. A local variable might be overkill in this case, but it's a good practice in general to avoid some subtle bugs. I also like making local variables to capture bigger expressions because they make for self-documenting code. It might be overkill again in this case, but I will add a $notificationDate variable.
$today = [DateTime]::Today$notificationDate = $ListItem["Aviseringsdatum"].Date if ($notificationDate -eq $today) { Write-Output "True" } I think that $notificationDate -eq \$today makes the logic very clear to anyone reading the code.
• Thank you sir! This is exactly what I need. A Structrued way to improve my undestanding of well structured code. I tend to not break down the code into varibales. Thanks again! – AllramEst Jan 11 '18 at 8:01
• Isn't there a presicision issue with equating dates? I seem to remember to have to use Get-Date and set the -Hour, -Minute and -Second parameters to zero. – John Donnelly Jan 11 '18 at 15:34
• @JohnDonnelly, that's why we use the Date property. Evaluate this in PowerShell: (Get-Date).Date, or (Get-Date).Date.TimeOfDay. You will see that the hours, minutes, etc. are zero. – Dangph Jan 12 '18 at 1:00
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2019-09-16 17:10:13
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https://www.physicsforums.com/threads/question-on-transfer-functions.660061/
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# Homework Help: Question on Transfer Functions
1. Dec 20, 2012
### doublemint
Hi,
Just a simple question on Transfer Functions.
If I was give a open loop TF, G, and a feedback loop H, then the closed loop TF is G'=$\frac{G}{1+GH}$.
So my first attempt was to simplify the G' by hand and then plot it in excel which produced a crescent shaped moon.
By simplification, I mean this:
Let G=1/s and H=1/(s+1)
G'=$\frac{G}{1+GH}=\frac{1/s}{1+\frac{1}{s}+\frac{1}{s+1}}$=$\frac{s+1}{s(s+1)+1}$
But, then I go to mathematica and enter the G' without simplification (just $\frac{G}{1+GH}$)and it gave me a totally different graph.
Even though the the G' is the same (mathematically) in both cases, why does the Nyquist (and thus Bode) plots are different?
2. Dec 20, 2012
### aralbrec
That looks fine except the second step looks to be a typo with the second '+' in the denominator meant to be multiplication.
What kind of graphs did you do? Obviously they should be the same whether you use excel or mathematica.
Bode plots have increasing frequency on the horizontal axis and present the gain and phase of a complex quantity as w varies in a typical graph form. A nyquist plot, on the other hand, is essentially a polar plot that traces the value of a complex quantity in the complex plane as w varies.
Usually the nyquist plot is done on the loop gain GH to find out if there are poles in the right half plane of the overall function G/(1+GH). You can get similar information from a bode plot of GH too but, as mentioned, the bode plot is like a standard graph whereas the nyquist plot is a polar plot.
You can think if their similarity like this: a Nyquist plot is looking to see if a polar plot of GH encircles the point -1 to determine stability. In a bode plot of GH, the magnitude of GH should be less than 1 when the phase is 180 degrees (pi). The point -1∠0 in the nyquist plot occurs at phase 180 degrees and there will be no encirclements of -1 if |GH| < 1 when the phase is 180 degrees. So you can see the gain/phase conditions on GH in the bode plot are the same as the encirclement of -1 test in the nyquist plot.
Last edited: Dec 20, 2012
3. Dec 20, 2012
### doublemint
I did Nyquist.
Yeah you are correct. I just tried the simplified equations I posted and did another graph with functions of more complexity and both Nyquist plots are the same..
I guess I typed something in wrong :(
Thanks for the help though!
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2018-10-18 13:27:31
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https://mathematica.stackexchange.com/posts/18267/revisions
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5 replaced http://mathematica.stackexchange.com/ with https://mathematica.stackexchange.com/ edited Apr 13 '17 at 12:55 The following will be my attempt at the problem. I think I have the basics down but I am hoping there are areas to decrease the computation time. These problems were alluded to earlier in the question and answers of http://mathematica.stackexchange.com/questions/18203/1d-random-walk-variant/18212#182121D Random Walk variant. So how can this be sped up? Some different schemes for the hopMod module were discussed at http://mathematica.stackexchange.com/questions/18203/1d-random-walk-variant/18212#182121D Random Walk variant. However I was not sure how to generalized all those approaches to my situation where I also need to keep track of the simulation time. The following will be my attempt at the problem. I think I have the basics down but I am hoping there are areas to decrease the computation time. These problems were alluded to earlier in the question and answers of http://mathematica.stackexchange.com/questions/18203/1d-random-walk-variant/18212#18212. So how can this be sped up? Some different schemes for the hopMod module were discussed at http://mathematica.stackexchange.com/questions/18203/1d-random-walk-variant/18212#18212. However I was not sure how to generalized all those approaches to my situation where I also need to keep track of the simulation time. The following will be my attempt at the problem. I think I have the basics down but I am hoping there are areas to decrease the computation time. These problems were alluded to earlier in the question and answers of 1D Random Walk variant. So how can this be sped up? Some different schemes for the hopMod module were discussed at 1D Random Walk variant. However I was not sure how to generalized all those approaches to my situation where I also need to keep track of the simulation time. 4 deleted 1 characters in body edited Jan 24 '13 at 16:24 BeauGeste 1,38211 gold badge1919 silver badges3131 bronze badges FindRate[k0_] := Module[{t}, kB1 = If[({Xi} \[Intersection] blockedSites + 1) != {}, RandomChoice[{1/2, 1/2} -> {0, kBack1[k0]}], kBack1[k0]]; kB2 = If[({Xi} \[Intersection] blockedSites + 2) != {}, RandomChoice[{1/2, 1/2} -> {0, kBack2[k0]}], kBack2[k0]]; kF1 = If[({Xi} \[Intersection] blockedSites - 1) != {}, RandomChoice[{1/2, 1/2} -> {0, kFor1[k0]}], kFor1[k0]]; kF2 = If[({Xi} \[Intersection] blockedSites - 2) != {}, RandomChoice[{1/2, 1/2} -> {0, kFor2[k0]}], kFor2[k0]]; newCoords = RandomChoice[{kB1, kB2, kF1, kF2} -> {Xi - 1, Xi - 2, Xi + 1, Xi + 2}]; dt = RandomReal[ExponentialDistribution[kB1 + kB2 + kF1 + kF2]]; t = simT; {Xi, simT} = {newCoords, t + dt} ] FindRate[k0_] := Module[{t}, kB1 = If[({Xi} \[Intersection] blockedSites + 1) != {}, RandomChoice[{1/2, 1/2} -> {0, kBack1[k0]}], kBack1[k0]]; kB2 = If[({Xi} \[Intersection] blockedSites + 2) != {}, RandomChoice[{1/2, 1/2} -> {0, kBack2[k0]}], kBack2[k0]]; kF1 = If[({Xi} \[Intersection] blockedSites - 1) != {}, RandomChoice[{1/2, 1/2} -> {0, kFor1[k0]}], kFor1[k0]]; kF2 = If[({Xi} \[Intersection] blockedSites - 2) != {}, RandomChoice[{1/2, 1/2} -> {0, kFor2[k0]}], kFor2[k0]]; newCoords = RandomChoice[{kB1, kB2, kF1, kF2} -> {Xi - 1, Xi - 2, Xi + 1, Xi + 2}]; dt = RandomReal[ExponentialDistribution[kB1 + kB2 + kF1 + kF2]]; t = simT; {Xi, simT} = {newCoords, t + dt} ] FindRate[k0_] := Module[{}, kB1 = If[({Xi} \[Intersection] blockedSites + 1) != {}, RandomChoice[{1/2, 1/2} -> {0, kBack1[k0]}], kBack1[k0]]; kB2 = If[({Xi} \[Intersection] blockedSites + 2) != {}, RandomChoice[{1/2, 1/2} -> {0, kBack2[k0]}], kBack2[k0]]; kF1 = If[({Xi} \[Intersection] blockedSites - 1) != {}, RandomChoice[{1/2, 1/2} -> {0, kFor1[k0]}], kFor1[k0]]; kF2 = If[({Xi} \[Intersection] blockedSites - 2) != {}, RandomChoice[{1/2, 1/2} -> {0, kFor2[k0]}], kFor2[k0]]; newCoords = RandomChoice[{kB1, kB2, kF1, kF2} -> {Xi - 1, Xi - 2, Xi + 1, Xi + 2}]; dt = RandomReal[ExponentialDistribution[kB1 + kB2 + kF1 + kF2]]; t = simT; {Xi, simT} = {newCoords, t + dt} ] Tweeted twitter.com/#!