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Thymidylate synthase inhibitors are chemical agents which inhibit the enzyme thymidylate synthase and have potential as an anticancer chemotherapy . [ 1 ] This inhibition prevents the methylation of C5 of deoxyuridine monophosphate (dUMP) thereby inhibiting the synthesis of deoxythymidine monophosphate (dTMP). The downstream effect is promotion of cell death because cells would not be able to properly undergo DNA synthesis if they are lacking dTMP, a necessary precursor to dTTP. [ 2 ] Five agents were in clinical trials in 2002: raltitrexed , pemetrexed , nolatrexed , Plevitrexed ( ZD9331 / BGC9331 ), and GS7904L . [ 3 ]
Examples include
This biochemistry article is a stub . You can help Wikipedia by expanding it .
This oncology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Thymidylate_synthase_inhibitor |
Thymol blue (thymolsulfonephthalein) is a brownish-green or reddish-brown crystalline powder that is used as a pH indicator . It is insoluble in water but soluble in alcohol and dilute alkali solutions.
It transitions from red to yellow at pH 1.2–2.8 and from yellow to blue at pH 8.0–9.6. It is usually a component of Universal indicator .
At wavelength (378 - 382) nm, extinction coefficient > 8000 and at wavelength (298 - 302) nm, the extinction coefficient > 12000. [ 3 ]
Thymol blue has different structures at different pH.
It may cause irritation. Its toxicological properties have not been fully investigated. [ 4 ] Harmful if swallowed, Acute Toxicity. Only Hazardous when percent values are above 10%. [ 5 ] | https://en.wikipedia.org/wiki/Thymol_blue |
Thymolphthalein is a phthalein dye used as an acid – base ( pH ) indicator . Its transition range is around pH 9.3–10.5. Below this pH, it is colorless; above, it is blue. The molar extinction coefficient for the blue thymolphthalein dianion is 38,000 M −1 cm −1 at 595 nm. [ 2 ]
Thymolphthalein is also known to have use as a laxative [ 3 ] and for disappearing ink . [ 4 ]
Thymolphthalein can be synthesized from thymol and phthalic anhydride .
This article about an organic compound is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Thymolphthalein |
Thymolphthalexone is a chemical compound from the group of iminodiacetic acid derivatives of thymolphthalein . [ 3 ] Its chemical formula is C 38 H 44 N 2 O 12 .
This is a metallochromic indicator widely used in complexometric titrations, particularly for the determination of transition metals . The compound features a thymolphthalein-derived core linked to aminopolycarboxylic acid functional groups. This hybrid architecture grants the compound the ability to preferentially bind specific metal ions through coordinated interactions.
Thymolphthalexone can be obtained by Mannich condensation of formaldehyde and iminodiacetic acid with thymolphthalein. [ 4 ]
Thymolphthalexone forms a white crystalline powder soluble in water and organic solvents. [ 2 ]
Thymolphthalexone and its sodium salt are used as an indicator or photometric reagent for alkaline metal ions, such as those of calcium, strontium, barium, and others. [ 5 ] [ 6 ] | https://en.wikipedia.org/wiki/Thymolphthalexone |
Thymus transplantation is a form of organ transplantation where the thymus is moved from one body to another. It is used in certain immunodeficiencies , such as DiGeorge Syndrome. [ 1 ]
Thymus transplantation is used to treat infants with DiGeorge syndrome , which results in an absent or hypoplastic thymus, in turn causing problems with the immune system 's T-cell mediated response. It is used in people with complete DiGeorge anomaly, which are entirely athymic. This subgroup represents less than 1% of DiGeorge syndrome patients. [ 2 ]
Nezelof syndrome is another thymus-related disease where it can be used. [ 3 ]
Thymus transplantation can also be used in pediatric patients with a Foxn1 deficiency. [ 4 ]
In the 2000s, promising animal experiments into transplanting thymic tissue and another organ at the same time were carried out, in order to improve the recipient's tolerance of the transplanted organ, and to reduce the need for immunosuppressing drugs like tacrolimus . Such trials have been performed with kidney and heart transplants, drastically extending the time the animals were surviving without immunosuppressing drugs. [ 5 ] The first human heart-and-thymus co-transplantation was performed on Easton Sinnamon in 2022, a newborn who suffered from both a lack of T cells, and a serious heart defect. Depending on the development, it is planned to wean him off immunosuppressant drugs, but it remains to be seen whether the same technique is viable in adults, as the thymus shrinks with age, with the bone marrow taking over T cell production. [ 6 ]
A study of 54 DiGeorge syndrome infants resulted in all tested subjects having developed polyclonal T-cell repertoires and proliferative responses to mitogens. The procedure was well tolerated and resulted in stable immunoreconstitution in these infants. It had a survival rate of 75%, having a follow-up as long as 13 years. [ 2 ]
Complications include an increased susceptibility to infections while the T cells have not yet developed, rashes and erythema. [ 2 ]
Theoretically, thymus transplantation could cause two types of graft-versus-host disease (GVHD): First, it could cause a donor T cell-related GVHD, because of T cells from the donor that are present in the transplanted thymus that recognizes the recipient as foreign. Donor T cells can be detected in the recipient after transplantation, but there is no evidence of any donor T cell-related graft-versus-host disease. [ 2 ] [ 7 ]
Second, a thymus transplantation can cause a non-donor T cell-related GVHD because the recipients thymocytes would use the donor thymus cells as models when going through the negative selection to recognize self-antigens, and could therefore still mistake own structures in the rest of the body for being non-self. This is a rather indirect GVHD because it is not directly cells in the graft itself that causes it, but cells in the graft that make the recipient's T cells act like donor T cells. It would also be of relatively late-onset because it requires the formation of new T cells. It can be seen as a multiple-organ autoimmunity in xenotransplantation experiments of the thymus between different species. [ 8 ] Autoimmune disease is a frequent complication after human allogeneic thymus transplantation, found in 42% of subjects over 1 year post transplantation. [ 9 ] However, this is partially explained by that the indication itself, that is, complete DiGeorge syndrome, increases the risk of autoimmune disease. [ 2 ] | https://en.wikipedia.org/wiki/Thymus_transplantation |
Thyroid's secretory capacity ( G T , also referred to as thyroid's incretory capacity , maximum thyroid hormone output , T4 output or, if calculated from serum levels of thyrotropin and thyroxine, as SPINA-GT [ a ] ) is the maximum stimulated amount of thyroxine that the thyroid can produce in a given time-unit (e.g. one second). [ 2 ] [ 3 ]
Experimentally, G T can be determined by stimulating the thyroid with a high thyrotropin concentration (e.g. by means of rhTSH , i.e. recombinant human thyrotropin) and measuring its output in terms of T4 production, or by measuring the serum concentration of protein-bound iodine-131 after administration of radioiodine . [ 4 ] These approaches are, however, costly and accompanied by significant exposure to radiation. [ 5 ]
In vivo , G T can also be estimated from equilibrium levels of TSH and T4 or free T4. In this case it is calculated with
G ^ T = β T ( D T + [ T S H ] ) ( 1 + K 41 [ T B G ] + K 42 [ T B P A ] ) [ F T 4 ] α T [ T S H ] {\displaystyle {\hat {G}}_{T}={{\beta _{T}(D_{T}+[TSH])(1+K_{41}[TBG]+K_{42}[TBPA])[FT_{4}]} \over {\alpha _{T}[TSH]}}}
or
G ^ T = β T ( D T + [ T S H ] ) [ T T 4 ] α T [ T S H ] {\displaystyle {\hat {G}}_{T}={{\beta _{T}(D_{T}+[TSH])[TT_{4}]} \over {\alpha _{T}[TSH]}}}
[ TSH ]: Serum thyrotropin concentration (in mIU/L or μIU/mL) [ FT4 ]: Serum free T4 concentration (in pmol/L) [ TT4 ]: Serum total T4 concentration (in nmol/L) G ^ T {\displaystyle {\hat {G}}_{T}} : Theoretical (apparent) secretory capacity (SPINA-GT) α T {\displaystyle \alpha _{T}} : Dilution factor for T4 (reciprocal of apparent volume of distribution, 0.1 L −1 ) β T {\displaystyle \beta _{T}} : Clearance exponent for T4 (1.1e-6 sec −1 ), i. e., reaction rate constant for degradation K 41 : Binding constant T4-TBG (2e10 L/mol) K 42 : Binding constant T4-TBPA (2e8 L/mol) D T : EC 50 for TSH (2.75 mU/L) [ 2 ] [ 6 ]
The method is based on mathematical models of thyroid homeostasis. [ 2 ] [ 3 ] Calculating the secretory capacity with one of these equations is an inverse problem . Therefore, certain conditions (e.g. stationarity) have to be fulfilled to deliver a reliable result.
The ratio of SPINA-GT and thyroid volume V T (as determined e.g. by ultrasonography )
G ^ T S = G ^ T V T {\displaystyle {\hat {G}}_{TS}={\frac {{\hat {G}}_{T}}{{V}_{T}}}} ,
i.e.
G ^ T S = β T ( D T + [ T S H ] ) ( 1 + K 41 [ T B G ] + K 42 [ T B P A ] ) [ F T 4 ] α T [ T S H ] V T {\displaystyle {\hat {G}}_{TS}={\frac {\beta _{T}(D_{T}+[TSH])(1+K_{41}[TBG]+K_{42}[TBPA])[FT_{4}]}{\alpha _{T}[TSH]{V}_{T}}}}
or
G ^ T S = β T ( D T + [ T S H ] ) [ T T 4 ] α T [ T S H ] V T {\displaystyle {\hat {G}}_{TS}={\frac {\beta _{T}(D_{T}+[TSH])[TT_{4}]}{\alpha _{T}[TSH]{V}_{T}}}}
is referred to as specific thyroid capacity (SPINA-GTs). [ 7 ] It is a measure for how much one millilitre of thyroid tissue can produce under conditions of maximum stimulation. Thereby, SPINA-GTs is an estimate for the endocrine quality of thyroid tissue. [ citation needed ]
The equations and their parameters are calibrated for adult humans with a body mass of 70 kg and a plasma volume of ca. 2.5 L. [ 2 ]
SPINA-GT is elevated in primary hyperthyroidism [ 8 ] [ 9 ] and reduced in both primary hypothyroidism [ 10 ] [ 11 ] [ 12 ] [ 9 ] and untreated autoimmune thyroiditis. [ 13 ] It has been observed to correlate (with positive direction) to resting energy expenditure , [ 14 ] resting heart rate , [ 15 ] the colour Doppler ultrasound pattern [ 16 ] and thyroid volume, [ 2 ] [ 7 ] and (with negative direction) to thyroid autoantibody titres, which reflect organ destruction due to autoimmunity. [ 17 ] Elevated SPINA-GT in Graves' disease is reversible with antithyroid treatment. [ 14 ] While SPINA-GT is significantly altered in primary thyroid disorders, it is insensitive to disorders of secondary nature (e.g. pure pituitary diseases). [ 3 ]
In silico experiments with Monte Carlo simulations demonstrated that both SPINA-GT and SPINA-GD can be estimated with sufficient reliability, even if laboratory assays have limited accuracy. [ 3 ] This was confirmed by longitudinal in vivo studies that showed that GT has lower intraindividual variation (i.e. higher reliability) than TSH , FT4 or FT3 . [ 18 ]
In clinical trials SPINA-GT was significantly elevated in patients with Graves' disease and toxic adenoma compared to normal subjects. [ 2 ] [ 8 ] [ 19 ] It is also elevated in diffuse and nodular goiters , and reduced in untreated autoimmune thyroiditis. [ 2 ] [ 13 ] In patients with toxic adenoma it has higher specificity and positive likelihood ratio for diagnosis of thyrotoxicosis than serum concentrations of thyrotropin , free T4 or free T3. [ 2 ] GT's specificity is also high in thyroid disorders of secondary or tertiary origin. [ 3 ]
Calculating SPINA-GT has proved to be useful in challenging clinical situations, e.g. for differential diagnosis of subclinical hypothyroidism and elevated TSH concentration due to type 2 allostatic load (as it is typical for obesity and certain psychiatric diseases). [ 20 ] For this purpose, its usage has been recommended in sociomedical assessment . [ 21 ]
In patients suffering from toxic adenoma, toxic multinodular goitre and Graves’ disease radioiodine therapy leads to a significant decrease of the initially elevated SPINA-GT. [ 19 ]
Correlation of SPINA-GT with creatinine clearance suggests a negative influence of uremic toxins on thyroid biology. [ 22 ] [ 23 ] In the initial phase of major non-thyroidal illness syndrome (NTIS) SPINA-GT may be temporarily elevated. [ 24 ] [ 25 ] In chronic NTIS [ 26 ] as well as in certain non-critical chronic diseases, e.g. chronic fatigue syndrome [ 27 ] [ 28 ] or asthma [ 29 ] SPINA-GT is slightly reduced.
According to the results of a community-based study in China it was associated to sleep duration and exercise habits. [ 30 ] With respect to iodine supply, it showed a complex U-shaped pattern, being reduced in subjects consuming iodine-rich food, but elevated in situations of iodine excess. [ 30 ] In two other studies from China, SPINA-GT correlated with negative direction to markers of obesity including body mass index, waist circumference and waist to hip ratio. [ 31 ] [ 32 ] This doesn't seem to be the case, however, in Western populations. [ 33 ]
In women, therapy with Metformin results in increased SPINA-GT, in parallel to improved insulin sensitivity . [ 34 ] [ 35 ] This observation was reproducible in men with hypogonadism, but not in men with normal testosterone concentrations,. [ 36 ] In postmenopausal women this effect was only observed in subjects on oestradiol replacement therapy. [ 37 ] Therefore, the described phenomenon seems to depend on an interaction of metformin with sex hormones. [ 36 ] [ 38 ] In hyperthyroid [ 8 ] men both SPINA-GT and SPINA-GD negatively correlate to erectile function , intercourse satisfaction, orgasmic function and sexual desire . Likewise, in women with thyrotoxicosis elevated thyroid's secretory capacity predicts depression and sexual dysfunction. [ 39 ] Conversely, in androgen-deficient men with concomitant autoimmune thyroiditis , substitution therapy with testosterone leads to a decrease in thyroid autoantibody titres and an increase in SPINA-GT. [ 40 ] In a large study from mainland China, SPINA-GT was elevated in certain psychiatric diseases including bipolar disorder and schizophrenia. [ 41 ] In bipolar disorder with manic or mixed episodes it was higher than in cases with depressive episodes. [ 41 ]
SPINA-GT is reduced in persons suffering from hidradenitis suppurativa compared to healthy controls with the same sex and age distribution. [ 42 ] This phenomenon has been ascribed to B-cell -mediated hypothyroidism , i.e. hypothyroid Graves' disease due to inhibiting TSH receptor autoantibodies (iTRAb). [ 42 ]
In patients with autoimmune thyroiditis a gluten-free diet results in increased SPINA-GT (in parallel to sinking autoantibody titres). [ 43 ] Statin therapy has the same effect, but only if supply with vitamin D is sufficient. [ 44 ] Accordingly, substitution therapy with 25-hydroxyvitamin D leads to rising secretory capacity. [ 45 ] [ 46 ] [ 47 ] [ 48 ] This effect is potentiated by substitution therapy with myo-inositol [ 49 ] and selenomethionine [ 45 ] [ 46 ] [ 50 ] or, in women, with dehydroepiandrosterone , [ 51 ] but impaired in males with early-onset androgenic alopecia. [ 52 ] The effects of vitamin D and selenomethionine are attenuated in hyperprolactinaemia , suggesting an inhibitory effect of prolactin . [ 53 ] Although both vitamin D supplementation and gluten-free diet result in increased SPINA-GT, there seems to be a complex interaction between both therapeutic measures, since vitamin D treatment is only able to elevate the thyroid's secretory capacity in subjects not following any dietary recommendation. [ 54 ]
On the other hand, men treated with spironolactone are faced with decreasing SPINA-GT (in addition to rising thyroid antibody titres). [ 55 ] It has, therefore, been concluded that spironolactone may aggravate thyroid autoimmunity in men. [ 55 ]
In subjects with type 2 diabetes , treatment with beta blockers resulted in decreased SPINA-GT, suggesting sympathetic innervation to contribute to the control of thyroid function. [ 56 ] In diabetic women, but not in men, SPINA-GT shows a positive correlation to the β-C-terminal cross-linked telopeptides of type I collagen (β-CTX), a marker of bone resorption. [ 57 ] In both diabetic and non-diabetic persons it correlates (negatively) with age and (positively) with the concentrations of troponin T and HbA1c . [ 58 ]
SPINA-GT correlates to mechanical pain sensitivity (MPS) in quantitative sensory testing (QST) and to measures of respiratory arrhythmia in the analysis of heart rate variability , indicating a potential link to both sensorimotor and autonomic neuropathy . [ 59 ]
A study in euthyroid subjects with structural heart disease found that increased SPINA-GT predicts the risk of malignant arrhythmia including ventricular fibrillation and ventricular tachycardia . [ 60 ] This applies to both incidence and event-free survival. [ 60 ] Likewise, SPINA-GT is elevated in a significant subgroup of patients with takotsubo syndrome , [ 61 ] especially in non-survivors. [ 62 ] A stress-mediated effect on SPINA-GT is also suggested by the observation that it is increased in persons with a history of psychological trauma. [ 63 ] On the other hand, two studies found negative correlation between SPINA-GT and markers of dispersion in cardiac repolarisation, including Tp-e interval, JT interval, Tp-e/ QT ratio and Tp-e/QTc ratio. These results suggest that reduced thyroid function may trigger cardiovascular mortality as well. [ 64 ] [ 9 ]
Among subjects with Parkinson's disease , SPINA-GT is significantly elevated in tremor -dominant and mixed subtypes compared to the akinetic-rigid type. [ 65 ]
Specific secretory capacity (SPINA-GTs) is reduced in obesity [ 2 ] and autoimmune thyroiditis . [ 7 ] [ 66 ]
Endocrine disruptors may affect stimulated thyroid output, as demonstrated by a positive correlation of SPINA-GT with exposure to 2-hydroxynaphthalene (2-NAP), [ 67 ] urinary mercury concentration [ 68 ] and the excretion of certain phthalate metabolites, [ 69 ] and negative correlation with combined exposure to polycyclic aromatic hydrocarbons (PAHs) [ 67 ] and nickel . [ 70 ] Additionally, SPINA-GT is altered in persons exposed to butylparaben and propylparaben . [ 71 ] [ 72 ]
In a longitudinal evaluation of a large sample of the general US population over 10 years, reduced SPINA-GT significantly predicted all-cause mortality. [ 73 ] | https://en.wikipedia.org/wiki/Thyroid's_secretory_capacity |
The Thyroid Feedback Quantile-based Index ( TFQI ) is a calculated parameter for thyrotropic pituitary function. It was defined to be more robust to distorted data than established markers including Jostel's TSH index (JTI) and the thyrotroph thyroid hormone sensitivity index (TTSI).
The TFQI can be calculated with
from quantiles of FT4 and TSH concentration (as determined based on cumulative distribution functions ). [ 1 ] Per definition the TFQI has a mean of 0 and a standard deviation of 0.37 in a reference population. [ 1 ] This explains the reference range of –0.74 to + 0.74.
Higher values of TFQI are associated with obesity , metabolic syndrome , impaired renal function, diabetes , and diabetes-related mortality . [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] In a large population of community-dwelling euthyroid subjects the thyroid feedback quantile-based index predicted all-cause mortality, even after adjustment for other established risk factors and comorbidities. [ 8 ]
A cross-sectional study from Spain observed increased prevalence of type 2 diabetes, atrial fibrillation, ischemic heart disease and hypertension in persons with elevated PTFQI. [ 9 ]
Serum Concentrations of Adipocyte Fatty Acid-Binding Protein (A-FABP) are significantly correlateted to TFQI, suggesting some form of cross-talk between adipose tissue and HPT axis. [ 10 ]
TFQI results are also elevated in takotsubo syndrome , [ 11 ] potentially reflecting type 2 allostatic load in the situation of psychosocial stress . Reductions have been observed in subjects with schizophrenia after initiation of therapy with oxcarbazepine [ 12 ] and quetiapine, [ 13 ] potentially reflecting declining allostatic load.
Despite positive association to metabolic syndrome and type 2 allostatic load a large population-based study failed to identify an association to risks of dyslipidemia and non-alcoholic fatty liver disease (NAFLD). [ 14 ] | https://en.wikipedia.org/wiki/Thyroid_Feedback_Quantile-based_Index |
Potassium iodide (KI) and potassium iodate (KIO 3 ) are called thyroid blockers when used in radiation protection . [ 1 ] [ 2 ] [ 3 ] [ 4 ]
If a person consumes a dose of one of these chemical compounds , his or her thyroid may saturate with stable iodine , preventing accumulation of radioactive iodine found after a nuclear meltdown or explosion .
This medical treatment –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Thyroid_blocker |
Thyroid function tests ( TFTs ) is a collective term for blood tests used to check the function of the thyroid . [ 1 ] TFTs may be requested if a patient is thought to suffer from hyperthyroidism (overactive thyroid) or hypothyroidism (underactive thyroid), or to monitor the effectiveness of either thyroid-suppression or hormone replacement therapy. It is also requested routinely in conditions linked to thyroid disease, such as atrial fibrillation and anxiety disorder .
A TFT panel typically includes thyroid hormones such as thyroid-stimulating hormone (TSH, thyrotropin) and thyroxine (T4), and triiodothyronine (T3) depending on local laboratory policy.
Thyroid-stimulating hormone (TSH, thyrotropin) is generally increased in hypothyroidism and decreased in hyperthyroidism, [ 2 ] making it the most important test for early detection of both of these conditions. [ 3 ] [ 4 ] The result of this assay is suggestive of the presence and cause of thyroid disease, since a measurement of elevated TSH generally indicates hypothyroidism , while a measurement of low TSH generally indicates hyperthyroidism . [ 2 ] However, when TSH is measured by itself, it can yield misleading results, so additional thyroid function tests must be compared with the result of this test for accurate diagnosis. [ 4 ] [ 5 ] [ 6 ]
TSH is produced in the pituitary gland . The production of TSH is controlled by thyrotropin-releasing hormone (TRH), which is produced in the hypothalamus . TSH levels may be suppressed by excess free T3 (fT3) or free T4 (fT4) in the blood. [ citation needed ]
First-generation TSH assays were done by radioimmunoassay and were introduced in 1965. [ 3 ] There were variations and improvements upon TSH radioimmunoassay, but their use declined as a new immunometric assay technique became available in the middle of the 1980s. [ 3 ] [ 4 ] The new techniques were more accurate, leading to the second, third, and even fourth generations of TSH assay, with each generation possessing ten times greater functional sensitivity than the last. [ 7 ] Third generation immunometric assay methods are typically automated. [ 3 ] Fourth generation TSH immunometric assay has been developed for use in research. [ 4 ]
Third generation TSH assay is the requirement for modern standards of care. TSH testing in the United States is typically carried out with automated platforms using advanced forms of immunometric assay. [ 3 ] Nonetheless, there is no international standard for measurement of thyroid-stimulating hormone. [ 4 ]
Accurate interpretation takes a variety of factors into account, such as the thyroid hormones i.e. thyroxine (T 4 ) and triiodothyronine (T 3 ), current medical status (such as pregnancy [ 3 ] ), [ 4 ] certain medications like propylthiouracil , [ 4 ] temporal effects including circadian rhythm [ 8 ] and hysteresis , [ 9 ] and other past medical history . [ 10 ]
Total thyroxine is rarely measured, having been largely superseded by free thyroxine tests. Total thyroxine (Total T 4 ) is generally elevated in hyperthyroidism and decreased in hypothyroidism . [ 2 ] It is usually slightly elevated in pregnancy secondary to increased levels of thyroid binding globulin (TBG). [ 2 ]
Total T4 is measured to see the bound and unbound levels of T4. The total T4 is less useful in cases where there could be protein abnormalities. The total T4 is less accurate due to the large amount of T4 that is bound. The total T3 is measured in clinical practice since the T3 has decreased amount that is bound as compared to T4. [ citation needed ]
Reference ranges depend on the method of analysis. Results should always be interpreted using the range from the laboratory that performed the test. Example values are:
Free thyroxine (fT 4 or free T4) is generally elevated in hyperthyroidism and decreased in hypothyroidism . [ 2 ]
Reference ranges depend on the method of analysis. Results should always be interpreted using the range from the laboratory that performed the test. Example values are:
Total triiodothyronine (Total T 3 ) is rarely measured, having been largely superseded by free T3 tests. Total T3 is generally elevated in hyperthyroidism and decreased in hypothyroidism. [ 2 ]
Reference ranges depend on the method of analysis. Results should always be interpreted using the range from the laboratory that performed the test. Example values are:
Free triiodothyronine (fT 3 or free T3) is generally elevated in hyperthyroidism and decreased in hypothyroidism. [ 2 ]
Reference ranges depend on the method of analysis. Results should always be interpreted using the range from the laboratory that performed the test. Example values are:
An increased thyroxine-binding globulin results in an increased total thyroxine and total triiodothyronine without an actual increase in hormonal activity of thyroid hormones.
Reference ranges:
Reference ranges:
Thyroid hormone uptake (T uptake or T 3 uptake ) is a measure of the unbound thyroxine binding globulins in the blood, that is, the TBG that is unsaturated with thyroid hormone. [ 2 ] Unsaturated TBG increases with decreased levels of thyroid hormones. It is not directly related to triiodothyronine, despite the name T 3 uptake . [ 2 ]
Reference ranges:
The Free Thyroxine Index (FTI or T7) is obtained by multiplying the total T 4 with T 3 uptake. [ 2 ] FTI is considered to be a more reliable indicator of thyroid status in the presence of abnormalities in plasma protein binding. [ 2 ] This test is rarely used now that reliable free thyroxine and free triiodothyronine assays are routinely available.
FTI is elevated in hyperthyroidism and decreased in hypothyroidism. [ 2 ]
Derived structure parameters that describe constant properties of the overall feedback control system may add useful information for special purposes, e.g. in diagnosis of nonthyroidal illness syndrome or central hypothyroidism . [ 20 ] [ 21 ] [ 22 ] [ 23 ]
Thyroid's secretory capacity ( G T , also referred to as SPINA-GT) is the maximum stimulated amount of thyroxine the thyroid can produce in one second. [ 24 ] G T is elevated in hyperthyroidism and reduced in hypothyroidism. [ 25 ]
G T is calculated with
G ^ T = β T ( D T + [ T S H ] ) ( 1 + K 41 [ T B G ] + K 42 [ T B P A ] ) [ F T 4 ] α T [ T S H ] {\displaystyle {\hat {G}}_{T}={{\beta _{T}(D_{T}+[TSH])(1+K_{41}[TBG]+K_{42}[TBPA])[FT_{4}]} \over {\alpha _{T}[TSH]}}}
or
G ^ T = β T ( D T + [ T S H ] ) [ T T 4 ] α T [ T S H ] {\displaystyle {\hat {G}}_{T}={{\beta _{T}(D_{T}+[TSH])[TT_{4}]} \over {\alpha _{T}[TSH]}}}
α T {\displaystyle \alpha _{T}} : Dilution factor for T4 (reciprocal of apparent volume of distribution, 0.1 l −1 ) β T {\displaystyle \beta _{T}} : Clearance exponent for T4 (1.1e-6 sec −1 ) K 41 : Dissociation constant T4-TBG (2e10 L/mol) K 42 : Dissociation constant T4-TBPA (2e8 L/mol) D T : EC 50 for TSH (2.75 mU/L) [ 24 ]
The sum activity of peripheral deiodinases ( G D , also referred to as SPINA-GD) is reduced in nonthyroidal illness with hypodeiodination. [ 21 ] [ 22 ] [ 26 ]
G D is obtained with
G ^ D = β 31 ( K M 1 + [ F T 4 ] ) ( 1 + K 30 [ T B G ] ) [ F T 3 ] α 31 [ F T 4 ] {\displaystyle {\hat {G}}_{D}={{\beta _{31}(K_{M1}+[FT_{4}])(1+K_{30}[TBG])[FT_{3}]} \over {\alpha _{31}[FT_{4}]}}}
or
G ^ D = β 31 ( K M 1 + [ F T 4 ] ) [ T T 3 ] α 31 [ F T 4 ] {\displaystyle {\hat {G}}_{D}={{\beta _{31}(K_{M1}+[FT_{4}])[TT_{3}]} \over {\alpha _{31}[FT_{4}]}}}
α 31 {\displaystyle \alpha _{31}} : Dilution factor for T3 (reciprocal of apparent volume of distribution, 0.026 L −1 ) β 31 {\displaystyle \beta _{31}} : Clearance exponent for T3 (8e-6 sec −1 ) K M 1 : Dissociation constant of type-1-deiodinase (5e-7 mol/L) K 30 : Dissociation constant T3-TBG (2e9 L/mol) [ 24 ]
Jostel's TSH index (JTI or TSHI) helps to determine thyrotropic function of anterior pituitary on a quantitative level. [ 27 ] It is reduced in thyrotropic insufficiency [ 27 ] and in certain cases of non-thyroidal illness syndrome. [ 26 ]
It is calculated with
T S H I = L N ( T S H ) + 0.1345 ∗ F T 4 {\displaystyle TSHI=LN(TSH)+0.1345*FT4} .
Additionally, a standardized form of TSH index may be calculated with
s T S H I = ( T S H I − 2.7 ) / 0.676 {\displaystyle sTSHI=(TSHI-2.7)/0.676} . [ 27 ]
The Thyrotroph Thyroid Hormone Sensitivity Index (TTSI, also referred to as Thyrotroph T4 Resistance Index or TT4RI) was developed to enable fast screening for resistance to thyroid hormone . [ 28 ] [ 29 ] Somewhat similar to the TSH Index it is calculated from equilibrium values for TSH and FT4, however with a different equation.
The Thyroid Feedback Quantile-based Index (TFQI) is another parameter for thyrotropic pituitary function. It was defined to be more robust to distorted data than JTI and TTSI. It is calculated with
T F Q I = F F T 4 ( F T 4 ) − ( 1 − F T S H ( T S H ) ) {\displaystyle TFQI=F_{FT4}(FT4)-(1-F_{TSH}(TSH))}
from quantiles of FT4 and TSH concentration (as determined based on cumulative distribution functions ). [ 30 ] Per definition the TFQI has a mean of 0 and a standard deviation of 0.37 in a reference population. [ 30 ] Higher values of TFQI are associated with obesity , metabolic syndrome , impaired renal function, diabetes , and diabetes-related mortality . [ 30 ] [ 31 ] [ 32 ] [ 33 ] [ 34 ] [ 35 ] [ 36 ] TFQI results are also elevated in takotsubo syndrome , [ 37 ] potentially reflecting type 2 allostatic load in the situation of psychosocial stress . Reductions have been observed in subjects with schizophrenia after initiation of therapy with oxcarbazepine , potentially reflecting declining allostatic load. [ 38 ]
In healthy persons, the intra-individual variation of TSH and thyroid hormones is considerably smaller than the inter-individual variation. [ 39 ] [ 40 ] [ 41 ] This results from a personal set point of thyroid homeostasis. [ 42 ] In hypothyroidism, it is impossible to directly access the set point, [ 43 ] but it can be reconstructed with methods of systems theory. [ 44 ] [ 45 ] [ 46 ]
A computerised algorithm, called Thyroid-SPOT, which is based on this mathematical theory, has been implemented in software applications. [ 47 ] In patients undergoing thyroidectomy it could be demonstrated that this algorithm can be used to reconstruct the personal set point with sufficient precision. [ 48 ]
Drugs can profoundly affect thyroid function tests. Listed below is a selection of important effects.
↓: reduced serum concentration or structure parameter; ↑: increased serum concentration or structure parameter; ↔: no change; TSH: Thyroid-stimulating hormone; T 3 : Total triiodothyronine; T 4 : Total thyroxine; fT 4 : Free thyroxine; fT 3 : Free triiodothyronine; rT 3 : Reverse triiodothyronine
The Centers for Disease Control and Prevention has published the following laboratory procedure manuals for measuring thyroid-stimulating hormone: | https://en.wikipedia.org/wiki/Thyroid_function_tests |
Thyroid hormone binding ratio (THBR) is a thyroid function test that measures the "uptake" of T3 or T4 tracer by thyroid-binding globulin (TBG) in a given serum sample. This provides an indirect and reciprocal estimate of the available binding sites on TBG within the sample.
The results are then reported as a ratio to normal serum.
Attempts to correct for changes in thyroid binding globulin due to liver disease, protein losing states, pregnancy or various drugs [ citation needed ]
It is used to calculate free thyroxine index (total T4 x T3 uptake), an estimate of free T4. Free thyroxine index may be calculated with increased diagnostic accuracy using direct TBG measurement when the total hormone concentration is abnormally elevated [ 1 ]
Invalid if other proteins or immunoglobulins compete with TBG, including familial dysalbuminemic hyperthyroxinemia [ 2 ] | https://en.wikipedia.org/wiki/Thyroid_hormone_binding_ratio |
Thyrotoxicosis factitia ( alimentary thyrotoxicosis , exogenous thyrotoxicosis ) [ 1 ] [ 2 ] is a condition of thyrotoxicosis caused by the ingestion [ 3 ] of exogenous thyroid hormone . [ 4 ] [ 5 ] It can be the result of mistaken ingestion of excess drugs, such as levothyroxine [ 6 ] and triiodothyronine , [ 7 ] or as a symptom of Munchausen syndrome . It is an uncommon form of hyperthyroidism .
Patients present with hyperthyroidism and may be mistaken for Graves’ disease , if TSH receptor positive, or thyroiditis because of absent uptake on a thyroid radionuclide uptake scan due to suppression of thyroid function by exogenous thyroid hormones. [ 8 ] Ingestion of thyroid hormone also suppresses thyroglobulin levels helping to differentiate thyrotoxicosis factitia from other causes of hyperthyroidism, in which serum thyroglobulin is elevated. Caution, however, should be exercised in interpreting thyroglobulin results without thyroglobulin antibodies, since thyroglobulin antibodies commonly interfere in thyroglobulin immunoassays causing false positive and negative results which may lead to clinical misdirection. In such cases, increased fecal thyroxine levels in thyrotoxicosis factitia may help differentiate it from other causes of hyperthyroidism. [ citation needed ]
This article about an endocrine, nutritional, or metabolic disease is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Thyrotoxicosis_factitia |
The Thyrotroph Thyroid Hormone Sensitivity Index (abbreviated TTSI , also referred to as Thyrotroph T4 Resistance Index or TT4RI ) is a calculated structure parameter of thyroid homeostasis . It was originally developed to deliver a method for fast screening for resistance to thyroid hormone . [ 1 ] [ 2 ] Today it is also used to get an estimate for the set point of thyroid homeostasis, [ 3 ] especially to assess dynamic thyrotropic adaptation of the anterior pituitary gland, including non-thyroidal illnesses . [ 4 ]
The TTSI can be calculated with
from equilibrium serum or plasma concentrations of thyrotropin (TSH), free T4 (FT4) and the assay -specific upper limit of the reference interval for FT4 concentration ( l u ). [ 4 ]
Some publications use a simpler form of this equation that doesn't correct for the reference range of free T4. It is calculated with
The disadvantage of this uncorrected version is that its numeric results are highly dependent on the used assays and their units of measurement . [ citation needed ]
In case of resistance to thyroid hormone, the magnitude of TTSI depends on which nucleotide in the THRB gene is mutated, but also on the genotype of coactivators. A systematic investigation in mice demonstrated a strong association of TT4RI to the genotypes of THRB and the steroid receptor coactivator ( SRC-1 ) gene. [ 5 ]
The TTSI is used as a screening parameter for resistance to thyroid hormone due to mutations in the THRB gene, where it is elevated. [ 4 ] It is also beneficial for assessing the severity of already confirmed thyroid hormone resistance, [ 6 ] even on replacement therapy with L-T4, [ 7 ] and for monitoring the pituitary response to substitution therapy with thyromimetics (e.g. TRIAC ) in RTH Beta. [ 8 ]
In autoimmune thyroiditis the TTSI is moderately elevated. [ 9 ]
A large cohort study demonstrated TTSI to be strongly influenced by genetic factors. [ 10 ] A variant of the TTSI that is not corrected for the upper limit of the FT4 reference range was shown to be significantly increased in offspring from long-lived siblings compared to their partners. [ 11 ]
Conversely, an elevated set point of thyroid homeostasis, as quantified by the TT4RI, is associated to higher prevalence of metabolic syndrome [ 3 ] and several harmonized criteria by the International Diabetes Federation , including triglyceride and HDL concentration and blood pressure . [ 12 ] [ 13 ]
In certain phenotypes of non-thyroidal illness syndrome , especially in cases with concomitant sepsis , the TTSI is reduced. [ 14 ] This reflects a reduced set point of thyroid homeostasis, as also experimentally predicted in rodent models of inflammation and sepsis. [ 15 ] [ 16 ] [ 17 ]
Negative correlation of the TTSI with the urinary excretion of certain phthalates suggests that endocrine disruptors may affect the central set point of thyroid homeostasis. [ 18 ] | https://en.wikipedia.org/wiki/Thyrotroph_Thyroid_Hormone_Sensitivity_Index |
Théâtrophone ( French pronunciation: [teatʁɔfɔn] , "the theatre phone") was a telephonic distribution system available in portions of Europe that allowed the subscribers to listen to opera and theatre performances over the telephone lines. The théâtrophone evolved from a Clément Ader invention, which was first demonstrated in 1881, in Paris . Subsequently, in 1890, the invention was commercialized by Compagnie du Théâtrophone, which continued to operate until 1932.
The origin of the théâtrophone can be traced to a telephonic transmission system demonstrated by Clément Ader at the 1881 International Exposition of Electricity in Paris. The system was inaugurated by the French President Jules Grévy , and allowed broadcasting of concerts or plays. Ader had arranged 80 telephone transmitters across the front of a stage to create a form of binaural stereophonic sound . [ 1 ] It was the first two-channel audio system, and consisted of a series of telephone transmitters connected from the stage of the Paris Opera to a suite of rooms at the Paris Electrical Exhibition, where the visitors could hear Comédie-Française and opera performances in stereo using two headphones; the Opera was located more than two kilometers away from the venue. [ 2 ] In a note dated 11 November 1881, Victor Hugo describes his first experience of théâtrophone as pleasant. [ 3 ] [ 4 ]
In 1884, the King Luís I of Portugal decided to use the system, when he could not attend an opera in person. The director of the Edison Gower Bell Company, who was responsible for this théâtrophone installation, was later awarded the Military Order of Christ . [ 5 ]
The théâtrophone technology was made available in Belgium in 1884, and in Lisbon in 1885. In Sweden, the first telephone transmission of an opera performance took place in Stockholm in May 1887. The British writer Ouida describes a female character in the novel Massarenes (1897) as "A modern woman of the world. As costly as an ironclad and as complicated as theatrophone." [ 5 ]
In 1890, the system became operational as a service under the name "théâtrophone" in Paris. The service was offered by Compagnie du Théâtrophone (The Théâtrophone Company), which was founded by MM. Marinovitch and Szarvady. [ 5 ] The théâtrophone offered theatre and opera performances to the subscribers. The service can be called a prototype of the telephone newspaper , as it included five-minute news programs at regular intervals. [ 7 ] The Théâtrophone Company set up coin-operated telephone receivers in hotels, cafés, clubs, and other locations, costing 50 centimes for five minutes of listening. [ 8 ] The subscription tickets were also issued at a reduced rate, in order to attract regular patrons. The service was also available to home subscribers.
French writer Marcel Proust was a keen follower of théâtrophone, as evident by his correspondence. He subscribed to the service in 1911. [ 9 ] [ 10 ]
Many technological improvements were gradually made to the original théâtrophone system. The Brown telephone relay, invented in 1913, yielded interesting results for amplification of the current. [ 5 ]
The théâtrophone finally succumbed to the rising popularity of radio broadcasting and the phonograph , and the Compagnie du Théâtrophone ceased its operations in 1932. [ 5 ]
Similar systems elsewhere in Europe included Telefon Hírmondó (est. 1893) of Budapest and Electrophone of London (est. 1895). In the United States , the systems similar to théâtrophone were limited to one-off experiments. Erik Barnouw reported a concert by telephone that was organized in the summer of 1890; around 800 people at the Grand Union Hotel in Saratoga listened to a telephonic transmission of The Charge of the Light Brigade conducted at Madison Square Garden . [ 5 ]
The Andrew Crumey novel Mr Mee (2000) has a chapter depicting the installation of a théâtrophone in the home of Marcel Proust .
The Eça de Queiroz novel A Cidade e as Serras (1901) mentions the device as one of the many technological commodities available for the distraction of the upper classes.
In his utopian science fiction novel Looking Backward (1888), Edward Bellamy predicted sermons and music being available in the home through a system like théâtrophone. | https://en.wikipedia.org/wiki/Théâtrophone |
Titanium(III) oxide is the inorganic compound with the formula Ti 2 O 3 . A black semiconducting solid, it is prepared by reducing titanium dioxide with titanium metal at 1600 °C. [ 3 ]
Ti 2 O 3 adopts the Al 2 O 3 (corundum) structure. [ 3 ] It is reactive with oxidising agents. [ 3 ] At around 200 °C, there is a transition from semiconducting to metallic conducting. [ 3 ] Titanium(III) oxide occurs naturally as the extremely rare mineral in the form of tistarite . [ 4 ]
Other titanium(III) oxides include LiTi 2 O 4 and LiTiO 2 . [ 5 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ti2O3 |
Titanium aluminide (chemical formula TiAl ), commonly gamma titanium , is an intermetallic chemical compound . It is lightweight and resistant to oxidation [ 1 ] and heat, but has low ductility . The density of γ-TiAl is about 4.0 g/cm 3 . It finds use in several applications including aircraft, jet engines, sporting equipment and automobiles. [ citation needed ] The development of TiAl based alloys began circa 1970. The alloys have been used in these applications only since about 2000.
Titanium aluminide has three major intermetallic compounds: gamma titanium aluminide ( gamma TiAl , γ-TiAl ), alpha 2-Ti 3 Al and TiAl 3 . Among the three, gamma TiAl has received the most interest and applications.
Gamma TiAl has excellent mechanical properties and oxidation and corrosion resistance at elevated temperatures (over 600 °C), which makes it a possible replacement for traditional Ni based superalloy components in aircraft turbine engines.
TiAl-based alloys have potential to increase the thrust-to-weight ratio in aircraft engines. This is especially the case with the engine's low-pressure turbine blades and the high-pressure compressor blades. These are traditionally made of Ni-based superalloy, which is nearly twice as dense as TiAl-based alloys. Some gamma titanium aluminide alloys retain strength and oxidation resistance to 1000 °C, which is 400 °C higher than the operating temperature limit of conventional titanium alloys. [ not specific enough to verify ] [ 3 ]
General Electric uses gamma TiAl for the low-pressure turbine blades on its GEnx engine, which powers the Boeing 787 and Boeing 747-8 aircraft. This was the first large-scale use of this material on a commercial jet engine [ 4 ] when it entered service in 2011. [ 5 ] The TiAl LPT blades are cast by Precision Castparts Corp. and Avio s.p.a. Machining of the Stage 6, and Stage 7 LPT blades is performed by Moeller Manufacturing. [ 6 ] [ citation needed ] An alternate pathway for production of the gamma TiAl blades for the GEnx and GE9x engines using additive manufacturing is being explored. [ 7 ]
In 2019 a new 55 g lightweight version of the Omega Seamaster wristwatch was made, using gamma titanium aluminide for the case, backcase and crown, and a titanium dial and mechanism in Ti 6/4 (grade 5). The retail price of this watch at £37,240 was nine times that of the basic Seamaster and comparable to the top of the range platinum-cased version with a moonphase complication . [ 8 ]
Alpha 2-Ti₃Al is an intermetallic compound of titanium and aluminum, belonging to the Ti-Al system of advanced high-temperature materials. It is primarily used in aerospace and other high-performance applications due to its balance of strength, lightweight properties, and oxidation resistance.
It has an ordered hexagonal (D0₁₉) crystal structure, which makes it distinct from the more commonly known γ-TiAl (gamma titanium aluminide).
Higher strength than conventional titanium alloys, especially at high temperatures. More brittle than pure titanium but tougher than γ-TiAl, making it useful in applications requiring a trade-off between toughness and lightweight properties.
Improved high-temperature oxidation resistance compared to pure titanium, but generally not as good as γ-TiAl or other high-temperature alloys like nickel-based superalloys. Often used with coatings to further enhance oxidation resistance.
Density and Lightweight Properties:
Lower density than traditional nickel-based superalloys, making it attractive for aerospace applications where weight reduction is crucial.
Operates effectively at 600–800°C, making it useful in jet engines, turbine components, and hypersonic vehicles.
Applications of Alpha 2-Ti₃Al:
Aerospace: Used in jet engine components, compressor blades, and airframe structures where high strength and lightweight properties are needed.
Automotive (High-Performance Vehicles): Some high-end applications in racing engines. Military and Defense: Structural components in hypersonic aircraft and advanced missiles.
Energy Sector : Potential use in turbine components for power generation.
Challenges and Limitations:
Brittleness : More brittle than conventional titanium alloys, requiring careful processing and potential use of composite materials.
Manufacturing Complexity: Difficult to process and fabricate due to its intermetallic nature, often requiring advanced techniques like powder metallurgy, additive manufacturing, or specialized forging methods.
Oxidation Resistance: While better than standard titanium, it still requires protective coatings for long-term use in extreme environments.
TiAl 3 has the lowest density of 3.4 g/cm 3 , the highest micro hardness of 465–670 kg/mm 2 and the best oxidation resistance even at 1 000 °C. However, the applications of TiAl 3 in the engineering and aerospace fields are limited by its poor ductility. In addition, the loss of ductility at ambient temperature is usually accompanied by a change of fracture mode from ductile transgranular to brittle intergranular or to brittle cleavage. Despite the fact that a lot of toughening strategies have been developed to improve their toughness, machining quality is still a difficult problem to tackle. Near-net shape manufacturing technology is considered as one of the best choices for preparing such materials. {date=July 2022} [ citation needed ] | https://en.wikipedia.org/wiki/Ti3Al |
Titanium silicon carbide , chemical formula Ti 3 SiC 2 , is a material with both metallic and ceramic properties. [ 1 ] [ 2 ] It is one of the MAX phases .
This material -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ti3SiC2 |
Titanium aluminide (chemical formula TiAl ), commonly gamma titanium , is an intermetallic chemical compound . It is lightweight and resistant to oxidation [ 1 ] and heat, but has low ductility . The density of γ-TiAl is about 4.0 g/cm 3 . It finds use in several applications including aircraft, jet engines, sporting equipment and automobiles. [ citation needed ] The development of TiAl based alloys began circa 1970. The alloys have been used in these applications only since about 2000.
Titanium aluminide has three major intermetallic compounds: gamma titanium aluminide ( gamma TiAl , γ-TiAl ), alpha 2-Ti 3 Al and TiAl 3 . Among the three, gamma TiAl has received the most interest and applications.
Gamma TiAl has excellent mechanical properties and oxidation and corrosion resistance at elevated temperatures (over 600 °C), which makes it a possible replacement for traditional Ni based superalloy components in aircraft turbine engines.
TiAl-based alloys have potential to increase the thrust-to-weight ratio in aircraft engines. This is especially the case with the engine's low-pressure turbine blades and the high-pressure compressor blades. These are traditionally made of Ni-based superalloy, which is nearly twice as dense as TiAl-based alloys. Some gamma titanium aluminide alloys retain strength and oxidation resistance to 1000 °C, which is 400 °C higher than the operating temperature limit of conventional titanium alloys. [ not specific enough to verify ] [ 3 ]
General Electric uses gamma TiAl for the low-pressure turbine blades on its GEnx engine, which powers the Boeing 787 and Boeing 747-8 aircraft. This was the first large-scale use of this material on a commercial jet engine [ 4 ] when it entered service in 2011. [ 5 ] The TiAl LPT blades are cast by Precision Castparts Corp. and Avio s.p.a. Machining of the Stage 6, and Stage 7 LPT blades is performed by Moeller Manufacturing. [ 6 ] [ citation needed ] An alternate pathway for production of the gamma TiAl blades for the GEnx and GE9x engines using additive manufacturing is being explored. [ 7 ]
In 2019 a new 55 g lightweight version of the Omega Seamaster wristwatch was made, using gamma titanium aluminide for the case, backcase and crown, and a titanium dial and mechanism in Ti 6/4 (grade 5). The retail price of this watch at £37,240 was nine times that of the basic Seamaster and comparable to the top of the range platinum-cased version with a moonphase complication . [ 8 ]
Alpha 2-Ti₃Al is an intermetallic compound of titanium and aluminum, belonging to the Ti-Al system of advanced high-temperature materials. It is primarily used in aerospace and other high-performance applications due to its balance of strength, lightweight properties, and oxidation resistance.
It has an ordered hexagonal (D0₁₉) crystal structure, which makes it distinct from the more commonly known γ-TiAl (gamma titanium aluminide).
Higher strength than conventional titanium alloys, especially at high temperatures. More brittle than pure titanium but tougher than γ-TiAl, making it useful in applications requiring a trade-off between toughness and lightweight properties.
Improved high-temperature oxidation resistance compared to pure titanium, but generally not as good as γ-TiAl or other high-temperature alloys like nickel-based superalloys. Often used with coatings to further enhance oxidation resistance.
Density and Lightweight Properties:
Lower density than traditional nickel-based superalloys, making it attractive for aerospace applications where weight reduction is crucial.
Operates effectively at 600–800°C, making it useful in jet engines, turbine components, and hypersonic vehicles.
Applications of Alpha 2-Ti₃Al:
Aerospace: Used in jet engine components, compressor blades, and airframe structures where high strength and lightweight properties are needed.
Automotive (High-Performance Vehicles): Some high-end applications in racing engines. Military and Defense: Structural components in hypersonic aircraft and advanced missiles.
Energy Sector : Potential use in turbine components for power generation.
Challenges and Limitations:
Brittleness : More brittle than conventional titanium alloys, requiring careful processing and potential use of composite materials.
Manufacturing Complexity: Difficult to process and fabricate due to its intermetallic nature, often requiring advanced techniques like powder metallurgy, additive manufacturing, or specialized forging methods.
Oxidation Resistance: While better than standard titanium, it still requires protective coatings for long-term use in extreme environments.
TiAl 3 has the lowest density of 3.4 g/cm 3 , the highest micro hardness of 465–670 kg/mm 2 and the best oxidation resistance even at 1 000 °C. However, the applications of TiAl 3 in the engineering and aerospace fields are limited by its poor ductility. In addition, the loss of ductility at ambient temperature is usually accompanied by a change of fracture mode from ductile transgranular to brittle intergranular or to brittle cleavage. Despite the fact that a lot of toughening strategies have been developed to improve their toughness, machining quality is still a difficult problem to tackle. Near-net shape manufacturing technology is considered as one of the best choices for preparing such materials. {date=July 2022} [ citation needed ] | https://en.wikipedia.org/wiki/TiAl3 |
Titanium tetrabromide is the chemical compound with the formula TiBr 4 . It is the most volatile transition metal bromide. The properties of TiBr 4 are an average of TiCl 4 and TiI 4 . Some key properties of these four-coordinated Ti(IV) species are their high Lewis acidity and their high solubility in nonpolar organic solvents. TiBr 4 is diamagnetic, reflecting the d 0 configuration of the metal centre. [ 2 ]
This four-coordinated complex adopts a tetrahedral geometry. It can be prepared via several methods: (i) from the elements, (ii) via the reaction of TiO 2 with carbon and bromine (see Kroll process ), and (iii) by treatment of TiCl 4 with HBr .
Titanium tetrabromide forms adducts such as TiBr 4 ( THF ) 2 and [TiBr 5 ] − . [ 3 ] With bulky donor ligands, such as 2-methylpyridine (2-Mepy), five-coordinated adducts form. TiBr 4 (2-MePy) is trigonal bipyramidal with the pyridine in the equatorial plane. [ 4 ]
TiBr 4 has been used as a Lewis-acid catalyst in organic synthesis . [ 5 ]
The tetrabromide and tetrachlorides of titanium react to give a statistical mixture of the mixed tetrahalides, TiBr 4−x Cl x (x = 0-4). The mechanism of this redistribution reaction is uncertain. One proposed pathway invokes the intermediacy of dimers . [ 6 ]
TiBr 4 hydrolyzes rapidly, potentially dangerously, to release hydrogen bromide , otherwise known as hydrobromic acid. | https://en.wikipedia.org/wiki/TiBr4 |
Titanium(II) chloride is the chemical compound with the formula TiCl 2 . The black solid has been studied only moderately, probably because of its high reactivity. [ 2 ] Ti(II) is a strong reducing agent: it has a high affinity for oxygen and reacts irreversibly with water to produce H 2 . The usual preparation is the thermal disproportionation of TiCl 3 at 500 °C. The reaction is driven by the loss of volatile TiCl 4 :
The method is similar to that for the conversion of VCl 3 into VCl 2 and VCl 4 .
TiCl 2 crystallizes as the layered CdI 2 structure. Thus, the Ti(II) centers are octahedrally coordinated to six chloride ligands. [ 3 ] [ 4 ]
Molecular complexes are known such as TiCl 2 (chel) 2 , where chel is DMPE (CH 3 ) 2 PCH 2 CH 2 P(CH 3 ) 2 and TMEDA ((CH 3 ) 2 NCH 2 CH 2 N(CH 3 ) 2 ). [ 5 ] Such species are prepared by reduction of related Ti(III) and Ti(IV) complexes.
Unusual electronic effects have been observed in these species: TiCl 2 [(CH 3 ) 2 PCH 2 CH 2 P(CH 3 ) 2 ] 2 is paramagnetic with a triplet ground state , but Ti(CH 3 ) 2 [(CH 3 ) 2 PCH 2 CH 2 P(CH 3 ) 2 ] 2 is diamagnetic. [ 6 ]
A solid-state derivative of TiCl 2 is Na 2 TiCl 4 , which has been prepared by the reaction of Ti metal with TiCl 3 in a NaCl flux . [ 7 ] This species adopts a linear chain structure wherein again the Ti(II) centers are octahedral with terminal, axial halides. [ 8 ] | https://en.wikipedia.org/wiki/TiCl2 |
Titanium(III) chloride is the inorganic compound with the formula TiCl 3 . At least four distinct species have this formula; additionally hydrated derivatives are known. TiCl 3 is one of the most common halides of titanium and is an important catalyst for the manufacture of polyolefins .
In TiCl 3 , each titanium atom has one d electron, rendering its derivatives paramagnetic , that is, the substance is attracted into a magnetic field. Solutions of titanium(III) chloride are violet, which arises from excitations of its d -electron . The colour is not very intense since the transition is forbidden by the Laporte selection rule .
Four solid forms or polymorphs of TiCl 3 are known. All feature titanium in an octahedral coordination sphere. These forms can be distinguished by crystallography as well as by their magnetic properties, which probes exchange interactions . β-TiCl 3 crystallizes as brown needles. Its structure consists of chains of TiCl 6 octahedra that share opposite faces such that the closest Ti–Ti contact is 2.91 Å. This short distance indicates strong metal–metal interactions (see figure in upper right). The three violet "layered" forms, named for their color and their tendency to flake, are called alpha (α), gamma (γ), and delta (δ). In α-TiCl 3 , the chloride anions are hexagonal close-packed . In γ-TiCl 3 , the chlorides anions are cubic close-packed . Finally, disorder in shift successions, causes an intermediate between alpha and gamma structures, called the δ form. The TiCl 6 share edges in each form, with 3.60 Å being the shortest distance between the titanium cations. This large distance between titanium cations precludes direct metal-metal bonding. In contrast, the trihalides of the heavier metals hafnium and zirconium engage in metal-metal bonding. Direct Zr–Zr bonding is indicated in zirconium(III) chloride . The difference between the Zr(III) and Ti(III) materials is attributed in part to the relative radii of these metal centers. [ 2 ]
Two hydrates of titanium(III) chloride are known, i.e. complexes containing aquo ligands . These include the pair of hydration isomers [Ti(H 2 O) 6 ]Cl 3 and [Ti(H 2 O) 4 Cl 2 ]Cl(H 2 O) 2 . The former is violet and the latter, with two molecules of water of crystallization, is green. [ 3 ]
TiCl 3 is produced usually by reduction of titanium(IV) chloride . Older reduction methods used hydrogen : [ 4 ]
More modern techniques prefer aluminum ; the product is sold as a mixture with aluminium trichloride , TiCl 3 ·AlCl 3 . [ 5 ]
TiCl 3 can also be produced by the reaction of titanium metal and hot, concentrated hydrochloric acid ; the reaction does not proceed at room temperature, as titanium is passivated against most mineral acids by a thin surface layer of titanium dioxide .
Treating TiCl 3 with tetrahydrofuran (THF) gives the light-blue colored, [ 6 ] meridional complex, TiCl 3 ( THF ) 3 : [ 7 ]
TiCl 3 ·AlCl 3 gives the same product. [ 5 ]
An analogous dark green complex arises from complexation with dimethylamine . In a reaction where all ligands are exchanged, TiCl 3 is a precursor to the blue-colored complex Ti(acac) 3 . [ 8 ]
The more reduced titanium(II) chloride is prepared by the thermal disproportionation of TiCl 3 at 500 °C. The reaction is driven by the loss of volatile TiCl 4 : [ 9 ]
The trichloride is a Lewis acid , forming ternary hexahalide complexes with stoichiometry M 3 TiCl 6 . These have structures that depend on the cation (M + ) added. [ 10 ] Caesium chloride treated with titanium(II) chloride and hexachlorobenzene produces crystalline CsTi 2 Cl 7 . In these structures Ti 3+ exhibits octahedral coordination geometry. [ 11 ]
TiCl 3 is the main Ziegler–Natta catalyst , responsible for most industrial production of polyethylene . The catalytic activities depend strongly on the polymorph of the TiCl 3 (α vs. β vs. γ vs. δ) and the method of preparation. [ 12 ]
TiCl 3 is also a specialized reagent in organic synthesis, useful for reductive coupling reactions, often in the presence of added reducing agents such as zinc. It reduces oximes to imines . [ 13 ] Titanium trichloride can reduce nitrate to ammonium ion thereby allowing for the sequential analysis of nitrate and ammonia. [ 14 ] Slow deterioration occurs in air-exposed titanium trichloride, often resulting in erratic results, such as in reductive coupling reactions . [ 15 ]
TiCl 3 and most of its complexes are typically handled under air-free conditions to prevent reactions with oxygen and moisture. Samples of TiCl 3 can be relatively air stable or pyrophoric . [ 16 ] [ 17 ] | https://en.wikipedia.org/wiki/TiCl3 |
Titanium tetrachloride is the inorganic compound with the formula TiCl 4 . It is an important intermediate in the production of titanium metal and the pigment titanium dioxide . TiCl 4 is a volatile liquid. Upon contact with humid air, it forms thick clouds of titanium dioxide ( TiO 2 ) and hydrochloric acid , a reaction that was formerly exploited for use in smoke machines. It is sometimes referred to as "tickle" or "tickle 4", as a phonetic representation of the symbols of its molecular formula ( TiCl 4 ). [ 7 ] [ 8 ]
TiCl 4 is a dense, colourless liquid, although crude samples may be yellow or even red-brown. It is one of the rare transition metal halides that is a liquid at room temperature, VCl 4 being another example. This property reflects the fact that molecules of TiCl 4 weakly self-associate. Most metal chlorides are polymers , wherein the chloride atoms bridge between the metals. Its melting point is similar to that of CCl 4 . [ 9 ] [ 10 ]
Ti 4+ has a "closed" electronic shell, with the same number of electrons as the noble gas argon . The tetrahedral structure for TiCl 4 is consistent with its description as a d 0 metal center ( Ti 4+ ) surrounded by four identical ligands. This configuration leads to highly symmetrical structures, hence the tetrahedral shape of the molecule. TiCl 4 adopts similar structures to TiBr 4 and TiI 4 ; the three compounds share many similarities. TiCl 4 and TiBr 4 react to give mixed halides TiCl 4− x Br x , where x = 0, 1, 2, 3, 4. Magnetic resonance measurements also indicate that halide exchange is also rapid between TiCl 4 and VCl 4 . [ 11 ]
TiCl 4 is soluble in toluene and chlorocarbons . Certain arenes form complexes of the type [(C 6 R 6 )TiCl 3 ] + . [ 12 ] TiCl 4 reacts exothermically with donor solvents such as THF to give hexacoordinated adducts . [ 13 ] Bulkier ligands (L) give pentacoordinated adducts TiCl 4 L .
TiCl 4 is produced by the chloride process , which involves the reduction of titanium oxide ores, typically ilmenite ( FeTiO 3 ), with carbon under flowing chlorine at 900 °C. Impurities are removed by distillation . [ 10 ]
The coproduction of FeCl 3 is undesirable, which has motivated the development of alternative technologies. Instead of directly using ilmenite, "rutile slag" is used. This material, an impure form of TiO 2 , is derived from ilmenite by removal of iron, either using carbon reduction or extraction with sulfuric acid . Crude TiCl 4 contains a variety of other volatile halides, including vanadyl chloride ( VOCl 3 ), silicon tetrachloride ( SiCl 4 ), and tin tetrachloride ( SnCl 4 ), which must be separated. [ 10 ]
The world's supply of titanium metal, about 250,000 tons per year, is made from TiCl 4 . The conversion involves the reduction of the tetrachloride with magnesium metal. This procedure is known as the Kroll process : [ 14 ]
In the Hunter process , liquid sodium is the reducing agent instead of magnesium. [ 15 ]
Around 90% of the TiCl 4 production is used to make the pigment titanium dioxide ( TiO 2 ). The conversion involves hydrolysis of TiCl 4 , a process that forms hydrogen chloride : [ 14 ]
In some cases, TiCl 4 is oxidised directly with oxygen :
It has been used to produce smoke screens since it produces a heavy, white smoke that has little tendency to rise. "Tickle" was the standard means of producing on-set smoke effects for motion pictures, before being phased out in the 1980s due to concerns about hydrated HCl 's effects on the respiratory system. [ citation needed ]
Titanium tetrachloride is a versatile reagent that forms diverse derivatives including those illustrated below. [ 16 ]
A characteristic reaction of TiCl 4 is its easy hydrolysis , signaled by the release of HCl vapors and titanium oxides and oxychlorides . Titanium tetrachloride has been used to create naval smokescreens , as the hydrochloric acid aerosol and titanium dioxide that is formed scatter light very efficiently. This smoke is corrosive, however. [ 10 ]
Alcohols react with TiCl 4 to give alkoxides with the formula [Ti(OR) 4 ] n (R = alkyl , n = 1, 2, 4). As indicated by their formula, these alkoxides can adopt complex structures ranging from monomers to tetramers. Such compounds are useful in materials science as well as organic synthesis . A well known derivative is titanium isopropoxide , which is a monomer. Titanium bis(acetylacetonate)dichloride results from treatment of titanium tetrachloride with excess acetylacetone : [ 17 ]
Organic amines react with TiCl 4 to give complexes containing amido ( R 2 N − -containing) and imido ( RN 2− -containing) complexes. With ammonia, titanium nitride is formed. An illustrative reaction is the synthesis of tetrakis(dimethylamido)titanium Ti(N(CH 3 ) 2 ) 4 , a yellow, benzene-soluble liquid: [ 18 ] This molecule is tetrahedral, with planar nitrogen centers. [ 19 ]
TiCl 4 is a Lewis acid as implicated by its tendency to hydrolyze . With the ether THF , TiCl 4 reacts to give yellow crystals of TiCl 4 (THF) 2 . With chloride salts, TiCl 4 reacts to form sequentially [Ti 2 Cl 9 ] − , [Ti 2 Cl 10 ] 2− (see figure above), and [TiCl 6 ] 2− . [ 20 ] The reaction of chloride ions with TiCl 4 depends on the counterion. [N(CH 2 CH 2 CH 2 CH 3 ) 4 ]Cl and TiCl 4 gives the pentacoordinate complex [N(CH 2 CH 2 CH 2 CH 3 ) 4 ][TiCl 5 ] , whereas smaller [N(CH 2 CH 3 ) 4 ] + gives [N(CH 2 CH 3 ) 4 ] 2 [Ti 2 Cl 10 ] . These reactions highlight the influence of electrostatics on the structures of compounds with highly ionic bonding.
Reduction of TiCl 4 with aluminium results in one-electron reduction. The trichloride ( TiCl 3 ) and tetrachloride have contrasting properties: the trichloride is a colored solid, being a coordination polymer , and is paramagnetic . When the reduction is conducted in THF solution, the Ti(III) product converts to the light-blue adduct TiCl 3 (THF) 3 .
The organometallic chemistry of titanium typically starts from TiCl 4 . An important reaction involves sodium cyclopentadienyl to give titanocene dichloride , TiCl 2 (C 5 H 5 ) 2 . This compound and many of its derivatives are precursors to Ziegler–Natta catalysts . Tebbe's reagent , useful in organic chemistry, is an aluminium-containing derivative of titanocene that arises from the reaction of titanocene dichloride with trimethylaluminium . It is used for the "olefination" reactions. [ 16 ]
Arenes , such as C 6 (CH 3 ) 6 react to give the piano-stool complexes [Ti(C 6 R 6 )Cl 3 ] + (R = H, CH 3 ; see figure above). This reaction illustrates the high Lewis acidity of the TiCl + 3 entity, which is generated by abstraction of chloride from TiCl 4 by AlCl 3 . [ 12 ]
TiCl 4 finds occasional use in organic synthesis , capitalizing on its Lewis acidity , its oxophilicity , and the electron-transfer properties of its reduced titanium halides. It is used in the Lewis acid catalysed aldol addition [ 21 ] Key to this application is the tendency of TiCl 4 to activate aldehydes (RCHO) by formation of adducts such as (RCHO)TiCl 4 OC(H)R . [ 22 ]
Hazards posed by titanium tetrachloride generally arise from its reaction with water that releases hydrochloric acid , which is severely corrosive itself and whose vapors are also extremely irritating. TiCl 4 is a strong Lewis acid , which exothermically forms adducts with even weak bases such as THF and water. | https://en.wikipedia.org/wiki/TiCl4 |
Titanium(III) fluoride is the inorganic compound with the formula Ti F 3 . A violet, paramagnetic solid, it is one of two titanium fluorides, the other being titanium tetrafluoride . [ 1 ] It adopts a defect perovskite -like structure such that each Ti center has octahedral coordination geometry , and each fluoride ligand is doubly bridging . [ 2 ]
Titanium(III) fluoride can be prepared by dissolution of titanium metal in hydrogen fluoride . In air, it slowly oxidizes to titanium(IV). [ 1 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/TiF3 |
Titanium(IV) fluoride is the inorganic compound with the formula Ti F 4 . It is a white hygroscopic solid. In contrast to the other tetrahalides of titanium, it adopts a polymeric structure. [ 2 ] In common with the other tetrahalides, TiF 4 is a strong Lewis acid .
The traditional method involves treatment of titanium tetrachloride with excess hydrogen fluoride : [ 3 ]
Purification is by sublimation, which involves reversible cracking of the polymeric structure. [ 4 ] X-ray crystallography reveals that the Ti centres are octahedral, but conjoined in an unusual columnar structure. [ 5 ]
TiF 4 forms adducts with many ligands. One example is the complex cis -TiF 4 (CH 3 CN) 2 , which is formed by treatment with acetonitrile . [ 6 ] It is also used as a reagent in the preparation of organofluorine compounds . [ 7 ] With fluoride, the cluster [Ti 4 F 18 ] 2- forms. It has an adamantane-like Ti 4 F 6 core. [ 8 ]
Related to its Lewis acidity , TiF 4 forms a variety of hexafluorides also called hexafluorotitanates. Hexafluorotitanic acid has been used commercially to clean metal surfaces. These salts are stable at pH<4 in the presence of hydrogen fluoride , otherwise they hydrolyze to give oxides . [ 3 ] | https://en.wikipedia.org/wiki/TiF4 |
Titanium hydride normally refers to the inorganic compound TiH 2 and related nonstoichiometric materials. [ 1 ] [ 2 ] It is commercially available as a stable grey/black powder, which is used as an additive in the production of Alnico sintered magnets, in the sintering of powdered metals, the production of metal foam , the production of powdered titanium metal and in pyrotechnics. [ 3 ]
Also known as titanium–hydrogen alloy , [ 4 ] [ 5 ] it is an alloy [ 6 ] of titanium , hydrogen , and possibly other elements. When hydrogen is the main alloying element, its content in the titanium hydride is between 0.02% and 4.0% by weight. Alloying elements intentionally added to modify the characteristics of titanium hydride include gallium , iron , vanadium , and aluminium .
In the commercial process for producing non-stoichiometric TiH 2− x , titanium metal sponge is treated with hydrogen gas at atmospheric pressure at between 300-500 °C. Absorption of hydrogen is exothermic and rapid, changing the color of the sponge grey/black. The brittle product is ground to a powder, which has a composition around TiH 1.95 . [ 3 ] In the laboratory, titanium hydride is produced by heating titanium powder under flowing hydrogen at 700 °C, the idealized equation being: [ 7 ]
Other methods of producing titanium hydride include electrochemical and ball milling methods. [ 8 ] [ 9 ]
TiH 1.95 is unaffected by water and air. [ citation needed ] It is slowly attacked by strong acids and is degraded by hydrofluoric and hot sulfuric acids. It reacts rapidly with oxidizing agents, this reactivity leading to the use of titanium hydride in pyrotechnics. [ 3 ]
The material has been used to produce highly pure hydrogen, which is released upon heating the solid. [ 7 ] Hydrogen release in TiH ~2 starts just above 400 °C but may not be complete until the melting point of titanium metal. [ 10 ] [ 3 ] Titanium tritide (Ti 3 H x ) has been proposed for long-term storage of tritium gas. [ 11 ]
As TiH x approaches stoichiometry, it adopts a distorted body-centered tetragonal structure, termed the ε-form with an axial ratio of less than 1. This composition is very unstable with respect to partial thermal decomposition, unless maintained under a pure hydrogen atmosphere. Otherwise, the composition rapidly decomposes at room temperature until an approximate composition of TiH 1.74 is reached. [ citation needed ] This composition adopts the fluorite structure, and is termed the δ-form, and only very slowly thermally decomposing at room temperature until an approximate composition of TiH 1.47 is reached, at which point, inclusions of the hexagonal close packed α-form, which is the same form as pure titanium, begin to appear.
The evolution of the dihydride from titanium metal and hydrogen has been examined in some detail. α-Titanium has a hexagonal close packed (hcp) structure at room temperature. Hydrogen initially occupies tetrahedral interstitial sites in the titanium. As the H/Ti ratio approaches 2, the material adopts the β-form to a face centred cubic (fcc) , δ-form, the H atoms eventually filling all the tetrahedral sites to give the limiting stoichiometry of TiH 2 . The various phases are described in the table below.
If titanium hydride contains 4.0% hydrogen at less than around 40 °C then it transforms into a body-centred tetragonal (bct) structure called ε-titanium. [ 12 ]
When titanium hydrides with less than 1.3% hydrogen, known as hypoeutectoid titanium hydride are cooled, the β-titanium phase of the mixture attempts to revert to the α-titanium phase, resulting in an excess of hydrogen. One way for hydrogen to leave the β-titanium phase is for the titanium to partially transform into δ-titanium, leaving behind titanium that is low enough in hydrogen to take the form of α-titanium, resulting in an α-titanium matrix with δ-titanium inclusions.
A metastable γ-titanium hydride phase has been reported. [ 13 ] When α-titanium hydride with a hydrogen content of 0.02-0.06% is quenched rapidly, it forms into γ-titanium hydride, as the atoms "freeze" in place when the cell structure changes from hcp to fcc. γ-Titanium takes a body centred tetragonal (bct) structure. Moreover, there is no compositional change so the atoms generally retain their same neighbours.
The absorption of hydrogen and the formation of titanium hydride are a source of damage to titanium and titanium alloys. This hydrogen embrittlement process is of particular concern when titanium and alloys are used as structural materials, as in nuclear reactors.
