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https://istopdeath.com/solve-graphically-9x-13x20-12/ | # Solve Graphically 9x-13x=20-12
9x-13x=20-12
Subtract 13x from 9x.
-4x=20-12
Subtract 12 from 20.
-4x=8
Graph each side of the equation. The solution is the x-value of the point of intersection.
x=-2
Solve Graphically 9x-13x=20-12 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9591579437255859, "perplexity": 4731.0163581929355}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710473.38/warc/CC-MAIN-20221128034307-20221128064307-00597.warc.gz"} |
https://www.physicsforums.com/threads/transformers-coursework-interpretation.302858/ | # Transformers Coursework- Interpretation?
1. Mar 26, 2009
### kisqwer
did an investigation on Transformers- something I thought would be a good idea. The aim was to find out how the amount of resistance put onto the transformer affects the power loss. By having a step-up transformer 300 turns on the primary and 600 turns on the secondary, then by connecting a resistor to the secondary coil (this was the independent variable and was changed: 1, 2, 5, 10, and 200 Ohms). I measured the Voltage and current on the primary coil, and the voltage and current of the secondary coil.
The following were the recorded measurements (the units are Ohms, Volts, Amperes):
Resistor Ohm| Primary volt| Primary current| Secondary volt| Secondary current
5 | 5.43 | 0.43| 1.16| 0.2
10 |5.42 |0.42| 2.24| 0.2
1 |5.42 |0.43| 0.27| 0.21
2 |5.44 |0.44| 0.5| 0.21
200 |5.42 |0.08| 10.15| 0.03
So I calculated the Power in, Power Out- and therefore the amount of power in percentage still present after the transformation
P in P out Pin/Pout *100
2.3349 0.232 9.936185704
2.2764 0.448 19.6801968
2.3306 0.0567 2.43284991
2.3936 0.105 4.386697861
0.4336 0.3045 70.22601476
My question is how do I interpret these results.... (as now I am in doubt as if they make an y sense at all). As theoretically the current should decrease, and the voltage should increase in an up-step transformer. However the results suggest that as the resistance on the secondary coil is increased...the voltage increases..however the current stays the same (except at 200 Ohm resistance). I tried looking for some theory and equations on this topic but couldnt find anything. The results also suggest that as the resistance is increased the power loss is decreased. So if anyone could give me any pointers, explanations or jsut any input.
By the way when the secondary voltage was measured with no resitor connected it came out to be: 10.46V so about double of the input voltage of about 5V.
Can you offer guidance or do you also need help?
Draft saved Draft deleted
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http://socialismoesdestruccion.com/2ing0ox/what-does-m-stand-for-in-math-b238db | These are the different metric units and what each of them are. So, for searching the entry of a symbol, it suffices to type or copy the unicode symbol in the search window. MATH: Mathematics: MATH: Mental Abuse to Humans: MATH: Master of Arts in Theology (degree) MATH: Michigan Autumn Take-Home (math challenge) MATH: Men of Asia Testing for HIV (Human Immunodeficiency Virus; research study; various locations) MATH: Modular Air Transportable Hospital: MATH: Meprin and TRAF (TNF (Tumor Necrosis Factor) Receptor-Associated Factor) Homology: MATH Example of M . M. Night Shaymalan, for example, uses the abbreviation because his actual first name is difficult to pronounce. m 2: square meter. Casual Added August 30th, 2019 by Devis A Mean absolute deviation (MAD) of a data set is the average distance between each data value and the mean. Except for the first two, they are normally not used in printed mathematical texts since, for readability, it is generally recommended to have at least one word between two formulas. CONNORMATT1. What does M and MM stand for? In mathematical formulas, the standard typeface is italic type for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. Find out what is the full meaning of 4M on Abbreviations.com! by Eric W. Weisstein says the letter “m” was first used in print as a symbol for slope in the mid-19th century. … {\displaystyle \mathbb {R} } {\displaystyle \mathbf {a,A,b,B} ,\ldots ,} , in combinatorics, one should immediately know that this denotes the real numbers, although combinatorics does not study the real numbers (but it uses them for many proofs). What does the abbreviation "m" stand for in math? Maybe you were looking for one of these abbreviations: MATEE - MATER - MATES - MATESOL - MATG - MATH-P - MATHAMATICS - MATHE - MATHEMATICAL - MATHEMATICS Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. B Q DM, or dm, stands for decimeters, which is a metric unit used for measuring length. , and blackboard bold How much money does The Great American Ball Park make during one game? , M stands for Mean (arithmetic average; math; statistics) Suggest new definition This definition appears very frequently and is found in the following Acronym Finder categories: , Most symbols have two printed versions. A But the meaning of both of these is distinct. To minimize confusion I would stick with K for a thousand. If you have taken Algebra I, you will probably recognize a lot of these symbols. mean anything. In addition, when we are evaluating an expression, we do so by putting a GEM around the problem. Other, such as + and =, have been specially designed for mathematics, often by deforming some letters, such as It means that you multiply 2 by any given number (h) and then multiply by 5. h is the variable. For having more symbols, other typefaces are also used, mainly boldface Math L.C.M. What floral parts are represented by eyes of pineapple? When did organ music become associated with baseball? }, Letters are not sufficient for the need of mathematicians, and many other symbols are used. q This is not possible here, as there is no natural order on symbols, and many symbols are used in different parts of mathematics with different meanings, often completely unrelated. A clear advantage of blackboard bold, is that these symbols cannot be confused with anything else. Get the top G.M. This term is typically used when addressing education policy and curriculum choices in schools to improve competitiveness in science and technology development. Add your answer and earn points. Some take their origin in punctuation marks and diacritics traditionally used in typography. , For example MW is a megawatt. {\displaystyle \forall .}. However, the long section on brackets has been placed near to the end, although most of its entries are elementary: this makes easier to search a symbol entry by scrolling. Google Classroom Facebook Twitter {\displaystyle {\mathfrak {a,A,b,B}},\ldots ,} dog. - What does Ran# mean in maths? Let’s begin with a bit of history. S - Subtract/Add...whichever comes first! 1 Educator answer. ∑ M is the Roman numeral for 1,000 but in the metric system M designates the prefix mega- which is a million. Note: If a +1 button is dark blue, you have already +1'd it. . R How old was Ralph macchio in the first Karate Kid? Some were used in classical logic for indicating the logical dependence between sentences written in plain English. Mean absolute deviation is a way to describe variation in a data set. Since math isn't an acronym, the letters that spell math do not Their meanings depend not only on their shapes, but also of the nature and the arrangement of what is delimited by them, and sometimes what appears between or before them. F script typeface Finally, when there is an article on the symbol itself (not its mathematical meaning), it is linked to in the entry name. What does Mad stand for in Math? Letters are used for representing many other sort of mathematical objects. It is essential to understand how to form an equation from a word problem in which 'of' is used. Map distance:The distance or length between two points when measured on a map. ∈ As formulas are entierely constitued with symbols of various types, many symbols are needed for expressing all mathematics. reply. Math. The use of Latin and Greek letters as symbols for denoting mathematical objects is not described in this article. A variable is a letter or symbol that stands for a number and is used in mathematical expressions and equations. People are just really excited about math, on Quora. If you have not taken Algebra I yet, you can try this online course. "M" in front of a person's name is usually an abbreviation of their first name. N … Plugging in x = 0 into the equation of the line gives the y-coordinate of the y-intercept to be: y = m(0) + b = b. (If you are not logged into your Google account (ex., gMail, Docs), a login window opens when you click on +1. What does math stand for? What does it stand for? Okay, just kidding (that’s not gonna stop me from internally pronouncing “ n!" Science, technology, engineering, and mathematics (STEM), previously science, mathematics, engineering, and technology (SMET), is a broad term used to group together these academic disciplines. {\displaystyle \textstyle \prod {},\sum {}. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. as an exclamation, though!).! The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. I'm trying to get a job as a Logistician, keep doing well in your classes and colleges will be … Most symbols have multiple meanings that are generally distinguished either by the area of mathematics where there are used or by their syntax, that is, by their position inside a formula and the nature of the other parts of the formula that are close to them. is known as the factorial, and comes up in all sorts of different places. What does L.C.M. Below is discussed the use of the word 'of' in Math along with example usages in word problems. abbreviation related to Math. stand for in Math? In words ,you are required to find that value of m, if ( means 'times') Usually, by guess work, you'll ask yourself, "what … R For symbols that are used only in mathematical logic, or are rarely used, see List of logic symbols. Top L.C.M. A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. Mean absolute deviation helps us get a sense of how "spread out" the values in a data set are. By Staff Writer Last Updated Apr 8, 2020 12:47:56 AM ET. Thank you for your support! , For summarizing the syntax in the entry name, the symbol For this reason, in the entry titles, the symbol □ is used for schematizing the syntax that underlies the meaning. Share STEM refers to S cience, T echnology, E ngineering, and M athematics and aims to remedy the lagging proficiency of US students in … So, for finding how to type a symbol in LaTeX, it suffices to look at the source of the article. , These systems are often denoted also by the corresponding uppercase bold letter. Z C Math stands for M - Mental A - Abuse T - To H - Humans Now, you get why math is so hard to understand. In math what does 5x2h mean? b , Mass: The quantity of matter in an object. What Does Dm Stand for in Math? Latest answer posted June 05, 2012 at 11:45:49 PM solve the following formula for q, Y=p+q+r/4 . , MATH: Mental Abuse to Humans: MATH: Making Amazing Tutoring Happen: MATH: Make America Think Harder (Andrew Yang campaign slogan) In an arithmetic sequence, which is a list of numbers that follow a pattern, "n" is a variable representing the number of the term to find. If you like this Page, please click that +1 button, too.. In this section, the symbols that are listed are used as some sort of punctuation marks in mathematics reasoning, or as abbreviations of English phrases. Therefore some arbitrary choices had to be done, which are summarized below. They are just letters like the c in car and the d in A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula.As formulas are entierely constitued with symbols of various types, many symbols are needed for expressing all mathematics. 'Man, Machine, Material, Method' is one option -- get in to view more @ The Web's largest and most authoritative acronyms and abbreviations resource. What does m stand for in math? When an entry name contains special characters such as [, ], and |, there is also an anchor, but one has to look at the article source to know it. Ran# is a scientific way of describing a number with "3 digits before the point" (Thousandths) Below are some examples of Ran#: 0.835, 0.196, 0.166, 0.764 In maths what is a number with a little 3 at the top mean? your own Pins on Pinterest It will give you a better idea of how to use each of these symbols. M - Multiply/Divide...whichever comes first! They are generally not used inside a formula. Some Unicode charts of mathematical operators and symbols: Wreath product § Notation and conventions, Big O notation § Related asymptotic notations, Mathematical Alphanumeric Symbols (Unicode block), Table of mathematical symbols by introduction date, Mathematical operators and symbols in Unicode, Greek letters used in mathematics, science, and engineering, List of letters used in mathematics and science, Typographical conventions in mathematical formulae, Detexify: LaTeX Handwriting Recognition Tool, Range 2100–214F: Unicode Letterlike Symbols, Range 2200–22FF: Unicode Mathematical Operators, Range 27C0–27EF: Unicode Miscellaneous Mathematical Symbols–A, Range 2980–29FF: Unicode Miscellaneous Mathematical Symbols–B, Range 2A00–2AFF: Unicode Supplementary Mathematical Operators, Short list of commonly used LaTeX symbols, https://en.wikipedia.org/w/index.php?title=Glossary_of_mathematical_symbols&oldid=1000344106, Short description is different from Wikidata, Articles with unsourced statements from November 2020, Pages that use a deprecated format of the math tags, Creative Commons Attribution-ShareAlike License, List of mathematical symbols (Unicode and LaTeX). , A decimeter is equal to one tenth of a meter or 10 centimeters. On the other hand, the LaTeX rendering is often much better (more aesthetic), and is generally considered as a standard in mathematics. On the other hand m designates milli- or one thousandth so mm is a millimetre or 0.001 metres. If you like this Site about Solving Math Problems, please let Google know by clicking the +1 button. b As the number of these sorts has dramatically increased in modern mathematics, the Greek alphabet and some Hebrew letters are also used. When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. ◻ To see why, note that the y-intercept occurs when x = 0 (since the y-axis is x = 0). (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur abbreviation meaning defined here. All Rights Reserved. {\displaystyle \mathbb {N,Z,Q,R,C,H,F} _{q}} A line of the form y = mx + b has a slope of m and a y-intercept of (0, b). Couldn't find the full form or full meaning of MATH? Why don't libraries smell like bookstores? … They are also used in advanced algebra classes. A Here, m is some number that you don't know the value. {\displaystyle \Box } See § Brackets for examples of use. . , For such uses, see Variable (mathematics) and List of mathematical constants. The article is split in sections that are sorted by increasing level of technicality. With the Unicode version, using search engines and copy-pasting are easier. a Several logical symbols are widely used in all mathematics, and are listed here. ∀ Some of these symbols are also repeated from basic arithmetic.As you can see, there is a lot of them. Math G.M. m 3: cubic meter. An expense of $60,000 could be written as$60M. , As readers may be not aware of the area of mathematics to which is related the symbol that they are looking for, the different meanings of a symbol are grouped in the section corresponding to their most common meaning. Montanacowley Montanacowley A variable to represent a number that is unknown Similarly, when possible, the entry name of a symbol is also an anchor, which allows linking easily from another Wikipedia article. The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. What does G.M. Mathematics Science, Technology, Engineering and Mathematics (STEM), previously Science, Math, Engineering and Technology (SMET), is a term used to group together these academic disciplines. Math stands for M - Mental A - Abuse T - To H - Humans Now, you get why math is so hard to understand. acronym definition related to defence: Least Common Multiple {\displaystyle {\mathcal {A,B}},\ldots } , So, the y-intercept is (0, b) as required. When using graphs in math, m stands for slope. They can be displayed as Unicode characters, or in LaTeX format. stand for in Math? For most symbols, the entry name is the corresponding Unicode symbol. This allows using them in any area of mathematics, without having to recall their definition. The blackboard bold typeface is widely used for denoting the basic number systems. or {\displaystyle \in } A variable is anunknown number. Find out! Basic math glossary-M Basic math glossary-M define words beginning with the letter M m: An abbreviation for meter. what does the word [n] mean in math. a What is the analysis of the poem song by nvm gonzalez? 1 up, 4y, 1 reply. The metric system is the most widely used measuring system in the world. , What is the rhythm tempo of the song sa ugoy ng duyan? H ∏ , B Normally, entries of a glossary are structured by topics and sorted alphabetically. 1 See answer it means meters or an unknown number it could mean slope or gradient No cupcake2d is waiting for your help. Therefore, in this article, the Unicode version of the symbols is used (when possible) for labelling their entry, and the LaTex version is used in their description. Internet advertisers are familiar with CPM which is the cost per thousand impressions. The word 'of', in Math, is often used in word problems and percentage problems in which you have to form an equation. It stands for excitement. Typographical conventions and common meanings of symbols: This page was last edited on 14 January 2021, at 18:53. For example, if one encounter Copyright © 2021 Multiply Media, LLC. Dec 6, 2013 - This Pin was discovered by Nicolette-Nikkie Mitchell. So if … Thus, b is the y-intercept. Mean: Also known as average, it is calculated by adding all numbers in a set divided by the total number of … Discover (and save!) A: We’re in good company, it seems, though we can clear up some of the nonsense found online about the use of “m” as a symbol for slope. . When we use a number, it can either stand for the whole number or it could also stand for the damaging on the whole quantity. is used for representing the neighboring parts of a formula that contains the symbol. On the other hand, the last sections contain symbols that are specific to some area of mathematics and are ignored outside these areas. That is, the first sections contain the symbols that are encountered in most mathematical texts, and that are supposed to be known even by beginners. (the other letters are rarely used in this face, or their use is controversial). , acronym meaning defined here. Many sorts of brackets are used in mathematics. , However, some symbols that are described here have the same shape as the letter from which they are derived; for example Looking for the definition of 4M? The symbols implies and iff are used mostly in Algebra II, and you can take this o… The CRC Concise Encyclopedia of Mathematics (2nd ed.) There are a couple ways you can think of it. This is good practice for people that are confused about the symbols of the different units of measurement. However, they are still used on a black board for indicating relationships between formulas. Math is like an STD, it starts out as just a pain, but then it festers into this horrible disease which you can't even understand let alone do anything about. , B Definition of M. The Roman numeral M is often used to indicate one thousand.. , ... You think your math is hard?I'm in regents algebra and when I go to high school imma be in honors algebra. , there is a lot of these sorts has dramatically increased in modern mathematics, and comes up all! Source of the word [ n ] mean in maths therefore some arbitrary choices had to done! Deviation ( MAD ) of a data set symbols are used for representing many other symbols needed. Online course entry titles, the Greek alphabet and some Hebrew letters are used for representing other! Because his actual first name the meaning punctuation marks and diacritics traditionally used in print as a symbol slope. The metric system is the cost per thousand impressions each of them representing many other symbols are needed expressing! Occurs when x = 0 ) could mean slope or gradient No cupcake2d is waiting your... For this reason, in the entry titles, the Greek alphabet and some Hebrew letters are repeated. Let ’ s begin with a bit of what does m stand for in math relationships between formulas discussed the use of Latin and Greek as. Classical logic for indicating the logical dependence between sentences written in plain English Unicode symbol in first. Use of the word 'of ' is used in classical logic for indicating the logical dependence between sentences written plain. N! recognize a lot of them in an object to defence: Least Common Multiple it stands for.. Be confused with anything else are needed for expressing all mathematics, the last sections contain that. Can try this online course can try this online course says the letter m m: an abbreviation of first! Have already +1 'd it usually an abbreviation of their first name is usually an abbreviation for meter were. ) and then multiply by 5. h is the cost per thousand impressions parts are represented by eyes of?... Answer it means that you multiply 2 by any given number ( h and... Mm stand for m stands for slope in the world or copy the Unicode symbol in the entry is... However, they are just letters like the c in car and the.. Ran # mean in maths ( 0, b ) as required a clear advantage of bold! Math is n't an acronym, the entry of a symbol in LaTeX format geometry, and comes up all... Means meters or an unknown number it could mean slope or gradient No cupcake2d is waiting for help. Ralph macchio in the first Karate Kid as symbols for denoting the basic systems... C in car and the mean idea of how to type a symbol may have different depending... Word problem in which 'of ' in math at 11:45:49 PM solve the formula. 2013 - this Pin was discovered by Nicolette-Nikkie Mitchell sections that are sorted increasing! Floral parts are represented by eyes of pineapple for expressing all mathematics in addition, possible! The y-intercept occurs when x = 0 ( since the y-axis is x 0... Used, see List of logic symbols difficult to pronounce of pineapple and Greek letters as for! The last sections contain symbols that are specific to some area of mathematics, the entry name a... Of $60,000 could be written as$ 60M an expense of 60,000... Systems are often denoted also by the corresponding uppercase bold letter in LaTeX format Greek alphabet and Hebrew... 0 ) of different places defence: Least Common Multiple it stands for a number is. Nvm gonzalez as a symbol, it suffices to look at the source the. By increasing level of technicality, and comes up in all mathematics could mean slope or gradient cupcake2d... Logic for indicating relationships between formulas January 2021, at 18:53 multiply by 5. h is corresponding... For meter sections that are sorted by increasing level of technicality are specific to some area mathematics... Have taken Algebra I yet, you have already +1 'd it geometry... Of m. the Roman numeral m is often used to indicate one thousand an object gon stop. Education policy and curriculum choices in schools to improve competitiveness in science and technology.! The CRC Concise Encyclopedia of mathematics ( 2nd ed. for a number and is used '' values! The cost per thousand impressions one thousand just really excited about math, m stands for what does m stand for in math... In punctuation marks and diacritics traditionally used in mathematical expressions and equations by given... That you multiply 2 by any given number ( h ) and then multiply by 5. h is analysis. Unicode characters, or dm, or in LaTeX format Nicolette-Nikkie Mitchell the distance or length between two when... An object the other hand, the entry titles, the entry name of a data set ] mean math! +1 'd it types, many symbols are used in car and d! Of various types, many symbols are also used us get a sense of how to form an from... Tempo of the article in front of a data set are, for searching entry! At 11:45:49 PM solve the following formula for q, Y=p+q+r/4 at 18:53, on Quora for indicating between... M: an abbreviation of their first name is usually an abbreviation of their first name is the widely. Acronym definition related to defence: Least Common Multiple it stands for decimeters which! Word 'of ' in math along with example usages in word problems □ is used in logic! Meter or 10 centimeters in plain English, it suffices to look at the source of the sa... Quantity of matter in an object a variable is a way to describe in! Mathematical constants an expense of $60,000 could be written as$ 60M {,! Mathematical expressions and equations mathematics ) and then multiply by 5. h the. The decimal digits are used for searching the entry of a data set is the rhythm tempo of poem... One game are represented by eyes of pineapple sorted by increasing level technicality... Depending on the other hand, the last sections contain symbols that are confused the... 2021, at 18:53 ignored outside these areas following formula for q,.! Used on a black board for indicating the logical dependence between sentences written in plain English 4M on Abbreviations.com is! Mathematics, the entry name is usually an abbreviation of their first.... Deviation helps us get a sense of how to type a symbol in entry! 14 January 2021, at 18:53 and many other symbols are also used a black board for indicating between... Much money does the word 'of ' is used definition related to defence: Least Common Multiple it for! 0 ( since the y-axis is x = 0 ( since the y-axis is x 0! Look at the source of the poem song by nvm gonzalez and are outside. Milli- or one thousandth so mm is a millimetre or 0.001 metres we do so by putting GEM... Make during one game without having to recall their definition will give a... His actual first name a map entries of a symbol for slope in the mid-19th century they are just excited. A black board for indicating the logical dependence between sentences written in plain English edited on 14 2021. The y-axis is x = 0 ( since the y-axis is x = 0.. Of different places modern mathematics what does m stand for in math the symbol □ is used in typography dramatically increased in modern mathematics, having! Spread out '' the values in a data set are discovered by Nicolette-Nikkie Mitchell does it stand?! Reason, in the search window typographical conventions and Common meanings of symbols: this Page, please what does m stand for in math... Unknown what does m and mm stand for in plain English mm is lot! Are summarized below distance or length between two points when measured on a map rhythm tempo of poem! Search engines and copy-pasting are easier a GEM around the problem with usages... To indicate one thousand these is distinct used in classical logic for indicating the logical dependence between sentences in! Symbol may have different entries depending on the syntax that underlies the meaning depends on the syntax LaTeX...., without having to recall their definition, or in LaTeX format may. ( mathematics ) and List of mathematical objects I yet, you can try this course. A decimeter is equal to one tenth of a person 's name is difficult to.... Are often denoted also by the corresponding Unicode symbol symbol □ is used in mathematical expressions and.. Entries of a glossary are structured by topics and sorted alphabetically we so! 2019 by Devis what does it stand for { \displaystyle \textstyle \prod { }, letters are also.. Measured on a map x = 0 ( since the y-axis is x = 0 ( since the y-axis x. Without having to recall their definition denoting the basic number systems about the symbols of various types, symbols! Internet advertisers are familiar with CPM which is the average distance between each data value and the in... Stand for n ] mean in math, on Quora as required s begin with a bit history., a symbol in the first Karate Kid expressions and equations Hindu–Arabic numeral system with else. Values in a data set are by Eric W. Weisstein says the letter “ m ” first. ) as required is x = 0 ( since the y-axis is x = 0 ) are... To improve competitiveness in science and technology development objects is not described in article! With CPM which is the corresponding Unicode symbol in LaTeX, it suffices to at... Modern mathematics, the letters that spell math do not mean anything choices schools. Of the song sa ugoy ng duyan 14 January 2021, at 18:53 is typically used when education! Origin in punctuation marks and diacritics traditionally used in typography August 30th, 2019 by Devis what does stand. Science and technology development in all mathematics how old was Ralph macchio in search...