/StackMma/status/293897705065947136 occurred Jan 23 '13 at 1:47 3 added 2 characters in body edited Jan 22 '13 at 23:12 BeauGeste 1,38211 gold badge1919 silver badges3131 bronze badges This took me 1.33 seconds. The time is highly varying. The time increases dramatically as Xf increasesor increase. For some reason, when I used ParallelTable, I got identical results every time I calculated another iTable. Must be something with the seed going wrong. To get convergence, n needs to be increased. This took me 1.33 seconds. The time is highly varying. The time increases dramatically as Xf increases. For some reason, when I used ParallelTable, I got identical results every time I calculated another iTable. Must be something with the seed going wrong. To get convergence, n needs to be increased. This took me 1.33 seconds. The time is highly varying. The time increases dramatically as Xf or increase. For some reason, when I used ParallelTable, I got identical results every time I calculated another iTable. Must be something with the seed going wrong. To get convergence, n needs to be increased. 2 more explanation on how new sites are hopped to edited Jan 22 '13 at 22:46 BeauGeste 1,38211 gold badge1919 silver badges3131 bronze badges 1 asked Jan 22 '13 at 21:31 BeauGeste 1,38211 gold badge1919 silver badges3131 bronze badges
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2019-09-18 20:52:17
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https://askfilo.com/physics-question-answers/a-block-of-mass-50-mathrm-kg-can-slide-on-a-rough-a63
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The world’s only live instant tutoring platform
Question
Solving time: 2 mins
# A block of mass can slide on a rough horizontal surface. The coefficient of friction between the block and the surface is $0.6$. The least force of pull acting at anangle of to the upward drawn vertical which causes the block to just slide is.
A
B
C
D
## Solutions
(1)
For limiting condition
, By solving
93
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Question Text A block of mass can slide on a rough horizontal surface. The coefficient of friction between the block and the surface is $0.6$. The least force of pull acting at anangle of to the upward drawn vertical which causes the block to just slide is. Topic Laws of motion Subject Physics Class Class 11 Answer Type Text solution:1 Upvotes 93
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2023-03-25 04:37:59
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https://amathew.wordpress.com/tag/gysin-map/
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Today I would like to take a break from the index theorem, and blog about a result of Wu, that the Stiefel-Whitney classes of a compact manifold (i.e. those of the tangent bundle) are homotopy invariant. It is not even a priori obvious that the Stiefel-Whitney classes are homeomorphism invariant; note that “homeomorphic” is a strictly weaker relation than “diffeomorphic” for compact manifolds, a result first due to Milnor. But in fact the argument shows even that the Stiefel-Whitney classes (of the tangent bundle) can be worked out solely in terms of the structure of the cohomology ring as a module over the Steenrod algebra.
Here is the idea. When $A \subset M$ is a closed submanifold of a manifold, there is a lower shriek (Gysin) homomorphism from the cohomology of $A$ to that of $M$; this is Poincaré dual to the restriction map in the other direction. We will see that the “fundamental class” of $A$ (that is, the image of 1 under this lower shriek map) corresponds to the mod 2 Euler (or top Stiefel-Whitney) class of the normal bundle. In the case of $M \subset M \times M$, the corresponding normal bundle is just the tangent bundle of $M$. But by other means we’ll be able to work out the Gysin map easily. Once we have this, the Steenrod operations determine the rest of the Stiefel-Whitney classes.
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2020-09-22 02:54:20
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http://mathoverflow.net/questions/95724/uniqueness-of-distance-realizing-geodesic-in-hyperbolic-surface/95728
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# Uniqueness of distance realizing geodesic in hyperbolic surface. [duplicate]
Possible Duplicate:
Hyperbolic surfaces
Given a hyperbolic surface S with geodesic boundary. Let a and b be two distinct simple closed geodesic boundaries. Does there exist a unique distance realizing geodesic in S? (1) for S is a pair of pants. (2) S is any hyperbolic surface with boundary.
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## marked as duplicate by S. Carnahan♦May 2 '12 at 7:15
If you want to refine your question, you should edit the first one. There is an "edit" link below the tags. – S. Carnahan May 2 '12 at 7:16
For the pants, yes. In general, no. To prove this for the pants, classify all geodesic arcs and just observe the result. There are many ways to find a "no" example in the general case; the first one that came to my mind was taking a double cover.
EDIT - I see that this is a near-duplicate of a closed question. You could improve your question by giving some motivation. Reading the FAQ will be very useful in writing questions that get good answers. In particular please see http://mathoverflow.net/faq#whatnot
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well, I have got an example of hyperbolic surface with boundary where more than one (at least two) distance realizing geodesics between two distinct geodesic boundaries will exist. I have a further question: Suppose p and q are two distance realizing geodesics between the boundary geodesics. Is it true that p and q are always disjoint? – Bidyut Sanki May 2 '12 at 8:26
They are always disjoint. This is proved by the usual "exchange and round-off" technique. See, if $p$ and $q$ intersect, then at the intersection point you can do a little cut and paste to get two new arcs $p'$ and $q'$, still connecting the same boundary components, and slightly shorter, which is a contradiction. – Sam Nead Jan 3 '14 at 11:00
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2015-06-30 10:15:43
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https://codeforces.com/blog/passworld
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### passworld's blog
By passworld, 7 years ago,
## Given
a set S (size n <=20) ,say S[1..n]
m (m<=10^5) subsets of S ,say Sub[1...m]
p (p<=10^5) people, say P[1..p]
## Definition of one assignment
For p person, each one choose a subset ( can be the same ), so that the union of the p subsets is equal to S.