Hydrogen embrittlement manifests as a reduction in ductility and eventually spalling of titanium surfaces. The effect of hydrogen is to a large extent determined by the composition, metallurgical history and handling of the Ti and Ti alloy. [ 14 ] CP-titanium ( commercially pure : ≤99.55% Ti content) is more susceptible to hydrogen attack than pure α-titanium. Embrittlement, observed as a reduction in ductility and caused by the formation of a solid solution of hydrogen, can occur in CP-titanium at concentrations as low as 30-40 ppm. Hydride formation has been linked to the presence of iron in the surface of a Ti alloy. Hydride particles are observed in specimens of Ti and Ti alloys that have been welded, and because of this welding is often carried out under an inert gas shield to reduce the possibility of hydride formation. [ 14 ]
Ti and Ti alloys form a surface oxide layer , composed of a mixture of Ti(II) , Ti(III) and Ti(IV) oxides, [ 15 ] which offers a degree of protection to hydrogen entering the bulk. [ 14 ] The thickness of this can be increased by anodizing , a process which also results in a distinctive colouration of the material. Ti and Ti alloys are often used in hydrogen containing environments and in conditions where hydrogen is reduced electrolytically on the surface. Pickling , an acid bath treatment which is used to clean the surface can be a source of hydrogen.
Common applications include ceramics , pyrotechnics , sports equipment , as a laboratory reagent , as a blowing agent , and as a precursor to porous titanium. When heated as a mixture with other metals in powder metallurgy , titanium hydride releases hydrogen which serves to remove carbon and oxygen, producing a strong alloy. [ 3 ]
The density of titanium hydride varies based on the alloying constituents, but for pure titanium hydride it ranges between 3.76 and 4.51 g/cm 3 .
Even in the narrow range of concentrations that make up titanium hydride, mixtures of hydrogen and titanium can form a number of different structures, with very different properties. Understanding such properties is essential to making quality titanium hydride. At room temperature , the most stable form of titanium is the hexagonal close-packed (HCP) structure α-titanium. It is a fairly hard metal that can dissolve only a small concentration of hydrogen, no more than 0.20 wt% at 464 °C (867 °F), and only 0.02% at 25 °C (77 °F). If titanium hydride contains more than 0.20% hydrogen at titanium hydride-making temperatures it transforms into a body-centred cubic (BCC) structure called β-titanium. It can dissolve considerably more hydrogen, more than 2.1% hydrogen at 636 °C (1,177 °F). If titanium hydride contains more than 2.1% at 636 °C (1,177 °F) then it transforms into a face-centred cubic (FCC) structure called δ-titanium. It can dissolve even more hydrogen, as much as 4.0% hydrogen 37 °C (99 °F), which reflects the upper hydrogen content of titanium hydride. [ 16 ]
There are many types of heat treating processes available to titanium hydride. The most common are annealing and quenching. Annealing is the process of heating the titanium hydride to a sufficiently high temperature to soften it. This process occurs through three phases: recovery , recrystallization , and grain growth . The temperature required to anneal titanium hydride depends on the type of annealing. Annealing must be done under a hydrogen atmosphere to prevent outgassing . | https://en.wikipedia.org/wiki/TiH2 |
Titanium(IV) hydride (systematically named titanium tetrahydride ) is an inorganic compound with the empirical chemical formula TiH 4 . It has not yet been obtained in bulk, hence its bulk properties remain unknown. However, molecular titanium(IV) hydride has been isolated in solid gas matrices. The molecular form is a colourless gas, and very unstable toward thermal decomposition. As such the compound is not well characterised, although many of its properties have been calculated via computational chemistry .
Titanium(IV) hydride was first produced in 1963 by the photodissociation of mixtures of TiCl 4 and H 2 , followed by immediate mass spectrometry . [ 1 ] Rapid analysis was required as titanium(IV) hydride is extremely unstable. Computational analysis of TiH 4 has given a theoretical bond dissociation energy (relative to M+4H) of 132 kcal/mole. [ 2 ] As the dissociation energy of H 2 is 104 kcal/mole the instability of TiH 4 can be expected to be thermodynamic ; with it dissociating to metallic titanium and hydrogen :
TiH 4 , along with other unstable molecular titanium hydrides, (TiH, TiH 2 , TiH 3 and polymeric species) has been isolated at low temperature following laser ablation of titanium. [ 3 ]
It is suspected that within solid titanium(IV) hydride, the molecules form aggregations ( polymers ), being connected by covalent bonds . [ 4 ] Calculations suggest that TiH 4 is prone to dimerisation . [ 3 ] This largely attributed to the electron deficiency of the monomer and the small size of the hydride ligands; which allows dimerisation to take place with a very low energy barrier as there is a negligible increase in inter-ligand repulsion.
The dimer is a calculated to be a fluxional molecule rapidly inter-converting between a number of forms, all of which display bridging hydrogens. [ 4 ] This is an example of three-center two-electron bonding .
Monomeric titanium(IV) hydride is the simplest transition metal molecule that displays sd 3 orbital hybridisation . [ 5 ] | https://en.wikipedia.org/wiki/TiH4 |
Titanium tetraiodide is an inorganic compound with the formula TiI 4 . It is a black volatile solid, first reported by Rudolph Weber in 1863. [ 2 ] It is an intermediate in the van Arkel–de Boer process for the purification of titanium.
TiI 4 is a rare molecular binary metal iodide, consisting of isolated molecules of tetrahedral Ti(IV) centers. The Ti-I distances are 261 pm . [ 3 ] Reflecting its molecular character, TiI 4 can be distilled without decomposition at one atmosphere; this property is the basis of its use in the van Arkel–de Boer process . The difference in melting point between TiCl 4 (m.p. -24 °C) and TiI 4 (m.p. 150 °C) is comparable to the difference between the melting points of CCl 4 (m.p. -23 °C) and CI 4 (m.p. 168 °C), reflecting the stronger intermolecular van der Waals bonding in the iodides.
Two polymorphs of TiI 4 exist, one of which is highly soluble in organic solvents. In the less soluble cubic form, the Ti-I distances are 261 pm . [ 3 ]
Three methods are well known:
1) From the elements, typically using a tube furnace at 425 °C: [ 4 ]
This reaction can be reversed to produce highly pure films of Ti metal. [ 5 ]
2) Exchange reaction from titanium tetrachloride and HI.
3) Oxide-iodide exchange from aluminium iodide .
Like TiCl 4 and TiBr 4 , TiI 4 forms adducts with Lewis bases, and it can also be reduced. When the reduction is conducted in the presence of Ti metal, one obtains polymeric Ti(III) and Ti(II) derivatives such as CsTi 2 I 7 and the chain CsTiI 3 , respectively. [ 6 ]
TiI 4 exhibits extensive reactivity toward alkenes and alkynes resulting in organoiodine derivatives. It also effects pinacol couplings and other C-C bond-forming reactions. [ 7 ] | https://en.wikipedia.org/wiki/TiI4 |
Titanium nitride ( TiN ; sometimes known as tinite ) is an extremely hard ceramic material, often used as a physical vapor deposition (PVD) coating on titanium alloys , steel , carbide , and aluminium components to improve the substrate's surface properties.
Applied as a thin coating, TiN is used to harden and protect cutting and sliding surfaces, for decorative purposes (for its golden appearance), and as a non-toxic exterior for medical implants . In most applications a coating of less than 5 micrometres (0.00020 in) is applied. [ 5 ]
TiN has a Vickers hardness of 1800–2100, hardness of 31 ± 4 GPa , [ 6 ] a modulus of elasticity of 550 ± 50 GPa , [ 6 ] a thermal expansion coefficient of 9.35 × 10 −6 K −1 , and a superconducting transition temperature of 5.6 K. [ 7 ] [ 6 ]
TiN oxidizes at 800 °C in a normal atmosphere. It is chemically stable at 20 °C, according to laboratory tests, but can be slowly attacked by concentrated acid solutions with rising temperatures. [ 7 ] TiN has a brown color and appears gold when applied as a coating. Depending on the substrate material and surface finish, TiN has a coefficient of friction ranging from 0.4 to 0.9 against another TiN surface (non-lubricated). The typical TiN formation has a crystal structure of NaCl type with a roughly 1:1 stoichiometry ; TiN x compounds with x ranging from 0.6 to 1.2 are, however, thermodynamically stable. [ 8 ]
TiN becomes superconducting at cryogenic temperatures, with critical temperature up to 6.0 K for single crystals. [ 9 ] Superconductivity in thin-film TiN has been studied extensively, with the superconducting properties strongly varying depending on sample preparation, up to complete suppression of superconductivity at a superconductor–insulator transition . [ 10 ] A thin film of TiN was chilled to near absolute zero , converting it into the first known superinsulator , with resistance suddenly increasing by a factor of 100,000. [ 11 ]
Osbornite is a very rare natural form of titanium nitride, found almost exclusively in meteorites. [ 12 ] [ 13 ]
A well-known use for TiN coating is for edge retention and corrosion resistance on machine tooling, such as drill bits and milling cutters , often improving their lifetime by a factor of three or more. [ 14 ]
Because of the metallic gold color of TiN, this material is used to coat costume jewelry and automotive trim for decorative purposes. TiN is also widely used as a top-layer coating, usually with nickel - or chromium -plated substrates, on consumer plumbing fixtures and door hardware. As a coating, it is used in aerospace and military applications and to protect the sliding surfaces of suspension forks of bicycles and motorcycles , as well as the shock shafts of radio-controlled cars . TiN is also used as a protective coating on the moving parts of many rifles and semi-automatic firearms, as it is extremely durable. As well as being durable, it is also extremely smooth, making removing the carbon build-up extremely easy. TiN is non-toxic, meets FDA guidelines, [ 15 ] and has seen use in medical devices such as scalpel blades and orthopedic bone-saw blades, where sharpness and edge retention are important. [ 16 ] TiN coatings have also been used in implanted prostheses (especially hip replacement implants) and other medical implants.
Though less visible, thin films of TiN are also used in microelectronics , where they serve as a conductive connection between the active device and the metal contacts used to operate the circuit, while acting as a diffusion barrier to block the diffusion of the metal into the silicon. In this context, TiN is classified as a "barrier metal" (electrical resistivity ~ 25 μΩ·cm [ 2 ] ), even though it is clearly a ceramic from the perspective of chemistry or mechanical behavior. Recent chip design in the 45 nm technology and beyond also makes use of TiN as a "metal" for improved transistor performance. In combination with gate dielectrics (e.g. HfSiO 4 ) that have a higher permittivity compared to standard SiO 2 , the gate length can be scaled down with low leakage , higher drive current and the same or better threshold voltage . [ 17 ] Additionally, TiN thin films are currently under consideration for coating zirconium alloys for accident-tolerant nuclear fuels. [ 18 ] [ 19 ] It is also used as a coating on some compression driver diaphragms to improve performance.
Owing to their high biostability, TiN layers may also be used as electrodes in bioelectronic applications [ 20 ] like in intelligent implants or in-vivo biosensors that have to withstand the severe corrosion caused by body fluids . TiN electrodes have already been applied in the subretinal prosthesis project [ 21 ] as well as in biomedical microelectromechanical systems ( BioMEMS ). [ 22 ]
The most common methods of TiN thin film creation are physical vapor deposition (PVD, usually sputter deposition , cathodic arc deposition or electron-beam heating ) and chemical vapor deposition (CVD). [ 23 ] In both methods, pure titanium is sublimed and reacted with nitrogen in a high-energy, vacuum environment. TiN film may also be produced on Ti workpieces by reactive growth (for example, annealing ) in a nitrogen atmosphere. PVD is preferred for steel parts because the deposition temperatures exceeds the austenitizing temperature of steel. TiN layers are also sputtered on a variety of higher-melting-point materials such as stainless steels , titanium and titanium alloys . [ 24 ] Its high Young's modulus (values between 450 and 590 GPa have been reported in the literature [ 25 ] ) means that thick coatings tend to flake away, making them much less durable than thin ones. Titanium-nitride coatings can also be deposited by thermal spraying whereas TiN powders are produced by nitridation of titanium with nitrogen or ammonia at 1200 °C. [ 7 ]
Bulk ceramic objects can be fabricated by packing powdered metallic titanium into the desired shape, compressing it to the proper density, then igniting it in an atmosphere of pure nitrogen. The heat released by the chemical reaction between the metal and gas is sufficient to sinter the nitride reaction product into a hard, finished item. See powder metallurgy .
There are several commercially used variants of TiN that have been developed since 2010, such as titanium carbonitride (TiCN), titanium aluminium nitride (TiAlN or AlTiN), and titanium aluminum carbon nitride, which may be used individually or in alternating layers with TiN. These coatings offer similar or superior enhancements in corrosion resistance and hardness, and additional colors ranging from light gray to nearly black, to a dark, iridescent , bluish-purple, depending on the exact process of application. These coatings are becoming common on sporting goods, particularly knives and handguns , where they are used for both aesthetic and functional reasons.
Titanium nitride is also produced intentionally, within some steels, by judicious addition of titanium to the alloy . TiN forms at very high temperatures because of its very low enthalpy of formation , and even nucleates directly from the melt in secondary steel-making. It forms discrete, micrometre-sized cubic particles at grain boundaries and triple points, and prevents grain growth by Ostwald ripening up to very high homologous temperatures . Titanium nitride has the lowest solubility product of any metal nitride or carbide in austenite, a useful attribute in microalloyed steel formulas. | https://en.wikipedia.org/wiki/TiN |
Titanium(II) oxide ( Ti O ) is an inorganic chemical compound of titanium and oxygen. It can be prepared from titanium dioxide and titanium metal at 1500 °C. [ 1 ] It is non-stoichiometric in a range TiO 0.7 to TiO 1.3 and this is caused by vacancies of either Ti or O in the defect rock salt structure. [ 1 ] In pure TiO 15% of both Ti and O sites are vacant, [ 1 ] as the vacancies allow metal-metal bonding between adjacent Ti centres. Careful annealing can cause ordering of the vacancies producing a monoclinic form which has 5 TiO units in the primitive cell that exhibits lower resistivity. [ 2 ] A high temperature form with titanium atoms with trigonal prismatic coordination is also known. [ 3 ] Acid solutions of TiO are stable for a short time then decompose to give hydrogen: [ 1 ]
Gas-phase TiO shows strong bands in the optical spectra of cool ( M-type ) stars. [ 4 ] [ 5 ] In 2017, TiO was claimed to be detected in an exoplanet atmosphere for the first time; a result which is still debated in the literature. [ 6 ] [ 7 ] Additionally, evidence has been obtained for the presence of the diatomic molecule TiO in the interstellar medium. [ 8 ] | https://en.wikipedia.org/wiki/TiO |
3.15 eV (rutile) [ 1 ]
Titanium dioxide , also known as titanium(IV) oxide or titania / t aɪ ˈ t eɪ n i ə / , is the inorganic compound derived from titanium with the chemical formula TiO 2 . When used as a pigment , it is called titanium white , Pigment White 6 ( PW6 ), or CI 77891 . [ 4 ] It is a white solid that is insoluble in water, although mineral forms can appear black. As a pigment, it has a wide range of applications, including paint , sunscreen , and food coloring . When used as a food coloring, it has E number E171. World production in 2014 exceeded 9 million tonnes. [ 5 ] [ 6 ] [ 7 ] It has been estimated that titanium dioxide is used in two-thirds of all pigments, and pigments based on the oxide have been valued at a price of $13.2 billion. [ 8 ]
In all three of its main dioxides, titanium exhibits octahedral geometry , being bonded to six oxide anions. The oxides in turn are bonded to three Ti centers. The overall crystal structures of rutile and anatase are tetragonal in symmetry whereas brookite is orthorhombic. The oxygen substructures are all slight distortions of close packing : in rutile, the oxide anions are arranged in distorted hexagonal close-packing, whereas they are close to cubic close-packing in anatase and to "double hexagonal close-packing" for brookite. The rutile structure is widespread for other metal dioxides and difluorides, e.g. RuO 2 and ZnF 2 .
Molten titanium dioxide has a local structure in which each Ti is coordinated to, on average, about 5 oxygen atoms. [ 9 ] This is distinct from the crystalline forms in which Ti coordinates to 6 oxygen atoms.
Synthetic TiO 2 is mainly produced from the mineral ilmenite . Rutile , and anatase , naturally occurring TiO 2 , occur widely also, e.g. rutile as a 'heavy mineral' in beach sand. Leucoxene , fine-grained anatase formed by natural alteration of ilmenite, is yet another ore. Star sapphires and rubies get their asterism from oriented inclusions of rutile needles. [ 10 ]
Titanium dioxide occurs in nature as the minerals rutile and anatase . Additionally two high-pressure forms are known minerals: a monoclinic baddeleyite -like form known as akaogiite , and the other has a slight monoclinic distortion of the orthorhombic α-PbO 2 structure and is known as riesite. Both of which can be found at the Ries crater in Bavaria . [ 11 ] [ 12 ] [ 13 ] It is mainly sourced from ilmenite , which is the most widespread titanium dioxide-bearing ore around the world. Rutile is the next most abundant and contains around 98% titanium dioxide in the ore. The metastable anatase and brookite phases convert irreversibly to the equilibrium rutile phase upon heating above temperatures in the range 600–800 °C (1,110–1,470 °F). [ 14 ]
Titanium dioxide has twelve known polymorphs – in addition to rutile, anatase, brookite, akaogiite and riesite, three metastable phases can be produced synthetically ( monoclinic , tetragonal , and orthorhombic ramsdellite-like), and four high-pressure forms (α-PbO 2 -like, cotunnite -like, orthorhombic OI, and cubic phases) also exist:
The cotunnite -type phase was claimed to be the hardest known oxide with the Vickers hardness of 38 GPa and the bulk modulus of 431 GPa (i.e. close to diamond's value of 446 GPa) at atmospheric pressure. [ 22 ] However, later studies came to different conclusions with much lower values for both the hardness (7–20 GPa, which makes it softer than common oxides like corundum Al 2 O 3 and rutile TiO 2 ) [ 23 ] and bulk modulus (~300 GPa). [ 24 ] [ 25 ]
Titanium dioxide (B) is found as a mineral in magmatic rocks and hydrothermal veins, as well as weathering rims on perovskite . TiO 2 also forms lamellae in other minerals. [ 26 ]
The largest TiO 2 pigment processors are Chemours , Venator , Kronos [ de ] , and Tronox . [ 27 ] [ 28 ] Major paint and coating company end users for pigment grade titanium dioxide include Akzo Nobel , PPG Industries , Sherwin Williams , BASF , Kansai Paints and Valspar . [ 29 ] Global TiO 2 pigment demand for 2010 was 5.3 Mt with annual growth expected to be about 3–4%. [ 30 ]
The production method depends on the feedstock. In addition to ores, other feedstocks include upgraded slag . Both the chloride process and the sulfate process (both described below) produce titanium dioxide pigment in the rutile crystal form, but the sulfate process can be adjusted to produce the anatase form. Anatase, being softer, is used in fiber and paper applications. The sulfate process is run as a batch process ; the chloride process is run as a continuous process . [ 31 ]
In chloride process , the ore is treated with chlorine and carbon to give titanium tetrachloride , a volatile liquid that is further purified by distillation. The TiCl4 is treated with oxygen to regenerate chlorine and produce the titanium dioxide.
In the sulfate process, ilmenite is treated with sulfuric acid to extract iron(II) sulfate pentahydrate . This process requires concentrated ilmenite (45–60% TiO 2 ) or pretreated feedstocks as a suitable source of titanium. [ 32 ] The resulting synthetic rutile is further processed according to the specifications of the end user, i.e. pigment grade or otherwise. [ 33 ]
Examples of plants using the sulfate process are the Sorel-Tracy plant of QIT-Fer et Titane and the Eramet Titanium & Iron smelter in Tyssedal Norway. [ 34 ]
The Becher process is another method for the production of synthetic rutile from ilmenite. It first oxidizes the ilmenite as a means to separate the iron component.
For specialty applications, TiO 2 films are prepared by various specialized chemistries. [ 35 ] Sol-gel routes involve the hydrolysis of titanium alkoxides such as titanium ethoxide :
A related approach that also relies on molecular precursors involves chemical vapor deposition . In this method, the alkoxide is volatilized and then decomposed on contact with a hot surface:
First mass-produced in 1916, [ 36 ] titanium dioxide is the most widely used white pigment because of its brightness and very high refractive index , in which it is surpassed only by a few other materials (see list of indices of refraction ). Titanium dioxide crystal size is ideally around 220 nm (measured by electron microscope) to optimize the maximum reflection of visible light. However, abnormal grain growth is often observed in titanium dioxide, particularly in its rutile phase. [ 37 ] The occurrence of abnormal grain growth brings about a deviation of a small number of crystallites from the mean crystal size and modifies the physical behaviour of TiO 2 . The optical properties of the finished pigment are highly sensitive to purity. As little as a few parts per million (ppm) of certain metals (Cr, V, Cu, Fe, Nb) can disturb the crystal lattice so much that the effect can be detected in quality control. [ 38 ] [ full citation needed ] Approximately 4.6 million tons of pigmentary TiO 2 are used annually worldwide, and this number is expected to increase as use continues to rise. [ 39 ]
TiO 2 is also an effective opacifier in powder form, where it is employed as a pigment to provide whiteness and opacity to products such as paints , coatings , plastics , papers , inks , foods , supplements , medicines (i.e. pills and tablets), and most toothpastes ; in 2019 it was present in two-thirds of toothpastes on the French market. [ 40 ] In paint, it is often referred to offhandedly as "brilliant white", "the perfect white", "the whitest white", or other similar terms. Opacity is improved by optimal sizing of the titanium dioxide particles.
Often used as color in food, [ 41 ] it is commonly found in ice creams, chocolates, all types of candy, creamers, desserts, marshmallows, chewing gum, pastries, spreads, dressings, cakes, some cheeses, and many other foods. [ 42 ]
When deposited as a thin film , its refractive index and colour make it an excellent reflective optical coating for dielectric mirrors ; it is also used in generating decorative thin films such as found in "mystic fire topaz". [ citation needed ]
Some grades of modified titanium based pigments as used in sparkly paints, plastics, finishes and cosmetics – these are man-made pigments whose particles have two or more layers of various oxides – often titanium dioxide, iron oxide or alumina – in order to have glittering, iridescent and or pearlescent effects similar to crushed mica or guanine -based products. In addition to these effects a limited colour change is possible in certain formulations depending on how and at which angle the finished product is illuminated and the thickness of the oxide layer in the pigment particle; one or more colours appear by reflection while the other tones appear due to interference of the transparent titanium dioxide layers. [ 43 ] In some products, the layer of titanium dioxide is grown in conjunction with iron oxide by calcination of titanium salts (sulfates, chlorates) around 800 °C [ 44 ] One example of a pearlescent pigment is Iriodin, based on mica coated with titanium dioxide or iron (III) oxide. [ 45 ]
The iridescent effect in these titanium oxide particles is unlike the opaque effect obtained with usual ground titanium oxide pigment obtained by mining, in which case only a certain diameter of the particle is considered and the effect is due only to scattering.
In cosmetic and skin care products, titanium dioxide is used as a pigment, sunscreen and a thickener . As a sunscreen, ultrafine TiO 2 is used, which is notable in that combined with ultrafine zinc oxide , it is considered to be an effective sunscreen that lowers the incidence of sun burns and minimizes the premature photoaging , photocarcinogenesis and immunosuppression associated with long term excess sun exposure. [ 46 ] Sometimes these UV blockers are combined with iron oxide pigments in sunscreen to increase visible light protection. [ 47 ]
Titanium dioxide and zinc oxide are generally considered to be less harmful to coral reefs than sunscreens that include chemicals such as oxybenzone , octocrylene and octinoxate . [ 48 ]
Nanosized titanium dioxide is found in the majority of physical sunscreens because of its strong UV light absorbing capabilities and its resistance to discolouration under ultraviolet light. This advantage enhances its stability and ability to protect the skin from ultraviolet light. Nano-scaled (particle size of 20–40 nm) [ 49 ] titanium dioxide particles are primarily used in sunscreen lotion because they scatter visible light much less than titanium dioxide pigments, and can give UV protection. [ 39 ] Sunscreens designed for infants or people with sensitive skin are often based on titanium dioxide and/or zinc oxide , as these mineral UV blockers are believed to cause less skin irritation than other UV absorbing chemicals. Nano-TiO 2 , which blocks both UV-A and UV-B radiation, is used in sunscreens and other cosmetic products.
The EU Scientific Committee on Consumer Safety considered nano sized titanium dioxide to be safe for skin applications, in concentrations of up to 25 percent based on animal testing. [ 50 ] The risk assessment of different titanium dioxide nanomaterials in sunscreen is currently evolving since nano-sized TiO 2 is different from the well-known micronized form. [ 51 ] The rutile form is generally used in cosmetic and sunscreen products due to it not possessing any observed ability to damage the skin under normal conditions [ 52 ] and having a higher UV absorption . [ 53 ] In 2016 Scientific Committee on Consumer Safety (SCCS) tests concluded that the use of nano titanium dioxide (95–100% rutile, ≦5% anatase) as a UV filter can be considered to not pose any risk of adverse effects in humans post-application on healthy skin, [ 54 ] except in the case the application method would lead to substantial risk of inhalation (ie; powder or spray formulations). This safety opinion applied to nano TiO 2 in concentrations of up to 25%. [ 55 ]
Initial studies indicated that nano-TiO 2 particles could penetrate the skin, causing concern over its use. These studies were later refuted, when it was discovered that the testing methodology couldn't differentiate between penetrated particles and particles simply trapped in hair follicles and that having a diseased or physically damaged dermis could be the true cause of insufficient barrier protection. [ 51 ]
SCCS research found that when nanoparticles had certain photostable coatings (e.g., alumina , silica , cetyl phosphate, triethoxycaprylylsilane , manganese dioxide ), the photocatalytic activity was attenuated and no notable skin penetration was observed; the sunscreen in this research was applied at amounts of 10 mg/cm2 for exposure periods of 24 hours. [ 55 ] Coating TiO 2 with alumina, silica, zircon or various polymers can minimize avobenzone degradation [ 56 ] and enhance UV absorption by adding an additional light diffraction mechanism. [ 53 ]
TiO 2 is used extensively in plastics and other applications as a white pigment or an opacifier and for its UV resistant properties where the powder disperses light – unlike organic UV absorbers – and reduces UV damage, due mostly to the particle's high refractive index. [ 57 ]
In ceramic glazes , titanium dioxide acts as an opacifier and seeds crystal formation.
It is used as a tattoo pigment and in styptic pencils . Titanium dioxide is produced in varying particle sizes which are both oil and water dispersible, and in certain grades for the cosmetic industry. It is also a common ingredient in toothpaste.
The exterior of the Saturn V rocket was painted with titanium dioxide; this later allowed astronomers to determine that J002E3 was likely the S-IVB stage from Apollo 12 and not an asteroid . [ 58 ]
Titanium dioxide is an n-type semiconductor and is used in dye-sensitized solar cells . [ 59 ] It is also used in other electronics components such as electrodes in batteries. [ 60 ]
Between 2002 and 2022, there were 459 patent families that describe the production of titanium dioxide from ilmenite . The majority of these patents describe pre-treatment processes, such as using smelting and magnetic separation to increase titanium concentration in low-grade ores, leading to titanium concentrates or slags. Other patents describe processes to obtain titanium dioxide, either by a direct hydrometallurgical process or through the main industrial production processes, the sulfate process and the chloride process . [ 61 ] The sulfate process represents 40% of the world’s titanium dioxide production and is protected in 23% of patent families. The chloride process is only mentioned in 8% of patent families, although it provides 60% of the worldwide industrial production of titanium dioxide. [ 61 ]
Key contributors to patents on the production of titanium dioxide are companies from China, Australia and the United States, reflecting the major contribution of these countries to industrial production. Chinese companies Pangang and Lomon Billions Groups hold major patent portfolios. [ 61 ]
Nanosized titanium dioxide, particularly in the anatase form, exhibits photocatalytic activity under ultraviolet (UV) irradiation. This photoactivity is reportedly most pronounced at the {001} planes of anatase, [ 62 ] [ 63 ] although the {101} planes are thermodynamically more stable and thus more prominent in most synthesised and natural anatase, [ 64 ] as evident by the often observed tetragonal dipyramidal growth habit . Interfaces between rutile and anatase are further considered to improve photocatalytic activity by facilitating charge carrier separation and as a result, biphasic titanium dioxide is often considered to possess enhanced functionality as a photocatalyst. [ 65 ] It has been reported that titanium dioxide, when doped with nitrogen ions or doped with metal oxide like tungsten trioxide, exhibits excitation also under visible light. [ 66 ] The strong oxidative potential of the positive holes oxidizes water to create hydroxyl radicals . It can also oxidize oxygen or organic materials directly. Hence, in addition to its use as a pigment, titanium dioxide can be added to paints, cements, windows, tiles, or other products for its sterilizing, deodorizing, and anti-fouling properties, and is used as a hydrolysis catalyst . It is also used in dye-sensitized solar cells , which are a type of chemical solar cell (also known as a Graetzel cell).
The photocatalytic properties of nanosized titanium dioxide were discovered by Akira Fujishima in 1967 [ 67 ] and published in 1972. [ 68 ] The process on the surface of the titanium dioxide was called the Honda-Fujishima effect [ ja ] . [ 67 ] In thin film and nanoparticle form, titanium dioxide has the potential for use in energy production: As a photocatalyst, it can break water into hydrogen and oxygen. With the hydrogen collected, it could be used as a fuel. The efficiency of this process can be greatly improved by doping the oxide with carbon. [ 69 ] Further efficiency and durability has been obtained by introducing disorder to the lattice structure of the surface layer of titanium dioxide nanocrystals, permitting infrared absorption. [ 70 ] Visible-light-active nanosized anatase and rutile has been developed for photocatalytic applications. [ 71 ] [ 72 ]
In 1995 Fujishima and his group discovered the superhydrophilicity phenomenon for titanium dioxide coated glass exposed to sun light. [ 67 ] This resulted in the development of self-cleaning glass and anti-fogging coatings.
Nanosized TiO 2 incorporated into outdoor building materials, such as paving stones in noxer blocks [ 73 ] or paints, could reduce concentrations of airborne pollutants such as volatile organic compounds and nitrogen oxides . [ 74 ] A TiO 2 -containing cement has been produced. [ 75 ]
Using TiO 2 as a photocatalyst, attempts have been made to mineralize pollutants (to convert into CO 2 and H 2 O) in waste water. [ 76 ] [ 77 ] [ 78 ] The photocatalytic destruction of organic matter could also be exploited in coatings with antimicrobial applications. [ 79 ]
Although nanosized anatase TiO 2 does not absorb visible light, it does strongly absorb ultraviolet (UV) radiation ( hv ), leading to the formation of hydroxyl radicals. [ 80 ] This occurs when photo-induced valence bond holes (h + vb ) are trapped at the surface of TiO 2 leading to the formation of trapped holes (h + tr ) that cannot oxidize water. [ 81 ]
Anatase can be converted into non-carbon nanotubes and nanowires . [ 82 ] Hollow TiO 2 nanofibers can be also prepared by coating carbon nanofibers by first applying titanium butoxide . [ 83 ]
Titanium dioxide is insoluble in water, organic solvents, and inorganic acids. It is slightly soluble in alkali , soluble in saturated potassium acid carbonate, and can be completely dissolved in strong sulfuric acid and hydrofluoric acid after boiling for a long time. [ 84 ]
Widely-occurring minerals and even gemstones are composed of TiO 2 . All natural titanium, comprising more than 0.5% of the Earth's crust, exists as oxides. [ 85 ]
As of 2024, titanium dioxide is considered safe by the US FDA as a color ingredient for oral human consumption as long as it is 1% or less of the total food composition. [ 86 ] A 2021 ban by the EU EFSA has been criticized as based on errors regarding the safety of titanium dioxide (E171) particles as a food additive, [ 87 ] and according to a 2022 review, existing evidence does not support a direct DNA damaging mechanism for titanium dioxide. [ 88 ]
TiO 2 whitener in food was banned in France from 2020, due to uncertainty about safe quantities for human consumption. [ 89 ]
In 2021, the European Food Safety Authority (EFSA) ruled that as a consequence of new understandings of nanoparticles , titanium dioxide could "no longer be considered safe as a food additive", and the EU health commissioner announced plans to ban its use across the EU, with discussions beginning in June 2021. EFSA concluded that genotoxicity —which could lead to carcinogenic effects—could not be ruled out, and that a "safe level for daily intake of the food additive could not be established". [ 90 ] In 2022, the UK Food Standards Agency and Food Standards Scotland announced their disagreement with the EFSA ruling, and did not follow the EU in banning titanium dioxide as a food additive. [ 91 ] Health Canada similarly reviewed the available evidence in 2022 and decided not to change their position on titanium dioxide as a food additive. [ 92 ]
The European Union removed the authorization to use titanium dioxide (E 171) in foods, effective 7 February 2022, with a six months grace period. [ 93 ]
As of May 2023, following the European Union 2022 ban, the U.S. states California and New York were considering banning the use of titanium dioxide in foods. [ 94 ]
As of 2024, the Food and Drug Administration (FDA) in the United States permits titanium dioxide as a food additive. [ 86 ] It may be used to increase whiteness and opacity in dairy products (some cheeses, ice cream, and yogurt), candies, frostings, fillings, and many other foods. The FDA regulates the labeling of products containing titanium dioxide, allowing the product's ingredients list to identify titanium dioxide either as "color added" or "artificial colors" or "titanium dioxide;" it does not require that titanium dioxide be explicitly named. [ 86 ] In 2023, the Consumer Healthcare Products Association , a manufacturer's trade group, defended the substance as safe at certain limits while allowing that additional studies could provide further insight, saying an immediate ban would be a "knee-jerk" reaction. [ 95 ]
Dunkin' Donuts dropped titanium dioxide from their merchandise in 2015 after public pressure. [ 96 ]
Size distribution analyses showed that batches of food-grade TiO₂, which is produced with a target particle size in the 200–300 nm range for optimal pigmentation qualities, include a nanoparticle-sized fraction as inevitable byproduct of the manufacturing processes. [ 97 ]
Titanium dioxide dust, when inhaled, has been classified by the International Agency for Research on Cancer (IARC) as an IARC Group 2B carcinogen , meaning it is possibly carcinogenic to humans . [ 98 ] [ 99 ] The US National Institute for Occupational Safety and Health recommends two separate exposure limits. NIOSH recommends that fine TiO 2 particles be set at an exposure limit of 2.4 mg/m 3 , while ultrafine TiO 2 be set at an exposure limit of 0.3 mg/m 3 , as time-weighted average concentrations up to 10 hours a day for a 40-hour work week. [ 100 ]
Although no evidence points to acute toxicity, recurring concerns have been expressed about nanophase forms of these materials. Studies of workers with high exposure to TiO 2 particles indicate that even at high exposure there is no adverse effect to human health. [ 85 ]
Titanium dioxide (TiO₂) is mostly introduced into the environment as nanoparticles via wastewater treatment plants. [ 101 ] Cosmetic pigments including titanium dioxide enter the wastewater when the product is washed off into sinks after cosmetic use. Once in the sewage treatment plants, pigments separate into sewage sludge which can then be released into the soil when injected into the soil or distributed on its surface. 99% of these nanoparticles wind up on land rather than in aquatic environments due to their retention in sewage sludge. [ 101 ] In the environment, titanium dioxide nanoparticles have low to negligible solubility and have been shown to be stable once particle aggregates are formed in soil and water surroundings. [ 101 ] In the process of dissolution, water-soluble ions typically dissociate from the nanoparticle into solution when thermodynamically unstable. TiO 2 dissolution increases when there are higher levels of dissolved organic matter and clay in the soil. However, aggregation is promoted by pH at the isoelectric point of TiO 2 (pH= 5.8) which renders it neutral and solution ion concentrations above 4.5 mM. [ 102 ] [ 103 ]
This article incorporates text from a free content work. Licensed under CC-BY. Text taken from Production of titanium and titanium dioxide from ilmenite and related applications , WIPO. | https://en.wikipedia.org/wiki/TiO2 |
Titanium disulfide is an inorganic compound with the formula Ti S 2 . A golden yellow solid with high electrical conductivity , [ 1 ] it belongs to a group of compounds called transition metal di chalcogenides , which consist of the stoichiometry M E 2 . TiS 2 has been employed as a cathode material in rechargeable batteries .
With a layered structure , TiS 2 adopts a hexagonal close packed (hcp) structure, analogous to cadmium iodide (CdI 2 ). In this motif, half of the octahedral holes are filled with a " cation ", in this case Ti 4+ . [ 1 ] [ 2 ] Each Ti centre is surrounded by six sulfide ligands in an octahedral structure. Each sulfide is connected to three Ti centres, the geometry at S being pyramidal. Several metal di chalcogenides adopt similar structures, but some, notably MoS 2 , do not. [ 2 ] The layers of TiS 2 consist of covalent Ti-S bonds. The individual layers of TiS 2 are bound together by van der Waals forces , which are relatively weak intermolecular forces. It crystallises in the space group P 3 m1. [ 3 ] The Ti-S bond lengths are 2.423 Å. [ 4 ]
The single most useful and most studied property of TiS 2 is its ability to undergo intercalation upon treatment with electropositive elements. The process is a redox reaction , illustrated in the case of lithium:
LiTiS 2 is generally described as Li + [TiS 2 − ]. During the intercalation and deintercalation, a range of stoichimetries are produced with the general formul Li x TiS 2 (x < 1). During intercalation, the interlayer spacing expands (the lattice "swells") and the electrical conductivity of the material increases. Intercalation is facilitated because of the weakness of the interlayer forces as well as the susceptibility of the Ti(IV) centers toward reduction. Intercalation can be conducted by combining a suspension of the disulfide material and a solution of the alkali metal in anhydrous ammonia. Alternatively solid TiS 2 reacts with the alkali metal upon heating.
The Rigid-Band Model (RBM), which assumes that electronic band structure does not change with intercalation, describes changes in the electronic properties upon intercalation.
Deintercalation is the opposite of intercalation; the cations diffuse out from between the layers. This process is associated with recharging a Li/TiS 2 battery. Intercalation and deintercalation can be monitored by cyclic voltammetry . The microstructure of the titanium disulfide greatly affects the intercalation and deintercalation kinetics . Titanium disulfide nanotubes have a higher uptake and discharge capacity than the polycrystalline structure. [ 5 ] The higher surface area of the nanotubes is postulated to provide more binding sites for the anode ions than the polycrystalline structure. [ 5 ]
Formally containing the d 0 ion Ti 4+ and closed shell dianion S 2− , TiS 2 is essentially diamagnetic. Its magnetic susceptibility is 9 x 10 −6 emu/mol, the value being sensitive to stoichiometry. [ 6 ] Titanium disulfide is a semimetal , meaning there is small overlap of the conduction band and valence band .
The properties of titanium disulfide powder have been studied by high pressure synchrotron x-ray diffraction (XRD) at room temperature. [ 3 ] At ambient pressure, TiS 2 behaves as semiconductor while at high pressures of 8 GPa the material behaves as a semimetal. [ 3 ] [ 7 ] At 15 GPa, the transport properties change. [ 7 ] There is no significant change in the density of states at the Fermi level up to 20 GPa and phase change does not occur until 20.7 GPa. A change in the structure of TiS 2 was observed at a pressure of 26.3 GPa, although the new structure of the high pressure phase has not been determined. [ 3 ]
The unit cell of titanium disulfide is 3.407 by 5.695 angstroms . The size of the unit cell decreased at 17.8 GPa. The decrease in unit cell size was greater than was observed for MoS 2 and WS 2 , indicating that titanium disulfide is softer and more compressible. The compression behavior of titanium disulfide is anisotropic . The axis parallel to S-Ti-S layers (c-axis) is more compressible than the axis perpendicular to S-Ti-S layers (a-axis) because of weak van der waals forces keeping S and Ti atoms together. At 17.8 GPa, the c-axis is compressed by 9.5% and the a-axis is compressed by 4%. The longitudinal sound velocity is 5284 m/s in the plane parallel to S-Ti-S layers. The longitudinal sound velocity perpendicular to the layers is 4383 m/s. [ 8 ]
Titanium disulfide is prepared by the reaction of the elements around 500 °C. [ 6 ]
It can be more easily synthesized from titanium tetrachloride , but this product is typically less pure than that obtained from the elements. [ 6 ]
This route has been applied to the formation of TiS 2 films by chemical vapor deposition. Thiols and organic disulfides can be employed in place of hydrogen sulfide. [ 9 ]
A variety of other titanium sulfides are known. [ 10 ]
Samples of TiS 2 are unstable in air. [ 6 ] Upon heating, the solid undergoes oxidation to titanium dioxide :
TiS 2 is also sensitive to water:
Upon heating, TiS 2 releases sulfur, forming the titanium(III) derivative:
Thin films of TiS 2 have been prepared by the sol-gel process from titanium isopropoxide (Ti(OPr i ) 4 ) followed by spin coating . [ 11 ] This method affords amorphous material that crystallised at high temperatures to hexagonal TiS 2 , which crystallization orientations in the [001], [100], and [001] directions. [ 11 ] Because of their high surface area, such films are attractive for battery applications. [ 11 ]
More specialized morphologies— nanotubes , nanoclusters , whiskers, nanodisks, thin films, fullerenes—are prepared by combining the standard reagents, often TiCl 4 in unusual ways. For example, flower-like morphologies were obtain by treating a solution of sulfur in 1-octadecene with titanium tetrachloride. [ 12 ]
A form of TiS 2 with a fullerene -like structure has been prepared using the TiCl 4 /H 2 S method. The resulting spherical structures have diameters between 30 and 80 nm. [ 13 ] Owing to their spherical shape, these fullerenes exhibit reduced friction coefficient and wear, which may prove useful in various applications.
Nanotubes of TiS 2 can be synthesized using a variation of the TiCl 4 /H 2 S route. According to transmission electron microscopy (TEM), these tubes have an outer diameter of 20 nm and an inner diameter of 10 nm. [ 14 ] The average length of the nanotubes was 2-5 μm and the nanotubes were proven to be hollow. [ 14 ] TiS 2 nanotubes with open ended tips are reported to store up to 2.5 weight percent hydrogen at 25 °C and 4 MPa hydrogen gas pressure. [ 15 ] Absorption and desorption rates are fast, which is an attractive for hydrogen storage. The hydrogen atoms are postulated to bind to sulfur. [ 15 ]
Nanoclusters, or quantum dots of TiS 2 have distinctive electronic and chemical properties due to quantum confinement and very large surface to volume ratios. Nanoclusters can be synthesized using micelle . The nanoclusters are prepared from a solution of TiCl 4 in tridodecylmethyl ammonium iodide (TDAI), which served as the inverse micelle structure and seeded the growth of nanoclusters in the same general reaction as nanotubes. [ 14 ] Nucleation only occurs inside the micelle cage due to the insolubility of the charged species in the continuous medium, which is generally a low dielectric constant inert oil. Like the bulk material, nanocluster-form of TiS 2 is a hexagonal layered structure. . Quantum confinement creates well separated electronic states and increases the band gap more than 1 eV in comparison to the bulk material. A spectroscopic comparison shows a large blueshift for the quantum dots of 0.85 eV.
Nanodisks of TiS 2 arise by treating TiCl 4 with sulfur in oleylamine . [ 16 ]
The promise of titanium disulfide as a cathode material in rechargeable batteries was described in 1973 by M. Stanley Whittingham . [ 17 ] The Group IV and V dichalcogenides attracted attention for their high electrical conductivities. The originally described battery used a lithium anode and a titanium disulfide cathode. This battery had high energy density and the diffusion of lithium ions into the titanium disulfide cathode was reversible, making the battery rechargeable. Titanium disulfide was chosen because it is the lightest and cheapest chalcogenide. Titanium disulfide also has the fastest rate of lithium ion diffusion into the crystal lattice. The main problem was degradation of the cathode after multiple recycles. This reversible intercalation process allows the battery to be rechargeable. Additionally, titanium disulfide is the lightest and the cheapest of all group IV and V layered dichalcogenides. [ 18 ] In the 1990s, titanium disulfide was replaced by other cathode materials (manganese and cobalt oxides) in most rechargeable batteries.
The use of TiS 2 cathodes remains of interest for use in solid-state lithium batteries, e.g., for hybrid electric vehicles and plug-in electric vehicles . [ 18 ]
In contrast to the all-solid state batteries, most lithium batteries employ liquid electrolytes, which pose safety issues due to their flammability. Many different solid electrolytes have been proposed to replace these hazardous liquid electrolytes. For most solid-state batteries, high interfacial resistance lowers the reversibility of the intercalation process, shortening the life cycle. These undesirable interfacial effects are less problematic for TiS 2 . One all-solid-state lithium battery exhibited a power density of 1000 W/kg over 50 cycles with a maximum power density of 1500 W/kg. Additionally, the average capacity of the battery decreased by less than 10% over 50 cycles. Although titanium disulfide has high electrical conductivity, high energy density, and high power, its discharge voltage is relatively low compared to other lithium batteries where the cathodes have higher reduction potentials. [ 18 ] | https://en.wikipedia.org/wiki/TiS2 |
Titanium diselenide (TiSe 2 ) also known as titanium(IV) selenide, is an inorganic compound of titanium and selenium . In this material selenium is viewed as selenide (Se 2− ) which requires that titanium exists as Ti 4+ . Titanium diselenide is a member of metal dichalcogenides , compounds that consist of a metal and an element of the chalcogen column within the periodic table. Many exhibit properties of potential value in battery technology, such as intercalation and electrical conductivity, although most applications focus on the less toxic and lighter disulfides, e.g. TiS 2 .
Within the titanium-selenium system, many stoichiometries have been identified. Titanium diselenide crystallizes with the CdI 2 -type structure, in which the octahedral holes between alternating hexagonal closely packed layer of Se 2− layers (that is half of the total number of octahedral holes) are occupied by Ti 4+ centers. The CdI 2 structure is often referred to as a layer structure as the repeating layers of atoms perpendicular to the close packed layer form the sequence Se-Ti-Se … Se-Ti-Se … Se-Ti-Se with weak van der Waals interactions between the selenium atoms in adjacent layers. The structure has (6,3)-coordination, being octahedral for the cation and trigonal pyramidal for the anions. The structure type is found commonly for many transition metal halides as well. [ 1 ] This layered structure is known to undergo intercalation by alkali metals (M), resulting in the formation of M x TiSe 2 (x ≤ 1), thereby expanding the weak van der Waals gaps between the 2D layered sheets. [ 2 ]
A mixture of titanium and selenium are heated under argon atmosphere to produce crude samples. The crude product is typically purified by chemical vapor transport using iodine as the transport agent. [ 3 ] | https://en.wikipedia.org/wiki/TiSe2 |
A tumour inducing (Ti) plasmid is a plasmid found in pathogenic species of Agrobacterium , including A. tumefaciens , A. rhizogenes , A. rubi and A. vitis .
Evolutionarily, the Ti plasmid is part of a family of plasmids carried by many species of Alphaproteobacteria . Members of this plasmid family are defined by the presence of a conserved DNA region known as the repABC gene cassette, which mediates the replication of the plasmid, the partitioning of the plasmid into daughter cells during cell division as well as the maintenance of the plasmid at low copy numbers in a cell. [ 1 ] The Ti plasmids themselves are sorted into different categories based on the type of molecule, or opine , they allow the bacteria to break down as an energy source. [ 2 ]
The presence of this Ti plasmid is essential for the bacteria to cause crown gall disease in plants. [ 1 ] This is facilitated via certain crucial regions in the Ti plasmid, including the vir region, which encodes for virulence genes, and the transfer DNA (T-DNA) region, which is a section of the Ti plasmid that is transferred via conjugation into host plant cells after an injury site is sensed by the bacteria. These regions have features that allow the delivery of T-DNA into host plant cells, and can modify the host plant cell to cause the synthesis of molecules like plant hormones (e.g. auxins , cytokinins ) and opines and the formation of crown gall tumours. [ 1 ]
Because the T-DNA region of the Ti plasmid can be transferred from bacteria to plant cells, it represented an exciting avenue for the transfer of DNA between kingdoms and spurred large amounts of research on the Ti plasmid and its possible uses in bioengineering.
The Ti plasmid is a member of the RepABC plasmid family found in Alphaproteobacteria. [ 3 ] These plasmids are often relatively large in size, ranging from 100kbp to 2Mbp. They are also often termed replicons , as their replication begins at a single site. Members of this family have a characteristic repABC gene cassette. [ 4 ] Another notable member of this family is the root inducing (Ri) plasmid carried by A. rhizogenes , which causes another plant disease known as hairy root disease. [ 1 ]
A key feature of Ti plasmids is their ability to drive the production of opines, which are derivatives of various amino acids or sugar phosphates , in host plant cells. These opines can then be used as a nutrient for the infecting bacteria, which catabolizes the respective opines using genes encoded in the Ti plasmid.
Accordingly, Ti plasmids have been classified based on the type of opine they catabolize, namely: nopaline- , octopine- or mannityl-types, which are amino acid derivatives, or agrocinopine-type, which are sugar phosphate derivatives. [ 1 ]
The identification of A. tumefaciens as the cause of gall tumours in plants paved the way for insights into the molecular basis of crown gall disease. [ 5 ]
The first indication of a genetic effect on host plant cells came in 1942-1943, where plant cells of secondary tumours were found to lack any bacterial cells within. However, these tumour cells did possess the ability to produce opines metabolized by the infecting bacterial strain. [ 6 ] Crucially, the production of the respective opines occurred regardless of the plant species and occasionally only within crown gall tissues, indicating that the bacteria had transferred some genetic material to the host plant cells in order to allow opine synthesis. [ 5 ]
However, how and to what extend did DNA transfer occur remained an open question. Adding A. tumefaciens DNA alone did not cause tumors in plants, [ 7 ] while very little A. tumefaciens DNA was found to be integrated into the host plant cell genome. [ 8 ] The addition of deoxyribonucleases (DNases) to degrade DNA also failed to prevent the formation and growth of the plant tumors. [ 9 ] These suggested that little, if any, of the A. tumefaciens DNA is transferred to the host plant cell to cause disease and, if DNA is indeed transferred from the bacteria to the plant, it must occur in a protected manner.
Subsequently, oncogenic bacterial strains were found to be able to convert non-pathogenic bacteria into pathogens via the process of conjugation, where the genes responsible for virulence were transferred to the non-pathogenic cells. [ 10 ] The role of a plasmid in this pathogenic ability was further supported when large plasmids were found only in pathogenic bacteria but not avirulent bacteria. [ 11 ] Eventually, the detection of parts of bacterial plasmids in host plant cells was established, confirming that this was the genetic material responsible for the genetic effect of infection. [ 12 ]
With the identification of the Ti plasmid, many studies were carried out to determine the characteristics of the Ti plasmid and how the genetic material is transferred from the Agrobacterium to the plant host. Some notable early milestones in the studies of Ti plasmids include the mapping of a Ti plasmid in 1978 and the studying of sequence similarity between different Ti plasmids in 1981. [ 13 ] [ 14 ]
Between 1980–2000, the characterization of the T-DNA region and the 'vir' region was also pursued. Studies into the T-DNA region determined their process of transfer and identified genes allowing the synthesis of plant hormones and opines. [ 15 ] Separately, early work aimed to determine the functions of the genes encoded in the 'vir' region - these were broadly categorized into those that allowed bacterial-host interactions and those that enabled T-DNA delivery. [ 2 ]
The replication, partitioning and maintenance of the Ti plasmid depends on the repABC gene cassette, which is mainly made up of three genes: repA , repB and repC . repA and repB each encode for proteins involved in plasmid partitioning, while repC encodes a replication initiator. [ 1 ] These genes are expressed from 4 different promoters located upstream of repA . repE encodes for a small antisense RNA and is located between repB and repC . [ 4 ] Additionally, there is a partitioning site ( parS ) and an origin of replication ( oriV ) present within the repABC cassette. [ 1 ]
The replication of the Ti plasmid is driven by the RepC initiator protein ( P05684 ), which possesses two protein domains : an N-terminal domain (NTD) that binds to DNA and a C-terminal domain (CTD). Mutational analyses have shown that without a functional RepC protein, the Ti plasmid is unable to replicate. [ 4 ] Meanwhile, the oriV sequence is around 150 nucleotides in length and is found within the repC gene. [ 3 ] Laboratory experiments have shown that the RepC protein binds to this region, suggesting its role as the origin of replication. [ 16 ] Therefore, while the complete process behind the replication of the Ti plasmid has not been fully described, the initial step of replication would likely depend on the expression of RepC and its binding to oriV . Of note, the RepC protein only acts in cis , where it only drives the replication of the plasmid it is encoded in and not any other plasmid also present in the bacterial cell. [ 16 ]
The partitioning system of the Ti plasmid is similar to the ParA/ParB system used in other plasmids and bacterial chromosomes and is thought to act in the same way. [ 17 ] Mutations in either of the proteins RepA or RepB have resulted in a decrease in plasmid stability, indicating their role and importance in plasmid partitioning. [ 4 ] The ability of RepA to form filaments allows it to create a physical bridge along which DNA can be pulled to opposite poles of a dividing cell. Meanwhile, the RepB protein can bind specifically to the parS sequence, forming a complex with DNA that can be recognized by RepA. [ 1 ] [ 4 ] This system is particularly important for the proper partitioning of the Ti plasmid, as the plasmid is only present in few copy numbers in the bacterial cell.
The Ti plasmid is maintained at low copy numbers within a bacterial cell. This is partly achieved by influencing the expression of the replication initiator RepC. [ 1 ] When bound to ADP , RepA is activated to work with RepB, acting as a negative regulator of the repABC cassette. [ 3 ] The levels of RepC is therefore kept low within a cell, preventing too many rounds of replication from occurring during each cell division cycle. Furthermore, there is a small RNA known as RepE encoded between repB and repC that lowers the expression of repC . [ 18 ] RepE is complementary to RepC and will bind with the repC mRNA to form a double-stranded molecule. This can then block the translational production of the RepC protein. [ 18 ]
Separately, the expression of the repABC cassette and hence the copy number of the Ti plasmid is also influenced via a quorum sensing system in Agrobacterium . [ 4 ] Quorum sensing systems respond to bacterial population densities by sensing a molecule, known as an autoinducer, that is produced by the bacterial cells at low levels and would build up to a threshold level when there is a high density of bacteria present. [ 18 ] In this case, the autoinducer is the N-3-oxooctanoyl-L-homoserine lactone (3-O-C 8 -AHL) molecule, which is sensed by a regulator known as TraR. [ 4 ] When activated, TraR will bind to regions known as tra boxes in the repABC gene cassette's promoter regions to drive expression. Therefore, a high level of population density increases the number of plasmids present within each bacterial cell, likely to support pathogenesis in the plant host. [ 4 ]
The expression of the vir region is usually repressed under normal conditions, and only becomes activated when the bacteria senses plant-derived signals from wound sites. This activation is necessary for the production of Vir proteins and the transfer of DNA and proteins into host plant cells. [ 1 ]
VirA and VirG form a two-component regulatory system within Agrobacterium . [ 19 ] This is a type of sensing and signalling system found commonly in bacteria; in this case, they act to sense plant-derived signals to drive the expression of the vir region. During the sensing, VirA, a histidine sensor kinase, will become phosphorylated before passing on this phosphate group to the response regulator VirG. [ 20 ] The activated response regulator VirG can then bind to a region of DNA known as the vir box, located upstream of each vir promoter, to activate the expression of the vir region. [ 1 ] [ 19 ] One possible downstream functions of the sensing mediated by VirA and VirG is the directional movement, or chemotaxis , of the bacteria towards plant-derived signals; this allows the Agrobacterium to move towards the wound site in plants. [ 21 ] Furthermore, with the induction of the vir region, the transfer of T-DNA can be mediated by the Vir proteins. [ 22 ]
The virB operon is the largest operon in the vir region, encoding for 11 VirB proteins involved in the transfer process of T-DNA and bacterial proteins into host plant cells (see transfer apparatus below). [ 23 ] [ 24 ]
The virC operon encodes for two proteins: VirC1 and VirC2. These proteins influence the pathogenesis of the Agrobacterium towards different plant hosts, and mutations can reduce but not remove the virulence of the bacteria. [ 25 ] Both the virC and virD operons can be repressed by a chromosomally encoded protein known as Ros. [ 26 ] [ 27 ] Ros binds to a region of DNA that overlaps with the binding site of the VirG regulator, and therefore competes with VirG to control their expression levels. [ 26 ] [ 27 ] Functionally, VirC1 and VirC2 promote the assembly of a relaxosome complex during the conjugative transfer of T-DNA from the bacteria to the host plant cell. [ 28 ] This is an energy-dependent process mediated via their NTPase activity, and occurs as they bind to a region of DNA known as overdrive . [ 28 ] As a result, they act to increase the amount of T-DNA strands produced. Following the production of the DNA strand to be transferred (transfer strand, T-strand), the VirC proteins can also help to direct the transfer strand to the transfer apparatus. [ 28 ]
The virD operon encodes for 4 proteins: VirD1-D4. [ 29 ] VirD1 and VirD2 are involved in the processing of T-DNA during conjugation to produce the T-strand; this is the single-stranded DNA molecule that is transported to the host plant cell (see transfer apparatus below). [ 30 ] During the processing, VirD1 will act as a topoisomerase to unwind the DNA strands. [ 30 ] VirD2, a relaxase , will then nick one of the DNA strands and remain bound to the DNA as it is transferred to the recipient cell. [ 31 ] [ 32 ] Within the recipient cell, VirD2 will also work together with VirE2 to direct the transferred DNA to the recipient cell's nucleus. There are suggestions that VirD2 may be phosphorylated and dephosphorylated by different proteins, affecting its ability to deliver DNA. [ 33 ] Conversely, little is known about VirD3, and mutational analyses have not provided any support for its role in the virulence of Agrobacterium . [ 34 ] Finally, VirD4 is a crucial part of the conjugation process, serving as a coupling factor that recognizes and transfers the T-strand to the transport channel. [ 35 ]
The virE operon encodes for 2 proteins: VirE1 and VirE2. [ 36 ] VirE2 is an effector protein translocated together with the T-strand into host plant cells. There, it binds to the T-strand to direct its delivery to the nucleus of the host plant cell. [ 37 ] [ 38 ] Part of this activity involves the presence of nuclear localization sequences within the protein, which marks the protein and the associated DNA for entry into the nucleus. It also protects the T-strand from nuclease attack. [ 39 ] There is some speculation regarding the role of VirE2 as a protein channel, allowing DNA to move through the plant cytoplasmic membrane . [ 40 ] On the other hand, VirE1 may be involved in promoting the transfer of the VirE2 protein into the host plant cell. [ 41 ] It binds to the ssDNA-binding domain of VirE2, therefore preventing the VirE2 protein from prematurely binding to the T-strand within the bacterial cell. [ 42 ]
virF is a host specificity factor found in some but not all types of Ti plasmids; for example, octopine-type Ti plasmids possess virF but nopaline-types do not. [ 43 ] [ 44 ] The ability of A. tumefaciens to induce crown gall tumours in certain plant species but not others has been attributed to the presence or absence of this virF gene. [ 43 ] [ 44 ]
The virH operon encodes for 2 proteins: VirH1 and VirH2. [ 45 ] A bioinformatics study of the amino acid sequences of the VirH protein showed similarities between them and a superfamily of proteins known as cytochrome P450 enzymes. [ 46 ] VirH2 was then discovered to metabolize certain phenolic compounds detected by VirA. [ 45 ]
The T-DNA of Agrobacterium is approximately 15-20 kbp in length and will become integrated into the host plant genome upon its transfer via a process known as recombination . This process utilizes preexisting gaps in the host plant cell's genome to allow the T-DNA to pair with short sequences in the genome, priming the process of DNA ligation , where the T-DNA is permanently joint to the plant genome. [ 37 ] The T-DNA region is flanked at both ends by 24bp sequences.
Within the host plant cell's genome, the T-DNA of Agrobacterium is expressed to produce two main groups of proteins. [ 1 ] One group is responsible for the production of plant growth hormones. As these hormones are produced, there will be an increase in the rate of cell division and therefore the formation of crown gall tumors. [ 47 ] The second group of proteins are responsible for driving the synthesis of opines in the host plant cells. The specific opines produced depends on the type of the Ti plasmid but not on the plant host. These opines cannot be utilized by the plant host, and will instead be exported out of the plant cell where it can be taken up by the Agrobacterium cells. The bacteria possess genes in other regions of the Ti plasmid that allows the catabolism of opines. [ 1 ]
Transfer apparatuses encoded within the Ti plasmid have to achieve two objectives: allow the conjugative transfer of the Ti plasmid between bacteria and allow the delivery of the T-DNA and certain effector proteins into host plant cells. These are achieved by the Tra/Trb system and the VirB/VirD4 system respectively, which are members of the type IV secretion system (T4SS). [ 47 ]
For the Ti plasmid and T-DNA to be transferred via conjugation, they must first be processed by different proteins, such as the relaxase enzyme (TraA/VirD2) and the DNA transfer and replication (Dtr) proteins. Together, these proteins will recognize and bind to a region known as the origin of transfer ( oriT ) in the Ti plasmid to form the relaxosome complex. For the T-DNA, a nick will be created at the T-DNA's border sequence, and the nicked T-strand will be transported to the cell membrane, where the rest of the transfer machinery is present. [ 31 ]
Within the VirB/VirD4 system, the VirD2 relaxase is aided by the accessory factors VirD1, VirC1 and VirC2 while it processes the DNA substrate. [ 48 ] Furthermore, the VirD2 relaxase and the VirC proteins will contribute to the delivery of the DNA strand to the VirD4 receptor at the cell membrane. [ 28 ] This receptor is an essential component of T4SSs, and is thought to energize and mediate the transfer of the DNA into the translocation channel between two cells. [ 49 ] The table below summarizes the proteins encoded in the virB operon that makes up the translocation channel of the VirB/VirD4 system. [ 1 ]
The ability of Agrobacterium to deliver DNA into plant cells opened new doors for plant genome engineering , allowing the production of genetically modified plants (transgenic plants). [ 57 ] Proteins involved in mediating the transfer of T-DNA will first recognize the border sequences of the T-DNA region. Therefore, it is possible for scientists to use T-DNA border sequences to flank any desired sequence of interest - such a product can then be inserted into a plasmid and introduced into Agrobacterium cells. [ 58 ] There, the border sequences will be recognized by the transfer apparatus of A. tumefaciens and delivered in a standard manner into the target plant cell. [ 1 ] Moreover, by leaving behind only the border sequences of the T-DNA, the resulting product will edit the plant genome without causing any tumours in plants. [ 59 ] This method has been used to modify several crop plants, including rice, [ 60 ] barley [ 61 ] and wheat. [ 62 ] Further work have since extended the targets of A. tumefaciens to include fungi and human cell lines. [ 63 ] [ 64 ] | https://en.wikipedia.org/wiki/Ti_plasmid |
The TianQin Project ( Chinese : 天琴计划 ) is a proposed space-borne gravitational-wave observatory (gravitational-wave detector) consisting of three spacecraft in Earth orbit. The TianQin project is being led by Professor Luo Jun ( Chinese : 罗俊 ), President of Sun Yat-sen University , and is based in the university's Zhuhai campus. Construction on project-related infrastructure, which will include a research building, ultra-quiet cave laboratory, and observation center, began in March 2016. The project is estimated to cost 15 billion RMB (US$2.3 billion), [ 1 ] [ 2 ] [ 3 ] [ 4 ] with a projected completion date in the mid-2030s. [ 5 ] [ 6 ] In December 2019, China launched Tianqin-1 , a technology demonstration . [ 7 ]
The project's name combines the Chinese words " Tianqin " ( Chinese : 天琴 ; pinyin : Tiān qín ) - the plucked string musical instrument of Zhuang people in China . This name refers to the metaphorical concept of gravitational waves "plucking the strings" by causing fluctuations in the 100,000 kilometer laser beams stretching between each of the three TianQin spacecraft.
The observatory will consist of three identical drag-free controlled spacecraft in high Earth orbits at an altitude of about 100,000 km. The nominal source of the observatory is a white-dwarf binary RX J0806.3+1527 (also known as HM Cancri ). [ 8 ] This could serve as a good calibration source for the TianQin gravitational wave observatory.
Similar configuration of geocentric orbit space-borne gravitational wave detectors have been developed since 2011, [ 9 ] [ 10 ] and was shown to have favorable properties for observing intermediate-mass and massive black-hole binaries. [ 10 ]
Apart from Galactic binaries, the TianQin observatory can also detect sources like massive black hole binaries, extreme mass ratio inspirals , stellar-mass black hole binary inspirals , and stochastic gravitational wave background , etc. [ 11 ]
The detection rate for massive black hole binaries is expected to be as high as about 60 per year, [ 12 ] and TianQin would have accurate estimate to the source's parameters, [ 13 ] which enable the potential for distinguishing the seed models for massive black holes, as well as issuing early warning for nearby mergers. [ 12 ] It can also be used to test the no-hair theorem [ 14 ] or constrain modified gravity. [ 15 ] | https://en.wikipedia.org/wiki/TianQin |
Tian yuan shu ( simplified Chinese : 天元术 ; traditional Chinese : 天元術 ; pinyin : tiān yuán shù ) is a Chinese system of algebra for polynomial equations. Some of the earliest existing writings were created in the 13th century during the Yuan dynasty . However, the tianyuanshu method was known much earlier, in the Song dynasty and possibly before.
The Tianyuanshu was explained in the writings of Zhu Shijie ( Jade Mirror of the Four Unknowns ) and Li Zhi ( Ceyuan haijing ), two Chinese mathematicians during the Mongol Yuan dynasty . [ 1 ]
However, after the Ming overthrew the Mongol Yuan, Zhu and Li's mathematical works went into disuse as the Ming literati became suspicious of knowledge imported from Mongol Yuan times.
Only recently, with the advent of modern mathematics in China, has the tianyuanshu been re-deciphered.
Meanwhile, tian yuan shu arrived in Japan, where it is called tengen-jutsu . Zhu's text Suanxue qimeng was deciphered and was important in the development of Japanese mathematics ( wasan ) in the 17th and 18th centuries.
Tian yuan shu means "method of the heavenly element" or "technique of the celestial unknown". The "heavenly element" is the unknown variable , usually written x in modern notation.
It is a positional system of rod numerals to represent polynomial equations . For example, 2 x 2 + 18 x − 316 = 0 is represented as
, which in Arabic numerals is
The 元 ( yuan ) denotes the unknown x , so the numerals on that line mean 18 x . The line below is the constant term ( -316 ) and the line above is the coefficient of the quadratic ( x 2 ) term. The system accommodates arbitrarily high exponents of the unknown by adding more lines on top and negative exponents by adding lines below the constant term. Decimals can also be represented.
In later writings of Li Zhi and Zhu Shijie, the line order was reversed so that the first line is the lowest exponent.
This article about the history of mathematics is a stub . You can help Wikipedia by expanding it .
This polynomial -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tian_yuan_shu |
Tianchi basins were meteorological measuring instruments used to gather and measure the amount of liquid precipitation over a period of time during the Song Dynasty. The instrument was devised by the Song Chinese mathematician and inventor Qin Jiushao in 1247. [ 1 ] [ 2 ]
As precipitation was for important agriculture and food production, the Song Chinese mathematician and inventor Qin Jiushao developed a precipitation gauge that was widely used in 1247 during the Southern Song dynasty to gather meteorological data. Qin Jiushao later records application of rainfall measurements in the mathematical treatise Mathematical Treatise in Nine Sections . The book also discusses problems using large snow gauges made from bamboo situated in mountain passes and uplands which are speculated to be first referenced to snow measurement. [ 3 ] [ 4 ]
Tianchi basins were installed at provincial and district capitals and bamboo snow gauges were situated in mountain passes. The rain gauges were bowl-shaped with one being installed at each provincial and district capital in China. In the treatise, Qin Jiushao also discusses how point measurements were converted to real averages. These averages were important as they postulated indicators of natural disasters such as flooding, since river flooding has always been a problem in China. [ 5 ] | https://en.wikipedia.org/wiki/Tianchi_basin |
The Tianjin animal cloning center was planned in 2015 and "to be put into use in the first half of 2016" [ 1 ] in the Tianjin Economic-Technological Development Area of Tianjin , China, [ 2 ] but as of 2022, no opening has been reported.
The factory was announced to be developed by Sinica, a subsidiary of the Chinese company Boyalife, along with the Institute of Molecular Medicine at Peking University , the Tianjin International Joint Academy of Biomedicine, and the Sooam Bioengineering Research Institute in South Korea. [ 1 ]
The 14,000-square-metre facility would have hosted a laboratory, a cloning center, a gene bank, and educational exhibits for the public. [ 2 ] The consortium planned to spend 200 million RMB (US$31 million) to produce 100,000 cloned cattle per year for China's rapidly growing beef market, and then expand to one million cattle per year [ 3 ] [ 1 ] (China planned to buy one million head of cattle from Australia in 2016 at a cost of US$2 billion [ 4 ] ). In addition to cows, the factory had planned to clone many different types of animals, including dogs, horses, and endangered and extinct animals. [ 5 ] [ 6 ] | https://en.wikipedia.org/wiki/Tianjin_animal_cloning_center |
The Tantra of Kalachakra is the basis of Tibetan astronomy . It explains some phenomena in a similar manner as modern astronomy science. Hence, Sun eclipse is described as the Moon passing between the Sun and the Earth. [ 1 ] [ dubious – discuss ] [ better source needed ]
In 1318, the 3rd Karmapa received vision of Kalachakra which he used to introduce a revised system of astronomy and astrology named the "Tsurphu Tradition of Astrology" ( Standard Tibetan : Tsur-tsi ) which is still used in the Karma Kagyu school for the calculation of the Tibetan calendar . [ 2 ] [ 3 ]
This astronomy -related article is a stub . You can help Wikipedia by expanding it .
This Tibet -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tibetan_astronomy |
Tibor Szele (21 June 1918 – 5 April 1955) Hungarian mathematician , working in combinatorics and abstract algebra .
Szele was born in Debrecen . After graduating at the Debrecen University , he became a researcher at the Szeged University in 1946, then he went back at the Debrecen University in 1948 where he became full professor in 1952. He worked especially in the theory of Abelian groups and ring theory . He generalized Hajós's theorem . He founded the Hungarian school of algebra. Tibor Szele received the Kossuth Prize in 1952. He died in Szeged .
A panorama of Hungarian Mathematics in the Twentieth Century , p. 601. | https://en.wikipedia.org/wiki/Tibor_Szele |
Tibovirus is a term often used to describe viruses that are transmitted by tick vectors. The word tibovirus is an acronym (TIck-BOrne virus). [ 1 ] This falls within the superorder arthropod thus tibovirus is classified under Arthropod Borne virus (Arborvirus). For a person to acquire infection the tick must bite and feed for a sufficient period of time. The tiboviruses that affect humans are limited to within 3 families: Flaviviridae, Reoviridae, and Bunyaviridae. [ 2 ] [ 3 ] | https://en.wikipedia.org/wiki/Tibovirus |
Titanium tetrachloride is the inorganic compound with the formula TiCl 4 . It is an important intermediate in the production of titanium metal and the pigment titanium dioxide . TiCl 4 is a volatile liquid. Upon contact with humid air, it forms thick clouds of titanium dioxide ( TiO 2 ) and hydrochloric acid , a reaction that was formerly exploited for use in smoke machines. It is sometimes referred to as "tickle" or "tickle 4", as a phonetic representation of the symbols of its molecular formula ( TiCl 4 ). [ 7 ] [ 8 ]
TiCl 4 is a dense, colourless liquid, although crude samples may be yellow or even red-brown. It is one of the rare transition metal halides that is a liquid at room temperature, VCl 4 being another example. This property reflects the fact that molecules of TiCl 4 weakly self-associate. Most metal chlorides are polymers , wherein the chloride atoms bridge between the metals. Its melting point is similar to that of CCl 4 . [ 9 ] [ 10 ]
Ti 4+ has a "closed" electronic shell, with the same number of electrons as the noble gas argon . The tetrahedral structure for TiCl 4 is consistent with its description as a d 0 metal center ( Ti 4+ ) surrounded by four identical ligands. This configuration leads to highly symmetrical structures, hence the tetrahedral shape of the molecule. TiCl 4 adopts similar structures to TiBr 4 and TiI 4 ; the three compounds share many similarities. TiCl 4 and TiBr 4 react to give mixed halides TiCl 4− x Br x , where x = 0, 1, 2, 3, 4. Magnetic resonance measurements also indicate that halide exchange is also rapid between TiCl 4 and VCl 4 . [ 11 ]
TiCl 4 is soluble in toluene and chlorocarbons . Certain arenes form complexes of the type [(C 6 R 6 )TiCl 3 ] + . [ 12 ] TiCl 4 reacts exothermically with donor solvents such as THF to give hexacoordinated adducts . [ 13 ] Bulkier ligands (L) give pentacoordinated adducts TiCl 4 L .