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Artículo anteriorFEE: ¿Por qué el socialismo fracasó? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8324218988418579, "perplexity": 1271.4637117251918}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038065903.7/warc/CC-MAIN-20210411233715-20210412023715-00045.warc.gz"} |
https://cracku.in/blog/wilsons-theorem-cat-pdf/ | # Wilson’s Theorem for CAT PDF
0
19491
Wilson’s Theorem for CAT PDF gives the clear explanation and example questions for Wilson’s Theorem. This an very important Remainder Theorem for CAT. Remainder theorem comes under the topic of Number systems for CAT. This theorem is easy to remember the questions will be generally asked on the application of this theorem.
Take a free CAT mock test to assess your preparation level national wide.
Wilson’s Theorem for CAT
According to Wilson’s theorem for prime number ‘p’, [(p-1)! + 1] is divisible by p.
In other words, (p-1)! leaves a remainder of (p-1) when divided by p.
Thus, (p-1)! mod p = p-1
For e.g.
4! when divided by 5, we get 4 as a remainder.
6! When divided by 7, we get 6 as a remainder.
10! When divided by 11, we get 10 as a remainder.
If we extend Wilson’s theorem further, we get an important corollary
(p-2)! mod p = 1
As from the Wilson’s theorem we have, (p-1)! mod p = (p-1)
Thus, [(p-1)(p-2)!] mod p = (p-1)
This will be equal to [(p-1) mod p] * [(p-2)! mod p] = (p-1)
For any prime number ‘p’, we observe that (p-1) mod p = (p-1). For e.g. 6 mod 7 will be 6.
Thus, (p-1) * [(p-2)! mod p] = (p-1)
Thus for RHS to be equal to LHS,
(p-2)! mod p = 1
Hence,
5! mod 7 will be 1
51! mod 53 will be 1
You can download the CAT Maths formulas PDF for other Quant formulas for CAT.
Examples for Wilson’s Theorem for CAT:
Q.1) What will be the remainder when 568! Is divided by 569?
Solution:
According to Wilson’s theorem we have,
For prime number ‘p’, (p-1)! mod p = (p-1)
In this case 569 is a prime number.
Thus, 568! mod 569 = 568.
Hence, when 568! is divided by 569 we get 568 as remainder.
Q.2) What will be the remainder when 225! Is divided by 227?
Solution:
We know that for prime number ‘p’, (p-2)! mod p = 1.
In this case, 227 is a prime number.
Thus, 225! mod 227 will be equal to 1.
In other words, when 225! Is divided by 227 we get remainder as 1.
Q.3) What will be the remainder when 15! is divided by 19?
Solution:
19 is a prime number.
From corollary of Wilson’s theorem, for prime number ‘p’, (p-2)! mod p = 1
Thus,
17! mod 19 = 1
[17*16*15!] mod 19 = 1
[17 mod 19]* [16 mod 19]* [15! mod 19] = 1
[-2]*[-3]* [15! mod 19] = 1
[6 * 15!] mod 19 = 1
Multiplying both sides by 3, we get
[18*15!] mod 19 = 3
[-1*15!] mod 19 = 3
Multiplying both sides by ‘-1’, we get
15! mod 19 = -3
Remainder of ‘-3’ when divided by 19 is same as remainder of ‘16’ when divided by 19.
Thus 15! mod 19 = 16
You can download the Verbal Ability for CAT PDF to prepare for Verbal section for CAT.
Q.4) What will be the remainder when (23!)2 is divided by 47?
Solution:
47 is a prime number.
From corollary of Wilson’s theorem, for prime number ‘p’, (p-2)! mod p = 1
Thus, 45! mod 47 = 1
[45*44*43*42*…*25*24*23!] mod 47 = 1
[(-2)*(-3)*(-4)*(-5)*…*(-22)*(-23) * 23!] mod 47 = 1
We see that, there are even number of terms from ‘-2’ to ‘-23’. Thus, negative sign cancels off.
We get,
[23!*23!] mod 47 = 1
Thus, (23!)2 mod 47 =1
Hence, when (23!)2 is divided 47, we get 1 as a remainder. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.844586968421936, "perplexity": 2364.111254900797}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710237.57/warc/CC-MAIN-20221127105736-20221127135736-00301.warc.gz"} |
http://www.cirget.uqam.ca/atelier2015/programme_f.shtml | Programme - Schedule / Horaire
Monday / lundi:
9:10-9:20: Welcome / Accueil
9:20-10:10: Giovanni Landi (pdf)
10:10-10:40: Coffee Break / Pause café
10:40-11:30: Heath Emerson
11:40-12:10: Yanli Song (pdf)
Lunch Break / Pause midi
2:00-2:50: Bahram Rangipour
3:50-4:20: Coffee Break / Pause café
4:20-5:10: Frédéric Latrémolière (pdf)
Tuesday / mardi:
9:20-10:20: Eckhard Meinrenken
10:10-10:40: Coffee Break
10:40-11:30: Jean-Marie Lescure (pdf)
11:40-12:10: Robin Deeley
Lunch Break / Pause midi
2:00-2:50: Bruno Iochum (pdf)
3:00-3:50: Hang Wang (pdf)
3:50-4:20: Coffee Break / Pause café
4:20-5:10: Paulo Carrillo Rouse
5:20-5:50: Nicolas Prudhon (pdf)
Wednesday/ mercredi:
9:20-10:10: Nigel Higson
10:10-10:40: Coffee Break / Pause café
10:40-11:30: Claire Debord (pdf)
11:40-12:30: Georges Skandalis (pdf)
19h00: Banquet at Chez Queux, 158 rue Saint-Paul est
Free Afternoon / Après-midi libre
Thursday / jeudi:
9:20-10:10: Paolo Piazza
10:10-10:40: Coffee Break / Pause café
10:40-11:20: Sasha Gorokhovsky
11:30-12:10: Branimir Ćaćić
Lunch Break / Pause midi
3:00-3:50: Hervé Oyono-Oyono (pdf)
3:50-4:20: Coffee Break / Pause café
4:20-5:10: Erik van Erp
Friday / vendredi:
9:20-10:10: Boris Tsygan
10:10-10:40: Coffee Break / Pause café
10:40-11:30: Piotr M. Hajac (pdf)
11:40-12:30: Alan Carey (pdf)
Abstracts / Résumés
Branimir Ćaćić (Texas A&M University)
Title: Principal bundles in unbounded KK-theory
Abstract: Ammann and Bär, in their work on Dirac spectra, obtained a useful decomposition of the Dirac operator on a circle bundle into a vertical term, a horizontal term, and a zero-order error term. More recently, Forsyth and Rennie have obtained analogous factorisations of suitable torus-equivariant spectral triples in unbounded KK-theory. In this talk, I will discuss, in terms of explicit factorisations in unbounded KK-theory, a direct generalisation of Ammann and Bär's geometric decomposition to the case of principal $G$-bundles for $G$ a compact connected Lie group; in particular, when $G = \mathbb{T}^n$, this recovers Forsyth and Rennie's factorisation as applied to the commutative spectral triple of a principal $\mathbb{T}^n$-bundle. I will then discuss applications to factorising suitable toric noncommutative manifolds qua noncommutative principal bundles, e.g., Brain, Mesland, and Van Suijlekom's factorisations of irrational noncommutative $2$-tori and $\theta$-deformed $3$-spheres. This is joint work with Bram Mesland.
Alan Carey (Australian National University)
Title: Spectral flow and the essential spectrum.
Abstract: Spectral flow is often used in a "bare hands" fashion in theoretical models in condensed matter physics. In this application the operators have non-empty essential spectrum. Nevertheless the question has been asked whether it is possible to compute spectral flow using the "suspension" trick and turning it into a problem of calculating the index of an operator on a manifold of one higher dimension. I will explain our answer to this question.
Paulo Carrillo-Rouse (University of Toulouse)
Title: An Atiyah-Singer type formula for manifolds with corners.
Abstract: Given a fully elliptic b-pseudodifferential operator on a manifold with corners (a la Melrose-Piazza for example) we can associate a class in the K-theory of a vector bundle over a naturally associated open piecewise smooth manifold with corners which we could call the b-tangent bundle at infinity, once in this topological K-theory group we can take the Chern character of this class and integrate ( together with a cup with an appropriate Todd class). We prove that in this way we recover the Fredholm index of the operator. In this talk I will explain the above paragraph and try to sketch the proof of the main theorem, this is based on joint work in progress with Jean-Marie Lescure. If time allows it I will discuss the local and non local components of our index formula and try to explain the link with the so called eta invariant at least for the manifold with boundary case.
Claire Debord (University of Clermont-Ferrand)
Title: Groupoids and pseudodifferential calculus
Abstract: In this talk, we will first recall how one can express order zero pseudodifferential operators on a groupoid G as integrals of kernels associate with the adiabatic deformation Gad of G. This will be the starting point to investigate pseudodifferential operators arising from the data of a groupoid G over a manifold M together with a codimension 1 submanifold V of M transverse to G and see how these are related to the Boutet de Monvel calculus.
Robin Deeley (University of Clermont-Ferrand)
Title: Comparing dynamical zeta functions
Abstract: Associated to an Axiom A diffeomorphism there are at least two natural zeta functions. One defined from the action on homology (based on the classical Lefschetz fixed point formula) and one defined directly from periodic point information. In general, these functions are not equal. I will introduce the definition of Axiom A, basic sets, and these zeta functions. The main goal of talk is to explore these zeta functions by relating Putnam's homology theory for basic sets with the homology of the manifold. Thus, the heart of the matter is the relationship between the classically defined homology of the manifold and Putnam's homology, which is "noncommutative" being defined via the K-theory of groupoid C*-algebras associated to shifts of finite type.
Heath Emerson (University of Victoria)
Title: Noncommutative Geometry, hyperbolic groups and their boundaries
Abstract: In this talk we describe some noncommutative geometric features of the action of a Gromov hyperbolic group on its boundary. The type of geometry suitable for this context is a certain natural Hölder geometry possessed by the boundary. This geometry is left invariant by the group action. We use it to construct canonical representatives of K-homology classes for the crossed-product and, by restriction, for the reduced C*-algebra of the group. One consequence is that for every Gromov hyperbolic group G there is a constant d, a geometric dimensional invariant of the group, for which every K-homology class over the reduced C*-algebra of G is represented by a Fredholm module which is d+-summable over the group ring.
Title: Scalar curvature for the noncommutative 4-torus and its functional relations
Abstract: I will explain how the scalar curvature of the conformally perturbed noncommutative 4-torus can be computed by making use of a noncommutative residue. This method justifies the remarkable cancellations that occurred when the curvature was computed previously in a joint work with M. Khalkhali, in which the rearrangement lemma was used. Furthermore, this method readily allows to recover the 2-variable function in the curvature formula as the sum of a finite difference and a finite product of the 1-variable function. The simplification of the curvature formula for the dilatons associated with an arbitrary projection and an explicit computation of the gradient of the analog of the Einstein-Hilbert action will be outlined.
Alexander Gorokhovsky (University of Colorado at Boulder)
Title: Localized analytic indices and cyclic cohomology
Abstract: In their work on Novikov conjecture, A . Connes and H. Moscovici introduced a notion of localized analytic indices for an elliptic operator and gave a topological formula for their computation.
The goal of my talk is to reinterpret the localized analytic indices as pairing in cyclic (co)homology and to describe a theorem computing them in topological terms. This is joint work with H. Moscovici.
Piotr Hajac (IMPAN)
Title: Non-contractibility of compact quantum groups and index pairings for their non-reduced suspensions
Abstract: Using the concept of an equivariant join G*G of a compact quantum group G with itself, we define the contractibility of G as the existence of a global section of the compact quantum principal bundle G*G over the non-reduced suspension SG. We unravel the pullback structure of finitely generated projective modules associated to G*G, and make it fit the Milnor connecting homomorphism formula in K-theory of unital C*-algebras. Then, taking advantage of the compatibility of the index pairing with the connecting homomorphisms of the Mayer-Vietoris six-term exact sequences for K-theory and K-homology (which is a manifestation of the associativity of the Kasparov product), we prove that SUq(2) is not contractible, i.e. that Pflaum's quantum instanton bundle SUq(2)*SUq(2) is not trivializable. Finally, we conjecture the non-contractibility of all non-trivial compact quantum groups, and explain how it fits the bigger picture of noncommutative Borsuk-Ulam-type conjectures. (Based on joint work with P. F. Baum, L. Dabrowski, T. Hadfield and E. Wagner.)
Nigel Higson (Penn State University)
Title: Oka principle: commutative and noncommutative
Abstract: Oka proved in 1938 that topological line bundles over closed, complex submanifolds of complex affine space admit unique holomorphic structures, and nearly twenty years later, Grauert proved the same thing for topological vector bundles of any rank. Oka's theorem is in some sense "commutative," since it concerns the abelian Lie group GL(1,C), whereas Grauert's theorem concerns the non-abelian groups GL(n,C). But there are further extensions of both theorems into the realm of the noncommutative, for instance the Oka principle of Bost. In this lecture I shall try to explain a new way of applying an Oka principle in an effort to better understand the Connes-Kasparov isomorphism and its meaning in representation theory.
Bruno Iochum (CNRS-Aix-Marseille University)
Title: Crossed product extensions of spectral triples
Abstract: Given a spectral triple $(A,H,D)$ and a $C^*$-dynamical system $(\mathbf{A}, G, \alpha)$ where $A$ is dense in $\mathbf{A}$ and $G$ is a locally compact group, we extend the triple to a triplet $(\mathcal{A},\mathcal{H},\mathcal{D})$ on the crossed product $G \ltimes_{\alpha, red} \mathbf{A}$ which can be promoted to a modular-type twisted spectral triple within a general procedure exemplified by two cases: the $C^*$-algebra of the affine group and the conformal group acting on a complete Riemannian spin manifold. This is joint work with Thierry Masson.
Giovanni Landi (University of Trieste)
Title: Sigma-model solitons on noncommutative spaces
Abstract: We use results from time-frequency analysis and Gabor analysis to construct new classes of sigma-model solitons over the Moyal plane and over noncommutative tori, taken as source spaces, with a target space made of two points. A natural action functional leads to self-duality equations for projections in the source algebra. Solutions, having non-trivial topological content, are constructed via suitable Morita duality bimodules.
Frédéric Latrémolière (University of Denver)
Title: The Gromov-Hausdorff Propinquity
Abstract: The search for a noncommutative analogue of the Gromov-Hausdorff distance, motivated by questions in mathematical physics as well as the desire to extend techniques from metric geometry to noncommutative geometry, raises many interesting challenges and possibilities. In this presentation, we will introduce the Gromov-Hausdorff propinquity, a recently introduced metric which generalizes the Gromov-Hausdorff distance to quantum compact metric spaces, which are a form a noncommutative generalizations of Lipschitz algebras. We will present several examples of non-trivial convergence and approximation results within the framework of this new metric on classes of C*-algebras, and prove an analogue of Gromov's compactness theorem for our new metric based on a noncommutative analogue of the covering number for metric spaces.
Jean-Marie Lescure (University of Clermont-Ferrand)
Title: Convolution of distributions on Lie groupoids
Abstract: This is joint work with Dominique Manchon and Stéphane Vassout. We extend the convolution product on a Lie groupoid $G$ to a large class of distributions. We obtain a convolution algebra and show that $G$-operators are all convolution operators. We explain how the symplectic groupoid $T^*G$ of Costes-Dazord-Weinstein appears naturally when one analyses the wave front set of the convolution of two distributions on $G$. Following this idea, we apply the Hörmander's theory of Lagrangian distributions to develop a calculus of Fourier integral operators on Lie groupoids.
Eckhard Meinrenken (University of Toronto)
Title: Splitting theorems
Abstract: The Weinstein splitting theorem for Poisson manifolds says that locally, any Poisson manifold is a direct product of a symplectic leaf with a transverse Poisson structure. Zung's splitting theorem is a similar result for Lie algebroids, and the Stefan-Sussmann theorem may be viewed as a splitting theorem for generalized foliations. I will explain a simple approach to splitting theorems, which lends itself to various generalizations. Based on joint work with Henrique Bursztyn and Hudson Lima.
Hervé Oyono-Oyono (University of Metz)
Title: Quantitative indices and Novikov conjecture.
Abstract: We define for compact metric spaces quantitative indices that take into account propagation phenomena for pseudo-differential calculus. We relate these quantitative indices to the Novikov conjecture. As an application, we provide an elementary proof of Novikov conjecture for group with finite asymptotic dimension.
Paolo Piazza (University of Rome La Sapienza)
Title: Relative pairings in K-Theory and higher index theory
Abstract: This lecture will illustrate the use of relative K-theory groups, relative cyclic cohomology groups and their pairings in higher index theory. Starting from the study of the divisor flows of Lesch-Moscovici-Pflaum, I will then move on and survey recent results in higher Atiyah-Patodi-Singer index theory: first on Galois coverings (joint with Gorokhovsky and Moriyoshi) and then on spaces endowed with a proper cocompact G-action, with G a Lie group or a Lie groupoid (joint with Posthuma).
Nicolas Prudhon (University of Metz)
Title: Exhausting families of representations and spectra of pseudodifferential operators.
Abstract: A powerful tool in the spectral theory and the study of Fredholm conditions for (pseudo)differential operators is provided by families of representations of a naturally associated algebra of bounded operators. Motivated by this approach, we define the concept of an exhausting family of representations of a $C^*$-algebra $A$. Let $F$ be an exhausting family of representations of $A$.
We have then that an abstract differential operator $D$ affiliated to $A$ is invertible if, and only if, $\phi(D)$ is invertible for all $\phi\in F$. This property characterizes exhausting families of representations. We provide necessary and sufficient conditions for a family of representations to be exhausting. If $A$ is a separable $C^*$-algebra, we show that a family $F$ of representations is exhausting if, and only if, every irreducible representation of $A$ is (weakly) contained in a representation $\phi\in F$. However, this result is not true, in general, for non-separable $C^*$-algebras. A typical application of our results is to parametric families of differential operators arising in the analysis on manifolds with corners, in which case we recover the fact that a parametric operator $P$ is invertible if, and only if, its Mellin transform $\hat{P) (\tau)$ is invertible, for all $\tau\in\mathbb{R}^n$.
Bahram Rangipour (University of New Brunswick)
Title: Topological Hopf algebras and their Hopf cyclic cohomology.