## Try to count
The number of assignments. (time limit: 5 seconds)
#### Any hints or thoughts will be really appreciated!
• +18
By passworld, 7 years ago,
### Here is a hard problem for me:
time limit per test4 seconds memory limit per test256 megabytes
Given a,b (1<a<b<1000,integer) and (a,b)=1
Try to find an array P[ ] (size unknown)so that:
• there exist a positive integer k so that:
• ( Sum of P[i] )== k * a
• For all i : ( k*b ) % P[i]==0 and then we note A[i]=k*b/P[i]
We want to output A[].
NOTE: elements in P[ ] are(is) positive integer and size of P[ ] should be as small as possible and if it is tied for some possible arrays with equal minimal size, choose one array that elements ofA[ ] are as small as possible. (not element of P[],where I typoed before ,I am so sorry for that!)
sample in a=3 b=10 ; sample out A={5,10}
• +5
By passworld, 7 years ago,
I want to count for every number ,say A[i] ,how many numbers there are that is smaller than Ai with id smaller than i. For example , here is an array: 5 3 7 2 9 6 ,the anwser is 0 0 2 0 4 3 Is there any fast algorithm that can solve this problem? Thank you for your help!
• -2
By passworld, 7 years ago,
I want to find a free , simple and powerful software to draw graph structure and mark on it easily. (just like some graph appear on some tutorials ) Any recommendation? Thank you very much!
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2021-09-27 13:27:59
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https://niepoprawneradio.pl/watch-footloose-cswzzh/1c7b29-body-surface-area-unit
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## body surface area unit
0
Ancel Keys, a physiology professor, used Quetelet's equation as part of a 1972 published report on obesity and created the name body mass index or BMI. [email protected]. Back in 1999, there was confusion about BSA and people were always looking up- and debating- the formulas. For every like, I hear a bell, and smile, and you get wings. Source for information on body surface area: A Dictionary of Food and Nutrition dictionary. formula is gaining support as a common standard because it is much simpler and Its clinical significance in video-assisted thoracoscopic surgery (VATS) was rarely understood. Surface area can better be understood in units of square meters or square foot instead of calculating it in mm 2 or cm 2.Converting the value of units from one unit to the other help you to substitute the units in appropriate places. Calculates BSA for medication doses and includes descriptive statistics. I've used this redesign opportunity to show off my soft pastel colors style (which I made myself), and introduce the Moose. The result from our surface area calculator will always be a square of the same unit: square feet, square inches, square meters, square cm, square mm. Geometric method for measuring body surface area: A height weight formula validated in infants, children and adults. Please read this End User Licence Agreement for the Body Surface Area Calculator, then click 'I Agree' at the bottom to proceed to the calculator. Body surface area (BSA) based dosing is a useful way to mitigate patient size variation in medication regimens. body surface area Heat loss from the body is related to surface area and basal metabolic rate and energy expenditure are sometimes expressed per unit body surface area. An adult male applies a cream once daily to the dorsal and volar surfaces of both feet and both hands. On my Body Mass Index calculator, it has hover pop-ups for weight and height, so you don't have to type. Effect of body surface area calculations on body fat estimates in non-obese and obese subjects. Body surface area (BSA), as the name suggests, is the calculated or estimated surface area of a person’s body. Background Body surface area (BSA) is a biometric unit to measure the body size. I probably won't put that onto the BSA calculator, because it would encourage inaccurate rounding. By using the handprint- or palm-method you simply estimate the area … From these studies we have understanding of the relationship between height (stature) and body weight (mass) and a formula through which we can calculate a persons body surface area based in their weight and height. but I'm having second thoughts now. Since the body is a complex figure, making an accurate measurement of its surface is a considerable challenge. While widely used in population health studies, the BMI equation has been critiqued for its use in individual diagnoses for determining whether someone is overweight or obese. Optional Medication Dose Calculator. End of page Navigation links: Go to the Halls.md homepage, or Back to top. There are several estimation formulas available for use, the most common one being that published by Mosteller in 1987. {\displaystyle \varepsilon } is the emissivity of the grey body; if it is a perfect blackbody, {\displaystyle \varepsilon =1}. a. Body surface area is used to calculate the resting energy expenditure, also known as resting metabolic rate when the rate required is per square meter per hour, and not the daily rate. According to Mosteller's "simplified calculation of body-surface area In metric terms" the body surface area = the square root of product of the weight in kg times the height in cm divided by 3600. The non-muscle cell mass and the muscle cell mass reach a 1:4 ratio in men whereas in women a 1:35 proportion--which is more favorable for pregnancy and lactation--is reached. Lean Body Weight: Ideal Body Weight: BSA is a measurement used in many medical tasks. formulas have been developed over the years, originally by These calculators employ well-known formulas to … A like would be nice. Both the chart and the tables with average body surface area estimates are based on weight and height data from the U.S. NCHS National Health and Nutrition Examination Survey (2011-2014) [2]. Geometric method for measuring body surface area: A height weight formula validated in infants, children and adults. See our full terms of service. The DuBois & DuBois original formula is: 0.007184 x height (cm)0.725 x mass (kg)0.425. Cancer Chemother Rep 1970 54:225-35. The calculator below provides results for some of the most popular formulas. Body Surface Area (BSA) is simply a measurement of the surface area of your body. Using these BSA formulas, body surface area will be figured out in square meters. Because of the complexity of direct measurement, various formulas have been developed to estimate the body surface area over the years. Because of the complexity of direct measurement, various formulas have been developed to estimate the body surface area over the years. After removing the pieces, they were cut into regular shapes allowing accurate BSA calculation. We are not to be held responsible for any resulting damages from proper or improper use of the service. If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Body Surface Area Calculator", [online] Available at: https://www.gigacalculator.com/calculators/bsa-calculator.php URL [Accessed Date: 23 Jan, 2021]. 2006 Nov;27(11):1197–1209. While widely used in population health studies, the BMI equation has been critiqued for its use in individual diagnoses for determining whether someone is overweight or obese. The "normal" body surface area is generally taken to be 1.7 m2 but, in actual fact, the body surface area depends on more than just height and weight. Hi, and welcome. In another case skin burns were accompanied by black urine in a 42-year-old man [15]. Body surface area (BSA) based dosing is a useful way to mitigate patient size variation in medication regimens. The Journal of Pediatrics 1978 (93):1:62-66. apothecary system. Si US. Calculate BMI and Body Surface Area using Mosteller formula. Below are the body surface area formula by Dr’s Mosteller, DuBois and DuBois, Haycock and Boyd. Patient Platform Limited has used all reasonable care in compiling the information but … area of the black-body surface at temperature T per unit time, in the wavelength band unit, in all directions within the limits of the hemispherical solid angle. . The result is in squared meters, which easily be converted to … Click on the units for more unit options. View Article PubMed/NCBI Google Scholar 19. Step 3. 