TiCl 4 is produced by the chloride process , which involves the reduction of titanium oxide ores, typically ilmenite ( FeTiO 3 ), with carbon under flowing chlorine at 900 °C. Impurities are removed by distillation . [ 10 ]
The coproduction of FeCl 3 is undesirable, which has motivated the development of alternative technologies. Instead of directly using ilmenite, "rutile slag" is used. This material, an impure form of TiO 2 , is derived from ilmenite by removal of iron, either using carbon reduction or extraction with sulfuric acid . Crude TiCl 4 contains a variety of other volatile halides, including vanadyl chloride ( VOCl 3 ), silicon tetrachloride ( SiCl 4 ), and tin tetrachloride ( SnCl 4 ), which must be separated. [ 10 ]
The world's supply of titanium metal, about 250,000 tons per year, is made from TiCl 4 . The conversion involves the reduction of the tetrachloride with magnesium metal. This procedure is known as the Kroll process : [ 14 ]
In the Hunter process , liquid sodium is the reducing agent instead of magnesium. [ 15 ]
Around 90% of the TiCl 4 production is used to make the pigment titanium dioxide ( TiO 2 ). The conversion involves hydrolysis of TiCl 4 , a process that forms hydrogen chloride : [ 14 ]
In some cases, TiCl 4 is oxidised directly with oxygen :
It has been used to produce smoke screens since it produces a heavy, white smoke that has little tendency to rise. "Tickle" was the standard means of producing on-set smoke effects for motion pictures, before being phased out in the 1980s due to concerns about hydrated HCl 's effects on the respiratory system. [ citation needed ]
Titanium tetrachloride is a versatile reagent that forms diverse derivatives including those illustrated below. [ 16 ]
A characteristic reaction of TiCl 4 is its easy hydrolysis , signaled by the release of HCl vapors and titanium oxides and oxychlorides . Titanium tetrachloride has been used to create naval smokescreens , as the hydrochloric acid aerosol and titanium dioxide that is formed scatter light very efficiently. This smoke is corrosive, however. [ 10 ]
Alcohols react with TiCl 4 to give alkoxides with the formula [Ti(OR) 4 ] n (R = alkyl , n = 1, 2, 4). As indicated by their formula, these alkoxides can adopt complex structures ranging from monomers to tetramers. Such compounds are useful in materials science as well as organic synthesis . A well known derivative is titanium isopropoxide , which is a monomer. Titanium bis(acetylacetonate)dichloride results from treatment of titanium tetrachloride with excess acetylacetone : [ 17 ]
Organic amines react with TiCl 4 to give complexes containing amido ( R 2 N − -containing) and imido ( RN 2− -containing) complexes. With ammonia, titanium nitride is formed. An illustrative reaction is the synthesis of tetrakis(dimethylamido)titanium Ti(N(CH 3 ) 2 ) 4 , a yellow, benzene-soluble liquid: [ 18 ] This molecule is tetrahedral, with planar nitrogen centers. [ 19 ]
TiCl 4 is a Lewis acid as implicated by its tendency to hydrolyze . With the ether THF , TiCl 4 reacts to give yellow crystals of TiCl 4 (THF) 2 . With chloride salts, TiCl 4 reacts to form sequentially [Ti 2 Cl 9 ] − , [Ti 2 Cl 10 ] 2− (see figure above), and [TiCl 6 ] 2− . [ 20 ] The reaction of chloride ions with TiCl 4 depends on the counterion. [N(CH 2 CH 2 CH 2 CH 3 ) 4 ]Cl and TiCl 4 gives the pentacoordinate complex [N(CH 2 CH 2 CH 2 CH 3 ) 4 ][TiCl 5 ] , whereas smaller [N(CH 2 CH 3 ) 4 ] + gives [N(CH 2 CH 3 ) 4 ] 2 [Ti 2 Cl 10 ] . These reactions highlight the influence of electrostatics on the structures of compounds with highly ionic bonding.
Reduction of TiCl 4 with aluminium results in one-electron reduction. The trichloride ( TiCl 3 ) and tetrachloride have contrasting properties: the trichloride is a colored solid, being a coordination polymer , and is paramagnetic . When the reduction is conducted in THF solution, the Ti(III) product converts to the light-blue adduct TiCl 3 (THF) 3 .
The organometallic chemistry of titanium typically starts from TiCl 4 . An important reaction involves sodium cyclopentadienyl to give titanocene dichloride , TiCl 2 (C 5 H 5 ) 2 . This compound and many of its derivatives are precursors to Ziegler–Natta catalysts . Tebbe's reagent , useful in organic chemistry, is an aluminium-containing derivative of titanocene that arises from the reaction of titanocene dichloride with trimethylaluminium . It is used for the "olefination" reactions. [ 16 ]
Arenes , such as C 6 (CH 3 ) 6 react to give the piano-stool complexes [Ti(C 6 R 6 )Cl 3 ] + (R = H, CH 3 ; see figure above). This reaction illustrates the high Lewis acidity of the TiCl + 3 entity, which is generated by abstraction of chloride from TiCl 4 by AlCl 3 . [ 12 ]
TiCl 4 finds occasional use in organic synthesis , capitalizing on its Lewis acidity , its oxophilicity , and the electron-transfer properties of its reduced titanium halides. It is used in the Lewis acid catalysed aldol addition [ 21 ] Key to this application is the tendency of TiCl 4 to activate aldehydes (RCHO) by formation of adducts such as (RCHO)TiCl 4 OC(H)R . [ 22 ]
Hazards posed by titanium tetrachloride generally arise from its reaction with water that releases hydrochloric acid , which is severely corrosive itself and whose vapors are also extremely irritating. TiCl 4 is a strong Lewis acid , which exothermically forms adducts with even weak bases such as THF and water. | https://en.wikipedia.org/wiki/Ticl4 |
A tidal bore , [ 1 ] often simply given as bore in context, is a tidal phenomenon in which the leading edge of the incoming tide forms a wave (or waves) of water that travels up a river or narrow bay, reversing the direction of the river or bay's current. It is a strong tide that pushes up the river, against the current.
Bores occur in relatively few locations worldwide, usually in areas with a large tidal range (typically more than 6 meters (20 ft) between high and low tide) and where incoming tides are funneled into a shallow, narrowing river or lake via a broad bay. [ 2 ] The funnel-like shape not only increases the tidal range, but it can also decrease the duration of the flood tide , down to a point where the flood appears as a sudden increase in the water level. A tidal bore takes place during the flood tide and never during the ebb tide .
A tidal bore may take on various forms, ranging from a single breaking wavefront with a roller — somewhat like a hydraulic jump [ 4 ] [ 5 ] — to undular bores , comprising a smooth wavefront followed by a train of secondary waves known as whelps . [ 6 ] Large bores can be particularly unsafe for shipping but also present opportunities for river surfing . [ 6 ]
Two key features of a tidal bore are the intense turbulence and turbulent mixing generated during the bore propagation, as well as its rumbling noise. The visual observations of tidal bores highlight the turbulent nature of the surging waters. The tidal bore induces a strong turbulent mixing in the estuarine zone, and the effects may be felt along considerable distances. The velocity observations indicate a rapid deceleration of the flow associated with the passage of the bore as well as large velocity fluctuations. [ 7 ] [ 8 ] A tidal bore creates a powerful roar that combines the sounds caused by the turbulence in the bore front and whelps, entrained air bubbles in the bore roller, sediment erosion beneath the bore front and of the banks, scouring of shoals and bars, and impacts on obstacles. The bore rumble is heard far away because its low frequencies can travel over long distances. The low-frequency sound is a characteristic feature of the advancing roller in which the air bubbles entrapped in the large-scale eddies are acoustically active and play the dominant role in the rumble-sound generation. [ 9 ]
The word bore derives through Old English from the Old Norse word bára , meaning "wave" or "swell."
Tidal bores can be dangerous. Certain rivers such as the Seine in France , the Petitcodiac River in Canada , and the Colorado River in Mexico to name a few, have had a sinister reputation in association with tidal bores. In China, despite warning signs erected along the banks of the Qiantang River , a number of fatalities occur each year by people who take too much risk with the bore. [ 2 ] The tidal bores affect the shipping and navigation in the estuarine zone, for example, in Papua New Guinea (in the Fly and Bamu Rivers ), Malaysia (the Benak in the Batang Lupar ), and India (the Hooghly River bore).
On the other hand, tidal bore-affected estuaries are rich feeding zones and breeding grounds of several forms of wildlife. [ 2 ] The estuarine zones are the spawning and breeding grounds of several native fish species, while the aeration induced by the tidal bore contributes to the abundant growth of many species of fish and shrimp (for example in the Rokan River , Indonesia ). The tidal bores also provide opportunity for recreational inland surfing , such as the Seven Ghosts bore on the Kampar River , Indonesia or the Severn Bore on the River Severn , England .
Scientific studies have been carried out at the River Dee [ 10 ] in Wales in the United Kingdom, the Garonne [ 11 ] [ 12 ] [ 13 ] [ 14 ] [ 15 ] and Sélune [ 16 ] in France, the Daly River [ 17 ] in Australia, and the Qiantang River estuary [ 18 ] in China. The force of the tidal bore flow often poses a challenge to scientific measurements, as evidenced by a number of field work incidents in the River Dee, [ 10 ] Rio Mearim, Daly River, [ 17 ] and Sélune River. [ 16 ]
Rivers and bays that have been known to exhibit bores include those listed below. [ 2 ] [ 19 ]
The phenomenon is generally named un mascaret in French. [ 23 ] but some other local names are preferred. [ 19 ]
With the Bay of Fundy having the highest tidal range in the world, most rivers draining into the upper bay between Nova Scotia and New Brunswick have significant tidal bores. They include:
Historically, there was a tidal bore on the Gulf of California in Mexico at the mouth of the Colorado River . It formed in the estuary about Montague Island and propagated upstream. It was once very strong, but diversions of the river for irrigation have weakened the flow of the river to the point the tidal bore has nearly disappeared.
Lakes with an ocean inlet can also exhibit tidal bores. [ citation needed ] | https://en.wikipedia.org/wiki/Tidal_bore |
A tidal disruption event ( TDE ) is a transient astronomical source produced when a star passes so close to a supermassive black hole (SMBH) that it is pulled apart by the black hole's tidal force . [ 2 ] [ 3 ] The star undergoes spaghettification , producing a tidal stream of material that loops around the black hole. Some portion of the stellar material is captured into orbit, forming an accretion disk around the black hole, which emits electromagnetic radiation . In a small fraction of TDEs, a relativistic jet is also produced. As the material in the disk is gradually consumed by the black hole, the TDE fades over several months or years.
TDEs were predicted in the 1970s and first observed in the 1990s. Over a hundred have since been observed, with detections at optical, infrared, radio and X-ray wavelengths. Sometimes a star can survive the encounter with an SMBH, leaving a remnant; those events are termed partial TDEs. [ 4 ] [ 5 ]
TDEs were first theorized by Jack G. Hills in 1975. [ 6 ] A consequence of a star getting sufficiently close to a SMBH that the tidal forces between the star will overcome the star's self-gravity . In 1988 Martin Rees described how approximately half of the disrupted stellar material will remain bound, eventually accreting onto the black hole and forming a luminous accretion disk. [ 7 ]
According to early [ when? ] studies, tidal disruption events are an inevitable consequence of massive black holes' activity hidden in galaxy nuclei. Later theorists concluded that the resulting explosion or flare of radiation from the accretion of the stellar debris could reveal the presence of a dormant black hole in the center of a normal galaxy. [ 8 ]
TDEs were first observed in the early 1990s using the X-ray ROSAT All-Sky Survey. [ 9 ]
As of May 2024 [update] , roughly 100 TDEs are known, [ 10 ] [ 11 ] [ 12 ] and have been discovered through several astronomical methods. such as optical transient surveys including Zwicky Transient Facility (ZTF) [ 12 ] and the All Sky Automated Survey for SuperNovae (ASAS-SN). [ 13 ] Other TDEs have been discovered in X-rays, using the ROSAT , XMM-Newton , and eROSITA . [ 14 ] TDEs have also been discovered in the ultraviolet . [ 15 ]
The light curves of TDEs have an initially sharp rise in brightness, as the disrupted stellar material falls towards the black hole, followed by a more gradual decline lasting months or years. During the declining phase, the luminosity is proportional to t − 5 / 3 {\displaystyle t^{-5/3}} , where t is time, [ 16 ] although some TDEs have been observed to deviate from the typical t − 5 / 3 {\displaystyle t^{-5/3}} rate. [ 17 ] These properties allow TDEs to be distinguished from other transient astronomical sources , such as supernovae . The peak luminosity of TDEs is proportional to the central black hole mass; it can approach or exceed that of their host galaxies, making them some of the brightest sources observed in the Universe. [ 18 ]
There are two broad classes of TDEs. The majority of TDEs consist of "non-relativistic" events, where the outflows from the TDE are akin to the energetics seen in Type Ib and Ic supernovae . [ 19 ]
Approximately 1% of TDEs, however, are relativistic TDEs, where an astrophysical jet is launched from the black hole shortly after the star is destroyed. This jet persists for several years before shutting off. [ 20 ] As of 2023 [update] only four TDEs with jets have been observed. [ 21 ]
A star gets tidally disrupted when the tidal force exerted by a black hole f t i d a l ≈ G M B H R ∗ R 3 {\displaystyle f_{tidal}\approx {\frac {GM_{BH}R^{*}}{R^{3}}}} exceeds the self-gravity of the star f s g ≈ G M ∗ R ∗ 2 {\displaystyle f_{sg}\approx {\frac {GM^{*}}{R^{*2}}}} .
The distance below which f t i d a l > f s g {\displaystyle f_{tidal}>f_{sg}} is called the tidal radius and is given approximately by: [ 22 ] [ 23 ]
This is identical to the Roche limit for disruptions of planetary bodies.
Usually, the tidal-disruption radius of a black hole is bigger than its Schwarzschild radius , R S = 2 G M c 2 {\displaystyle R_{S}={\frac {2GM}{c^{2}}}} , but considering the radius and mass of the star fixed there is a mass for the black hole where both radii become equal meaning that at this point the star would simply disappear before being torn apart. [ 24 ] [ 7 ] | https://en.wikipedia.org/wiki/Tidal_disruption_event |
Tidal downsizing is a hypothetical mechanism for the formation of planets . [ 1 ] [ 2 ] The process begins with the formation of large clumps of gas, of roughly 10 Jupiter masses, via gravitational instability in the outer parts of the protoplanetary disk . The clumps migrate inward due to gravitational interactions with the gas disk. Solid grains within the clump collide and grow and settle toward the center forming a massive core . The clump is disrupted due to tidal forces or heating from the star when it approaches within a few AU of the star leaving behind a smaller object. Depending on the extent and timing of the mass loss the remnant may be a terrestrial planet , an ice giant or a gas giant . [ 3 ] | https://en.wikipedia.org/wiki/Tidal_downsizing |
The tidal force or tide-generating force is the difference in gravitational attraction between different points in a gravitational field , causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is the differential force of gravity, the net between gravitational forces , the derivative of gravitational potential , the gradient of gravitational fields. Therefore tidal forces are a residual force , a secondary effect of gravity, highlighting its spatial elements, making the closer near-side more attracted than the more distant far-side.
This produces a range of tidal phenomena , such as ocean tides. Earth's tides are mainly produced by the relative close gravitational field of the Moon
and to a lesser extend by the stronger, but further away gravitational field of the Sun. The ocean on the side of Earth facing the Moon is being pulled by the gravity of the Moon away from Earth's crust , while on the other side of Earth there the crust is being pulled away from the ocean, resulting in Earth being stretched, bulging on both sides, and having opposite high-tides . Tidal forces viewed from Earth, that is from a rotating reference frame , appear as centripedal and centrifugal forces , but are not caused by the rotation. [ 2 ]
Further tidal phenomena include solid-earth tides , tidal locking , breaking apart of celestial bodies and formation of ring systems within the Roche limit , and in extreme cases, spaghettification of objects. Tidal forces have also been shown to be fundamentally related to gravitational waves . [ 3 ]
In celestial mechanics , the expression tidal force can refer to a situation in which a body or material (for example, tidal water) is mainly under the gravitational influence of a second body (for example, the Earth), but is also perturbed by the gravitational effects of a third body (for example, the Moon). The perturbing force is sometimes in such cases called a tidal force [ 4 ] (for example, the perturbing force on the Moon ): it is the difference between the force exerted by the third body on the second and the force exerted by the third body on the first. [ 5 ]
When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. Figure 2 shows the differential force of gravity on a spherical body (body 1) exerted by another body (body 2).
These tidal forces cause strains on both bodies and may distort them or even, in extreme cases, break one or the other apart. [ 6 ] The Roche limit is the distance from a planet at which tidal effects would cause an object to disintegrate because the differential force of gravity from the planet overcomes the attraction of the parts of the object for one another. [ 7 ] These strains would not occur if the gravitational field were uniform, because a uniform field only causes the entire body to accelerate together in the same direction and at the same rate.
The relationship of an astronomical body's size, to its distance from another body, strongly influences the magnitude of tidal force. [ 8 ] The tidal force acting on an astronomical body, such as the Earth, is directly proportional to the diameter of the Earth and inversely proportional to the cube of the distance from another body producing a gravitational attraction, such as the Moon or the Sun. Tidal action on bath tubs, swimming pools, lakes, and other small bodies of water is negligible. [ 9 ]
Figure 3 is a graph showing how gravitational force declines with distance. In this graph, the attractive force decreases in proportion to the square of the distance ( Y = 1/ X 2 ), while the slope ( Y ′ = −2/ X 3 ) is inversely proportional to the cube of the distance.
The tidal force corresponds to the difference in Y between two points on the graph, with one point on the near side of the body, and the other point on the far side. The tidal force becomes larger, when the two points are either farther apart, or when they are more to the left on the graph, meaning closer to the attracting body.
For example, even though the Sun has a stronger overall gravitational pull on Earth, the Moon creates a larger tidal bulge because the Moon is closer. This difference is due to the way gravity weakens with distance: the Moon's closer proximity creates a steeper decline in its gravitational pull as you move across Earth (compared to the Sun's very gradual decline from its vast distance). This steeper gradient in the Moon's pull results in a larger difference in force between the near and far sides of Earth, which is what creates the bigger tidal bulge.
Gravitational attraction is inversely proportional to the square of the distance from the source. The attraction will be stronger on the side of a body facing the source, and weaker on the side away from the source. The tidal force is proportional to the difference. [ 9 ]
The Earth is 81 times more massive than the Moon, the Earth has roughly 4 times the Moon's radius. As a result, at the same distance, the tidal force of the Earth at the surface of the Moon is about 20 times stronger than that of the Moon at the Earth's surface. [ 10 ]
In the case of an infinitesimally small elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an ellipsoid with two bulges, pointing towards and away from the other body. Larger objects distort into an ovoid , and are slightly compressed, which is what happens to the Earth's oceans under the action of the Moon. All parts of the Earth are subject to the Moon's gravitational forces, causing the water in the oceans to redistribute, forming bulges on the sides near the Moon and far from the Moon. [ 13 ]
When a body rotates while subject to tidal forces, internal friction results in the gradual dissipation of its rotational kinetic energy as heat. In the case for the Earth, and Earth's Moon, the loss of rotational kinetic energy results in a gain of about 2 milliseconds per century. If the body is close enough to its primary, this can result in a rotation which is tidally locked to the orbital motion, as in the case of the Earth's moon. Tidal heating produces dramatic volcanic effects on Jupiter's moon Io . Stresses caused by tidal forces also cause a regular monthly pattern of moonquakes on Earth's Moon. [ 8 ]
Tidal forces contribute to ocean currents, which moderate global temperatures by transporting heat energy toward the poles. It has been suggested that variations in tidal forces correlate with cool periods in the global temperature record at 6- to 10-year intervals, [ 14 ] and that harmonic beat variations in tidal forcing may contribute to millennial climate changes. No strong link to millennial climate changes has been found to date. [ 15 ]
Tidal effects become particularly pronounced near small bodies of high mass, such as neutron stars or black holes , where they are responsible for the " spaghettification " of infalling matter. Tidal forces create the oceanic tide of Earth 's oceans, where the attracting bodies are the Moon and, to a lesser extent, the Sun . Tidal forces are also responsible for tidal locking , tidal acceleration , and tidal heating. Tides may also induce seismicity .
By generating conducting fluids within the interior of the Earth, tidal forces also affect the Earth's magnetic field . [ 16 ]
For a given (externally generated) gravitational field, the tidal acceleration at a point with respect to a body is obtained by vector subtraction of the gravitational acceleration at the center of the body (due to the given externally generated field) from the gravitational acceleration (due to the same field) at the given point. Correspondingly, the term tidal force is used to describe the forces due to tidal acceleration. Note that for these purposes the only gravitational field considered is the external one; the gravitational field of the body (as shown in the graphic) is not relevant. (In other words, the comparison is with the conditions at the given point as they would be if there were no externally generated field acting unequally at the given point and at the center of the reference body. The externally generated field is usually that produced by a perturbing third body, often the Sun or the Moon in the frequent example-cases of points on or above the Earth's surface in a geocentric reference frame.)
Tidal acceleration does not require rotation or orbiting bodies; for example, the body may be freefalling in a straight line under the influence of a gravitational field while still being influenced by (changing) tidal acceleration.
By Newton's law of universal gravitation and laws of motion, a body of mass m at distance R from the center of a sphere of mass M feels a force F → g {\textstyle {\vec {F}}_{g}} ,
F → g = − r ^ G M m R 2 {\displaystyle {\vec {F}}_{g}=-{\hat {r}}~G~{\frac {Mm}{R^{2}}}}
equivalent to an acceleration a → g {\textstyle {\vec {a}}_{g}} ,
a → g = − r ^ G M R 2 {\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}}
where r ^ {\textstyle {\hat {r}}} is a unit vector pointing from the body M to the body m (here, acceleration from m towards M has negative sign).
Consider now the acceleration due to the sphere of mass M experienced by a particle in the vicinity of the body of mass m . With R as the distance from the center of M to the center of m , let ∆ r be the (relatively small) distance of the particle from the center of the body of mass m . For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of mass M . If the body of mass m is itself a sphere of radius ∆ r , then the new particle considered may be located on its surface, at a distance ( R ± ∆r ) from the centre of the sphere of mass M , and ∆r may be taken as positive where the particle's distance from M is greater than R . Leaving aside whatever gravitational acceleration may be experienced by the particle towards m on account of m ' s own mass, we have the acceleration on the particle due to gravitational force towards M as:
a → g = − r ^ G M ( R ± Δ r ) 2 {\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{(R\pm \Delta r)^{2}}}}
Pulling out the R 2 term from the denominator gives:
a → g = − r ^ G M R 2 1 ( 1 ± Δ r R ) 2 {\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}~{\frac {1}{\left(1\pm {\frac {\Delta r}{R}}\right)^{2}}}}
The Maclaurin series of 1 / ( 1 ± x ) 2 {\textstyle 1/(1\pm x)^{2}} is 1 ∓ 2 x + 3 x 2 ∓ ⋯ {\textstyle 1\mp 2x+3x^{2}\mp \cdots } which gives a series expansion of:
a → g = − r ^ G M R 2 ± r ^ G 2 M R 2 Δ r R + ⋯ {\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}\pm {\hat {r}}~G~{\frac {2M}{R^{2}}}~{\frac {\Delta r}{R}}+\cdots }
The first term is the gravitational acceleration due to M at the center of the reference body m {\textstyle m} , i.e., at the point where Δ r {\textstyle \Delta r} is zero. This term does not affect the observed acceleration of particles on the surface of m because with respect to M , m (and everything on its surface) is in free fall. When the force on the far particle is subtracted from the force on the near particle, this first term cancels, as do all other even-order terms. The remaining (residual) terms represent the difference mentioned above and are tidal force (acceleration) terms. When ∆ r is small compared to R , the terms after the first residual term are very small and can be neglected, giving the approximate tidal acceleration a → t , axial {\textstyle {\vec {a}}_{t,{\text{axial}}}} for the distances ∆ r considered, along the axis joining the centers of m and M :
a → t , axial ≈ ± r ^ 2 Δ r G M R 3 {\displaystyle {\vec {a}}_{t,{\text{axial}}}\approx \pm {\hat {r}}~2\Delta r~G~{\frac {M}{R^{3}}}}
When calculated in this way for the case where ∆ r is a distance along the axis joining the centers of m and M , a → t {\textstyle {\vec {a}}_{t}} is directed outwards from to the center of m (where ∆ r is zero).
Tidal accelerations can also be calculated away from the axis connecting the bodies m and M , requiring a vector calculation. In the plane perpendicular to that axis, the tidal acceleration is directed inwards (towards the center where ∆ r is zero), and its magnitude is 1 2 | a → t , axial | {\textstyle {\frac {1}{2}}\left|{\vec {a}}_{t,{\text{axial}}}\right|} in linear approximation as in Figure 2.
The tidal accelerations at the surfaces of planets in the Solar System are generally very small. For example, the lunar tidal acceleration at the Earth's surface along the Moon–Earth axis is about 1.1 × 10 −7 g , while the solar tidal acceleration at the Earth's surface along the Sun–Earth axis is about 0.52 × 10 −7 g , where g is the gravitational acceleration at the Earth's surface. Hence the tide-raising force (acceleration) due to the Sun is about 45% of that due to the Moon. [ 18 ] The solar tidal acceleration at the Earth's surface was first given by Newton in the Principia . [ 19 ] | https://en.wikipedia.org/wiki/Tidal_force |
Tidal heating (also known as tidal working or tidal flexing ) occurs through the tidal friction processes: orbital and rotational energy is dissipated as heat in either (or both) the surface ocean or interior of a planet or satellite. When an object is in an elliptical orbit , the tidal forces acting on it are stronger near periapsis than near apoapsis. Thus the deformation of the body due to tidal forces (i.e. the tidal bulge) varies over the course of its orbit, generating internal friction which heats its interior. This energy gained by the object comes from its orbital energy and/or rotational energy , so over time in a two-body system, the initial elliptical orbit decays into a circular orbit ( tidal circularization ) and the rotational periods of the two bodies adjust towards matching the orbital period ( tidal locking ). Sustained tidal heating occurs when the elliptical orbit is prevented from circularizing due to additional gravitational forces from other bodies that keep tugging the object back into an elliptical orbit. In this more complex system, orbital and rotational energy still is being converted to thermal energy; however, now the orbit's semimajor axis would shrink rather than its eccentricity .
Tidal heating is responsible for the geologic activity of the most volcanically active body in the Solar System : Io , a moon of Jupiter . Io's eccentricity persists as the result of its orbital resonances with the Galilean moons Europa and Ganymede . [ 1 ] The same mechanism has provided the energy to melt the lower layers of the ice surrounding the rocky mantle of Jupiter's next-closest large moon, Europa. However, the heating of the latter is weaker, because of reduced flexing—Europa has half Io's orbital frequency and a 14% smaller radius; also, while Europa's orbit is about twice as eccentric as Io's, tidal force falls off with the cube of distance and is only a quarter as strong at Europa. Jupiter maintains the moons' orbits via tides they raise on it and thus its rotational energy ultimately powers the system. [ 1 ] Saturn's moon Enceladus is similarly thought to have a liquid water ocean beneath its icy crust, due to tidal heating related to its resonance with Dione . The water vapor geysers which eject material from Enceladus are thought to be powered by friction generated within its interior. [ 2 ]
Munk & Wunsch (1998) estimated that Earth experiences 3.7 TW (0.0073 W/m 2 ) of tidal heating, of which 95% (3.5 TW or 0.0069 W/m 2 ) is associated with ocean tides and 5% (0.2 TW or 0.0004 W/m 2 ) is associated with Earth tides , with 3.2 TW being due to tidal interactions with the Moon and 0.5 TW being due to tidal interactions with the Sun. [ 3 ] Egbert & Ray (2001) confirmed that overall estimate, writing "the total amount of tidal energy dissipated in the Earth-Moon-Sun system is now well-determined. The methods of space geodesy—altimetry, satellite laser ranging, lunar laser ranging—have converged to 3.7 TW ..." [ 4 ]
Heller et al. (2021) estimated that shortly after the Moon was formed, when the Moon orbited 10-15 times closer to Earth than it does now, tidal heating might have contributed ~10 W/m 2 of heating. This heating happened over perhaps 100 million years, and could have accounted for a temperature increase of up to 5°C on the early Earth. [ 5 ] [ 6 ]
Harada et al. (2014) proposed that tidal heating may have created a molten layer at the core-mantle boundary within Earth's Moon. [ 7 ]
The tidal heating rate, E ˙ Tidal {\displaystyle {\dot {E}}_{\text{Tidal}}} , in a satellite that is spin-synchronous , coplanar ( I = 0 {\displaystyle I=0} ), and has an eccentric orbit is given by: E ˙ Tidal = − Im ( k 2 ) 21 2 G M h 2 R 5 n e 2 a 6 {\displaystyle {\dot {E}}_{\text{Tidal}}=-\operatorname {Im} (k_{2}){\frac {21}{2}}{\frac {GM_{h}^{2}R^{5}ne^{2}}{a^{6}}}} where R {\displaystyle R} , n {\displaystyle n} , a {\displaystyle a} , and e {\displaystyle e} are respectively the satellite's mean radius, mean orbital motion , orbital distance , and eccentricity. [ 8 ] M h {\displaystyle M_{h}} is the host (or central) body's mass and Im ( k 2 ) {\displaystyle \operatorname {Im} (k_{2})} represents the imaginary portion of the second-order Love Number which measures the efficiency at which the satellite dissipates tidal energy into frictional heat. This imaginary portion is defined by interplay of the body's rheology and self-gravitation. It, therefore, is a function of the body's radius, density, and rheological parameters (the shear modulus , viscosity , and others – dependent upon the rheological model). [ 9 ] [ 10 ] The rheological parameters' values, in turn, depend upon the temperature and the concentration of partial melt in the body's interior. [ 11 ]
The tidally dissipated power in a nonsynchronised rotator is given by a more complex expression. [ 12 ] | https://en.wikipedia.org/wiki/Tidal_heating |
A tide pool or rock pool is a shallow pool of seawater that forms on the rocky intertidal shore . These pools typically range from a few inches to a few feet deep and a few feet across. [ 1 ] Many of these pools exist as separate bodies of water only at low tide , as seawater gets trapped when the tide recedes. Tides are caused by the gravitational pull of the sun and moon. A tidal cycle is usually about 25 hours and consists of two high tides and two low tides. [ 2 ]
Tide pool habitats are home to especially adaptable animals , like snails, barnacles, mussels, anemones, urchins, sea stars, crustaceans, seaweed, and small fish. [ 1 ] Inhabitants must be able to cope with constantly changing water levels, water temperatures, salinity , and oxygen content. [ 2 ] At low tide, there is the risk of predators like seabirds. These pools have engaged the attention of naturalists and marine biologists , as well as philosophical essayists: John Steinbeck wrote in The Log from the Sea of Cortez , "It is advisable to look from the tide pool to the stars and then back to the tide pool." [ 2 ]
Some examples have been artificially augmented to enable safer swimming (for example without waves or without sharks) in seawater at certain states of the tide. [ 3 ]
The rocky shoreline exhibits distinct zones with unique characteristics. These zones are created by the tidal movements of water along the rocky shores from high to low-tide. They are:
The presence and abundance of flora and fauna vary between zones along the rocky shore. This is due to niche adaptations in response to the varying tides and solar exposure.
Tide pools exist in the intertidal zone (the area within the tidal range ), which is submerged by the sea at high tides and during storms . At other times, the rocks may undergo other extreme conditions, such as baking in the sun or being exposed to cold winds. Few organisms can survive such harsh conditions.
The high tide zone is flooded during each high tide, which occurs once or twice daily. Organisms must survive wave action , currents , and long exposure to the sun and open air. [ 4 ] This zone is predominantly inhabited by seaweed and invertebrates , such as sea anemones , sea star , chitons , crabs , green algae , and mussels . Marine algae provide shelter for nudibranchs and hermit crabs . The same waves and currents that make life in the high tide zone difficult bring food to filter feeders and other intertidal organisms.
This zone is constantly covered and uncovered by water, so its inhabitants have adapted to surviving in these conditions. More plants and animals live here, compared to the high tide zone, because they are not exposed to drying conditions for so long. [ 4 ] During low tide, anemones close up and mussels close their shells to keep in moisture. They reopen when the tide returns and brings them food. [ 2 ]
This area is mostly submerged and is exposed only during unusually low tide. [ 2 ] It usually teems with life and has far more marine vegetation, especially seaweeds. Organisms in this zone do not have to be as well adapted to drying out and temperature extremes. Low tide zone organisms include abalone , anemones, brown seaweed, chitons, crabs, green algae, hydroids , isopods , limpets , mussels, and sometimes even small vertebrates such as fish. Seaweeds provide shelter for many animals, like sea slugs and urchins that are too fragile for other zones. [ 2 ] These creatures can grow to larger sizes because there is more available energy and better water coverage: the water is shallow enough to allow additional sunlight for photosynthetic activity, with almost normal levels of salinity . This area is also relatively protected from large predators because of the wave action and shallow water.
Tide pools provide a home for many organisms such as sea stars , mussels and clams . Inhabitants deal with a frequently changing environment : fluctuations in water temperature , salinity, and oxygen content. Hazards include waves , strong currents , exposure to midday sun and predators.
Waves can dislodge mussels and draw them out to sea. Gulls pick up and drop sea urchins to break them open. Sea stars prey on mussels and are eaten by gulls themselves. Black bears are known to sometimes feast on intertidal creatures at low tide. [ 5 ] Although tide pool organisms must avoid getting washed away into the ocean , drying up in the sun, or being eaten, they depend on the tide pool's constant changes for food. [ 2 ] Tide pools contain complex food webs that can vary based on the climate. [ 6 ]
The sea anemone Anthopleura elegantissima reproduces clones of itself through a process of longitudinal fission , in which the animal splits into two parts along its length. [ 7 ] The sea anemone Anthopleura sola often engages in territorial fights. The white tentacles (acrorhagi), which contain stinging cells, are for fighting. The sea anemones sting each other repeatedly until one of them moves. [ 8 ]
Some species of sea stars can regenerate lost arms. Most species must retain an intact central part of the body to be able to regenerate, but a few can regrow from a single ray. The regeneration of these stars is possible because the vital organs are in the arms. [ 9 ]
Sea urchins (" Echinoidia ") move around tide pools with tube like feet. Different species of urchin have different colors, and many are seen in tide pools. With spines , some filled with poison like with " Toxopnesutes pileolus ", that protect them from predators they feed almost undisturbed in tide pools. Algae and other microorganism are the food sources that attract them to the tide pools. [ 10 ]
The presence of the California mussel increases the supply of inorganic nitrogen and phosphorus in coastal marine tide pools which allows the ecosystem the nutrients to be more productive. [ 11 ] The shell of a California mussel is primarily composed of Aragonite and Calcite which are both polymorphs of Calcium carbonate . [ 12 ] Climate change and ocean acidification has led to a decrease in these amounts important compounds in California Mussel shells over many years. [ 12 ]
Lichens and barnacles live in the splash zone. [ 2 ] Different barnacle species live at very tightly constrained elevations, with tidal conditions precisely determining the exact height of an assemblage relative to sea level. The intertidal zone is periodically exposed to sun and wind, conditions that can cause barnacles to become desiccated . These animals, therefore, need to be well adapted to water loss. Their calcite shells are impermeable, and they possess two plates which they slide across their mouth opening when not feeding. These plates also protect against predation. [ 13 ]
Many species of Hermit crab are commonly found in tide pool environments. The long-wristed hermit crab ( Pagurus longicarpus ) has been found to become stranded in tide pools and are forced to inhabit gastropod shells in response to the rapidly changing temperature of the pools. [ 14 ] Hermit crabs of different or the same species compete for the snail shells that are available. [ 15 ]
Many fish species can live in tidepools. Tidepool fishes are those inhabiting the intertidal zone during part or the entirety of their life cycle, including residents displaying morphological, physiological and behavioral adaptations to withstand the fluctuating environment and non-residents that use the intertidal as juvenile habitat, feeding or refuge ground, or as transient space between nearshore areas. [ 16 ] [ 17 ] Tidepools fishes can be classified as residents and non-residents (sometimes called transients or visitors). [ 18 ] [ 19 ] Residents are those that spend the whole lifetime in the tidepools. [ 16 ] [ 20 ] Non-resident species are commonly divided into two groups: secondary residents (also known as partial residents or opportunists) and transients (which can be further classified as tidal and seasonal transients). Secondary residents are species that spend only a portion of their life history in tidepools, typically during their juvenile stage, before moving on to adult subtidal habitats. [ 16 ] [ 17 ] Transients, on the other hand, may temporarily inhabit tidepools for various reasons such as foraging, seeking refuge, or transit. Unlike residents, transients lack specialized adaptations for intertidal life and typically occupy large tidepools for a relatively short period, ranging from a single tidal cycle to a few months. [ 17 ] The Tidepool sculpin is a species of fish that is named for its tide pool habitat. The Tidepool Sculpin has been found to show preferences for certain tide pools and will return to their tide pool of choice after being removed from it. [ 21 ] This is a behavior known as Homing (biology) . These fish crawl on the floor of tide pools using a back and forth movement of their tail fin and a rotating motion of their pectoral fins. [ 22 ]
Multiple species of Amphipods ( Amphipoda ) can be found in coastal tide pools. These small crustaceans provide an important food source for predator species as well as limiting the growth of algae attached to vegetation. [ 23 ]
Sea palms ( Postelsia ) look similar to miniature palm trees . They live in the middle to upper intertidal zones in areas with greater wave action. High wave action may increase nutrient availability and moves the blades of the thallus , allowing more sunlight to reach the organism so that it can photosynthesize. In addition, the constant wave action removes competitors, such as the mussel species Mytilus californianus .
Recent studies have shown that Postelsia grows in greater numbers when such competition exists; a control group with no competition produced fewer offspring than an experimental group with mussels; from this it is thought that the mussels provide protection for the developing gametophytes . [ 24 ] Alternatively, the mussels may prevent the growth of competing algae such as Corallina or Halosaccion , allowing Postelsia to grow freely after wave action has eliminated the mussels. [ 25 ]
Coralline algae "Corallinales" are predominant features of mid and low intertidal tide pools . Calcium carbonate (CaCO 3 ) takes the form of calcite in their cell walls providing them with a hard outer shell. This shell protects from herbivores and desiccation due to lack of water and evaporation. Many forms of the Coralline algae bring herbivores, such as mollusks "Notoacmea", to the tide pools during high tides, increasing the biomass of the area. Once low tides comes, these herbivores are exposed to carnivores in the areas, fueling the food web. [ 26 ]
Tide pools are often surrounded by coastal predators who feed on tide pool flora and fauna. These predators play an important role in the tide pool food web and create competition for resources. | https://en.wikipedia.org/wiki/Tide_pool |
The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans (especially Earth's oceans) under the gravitational loading of another astronomical body or bodies (especially the Moon and Sun ).
The tides received relatively little attention in the civilizations around the Mediterranean Sea , as the tides there are relatively small, and the areas that experience tides do so unreliably. [ 1 ] [ 2 ] [ 3 ] A number of theories were advanced, however, from comparing the movements to breathing or blood flow to theories involving whirlpools or river cycles. [ 2 ] A similar "breathing earth" idea was considered by some Asian thinkers. [ 4 ] Plato reportedly believed that the tides were caused by water flowing in and out of undersea caverns. [ 1 ] Crates of Mallus attributed the tides to "the counter-movement (ἀντισπασμός) of the sea” and Apollodorus of Corcyra to "the refluxes from the Ocean". [ 5 ] An ancient Indian Purana text dated to 400-300 BC refers to the ocean rising and falling because of heat expansion from the light of the Moon. [ a ] [ 6 ] The Yolngu people of northeastern Arnhem Land in the Northern Territory of Australia identified a link between the Moon and the tides, which they mythically attributed to the Moon filling with water and emptying out again. [ 7 ] [ 8 ]
Ultimately the link between the Moon (and Sun ) and the tides became known to the Greeks , although the exact date of discovery is unclear; references to it are made in sources such as Pytheas of Massilia in 325 BC and Pliny the Elder 's Natural History in 77 AD. Although the schedule of the tides and the link to lunar and solar movements was known, the exact mechanism that connected them was unclear. [ 2 ] Classicists Thomas Little Heath claimed that both Pytheas and Posidonius connected the tides with the moon, "the former directly, the latter through the setting up of winds". [ 5 ] Seneca mentions in De Providentia the periodic motion of the tides controlled by the lunar sphere. [ 9 ] Eratosthenes (3rd century BC) and Posidonius (1st century BC) both produced detailed descriptions of the tides and their relationship to the phases of the Moon , Posidonius in particular making lengthy observations of the sea on the Spanish coast, although little of their work survived. The influence of the Moon on tides was mentioned in Ptolemy 's Tetrabiblos as evidence of the reality of astrology . [ 1 ] [ 10 ] Seleucus of Seleucia is thought to have theorized around 150 BC that tides were caused by the Moon as part of his heliocentric model. [ 11 ] [ 12 ]
Aristotle , judging from discussions of his beliefs in other sources, is thought to have believed the tides were caused by winds driven by the Sun's heat, and he rejected the theory that the Moon caused the tides. An apocryphal legend claims that he committed suicide in frustration with his failure to fully understand the tides. [ 1 ] Heraclides also held "the sun sets up winds, and that these winds, when they blow, cause the high tide and, when they cease, the low tide". [ 5 ] Dicaearchus also "put the tides down to the direct action of the sun according to its position". [ 5 ] Philostratus discusses tides in Book Five of Life of Apollonius of Tyana (circa 217-238 AD); he was vaguely aware of a correlation of the tides with the phases of the Moon but attributed them to spirits moving water in and out of caverns, which he connected with the legend that spirits of the dead cannot move on at certain phases of the Moon. [ b ]
The Venerable Bede discusses the tides in The Reckoning of Time and shows that the twice-daily timing of tides is related to the Moon and that the lunar monthly cycle of spring and neap tides is also related to the Moon's position. He goes on to note that the times of tides vary along the same coast and that the water movements cause low tide at one place when there is high tide elsewhere. [ 13 ] However, he made no progress regarding the question of how exactly the Moon created the tides. [ 2 ]
Medieval rule-of-thumb methods for predicting tides were said to allow one "to know what Moon makes high water" from the Moon's movements. [ 14 ] Dante references the Moon's influence on the tides in his Divine Comedy . [ 15 ] [ 1 ]
Medieval European understanding of the tides was often based on works of Muslim astronomers that became available through Latin translation starting from the 12th century. [ 16 ] Abu Ma'shar al-Balkhi , in his Introductorium in astronomiam , taught that ebb and flood tides were caused by the Moon. [ 16 ] Abu Ma'shar discussed the effects of wind and Moon's phases relative to the Sun on the tides. [ 16 ] In the 12th century, al-Bitruji contributed the notion that the tides were caused by the general circulation of the heavens. [ 16 ] Medieval Arabic astrologers frequently referenced the Moon's influence on the tides as evidence for the reality of astrology; some of their treatises on the topic influenced western Europe. [ 10 ] [ 1 ] Some theorized that the influence was caused by lunar rays heating the ocean's floor. [ 3 ]
Simon Stevin in his 1608 De spiegheling der Ebbenvloet (The Theory of Ebb and Flood ) dismisses a large number of misconceptions that still existed about ebb and flood. Stevin pleads for the idea that the attraction of the Moon was responsible for the tides and writes in clear terms about ebb, flood, spring tide and neap tide, stressing that further research needed to be made. [ 17 ] [ 18 ] In 1609, Johannes Kepler correctly suggested that the gravitation of the Moon causes the tides, [ c ] which he compared to magnetic attraction [ 20 ] [ 2 ] [ 21 ] [ 22 ] basing his argument upon ancient observations and correlations.
In 1616, Galileo Galilei wrote Discourse on the Tides . [ 23 ] He strongly and mockingly rejects the lunar theory of the tides, [ 21 ] [ 2 ] and tries to explain the tides as the result of the Earth 's rotation and revolution around the Sun , believing that the oceans moved like water in a large basin: as the basin moves, so does the water. [ 24 ] But his contemporaries noticed that this made predictions that did not fit observations. [ 25 ]
René Descartes theorized that the tides (alongside the movement of planets, etc.) were caused by aetheric vortices , without reference to Kepler's theories of gravitation by mutual attraction; this was extremely influential, with numerous followers of Descartes expounding on this theory throughout the 17th century, particularly in France. [ 26 ] However, Descartes and his followers acknowledged the influence of the Moon, speculating that pressure waves from the Moon via the aether were responsible for the correlation. [ 3 ] [ 27 ] [ 4 ] [ 28 ]
Newton , in the Principia , provides a correct explanation for the tidal force , which can be used to explain tides on a planet covered by a uniform ocean but which takes no account of the distribution of the continents or ocean bathymetry . [ 29 ]
While Newton explained the tides by describing the tide-generating forces and Daniel Bernoulli gave a description of the static reaction of the waters on Earth to the tidal potential, the dynamic theory of tides , developed by Pierre-Simon Laplace in 1775, [ 30 ] describes the ocean's real reaction to tidal forces. [ 31 ] Laplace's theory of ocean tides takes into account friction , resonance and natural periods of ocean basins. It predicts the large amphidromic systems in the world's ocean basins and explains the oceanic tides that are actually observed. [ 32 ]
The equilibrium theory—based on the gravitational gradient from the Sun and Moon but ignoring the Earth's rotation, the effects of continents, and other important effects—could not explain the real ocean tides. [ 33 ] Since measurements have confirmed the dynamic theory, many things have possible explanations now, like how the tides interact with deep sea ridges, and chains of seamounts give rise to deep eddies that transport nutrients from the deep to the surface. [ 34 ] The equilibrium tide theory calculates the height of the tide wave of less than half a meter, while the dynamic theory explains why tides are up to 15 meters. [ 35 ]
Satellite observations confirm the accuracy of the dynamic theory, and the tides worldwide are now measured to within a few centimeters. [ 36 ] [ 37 ] Measurements from the CHAMP satellite closely match the models based on the TOPEX data. [ 38 ] [ 39 ] [ 40 ] Accurate models of tides worldwide are essential for research since the variations due to tides must be removed from measurements when calculating gravity and changes in sea levels. [ 41 ]
In 1776, Laplace formulated a single set of linear partial differential equations for tidal flow described as a barotropic two-dimensional sheet flow. Coriolis effects are introduced as well as lateral forcing by gravity . Laplace obtained these equations by simplifying the fluid dynamics equations, but they can also be derived from energy integrals via Lagrange's equation .
For a fluid sheet of average thickness D , the vertical tidal elevation ζ , as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively) satisfy Laplace's tidal equations : [ 42 ]
where Ω is the angular frequency of the planet's rotation, g is the planet's gravitational acceleration at the mean ocean surface, a is the planetary radius, and U is the external gravitational tidal-forcing potential .
William Thomson (Lord Kelvin) rewrote Laplace's momentum terms using the curl to find an equation for vorticity . Under certain conditions this can be further rewritten as a conservation of vorticity.
Laplace's improvements in theory were substantial, but they still left prediction in an approximate state. This position changed in the 1860s when the local circumstances of tidal phenomena were more fully brought into account by Lord Kelvin 's application of Fourier analysis to the tidal motions as harmonic analysis . Thomson's work in this field was further developed and extended by George Darwin , applying the lunar theory current in his time. Darwin's symbols for the tidal harmonic constituents are still used, for example: M : moon/lunar; S : sun/solar; K : moon-sun/lunisolar.
Darwin's harmonic developments of the tide-generating forces were later improved when A.T. Doodson , applying the lunar theory of E.W. Brown , [ 43 ] developed the tide-generating potential (TGP) in harmonic form, distinguishing 388 tidal frequencies. [ 44 ] Doodson's work was carried out and published in 1921. [ 45 ] Doodson devised a practical system for specifying the different harmonic components of the tide-generating potential, the Doodson numbers , a system still in use.
Since the mid-twentieth century further analysis has generated many more terms than Doodson's 388. About 62 constituents are of sufficient size to be considered for possible use in marine tide prediction, but sometimes many fewer can predict tides to useful accuracy. The calculations of tide predictions using the harmonic constituents are laborious, and from the 1870s to about the 1960s they were carried out using a mechanical tide-predicting machine , a special-purpose form of analog computer . More recently digital computers, using the method of matrix inversion, are used to determine the tidal harmonic constituents directly from tide gauge records.
Tidal constituents combine to give an endlessly varying aggregate because of their different and incommensurable frequencies: the effect is visualized in an animation of the American Mathematical Society illustrating the way in which the components used to be mechanically combined in the tide-predicting machine. Amplitudes (half of peak-to-peak amplitude ) of tidal constituents are given below for six example locations: Eastport, Maine (ME), [ 46 ] Biloxi, Mississippi (MS), San Juan, Puerto Rico (PR), Kodiak, Alaska (AK), San Francisco, California (CA), and Hilo, Hawaii (HI).
In order to specify the different harmonic components of the tide-generating potential, Doodson devised a practical system which is still in use, involving what are called the Doodson numbers based on the six Doodson arguments or Doodson variables. The number of different tidal frequency components is large, but each corresponds to a specific linear combination of six frequencies using small-integer multiples, positive or negative. In principle, these basic angular arguments can be specified in numerous ways; Doodson's choice of his six "Doodson arguments" has been widely used in tidal work. In terms of these Doodson arguments, each tidal frequency can then be specified as a sum made up of a small integer multiple of each of the six arguments. The resulting six small integer multipliers effectively encode the frequency of the tidal argument concerned, and these are the Doodson numbers: in practice all except the first are usually biased upwards by +5 to avoid negative numbers in the notation. (In the case that the biased multiple exceeds 9, the system adopts X for 10, and E for 11.) [ 47 ]
The Doodson arguments are specified in the following way, in order of decreasing frequency: [ 47 ]
In these expressions, the symbols l {\displaystyle l} , l ′ {\displaystyle l'} , F {\displaystyle F} and D {\displaystyle D} refer to an alternative set of fundamental angular arguments (usually preferred for use in modern lunar theory), in which:-
It is possible to define several auxiliary variables on the basis of combinations of these.
In terms of this system, each tidal constituent frequency can be identified by its Doodson numbers. The strongest tidal constituent "M 2 " has a frequency of 2 cycles per lunar day, its Doodson numbers are usually written 255.555, meaning that its frequency is composed of twice the first Doodson argument, and zero times all of the others. The second strongest tidal constituent "S 2 " is influenced by the sun, and its Doodson numbers are 273.555, meaning that its frequency is composed of twice the first Doodson argument, +2 times the second, -2 times the third, and zero times each of the other three. [ 48 ] This aggregates to the angular equivalent of mean solar time +12 hours. These two strongest component frequencies have simple arguments for which the Doodson system might appear needlessly complex, but each of the hundreds of other component frequencies can be briefly specified in a similar way, showing in the aggregate the usefulness of the encoding. | https://en.wikipedia.org/wiki/Tide_spectrum |
Tides in marginal seas are tides affected by their location in semi-enclosed areas along the margins of continents and differ from tides in the open oceans. Tides are water level variations caused by the gravitational interaction between the Moon, the Sun and the Earth. The resulting tidal force is a secondary effect of gravity: it is the difference between the actual gravitational force and the centrifugal force . While the centrifugal force is constant across the Earth, the gravitational force is dependent on the distance between the two bodies and is therefore not constant across the Earth. The tidal force is thus the difference between these two forces on each location on the Earth. [ 1 ]
In an idealized situation, assuming a planet with no landmasses (an aqua planet ), the tidal force would result in two tidal bulges on opposite sides of the earth. This is called the equilibrium tide. However, due to global and local ocean responses different tidal patterns are generated. The complicated ocean responses are the result of the continental barriers, resonance due to the shape of the ocean basin, the tidal waves impossibility to keep up with the Moons tracking, the Coriolis acceleration and the elastic response of the solid earth. [ 2 ]
In addition, when the tide arrives in the shallow seas it interacts with the sea floor which leads to the deformation of the tidal wave. As a results, tides in shallow waters tend to be larger, of shorter wavelength, and possibly nonlinear relative to tides in the deep ocean. [ 3 ]
The transition from the deep ocean to the continental shelf , known as the continental slope, is characterized by a sudden decrease in water depth. In order to apply to the conservation of energy , the tidal wave has to deform as a result of the decrease in water depth. The total energy of a linear progressive wave per wavelength is the sum of the potential energy (PE) and the kinetic energy (KE). The potential and kinetic energy integrated over a complete wavelength are the same, under the assumption that the water level variations are small compared to the water depth ( η << H {\displaystyle \eta <<H} ). [ 4 ]
∫ 0 λ P E = ∫ 0 λ K E = 1 2 ρ g ∫ 0 λ η 2 d x {\displaystyle \int _{0}^{\lambda }PE=\int _{0}^{\lambda }KE={\frac {1}{2}}\rho g\int _{0}^{\lambda }\eta ^{2}dx}
where ρ {\displaystyle \rho } is the density , g {\displaystyle g} the gravitation acceleration and η {\displaystyle \eta } the vertical tidal elevation. The total wave energy becomes:
E = ρ g ∫ 0 λ η 2 d x {\displaystyle E=\rho g\int _{0}^{\lambda }\eta ^{2}dx}
If we now solve for a harmonic wave η ( x ) = A c o s ( k x ) {\displaystyle \eta (x)=Acos(kx)} , where k {\displaystyle k} is the wave number and A {\displaystyle A} the amplitude , the total energy per unit area of surface becomes: [ 5 ]
E s = 1 2 ρ g A 2 {\displaystyle E_{s}={\frac {1}{2}}\rho gA^{2}}
A tidal wave has a wavelength that is much larger than the water depth. And thus according to the dispersion of gravity waves , they travel with the phase and group velocity of a shallow water wave: c p = c g = g h {\displaystyle c_{p}=c_{g}={\sqrt {gh}}} . The wave energy is transmitted by the group velocity of a wave [ 6 ] and thus the energy flux ( F E {\displaystyle F_{E}} ) is given by:
F E = 1 2 ρ g A 2 g h {\displaystyle F_{E}={\frac {1}{2}}\rho gA^{2}{\sqrt {gh}}}
The energy flux needs to be conserved and with ρ {\displaystyle \rho } and g {\displaystyle g} constant, this leads to:
F E , 1 = F E , 2 ⟹ A 1 2 g h 1 = A 2 2 g h 2 {\displaystyle F_{E,1}=F_{E,2}\Longrightarrow {A_{1}}^{2}{\sqrt {gh_{1}}}={A_{2}}^{2}{\sqrt {gh_{2}}}}
where h 2 < h 1 {\displaystyle h2<h1} and thus A 2 > A 1 {\displaystyle A_{2}>A_{1}} .
When the tidal wave propagates onto the continental shelf, the water depth ( h ) {\displaystyle (h)} decreases. In order to conserve the energy flux, the amplitude of the wave needs to increase (see figure 1).
The above explanation is a simplification as not all tidal wave energy is transmitted, but it is partly reflected at the continental slope. The transmission coefficient of the tidal wave is given by:
A 2 A 1 = 2 c 1 ( c 1 + c 2 ) {\displaystyle {\frac {A_{2}}{A_{1}}}={\frac {2c_{1}}{(c_{1}+c_{2})}}} [ 4 ]
This equation indicates that when c 1 = c 2 {\displaystyle c_{1}=c_{2}} the transmitted tidal wave has the same amplitude as the original wave. Furthermore, the transmitted wave will be larger than the original wave when c 1 > c 2 {\displaystyle c_{1}>c_{2}} as is the case for the transition to the continental shelf.
The reflected wave amplitude ( A ′ {\displaystyle A^{'}} ) is determined by the reflection coefficient of the tidal wave:
A ′ A 1 = c 1 − c 2 ( c 1 + c 2 ) {\displaystyle {\frac {A^{'}}{A_{1}}}={\frac {c_{1}-c_{2}}{(c_{1}+c_{2})}}} [ 4 ]
This equation indicates that when c 1 = c 2 {\displaystyle c_{1}=c_{2}} there is no reflected wave and if c 1 > c 2 {\displaystyle c_{1}>c_{2}} the reflected tidal wave will be smaller than the original tidal wave.
At the continental shelf the reflection and transmission of the tidal wave can lead to the generation of internal tides on the pycnocline . The surface (i.e. barotropic ) tide generates these internal tides where stratified waters are forced upwards over a sloping bottom topography. [ 7 ] The internal tide extracts energy from the surface tide and propagates both in shoreward and seaward direction. [ 8 ] The shoreward propagating internal waves shoals when reaching shallower water where the wave energy is dissipated by wave breaking . The shoaling of the internal tide drives mixing across the pycnocline, high levels carbon sequestration and sediment resuspension. [ 9 ] [ 10 ] Furthermore, through nutrient mixing the shoaling of the internal tide has a fundamental control on the functioning of ecosystems on the continental margin. [ 11 ]
After entering the continental shelf, a tidal wave quickly faces a boundary in the form of a landmass . When the tidal wave reaches a continental margin , it continues as a boundary trapped Kelvin wave . Along the coast, a boundary trapped Kelvin is also known as a coastal Kelvin wave or Edge wave . A Kelvin wave is a special type of gravity wave that can exist when there is (1) gravity and stable stratification , (2) sufficient Coriolis force and (3) the presence of a vertical boundary. [ 12 ] Kelvin waves are important in the ocean and shelf seas, they form a balance between inertia , the Coriolis force and the pressure gradient force . The simplest equations that describe the dynamics of Kelvin waves are the linearized shallow water equations for homogeneous , in-viscid flows . These equations can be linearized for a small Rossby number , no frictional forces and under the assumption that the wave height is small compared to the water depth ( η << h {\displaystyle \eta <<h} ). The linearized depth-averaged shallow water equations become:
u momentum equation:
v momentum equation:
the continuity equation:
where u {\displaystyle u} is the zonal velocity ( x {\displaystyle x} direction), v {\displaystyle v} the meridional velocity ( y {\displaystyle y} direction), t {\displaystyle t} is time and f {\displaystyle f} is the Coriolis frequency.
Kelvin waves are named after Lord Kelvin , who first described them after finding solutions to the linearized shallow water equations with the boundary condition u ( x , y , t ) = 0 {\displaystyle u(x,y,t)=0} . [ 13 ] When this assumption is made the linearized depth-averaged shallow water equations that can describe a Kelvin wave become:
u momentum equation:
v momentum equation:
the continuity equation :
Now it is possible to get an expression for η {\displaystyle \eta } , by taking the time derivative of the continuity equation and substituting the momentum equation:
The same can be done for v {\displaystyle v} , by taking the time derivative of the v momentum equation and substituting the continuity equation
Both of these equations take the form of the classical wave equation , where c = g h {\displaystyle c={\sqrt {gh}}} . Which is the same velocity as the tidal wave and thus of a shallow water wave. These preceding equations govern the dynamics of a one-dimensional non-dispersive wave, for which the following general solution exist:
where length R = g h f {\displaystyle R={\frac {\sqrt {gh}}{f}}} is the Rossby radius of deformation and F ( y + c t ) {\displaystyle F(y+ct)} is an arbitrary function describing the wave motion. In the most simple form F {\displaystyle F} is a cosine or sine function which describes a wave motion in the positive and negative direction. The Rossby radius of deformation is a typical length scale in the ocean and atmosphere that indicates when rotational effects become important. The Rossby radius of deformation is a measure for the trapping distance of a coastal Kelvin wave. [ 14 ] The exponential term results in an amplitude that decays away from the coast.
The expression of tides as a bounded Kelvin wave is well observable in enclosed shelf seas around the world (e.g. the English channel , the North Sea or the Yellow sea ). Animation 1 shows the behaviour of a simplified case of a Kelvin wave in an enclosed shelf sea for the case with (lower panel) and without friction (upper panel). The shape of an enclosed shelf sea is represented as a simple rectangular domain in the Northern Hemisphere which is open on the left hand side and closed on the right hand side. The tidal wave, a Kelvin wave, enters the domain in the lower left corner and travels to the right with the coast on its right. The sea surface height (SSH, left panels of animation 1), the tidal elevation, is maximum at the coast and decreases towards the centre of the domain. The tidal currents (right panels of animation 1) are in the direction of wave propagation under the crest and in the opposite direction under the through. They are both maximum under the crest and the trough of the waves and decrease towards the centre. This was expected as the equations for η {\displaystyle \eta } and v {\displaystyle v} are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term. Therefore this set of equations describes a wave that travels along the coast with a maximum amplitude at the coast which declines towards the ocean. These solutions also indicate that a Kelvin wave always travels with the coast on their right hand side in the Northern Hemisphere and with the coast at their left hand side in the Southern Hemisphere. In the limit of no rotation where f → 0 {\displaystyle f\rightarrow 0} , the exponential term increase without a bound and the wave will become a simple gravity wave orientated perpendicular to the coast. [ 14 ] In the next section, it will be shown how these Kelvin waves behaves when traveling along a coast, in an enclosed shelf seas or in estuaries and basins.
The expression of tides as a bounded Kelvin wave is well observable in enclosed shelf seas around the world (e.g. the English channel , the North Sea or the Yellow sea ). Animation 1 shows the behaviour of a simplified case of a Kelvin wave in an enclosed shelf sea for the case with (lower panel) and without friction (upper panel). The shape of an enclosed shelf sea is represented as a simple rectangular domain in the Northern Hemisphere which is open on the left hand side and closed on the right hand side. The tidal wave, a Kelvin wave, enters the domain in the lower left corner and travels to the right with the coast on its right. The sea surface height (SSH, left panels of animation 1), the tidal elevation, is maximum at the coast and decreases towards the centre of the domain. The tidal currents (right panels of animation 1) are in the direction of wave propagation under the crest and in the opposite direction under the through. They are both maximum under the crest and the trough of the waves and decrease towards the centre. This was expected as the equations for η {\displaystyle \eta } and v {\displaystyle v} are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term.
On the enclosed right hand side, the Kelvin wave is reflected and because it always travels with the coast on its right, it will now travel in the opposite direction. The energy of the incoming Kelvin wave is transferred through Poincare waves along the enclosed side of the domain to the outgoing Kelvin wave. The final pattern of the SSH and the tidal currents is made up of the sum of the two Kelvin waves. These two can amplify each other and this amplification is maximum when the length of the shelf sea is a quarter wavelength of the tidal wave. [ 2 ] Next to that, the sum of the two Kelvin waves result in several static minima's in the centre of the domain which hardly experience any tidal motion, these are called Amphidromic points . In the upper panel of figure 2, the absolute time averaged SSH is shown in red shading and the dotted lines show the zero tidal elevation level at roughly hourly intervals, also known as cotidal lines. Where these lines intersect the tidal elevation is zero during a full tidal period and thus this is the location of the Amphidromic points.
In the real world, the reflected Kelvin wave has a lower amplitude due to energy loss as a result of friction and through the transfer via Poincare waves (lower left panel of animation 1). The tidal currents are proportional to the wave amplitude and therefore also decrease on the side of the reflected wave (lower right panel of animation 1). Finally, the static minima's are no longer in the centre of the domain as wave amplitude is no longer symmetric. Therefore, the Amphidromic points shift towards the side of the reflected wave (lower panel figure 2).
The dynamics of a tidal Kelvin wave in enclosed shelf sea is well manifested and studied in the North Sea. [ 15 ]
When tides enter estuaries or basins, the boundary conditions change as the geometry changes drastically. The water depth becomes shallower and the width decreases, next to that the depth and width become significantly variable over the length and width of the estuary or basin. As a result the tidal wave deforms which affects the tidal amplitude, phase speed and the relative phase between tidal velocity and elevation. The deformation of the tide is largely controlled by the competition between bottom friction and channel convergence. [ 16 ] Channel convergence increases the tidal amplitude and phase speed as the energy of the tidal wave is traveling through a smaller area while bottom friction decrease the amplitude through energy loss. [ 17 ] The modification of the tide leads to the creation of overtides (e.g. M 4 {\displaystyle M_{4}} tidal constituents) or higher harmonics. These overtides are multiples, sums or differences of the astronomical tidal constituents and as a result the tidal wave can become asymmetric. [ 18 ] A tidal asymmetry is a difference between the duration of the rise and the fall of the tidal water elevation and this can manifest itself as a difference in flood/ebb tidal currents. [ 19 ] The tidal asymmetry and the resulting currents are important for the sediment transport and turbidity in estuaries and tidal basins. [ 20 ] Each estuary and basin has its own distinct geometry and these can be subdivided in several groups of similar geometries with its own tidal dynamics. [ 16 ] | https://en.wikipedia.org/wiki/Tides_in_marginal_seas |
The tie in a cavity wall [ 1 ] [ 2 ] is a component used to tie the internal and external walls (or leaves)—constructed of bricks or cement blocks —together, making the two parts to act as a homogeneous unit. It is placed in the cavity wall during construction and spans the cavity. The ends of the tie are designed to lock into the mortar . Also incorporated into the design of the tie is means of preventing water transfer from the outer to the inner leaves. In flat ties, this can be a twist. In wire ties , this can be corrugations formed in the wire or again a twist.
Cavity walls often have insulation in the cavity which may either partially or fully fill the cavity. Partial fill insulation systems require specialized ties or clips to keep the insulation in position. A vapour barrier may be necessary on the inner wall to prevent interstitial condensation . This is often incorporated into the cavity wall insulation system. The spacing of ties is laid down in building regulations , though there may be variations with specialised blocks. Additional ties are used around window and door openings. Improper installation may lead to water damage or fungus formation within the cavity, leading to structural and health hazards.
Ties are exposed to water and chemical attack from cement . They were traditionally made of galvanized steel , the fishtail tie being the most common. On high quality work, ties were occasionally made of bronze . In the mid-twentieth century, wire ties were widely used, again made from galvanized steel wire. As time has passed, many galvanized steel ties have deteriorated due to moisture in the outer leaf of brickwork. The corrosion may force apart the cement joints and even result in the collapse of walls if no remedial action is taken. Any cracks appearing in cavity walls dating from the twentieth century need to be investigated before irremediable damage ensues. Horizontal cracking is especially suspect. Failed ties have to be isolated and substitute specialist ties installed by drilling through inner and outer leaves from outside the building. The replacement ties may be fixed mechanically or with special adhesives .
Galvanized steel ties are no longer in use for this reason. For a brief period, plastic ties were used but were not satisfactory. Modern practice is to use stainless steel ties.
Cavity walls were traditionally spaced 2"(50mm) apart. Due to the need for thicker insulation in exterior walls these days, a range of longer ties are now available so that cavities of up to 6"(150mm) can be constructed.
Ties in a cavity wall are typically made of iron, steel, or plastic; though figures are various. Basically, a tie has ring to fasten with mortar on both end like a bow. Ties would be selected by type of masonry , the cavity width, and so on.
Typically, ties in a cavity wall are double triangular shape (like a bow), though, depending on the existence of another layer (e.g. insulation sheet) at the cavity, which fills the cavity partially, ties include retaining discs or rods.
Failure of ties is an increasing problem with cavity wall ties made from galvanized steel. It arises when the galvanizing is not of sufficient quality and the outer leaf of the cavity wall allows water penetration, usually due to porous brick/blockwork. If the tie rusts, the swelling effect may cause horizontal, external cracks to appear in the wall. Frost action can swiftly enlarge these cracks to cause damage.
Since ties in a cavity wall are typically made of metal (iron or steel), they are prone to corrode over time. When metal corrodes, it expands its size, causing ties to lift up from the brickwork . Cracks caused by vertical loads leave parts of buildings vulnerable to corrosion, such as eaves and gable walls above purlin positions, or placed directly beneath openings, where the weight on brickwork is light. Over the time, cracks appear from the top of the wall and extend downward.
Cracks due to tie corrosion at the cavity wall are horizontal and tend to occur at the location of wall ties, normally 6 courses apart. The corrosion causes the walls to detach and tilt, resulting in the outer wall snapping outward. The inner wall is held in place by the added support of floor joists. At the gable wall, inner walls without these supporting joists, bow inwards. By avoiding several factors accelerating corrosion, it will last more than 60 years.
Due to its materiality of the tie, its corrosion is related to the extent of the exposure at acidic substances . The major factors of the corrosion are:
Cement -based mortar, giving an alkaline environment for cavity wall ties, has been used since the Second World War. Alkaline environment is beneficial for ties protecting from acid and procrastinating the corrosion process. Before cement-based mortar , lime -based one was used, which does not present an alkaline environment. Therefore, the majority buildings with wall tie corrosion are built before the Second World War. An alkaline degree gets weaker while exposed at acidic rain over the time.
Moreover, metal corrosion is a chemical reaction that the reaction rate is triggered by water and heat. Thus, some [ who? ] say that cavity wall insulation enhance the ability to grab water longer, so that the environment of the ties damper. Accordingly, the South or South West [ where? ] elevations warmed by the sunlight are warmer, letting ties more at risk for this type of corrosion. | https://en.wikipedia.org/wiki/Tie_(cavity_wall) |
In organic chemistry , an oxime is an organic compound belonging to the imines , with the general formula RR’C=N−OH , where R is an organic side-chain and R' may be hydrogen , forming an aldoxime , or another organic group , forming a ketoxime . O-substituted oximes form a closely related family of compounds. Amidoximes are oximes of amides ( R 1 C(=O)NR 2 R 3 ) with general structure R 1 C(=NOH)NR 2 R 3 .
Oximes are usually generated by the reaction of hydroxylamine with aldehydes ( R−CH=O ) or ketones ( RR’C=O ). The term oxime dates back to the 19th century, a combination of the words oxygen and imine . [ 1 ]
If the two side-chains on the central carbon are different from each other—either an aldoxime, or a ketoxime with two different "R" groups—the oxime can often have two different geometric stereoisomeric forms according to the E / Z configuration . An older terminology of syn and anti was used to identify especially aldoximes according to whether the R group was closer or further from the hydroxyl. Both forms are often stable enough to be separated from each other by standard techniques.
Oximes have three characteristic bands in the infrared spectrum , whose wavelengths corresponding to the stretching vibrations of its three types of bonds: 3600 cm −1 (O−H), 1665 cm −1 (C=N) and 945 cm −1 (N−O). [ 2 ]
In aqueous solution, aliphatic oximes are 10 2 - to 10 3 -fold more resistant to hydrolysis than analogous hydrazones. [ 3 ]
Oximes can be synthesized by condensation of an aldehyde or a ketone with hydroxylamine . The condensation of aldehydes with hydroxylamine gives aldoximes, and ketoximes are produced from ketones and hydroxylamine. In general, oximes exist as colorless crystals or as thick liquids and are poorly soluble in water. Therefore, oxime formation can be used for the identification of ketone or aldehyde functional groups.
Oximes can also be obtained from reaction of nitrites such as isoamyl nitrite with compounds containing an acidic hydrogen atom. Examples are the reaction of ethyl acetoacetate and sodium nitrite in acetic acid , [ 4 ] [ 5 ] the reaction of methyl ethyl ketone with ethyl nitrite in hydrochloric acid . [ 6 ] and a similar reaction with propiophenone , [ 7 ] the reaction of phenacyl chloride , [ 8 ] the reaction of malononitrile with sodium nitrite in acetic acid [ 9 ]
A conceptually related reaction is the Japp–Klingemann reaction .