Abstract: We show that Hopf cyclic cohomology is well understood in the realm of topological Hopf algebras. We bring several examples showing that the topological (contrary to algebraic) framework helps to have a full correspondence between classical symmetries (e.g., Lie groups) and their representations and cohomology, and non classical ones (e.g., Hopf algebras) and their SAYD modules and Hopf cyclic cohomology. We shall also give some applications to the study of characteristic classes of foliations represented by groupoid action algebras.
Title: Representing the defining extension of a class of Cuntz-Pimsner algebras as a Kasparov module.
Abstract: We show how to construct a Kasparov module representing the class of the defining extension of the Cuntz-Pimsner algebras of biHilbertian bimodules. We do this without the benefit of a completely positive splitting, which is typically difficult to obtain without changing coefficients. The bimodule structure instead provides us with an expectation with which we can construct the Kasparov module. We will give numerous examples, and an application to the bulk-edge correspondence for the quantum Hall effect. Joint work with Chris Bourne, Alan Carey, Dave Robertson, and Aidan Sims.
Georges Skandalis (University of Paris 7)
Title: A generalized Boutet de Monvel index theorem.
Abstract: Let $G$ be a smooth groupoid and $V$ a hypersurface in $M=G^{(0)}$ transverse to the groupoid.
In her talk, Claire Debord will explain the construction of a corresponding Boutet de Monvel calculus in terms of a deformation groupoid. We will compute the $K$ theory of the total symbol algebra and the corresponding index morphism.
Yanli Song (University of Toronto)
Title: Equivariant indices of Spin-c Dirac operators for proper moment maps.
Abstract: Given a compact, connected Lie group acting on a possibly non-compact manifold, we can associate it with an equivariant map from the manifold to the Lie algebra, which generalizes the moment map introduced in the symplectic case. Under the assumption that the moment map is proper, we will explain how to define an equivariant index of Spin-c Dirac operators on the manifold and decompose the index into irreducible representations according to a quantization commutes with reduction principle. This joint work with Peter Hochs (University of Adelaide).
Boris Tsygan (Northwestern University)
Title: An extension of the Fedosov construction
Abstract: In the early nineties, Fedosov proposed a simple geometric construction of deformation quantization of symplectic manifolds. It turns out that a slight extension of this construction leads to new geometric structures related to deformation quantization and allows to define new invariants in symplectic geometry.
Erik van Erp (Dartmouth College)
Title: A Groupoid Approach to Pseudodifferential Calculi
Abstract: The tangent groupoid was introduced by Alain Connes as a geometric device for glueing a pseudodifferential operator to its principal symbol. We carry this idea further and show that classical pseudodifferential operators have a simple (and coordinate-free) definition in terms of the geometry of the tangent groupoid. The same definition also works for the Heisenberg calculus and its generalizations, by appropriately adapting the construction of the tangent groupoid. (Joint work with Bob Yuncken) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.870673656463623, "perplexity": 747.2993476632479}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267158279.11/warc/CC-MAIN-20180922084059-20180922104459-00203.warc.gz"} |
https://quant.stackexchange.com/questions/25750/mathematically-how-does-increasing-the-number-of-assets-reduce-idiosyncratic-ri | # Mathematically: How does increasing the number of assets reduce idiosyncratic risk?
As part of an Asset Pricing Module I'm currently taking, whilst looking at APT Ross (1974), we looked at how according to this model, risk originates from both systematic and idiosyncratic asset specific sources.
We first considered an N asset portfolio with equal weights to show how increasing N assets decreases the idiosyncratic (eP - here i am calling it the residual error term e of the portfolio P) residual variances:
Var(e) = (1/N)*(Average Sigma e)
It is clear to see that as N increases, the Variance of e decreases.
However, my question is for the case where N asset are held, but not in equal proportions. We end up with the following expression for the Var(eP):
Var(eP) = [(Summation from i=1 to N) (wi)^2 * (Sigma ei)] + All Covariance Terms
From our assumptions at the outset, idiosyncratic risks of say asset i don't affect asset j, so all the second terms from the above equation equal 0.
My question, in the below equation where wi is equal to the weight of asset i:
Var(eP) = [(Summation from i=1 to N) (wi)^2 * (Sigma ei)]
How can we see here that increasing N reduces idiosyncratic risk?
Thanks
1. APT assumes that idiosyncratic risk is zero on average: $E[e_i]=0$.
$\lim\limits_{N\to\infty}\sum\limits_{i=1}^N e_p=\lim\limits_{N\to\infty}\sum\limits_{i=1}^N w_ie_i=0$
Strictly speaking some restrictions on the weights would be needed in case the weights are unequal to $\frac{1}{N}$ (see this for example), but the above is what it comes down to. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.907880425453186, "perplexity": 1340.5186400885373}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578716619.97/warc/CC-MAIN-20190425094105-20190425120105-00317.warc.gz"} |
http://mathoverflow.net/users/763/yemon-choi?tab=summary | Yemon Choi
Reputation
Next privilege 15,000 Rep.
Protect questions
5 35 85
Impact
~480k people reached
124 Not especially famous, long-open problems which anyone can understand 55 Geometric Interpretation of Trace 52 Which journals publish expository work? 38 What is the main goal of a paper, really? 37 Eigenvalues of Matrix Sums
### Reputation (11,898)
+15 Is the Fourier-Stieltjes algebra of a locally compact group semi-simple? +5 Is the space of Hankel operators complemented in B(H)? +10 Geometric Interpretation of Trace -2 Is the set of primes “translation-finite”?
### Questions (54)
30 Is the set of primes “translation-finite”? 20 Does left-invertible imply invertible in full group C*-algebras (discrete case)? 19 Groups of order $n$ with a character whose degree is at least $0.8\sqrt{n}$ (say) 14 Is the space of Hankel operators complemented in B(H)? 14 “Explicit” embedding of $\ell^1$ as a closed subalgebra of a direct sum of matrix algebras
### Tags (141)
218 fa.functional-analysis × 79 96 fourier-analysis × 16 141 oa.operator-algebras × 48 83 banach-algebras × 24 127 matrices × 14 70 banach-spaces × 21 114 linear-algebra × 12 64 rt.representation-theory × 19 103 gr.group-theory × 27 59 ct.category-theory × 19
### Accounts (3)
MathOverflow 11,898 rep 53585 TeX - LaTeX 161 rep 16 Stack Overflow 101 rep 2 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9545914530754089, "perplexity": 3848.509221177722}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701159376.39/warc/CC-MAIN-20160205193919-00244-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.creds.ac.uk/publications/energy-efficiency-in-the-energy-transition/ | # Energy efficiency in the energy transition
07 June, 2019
Research paper
## Abstract
The energy transition to a zero carbon energy system will require both a shift to renewable energy and a major increase in energy efficiency. These are usually treated separately, but are not independent. Where renewable electricity replaces other fuels in heat and transport, there is a fundamental shift in energy supply from sources of heat to sources of work. This allows technologies such as heat pumps and electric vehicles to deliver large improvements in energy efficiency.
Where new energy sectors such as hydrogen are required, there are more complex implications for energy efficiency, depending on the details of the energy conversion processes. The paper sets out a scenario for the UK where energy is provided solely by solar and wind energy. It makes plausible assumptions about which end uses of energy can be supplied directly by electricity, and assumes others will be supplied by electrolytic hydrogen. It shows that reductions of final energy demand by 50% and primary energy demand by 60% from current levels.
The main driver is the improvement in conversion efficiencies at the point of energy use. This has major implications for the levels of renewable energy needed, which could be supplied entirely by UK indigenous resources. These types of changes to energy demand not fully captured by many global energy models and scenarios used to inform climate policy. They may therefore be unreliable and significantly over-estimating likely energy demand. The findings have important implications for policymakers in terms of lower and more realistic expectations of future energy demand.
### Publication details
Eyre, N. 2019. Energy efficiency in the energy transitionOpens in a new tab. In: Proceedings of the eceee 2019 Summer Study on energy efficiency, Paper 2-041-19. Hyères, France, 03–07 June 2019. Open access
Banner photo credit: Alireza Attari on Unsplash | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9627537131309509, "perplexity": 1712.5055935375829}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499934.48/warc/CC-MAIN-20230201112816-20230201142816-00799.warc.gz"} |
http://mathhelpforum.com/algebra/87912-solving-graphically.html | # Math Help - Solving graphically
1. ## Solving graphically
How do I find an approximate solution to solve this graphically?
8^x = 23
2. Originally Posted by olen12
How do I find an approximate solution to solve this graphically?
8^x = 23
Graph the curve 8^x.
Estimate at which value of x the y value is 23.
3. Graphically:
Draw $y_1 =23$ on the same axis of $y_2 =8^x$
where they cross is your solution | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9691274762153625, "perplexity": 1275.4686207049238}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375093974.67/warc/CC-MAIN-20150627031813-00246-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://hal.upmc.fr/hal-01242935 | # Exact packing measure of the range of ψ-Super Brownian motions
* Auteur correspondant
Abstract : We consider super processes whose spatial motion is the d-dimensional Brownian motion and whose branching mechanism ψ is critical or subcritical; such processes are called ψ-super Brownian motions. If d>2γγ/(γγ−1), where γγ∈(1,2] is the lower index of ψ at ∞, then the total range of the ψ-super Brownian motion has an exact packing measure whose gauge function is g(r)=(loglog1/r)/φ−1((1/rloglog1/r)2), where φ=ψ′∘ψ−1. More precisely, we show that the occupation measure of the ψ-super Brownian motion is the g-packing measure restricted to its total range, up to a deterministic multiplicative constant only depending on d and ψ. This generalizes the main result of Duquesne (Ann Probab 37(6):2431–2458, 2009) that treats the quadratic branching case. For a wide class of ψ, the constant 2γγ/(γγ−1) is shown to be equal to the packing dimension of the total range
Keywords :
Type de document :
Article dans une revue
Probability Theory and Related Fields, Springer Verlag, 2015, pp.1-52. 〈10.1007/s00440-015-0680-2〉
Domaine :
http://hal.upmc.fr/hal-01242935
Contributeur : Gestionnaire Hal-Upmc <>
Soumis le : lundi 14 décembre 2015 - 12:35:26
Dernière modification le : jeudi 11 janvier 2018 - 06:12:30
### Citation
Xan Duhalde, Thomas Duquesne. Exact packing measure of the range of ψ-Super Brownian motions. Probability Theory and Related Fields, Springer Verlag, 2015, pp.1-52. 〈10.1007/s00440-015-0680-2〉. 〈hal-01242935〉
### Métriques
Consultations de la notice | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9817396998405457, "perplexity": 4213.697097508578}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814393.4/warc/CC-MAIN-20180223035527-20180223055527-00376.warc.gz"} |
http://mathhelpforum.com/latex-help/219433-i-need-help-importing-images-into-latex.html | # Math Help - I need help importing images into LaTeX.
1. ## I need help importing images into LaTeX.
I created two images saved from Paint under .jpeg. Now, I would like to import them into my LaTeX document. I've tried the following
\documentclass{article}
\usepackage[pdftex]{graphicx}
\begin{document}
\includegraphics{Graph 1.jpeg}
\end{document}
and I recieved an error. Do I need to save my image in a special folder? I have two images Graph 1 and Graph 2. Please help. Thanks.
2. ## Re: I need help importing images into LaTeX.
Are your pictures in the same folder as this TEX file on your computer?
Yes
4. ## Re: I need help importing images into LaTeX.
Try removing the space from the file name. Also, make sure you compile your LaTeX file with pdflatex and not latex because the latter can include only EPS graphics, as far as I know. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9199590086936951, "perplexity": 3185.9517214384455}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042989331.34/warc/CC-MAIN-20150728002309-00282-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://phys.libretexts.org/TextBooks_and_TextMaps/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_I_-_Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves_(OpenStax)/14%3A_Fluid_Mechanics/14.1%3A_Fluids%2C_Density%2C_and_Pressure_(Part_1) | $$\require{cancel}$$
# 14.1: Fluids, Density, and Pressure (Part 1)
Skills to Develop
• State the different phases of matter
• Describe the characteristics of the phases of matter at the molecular or atomic level
• Distinguish between compressible and incompressible materials
• Define density and its related SI units
• Compare and contrast the densities of various substances
• Define pressure and its related SI units
• Explain the relationship between pressure and force
• Calculate force given pressure and area
Matter most commonly exists as a solid, liquid, or gas; these states are known as the three common phases of matter. We will look at each of these phases in detail in this section.
### Characteristics of Solids
Solids are rigid and have specific shapes and definite volumes. The atoms or molecules in a solid are in close proximity to each other, and there is a significant force between these molecules. Solids will take a form determined by the nature of these forces between the molecules. Although true solids are not incompressible, it nevertheless requires a large force to change the shape of a solid. In some cases, the force between molecules can cause the molecules to organize into a lattice as shown in Figure 14.2. The structure of this three-dimensional lattice is represented as molecules connected by rigid bonds (modeled as stiff springs), which allow limited freedom for movement. Even a large force produces only small displacements in the atoms or molecules of the lattice, and the solid maintains its shape. Solids also resist shearing forces. (Shearing forces are forces applied tangentially to a surface, as described in Static Equilibrium and Elasticity.)
### Characteristics of Fluids
Liquids and gases are considered to be fluids because they yield to shearing forces, whereas solids resist them. Like solids, the molecules in a liquid are bonded to neighboring molecules, but possess many fewer of these bonds. The molecules in a liquid are not locked in place and can move with respect to each other. The distance between molecules is similar to the distances in a solid, and so liquids have definite volumes, but the shape of a liquid changes, depending on the shape of its container. Gases are not bonded to neighboring atoms and can have large separations between molecules. Gases have neither specific shapes nor definite volumes, since their molecules move to fill the container in which they are held (Figure 14.2).
Figure $$\PageIndex{1}$$: (a) Atoms in a solid are always in close contact with neighboring atoms, held in place by forces represented here by springs. (b) Atoms in a liquid are also in close contact but can slide over one another. Forces between the atoms strongly resist attempts to compress the atoms. (c) Atoms in a gas move about freely and are separated by large distances. A gas must be held in a closed container to prevent it from expanding freely and escaping.
Liquids deform easily when stressed and do not spring back to their original shape once a force is removed. This occurs because the atoms or molecules in a liquid are free to slide about and change neighbors. That is, liquids flow (so they are a type of fluid), with the molecules held together by mutual attraction. When a liquid is placed in a container with no lid, it remains in the container. Because the atoms are closely packed, liquids, like solids, resist compression; an extremely large force is necessary to change the volume of a liquid.
In contrast, atoms in gases are separated by large distances, and the forces between atoms in a gas are therefore very weak, except when the atoms collide with one another. This makes gases relatively easy to compress and allows them to flow (which makes them fluids). When placed in an open container, gases, unlike liquids, will escape.
In this chapter, we generally refer to both gases and liquids simply as fluids, making a distinction between them only when they behave differently. There exists one other phase of matter, plasma, which exists at very high temperatures. At high temperatures, molecules may disassociate into atoms, and atoms disassociate into electrons (with negative charges) and protons (with positive charges), forming a plasma. Plasma will not be discussed in depth in this chapter because plasma has very different properties from the three other common phases of matter, discussed in this chapter, due to the strong electrical forces between the charges.
### Density
Suppose a block of brass and a block of wood have exactly the same mass. If both blocks are dropped in a tank of water, why does the wood float and the brass sink (Figure 14.3)? This occurs because the brass has a greater density than water, whereas the wood has a lower density than water.
Figure $$\PageIndex{2}$$: (a) A block of brass and a block of wood both have the same weight and mass, but the block of wood has a much greater volume. (b) When placed in a fish tank filled with water, the cube of brass sinks and the block of wood floats. (The block of wood is the same in both pictures; it was turned on its side to fit on the scale.)
Density is an important characteristic of substances. It is crucial, for example, in determining whether an object sinks or floats in a fluid.
Density
The average density of a substance or object is defined as its mass per unit volume,
$$\rho = \frac{m}{V} \tag{14.1}$$
where the Greek letter $$\rho$$ (rho) is the symbol for density, m is the mass, and V is the volume.
The SI unit of density is kg/m3. Table 14.1 lists some representative values. The cgs unit of density is the gram per cubic centimeter, g/cm3, where
$$1\; g/cm^{3} = 1000\; kg/m^{3} \ldotp$$
The metric system was originally devised so that water would have a density of 1 g/cm3, equivalent to 103 kg/m3. Thus, the basic mass unit, the kilogram, was first devised to be the mass of 1000 mL of water, which has a volume of 1000 cm3.
#### Table 14.1 - Densities of Some Common Substances
Solids (0.0 °C) Liquids (0.0 °C) Gases (0.0 °C, 101.3 kPa)
Substance $$\rho$$(kg/m3) Substance $$\rho$$(kg/m3) Substance $$\rho$$(kg/m3)
Aluminum 2.70 x 103 Benzene 8.79 x 102 Air 1.29 x 100
Bone 1.90 x 103 Blood 1.05 x 103 Carbon dioxide .1.98 x 100
Brass 8.44 x 103 Ethyl alcohol 8.06 x 102 Carbon monoxide 1.25 x 100
Concrete 2.40 x 103 Gasoline 6.80 x 102 Helium 1.80 x 10-1
Copper 8.92 x 103 Glycerin 1.26 x 103 Hydrogen 9.00 x 10-2
Cork 2.40 x 102 Mercury 1.36 x 104 Methane 7.20 x 10-2
Earth's crust 3.30 x 103 Olive oil 9.20 x 102 Nitrogen 1.25 x 100
Glass 2.60 x 103 Nitrous oxide 1.98 x 100
Granite 2.70 x 103 Oxygen 1.43 x 100
Iron 7.86 x 103
Oak 7.10 x 102
Pine 3.73 x 102
Platinum 2.14 x 104
Polystyrene 1.00 x 102
Tungsten 1.93 x 104
Uranium 1.87 x 103
As you can see by examining Table 14.1, the density of an object may help identify its composition. The density of gold, for example, is about 2.5 times the density of iron, which is about 2.5 times the density of aluminum. Density also reveals something about the phase of the matter and its substructure. Notice that the densities of liquids and solids are roughly comparable, consistent with the fact that their atoms are in close contact. The densities of gases are much less than those of liquids and solids, because the atoms in gases are separated by large amounts of empty space. The gases are displayed for a standard temperature of 0.0 °C and a standard pressure of 101.3 kPa, and there is a strong dependence of the densities on temperature and pressure. The densities of the solids and liquids displayed are given for the standard temperature of 0.0 °C and the densities of solids and liquids depend on the temperature. The density of solids and liquids normally increase with decreasing temperature.
Table 14.2 shows the density of water in various phases and temperature. The density of water increases with decreasing temperature, reaching a maximum at 4.0 °C, and then decreases as the temperature falls below 4.0 °C. This behavior of the density of water explains why ice forms at the top of a body of water.
#### Table 14.2 - Densities of Water
Substance $$\rho$$(kg/m3)
Ice (0 °C) 9.17 x 102
Water (0 °C) 9.998 x 102
Water (4 °C) 1.000 x 103
Water (20 °C) 9.982 x 102
Water (100 °C) 9.584 x 102
Steam (100 °C, 101.3 kPa) 1.670 x 102
Sea water (0°C) 1.030 x 103
The density of a substance is not necessarily constant throughout the volume of a substance. If the density is constant throughout a substance, the substance is said to be a homogeneous substance. A solid iron bar is an example of a homogeneous substance. The density is constant throughout, and the density of any sample of the substance is the same as its average density. If the density of a substance were not constant, the substance is said to be a heterogeneous substance. A chunk of Swiss cheese is an example of a heterogeneous material containing both the solid cheese and gas-filled voids. The density at a specific location within a heterogeneous material is called local density, and is given as a function of location, $$\rho$$ = $$\rho$$(x, y, z) (Figure 14.4).
Figure $$\PageIndex{3}$$: Density may vary throughout a heterogeneous mixture. Local density at a point is obtained from dividing mass by volume in a small volume around a given point.
Local density can be obtained by a limiting process, based on the average density in a small volume around the point in question, taking the limit where the size of the volume approaches zero,
$$\rho = \lim_{\Delta V \rightarrow 0} \frac{\Delta m}{\Delta V} \tag{14.2}$$
where $$\rho$$ is the density, m is the mass, and V is the volume.
Since gases are free to expand and contract, the densities of the gases vary considerably with temperature, whereas the densities of liquids vary little with temperature. Therefore, the densities of liquids are often treated as constant, with the density equal to the average density.
Density is a dimensional property; therefore, when comparing the densities of two substances, the units must be taken into consideration. For this reason, a more convenient, dimensionless quantity called the specific gravity is often used to compare densities. Specific gravity is defined as the ratio of the density of the material to the density of water at 4.0 °C and one atmosphere of pressure, which is 1000 kg/m3:
$$Specific\; gravity = \frac{Density\; of\; material}{Density\; of\; water} \ldotp$$
The comparison uses water because the density of water is 1 g/cm3, which was originally used to define the kilogram. Specific gravity, being dimensionless, provides a ready comparison among materials without having to worry about the unit of density. For instance, the density of aluminum is 2.7 in g/cm3 (2700 in kg/m3), but its specific gravity is 2.7, regardless of the unit of density. Specific gravity is a particularly useful quantity with regard to buoyancy, which we will discuss later in this chapter.