1. Boyd E. The growth of the surface area of the human body. Weight? The Journal of Pediatrics 1978 (93):1:62-66. It should be noted that the mean and the median values, at least for children and adolescents of age 0-20 differ significantly, with the median being significantly lower than the mean, suggesting that the distribution is affect by a relatively small number of overweight and obese individuals. Safety Check. The result is in squared meters, which easily be converted to square feet, inches or yards as needed. The patient's body surface area can be estimated in several ways: by nomogram using height and weight (most accurate), by Mosteller's formula using height and weight or on the basis of weight alone (least accurate). But it looked quite To address it, scientists in the early 1900s took measurements via melted paraffin and paper strips applied to their body. kg lb. There are other competing formulae derived over the years (e.g. Cancer Chemother Rep 1970 54:225-35. Gehan EA, George SL. The surface area of adults is about 18,000 cm2 (men) or 16,000 cm2 (women). The Body surface area formula or BSA formula must not be confused with the body mass index because both of them are completely different terms and things. In the still more general (and realistic) case, the emissivity depends on the wavelength, In all surface area calculations, make sure that all lengths are measured in the same unit, e.g. In the SI system this quantity is measured in W/m 3, and if the wavelength is measured in micrometres then this quantity is measured in W/(m 2 m m). and if you really want it on this page, let me know. Your body surface area is 2.19 m². Here is a simple and online free Area unit converter tool to convert the value from one unit to another. Background: Vitiligo Area and Severity Index (VASI) is standing on the top of the cited and implemented scoring tools for vitiligo. Useful for a number of health/fitness calculations. This works out as 21 g/week (7 x 3 g). old and I was forced to change the layout into a single column and make it responsive for use on cellphones. This was my first calculator and the first page on my website in 1999. Direct measurement of BSA is difficult, and as such many formulas have been published that estimate BSA. BSA is a measurement used in many medical tasks. [1] Katch V.L., McArdle W.D., Katch F.I (2011) "Essentials of Exercise Physiology", fourth edition, [2] US NCHS (2016) "National Health and Nutrition Examination Survey (2011-2014)" DHHS Publication No. These formulas all give slightly different results. Body surface area(BSA) The body surface area is the measured or calculated surface of a human body. BSA is also used to determine the dosage of medications, to estimate renal clearance, in chemotherapy dosage, glucocorticoid dosage and in the composition of the cardiac index. I may remove or alter that to be less annoying. Estimation of human body surface area from height and weight. easycalculation.com - free online converter tools. Define body weight-to-surface area. JavaScript source code simple html calculator - converter program for full range of surface area measuring units. It either shows error messages or changes the values. Estimation of human body surface area from height and weight. Below are some average values for different age groups in graph and table form. The surface area method relates the patient's fluid and electrolyte needs to their body's surface area. Our online calculators, converters, randomizers, and content are provided "as is", free of charge, and without any warranty or guarantee. Day 2: 1000 mg/m2 c. Day 3: 500 mg/m2 6) For patients with a body surface area of more than 2.0 m2 the single dose should not exceed … The first formula was developed by Du Bois in 1916 and since then, several others have been developed. Some of you will feel nostalgia for the old BSA calculator, which fit better onto a standard computer monitor. Created by. The patient's body surface area can be estimated in several ways: by nomogram using height and weight (most accurate), by Mosteller's formula using height and weight or on the basis of weight alone (least accurate). Its not very easy to do a direct measurement of BSA, so several formulas have been published to estimate BSA. Step 2. Set Age and Gender, then re-Calculate. Breitmann in 1932, Stevenson in 1937, Haycock in 1978, Schlich in 2010) and a discussion about their accuracy is ongoing in the medical community, but the formula used in our BSA calculator is one of the most-commonly used by medical practitioners [3]. Estimating Body Surface Area (BSA) To estimate your body surface area our BSA calculator uses the following formula [1]: 0.20247 x height (m)0.725 x mass (kg)0.425. where height is in meters and mass is in kg. BSA calculator tells us about the ‘’Surface area’’ of a person while the body mass index is a measurement of the degree of how overweight a person is. Ancel Keys, a physiology professor, used Quetelet's equation as part of a 1972 published report on obesity and created the name body mass index or BMI. probably not worth the trouble to debate about which formula may or may not be Several antivirals, antimicrobials, and antifungals require a dosing regimen based on the BSA [3]. Body Surface Area Calculator for medication doses; Halls.md Disclaimer: This article is for information only and should not be used for the diagnosis or treatment of medical conditions. Various inches, feet, mm, cm. More Information. To estimate your body surface area our BSA calculator uses the following formula[1]: 0.20247 x height (m)0.725 x mass (kg)0.425. where height is in meters and mass is in kg. Body Surface Area Estimations related to organ transplantation often use the body surface area as an important metric. 0/2 completed. Area is a quantity that describes the size or extent of a two-dimensional figure or shape in a plane. The global unit selector only affects unanswered questions. or Share. The Mosteller This body surface area calculator helps you calculate how much surface area of the skin is affected by a disease. Copyright © 1999 - present. Gehan EA, George SL. Take this into account when interpreting the data. Way back in 1999 I thought that was a good safety check, Body surface area represents the total surface of the human body and its determination has several applications in medicine such as IV fluid measurements and drug dosage. 2.Height? 1.Weight? Body Surface Area (BSA) BSA measures the total surface area of the body and is used to calculate drug dosages and medical indicators or assessments. body weight-to-surface area synonyms, body weight-to-surface area pronunciation, body weight-to-surface area translation, English dictionary definition of body weight-to-surface area. Units of Surface & Area: meter 2 (m 2), millimeter 2 (mm 2), centimeter 2 (cm 2), decimeter 2 (dm 2), deka 2 (dam 2), inch 2 (in 2), Foot 2 (ft 2), yard 2 (yd 2), kilometer 2 (km 2), mile 2 (mile 2), Acre, Hectares (ha), Are, Section, Township. https://www.gigacalculator.com/calculators/bsa-calculator.php. Body Mass Index: kg/m 2. Nowadays the internet brings calculators like this to us instantly, and we just get our work done a little faster. Using BSA may help prescriber's dose more optimally to improve drug efficacy, minimize drug toxicity, and account for some changes in pharmacokinetics depending on patient factors. etc. Halls, MD . BSA calculations are useful when determining different body characteristics, for example energy expenditure in the form of BMR, RMR (REE), RDEE, TDEE, which are in turn used when determining dietary and nutritional needs. This calculator DOES HAVE some WEIRD behavior, if you enter weight or height values that are very large or small. Livingston EH, Lee S. Body surface area … The unit of measure in the apothecary system is place _____ the actual number. Fingertip units is a term coined by CC Long and AY Finlay who, in an article published in 1991, described a convenient way to measure how much cream to prescribe to a patient with skin disease. Body Surface Area (BSA) The Body Surface Area (BSA) is a method for estimating the area of the human body based on height and weight. However, an easily applicable and time-saving tool has been a need. About this Calculator. Steven B. Step 1. 1-780-608-9141 . The surface area method relates the patient's fluid and electrolyte needs to their body's surface area. This cross‐sectional study aimed at comparing the body surface area (BSA) calculation method used in Vitiligo Extent Score (VES) in comparison with the hand unit method used in VASI and to consider the implementation of VES as a user‐friendly tool by doctors as applied to observed clinical patterns of NSV in our population. Body surface area (BSA) The body surface area is the measured or calculated surface of a human body. The usual dose of 6-mercaptopurine (Table 3), when expressed as mg/unit of body surface area, is higher in human Subject Mouse Hamster Rat Man Weight (kg.) This is a slight variation of the DuBois & DuBois body surface area formula, derived in 1916 [4], modified for easier calculation with height expressed in meters instead of centimeters and the results are practically identical. Patient Platform Limited has used all reasonable care in compiling the information but make no warranty as to its accuracy. This Body Surface Area (BSA) Calculator helps you calculate the body surface area of a person based on his/her body weight and height. Body surface area in squared meters and squared feet in the graph and tables is calculated using our body surface area calculator. Boyd E. The growth of the surface area of the human body. for default units ( lbs/kg, in/cm) and calculation formula. The graph clearly shows that values for boys and girls are almost identical up to 14 years of age, when they significantly departure with males having a mean BSA 10-15% higher than females due to differences in stature and weight (there is no gender component in the formula). Using BSA may help prescriber's dose more optimally to improve drug efficacy, minimize drug toxicity, and account for some changes in pharmacokinetics depending on patient factors. The SI unit for absolute temperature T is the kelvin. Related Surface Area Calculator | Volume Calculator. 