The hydrolysis of oximes proceeds easily by heating in the presence of various inorganic acids , and the oximes decompose into the corresponding ketones or aldehydes, and hydroxylamines. The reduction of oximes by sodium metal, [ 10 ] sodium amalgam , hydrogenation , or reaction with hydride reagents produces amines . [ 11 ] Typically the reduction of aldoximes gives both primary amines and secondary amines; however, reaction conditions can be altered (such as the addition of potassium hydroxide in a 1/30 molar ratio) to yield solely primary amines. [ 12 ]
In general, oximes can be changed to the corresponding amide derivatives by treatment with various acids. This reaction is called Beckmann rearrangement . [ 13 ] In this reaction, a hydroxyl group is exchanged with the group that is in the anti position of the hydroxyl group. The amide derivatives that are obtained by Beckmann rearrangement can be transformed into a carboxylic acid by means of hydrolysis (base or acid catalyzed). Beckmann rearrangement is used for the industrial synthesis of caprolactam (see applications below).
The Ponzio reaction (1906) [ 14 ] concerning the conversion of m -nitrobenzaldoxime to m -nitrophenyldinitromethane using dinitrogen tetroxide was the result of research into TNT analogues: [ 15 ]
Gentler oxidants give mono-nitro compounds. [ 16 ]
In the Neber rearrangement certain oximes are converted to the corresponding alpha-amino ketones.
Oximes can be dehydrated using acid anhydrides to yield corresponding nitriles .
Certain amidoximes react with benzenesulfonyl chloride to make substituted ureas in the Tiemann rearrangement : [ 17 ] [ 18 ]
In their largest application, an oxime is an intermediate in the industrial production of caprolactam , a precursor to Nylon 6 . About half of the world's supply of cyclohexanone , more than a million tonnes annually, is converted to the oxime. In the presence of sulfuric acid catalyst , the oxime undergoes the Beckmann rearrangement to give the cyclic amide caprolactam: [ 19 ]
Oximes are commonly used as ligands and sequestering agents for metal ions. Dimethylglyoxime (dmgH 2 ) is a reagent for the analysis of nickel and a popular ligand in its own right. In the typical reaction, a metal reacts with two equivalents of dmgH 2 concomitant with ionization of one proton. Salicylaldoxime is a chelator in hydrometallurgy . [ 20 ]
Amidoximes such as polyacrylamidoxime can be used to capture trace amounts of uranium from sea water. [ 21 ] [ 22 ] In 2017 researchers announced a configuration that absorbed up to nine times as much uranyl as previous fibers without saturating. [ 23 ] | https://en.wikipedia.org/wiki/Tiemann_rearrangement |
Tierra is a computer simulation developed by ecologist Thomas S. Ray in the early 1990s in which computer programs compete for time (central processing unit ( CPU ) time) and space (access to main memory ). In this context, the computer programs in Tierra are considered to be evolvable and can mutate , self-replicate and recombine . Tierra's virtual machine is written in C . [ 1 ] It operates on a custom instruction set designed to facilitate code changes and reordering, including features such as jump to template [ 2 ] (as opposed to the relative or absolute jumps common to most instruction sets).
The basic Tierra model has been used to experimentally explore in silico the basic processes of evolutionary and ecological dynamics. Processes such as the dynamics of punctuated equilibrium , host-parasite co-evolution and density-dependent natural selection are amenable to investigation within the Tierra framework. A notable difference between Tierra and more conventional models of evolutionary computation , such as genetic algorithms , is that there is no explicit, or exogenous fitness function built into the model. Often in such models there is the notion of a function being "optimized"; in the case of Tierra, the fitness function is endogenous: there is simply survival and death.
According to Thomas S. Ray and others, this may allow for more "open-ended" evolution, in which the dynamics of the feedback between evolutionary and ecological processes can itself change over time (see evolvability ), although this claim has not been realized – like other digital evolution systems, it eventually reaches a point where novelty ceases to be created, and the system at large begins either looping or ceases to 'evolve'. The issue of how true open-ended evolution can be implemented in an artificial system is still an open question in the field of artificial life . [ 3 ]
Mark Bedau and Norman Packard developed a statistical method of classifying evolutionary systems and in 1997, Bedau et al. applied these statistics to Evita, an Artificial life model similar to Tierra and Avida , but with limited organism interaction and no parasitism, and concluded that Tierra-like systems do not exhibit the open-ended evolutionary signatures of naturally evolving systems. [ 4 ]
Russell K. Standish has measured the informational complexity of Tierran 'organisms', and has similarly not observed complexity growth in Tierran evolution. [ 5 ]
Tierra is an abstract model, but any quantitative model is still subject to the same validation and verification techniques applied to more traditional mathematical models , and as such, has no special status. The creation of more detailed models in which more realistic dynamics of biological systems and organisms are incorporated is now an active research field (see systems biology ). | https://en.wikipedia.org/wiki/Tierra_(computer_simulation) |
In topology , the Tietze extension theorem (also known as the Tietze– Urysohn – Brouwer extension theorem or Urysohn-Brouwer lemma [ 1 ] ) states that any real-valued , continuous function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
If X {\displaystyle X} is a normal space and f : A → R {\displaystyle f:A\to \mathbb {R} } is a continuous map from a closed subset A {\displaystyle A} of X {\displaystyle X} into the real numbers R {\displaystyle \mathbb {R} } carrying the standard topology , then there exists a continuous extension of f {\displaystyle f} to X ; {\displaystyle X;} that is, there exists a map F : X → R {\displaystyle F:X\to \mathbb {R} } continuous on all of X {\displaystyle X} with F ( a ) = f ( a ) {\displaystyle F(a)=f(a)} for all a ∈ A . {\displaystyle a\in A.} Moreover, F {\displaystyle F} may be chosen such that sup { | f ( a ) | : a ∈ A } = sup { | F ( x ) | : x ∈ X } , {\displaystyle \sup\{|f(a)|:a\in A\}~=~\sup\{|F(x)|:x\in X\},} that is, if f {\displaystyle f} is bounded then F {\displaystyle F} may be chosen to be bounded (with the same bound as f {\displaystyle f} ).
The function F {\displaystyle F} is constructed iteratively. Firstly, we define c 0 = sup { | f ( a ) | : a ∈ A } E 0 = { a ∈ A : f ( a ) ≥ c 0 / 3 } F 0 = { a ∈ A : f ( a ) ≤ − c 0 / 3 } . {\displaystyle {\begin{aligned}c_{0}&=\sup\{|f(a)|:a\in A\}\\E_{0}&=\{a\in A:f(a)\geq c_{0}/3\}\\F_{0}&=\{a\in A:f(a)\leq -c_{0}/3\}.\end{aligned}}} Observe that E 0 {\displaystyle E_{0}} and F 0 {\displaystyle F_{0}} are closed and disjoint subsets of A {\displaystyle A} . By taking a linear combination of the function obtained from the proof of Urysohn's lemma , there exists a continuous function g 0 : X → R {\displaystyle g_{0}:X\to \mathbb {R} } such that g 0 = c 0 3 on E 0 g 0 = − c 0 3 on F 0 {\displaystyle {\begin{aligned}g_{0}&={\frac {c_{0}}{3}}{\text{ on }}E_{0}\\g_{0}&=-{\frac {c_{0}}{3}}{\text{ on }}F_{0}\end{aligned}}} and furthermore − c 0 3 ≤ g 0 ≤ c 0 3 {\displaystyle -{\frac {c_{0}}{3}}\leq g_{0}\leq {\frac {c_{0}}{3}}} on X {\displaystyle X} . In particular, it follows that | g 0 | ≤ c 0 3 | f − g 0 | ≤ 2 c 0 3 {\displaystyle {\begin{aligned}|g_{0}|&\leq {\frac {c_{0}}{3}}\\|f-g_{0}|&\leq {\frac {2c_{0}}{3}}\end{aligned}}} on A {\displaystyle A} . We now use induction to construct a sequence of continuous functions ( g n ) n = 0 ∞ {\displaystyle (g_{n})_{n=0}^{\infty }} such that | g n | ≤ 2 n c 0 3 n + 1 | f − g 0 − . . . − g n | ≤ 2 n + 1 c 0 3 n + 1 . {\displaystyle {\begin{aligned}|g_{n}|&\leq {\frac {2^{n}c_{0}}{3^{n+1}}}\\|f-g_{0}-...-g_{n}|&\leq {\frac {2^{n+1}c_{0}}{3^{n+1}}}.\end{aligned}}} We've shown that this holds for n = 0 {\displaystyle n=0} and assume that g 0 , . . . , g n − 1 {\displaystyle g_{0},...,g_{n-1}} have been constructed. Define c n − 1 = sup { | f ( a ) − g 0 ( a ) − . . . − g n − 1 ( a ) | : a ∈ A } {\displaystyle c_{n-1}=\sup\{|f(a)-g_{0}(a)-...-g_{n-1}(a)|:a\in A\}} and repeat the above argument replacing c 0 {\displaystyle c_{0}} with c n − 1 {\displaystyle c_{n-1}} and replacing f {\displaystyle f} with f − g 0 − . . . − g n − 1 {\displaystyle f-g_{0}-...-g_{n-1}} . Then we find that there exists a continuous function g n : X → R {\displaystyle g_{n}:X\to \mathbb {R} } such that | g n | ≤ c n − 1 3 | f − g 0 − . . . − g n | ≤ 2 c n − 1 3 . {\displaystyle {\begin{aligned}|g_{n}|&\leq {\frac {c_{n-1}}{3}}\\|f-g_{0}-...-g_{n}|&\leq {\frac {2c_{n-1}}{3}}.\end{aligned}}} By the inductive hypothesis, c n − 1 ≤ 2 n c 0 / 3 n {\displaystyle c_{n-1}\leq 2^{n}c_{0}/3^{n}} hence we obtain the required identities and the induction is complete. Now, we define a continuous function F n : X → R {\displaystyle F_{n}:X\to \mathbb {R} } as F n = g 0 + . . . + g n . {\displaystyle F_{n}=g_{0}+...+g_{n}.} Given n ≥ m {\displaystyle n\geq m} , | F n − F m | = | g m + 1 + . . . + g n | ≤ ( ( 2 3 ) m + 1 + . . . + ( 2 3 ) n ) c 0 3 ≤ ( 2 3 ) m + 1 c 0 . {\displaystyle {\begin{aligned}|F_{n}-F_{m}|&=|g_{m+1}+...+g_{n}|\\&\leq \left(\left({\frac {2}{3}}\right)^{m+1}+...+\left({\frac {2}{3}}\right)^{n}\right){\frac {c_{0}}{3}}\\&\leq \left({\frac {2}{3}}\right)^{m+1}c_{0}.\end{aligned}}} Therefore, the sequence ( F n ) n = 0 ∞ {\displaystyle (F_{n})_{n=0}^{\infty }} is Cauchy . Since the space of continuous functions on X {\displaystyle X} together with the sup norm is a complete metric space , it follows that there exists a continuous function F : X → R {\displaystyle F:X\to \mathbb {R} } such that F n {\displaystyle F_{n}} converges uniformly to F {\displaystyle F} . Since | f − F n | ≤ 2 n c 0 3 n + 1 {\displaystyle |f-F_{n}|\leq {\frac {2^{n}c_{0}}{3^{n+1}}}} on A {\displaystyle A} , it follows that F = f {\displaystyle F=f} on A {\displaystyle A} . Finally, we observe that | F n | ≤ ∑ n = 0 ∞ | g n | ≤ c 0 {\displaystyle |F_{n}|\leq \sum _{n=0}^{\infty }|g_{n}|\leq c_{0}} hence F {\displaystyle F} is bounded and has the same bound as f {\displaystyle f} . ◻ {\displaystyle \square }
L. E. J. Brouwer and Henri Lebesgue proved a special case of the theorem, when X {\displaystyle X} is a finite-dimensional real vector space . Heinrich Tietze extended it to all metric spaces , and Pavel Urysohn proved the theorem as stated here, for normal topological spaces. [ 2 ] [ 3 ]
This theorem is equivalent to Urysohn's lemma (which is also equivalent to the normality of the space) and is widely applicable, since all metric spaces and all compact Hausdorff spaces are normal. It can be generalized by replacing R {\displaystyle \mathbb {R} } with R J {\displaystyle \mathbb {R} ^{J}} for some indexing set J , {\displaystyle J,} any retract of R J , {\displaystyle \mathbb {R} ^{J},} or any normal absolute retract whatsoever.
If X {\displaystyle X} is a metric space, A {\displaystyle A} a non-empty subset of X {\displaystyle X} and f : A → R {\displaystyle f:A\to \mathbb {R} } is a Lipschitz continuous function with Lipschitz constant K , {\displaystyle K,} then f {\displaystyle f} can be extended to a Lipschitz continuous function F : X → R {\displaystyle F:X\to \mathbb {R} } with same constant K . {\displaystyle K.} This theorem is also valid for Hölder continuous functions , that is, if f : A → R {\displaystyle f:A\to \mathbb {R} } is Hölder continuous function with constant less than or equal to 1 , {\displaystyle 1,} then f {\displaystyle f} can be extended to a Hölder continuous function F : X → R {\displaystyle F:X\to \mathbb {R} } with the same constant. [ 4 ]
Another variant (in fact, generalization) of Tietze's theorem is due to H.Tong and Z. Ercan: [ 5 ] Let A {\displaystyle A} be a closed subset of a normal topological space X . {\displaystyle X.} If f : X → R {\displaystyle f:X\to \mathbb {R} } is an upper semicontinuous function, g : X → R {\displaystyle g:X\to \mathbb {R} } a lower semicontinuous function, and h : A → R {\displaystyle h:A\to \mathbb {R} } a continuous function such that f ( x ) ≤ g ( x ) {\displaystyle f(x)\leq g(x)} for each x ∈ X {\displaystyle x\in X} and f ( a ) ≤ h ( a ) ≤ g ( a ) {\displaystyle f(a)\leq h(a)\leq g(a)} for each a ∈ A {\displaystyle a\in A} , then there is a continuous
extension H : X → R {\displaystyle H:X\to \mathbb {R} } of h {\displaystyle h} such that f ( x ) ≤ H ( x ) ≤ g ( x ) {\displaystyle f(x)\leq H(x)\leq g(x)} for each x ∈ X . {\displaystyle x\in X.} This theorem is also valid with some additional hypothesis if R {\displaystyle \mathbb {R} } is replaced by a general locally solid Riesz space . [ 5 ]
Dugundji (1951) extends the theorem as follows: If X {\displaystyle X} is a metric space, Y {\displaystyle Y} is a locally convex topological vector space , A {\displaystyle A} is a closed subset of X {\displaystyle X} and f : A → Y {\displaystyle f:A\to Y} is continuous, then it could be extended to a continuous function f ~ {\displaystyle {\tilde {f}}} defined on all of X {\displaystyle X} . Moreover, the extension could be chosen such that f ~ ( X ) ⊆ conv f ( A ) {\displaystyle {\tilde {f}}(X)\subseteq {\text{conv}}f(A)} | https://en.wikipedia.org/wiki/Tietze_extension_theorem |
The Tiffeneau–Demjanov rearrangement is the chemical reaction of a 1-aminomethyl-cycloalkanol with nitrous acid to form an enlarged cycloketone.
The Tiffeneau–Demjanov ring expansion, Tiffeneau–Demjanov rearrangement, or TDR, provides an easy way to increase amino-substituted cycloalkanes and cycloalkanols in size by one carbon. Ring sizes from cyclopropane through cyclooctane are able to undergo Tiffeneau–Demjanov ring expansion with some degree of success. Yields decrease as initial ring size increases, and the ideal use of TDR is for synthesis of five, six, and seven membered rings. A principal synthetic application of Tiffeneau–Demjanov ring expansion is to bicyclic or polycyclic systems. Several reviews on this reaction have been published. [ 1 ] [ 2 ] [ 3 ]
The reaction now known as the Tiffeneau–Demjanov rearrangement (TDR) was discovered in two steps. The first step of occurred in 1901 when Russian chemist Nikolai Demyanov discovered that aminomethylcycloalkanes produce novel products upon treatment with nitrous acid. When this product was identified as the expanded alcohol in 1903, the Demjanov rearrangement was coined.
The Demjanov rearrangement itself has since been successfully used in industry and synthetical organic chemistry. However, its scope is limited. The Demjanov rearrangement is only best suited for expanding four, five, and six member aminomethylcycloalkanes. Moreover, alkenes and un-expanded cycloalcohols form as by-products. Yields diminish as the starting cycloalkane becomes larger.
A discovery by French scientists a few years before World War II would result in the modern TDR reaction. In 1937, Marc Tiffeneau , Weill, and Tchoubar published in Comptes Rendus their finding that 1-aminomethylcycloahexanol converts readily to cycloheptanone upon treatment with nitrous acid. [ 4 ] Perhaps due to such a large ring being expanded, the authors did not immediately relate it to the Demjanov rearrangement. Instead, they envisioned that their reaction was similar to one discovered by Wallack in 1906. Upon oxidation with permanganate , cycloglycols will dehydrate to yield an aldehyde via an epoxide intermediate. The authors postulated that deamination resulted in a similar epoxide intermediate that subsequently formed a ring enlarge cycloketone. However, in the time that followed, scientists began to realize that these reactions were related. By the early 1940s, TDR was in organic vernacular. Tiffeneau's discovery enlarged the synthetic scope of the Demjanov rearrangement as now seven and eight carbon rings could be enlarged. Since the resulting cycloketone could be easily converted to a cycloaminoalcohol again, this new method quickly became popular among organic chemists.
The basic reaction mechanism is a diazotation of the amino group by nitrous acid followed by expulsion of nitrogen and formation of a primary carbocation . A rearrangement reaction with ring expansion forms a more stable oxonium ion which is deprotonated . [ 5 ]
Although chemists at the time knew very well what the product of a symmetrical 1-aminomethylcycloalcohol would be when exposed to nitrous acid, there was significant debate on the reaction's mechanism that lasted up until the 1980s. Scientists were puzzled over the array of products they would obtain when the reaction was performed on an unsymmetrical 1-aminomethylcycloalcohols and bridged cyclo-systems. Even today, experiments continue that are designed to shed light into the more subtle mechanistic features of this reaction and increase yields of desired expanded products.
In 1960, Peter A.S. Smith and Donald R. Baer , both of the University of Michigan , published a treatise on the TDR. Their proposed mechanism contained within provides an excellent perspective of scientist's understanding of the TDR at that time.
The mechanism proposed by Baer and Smith was the summation of several sources of information. Since the early 1950s, it had been postulated by many that the TDR mechanism involved a carbonium ion. However, a major breakthrough in the development of the TDR mechanism came with the improved understanding of the phenomenon behind amine groups reacting with nitrous acid. Meticulous kinetic studies throughout the late 1950s led scientists to believe that nitrous acid reacts with an amine by first producing a nitrous acid derivative, potentially N2O3. While this derivative would prove incorrect as it relates to TDR, scientists of the time still correctly came to the conclusion that the derivative would react with the amine to produce the diazonium ion. The inferred instability of this diazonium ion gave solid evidence for the existence of a carbocation in the TDR mechanism.
Another piece of information that had implications in the mechanism of the TDR was the simple fact that the reaction proceeds more easily with the conditions discovered by Tiffeneau. By placing an alcohol on the carbon on the reagent, reaction rates and yields are much improved to those of the simple Demjanov rearrangement. Moreover, few unwanted by products are formed, such as olefins. These aforementioned observations were the center around which Smith and Baer's mechanism was constructed. It is easy to see that hydrogen's presence would mean that hydride shifts would occur in competition with carbon shifts during the course of the reaction. Moreover, this shift is likely as it would move place positive charge from a 1° carbon to a 3° carbon. In a mildly basic solvent such as water, this new intermediate could easily produce an olefin by an E1-like reaction.
Such olefins are typically seen in simple Demjanov rearrangements but are not seen in the TDR. The alcohol's presence explains how this E1 reaction does not occur. Moreover, having an alcohol present puts the developing positive charge of the ring enlarged intermediate next to an oxygen. This would be more favorable than hydrogen as oxygen can lend electron density to the carbonium ion via resonance.
This again favors ring expansion and is another caveat that shows how it incorporates higher yields of the TDR over the Demjanov rearrangement.
Smith and Baer' mechanism was also able to account for other observations of the time. Tiffeneau–Demjanov rearrangements of1-aminomethylcycloalkanols with alkyl substitutions on the side aminomethyl chain had been accomplished by many scientists before 1960. Smith and Baer investigated how such substitution affects the TDR by synthesizing various 1-hydroxycyclohexylbenzyl-amines and exposing them to TDR conditions.
Seeing as six member rings are routinely enlarged by the TDR, one might expect the reaction to occur. However instead of the anticipated ring enlargements, only diols are seen as products. Five member analogues to the above substituted reagents enlarge under TDR conditions. Alkyl substitutions as opposed to aryl substitutions result in diminished TDRs. Smith and Baer assert that these observations support their mechanism. Since substitution stabilizes the carbonium ion after damnification, the resulting carbonium ion is more likely to react with a nucleophile present (water in this case) and not undergo rearrangement. Five member rings rearrange due to the ring strain encouraging the maneuver. This strain makes the carbocation unstable enough to cause a carbon to shift.
As definitive as Smith and Baer's early mechanism seems, there are several phenomena that it did not account for. The problem with their mechanism mainly focused around TDR precursors that have alkyl substituents on the ring. When said substituent is placed on the ring as to make the molecule still symmetric, one product is formed upon exposure to TDR conditions. However, if the alkyl is placed on the ring as to make the molecule unsymmetric, several products could form.
The principal method for synthesizing the starting amino alcohols is through the addition of cyanide anion to a cyclic ketone. The resulting hydroxynitrile is then reduced, forming the desired amino alcohol. This method forms diastereomers, possibly affecting the regioselectivity of the reaction. For nearly all asymmetric precursors, one product isomer is formed preferentially to another. As TDR was routinely being used to synthesize various steroids and bicyclic compounds, their precursors were rarely symmetric. As a result, a lot of time was spent identifying and separating products. At the time, this phenomenon baffled chemists. Due to spectroscopic and separation limitations, it was very difficult for scientists to probe this caveat of the TDR in a sophisticated way. However, most believed that what was governing preferential product formation involved the migratory aptitudes of competing carbons and/or steric control. Migratory aptitude made reference to the possibility that the preferred product of the reaction was the result of an initial stability of one carbon migrating in preference to another. This possibility was more the belief and subject of research of earlier scientists, including Marc Tiffeneau himself. However, in the early 1960s, more and more scientists were starting to think that steric factors were the driving force behind the selectivity for this reaction.
As chemists continued to probe this reaction with more and more advanced technology and methods, other factors began to be tabled as possibilities for what was controlling product formation of unsymmetrical amino alcohols. In 1963, Jones and Price of the University of Toronto demonstrated how remote substituents in steroids play a role in product distribution. In 1968, Carlson and Behn of the University of Kansas discovered that experimental conditions also play a role. These latter scientists established that in ring extension via the TDR, initial temperature and concentration of reagents all played a role in eventual product distribution. Indeed, other avenues of the TDR were being explored and charted.
However, Carlson and Behn did manage to report a significant breakthrough in the realm of sterics and migratory aptitudes as they relate to the TDR. As it might be expected based on electronic reasoning, the more highly substituted carbon should migrate preferentially to a less substituted carbon. However, this is not always seen and often accounts of migratory aptitudes show fickle preferences. Thus, the authors assert that such aptitudes are of little importance. Sterically, thanks chiefly to improved spectroscopic methods, they were able to confirm that having the amine group equatorial to the alkane ring corresponded to drastically different product yields.
According to the authors, the preferential formation of D from A does not reflect a preferred conformation of A. Their modeling indicates that both A and B are initially just as likely to become C. He concludes that there must be a steric interaction to develop in the transition state during migration that makes A preferentially form D when exposed to the TDR conditions. The idea that sterics played a factor during migration and was not a factor just at the beginning to the reaction, was new. Carlson and Behn speculate that the factor might lay in transannular hydrogen interactions along the path of migration. Their modeling suggested that this interaction may be more severe for A forming C. However, they are not certain enough to offer this as a definitive explanation as they concede that more subtle conformational and/or electronic effects could be at work as well.
At this point, the mechanism proposed by Smith and Baer seemed to be on its way out. If steric interactions relating to carbon migration during the reaction's transition state were important, this did not support the carbocation envisioned by Smith and Baer. Research around bi-cyclics during the 1970s would shed even more light into the TDR mechanism. In 1973, McKinney and Patel of Marquette University published an article in which they used the TDR for expanding norcamphor and dehydronorcamphor . Two of their observations are important. One centers on the expansion of exo and endo-2-norbornylcarbinyl systems.
One might expect in (I) that A would migrate in preference to B seeing as such a migration would place the developing charge on a 2° carbon and pass the specie through a more favorable chair-like intermediate. This is not seen. Only 38% of the product exhibits A migration. To account for why A migration is not dominant in the expansion of I, the authors assert a least movement argument. Simply put, the migration of the non-bridgehead carbon provides for the least amount of total atom movement, something that plays into the energetics of the reaction. This least movement consideration would prove important in the TDR mechanism as it accounts for products with intermediates passing through unfavorable conformations.
However, McKinney and Patel also confirm that traditional factors such as developing positive charge stability still play a crucial role in the direction of expansion. They accomplish this by expanding 2-norbornenyl carbinyl systems.
By adding a simple double bond to these systems, the authors see a significant increase in the migration of the bridgehead carbon A (50% in this case.) The authors attribute this jump in migration to the fact that this bride carbon migrating allows the developing positive charge to be stabilizing by resonance contributed by the double bond. Therefore, carbocation/ positive charge effects can not be ignored in the discussion of the factors influencing product distribution.
As evidence continued to mount during the years after Smith and Baer's publication in 1960, it was obvious that the TDR mechanism would need revisiting. This new mechanism would have to de-stress the carbocation as there are other factors that influence ring expansion. Orientation of the developing diazonium ion, the possibility of steric interactions during the reaction, and atomic movement would all have to be included. In 1982, Cooper and Jenner published such a mechanism. [ 6 ] Their mechanism has stood to this day as the current understanding of the TDR.
The most obvious departure from Smith and Baer's mechanism is that Cooper and Jenner represent the diazonium departure and subsequent alkyl shift as a concerted step. Such a feature allows for sterics, orientations, and atomic movement to be factors. However, distribution of positive charge is still important in this mechanism as it does explain much of the observed behavior of the TDR. Another observation that should be made is that there is no preference given to these aforementioned factors in the mechanism. That is to say, even today it is very difficult to predict which carbon will migrate preferentially. Indeed, the TDR has become more useful as spectroscopic and separation techniques have advanced. Such advancements allows for the quick identification and isolation of desired products.
Since the mid-1980s, most organic chemists have resigned themselves to accepting the fact that the TDR is governed by several factors that often seem fickle in importance. As a result, much research is now being directed towards the development of techniques to increase migration of a specific carbon. One example of such an effort has recently come out of the University of Melbourne .
Noting that group 4 metal substituents can stabilize positive charge that is β to them, Chow, McClure, and White attempted to use this to direct TDRs in 2004. [ 7 ] They hypothesized that placing a silicon trimethyl group β to a carbon that can migrate would increase such migration.
Their results show that this does occur to a small extent. The authors believe that the reason why the carbon migration increases only slightly is that positive charge is not a large factor in displacing the diazonium ion. Since this ion is such a good leaving group, it requires very little 'push' from the developing carbon-carbon bond. Their results again highlight the fact that multiple factors determine the direction of carbon migration. | https://en.wikipedia.org/wiki/Tiffeneau–Demjanov_rearrangement |
Tiger bush , or brousse tigrée in the French language, is a patterned vegetation community and ground consisting of alternating bands of trees , shrubs , or grass separated by bare ground or low herb cover, that run roughly parallel to contour lines of equal elevation. The patterns occur on low slopes in arid and semi-arid regions, [ 1 ] such as in Australia , Sahelian West Africa, and North America . [ 2 ] [ 3 ]
Due to the natural water harvesting capacity, many species in tiger bush usually occur only under a higher rainfall regime.
The alternating pattern arises from the interplay of hydrological , ecological , and erosional phenomena. In the regions where tiger bush is present, plant growth is water-limited - the shortage of rainfall prevents vegetation from covering the entire landscape. Instead, trees and shrubs are able to establish by either tapping soil moisture reserves laterally or by sending roots to deeper, wetter soil depths. By a combination of plant litter , root macropores, and increased surface roughness, infiltration into the soil around the base of these plants is enhanced. Surface runoff arriving at these plants will thus likely to become run-on , and infiltrate into the soil.
By contrast, the areas between these larger plants contain a greater portion of bare ground and herbaceous plants. Both bare soil, with its smoother surface and soil crusts , and herbaceous plants, with fewer macropores, inhibit infiltration. This causes much of the rainfall that falls in the inter-canopy areas to flow downslope, and infiltrate beneath the larger plants. The larger plants are in effect harvesting rainfall from the ground immediately up-slope. [ 4 ]
Although these vegetation patterns may seem very stable through time, such patterning requires specific climatic conditions. For instance, a decrease in rainfall is able to trigger patterning in formerly homogeneous vegetation within a few decades. [ 5 ]
More water will infiltrate at the up-slope edge of the canopies than down-slope. [ 6 ] This favours the establishment and growth of plants at the up-slope edge, and mortality of those down-slope. Differences in growth and mortality across the vegetation band result in the band moving gradually upslope. [ 7 ] [ 8 ] [ 9 ]
Tiger bush never develops on moderate to steep slopes, because in these cases surface runoff concentrates into narrow threads or rills instead of flowing over the surface as sheet flow. Sheet flow distributes water more evenly across a hillslope, allowing a continuous vegetation band to form.
The exact roles and importance of the different phenomena is still the subject of research, especially of research in physics since the 1990s.
The woody plants which make up tiger bush are used for fire wood and as a source of foliage for grazers . The extensive loss of tiger bush around Niamey , Niger , now threatens local giraffe populations. In neighbouring Burkina Faso , the tiger bush vegetation is also declining.
The pattern was first described in 1950 in British Somaliland by W.A. Macfadyen. [ 10 ] The term tiger bush was first coined by Albert Clos-Arceduc in 1956. [ 11 ] | https://en.wikipedia.org/wiki/Tiger_bush |
Tiger eye or goat eye is a gene causing diluted eye color in horses. There are two variants, Tiger-eye 1 (TE1) and Tiger-eye 2 (TE2), which are both recessive . [ 1 ] Horses displaying tiger eye typically have a yellow, orange, or amber iris . Tiger eye has only been found in Puerto Rican Paso Fino horses. Horses of related breeds were tested (90 Colombian Pasos , 20 Mangalargas , 44 Lusitanos , and 42 Andalusian horses ), and none were found to have either tiger eye allele. No obvious link between eye shade and coat color was seen, making this the first studied gene in horses to affect eye color but not coat color. Tiger eye does not appear to affect vision, and there were no signs of reduced pigment on the retina or retinal pigment epithelium . [ 2 ]
The gene involved codes for SLC24A5 , a solute carrier known to be involved in pigmentation in other species. SLC24A5 is found on equine chromosome 1 base pairs 141,657,837–141,678,329 and the protein is a potassium-dependent sodium–calcium ion exchanger involved in melanocyte maturation. The protein is believed to be located in the trans -golgi network of melanocytes. Tiger-eye 1 is a missense mutation (c.272A>T and p.Phe91Tyr) in which a single adenine is replaced with a thymine in exon 2, changing a phenylalanine to a tyrosine in the resulting protein. Tiger-eye 2 is a deletion (c.875-340_1081+82del) in which the entirety of exon 7, and a bit of the introns on either side, are removed, resulting in a protein that is 69 amino acids shorter. Both mutations are predicted to be deleterious to protein function. [ 2 ] [ 3 ]
Another mutation at this same gene is associated with the shade of black horses. One variant is mainly found in jet black horses, while the other variant is mainly found in fading black horses. However, the association is not perfect and there are probably other mutations that can also cause the fading black appearance. [ 4 ]
SLC24A5 is involved in pigmentation in humans, mice, and zebrafish. In mice a targeted mutation was found which diluted the eye color without visibly affecting coat color, though a closer examination found that the melanosomes were smaller and paler than the wild type . [ 5 ] Unlike tiger eye in horses, the mice showed reduced pigment in the retinal pigment epithelium. In humans a widespread mutation to the homologous gene plays a large role in the light skin color of European humans, [ 6 ] and another mutation can cause oculocutaneous albinism (OCA) type 6 (OCA6), which impairs vision. No vision impairment is seen in horses, and the coat color is not visibly affected. [ 2 ] | https://en.wikipedia.org/wiki/Tiger_eye |
A tiger team is a team of specialists assembled to work on a specific goal, [ 1 ] or to solve a particular problem. [ 2 ]
A 1964 paper entitled Program Management in Design and Development used the term tiger teams and defined it as "a team of undomesticated and uninhibited technical specialists, selected for their experience, energy, and imagination, and assigned to track down relentlessly every possible source of failure in a spacecraft subsystem or simulation". [ 2 ] Walter C. Williams gave this definition in response to the question "How best can advancements in reliability/maintainability state-of-the-art be attained and used with compressed schedules?" Williams was an engineer at the Manned Spacecraft Center and part of the Edwards Air Force Base National Advisory Committee for Aeronautics.
The paper consists of anecdotes and answers to questions from a panel on improving issues in program management concerning testing and quality assurance in aerospace vehicle development and production. [ 3 ] The panel consisted of Williams, Col. J. R. Dempsey of General Dynamics , [ 4 ] Lt. Gen. W. A. Davis [ 5 ] from the Ballistic Systems Div., Norton Air Force Base, A. S. Crossfield from North American Aviation . | https://en.wikipedia.org/wiki/Tiger_team |
Tight junction proteins ( TJ proteins ) are molecules situated at the tight junctions of epithelial , endothelial and myelinated cells. This multiprotein junctional complex has a regulatory function in passage of ions, water and solutes through the paracellular pathway. It can also coordinate the motion of lipids and proteins between the apical and basolateral surfaces of the plasma membrane . Thereby tight junction conducts signaling molecules, that influence the differentiation, proliferation and polarity of cells. So tight junction plays a key role in maintenance of osmotic balance and trans-cellular transport of tissue specific molecules. Currently, more than 40 different proteins involved in these selective TJ channels have been identified. [ 1 ]
The morphology of tight junction is formed by transmembrane strands in the inner side of plasma membrane with complementary grooves on the outer side. This TJ strand network is composed by transmembrane proteins , that interact with the actin in cytoskeleton and with submembrane proteins, which send a signal into the cell. The complexity of the network structure depends on the cell type and it can be visualized and analyzed by freeze-fracture electron microscopy , which shows the individual strands of the tight junction. [ 2 ] [ 3 ]
TJ proteins could be divided in different groups according to their function or localization in tight junction. TJ proteins are mostly described in the epithelia and endothelia but also in myelinated cells. In the central and peripheral nervous system are TJ localized between a glia and an axon and within myelin sheaths , where they facilitate the signaling. Some of TJ proteins act as a scaffolds, that connect integral proteins with the actin in a cytoskeleton. Others have an ability to crosslink junctional molecules or transport vesicles through the tight junction. Some submembrane proteins are involved in the cell signaling and gene expression due to their specific binding to the transcription factor . The most important tight junction proteins are occludin , claudin and JAM family, that establish the backbone of tight junction and allow to passing of immune cells through the tissue. [ 1 ]
Proteins in epithelial and endothelial cells are occludin, claudin and tetraspanin , that each has a one or two different types of the conformation. All of them are created by four transmembrane regions with two (amino-, carboxyl-) extracellular domains, that are orientated towards the cytoplasm . But occludin has a structure with two similar extracellular loops compared to claudin and tetraspanin, which have one extracellular loop significantly longer than the other one. [ 1 ]
Occludin (60kDa) was the first identified component of tight junction. The tetraspan membrane protein is established by two extracellular loops, two extracellular domains and one short intracellular domain. The C-terminal domain of occludin is directly bound to ZO-1 , which interacts with actin filaments in cytoskeleton. It works as a transmitter from and to the tight junction, because of its association with signaling molecules ( PI3-kinase , PKC , YES , protein phosphases 2A , 1). [ 4 ] This TJ protein also participate in a selective diffusion of solutes along concentration gradient and transmigration of leukocytes across the endothelium and epithelium. Therefore the result of the overexpression of mutant occludin in epithelial cells leads to break down the barrier function of tight junction and changes in a migration of neutrophils . Occludin cooperates with members of the claudin family directly or indirectly and together they form the long strands of tight junction. [ 3 ]
The claudin family is composed by 24 members. Some of them haven't been well characterized yet but all members are encoded by 20-27kDa tetraspan proteins with two extracellular domains, one short intracellular domain and two extracellular loops, where is the first one notably larger than the second one. [ 1 ] The C-terminal domain of claudins is required for their stability and targeting. This domain contains PDZ-binding motif, that facilitate to bind them to the PDZ membrane proteins, like a ZO-1 , ZO-2 , ZO-3 , MUPP1. Each claudin has a specific variation and amount of charged aminoacids in the first extracellular loop. So through the repolarization of aminoacids could claudins selectively regulate the molecule transfer. In contrast to occludin, which makes paracellular holes for ion-trafficking between neighbour cells. [ 4 ] Claudins seem to be on a tissue specific manner, because some of them are expressed only in a specific cell type. Claudin 11 is expressed in oligodendrocytes and Sertoli cells or Claudin 5 is expressed in the vascular endothelial cells . [ 3 ]
Claudin 2,3,4,7,8,12,15 are present in epithelial cells throughout the segments of intestinal tract. Claudin 7 is occurred also in epithelial cells of the lung and kidney. Claudin-18 is expressed in the alveolar epithelial cells of the lung. [ 5 ] Most types of claudins have more than two isoforms , that have a distinguish size or function. The specific combination of these isoforms creates tight junction strands, while the occulin is not required for. Occludin play a role in selective regulation by an incorporating itself into the claudin-based strands. The different proportion of claudin species in the cell gives them specific barrier properties. Claudins also have a function in a signaling of the cell adhesion , for example Cldn 7 binds directly to adhesion molecule EpCAM on the cell membrane. And Cldn 16 is associated with reabsorption of divalent cations, because it locates in epithelial cells of thick ascending loop of Henle . [ 4 ]
OSP/ Claudin 11 is occurred in a myelin of nerve cells and between Sertoli cells, so it forms tight junctions in the CNS . This protein in a cooperation with the second loop of occludin maintains the blood-testis barrier and spermatogenesis . [ 1 ]
PMP22/gas-3 , called peripheral myelin protein, is located in the myelin sheath. The expression of this protein is associated with a differentiation of Schwann cells , an establishment of tight junction in the Schwamm cell membrane or a compact formation of myelin. It is also present in epithelial cells of lungs and intestine, where interacts with occludin and ZO-1, that together create the TJ in the epithelia. PMP22/gas-3 belongs to the epithelial membrane protein family ( EMP1 -3), which conducts a growth and differentiation of cells. [ 1 ]
OAP-1/TSPAN-3 cooperates with β1-integrin and OSP/Claudin11 within myelin sheaths of oligodendrocytes, thereby affects the proliferation and migration. [ 1 ]
Junctional adhesion molecules are divided in subgroups according to their composition and binding motif.
Glycosylated transmembrane proteins JAMs are classified in the immunoglobulin superfamily, that are formed by two extracellular Ig-like domains: the transmembrane region and the C-terminal cytoplasmatic domain. Members of this JAM family could express two distinguish binding motifs. First subgroup composed by JAM-A, JAM-B , JAM-C has a PDZ-domain binding motif type II at their C-termini, which interacts with the PDZ domain of ZO-1, AF-6, PAR-3 and MUPP1. [ 3 ] [ 4 ] JAM proteins are not a part of tight junction strands but they participate in a signalization that leads to an adhesion of monocytes and neutrophils and their transmigration through the epithelium. JAMs in epithelial cells can aggregate with TJ strands, that are made of polymers of claudin and occludin. JAM-A maintains barrier properties in the endothelium and the epithelium as well as JAM-B and -C in Sertoli cells and spermatids . [ 1 ]
The second subgroup of CAR , ESAM, CLMP and JAM4 proteins contains a PDZ-domain binding motif type I at their C-termini.
CAR ( coxsackie and adenovirus receptor) also belongs to the immunoglobulin superfamily , same like JAM proteins. CAR is expressed in the epithelia of trachea , bronchi , kidney, liver and intestine, where positively contributes to the barrier function of the tight junction. This protein mediates a neutrophil migration, cells contacts and an aggregation. It´s necessary for the embryonal heart development, especially for the organization of myofibrils in cardiomyocytes . CAR is associated with PDZ-scaffolding proteins MAGI-1b , PICK, PSD-95, MUPP1 and LNX. [ 6 ]
ESAM (endothelial cell selective adhesion molecule) is an immunoglobulin-transmembrane protein, which influences properties of the endothelial TJ. ESAM is present in endothelial cells and platelets but not in the epithelium and leukocytes . There, it directly binds to the MAGI-1 molecules through the ligation of C-terminal domain and PDZ-domain. This cooperation provides the formation of large molecular complex at tight junctions in the endothelium. [ 7 ]
JAM4 is a component of immunoglobulin superfamily JAM but it expresses a PDZ-domain binding motif class I (doesn´t express a class II like members JAM-A,-B,-C). The JAM4 is situated in a kidney glomeruli and an intestine epithelium, where cooperates with MAGI-1, ZO-1, occludin and effectively regulates the permeability of these cells. JAM4 has a cell adhesion activity, which is conducted by MAGI-1. [ 8 ]
Protein 0 is a major myelin protein of the peripheral nervous system, which integrates with PMP22 . Together they form and compact myelin sheaths of nerve cells. [ 1 ]
Plaque proteins are molecules, that are required for the coordination of signals coming from the plasma membrane. In recent years exist about 30 different proteins associated with cytoplasmatic properties of the tight junction.
One group of these proteins are attended in the organization of transmembrane proteins and the interaction with actin filaments . This PDZ-containing group is composed by ZO-1 , ZO-2 , ZO-3, AF-6, MAGI , MUPP1, PAR, PATJ, and the PDZ domain gives them a scaffolding function. PDZ domains are important for a clustering and an anchoring of transmembrane proteins. With the first group interacts one plaque protein without PDZ domain , called cingulin , which plays a key role in the cell adhesion.
The second group of plague proteins are used for a vesicular trafficking , barrier regulation and gene transcription , because certain of them are transcription factors or proteins with nuclear functions. Members of this second group are ZONAB, Ral-A, Raf-1 , PKC , symplekin , cingulin and some more. They are characterized by lacking of the PDZ domain. [ 1 ] | https://en.wikipedia.org/wiki/Tight_junction_proteins |
Tijana Rajh ( Serbian : Тијана Рајх ; born 1957) is an American materials scientist who is a professor and director of the Arizona State University School of Molecular Sciences. Her research considers the development of nanomaterials and materials for quantum technologies. She was awarded the Association for Women in Science Innovator Award in 2009, and named a Fellow of the American Association for the Advancement of Science in 2014.
Rajh was born in 1957, Belgrade, former Yugoslavia in a Serbo - Jewish family . [ 1 ] [ 2 ] Her father Zdenko Rajh (Reich, 1905–1990) was a publicist and lawyer, and her mother Gordana Nikolić Rajh was a linguist. [ 3 ] Despite being interested in Aristotle , Rajh became fascinated by better understanding the natural world. [ citation needed ] She completed her undergraduate and graduate degrees at the University of Belgrade , where she specialized in physical chemistry. After earning her doctorate, Rajh worked in solar energy research between National Renewable Energy Laboratory and the Boris Kidrič Institute in Belgrade, where she worked on photo-electrochemistry and semiconductors. [ 4 ]
Rajh worked at the Argonne National Laboratory , where she was eventually made an Argonne Distinguished Fellow. [ citation needed ] She worked on semiconducting nanocrystals for water splitting and electrochemistry. In particular, Rajh worked on the synthesis of the nanocrystals, and developed strategies to assemble them. She developed electron paramagnetic spectroscopy and other electron resonance techniques to understand spin effects during electron transfer. [ citation needed ] In 2009, she was awarded the Association for Women in Science Innovator Award, and she was named an American Association for the Advancement of Science Fellow in 2014. [ 5 ] [ 6 ] [ 7 ]
Alongside her work on nanomaterials, Rajh developed quantum-enabled strategies for sensing. [ 8 ] She showed that the high surface areas of metal–organic frameworks could be used to maximize sensitivity, permitting quantitive analysis using electron paramagnetic resonance . She has proposed that carbon nanotubes with highly confined electron spins could be used as qubits with record long coherence times. [ 9 ]
In 2021, Rajh was named Director of the Arizona State University School of Molecular Sciences. [ 2 ] [ 10 ] | https://en.wikipedia.org/wiki/Tijana_Rajh |
In number theory , Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x , y , n , m of the exponential diophantine equation
for exponents n and m greater than one, is finite. [ 1 ] [ 2 ]
The theorem was proven by Dutch number theorist Robert Tijdeman in 1976, [ 3 ] making use of Baker's method in transcendental number theory to give an effective upper bound for x , y , m , n . Michel Langevin computed a value of exp exp exp exp 730 for the bound. [ 1 ] [ 4 ] [ 5 ]
Tijdeman's theorem provided a strong impetus towards the eventual proof of Catalan's conjecture by Preda Mihăilescu . [ 6 ] Mihăilescu's theorem states that there is only one member of the set of consecutive power pairs, namely 9=8+1. [ 7 ]
That the powers are consecutive is essential to Tijdeman's proof; if we replace the difference of 1 by any other difference k and ask for the number of solutions
of
with n and m greater than one we have an unsolved problem, [ 8 ] called the generalized Tijdeman problem. It is conjectured that this set also will be finite. This would follow from a yet stronger conjecture of Subbayya Sivasankaranarayana Pillai (1931), see Catalan's conjecture , stating that the equation A y m = B x n + k {\displaystyle Ay^{m}=Bx^{n}+k} only has a finite number of solutions. The truth of Pillai's conjecture, in turn, would follow from the truth of the abc conjecture . [ 9 ] | https://en.wikipedia.org/wiki/Tijdeman's_theorem |
In applied mathematics, Tikhonov's theorem on dynamical systems is a result on stability of solutions of systems of differential equations . It has applications to chemical kinetics . [ 1 ] [ 2 ] The theorem is named after Andrey Nikolayevich Tikhonov .
Consider this system of differential equations:
Taking the limit as μ → 0 {\displaystyle \mu \to 0} , this becomes the "degenerate system":
where the second equation is the solution of the algebraic equation
Note that there may be more than one such function φ {\displaystyle \varphi } .
Tikhonov's theorem states that as μ → 0 , {\displaystyle \mu \to 0,} the solution of the system of two differential equations above approaches the solution of the degenerate system if z = φ ( x , t ) {\displaystyle \mathbf {z} =\varphi (\mathbf {x} ,t)} is a stable root of the "adjoined system" | https://en.wikipedia.org/wiki/Tikhonov's_theorem_(dynamical_systems) |
The tilde ( / ˈ t ɪ l d ə / , also / ˈ t ɪ l d , - d i , - d eɪ / ) [ 1 ] is a grapheme ⟨ ˜ ⟩ or ⟨ ~ ⟩ with a number of uses. The name of the character came into English from Spanish tilde , which in turn came from the Latin titulus , meaning 'title' or 'superscription'. [ 2 ] Its primary use is as a diacritic (accent) in combination with a base letter. Its freestanding form is used in modern texts mainly to indicate approximation .
The tilde was originally one of a variety of marks written over an omitted letter or several letters as a scribal abbreviation (a "mark of contraction"). [ 3 ] Thus, the commonly used words Anno Domini were frequently abbreviated to A o Dñi , with an elevated terminal with a contraction mark placed over the "n". Such a mark could denote the omission of one letter or several letters. This saved on the expense of the scribe's labor and the cost of vellum and ink. Medieval European charters written in Latin are largely made up of such abbreviated words with contraction marks and other abbreviations; only uncommon words were given in full.
The text of the Domesday Book of 1086, relating for example, to the manor of Molland in Devon (see adjacent picture), is highly abbreviated as indicated by numerous tildes.
The text with abbreviations expanded is as follows:
Mollande tempore regis Eduardi geldabat pro quattuor hidis et uno ferling. Terra est quadraginta carucae. In dominio sunt tres carucae et decem servi et triginta villani et viginti bordarii cum sedecim carucis. Ibi duodecim acrae prati et quindecim acrae silvae. Pastura tres leugae in longitudine et latitudine. Reddit quattuor et viginti libras ad pensam. Huic manerio est adjuncta Blachepole. Elwardus tenebat tempore regis Edwardi pro manerio et geldabat pro dimidia hida. Terra est duae carucae. Ibi sunt quinque villani cum uno servo. Valet viginti solidos ad pensam et arsuram. Eidem manerio est injuste adjuncta Nimete et valet quindecim solidos. Ipsi manerio pertinet tercius denarius de Hundredis Nortmoltone et Badentone et Brantone et tercium animal pasturae morarum.
On typewriters designed for languages that routinely use diacritics (accent marks), there are two possible solutions. Keys can be dedicated to precomposed characters or alternatively a dead key mechanism can be provided. With the latter, a mark is made when a dead key is typed, but unlike normal keys, the paper carriage does not move on and thus the next letter to be typed is printed under that accent. Typewriters for Spanish typically have a dedicated key for Ñ /ñ but, as Portuguese uses à /ã and Õ /õ, a single dead-key (rather than take two keys to dedicate) is the most practical solution.
The tilde symbol did not exist independently as a movable type or hot-lead printing character since the type cases for Spanish or Portuguese would include sorts for the accented forms.
The first ASCII standard (X3.64-1963) did not have a tilde. [ 4 ] : 246 Like Portuguese and Spanish, the French, German and Scandinavian languages also needed symbols in excess of the basic 26 needed for English. The ASA worked with and through the CCITT to internationalize the code-set, to meet the basic needs of at least the Western European languages.
It appears to have been at their May 13–15, 1963 meeting that the CCITT decided that the proposed ISO 7-bit code standard would be suitable for their needs if a lower case alphabet and five diacritical marks [...] were added to it. [ 5 ] At the October 29–31 meeting, then, the ISO subcommittee altered the ISO draft to meet the CCITT requirements, replacing the up-arrow and left-arrow with diacriticals, adding diacritical meanings to the apostrophe and quotation mark, and making the number sign a dual [ a ] for the tilde. [ 6 ]
Thus ISO 646 was born (and the ASCII standard updated to X3.64-1967), providing the tilde and other symbols as optional characters. [ 4 ] : 247 [ b ]
ISO 646 and ASCII incorporated many of the overprinting lower-case diacritics from typewriters, including tilde. Overprinting was intended to work by putting a backspace code between the codes for letter and diacritic. [ 8 ] However even at that time, mechanisms that could do this or any other overprinting were not widely available, did not work for capital letters, and were impossible on video displays, with the result that this concept failed to gain significant acceptance. Consequently, many of these free-standing diacritics (and the underscore ) were quickly reused by software as additional syntax, basically becoming new types of syntactic symbols that a programming language could use. As this usage became predominant, type design gradually evolved so these diacritic characters became larger and more vertically centered, making them useless as overprinted diacritics but much easier to read as free-standing characters that had come to be used for entirely different and novel purposes. Most modern fonts align the plain ASCII " spacing " (free-standing) tilde at the same level as dashes , or only slightly higher. [ citation needed ]
The free-standing tilde is at code 126 in ASCII, where it was inherited into Unicode as U+007E.
A similar shaped mark ( ⁓ ) is known in typography and lexicography as a swung dash : these are used in dictionaries to indicate the omission of the entry word. [ 9 ]
As indicated by the etymological origin of the word "tilde" in English, this symbol has been closely associated with the Spanish language . The connection stems from the use of the tilde above the letter ⟨n⟩ to form the (different) letter ⟨ñ⟩ in Spanish, a feature shared by only a few other languages , most of which are historically connected to Spanish. This peculiarity can help non-native speakers quickly identify a text as being written in Spanish with little chance of error. Particularly during the 1990s, Spanish-speaking intellectuals and news outlets demonstrated support for the language and the culture by defending this letter against globalisation and computerisation trends that threatened to remove it from keyboards and other standardised products and codes. [ 10 ] [ 11 ] The Instituto Cervantes , founded by Spain's government to promote the Spanish language internationally, chose as its logo a highly stylised Ñ with a large tilde. The 24-hour news channel CNN in the US later adopted a similar strategy on its existing logo for the launch of its Spanish-language version , therefore being written as CN͠N. And similarly to the National Basketball Association (NBA), the Spain men's national basketball team is nicknamed "ÑBA".
In Spanish itself the word tilde is used more generally for diacritics, including the stress-marking acute accent. [ 12 ] The diacritic ~ is more commonly called virgulilla or la tilde de la eñe , and is not considered an accent mark in Spanish, but rather simply a part of the letter ñ (much like the dot over ı makes an i character that is familiar to readers of English).
The English language does not use the tilde as a diacritic, though it is used in some loanwords . The standalone form of the symbol is used more widely. Informally, [ 13 ] it means "approximately" , "about", or "around", such as "~30 minutes before", meaning " approximately 30 minutes before". [ 14 ] [ 15 ] It may also mean "similar to", [ 16 ] including "of the same order of magnitude as", [ 13 ] such as " x ~ y " meaning that x and y are of the same order of magnitude. Another approximation symbol is the double tilde ≈ , meaning "approximately/almost equal to". [ 14 ] [ 16 ] [ 17 ] The tilde is also used to indicate congruence of shapes by placing it over an = symbol, thus ≅ .
In more recent digital usage, tildes on either side of a word or phrase have sometimes come to convey a particular tone that "let[s] the enclosed words perform both sincerity and irony", which can pre-emptively defuse a negative reaction. [ 18 ] For example, BuzzFeed journalist Joseph Bernstein interprets the tildes in the following tweet :
as a way of making it clear that both the author and reader are aware that the enclosed phrase – "spirit of the season" – "is cliche and we know this quality is beneath our author, and we don't want you to think our author is a cliche person generally". [ 18 ] [ c ] More uses are in the text messaging app Whatsapp used by the side of a username.
Among other uses, the symbol has been used on social media to indicate sarcasm . [ 19 ] It may also be used online, especially in informal writing such as fanfiction , to convey a cutesy, playful, or flirtatious tone. [ 20 ]
In some languages, the tilde is a diacritic mark placed over a letter to indicate a change in its pronunciation:
The tilde was firstly used in the polytonic orthography of Ancient Greek , as a variant of the circumflex , representing a rise in pitch followed by a return to standard pitch. [ 21 ]
Later, it was used to make abbreviations in medieval Latin documents. When an ⟨n⟩ or ⟨m⟩ followed a vowel, it was often omitted, and a tilde (physically, a small ⟨N⟩ ) was placed over the preceding vowel to indicate the missing letter; this is the origin of the use of tilde to indicate nasalization (compare the development of the umlaut as an abbreviation of ⟨e⟩ .) [ citation needed ] A tilde represented an omitted ⟨a⟩ or a syllable containing it. [ 22 ] The practice of using the tilde over a vowel to indicate omission of an ⟨n⟩ or ⟨m⟩ continued in printed books in French as a means of reducing text length until the 17th century. It was also used in Portuguese and Spanish . [ citation needed ]
The tilde was also used occasionally to make other abbreviations, such as over the letter ⟨q⟩ , making q̃ , to signify the word que ("that") [ citation needed ] . It also appears for qua and together with the letter ⟨p⟩ to form p̃ for pra . [ 22 ]
It is also as a small ⟨n⟩ that the tilde originated when written above other letters, marking a Latin ⟨n⟩ which had been elided in old Galician-Portuguese. In modern Portuguese it indicates nasalization of the base vowel: mão "hand", from Lat. manu- ; razões "reasons", from Lat. rationes . [ citation needed ] This usage has been adopted in the orthographies of several native languages of South America , such as Guarani and Nheengatu , as well as in the International Phonetic Alphabet (IPA) and many other phonetic alphabets. For example, [ljɔ̃] is the IPA transcription of the pronunciation of the French place-name Lyon .
In Breton , the symbol ⟨ñ⟩ after a vowel means that the letter ⟨n⟩ serves only to give the vowel a nasalised pronunciation, without being itself pronounced, as it normally is. For example, ⟨an⟩ gives the pronunciation [ãn] whereas ⟨añ⟩ gives [ã] .
In the DMG romanization of Tunisian Arabic , the tilde is used for nasal vowels õ and ṏ.
The tilded ⟨n⟩ ( ⟨ñ⟩ , ⟨Ñ⟩ ) developed from the digraph ⟨nn⟩ in Spanish. In this language, ⟨ñ⟩ is considered a separate letter called eñe ( IPA: [ˈeɲe] ), rather than a letter-diacritic combination; it is placed in Spanish dictionaries between the letters ⟨n⟩ and ⟨o⟩ . In Spanish, the word tilde actually refers to diacritics in general, e.g. the acute accent in José , [ 23 ] while the diacritic in ⟨ñ⟩ is called "virgulilla" ( IPA: [birɣuˈliʝa] ) ( yeísta ) or ( IPA: [birɣuˈliʎa] ) (non-yeísta). [ 24 ] Current languages in which the tilded ⟨n⟩ ( ⟨ñ⟩ ) is used for the palatal nasal consonant /ɲ/ include
In Vietnamese , a tilde over a vowel represents a creaky rising tone ( ngã ). Letters with the tilde are not considered separate letters of the Vietnamese alphabet .
In phonetics , a tilde is used as a diacritic that is placed above a letter, below it or superimposed onto the middle of it:
A tilde between two phonemes indicates optionality, or "alternates with". E.g. ⟨ ɕ ~ ʃ ⟩ could indicate that the sounds may alternate depending on context ( free variation ), or that they vary based on region or speaker, or some other variation.
In Estonian , the symbol ⟨õ⟩ stands for the close-mid back unrounded vowel , and it is considered an independent letter.
Some languages and alphabets use the tilde for other purposes, such as:
The tilde is used in various ways in punctuation, including:
In some languages (such as in French), [ citation needed ] a tilde or a tilde-like wave dash (Unicode: U+301C 〜 WAVE DASH ) may be used as a punctuation mark (instead of an unspaced hyphen , en dash or em dash ) between two numbers , to indicate a range . Doing so avoids the risk of confusion with subtraction or a hyphenated number (such as a part number or model number). For example, "12~15" means "12 to 15", "~3" means "up to three", and "100~" means "100 and greater". [ citation needed ] East Asian languages almost always use this convention, but it is sometimes done for clarity in some other languages as well. Chinese uses the wave dash and full-width em dash interchangeably for this purpose. In English, the tilde is often used to express ranges and model numbers in electronics , but rarely in formal grammar or in type-set documents, as a wavy dash preceding a number sometimes represents an approximation (see below).
The range tilde is used for various purposes in French , but only to denote ranges of numbers (e.g., « 21~32 degrés Celsius » " means "21 to 32 degrees Celsius") [ citation needed ]
(The symbol U+2248 ≈ ALMOST EQUAL TO (a double tilde ) is also used in French, for example, « ≈400 mètres » means "approximately 400 meters" [ citation needed ] .)
Before a number the tilde can mean 'approximately'; '~42' means 'approximately 42'. [ 28 ] When used with currency symbols that precede the number (national conventions differ), the tilde precedes the symbol, thus for example '~$10' means 'about ten dollars'. [ 29 ] [ better source needed ]
The symbols ≈ (almost equal to) and ≅ (approximately equal to) are among the other symbols used to express approximation .
The wave dash ( 波ダッシュ , nami dasshu ) is used for various purposes in Japanese, including to denote ranges of numbers (e.g., 5〜10 means between 5 and 10) in place of dashes or brackets, and to indicate origin. The wave dash is also used to separate a title and a subtitle in the same line, as a colon is used in English.
When used in conversations via email or instant messenger it may be used as a sarcasm mark [ citation needed ] .
The sign is used as a replacement for the chōon , katakana character, in Japanese, extending the final syllable.
WeChat users frequently replace final punctuations with tildes in messages. An analysis of such "innovative uses" of tildes found that final tildes are most used to make the message friendlier and polite. They make expressives more sincere and directives less abrupt. Less commonly, final tildes imply sounds, i.e. otomatopeas and sound extensions. This use is compared to sajiao ( Chinese : 撒娇 ), a child-like acting seen in East Asian cultures that are also vocalized by raising or extending tone. [ 30 ]
A tilde in front of a single quantity can mean "approximately", "about" [ 14 ] or "of the same order of magnitude as."
In written mathematical logic , the tilde represents negation : "~ p " means "not p ", where " p " is a proposition . Modern use often replaces the tilde with the negation symbol (¬) for this purpose, to avoid confusion with equivalence relations .
In mathematics , the tilde operator (which can be represented by a tilde or the dedicated character U+223C ∼ TILDE OPERATOR ), sometimes called "twiddle", is often used to denote an equivalence relation between two objects. Thus " x ~ y " means " x is equivalent to y ". It is a weaker statement than stating that x equals y . The expression " x ~ y " is sometimes read aloud as " x twiddles y ", perhaps as an analogue to the verbal expression of " x = y ". [ 31 ]
The tilde can indicate approximate equality in a variety of ways. It can be used to denote the asymptotic equality of two functions. For example, f ( x ) ~ g ( x ) means that lim x → ∞ f ( x ) g ( x ) = 1 {\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1} . [ 13 ]
A tilde is also used to indicate " approximately equal to" (e.g. 1.902 ~= 2). This usage probably developed as a typed alternative to the libra symbol used for the same purpose in written mathematics, which is an equal sign with the upper bar replaced by a bar with an upward hump, bump, or loop in the middle (︍︍♎︎) or, sometimes, a tilde (≃). [ citation needed ] The symbol "≈" is also used for this purpose.
In physics and astronomy , a tilde can be used between two expressions (e.g. h ~ 10 −34 J s ) to state that the two are of the same order of magnitude . [ 13 ]
In statistics and probability theory , the tilde means "is distributed as"; [ 13 ] see random variable (e.g. X ~ B ( n , p ) for a binomial distribution ).
A tilde can also be used to represent geometric similarity (e.g. ∆ ABC ~ ∆ DEF , meaning triangle ABC is similar to DEF ). A triple tilde ( ≋ ) is often used to show congruence , an equivalence relation in geometry. [ citation needed ]
In graph theory , the tilde can be used to represent adjacency between vertices. The edge ( x , y ) {\displaystyle (x,y)} connects vertices x {\displaystyle x} and y {\displaystyle y} which can be said to be adjacent, and this adjacency can be denoted x ∼ y {\displaystyle x\sim y} .
The symbol " f ~ {\displaystyle {\tilde {f}}} " is pronounced as "eff tilde" or, informally, as "eff twiddle". [ 32 ] [ 33 ] This can be used to denote the Fourier transform of f , or a lift of f , and can have a variety of other meanings depending on the context.
A tilde placed below a letter in mathematics can represent a vector quantity (e.g. ( x 1 , x 2 , x 3 , … , x n ) = x ∼ {\displaystyle (x_{1},x_{2},x_{3},\ldots ,x_{n})={\underset {^{\sim }}{\mathbf {x} }}} ).
In statistics and probability theory , a tilde placed on top of a variable is sometimes used to represent the median of that variable; thus y ~ {\displaystyle {\tilde {\mathbf {y} }}} would indicate the median of the variable y {\displaystyle \mathbf {y} } . A tilde over the letter n ( n ~ {\displaystyle {\tilde {n}}} ) is sometimes used to indicate the harmonic mean .
In machine learning, a tilde may represent a candidate value for a cell state in GRUs or LSTM units. (e.g. c̃)
Often in physics , one can consider an equilibrium solution to an equation, and then a perturbation to that equilibrium. For the variables in the original equation (for instance X {\displaystyle X} ) a substitution X → x + x ~ {\displaystyle X\to x+{\tilde {x}}} can be made, where x {\displaystyle x} is the equilibrium part and x ~ {\displaystyle {\tilde {x}}} is the perturbed part.
A tilde is also used in particle physics to denote the hypothetical supersymmetric partner. For example, an electron is referred to by the letter e , and its superpartner the selectron is written ẽ .
In multibody mechanics, the tilde operator maps three-dimensional vectors ω ∈ R 3 {\displaystyle {\boldsymbol {\omega }}\in \mathbb {R} ^{3}} to skew-symmetrical matrices ω ~ = [ 0 − ω 3 ω 2 ω 3 0 − ω 1 − ω 2 ω 1 0 ] {\displaystyle {\tilde {\boldsymbol {\omega }}}={\begin{bmatrix}0&-\omega _{3}&\omega _{2}\\\omega _{3}&0&-\omega _{1}\\-\omega _{2}&\omega _{1}&0\end{bmatrix}}} (see [ 34 ] or [ 35 ] ).
For relations involving preference, economists sometimes use the tilde to represent indifference between two or more bundles of goods. For example, to say that a consumer is indifferent between bundles x and y , an economist would write x ~ y .
It can approximate the sine wave symbol (∿, U+ 223F), which is used in electronics to indicate alternating current , in place of +, −, or ⎓ for direct current .
The tilde may indicate alternating allomorphs or morphological alternation , as in //ˈniː~ɛl+t// for kneel~knelt (the plus sign '+' indicates a morpheme boundary). [ 36 ] [ 37 ]
The tilde may represent some sort of phonetic or phonemic variation between two sounds, which might be allophones or in free variation . For example, [χ ~ x] can represent "either [χ] or [x] ".
In formal semantics , it is also used as a notation for the squiggle operator which plays a key role in many theories of focus . [ 38 ]
In interlinear gloss , a tilde sets off an element added to a word by reduplication ; were a hyphen or double hyphen used instead, confusion would arise because that element would be notated in the same way as an independent morpheme requiring an independent gloss.
Computer programmers use the tilde in various ways and sometimes call the symbol (as opposed to the diacritic) a squiggle , squiggly , swiggle , or twiddle . According to the Jargon File , other synonyms sometimes used in programming include not , approx , wiggle , enyay (after eñe ) and (humorously) sqiggle / ˈ s k ɪ ɡ əl / . [ 39 ]
On Unix -like operating systems (including AIX , BSD , Linux and macOS ), tilde normally indicates the current user's home directory . For example, if the current user's home directory is /home/user , then the command cd ~ is equivalent to cd /home/user , cd $HOME , or cd . [ 39 ] This convention derives from the Lear-Siegler ADM-3A terminal in common use during the 1970s, which happened to have the tilde symbol and the word "Home" (for moving the cursor to the upper left) on the same key. [ 40 ] When prepended to a particular username, the tilde indicates that user's home directory (e.g., ~janedoe for the home directory of user janedoe , such as /home/janedoe ). [ 41 ]
Used in URLs on the World Wide Web , it often denotes a personal website on a Unix -based server. For example, http://www.example.com/~johndoe/ might be the personal website of John Doe. This mimics the Unix shell usage of the tilde. However, when accessed from the web, file access is usually directed to a subdirectory in the user's home directory, such as /home/ username /public_html or /home/ username /www . [ 42 ]
In URLs, the characters %7E (or %7e ) may substitute for a tilde if an input device lacks a tilde key. [ 43 ] Thus, http://www.example.com/~johndoe/ and http://www.example.com/%7Ejohndoe/ will behave in the same manner.
The tilde is used in the AWK programming language as part of the pattern match operators for regular expressions : [ 44 ]
The operators are also used in the SQL variant of the database PostgreSQL . [ 45 ]
A variant of this, with the plain tilde replaced with =~ , was adopted in Perl [ 46 ] . Ruby also uses this variant without the negated operator. [ 47 ]
In APL [ 48 ] : 68 and MATLAB , [ 49 ] tilde represents the monadic logical function NOT. and in APL it additionally represents the dyadic multiset function without ( set difference ). [ 48 ] : 258
In C the tilde character is used as bitwise NOT unary operator , following the notation in logic (an ! causes a logical NOT, instead). [ 50 ] This is also used by many languages based on or influenced by C, such as C++ , C# , D , Java , JavaScript , Perl , PHP , and Python . [ 51 ] The MySQL database also use tilde as bitwise invert [ 52 ] as does Microsoft's SQL Server Transact-SQL (T-SQL) language.
JavaScript also uses tilde as bitwise NOT. Because bitwise operators work on integers, and numbers in JavaScript are 64 bit floating point numbers, the operator converts numbers to a 32-bit signed integer before it performing the negation. [ 53 ] The conversion truncates the fractional part and most significant bits. This lets two tildes ~~x to be used as a short syntax to cast to integer. However, it is not recommended as use for truncation. In contrast, it does not truncate BigInts, which are arbitrarily large integers. [ 54 ]
In C++ [ 55 ] and C#, [ 56 ] the tilde is also used as the first character in a class 's method name (where the rest of the name must be the same name as the class) to indicate a destructor – a special method which is called at the end of the object's life .
In ASP.NET applications, tilde ('~') is used as a shortcut to the root of the application's virtual directory. [ 57 ]
In the CSS stylesheet language, the tilde finds the element selected by the right-hand side that shares the parent with an element selected by the left-hand side. [ 58 ]
In the D programming language , the tilde is used as bitwise not operator, concatenation operator such as those of arrays , [ 59 ] and to indicate an object destructor. [ 60 ] [ 61 ] Tilde operator can be overloaded for user types, [ 62 ] and binary tilde operator is mostly used to merging two objects, or adding some objects to set of objects. It was introduced because plus operator can have different meaning in many situations. For example, "120" + "14" may produce "134" (addition of two numbers), "12014" (concatenation of strings), or something else. [ 63 ] D disallows + operator for arrays (and strings), and provides separate operator for concatenation (similarly PHP programming language solved this problem by using dot operator for concatenation, and + for number addition, which will also work on strings containing numbers).
In Eiffel , the tilde is used for object comparison. If a and b denote objects, the Boolean expression a ~ b has value true if and only if these objects are equal, as defined by the applicable version of the library routine is_equal , which by default denotes field-by-field object equality but can be redefined in any class to support a specific notion of equality. [ 64 ] : 114–115 If a and b are references, the object equality expression a ~ b is to be contrasted with a = b which denotes reference equality. Unlike the call a . is_equal ( b ), the expression a ~ b is type-safe even in the presence of covariance .
In the Apache Groovy programming language the tilde character is used as an operator mapped to the bitwiseNegate() method. [ 65 ] Given a String the method will produce a java.util.regex.Pattern. Given an integer it will negate the integer bitwise like in C. =~ and ==~ can in Groovy be used to match a regular expression. [ 66 ] [ 67 ]
In Haskell , the tilde is used in type constraints to indicate type equality. [ 68 ] Also, in pattern-matching, the tilde is used to indicate a lazy pattern match. [ 69 ]
In the Inform 6 programming language, the tilde is used to indicate a quotation mark inside a quoted string. Tilde itself is created by @@126 . [ 70 ]
In "text mode" of the LaTeX typesetting language a tilde diacritic can be obtained using, e.g., \~{n} , yielding "ñ". A stand-alone tilde can be obtained by using \textasciitilde or \string~ .
In "math mode" a tilde diacritic can be written as, e.g., \tilde{x} . For a wider tilde \widetilde can be used. The \sim command produce a tilde-like binary relation symbol that is often used in mathematical expressions, and the double-tilde ≈ is obtained with \approx .In both text and math mode, a tilde on its own ( ~ ) renders a white space with no line breaking.In both text and math mode, a tilde on its own ( ~ ) renders a white space with no line breaking. [ 71 ] The url package also supports entering tildes directly, e.g., \url{http://server/~name} . [ citation needed ] .