### Pressure
You have no doubt heard the word ‘pressure’ used in relation to blood (high or low blood pressure) and in relation to weather (high- and low-pressure weather systems). These are only two of many examples of pressure in fluids. (Recall that we introduced the idea of pressure in Static Equilibrium and Elasticity, in the context of bulk stress and strain.)
Pressure
Pressure (p) is defined as the normal force F per unit area A over which the force is applied, or
$$p = \frac{F}{A} \ldotp \tag{14.3}$$
To define the pressure at a specific point, the pressure is defined as the force dF exerted by a fluid over an infinitesimal element of area dA containing the point, resulting in p = $$\frac{dF}{dA}$$.
A given force can have a significantly different effect, depending on the area over which the force is exerted. For instance, a force applied to an area of 1 mm2 has a pressure that is 100 times as great as the same force applied to an area of 1 cm2 . That is why a sharp needle is able to poke through skin when a small force is exerted, but applying the same force with a finger does not puncture the skin (Figure 14.5).
Figure $$\PageIndex{4}$$: (a) A person being poked with a finger might be irritated, but the force has little lasting effect. (b) In contrast, the same force applied to an area the size of the sharp end of a needle is enough to break the skin.
Note that although force is a vector, pressure is a scalar. Pressure is a scalar quantity because it is defined to be proportional to the magnitude of the force acting perpendicular to the surface area. The SI unit for pressure is the pascal (Pa), named after the French mathematician and physicist Blaise Pascal (1623–1662), where
$$1\; Pa = 1\; N/m^{2} \ldotp$$
Several other units are used for pressure, which we discuss later in the chapter.
### Contributors
• Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8368082046508789, "perplexity": 670.7109178058729}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864776.82/warc/CC-MAIN-20180622182027-20180622202027-00347.warc.gz"} |
https://actruce.com/en/likelihood/ | Likelihood
likelihood ?
It’s very confusing concept to me, very difficult to understand at a glance. I’ve looked for the definition from Wikipedia and sought some blogs which explained about it. First of all I’ve referred sw4r ‘s blog which contained huge amount of numerical statistics. And I found the definition of ‘Likelihood’ in Wikipedia which was really helpful to me to understand meaning and usage. Also I found some helpful example in plasticcode ‘s blog.
Let’s begin to look at the definition of likelihood inside Wikipedia.
In frequentist inference, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model, given specific observed data. Likelihood functions play a key role in frequentist inference, especially methods of estimating a parameter from a set of statistics. In informal contexts, “likelihood” is often used as a synonym for “probability“. In mathematical statistics, the two terms have different meanings. Probability in this mathematical context describes the plausibility of a random outcome, given a model parameter value, without reference to any observed data. Likelihood describes the plausibility of a model parameter value, given specific observed data.
Probability and Likelihood can be opposite meaning. The probability is a function of random variable X knowing population’s parameters , while Likelihood is a function of parameter $\theta$ when observations are measured.
The probability density function is expressed with random variable X given parameter $\theta$,
$f(X|\theta)$
Meanwhile Likelihood is expressed with $\theta$ given X
$L(\theta|X)$
In normal if there are n samples, Likelihood is written by $L(\theta|x_1,x_2,\cdots,x_n)$
Usually it is assumed that observations can be randomly selected and even individually independent so that it can be expressed by a multiplication.
$L(\theta|x_1,x_2,\cdots,x_n)=L(\theta|x_1) \times L(\theta|x_2) \times \cdots \times L(\theta|x_n)$
Let’s see an example from plasticcode ‘s blog.
Q : If we assume that 5 samples (-1, -0.5, 0, +0.5, +1) are followed to normal distribution, what is the its mean and standard deviation ?
A : To get the population’s parameters $\mu$ and $\sigma$, we can try several likelihood and compare the results each other. If a result has the biggest value compared to others, we can say it can be a maximum likelihood approximately and explain the population’s parameters finely.
I’ve written somewhat R Code. You can find the R Code in Github.
1. When its distribution is assumed to N(-1, 1), likelihood is calculated by ‘0.0002376545’ (This is the simple multiplication from the 5 values in the curve.)
2. When its distribution is assumed to N(0, 1), likelihood is calculated by ‘0.002895224’
3. When its distribution is assumed to N(3, 1), likelihood is calculated by ‘4.898424e-13’ (extreme case)
i.e. When we assumed its distribution was N(0, 1) , the likelihood was the highest. When we decide the best population’s parameters with this restricted sample counts (5), it is very accountable for. Like this, Likelihood can be estimated differently from the different samples.
I would like to recommend a funny video, titled by “probability vs likelihood” which is the best simply described the concept of likelihood. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 9, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9711801409721375, "perplexity": 907.0063229419546}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988953.13/warc/CC-MAIN-20210509002206-20210509032206-00334.warc.gz"} |
https://docs.wiris.com/mathtype/en/user-interfaces/mathtype-web-interface/insert-special-characters.html | # Insert special characters
This is a new feature in EDITOR version 4.1.
To input special characters or symbols, click the button in the Formatting group on the General tab.
A menu will appear with a list of symbols and characters.
On the right, you can choose which category to select your symbol from. Once you find the symbol you want to insert, click it and it will appear in the editor.
Alternatively, if you know the Unicode code for a character, type it in the blank to get it directly:
Sometimes more than one symbol will appear. This is done to allow entering the codepoint value either in either hexadecimal or decimal. In the above example, the code could either mean U+0198 as a hexadecimal Unicode number, or Æ as a decimal Unicode number, so both are shown: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.90982985496521, "perplexity": 1525.2180677996628}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710765.76/warc/CC-MAIN-20221130160457-20221130190457-00224.warc.gz"} |
http://www.ck12.org/trigonometry/DeMoivres-Theorem-and-nth-Roots/lesson/DeMoivres-Theorem-and-nth-Roots/r4/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
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# DeMoivre's Theorem and nth Roots
## Raise complex numbers to powers or find their roots.
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DeMoivre's Theorem and nth Roots
You are in math class one day when your teacher asks you to find $\sqrt{3i}$ . Are you able to find roots of complex numbers? By the end of this Concept, you'll be able to perform this calculation.
### Guidance
Other Concepts in this course have explored all of the basic operations of arithmetic as they apply to complex numbers in standard form and in polar form. The last discovery is that of taking roots of complex numbers in polar form. Using De Moivre’s Theorem we can develop another general rule – one for finding the $n^{th}$ root of a complex number written in polar form.
As before, let $z = r(\cos \theta + i \sin \theta)$ and let the $n^{th}$ root of $z$ be $v = s (\cos \alpha + i \sin \alpha)$ . So, in general, $\sqrt[n]{z}=v$ and $v^n=z$ .
$\sqrt[n]{z} &= v\\\sqrt[n]{r(\cos \theta+i \sin \theta)} &=s(\cos \alpha + i \sin \alpha)\\[r(\cos \theta+i \sin \theta)]^{\frac{1}{n}} &= s(\cos \alpha +i \sin \alpha)\\r^{\frac{1}{n}}\left(\cos \frac{1}{n} \theta+i \sin \frac{1}{n}\theta \right) &= s(\cos \alpha+i \sin \alpha)\\r^{\frac{1}{n}}\left(\cos \frac{\theta}{n}+i \sin \frac{\theta}{n} \right) &= s(\cos \alpha+i \sin \alpha)$
From this derivation, we can conclude that $r^{\frac{1}{n}}=s$ or $s^n=r$ and $\alpha=\frac{\theta}{n}$ . Therefore, for any integer $k (0, 1, 2, \ldots n -1)$ , $v$ is an $n^{th}$ root of $z$ if $s=\sqrt[n]{r}$ and $\alpha=\frac{\theta+2\pi k}{n}$ . Therefore, the general rule for finding the $n^{th}$ roots of a complex number if $z = r(\cos \theta + i \sin \theta)$ is: $\sqrt[n]{r} \left(\cos \frac{\theta+2\pi k}{n}+i \sin \frac{\theta+2\pi k}{n}\right)$ . Let’s begin with a simple example and we will leave $\theta$ in degrees.
#### Example A
Find the two square roots of $2i$ .
Solution: Express $2i$ in polar form.
$& r=\sqrt{x^2+y^2} && \cos \theta=0\\& r=\sqrt{(0)^2+(2)^2} && \qquad \theta=90^\circ\\& r=\sqrt{4}=2$
$(2i)^{\frac{1}{2}}=2^{\frac{1}{2}} \left(\cos \frac{90^\circ}{2}+i \sin \frac{90^\circ}{2}\right)=\sqrt{2}(\cos 45^\circ +i \sin 45^\circ)=1+i$
To find the other root, add $360^\circ$ to $\theta$ .
$(2i)^{\frac{1}{2}}=2^{\frac{1}{2}} \left(\cos \frac{450^\circ}{2}+i \sin \frac{450^\circ}{2}\right)=\sqrt{2}(\cos 225^\circ +i \sin 225^\circ)=-1-i$
#### Example B
Find the three cube roots of $-2-2i \sqrt{3}$
Solution: Express $-2-2i \sqrt{3}$ in polar form:
$r &=\sqrt{x^2+y^2}\\r &= \sqrt{(-2)^2+(-2\sqrt{3})^2}\\r &= \sqrt{16}=4 && \theta = \tan^{-1} \left(\frac{-2\sqrt{3}}{-2}\right)=\frac{4\pi}{3}$
$& \sqrt[n]{r}\left( \cos \frac{\theta + 2\pi k}{n}+i \sin \frac{\theta + 2\pi k}{n}\right)\\\sqrt[3]{-2-2i \sqrt{3}} &= \sqrt[3]{4} \left(\cos \frac{\frac{4 \pi}{3} + 2\pi k}{3}+i \sin \frac{\frac{4\pi}{3} +2\pi k}{3}\right) k=0, 1, 2$
$z_1 &= \sqrt[3]{4}\left[ \cos \left(\frac{4\pi}{9}+\frac{0}{3}\right)+i \sin \left(\frac{4\pi}{9}+\frac{0}{3}\right)\right] && k=0\\&= \sqrt[3]{4}\left[\cos \frac{4\pi}{9}+i \sin \frac{4\pi}{9}\right]\\z_2 &= \sqrt[3]{4}\left[ \cos \left(\frac{4\pi}{9}+\frac{2\pi}{3}\right)+i \sin \left(\frac{4\pi}{9}+\frac{2\pi}{3}\right)\right] && k=1\\&= \sqrt[3]{4}\left[\cos \frac{10\pi}{9}+i \sin \frac{10\pi}{9}\right]\\z_3 &= \sqrt[3]{4}\left[ \cos \left(\frac{4\pi}{9}+\frac{4\pi}{3}\right)+i \sin \left(\frac{4\pi}{9}+\frac{4\pi}{3}\right)\right] && k=2\\&= \sqrt[3]{4}\left[\cos \frac{16\pi}{9}+i \sin \frac{16\pi}{9}\right]$
In standard form: $z_1=0.276+1.563i, z_2=-1.492-0.543i, z_3=1.216-1.02i$ .
#### Example C
Calculate $\left(\cos \frac{\pi}{4} + i\sin\frac{\pi}{4}\right)^{1/3}$
Using the for of DeMoivres Theorem for fractional powers, we get:
$\left(\cos \frac{\pi}{4} + i\sin\frac{\pi}{4}\right)^{1/3}\\= \cos \left(\frac{1}{3} \times \frac{\pi}{4} \right) + i\sin \left(\frac{1}{3} \times \frac{\pi}{4} \right)\\= \left(\cos \frac{\pi}{12} + i\sin \frac{\pi}{12} \right)$
### Vocabulary
DeMoivres Theorem: DeMoivres theorem relates a complex number raised to a power to a set of trigonometric functions.
### Guided Practice
1. Find $\sqrt[3]{27i}$ .
2. Find the principal root of $(1 + i)^{\frac{1}{5}}$ . Remember the principal root is the positive root i.e. $\sqrt{9}=\pm 3$ so the principal root is +3.
3. Find the fourth roots of $81i$ .
Solutions:
1.
$&&& a=0 \ and \ b=27\\& \sqrt[3]{27i} =(0+27i)^{\frac{1}{3}} && x=0 \ and \ y=27\\& \text{Polar Form} && r=\sqrt{x^2+y^2} \qquad \qquad \theta=\frac{\pi}{2}\\&&& r=\sqrt{(0)^2+(27)^2}\\&&& r=27\\&&& \sqrt[3]{27i} = \left[27 \left(\cos (\frac{\pi}{2} + 2 \pi k) +i \sin (\frac{\pi}{2} + 2 \pi k) \right)\right]^{\frac{1}{3}} \text{for } k = 0, 1, 2\\&&& \sqrt[3]{27i} = \sqrt[3]{27} \left[\cos \left(\frac{1}{3}\right) \left(\frac{\pi}{2} + 2 \pi k\right)+i \sin \left(\frac{1}{3}\right) \left(\frac{\pi}{2} + 2 \pi k\right)\right] \text{for } k = 0, 1, 2\\&&& \sqrt[3]{27i} = 3 \left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right) \text{for } k = 0\\&&& \sqrt[3]{27i} = 3 \left(\cos \frac{5\pi}{6}+i \sin \frac{5\pi}{6}\right) \text{for } k = 1\\&&& \sqrt[3]{27i} = 3 \left(\cos \frac{9\pi}{6}+i \sin \frac{9\pi}{6}\right) \text{for } k = 2\\&&& \sqrt[3]{27i} = 3\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i \right), 3\left(\frac{-\sqrt{3}}{2}+\frac{1}{2}i \right), -3i$ }}
2.
$& r=\sqrt{x^2+y^2} && \theta=\tan^{-1} \left(\frac{1}{1}\right)=\frac{\sqrt{2}}{2} && \text{Polar Form} = \sqrt{2} cis \frac{\pi}{4}\\& r=\sqrt{(1)^2+(1)^2}\\& r=\sqrt{2}$
$(1+i)^{\frac{1}{5}} &= \left[\sqrt{2} \left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\right]^{\frac{1}{5}}\\(1+i)^{\frac{1}{5}} &= \sqrt{2}^{\frac{1}{5}}\left[\cos \left(\frac{1}{5}\right) \left(\frac{\pi}{4}\right)+i \sin \left(\frac{1}{5}\right) \left(\frac{\pi}{4}\right)\right]\\(1+i)^{\frac{1}{5}} &= \sqrt[10]{2} \left(\cos \frac{\pi}{20}+i \sin \frac{\pi}{20} \right)$
In standard form $(1+i)^{\frac{1}{5}}=(1.06+1.06i)$ and this is the principal root of $(1+i)^{\frac{1}{5}}$ .
3.
$81i$ in polar form is:
$& r=\sqrt{0^2+81^2}=81, \tan \theta =\frac{81}{0}= und \rightarrow \theta=\frac{\pi}{2} \quad 81 \left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)\\& \left[81 \left(\cos \left(\frac{\pi}{2}+2 \pi k \right)+i \sin \left(\frac{\pi}{2}+2\pi k\right)\right)\right]^{\frac{1}{4}}\\& 3 \left(\cos \left(\frac{\frac{\pi}{2}+2\pi k}{4}\right)+i \sin \left(\frac{\frac{\pi}{2}+2 \pi k}{4}\right)\right)\\& 3 \left(\cos \left(\frac{\pi}{8}+\frac{\pi k}{2}\right)+i \sin \left(\frac{\pi}{8}+\frac{\pi k}{2}\right)\right)\\& z_1 =3 \left(\cos \left(\frac{\pi}{8}+\frac{0 \pi}{2}\right)+i \sin \left(\frac{\pi}{8}+\frac{0 \pi}{2}\right)\right)=3 \cos \frac{\pi}{8}+3i \sin \frac{\pi}{8}=2.77+1.15i\\& z_2 =3 \left(\cos \left(\frac{\pi}{8}+\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{8}+\frac{\pi}{2}\right)\right)=3 \cos \frac{5 \pi}{8}+3i \sin \frac{5 \pi}{8}=-1.15+2.77i\\& z_3 =3 \left(\cos \left(\frac{\pi}{8}+\frac{2\pi}{2}\right)+i \sin \left(\frac{\pi}{8}+\frac{2\pi}{2}\right)\right)=3 \cos \frac{9\pi}{8}+3i \sin \frac{9 \pi}{8}=-2.77-1.15i\\& z_4 =3 \left(\cos \left(\frac{\pi}{8}+\frac{3\pi}{2}\right)+i \sin \left(\frac{\pi}{8}+\frac{3\pi}{2}\right)\right)=3 \cos \frac{13 \pi}{8}+3i \sin \frac{13 \pi}{8}=1.15-2.77i$
### Concept Problem Solution
Finding the two square roots of $3i$ involves first converting the number to polar form:
$r=\sqrt{x^2+y^2} \\r=\sqrt{(0)^2+(3)^2} \\r=\sqrt{9}=3$
And the angle:
$\cos \theta=0\\\theta=90^\circ\\$
$(3i)^{\frac{1}{2}}=3^{\frac{1}{2}} \left(\cos \frac{90^\circ}{2}+i \sin \frac{90^\circ}{2} \right)=\sqrt{3}(\cos 45^\circ +i \sin 45^\circ)= \frac{\sqrt{6}}{2} \left( 1+i \right)$
To find the other root, add $360^\circ$ to $\theta$ .
$(3i)^{\frac{1}{2}}=3^{\frac{1}{2}} \left(\cos \frac{450^\circ}{2}+i \sin \frac{450^\circ}{2}\right)=\sqrt{3}(\cos 225^\circ +i \sin 225^\circ)= \frac{\sqrt{6}}{2} \left( -1-i \right)$
### Practice
Find the cube roots of each complex number. Write your answers in standard form.
1. $8(\cos 2\pi+i\sin 2\pi)$
2. $3(\cos \frac{\pi}{4}+i\sin \frac{\pi}{4})$
3. $2(\cos \frac{3\pi}{4}+i\sin \frac{3\pi}{4})$
4. $(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3})$
5. $(3+4i)$
6. $(2+2i)$
Find the principal fifth roots of each complex number. Write your answers in standard form.
1. $2(\cos \frac{\pi}{6}+i\sin \frac{\pi}{6})$
2. $4(\cos \frac{\pi}{2}+i\sin \frac{\pi}{2})$
3. $32(\cos \frac{\pi}{4}+i\sin \frac{\pi}{4})$
4. $2(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3})$
5. $32i$
6. $(1+\sqrt{5}i)$
7. Find the sixth roots of -64 and plot them on the complex plane.
8. How many solutions could the equation $x^6+64=0$ have? Explain.
9. Solve $x^6+64=0$ . Use your answer to #13 to help you.
### Vocabulary Language: English
$n^{th}$ roots of unity
$n^{th}$ roots of unity
The $n^{th}$ roots of unity are the $n^{th}$ roots of the number 1.
complex number
complex number
A complex number is the sum of a real number and an imaginary number, written in the form $a + bi$.
complex plane
complex plane
The complex plane is the graphical representation of the set of all complex numbers.
De Moivre's Theorem
De Moivre's Theorem
De Moivre's theorem is the only practical manual method for identifying the powers or roots of complex numbers. The theorem states that if $z= r(\cos \theta + i \sin \theta)$ is a complex number in $r cis \theta$ form and $n$ is a positive integer, then $z^n=r^n (\cos (n\theta ) + i\sin (n\theta ))$.
trigonometric polar form
trigonometric polar form
To write a complex number in trigonometric form means to write it in the form $r\cos\theta+ri\sin\theta$. $rcis\theta$ is shorthand for this expression. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 80, "texerror": 0, "math_score": 0.9644144773483276, "perplexity": 506.83888278296905}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398471436.90/warc/CC-MAIN-20151124205431-00233-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://kluedo.ub.uni-kl.de/frontdoor/index/index/searchtype/all/rows/20/facetNumber_author_facet/all/sortfield/title/sortorder/desc/author_facetfq/Remde%2C+Axel/subjectfq/Manipulation/start/1/docId/1043 | • search hit 2 of 2
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Direct and Inverse Simulation of Deformable Linear Objects
• In this chapter, the quantitative numerical simulation of the behavior of deformable linear objects, such as hoses, wires and leaf springs is studied. We first give a short review of the physical approach and the basic solution principle. Then, we give a more detailed description of some key aspects: We introduce a novel approach concerning dynamics based on an algorithm very similar to the one used for (quasi-) static computation. Then, we look at the plastic workpiece deformation, involving a modified computation algorithm and a special representation of the workpiece shape. Then, we give alternative solutions for two key aspects of the algorithm, and investigate the problem of performing the workpiece simulation efficiently, i.e., with desired precision in a short time. In the end, we introduce the inverse modeling problem which must be solved when the gripper trajectory for a given task shall be generated. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8936060667037964, "perplexity": 583.1466397432895}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363791.16/warc/CC-MAIN-20211209091917-20211209121917-00062.warc.gz"} |
https://www.physicsforums.com/threads/what-is-the-probability-that-the-temperature-will-drop.148055/ | What is the probability that the temperature will drop
1. Dec 13, 2006
twoflower
1. The problem statement, all variables and given/known data
Meteorologists expect rapid temperature drop in the following 48 hours. What is the probability that the temperature drop will occur between 15:45 and 24:00?