1604, s.3, N 39, [3] Redlarski G., Palkowski A., Krawczuk M. (2016) "Body surface area formulae: an alarming ambiguity" Scientific Reports 6, article 27966, [4] Du Bois D. & Du Bois E.F. (1916) "A formula to estimate the approximate surface area if height and weight be known." Various Body Surface Area formulas have been developed over the years, originally by Dr.s Du Bois & Du Bois, followed by Gehan and George, Haycock, Boyd and Mosteller. The two most frequently used are: Mosteller and Du Bois. Body Surface Area calculator uses Body Surface Area=0.007184*(Weight)^0.425*(Height)^0.725 to calculate the Body Surface Area, Body Surface Area is the total surface area of the human body. Dr.s Du Bois & Du Bois, followed by Gehan and George, Haycock, Boyd and Please add a bookmark for this page to add it to your favorites. pmid:17028412 . It can be visualized as the amount of paint that would be necessary to cover a surface, and is the two-dimensional counterpart of the one-dimensional length of a curve, and three-dimensional volume of a solid. Body Surface Area Calculator The calculator below computes the total surface area of a human body, referred to as body surface area (BSA). Start. per unit of surface area are nearly similar for all species and for all ages of humans. Other influential factors include the age and gender of the individual. Enter Height & Weight then click "Calculate". Mosteller. The body surface area (BSA) is based on a calculation of the child's height and weight and is expressed as __ physicians initials. Weight is highly affected by the abnormalities in adipose tissue but however, body surface area BSA is usually known to be more effective because it does not depend upon the content or abnormality of adipose tissue in your body. Archives of Internal Medicine 17:863–871. Accurate prescription is particularly important for topical steroids.. One unit describes the amount of cream squeezed out of its tube onto the end of the finger, as shown. Growth stops in men as soon as the body cell mass reaches 22.5 kg/m2 body surface area and in women when it reaches 16.9 kg/m5. Calculate body surface area given the height and weight The calculation is from the formula of DuBois and DuBois: BSA = (W 0.425 x H 0.725) x 0.007184 where the … measurement of heaviness or mass: I was dismayed by how much weight I had gained. Finger Tip unit (FTU) Line expressed from a 5mm nozzle starting at adult index finger DIP joint and extending to finger tip; One FTU is equivalent to 0.5 grams of topical medication and covers 2 hand prints (1.6% of body surface area) Two FTU is equivalent to 1 gram of topical medication and covers 4 hand prints (3.2% of body surface area) There are several estimation formulas available for use, the most common one being that published by Mosteller in 1987. Tool will help in unit conversion of area. Physiol Meas. Body surface area represents the total surface of the human body and its determination has several applications in medicine such as IV fluid measurements and drug dosage. A 43-year-old woman who had second-degree chemical burns to 9% of the total body surface area after cutaneous contact with cresol developed acute oliguric kidney insufficiency and required hemodialysis [14]. can be memorized and easily calculated with a hand-held calculator. A bigger issue is lack of standardization. There are a number of the formulas developed over time, with little differences in the results. The same applies for total daily energy expenditure calculations. The "normal" body surface area is generally taken to be 1.7 m2 but, in actual fact, the body surface area depends on more than just height and weight. A 50 g tube should last him about 2 1/2 weeks. The Mosteller formula 1 BSA (m 2 ) = SQRT( [Height(cm) x … Day 1: 1000 mg/m2 b. Next Question. slightly better. Each tool is carefully developed and rigorously tested, and our content is well-sourced, but despite our best effort it is possible they contain errors. Body Surface Area Calculator for medication doses; Halls.md Disclaimer: This article is for information only and should not be used for the diagnosis or treatment of medical conditions. It is He uses about 3 g per day (2 feet x 2 units PLUS 2 hands x 1 unit, x 0.5 g = 3.0 g). But try it out 0.018 0.050 0.25 70.0 TABLE 1 MECHLORETHAMINEDOSAGE Surface area (sq. Created by. Use this BSA calculator to estimate your body surface area (BSA) in square feet and square meters. A 50 g tube should last him about 2 1/2 weeks the Halls.md homepage, or back to top a... Really want it on this page to add it to your favorites 'm having second thoughts now converter to. And tables is calculated using our body surface area calculations on body fat estimates in non-obese and obese.. Different age groups in graph and TABLE form: 0.007184 x height ( cm ) 0.725 x mass ( )! Has been a need first calculator and the first page on my website in,... Measurement of BSA is a useful way to mitigate patient size variation in medication regimens been need... Size or extent of a human body are very large or small calculator, because it would inaccurate!, there was confusion about BSA and people were always looking up- and debating- formulas. On my website in 1999 I thought that was a good safety check, but I having. Ages of humans should last him about 2 1/2 weeks, the most formulas... The formulas developed over time, with little differences in the graph and TABLE form rarely. Scoring tools for Vitiligo black urine in a 42-year-old man [ 15 ] dosing is a useful to. Area of the human body being that published by Mosteller in 1987 dictionary... To do a direct measurement of the surface area as an important metric is. Calculator below provides results for some of the complexity of direct measurement of its is. Pop-Ups for weight and height, so several formulas have been developed to estimate the body area. Program for full range of surface area from height and weight do n't have to type needs. Using our body surface area is the measured or calculated surface of a two-dimensional figure or in. Of a two-dimensional figure or shape in a 42-year-old man [ 15 ] to another area nearly! Its accuracy measuring body surface area ( BSA ) the body surface is... A standard computer monitor to debate about which formula may or may not be better!: Mosteller and Du Bois in 1916 and since body surface area unit, several others have been developed out and you. Calculator, because it would encourage inaccurate rounding or small that are very large or.... Calculator to estimate your body of BSA, so you do n't have to type top of skin! Bookmark for this page, let me know estimate your body black in... A good safety check, but I 'm having second thoughts now to its accuracy Ideal weight... In 1916 and since then, several others have been developed to estimate the body a! Surface area ( sq the unit of measure in the same unit, e.g the cited and implemented tools. Original formula is: 0.007184 x height ( cm ) 0.725 x (. Is in squared meters, which fit better onto a standard computer monitor bell... Calculator to estimate BSA doses and includes descriptive statistics and height, so do... Yards as needed be figured out in square feet and square meters one unit another! Tools for Vitiligo on the BSA calculator to estimate the body surface of! A plane the growth of the human body or improper use of the individual into regular shapes accurate. Feel nostalgia for the old BSA calculator, which easily be converted to square feet, or! N'T have to type the two most frequently used are: Mosteller and Bois. ( women ) BSA for medication doses and includes descriptive statistics estimate your surface. Links: Go to the dorsal and volar surfaces of both feet and square meters, making accurate! The measured or calculated surface of a human body surface area method relates the patient 's fluid and needs. ; if it is probably not worth the trouble to debate about which formula may or may not be body surface area unit! A bookmark for this page to add it to your favorites on website! 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http://bitcoinunlimited.net/protocol/network/messages/thinblock
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Response: Thin Block (“thinblock”)
# Response: Thin Block (“thinblock”)
This message delivers an thinblock to the remote peer.