In MediaWiki syntax, four tildes are a shortcut for a user's signature. Three and five tildes puts the signature without timestamp and only the timestamp, respectively. [ 72 ]
In Common Lisp , the tilde is used as the prefix for format specifiers in format strings. [ 73 ]
In Max/MSP , MSP objects have names ending with a tilde. MSP objects process at the computer's sampling rate and mainly deal with sound. [ 74 ]
In Standard ML , the tilde is used as the prefix for negative numbers and as the unary negation operator. [ 75 ]
In OCaml , the tilde is used to specify the label for a labeled parameter. [ 76 ]
In R , the tilde operator is used to separate the left- and right-hand sides in a model formula. [ 77 ]
In Object REXX , the twiddle is used as a "message send" symbol. For example, Employee.name~lower() would cause the lower() method to act on the object Employee 's name attribute, returning the result of the operation. ~~ returns the object that received the method rather than the result produced. Thus, it can be used when the result need not be returned or when cascading methods are to be used. team~~insert("Jane")~~insert("Joe")~~insert("Steve") would send multiple concurrent insert messages, thus invoking the insert method three consecutive times on the team object. [ 78 ]
In Raku , a prefixing tilde converts a value to a string. An infix tilde concatenates strings, [ 79 ] taking place of the dot operator in Perl, as the dot is used for member access instead of -> . [ 80 ] ~~ is called "the smartmatch operator" and its semantics depend on the type of the right-side argument. Namely, it checks numeric and string equalities, performs regular expression match tests (as opposed to =~ in Perl [ 80 ] ), and type checking . [ 79 ]
In YAML , the "Core schema," a set of aliases that processors are recommended to use, resolves a tilde as null. [ 81 ]
The presence (or absence) of a tilde engraved on the keyboard depends on the territory where it was sold. In either case, computer's system settings determine the keyboard mapping and the default setting will match the engravings on the keys. Even so, it certainly possible to configure a keyboard for a different locale than that supplied by the retailer. On American and British keyboards, the tilde is a standard keytop and pressing it produces a free-standing "ASCII Tilde". To generate a letter with a tilde diacritic requires the US international or UK extended keyboard setting.
Instructions for other national languages and keyboards are beyond the scope of this article.
The dominant Unix convention for naming backup copies of files is appending a tilde to the original file name. It originated with the Emacs text editor [ 82 ] and was adopted by many other editors and some command-line tools.
Emacs also introduced an elaborate numbered backup scheme, with files named filename.~1~ , filename.~2~ and so on. [ 83 ] It didn't catch on, as the rise of version control software eliminates the need for this usage. [ citation needed ]
The tilde was part of Microsoft 's filename mangling scheme when it extended the FAT file system standard to support long filenames for Microsoft Windows . Programs written prior to this development could only access filenames in the so-called 8.3 format —the filenames consisted of a maximum of eight characters from a restricted character set (e.g. no spaces), followed by a period, followed by three more characters. In order to permit these legacy programs to access files in the FAT file system, each file had to be given two names—one long, more descriptive one, and one that conformed to the 8.3 format. This was accomplished with a name-mangling scheme in which the first six characters of the filename are followed by a tilde and a digit. For example, " Program Files " might become " PROGRA~1 ". [ 84 ]
The tilde symbol is also often used to prefix hidden temporary files that are created when a document is opened in Windows. [ citation needed ] For example, when a document "Document1.doc" is opened in Word, a file called "~$cument1.doc" is created in the same directory. This file contains information about which user has the file open, to prevent multiple users from attempting to change a document at the same time. [ 85 ]
In the juggling notation system Beatmap, tilde can be added to either "hand" in a pair of fields to say "cross the arms with this hand on top". Mills' Mess is thus represented as (~2x,1)(1,2x)(2x,~1)*. [ 86 ]
Unicode encodes a number of cases of "letter with tilde" as precomposed characters and these are displayed below. In addition, many more symbols may be composed using the combining character facility ( U+0303 ◌̃ COMBINING TILDE , U+0330 ◌̰ COMBINING TILDE BELOW and others) that may be used with any letter or other diacritic to create a customised symbol but this does not mean that the result has any real-world application and are not shown in the table.
A tilde diacritic can be added to almost any character by using a combining tilde. Greek and Cyrillic letters with tilde ( Α͂ ᾶ , Η͂ ῆ , Ι͂ ῖ , ῗ, Υ͂ ῦ , ῧ and А̃ а̃ , Ә̃ ә̃ , Е̃ е̃ , И̃ и̃ , О̃ о̃ , У̃ у̃ , Ј̃ j̃ ) are formed using this method.
In practice the full-width tilde ( 全角チルダ , zenkaku chiruda ) (Unicode U+FF5E ～ FULLWIDTH TILDE ), is often used instead of the wave dash ( 波ダッシュ , nami dasshu ) (Unicode U+301C 〜 WAVE DASH ), because the Shift JIS code for the wave dash, 0x8160, which should be mapped to U+301C, [ 87 ] [ 88 ] is instead mapped to U+FF5E [ 89 ] in Windows code page 932 ( Microsoft 's code page for Japanese), a widely used extension of Shift JIS.
This decision avoided a shape definition error in the original (6.2) Unicode code charts: [ 90 ] the wave dash reference glyph in JIS / Shift JIS [ 91 ] [ 92 ] matches the Unicode reference glyph for U+FF5E FULLWIDTH TILDE , [ 93 ] while the original reference glyph for U+301C [ 90 ] was reflected, incorrectly, [ 94 ] when Unicode imported the JIS wave dash. In other platforms such as the classic Mac OS and macOS , 0x8160 is correctly mapped to U+301C. It is generally difficult, if not impossible, for users of Japanese Windows to type U+301C, especially in legacy, non-Unicode applications.
A similar situation exists regarding the Korean KS X 1001 character set, in which Microsoft maps the EUC-KR or UHC code for the wave dash (0xA1AD) to U+223C ∼ TILDE OPERATOR , [ 95 ] [ 96 ] while IBM and Apple map it to U+301C. [ 97 ] [ 98 ] [ 99 ] Microsoft also uses U+FF5E to map the KS X 1001 raised tilde (0xA2A6), [ 96 ] while Apple uses U+02DC ˜ SMALL TILDE . [ 99 ]
The current Unicode reference glyph for U+301C has been corrected [ 94 ] to match the JIS standard [ 100 ] in response to a 2014 proposal, which noted that while the existing Unicode reference glyph had been matched by fonts from the discontinued Windows XP , all other major platforms including later versions of Microsoft Windows shipped with fonts matching the JIS reference glyph for U+301C. [ 101 ]
The JIS / Shift JIS wave dash is still formally mapped to U+301C as of JIS X 0213 , [ 102 ] whereas the WHATWG Encoding Standard used by HTML5 follows Microsoft in mapping 0x8160 to U+FF5E. [ 103 ] These two code points have a similar or identical glyph in several computer fonts , reducing the confusion and incompatibility. | https://en.wikipedia.org/wiki/Tilde |
Tile processors [ 1 ] for computer hardware , are multi-core or manycore chips that contain one-dimensional, or more commonly, two-dimensional arrays of identical tiles. Each tile comprises a compute unit (or a processing engine or CPU ), caches and a switch. Tiles can be viewed as adding a switch to each core, where a core comprises a compute unit and caches.
In a typical Tile Processor configuration, the switches in each of the tiles are connected to each other using one or more mesh networks . [ 2 ] The Tilera TILE Pro 64 , for example, contains 64 tiles. Each of the tiles comprises a CPU , L1 and L2 caches , and switches for several mesh networks.
Other processors in a tile configuration include SEAforth24, Kilocore KC256, XMOS xCORE microcontrollers , and some massively parallel processor arrays .
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tile_processor |
Tiling arrays are a subtype of microarray chips. Like traditional microarrays, they function by hybridizing labeled DNA or RNA target molecules to probes fixed onto a solid surface.
Tiling arrays differ from traditional microarrays in the nature of the probes. Instead of probing for sequences of known or predicted genes that may be dispersed throughout the genome , tiling arrays probe intensively for sequences which are known to exist in a contiguous region. This is useful for characterizing regions that are sequenced, but whose local functions are largely unknown. Tiling arrays aid in transcriptome mapping as well as in discovering sites of DNA/ protein interaction ( ChIP-chip , DamID ), of DNA methylation (MeDIP-chip) and of sensitivity to DNase (DNase Chip) and array CGH. [ 1 ] In addition to detecting previously unidentified genes and regulatory sequences, improved quantification of transcription products is possible. Specific probes are present in millions of copies (as opposed to only several in traditional arrays) within an array unit called a feature, with anywhere from 10,000 to more than 6,000,000 different features per array. [ 2 ] Variable mapping resolutions are obtainable by adjusting the amount of sequence overlap between probes, or the amount of known base pairs between probe sequences, as well as probe length. For smaller genomes such as Arabidopsis , whole genomes can be examined. [ 3 ] Tiling arrays are a useful tool in genome-wide association studies .
The two main ways of synthesizing tiling arrays are photolithographic manufacturing and mechanical spotting or printing.
The first method involves in situ synthesis where probes, approximately 25bp, are built on the surface of the chip. These arrays can hold up to 6 million discrete features, each of which contains millions of copies of one probe.
The other way of synthesizing tiling array chips is via mechanically printing probes onto the chip. This is done by using automated machines with pins that place the previously synthesized probes onto the surface. Due to the size restriction of the pins, these chips can hold up to nearly 400,000 features. [ 4 ] Three manufacturers of tiling arrays are Affymetrix , NimbleGen and Agilent . Their products vary in probe length and spacing. ArrayExplorer.com is a free web-server to compare tiling arrays.
ChIP-chip is one of the most popular usages of tiling arrays. Chromatin immunoprecipitation allows binding sites of proteins to be identified. A genome-wide variation of this is known as ChIP-on-chip. Proteins that bind to chromatin are cross-linked in vivo , usually via fixation with formaldehyde . The chromatin is then fragmented and exposed to antibodies specific to the protein of interest. These complexes are then precipitated. The DNA is then isolated and purified. With traditional DNA microarrays, the immunoprecipitated DNA is hybridized to the chip, which contains probes that are designed to cover representative genome regions. Overlapping probes or probes in very close proximity can be used. This gives an unbiased analysis with high resolution. Besides these advantages, tiling arrays show high reproducibility and with overlapping probes spanning large segments of the genome, tiling arrays can interrogate protein binding sites, which harbor repeats. ChIP-chip experiments have been able to identify binding sites of transcription factors across the genome in yeast, drosophila and a few mammalian species. [ 5 ]
Another popular use of tiling arrays is in finding expressed genes. Traditional methods of gene prediction for annotation of genomic sequences have had problems when used to map the transcriptome, such as not producing an accurate structure of the genes and also missing transcripts entirely. The method of sequencing cDNA to find transcribed genes also runs into problems, such as failing to detect rare or very short RNA molecules, and so do not detect genes that are active only in response to signals or specific to a time frame. Tiling arrays can solve these issues. Due to the high resolution and sensitivity, even small and rare molecules can be detected. The overlapping nature of the probes also allows detection of non-polyadenylated RNA and can produce a more precise picture of gene structure. [ 6 ] Earlier studies on chromosome 21 and 22 showed the power of tiling arrays for identifying transcription units. [ 7 ] [ 8 ] [ 9 ] The authors used 25-mer probes that were 35bp apart, spanning the entire chromosomes. Labeled targets were made from polyadenylated RNA. They found many more transcripts than predicted and 90% were outside of annotated exons . Another study with Arabidopsis used high-density oligonucleotide arrays that cover the entire genome. More than 10 times more transcripts were found than predicted by ESTs [ clarification needed ] and other prediction tools. [ 3 ] [ 10 ] Also found were novel transcripts in the centromeric regions where it was thought that no genes are actively expressed. Many noncoding and natural antisense RNA have been identified using tiling arrays. [ 9 ]
Methyl-DNA immunoprecipitation followed by tiling array allows DNA methylation mapping and measurement across the genome. DNA is methylated on cytosine in CG di-nucleotides in many places in the genome. This modification is one of the best-understood inherited epigenetic changes and is shown to affect gene expression. Mapping these sites can add to the knowledge of expressed genes and also epigenetic regulation on a genome-wide level. Tiling array studies have generated high-resolution methylation maps for the Arabidopsis genome to generate the first "methylome".
DNase chip is an application of tiling arrays to identify hypersensitive sites, segments of open chromatin that are more readily cleaved by DNaseI. DNaseI cleaving produces larger fragments of around 1.2kb in size. These hypersensitive sites have been shown to accurately predict regulatory elements such as promoter regions, enhancers and silencers. [ 11 ] Historically, the method uses Southern blotting to find digested fragments. Tiling arrays have allowed researchers to apply the technique on a genome-wide scale.
Array-based CGH is a technique often used in diagnostics to compare differences between types of DNA, such as normal cells vs. cancer cells. Two types of tiling arrays are commonly used for array CGH, whole genome and fine tiled. The whole genome approach would be useful in identifying copy number variations with high resolution. On the other hand, fine-tiled array CGH would produce ultrahigh resolution to find other abnormalities such as breakpoints. [ 12 ]
Several different methods exist for tiling an array. One protocol for analyzing gene expression involves first isolating total RNA. This is then purified of rRNA molecules. The RNA is copied into double stranded DNA, which is subsequently amplified and in vitro transcribed to cRNA. The product is split into triplicates to produce dsDNA, which is then fragmented and labeled. Finally, the samples are hybridized to the tiling array chip. The signals from the chip are scanned and interpreted by computers.
Various software and algorithms are available for data analysis and vary in benefits depending on the manufacturer of the chip. For Affymetrix chips, the model-based analysis of tiling array (MAT) or hypergeometric analysis of tiling-arrays (HAT [ 13 ] ) are effective peak-seeking algorithms. For NimbleGen chips, TAMAL is more suitable for locating binding sites. Alternative algorithms include MA2C and TileScope, which are less complicated to operate. The Joint binding deconvolution algorithm is commonly used for Agilent chips. If sequence analysis of binding site or annotation of the genome is required then programs like MEME, Gibbs Motif Sampler, Cis-regulatory element annotation system and Galaxy are used. [ 4 ]
Tiling arrays provide an unbiased tool to investigate protein binding, gene expression and gene structure on a genome-wide scope. They allow a new level of insight in studying the transcriptome and methylome.
Drawbacks include the cost of tiling array kits. Although prices have fallen in the last several years, the price makes it impractical to use genome-wide tiling arrays for mammalian and other large genomes. Another issue is the "transcriptional noise" produced by its ultra-sensitive detection capability. [ 2 ] Furthermore, the approach provides no clearly defined start or stop to regions of interest identified by the array. Finally, arrays usually give only chromosome and position numbers, often necessitating sequencing as a separate step (although some modern arrays do give sequence information. [ 14 ] ) | https://en.wikipedia.org/wiki/Tiling_array |
A tiling with rectangles is a tiling which uses rectangles as its parts. The domino tilings are tilings with rectangles of 1 × 2 side
ratio. The tilings with straight polyominoes of shapes such as 1 × 3 , 1 × 4 and
tilings with polyominoes of shapes such as 2 × 3 fall also into this category.
Some tiling of rectangles include:
The smallest square that can be cut into (m × n) rectangles, such that all m and n are different integers, is the 11 × 11 square, and the tiling uses five rectangles. [ 1 ]
The smallest rectangle that can be cut into (m × n) rectangles, such that all m and n are different integers, is the 9 × 13 rectangle, and the tiling uses five rectangles. [ 1 ] [ 2 ]
This geometry-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tiling_with_rectangles |
A tiller is a shoot that arises from the base of a grass plant. The term refers to all shoots that grow after the initial parent shoot grows from a seed. [ 1 ] [ 2 ] Tillers are segmented, each segment possessing its own two-part leaf. They are involved in vegetative propagation and, in some cases, also seed production. [ 3 ]
"Tillering" refers to the production of side shoots and is a property possessed by many species in the grass family. This enables them to produce multiple stems (tillers) starting from the initial single seedling. This ensures the formation of dense tufts and multiple seed heads. Tillering rates are heavily influenced by soil water quantity. When soil moisture is low, grasses tend to develop more sparse and deep root systems (as opposed to dense, lateral systems). Thus, in dry soils, tillering is inhibited: the lateral nature of tillering is not supported by lateral root growth.
This plant morphology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tiller_(botany) |
Tilomisole ( WY-18,251 ) is an experimental drug which acts as an immunomodulator and has been studied for the treatment of some forms of cancer. [ 1 ] [ 2 ]
This antineoplastic or immunomodulatory drug article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tilomisole |
A tilt-tray sorter is a mechanical assembly similar to a conveyor belt but instead of a continuous belt, it consists of individual trays traveling in the same direction.
A tilt-tray sorter can be configured in an inline (AKA over/under) formation, or in a continuous-loop.
Items are loaded onto the passing trays at the front end of the sorter and travel towards a series of destinations on either side of the sorter. Items are loaded on to trays individually and their sort destination is determined in advance.
As the tray with an item approaches its destination the tray is tilted to slide the object into the chute. The empty tray will then return to the load section before it is loaded again with a new item.
[ 1 ]
A tilt-tray sorter is a continuous-loop sortation conveyor that uses a technique of tilting a tray at a chute to slide the object into the chute. [ 2 ]
This industry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tilt_tray_sorter |
Tilt-up, tilt-slab or tilt-wall is a type of building and a construction technique using concrete . Though it is a cost-effective technique with a shorter completion time, [ 1 ] poor performance in earthquakes has mandated significant seismic retrofit requirements in older buildings. [ 2 ]
With the tilt-up method, concrete elements (walls, columns, structural supports, etc.) are formed horizontally on a concrete slab ; this normally requires the building floor as a building form but may be a temporary concrete casting surface near the building footprint. After the concrete has cured, the elements are "tilted" to the vertical position with a crane and braced into position until the remaining building structural components (roofs, intermediate floors and walls) are secured. [ 3 ] [ 4 ]
Tilt-up construction is a common method of construction throughout North America, several Caribbean nations, Australia , and New Zealand . It is not significantly used in Europe or the northern two thirds of Asia. It is gaining popularity in southern Asia, the Middle East, parts of Africa, Central and South America.
Concrete elements can also be formed at factories away from the building site. [ 5 ] Tilt-up differs from prefabrication , or plant cast construction, in that all elements are constructed on the job site. This eliminates the size limitation imposed by transporting elements from a factory to the project site.
Tilt-up construction requires significant organization and collaboration on the building site. The chronological steps that need to be taken for a tilt-up project are: site evaluation, engineering, footings and floor slabs, forming tilt-up panels, steel placement, embeds and inserts, concrete placement, panel erection and panel finishing. [ 1 ] [ 6 ] Once the pad (casting surface or floor slab) has cured, forms are built on top. Dimensional lumber, a high quality plywood or fiber board that has at least one smooth face is typically used, although aluminum or steel forms are also common. Carpenters work from engineered drawings designed for each panel or element to construct on site. They incorporate all door and window openings, as well as architectural features and other desired shapes that can be molded into the concrete. Studs , gussets and attachment plates are located within the form for embedding in the concrete. The forms are usually anchored to the casting surface with masonry nails or otherwise adhered to prevent damage to the floor slab. [ 7 ]
Next, a chemically reactive bondbreaker is sprayed on the form's surfaces to prevent the cast concrete from bonding with the slab. This allows the cast element to separate from the casting surface once it has cured. This is a critical step, as improper chemical selection or application will prevent the lifting of the panels, and will entail costly demolition and rework.
A rebar grid is constructed inside the forms, after the form release is applied, spaced off the casting surface the desired distance with plastic "chairs". The rebar size and spacing is generally specified by the engineer of record. [ 8 ]
Concrete is then poured, filling the desired thickness and surrounding all steel inserts, embedded features and rebar. The concrete is then settled through vibration to prevent any voids or honeycomb effects. The forms are removed when the concrete is cured; rigging is attached and a crane tilts the panel or lifts the element into place. In circumstances when space is at a premium, concrete elements can be cast one on top of the other, or stack cast. Quite often a separate casting pad is poured for this purpose and is removed when the panels are erected. [ 9 ]
Cranes are used to tilt the concrete elements from the casting slab to a vertical position. The slabs are then most often set onto a foundation and secured with braces until the structural steel and the roof diaphragm is in place.
Concrete tilt-up walls can be very heavy, sometimes over 300,000 pounds (140 t). [ 10 ] Most tilt-up wall panels are engineered to work with the roof structure and/or floor structures to resist all forces; that is, to function as load-bearing walls. The connections to the roof and floors are usually steel plates with headed studs that were secured into the forms prior to concrete placement. These attachment points are bolted or welded. The upper attachment points are made to the roof trusses . Interior walls may be present for additional stiffness in the building structure as necessary, known as shear walls.
Insulation can be applied to either side of the panels or cast as an integral part of the panel between two layers of concrete to create sandwich panels. Concrete has the ability to absorb and store energy and is high mass, which regulates interior temperature ( thermal mass ) and provides soundproofing and durability. [ 1 ]
Like all concrete construction, tilt-up buildings are fire-resistant . In addition, wall panels can be designed to sag inward when damaged, which minimizes collapse (this can also be done with prefabricated panels). [ 1 ]
Tilt-up was first used in America circa 1905. In 1908 Robert Akin patented the tilt-slab method of concrete construction used in the construction of the Schindler House . [ 11 ] Early erection was done using tilt tables, but the development of the mobile crane and truck mixers allowed tilt-up construction to grow. Tilt-up gained widespread popularity in the post World War II construction boom. [ 12 ] [ 13 ] Tilt-up was not used successfully in Australia until 1969. [ 14 ] [ 15 ]
Most early tilt-up buildings were warehouses . Today the method is used in nearly every type of building from schools to office structures, houses to hotels. They range from single story to more than seven and can be more than 29 metres (96 feet) in height. [ 16 ]
An early example of this method is found in the innovative Schindler House, built in 1922 in West Hollywood, California . Architect Rudolf Schindler claimed that with the assistance of a small hand-operated crane, just two workmen were needed to raise and attach the tilt-up walls. [ citation needed ]
The first tilt-up built home in Ontario, Canada was built in 2017. [ 17 ]
Early tilt-up architecture was very minimalist and boxy. Recent techniques have expanded the range of appearance and shape.
Many finish options are available to the tilt-up contractor, from paints and stains to pigmented concrete, cast-in features like brick and stone to aggressive erosion finishes like sandblasting and acid-etching. Shapes are also a feature that have become dominant in the tilt-up market, with many panels configured with circular or elliptical openings, panel tops that are pedimented or curved, facades that are curved or segmented and featured with significant areas of glazing or other materials.
The Tilt-Up Concrete Association (TCA) is the international trade association for tilt-up concrete construction. TCA is a membership-based association, with nearly 500 members worldwide. [ 18 ] TCA members can be contractors (general contractors or tilt-up subcontractors), engineers, architects, developers, consultants, suppliers, specialty trade firms, educators and students.
TCA offers primarily educational, networking and exposure benefits to its members. TCA also offers an Achievement Awards program annually, recognizing the best examples of tilt-up construction over a variety of end uses. [ 19 ]
In the wake of the 2011 Joplin tornado in which seven people were killed in a Home Depot when the 100,000-pound (45 t) panel walls collapsed after the store was hit by an EF5 tornado , engineers in an article published in The Kansas City Star criticized the practice. They said that once one wall falls, it creates a domino effect . Twenty-eight people in an un-reinforced training room in the back of the building survived. According to a study of the collapse, the tornado hit the south corner of the store and lifted the roof up causing the west walls to collapse into the store. The walls on the east side (where the people survived) collapsed out. Only two walls remained standing. Engineers said that stronger roof-to-wall connections might have tempered the collapse. Two other big box stores at the corner that had concrete block construction (an Academy Sports + Outdoors and Walmart ) lost their roofs but the walls remained intact. Those buildings were not directly hit by the tornado but the Home Depot building suffered a direct hit. Three people died in the Walmart, but 200 survived. The engineers told the Star that when concrete blocks fail they usually break apart, and do not come down in huge slabs. Home Depot, which has hundreds of stores built with tilt-up, said it disagreed with the finding and that it would use tilt-up when it rebuilt the Joplin store. [ 20 ]
Shortly after publication of the Kansas City Star article, the technical committee of the Tilt-Up Concrete Association (TCA) formed a task force to investigate the claims presented in the article. With the cooperation of Home Depot, the task group performed detailed engineering calculations, research and investigation of the claims posed in the article. This task force consisted of a nationwide group of practicing structural engineers with a diverse range of experience in tilt-up construction and " big box " buildings. The final report was published on January 12, 2012. "The information provided in these findings will help Association efforts to promote the benefits of site cast Tilt-Up construction and dispute many of the claims presented in The Kansas City Star article." [ 21 ] [ non-primary source needed ]
"The Task Force's findings to date include:
One of the conclusions of the Task Force's report was "Recommend to ICC and direct
to building owners the use of storm shelters in lieu of designing buildings for high
winds. The TCA should develop specific Tilt-Up based storm shelter designs for winds
of up to 200 MPH that would compete against alternative masonry, precast, or
cast-in-place designs. Storm shelter design is addressed in 2009 IBC, section 423 and ICC-500." [ 22 ] [ non-primary source needed ] | https://en.wikipedia.org/wiki/Tilt_up |
In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by exponential tilting , i.e. integration against an exponential functional . It can be seen as an alternative formulation of Varadhan's lemma .
Let X be a Polish space (i.e., a separable , completely metrizable topological space ), and let ( μ ε ) ε >0 be a family of probability measures on X that satisfies the large deviation principle with rate function I : X → [0, +∞]. Let F : X → R be a continuous function that is bounded from above. For each Borel set S ⊆ X , let
and define a new family of probability measures ( ν ε ) ε >0 on X by
Then ( ν ε ) ε >0 satisfies the large deviation principle on X with rate function I F : X → [0, +∞] given by | https://en.wikipedia.org/wiki/Tilted_large_deviation_principle |
Tilth is a physical condition of soil, especially in relation to its suitability for planting or growing a crop. Factors that determine tilth include the formation and stability of aggregated soil particles , moisture content, degree of aeration, soil biota , rate of water infiltration and drainage. Tilth can change rapidly, depending on environmental factors, such as changes in moisture, tillage and soil amendments . The objective of tillage (mechanical manipulation of the soil) is to improve tilth, thereby increasing crop production; in the long term, however, conventional tillage, especially plowing, often has the opposite effect, causing the soil carbon sponge to oxidize, break down and become compacted. [ 1 ]
Soil with good tilth is spongy with large pore spaces for air infiltration and water movement. Roots grow only where the soil tilth allows for adequate levels of soil oxygen. Such soil also holds a reasonable supply of water and nutrients. [ 2 ]
Tillage , organic matter amendments, fertilization and irrigation can each improve tilth, but when used excessively, can have the opposite effect. [ 2 ] Crop rotation and cover crops can rebuild the soil carbon sponge and positively affect tilth. A combined approach can produce the greatest improvement. [ citation needed ]
Good tilth shares a balanced relation between soil-aggregate tensile strength and friability , in which it has a stable mixture of aggregate soil particles that can be readily broken up by shallow, non-abrasive tilling. A high tensile strength will result in large cemented clods of compacted soil with low friability. Proper management of agricultural soils can positively affect soil aggregation and improve tilth quality. [ 3 ]
Aggregation is positively associated with tilth. With finer-textured soils, aggregates may in turn be made up of smaller aggregates. Aggregation implies substantial pores between individual aggregates. [ 4 ]
Aggregation is important in the subsoil, the layer below tillage. Such aggregates involve larger (2- to 6-inch) blocks of soil that are more angular and not as distinctive. These aggregates are less affected by biological activity than the tillage layer. Subsurface aggregates are important for root growth deep into the profile. Deep roots allow greater access to moisture, which helps in drought periods. Subsoil aggregates can also be compacted, mainly by heavy equipment on wet soil. Another significant source of subsoil compaction is the practice of plowing with tractor wheels in the open furrow. [ 4 ]
Soil that is well aggregated has a range of pore sizes. Each pore size plays a role in soil's physical functioning. Large pores drain rapidly and are needed for good air exchange during wet periods, preventing oxygen deficiency that can drown plants and increase pest problems. Oxygen-deficient wet soils increase denitrification – conversion of nitrogen to gaseous forms. In degraded soil, large pores are compressed into small ones. [ 4 ]
Small pores are critical for water retention and help a crop endure dry periods with minimal yield loss. [ 4 ]
Soil tilth is naturally maintained by the interaction of plant roots with the soil biota . [ 5 ]
Short lived tilth can be obtained through mechanical and biological manipulation.
In 2021, the globally tilled soil volume was estimated at 1840 km 3 /yr. This value exceeds by two orders of magnitude the global total of all engineering earthworks. [ 6 ] For comparison globally, the natural process of soil bioturbation by plant roots and earthworms , was estimated at 960 km 3 /yr. [ 7 ]
Mechanical soil cultivation practices, including primary tillage (mold-board or chisel plowing) followed by secondary tillage (disking, harrowing , etc.), break up and aerate soil. Mechanical traffic and intensive tilling methods have a negative impact on soil aggregates, friability, soil porosity, and soil-bulk density. When soils become degraded and compacted, such tillage practices are often deemed necessary. The tilth created by tillage, however, tends to be unstable , because the aggregation is obtained through the physical manipulation of the soil, which is short lived, especially after years of intensive tillage. [ 4 ] The compaction of soil aggregates can also decrease soil biota due to the low levels of oxygen in the top-soil. The resulting high soil-bulk density results in lower water infiltration from rainfall or conventional irrigation (surface, sprinkler, center-pivot); in turn, the series of processes will naturally erode and dissolve small soil particles and organic matter. [ 8 ] The consequences from these processes cyclically require more tilling and intervention, thus tillage practices have the capability to disrupt biological mechanisms that stabilize soil structure, the soil carbon sponge and tilth quality. [ 9 ]
The preferred scenario for good tilth is as the result of natural soil-building processes, provided by the activity of plant roots, microorganisms, earthworms and other beneficial organisms. Such stable aggregates break apart during tillage/planting and readily provide good tilth. [ 4 ] Soil biota and organic matter work in unison to bind soil aggregates and establish a natural soil stability – a soil carbon sponge. Plant root exudates feed bacteria that emit extracellular polysaccharides (EPS), and feed the growth of fungal hyphae, to form a soil carbon sponge with the dispersed clay particles. These active tilth-forming processes contribute to the formation and stabilization of soil structure. [ 3 ] The resulting soil structure reduces tensile strength and soil-bulk density while still forming soil aggregates through their abiotic/biotic binding mechanisms that resist breakdown during water saturation. The fungal hyphae networks can establish a role of enmeshment with EPS and rhizodeposition, thus improving aggregate stability. [ 3 ] However, these organic materials are themselves subject to biological degradation, requiring active amendments with organic material, and minimal mechanical tillage. [ 4 ] Tilth quality is heavily dependent on these naturally binding processes between biotic microorganisms and abiotic soil particles, as well as the necessary input of organic matter. All constituents in this naturally binding network must be supplied or managed in agriculture to ensure the sustainability of their presence through growing seasons.
Crop rotation can help restore tilth in compacted soils . Two processes contribute to this gain. First, accelerated organic matter decomposition from tillage ends under the sod crop. Another way to achieve this is via no-till farming . Second, grass and legume sods develop extensive root systems that continually grow and die off. The dead roots supply a source of active organic matter, which feeds soil organisms that create aggregation – the soil carbon sponge. Beneficial organisms need continual supplies of organic matter to sustain themselves and they deposit the digested materials on soil aggregates and thereby stabilize them. Also, the living roots and symbiotic microorganisms (for example, mycorrhizal fungi) can exude organic materials that nourish soil organisms and help with aggregation. Grass and legume sod crops therefore deposit more organic matter in the soil than most other crops. [ 4 ]
Some annual rotation crops, such as buckwheat , also have dense, fibrous, root systems and can improve tilth. Crop mixtures with different rooting systems can be beneficial. For example, red clover seeded into winter wheat provides additional roots and a more protein-rich soil organic matter. [ 4 ]
Other rotation crops are more valuable for improving subsoils. Perennial crops, such as alfalfa , have strong, deep, penetrating tap roots that can push through hard layers, especially during wet periods when the soil is soft. These deep roots establish pathways for water and future plant roots, and produce soil organic matter. [ 4 ]
Crops rotation can extend the period of active growth compared to conventional row crops, leaving more organic material behind. For example, in a corn–soybean rotation, active growth occurs 32% of the time, while a dry bean–winter wheat–corn rotation is active 72% of the time. Crops such as rye , wheat , oat , barley , pea and cool-season grasses grow actively in the late fall and early spring when other crops are inactive. They are beneficial both as rotation and cover crops, although intensive tillage can negate their effects. [ 4 ]
The soil management practices required to maintain soil tilth are a function of the soil type. Sandy and gravelly soils are naturally deficient in small pores and are therefore drought-prone, whereas loams and clays can retain and thus supply crops with more water. [ 4 ]
Sandy soil has a lower capacity to hold water and nutrients. Water is frequently applied in smaller amounts to avoid leaching and carrying nutrients below the root zone. Routine application of organic matter increases sandy soil's ability to hold water and nutrients by 10 times or more. [ 2 ]
Clay soils lack large pores, restricting both water and air movement. During irrigation or rain events, the limited large pore space in fine-textured soils quickly fills with water, reducing soil oxygen levels. In addition to routine application of organic matter, microorganisms and earthworms perform a crucial role in soil tilth. As microorganisms decompose the organic matter, soil particles bind into larger aggregates, increasing large pore space. Clay soils are more subject to soil compaction, which reduces large pore spaces. [ 2 ]
Such soils natively have little tilth, especially once they have been disturbed. Adding organic matter up to 25% by volume can help compensate. For example, if tilling to a depth of eight inches, add two inches of organic materials. [ 2 ] | https://en.wikipedia.org/wiki/Tilth |
A tilting pan filter is a chemical equipment used in continuous solid - liquid filtration .
It is formed by a number of trapezoidal pans arranged in circle. At the center of the equipment there is the main valve which is connected to every pan through pipes . The pans are rotating continuously around the main valve, which provides the air or the vacuum necessary for the operation. In each pan it is carried out the filtration in a cyclic process that involves these stages: [ 1 ]
This chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Tilting_pan_filter |
Tim9 and Tim10 make up the group of essential small Tim proteins that assist in transport of hydrophobic precursors across the intermembrane space in mammalian cells. Both Tim9 and Tim10 form a hexamer, the Tim9-Tim10 complex, that when associated, functions as a chaperone to assist translocation of preproteins from the outer mitochondrial membrane to the translocase of the inner membrane . The functional Tim9-Tim10 complex not only directs preproteins to the inner mitochondrial membrane in order to interact with the TIM22 complex, but also guides β-barrel precursor proteins to the sorting and assembly machinery (SAM) of the outer membrane.
The Tim9-Tim10 complex is made up of three Tim9 molecules and three Tim10 molecules. Each Tim9 and Tim10 subunit consists of 80-110 amino acid residues with four conserved cysteine residues that form two intramolecular disulfide bonds . [ 1 ] Each subunit folds into a helix-loop-helix structure, with each loop forming a donut shape that comprises the upper face of the complex. The structure of the Tim9-Tim10 complex takes on the form of an α-propeller, with two helical blades radiating from a narrow central pore. [ 1 ]
Small Tim proteins are synthesised lacking a cleavable presequence, but instead containing internal targeting information and need to be imported to the intermembrane space. The intermembrane space import and assembly machinery (MIA) is believed to mediate transport of the small Tim precursors into the intermembrane space subcompartment. [ 2 ] MIA is composed of two main essential cysteine-rich proteins; Mia40 and Erv1. Mia40 is also referred to as Tim40 in yeast and deficiency of Mia40 has been reported to affect import of the small Tims. [ 2 ] Mia40 is anchored to the mitochondrial inner membrane via an N-terminal hydrophobic segment, exposing a large domain to the intermembrane space. It contains 6 conserved cysteine residues, which allow the binding of incoming Tim precursor proteins. [ 3 ] Following import of small Tim proteins into the intermembrane space Mia40 interacts with small Tim proteins via disulfide bonds. Following isomerisation of the disulfide bridge, the polypeptide is released. Mia40 which is now in reduced state, is then oxidised by Erv1. [ 4 ] This oxidation step is vital to facilitate further rounds of precursor protein import. Without Erv1 activity, reduced Mia40 accumulates and is in inactive conformation. Interaction between the incoming precursor proteins, Mia40 and Erv1, is maintained as a result of a flow of electrons that are transferred from the incoming protein to Mia40 and from reduced Mia40 to oxidised Erv1. Precursors are then released in oxidised state and form disulfide bridges which prevents their escape out of the intermembrane space. Small Tim proteins are then maintained in active conformation within the intermembrane space by Hot13 (helper of Tims). [ 5 ] It is possible that Hot13 may have reducing effects on small Tim proteins as they counterbalance the harmful effects that oxidative agents exhibit. [ 2 ] | https://en.wikipedia.org/wiki/Tim9-Tim10_complex |
Tim Cannon is an American software developer , entrepreneur, and biohacker based in Pittsburgh, Pennsylvania . He is best known as Chief Information Officer of Grindhouse Wetware , a biotechnology startup company that creates technology to augment human capabilities. Grindhouse was co-founded by Cannon and Shawn Sarver in 2012. Cannon himself has had a variety of body modification implants, and has been referred to in the media as a cyborg . [ 1 ] [ 2 ] [ 3 ]
Cannon has spoken at conferences around the world on the topics of human enhancement, futurism, and citizen science, including at TEDx Rosslyn, [ 4 ] FITUR, [ 5 ] the University of Maryland , [ 6 ] the World Business Dialogue , [ 7 ] the Medical Entrepreneur Startup Hospital, [ 8 ] and others. He has been published in Wired [ 9 ] and featured in television shows such as National Geographic Channel 's The Big Picture with Kal Penn . [ 10 ] Cannon has been featured on podcasts including Ryan O'Shea's Future Grind [ 11 ] and Roderick Russell's Remarkably Human . [ 12 ]
Cannon has had a variety of body modification implants, including a radio-frequency identification (RFID) tag in his hand and magnetic implants in a finger, wrist, and tragus , [ 13 ] causing him to be labelled a cyborg by media outlets including Business Insider , [ 14 ] Newsweek , [ 15 ] The Awl , [ 16 ] and others. Because of legal and ethical restrictions on the types of surgery that can be done on humans, most of these modifications cannot be done by doctors or anesthetists. Instead they are done by body modification experts or on a "DIY" basis. [ 16 ]
In May 2012, inspired by Lepht Anonym , Cannon had finger magnets implanted to give him an "extra sense", the ability to feel electromagnetism . [ 17 ] [ 18 ]
In October 2013, Cannon became the first person to be implanted with the Grindhouse-designed biometric sensor known as Circadia, a procedure which was performed by body modification artist Steve Haworth in Essen, Germany . [ 19 ] [ 14 ] [ 20 ] The device, approximately the size of a deck of playing cards , automatically sent Cannon's temperature to his phone, was powered wirelessly through inductive charging, and mimicked bioluminescence with subdermal LEDs . [ 21 ] [ 22 ] After a few months as an initial proof-of-concept test, a series of panic attacks led to the device's removal. [ 23 ] Cannon is currently working to design an improved, consumer-friendly version of his Circadia implant that measures additional biometrics such as blood glucose, blood oxygen, blood pressure, and heart rate data. [ 24 ]
In November 2015, Tim had a prototype of Grindhouse's Northstar device implanted into his right forearm during a procedure at the “Cyborg Fair” in Düsseldorf, Germany . [ 25 ] A little larger than a coin, Northstar contained five LED lights, creating a bioluminescent effect when touched with a magnet (such as the ones implanted in Cannon's fingertips.) [ 26 ] Its purpose is solely aesthetic. [ 27 ] Capable of blinking around 10,000 times before the battery runs down, the device has been presented as a way to "light" tattoos. [ 26 ] | https://en.wikipedia.org/wiki/Tim_Cannon |
Timber framing ( German : Fachwerkbauweise ) and "post-and-beam" construction are traditional methods of building with heavy timbers , creating structures using squared-off and carefully fitted and joined timbers with joints secured by large wooden pegs. If the structural frame of load-bearing timber is left exposed on the exterior of the building it may be referred to as half-timbered , and in many cases the infill between timbers will be used for decorative effect. The country most known for this kind of architecture is Germany, where timber-framed houses are spread all over the country. [ 1 ] [ 2 ]
The method comes from working directly from logs and trees rather than pre-cut dimensional lumber . Hewing this with broadaxes , adzes , and draw knives and using hand-powered braces and augers (brace and bit) and other woodworking tools, artisans or framers could gradually assemble a building.
Since this building method has been used for thousands of years in many parts of the world like Europe [ 3 ] (Germany, France, Norway, Switzerland, etc.) and Asia , [ 4 ] many styles of historic framing have developed. These styles are often categorized by the type of foundation, walls, how and where the beams intersect, the use of curved timbers, and the roof framing details.
A simple timber frame made of straight vertical and horizontal pieces with a common rafter roof without purlins . The term box frame is not well defined and has been used for any kind of framing (with the usual exception of cruck framing). The distinction presented here is that the roof load is carried by the exterior walls. Purlins are also found even in plain timber frames.
A cruck is a pair of crooked or curved timbers [ 5 ] which form a bent (U.S.) or crossframe (UK); the individual timbers are each called a blade. More than 4,000 cruck frame buildings have been recorded in the UK. Several types of cruck frames are used; more information follows in English style below and at the main article Cruck .
Aisled frames have one or more rows of interior posts. These interior posts typically carry more structural load than the posts in the exterior walls. This is the same concept of the aisle in church buildings, sometimes called a hall church , where the center aisle is technically called a nave . However, a nave is often called an aisle, and three-aisled barns are common in the U.S., the Netherlands , and Germany. Aisled buildings are wider than the simpler box-framed or cruck-framed buildings, and typically have purlins supporting the rafters. In northern Germany, this construction is known as variations of a Ständerhaus .
Half-timbering refers to a structure with a frame of load-bearing timber, creating spaces between the timbers called panels (in German Gefach or Fächer = partitions), which are then filled-in with some kind of nonstructural material known as infill . The frame is often left exposed on the exterior of the building. [ 6 ]
Gallery of infill types:
The earliest known type of infill, called opus craticum by the Romans, was a wattle and daub type construction. [ 7 ] Opus craticum is now confusingly applied to a Roman stone/mortar infill as well. Similar methods to wattle and daub were also used and known by various names, such as clam staff and daub, cat-and-clay, or torchis (French), to name only three.
Wattle and daub was the most common infill in ancient times. The sticks were not always technically wattlework (woven), but also individual sticks installed vertically, horizontally, or at an angle into holes or grooves in the framing. The coating of daub has many recipes, but generally was a mixture of clay and chalk with a binder such as grass or straw and water or urine . [ 8 ] When the manufacturing of bricks increased, brick infill replaced the less durable infills and became more common. Stone laid in mortar as an infill was used in areas where stone rubble and mortar were available.
Other infills include bousillage , fired brick , unfired brick such as adobe or mudbrick , stones sometimes called pierrotage , planks as in the German ständerbohlenbau , timbers as in ständerblockbau , or rarely cob without any wooden support. [ 9 ] The wall surfaces on the interior were often "ceiled" with wainscoting and plastered for warmth and appearance.
Brick infill sometimes called nogging became the standard infill after the manufacturing of bricks made them more available and less expensive. Half-timbered walls may be covered by siding materials including plaster , weatherboarding , tiles , or slate shingles. [ 10 ]
The infill may be covered by other materials, including weatherboarding or tiles , [ 10 ] or left exposed. When left exposed, both the framing and infill were sometimes done in a decorative manner. Germany is famous for its decorative half-timbering and the figures sometimes have names and meanings. The decorative manner of half-timbering is promoted in Germany by the German Timber-Frame Road , several planned routes people can drive to see notable examples of Fachwerk buildings.
Gallery of some named figures and decorations:
The collection of elements in half timbering are sometimes given specific names:
According to Craven (2019), [ 11 ] the term:
was used informally to mean timber-framed construction in the Middle Ages. For economy, cylindrical logs were cut in half, so one log could be used for two (or more) posts. The shaved side was traditionally on the exterior and everyone knew it to be half the timber.
The term half-timbering is not as old as the German name Fachwerk or the French name colombage , but it is the standard English name for this style. One of the first people to publish the term "half-timbered" was Mary Martha Sherwood (1775–1851), who employed it in her book, The Lady of the Manor , published in several volumes from 1823 to 1829. She uses the term picturesquely: "...passing through a gate in a quickset hedge, we arrived at the porch of an old half-timbered cottage, where an aged man and woman received us." [ 12 ] By 1842, half-timbered had found its way into The Encyclopedia of Architecture by Joseph Gwilt (1784–1863). This juxtaposition of exposed timbered beams and infilled spaces created the distinctive "half-timbered", or occasionally termed, " Tudor " style, or "black-and-white".
The most ancient known half-timbered building is called the House of opus craticum . It was buried by the eruption of Mount Vesuvius in 79 AD in Herculaneum, Italy. Opus craticum was mentioned by Vitruvius in his books on architecture as a timber frame with wattlework infill. [ 13 ] However, the same term is used to describe timber frames with an infill of stone rubble laid in mortar the Romans called opus incertum . [ 14 ]
A less common meaning of the term "half-timbered" is found in the fourth edition of John Henry Parker's Classic Dictionary of Architecture (1873) which distinguishes full-timbered houses from half-timbered, with half-timber houses having a ground floor in stone [ 15 ] or logs such as the Kluge House which was a log cabin with a timber-framed second floor.
Traditional timber framing is the method of creating framed structures of heavy timber jointed together with various joints, commonly and originally with lap jointing , and then later pegged mortise and tenon joints. Diagonal bracing is used to prevent "racking", or movement of structural vertical beams or posts. [ 16 ]
Originally, German (and other) master carpenters would peg the joints with allowance of about 1 inch (25 mm), enough room for the wood to move as it ' seasoned ', then cut the pegs, and drive the beam home fully into its socket. [ citation needed ]
To cope with variable sizes and shapes of hewn (by adze or axe) and sawn timbers, two main carpentry methods were employed: scribe carpentry and square rule carpentry.
Scribing or coping was used throughout Europe, especially from the 12th century to the 19th century, and subsequently imported to North America, where it was common into the early 19th century. In a scribe frame, timber sockets are fashioned or "tailor-made" to fit their corresponding timbers; thus, each timber piece must be numbered (or "scribed").
Square-rule carpentry was developed in New England in the 18th century. It used housed joints in main timbers to allow for interchangeable braces and girts. Today, standardized timber sizing means that timber framing can be incorporated into mass-production methods as per the joinery industry, especially where timber is cut by precision computer numerical control machinery.
A jetty is an upper floor which sometimes historically used a structural horizontal beam, supported on cantilevers, called a bressummer or 'jetty bressummer' to bear the weight of the new wall, projecting outward from the preceding floor or storey.
In the city of York in the United Kingdom, the famous street known as The Shambles exemplifies this, where jettied houses seem to almost touch above the street.
Historically, the timbers would have been hewn square using a felling axe and then surface-finished with a broadaxe . If required, smaller timbers were ripsawn from the hewn baulks using pitsaws or frame saws. Today, timbers are more commonly bandsawn, and the timbers may sometimes be machine- planed on all four sides.
The vertical timbers include:
The horizontal timbers include:
When jettying, horizontal elements can include:
The sloping timbers include:
Historically were two different systems of the position of posts and studs:
Ridge-post framing is a structurally simple and ancient post and lintel framing where the posts extend all the way to the ridge beams. Germans call this Firstsäule or Hochstud .
In the 1930s a system of timber framing referred to as the "modern timber connector method" [ 17 ] was developed. It was characterized by the use of timber members assembled into trusses and other framing systems and fastened using various types of metal timber connectors. This type of timber construction was used for various building types including warehouses, factories, garages, barns, stores/markets, recreational buildings, barracks, bridges, and trestles. [ 18 ] The use of these structures was promoted because of their low construction costs, easy adaptability, and performance in fire as compared to unprotected steel truss construction.
During World War II, the United States Army Corps of Engineers and the Canadian Military Engineers undertook to construct airplane hangars using this timber construction system in order to conserve steel. Wood hangars were constructed throughout North America and employed various technologies including bowstring , Warren , and Pratt trusses, glued laminated arches, and lamella roof systems. Unique to this building type is the interlocking of the timber members of the roof trusses and supporting columns and their connection points. The timber members are held apart by "fillers" (blocks of timber). This leaves air spaces between the timber members which improves air circulation and drying around the members which improves resistance to moisture borne decay.
Timber members in this type of framing system were connected with ferrous timber connectors of various types. Loads between timber members were transmitted using split-rings (larger loads), toothed rings (lighter loads), or spiked grid connectors. [ 19 ] Split-ring connectors were metal rings sandwiched between adjacent timber members to connect them together. The rings were fit into circular grooves on in both timber members then the assembly was held together with through-bolts. The through-bolts only held the assembly together but were not load-carrying. [ 18 ] Shear plate connectors were used to transfer loads between timber members and metal. Shear plate connectors resembled large washers, deformed on the side facing the timber in order to grip it, and were through-fastened with long bolts or lengths of threaded rod. A leading manufacturer of these types of timber connectors was the Timber Engineering Company, or TECO, of Washington, DC. The proprietary name of their split-ring connectors was the "TECO Wedge-Fit".
Timber-framed structures differ from conventional wood-framed buildings in several ways. Timber framing uses fewer, larger wooden members, commonly timbers in the range of 15 to 30 cm (6 to 12 in), while common wood framing uses many more timbers with dimensions usually in the 5- to 25-cm (2- to 10-in) range. The methods of fastening the frame members also differ. In conventional framing, the members are joined using nails or other mechanical fasteners, whereas timber framing uses the traditional mortise and tenon or more complex joints that are usually fastened using only wooden pegs. [ citation needed ] Modern complex structures and timber trusses often incorporate steel joinery such as gusset plates, for both structural and architectural purposes.
Recently, it has become common practice to enclose the timber structure entirely in manufactured panels such as structural insulated panels (SIPs). Although the timbers can only be seen from inside the building when so enclosed, construction is less complex and insulation is greater than in traditional timber building. SIPs are "an insulating foam core sandwiched between two structural facings, typically oriented strand board" according to the Structural Insulated Panel Association. [ 20 ] SIPs reduce dependency on bracing and auxiliary members, because the panels span considerable distances and add rigidity to the basic timber frame.
An alternate construction method is with concrete flooring with extensive use of glass. This allows a solid construction combined with open architecture. Some firms have specialized in industrial prefabrication of such residential and light commercial structures such as Huf Haus as low-energy houses or – dependent on location – zero-energy buildings .
Straw-bale construction is another alternative where straw bales are stacked for nonload-bearing infill with various finishes applied to the interior and exterior such as stucco and plaster. This appeals to the traditionalist and the environmentalist as this is using "found" materials to build.
Mudbricks also called adobe are sometimes used to fill in timber-frame structures. They can be made on site and offer exceptional fire resistance. Such buildings must be designed to accommodate the poor thermal insulating properties of mudbrick, however, and usually have deep eaves or a veranda on four sides for weather protection.
Timber design or wood design is a subcategory of structural engineering that focuses on the engineering of wood structures. Timber is classified by tree species (e.g., southern pine, douglas fir, etc.) and its strength is graded using numerous coefficients that correspond to the number of knots, the moisture content, the temperature, the grain direction, the number of holes, and other factors. There are design specifications for sawn lumber, glulam members, prefabricated I-joists , composite lumber , and various connection types. In the United States, structural frames are then designed according to the Allowable Stress Design method or the Load Reduced Factor Design method (the latter being preferred). [ 21 ]
The techniques used in timber framing date back to Neolithic times, and have been used in many parts of the world during various periods such as ancient Japan, continental Europe, and Neolithic Denmark, England, France, Germany , Spain, parts of the Roman Empire , and Scotland. [ 22 ] The timber-framing technique has historically been popular in climate zones which favour deciduous hardwood trees, such as oak . Its northernmost areas are Baltic countries and southern Sweden. Timber framing is rare in Russia, Finland, northern Sweden, and Norway, where tall and straight lumber, such as pine and spruce, is readily available and log houses were favored, instead.
Half-timbered construction in the Northern European vernacular building style is characteristic of medieval and early modern Denmark, England, Germany, and parts of France and Switzerland, where timber was in good supply yet stone and associated skills to dress the stonework were in short supply. In half-timbered construction, timbers that were riven (split) in half provided the complete skeletal framing of the building.
Europe is full of timber-framed structures dating back hundreds of years, including manors, castles, homes, and inns, whose architecture and techniques of construction have evolved over the centuries. In Asia, timber-framed structures are found, many of them temples.
Some Roman carpentry preserved in anoxic layers of clay at Romano-British villa sites demonstrate that sophisticated Roman carpentry had all the necessary techniques for this construction. The earliest surviving (French) half-timbered buildings date from the 12th century. [ citation needed ]
Important resources for the study and appreciation of historic building methods are open-air museums .
The topping out ceremony is a builders' rite , an ancient tradition thought to have originated in Scandinavia by 700 AD. [ 23 ] In the U.S., a bough or small tree is attached to the peak of the timber frame after the frame is complete as a celebration. Historically, it was common for the master carpenter to give a speech, make a toast, and then break the glass. In Northern Europe, a wreath made for the occasion is more commonly used rather than a bough. In Japan, the "ridge raising" is a religious ceremony called the jotoshiki . [ 24 ] In Germany, it is called the Richtfest .
Carpenters' marks are markings left on the timbers of wooden buildings during construction.
Many historic hand tools used by timber framers for thousands of years have similarities, but vary in shape. Electrically powered tools first became available in the 1920s in the U.S. and continue to evolve. See the list of timber framing tools for basic descriptions and images of unusual tools (The list is incomplete at this time).
Some of the earliest known timber houses in Europe have been found in Great Britain , dating to Neolithic times; Balbridie and Fengate are some of the rare examples of these constructions.
Molded plaster ornamentation, pargetting [ 25 ] further enriched some English Tudor architecture houses. Half-timbering is characteristic of English vernacular architecture in East Anglia, [ 26 ] Warwickshire, [ 27 ] [ 28 ] Worcestershire, [ 29 ] Herefordshire, [ 30 ] [ 31 ] Shropshire, [ 32 ] [ 33 ] and Cheshire, [ 34 ] where one of the most elaborate surviving English examples of half-timbered construction is Little Moreton Hall . [ 35 ]
In South Yorkshire , the oldest timber house in Sheffield , the " Bishops' House " (c. 1500), shows traditional half-timbered construction.
In the Weald of Kent and Sussex, [ 36 ] the half-timbered structure of the Wealden hall house , [ 37 ] consisted of an open hall with bays on either side and often jettied upper floors.
Half-timbered construction traveled with British colonists to North America in the early 17th century but was soon abandoned in New England and the mid-Atlantic colonies for clapboard facings (an East Anglia tradition). The original English colonial settlements, such as Plymouth, Massachusetts and Jamestown, Virginia had timber-framed buildings, rather than the log cabins often associated with the American frontier. Living history programs demonstrating the building technique are available at both these locations.
One of the surviving streets lined with almost-touching houses is known as The Shambles , York , and is a popular tourist attraction.
For Timber-framed houses in Wales see: Architecture of Wales
Historic timber-frame construction in England (and the rest of the United Kingdom) showed regional variation [ 38 ] which has been divided into the "eastern school", the "western school", and the "northern school", although the characteristic types of framing in these schools can be found in the other regions (except the northern school). [ 39 ] A characteristic of the eastern school is close studding which is a half-timbering style of many studs spaced about the width of the studs apart (for example six-inch studs spaced six inches apart) until the middle of the 16th century and sometimes wider spacing after that time. Close studding was an elite style found mostly on expensive buildings. A principal style of the western school is the use of square panels of roughly equal size and decorative framing utilizing many shapes such as lozenges , stars, crosses, quatrefoils , cusps , and many other shapes. [ 39 ] The northern school sometimes used posts which landed on the foundation rather than on a sill beam, the sill joining to the sides of the posts and called an interrupted sill. Another northern style was to use close studding but in a herring-bone or chevron pattern. [ 39 ]
As houses were modified to cope with changing demands there sometimes were a combination of styles within a single timber-frame construction. [ 40 ] The major types of historic framing in England are 'cruck frame' , [ 40 ] box frame, [ 40 ] and aisled construction. From the box frame, more complex framed buildings such as the Wealden House and Jettied house developed. [ citation needed ]
The cruck frame design is among the earliest, and was [ 40 ] in use by the early 13th century, with its use continuing to the present day, although rarely after the 18th century. [ 40 ] Since the 18th century however, many existing cruck structures have been modified, with the original cruck framework becoming hidden. [ citation needed ] Aisled barns are of two or three aisled types, the oldest surviving aisled barn being the barley barn at Cressing Temple [ 39 ] dated to 1205–1235. [ 41 ]
Jettying was introduced in the 13th century and continued to be used through the 16th century. [ 39 ]
Generally speaking, the size of timbers used in construction, and the quality of the workmanship reflect the wealth and status of their owners. Small cottages often used quite small cross-section timbers which would have been deemed unsuitable by others. Some of these small cottages also have a 'home-made' – even temporary – appearance. Many such example can be found in the English shires. Equally, some relatively small buildings can be seen to incorporate substantial timbers and excellent craftsmanship, reflecting the relative wealth and status of their original owners. Important resources for the study of historic building methods in the UK are open-air museums .
It is often claimed that timber-framed buildings in Britain contain reused ships' timbers. This belief is dismissed by experts, who point out that curved timbers are rarely suitable, that salt is destructive to cellulose in the wood, and that ships' timbers are generally slight compared to cruck trusses. [ 42 ]
Elaborately half-timbered houses of the 13th through 18th centuries still remain in Bourges , Tours , Troyes , Rouen , Thiers , Dinan , Rennes , and many other cities, except in Provence and Corsica . Timber framing in French is known colloquially as pan de bois and half-timbering as colombage . Alsace is the region with the most timbered houses in France.
The Normandy tradition features two techniques: frameworks were built of four evenly spaced regularly hewn timbers set into the ground ( poteau en terre ) or into a continuous wooden sill ( poteau de sole ) and mortised at the top into the plate. The openings were filled with many materials including mud and straw, wattle and daub, or horsehair and gypsum. [ 43 ]
Germany has several styles of timber framing, but probably the greatest number of half-timbered buildings in the world are to be found in Germany and in Alsace (France). There are many small towns which escaped both war damage and modernisation and consist mainly, or even entirely, of half-timbered houses.
The German Timber-Frame Road ( Deutsche Fachwerkstraße ) is a tourist route that connects towns with remarkable fachwerk . It is more than 2,000 km (1,200 mi) long, crossing Germany through the states of Lower Saxony , Saxony-Anhalt , Hesse , Thuringia , Bavaria , and Baden-Württemberg . [ 16 ] [ 44 ]
Some of the more prominent towns (among many) include: Quedlinburg , a UNESCO -listed town, which has over 1200 half-timbered houses spanning five centuries; Goslar , another UNESCO-listed town; Hanau-Steinheim (home of the Brothers Grimm ); Bad Urach ; Eppingen ("Romance city" with a half-timbered church dating from 1320); Mosbach ; Vaihingen an der Enz and nearby UNESCO-listed Maulbronn Abbey ; Schorndorf (birthplace of Gottlieb Daimler ); Calw ; Celle ; and Biberach an der Riß with both the largest medieval complex, the Holy Spirit Hospital and one of Southern Germany's oldest buildings, now the Braith-Mali-Museum , dated to 1318.
German fachwerk building styles are extremely varied with a huge number of carpentry techniques which are highly regionalized. German planning laws for the preservation of buildings and regional architecture preservation dictate that a half-timbered house must be authentic to regional or even city-specific designs before being accepted. [ 45 ] [ 46 ]
A brief overview of styles follows, as a full inclusion of all styles is impossible.
In general the northern states have fachwerk similar to that of the nearby Netherlands and England while the more southerly states (most notably Bavaria and Switzerland) have more decoration using timber because of greater forest reserves in those areas. During the 19th century, a form of decorative timber-framing called bundwerk became popular in Bavaria, Austria and South Tyrol .
The German fachwerkhaus usually has a foundation of stone, or sometimes brick, perhaps up to several feet (a couple of metres) high, which the timber framework is mortised into or, more rarely, supports an irregular wooden sill.
The three main forms may be divided geographically:
The most characteristic feature is the spacing between the posts and the high placement of windows. Panels are enclosed by a sill , posts , and a plate , and are crossed by two rails between which the windows are placed—like "two eyes peering out". [ 45 ] [ 46 ]
In addition there is a myriad of regional scrollwork and fretwork designs of the non-loadbearing large timbers (braces) peculiar to particularly wealthy towns or cities.
A unique type of timber-frame house can be found in the region where the borders of Germany, the Czech Republic, and Poland meet – it is called the Upper Lusatian house (Umgebindehaus, translates as round-framed house ). This type has a timber frame surrounding a log structure on part of the ground floor. [ citation needed ]
Several half-timbered houses can be found in Northern Italy, especially in Piedmont , Lombardy , in the city of Bologna , in Sardinia in the Barbagia region and in the Iglesiente mining region.
Historically, the majority of Polish cities as well as their central marketplaces possessed timber-framed dwellings and housing. [ 47 ] Throughout the Middle Ages it was customary in Poland to use either bare brick or wattle and daub ( Polish : szachulec ) as filling in-between the timber frame. [ 47 ] However, the half-timbered houses which can be observed nowadays have been built in regions that were historically German or had significant German cultural influence. As these regions were at some point parts of German Prussia , half-timbered walls are often called mur pruski (lit. Prussian wall) in Polish. A distinctive type of house associated with mostly Mennonite immigrant groups from Frisia and the Netherlands, known as the Olędrzy , is called an "arcade house" ( dom podcieniowy ). The biggest timber-framed religious buildings in Europe are the Churches of Peace in southwestern Poland. [ 48 ] There are also numerous examples of timber-framed secular structures such as the granaries in Bydgoszcz .
The Umgebindehaus rural housing tradition of south Saxony (Germany) is also found in the neighboring areas of Poland, particularly in the Silesian region.
Another world-class type of wooden building Poland shares with some neighboring countries are its wooden church buildings .
The Spanish generally follow the Mediterranean forms of architecture with stone walls and shallow roof pitch. Timber framing is often of the post and lintel style. Castile and León , par example La Alberca , and the Basque Country have the most representative examples of the use of timber framing in the Iberian Peninsula.
Most traditional Basque buildings with half-timbering elements are detached farm houses (in Basque: baserriak ). Their upper floors were built with jettied box frames in close studding . In the oldest farmsteads and, if existing, in the third floor the walls were sometimes covered with vertical weatherboards . Big holes were left in the gable of the main façade for ventilation. The wooden beams were painted over, mostly in dark red. The vacancies were filled in with wattle and daub or rubble laid in a clay mortar and then plastered over with white chalk or nogged with bricks. Although the entire supporting structure is made of wood, the timbering is only visible on the main façade, which is generally oriented to the southeast.
Although the typical Basque house is now mostly associated with half-timbering, the outer walls and the fire-walls were built in masonry (rubble stone, bricks or, ideally, ashlars ) whenever it could be afforded. Timber was a sign of poverty. Oak-wood was cheaper than masonry: that is why, when the money was running out, the upper floor walls were mostly built timbered. Extant baserriak with half-timbered upper-floor façades were built from the 15th to 19th centuries and are found in all Basque regions with oceanic climate , except in Zuberoa (Soule), but are concentrated in Lapurdi (Labourd).
Some medieval Basque tower houses ( Dorretxe [ eu ] ) feature an overhanged upper floor in half-timbering.
To a lesser extent timbered houses are also found in villages and towns as row houses , as the photo from the Uztaritz village shows.
Currently, it has again become popular to build Néobasque [ fr ] houses resembling old Basque farmsteads, with more or less respect for the principles of traditional half-timbered building.
Switzerland has many styles of timber framing which overlap with its neighboring countries.
Nowadays, timber framing is primarily found in the provinces of Limburg , Liège , and Luxembourg . In urban areas, the ground floor was formerly built in stone and the upper floors in timber framing. Also, as timber framing was seen as a cheaper way of building, often the visible structures of noble houses were in stone and bricks, and the invisible or lateral walls in timber framing. The open-air museums of Bokrijk and Saint-Hubert ( Fourneau Saint-Michel ) show many examples of Belgian timber framing. Many post-and-beam houses can be found in cities and villages, but, unlike France, the United Kingdom, and Germany, there are few fully timber framed cityscapes.
Timber frame ( bindingsværk , literally "binding work") is the traditional building style in almost all of Denmark, making it the only Nordic country where this style is prevalent in all regions. Along the west coast of Jutland, houses built entirely of bricks were traditionally more common due to lack of suitable wood. In the 19th and especially in the 20th century, bricks have been the preferred building material in all of Denmark, but traditional timber-frame houses remain common both in the towns and in the countryside. Different regions have different traditions as to whether the timber frame should be tarred and thus clearly visible or be limewashed or painted in the same colour as the infills.
The Swedish mostly built log houses but they do have traditions of several types of timber framing: Some of the following links are written in Swedish. Most of the half-timbered houses in Sweden were built during the Danish time and are located in what until 1658 used to be Danish territory in southern Sweden, primarily in the province Skåne and secondarily in Blekinge and Halland . In Swedish half-timber is known as korsvirke .
Norway has at least two significant types of timber-framed structures: the stave church and Grindverk [ no ] . The term stave (a post or pole) indicates that a stave church essentially means a framed church, a distinction made in a region where log building is common. All but one surviving stave churches are in Norway, one in Sweden. Replicas of stave churches and other Norwegian building types have been reproduced elsewhere, e.g. at the Scandinavian Heritage Park in North Dakota , United States.
Grindverk translates as trestle construction, consisting of a series of transversal frames of two posts and a connecting beam, supporting two parallel wall plates bearing the rafters . Unlike other types of timber framing in Europe, the trestle frame construction uses no mortise and tenon joints. Archaeological excavations have uncovered similar wooden joints from more than 3,000 years ago, suggesting that this type of framing is an ancient unbroken tradition. Grindverk buildings are only found on part of the western coast of Norway, and most of them are boathouses and barns.
Log building was the common construction used for housing humans and livestock in Norway from the Middle Ages until the 18th century. Timber framing of the type used in large parts of Europe appeared occasionally in late medieval towns, but never became common, except for the capital Christiania . After a fire in 1624 in Oslo, King Christian IV ordered the town to be relocated to a new site. He outlawed log building to prevent future conflagrations and required wealthy burghers to use brickwork and the less affluent to use timber framing in the Danish manner. During the next two centuries, 50 per cent of the houses were timber framed.
All of these buildings disappeared as a consequence of this small provincial town of Christiania becoming the capital of independent Norway in 1814. This caused a rapid growth, with the population rising from 10 000 to 250 000 by 1900. Increasing prices caused a massive urban renewal , which resulted in all wooden structures being replaced with office blocks.
The Netherlands is often overlooked for its timbered houses, yet many exist, including windmills. It was in North Holland where the import of cheaper timber, combined with the Dutch innovation of windmill -powered sawmills , allowed economically viable widespread use of protective wood covering over framework. In the late 17th century the Dutch introduced vertical cladding also known in Eastern England as clasp board and in western England as weatherboard, then as more wood was available more cheaply, horizontal cladding in the 17th century. Perhaps owing to economic considerations, vertical cladding returned to fashion. [ 49 ] Dutch wall framing is virtually always built in bents and the three basic types of roof framing are the rafter roof, purlin roof, and ridge-post roof. [ 50 ]
Half-timbered houses can be found in Romania mostly in areas once inhabited by Transylvanian Saxons , in cities, towns and villages with Germanic influence such as Bistrița , Brașov , Mediaș , Sibiu and Sighișoara . However the number of half-timbered houses is small. In Wallachia there are few examples of this type of architecture, most of those buildings being located in Sinaia , such as the Peleș Castle .
As the result of centuries of German settlement and cultural influence, towns in the Baltic states such as Klaipėda and Riga also preserve German-style Fachwerkhäuser.
Most "haft-timbered" houses existing in Missouri, Pennsylvania, and Texas were built by German settlers. [ 43 ] Old Salem North Carolina has fine examples of German fachwerk buildings. [ 51 ] : 42–43 Many are still present in Colonia Tovar ( Venezuela ), Santa Catarina and Rio Grande do Sul (Brazil), where Germans settled. Later, they chose more suitable building materials for local conditions (most likely because of the great problem of tropical termites.)
In the historical region of North America known as New France , colombage pierroté , also called maçonnerie entre poteaux , [ 52 ] half-timbered construction with the infill between the posts and studs of stone rubble and lime plaster or bousillage [ 52 ] and simply called colombage in France. Colombage was used from the earliest settlement until the 18th century but was known as bousillage entre poteaus sur solle in Lower Louisiana . The style had its origins in Normandy, and was brought to Canada by early Norman settlers. The Men's House at Lower Fort Garry is a good example. The exterior walls of such buildings were often covered over with clapboards to protect the infill from erosion. Naturally, this required frequent maintenance, and the style was abandoned as a building method in the 18th century in Québec. For the same reasons, half-timbering in New England, which was originally employed by the English settlers, fell out of favour soon after the colonies had become established.
Other variations of half-timbering are colombage à teurques (torchis), straw coated with mud and hung over horizontal staves (or otherwise held in place), colombage an eclisses, and colombage a lattes. [ 52 ]
Poteaux-en-terre (posts in ground) is a type of timber framing with the many vertical posts or studs buried in the ground called post in ground or "earthfast" construction. The tops of the posts are joined to a beam and the spaces between are filled in with natural materials called bousillage or pierrotage .
Poteaux-sur-sol (posts on a sill) is a general term for any kind of framing on a sill. However, sometimes it specifically refers to "vertical log construction" like poteaux-en-terre placed on sills with the spaces between the timbers infilled.
Piece-sur-piece , also known as Post-and-plank style or "corner post construction" (and many other names) in which wood is used both for the frame and horizontal infill; for this reason it may be incorrect to call it "half-timbering". It is sometimes a blend of framing and log building with two styles: the horizontal pieces fit into groves in the posts and can slide up and down or the horizontal pieces fit into individual mortises in the posts and are pegged and the gaps between the pieces chinked (filled in with stones or chips of wood covered with mud or moss briefly discussed in Log cabin .)
This technique of a timber frame walls filled in with horizontal planks or logs proved better suited to the harsh climates of Québec and Acadia, which at the same time had abundant wood. It became popular throughout New France, as far afield as southern Louisiana. The Hudson's Bay Company used this technique for many of its trading posts, and this style of framing becoming known as Hudson Bay style or Hudson Bay corners. Also used by the Red River Colony this style also became known as "Red River Framing". "The support of horizontal timbers by corner posts is an old form of construction in Europe. It was apparently carried across much of the continent from Silesia by the Lausitz urnfield culture in the late Bronze Age." [ 53 ] Similar building techniques are apparently not found in France [ 51 ] : 121 but exist in Germany and Switzerland known as Bohlenstanderbau when planks are used or blockstanderbau when beams are used as the infill. In Sweden the technique is known as sleppvegg or skiftesverk and in Denmark as bulhus .
A particularly interesting example in the U.S. is the Golden Plough Tavern (c. 1741), York, York County, Pennsylvania, which has the ground level of corner-post construction with the second floor of fachwerk (half timbered) and was built for a German with other Germanic features. [ 54 ]
Settlers in New France also built horizontal log, brick, and stone buildings.
Characteristics of traditional timber framing in the parts of the U.S. formerly known as New Netherland are H-framing also known as dropped-tie framing in the U.S. and the similar anchor beam framing as found in the New World Dutch barn .
Some time periods/regions within New England contain certain framing elements such as common purlin roofs, five sided ridge beams, plank-frame construction and plank-wall construction. The English barn always contains an "English tying joint" and the later New England style barn were built using bents .
Japanese timber framing is believed to be descended from Chinese framing (see Ancient Chinese wooden architecture ). Asian framing is significantly different from western framing, with its predominant use of post and lintel framing and an almost complete lack of diagonal bracing.
When half-timbering regained popularity in Britain after 1860 in the various revival styles, such as the Queen Anne style houses by Richard Norman Shaw and others, it was often used to evoke a "Tudor" atmosphere ( see Tudorbethan ), though in Tudor times half-timbering had begun to look rustic and was increasingly limited to village houses ( illustration, above left ).
In 1912, Allen W. Jackson published The Half-Timber House: Its Origin, Design, Modern Plan, and Construction, and rambling half-timbered beach houses appeared on dune-front properties in Rhode Island or under palm-lined drives of Beverly Hills . During the 1920s increasingly minimal gestures towards some half-timbering in commercial speculative housebuilding saw the fashion diminish.
In the revival styles, such as Tudorbethan (Mock Tudor), the half-timbered appearance is superimposed on the brickwork or other material as an outside decorative façade rather than forming the main frame that supports the structure.
The style was used in many of the homes built in Lake Mohawk, New Jersey , as well as all of the clubhouse, shops, and marina.