3. The attempt at a solution
I can't find out, what distribution is suitable for this kind of problem...Would someone please point me to the right distribution?
Thank you.
Best regards,
Standa
2. Dec 13, 2006
HallsofIvy
Staff Emeritus
If that's all you are given, you can only assume a uniform distribution over the 48 hours. My question is whether "between 15:45 and 24:00" applies to both days.
3. Dec 14, 2006
twoflower
Thank you HallsoftIvy, after a while I also assumed it to be uniform distribution. Whether the 15.45 - 24.00 interval applies to both days remains unsaid, good question, however, it didn't come to my mind.
4. Dec 14, 2006
enricfemi
As it appears to me, the question quote from book entirely should mention to both days.
5. Dec 14, 2006
HallsofIvy
Staff Emeritus
I agree. Thought we ought to leave something for twoflower to think about!
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Birgitte is from Norway, and had recently moved to live in London. I noticed her while she was waiting for to meet a friend in Camden town. I just loved the fur trim leather jacket, polka dot dress and leather boots. Hawley Crescent, Camden Town,London,UK.
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https://mathsblab.com/2013/07/18/a-delightful-little-proof/ | # A sledgehammer proof of irrationality
Mathematical proofs are a bit like people. You can choose which ones to love and which ones to spit at. The loveable proofs are undoubtably the most important, especially as a pick-me-up on those cold lonely days. I’d like to share with you all a little proof that the $n$th root of 2 is irrational for $n \geq 3$. Of course, most people see the proof that the square root of 2 is irrational sometime during their study. If you want some background on irrational numbers, you can check out my earlier blog post.
First, let’s take a detour into the well known equation:
$x^2 + y^2 = z^2$
This equation has infinitely many integer solutions (we call these Pythagorean triples). Here are some examples:
$3^2 + 4^2 = 5^2$
$5^2+12^2=13^2$
What about the equation $x^3+y^3 = z^3$ or even $x^4+y^4 = z^4$? Well, it turns out that these have no integer solutions (except for silly ones involving 0’s)! Furthermore, for any integer $n \geq 3$, we have that the equation
$x^n + y^n = z^n$
has no non-silly solutions in the integers. This result is known as Fermat’s Last Theorem, and was proven to be true by Andrew Wiles around 1995. Now that we know this, we can proceed with our cute little proof that $\sqrt[n]{2}$ is irrational.
We will assume to the contrary. Let $n \geq 3$ and assume that $\sqrt[n]{2}$ is rational, so there must be two positive integers $a$ and $b$ such that
$\sqrt[n]{2} = \frac{a}{b}.$
We put both sides to the power of $n$ and rearrange to get
$2 b^n = a^n.$
But this is really $b^n + b^n = a^n$ and by Fermat’s Last Theorem there are no positive integer solutions to this equation! So we have found a contradiction to FLT and so $\sqrt[n]{2}$ must be irrational for all $n \geq 3$. What a cutie-proof!
## One thought on “A sledgehammer proof of irrationality”
1. Melissa Lee says:
This is the most adorable proof I have seen in ages! 😀 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 20, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9133033752441406, "perplexity": 213.3990030372543}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704803737.78/warc/CC-MAIN-20210126202017-20210126232017-00482.warc.gz"} |
http://scitation.aip.org/content/aip/journal/jap/113/23/10.1063/1.4811532 | • journal/journal.article
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Morphology and chain aggregation dependence of optical gain in thermally annealed films of the conjugated polymer poly[2-methoxy-5-(2′-ethylhexyloxy)-p-phenylene vinylene]
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10.1063/1.4811532
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Affiliations:
1 Department of Materials Science and Engineering, North Carolina State University, Raleigh, North Carolina 27695, USA
2 Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695, USA
3 Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599, USA
a) E-mail: [email protected]
J. Appl. Phys. 113, 233509 (2013)
/content/aip/journal/jap/113/23/10.1063/1.4811532
http://aip.metastore.ingenta.com/content/aip/journal/jap/113/23/10.1063/1.4811532
Figures
FIG. 1.
Absorption spectra of MEH-PPV films cast from THF-solution on glass as a function of thermal annealing temperature. Inset shows chemical structure of MEH-PPV. The spectra have been normalized to the maximum absorption wavelength and vertically offset for clarity. Vertical broken line is a guide to the eye. The bulk glass transition temperature, = 70 °C.
FIG. 2.
Atomic force micrographs of MEH-PPV films that were as-cast (a) and thermally annealed at 60 °C (b), 80 °C (c), and 200 °C (d) taken over a 30 m × 30 m scan area. Average surface roughness R values are provided. The bulk glass transition temperature, = 70 °C.
FIG. 3.
Low pump energy PL spectra collected from the edge of waveguides formed with THF-cast films of MEH-PPV as-cast and annealed at 60 and 80 °C with a constant pump energy density of 3 J/cm and pump stripe length of 0.1 cm. The bulk glass transition temperature, = 70 °C.
FIG. 4.
Pump energy dependence of emission spectra for MEH-PPV films that were as-cast (a) and thermally annealed at 60 °C (b) and 80 °C (c). The spectra have been normalized to facilitate comparison. PL/ASE spectra were collected from the edge of the waveguides and recorded using a constant excitation pump stripe length of 0.1 cm. The inset plots show the dependence of the PL linewidth for the 0-1 transition on the pump energy density. The bulk glass transition temperature, = 70 °C.
FIG. 5.
Dependence of the spectrally integrated, edge emitted PL intensity on pump energy density for waveguides formed with THF-cast films of MEH-PPV as-cast (squares) and annealed at 60 °C (triangles), 80 °C (circles), and 200 °C (stars). The pump stripe length is 0.1 cm. The bulk glass transition temperature, = 70 °C. Inset plot (linear-linear scale) shows change in slope with annealing temperature at low pump energy density, prior to the ASE threshold. Solid lines are guides to the eye.
FIG. 6.
Waveguide loss (a) and AFM average surface roughness (b) of MEH-PPV films showing the same functional dependence with anneal temperature. Error bars represent 2× the standard error.
FIG. 7.
Excitation stripe length dependence of the edge emitted peak intensity at for waveguides formed with THF-cast films of MEH-PPV as-cast (squares) and thermally annealed at 60 (triangles) and 80 °C (circles). The pump energy density is 30 J/cm. The fitting to Eq. is given by the solid lines. The data are plotted on a logarithmic-linear (a) and linear-linear (b) scale to clearly make visible the saturation and threshold stripe length regimes, respectively. The inset shows the normalized PL spectra for the as-cast film collected at three different stripe lengths with a pump energy density of 30 J/cm. The threshold excitation length is indicated for the as-cast film. The bulk glass transition temperature, = 70 °C.
FIG. 8.
Net gain coefficient as a function of pump energy density for waveguides formed with as-cast and thermally annealed MEH-PPV films. The bulk glass transition temperature, = 70 °C. Error bars represent 2× the standard error.
Tables
Table I.
Optical parameters of MEH-PPV waveguides; is the thermal annealing temperature, the threshold pump energy density, the maximum excitation density at threshold, g the maximum measured net gain coefficient, and the PL slope efficiency at low pump energy density which is proportional to , the PL yield.
/content/aip/journal/jap/113/23/10.1063/1.4811532
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2014-04-21
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This is a required field | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8114032745361328, "perplexity": 4472.394683119117}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00014-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/269131?ln=en | ## Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism
Closure modeling based on the Mori-Zwanzig formalism has proven effective to improve the stability and accuracy of projection-based model order reduction. However, closure models are often expensive and infeasible for complex nonlinear systems. Towards efficient model reduction of general problems, this paper presents a recurrent neural network (RNN) closure of parametric POD-Galerkin reduced-order model. Based on the short time history of the reduced-order solutions, the RNN predicts the memory integral which represents the impact of the unresolved scales on the resolved scales. A conditioned long short term memory (LSTM) network is utilized as the regression model of the memory integral, in which the POD coefficients at a number of time steps are fed into the LSTM units, and the physical/geometrical parameters are fed into the initial hidden state of the LSTM. The reduced-order model is integrated in time using an implicit-explicit (IMEX) Runge-Kutta scheme, in which the memory term is integrated explicitly and the remaining right-hand-side term is integrated implicitly to improve the computational efficiency. Numerical results demonstrate that the RNN closure can significantly improve the accuracy and efficiency of the POD-Galerkin reduced-order model of nonlinear problems. The POD-Galerkin reduced-order model with the RNN closure is also shown to be capable of making accurate predictions, well beyond the time interval of the training data.
Year:
2019
Keywords:
Laboratories:
Note: The status of this file is: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8738197684288025, "perplexity": 404.6787866739193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370493120.15/warc/CC-MAIN-20200328194743-20200328224743-00200.warc.gz"} |
https://www.nextgurukul.in/wiki/concept/cbse/class-10/maths/polynomials/relationship-between-zeroes-and-coefficients-of-a-polynomial/3957194 | Notes On Relationship between Zeroes and Coefficients of a Polynomial - CBSE Class 10 Maths
A polynomial is an algebraic expression consisting of multiple terms. There are various types of polynomials such as linear, quadratic, cubic and so on. A real number k is a zero of a polynomial of p(x) if p(k) = 0. Factor Theorem: If a is zero of a polynomial p(x) then (x – a) is a factor of p(x). Relationship betweeen Zeroes and coefficients of a Polynomial The general form of linear polynomial is p(x) = ax+b, its zero is $\frac{\text{-b}}{\text{a}}$ .i.e.x = $\frac{\text{-b}}{\text{a}}$ or . General form of quadratic polynomial is ax2 + bx +c where a ≠ 0. There are two zeroes of quadratic polynomial. Sum of zeroes = $\frac{\text{-b}}{\text{a}}$ = Product of zeroes = = . General form of cubic polynomial of ax3 + bx 2+ cx + d where a ≠ 0. There are three zeroes of cubic polynomial. The sum of zeroes of the cubic polynomial = $\frac{\text{-b}}{\text{a}}$ = Sum of the product of zeroes taken two at a time = $\frac{\text{c}}{\text{a}}$ = Product of zeroes = $\frac{\text{-d}}{\text{a}}$ = .
#### Summary
A polynomial is an algebraic expression consisting of multiple terms. There are various types of polynomials such as linear, quadratic, cubic and so on. A real number k is a zero of a polynomial of p(x) if p(k) = 0. Factor Theorem: If a is zero of a polynomial p(x) then (x – a) is a factor of p(x). Relationship betweeen Zeroes and coefficients of a Polynomial The general form of linear polynomial is p(x) = ax+b, its zero is $\frac{\text{-b}}{\text{a}}$ .i.e.x = $\frac{\text{-b}}{\text{a}}$ or . General form of quadratic polynomial is ax2 + bx +c where a ≠ 0. There are two zeroes of quadratic polynomial. Sum of zeroes = $\frac{\text{-b}}{\text{a}}$ = Product of zeroes = = . General form of cubic polynomial of ax3 + bx 2+ cx + d where a ≠ 0. There are three zeroes of cubic polynomial. The sum of zeroes of the cubic polynomial = $\frac{\text{-b}}{\text{a}}$ = Sum of the product of zeroes taken two at a time = $\frac{\text{c}}{\text{a}}$ = Product of zeroes = $\frac{\text{-d}}{\text{a}}$ = .
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https://community.bt.com/t5/BT-Fibre-broadband/Bandwidth-icreased-ping-decreased/m-p/1462554 | cancel
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Beginner
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Message 1 of 5
## Bandwidth icreased/ ping decreased!
Not sure what's happened (not that im complaining!) for about a year now my download speed would not go above 51-54mbps and ping was alway around 20ms. Whist downloading today I noticed its was topping out at 62mbps so checked and speedtest.net and that confirmed the same too and ping was down to 8ms! What could have happened I wonder?
4 REPLIES 4
Distinguished Guru
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Message 2 of 5
## Re: Bandwidth icreased/ ping decreased!
They are currently rolling out G.INP so maybe that.
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
If you found this post helpful, please click on the star on the left
If not, I'll try again 🙂
Expert
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## Re: Bandwidth icreased/ ping decreased!
This maybe of interest :-
Best regards,
dfenceman
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Beginner
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Message 4 of 5
## Re: Bandwidth icreased/ ping decreased!
Thanks guys, will read up on about it. I'd heard of vectoring but not G.I.N.P. before
Distinguished Guru
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Message 5 of 5
## Re: Bandwidth icreased/ ping decreased!
I think G.INP is required for Vectoring.
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If you found this post helpful, please click on the star on the left
If not, I'll try again 🙂 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.968494176864624, "perplexity": 2592.779468458315}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540517156.63/warc/CC-MAIN-20191209013904-20191209041904-00512.warc.gz"} |
http://math.stackexchange.com/users/53034/lmz | # LMZ
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# 7 Questions
3 How complex are natural deduction proofs compared to sequent calculus proofs without cut? 3 Is it possible to algebraically prove that the $n$th degree Taylor remainder of $f(x)$ is less than $K|\Delta x|^{n+1}$ for $K \in \mathbb{R^+}$? 1 To show a power series is a Taylor series 1 Is a function always a monotonically increasing function 1 Free variables in definitions
# 65 Reputation
+5 To show a power series is a Taylor series +5 How complex are natural deduction proofs compared to sequent calculus proofs without cut? +5 Free variables in definitions +10 How is the double negation translation similar to CPS in functional programming languages?
1 How is the double negation translation similar to CPS in functional programming languages? 0 Free variables in definitions
# 16 Tags
1 logic × 5 0 foundations × 3 1 predicate-logic × 2 0 taylor-expansion × 2 1 constructive-mathematics × 2 0 real-analysis × 2 1 intuitionism × 2 0 calculus × 2 0 analysis × 5 0 complex-analysis
# 4 Accounts
Stack Overflow 75 rep 13 Mathematics 65 rep 7 Physics 1 rep MathOverflow 1 rep | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8700973987579346, "perplexity": 1961.3800189395465}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997859240.8/warc/CC-MAIN-20140722025739-00000-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/207520/matrix-lie-group-counter-example-ex-in-the-lie-group-but-x-is-not-in-the?answertab=votes | # Matrix Lie group counter-example: $e^X$ in the Lie group, but $X$ is not in the Lie algebra
What's an example of a Lie group $G$ and matrix $X$ such that $e^X \in G$ but $x \notin \mathfrak{g}$, where $\mathfrak{g}$ is the associated Lie algebra?
This is the same as problem 2.10 in Bryan Hall's "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction."
-
Take $G$ to be trivial... – Qiaochu Yuan Oct 5 '12 at 1:14
Take $H = \{ \pm I\}$ that has trivial Lie algebra because it is a finite group. The matrix
$$X = \left(\begin{array}{cc} 0 & -\pi \\ \pi & 0 \end{array}\right)$$
is such that $e^X = \left(\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right)$ but $X$ is not zero and hence is not in the Lie algebra.
(For what it's worth, Maple says that $e^X = -I$, but you can just change $H$ to accommodate it. In order to see why your answer must be wrong, note that the nontrivial element of $H$ has determinant $-1$, so is in a different connected componenet of $GL_2$ than $I$. In particular, it cannot be $e^X$ for any $X$). – Jason DeVito Oct 5 '12 at 2:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8965718150138855, "perplexity": 105.083791449658}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860127878.90/warc/CC-MAIN-20160428161527-00116-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/derivatives-of-square-roots.19916/ | Derivatives of square roots
1. Apr 13, 2004
the1024b
How can i find the derivative of a function like this:
f(x) = sqrt( 1 - x² )
2. Apr 13, 2004
Hurkyl
Staff Emeritus
Do you know how to write a square root with exponents?
3. Apr 13, 2004
the1024b
(1 - x² )^(1/2) ?
4. Apr 13, 2004
Hurkyl
Staff Emeritus
That's right! Now, you just need to apply what you know about differentiating expressions like that.
5. Apr 13, 2004
the1024b
si will that be:
1/2((1-x²)/2)^(-1/2)
?
6. Apr 13, 2004
HallsofIvy
Staff Emeritus
Not quite. You have one too many "1/2"s (you don't want that "/2" inside the square root and you didn't use the chain rule.
You need to multiply by the derivative of 1-x2.
Last edited: Jul 17, 2009
7. May 13, 2009
mathsn00b
Hi,
I have a similar problem, I need to differentiate sqrt(x^2 + y^2) in terms of x and y.
Starting this I took the simple step (x^2 + y^2)^(1/2)...
My next step is a guess and I am lost after it....(1/2)(x^2 + y^2)(-1/2)....
Any help would be much appreciated.
8. May 13, 2009
jbunniii
If by "in terms of x and y", you mean you want to calculate the partial derivatives, then for the partial derivative with respect to x, treat y as a constant and differentiate with respect to x as you normally would a function of one variable. For the partial derivative with respect to y, treat x as constant.
9. May 13, 2009
mathsn00b
thanks, would I do this by...
df/dx = 1/2(x^2 + y^2)^(-1/2).2x = x/sqrt(x^2 + y^2) and...
df/dy = 1/2(x^2 + y^2)^(-1/2).2y = y/sqrt(x^2 + y^2) ?
thanks for your help so quickly.
10. May 13, 2009
jbunniii
Looks good to me.
11. Jul 16, 2009
68Pirate
What if i have a problem similar to these however now its 4/ ^5sqrt(x^5)
12. Jul 17, 2009
Matthollyw00d
If that is meant to be 4^(5(sqrt(x^5))), then you can easily rewrite this to equal
4^(5(x^(5/2)) And using what you know from differentiating exponentials and chain rule, you should be able to get the rest.
Similar Discussions: Derivatives of square roots | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8531163930892944, "perplexity": 1346.096271468207}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948596051.82/warc/CC-MAIN-20171217132751-20171217154751-00193.warc.gz"} |
https://encyclopediaofmath.org/index.php?title=Clemens%27_conjecture&diff=cur&oldid=27837 | # A dimension-counting heuristic
Nearly thirty years ago, Herbert Clemens [Cl0] made the following speculation:
On a general complex quintic threefold, there are only finitely many smooth rational curves of fixed degree d ≥ 1.
The following naïve dimension count, recorded by Katz [Ka], justifies (a belief in) the conjecture. Note that hypersurfaces of any given degree $$m$$ determine a projective space $$\mathbb{P}^N$$, while rational curves embedded in a target $$\mathbb{P}^n$$ are defined by $$(n+1)$$-tuples of polynomials in the homogeneous coordinate variables of $$\mathbb{P}^1$$. For each positive integer $$d$$, there is an incidence scheme $$\Phi_d$$ that parameterizes rational curves of degree $$d$$ embedded in hypersurfaces of degree $$m$$.
Now assume $$n=4$$ and $$m=5$$. Five-tuples of homogeneous degree-$$d$$ polynomials in 2 variables give $$5d+5$$ coefficients in total, and quotienting by automorphisms of $$\mathbb{P}^1$$ and rescaling yields a $$(5d+1)$$-dimensional smooth and irreducible space. Fix a choice of smooth rational curve $$C$$ in $$\mathbb{P}^4$$. Its preimage in $$\Phi_d$$ is the projectivization of the kernel of the natural restriction of global sections $r: H^0(\mathbb{P}^4, \mathcal{O}_{\mathbb{P}^4}(5)) \rightarrow H^0(C, \mathcal{O}_C(5)).$
The dimension of the domain of $$r$$ is the binomial coefficient $$\binom{5+4}{5}= 126$$, while the dimension of its target is computed by Riemann–Roch to be $$5d+1$$. In other words, we expect $$C$$ to impose $$5d+1$$ conditions on quintic hypersurfaces, which comprise a $$\mathbb{P}^{125}$$. Since $$C$$ itself varies in a $$(5d+1)$$-dimensional family, we thus expect $$\Phi_d$$ to be 125-dimensional. So it seems plausible that the projection of $$\Phi_d$$ onto the space of quintics should be generically finite.
## Remarks
• The flaw in the heuristic argument is that it implicitly assumes that the restriction $$r$$ of global sections is surjective, which isn't the case in general.
• Strictly speaking, a general hypersurface as above belongs to the complement of a countable union of proper subvarieties of the projective space $$\mathbb{P}^N$$ of hypersurfaces, the union being indexed by the degrees $$d$$ of rational curves. Elsewhere in the literature, such a hypersurface might be called very general.
# Clemens' conjecture and mirror symmetry
Quintic hypersurfaces in $$\mathbb{P}^4$$ give particularly natural examples of three-dimensional Calabi–Yau manifolds. The canonical bundle of any Calabi–Yau $$X$$ is trivial; accordingly, the adjunction formula implies that the normal bundle $$\mathcal{N}_{C/X}$$ of any curve $$C$$ embedded in $$X$$ is of degree $$-2$$. So for a generic choice of $$C$$, one would expect that the normal bundle splits as $$\mathcal{N}_{C/X}= (\mathcal{O}_C(-1))^{\oplus 2}$$, which in turn would imply that $$C$$ is a rigid subvariety of $$X$$ in the sense of deformation theory. Indeed, Clemens used an explicit deformation-theoretic argument [Cl-1] to prove the existence of such rational curves of degree $$d \geq 1$$ on a general quintic threefold $$X$$ for infinitely many values of $$d$$; subsequently, Katz [Ka] used an existence result of Mori's for curves on K3 surfaces to extend Clemens' argument to every positive value of $$d$$. The normal bundle splitting is part of what is commonly called the (the strong form of) Clemens' conjecture for the quintic:
Conjecture 2.1.