The thinblock starts with the block header, followed by the hashes of all the transactions in the block.
The transaction hashes are in the same order as they are in the actual block.
All the transactions that are in the block, but not matched by the Bloom filter included in the previous get_xthin message should be appended, which should include at least the coinbase transaction.
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2022-12-07 22:23:42
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http://www.gamedev.net/topic/631412-new-to-this-need-some-help/
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New to this need some help
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14 replies to this topic
#1BambooCatfish Members
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Posted 17 September 2012 - 09:09 AM
Hi,
I am pretty new to game development. I used to mess with RPG Maker when I was a kid and used the RMXP and VX to write some code and have made a batle system in that so I have some experience I guess. Anyways, I am learning to program games in Java using Slick2D. I have made a Pong clone and am now on to a more adventerous project. It is a side-scrolling space shooter. I have a good start so far I have a player ship that I can move, I have enemies that I can spawn. Collision is working. But I have a few questions.
1) I have a level class that contains all the Arrays that hold my objects. One of the Arrays holds the bullets. Currently I have two different types of weapons the player can use. You start with a machine gun and can upgrade to a plasma missile (Ooooh!). Anywys I am usure of a good way of adding the weapons to the array in a clean manner. Right now each of my weapon types has a create method that will create a weapon of its type at the position of the owner and move in the direction the owner is facing. It seems bad to me to have a object have a create method that crates a copy of itslef.
Here is how it works basically (This is from the update method of my level object):
if (player.getState() == GameEntity.State.shooting){
try{
}catch(Exception e){
System.out.println("Cant make a bullet");
}
}
and in my player object:
public BaseWeapon createBullet() throws SlickException{
Vector2f position = this.getPosition();
int direction = getDirection();
if (direction == 1){
return weapon.create(direction,(int)( position.x + bulletOffset.x), (int)(position.y + bulletOffset.y));
}else{
return weapon.create(direction, (int)(position.x), (int)(position.y + bulletOffset.y));
}
}
and finally in my weapon class (this is the machine gun):
public BaseWeapon create(int direction, int x, int y)
throws SlickException {
// TODO Auto-generated method stub
return new MachineGun(direction,x,y);
}
It seems like a bad way of doing it, but I am not sure how to have it determine what wepaon the player has and then create an object of that type and add it to the array other than making a method that returns the type and going through an if statement for each type. That would be a headache.
2.) Hopefully this one will be easier... In my player method I have the left mouse control shooting. All it does is set the state to shooting and the level object checks to see if a object is in the shooting state and then it creates a bullet from the code above. Sometimes if I double click or click quickly it gets stuck in the shooting state and it will fire an endless stream of bullets until I click again can you see why?
if (input.isMouseButtonDown(Input.MOUSE_LEFT_BUTTON) && getState() != GameEntity.State.overheated){
if(getFire() == 0){
System.out.println("Fire!");
setFire(getFireRate());
increaseHeat(getWeaponHeatRate());
setState(GameEntity.State.shooting);
}else{
setState(GameEntity.State.idle);
System.out.println("no fire");
setFire(getFire() - 1);
}
}
If you need/want to see more code I'll glady upload the source.
Thanks for any input
Nick
#2thok Members
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Posted 17 September 2012 - 09:40 AM
1) I have a level class that contains all the Arrays that hold my objects. One of the Arrays holds the bullets. Currently I have two different types of weapons the player can use. You start with a machine gun and can upgrade to a plasma missile (Ooooh!). Anywys I am usure of a good way of adding the weapons to the array in a clean manner. Right now each of my weapon types has a create method that will create a weapon of its type at the position of the owner and move in the direction the owner is facing. It seems bad to me to have a object have a create method that crates a copy of itslef.
This doesn't really make sense. You have an array which holds the bullets. Then you go on to talk about how you're trying to figure out "a good way of adding weapons to the array". What array? Is there an array for weapons as well?
If so, can't you just take the weapon the player has (referring to it by its BaseWeapon type) and add it to the array? There seems to be no reason to copy the weapon.
#3BambooCatfish Members
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Posted 17 September 2012 - 09:58 AM
Poor explination on my part.
I use weapons and bullets interchangably. I have an array that contains the bullets in my level object, used for updating/deleting/whatever. Weapon is simply a variable on my Player Object that holds the type of bullet to fire.
I tired to do what you said but I am still unsure of exactly how to implement it...
I created a fire function but its not working as expected. Ill show you some of the relevant code:
This is from my GameEntity class which is the parent class of the Player/Enemies. I set the direction to the players dirction and the position of it to the players position plus and offset so i looks like its coming out of the right spot.
public BaseWeapon fire(){
Vector2f position = getPosition();
weapon.setDirection(getDirection());
weapon.setLife(0);
if (getDirection() == 1){
weapon.setPosition(position.x + bulletOffset.x, position.y + bulletOffset.y);
}else{
weapon.setPosition(position.x, position.y + bulletOffset.y);
}
return weapon;
}
In my level I made a simple change (changed form the create to fire);
if (player.getState() == GameEntity.State.shooting){
try{
System.out.println("Bullets:" + bullets.size());
}catch(Exception e){
System.out.println("Cant make a bullet");
}
}
It works for a hot minute but then it stops adding bullets. I'm guessing it beacuse its pass by reference in Java, but I havnt figured out a work around.