For information about "roundwood framing" see the book Roundwood Timber Framing: Building Naturally Using Local Resources by Ben Law (East Meon, Hampshire: Permanent Publications; 2010. ISBN 1856230414 )
The use of timber framing in buildings offers various aesthetic and structural benefits, as the timber frame lends itself to open plan designs and allows for complete enclosure in effective insulation for energy efficiency. In modern construction, a timber-frame structure offers many benefits:
In North America, heavy timber construction is classified Building Code Type IV: a special class reserved for timber framing which recognizes the inherent fire resistance of large timber and its ability to retain structural capacity in fire situations. In many cases this classification can eliminate the need and expense of fire sprinklers in public buildings. [ 58 ]
In terms of the traditional half-timber or fachwerkhaus there are maybe more disadvantages than advantages today. Such houses are notoriously expensive to maintain let alone renovate and restore, most commonly owing to local regulations that do not allow divergence from the original, modification or incorporation of modern materials.
Additionally, in such nations as Germany, where energy efficiency is highly regulated, the renovated building may be required to meet modern energy efficiencies, if it is to be used as a residential or commercial structure (museums and significant historic buildings have no semi-permanent habitade exempt). Many framework houses of significance are treated merely to preserve, rather than render inhabitable – most especially as the required heavy insecticidal fumigation is highly poisonous.
In some cases, it is more economical to build anew using authentic techniques and correct period materials than restore. One major problem with older structures is the phenomenon known as mechano-sorptive creep or slanting: where wood beams absorb moisture whilst under compression or tension strains and deform, shift position or both. This is a major structural issue as the house may deviate several degrees from perpendicular to its foundations (in the x-axis, y-axis, and even z-axis) and thus be unsafe and unstable or so out of square it is extremely costly to remedy. [ 59 ]
A summary of problems with Fachwerkhäuser or half-timbered houses includes the following, though many can be avoided by thoughtful design and application of suitable paints and surface treatments and routine maintenance. Often, though when dealing with a structure of a century or more old, it is too late. [ 49 ] | https://en.wikipedia.org/wiki/Timber_framing |
Timber grading is the process of evaluating and categorizing timber based on its physical characteristics, strength, and suitability for specific applications. [ 1 ] [ 2 ] This classification ensures that timber meets industry standards and is appropriate for its intended use in construction, furniture making, and other applications. [ 3 ]
Timber grading (or wood grading) is primarily conducted through two methods:
Visual grading involves the manual inspection of timber by trained graders who assess characteristics such as knots, grain patterns, and defects. This method is widely used due to its simplicity and cost-effectiveness. [ 4 ]
Machine grading utilizes mechanical devices to assess the strength and stiffness of timber. This method provides more consistent and objective results compared to visual grading. [ 5 ]
Different regions in the globe have established standards to ensure uniformity in timber grading:
Properly graded timber ensures safety, durability, and performance in various applications, such as:
In Europe, strength grading classifies the structural performance of individual timber boards. In accordance with the standards outlined in standards BS EN 338 and EN 14081, timber is subjected to various assessment methods to determine its mechanical properties. Based on these results, a strength class is assigned, providing an at-a-glance indication of the timber’s load-bearing capability. This classification system helps ensure that the right type of timber is selected for specific structural applications. Timber strength classes are categorized based on the type of wood — hardwoods or softwoods. For hardwoods , the classification begins with the letter ‘D’, representing their origin from deciduous trees. The number following the ‘D’ indicates the strength level, with higher numbers denoting greater strength. Available hardwood strength classes include D24, D30, D40, D50, D60, and D70. Softwoods , which come from coniferous trees, are labeled with a ‘C’ . The numbering system follows the same principle as hardwoods: the higher the number, the stronger the timber. Common softwood grades include C14, C16, C18, and C24, or (rarely) higher. [ 9 ] | https://en.wikipedia.org/wiki/Timber_grading |
Time-Sensitive Networking ( TSN ) is a set of standards under development by the Time-Sensitive Networking task group of the IEEE 802.1 working group. [ 1 ] The TSN task group was formed in November 2012 by renaming the existing Audio Video Bridging Task Group [ 2 ] and continuing its work. The name changed as a result of the extension of the working area of the standardization group. The standards define mechanisms for the time-sensitive transmission of data over deterministic Ethernet networks.
The majority of projects define extensions to the IEEE 802.1Q – Bridges and Bridged Networks, which describes virtual LANs and network switches . [ 3 ] These extensions in particular address transmission with very low latency and high availability. Applications include converged networks with real-time audio/video streaming and real-time control streams which are used in automotive applications and industrial control facilities.
Standard information technology network equipment has no concept of "time" and cannot provide synchronization and precision timing. Delivering data reliably is more important than delivering within a specific time, so there are no constraints on delay or synchronization precision. Even if the average hop delay is very low, individual delays can be unacceptably high. Network congestion is handled by throttling and retransmitting dropped packets at the transport layer, but there are no means to prevent congestion at the link layer. Data can be lost when the buffers are too small or the bandwidth is insufficient, but excessive buffering adds to the delay, which is unacceptable when low deterministic delays are required.
The different AVB/TSN standards documents specified by IEEE 802.1 can be grouped into three basic key component categories that are required for a complete real-time communication solution based on switched Ethernet networks with deterministic quality of service (QoS) for point-to-point connections. Each and every standard specification can be used on its own and is mostly self-sufficient. However, only when used together in a concerted way, TSN as a communication system can achieve its full potential. The three basic components are:
Applications which need a deterministic network that behaves in a predictable fashion include audio and video, initially defined in Audio Video Bridging (AVB); control networks that accept inputs from sensors, perform control loop processing, and initiate actions; safety-critical networks that implement packet and link redundancy; and mixed media networks that handle data with varying levels of timing sensitivity and priority, such as vehicle networks that support climate control, infotainment, body electronics, and driver assistance. The IEEE AVB/TSN suite serves as the foundation for deterministic networking to satisfy the common requirements of these applications.
AVB/TSN can handle rate-constrained traffic, where each stream has a bandwidth limit defined by minimum inter-frame intervals and maximal frame size, and time-trigger traffic with an exact accurate time to be sent. Low-priority traffic is passed on best-effort base, with no timing and delivery guarantees.
In contrast to standard Ethernet according to IEEE 802.3 and Ethernet bridging according to IEEE 802.1Q , time is very important in TSN networks. For real-time communication with hard, non-negotiable time boundaries for end-to-end transmission latencies, all devices in this network need to have a common time reference and therefore, need to synchronize their clocks among each other. This is not only true for the end devices of a communication stream, such as an industrial controller and a manufacturing robot, but also true for network components, such as Ethernet switches . Only through synchronized clocks, it is possible for all network devices to operate in unison and execute the required operation at exactly the required point in time.
Although time synchronization in TSN networks can be achieved with GPS clock , this is costly and there is no guarantee that the endpoint device has access to the radio or satellite signal at all times. Due to these constraints, time in TSN networks is usually distributed from one central time source directly through the network itself using the IEEE 1588 Precision Time Protocol , which utilizes Ethernet frames to distribute time synchronization information. IEEE 802.1AS is a tightly constrained subset of IEEE 1588 with sub-microsecond precision and extensions to support synchronisation over WiFi radio ( IEEE 802.11 ). The idea behind this profile is to narrow the huge list of different IEEE 1588 options down to a manageable few critical options that are applicable to home networks or networks in automotive or industrial automation environments.
IEEE 802.1AS-2011 defines the Generalized Precision Time Protocol (gPTP) profile which, like all profiles of IEEE 1588 , selects among the options of 1588, but also generalizes the architecture to allow PTP to apply beyond wired Ethernet networks.
To account for data path delays, the gPTP protocol measures the frame residence time within each bridge (the time required for receiving, processing, queuing and transmission of timing information from the ingress to egress ports), and the link latency of each hop (a propagation delay between two adjacent bridges). These calculated delays are then referenced to the GrandMaster (GM) clock in a bridge elected by the Best Master Clock Algorithm, a clock spanning tree protocol to which all Clock Master (CM) and endpoint devices attempt to synchronize. Any device which does not synchronize to timing messages is outside of the timing domain boundaries (Figure 2).
Synchronization accuracy depends on precise measurements of link delay and frame residence time. 802.1AS uses 'logical syntonization', where a ratio between local clock and GM clock frequencies is used to calculate synchronized time, and a ratio between local and GM clock frequencies to calculate propagation delay.
IEEE802.1AS-2020 introduces improved time measurement accuracy and support for multiple time domains for redundancy.
Scheduling and traffic shaping allows for the coexistence of different traffic classes with different priorities on the same network - each with different requirements to available bandwidth and end-to-end latency.
Traffic shaping refers to the process of distributing frames/packets evenly in time to smooth out the traffic. Without traffic shaping at sources and bridges, the packets will "bunch", i.e. agglomerate into bursts of traffic, overwhelming the buffers in subsequent bridges/switches along the path.
Standard bridging according to IEEE 802.1Q uses a strict priority scheme with eight distinct priorities. On the protocol level, these priorities are visible in the Priority Code Point (PCP) field in the 802.1Q VLAN tag of a standard Ethernet frame . These priorities already distinguish between more important and less important network traffic, but even with the highest of the eight priorities, no absolute guarantee for an end-to-end delivery time can be given. The reason for this is buffering effects inside the Ethernet switches. If a switch has started the transmission of an Ethernet frame on one of its ports, even the highest priority frame has to wait inside the switch buffer for this transmission to finish. With standard Ethernet switching, this non-determinism cannot be avoided. This is not an issue in environments where applications do not depend on the timely delivery of single Ethernet frames - such as office IT infrastructures. In these environments, file transfers, emails or other business applications have limited time sensitivity themselves and are usually protected by other mechanisms further up the protocol stack, such as the Transmission Control Protocol . In industrial automation (Programmable Logic Controller ( PLC ) with an industrial robot ) and automotive car environments, where closed loop control or safety applications are using the Ethernet network, reliable and timely delivery is of utmost importance. AVB/TSN enhances standard Ethernet communication by adding mechanisms to provide different time slices for different traffic classes and ensure timely delivery with soft and hard real-time requirements of control system applications. The mechanism of utilizing the eight distinct VLAN priorities is retained, to ensure complete backward compatibility to non-TSN Ethernet. To achieve transmission times with guaranteed end-to-end latency, one or several of the eight Ethernet priorities can be individually assigned to already existing methods (such as the IEEE 802.1Q strict priority scheduler) or new processing methods, such as the IEEE 802.1Qav credit-based traffic shaper, IEEE 802.1Qbv time-aware shaper, [ 4 ] or IEEE 802.1Qcr asynchronous shaper.
Time-sensitive traffic has several priority classes. For credit-based shaper 802.1Qav, Stream Reservation Class A is the highest priority with a transmission period of 125 μs ; Class B has the second-highest priority with a maximum transmission period of 250 μs . Traffic classes shall not exceed their preconfigured maximum bandwidth (75% for audio and video applications). The maximum number of hops is 7. The worst-case latency requirement is defined as 2 ms for Class A and 50 ms for Class B, but has been shown to be unreliable. [ 5 ] [ 6 ] The per-port peer delay provided by gPTP and the network bridge residence delay are added to calculate the accumulated delays and ensure the latency requirement is met. Control traffic has the third-highest priority and includes gPTP and SRP traffic. Time-aware scheduler 802.1Qbv introduces Class CDT for realtime control data from sensors and command streams to actuators, with worst-case latency of 100 μs over 5 hops, and a maximum transmission period of 0.5 ms. Class CDT takes the highest priority over classes A, B, and control traffic.
IEEE 802.1Qav Forwarding and Queuing Enhancements for Time-Sensitive Streams defines traffic shaping using priority classes, which is based on a simple form of "leaky bucket" credit-based fair queuing . 802.1Qav is designed to reduce buffering in receiving bridges and endpoints.
The credit-based shaper defines credits in bits for two separate queues, dedicated to Class A and Class B traffic. Frame transmission is only allowed when credit is non-negative; during transmission the credit decreases at a rate called sendSlope:
The credit increases at a rate idleSlope if frames are waiting for other queues to be transmitted:
Thus the idleSlope is the bandwidth reserved for the queue by the bridge, and the sendSlope is the transmission rate of the port MAC service.
If the credit is negative and no frames are transmitted, credit increases at idleSlope rate until zero is reached. If an AVB frame cannot be transmitted because a non-AVB frame is in transmission, credit accumulates at idleSlope rate but positive credit is allowed.
Additional limits hiCredit and loCredit are derived from the maximum frame size and maximum interference size, the idleSlope/sendSlope, and the maximum port transmission rate.
Reserved AV stream traffic frames are forwarded with high priority over non-reserved best-effort traffic, subject to credit-based traffic shaping rules which may require them to wait for certain amount of credits. This protects best-effort traffic by limiting maximum AV stream burst. The frames are scheduled very evenly, though only on an aggregate basis, to smooth out the delivery times and reduce bursting and bunching, which can lead to buffer overflows and packet drops that trigger retransmissions. The increased buffering delay makes re-transmitted packets obsolete by the time they arrive, resulting in frame drops which reduces the quality of AV applications.
Though credit-based shaper provides fair scheduling for low-priority packets and smooths out traffic to eliminate congestion, unfortunately, average delay increases up to 250 μs per hop, which is too high for control applications, whereas a time-aware shaper (IEEE 802.1Qbv) has a fixed cycle delay from 30 μs to several milliseconds, and typical delay of 125 μs. Deriving guaranteed upper bounds on delays in TSN is non-trivial and is currently being researched, e.g., by using the mathematical framework Network Calculus. [ 7 ]
IEEE 802.1Qat Stream Reservation Protocol (SRP) is a distributed peer-to-peer protocol that specifies admission controls based on resource requirements of the flow and available network resources.
SRP reserves resources and advertises streams from the sender/source (talker) to the receivers/destinations (listeners); it works to satisfy QoS requirements for each stream and guarantee the availability of sufficient network resources along the entire flow transmission path.
The traffic streams are identified and registered with a 64-bit StreamID, made up of the 48-bit MAC address (EUI) and 16-bit UniqueID to identify different streams from the one source.
SRP employs variants of Multiple Registration Protocol (MRP) to register and de-register attribute values on switches/bridges/devices - the Multiple MAC Registration Protocol (MMRP), the Multiple VLAN Registration Protocol (MVRP), and the Multiple Stream Registration Protocol (MSRP).
The SRP protocol essentially works in the following sequence:
Resources are allocated and configured in both the end nodes of the data stream and the transit nodes along the data flow path, with an end-to-end signaling mechanism to detect the success/failure. Worst-case latency is calculated by querying every bridge.
Reservation requests use the general MRP application with MRP attribute propagation mechanism. All nodes along the flow path pass the MRP Attribute Declaration (MAD) specification which describes the stream characteristics so that bridges could allocate the necessary resources.
If a bridge is able to reserve the required resources, it propagates the advertisement to the next bridge; otherwise, a 'talker failed' message is raised. When the advertise message reaches the listener, it replies with 'listener ready' message that propagates back to the talker.
Talker advertise and listener ready messages can be de-registered, which terminates the stream.
Successful reservation is only guaranteed when all intermediate nodes support SRP and respond to advertise and ready messages; in Figure 2 above, AVB domain 1 is unable to connect with AVB domain 2.
SRP is also used by TSN/AVB standards for frame priorities, frame scheduling, and traffic shaping
SRP uses decentralized registration and reservation procedure, multiple requests can introduce delays for critical traffic. IEEE 802.1Qcc-2018 "Stream Reservation Protocol (SRP) Enhancements and Performance Improvements" amendment reduces the size of reservation messages and redefines timers so they trigger updates only when link state or reservation is changed. To improve TSN administration on large scale networks, each User Network Interface (UNI) provides methods for requesting Layer 2 services, supplemented by Centralized Network Configuration (CNC) to provide centralized reservation and scheduling, and remote management using NETCONF/RESTCONF protocols and IETF YANG/NETCONF data modeling.
CNC implements a per-stream request-response model, where SR class is not explicitly used: end-stations send requests for a specific stream (via edge port) without knowledge of the network configuration, and CNC performs stream reservation centrally. MSRP only runs on the link to end-stations as an information carrier between CNC and end-stations, not for stream reservation. Centralized User Configuration (CUC) is an optional node that discovers end stations, their capabilities and user requirements, and configures delay-optimized TSN features (for closed-loop IACS applications). Seamless interop with Resource Reservation Protocol (RSVP) transport is provided.
802.1Qcc allows centralized configuration management to coexist with decentralized, fully distributed configuration of the SRP protocol, and also supports hybrid configurations for legacy AVB devices.
802.1Qcc can be combined with IEEE 802.1Qca Path Control and Reservation (PCR) and TSN traffic shapers.
While the 802.1Qav FQTSS/CBS works very well with soft real-time traffic, worst-case delays are both hop count and network topology dependent. Pathological topologies introduce delays, so buffer size requirements have to consider network topology.
IEEE 802.1Qch Cyclic Queuing and Forwarding (CQF), also known as the Peristaltic Shaper (PS), introduces double buffering which allows bridges to synchronize transmission (frame enqueue/dequeue operations) in a cyclic manner, with bounded latency depending only on the number of hops and the cycle time, completely independent of the network topology.
CQF can be used with the IEEE 802.1Qbv time-aware scheduler, IEEE 802.1Qbu frame preemption, and IEEE 802.1Qci ingress traffic policing.
IEEE 802.1Qci Per-Stream Filtering and Policing (PSFP) improves network robustness by filtering individual traffic streams. It prevents traffic overload conditions that may affect bridges and the receiving endpoints due to malfunction or Denial of Service (DoS) attacks.
The stream filter uses rule matching to allow frames with specified stream IDs and priority levels and apply policy actions otherwise. All streams are coordinated at their gates, similarly to the 802.1Qch signaling.
The flow metering applies predefined bandwidth profiles for each stream.
The IEEE 802.1Qbv time-aware scheduler is designed to separate the communication on the Ethernet network into fixed length, repeating time cycles. Within these cycles, different time slices can be configured that can be assigned to one or several of the eight Ethernet priorities. By doing this, it is possible to grant exclusive use - for a limited time - to the Ethernet transmission medium for those traffic classes that need transmission guarantees and can't be interrupted. The basic concept is a time-division multiple access (TDMA) scheme. By establishing virtual communication channels for specific time periods, time-critical communication can be separated from non-critical background traffic.
Time-aware scheduler introduces Stream Reservation Class CDT for time-critical control data, with worst-case latency of 100 μs over 5 hops, and maximum transmission period of 0.5 ms, in addition to classes A and B defined for IEEE 802.1Qav credit-based traffic shaper. By granting exclusive access to the transmission medium and devices to time-critical traffic classes, the buffering effects in the Ethernet switch transmission buffers can be avoided and time-critical traffic can be transmitted without non-deterministic interruptions. One example for an IEEE 802.1Qbv scheduler configuration is visible in figure 1:
In this example, each cycle consists of two time slices. Time slice 1 only allows the transmission of traffic tagged with VLAN priority 3, and time slice 2 in each cycle allows for the rest of the priorities to be sent. Since the IEEE 802.1Qbv scheduler requires all clocks on all network devices (Ethernet switches and end devices) to be synchronized and the identical schedule to be configured, all devices understand which priority can be sent to the network at any given point in time. Since time slice 2 has more than one priority assigned to it, within this time slice, the priorities are handled according to standard IEEE 802.1Q strict priority scheduling.
This separation of Ethernet transmissions into cycles and time slices can be enhanced further by the inclusion of other scheduling or traffic shaping algorithms, such as the IEEE 802.1Qav credit-based traffic shaper. IEEE 802.1Qav supports soft real-time. In this particular example, IEEE 802.1Qav could be assigned to one or two of the priorities that are used in time slice two to distinguish further between audio/video traffic and background file transfers. The Time-Sensitive Networking Task Group specifies a number of different schedulers and traffic shapers that can be combined to achieve the nonreactive coexistence of hard real-time, soft real-time and background traffic on the same Ethernet infrastructure.
When an Ethernet interface has started the transmission of a frame to the transmission medium, this transmission has to be completely finished before another transmission can take place. This includes the transmission of the CRC32 checksum at the end of the frame to ensure a reliable, fault-free transmission. This inherent property of Ethernet networks - again- poses a challenge to the TDMA approach of the IEEE 802.1Qbv scheduler. This is visible in figure 2:
Just before the end of time slice 2 in cycle n, a new frame transmission is started. Unfortunately, this frame is too large to fit into its time slice. Since the transmission of this frame cannot be interrupted, the frame infringes the following time slice 1 of the next cycle n+1. By partially or completely blocking a time-critical time slice, real-time frames can be delayed up to the point where they cannot meet the application requirements any longer. This is very similar to the actual buffering effects that happen in non-TSN Ethernet switches, so TSN has to specify a mechanism to prevent this from happening.
The IEEE 802.1Qbv time-aware scheduler has to ensure that the Ethernet interface is not busy with the transmission of a frame when the scheduler changes from one-time slice into the next. The time-aware scheduler achieves this by putting a guard band in front of every time slice that carries time-critical traffic. During this guard band time, no new Ethernet frame transmission may be started, only already ongoing transmissions may be finished. The duration of this guard band has to be as long as it takes the maximum frame size to be safely transmitted. For an Ethernet frame according to IEEE 802.3 with a single IEEE 802.1Q VLAN tag and including interframe spacing , the total length is: 1500 byte (frame payload) + 18 byte (Ethernet addresses, EtherType and CRC) + 4 byte (VLAN Tag) + 12 byte (Interframe spacing) + 8 byte (preamble and SFD) = 1542 byte.
The total time needed for sending this frame is dependent on the link speed of the Ethernet network. With Fast Ethernet and 100 Mbit/s transmission rate, the transmission duration is as follows:
In this case, the guard band has to be at least 123.36 μs long. With the guard band, the total bandwidth or time that is usable within a time slice is reduced by the length of the guard band. This is visible in figure 3
Note: to facilitate the presentation of the topic, the actual size of the guard band in figure 3 is not to scale, but is significantly smaller than indicated by the frame in figure 2.
In this example, the time slice 1 always contains high priority data (e.g. for motion control), while time slice 2 always contains best-effort data. Therefore, a guard band needs to be placed at every transition point into time slice 1 to protect the time slice of the critical data stream(s).
While the guard bands manage to protect the time slices with high priority, critical traffic, they also have some significant drawbacks:
To partially mitigate the loss of bandwidth through the guard band, the standard IEEE 802.1Qbv includes a length-aware scheduling mechanism. This mechanism is used when store-and-forward switching is utilized: after the full reception of an Ethernet frame that needs to be transmitted on a port where the guard band is in effect, the scheduler checks the overall length of the frame. If the frame can fit completely inside the guard band, without any infringement of the following high priority slice, the scheduler can send this frame, despite an active guard band, and reduce the waste of bandwidth. This mechanism, however, cannot be used when cut-through switching is enabled, since the total length of the Ethernet frame needs to be known a priori. Therefore, when cut-through switching is used to minimize end-to-end latency, the waste of bandwidth will still occur. Also, this does not help with the minimum achievable cycle time. Therefore, length-aware scheduling is an improvement, but cannot mitigate all drawbacks that are introduced by the guard band.
To further mitigate the negative effects from the guard bands, the IEEE working groups 802.1 and 802.3 have specified the frame pre-emption technology. The two working groups collaborated in this endeavour since the technology required both changes in the Ethernet Media Access Control (MAC) scheme that is under the control of IEEE 802.3, as well as changes in the management mechanisms that are under the control of IEEE 802.1. Due to this fact, frame pre-emption is described in two different standards documents: IEEE 802.1Qbu [ 8 ] for the bridge management component and IEEE 802.3br [ 9 ] for the Ethernet MAC component.
Frame preemption defines two MAC services for an egress port, preemptable MAC (pMAC) and express MAC (eMAC). Express frames can interrupt transmission of preemptable frames. On resume, MAC merge sublayer re-assembles frame fragments in the next bridge.
Preemption causes computational overhead in the link interface, as the operational context shall be transitioned to the express frame.
Figure 4 gives a basic example how frame pre-emption works. During the process of sending a best effort Ethernet frame, the MAC interrupts the frame transmission just before the start of the guard band. The partial frame is completed with a CRC and will be stored in the next switch to wait for the second part of the frame to arrive. After the high priority traffic in time slice 1 has passed and the cycle switches back to time slice 2, the interrupted frame transmission is resumed. Frame pre-emption always operates on a pure link-by-link basis and only fragments from one Ethernet switch to the next Ethernet switch, where the frame is reassembled. In contrast to fragmentation with the Internet Protocol (IP) , no end-to-end fragmentation is supported.
Each partial frame is completed by a CRC32 for error detection. In contrast to the regular Ethernet CRC32, the last 16 bits are inverted to make a partial frame distinguishable from a regular Ethernet frame. In addition, the start of frame delimiter (SFD) is changed.
The support for frame pre-emption has to be activated on each link between devices individually. To signal the capability for frame pre-emption on a link, an Ethernet switch announces this capability through the LLDP (Link Layer Discovery Protocol) . When a device receives such an LLDP announcement on a network port and supports frame pre-emption itself, it may activate the capability. There is no direct negotiation and activation of the capability on adjacent devices. Any device that receives the LLDP pre-emption announcement assumes that on the other end of the link, a device is present that can understand the changes in the frame format (changed CRC32 and SFD).
Frame pre-emption allows for a significant reduction of the guard band. The length of the guard band is now dependent on the precision of the frame pre-emption mechanism: how small is the minimum size of the frame that the mechanism can still pre-empt. IEEE 802.3br specifies the best accuracy for this mechanism at 64 byte - due to the fact that this is the minimum size of a still valid Ethernet frame. In this case, the guard band can be reduced to a total of 127 byte: 64 byte (minimum frame) + 63 byte (remaining length that cannot be pre-empted). All larger frames can be pre-empted again and therefore, there is no need to protect against this size with a guard band.
This minimizes the best effort bandwidth that is lost and also allows for much shorter cycle times at slower Ethernet speeds, such as 100 Mbit/s and below. Since the pre-emption takes place in hardware in the MAC, as the frame passes through, cut-through switching can be supported as well, since the overall frame size is not needed a priori. The MAC interface just checks in regular 64 byte intervals whether the frame needs to be pre-empted or not.
The combination of time synchronization, the IEEE 802.1Qbv scheduler and frame pre-emption already constitutes an effective set of standards that can be utilized to guarantee the coexistence of different traffic categories on a network while also providing end-to-end latency guarantees. This will be enhanced further as new IEEE 802.1 specifications, such as 802.1Qch are finalized.
Overall, the time-aware scheduler has high implementation complexity and its use of bandwidth is not efficient.
Task and event scheduling in endpoints has to be coupled with the gate scheduling of the traffic shaper in order to lower the latencies.
A critical shortcoming is some delay incurred when an end-point streams unsynchronized data, due to the waiting time for the next time-triggered window.
The time-aware scheduler requires tight synchronization of its time-triggered windows, so all bridges on the stream path must be synchronized. However synchronizing TSN bridge frame selection and transmission time is nontrivial even in moderately sized networks and requires a fully managed solution.
Frame preemption is hard to implement and has not seen wide industry support.
Credit-based, time-aware and cyclic (peristaltic) shapers require network-wide coordinated time and utilize network bandwidth inefficiently, as they enforce packet transmission at periodic cycles. The IEEE 802.1Qcr Asynchronous Traffic Shaper (ATS) operates asynchronously based on local clocks in each bridge, improving link utilization for mixed traffic types, such as periodic with arbitrary periods, sporadic (event driven), and rate-constrained.
ATS employs the urgency-based scheduler (UBS) which prioritizes urgent traffic using per-class queuing and per-stream reshaping. Asynchronicity is achieved by interleaved shaping with traffic characterization based on Token Bucket Emulation, a token bucket emulation model, to eliminate the burstiness cascade effects of per-class shaping. The TBE shaper controls the traffic by average transmission rate, but allows a certain level of burst traffic. When there is a sufficient number of tokens in the bucket, transmission starts immediately; otherwise the queue's gate closes for the time needed to accumulate enough tokens.
The UBS is an improvement on Rate-Controlled Service Disciplines (RCSDs) to control selection and transmission of each individual frame at each hop, decoupling stream bandwidth from the delay bound by separation of rate control and packet scheduling, and using static priorities and First Come - First Serve and Earliest Due - Date First queuing.
UBS queuing has two levels of hierarchy: per-flow shaped queues, with fixed priority assigned by the upstream sources according to application-defined packet transmission times, allowing arbitrary transmission period for each stream, and shared queues that merge streams with the same internal priority from several shapers. This separation of queuing has low implementation complexity while ensuring that frames with higher priority will bypass the lower priority frames.
The shared queues are highly isolated, with policies for separate queues for frames from different transmitters, the same transmitter but different priority, and the same transmitter and priority but a different priority at the receiver. Queue isolation prevents propagation of malicious data, assuring that ordinary streams will get no interference, and enables flexible stream or transmitter blocking by administrative action.
The minimum number of shared queues is the number of ports minus one, and more with additional isolation policies. Shared queues have scheduler internal fixed priority, and frames are transmitted on the First Come First Serve principle.
Worst case clock sync inaccuracy does not decrease link utilization, contrary to time-triggered approaches such as TAS (Qbv) and CQF (Qch).
IEEE 802.1Qca Path Control and Reservation (PCR) specifies extensions to the Intermediate Station to Intermediate Station (IS-IS) protocol to configure multiple paths in bridged networks.
The IEEE 802.1Qca standard uses Shortest Path Bridging (SPB) with a software-defined networking (SDN) hybrid mode - the IS-IS protocol handles basic functions, while the SDN controller manages explicit paths using Path Computation Elements (PCEs) at dedicated server nodes.
IEEE 802.1Qca integrates control protocols to manage multiple topologies, configure an explicit forwarding path (a predefined path for each stream), reserve bandwidth, provides data protection and redundancy, and distribute flow synchronization and flow control messages. These are derived from Equal Cost Tree (ECT), Multiple Spanning Tree Instance (MSTI) and Internal Spanning Tree (IST), and Explicit Tree (ET) protocols.
IEEE 802.1CB Frame Replication and Elimination for Reliability (FRER) sends duplicate copies of each frame over multiple disjoint paths, to provide proactive seamless redundancy for control applications that cannot tolerate packet losses.
The packet replication can use traffic class and path information to minimize network congestion. Each replicated frame has a sequence identification number, used to re-order and merge frames and to discard duplicates.
FRER requires centralized configuration management and needs to be used with 802.1Qcc and 802.1Qca. Industrial fault-tolerance HSR and PRP specified in IEC 62439-3 are supported.
MRP state data for a stream takes 1500 bytes. With additional traffic streams and larger networks, the size of the database proportionally increases and MRP updates between bridge neighbors significantly slow down. The Link-Local Registration Protocol (LRP) is optimized for a larger database size of about 1 Mbyte with efficient replication that allows incremental updates. Unresponsive nodes with stale data are automatically discarded. While MRP is application specific, with each registered application defining its own set of operations, LRP is application neutral.
SRP and MSRP are primarily designed for AV applications - their distributed configuration model is limited to Stream Reservation (SR) Classes A and B defined by the Credit-Based Shaper (CBS), whereas IEEE 802.1Qcc includes a more centralized CNC configuration model supporting all new TSN features such as additional shapers, frame preemption, and path redundancy.
IEEE P802.1Qdd project updates the distributed configuration model by defining new peer-to-peer Resource Allocation Protocol signaling built upon P802.1CS Link-local Registration Protocol. RAP will improve scalability and provide dynamic reservation for a larger number of streams with support for redundant transmission over multiple paths in 802.1CB FRER, and autoconfiguration of sequence recovery.
RAP supports the 'topology-independent per-hop latency calculation' capability of TSN shapers such as 802.1Qch Cyclic Queuing and Forwarding (CQF) and P802.1Qcr Asynchronous Traffic Shaping (ATS). It will also improve performance under high load and support proxying and enhanced diagnostics, all while maintaining backward compatibility and interoperability with MSRP.
IEEE P802.1ABdh Station and Media Access Control Connectivity Discovery - Support for Multiframe Protocol Data Units (LLDPv2) [ 10 ] [ 11 ] updates LLDP to support IETF Link State Vector Routing protocol [ 12 ] and improve efficiency of protocol messages.
The IEEE 802.1Qcp standard implements the YANG data model to provide a Universal Plug-and-Play (uPnP) framework for status reporting and configuration of equipment such as Media Access Control (MAC) Bridges, Two-Port MAC Relays (TPMRs), Customer Virtual Local Area Network (VLAN) Bridges, and Provider Bridges, and to support the 802.1X Security and 802.1AX Datacenter Bridging standards.
YANG is a Unified Modeling Language (UML) for configuration and state data, notifications, and remote procedure calls, to set up device configuration with network management protocols such as NETCONF/RESTCONF.
The IETF Deterministic Networking (DetNet) Working Group is focusing to define deterministic data paths with high reliability and bounds on latency, loss, and packet delay variation (jitter), such as audio and video streaming, industrial automation, and vehicle control.
The goals of Deterministic Networking are to migrate time-critical, high-reliability industrial and audio-video applications from special-purpose Fieldbus networks to IP packet networks. To achieve these goals, DetNet uses resource allocation to manage buffer sizes and transmission rates in order to satisfy end-to-end latency requirements. Service protection against failures with redundancy over multiple paths and explicit routes to reduce packet loss and reordering. The same physical network shall handle both time-critical reserved traffic and regular best-effort traffic, and unused reserved bandwidth shall be released for best-effort traffic.
DetNet operates at the IP Layer 3 routed segments using a software-defined networking layer to provide IntServ and DiffServ integration, and delivers services over lower Layer 2 bridged segments using technologies such as MPLS and IEEE 802.1 AVB/TSN. [ 13 ]
Traffic Engineering (TE) routing protocols translate DetNet flow specification to AVB/TSN controls for queuing, shaping, and scheduling algorithms, such as IEEE 802.1Qav credit-based shaper, IEEE802.1Qbv time-triggered shaper with a rotating time scheduler, IEEE802.1Qch synchronized double buffering, 802.1Qbu/802.3br Ethernet packet pre-emption, and 802.1CB frame replication and elimination for reliability. Also protocol interworking defined by IEEE 802.1CB is used to advertise TSN sub-network capabilities to DetNet flows via the Active Destination MAC and VLAN Stream identification functions. DetNet flows are matched by destination MAC address, VLAN ID and priority parameters to Stream ID and QoS requirements for talkers and listeners in the AVB/TSN sub-network. [ 14 ]
Related projects: | https://en.wikipedia.org/wiki/Time-Sensitive_Networking |
Time-dependent density-functional theory ( TDDFT ) is a quantum mechanical theory used in physics and chemistry to investigate the properties and dynamics of many-body systems in the presence of time-dependent potentials, such as electric or magnetic fields . The effect of such fields on molecules and solids can be studied with TDDFT to extract features like excitation energies , frequency-dependent response properties, and photoabsorption spectra .
TDDFT is an extension of density-functional theory (DFT), and the conceptual and computational foundations are analogous – to show that the (time-dependent) wave function is equivalent to the (time-dependent) electronic density , and then to derive the effective potential of a fictitious non-interacting system which returns the same density as any given interacting system. The issue of constructing such a system is more complex for TDDFT, most notably because the time-dependent effective potential at any given instant depends on the value of the density at all previous times. Consequently, the development of time-dependent approximations for the implementation of TDDFT is behind that of DFT, with applications routinely ignoring this memory requirement.
The formal foundation of TDDFT is the Runge–Gross (RG) theorem (1984) [ 1 ] – the time-dependent analogue of the Hohenberg–Kohn (HK) theorem (1964). [ 2 ] The RG theorem shows that, for a given initial wavefunction, there is a unique mapping between the time-dependent external potential of a system and its time-dependent density. This implies that the many-body wavefunction, depending upon 3 N variables, is equivalent to the density, which depends upon only 3, and that all properties of a system can thus be determined from knowledge of the density alone. Unlike in DFT, there is no general minimization principle in time-dependent quantum mechanics. Consequently, the proof of the RG theorem is more involved than the HK theorem.
Given the RG theorem, the next step in developing a computationally useful method is to determine the fictitious non-interacting system which has the same density as the physical (interacting) system of interest. As in DFT, this is called the (time-dependent) Kohn–Sham system. This system is formally found as the stationary point of an action functional defined in the Keldysh formalism . [ 3 ]
The most popular application of TDDFT is in the calculation of the energies of excited states of isolated systems and, less commonly, solids. Such calculations are based on the fact that the linear response function – that is, how the electron density changes when the external potential changes – has poles at the exact excitation energies of a system. Such calculations require, in addition to the exchange-correlation potential, the exchange-correlation kernel – the functional derivative of the exchange-correlation potential with respect to the density. [ 4 ] [ 5 ]
The approach of Runge and Gross considers a single-component system in the presence of a time-dependent scalar field for which the Hamiltonian takes the form
where T is the kinetic energy operator, W the electron-electron interaction, and V ext ( t ) the external potential which along with the number of electrons defines the system. Nominally, the external potential contains the electrons' interaction with the nuclei of the system. For non-trivial time-dependence, an additional explicitly time-dependent potential is present which can arise, for example, from a time-dependent electric or magnetic field. The many-body wavefunction evolves according to the time-dependent Schrödinger equation under a single initial condition ,
Employing the Schrödinger equation as its starting point, the Runge–Gross theorem shows that at any time, the density uniquely determines the external potential. This is done in two steps:
For a given interaction potential, the RG theorem shows that the external potential uniquely determines the density. The Kohn–Sham approaches chooses a non-interacting system (that for which the interaction potential is zero) in which to form the density that is equal to the interacting system. The advantage of doing so lies in the ease in which non-interacting systems can be solved – the wave function of a non-interacting system can be represented as a Slater determinant of single-particle orbitals , each of which are determined by a single partial differential equation in three variable – and that the kinetic energy of a non-interacting system can be expressed exactly in terms of those orbitals. The problem is thus to determine a potential, denoted as v s ( r , t ) or v KS ( r , t ), that determines a non-interacting Hamiltonian, H s ,
which in turn determines a determinantal wave function
which is constructed in terms of a set of N orbitals which obey the equation,
and generate a time-dependent density
such that ρ s is equal to the density of the interacting system at all times:
Note that in the expression of density above, the summation is over all N b {\displaystyle N_{\textrm {b}}} Kohn–Sham orbitals and f j ( t ) {\displaystyle f_{j}(t)} is the time-dependent occupation number for orbital j {\displaystyle j} . If the potential v s ( r , t ) can be determined, or at the least well-approximated, then the original Schrödinger equation, a single partial differential equation in 3 N variables, has been replaced by N differential equations in 3 dimensions, each differing only in the initial condition.
The problem of determining approximations to the Kohn–Sham potential is challenging. Analogously to DFT, the time-dependent KS potential is decomposed to extract the external potential of the system and the time-dependent Coulomb interaction, v J . The remaining component is the exchange-correlation potential:
In their seminal paper, Runge and Gross approached the definition of the KS potential through an action-based argument starting from the Dirac action
Treated as a functional of the wave function, A [Ψ], variations of the wave function yield the many-body Schrödinger equation as the stationary point. Given the unique mapping between densities and wave function, Runge and Gross then treated the Dirac action as a density functional,
and derived a formal expression for the exchange-correlation component of the action, which determines the exchange-correlation potential by functional differentiation. Later it was observed that an approach based on the Dirac action yields paradoxical conclusions when considering the causality of the response functions it generates. [ 6 ] The density response function, the functional derivative of the density with respect to the external potential, should be causal: a change in the potential at a given time can not affect the density at earlier times. The response functions from the Dirac action however are symmetric in time so lack the required causal structure. An approach which does not suffer from this issue was later introduced through an action based on the Keldysh formalism of complex-time path integration. An alternative resolution of the causality paradox through a refinement of the action principle in real time has been recently proposed by Vignale . [ 7 ]
Linear-response TDDFT can be used if the external perturbation is small in the sense that it does not completely destroy the ground-state structure of the system. In this case one can analyze the linear response of the system. This is a great advantage as, to first order, the variation of the system will depend only on the ground-state wave-function so that we can simply use all the properties of DFT.
Consider a small time-dependent external perturbation δ V e x t ( t ) {\displaystyle \delta V^{ext}(t)} .
This gives
and looking at the linear response of the density
where δ V e f f [ ρ ] ( t ) = δ V e x t ( t ) + δ V H [ ρ ] ( t ) + δ V x c [ ρ ] ( t ) {\displaystyle \delta V^{eff}[\rho ](t)=\delta V^{ext}(t)+\delta V_{H}[\rho ](t)+\delta V_{xc}[\rho ](t)} Here and in the following it is assumed that primed variables are integrated.
Within the linear-response domain, the variation of the Hartree (H) and the exchange-correlation (xc) potential to linear order may be expanded with respect to the density variation
and
Finally, inserting this relation in the response equation for the KS system and comparing
the resultant equation with the response equation for the physical system yields the Dyson
equation of TDDFT:
From this last equation it is possible to derive the excitation energies of the system, as these are simply the poles of the response function.
Other linear-response approaches include the Casida formalism (an expansion in electron-hole pairs) and the Sternheimer equation (density-functional perturbation theory). | https://en.wikipedia.org/wiki/Time-dependent_density_functional_theory |
Time-domain astronomy is the study of how astronomical objects change with time. Said to have begun with Galileo's Letters on Sunspots , the field has now naturally expanded to encompass variable objects beyond the Solar System . Temporal variation may originate from movement of the source, or changes in the object itself. Common targets include novae , supernovae , pulsating stars , flare stars , blazars and active galactic nuclei . Optical time domain surveys include OGLE , HAT-South , PanSTARRS , SkyMapper , ASAS , WASP , CRTS , GOTO , and the forthcoming LSST at the Vera C. Rubin Observatory .
Time-domain astronomy studies transient astronomical events (" transients "), which include various types of variable stars, including periodic , quasi-periodic , high proper motion stars, and lifecycle events ( supernovae , kilonovae ) or other changes in behavior or type. Non-stellar transients include asteroids , planetary transits and comets .
Transients characterize astronomical objects or phenomena whose duration of presentation may be from milliseconds to days, weeks, or even several years. This is in contrast to the timescale of the millions or billions of years during which the galaxies and their component stars in the universe have evolved. The term is used for violent deep-sky events, such as supernovae , novae , dwarf nova outbursts, gamma-ray bursts , and tidal disruption events , as well as gravitational microlensing . [ 1 ]
Time-domain astronomy also involves long-term studies of variable stars and their changes on the timescale of minutes to decades. Variability studied can be intrinsic , including periodic or semi-regular pulsating stars , young stellar objects , stars with outbursts , asteroseismology studies; or extrinsic , which results from eclipses (in binary stars , planetary transits ), stellar rotation (in pulsars , spotted stars), or gravitational microlensing events .
Modern time-domain astronomy surveys often uses robotic telescopes , automatic classification of transient events, and rapid notification of interested people. Blink comparators have long been used to detect differences between two photographic plates, and image subtraction became more used when digital photography eased the normalization of pairs of images. [ 2 ] Due to large fields of view required, the time-domain work involves storing and transferring a huge amount of data. This includes data mining techniques, classification, and the handling of heterogeneous data. [ 3 ]
The importance of time-domain astronomy was recognized in 2018 by German Astronomical Society by awarding a Karl Schwarzschild Medal to Andrzej Udalski for "pioneering contribution to the growth of a new field of astrophysics research, time-domain astronomy , which studies the variability of brightness and other parameters of objects in the universe in different time scales." [ 4 ] Also the 2017 Dan David Prize was awarded to the three leading researchers in the field of time-domain astronomy: Neil Gehrels ( Swift Gamma-Ray Burst Mission ), [ 5 ] Shrinivas Kulkarni ( Palomar Transient Factory ), [ 6 ] Andrzej Udalski ( Optical Gravitational Lensing Experiment ). [ 7 ]
Before the invention of telescopes , transient events that were visible to the naked eye , from within or near the Milky Way Galaxy, were very rare, and sometimes hundreds of years apart. However, such events were recorded in antiquity, such as the supernova in 1054 observed by Chinese, Japanese and Arab astronomers, and the event in 1572 known as " Tycho's Supernova " after Tycho Brahe , who studied it until it faded after two years. [ 8 ] Even though telescopes made it possible to see more distant events, their small fields of view – typically less than 1 square degree – meant that the chances of looking in the right place at the right time were low. Schmidt cameras and other astrographs with wide field were invented in the 20th century, but mostly used to survey the unchanging heavens.
Historically time domain astronomy has come to include appearance of comets and variable brightness of Cepheid-type variable stars . [ 2 ] Old astronomical plates exposed from the 1880s through the early 1990s held by the Harvard College Observatory are being digitized by the DASCH project. [ 9 ]
The interest in transients has intensified when large CCD detectors started to be available to the astronomical community. As telescopes with larger fields of view and larger detectors come into use in the 1990s, first massive and regular survey observations were initiated - pioneered by the gravitational microlensing surveys such as Optical Gravitational Lensing Experiment and the MACHO Project . These efforts, beside the discovery of the microlensing events itself, resulted in the orders of magnitude more variable stars known to mankind. [ 10 ] [ 11 ] Subsequent, dedicated sky surveys such as the Palomar Transient Factory , the spacecraft Gaia and the LSST , focused on expanding the coverage of the sky monitoring to fainter objects, more optical filters and better positional and proper motions measurement capabilities. In 2022, the Gravitational-wave Optical Transient Observer (GOTO) began looking for collisions between neutron stars. [ 12 ]
The ability of modern instruments to observe in wavelengths invisible to the human eye ( radio waves , infrared , ultraviolet , X-ray ) increases the amount of information that may be obtained when a transient is studied.
In radio astronomy the LOFAR is looking for radio transients. Radio time domain studies have long included pulsars and scintillation. Projects to look for transients in X-ray and gamma rays include Cherenkov Telescope Array , eROSITA , AGILE , Fermi , HAWC , INTEGRAL , MAXI , Swift Gamma-Ray Burst Mission and Space Variable Objects Monitor . Gamma ray bursts are a well known high energy electromagnetic transient. [ 13 ] The proposed ULTRASAT satellite will observe a field of more than 200 square degrees continuously in an ultraviolet wavelength that is particularly important for detecting supernovae within minutes of their occurrence. | https://en.wikipedia.org/wiki/Time-domain_astronomy |
Time-domain diffuse optics [ 1 ] or time-resolved functional near-infrared spectroscopy is a branch of functional near-Infrared spectroscopy which deals with light propagation in diffusive media. There are three main approaches to diffuse optics namely continuous wave [ 2 ] (CW), frequency domain [ 3 ] (FD) and time-domain [ 4 ] (TD). Biological tissue in the range of red to near-infrared wavelengths are transparent to light and can be used to probe deep layers of the tissue thus enabling various in vivo applications and clinical trials.
In this approach, a narrow pulse of light (< 100 picoseconds) is injected into the medium. The injected photons undergo multiple scattering and absorption events and the scattered photons are then collected at a certain distance from the source and the photon arrival times are recorded. The photon arrival times are converted into the histogram of the distribution of time-of-flight (DTOF) of photons or temporal point spread function. This DTOF is delayed, attenuated and broadened with respect to the injected pulse. The two main phenomena affecting photon migration in diffusive media are absorption and scattering. Scattering is caused by microscopic refractive index changes due to the structure of the media. Absorption, on the other hand, is caused by a radiative or non-radiative transfer of light energy on interaction with absorption centers such as chromophores. Both absorption and scattering are described by coefficients μ a {\displaystyle \mu _{a}} and μ s {\displaystyle \mu _{s}} respectively.
Multiple scattering events broaden the DTOF and the attenuation of a result of both absorption and scattering as they divert photons from the direction of the detector. Higher scattering leads to a more delayed and a broader DTOF and higher absorption reduces the amplitude and changes the slope of the tail of the DTOF. Since absorption and scattering have different effects on the DTOF, they can be extracted independently while using a single source-detector separation. Moreover, the penetration depth in TD depends solely on the photon arrival times and is independent of the source-detector separation unlike in CW approach .
The theory of light propagation in diffusive media is usually dealt with using the framework of radiative transfer theory under the multiple scattering regime. It has been demonstrated that radiative transfer equation under the diffusion approximation yields sufficiently accurate solutions for practical applications. [ 5 ] For example, it can be applied for the semi-infinite geometry or the infinite slab geometry, using proper boundary conditions. The system is considered as a homogeneous background and an inclusion is considered as an absorption or scattering perturbation.
The time-resolved reflectance curve at a point ρ {\displaystyle \rho } from the source for a semi-infinite geometry is given by
R ( ρ , t ) = k t − 5 / 2 exp ( − μ a ν t ) exp ( − ρ 2 4 D ν t ) S ( D , s 0 , t ) {\textstyle R(\rho ,t)=kt^{-5/2}\exp \left(-\mu _{a}\nu t\right)\exp \left(-{\frac {\rho ^{2}}{4D\nu t}}\right)S\left(D,s_{0},t\right)}
where D = 1 3 μ s ′ {\displaystyle D={\frac {1}{3\mu _{s}^{\prime }}}} is the diffusion coefficient, μ s ′ = μ s ( 1 − g ) {\displaystyle \mu _{s}^{\prime }=\mu _{s}(1-g)} is the reduced scattering coefficient and g {\displaystyle g} is asymmetry factor, ν {\displaystyle \nu } is the photon velocity in the medium, S ( D , s 0 , t ) {\displaystyle S(D,s_{0},t)} takes into account the boundary conditions and k {\displaystyle k} is a constant.
The final DTOF is a convolution of the instrument response function (IRF) of the system with the theoretical reflectance curve.
When applied to biological tissues estimation of μ a {\displaystyle \mu _{a}} and μ s ′ {\displaystyle \mu '_{s}} allows us to then estimate the concentration of the various tissue constituents as well as provides information about blood oxygenation (oxy and deoxy-hemoglobin) as well as saturation and total blood volume. These can then be used as biomarkers for detecting various pathologies.
Instrumentation in time-domain diffuse optics consists of three fundamental components namely, a pulsed laser source, a single photon detector and a timing electronics.
Time-domain diffuse optical sources must have the following characteristics; emission wavelength in the optical window i.e. between 650 and 1350 nanometre (nm); a narrow full width at half maximum (FWHM), ideally a delta function ; high repetition rate (>20 MHz) and finally, sufficient laser power (>1 mW) to achieve good signal to noise ratio .
In the past bulky tunable Ti:sapphire Lasers [ 6 ] were used. They provided a wide wavelength range of 400 nm, a narrow FWHM (< 1 ps) high average power (up to 1W) and high repetition (up to 100 MHz) frequency. However, they are bulky, expensive and take a long time for wavelength swapping.
In recent years, pulsed fiber lasers based on super continuum generation have emerged. [ 7 ] They provide a wide spectral range (400 to 2000 ps), typical average power of 5 to 10 W, a FWHM of < 10ps and a repetition frequency of tens of MHz. However, they are generally quite expensive and lack stability in super continuum generation and hence, have been limited in there use.
The most wide spread sources are the pulsed diode lasers. [ 8 ] They have a FWHM of around 100 ps and repetition frequency of up to 100 MHz and an average power of about a few milliwatts. Even though they lack tunability, their low cost and compactness allows for multiple modules to be used in a single system.
Single photon detector used in time-domain diffuse optics require not only a high photon detection efficiency in the wavelength range of optical window, but also a large active area as well as large numerical aperture (N.A.) to maximize the overall light collection efficiency. They also require narrow timing response and a low noise background.
Traditionally, fiber coupled photomultiplier tubes (PMT) have been the detector of choice for diffuse optical measurements, thanks mainly due to the large active area, low dark count and excellent timing resolution. However, they are intrinsically bulky, prone to electromagnetic disturbances and they have a quite limited spectral sensitivity. Moreover, they require a high biasing voltage and they are quite expensive. Single photon avalanche diodes have emerged as an alternative to PMTS. They are low cost, compact and can be placed in contact, while needing a much lower biasing voltage. Also, they offer a wider spectral sensitivity and they are more robust to bursts of light. However, they have a much lower active area and hence a lower photon collection efficiency and a larger dark count. Silicon photomultipliers (SiPM) are an arrays of SPADs with a global anode and a global cathode and hence have a larger active area while maintaining all the advantages offered by SPADs. However, they suffer from a larger dark count and a broader timing response. [ 9 ]
The timing electronics is needed to losslessly reconstruct the histogram of the distribution of time of flight of photons. This is done by using the technique of time-correlated single photon counting [ 10 ] (TCSPC), where the individual photon arrival times are marked with respect to a start/stop signal provided by the periodic laser cycle. These time-stamps can then be used to build up histograms of photon arrival times.
The two main types of timing electronics are based on a combination of time-to-analog converter (TAC) and an analog-to-digital converter (ADC), and time-to-digital converter [ 11 ] (TDC), respectively. In the first case, the difference between the start and the stop signal is converted into an analog voltage signal, which is then processed by the ADC. In the second method, the delay is directly converted into a digital signal. Systems based on ADCs generally have a better timing resolution and linearity while being expensive and the capability of being integrated. TDCs, on the other hand, can be integrated into a single chip and hence are better suited in multi-channel systems. [ 9 ] However, they have a worse timing performance and can handle much lower sustained count-rates.
The usefulness of TD Diffuse optics lies in its ability to continually and noninvasive monitor optical properties of tissue. Making it a powerful diagnostic tool for long-term bedside monitoring in infants and adults alike. It has already been demonstrated that TD diffuse optics can be successfully applied to various biomedical applications such as cerebral monitoring, [ 12 ] optical mammography , [ 13 ] muscle monitoring, [ 14 ] etc. | https://en.wikipedia.org/wiki/Time-domain_diffuse_optics |
Time-domain harmonic scaling (TDHS) is a method for time-scale modification of speech (or other audio signals), [ 1 ] allowing the apparent rate of speech articulation to be changed without affecting the pitch-contour and the time-evolution of the formant structure. [ 2 ] TDHS differs from other time-scale modification algorithms in that time-scaling operations are performed in the time domain (not the frequency domain ). [ 3 ] TDHS was proposed by D. Malah in 1979. [ 4 ]
This signal processing -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Time-domain_harmonic_scaling |
Time-domain thermoreflectance ( TDTR ) is a method by which the thermal properties of a material can be measured, most importantly thermal conductivity . This method can be applied most notably to thin film materials (up to hundreds of nanometers thick), which have properties that vary greatly when compared to the same materials in bulk. The idea behind this technique is that once a material is heated up, the change in the reflectance of the surface can be utilized to derive the thermal properties. The reflectivity is measured with respect to time, and the data received can be matched to a model with coefficients that correspond to thermal properties.
The technique of this method is based on the monitoring of acoustic waves that are generated with a pulsed laser . Localized heating of a material will create a localized temperature increase, which induces thermal stress . This stress build in a localized region causes an acoustic strain pulse. At an interface, the pulse will be subjected to a transmittance/reflectance state, and the characteristics of the interface may be monitored with the reflected waves. A probe laser will detect the effects of the reflecting acoustic waves by sensing the piezo-optic effect .
The amount of strain is related to the optical laser pulse as follows. Take the localized temperature increase due to the laser,
where R is the sample reflectivity, Q is the optical pulse energy, C is the specific heat (per unit volume), A is the optical spot area, ζ is the optical absorption length, and z is the distance into the sample. [ 1 ] This temperature increase results in a strain that can be estimated by multiplying it with the linear coefficient of thermal expansion of the film. Usually, a typical magnitude value of the acoustic pulse will be small, and for long propagation nonlinear effects could become important. But propagation of such short duration pulses will suffer acoustic attenuation if the temperature is not very low. [ 2 ] Thus, this method is most efficient with the utilization of surface acoustic waves, and studies on investigation of this method toward lateral structures are being conducted.
To sense the piezo-optic effect of the reflected waves, fast monitoring is required due to the travel time of the acoustic wave and heat flow . Acoustic waves travel a few nanometers in a picosecond, where heat flows about a hundred nanometers in a second. [ 1 ] [ 3 ] Thus, lasers such as titanium sapphire (Ti:Al 2 O 3 ) laser , with pulse width of ~200 fs, are used to monitor the characteristics of the interface. Other type of lasers include Yb:fiber, Yb:tungstate, Er:fiber, Nd:glass. Second-harmonic generation may be utilized to achieve frequency of double or higher.
The output of the laser is split into pump and probe beams by a half-wave plate followed by a polarizing beam splitter leading to a cross-polarized pump and probe. The pump beam is modulated on the order of a few megahertz by an acousto-optic or electro-optic modulator and focused onto the sample with a lens. The probe is directed into an optical delay line . The probe beam is then focused with a lens onto the same spot on the sample as the pulse. Both pump and probe have a spot size on the order of 10–50 μm. The reflected probe light is input to a high bandwidth photodetector. The output is fed into a lock-in amplifier whose reference signal has the same frequency used to modulate the pump. The voltage output from the lock-in will be proportional to the change in reflectivity (ΔR). Recording this signal as the optical delay line is changed provides a measurement of ΔR as a function of optical probe-pulse time delay. [ 4 ]
The surface temperature of a single layer
The frequency domain solution for a semi-infinite solid which is heated by a point source with angular frequency ω {\displaystyle \omega } can be expressed by the following equation: [ 2 ]
Here, Λ is the thermal conductivity of the solid, D is the thermal diffusivity of the solid, and r is the radial coordinate. In a typical time-domain thermoreflectance experiment, the co-aligned laser beams have cylindrical symmetry, therefore the Hankel transform can be used to simplify the computation of the convolution of the equation with the distributions of the laser intensities.
Here g ( r ) {\displaystyle g(r)} is radially symmetric and by the definition of Hankel transform,
Since the pump and probe beams used here have Gaussian distribution , the 1 / e 2 {\displaystyle 1/e^{2}} radius of the pump and probe beam are w 0 {\displaystyle w_{0}} and w 1 {\displaystyle w_{1}} respectively.
The surface is heated by the pump laser beam with the intensity p ( r ) {\displaystyle p(r)} , i.e.
where A {\displaystyle A} is the amplitude of the heat absorbed by the sample at frequency ω {\displaystyle \omega } .
Then the Hankel transform of p ( r ) {\displaystyle p(r)} is
Then the distributions of temperature oscillations at the surface θ ( r ) {\displaystyle \theta (r)} is the inverse Hankel transforms of the product G ( k ) {\displaystyle G(k)} and P ( k ) {\displaystyle P(k)} , i.e.
The surface temperatures are measured due to the change in the reflectivity R {\displaystyle R} with the temperature T {\displaystyle T} , i.e. d R / d T {\displaystyle dR/dT} ,
while this change is measured by the changes in the reflected intensity of a probe laser beam.
The probe laser beam measures a weighted average of the temperature θ ( r ) {\displaystyle \theta (r)} , i.e.
This last integral can be simplified to an integral over k {\displaystyle k} :
The surface temperature of a layered structure
In the similar way, frequency domain solution for the surface temperature of a layered structure can be acquired. G ( k ) {\displaystyle G(k)} for a layered structure is
where
Here Λ n is the thermal conductivity of nth layer, D n is the thermal diffusivity of nth layer, and L n is the thickness of nth layer. Then we can calculate the changes of temperature of a layered structure as before using the updated G ( k ) {\displaystyle G(k)} .
Modeling of data acquired in time-domain thermoreflectance
The acquired data from time-domain thermoreflectance experiments are required to be compared with the model.
where Q is the quality factor of the resonant circuit. This calculated V f / V 0 {\displaystyle V_{f}/V_{0}} would be compared with the measured one.
Through this process of time-domain thermoreflectance, the thermal properties of many materials can be obtained. Common test setups include having multiple metal blocks connected together in a diffusion multiple, where once subjected to high temperatures various compounds can be created as a result of the diffusion of two adjacent metal blocks. An example would be a Ni-Cr-Pd-Pt-Rh-Ru diffusion multiple which would have diffusion zones of Ni-Cr, Ni-Pd, Ni-Pt and so on. In this way, many different materials can be tested at the same time. [ 5 ] Lowest thermal conductivity for a thin film of solid, fully dense material (i.e. not porous) was also recently reported with measurements using this method. [ 6 ]
Once this test sample is obtained, time-domain thermoreflectance measurements can take place, with laser pulses of very short duration for both the pump and the probe lasers (<1 ps). The thermoreflected signal is then measured by a photodiode which is connected to a RF lock-in amplifier. The signals that come out of the amplifier consist of an in phase and out of phase component, and the ratio of these allow thermal conductivity data to be measured for a specific delay time.
The data received from this process can then be compared to a thermal model, and the thermal conductivity and thermal conductance can then be derived. It is found that these two parameters can be derived independently based on the delay times, with short delay times (0.1–0.5 ns) resulting in the thermal conductivity and longer delay times (> 2ns) resulting in the thermal conductance.
There is much room for error involved due to phase errors in the RF amplifier in addition to noise from the lasers. Typically, however, accuracy can be found to be within 8%. | https://en.wikipedia.org/wiki/Time-domain_thermoreflectance |
In telecommunication and computer networking , time-driven switching (TDS) is a node by node time variant implementation of circuit switching , where the propagating datagram is shorter in space than the distance between source and destination. With TDS it is no longer necessary to own a complete circuit between source and destination, but only the fraction of circuit where the propagating datagram is temporarily located.
TDS adds flexibility and capacity to circuit-switched networks but requires precise synchronization among nodes and propagating datagrams.
Datagrams are formatted according to schedules that depend on quality of service and availability of switching nodes and physical links. In respect to circuit switching, the added time dimension introduces additional complexity to network management. Like circuit switching, TDS operates without buffers and header processing according to the pipeline forwarding principle; therefore an all optical implementation with optical fibers and optical switches is possible with low cost. The TDS concept itself pervades and is applicable with advantage to existing data switching technologies, including packet switching , where packets, or sets of packets become the datagrams that are routed through the network. [ citation needed ]
TDS has been invented in 2002 by Prof. Mario Baldi and Prof. Yoram Ofek of Synchrodyne Networks that is the assignee of several patents issued by both the United States Patent and Trademark Office and the European Patent Office . [ 1 ] | https://en.wikipedia.org/wiki/Time-driven_switching |
The time-evolving block decimation ( TEBD ) algorithm is a numerical scheme used to simulate one-dimensional quantum many-body systems, characterized by at most nearest-neighbour interactions. It is dubbed Time-evolving Block Decimation because it dynamically identifies the relevant low-dimensional Hilbert subspaces of an exponentially larger original Hilbert space . The algorithm, based on the Matrix Product States formalism, is highly efficient when the amount of entanglement in the system is limited, a requirement fulfilled by a large class of quantum many-body systems in one dimension.
Time-evolving block decimation (TEBD) is a numerical algorithm that can efficiently simulate the time evolution of one dimensional quantum systems having limited entanglement entropy. Naively, to time evolve a system characterized by a Hamiltonian H ^ {\displaystyle {\hat {H}}} one would directly exponentiate the Hamiltonian to get the time evolution operator U ^ ( t ) = e − i H ^ t / ℏ {\displaystyle {\hat {U}}(t)=e^{-i{\hat {H}}t/\hbar }} and apply this to an initial state: ψ ( t ) = U ^ ( t ) ψ ( t = 0 ) {\displaystyle \psi (t)={\hat {U}}(t)\psi (t=0)} However, as the number of degrees of freedom of the system grows it quickly becomes computationally infeasible to perform the associated matrix exponentiation and matrix-vector multiplication. For example, if ψ {\displaystyle \psi } represents a system of n {\displaystyle n} qubits then the Hilbert space in which ψ {\displaystyle \psi } resides has dimension 2 n {\displaystyle 2^{n}} , meaning matrix operations are effectively intractable for all but the smallest values of n {\displaystyle n} . TEBD presents an efficient scheme for performing time evolution by limiting itself to a much smaller subspace of the configuration space. There are several other noteworthy examples of ways to get around this exponential scaling, including quantum Monte Carlo and the density matrix renormalization group .
Guifré Vidal proposed the scheme while at the Institute for Quantum Information, Caltech . [ 1 ] He asserts that "any quantum computation with pure states can be efficiently simulated with a classical computer provided the amount of entanglement involved is sufficiently restricted" .
This happens to be the case for a wide suite of Hamiltonians characterized by local interactions, for example, Hubbard -like Hamiltonians. The method exhibits a low-degree polynomial behavior in the increase of computational time with respect to the amount of entanglement present in the system. The algorithm is based on a scheme that exploits the fact that in these one-dimensional systems the eigenvalues of the reduced density matrix on a bipartite split of the system are exponentially decaying, thus allowing one to work in a re-sized space spanned by the eigenvectors corresponding to the selected eigenvalues .
The numerical method is efficient in simulating real-time dynamics or calculations of ground states using imaginary-time evolution or isentropic interpolations between a target Hamiltonian and a Hamiltonian with an already-known ground state. The computational time scales linearly with the system size, hence many-particles systems in 1D can be investigated.
A useful feature of the TEBD algorithm is that it can be reliably employed for time evolution simulations of time-dependent Hamiltonians, describing systems that can be realized with cold atoms in optical lattices , or in systems far from equilibrium in quantum transport. From this point of view, TEBD had a certain ascendance over DMRG, a very powerful technique, but until recently not very well suited for simulating time-evolutions. With the Matrix Product States formalism being at the mathematical heart of DMRG, the TEBD scheme was adopted by the DMRG community, thus giving birth to the time dependent DMRG [2] [ permanent dead link ] , t-DMRG for short.
Other groups have developed similar approaches in which quantum information plays a predominant role: for example, in DMRG implementations for periodic boundary conditions [3] , and for studying mixed-state dynamics in one-dimensional quantum lattice systems,. [ 2 ] [ 3 ] Those last approaches actually provide a formalism that is more general than the original TEBD approach, as it also allows to deal with evolutions with matrix product operators; this enables the simulation of nontrivial non-infinitesimal evolutions as opposed to the TEBD case, and is a crucial ingredient to deal with higher-dimensional analogues of matrix product states.
Consider a chain of N qubits , described by the function | Ψ ⟩ ∈ H ⊗ N {\displaystyle |\Psi \rangle \in H^{{\otimes }N}} . The most natural way of describing | Ψ ⟩ {\displaystyle |\Psi \rangle } would be using the local M N {\displaystyle M^{N}} -dimensional basis | i 1 , i 2 , . . , i N − 1 , i N ⟩ {\displaystyle |i_{1},i_{2},..,i_{N-1},i_{N}\rangle } : | Ψ ⟩ = ∑ i = 1 M c i 1 i 2 . . i N | i 1 , i 2 , . . , i N − 1 , i N ⟩ {\displaystyle |\Psi \rangle =\sum \limits _{i=1}^{M}c_{i_{1}i_{2}..i_{N}}|{i_{1},i_{2},..,i_{N-1},i_{N}}\rangle } where M is the on-site dimension.
The trick of TEBD is to re-write the coefficients c i 1 i 2 . . i N {\displaystyle c_{i_{1}i_{2}..i_{N}}} : c i 1 i 2 . . i N = ∑ α 1 , . . , α N − 1 = 0 χ Γ α 1 [ 1 ] i 1 λ α 1 [ 1 ] Γ α 1 α 2 [ 2 ] i 2 λ α 2 [ 2 ] Γ α 2 α 3 [ 3 ] i 3 λ α 3 [ 3 ] ⋅ . . ⋅ Γ α N − 2 α N − 1 [ N − 1 ] i N − 1 λ α N − 1 [ N − 1 ] Γ α N − 1 [ N ] i N {\displaystyle c_{i_{1}i_{2}..i_{N}}=\sum \limits _{\alpha _{1},..,\alpha _{N-1}=0}^{\chi }\Gamma _{\alpha _{1}}^{[1]i_{1}}\lambda _{\alpha _{1}}^{[1]}\Gamma _{\alpha _{1}\alpha _{2}}^{[2]i_{2}}\lambda _{\alpha _{2}}^{[2]}\Gamma _{\alpha _{2}\alpha _{3}}^{[3]i_{3}}\lambda _{\alpha _{3}}^{[3]}\cdot ..\cdot \Gamma _{\alpha _{N-2}\alpha _{N-1}}^{[{N-1}]i_{N-1}}\lambda _{\alpha _{N-1}}^{[N-1]}\Gamma _{\alpha _{N-1}}^{[N]i_{N}}}
This form, known as a Matrix product state , simplifies the calculations greatly.
To understand why, one can look at the Schmidt decomposition of a state, which uses singular value decomposition to express a state with limited entanglement more simply.
Consider the state of a bipartite system | Ψ ⟩ ∈ H A ⊗ H B {\displaystyle \vert \Psi \rangle \in {H_{A}\otimes H_{B}}} . Every such state | Ψ ⟩ {\displaystyle |{\Psi }\rangle } can be represented in an appropriately chosen basis as: | Ψ ⟩ = ∑ i = 1 M A | B a i | Φ i A Φ i B ⟩ {\displaystyle \left\vert \Psi \right\rangle =\sum \limits _{i=1}^{M_{A|B}}a_{i}\left\vert {\Phi _{i}^{A}\Phi _{i}^{B}}\right\rangle } where | Φ i A Φ i B ⟩ = | Φ i A ⟩ ⊗ | Φ i B ⟩ {\displaystyle |{\Phi _{i}^{A}\Phi _{i}^{B}}\rangle =|{\Phi _{i}^{A}}\rangle \otimes |{\Phi _{i}^{B}}\rangle } are formed with vectors | Φ i A ⟩ {\displaystyle |{\Phi _{i}^{A}}\rangle } that make an orthonormal basis in H A {\displaystyle H_{A}} and, correspondingly, vectors | Φ i B ⟩ {\displaystyle |{\Phi _{i}^{B}}\rangle } , which form an orthonormal basis in H B {\displaystyle {H_{B}}} , with the coefficients a i {\displaystyle a_{i}} being real and positive, ∑ i = 1 M A | B a i 2 = 1 {\textstyle \sum \limits _{i=1}^{M_{A|B}}a_{i}^{2}=1} . This is called the Schmidt decomposition (SD) of a state. In general the summation goes up to M A | B = min ( dim ( H A ) , dim ( H B ) ) {\displaystyle M_{A|B}=\min(\dim({H_{A}}),\dim({H_{B}}))} . The Schmidt rank of a bipartite split is given by the number of non-zero Schmidt coefficients. If the Schmidt rank is one, the split is characterized by a product state. The vectors of the SD are determined up to a phase and the eigenvalues and the Schmidt rank are unique.
For example, the two-qubit state: | Ψ ⟩ = 1 2 2 ( | 00 ⟩ + 3 | 01 ⟩ + 3 | 10 ⟩ + | 11 ⟩ ) {\displaystyle |{\Psi }\rangle ={\frac {1}{2{\sqrt {2}}}}\left(|{00}\rangle +{\sqrt {3}}|{01}\rangle +{\sqrt {3}}|{10}\rangle +|{11}\rangle \right)} has the following SD: | Ψ ⟩ = 3 + 1 2 2 | ϕ 1 A ϕ 1 B ⟩ + 3 − 1 2 2 | ϕ 2 A ϕ 2 B ⟩ {\displaystyle \left|{\Psi }\right\rangle ={\frac {{\sqrt {3}}+1}{2{\sqrt {2}}}}\left|{\phi _{1}^{A}\phi _{1}^{B}}\right\rangle +{\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}\left|{\phi _{2}^{A}\phi _{2}^{B}}\right\rangle } with | ϕ 1 A ⟩ = 1 2 ( | 0 A ⟩ + | 1 A ⟩ ) , | ϕ 1 B ⟩ = 1 2 ( | 0 B ⟩ + | 1 B ⟩ ) , | ϕ 2 A ⟩ = 1 2 ( | 0 A ⟩ − | 1 A ⟩ ) , | ϕ 2 B ⟩ = 1 2 ( | 1 B ⟩ − | 0 B ⟩ ) {\displaystyle |{\phi _{1}^{A}}\rangle ={\frac {1}{\sqrt {2}}}(|{0_{A}}\rangle +|{1_{A}}\rangle ),\ \ |{\phi _{1}^{B}}\rangle ={\frac {1}{\sqrt {2}}}(|{0_{B}}\rangle +|{1_{B}}\rangle ),\ \ |{\phi _{2}^{A}}\rangle ={\frac {1}{\sqrt {2}}}(|{0_{A}}\rangle -|{1_{A}}\rangle ),\ \ |{\phi _{2}^{B}}\rangle ={\frac {1}{\sqrt {2}}}(|{1_{B}}\rangle -|{0_{B}}\rangle )}
On the other hand, the state: | Φ ⟩ = 1 3 | 00 ⟩ + 1 6 | 01 ⟩ − i 3 | 10 ⟩ − i 6 | 11 ⟩ {\displaystyle |{\Phi }\rangle ={\frac {1}{\sqrt {3}}}|{00}\rangle +{\frac {1}{\sqrt {6}}}|{01}\rangle -{\frac {i}{\sqrt {3}}}|{10}\rangle -{\frac {i}{\sqrt {6}}}|{11}\rangle } is a product state: | Φ ⟩ = ( 1 3 | 0 A ⟩ − i 3 | 1 A ⟩ ) ⊗ ( | 0 B ⟩ + 1 2 | 1 B ⟩ ) {\displaystyle \left|\Phi \right\rangle =\left({\frac {1}{\sqrt {3}}}\left|0_{A}\right\rangle -{\frac {i}{\sqrt {3}}}\left|1_{A}\right\rangle \right)\otimes \left(\left|0_{B}\right\rangle +{\frac {1}{\sqrt {2}}}\left|1_{B}\right\rangle \right)}
At this point we know enough to try to see how we explicitly build the decomposition (let's call it D ).