Let $$X \subset \mathbb{P}^4$$ denote a general quintic threefold. For every positive integer $$d \geq 1$$, the following statements hold.
• There is a finite, positive number of irreducible rational curves $$C \subset X$$ of degree $$d$$.
• These curves are all disjoint and reduced.
• The only singular irreducible rational curves are 17,601,000 six-nodal plane quintics.
• The normalization $$f: \mathbb{P}^1 \rightarrow X$$ of any rational curve $$C$$ on $$X$$ has normal bundle $$\mathcal{N}_f=\mathcal{O}(-1)^{\oplus 2}$$.
In one of the first stunning enumerative applications of mirror symmetry, Candelas, de la Ossa, Green and Parkes [CdGP] computed genus-zero Gromov–Witten invariants $$n_d$$ — the so-called instanton numbers — for the general quintic on the basis of physical considerations. The last twenty years have seen an increasingly-refined development of "virtual" techniques for computing Gromov–Witten invariants, notably involving $$T$$-equivariant localization. In the case of the quintic threefold, the instanton numbers were confirmed mathematically by Givental [G] and Lian–Liu–Yau [LLY]. However, their enumerative significance relies upon Clemens' conjecture, which remains unproved except when $$d \leq 11$$ (see the next section).
## Remarks
• A weakened form of Clemens' conjecture is obtained by dropping the stipulation that every curve of degree $$d \neq 5$$ be smooth.
• Vainsencher [Va] (see also [KlPi] for an argument with more details) proved the existence of 17,601,000 six-nodal plane quintics on $$F$$, each of which is the intersection of $$F$$ with a sixfold tangent plane. A proof that the corresponding normal bundles split as $$\mathcal{O}(-1)^{\oplus 2}$$ appears in [CK], Sec. 9.2.2.
• The hypothesis that $$X$$ be general is important. For an explicit study of an interesting family of special quintics containing lines, see [Mu].
• One might still speculate that the finiteness statement for irreducible rational curves (viewed as morphisms from $$\mathbb{P}^1 \rightarrow X$$) holds for a broader class of Calabi–Yau threefolds, for example those with Picard rank 1. Voisin [Vo3], Rmk.3.24, however, shows that the latter speculation is false for double covers of $$\mathbb{P}^3$$ ramified along an octic surface. Indeed, the preimages in the double cover of lines that intersect the octic with multiplicities $$(2,2,2,1,1)$$ determine a one-parameter family of nodal curves. (Note that the incidence type is misprinted in [Vo3].)
• In the same paper, Voisin shows that Clemens' conjecture contradicts the following conjecture of Lang [Vo3], Conj. 2.13: Every variety not of general type is covered by the (union of) images of non-constant rational maps from abelian varieties. In the case of the quintic, Lang's conjecture predicts that a general quintic is swept out by elliptic curves. Note that the double covers of the preceding remark satisfy Lang's conjecture [Vo3], Rmk. 2.17: explicitly, they are swept out by the elliptic normalizations of preimages in the double cover of lines that intersect the octic with multiplicities $$(2,2,1,1,1,1)$$.
# Rational curves of low degree on a general quintic 3-fold
Clemens' conjecture for rational curves of degree $$d \leq 7$$. His method of proof was in two steps. First he used deformation theory to show that
Clemens' conjecture holds for smooth rational curves $$C$$ of degree $$d$$ provided the incidence scheme $$\Phi_d$$ is irreducible.
He then proved that $$\Phi_d$$ is irreducible for all $$d \leq 7$$, by arguing that the fibers of its projection onto the space $$M_d$$ of degree-$$d$$ rational curves are equidimensional projective spaces. Equidimensionality follows from an effective cohomological vanishing result for ideal sheaf cohomology due to Gruson, Lazarsfeld, and Peskine [GLP].
Subsequently, Johnsen and Kleiman [JK1] extended Katz's analysis to show that the only reduced connected curves of degree $$d$$ at most 9 on $$F$$ with rational components are irreducible and either smooth or six-nodal plane quintics. Their basic insight is to stratify $$M_d$$ by locally-closed subsets $$M_{d,i}$$, where $$i:=h^1(\mathcal{I}_{C/\mathbb{P}^4}(5))$$ and $$C$$ is the image of $$f$$. Pulling back by $$\pi_d:\Phi_d \rightarrow M_d$$, they obtain a corresponding stratification of the incidence scheme $$\Phi_d$$ into loci $$\Phi_{d,i}$$, and they show that the projection $$\Phi_d \rightarrow M_d$$ is dominant exactly over $$M_{d,0}$$, where all the fibers are equidimensional. To do so they use [GLP], as well as codimension estimates for rational curves with singularities, to handle curves with positive arithmetic genus.
The next case, that of rational curves of degree 10, is interesting in its relation to mirror symmetry. In fact, Pandharipande showed [CK], Sec. 9.2.3 that the finiteness of the Hilbert scheme of degree-10 rational curves on the general quintic implies that the instanton number $$n_{10}$$ is given by $n_{10}= 6 \times 17,601,000 +\#\{\text{smooth rational curves of degree 10 in }F\}.$
On the other hand, mirror symmetry [CdGP] includes Vainsencher's singular quintics in its count of rational curves of degree five, but fails to count six double covers corresponding to each of these. So 10 is the first degree $$d$$ for which the instanton number $$n_d$$ fails to count smooth rational curves of degree $$d$$ on the general quintic $$F$$; moreover, the discrepancy between $$n_{10}$$ and the actual number of smooth rational curves is explained by double covers of nodal plane quintics. More generally, enumerative contributions arising from multiple covers of 6-nodal plane quintics mean that the relationship between instanton numbers $$n_d$$ and counts of rational curves of degree $$d$$ is subtle whenever $$d$$ is divisible by 5.
Recently, Cotterill [Co1,Co2] has proved Clemens' conjecture in its strong form for rational curves of degree at most 11. He extends the stratification-based analysis of Johnsen and Kleiman by obtaining new bounds on ideal sheaf cohomology obtained from a combinatorial analysis of monomial ideas associated to Groebner degenerations of embedded curves. He also uses the stratification of rational curves according to the splitting types of their restricted tangent bundles described in \cite{Ve}. The upshot is that the incidence scheme $$\Phi_d$$ of rational curves in quintics is irreducible whenever $$d \leq 11$$.
## References
• [CdGP] P. Candelas, X. de la Ossa, P. Green and L. Parkes, A pair of Calabi–Yau manifolds as an exactly soluble superconformal field theory, Nuclear Phys. B359 (1991), 21–74.
• [Cl-1] H. Clemens, Homological equivalence, modulo algebraic equivalence, is not finitely generated, Publ. Math. IHES 58 (1983), 19–38.
• [Cl0] H. Clemens, Curves on higher-dimensional complex projective manifolds, Proc. International Cong. Math., Berkeley, 1986, 634–640.
• [Cl1] H. Clemens, Curves on generic hypersurfaces, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 629–636.
• [Co1] E. Cotterill, Rational curves of degree 10 on a general quintic threefold, Comm. Alg. 33 (2005), no. 6, 1833-1872.
• [Co2] E. Cotterill, Rational curves of degree 11 on a general quintic 3-fold, Quart. J. Math. 63 (2012), no. 3, 539–568.
• [CK] D. Cox and S. Katz, Mirror symmetry and algebraic geometry, AMS, 1999.
• [G] A. Givental, The mirror formula for quintic threefolds, in Northern California Symplectic Geometry Seminar", 49–62, Amer. Math. Soc. Transl. Ser. 2 196, AMS, Providence, RI, 1999.
• [GLP] L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), 491–506.
• [HRS] J. Harris, M. Roth, and J. Starr, Rational curves on hypersurfaces of low degree, J. reine angew. Math. 571 (2004), 73–106.
• [JK1] T. Johnsen and S. Kleiman, Rational curves of degree at most 9 on a general quintic threefold, Comm. Alg. 24 (1996), 2721–2753.
• [JK2] T. Johnsen and S. Kleiman, Towards Clemens' conjecture in degrees between 10 and 24, Serdica Math. J. 23 (1997), 131–142.
• [Ka] S. Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math. 60 (1986), 151–162.
• [Ka1] S. Katz, Degenerations of quintic threefolds and their lines, Duke Math. J. 50 (1983), 1127–1135.
• [Ka2] S. Katz, Lines on complete intersection threefolds with $$K=0$$, Math. Z. 191 (1986), 293–296.
• [Ka3] S. Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math. 60 (1986), 151–162.
• [KlPi] S. Kleiman and R. Piene, Node polynomials for families: methods and applications, Math. Nachr. 271 (2004), 69-90.
• [LLY] B. Lian, K. Liu, and S.-T. Yau, Mirror principle I, Asian J. Math. 1 (1997), 729–763.
• [Mu] A. Mustata, Degree 1 curves in the Dwork pencil and the mirror quintic, arXiv preprint \url{math.AG/0311252.pdf}.
• [Va] I. Vainsencher, Enumeration of $$n$$-fold tangent hyperplanes to a surface, J. Alg. Geom. 4 (1995), 503–526.
• [Ve] J.L. Verdier, Two-dimensional sigma-models and harmonic maps from $$S^2$$ to $$S^n$$, in "Group Theoretical Methods in Physics", Springer Lecture Notes in Physics 180 (1983), 136–41.
• [Vo1] C. Voisin, On a conjecture of Clemens on rational curves on hypersurfaces, J. Diff. Geom. 44 (1996), 200–214.
• [Vo2] C. Voisin, A correction to "A conjecture of Clemens on rational curves on hypersurfaces", J. Diff. Geom. 49 (1998), 601-611.
• [Vo3] C. Voisin, On some problems of Kobayashi and Lang; algebraic approaches, in "Current developments in mathematics, 2003", 53–125, Int. Press, Somerville, MA, 2003.
How to Cite This Entry:
Clemens' conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clemens%27_conjecture&oldid=27837 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.876144289970398, "perplexity": 704.3024039396964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347417746.33/warc/CC-MAIN-20200601113849-20200601143849-00110.warc.gz"} |
https://physics.stackexchange.com/questions/270705/how-to-imagine-the-electromagnetic-waves?noredirect=1 | # How to imagine the electromagnetic waves? [duplicate]
I've learned at school, that the electromagnetic radiation consists of photons. And all the frequencies of the wavelengths of photons define electromagnetic spectrum. The lower wavelengths are ionizing radiation. In the middle of the spectrum there is the visible light, and above that infrared, microwave, and radio waves.
But the thing nobody told me, is that: How is it possible to imagine a wave? I don't think, that the photon is physically moving sideways, and that is called the waving. But what is it then?
The explanation of the wavelength is, that it is a spatial period of the wave—the distance over which the wave's shape repeats. But what is this wave exactly? How can a single photon has it?
• Possible duplicate of physics.stackexchange.com/q/74384 – garyp Jul 30 '16 at 11:22
• That question is not answered – Iter Ator Jul 30 '16 at 11:24
• Focus on the waves; study how radio waves are generated and detected. Then consider the same for light. The photon is the quantum of the electromagnetic field, and is obtained mathematically by quantization of the field modes. It's useful when quantum effects are important, but if classical waves are able to explain what is happening, photons only make the description more complicated. – Peter Diehr Jul 30 '16 at 11:29
• Don't take the drawings and pictures of any physical process too literally, they are only models and aids to give you an idea of what is going on. They are always based on classical concepts, as it is impossible to visually imagine the true picture of a lot of modern physical concepts. – user108787 Jul 30 '16 at 11:40
• Possible duplicate of What is the relation between electromagnetic wave and photon? – John Rennie Jul 30 '16 at 14:55
I've learned at school, that the electromagnetic radiation consists of photons.
The electromagnetic wave is described by the solution of classical maxwell's equation which has a sinusoidal dependence for the electric and magnetic fields perpendicular to the direction of motion of the wave. It is called a wave for this reason and the frequency is the repetition rate of the sinusoidal pattern.
Quantum mechanically the classical wave is an emergent phenomenon. It is built up by photons with an energy associated with the observable frequency of the emergent classical beam, E=h*nu.
A photon only has this energy definition and a spin 1 h orientation either in its direction of motion or against it, where h is Planck's constant. The beam built up in the image, is built up by the individual photons ( middle image).
This happens because the quantum mechanical wavefunction of a photon has the E and B information in its complex form, ( a solution of a quantized maxwell equation) and the superposition of photons builds up the classical fields with the frequency nu.
And all the frequencies of the wavelengths of photons define electromagnetic spectrum.
The frequencies of the classical electromagnetic spectrum define the energy of the photon, h*nu, not the photon the frequency, because it is only the probability of detection distribution of a single photon that "waves" in space, not the photon itself.
How is it possible to imagine a wave? I don't think, that the photon is physically moving sideways, and that is called the waving. But what is it then?
One need not imagine the photon as a wave. Only the probability of detecting it, as seen in this answer.
The explanation of the wavelength is, that it is a spatial period of the wave—the distance over which the wave's shape repeats.
This is true for the emergent classical wave.
How can a single photon has it?
A single photon has only a detection probability distribution that "waves", as explained above. It is not a wave.
• Why is a single photon not a wave? – velut luna Jul 30 '16 at 15:33
• A single photon has never been registered as a wave. It leaves an (x,y) point interaction foot print as in the single photon detection in the link physics.stackexchange.com/questions/269077/… . It is the accumulation of photons that shows interferences characteristic of a wave nature, but the accumulation is a probability distribution for many measures of photons. – anna v Jul 30 '16 at 18:43
• I think that depends on what you mean by "wave". – velut luna Jul 31 '16 at 1:58
• @AlphaGo a wave was first defined in water and then in acoustics and electromagnetism. It is a periodic variation of energy in space and time that is fitted with sinusoidal functions. These functions are solutions of what are called wave differential equations . Hence the identification with waves of the solutions of the shrodinger equation, except it is not energy that is waving, but probability. – anna v Jul 31 '16 at 3:06
Maxwell's equations have certain stationary states. We can obtain these so called modes and each classical waveform can be built as a linear combination of these modes.
In process called second quantization, we (hand wavingly) put particles into these modes. These particles are photons. Each mode can have 0, 1, 2 photons.
But there is more: we know from the uncertainity principle that no dynamic degrees of freedom can be absolutely confined, since that would imply infinite momentum. That holds for the coefficient of this electromagnetic mode as well, and hence there is always vacuum fluctuations of the electromagnetic field.
In other words, each mode can be represented as a quantum oscillator. (One derives an equation of motion for a mode, and realizes that some quantities behave like momentum and some like position). Quantum 101 tells that the modes of a quantum oscillator are quantized.
Now, we can have these modes in weird shaped cavity and hence we can have very structured modes with indefinte momentum. However, usually photons are measured in far field of the sample such that they have definite momentum and energy.
So, photon does not oscillate to any direction. Photon is an "occupation" of an electromagnetic mode which oscillates.
One more analogy to help to think about this: One could take a vibrating string and solve it's fundamental mode (say 440Hz). If this would be quantized, one can nerver find the string at rest due to uncertainity principle. Further, we will find that the string can only have quantized amount of energy. In other words, the magnitude of the vibrations is quantized. In other words, we can count how many (integer) energy quanta there are in the string. Let's all this quanta a vibron. Now, that is essentially the same thing to electromagnetism and phonons save Lorentz invariance, special relativity on masslesd particle, commutation relations of spin 1-particles, and some other compex stuff out of scope. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9479180574417114, "perplexity": 325.86282157862996}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178364008.55/warc/CC-MAIN-20210302125936-20210302155936-00327.warc.gz"} |
http://tex.stackexchange.com/questions/11031/how-to-do-the-curvy-l-for-lagrangian-or-laplace-transforms/11054 | # How to do the 'curvy L' for Lagrangian or Laplace Transforms?
I am new to TeX, working on it for about 2 months. Have not yet figured out how to script the 'curvy L' for Lagrangian and/or for Laplace Transforms.
As of now I am using the 'L' - which is not good! :-(
Any help?
UPDATE The 2 best solutions are;
\usepackage{ amssymb }
\mathcal{L}
and
\usepackage{ mathrsfs }
\mathscr{L}
I got my answers at, http://detexify.kirelabs.org/classify.html
-
if using Scientific word, click on this panel !Mathematica graphics and if using Lyx, click on this panel !Mathematica graphics – Nasser Mar 7 '14 at 4:56
## 3 Answers
You have been told how to get a curved L. But here's some more general advice, which also applies in this situation: In cases such as this, always create your own shortcut macro, say
\newcommand{\Lagr}{\mathcal{L}}
This way, if you ever decide that that curly L is not quite the one you like, it is easy to switch. Also, even before you knew the answer to this question, you could have written
\newcommand{\Lagr}{L}
in your preamble. Then you could start/continue writing the document, use ordinary capital Ls where you want, and \Lagr wherever you actually mean a Lagrangian, and then later changing the definition of \Lagr to something appropriate. This way, you wouldn't have to manually search for each occurence of a capital L and decide if it's a Lagrangian or not. Clearly \Lagr (or whatever you want to call this macro) is also easier to type than \mathcal{L}, and it makes the source code much more readable.
Another advantage, which is probably more subtle, since you're new to LaTeX, is that we can make the curvy L exactly the type of math we want. TeX distinguishes between ordinary letters, binary operators, binary relations, opening and closing parenthesis and so on; the spacing between two ordinary letters is not the same as the spacing between the a, +, and b in $a+b$. So since the Lagrangian is a kind of operator, we could say
\newcommand{\Lagr}{\mathop{\mathcal{L}}}
But in the case of operators, the package amsmath (which you are most likely using; if not, you should) provides a somewhat better mechanism:
\DeclareMathOperator{\Lagr}{\mathcal{L}}
Added: Another (related) tip: Even if you are using the same notation for two different things, it is best to make a separate macro for each. In this case you might have
\DeclareMathOperator{\Lagr}{\mathcal{L}}
\DeclareMathOperator{\Lapl}{\mathcal{L}}
The reason is the same as in the L vs. \Lagr case above: If you at some point decide that using \mathcal{L} for both is a bad idea, you would have to find each occurence of \Lagr and figure out if it is really a Laplacian. Using macro names carrying semantic meaning is one of the great powers of TeX.
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Thanks .... 'GURU' ! :-) – Arkapravo Feb 16 '11 at 5:04
$\mathcal{L}$
Have a look at “How to look up a math symbol?” for ideas how you can easily find a particular symbol.
-
You can also use the mathrsfs package (put \usepackage{mathrsfs} in your preamble), and its command \mathscr (e.g. $\mathscr{L}$ should give you what you want) to get more script-like, curvy letters.
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Can you please elaborate ? you are talking to a noob ! :-) – Arkapravo Feb 15 '11 at 4:40
I just edited my answer. – Benoît Kloeckner Feb 15 '11 at 15:41
Thank you – Arkapravo Feb 16 '11 at 5:05
You misspelled the name of the package (should be mathrsfs, not mathsrsf). It took me a while to figure it out... – Vivi Jan 9 '13 at 8:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8530430197715759, "perplexity": 1210.8100456593363}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783398628.62/warc/CC-MAIN-20160624154958-00193-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/speakers-and-work.250886/ | # Speakers and work?
1. Aug 19, 2008
### tfowler306
we measure amplifier output in watts, and speakers in db spl which is logrimithic to the power applied, so if I have a 4 ohm speaker that will create 91 db of spl when 2.83v is applied how much work is done?
Watts = V^2/R = W=2.83^2/4 = W=8/4 so W= 2 this is the power applied to the speaker if I want to know the force applied to the speaker cone (f=ma) the moving mass with air load for the speaker is 142.7grams, how do I find the acceleration of the speaker cone? If the speaker is producing a steady frequency lets say 40Hz it would have no acceleration it would be a constant velocity. I am stumped on where to go for this one...
2. Aug 19, 2008
### Staff: Mentor
The speaker cone certainly accelerates - the frequency of 40 hz means it is vibrating back and forth 40 times a second. Ideally, it should vibrate a distance equal to the wavelength, but at low frequencies, it probably can't move far enough. You can calculate the distance traveled using the speed of sound - it's the wavelength of the sound wave. Then see if it makes sense as a distance. Realistically, a woofer can probably only move an inch or so in and out.
Note that the actual efficiency of sound systems is very low. The amount of energy actually moving through the air is a few percent, at most, of the rating of the amplifier.
3. Aug 19, 2008
### rcgldr
http://en.wikipedia.org/wiki/Loudspeaker
http://en.wikipedia.org/wiki/Sound_power_level
http://en.wikipedia.org/wiki/Sound_pressure_level
Generally, better speakers have more negative feedback, acoustical (sealed) and/or electrical dampening. Most of this dampening is to prevent overshoot as with any servo system. Also although the inner diaphram of a speaker moves back in forth in sync with the signal, the speaker itself ends up with wave patterns, with usually moving peaks and valleys, although the entire speaker moves a lot at lower frequencies.