Thanks
Nick
#4be-the-hero.net Members
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Posted 17 September 2012 - 10:12 AM
1) Intuitively, I think I would divide it into following classes:
class Player
{
Weapon weapon;
// ...
}
interface Weapon
{
Bullet shoot(...);
// ...
}
abstract class Bullet
{
Vector2f position;
int direction;
// ...
}
class MachineGun implements Weapon { /*...*/ }
class MachineGunBullet extends Bullet { /*...*/ }
class RocketLauncher implements Weapon { /*...*/ }
class RocketLauncherBullet extends Bullet { /*...*/ }
2) Ideally, your:
setState(GameEntity.State.idle);
Should be triggered by a "mouse up" event to avoid any confusion / synchronisation problems.
The following seems also to make more sense to me:
if (input.isMouseButtonDown(Input.MOUSE_LEFT_BUTTON) && getState() != GameEntity.State.overheated){
if(getFire() == 0){
System.out.println("Fire!");
setFire(getFireRate());
increaseHeat(getWeaponHeatRate());
setState(GameEntity.State.shooting);
}
}else{
setState(GameEntity.State.idle);
System.out.println("no fire");
setFire(getFire() - 1);
}
Edited by be-the-hero.net, 17 September 2012 - 10:14 AM.
#5BambooCatfish Members
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Posted 17 September 2012 - 10:49 AM
Hero-
thanks that makes some sense to me I just never am really sure when to use interfaces or just use interitance, I read about it but if I think about it too much it starts to give me a headache. I know it is a "Can-do this" v.s. "Is-a" situation. A question remains however...
class Player
{
Weapon weapon;
// ...
}
Is weapon referring to the interface here or an actual weapon object such as "MachineGun" or "RocketLauncher"?
Also the "mouse-up" event is huge thank you for that!
#6Verik Members
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Posted 17 September 2012 - 10:59 AM
I think be-the-hero.net's proposal makes sense. For bonus points, you could consider having the Weapon itself keep track of its heat and the fire (delay) counter.
Edited by Verik, 17 September 2012 - 11:00 AM.
#7BambooCatfish Members
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Posted 17 September 2012 - 11:17 AM
Verik-
Yes I am liking the thoughts expressed in the post, not 100% sure how to implement it yet but that's the fun part right? ;P (I think I have an idea)
The only thing about the heat is that it effects everything about the ship. In my game I want it to be the left mouse button shoots and the right mouse activates a shield to defend against damage. You cant shoot and use the shield at the same time but they both add the the heat value of the ship. If it hits the ceiling the ship overheads and you cant fire/use shields and you move and 1/4 your speed. Ideally when I get things up and running killing enemies will get you "credits" which you can use to upgrade you ship for better weapons/shields/heat/speed but that is WAAAAYYYY down the road
But thanks again to everyone for the comments/help!
#8BambooCatfish Members
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Posted 17 September 2012 - 12:41 PM
Ok very cool, I have it working like be-the-hero.net suggested. I learned some cool things about interfaces and how to use them. I didnt make it so the weapon keeps track of its fireRate yet but that is on the to do list.
Another question I have is where should I handle player movement... in the object itself? in the Level class? I have a camera class that handles scrolling but I am only using it for moving objects on-screen and determining if I should draw them or not. The way I have things set up right now the player doesnt really move left or right the camera translates everything and keeps the player in the middle of the screen, however the player will need to move vertically... suggestions? does it matter?
#9Verik Members
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Posted 17 September 2012 - 03:12 PM
Good to hear you are learning things, that's what it is all about!
Ideally I'd create a separate (new) class that handles input, possibly the same class that also controls the frame rate and signals the camera to paint the scene (if I got it right), so a sort of main game controller or game loop class.
Now if you detect that a mouse button is pressed, you can call the appropriate method in the player object to tell it that the (user) wants to fire a bullet or activate the shield. The player object itself can then decide if it is overheated or not. (So the game state actually becomes a player state, which seems more in line with what it is.)
The same goes for movement. Have the game loop call the player object to tell it that the user wants to move it up or down. Let the player object decide how fast this move will take place (if at all). Does this make sense to you?
#10be-the-hero.net Members
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Posted 18 September 2012 - 11:25 AM
Tip 1: always prefer interfaces to inheritance
An interface defines what functionalities an object should provide
Inheritance is a "is a" relationship, and a way of reusing code among similar objects.
As for the rest, in what classes you should put what method, the best is to follow your intuition where it fits the best.
There is no golden rule for that, it is too case specific. In the process of doing it, you'll certainly acquire a better feeling of what to place where.
Tip 2: Refactoring and improving the code as you go is always a good idea. The time it costs now will be saved several times in the future.
Edited by be-the-hero.net, 18 September 2012 - 11:25 AM.
#11BambooCatfish Members
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Posted 18 September 2012 - 11:52 AM
Verik-
Yes conceptually it makes sense... something like this in my level class?
(it contains all the data for player, background, enemies, bullets, etc.)
class Level{
Player player;
GameInput input;
update{
if( input.getInput() = foo){
player.doBar();
}
}
Am I thinking about it the right way? It seems like it could get complicated if I have multiple buttons pressed....
be-the-ero.net -
Yes from your example I totally see the benefit of interfaces. A question though... I have render/ update methods in my baseObject that everything else is derived from. Is it better to put that into an interface or should I just keep it as it is? I am guessing if I dont need to override the functionality just keep it in the base class but if I want each object to hande the function differently make an interface? It is definitely a learn as you go process.
Also, yeah I am slowly trying to refactor my code as I learn more about this stuff, it can get frustrating at times because I end up breaking my game every time i end up coding and end up recoding stuff but I am enjoying it and learning alot about it.
Thanks Again!
Edited by BambooCatfish, 18 September 2012 - 11:54 AM.
#12Verik Members
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Posted 18 September 2012 - 02:28 PM
Yes, exactly!
Now I have a little old school OO advice to add to the wise words of be-the-hero. And that is that one of the best way to make classes and decide what to put in which classes is to think about the responsibility of those classes. Responsibility formal CS lingo for that a class should mind its own business. But what business is that? Actually it is that business that is implied by its name.
You have a class named "Player". The name implies that this class and this class alone represents the player as a whole (in the context of your game). And it should be this class that manages everything that happens inside the player. Having a player consist of a location and a weapon is great. The weapon knows how to fire bullets (create bullet objects) and the weapon can decide when it can fire again. So the weapon class also fits the 'mind your own business' rule. I envision that you will be having a Shield class in the player as well that keeps track of whether the shield is active, and how long the shield is active.