Consider the bipartite splitting [ 1 ] : [ 2.. N ] {\displaystyle [1]:[2..N]} . The SD has the coefficients λ α 1 [ 1 ] {\displaystyle \lambda _{{\alpha }_{1}}^{[1]}} and eigenvectors | Φ α 1 [ 1 ] ⟩ | Φ α 1 [ 2.. N ] ⟩ {\displaystyle \left|{\Phi _{\alpha _{1}}^{[1]}}\right\rangle \left|{\Phi _{\alpha _{1}}^{[2..N]}}\right\rangle } .
By expanding the | Φ α 1 [ 1 ] ⟩ {\displaystyle \left|{\Phi _{\alpha _{1}}^{[1]}}\right\rangle } 's in the local basis, one can write:
| Ψ ⟩ = ∑ i 1 , α 1 = 1 M , χ Γ α 1 [ 1 ] i 1 λ α 1 [ 1 ] | i 1 ⟩ | Φ α 1 [ 2.. N ] ⟩ {\displaystyle |{\Psi }\rangle =\sum \limits _{i_{1},{\alpha _{1}=1}}^{M,\chi }\Gamma _{\alpha _{1}}^{[1]i_{1}}\lambda _{\alpha _{1}}^{[1]}|{i_{1}}\rangle |{\Phi _{\alpha _{1}}^{[2..N]}}\rangle }
The process can be decomposed in three steps, iterated for each bond (and, correspondingly, SD) in the chain: Step 1 : express the | Φ α 1 [ 2.. N ] ⟩ {\displaystyle |{\Phi _{\alpha _{1}}^{[2..N]}}\rangle } 's in a local basis for qubit 2: | Φ α 1 [ 2.. N ] ⟩ = ∑ i 2 | i 2 ⟩ | τ α 1 i 2 [ 3.. N ] ⟩ {\displaystyle |{\Phi _{\alpha _{1}}^{[2..N]}}\rangle =\sum _{i_{2}}|{i_{2}}\rangle |{\tau _{\alpha _{1}i_{2}}^{[3..N]}}\rangle }
The vectors | τ α 1 i 2 [ 3.. N ] ⟩ {\displaystyle |{\tau _{\alpha _{1}i_{2}}^{[3..N]}}\rangle } are not necessarily normalized .
Step 2 : write each vector | τ α 1 i 2 [ 3.. N ] ⟩ {\displaystyle |{\tau _{\alpha _{1}i_{2}}^{[3..N]}}\rangle } in terms of the at most (Vidal's emphasis) χ {\displaystyle \chi } Schmidt vectors | Φ α 2 [ 3.. N ] ⟩ {\displaystyle |{\Phi _{\alpha _{2}}^{[3..N]}}\rangle } and, correspondingly, coefficients λ α 2 [ 2 ] {\displaystyle \lambda _{{\alpha }_{2}}^{[2]}} : | τ α 1 i 2 [ 3.. N ] ⟩ = ∑ α 2 Γ α 1 α 2 [ 2 ] i 2 λ α 2 [ 2 ] | Φ α 2 [ 3.. N ] ⟩ {\displaystyle |\tau _{\alpha _{1}i_{2}}^{[3..N]}\rangle =\sum _{\alpha _{2}}\Gamma _{\alpha _{1}\alpha _{2}}^{[2]i_{2}}\lambda _{{\alpha }_{2}}^{[2]}|{\Phi _{\alpha _{2}}^{[3..N]}}\rangle }
Step 3 : make the substitutions and obtain: | Ψ ⟩ = ∑ i 1 , i 2 , α 1 , α 2 Γ α 1 [ 1 ] i 1 λ α 1 [ 1 ] Γ α 1 α 2 [ 2 ] i 2 λ α 2 [ 2 ] | i 1 i 2 ⟩ | Φ α 2 [ 3.. N ] ⟩ {\displaystyle |{\Psi }\rangle =\sum _{i_{1},i_{2},\alpha _{1},\alpha _{2}}\Gamma _{\alpha _{1}}^{[1]i_{1}}\lambda _{\alpha _{1}}^{[1]}\Gamma _{\alpha _{1}\alpha _{2}}^{[2]i_{2}}\lambda _{{\alpha }_{2}}^{[2]}|{i_{1}i_{2}}\rangle |{\Phi _{\alpha _{2}}^{[3..N]}}\rangle }
Repeating the steps 1 to 3, one can construct the whole decomposition of state D . The last Γ {\displaystyle \Gamma } 's are a special case, like the first ones, expressing the right-hand Schmidt vectors at the ( N − 1 ) t h {\displaystyle (N-1)^{th}} bond in terms of the local basis at the N t h {\displaystyle N^{th}} lattice place. As shown in, [ 1 ] it is straightforward to obtain the Schmidt decomposition at k t h {\displaystyle k^{th}} bond, i.e. [ 1.. k ] : [ k + 1.. N ] {\displaystyle [1..k]:[k+1..N]} , from D .
The Schmidt eigenvalues, are given explicitly in D :
| Ψ ⟩ = ∑ α k λ α k [ k ] | Φ α k [ 1.. k ] ⟩ | Φ α k [ k + 1.. N ] ⟩ {\displaystyle |{\Psi }\rangle =\sum _{\alpha _{k}}\lambda _{{\alpha }_{k}}^{[k]}|{\Phi _{\alpha _{k}}^{[1..k]}}\rangle |{\Phi _{\alpha _{k}}^{[k+1..N]}}\rangle }
The Schmidt eigenvectors are simply:
| Φ α k [ 1.. k ] ⟩ = ∑ α 1 , α 2 . . α k − 1 Γ α 1 [ 1 ] i 1 λ α 1 [ 1 ] ⋅ ⋅ Γ α k − 1 α k [ k ] i k | i 1 i 2 . . i k ⟩ {\displaystyle |{\Phi _{\alpha _{k}}^{[1..k]}}\rangle =\sum _{\alpha _{1},\alpha _{2}..\alpha _{k-1}}\Gamma _{\alpha _{1}}^{[1]i_{1}}\lambda _{\alpha _{1}}^{[1]}\cdot \cdot \Gamma _{\alpha _{k-1}\alpha _{k}}^{[k]i_{k}}|{i_{1}i_{2}..i_{k}}\rangle } and
| Φ α k [ k + 1.. N ] ⟩ = ∑ α k + 1 , α k + 2 . . α N Γ α k α k + 1 [ k + 1 ] i k + 1 λ α k + 1 [ k + 1 ] ⋅ ⋅ λ α N − 1 N − 1 Γ α N − 1 [ N ] i N | i k + 1 i k + 2 . . i N ⟩ {\displaystyle |{\Phi _{\alpha _{k}}^{[k+1..N]}}\rangle =\sum _{\alpha _{k+1},\alpha _{k+2}..\alpha _{N}}\Gamma _{\alpha _{k}\alpha _{k+1}}^{[k+1]i_{k+1}}\lambda _{\alpha _{k+1}}^{[k+1]}\cdot \cdot \lambda _{\alpha _{N-1}}^{N-1}\Gamma _{\alpha _{N-1}}^{[N]i_{N}}|{i_{k+1}i_{k+2}..i_{N}}\rangle }
Now, looking at D , instead of M N {\displaystyle M^{N}} initial terms, there are χ 2 ⋅ M ( N − 2 ) + 2 χ M + ( N − 1 ) χ {\displaystyle {\chi }^{2}{\cdot }M(N-2)+2{\chi }M+(N-1)\chi } . Apparently this is just a fancy way of rewriting the coefficients c i 1 i 2 . . i N {\displaystyle c_{i_{1}i_{2}..i_{N}}} , but in fact there is more to it than that. Assuming that N is even, the Schmidt rank χ {\displaystyle \chi } for a bipartite cut in the middle of the chain can have a maximal value of M N / 2 {\displaystyle M^{N/2}} ; in this case we end up with at least M N + 1 ⋅ ( N − 2 ) {\displaystyle M^{N+1}{\cdot }(N-2)} coefficients, considering only the χ 2 {\displaystyle {\chi }^{2}} ones, slightly more than the initial M N {\displaystyle M^{N}} ! The truth is that the decomposition D is useful when dealing with systems that exhibit a low degree of entanglement, which fortunately is the case with many 1D systems, where the Schmidt coefficients of the ground state decay in an exponential manner with α {\displaystyle \alpha } :
λ α l [ l ] ∼ e − K α l , K > 0. {\displaystyle \lambda _{{\alpha }_{l}}^{[l]}{\sim }e^{-K\alpha _{l}},\ K>0.}
Therefore, it is possible to take into account only some of the Schmidt coefficients (namely the largest ones), dropping the others and consequently normalizing again the state:
| Ψ ⟩ = 1 ∑ α l = 1 χ c | λ α l [ l ] | 2 ⋅ ∑ α l = 1 χ c λ α l [ l ] | Φ α l [ 1.. l ] ⟩ | Φ α l [ l + 1.. N ] ⟩ , {\displaystyle |{\Psi }\rangle ={\frac {1}{\sqrt {\sum \limits _{{\alpha _{l}}=1}^{{\chi }_{c}}{|\lambda _{{\alpha }_{l}}^{[l]}|}^{2}}}}\cdot \sum \limits _{{{\alpha }_{l}}=1}^{{\chi }_{c}}\lambda _{{\alpha }_{l}}^{[l]}|{\Phi _{\alpha _{l}}^{[1..l]}}\rangle |{\Phi _{\alpha _{l}}^{[l+1..N]}}\rangle ,}
where χ c {\displaystyle \chi _{c}} is the number of kept Schmidt coefficients.
Let's get away from this abstract picture and refresh ourselves with a concrete example, to emphasize the advantage of making this decomposition. Consider for instance the case of 50 fermions in a ferromagnetic chain, for the sake of simplicity. A dimension of 12, let's say, for the χ c {\displaystyle \chi _{c}} would be a reasonable choice, keeping the discarded eigenvalues at 0.0001 {\displaystyle 0.0001} % of the total, as shown by numerical studies, [ 4 ] meaning roughly 2 14 {\displaystyle 2^{14}} coefficients, as compared to the originally 2 50 {\displaystyle 2^{50}} ones.
Even if the Schmidt eigenvalues don't have this exponential decay, but they show an algebraic decrease, we can still use D to describe our state ψ {\displaystyle \psi } . The number of coefficients to account for a faithful description of ψ {\displaystyle \psi } may be sensibly larger, but still within reach of eventual numerical simulations.
One can proceed now to investigate the behaviour of the decomposition D when acted upon with one-qubit gates (OQG) and two-qubit gates (TQG) acting on neighbouring qubits. Instead of updating all the M N {\displaystyle M^{N}} coefficients c i 1 i 2 . . i N {\displaystyle c_{i_{1}i_{2}..i_{N}}} , we will restrict ourselves to a number of operations that increase in χ {\displaystyle \chi } as a polynomial of low degree, thus saving computational time .
The OQGs are affecting only the qubit they are acting upon, the update of the state | ψ ⟩ {\displaystyle |{\psi }\rangle } after a unitary operator at qubit k does not modify the Schmidt eigenvalues or vectors on the left, consequently the Γ [ k − 1 ] {\displaystyle \Gamma ^{[k-1]}} 's, or on the right, hence the Γ [ k + 1 ] {\displaystyle \Gamma ^{[k+1]}} 's. The only Γ {\displaystyle \Gamma } 's that will be updated are the Γ [ k ] {\displaystyle \Gamma ^{[k]}} 's (requiring only at most O ( M 2 ⋅ χ 2 ) {\displaystyle {O}(M^{2}\cdot \chi ^{2})} operations), as
Γ α k − 1 α k ′ [ k ] i k = ∑ j U j k i k Γ α k − 1 α k [ k ] j k . {\displaystyle \Gamma _{\alpha _{k-1}\alpha _{k}}^{'[k]i_{k}}=\sum _{j}U_{j_{k}}^{i_{k}}\Gamma _{\alpha _{k-1}\alpha _{k}}^{[k]j_{k}}.}
The changes required to update the Γ {\displaystyle \Gamma } 's and the λ {\displaystyle \lambda } 's, following a unitary operation V on qubits k , k +1, concern only Γ [ k ] {\displaystyle \Gamma ^{[k]}} , and Γ [ k + 1 ] {\displaystyle \Gamma ^{[k+1]}} .
They consist of a number of O ( M ⋅ χ 3 ) {\displaystyle {O}({M\cdot \chi }^{3})} basic operations.
Following Vidal's original approach, | ψ ⟩ {\displaystyle |{\psi }\rangle } can be regarded as belonging to only four subsystems:
H = J ⊗ H C ⊗ H D ⊗ K . {\displaystyle {{H}=J{\otimes }H_{C}{\otimes }H_{D}{\otimes }K}.\,}
The subspace J is spanned by the eigenvectors of the reduced density matrix ρ J = T r C D K | ψ ⟩ ⟨ ψ | {\displaystyle \rho ^{J}=Tr_{CDK}|\psi \rangle \langle \psi |} :
ρ [ 1.. k − 1 ] = ∑ α ( λ α [ k − 1 ] ) 2 | Φ α [ 1.. k − 1 ] ⟩ ⟨ Φ α [ 1.. k − 1 ] | = ∑ α ( λ α [ k − 1 ] ) 2 | α ⟩ ⟨ α | . {\displaystyle \rho ^{[1..{k-1}]}=\sum _{\alpha }{(\lambda _{\alpha }^{[k-1]})}^{2}|{\Phi _{\alpha }^{[1..{k-1}]}}\rangle \langle {\Phi _{\alpha }^{[1..{k-1}]}}|=\sum _{\alpha }{(\lambda _{\alpha }^{[k-1]})^{2}}|{\alpha }\rangle \langle {\alpha }|.}
In a similar way, the subspace K is spanned by the eigenvectors of the reduced density matrix:
ρ [ k + 2 . . N ] = ∑ γ ( λ γ [ k + 1 ] ) 2 | Φ γ [ k + 2 . . N ] ⟩ ⟨ Φ γ [ k + 2 . . N ] | = ∑ γ ( λ γ [ k + 1 ] ) 2 | γ ⟩ ⟨ γ | . {\displaystyle \rho ^{[{k+2}..{N}]}=\sum _{\gamma }{(\lambda _{\gamma }^{[k+1]})^{2}}|{\Phi _{\gamma }^{[{k+2}..N]}}\rangle \langle {\Phi _{\gamma }^{[{k+2}..N]}}|=\sum _{\gamma }{(\lambda _{\gamma }^{[k+1]})^{2}}|{\gamma }\rangle \langle {\gamma }|.}
The subspaces H C {\displaystyle H_{C}} and H D {\displaystyle H_{D}} belong to the qubits k and k + 1.
Using this basis and the decomposition D , | ψ ⟩ {\displaystyle |{\psi }\rangle } can be written as:
| ψ ⟩ = ∑ α , β , γ = 1 χ ∑ i , j = 1 M λ α [ C − 1 ] Γ α β [ C ] i λ β [ C ] Γ β γ [ D ] j λ γ [ D ] | α i j γ ⟩ {\displaystyle |{\psi }\rangle =\sum \limits _{\alpha ,\beta ,\gamma =1}^{\chi }\sum \limits _{i,j=1}^{M}\lambda _{\alpha }^{[C-1]}\Gamma _{\alpha \beta }^{[C]i}\lambda _{\beta }^{[C]}\Gamma _{\beta \gamma }^{[D]j}\lambda _{\gamma }^{[D]}|{{\alpha }ij{\gamma }}\rangle }
Using the same reasoning as for the OQG, the applying the TQG V to qubits k , k + 1 one needs only to update Γ [ C ] {\displaystyle \Gamma ^{[C]}} , λ {\displaystyle \lambda } and Γ [ D ] . {\displaystyle \Gamma ^{[D]}.}
We can write | ψ ′ ⟩ = V | ψ ⟩ {\displaystyle |{\psi '}\rangle =V|{\psi }\rangle } as: | ψ ′ ⟩ = ∑ α , γ = 1 χ ∑ i , j = 1 M λ α Θ α γ i j λ γ | α i j γ ⟩ {\displaystyle |{\psi '}\rangle =\sum \limits _{\alpha ,\gamma =1}^{\chi }\sum \limits _{i,j=1}^{M}\lambda _{\alpha }\Theta _{\alpha \gamma }^{ij}\lambda _{\gamma }|{{\alpha }ij\gamma }\rangle } where Θ α γ i j = ∑ β = 1 χ ∑ m , n = 1 M V m n i j Γ α β [ C ] m λ β Γ β γ [ D ] n . {\displaystyle \Theta _{\alpha \gamma }^{ij}=\sum \limits _{\beta =1}^{\chi }\sum \limits _{m,n=1}^{M}V_{mn}^{ij}\Gamma _{\alpha \beta }^{[C]m}\lambda _{\beta }\Gamma _{\beta \gamma }^{[D]n}.}
To find out the new decomposition, the new λ {\displaystyle \lambda } 's at the bond k and their corresponding Schmidt eigenvectors must be computed and expressed in terms of the Γ {\displaystyle {\Gamma }} 's of the decomposition D . The reduced density matrix ρ ′ [ D K ] {\displaystyle \rho ^{'[DK]}} is therefore diagonalized : ρ ′ [ D K ] = T r J C | ψ ′ ⟩ ⟨ ψ ′ | = ∑ j , j ′ , γ , γ ′ ρ γ γ ′ j j ′ | j γ ⟩ ⟨ j ′ γ ′ | . {\displaystyle \rho ^{'[DK]}=Tr_{JC}|{\psi '}\rangle \langle {\psi '}|=\sum _{j,j',\gamma ,\gamma '}\rho _{\gamma \gamma '}^{jj'}|{j\gamma }\rangle \langle {j'\gamma '}|.}
The square roots of its eigenvalues are the new λ {\displaystyle \lambda } 's.
Expressing the eigenvectors of the diagonalized matrix in the basis: { | j γ ⟩ } {\displaystyle \{|{j\gamma }\rangle \}} the Γ [ D ] {\displaystyle \Gamma ^{[{D]}}} 's are obtained as well: | Φ ′ [ D K ] ⟩ = ∑ j , γ Γ β γ ′ [ D ] j λ γ | j γ ⟩ . {\displaystyle |{\Phi ^{'[{DK}]}}\rangle =\sum _{j,\gamma }\Gamma _{\beta \gamma }^{'[{D}]j}\lambda _{\gamma }|{j\gamma }\rangle .}
From the left-hand eigenvectors, λ β ′ | Φ β ′ [ J C ] ⟩ = ⟨ Φ β ′ [ D K ] | ψ ′ ⟩ = ∑ i , j , α , γ ( Γ β γ ′ [ D ] j ) ∗ Θ α γ i j ( λ γ ) 2 λ α | α i ⟩ {\displaystyle \lambda _{\beta }^{'}|{\Phi _{\beta }^{'[{JC}]}}\rangle =\langle {\Phi _{\beta }^{'[{DK}]}}|{\psi '}\rangle =\sum _{i,j,\alpha ,\gamma }(\Gamma _{\beta \gamma }^{'[{D}]j})^{*}\Theta _{\alpha \gamma }^{ij}(\lambda _{\gamma })^{2}\lambda _{\alpha }|{{\alpha }i}\rangle } after expressing them in the basis { | i α ⟩ } {\displaystyle \{|{i\alpha }\rangle \}} , the Γ [ C ] {\displaystyle \Gamma ^{[{C}]}} 's are: | Φ ′ [ J C ] ⟩ = ∑ i , α Γ α β ′ [ C ] i λ α | α i ⟩ . {\displaystyle |{\Phi ^{'[{JC}]}}\rangle =\sum _{i,\alpha }\Gamma _{\alpha \beta }^{'[{C}]i}\lambda _{\alpha }|{{\alpha }i}\rangle .}
The dimension of the largest tensors in D is of the order O ( M ⋅ χ 2 ) {\displaystyle {O}(M{\cdot }{\chi }^{2})} ; when constructing the Θ α γ i j {\displaystyle \Theta _{\alpha \gamma }^{ij}} one makes the summation over β {\displaystyle \beta } , m {\displaystyle {\it {m}}} and n {\displaystyle {\it {n}}} for each γ , α , i , j {\displaystyle \gamma ,\alpha ,{\it {i,j}}} , adding up to a total of O ( M 4 ⋅ χ 3 ) {\displaystyle {O}(M^{4}{\cdot }{\chi }^{3})} operations. The same holds for the formation of the elements ρ γ γ ′ j j ′ {\displaystyle \rho _{\gamma \gamma '}^{jj'}} , or for computing the left-hand eigenvectors λ β ′ | Φ β ′ [ J C ] ⟩ {\displaystyle \lambda _{\beta }^{'}|{\Phi _{\beta }^{'[{\it {JC}}]}}\rangle } , a maximum of O ( M 3 ⋅ χ 3 ) {\displaystyle {\it {O}}(M^{3}{\cdot }{\chi }^{3})} , respectively O ( M 2 ⋅ χ 3 ) {\displaystyle {\it {O}}(M^{2}{\cdot }{\chi }^{3})} basic operations. In the case of qubits, M = 2 {\displaystyle M=2} , hence its role is not very relevant for the order of magnitude of the number of basic operations, but in the case when the on-site dimension is higher than two it has a rather decisive contribution.
The numerical simulation is targeting (possibly time-dependent) Hamiltonians of a system of N {\displaystyle N} particles arranged in a line, which are composed of arbitrary OQGs and TQGs:
H N = ∑ l = 1 N K 1 [ l ] + ∑ l = 1 N K 2 [ l , l + 1 ] . {\displaystyle H_{N}=\sum \limits _{l=1}^{N}K_{1}^{[l]}+\sum \limits _{l=1}^{N}K_{2}^{[l,l+1]}.}
It is useful to decompose H N {\displaystyle H_{N}} as a sum of two possibly non-commuting terms, H N = F + G {\displaystyle H_{N}=F+G} , where
F ≡ ∑ even l ( K 1 l + K 2 l , l + 1 ) = ∑ even l F [ l ] , {\displaystyle F\equiv \sum _{{\text{even }}l}(K_{1}^{l}+K_{2}^{l,l+1})=\sum _{{\text{even }}l}F^{[l]},} G ≡ ∑ odd l ( K 1 l + K 2 l , l + 1 ) = ∑ odd l G [ l ] . {\displaystyle G\equiv \sum _{{\text{odd }}l}(K_{1}^{l}+K_{2}^{l,l+1})=\sum _{{\text{odd }}l}G^{[l]}.}
Any two-body terms commute: [ F [ l ] , F [ l ′ ] ] = 0 {\displaystyle [F^{[l]},F^{[l']}]=0} , [ G [ l ] , G [ l ′ ] ] = 0 {\displaystyle [G^{[l]},G^{[l']}]=0} This is done to make the Suzuki–Trotter expansion (ST) [ 5 ] of the exponential operator, named after Masuo Suzuki and Hale Trotter .
The Suzuki–Trotter expansion of the first order (ST1) represents a general way of writing exponential operators: e A + B = lim n → ∞ ( e A n e B n ) n {\displaystyle e^{A+B}=\lim _{n\to \infty }\left(e^{\frac {A}{n}}e^{\frac {B}{n}}\right)^{n}} or, equivalently e δ ( A + B ) = e δ A e δ B + O ( δ 2 ) . {\displaystyle e^{{\delta }(A+B)}=e^{{\delta }A}e^{{\delta }B}+{\it {O}}(\delta ^{2}).}
The correction term vanishes in the limit δ → 0 {\displaystyle \delta \to 0}
For simulations of quantum dynamics it is useful to use operators that are unitary , conserving the norm (unlike power series expansions), and there's where the Trotter-Suzuki expansion comes in. In problems of quantum dynamics the unitarity of the operators in the ST expansion proves quite practical, since the error tends to concentrate in the overall phase , thus allowing us to faithfully compute expectation values and conserved quantities. Because the ST conserves the phase-space volume, it is also called a symplectic integrator.
The trick of the ST2 is to write the unitary operators e − i H t {\displaystyle e^{-iHt}} as: e − i H N T = [ e − i H N δ ] T / δ = [ e δ 2 F e δ G e δ 2 F ] n {\displaystyle e^{-iH_{N}T}=[e^{-iH_{N}\delta }]^{T/{\delta }}=[e^{{\frac {\delta }{2}}F}e^{{\delta }G}e^{{\frac {\delta }{2}}F}]^{n}} where n = T δ {\displaystyle n={\frac {T}{\delta }}} . The number n {\displaystyle n} is called the Trotter number.
The operators e δ 2 F {\displaystyle e^{{\frac {\delta }{2}}F}} , e δ G {\displaystyle e^{{\delta }G}} are easy to express, as:
e δ 2 F = ∏ even l e δ 2 F [ l ] {\displaystyle e^{{\frac {\delta }{2}}F}=\prod _{{\text{even }}l}e^{{\frac {\delta }{2}}F^{[l]}}} e δ G = ∏ odd l e δ G [ l ] {\displaystyle e^{{\delta }G}=\prod _{{\text{odd }}l}e^{{\delta }G^{[l]}}}
since any two operators F [ l ] {\displaystyle F^{[l]}} , F [ l ′ ] {\displaystyle F^{[l']}} (respectively, G [ l ] {\displaystyle G^{[l]}} , G [ l ′ ] {\displaystyle G^{[l']}} ) commute for l ≠ l ′ {\displaystyle l{\neq }l'} and an ST expansion of the first order keeps only the product of the exponentials, the approximation becoming, in this case, exact.
The time-evolution can be made according to
| ψ ~ t + δ ⟩ = e − i δ 2 F e − i δ G e − i δ 2 F | ψ ~ t ⟩ . {\displaystyle |{{\tilde {\psi }}_{t+\delta }}\rangle =e^{-i{\frac {\delta }{2}}F}e^{{-i\delta }G}e^{{\frac {-i\delta }{2}}F}|{{\tilde {\psi }}_{t}}\rangle .}
For each "time-step" δ {\displaystyle \delta } , e − i δ 2 F [ l ] {\displaystyle e^{-i{\frac {\delta }{2}}F^{[l]}}} are applied successively to all odd sites, then e − i δ G [ l ] {\displaystyle e^{{-i\delta }G^{[l]}}} to the even ones, and e − i δ 2 F [ l ] {\displaystyle e^{-i{\frac {\delta }{2}}F^{[l]}}} again to the odd ones; this is basically a sequence of TQG's, and it has been explained above how to update the decomposition D {\displaystyle {\it {D}}} when applying them.
Our goal is to make the time evolution of a state | ψ 0 ⟩ {\displaystyle |{\psi _{0}}\rangle } for a time T, towards the state | ψ T ⟩ {\displaystyle |{\psi _{T}}\rangle } using the n-particle Hamiltonian H n {\displaystyle H_{n}} .
It is rather troublesome, if at all possible, to construct the decomposition D {\displaystyle {\it {D}}} for an arbitrary n-particle state, since this would mean one has to compute the Schmidt decomposition at each bond, to arrange the Schmidt eigenvalues in decreasing order and to choose the first χ c {\displaystyle \chi _{c}} and the appropriate Schmidt eigenvectors. Mind this would imply diagonalizing somewhat generous reduced density matrices, which, depending on the system one has to simulate, might be a task beyond our reach and patience.
Instead, one can try to do the following:
| ψ g r ⟩ = lim τ → ∞ e − H ~ τ | ψ P ⟩ ‖ e − H ~ τ | ψ P ⟩ ‖ , {\displaystyle |{\psi _{gr}}\rangle =\lim _{\tau \rightarrow \infty }{\frac {e^{-{\tilde {H}}\tau }|{\psi _{P}}\rangle }{\|e^{-{\tilde {H}}\tau }|{\psi _{P}}\rangle \|}},}
The errors in the simulation are resulting from the Suzuki–Trotter approximation and the involved truncation of the Hilbert space.
In the case of a Trotter approximation of p t h {\displaystyle {\it {p^{th}}}} order, the error is of order δ p + 1 {\displaystyle {\delta }^{p+1}} . Taking into account n = T δ {\displaystyle n={\frac {T}{\delta }}} steps, the error after the time T is: ϵ = T δ δ p + 1 = T δ p {\displaystyle \epsilon ={\frac {T}{\delta }}\delta ^{p+1}=T\delta ^{p}}
The unapproximated state | ψ ~ T r ⟩ {\displaystyle |{{\tilde {\psi }}_{Tr}}\rangle } is:
| ψ ~ T r ⟩ = 1 − ϵ 2 | ψ T r ⟩ + ϵ | ψ T r ⊥ ⟩ {\displaystyle |{{\tilde {\psi }}_{Tr}}\rangle ={\sqrt {1-{\epsilon }^{2}}}|{\psi _{Tr}}\rangle +{\epsilon }|{\psi _{Tr}^{\bot }}\rangle }
where | ψ T r ⟩ {\displaystyle |{\psi _{Tr}}\rangle } is the state kept after the Trotter expansion and | ψ T r ⊥ ⟩ {\displaystyle |{\psi _{Tr}^{\bot }}\rangle } accounts for the part that is neglected when doing the expansion.
The total error scales with time T {\displaystyle T} as: ϵ ( T ) = 1 − | ⟨ ψ T r ~ | ψ T r ⟩ | 2 = 1 − 1 + ϵ 2 = ϵ 2 {\displaystyle \epsilon (T)=1-|\langle {\tilde {\psi _{Tr}}}|{\psi _{Tr}}\rangle |^{2}=1-1+\epsilon ^{2}=\epsilon ^{2}}
The Trotter error is independent of the dimension of the chain.
Considering the errors arising from the truncation of the Hilbert space comprised in the decomposition D , they are twofold.
First, as we have seen above, the smallest contributions to the Schmidt spectrum are left away, the state being faithfully represented up to: ϵ ( D ) = 1 − ∏ n = 1 N − 1 ( 1 − ϵ n ) {\displaystyle \epsilon ({\it {D}})=1-\prod \limits _{n=1}^{N-1}(1-\epsilon _{n})} where ϵ n = ∑ α = χ c χ ( λ α [ n ] ) 2 {\displaystyle \epsilon _{n}=\sum \limits _{\alpha =\chi _{c}}^{\chi }(\lambda _{\alpha }^{[n]})^{2}} is the sum of all the discarded eigenvalues of the reduced density matrix, at the bond n {\displaystyle {\it {n}}} .
The state | ψ ⟩ {\displaystyle |{\psi }\rangle } is, at a given bond n {\displaystyle {\it {n}}} , described by the Schmidt decomposition: | ψ ⟩ = 1 − ϵ n | ψ D ⟩ + ϵ n | ψ D ⊥ ⟩ {\displaystyle |{\psi }\rangle ={\sqrt {1-\epsilon _{n}}}|{\psi _{D}}\rangle +{\sqrt {\epsilon _{n}}}|{\psi _{D}^{\bot }}\rangle } where | ψ D ⟩ = 1 1 − ϵ n ∑ α n = 1 χ c λ α n [ n ] | Φ α n [ 1.. n ] ⟩ | Φ α n [ n + 1.. N ] ⟩ {\displaystyle |{\psi _{D}}\rangle ={\frac {1}{\sqrt {1-\epsilon _{n}}}}\sum \limits _{{{\alpha }_{n}}=1}^{{\chi }_{c}}\lambda _{{\alpha }_{n}}^{[n]}|{\Phi _{\alpha _{n}}^{[1..n]}}\rangle |{\Phi _{\alpha _{n}}^{[n+1..N]}}\rangle } is the state kept after the truncation and | ψ D ⊥ ⟩ = 1 ϵ n ∑ α n = χ c χ λ α n [ n ] | Φ α n [ 1.. n ] ⟩ | Φ α n [ n + 1.. N ] ⟩ {\displaystyle |{\psi _{D}^{\bot }}\rangle ={\frac {1}{\sqrt {\epsilon _{n}}}}\sum \limits _{{{\alpha }_{n}}={\chi }_{c}}^{\chi }\lambda _{{\alpha }_{n}}^{[n]}|{\Phi _{\alpha _{n}}^{[1..n]}}\rangle |{\Phi _{\alpha _{n}}^{[n+1..N]}}\rangle } is the state formed by the eigenfunctions corresponding to the smallest, irrelevant Schmidt coefficients, which are neglected.
Now, ⟨ ψ D ⊥ | ψ D ⟩ = 0 {\displaystyle \langle \psi _{D}^{\bot }|\psi _{D}\rangle =0} because they are spanned by vectors corresponding to orthogonal spaces. Using the same argument as for the Trotter expansion, the error after the truncation is: ϵ n = 1 − | ⟨ ψ | ψ D ⟩ | 2 = ∑ α = χ c χ ( λ α [ n ] ) 2 {\displaystyle \epsilon _{n}=1-|\langle {\psi }|\psi _{D}\rangle |^{2}=\sum \limits _{\alpha =\chi _{c}}^{\chi }(\lambda _{\alpha }^{[n]})^{2}}
After moving to the next bond, the state is, similarly: | ψ D ⟩ = 1 − ϵ n + 1 | ψ ′ D ⟩ + ϵ n + 1 | ψ ′ D ⊥ ⟩ {\displaystyle |{\psi _{D}}\rangle ={\sqrt {1-\epsilon _{n+1}}}|{{{\psi }'}_{D}}\rangle +{\sqrt {\epsilon _{n+1}}}|{{\psi '}_{D}^{\bot }}\rangle } The error, after the second truncation, is: ϵ = 1 − | ⟨ ψ | ψ D ′ ⟩ | 2 = 1 − ( 1 − ϵ n + 1 ) | ⟨ ψ | ψ D ⟩ | 2 = 1 − ( 1 − ϵ n + 1 ) ( 1 − ϵ n ) {\displaystyle \epsilon =1-|\langle {\psi }|\psi '_{D}\rangle |^{2}=1-(1-\epsilon _{n+1})|\langle {\psi }|\psi _{D}\rangle |^{2}=1-(1-\epsilon _{n+1})(1-\epsilon _{n})} and so on, as we move from bond to bond.
The second error source enfolded in the decomposition D {\displaystyle D} is more subtle and requires a little bit of calculation.
As we calculated before, the normalization constant after making the truncation at bond l {\displaystyle l} ( [ 1.. l ] : [ l + 1.. N ] ) {\displaystyle ([1..l]:[l+1..N])} is: R = ∑ α l = 1 χ c | λ α l [ l ] | 2 = 1 − ϵ l {\displaystyle R={\sum \limits _{{\alpha _{l}}=1}^{{\chi }_{c}}{|\lambda _{{\alpha }_{l}}^{[l]}|}^{2}}={1-\epsilon _{l}}}
Now let us go to the bond l − 1 {\displaystyle {\it {l}}-1} and calculate the norm of the right-hand Schmidt vectors ‖ Φ α l − 1 [ l − 1.. N ] ‖ {\displaystyle \|{\Phi _{\alpha _{l-1}}^{[l-1..N]}}\|} ; taking into account the full Schmidt dimension, the norm is: n 1 = 1 = ∑ α l = 1 χ c ( c α l − 1 α l ) 2 ( λ α l [ l ] ) 2 + ∑ α l = χ c χ ( c α l − 1 α l ) 2 ( λ α l [ l ] ) 2 = S 1 + S 2 , {\displaystyle n_{1}=1=\sum \limits _{\alpha _{l}=1}^{\chi _{c}}(c_{\alpha _{l-1}\alpha _{l}})^{2}(\lambda _{\alpha _{l}}^{[l]})^{2}+\sum \limits _{\alpha _{l}=\chi _{c}}^{\chi }(c_{\alpha _{l-1}\alpha _{l}})^{2}(\lambda _{\alpha _{l}}^{[l]})^{2}=S_{1}+S_{2},}
where ( c α l − 1 α l ) 2 = ∑ i l = 1 d ( Γ α l − 1 α l [ l ] i l ) ∗ Γ α l − 1 α l [ l ] i l {\displaystyle (c_{\alpha _{l-1}\alpha _{l}})^{2}=\sum \limits _{i_{l}=1}^{d}(\Gamma _{\alpha _{l-1}\alpha _{l}}^{[l]i_{l}})^{*}\Gamma _{\alpha _{l-1}\alpha _{l}}^{[l]i_{l}}} .
Taking into account the truncated space, the norm is: n 2 = ∑ α l = 1 χ c ( c α l − 1 α l ) 2 ⋅ ( λ ′ α l [ l ] ) 2 = ∑ α l = 1 χ c ( c α l − 1 α l ) 2 ( λ α l [ l ] ) 2 R = S 1 R {\displaystyle n_{2}=\sum \limits _{\alpha _{l}=1}^{\chi _{c}}(c_{\alpha _{l-1}\alpha _{l}})^{2}\cdot ({\lambda '}_{\alpha _{l}}^{[l]})^{2}=\sum \limits _{\alpha _{l}=1}^{\chi _{c}}(c_{\alpha _{l-1}\alpha _{l}})^{2}{\frac {(\lambda _{\alpha _{l}}^{[l]})^{2}}{R}}={\frac {S_{1}}{R}}}
Taking the difference, ϵ = n 2 − n 1 = n 2 − 1 {\displaystyle \epsilon =n_{2}-n_{1}=n_{2}-1} , we get: ϵ = S 1 R − 1 ≤ 1 − R R = ϵ l 1 − ϵ l → 0 a s ϵ l → 0 {\displaystyle \epsilon ={\frac {S_{1}}{R}}-1\leq {\frac {1-R}{R}}={\frac {\epsilon _{l}}{1-\epsilon _{l}}}{\to }0\ \ as\ \ {\epsilon _{l}{\to }{0}}}
Hence, when constructing the reduced density matrix, the trace of the matrix is multiplied by the factor: | ⟨ ψ D | ψ D ⟩ | 2 = 1 − ϵ l 1 − ϵ l = 1 − 2 ϵ l 1 − ϵ l {\displaystyle |\langle {\psi _{D}}|\psi _{D}\rangle |^{2}=1-{\frac {\epsilon _{l}}{1-\epsilon _{l}}}={\frac {1-2\epsilon _{l}}{1-\epsilon _{l}}}}
The total truncation error, considering both sources, is upper bounded by: ϵ ( D ) = 1 − ∏ n = 1 N − 1 ( 1 − ϵ n ) ∏ n = 1 N − 1 1 − 2 ϵ n 1 − ϵ n = 1 − ∏ n = 1 N − 1 ( 1 − 2 ϵ n ) {\displaystyle \epsilon ({D})=1-\prod \limits _{n=1}^{N-1}(1-\epsilon _{n})\prod \limits _{n=1}^{N-1}{\frac {1-2\epsilon _{n}}{1-\epsilon _{n}}}=1-\prod \limits _{n=1}^{N-1}(1-2\epsilon _{n})}
When using the Trotter expansion, we do not move from bond to bond, but between bonds of same parity; moreover, for the ST2, we make a sweep of the even ones and two for the odd. But nevertheless, the calculation presented above still holds. The error is evaluated by successively multiplying with the normalization constant, each time we build the reduced density matrix and select its relevant eigenvalues.
One thing that can save a lot of computational time without loss of accuracy is to use a different Schmidt dimension for each bond instead of a fixed one for all bonds, keeping only the necessary amount of relevant coefficients, as usual. For example, taking the first bond, in the case of qubits, the Schmidt dimension is just two. Hence, at the first bond, instead of futilely diagonalizing, let us say, 10 by 10 or 20 by 20 matrices, we can just restrict ourselves to ordinary 2 by 2 ones, thus making the algorithm generally faster. What we can do instead is set a threshold for the eigenvalues of the SD, keeping only those that are above the threshold.
TEBD also offers the possibility of straightforward parallelization due to the factorization of the exponential time-evolution operator using the Suzuki–Trotter expansion. A parallel-TEBD has the same mathematics as its non-parallelized counterpart, the only difference is in the numerical implementation. | https://en.wikipedia.org/wiki/Time-evolving_block_decimation |
Time-hopping ( TH ) is a communications signal technique which can be used to achieve anti-jamming (AJ) or low probability of intercept (LPI). It can also refer to pulse-position modulation , which in its simplest form employs 2 k discrete pulses (referring to the unique positions of the pulse within the transmission window) to transmit k bit(s) per pulse.
To achieve LPI, the transmission time is changed randomly by varying the period and duty cycle of the pulse (carrier) using a pseudo-random sequence. The transmitted signal will then have intermittent start and stop times. Although often used to form hybrid spread-spectrum (SS) systems, TH is strictly speaking a non-SS technique. Spreading of the spectrum is caused by other factors associated with TH, such as using pulses with low duty cycle having a wide frequency response. An example of hybrid SS is TH-FHSS or hybrid TDMA (time division multiple access).
This technology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Time-hopping |
Time-lapse microscopy is time-lapse photography applied to microscopy . Microscope image sequences are recorded and then viewed at a greater speed to give an accelerated view of the microscopic process.
Before the introduction of the video tape recorder in the 1960s, time-lapse microscopy recordings were made on photographic film . During this period, time-lapse microscopy was referred to as microcinematography . With the increasing use of video recorders, the term time-lapse video microscopy was gradually adopted. Today, the term video is increasingly dropped, reflecting that a digital still camera is used to record the individual image frames, instead of a video recorder.
Time-lapse microscopy can be used to observe any microscopic object over time. However, its main use is within cell biology to observe artificially cultured cells . Depending on the cell culture, different microscopy techniques can be applied to enhance characteristics of the cells as most cells are transparent. [ 1 ]
To enhance observations further, cells have therefore traditionally been stained before observation. Unfortunately, the staining process kills the cells. The development of less destructive staining methods and methods to observe unstained cells has led to that cell biologists increasingly observe living cells. This is known as live-cell imaging . A few tools have been developed to identify and analyze single cells during live-cell imaging. [ 2 ] [ 3 ] [ 4 ]
Time-lapse microscopy is the method that extends live-cell imaging from a single observation in time to the observation of cellular dynamics over long periods of time. [ 5 ] [ 6 ] Time-lapse microscopy is primarily used in research, but is clinically used in IVF clinics as studies has proven it to increase pregnancy rates, lower abortion rates and predict aneuploidy [ 7 ] [ 8 ]
Modern approaches are further extending time-lapse microscopy observations beyond making movies of cellular dynamics.
Traditionally, cells have been observed in a microscope and measured in a cytometer . Increasingly this boundary is blurred as cytometric techniques are being integrated with imaging techniques for monitoring and measuring dynamic activities of cells and subcellular structures. [ 5 ]
The Cheese Mites by Martin Duncan from 1903 is one of the earliest microcinematographic films. [ 9 ] However, the early development of scientific microcinematography took place in Paris. The first reported time-lapse microscope was assembled in the late 1890s at the Marey Institute, founded by the pioneer of chronophotography , Étienne-Jules Marey . [ 10 ] [ 11 ] [ 12 ] It was, however, Jean Comandon who made the first significant scientific contributions around 1910. [ 13 ] [ 14 ]
Comandon was a trained microbiologist specializing in syphilis research.
Inspired by Victor Henri's microcinematic work on Brownian motion , [ 15 ] [ 16 ] [ 17 ] he used the newly invented ultramicroscope to study the movements of the syphilis bacteria . [ 18 ] At the time, the ultramicroscope was the only microscope in which the thin spiral shaped bacteria was visible.
Using an enormous cinema camera bolted to the fragile microscope, he demonstrated visually that the
movement of the disease-causing bacteria is uniquely different from the non-disease-causing form.
Comandon's films proved instrumental in teaching doctors how to distinguish the two forms. [ 19 ] [ 20 ]
Comandon's extensive pioneering work inspired others to adopt microcinematography. Heniz Rosenberger builds a microcinematograph in the mid-1920s. In collerboration with Alexis Carrel , they used the device to further develop Carrel's cell culturing techniques . [ 21 ] Similar work was conducted by Warren Lewis. [ 22 ]
During World War II, Carl Zeiss AG released the first phase-contrast microscope on the market.
With this new microscope, cellular details could for the first time be observed without using lethal stains. [ 1 ] By setting up some of the first time-lapse experiments with chicken fibroblasts and a phase-contrast microscope, Michael Abercrombie described the basis of our current understanding of cell migration in 1953. [ 23 ] [ 24 ]
With the broad introduction of the digital camera at the beginning of this century, time-lapse microscopy has been made dramatically more accessible and is currently experiencing an unrepresented raise in scientific publications. [ 5 ] | https://en.wikipedia.org/wiki/Time-lapse_microscopy |
In software development , time-of-check to time-of-use ( TOCTOU , TOCTTOU or TOC/TOU ) is a class of software bugs caused by a race condition involving the checking of the state of a part of a system (such as a security credential) and the use of the results of that check.
TOCTOU race conditions are common in Unix between operations on the file system , [ 1 ] but can occur in other contexts, including local sockets and improper use of database transactions . In the early 1990s, the mail utility of BSD 4.3 UNIX had an exploitable race condition for temporary files because it used the mktemp() [ 2 ] function. [ 3 ] Early versions of OpenSSH had an exploitable race condition for Unix domain sockets . [ 4 ] They remain a problem in modern systems; as of 2019, a TOCTOU race condition in Docker allows root access to the filesystem of the host platform. [ 5 ] In the 2023 Pwn2Own competition in Vancouver, a team of hackers were able to compromise the gateway in an updated Tesla Model 3 using this bug. [ 6 ]
In Unix , the following C code, when used in a setuid program, has a TOCTOU bug:
Here, access is intended to check whether the real user who executed the setuid program would normally be allowed to write the file (i.e., access checks the real userid rather than effective userid ).
This race condition is vulnerable to an attack:
In this example, an attacker can exploit the race condition between the access and open to trick the setuid victim into overwriting an entry in the system password database. TOCTOU races can be used for privilege escalation to get administrative access to a machine.
Although this sequence of events requires precise timing, it is possible for an attacker to arrange such conditions without too much difficulty.
The implication is that applications cannot assume the state managed by the operating system (in this case the file system namespace) will not change between system calls.
Exploiting a TOCTOU race condition requires precise timing to ensure that the attacker's operations interleave properly with the victim's. In the example above, the attacker must execute the symlink system call precisely between the access and open . For the most general attack, the attacker must be scheduled for execution after each operation by the victim, also known as "single-stepping" the victim.
In the case of BSD 4.3 mail utility and mktemp() , [ 2 ] the attacker can simply keep launching mail utility in one process, and keep guessing the temporary file names and keep making symlinks in another process. The attack can usually succeed in less than one minute.
Techniques for single-stepping a victim program include file system mazes [ 7 ] and algorithmic complexity attacks. [ 8 ] In both cases, the attacker manipulates the OS state to control scheduling of the victim.
File system mazes force the victim to read a directory entry that is not in the OS cache, and the OS puts the victim to sleep while it is reading the directory from disk. Algorithmic complexity attacks force the victim to spend its entire scheduling quantum inside a single system call traversing the kernel's hash table of cached file names. The attacker creates a very large number of files with names that hash to the same value as the file the victim will look up.
Despite conceptual simplicity, TOCTOU race conditions are difficult to avoid and eliminate. One general technique is to use error handling instead of pre-checking, under the philosophy of EAFP – "It is easier to ask for forgiveness than permission" – rather than LBYL – "look before you leap". In this case there is no check, and failure of assumptions to hold are signaled by an error being returned. [ 9 ]
In the context of file system TOCTOU race conditions, the fundamental challenge is ensuring that the file system cannot be changed between two system calls. In 2004, an impossibility result was published, showing that there was no portable, deterministic technique for avoiding TOCTOU race conditions when using the Unix access and open filesystem calls. [ 10 ]
Since this impossibility result, libraries for tracking file descriptors and ensuring correctness have been proposed by researchers. [ 11 ]
An alternative solution proposed in the research community is for Unix systems to adopt transactions in the file system or the OS kernel. Transactions provide a concurrency control abstraction for the OS, and can be used to prevent TOCTOU races. While no production Unix kernel has yet adopted transactions, proof-of-concept research prototypes have been developed for Linux, including the Valor file system [ 12 ] and the TxOS kernel. [ 13 ] Microsoft Windows has added transactions to its NTFS file system, [ 14 ] but Microsoft discourages their use, and has indicated that they may be removed in a future version of Windows. [ 15 ]
File locking is a common technique for preventing race conditions for a single file, but it does not extend to the file system namespace and other metadata, nor does locking work well with networked filesystems, and cannot prevent TOCTOU race conditions.
For setuid binaries, a possible solution is to use the seteuid() system call to change the effective user and then perform the open() call. Differences in setuid() between operating systems can be problematic. [ 16 ] | https://en.wikipedia.org/wiki/Time-of-check_to_time-of-use |
Time-resolved fluorescence energy transfer ( TR-FRET ) is the practical combination of time-resolved fluorometry (TRF) with Förster resonance energy transfer (FRET) that offers a powerful tool for drug discovery researchers. TR-FRET combines the low background aspect of TRF with the homogeneous assay format of FRET. The resulting assay provides an increase in flexibility, reliability and sensitivity in addition to higher throughput and fewer false positive/false negative results. FRET involves two fluorophores , a donor and an acceptor. [ 1 ] Excitation of the donor by an energy source (e.g. flash lamp or laser) produces an energy transfer to the acceptor if the two are within a given proximity to each other. The acceptor in turn emits light at its characteristic wavelength.
The FRET aspect of the technology is driven by several factors, including spectral overlap and the proximity of the fluorophores involved, wherein energy transfer occurs only when the distance between the donor and the acceptor is small enough. In practice, FRET systems are characterized by the Förster's radius (R 0 ): the distance between the fluorophores at which FRET efficiency is 50%. For many FRET fluorophore pairs, R 0 lies between 20 and 90 Å, depending on the acceptor used and the spatial arrangements of the fluorophores within the assay. [ 1 ] Through measurement of this energy transfer, interactions between biomolecules can be assessed by coupling each partner with a fluorescent label and detecting the level of energy transfer. Acceptor emission as a measure of energy transfer can be detected without needing to separate bound from unbound assay components (e.g. a filtration or wash step) resulting in reduced assay time and cost. [ 2 ]
Homogeneous, mix-and-read TR-FRET assays offer advantages over other biomolecular screening assays, such as fluorescence polarization (FP) or TRF assays. [ 3 ] In FP assays, background fluorescence due to library compounds is normally depolarized and background signal due to scattered light (e.g. precipitated compounds) is normally polarized. Depending on the assay configuration, either case can lead to a false positive or false negative result. However, because the donor species used in a TR-FRET assay has a fluorescent lifetime that is many orders of magnitude longer than background fluorescence or scattered light, emission signal resulting from energy transfer can be measured after any interfering signal has completely decayed. TR-FRET assays can also be formatted to use limiting receptor and excess tracer concentrations (unlike FP assays), which can provide further cost savings. [ 4 ] In the case of TRF assays, a wash step is required to remove unbound fluorescent reagents prior to measuring the activity signal of the assay. This increases reagent use, time to complete the assay, and limits the ability to miniaturize the system (e.g. converting from a 384-well microtiter plate to a 1536-well plate ). [ 5 ] TR-FRET assays take advantage of the required proximity of the donor and acceptor species for generation of signal.
Additionally, this method is preferred by some researchers as it does not rely on radioactive materials to generate the signal to be detected. This avoids both the hazards of using the materials and the cost and logistics of storage, use, and disposal. [ 6 ]
Although TR-FRET can be accomplished with a variety of fluorophore combinations, lanthanide metals are particularly useful. Certain life science applications take advantage of the unique fluorescence properties of lanthanide ion complexes (Ln(III) chelates or cryptates). These are well-suited for this application due to their large Stokes shifts and extremely long emission lifetimes (from microseconds to milliseconds) compared to more traditional fluorophores (e.g. fluorescein, allophycoyanin, phycoerythrin, and rhodamine). The biological fluids or serum commonly used in these research applications contain many compounds and proteins which are naturally fluorescent. Therefore, the use of conventional, steady-state fluorescence measurement presents serious limitations in assay sensitivity. Long-lived fluorophores, such as lanthanides, combined with time-resolved detection (a delay between excitation and emission detection) minimizes prompt fluorescence interference. This method (commonly referred to as time-resolved fluorometry or TRF) involves two fluorophores: a donor and an acceptor. Excitation of the donor fluorophore (in this case, the lanthanide ion complex) by an energy source (e.g. flash lamp or laser) produces an energy transfer to the acceptor fluorophore if they are within a given proximity to each other (known as the Förster's radius). The acceptor fluorophore in turn emits light at its characteristic wavelength.
The two most commonly used lanthanides in life science assays are shown below along with their corresponding acceptor dye as well as their excitation and emission wavelengths and resultant Stokes shift (separation of excitation and emission wavelengths).
(donor excitation ⇒ acceptor emission)
As noted in the table above, fluorescent energy transfer from Europium to allophycocyanin can be used in a time resolved manner, particularly in biomolecular screening assays. The figure at right shows the intersection of the emission from Europium with the excitation of allophycocyanin (APC) where energy transfer occurs when Europium and APC are brought into proximity via biomolecular interactions.
When these two fluorophores are brought together by a biomolecular interaction, a portion of the energy captured by the Europium during excitation is released through fluorescence emission at 620 nm, while the remaining energy is transferred to the APC. This energy is then released by APC as specific fluorescence at 665 nm only via FRET with Europium.
Through the design of the high-throughput screening assay, the materials are mixed, and if the enzyme does act on the peptide, all components will bind their respective targets and FRET will occur. [ 8 ]
The instrument used to measure the assay then delays the reading of the emitted light by several hundred milliseconds after the incident/excitation light (the light energy pulse supplied by the instrument to excite the donor molecule) in order to eliminate any 'cross-talk' between the excitation and emission signals. ('cross-talk' in this instance refers to overlapping spectral profiles, which could result in false-positives, false-negatives, or reduced sensitivity depending on the assay design. [ 9 ] ) This process comprises the 'time-resolved' aspect of the assay. | https://en.wikipedia.org/wiki/Time-resolved_fluorescence_energy_transfer |
Time-resolved mass spectrometry ( TRMS ) is a strategy in analytical chemistry that uses mass spectrometry platform to collect data with temporal resolution . [ 1 ] [ 2 ] [ 3 ] Implementation of TRMS builds on the ability of mass spectrometers to process ions within sub-second duty cycles. It often requires the use of customized experimental setups. However, they can normally incorporate commercial mass spectrometers. As a concept in analytical chemistry , TRMS encompasses instrumental developments (e.g. interfaces, ion sources, mass analyzers), methodological developments, and applications.
An early application of TRMS was in the observation of flash photolysis process. [ 4 ] It took advantage of a time-of-flight mass analyzer. [ 5 ] TRMS currently finds applications in the monitoring of organic reactions, [ 6 ] formation of reactive intermediates, [ 7 ] enzyme -catalyzed reactions, [ 8 ] convection , [ 9 ] protein folding , [ 10 ] extraction , [ 11 ] and other chemical and physical processes.
TRMS is typically implemented to monitor processes that occur on second to millisecond time scale. However, there exist reports from studies in which sub-millisecond resolutions were achieved. [ 4 ] [ 5 ] [ 6 ] | https://en.wikipedia.org/wiki/Time-resolved_mass_spectrometry |
In mathematics , time-scale calculus is a unification of the theory of difference equations with that of differential equations , unifying integral and differential calculus with the calculus of finite differences , offering a formalism for studying hybrid systems . It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers then it is equivalent to the forward difference operator.
Time-scale calculus was introduced in 1988 by the German mathematician Stefan Hilger . [ 1 ] However, similar ideas have been used before and go back at least to the introduction of the Riemann–Stieltjes integral , which unifies sums and integrals.
Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. [ 2 ] The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a Cantor set .
The three most popular examples of calculus on time scales are differential calculus , difference calculus , and quantum calculus . Dynamic equations on a time scale have a potential for applications such as in population dynamics . For example, they can model insect populations that evolve continuously while in season, die out in winter while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a non-overlapping population.
A time scale (or measure chain ) is a closed subset of the real line R {\displaystyle \mathbb {R} } . The common notation for a general time scale is T {\displaystyle \mathbb {T} } .
The two most commonly encountered examples of time scales are the real numbers R {\displaystyle \mathbb {R} } and the discrete time scale h Z {\displaystyle h\mathbb {Z} } .
A single point in a time scale is defined as:
The forward jump and backward jump operators represent the closest point in the time scale on the right and left of a given point t {\displaystyle t} , respectively. Formally:
The graininess μ {\displaystyle \mu } is the distance from a point to the closest point on the right and is given by:
For a right-dense t {\displaystyle t} , σ ( t ) = t {\displaystyle \sigma (t)=t} and μ ( t ) = 0 {\displaystyle \mu (t)=0} . For a left-dense t {\displaystyle t} , ρ ( t ) = t . {\displaystyle \rho (t)=t.}
For any t ∈ T {\displaystyle t\in \mathbb {T} } , t {\displaystyle t} is:
As illustrated by the figure at right:
Continuity of a time scale is redefined as equivalent to density. A time scale is said to be right-continuous at point t {\displaystyle t} if it is right dense at point t {\displaystyle t} . Similarly, a time scale is said to be left-continuous at point t {\displaystyle t} if it is left dense at point t {\displaystyle t} .
Take a function:
(where R could be any Banach space , but is set to the real line for simplicity).
Definition: The delta derivative (also Hilger derivative) f Δ ( t ) {\displaystyle f^{\Delta }(t)} exists if and only if:
For every ε > 0 {\displaystyle \varepsilon >0} there exists a neighborhood U {\displaystyle U} of t {\displaystyle t} such that:
for all s {\displaystyle s} in U {\displaystyle U} .
Take T = R . {\displaystyle \mathbb {T} =\mathbb {R} .} Then σ ( t ) = t {\displaystyle \sigma (t)=t} , μ ( t ) = 0 {\displaystyle \mu (t)=0} , f Δ = f ′ {\displaystyle f^{\Delta }=f'} ; is the derivative used in standard calculus . If T = Z {\displaystyle \mathbb {T} =\mathbb {Z} } (the integers ), σ ( t ) = t + 1 {\displaystyle \sigma (t)=t+1} , μ ( t ) = 1 {\displaystyle \mu (t)=1} , f Δ = Δ f {\displaystyle f^{\Delta }=\Delta f} is the forward difference operator used in difference equations.
The delta integral is defined as the antiderivative with respect to the delta derivative. If F ( t ) {\displaystyle F(t)} has a continuous derivative f ( t ) = F Δ ( t ) {\displaystyle f(t)=F^{\Delta }(t)} one sets
A Laplace transform can be defined for functions on time scales, which uses the same table of transforms for any arbitrary time scale. This transform can be used to solve dynamic equations on time scales. If the time scale is the non-negative integers then the transform is equal [ 2 ] to a modified Z-transform : Z ′ { x [ z ] } = Z { x [ z + 1 ] } z + 1 {\displaystyle {\mathcal {Z}}'\{x[z]\}={\frac {{\mathcal {Z}}\{x[z+1]\}}{z+1}}}
Partial differential equations and partial difference equations are unified as partial dynamic equations on time scales. [ 3 ] [ 4 ] [ 5 ]
Multiple integration on time scales is treated in Bohner (2005). [ 6 ]
Stochastic differential equations and stochastic difference equations can be generalized to stochastic dynamic equations on time scales. [ 7 ]
Associated with every time scale is a natural measure [ 8 ] [ 9 ] defined via
where λ {\displaystyle \lambda } denotes Lebesgue measure and ρ {\displaystyle \rho } is the backward shift operator defined on R {\displaystyle \mathbb {R} } . The delta integral turns out to be the usual Lebesgue–Stieltjes integral with respect to this measure
and the delta derivative turns out to be the Radon–Nikodym derivative with respect to this measure [ 10 ]
The Dirac delta and Kronecker delta are unified on time scales as the Hilger delta : [ 11 ] [ 12 ]
Fractional calculus on time scales is treated in Bastos, Mozyrska, and Torres. [ 13 ] | https://en.wikipedia.org/wiki/Time-scale_calculus |
This article covers the evolution of time-sharing systems , providing links to major early time-sharing operating systems, showing their subsequent evolution.
The meaning of the term time-sharing has shifted from its original usage. From 1949 to 1960, time-sharing was used to refer to multiprogramming; it evolved to mean multi-user interactive computing.
Time-sharing was first proposed in the mid- to late-1950s and first implemented in the early 1960s. The concept was born out of the realization that a single expensive computer could be efficiently utilized by enabling multiprogramming , and, later, by allowing multiple users simultaneous interactive access . [ 1 ] In 1984, Christopher Strachey wrote he considered the change in the meaning of the term time-sharing to be a source of confusion and not what he meant when he wrote his original paper in 1959. [ 2 ] [ 3 ]
Without time-sharing, an individual user would enter bursts of information followed by long pauses; but with a group of users working at the same time, the pauses of one user would be filled by the activity of the others. Similarly, small slices of time spent waiting for disk, tape, or network input could be granted to other users. Given an optimal group size, the overall process could be very efficient. [ note 1 ]
Each user would use their own computer terminal , initially electromechanical teleprinters such as the Teletype Model 33 ASR or the Friden Flexowriter ; from about 1970 these were progressively superseded by CRT -based units such as the DEC VT05 , Datapoint 2200 and Lear Siegler ADM-3A .
Terminals were initially linked to a nearby computer via current loop or serial cables , by conventional telegraph circuits provided by PTTs and over specialist digital leased lines such T1 . Modems such as the Bell 103 and successors, allowed remote and higher-speed use over the analogue voice telephone network .
See details and additional systems in the table below. Relationships shown here are for the purpose of grouping entries and do not reflect all influences. The Cambridge Multiple-Access System [ 6 ] [ 7 ] was the first time-sharing system developed outside the United States. | https://en.wikipedia.org/wiki/Time-sharing_system_evolution |
The time-stretch analog-to-digital converter ( TS-ADC ), [ 1 ] [ 2 ] [ 3 ] also known as the time-stretch enhanced recorder ( TiSER ), is an analog-to-digital converter (ADC) system that has the capability of digitizing very high bandwidth signals that cannot be captured by conventional electronic ADCs. [ 4 ] Alternatively, it is also known as the photonic time-stretch (PTS) digitizer, [ 5 ] since it uses an optical frontend . It relies on the process of time-stretch, which effectively slows down the analog signal in time (or compresses its bandwidth) before it can be digitized by a standard electronic ADC.
There is a huge demand for very high-speed analog-to-digital converters (ADCs), as they are needed for test and measurement equipment in laboratories and in high speed data communications systems . [ citation needed ] Most of the ADCs are based purely on electronic circuits, which have limited speeds and add a lot of impairments, limiting the bandwidth of the signals that can be digitized and the achievable signal-to-noise ratio . In the TS-ADC, this limitation is overcome by time-stretching the analog signal, which effectively slows down the signal in time prior to digitization. By doing so, the bandwidth (and carrier frequency ) of the signal is compressed. Electronic ADCs that would have been too slow to digitize the original signal can now be used to capture and process this slowed down signal.
The time-stretch processor, which is generally an optical frontend , stretches the signal in time. It also divides the signal into multiple segments using a filter , for example, a wavelength-division multiplexing (WDM) filter , to ensure that the stretched replica of the original analog signal segments do not overlap each other in time after stretching. The time-stretched and slowed down signal segments are then converted into digital samples by slow electronic ADCs. Finally, these samples are collected by a digital signal processor (DSP) and rearranged in a manner such that output data is the digital representation of the original analog signal. Any distortion added to the signal by the time-stretch preprocessor is also removed by the DSP.
An optical front-end is commonly used to accomplish this process of time-stretching. An ultrashort optical pulse (typically 100 to 200 femtoseconds long), also called a supercontinuum pulse, which has a broad optical bandwidth, is time-stretched by dispersing it in a highly dispersive medium (such as a dispersion compensating fiber). This process results in (an almost) linear time-to- wavelength mapping in the stretched pulse, because different wavelengths travel at different speeds in the dispersive medium. The obtained pulse is called a chirped pulse as its frequency is changing with time, and it is typically a few nanoseconds long. The analog signal is modulated onto this chirped pulse using an electro-optic intensity modulator . Subsequently, the modulated pulse is stretched further in the second dispersive medium which has much higher dispersion value. Finally, this obtained optical pulse is converted to the electrical domain by a photodetector , giving the stretched replica of the original analog signal.
For continuous operation, a train of supercontinuum pulses is used. The chirped pulses arriving at the electro-optic modulator should be wide enough (in time) such that the trailing edge of one pulse overlaps the leading edge of the next pulse. For segmentation, optical filters separate the signal into multiple wavelength channels at the output of the second dispersive medium. For each channel, a separate photodetector and backend electronic ADC is used. Finally the output of these ADCs are passed on to the DSP which generates the desired digital output.
The PTS processor is based on specialized analog optical (or microwave photonic) fiber links [ 5 ] such as those used in cable TV distribution. While the dispersion of fiber is a nuisance in conventional analog optical links , time-stretch technique exploits it to slow down the electrical waveform in the optical domain. In the cable TV link, the light source is a continuous-wave (CW) laser . In PTS, the source is a chirped pulse laser.
In a conventional analog optical link, dispersion causes the upper and lower modulation sidebands , f optical ± f electrical , to slip in relative phase . At certain frequencies, their beats with the optical carrier interfere destructively, creating nulls in the frequency response of the system. For practical systems the first null is at tens of GHz , which is sufficient for handling most electrical signals of interest. Although it may seem that the dispersion penalty places a fundamental limit on the impulse response (or the bandwidth) of the time-stretch system, it can be eliminated. The dispersion penalty vanishes with single-sideband modulation . [ 5 ] Alternatively, one can use the modulator's secondary (inverse) output port to eliminate the dispersion penalty, [ 5 ] in much the same way as two antennas can eliminate spatial nulls in wireless communication (hence the two antennas on top of a WiFi access point ). This configuration is termed phase-diversity. [ 6 ] Combining the complementary outputs using a maximal ratio combining (MRC) algorithm results in a transfer function with a flat response in the frequency domain. Thus, the impulse response (bandwidth) of a time-stretch system is limited only by the bandwidth of the electro-optic modulator, which is about 120 GHz—a value that is adequate for capturing most electrical waveforms of interest.
Extremely large stretch factors can be obtained using long lengths of fiber, but at the cost of larger loss—a problem that has been overcome by employing Raman amplification within the dispersive fiber itself, leading to the world's fastest real-time digitizer. [ 7 ] Also, using PTS, capture of very high-frequency signals with a world record resolution in 10-GHz bandwidth range has been achieved. [ 8 ]
Another technique, temporal imaging using a time lens , can also be used to slow down (mostly optical) signals in time. The time-lens concept relies on the mathematical equivalence between spatial diffraction and temporal dispersion, the so-called space-time duality . [ 9 ] A lens held at a distance from an object produces a magnified image of the object. The lens imparts a quadratic phase shift to the spatial frequency components of the optical waves; in conjunction with the free space propagation (object to lens, lens to eye), this generates a magnified image. Owing to the mathematical equivalence between paraxial diffraction and temporal dispersion, an optical waveform can be temporally imaged by a three-step process of dispersing it in time, subjecting it to a phase shift that is quadratic in time (the time lens itself), and dispersing it again. Theoretically, a focused aberration -free image is obtained under a specific condition when the two dispersive elements and the phase shift satisfy the temporal equivalent of the classic lens equation. Alternatively, the time lens can be used without the second dispersive element to transfer the waveform's temporal profile to the spectral domain, analogous to the property that an ordinary lens produces the spatial Fourier transform of an object at its focal points . [ 10 ]
In contrast to the time-lens approach, PTS is not based on the space-time duality – there is no lens equation that needs to be satisfied to obtain an error-free slowed-down version of the input waveform. Time-stretch technique also offers continuous-time acquisition performance, a feature needed for mainstream applications of oscilloscopes .
Another important difference between the two techniques is that the time lens requires the input signal to be subjected to high amount of dispersion before further processing. For electrical waveforms, the electronic devices that have the required characteristics: (1) high dispersion to loss ratio, (2) uniform dispersion, and (3) broad bandwidths, do not exist. This renders time lens not suitable for slowing down wideband electrical waveforms. In contrast, PTS does not have such a requirement. It was developed specifically for slowing down electrical waveforms and enable high speed digitizers.
The phase stretch transform or PST is a computational approach to signal and image processing. One of its utilities is for feature detection and classification. phase stretch transform is a spin-off from research on the time stretch dispersive Fourier transform . It transforms the image by emulating propagation through a diffractive medium with engineered 3D dispersive property (refractive index).
In addition to wideband A/D conversion, photonic time-stretch (PTS) is also an enabling technology for high-throughput real-time instrumentation such as imaging [ 11 ] and spectroscopy . [ 12 ] [ 13 ] The first artificial intelligence facilitated high-speed phase microscopy is demonstrated to improve the diagnosis accuracy of cancer cells out of blood cells by simultaneous measurement of phase and intensity spatial profiles. [ 14 ] The world's fastest optical imaging method called serial time-encoded amplified microscopy (STEAM) makes use of the PTS technology to acquire image using a single-pixel photodetector and commercial ADC.
Wavelength-time spectroscopy, which also relies on photonic time-stretch technique, permits real-time single-shot measurements of rapidly evolving or fluctuating spectra.
Time stretch quantitative phase imaging ( TS-QPI ) is an imaging technique based on time-stretch technology for simultaneous measurement of phase and intensity spatial profiles. In time stretched imaging, the object's spatial information is encoded in the spectrum of laser pulses within a pulse duration of sub-nanoseconds. Each pulse representing one frame of the camera is then stretched in time so that it can be digitized in real-time by an electronic analog-to-digital converter (ADC). The ultra-fast pulse illumination freezes the motion of high-speed cells or particles in flow to achieve blur-free imaging. [ 15 ] [ 16 ] | https://en.wikipedia.org/wiki/Time-stretch_analog-to-digital_converter |
In electronic instrumentation and signal processing , a time-to-digital converter ( TDC ) or time digitizer ( TD ) is a device for recognizing events and providing a digital representation of the time they occurred. For example, a TDC might output the time of arrival for each incoming pulse. Some applications wish to measure the time interval between two events rather than some notion of an absolute time, and the digitizer is then used to measure a time interval and convert it into digital (binary) output. In some cases, [ 1 ] an interpolating TDC is also called a time counter ( TC ).
When TDCs are used to determine the time interval between two signal pulses (known as start and stop pulse), measurement is started and stopped when the rising or falling edge of a signal pulse crosses a set threshold. This pattern is seen in many physical experiments, like time-of-flight and lifetime measurements in atomic and high energy physics , experiments that involve laser ranging and electronic research involving the testing of integrated circuits and high-speed data transfer. [ 1 ]
Several methods exist for time digitization. Some types allow for nanosecond accuracy, while other are capable of picosecond accuracy [ citation needed ] [ clarification needed ] (see Coarse measurement and Fine measurement sections below, respectively).
TDCs are used to timestamp events and measure time differences between events, especially where picosecond precision and high accuracy is required, such as the measurement of events in high energy physics experiments , where particles (e.g. electrons, photons, and ions) are detected.
Another application is cost-effective and non-mechanical water flow metering by measuring the time difference between ultrasound pulses that travel through the flow and arrive at different times depending on the flow speed and direction. [ 2 ] [ 3 ]
In an all-digital phase-locked loop (ADPLL), a TDC measures the phase shift and its result is used to adjust the digital controlled oscillator (DCO). [ 4 ]
If the required time resolution is not high (nanosecond resolution? [ clarification needed ] ), then counters can be used to make the conversion.