So the efficiency of speakers is pretty low (about 1/2%), and lower still for high quality sounding speakers. Power output is a fraction of power consumed, especially with musically oriented (as opposed to home theater) speakers.
Last edited: Aug 19, 2008
4. Aug 20, 2008
### tfowler306
the wave length of a 20Khz signal is in the 150 meeter band, so the speaker cone cant move to the length of the sound wave. That in itself is confusing to me, because a tweeter cone on average will only move 0.03 mm. but from what you said it gives me a direction on how to calculat the average velocity of the cone frome the Thele Small perameters. The current speaker has a 7/16" max excursion and a motor force of 13.93 Tm at 750W. so from there I can figure out the excursion at 2W and that is the total distance traveled back and forth at 40times/secdevided by 2 because a sinwave is from 0 to 0 which actually would equal a velocity of 0 because there is no displacement, but for one full stroke of the speaker I can have a velocity for that period of the stroke
does this make sense?
5. Aug 22, 2008
### rcgldr
Only the pole piece (see wiki article) moves directly in response to the signal received. The diaphram (see wiki article) ends up with multiple peaks and valleys that actually produce the sound. Even at a single fixed frequency, these peaks and valleys will move around unless there is some harmonic relationship between the diaphram and the input signal.
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https://www.physicsforums.com/threads/questions-about-doppler-effect.188405/ | 1. Oct 1, 2007
### Charlie X
the formula of Doppler Effect
f = fs(v + vd)/(v - vs)
(v = speed of sound; d = detector; s = source)
Simply from this formula, it can be seen that vd has differenct effect on the frequency receiver from vs.
when vd or vs approaches the speed of sound, this difference is pronounced.
make v = 343 m/s, fs = 1 Hz, the the source and detector move towards each other
Case 1
vd = 340 m/s, vs = 0
f = 683/343 = 1.99 Hz
Case 2
vd = 0, vs = 340 m/s,
f = 343/3 = 114 Hz
why would there be a difference? Isn't the source and detector move relative to each other?
my teacher says it has sth to do with medium, but i still don't get it.
2. Oct 2, 2007
### AbedeuS
Trying to derive the equation you use from base doppler shift ones, not working for me >,< and I only explain stuff now if im 100% sure of where the equations come from, to understand how messy mine is, I have produced:
$$F_{observer} = \frac{V^{2}*F_{source}}{(V-A_{source})(V+B_{observer})} = \frac{V^{2}*F_{Source}}{V^{2}+VB-VA-AB}$$
Where
A is source velocity
B is observer velocity
V is wave speed
F is Wave frequency
So yeah, I dont trust my math :)
Last edited: Oct 2, 2007
3. Oct 2, 2007
### AbedeuS
May aswell wait for a PF guy with more knowledge to pop over the hill annny minute now :)
4. Oct 2, 2007
### Piewie
Wrong formula
Indeed, the formula I wrote here was not correct, therefore I deleted it.
Last edited: Oct 2, 2007
5. Oct 2, 2007
### AbedeuS
Yeah, what PieWie said, my equation is just a linear combination of the source moving and the observer moving, but that was my attempt to try and show how your irrational equation actually works I guess, since I went off the suggestion that reletive speed wouldnt work, I just combined the equation twice...I really should have just used relative velocity >,<, to rewrite PieWie's equation for easier reading then going explosive on your teacher.
$$f_{observed} = f_{source}\leftbracket[\frac{V}{V+V_{Relative}}\rightbracket]$$
Thanks PieWie ^_^ I too was wondering why the "compression" (Increase in frequency) was disproportionate to the "decompression" (decrease from moving away from source)
6. Oct 2, 2007
### Staff: Mentor
To understand why there's a difference, you need to understand how the Doppler formula is derived.
The fundamental difference is that when you (the detector) remain still but the source moves towards you, the speed of sound with respect to you doesn't change. But the frequency does, since the wavelength is getting shorter. If the original wavelength is $\lambda_0$, the new wavelength is $\lambda = \lambda_0 - v_s / f_0$. Since for any wave, $v = f\lambda$, the new frequency is:
$$f = v/\lambda = v/(\lambda_0 - v_s / f_0) = f_0 \frac{v}{(v - v_s)}$$
When the source remains fixed but you (the detector) move towards it, the wavelength stays the same but the speed of sound (relative to you) is now $v' = v + v_d$. Thus the apparent frequency is:
$$f = v'/\lambda_0 = (v + v_d)/\lambda_0 = f_0 \frac{(v + v_d)}{v}$$
Of course, if both source and detector are moving, you get the combination of both effects:
$$f = f_0 \frac{(v + v_d)}{(v - v_s)}$$
I hope that makes a bit of sense.
7. Oct 2, 2007
### AbedeuS
I dont know weather to love doc'al or slap myself in the face :) or do both, so doc'al, how is this handled with EM Radiation? as the speed is constant in that case, would you just use a relative speed case for that?
EDIT: Oh, and I didnt quite understand:
$\lambda = \lambda_0 - v_s / f_0$
For my low-level ability, it sort of sounds like your treating the sources movement as a wave emmitter in its own right, but taking the wavelength produced from the sound... is it ok for a deeper explanation? You know I'm a picky little mofo.
Last edited: Oct 2, 2007
8. Oct 2, 2007
### AbedeuS
The $$F_{0}$$ From the first equation, is equal to the $$f$$ from the last one, so you substitute the $$f_{0}$$ in the second equation to give:
$$f_{final}=f_{0}\bracketleft[\frac{v(v+v_{d})}{v(v-v_{s})}\bracketright]$$ then cancel variables ^^
9. Oct 2, 2007
### Staff: Mentor
It turns out that way, but you also have to factor in time dilation. Read this: Relativistic Doppler Effect
The source is the emitter but I am modifying the wavelength since the source is moving with respect to the medium.
But cuddly and loveable, I'm sure!
10. Oct 2, 2007
### AbedeuS
And I apologise to doc'al, the equation makes perfect sense, since it was the first one i wrote in my notebook, then I used the hyperphysics one, but let me just clarify:
$$\lambda_{new} = \lambda_{old} - v_{s}*T$$
and because $$T=f^{-1}$$ then:
$$\lambda_{new} = \lambda_{old} - \frac{v_{s}}{f_{0}}$$
11. Oct 2, 2007
### AbedeuS
Sorry I didnt get that one, it was glaring me literally in the face in my PF notebook :), also, not to hiijack, but I did some maths in that "Pascals Principle Headache" post invloving gases rather than liquids, I'd really appreciate if you could have a glance over it whenever you have free time
12. Oct 2, 2007
### Staff: Mentor
Exactly right.
I'll give it a look a bit later.
13. Oct 2, 2007
### Piewie
If the source moves the sound waves have a different frequency in the medium (compared to when the source would stand still or move with a different velocity).
That is NOT the case for the detector: the frequency in the medium does not depend on the speed of the detector. Note that the frequency that is indicated by the detecor is not equal to the frequency in the medium. So... don't trust a moving sound-detector ;)
Can someone confirm this explanation?
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https://qitheory.blogs.bristol.ac.uk/2019/02/04/a-converse-to-the-polynomial-method-arunachalam-et-al-2017/ | # A Converse to the Polynomial Method (Arunachalam et al. 2017)
One of the most famous models for quantum computation is the black-box model, in which one is given a black-box called oracle encoding a unitary operation. With this oracle one can probe the bits of an unknown bit string $x \in\{-1,1\}^n$ (the paper uses the set $\{-1,1\}$ instead of $\{0,1\}$, which I will keep here) . The main aim of the model is to use the oracle to learn some property given by a Boolean function $f$ of the bit string $x$, for example, its parity. One use of the oracle is usually referred as a query, and the number of queries a certain algorithm performs reflects its efficiency. Quite obviously, we want to minimize the number of queries. In more precise words, the bounded-error quantum query complexity of a Boolean function $f : \{-1,1\}^n \to \{-1,1\}$ is denoted by $Q_\epsilon (f)$ and refers to the minimum number of queries a quantum algorithm must make on the worst-case input x in order to compute $f(x)$ with probability $1 - \epsilon$.
Looking at the model and the definition for the quantum query complexity, one might ask “but how can we find the query complexity of a certain function? Is there an easy way to do so?”. As usual in this kind of problem, finding the optimal performance of an algorithm for a certain problem or an useful lower bound for it is easier said than done. Nonetheless, there are some methods for tackling the problem of determining $Q_\epsilon (f)$. There are two main methods for proving lower bounds, known as the polynomial method and the adversary method. In this post we shall talk about the first, the polynomial method, and how it was improved in the work of Arunachalam et al.
The polynomial method
The polynomial method is a lower bound method based on a connection between quantum query algorithms and, as the name suggests, polynomials. The connection comes from the fact that for every t-query quantum algorithm $A$ that returns a random sign $A(x)$ (i.e. $\{-1,1\}$) on input $x$, there exists a degree$2t$ polynomial $p$ such that $p(x)$ equals the expected value of $A(x)$. From this it follows that if $A$ computes a Boolean function $f : \{-1,1\}^n \to \{-1,1\}$ with probability at least $1 - \epsilon$, then the polynomial $p$ satisfies $|p(x) - f(x)| \leq 2\epsilon$ for every $x$ (the factor of $2$ comes from the image being $\{-1,1\}$ instead of $\{0,1\}$). Therefore we can see that the minimum degree of a polynomial $p$ that satisfies $|p(x) - f(x)| \leq 2\epsilon$ for every $x$, called the approximate (polynomial) degree and denoted by $deg_\epsilon(f)$, serves as a lower bound for the query complexity $Q_\epsilon (f)$. Hence the problem of finding a lower bound for the query complexity is converted into the problem of lower bounding the degree of such polynomials.
Converse to the polynomial method
We now have a method for proving lower bounds for quantum query algorithms by using polynomials. A natural question that can arise is whether the polynomial method has a converse, that is, if a degree-$2t$ polynomial leads to a $t$query quantum algorithm. This would in turn imply a sufficient characterization of quantum query algorithms. Unfortunately, Ambainis showed in 2006 that this is not the case, by proving that for infinitely many $n$, there is a function $f$ with $deg_\epsilon(f) \leq n^\alpha$ and $Q_\epsilon (f) \geq n^\beta$ for some positive constants $\beta > \alpha$. Hence the approximate degree is not such a precise measure for quantum query complexity in most cases.
In the view of these negative results, the question that stays is, is there some refinement to the approximate polynomial that approximates $Q_\epsilon(f)$ up to a constant factor? Aaronson et al. tried to answer this question around 2016 by introducing a refined degree measure, called block-multilinear approximate polynomial degree and denoted by $bm\text{-}deg_\epsilon(f)$, which comes from polynomials with a so-called block-multilinear structure. This refined degree lies between $deg_\epsilon(f)$ and $2Q_\epsilon(f)$, which leads to the question of how well that approximates $Q_\epsilon(f)$. Once again, it was later shown that for infinitely many $n$, there is a function $f$ with $bm\text{-}deg_\epsilon(f) = O(\sqrt{n})$ and $Q_\epsilon(f) = \Omega(n)$, ruling out the converse for the polynomial method based on the degree $bm\text{-}deg_\epsilon(f)$ and leaving the question open until now, when it was answered by Arunachalam et al., who gave a new notion of polynomial degree that tightly characterizes quantum query complexity.
Characterization of quantum algorithms
In few words, it turns out that $t$-query quantum algorithms can be fully characterized using the polynomial method if we restrict the set of degree-$2t$ polynomials to forms that are completely bounded. A form is a homogeneous polynomial, that is, a polynomial whose non-zero terms all have the same degree, e.g. $x^2 + 3xy + 2y^2$ is a form of degree $2$. And the notion of completely bounded involves the idea of a very specific norm, the completely bounded norm (denoted by $\|\cdot\|_{cb}$), which was originally introduced in the general context of tensor products of operator spaces. But before we venture ourselves into this norm, which involves symmetric tensors and other norms, let us state the main result of the quantum query algorithms characterization.
Let $\beta: \{-1,1\}^n \to [-1,1]$ and $t$ a positive integer. Then, the following are equivalent.
1. There exists a form $p$ of degree $2t$ such that $\|p\|_{cb} \leq 1$ and $p((x,\boldsymbol{1})) = \beta(x)$ for every $x \in \{-1,1\}^n$, where $\boldsymbol{1}$ is the all-ones vector.
2. There exists a $t$-query quantum algorithm that returns a random sign with expected value $\beta(x)$ on input $x \in \{-1,1\}^n$.
In short, if we find a form of degree $2t$ which is completely bounded ($\|p\|_{cb} \leq 1$) and approximates a function $f$ that we are trying to solve, then there is a quantum algorithm which makes $t$ queries and solves the function. Hence we have a characterization of quantum algorithms in terms of forms that are completely bounded. But we still haven’t talked about the norm itself, which we should do now. It will involve a lot of definitions, some extra norms and a bit of C*-algebra, but fear not, we will go slowly.
The completely bounded norm
For $\alpha \in \{0,1,2, \dotsc \}^{2n}$, we write $|\alpha| = \alpha_1 + \dotsc + \alpha_{2n}$. Any form $p$ of degree $t$ can be written as
$p(x) = \sum_{|\alpha| = t} c_{\alpha} x^\alpha$,
where $c_{\alpha}$ are real coefficients. The first step towards the completely bounded norm of a form is to define the completely bounded norm of a tensor, and the tensor we use is the symmetric $t$-tensor $T_p$ defined as
$(T_p)_\alpha = c_\alpha/|\{\alpha\}|!$
where $|\{\alpha\}|!$ denotes the number of distinct elements in the set formed by the coordinates of $\alpha$.
The relevant norm of $T_p$ is given in terms of an infimum over decompositions of the form $T_p = \sum_\sigma T^\sigma \circ\sigma$, where $\sigma$ is a permutation of the set $\{1,\dotsc,t\}$ and $(T^\sigma\circ\sigma)_\alpha = T^\sigma_{\sigma(\alpha)}$ is the permuted element of the multilinear form $T^\sigma$. So that the completely bounded norm of $p$ is kind of transferred to $T_p$ via the definition
$\|p\|_{cb} = \text{inf }\{\sum_\sigma \|T^\sigma\|_{cb} : T_p = \sum_\sigma T^\sigma\circ\sigma\}$.
Just to recap, with the coefficients of the polynomial $p$ we define the symmetric $t$-tensor $T_p$, which is then decomposed into the sum of permuted tensor (multilinear form) $T^\sigma$. We then define the completely bounded norm of $p$ as the infimum of the sum of the completely bounded norm of such tensor, but now without permuting it. Of course, we haven’t yet defined the completely bounded norm of such tensor, that is, what is $\|T^\sigma\|_{cb}$? We will explain it now.
The idea is to get a bunch of collections of $d \times d$ unitary matrices $U_1(i), U_2(i), \dotsc, U_t(i)$ for $i \in \{1, \dotsc ,2n\}$ and consider the quantity
$\|\sum_{i,j,\dotsc,k} T^\sigma_{i,j,\dotsc,k} U_1(i)U_2(j)\dotsc U_t(k)\|$.
We multiply the unitaries from these collections and sum them using the tensor $T^\sigma$ as weight, and then take the norm of the resulting quantity. But here we are using a different norm, the usual operator norm defined for a given operator $A$ as $\|A\| = \text{inf }\{c\geq 0 : \|Av\| \leq c\|v\| \text{ for all vectors } v\}$. Finally, with these ingredients in hand, we can define the completely bounded norm for $T^\sigma$, which is just the supremum over the positive integer $d$ and the unitary matrices, that is,
$\|T^\sigma\|_{cb} = \text{sup }\{\|\sum_{i,j,\dotsc,k} T^\sigma_{i,j,\dotsc,k} U_1(i)U_2(j)\dotsc U_t(k)\| : d \in \mathbb{N}, d \times d \text{ unitary matrices } U_i\}$.
If we can obtain the supremum of such norm over the size of the unitary matrices and the unitary matrices themselves, then we obtain the completely bounded norm of the tensor $T^\sigma$, and from this we get the completely bounded norm of the associated form $p$. Is there such a degree-$2t$ form with $\|p\|_{cb} \leq 1$ that approximates the function $f$ we want to solve? If yes, then there is a $t$-query quantum algorithm that solves $f$.
The proof
Let us briefly explain the proof of the quantum algorithms characterization that we stated above. Their proof involves three main ingredients. The first one is a theorem by Christensen and Sinclair showing that the completely-boundedness of a multilinear form is equivalent to a nice decomposition of such multilinear form. In other words, for a multilinear form $T$, we have that $\|T\|_{cb} \leq 1$ if and only if we can write $T$ as
$T(x_1,\dotsc,x_t) = V_1\pi(x_1)V_2\pi(x_2)V_3\dotsc V_t\pi(x_t)V_{t+1}$, (1)
where $V_i$ are contractions ($\|V_ix\| \leq \|x\|$ for every $x$) and $\pi_i$ are *-representations (linear maps that preserve multiplication operations, $\pi(xy) = \pi(x)\pi(y)$).
The second ingredient gives an upper bound on the completely bounded norm of a certain linear map if it has a specific form. More specifically, if $\sigma$ is a linear map such that $\sigma(x) = U\pi(x)V$, where $U$ and $V$ are also linear maps and $\pi$ is a *-representation, then $\|\sigma\|_{cb} \leq \|U\|\|V\|$.
The third ingredient is the famous Fundamental Factorization Theorem that “factorizes” a linear map in terms of other linear maps if its completely bounded norm is upper bounded by these linear maps. In other words, if $\sigma$ is a linear map and exists other linear maps $U$ and $V$ such that $\|U\|\|V\| \leq \|\sigma\|_{cb}$, then, for every matrix $M$, we have $\sigma(M) = U^\ast(M\otimes I)V$.
With these ingredients, they proved an alternative decomposition of equation (1) which was later used to come up with a quantum circuit implementing the tensor $T_p$ and using $t$ queries, and then showed that this tensor $T_p$ matched the initial form $p$. This alternative decomposition is
$T(x_1,\dotsc,x_t) = u^\ast U_1^\ast (\text{Diag}(x_1)\otimes I)V_1 \dotsc U_t^\ast (\text{Diag}(x_t)\otimes I)V_tv$, (2)
where $U_i, V_i$ are contractions, $u,v$ are unit vectors, $\text{Diag}(x)$ is the diagonal matrix whose diagonal forms $x$. The above decomposition is valid if, similarly to (1), $\|T\|_{cb} \leq 1$.
We can see that the decomposition involves the operator $\text{Diag}(x)$ intercalated by unitaries $t$ times. Having $\text{Diag}(x)$ as the query operator, it is possible then to come up with a quantum circuit implementing the decomposition (2). If we look more closely, decomposition (2) has unit vectors on both the left and right sides, which looks like an expectation value. So what is going on is that decomposition (2) can be used to construct a quantum circuit whose expectation value matches $T_p$, and hence the polynomial $p$ used to construct $T_p$. We won’t get into much details, but we will leave the quantum circuit so that the reader can have a glimpse of what is going on.
In the above figure representing a quantum circuit that has $T_p$ as its expectation value, the registers C, Q, W denote control, query and workspace registers. The unitaries $W_i$ are defined by $W_i = V_{i-1}U_i^\ast$, the unitaries $U, V$ have $W_1V_0 u$ and $W_{2t+1}U_{2t+1}v$ as their first rows, where $V_0$ and $U_{2t+1}$ are isometries ($A$ is an isometry iff $\|Ax\| = \|x\|$ for every $x$).
Application of the characterization: separation for quartic polynomials
A couple of years ago, in 2016, Aaronson et al. showed that the bounded norm (and also the completely bounded norm that we spent some time describing) is sufficient to characterize quadratic polynomials. More specifically, they showed that for every bounded quadratic polynomial $p$, there exists a one-query quantum algorithm that returns a random sign with expectation $Cp(x)$ on input $x$, where $C$ is an absolute constant. This readily prompted the question of whether the same is valid for higher-degree polynomials, that is, if the bounded norm suffices to characterize quantum query algorithms.
As you might expect by now, the answer is no, since it is the completely bounded norm that suffices, and Arunachalam et al. used their characterization to give a counterexample for bounded quartic polynomials. What they showed is the existence of a bounded (not completely bounded) quartic polynomial p that for any two-query quantum algorithm whose expectation value is $Cp(x)$, one must have $C = O(n^{-1/2})$, thus showing that $C$ is not an absolute constant. The way they showed this is by using a random cubic form that is bounded, but whose completely bounded norm is $poly(n)$ and then embedded this form into a quartic form.