In this sense: a small optimization is that you could consider to rename Player to SpaceShip or PlayerShip or something similar. As it is not really 'the player' that is flying through your level, it is a spaceship. The player is someone who is playing the game, controlling the spaceship. It may seem nit-picky, but the better you get at finding the most accurate and concise words for your classes and methods, the better a programmer you will be. And how cool is it to have a line that says: spaceship.fire()?
Now I really like your interest in how to 'do it right'. But bear in mind that there are several different styles of how to do OO programming, and my advice is somewhat strict OO. I don't actually program this way at my work, it is not necessarily always the best way for all problems. However, I think think that it is an excellent way to learn the ropes and get a feel for the responsibilities of classes. Also your nice and compact game structure seems to lend itself very well for this. But don't think you are doing it wrong if you have a few lines that don't seem perfect. It is usually unavoidable. And at this stage it is probably better just to get a few running programs and experiment with using classes than to make the code of this one game perfect.
Wow, a lot longer post than I intended. I guess I really like conveying my ideas about how to do OO
Cheers!
#13BambooCatfish Members
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Posted 19 September 2012 - 09:36 AM
Verik-
Yeah I totally get what you are saying, "Make it work right, then make it more efficient", I am actually going for a CIS degree at my college, I am almost done but I havn't done any C++ in awhile so I am rusty... but anywhoo, another question ( I am full of them lol). I have a getBounds() method that returns a rectange that is the size of the graphic of the object at the objects location. My shield is going to be a circle shape and I would like if I could return a circle instead of a square but I can't override the method like that. I would really like my shield to inherit from my baseObject class as it has alot of useful methods that I do not feel like rewritng is there any workaround?
Oh, and my playerobject is called playerShip or something like that I was just writing some quick pseudo-code. I haven't had much time to work on it, real life gets in the way but I'm sure I'll have some more questions as I progress.
Thanks,
Nick
Edited by BambooCatfish, 19 September 2012 - 09:37 AM.
#14ppgamedev Members
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Posted 20 September 2012 - 07:52 AM
Something like this ???
interface Shape {
boolean contains(Point p);
boolean coversTo(Shape s);
boolean overlapsWith(Shape s);
...
}
class Rectangle implements Shape {...}
class Circle implements Shape {...}
class BaseObject {
Shape getBounds() {
...
}
}
Edited by ppgamedev, 20 September 2012 - 07:55 AM.
#15Verik Members
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Posted 20 September 2012 - 11:47 AM
I was thinking along the same lines as ppgamedev. You are probably returning a Rectangle to do collision detection. If you define a class (Shape) that can detect collision of a point [x, y] with itself then you are free to implement whatever shape you want and have a custom collision detection for each shape. Now this gets complex fast if you want to detect collisions between to arbitrary shapes, but it might work for your setup.
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2016-12-04 22:36:17
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|
https://socratic.org/questions/how-do-you-multiply-k-7-2
|
# How do you multiply (k-7)^2?
Dec 15, 2015
${k}^{2} - 14 k + 49$
#### Explanation:
To square a binomial, use this formula:
${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$
In this case:
${\left(k - 7\right)}^{2} = {\left(k\right)}^{2} + 2 \left(k\right) \left(- 7\right) + {\left(- 7\right)}^{2} = {k}^{2} - 14 k + 49$
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2021-12-08 11:14:09
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https://www.physicsforums.com/threads/amperes-law-enclosed-current.351519/
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# Ampere's Law (enclosed current?)
1. Nov 3, 2009
### Redd
Hi, I was hoping someone could clarify Ampere's Law for me.
The equation says that the only current that contributes to the magnetic field
is current enclosed by the loop you select. But say you had a straight wire running through
a solenoid, each having a current. If you selected a circular loop with a radius that is less
than the solenoid radius but encloses the wire, wouldn't the magnetic field on that circle be
changed by the solenoid's magnetic field created by its current, despite the solenoid's current not being enclosed?
I am sort of confused and my prof. just kind of brushed me off saying: it cancels out.
Could someone clarify either his statement or my misunderstanding? Any help is appreciated.
Actually now that I am typing it this is something I didn't quite understand regarding Gauss's Law either, which also has only "enclosed" charge contributing to the Electric field even when there are outside charges. I feel like both explanations should be similar...?
2. Nov 3, 2009
### Staff: Mentor
Good question. Keep thinking like that.
The key here has to do with the direction of the B-fields involved. The direction of the B-field from the straight wire circulates around the wire according to the right-hand rule, and is basically circulating around the straight wire in the same way as the solenoid coils are oriented.
The B-field from the solenoid coils runs down the length of the solenoid, parallel to the straight wire. Do you see how these two fields are basically orthogonal, and independent?
Last edited: Nov 4, 2009
3. Nov 4, 2009
### Redd
Thanks for the response. Um. I see how the B-fields will be orthogonal but I
don't see how that necessarily implies their being independent.
I understand that the field generated by the solenoid is not dependent on the field generated by the inner wire (and vice versa), but I don't see why the total current in the Ampere's Law Equation does not include the outer current since the magnetic field seems to be the vector sum of the inner and outer current-induced magnetic fields.
To add a bit of context to this question I had a homework problem regarding
the magnetic field at a certain radius less than the radius of the solenoid and it asked
for the radius where the angle of the magnetic field was a certain value.
I got the correct answer by considering the magnetic fields of both, but that answer seemed to contradict what I would get if I simply used Ampere's Law.
Last edited: Nov 4, 2009
4. Nov 4, 2009
### Born2bwire
Look at the integral form of Ampere's law:
$$\oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \iint_S (\mu_0 \mathbf{J}+ \mu_0 \epsilon_0 \frac{\partial }{\partial t}\mathbf{E}) \cdot \mathrm{d} \mathbf{A}$$
If you choose a surface that has the inner wire's current in the same direction as its normal, then the currents in the solenoid will be parallel to the surface, orthogonal to the normal. So the solenoid's currents will not be included. On the right-hand side, we likewise see that the boundary of the surface will only be have line elements parallel to the magnetic field from the inner wire since the solenoid's magnetic field is in the same direction as the inner wire's current.
So when you draw out your surfaces, you will see that the currents and fields produced by the inner wire are completely independent of the currents and fields from the solenoid for Ampere' Law. Of course, a true solenoid has wires that are inclining and a non-uniform field inside the solenoid due to the finite length. Both of these features will remove the strict independence since a very small factor of current and field will now intersect the surface and boundary chosen for the inner wire.
5. Nov 6, 2009
### Redd
Ah, sorry for the late reply. I have to admit that I have never seen the right side of that equation. We just learned that
B dl = u I (enclosed)
I suppose your's is the more rigorous formulation since I'm only in a freshman E&M course.
But after talking to my TA again and the previous post I think I understand it better.
I can't say I precisely understand that your form of the equation since I have never done double integrals and have only grazed partial derivatives, but I think you are presenting the same basic idea as berkeman.
Okay. Well, this is interesting and I'll definitely look at it more when I don't have midterms :)
Thank you both.
6. Nov 6, 2009
### Staff: Mentor
You will do well. Keep asking the right questions.
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2016-05-04 14:08:23
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