In its simplest implementation, a TDC is simply a high- frequency counter that increments every clock cycle. The current contents of the counter represents the current time. When an event occurs, the counter's value is captured in an output register.
In that approach, the measurement is an integer number of clock cycles, so the measurement is quantized to a clock period. To get finer resolution, a faster clock is needed. The accuracy of the measurement depends upon the stability of the clock frequency.
Typically a TDC uses a crystal oscillator reference frequency for good long term stability. High stability crystal oscillators are usually relative low frequency such as 10 MHz (or 100 ns resolution). [ 5 ] To get better resolution, a phase-locked loop frequency multiplier can be used to generate a faster clock. One might, for example, multiply the crystal reference oscillator by 100 to get a clock rate of 1 GHz (1 ns resolution).
High clock rates impose additional design constraints on the counter: if the clock period is short, it is difficult to update the count. Binary counters, for example, need a fast carry architecture because they essentially add one to the previous counter value. A solution is using a hybrid counter architecture. A Johnson counter , for example, is a fast non-binary counter. It can be used to count very quickly the low order count; a more conventional binary counter can be used to accumulate the high order count. The fast counter is sometime called a prescaler .
The speed of counters fabricated in CMOS -technology is limited by the capacitance between the gate and the channel and by the resistance of the channel and the signal traces. The product of both is the cut-off-frequency. Modern chip technology allows multiple metal layers and therefore coils with a large number of windings to be inserted into the chip.
This allows designers to peak the device for a specific frequency , which may lie above the cut-off-frequency of the original transistor. [ citation needed ]
A peaked variant of the Johnson counter is the traveling-wave counter which also achieves sub-cycle resolution. Other methods to achieve sub-cycle resolution include analog-to-digital converters and vernier Johnson counters . [ citation needed ]
In most situations, the user does not want to just capture an absolute time that an event occurs, but wants to measure a time interval, i.e the time between a start event and a stop event.
That can be done by measuring an arbitrary time of both the start and stop events and subtracting. The measurement can be off by two counts.
The subtraction can be avoided if the counter is held at zero until the start event, counts during the interval, and then stops counting after the stop event.
Coarse counters base on a reference clock with signals generated at a stable frequency f 0 {\displaystyle f_{0}} . [ 1 ] When the start signal is detected the counter starts counting clock signals and terminates counting after the stop signal is detected. The time interval T {\displaystyle T} between start and stop is then
with n {\displaystyle n} , the number of counts and T 0 = 1 / f 0 {\displaystyle T_{0}=1/f_{0}} , the period of the reference clock .
Since start, stop and clock signal are asynchronous, there is a uniform probability distribution of the start and stop signal-times between two subsequent clock pulses. This detuning of the start and stop signal from the clock pulses is called quantization error .
For a series of measurements on the same constant and asynchronous time interval one measures two different numbers of counted clock pulses n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} (see picture). These occur with probabilities
with c = F r c ( T / T 0 ) {\displaystyle c=Frc(T/T_{0})} the fractional part of T / T 0 {\displaystyle T/T_{0}} . The value for the time interval is then obtained by
Measuring a time interval using a coarse counter with the averaging method described above is relatively time consuming because of the many repetitions that are needed to determine the probabilities p {\displaystyle p} and q {\displaystyle q} . In comparison to the other methods described later on, a coarse counter has a very limited resolution (1 ns in case of a 1 GHz reference clock ), but satisfies with its theoretically unlimited measuring range.
In contrast to the coarse counter in the previous section, fine measurement methods with much better accuracy (picosecond resolution? [ clarification needed ] ), but far smaller measuring range are presented here. [ 1 ] Analogue methods like time interval stretching or double conversion as well as digital methods like tapped delay lines and the Vernier method are under examination. Though the analogue methods still obtain better accuracies, digital time interval measurement is often preferred due to its flexibility in integrated circuit technology and its robustness against external perturbations like temperature changes.
The counter implementation's accuracy is limited by the clock frequency. If time is measured by whole counts, then the resolution is limited to the clock period. For example, a 10 MHz clock has a resolution of 100 ns. To get resolution finer than a clock period, there are time interpolation circuits. [ 6 ] These circuits measure the fraction of a clock period: that is, the time between a clock event and the event being measured. The interpolation circuits often require a significant amount of time to perform their function; consequently, the TDC needs a quiet interval before the next measurement.
When counting is not feasible because the clock rate would be too high, analog methods can be used. Analog methods are often used to measure intervals that are between 10 and 200 ns. [ 7 ] These methods often use a capacitor that is charged during the interval being measured. [ 8 ] [ 9 ] [ 10 ] [ 11 ] Initially, the capacitor is discharged to zero volts. When the start event occurs, the capacitor is charged with a constant current I 1 ; the constant current causes the voltage v on the capacitor to increase linearly with time. The rising voltage is called the fast ramp. When the stop event occurs, the charging current is stopped. The voltage on the capacitor v is directly proportional to the time interval T and can be measured with an analog-to-digital converter (ADC). The resolution of such a system is in the range of 1 to 10 ps. [ 12 ]
Although a separate ADC can be used, the ADC step is often integrated into the interpolator. A second constant current I 2 is used to discharge the capacitor at a constant but much slower rate (the slow ramp). The slow ramp might be 1/1000 of the fast ramp. This discharge effectively "stretches" the time interval; [ 13 ] it will take 1000 times as long for the capacitor to discharge to zero volts. The stretched interval can be measured with a counter. The measurement is similar to a dual-slope analog converter .
The dual-slope conversion can take a long time: a thousand or so clock ticks in the scheme described above. That limits how often a measurement can be made ( dead time ). Resolution of 1 ps with a 100 MHz (10 ns) clock requires a stretch ratio of 10,000 and implies a conversion time of 150 μs. [ 13 ] To decrease the conversion time, the interpolator circuit can be used twice in a residual interpolator technique . [ 13 ] The fast ramp is used initially as above to determine the time. The slow ramp is only at 1/100. The slow ramp will cross zero at some time during the clock period. When the ramp crosses zero, the fast ramp is turned on again to measure the crossing time ( t residual ). Consequently, the time can be determined to 1 part in 10,000.
Interpolators are often used with a stable system clock. The start event is asynchronous, but the stop event is a following clock. [ 9 ] [ 11 ] For convenience, imagine that the fast ramp rises exactly 1 volt during a 100 ns clock period. Assume the start event occurs at 67.3 ns after a clock pulse; the fast ramp integrator is triggered and starts rising. The asynchronous start event is also routed through a synchronizer that takes at least two clock pulses. By the next clock pulse, the ramp has risen to .327 V. By the second clock pulse, the ramp has risen to 1.327 V and the synchronizer reports the start event has been seen. The fast ramp is stopped and the slow ramp starts. The synchronizer output can be used to capture system time from a counter. After 1327 clocks, the slow ramp returns to its starting point, and interpolator knows that the event occurred 132.7 ns before the synchronizer reported.
The interpolator is actually more involved because there are synchronizer issues and current switching is not instantaneous. [ 14 ] Also, the interpolator must calibrate the height of the ramp to a clock period. [ 15 ]
The vernier method is more involved. [ 16 ] The method involves a triggerable oscillator [ 17 ] and a coincidence circuit. At the event, the integer clock count is stored and the oscillator is started. The triggered oscillator has a slightly different frequency than the clock oscillator. For sake of argument, say the triggered oscillator has a period that is 1 ns faster than the clock. If the event happened 67 ns after the last clock, then the triggered oscillator transition will slide by −1 ns after each subsequent clock pulse. The triggered oscillator will be at 66 ns after the next clock, at 65 ns after the second clock, and so forth. A coincidence detector looks for when the triggered oscillator and the clock transition at the same time, and that indicates the fraction time that needs to be added.
The interpolator design is more involved. The triggerable clock must be calibrated to clock. It must also start quickly and cleanly.
The Vernier method is a digital version of the time stretching method. Two only slightly detuned oscillators (with frequencies f 1 {\displaystyle f_{1}} and f 2 {\displaystyle f_{2}} ) start their signals with the arrival of the start and the stop signal. As soon as the leading edges of the oscillator signals coincide the measurement ends and the number of periods of the oscillators ( n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} respectively) lead to the original time interval T {\displaystyle T} :
Since highly reliable oscillators with stable and accurate frequency are still quite a challenge one also realizes the vernier method via two tapped delay lines using two slightly different cell delay times τ {\displaystyle \tau } . This setting is called differential delay line or vernier delay line . [ 18 ]
In the example presented here the first delay line affiliated with the start signal contains cells of D-flip-flops with delay τ L {\displaystyle \tau _{L}} which are initially set to transparent. During the transition of the start signal through one of those cells, the signal is delayed by τ L {\displaystyle \tau _{L}} and the state of the flip-flop is sampled as transparent. The second delay line belonging to the stop signal is composed of a series of non-inverting buffers with delay τ B < τ L {\displaystyle \tau _{B}<\tau _{L}} . Propagating through its channel the stop signal latches the flip-flops of the start signal's delay line. As soon as the stop signal passes the start signal, the latter is stopped and all leftover flip-flops are sampled opaque. Analogous to the above case of the oscillators the wanted time interval T {\displaystyle T} is then
with n the number of cells marked as transparent.
In general a digital delay-line based TDC , [ 19 ] also known as tapped delay line , contains a chain of cells (e.g. using D-latches in the figure) with well defined delay times τ {\displaystyle \tau } . The start signal propagates through this chain and is successively delayed by each cell. The number of cells that the start signal propagated through when the stop signal happens will be the ( rounded ) time interval between the start and stop signal divided by τ {\displaystyle \tau } .
Counters can measure long intervals but have limited resolution. Interpolators have high resolution but they cannot measure long intervals. A hybrid approach can achieve both long intervals and high resolution. [ 1 ] The long interval can be measured with a counter. The counter information is supplemented with two time interpolators: one interpolator measures the (short) interval between the start event and a following clock event, and the second interpolator measure the interval between the stop event and a following clock event. The basic idea has some complications: the start and stop events are asynchronous, and one or both might happen close to a clock pulse. The counter and interpolators must agree on matching the start and end clock events. To accomplish that goal, synchronizers are used.
The common hybrid approach is the Nutt method . [ 20 ] In this example the fine measurement circuit measures the time between start and stop pulse and the respective second nearest clock pulse of the coarse counter ( T start , T stop ), detected by the synchronizer (see figure). Thus the wanted time interval is
with n the number of counter clock pulses and T 0 the period of the coarse counter.
Time measurement has played a crucial role in the understanding of nature from the earliest times. Starting with sun, sand or water driven clocks we are able to use clocks today, based on the most precise caesium resonators.
The first direct predecessor of a TDC was invented in the year 1942 by Bruno Rossi for the measurement of muon lifetimes. [ 21 ] It was designed as a time-to-amplitude-converter , constantly charging a capacitor during the measured time interval. The corresponding voltage is directly proportional to the time interval under examination.
While the basic concepts (like Vernier methods ( Pierre Vernier 1584-1638) and time stretching) of dividing time into measurable intervals are still up-to-date, the implementation changed a lot during the past 50 years. [ when? ] Starting with vacuum tubes and ferrite pot-core transformers those ideas are implemented in complementary metal–oxide–semiconductor ( CMOS ) design today. [ 22 ]
Regarding even the fine measuring methods presented, there are still errors one may wish remove or at least to consider. Non-linearities of the time-to-digital conversion for example can be identified by taking a large number of measurements of a poissonian distributed source (statistical code density test). [ 23 ] Small deviations from the uniform distribution reveal the non-linearities.
Inconveniently the statistical code density method is quite sensitive to external temperature changes. Thus stabilizing delay or phase-locked loop (DLL or PLL) circuits are recommended.
In a similar way, offset errors (non-zero readouts at T = 0) can be removed.
For long time intervals, the error due to instabilities in the reference clock ( jitter ) plays a major role. Thus clocks of superior quality are needed for such TDCs.
Furthermore, external noise sources can be eliminated in postprocessing by robust estimation methods . [ 24 ]
TDCs are currently built as stand-alone measuring devices in physical experiments or as system components like PCI cards. They can be made up of either discrete or integrated circuits.
Circuit design changes with the purpose of the TDC, which can either be a very good solution for single-shot TDCs with long dead times or some trade-off between dead-time and resolution for multi-shot TDCs.
The time-to-digital converter measures the time between a start event and a stop event. There is also a digital-to-time converter or delay generator . The delay generator converts a number to a time delay. When the delay generator gets a start pulse at its input, then it outputs a stop pulse after the specified delay. The architectures for TDC and delay generators are similar. Both use counters for long, stable, delays. Both must consider the problem of clock quantization errors.
For example, the Tektronix 7D11 Digital Delay uses a counter architecture. [ 25 ] A digital delay may be set from 100 ns to 1 s in 100 ns increments. An analog circuit provides an additional fine delay of 0 to 100 ns. A 5 MHz reference clock drives a phase-locked loop to produce a stable 500 MHz clock. It is this fast clock that is gated by the (fine-delayed) start event and determines the main quantization error. The fast clock is divided down to 10 MHz and fed to main counter. [ 26 ] The instrument quantization error depends primarily on the 500 MHz clock (2 ns steps), but other errors also enter; the instrument is specified to have 2.2 ns of jitter . The recycle time is 575 ns.
Just as a TDC may use interpolation to get finer than one clock period resolution, a delay generator may use similar techniques. The Hewlett-Packard 5359A High Resolution Time Synthesizer provides delays of 0 to 160 ms, has an accuracy of 1 ns, and achieves a typical jitter of 100 ps. [ 27 ] The design uses a triggered phase-locked oscillator that runs at 200 MHz. Interpolation is done with a ramp, an 8-bit digital-to-analog converter, and a comparator. The resolution is about 45 ps.
When the start pulse is received, then counts down and outputs a stop pulse. For low jitter the synchronous counter has to feed a zero flag from the most significant bit down to the least significant bit and then combine it with the output from the Johnson counter.
A digital-to-analog converter (DAC) could be used to achieve sub-cycle resolution, but it is easier to either use vernier Johnson counters or traveling-wave Johnson counters.
The delay generator can be used for pulse-width modulation , e.g. to drive a MOSFET to load a Pockels cell within 8 ns with a specific charge.
The output of a delay generator can gate a digital-to-analog converter and so pulses of a variable height can be generated. This allows matching to low levels needed by analog electronics, higher levels for ECL and even higher levels for TTL . If a series of DACs is gated in sequence, variable pulse shapes can be generated to account for any transfer function. | https://en.wikipedia.org/wiki/Time-to-digital_converter |
Time-tracking software are computer programs that allows users to record time spent on tasks or projects. Time-tracking software may include time-recording software, which uses user activity monitoring to record the activities performed on a computer and the time spent on each project and task .
Timesheet software is software used to maintain timesheets . It was popularized when computers were first introduced to the office environment with the goal of automating heavy paperwork for big organizations. [ 1 ]
This software article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Time-tracking_software |
Time-translation symmetry or temporal translation symmetry ( TTS ) is a mathematical transformation in physics that moves the times of events through a common interval. Time-translation symmetry is the law that the laws of physics are unchanged (i.e. invariant) under such a transformation. Time-translation symmetry is a rigorous way to formulate the idea that the laws of physics are the same throughout history. Time-translation symmetry is closely connected, via Noether's theorem , to conservation of energy . [ 1 ] In mathematics, the set of all time translations on a given system form a Lie group .
There are many symmetries in nature besides time translation, such as spatial translation or rotational symmetries . These symmetries can be broken and explain diverse phenomena such as crystals , superconductivity , and the Higgs mechanism . [ 2 ] However, it was thought until very recently that time-translation symmetry could not be broken. [ 3 ] Time crystals , a state of matter first observed in 2017, break time-translation symmetry. [ 4 ]
Symmetries are of prime importance in physics and are closely related to the hypothesis that certain physical quantities are only relative and unobservable . [ 5 ] Symmetries apply to the equations that govern the physical laws (e.g. to a Hamiltonian or Lagrangian ) rather than the initial conditions, values or magnitudes of the equations themselves and state that the laws remain unchanged under a transformation. [ 1 ] If a symmetry is preserved under a transformation it is said to be invariant . Symmetries in nature lead directly to conservation laws, something which is precisely formulated by Noether's theorem . [ 6 ]
To formally describe time-translation symmetry we say the equations, or laws, that describe a system at times t {\displaystyle t} and t + τ {\displaystyle t+\tau } are the same for any value of t {\displaystyle t} and τ {\displaystyle \tau } .
For example, considering Newton's equation:
One finds for its solutions x = x ( t ) {\displaystyle x=x(t)} the combination:
does not depend on the variable t {\displaystyle t} . Of course, this quantity describes the total energy whose conservation is due to the time-translation invariance of the equation of motion. By studying the composition of symmetry transformations, e.g. of geometric objects, one reaches the conclusion that they form a group and, more specifically, a Lie transformation group if one considers continuous, finite symmetry transformations. Different symmetries form different groups with different geometries. Time independent Hamiltonian systems form a group of time translations that is described by the non-compact, abelian , Lie group R {\displaystyle \mathbb {R} } . TTS is therefore a dynamical or Hamiltonian dependent symmetry rather than a kinematical symmetry which would be the same for the entire set of Hamiltonians at issue. Other examples can be seen in the study of time evolution equations of classical and quantum physics.
Many differential equations describing time evolution equations are expressions of invariants associated to some Lie group and the theory of these groups provides a unifying viewpoint for the study of all special functions and all their properties. In fact, Sophus Lie invented the theory of Lie groups when studying the symmetries of differential equations. The integration of a (partial) differential equation by the method of separation of variables or by Lie algebraic methods is intimately connected with the existence of symmetries. For example, the exact solubility of the Schrödinger equation in quantum mechanics can be traced back to the underlying invariances. In the latter case, the investigation of symmetries allows for an interpretation of the degeneracies , where different configurations to have the same energy, which generally occur in the energy spectrum of quantum systems. Continuous symmetries in physics are often formulated in terms of infinitesimal rather than finite transformations, i.e. one considers the Lie algebra rather than the Lie group of transformations
The invariance of a Hamiltonian H ^ {\displaystyle {\hat {H}}} of an isolated system under time translation implies its energy does not change with the passage of time. Conservation of energy implies, according to the Heisenberg equations of motion, that [ H ^ , H ^ ] = 0 {\displaystyle [{\hat {H}},{\hat {H}}]=0} .
or:
Where T ^ ( t ) = e i H ^ t / ℏ {\displaystyle {\hat {T}}(t)=e^{i{\hat {H}}t/\hbar }} is the time-translation operator which implies invariance of the Hamiltonian under the time-translation operation and leads to the conservation of energy.
In many nonlinear field theories like general relativity or Yang–Mills theories , the basic field equations are highly nonlinear and exact solutions are only known for 'sufficiently symmetric' distributions of matter (e.g. rotationally or axially symmetric configurations). Time-translation symmetry is guaranteed only in spacetimes where the metric is static: that is, where there is a coordinate system in which the metric coefficients contain no time variable. Many general relativity systems are not static in any frame of reference so no conserved energy can be defined.
Time crystals , a state of matter first observed in 2017, break discrete time-translation symmetry. [ 4 ] | https://en.wikipedia.org/wiki/Time-translation_symmetry |
Time-triggered architecture (abbreviated as TTA ), also known as a time-triggered system , is a computer system that executes one or more sets of tasks according to a predetermined and set task schedule. [ 1 ] Implementation of a TT system will typically involve use of a single interrupt that is linked to the periodic overflow of a timer. This interrupt may drive a task scheduler (a restricted form of real-time operating system ). The scheduler will—in turn—release the system tasks at predetermined points in time. [ 1 ]
Because they have highly deterministic timing behavior, TT systems have been used for many years to develop safety-critical aerospace and related systems. [ 2 ]
An early text that sets forth the principles of time triggered architecture, communications, and sparse time approaches is Real-Time Systems: Design Principles for Distributed Embedded Applications in 1997. [ 3 ]
Use of TT systems was popularized by the publication of Patterns for Time-Triggered Embedded Systems (PTTES) in 2001 [ 1 ] and the related introductory book Embedded C in 2002. [ 4 ] The PTTES book also introduced the concepts of time-triggered hybrid schedulers (an architecture for time-triggered systems that require task pre-emption) and shared-clock schedulers (an architecture for distributed time-triggered systems involving multiple, synchronized, nodes). [ 1 ]
Since publication of PTTES, extensive research work on TT systems has been carried out. [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ]
Time-triggered systems are now commonly associated with international safety standards such as IEC 61508 (industrial systems), ISO 26262 (automotive systems), IEC 62304 (medical systems) and IEC 60730 (household goods).
Time-triggered systems can be viewed as a subset of a more general event-triggered (ET) system architecture (see event-driven programming ).
Implementation of an ET system will typically involve use of multiple interrupts, each associated with specific periodic events (such as timer overflows) or aperiodic events (such as the arrival of messages over a communication bus at random points in time). ET designs are traditionally associated with the use of what is known as a real-time operating system (or RTOS), though use of such a software platform is not a defining characteristic of an ET architecture. [ 1 ] | https://en.wikipedia.org/wiki/Time-triggered_architecture |
In communication theory, time-varying phasors are used for analyzing narrow-band signals, whose signal bandwidths in the frequency domain are considerably smaller than the carrier frequency. [ 1 ] [ 2 ] Time-varying phasors are mostly used for analysis of frequency domain of band-pass systems. [ 2 ] [ 1 ] The method uses classical impulse response. [ 1 ]
In electrical power system, phasors are used for transient analysis of the power system keeping the quasi-stationary conditions. [ 1 ] [ 3 ] [ 4 ] They were introduced to facilitate the computation and analysis of power systems in stationary operation. [ 3 ] Time-varying phasors are used in dynamic analysis of a large power system. [ 1 ] [ 5 ] The phasor representation of sinusoidal voltages and currents is generalized to arbitrary waveforms . [ 2 ] This mathematical transformation eliminates the 60 Hertz (Hz) carrier which is the only time-varying element in the stationary case. [ 3 ] The longer usage of time-varying phasors in large power systems since 1920s have created many misconceptions. One of the misuses suggest that quasi-stationary models are always accurate, but only when the system dynamics are slow as compared to nominal system frequency which is usually 60 Hz. [ 4 ]
The concern to study time-varying phasors is raised to understand in-depth the fast amplitude and phase variations of emerging electrical power generator technologies. [ 4 ] This is because current and voltage signals of latest machines may have harmonic components and they can damage the entire transmission system which is coupled with the machine. [ 3 ] [ 4 ] However, if we employ quasi-static model, we can accurately model AC signals by using time-varying phasors as opposed to traditional quasi-static model which supports constant voltage and current signals throughout the network. [ 5 ] | https://en.wikipedia.org/wiki/Time-varying_phasor |
A time and motion study (or time–motion study ) is a business efficiency technique combining the time study work of Frederick Winslow Taylor with the motion study work of Frank and Lillian Gilbreth (the same couple as is best known through the biographical 1950 film and book Cheaper by the Dozen ). It is a major part of scientific management (Taylorism). After its first introduction, time study developed in the direction of establishing standard times , while motion study evolved into a technique for improving work methods. The two techniques became integrated and refined into a widely accepted method applicable to the improvement and upgrading of work systems. This integrated approach to work system improvement is known as methods engineering [ 1 ] and it is applied today to industrial as well as service organizations, including banks, schools and hospitals. [ 2 ]
Time study is a direct and continuous observation of a task, using a timekeeping device (e.g., decimal minute stopwatch , computer-assisted electronic stopwatch, and videotape camera) to record the time taken to accomplish a task. [ 3 ] It is often used if at least one of the following applies: [ 4 ]
The Industrial Engineering Terminology Standard defines time study as "a work measurement technique consisting of careful time measurement of the task with a time measuring instrument, adjusted for any observed variance from normal effort or pace and to allow adequate time for such items as foreign elements, unavoidable or machine delays, rest to overcome fatigue, and personal needs." [ 5 ]
The systems of time and motion studies are frequently assumed to be interchangeable terms that are descriptive of equivalent theories. However, the underlying principles and the rationale for the establishment of each respective method are dissimilar, despite originating within the same school of thought.
The application of science to business problems and the use of time-study methods in standard setting and the planning of work were pioneered by Frederick Winslow Taylor. [ 6 ] Taylor liaised with factory managers and from the success of these discussions wrote several papers proposing the use of wage-contingent performance standards based on scientific time study. [ 7 ] At the most basic level time studies involved breaking down each job into component parts, timing each part and rearranging the parts into the most efficient method of working. [ 8 ] By counting and calculating, Taylor wanted to transform management, which was essentially an oral tradition, into a set of calculated and written techniques. [ 9 ] [ 10 ]
Taylor and his colleagues placed emphasis on the content of a fair day's work and sought to maximize productivity irrespective of the physiological cost to the worker. [ 11 ] For example, Taylor thought unproductive time usage ( soldiering ) to be the deliberate attempt of workers to promote their best interests and to keep employers ignorant of how fast work could be carried out. [ 12 ] This instrumental view of human behavior by Taylor prepared the path for human relations to supersede scientific management in terms of literary success and managerial application.
Following is the procedure developed by Mikell Groover for a direct time study: [ 13 ]
According to good practice guidelines for production studies [ 14 ] a comprehensive time study consists of:
Easy analysis of working areas
The collection of time data can be done in several ways, depending on study goal and environmental conditions. Time and motion data can be captured with a common stopwatch, a handheld computer or a video recorder. There are a number of dedicated software packages used to turn a palmtop or a handheld PC into a time study device. As an alternative, time and motion data can be collected automatically from the memory of computer-control machines (i.e. automated time studies).
In response to Taylor's time studies and view of human nature, many strong criticisms and reactions were recorded. Unions, for example, regarded time study as a disguised tool of management designed to standardize and intensify the pace of production. Similarly, individuals such as Gilbreth (1909), Cadbury [ 15 ] and Marshall [ 16 ] heavily criticized Taylor and pervaded his work with subjectivity. For example, Cadbury [ 17 ] in reply to Thompson [ 18 ] stated that under scientific management employee skills and initiatives are passed from the individual to management, [ 19 ] a view reiterated by Nyland. [ 20 ] In addition, Taylor's critics condemned the lack of scientific substance in his time studies, [ 21 ] in the sense that they relied heavily on individual interpretations of what workers actually do. [ 22 ] However, the value in rationalizing production is indisputable and supported by academics such as Gantt, Ford and Munsterberg, and Taylor society members Mr C.G. Renold, Mr W.H. Jackson and Mr C.B. Thompson. [ 23 ] Proper time studies are based on repeated observation, so that motions performed on the same part differently by one or many workers can be recorded, to determine those values that are truly repetitive and measurable.
In contrast to, and motivated by, Taylor's time study methods, the Gilbreths proposed a technical language, allowing for the analysis of the labor process in a scientific context. [ 24 ] The Gilbreths made use of scientific insights to develop a study method based upon the analysis of "work motions", consisting in part of filming the details of a worker's activities and their body posture while recording the time. [ 25 ] The films served two main purposes. One was the visual record of how work had been done, emphasizing areas for improvement. Secondly, the films also served the purpose of training workers about the best way to perform their work. [ 26 ] This method allowed the Gilbreths to build on the best elements of these workflows and to create a standardized best practice. [ 27 ]
Although for Taylor, motion studies remained subordinate to time studies, the attention he paid to the motion study technique demonstrated the seriousness with which he considered the Gilbreths' method. The split with Taylor in 1914, on the basis of attitudes to workers, meant the Gilbreths had to argue contrary to the trade unionists, government commissions and Robert F. Hoxie [ 28 ] who believed scientific management was unstoppable. [ 29 ] The Gilbreths were charged with the task of proving that motion study particularly, and scientific management generally, increased industrial output in ways which improved and did not detract from workers' mental and physical strength. This was no simple task given the propaganda fuelling the Hoxie report and the consequent union opposition to scientific management. In addition, the Gilbreths credibility and academic success continued to be hampered by Taylor who held the view that motion studies were nothing more than a continuation of his work.
Both Taylor and the Gilbreths continue to be criticized for their respective work, but it should be remembered that they were writing at a time of industrial reorganization and the emergence of large, complex organizations with new forms of technology. Furthermore, to equate scientific management merely with time and motion study and consequently labor control not only misconceives the scope of scientific management but also misinterprets Taylor's incentives for proposing a different style of managerial thought. [ 30 ]
A health care time and motion study is used to research and track the efficiency and quality of health care workers. [ 31 ] In the case of nurses, numerous programs have been initiated to increase the percent of a shift nurses spend providing direct care to patients. Prior to interventions nurses were found to spend ~20% of their time doing direct care. After focused intervention, some hospitals doubled that number, with some even exceeding 70% of shift time with patients, resulting in reduced errors, codes, and falls. [ 32 ] [ 33 ] | https://en.wikipedia.org/wiki/Time_and_motion_study |
In physics and engineering , the time constant , usually denoted by the Greek letter τ (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system. [ 1 ] [ note 1 ] The time constant is the main characteristic unit of a first-order LTI system. It gives speed of the response.
In the time domain, the usual choice to explore the time response is through the step response to a step input , or the impulse response to a Dirac delta function input. [ 2 ] In the frequency domain (for example, looking at the Fourier transform of the step response, or using an input that is a simple sinusoidal function of time) the time constant also determines the bandwidth of a first-order time-invariant system, that is, the frequency at which the output signal power drops to half the value it has at low frequencies.
The time constant is also used to characterize the frequency response of various signal processing systems – magnetic tapes , radio transmitters and receivers , record cutting and replay equipment, and digital filters – which can be modelled or approximated by first-order LTI systems. Other examples include time constant used in control systems for integral and derivative action controllers, which are often pneumatic , rather than electrical.
Time constants are a feature of the lumped system analysis (lumped capacity analysis method) for thermal systems, used when objects cool or warm uniformly under the influence of convective cooling or warming. [ 3 ]
Physically, the time constant represents the elapsed time required for the system response to decay to zero if the system had continued to decay at the initial rate, because of the progressive change in the rate of decay the response will have actually decreased in value to 1 / e ≈ 36.8% in this time (say from a step decrease). In an increasing system, the time constant is the time for the system's step response to reach 1 − 1 / e ≈ 63.2% of its final (asymptotic) value (say from a step increase). In radioactive decay the time constant is related to the decay constant ( λ ), and it represents both the mean lifetime of a decaying system (such as an atom) before it decays, or the time it takes for all but 36.8% of the atoms to decay. For this reason, the time constant is longer than the half-life , which is the time for only 50% of the atoms to decay.
First order LTI systems are characterized by the differential equation τ d V d t + V = f ( t ) {\displaystyle \tau {\frac {dV}{dt}}+V=f(t)}
where τ represents the exponential decay constant and V is a function of time t V = V ( t ) . {\displaystyle V=V(t).} The right-hand side is the forcing function f ( t ) describing an external driving function of time, which can be regarded as the system input , to which V ( t ) is the response , or system output. Classical examples for f ( t ) are:
The Heaviside step function , often denoted by u ( t ) : u ( t ) = { 0 , t < 0 1 , t ≥ 0 {\displaystyle u(t)={\begin{cases}0,&t<0\\1,&t\geq 0\end{cases}}} the impulse function , often denoted by δ ( t ) , and also the sinusoidal input function: f ( t ) = A sin ( 2 π f t ) {\displaystyle f(t)=A\sin(2\pi ft)} or f ( t ) = A e j ω t , {\displaystyle f(t)=Ae^{j\omega t},} where A is the amplitude of the forcing function, f is the frequency in Hertz, and ω = 2 π f is the frequency in radians per second.
An example solution to the differential equation with initial value V 0 and no forcing function is V ( t ) = V 0 e − t / τ {\displaystyle V(t)=V_{0}e^{-t/\tau }}
where V 0 = V ( t = 0 ) {\displaystyle V_{0}=V(t=0)}
is the initial value of V . Thus, the response is an exponential decay with time constant τ . The time constant indicates how rapidly an exponential function decays.
Suppose the forcing function is chosen as sinusoidal so:
τ d V d t + V = f ( t ) = A e j ω t . {\displaystyle \tau {\frac {dV}{dt}}+V=f(t)=Ae^{j\omega t}.}
(Response to a real cosine or sine wave input can be obtained by taking the real or imaginary part of the final result by virtue of Euler's formula .) The general solution to this equation for times t ≥ 0 s , assuming V ( t = 0) = V 0 is:
V ( t ) = V 0 e − t / τ + A e − t / τ τ ∫ 0 t d t ′ e t ′ / τ e j ω t ′ = V 0 e − t / τ + 1 τ j ω + 1 τ A ( e j ω t − e − t / τ ) . {\displaystyle {\begin{aligned}V(t)&=V_{0}e^{-t/\tau }+{\frac {Ae^{-t/\tau }}{\tau }}\int _{0}^{t}\,dt'\ e^{t'/\tau }e^{j\omega t'}\\[1ex]&=V_{0}e^{-t/\tau }+{\frac {\frac {1}{\tau }}{j\omega +{\frac {1}{\tau }}}}A\left(e^{j\omega t}-e^{-t/\tau }\right).\end{aligned}}}
For long times the decaying exponentials become negligible and the steady-state solution or long-time solution is:
V ∞ ( t ) = 1 / τ j ω + 1 / τ A e j ω t . {\displaystyle V_{\infty }(t)={\frac {1/\tau }{j\omega +1/\tau }}Ae^{j\omega t}.}
The magnitude of this response is: | V ∞ ( t ) | = A 1 τ ( ω 2 + ( 1 / τ ) 2 ) 1 / 2 = A 1 1 + ( ω τ ) 2 . {\displaystyle |V_{\infty }(t)|=A{\frac {1}{\tau \left(\omega ^{2}+(1/\tau )^{2}\right)^{1/2}}}=A{\frac {1}{\sqrt {1+(\omega \tau )^{2}}}}.} By convention, the bandwidth of this system is the frequency where | V ∞ | 2 drops to half-value, or where ωτ = 1 . This is the usual bandwidth convention, defined as the frequency range where power drops by less than half (at most −3 dB). Using the frequency in hertz, rather than radians/s ( ω = 2 πf ):
f 3 d B = 1 2 π τ . {\displaystyle f_{\mathrm {3dB} }={\frac {1}{2\pi \tau }}.}
The notation f 3dB stems from the expression of power in decibels and the observation that half-power corresponds to a drop in the value of | V ∞ | by a factor of 1/2 or by 3 decibels.
Thus, the time constant determines the bandwidth of this system.
Suppose the forcing function is chosen as a step input so:
d V d t + 1 τ V = f ( t ) = A u ( t ) , {\displaystyle {\frac {dV}{dt}}+{\frac {1}{\tau }}V=f(t)=Au(t),}
with u ( t ) the unit step function . The general solution to this equation for times t ≥ 0 s , assuming V ( t = 0) = V 0 is:
V ( t ) = V 0 e − t / τ + A τ ( 1 − e − t / τ ) . {\displaystyle V(t)=V_{0}e^{-t/\tau }+A\tau \left(1-e^{-t/\tau }\right).}
(It may be observed that this response is the ω → 0 limit of the above response to a sinusoidal input.)
The long-time solution is time independent and independent of initial conditions: V ∞ = A τ . {\displaystyle V_{\infty }=A\tau .}
The time constant remains the same for the same system regardless of the starting conditions. Simply stated, a system approaches its final, steady-state situation at a constant rate, regardless of how close it is to that value at any arbitrary starting point.
For example, consider an electric motor whose startup is well modelled by a first-order LTI system. Suppose that when started from rest, the motor takes 1 / 8 of a second to reach 63% of its nominal speed of 100 RPM, or 63 RPM—a shortfall of 37 RPM. Then it will be found that after the next 1 / 8 of a second, the motor has sped up an additional 23 RPM, which equals 63% of that 37 RPM difference. This brings it to 86 RPM—still 14 RPM low. After a third 1 / 8 of a second, the motor will have gained an additional 9 RPM (63% of that 14 RPM difference), putting it at 95 RPM.
In fact, given any initial speed s ≤ 100 RPM, 1 / 8 of a second later this particular motor will have gained an additional 0.63 × (100 − s ) RPM.
In an RL circuit composed of a single resistor and inductor, the time constant τ {\displaystyle \tau } (in seconds ) is τ = L R {\displaystyle \tau ={\frac {L}{R}}}
where R is the resistance (in ohms ) and L is the inductance (in henrys ).
Similarly, in an RC circuit composed of a single resistor and capacitor, the time constant τ {\displaystyle \tau } (in seconds) is: τ = R C {\displaystyle \tau =RC}
where R is the resistance (in ohms ) and C is the capacitance (in farads ).
Electrical circuits are often more complex than these examples, and may exhibit multiple time constants (See Step response and Pole splitting for some examples.) In the case where feedback is present, a system may exhibit unstable, increasing oscillations. In addition, physical electrical circuits are seldom truly linear systems except for very low amplitude excitations; however, the approximation of linearity is widely used.
In digital electronic circuits another measure, the FO4 is often used. This can be converted to time constant units via the equation 5 τ = FO4 {\displaystyle 5\tau ={\text{FO4}}} . [ 4 ]
Time constants are a feature of the lumped system analysis (lumped capacity analysis method) for thermal systems, used when objects cool or warm uniformly under the influence of convective cooling or warming . In this case, the heat transfer from the body to the ambient at a given time is proportional to the temperature difference between the body and the ambient: [ 5 ]
F = h A s ( T ( t ) − T a ) , {\displaystyle F=hA_{s}\left(T(t)-T_{a}\right),}
where h is the heat transfer coefficient , and A s is the surface area, T is the temperature function, i.e., T ( t ) is the body temperature at time t , and T a is the constant ambient temperature. The positive sign indicates the convention that F is positive when heat is leaving the body because its temperature is higher than the ambient temperature ( F is an outward flux). As heat is lost to the ambient, this heat transfer leads to a drop in temperature of the body given by: [ 5 ]
ρ c p V d T d t = − F , {\displaystyle \rho c_{p}V{\frac {dT}{dt}}=-F,}
where ρ = density, c p = specific heat and V is the body volume. The negative sign indicates the temperature drops when the heat transfer is outward from the body (that is, when F > 0). Equating these two expressions for the heat transfer,
ρ c p V d T d t = − h A s ( T ( t ) − T a ) . {\displaystyle \rho c_{p}V{\frac {dT}{dt}}=-hA_{s}\left(T(t)-T_{a}\right).}
Evidently, this is a first-order LTI system that can be cast in the form:
d T d t + 1 τ T = 1 τ T a , {\displaystyle {\frac {dT}{dt}}+{\frac {1}{\tau }}T={\frac {1}{\tau }}T_{a},}
with
τ = ρ c p V h A s . {\displaystyle \tau ={\frac {\rho c_{p}V}{hA_{s}}}.}
In other words, larger masses ρV with higher heat capacities c p lead to slower changes in temperature (longer time constant τ ), while larger surface areas A s with higher heat transfer h lead to more rapid temperature change (shorter time constant τ ).
Comparison with the introductory differential equation suggests the possible generalization to time-varying ambient temperatures T a . However, retaining the simple constant ambient example, by substituting the variable Δ T ≡ ( T − T a ), one finds:
d Δ T d t + 1 τ Δ T = 0. {\displaystyle {\frac {d\Delta T}{dt}}+{\frac {1}{\tau }}\Delta T=0.}
Systems for which cooling satisfies the above exponential equation are said to satisfy Newton's law of cooling . The solution to this equation suggests that, in such systems, the difference between the temperature of the system and its surroundings Δ T as a function of time t , is given by:
where Δ T 0 is the initial temperature difference, at time t = 0. In words, the body assumes the same temperature as the ambient at an exponentially slow rate determined by the time constant.
In an excitable cell such as a muscle or neuron , the time constant of the membrane potential τ {\displaystyle \tau } is τ = r m c m {\displaystyle \tau =r_{m}c_{m}}
where r m is the resistance across the membrane and c m is the capacitance of the membrane.
The resistance across the membrane is a function of the number of open ion channels and the capacitance is a function of the properties of the lipid bilayer .
The time constant is used to describe the rise and fall of membrane voltage, where the rise is described by V ( t ) = V max ( 1 − e − t / τ ) {\displaystyle V(t)=V_{\textrm {max}}\left(1-e^{-t/\tau }\right)}
and the fall is described by V ( t ) = V max e − t / τ {\displaystyle V(t)=V_{\textrm {max}}e^{-t/\tau }}
where voltage is in millivolts, time is in seconds, and τ {\displaystyle \tau } is in seconds.
V max is defined as the maximum voltage change from the resting potential , where V max = r m I {\displaystyle V_{\textrm {max}}=r_{m}I}
where r m is the resistance across the membrane and I is the membrane current.
Setting for t = τ {\displaystyle \tau } for the rise sets V ( t ) equal to 0.63 V max . This means that the time constant is the time elapsed after 63% of V max has been reached
Setting for t = τ {\displaystyle \tau } for the fall sets V ( t ) equal to 0.37 V max , meaning that the time constant is the time elapsed after it has fallen to 37% of V max .
The larger a time constant is, the slower the rise or fall of the potential of a neuron. A long time constant can result in temporal summation , or the algebraic summation of repeated potentials. A short time constant rather produces a coincidence detector through spatial summation .
In exponential decay , such as of a radioactive isotope, the time constant can be interpreted as the mean lifetime . The half-life T HL or T 1/2 is related to the exponential decay constant τ {\displaystyle \tau } by T 1 / 2 = T HL = τ ln 2. {\displaystyle T_{1/2}=T_{\text{HL}}=\tau \ln 2.} The reciprocal of the time constant is called the decay constant , and is denoted λ = 1 / τ {\displaystyle \lambda =1/\tau } .
A time constant is the amount of time it takes for a meteorological sensor to respond to a rapid change in a measure, and until it is measuring values within the accuracy tolerance usually expected of the sensor.
This most often applies to measurements of temperature, dew-point temperature, humidity and air pressure. Radiosondes are especially affected due to their rapid increase in altitude. | https://en.wikipedia.org/wiki/Time_constant |
In condensed matter physics , a time crystal is a quantum system of particles whose lowest-energy state is one in which the particles are in repetitive motion. The system cannot lose energy to the environment and come to rest because it is already in its quantum ground state . Time crystals were first proposed theoretically by Frank Wilczek in 2012 as a time-based analogue to common crystals – whereas the atoms in crystals are arranged periodically in space, the atoms in a time crystal are arranged periodically in both space and time. [ 1 ] Several different groups have demonstrated matter with stable periodic evolution in systems that are periodically driven. [ 2 ] [ 3 ] [ 4 ] [ 5 ] In terms of practical use, time crystals may one day be used as quantum computer memory . [ 6 ]
The existence of crystals in nature is a manifestation of spontaneous symmetry breaking , which occurs when the lowest-energy state of a system is less symmetrical than the equations governing the system. In the crystal ground state, the continuous translational symmetry in space is broken and replaced by the lower discrete symmetry of the periodic crystal. As the laws of physics are symmetrical under continuous translations in time as well as space, the question arose in 2012 as to whether it is possible to break symmetry temporally, and thus create a "time crystal" that is resistant to entropy . [ 1 ]
If a discrete time-translation symmetry is broken (which may be realized in periodically driven systems), then the system is referred to as a discrete time crystal . A discrete time crystal never reaches thermal equilibrium , as it is a type (or phase) of non-equilibrium matter. Breaking of time symmetry can occur only in non-equilibrium systems. [ 5 ] Discrete time crystals have in fact been observed in physics laboratories as early as 2016. One example of a time crystal, which demonstrates non-equilibrium, broken time symmetry is a constantly rotating ring of charged ions in an otherwise lowest-energy state. [ 6 ]
Ordinary (non-time) crystals form through spontaneous symmetry breaking related to spatial symmetry. Such processes can produce materials with interesting properties, such as diamonds , salt crystals , and ferromagnetic metals. By analogy, a time crystal arises through the spontaneous breaking of a time-translation symmetry. A time crystal can be informally defined as a time-periodic self-organizing structure. While an ordinary crystal is periodic (has a repeating structure) in space, a time crystal has a repeating structure in time. A time crystal is periodic in time in the same sense that the pendulum in a pendulum-driven clock is periodic in time. Unlike a pendulum, a time crystal "spontaneously" self-organizes into robust periodic motion (breaking a temporal symmetry). [ 7 ]
Symmetries in nature lead directly to conservation laws, something which is precisely formulated by Noether's theorem . [ 8 ]
The basic idea of time-translation symmetry is that a translation in time has no effect on physical laws, i.e. that the laws of nature that apply today were the same in the past and will be the same in the future. [ 9 ] This symmetry implies the conservation of energy . [ 10 ]
Common crystals exhibit broken translation symmetry : they have repeated patterns in space and are not invariant under arbitrary translations or rotations. The laws of physics are unchanged by arbitrary translations and rotations. However, if we hold fixed the atoms of a crystal, the dynamics of an electron or other particle in the crystal depend on how it moves relative to the crystal, and particle momentum can change by interacting with the atoms of a crystal—for example in Umklapp processes . [ 11 ] Quasimomentum , however, is conserved in a perfect crystal. [ 12 ]
Time crystals show a broken symmetry analogous to a discrete space-translation symmetry breaking. For example, [ citation needed ] the molecules of a liquid freezing on the surface of a crystal can align with the molecules of the crystal, but with a pattern less symmetric than the crystal: it breaks the initial symmetry. This broken symmetry exhibits three important characteristics: [ citation needed ]
Time crystals seem to break time-translation symmetry and have repeated patterns in time even if the laws of the system are invariant by translation of time. The time crystals that are experimentally realized show discrete time-translation symmetry breaking, not the continuous one: they are periodically driven systems oscillating at a fraction of the frequency of the driving force. (According to Philip Ball , DTC are so-called because "their periodicity is a discrete, integer multiple of the driving period". [ 13 ] )
The initial symmetry, which is the discrete time-translation symmetry ( t → t + n T {\displaystyle t\to t+nT} ) with n = 1 {\displaystyle n=1} , is spontaneously broken to the lower discrete time-translation symmetry with n > 1 {\displaystyle n>1} , where t {\displaystyle t} is time, T {\displaystyle T} the driving period, n {\displaystyle n} an integer. [ 14 ]
Many systems can show behaviors of spontaneous time-translation symmetry breaking but may not be discrete (or Floquet) time crystals: convection cells , oscillating chemical reactions , aerodynamic flutter , and subharmonic response to a periodic driving force such as the Faraday instability , NMR spin echos , parametric down-conversion , and period-doubled nonlinear dynamical systems. [ 14 ]
However, discrete (or Floquet) time crystals are unique in that they follow a strict definition of discrete time-translation symmetry breaking : [ 15 ]
Moreover, the broken symmetry in time crystals is the result of many-body interactions : the order is the consequence of a collective process , just like in spatial crystals. [ 14 ] This is not the case for NMR spin echos.
These characteristics makes discrete time crystals analogous to spatial crystals as described above and may be considered a novel type or phase of nonequilibrium matter. [ 14 ]
Time crystals do not violate the laws of thermodynamics : energy in the overall system is conserved, such a crystal does not spontaneously convert thermal energy into mechanical work, and it cannot serve as a perpetual store of work. But it may change perpetually in a fixed pattern in time for as long as the system can be maintained. They possess "motion without energy" [ 16 ] —their apparent motion does not represent conventional kinetic energy. [ 17 ] Recent experimental advances in probing discrete time crystals in their periodically driven nonequilibrium states have led to the beginning exploration of novel phases of nonequilibrium matter. [ 14 ]
Time crystals do not evade the second law of thermodynamics, [ 18 ] although they spontaneously break "time-translation symmetry", the usual rule that a stable object will remain the same throughout time. In thermodynamics, a time crystal's entropy, understood as a measure of disorder in the system, remains stationary over time, marginally satisfying the second law of thermodynamics by not decreasing. [ 19 ] [ 20 ]
The idea of a quantized time crystal was theorized in 2012 by Frank Wilczek , [ 21 ] [ 22 ] a Nobel laureate and professor at MIT . In 2013, Xiang Zhang , a nanoengineer at University of California, Berkeley , and his team proposed creating a time crystal in the form of a constantly rotating ring of charged ions. [ 23 ] [ 24 ]
In response to Wilczek and Zhang, Patrick Bruno ( European Synchrotron Radiation Facility ) and Masaki Oshikawa ( University of Tokyo ) published several articles stating that space–time crystals were impossible. [ 25 ] [ 26 ]
Subsequent work developed more precise definitions of time-translation symmetry -breaking, which ultimately led to the Watanabe–Oshikawa "no-go" statement that quantum space–time crystals in equilibrium are not possible. [ 27 ] [ 28 ] Later work restricted the scope of Watanabe and Oshikawa: strictly speaking, they showed that long-range order in both space and time is not possible in equilibrium, but breaking of time-translation symmetry alone is still possible. [ 29 ] [ 30 ] [ 31 ]
Several realizations of time crystals, which avoid the equilibrium no-go arguments, were later proposed. [ 32 ] In 2014 Krzysztof Sacha at Jagiellonian University in Kraków predicted the behaviour of discrete time crystals in a periodically driven system with "an ultracold atomic cloud bouncing on an oscillating mirror". [ 33 ] [ 34 ]
In 2016, research groups at Princeton and at Santa Barbara independently suggested that periodically driven quantum spin systems could show similar behaviour. [ 35 ] Also in 2016, Norman Yao at Berkeley and colleagues proposed a different way to create discrete time crystals in spin systems. [ 36 ] These ideas were successful and independently realized by two experimental teams: a group led by Harvard 's Mikhail Lukin [ 37 ] and a group led by Christopher Monroe at University of Maryland . [ 38 ] Both experiments were published in the same issue of Nature in March 2017.
Later, time crystals in open systems, so called dissipative time crystals, were proposed in several platforms breaking a discrete [ 39 ] [ 40 ] [ 41 ] [ 42 ] and a continuous [ 43 ] [ 44 ] time-translation symmetry. A dissipative time crystal was experimentally realized for the first time in 2021 by the group of Andreas Hemmerich at the Institute of Laser Physics at the University of Hamburg . [ 45 ] The researchers used a Bose–Einstein condensate strongly coupled to a dissipative optical cavity and the time crystal was demonstrated to spontaneously break discrete time-translation symmetry by periodically switching between two atomic density patterns. [ 45 ] [ 46 ] [ 47 ] In an earlier experiment in the group of Tilman Esslinger at ETH Zurich , limit cycle dynamics [ 48 ] was observed in 2019, [ 49 ] but evidence of robustness against perturbations and the spontaneous character of the time-translation symmetry breaking were not addressed.
In 2019, physicists Valerii Kozin and Oleksandr Kyriienko proved that, in theory, a permanent quantum time crystal can exist as an isolated system if the system contains unusual long-range multiparticle interactions. The original "no-go" argument only holds in the presence of typical short-range fields that decay as quickly as r − α for some α > 0 . Kozin and Kyriienko instead analyzed a spin-1/2 many-body Hamiltonian with long-range multispin interactions, and showed it broke continuous time-translational symmetry. Certain spin correlations in the system oscillate in time, despite the system being closed and in a ground energy state . However, demonstrating such a system in practice might be prohibitively difficult, [ 50 ] [ 51 ] and concerns about the physicality of the long-range nature of the model have been raised. [ 52 ]
In October 2016, Christopher Monroe at the University of Maryland claimed to have created the world's first discrete time crystal. Using the ideas proposed by Yao et al., [ 36 ] his team trapped a chain of 171 Yb + ions in a Paul trap , confined by radio-frequency electromagnetic fields. One of the two spin states was selected by a pair of laser beams. The lasers were pulsed, with the shape of the pulse controlled by an acousto-optic modulator , using the Tukey window to avoid too much energy at the wrong optical frequency. The hyperfine electron states in that setup, 2 S 1/2 | F = 0, m F = 0⟩ and | F = 1, m F = 0⟩ , have very close energy levels, separated by 12.642831 GHz. Ten Doppler-cooled ions were placed in a line 0.025 mm long and coupled together.
The researchers observed a subharmonic oscillation of the drive. The experiment showed "rigidity" of the time crystal, where the oscillation frequency remained unchanged even when the time crystal was perturbed, and that it gained a frequency of its own and vibrated according to it (rather than only the frequency of the drive). However, once the perturbation or frequency of vibration grew too strong, the time crystal "melted" and lost this subharmonic oscillation, and it returned to the same state as before where it moved only with the induced frequency. [ 38 ]
Also in 2016, Mikhail Lukin at Harvard also reported the creation of a driven time crystal. His group used a diamond crystal doped with a high concentration of nitrogen-vacancy centers , which have strong dipole–dipole coupling and relatively long-lived spin coherence . This strongly interacting dipolar spin system was driven with microwave fields, and the ensemble spin state was determined with an optical (laser) field. It was observed that the spin polarization evolved at half the frequency of the microwave drive. The oscillations persisted for over 100 cycles. This subharmonic response to the drive frequency is seen as a signature of time-crystalline order. [ 37 ]
In May 2018, a group in Aalto University reported that they had observed the formation of a time quasicrystal and its phase transition to a continuous time crystal in a Helium-3 superfluid cooled to within one ten thousandth of a kelvin from absolute zero (0.0001 K). [ 53 ] On August 17, 2020 Nature Materials published a letter from the same group saying that for the first time they were able to observe interactions and the flow of constituent particles between two time crystals. [ 54 ]
In February 2021, a team at Max Planck Institute for Intelligent Systems described the creation of time crystal consisting of magnons and probed them under scanning transmission X-ray microscopy to capture the recurring periodic magnetization structure in the first known video record of such type. [ 55 ] [ 56 ]
In July 2021, a team led by Andreas Hemmerich at the Institute of Laser Physics at the University of Hamburg presented the first realization of a time crystal in an open system, a so-called dissipative time crystal using ultracold atoms coupled to an optical cavity . The main achievement of this work is a positive application of dissipation – actually helping to stabilise the system's dynamics. [ 45 ] [ 46 ] [ 47 ]
In November 2021, a collaboration between Google and physicists from multiple universities reported the observation of a discrete time crystal on Google's Sycamore processor , a quantum computing device. A chip of 20 qubits was used to obtain a many-body localization configuration of up and down spins and then stimulated with a laser to achieve a periodically driven " Floquet " system where all up spins are flipped for down and vice-versa in periodic cycles which are multiples of the laser's frequency. While the laser is necessary to maintain the necessary environmental conditions, no energy is absorbed from the laser, so the system remains in a protected eigenstate order . [ 20 ] [ 57 ]
Previously in June and November 2021 other teams had obtained virtual time crystals based on floquet systems under similar principles to those of the Google experiment, but on quantum simulators rather than quantum processors: first a group at the University of Maryland obtained time crystals on trapped-ions qubits using high frequency driving rather than many-body localization [ 58 ] [ 59 ] and then a collaboration between TU Delft and TNO in the Netherlands called Qutech created time crystals from nuclear spins in carbon-13 nitrogen-vacancy (NV) centers on a diamond, attaining longer times but fewer qubits. [ 60 ] [ 61 ]
In February 2022, a scientist at UC Riverside reported a dissipative time crystal akin to the system of July 2021 but all-optical, which allowed the scientist to operate it at room temperature. In this experiment injection locking was used to direct lasers at a specific frequency inside a microresonator creating a lattice trap for solitons at subharmonic frequencies. [ 62 ] [ 63 ]
In March 2022, a new experiment studying time crystals on a quantum processor was performed by two physicists at the University of Melbourne , this time using IBM's Manhattan and Brooklyn quantum processors observing a total of 57 qubits. [ 64 ] [ 65 ] [ 66 ]
In June 2022, the observation of a continuous time crystal was reported by a team at the Institute of Laser Physics at the University of Hamburg , supervised by Hans Keßler and Andreas Hemmerich. In periodically driven systems, time-translation symmetry is broken into a discrete time-translation symmetry due to the drive. Discrete time crystals break this discrete time-translation symmetry by oscillating at a multiple of the drive frequency. In the new experiment, the drive (pump laser) was operated continuously, thus respecting the continuous time-translation symmetry. Instead of a subharmonic response, the system showed an oscillation with an intrinsic frequency and a time phase taking random values between 0 and 2π, as expected for spontaneous breaking of continuous time-translation symmetry. Moreover, the observed limit cycle oscillations were shown to be robust against perturbations of technical or fundamental character, such as quantum noise and, due to the openness of the system, fluctuations associated with dissipation. The system consisted of a Bose–Einstein condensate in an optical cavity , which was pumped with an optical standing wave oriented perpendicularly with regard to the cavity axis and was in a superradiant phase localizing at two bistable ground states between which it oscillated. [ 67 ] [ 68 ] [ 69 ] [ 70 ]
In February 2024, a team from Dortmund University in Germany built a time crystal from indium gallium arsenide that lasted for 40 minutes, nearly 10 million times longer than the previous record of around 5 milliseconds. In addition, the lack of any decay suggests the crystal could have lasted even longer, stating that it could last "at least a few hours, perhaps even longer". [ 71 ] [ 72 ] [ 73 ] [ 74 ] [ 75 ]
In March 2025, researchers at TU Dortmund University observed complex nonlinear behavior in a semiconductor -based time crystal made of indium gallium arsenide . By periodically driving the system with laser pulses, they uncovered transitions from synchronized oscillations to chaotic motion. The system exhibited structures such as the Farey tree sequence and the devil's staircase —patterns never before seen in semiconductor time crystals—offering new insights into dynamic phase transitions and chaos in driven quantum systems. [ 76 ] | https://en.wikipedia.org/wiki/Time_crystal |
Time dilation is the difference in elapsed time as measured by two clocks , either because of a relative velocity between them ( special relativity ), or a difference in gravitational potential between their locations ( general relativity ). When unspecified, "time dilation" usually refers to the effect due to velocity. The dilation compares "wristwatch" clock readings between events measured in different inertial frames and is not observed by visual comparison of clocks across moving frames.
These predictions of the theory of relativity have been repeatedly confirmed by experiment, and they are of practical concern, for instance in the operation of satellite navigation systems such as GPS and Galileo . [ 1 ]
Time dilation is a relationship between clock readings. Visually observed clock readings involve delays due to the propagation speed of light from the clock to the observer. Thus there is no direct way to observe time dilation. As an example of time dilation, two experimenters measuring a passing train traveling at .86 light speed may see a 2 second difference on their clocks while on the train the engineer reports only one second elapsed when the experimenters went by. Observations of a clock on the front of the train would give completely different results: the light from the train would not reach the second experimenter only 0.27s before the train passed. This effect of moving objects on observations is associated with the Doppler effect . [ 2 ]
Time dilation by the Lorentz factor was predicted by several authors at the turn of the 20th century. [ 3 ] [ 4 ] Joseph Larmor (1897) wrote that, at least for those orbiting a nucleus, individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio: 1 − v 2 c 2 {\textstyle {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}} . [ 5 ] Emil Cohn (1904) specifically related this formula to the rate of clocks. [ 6 ] In the context of special relativity it was shown by Albert Einstein (1905) that this effect concerns the nature of time itself, and he was also the first to point out its reciprocity or symmetry. [ 7 ] Subsequently, Hermann Minkowski (1907) introduced the concept of proper time which further clarified the meaning of time dilation. [ 8 ]
Special relativity indicates that, for an observer in an inertial frame of reference , a clock that is moving relative to the observer will be measured to tick more slowly than a clock at rest in the observer's frame of reference. This is sometimes called special relativistic time dilation. The faster the relative velocity , the greater the time dilation between them, with time slowing to a stop as one clock approaches the speed of light (299,792,458 m/s).
In theory, time dilation would make it possible for passengers in a fast-moving vehicle to advance into the future in a short period of their own time. With sufficiently high speeds, the effect would be dramatic. For example, one year of travel might correspond to ten years on Earth. Indeed, a constant 1 g acceleration would permit humans to travel through the entire known Universe in one human lifetime. [ 10 ]
With current technology severely limiting the velocity of space travel, the differences experienced in practice are minuscule. After 6 months on the International Space Station (ISS), orbiting Earth at a speed of about 7,700 m/s, an astronaut would have aged about 0.005 seconds less than he would have on Earth. [ 11 ] The cosmonauts Sergei Krikalev and Sergey Avdeev both experienced time dilation of about 20 milliseconds compared to time that passed on Earth. [ 12 ] [ 13 ]
Time dilation can be inferred from the observed constancy of the speed of light in all reference frames dictated by the second postulate of special relativity . This constancy of the speed of light means that, counter to intuition, the speeds of material objects and light are not additive. It is not possible to make the speed of light appear greater by moving towards or away from the light source. [ 14 ] [ 15 ] [ 16 ] [ 17 ]
Consider then, a simple vertical clock consisting of two mirrors A and B , between which a light pulse is bouncing. The separation of the mirrors is L and the clock ticks once each time the light pulse hits mirror A .
In the frame in which the clock is at rest (see left part of the diagram), the light pulse traces out a path of length 2 L and the time period between the ticks of the clock Δ t {\displaystyle \Delta t} is equal to 2 L divided by the speed of light c :
From the frame of reference of a moving observer traveling at the speed v relative to the resting frame of the clock (right part of diagram), the light pulse is seen as tracing out a longer, angled path 2 D . Keeping the speed of light constant for all inertial observers requires a lengthening (that is dilation) of the time period between the ticks of this clock Δ t ′ {\displaystyle \Delta t'} from the moving observer's perspective. That is to say, as measured in a frame moving relative to the local clock, this clock will be running (that is ticking) more slowly, since tick rate equals one over the time period between ticks 1/ Δ t ′ {\displaystyle \Delta t'} .
Straightforward application of the Pythagorean theorem leads to the well-known prediction of special relativity:
The total time for the light pulse to trace its path is given by:
The length of the half path can be calculated as a function of known quantities as:
Elimination of the variables D and L from these three equations results in:
Δ t ′ = Δ t 1 − v 2 c 2 = γ Δ t {\displaystyle \Delta t'={\frac {\Delta t}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\gamma }{\Delta t}}
which expresses the fact that the moving observer's period of the clock Δ t ′ {\displaystyle \Delta t'} is longer than the period Δ t {\displaystyle \Delta t} in the frame of the clock itself. The Lorentz factor gamma ( γ ) is defined as [ 18 ]
Because all clocks that have a common period in the resting frame should have a common period when observed from the moving frame, all other clocks—mechanical, electronic, optical (such as an identical horizontal version of the clock in the example)—should exhibit the same velocity-dependent time dilation. [ 19 ]
Given a certain frame of reference, and the "stationary" observer described earlier, if a second observer accompanied the "moving" clock, each of the observers would measure the other's clock as ticking at a slower rate than their own local clock, due to them both measuring the other to be the one that is in motion relative to their own stationary frame of reference.
Common sense would dictate that, if the passage of time has slowed for a moving object, said object would observe the external world's time to be correspondingly sped up. Counterintuitively, special relativity predicts the opposite. When two observers are in motion relative to each other, each will measure the other's clock slowing down, in concordance with them being in motion relative to the observer's frame of reference.
While this seems self-contradictory, a similar oddity occurs in everyday life. If two persons A and B observe each other from a distance, B will appear small to A, but at the same time, A will appear small to B. Being familiar with the effects of perspective , there is no contradiction or paradox in this situation. [ 20 ]
The reciprocity of the phenomenon also leads to the so-called twin paradox where the aging of twins, one staying on Earth and the other embarking on space travel, is compared, and where the reciprocity suggests that both persons should have the same age when they reunite. On the contrary, at the end of the round-trip, the traveling twin will be younger than the sibling on Earth. The dilemma posed by the paradox can be explained by the fact that situation is not symmetric. The twin staying on Earth is in a single inertial frame, and the traveling twin is in two different inertial frames: one on the way out and another on the way back. See also Twin paradox § Role of acceleration .
In the Minkowski diagram from the first image on the right, clock C resting in inertial frame S′ meets clock A at d and clock B at f (both resting in S). All three clocks simultaneously start to tick in S. The worldline of A is the ct-axis, the worldline of B intersecting f is parallel to the ct-axis, and the worldline of C is the ct′-axis. All events simultaneous with d in S are on the x-axis, in S′ on the x′-axis.
The proper time between two events is indicated by a clock present at both events. [ 27 ] It is invariant, i.e., in all inertial frames it is agreed that this time is indicated by that clock. Interval df is, therefore, the proper time of clock C, and is shorter with respect to the coordinate times ef=dg of clocks B and A in S. Conversely, also proper time ef of B is shorter with respect to time if in S′, because event e was measured in S′ already at time i due to relativity of simultaneity, long before C started to tick.
From that it can be seen, that the proper time between two events indicated by an unaccelerated clock present at both events, compared with the synchronized coordinate time measured in all other inertial frames, is always the minimal time interval between those events. However, the interval between two events can also correspond to the proper time of accelerated clocks present at both events. Under all possible proper times between two events, the proper time of the unaccelerated clock is maximal , which is the solution to the twin paradox . [ 27 ]
In addition to the light clock used above, the formula for time dilation can be more generally derived from the temporal part of the Lorentz transformation . [ 28 ] Let there be two events at which the moving clock indicates t a {\displaystyle t_{a}} and t b {\displaystyle t_{b}} , thus:
Since the clock remains at rest in its inertial frame, it follows x a = x b {\displaystyle x_{a}=x_{b}} , thus the interval Δ t ′ = t b ′ − t a ′ {\displaystyle \Delta t^{\prime }=t_{b}^{\prime }-t_{a}^{\prime }} is given by:
where Δ t is the time interval between two co-local events (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on their clock), known as the proper time , Δ t′ is the time interval between those same events, as measured by another observer, inertially moving with velocity v with respect to the former observer, v is the relative velocity between the observer and the moving clock, c is the speed of light, and the Lorentz factor (conventionally denoted by the Greek letter gamma or γ) is:
Thus the duration of the clock cycle of a moving clock is found to be increased: it is measured to be "running slow". The range of such variances in ordinary life, where v ≪ c , even considering space travel, are not great enough to produce easily detectable time dilation effects and such vanishingly small effects can be safely ignored for most purposes. As an approximate threshold, time dilation of 0.5% may become important when an object approaches speeds on the order of 30,000 km/s (1/10 the speed of light). [ 29 ]
In special relativity, time dilation is most simply described in circumstances where relative velocity is unchanging. Nevertheless, the Lorentz equations allow one to calculate proper time and movement in space for the simple case of a spaceship which is applied with a force per unit mass, relative to some reference object in uniform (i.e. constant velocity) motion, equal to g throughout the period of measurement.
Let t be the time in an inertial frame subsequently called the rest frame. Let x be a spatial coordinate, and let the direction of the constant acceleration as well as the spaceship's velocity (relative to the rest frame) be parallel to the x -axis. Assuming the spaceship's position at time t = 0 being x = 0 and the velocity being v 0 and defining the following abbreviation:
the following formulas hold: [ 30 ]
Position:
Velocity:
Proper time as function of coordinate time:
In the case where v (0) = v 0 = 0 and τ (0) = τ 0 = 0 the integral can be expressed as a logarithmic function or, equivalently, as an inverse hyperbolic function :
As functions of the proper time τ {\displaystyle \tau } of the ship, the following formulae hold: [ 31 ]
Position:
Velocity:
Coordinate time as function of proper time:
The clock hypothesis is the assumption that the rate at which a clock is affected by time dilation does not depend on its acceleration but only on its instantaneous velocity. This is equivalent to stating that a clock moving along a path P {\displaystyle P} measures the proper time , defined by:
The clock hypothesis was implicitly (but not explicitly) included in Einstein's original 1905 formulation of special relativity. Since then, it has become a standard assumption and is usually included in the axioms of special relativity, especially in light of experimental verification up to very high accelerations in particle accelerators . [ 32 ] [ 33 ]
Gravitational time dilation is experienced by an observer that, at a certain altitude within a gravitational potential well, finds that their local clocks measure less elapsed time than identical clocks situated at higher altitude (and which are therefore at higher gravitational potential).
Gravitational time dilation is at play e.g. for ISS astronauts. While the astronauts' relative velocity slows down their time, the reduced gravitational influence at their location speeds it up, although to a lesser degree. Also, a climber's time is theoretically passing slightly faster at the top of a mountain compared to people at sea level. It has also been calculated that due to time dilation, the core of the Earth is 2.5 years younger than the crust . [ 34 ] "A clock used to time a full rotation of the Earth will measure the day to be approximately an extra 10 ns/day longer for every km of altitude above the reference geoid." [ 35 ] Travel to regions of space where extreme gravitational time dilation is taking place, such as near (but not beyond the event horizon of) a black hole , could yield time-shifting results analogous to those of near-lightspeed space travel.
Contrarily to velocity time dilation, in which both observers measure the other as aging slower (a reciprocal effect), gravitational time dilation is not reciprocal. This means that with gravitational time dilation both observers agree that the clock nearer the center of the gravitational field is slower in rate, and they agree on the ratio of the difference.
High-accuracy timekeeping, low-Earth-orbit satellite tracking, and pulsar timing are applications that require the consideration of the combined effects of mass and motion in producing time dilation. Practical examples include the International Atomic Time standard and its relationship with the Barycentric Coordinate Time standard used for interplanetary objects.
Relativistic time dilation effects for the Solar System and the Earth can be modeled very precisely by the Schwarzschild solution to the Einstein field equations. In the Schwarzschild metric, the interval d t E {\displaystyle dt_{\text{E}}} is given by: [ 38 ] [ 39 ]
where:
The coordinate velocity of the clock is given by:
The coordinate time t c {\displaystyle t_{\text{c}}} is the time that would be read on a hypothetical "coordinate clock" situated infinitely far from all gravitational masses ( U = 0 {\displaystyle U=0} ), and stationary in the system of coordinates ( v = 0 {\displaystyle v=0} ). The exact relation between the rate of proper time and the rate of coordinate time for a clock with a radial component of velocity is:
where:
The above equation is exact under the assumptions of the Schwarzschild solution. It reduces to velocity time dilation equation in the presence of motion and absence of gravity, i.e. β e = 0 {\displaystyle \beta _{e}=0} . It reduces to gravitational time dilation equation in the absence of motion and presence of gravity, i.e. β = 0 = β ∥ {\displaystyle \beta =0=\beta _{\shortparallel }} .
Velocity and gravitational time dilation have been the subject of science fiction works in a variety of media. Some examples in film are the movies Interstellar and Planet of the Apes . [ 43 ] In Interstellar , a key plot point involves a planet, which is close to a rotating black hole and on the surface of which one hour is equivalent to seven years on Earth due to time dilation. [ 44 ] Physicist Kip Thorne collaborated in making the film and explained its scientific concepts in the book The Science of Interstellar . [ 45 ] [ 46 ]
The Queen song '39 was written by astrophysicist as well as musician Brian May , and is centred around the time dilation effect on spacefarers searching for a new home for mankind, as we gradually ruin planet Earth. They return successful, only to find that all and everything they knew has long since passed away.
Time dilation was used in the Doctor Who episodes " World Enough and Time " and " The Doctor Falls ", which take place on a spaceship in the vicinity of a black hole. Due to the immense gravitational pull of the black hole and the ship's length (400 miles), time moves faster at one end than the other. When The Doctor's companion, Bill, gets taken away to the other end of the ship, she waits years for him to rescue her; in his time, only minutes pass. [ 47 ] Furthermore, the dilation allows the Cybermen to evolve at a "faster" rate than previously seen in the show.
Tau Zero , a novel by Poul Anderson , is an early example of the concept in science fiction literature. In the novel, a spacecraft uses a Bussard ramjet to accelerate to high enough speeds that the crew spends five years on board, but thirty-three years pass on the Earth before they arrive at their destination. The velocity time dilation is explained by Anderson in terms of the tau factor which decreases closer and closer to zero as the ship approaches the speed of light—hence the title of the novel. [ 48 ] Due to an accident, the crew is unable to stop accelerating the spacecraft, causing such extreme time dilation that the crew experiences the Big Crunch at the end of the universe. [ 49 ] Other examples in literature, such as Rocannon's World , Hyperion and The Forever War , similarly make use of relativistic time dilation as a scientifically plausible literary device to have certain characters age slower than the rest of the universe. [ 50 ] [ 51 ] | https://en.wikipedia.org/wiki/Time_dilation |
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