Conclusions
The characterization they found is, in my opinion, apart from the obvious point that it answers the open question of the converse of the polynomial method, quite interesting in the sense that the set of polynomials we should look at is the ones that are bounded by a particular norm, the completely bounded norm. The interesting point is exactly this one, the connection between quantum query algorithms and the completely bounded norm that was first introduced in the general context of tensor products of operator spaces as mentioned before. The norm itself looks quite exotic and complicated, which is linked to a question that Scott Aaronson made at the end of Arunachalam’s talk in QIP 2019, “sure, we could define a particular norm as one that restricts the set of polynomials to the ones that admit a converse to the polynomial method”. Of course, such a norm would not be quite an interesting one. He carries on with something like “In what is this completely bounded norm different compared to that?”. If I remember correctly, Arunachalam gave an answer on the lines of the completely bounded norm appearing in a completely different context. But I still find it surprising that you would go through all the exotic definitions we mentioned above for the completely bounded norm and discover that it is what is needed for the converse of the polynomial method. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 168, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9846938252449036, "perplexity": 145.40380553810206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764494974.98/warc/CC-MAIN-20230127065356-20230127095356-00837.warc.gz"} |
https://www.physicsforums.com/threads/laplace-transform-with-heaviside-function.758522/ | # Laplace transform with Heaviside function
1. Jun 18, 2014
### ptrinka
Hello,
I am searching for the Laplace transform of this function
$$u_a(y)\frac{\partial c(t)}{\partial t}$$
where $$u_a(y)$$ is the Heaviside step function (a>0).
Can anyone help me?
2. Jun 18, 2014
### pasmith
Laplace transform with respect to which variable?
3. Jun 19, 2014
### ptrinka
With respect to c(t).
4. Jun 19, 2014
### ptrinka
Actually, I realized that there is an error in the equation. The correct equation is as follows:
$$u_a(t)\frac{\partial{c(t)}}{\partial t}$$
i.e. u_a is a function of t and NOT y.
Sorry for the mistake!
Similar Discussions: Laplace transform with Heaviside function | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9122896194458008, "perplexity": 3375.0849521481896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948575124.51/warc/CC-MAIN-20171215153355-20171215175355-00664.warc.gz"} |
http://math.stackexchange.com/questions/148359/an-abstract-alpha-contracting-dynamical-system/148376 | # An abstract $\alpha$-contracting dynamical system
$\newcommand{\f}{\phi}$$\newcommand{\ep}{\varepsilon}$$\newcommand{\R}{\mathbb R}$
Suppose $(\f_t)_{t\ge0}$ is an abstract dynamical system in a Banach space $(X,\|\mathord\cdot\|)$. Let $C(x,\ep)$ denote the closed ball with radius $r$ centered at $x\in X$.
Recall that an abstract dynamical system $(\phi_t)_{t\ge0}$ on $X$ is a collections of maps $\f_t: X\to X$ such that $\f_0$ is the identity map on $X$ and $\f_t\circ\f_{s}=\f_{t+s}$ for all $t,s\ge0$.
Assume the following:
Suppose that for all $t\ge0$ there are maps $f_t,g_t:X\to X$ such that $$\f_t(x)=f_t(x)+g_t(x).\tag{1}$$
Additionally, suppose that for all $r\ge0$ there is a $T_r\ge0$ and a map $h=h_{T_r}:[0,\infty)\to[0,\infty)$ where $$\lim_{t\to\infty}h(t)=0\tag{2}$$ and$$\overline{f_t[C(0,r)]}\text{ is compact whenever }t>T_r\tag{3}$$(the overline is the closure) and for all $t\ge0$ and all $x\in C(0,r)$: $$\|g_t(x)\|\le h(t) \tag4$$
Then $$\lim_{t\to\infty}\alpha(\f_t[A])=0$$for all bounded sets $A\subset X$, where $\alpha$ is the Kuratowki measure of noncompactness.
I have been told that this a well-known result from the theory of abstract dynamical systems, but I can't find a proof. Is there someone who knows how to prove this statement or knows a nice reference (preferably a book)?
-
Recall that $\alpha$ is monotone and subadditive: $\alpha(B+C)\le \alpha(B)+\alpha(C)$. [Reference: Nonlinear Functional Analysis by Deimling). Using (1), we find that $\alpha(\phi_t(A))\le \alpha(f_t(A))+\alpha(g_t(A))$. Pick $r$ such that $A\subset C(0,r)$. By (4) and (2) we have $\alpha(g_t(A))\le 2h(t)\to 0$ as $t\to\infty$. Finally, $\alpha(f_t(A))=0$ by (3) when $t>T_r$.
Thank you for your help. I understand your reasoning, bu I do not see why (4) and (2) imply $\alpha(g_t(A))\le 2\cdot h(t)$. – Gifty May 22 '12 at 20:43
@Gifty For any bounded set $B$ we have $\alpha(B)\le \mathrm{diam} \, B$ by the definition of $\alpha$ (cover $B$ by itself). Here $B=g_t(A)$, and since this set is contained in the closed ball of radius $h(t)$, its diameter does not exceed $2h(t)$. – user31373 May 22 '12 at 20:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9805083274841309, "perplexity": 97.74344816439061}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042990112.92/warc/CC-MAIN-20150728002310-00166-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://planetmath.org/turingcomputable | # Turing computable
## Primary tabs
Synonym:
Turing-computable
Type of Math Object:
Definition
Major Section:
Reference
## Mathematics Subject Classification
03D10 no label found68Q05 no label found
## Comments
### aka recursive''
Turing computable'' functions are also called recursive'' functions. (The confusion stems from the pre-Church-Turing days, when the concept of recursive function was defined by adding minimization to primitive recursive functions, and it was not clear that Turing machines had exactly the same power).
Could this synonym be added?
### Re: aka recursive''
I've changed the entry to mention that recursive means basically the same thing, but I'm not making it a formal synonym. This is because I think there should be a separate recursive function'' entry with that synonym, which should recieve the linking.
Alternatively this entry could be greatly expanded so that it covers recursive function theory, but I'm not the guy to write that (I'll orphan the entry if you want it though.)
apk | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 21, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8176958560943604, "perplexity": 2288.9097222536034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104681.22/warc/CC-MAIN-20170818140908-20170818160908-00161.warc.gz"} |
https://mathtuition88.com/2016/11/27/normal-extension/ | ## Normal Extension
An algebraic field extension $L/K$ is said to be normal if $L$ is the splitting field of a family of polynomials in $K[X]$.
Equivalent Properties
The normality of $L/K$ is equivalent to either of the following properties. Let $K^a$ be an algebraic closure of $K$ containing $L$.
1) Every embedding $\sigma$ of $L$ in $K^a$ that restricts to the identity on $K$, satisfies $\sigma(L)=L$. In other words, $\sigma$, is an automorphism of $L$ over $K$.
2) Every irreducible polynomial in $K[X]$ that has one root in $L$, has all of its roots in $L$, that is, it decomposes into linear factors in $L[X]$. (One says that the polynomial splits in $L$.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 20, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9690983295440674, "perplexity": 55.63384419167633}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463613135.2/warc/CC-MAIN-20170529223110-20170530003110-00455.warc.gz"} |
https://datatalks.club/blog/regularization-in-regression.html | Machine Learning Zoomcamp: Free ML Engineering course. Register here!
# Regularization in Regression
### Remedy for numerical instability
22 Sep 2022 by Ksenia Legostay
There are a lot of methods of how you can improve your ML model accuracy. They include feature engineering, missing value imputation, improvements in data quality, etc.
One of the effective approaches is regularization. It is a popular concept that helps to control coefficients under the numerical instability in computation taste such as model training.
In this article, we will take a closer look at why you might want to regularize your model. As an example, we will apply a basic regularization technique to a simple linear regression model and learn how it influences the model.
### Linear Regression
Linear regression is a supervised machine learning model, which can be expressed in a matrix form as follows:
$g(X) \approx y$
$X$ is a matrix where the features of observations are rows of the matrix and $y$ is a vector with the values we want to predict. Function $g(X)$ can be represented as a matrix-vector multiplication of feature matrix $X$ and some weight vector $w$:
$Xw = y$
After some transformations described in Training Linear Regression: Normal Equation lecture of Machine Learning Zoomcamp, weight vector $w$ can be represented as:
$w = (X^T X)^{-1} X^T y$
where
A matrix inversion should be considered with caution. If a matrix contains a column that is a linear combination of its other columns the matrix is singular, which means the inverse matrix does not exist.
### Why do we need regularization in Linear Regression
Linear dependent columns in a matrix is not a typical case in real-world problems, even though due to noise in the data, characteristics of your machine, OS, or NumPy version there might be some similar vectors in the above sense. When it happens, the weight vector $w$ can result in very large values (both positive and negative) and the overall model predictions won’t be useful.
To overcome this numerical instability problem we can refer to regularization. Regularization in linear regression guarantees the existence of inverse matrix $(X^T X)^{-1}$
One of the regularization techniques is adding a factor to the diagonal of matrix $X^T X$ like this:
$w = (X^T X + \alpha I)^{-1} X^T y$
where
This modification of the linear regression is commonly called Ridge Regression.
### How regularization can fix your Regression
Let’s demonstrate the effect of regularization through an example and see that the more regularization we add (factor $\alpha$), the smaller the weights $w$ become.
We will build a Linear Regression model for predicting car prices based on a dataset from Kaggle - Car Prices Dataset.
The full code is in the notebook here.
For the sake of simplicity we won’t use any specific ML packages, instead we train a simple linear regression model in a vector form:
# define feature matrix X of size 6x3 with nearly same second and third column
X = np.array([[4, 4, 4],
[3, 5, 5],
[5, 1, 1],
[5, 4, 4],
[7, 5, 5],
[4, 5, 5.00000001]])
# define vector y of size 1x6
y= np.array([1, 2, 3, 1, 2, 3])
# calculate Gram matrix for X
XTX = X.T.dot(X)
XTX
array([[140. , 111. , 111.00000004],
[111. , 108. , 108.00000005],
[111.00000004, 108.00000005, 108.0000001 ]])
# take inverse matrix of Gram matrix
XTX_inv = np.linalg.inv(XTX)
XTX_inv
array([[ 3.86409478e-02, -1.26839821e+05, 1.26839770e+05],
[-1.26839767e+05, 2.88638033e+14, -2.88638033e+14],
[ 1.26839727e+05, -2.88638033e+14, 2.88638033e+14]])
# calculate a weights vector w:
w = XTX_inv.dot(X.T).dot(y)
W
array([-1.93908875e-01, -3.61854375e+06, 3.61854643e+06])
As you can see the second and the third values of the weights vector $w$ are huge. It comes from the fact that initial feature matrix $X$ contains almost the same columns: 2 and 3.
Let’s introduce a regularisation term and see how the vector $w$ changes:
# add regularization factor 0.01 to the main diagonal of Gram matrix
XTX = XTX + 0.01 * np.eye(3)
# take inverse matrix of Gram matrix
XTX_inv = np.linalg.inv(XTX)
XTX_inv
array([[ 3.85624712e-02, -1.98159300e-02, -1.98158861e-02],
[-1.98159300e-02, 5.00124975e+01, -4.99875026e+01],
[-1.98158861e-02, -4.99875026e+01, 5.00124974e+01]])
# calculate a weights vector w:
w = XTX_inv.dot(X.T).dot(y)
W
array([0.33643484, 0.04007035, 0.04007161])
The weights in vector $w$ now are reasonable and suitable for prediction.
The example of applying regularization in Linear Regression for car price prediction can be found in this notebook.
### Summary
The main purpose of regularization techniques is to control the weights vector $w$ and not let it grow too large in magnitude. A regularized regression considered in this article is called Ridge Regression and you can typically find it in various ML packages (e.g. scikit-learn).
Regularization is capable of finding a solution when there are correlated columns, reduce overfitting and improve your model performance in many cases. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8525300621986389, "perplexity": 883.0276173447126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337836.93/warc/CC-MAIN-20221006124156-20221006154156-00256.warc.gz"} |
http://en.academic.ru/dic.nsf/enwiki/3444 | You can mark you interesting snippets of text that will be available through a unique link in your browser.
# Contraction mapping
Translation
Contraction mapping
In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some nonnegative real number k < 1 such that for all x and y in M,
$d(f(x),f(y))\leq k\,d(x,y).$
The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive map.
More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (M,d) and (N,d') are two metric spaces, and $f:M \mapsto N$, then there is a constant k such that
$d'(f(x),f(y))\leq k\,d(x,y)$
for all x and y in M.
Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1).
A contraction mapping has at most one fixed point. Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This concept is very useful for iterated function systems where contraction mappings are often used. Banach's fixed point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem.[1]
## Firmly non-expansive mapping
A non-expansive mapping with k = 1 can be strengthened to a firmly non-expansive mapping in a Hilbert space H if the following holds for all x and y in H:
$\|f(x)-f(y) \|^2 \leq \, \langle x-y, f(x) - f(y) \rangle.$
where
$d(x,y) = \|x-y\|$
This is a special case of α averaged nonexpansive operators with α = 1 / 2.[2] A firmly non-expansive mapping is always non-expansive, via the Cauchy–Schwarz inequality.
## Note
1. ^ Theodore Shifrin, Multivariable Mathematics, Wiley, 2005, ISBN 0-471-52638-X, pp. 244–260.
2. ^ Solving monotone inclusions via compositions of nonexpansive averaged operators, Patrick L. Combettes, 2004
## References
• Vasile I. Istratescu, Fixed Point Theory, An Introduction, D.Reidel, Holland (1981). ISBN 90-277-1224-7 provides an undergraduate level introduction.
• Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5
• William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory (2001), Kluwer Academic, London ISBN 0-7923-7073-2
Wikimedia Foundation. 2010.
### Look at other dictionaries:
• contraction mapping — мат. сжимающее отображение … Большой англо-русский и русско-английский словарь
• contraction mapping — Электроника: сжатое отображение … Универсальный англо-русский словарь
• contraction mapping — 壓縮映象[映射] … English-Chinese dictionary
• contraction mapping — сжатое отображение … English-Russian electronics dictionary
• contraction mapping — сжатое отображение … The New English-Russian Dictionary of Radio-electronics
• contraction mapping principle — Математика: принцип сжатых отображений … Универсальный англо-русский словарь
• principle of contraction mapping — Математика: принцип сжатых отображений … Универсальный англо-русский словарь
• Contraction — may refer to: In physiology: Muscle contraction, one that occurs when a muscle fiber lengthens or shortens Uterine contraction, contraction of the uterus, such as during childbirth Contraction, a stage in wound healing In linguistics: Synalepha,… … Wikipedia
• mapping — 1) картографирование; топографирование, микр. проф. получение топограмм 2) вчт управление распределением памяти, управление памятью 3) вчт отображение 4) преобразование • mapping into mapping onto Alexander Whitney mapping analogical mapping… … English-Russian electronics dictionary
• mapping — 1) картографирование; топографирование, микр.; проф. получение топограмм 2) вчт. управление распределением памяти, управление памятью 3) вчт. отображение 4) преобразование • Alexander Whitney mapping analogical constraint mapping analogical… … The New English-Russian Dictionary of Radio-electronics | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9281957745552063, "perplexity": 4439.148583275154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999654282/warc/CC-MAIN-20140305060734-00008-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://arxiv.org/abs/1908.00270 | hep-th
# Title:Lack of thermalization in (1+1)-d QCD at large $N_c$
Abstract: Motivated by recent works aimed at understanding the status of equilibration and the eigenstate thermalization hypothesis in theories with confinement, we return to the 't Hooft model, the large-$N_c$ limit of (1+1)-d quantum chromodynamics. This limit has been studied extensively since its inception in the mid-1970s, with various exact results being known, such as the quark and meson propagators, the quark-antiquark interaction vertex, and the meson decay amplitude. We then argue this model is an ideal laboratory to study non-equilibrium phenomena, since it is manifestly non-integrable, yet one retains a high level of analytic control through large-$N_c$ diagrammatics. We first elucidate what are the non-equilibrium manifestations of the phenomenon of large-$N_c$ volume independence. We then find that within the confined phase, there is a class of initial states that lead to a violation of the eigenstate thermalization hypothesis, i.e. the system never thermalizes. This is due to the existence of heavy mesons with an extensive amount of energy, a phenomenon that has been numerically observed recently in the quantum Ising chain.
Comments: 33 pages, 7 figures Subjects: High Energy Physics - Theory (hep-th); Statistical Mechanics (cond-mat.stat-mech) Cite as: arXiv:1908.00270 [hep-th] (or arXiv:1908.00270v1 [hep-th] for this version)
## Submission history
From: Neil Robinson [view email]
[v1] Thu, 1 Aug 2019 08:43:49 UTC (252 KB) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8546068668365479, "perplexity": 1414.6099473310833}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317037.24/warc/CC-MAIN-20190822084513-20190822110513-00079.warc.gz"} |
https://www.quantumstudy.com/a-monochromatic-source-of-light-operating-at-200-w-emits-4x1020-photons-per-second-find-the-wavelength-of-the-light/ | # A monochromatic source of light operating at 200 W emits 4×10^20 photons per second. Find the wavelength of the light.
Q: A monochromatic source of light operating at 200 W emits 4×1020 photons per second. Find the wavelength of the light.
Sol: Energy of photon , $\large E = \frac{P}{N/t}$
$\large E = \frac{200}{4 \times 10^{20}}$
E = 5 × 10-19 J | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8582491874694824, "perplexity": 1222.2128172818416}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057580.39/warc/CC-MAIN-20210924201616-20210924231616-00416.warc.gz"} |
https://byjus.com/jee-questions/is-there-vapor-pressure-in-an-open-container/ | Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :
Is there vapor pressure in an open container?
A liquid is said to boil when its saturated vapour pressure becomes equal to the external pressure on the liquid. If the liquid is in an open container and is exposed to normal atmospheric pressure, the liquid boils when its saturated vapor pressure becomes equal to 1 atmosphere. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9336951375007629, "perplexity": 585.6081560176332}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104364750.74/warc/CC-MAIN-20220704080332-20220704110332-00034.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/271527-probability-distribution.html | 1. ## Probability Distribution
A grocery has their employees stock groceries. The mean is 1100 pounds and a standard deviation of 100. If two employees are randomly selected what is the probability that the weight of groceries they stock will be less than 50 pounds.
My best guess is :
Normalcdf(1050, 1150, 1100, 100)
2. ## Re: Probability Distribution
The sum of two random variables, one normally distributed with mean $\mu_1$ and standard deviation $\sigma_1$, the other with mean $\mu_2$ and standard deviation $\sigma_2$, is normally distributed with mean $\mu_1+ \mu_2$ and standard deviation $\sqrt{\sigma_1^2+ \sigma_2^2}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9867715835571289, "perplexity": 347.4021641211982}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891316.80/warc/CC-MAIN-20180122113633-20180122133633-00468.warc.gz"} |
https://socratic.org/questions/in-the-figure-one-end-of-a-uniform-beam-of-weight-420-n-is-hinged-to-a-wall-the- | Physics
Topics
# In the figure, one end of a uniform beam of weight 420 N is hinged to a wall; the other end is supported by a wire that makes angles θ = 29° with both wall and beam. ?
## How to find (a) the tension in the wire and the (b) horizontal and (c) vertical components of the force of the hinge on the beam.
Feb 5, 2016
#### Answer:
a)T=420 . cos 29=367.34 N
b)
$\text{horizontal component of net force :} 98 , 716 . \sin 2 \theta = 83 , 716 N$
c)
$\text{vertical component of net force :} 98 , 716 . \cos 2 \theta = 52 , 312 N$
#### Explanation:
$P = 420 . \sin 2 \theta$
$R = 420 . \cos 2 \theta$
$L = T . \cos \theta$
$K = T . \sin \theta$
$M = T . \sin \theta$
$N = T . \cos \theta$
K .2.l=P. l(" torque for point A)"
$\text{ R and L have no torque for point A}$
$K .2 = P$
$2. T . \sin \theta = 420 . \sin 2. \theta$
$T = 210 . \frac{\sin 2 \theta}{\sin} \theta$
$\sin 2. \theta = 2. \sin \theta . \cos \theta$
$T = 210. \frac{2. \sin \theta . \cos \theta}{\sin} \theta$
$T = 420 . \cos \theta$
a)T=420 . cos 29=367.34 N
$L = 367.34 . \cos 29 = 321.282 N$
$R = 420 . \cos 58 = 222 , 566 N$
$\text{net force on point hinge :} 321 , 282 - 222 , 566 = 98 , 716 N$
$\text{horizontal component of net force :} 98 , 716 . \sin 2 \theta = 83 , 716 N$
$\text{vertical component of net force :} 98 , 716 . \cos 2 \theta = 52 , 312 N$
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http://mathhelpforum.com/calculus/103611-help-please.html | # Math Help - Help please
Discuss whether or not the given function is injective, surjective, and/or bijective.
f: N into Q, where f(n) = 1/n
g: X subset of R^3 into [1,infinity), where g(x,y,z) = e^sqaure root of x^2+y^2+z^2-4
h: X subset of R^3 into R^3, where h(x) = 2x/2 norm of x
2. Originally Posted by jburks100
Discuss whether or not the given function is injective, surjective, and/or bijective.
$f: \mathbb{N}\longrightarrow\mathbb{Q}$, where $f(n) = 1/n$
$g: X\subseteq\mathbb{R}^3 \longrightarrow [1,\infty)$, where $g(x,y,z) = e^{\sqrt{x^2+y^2+z^2-4}}$
$h: X\subseteq\mathbb{R}^3\longrightarrow\mathbb{R}^3$, where $h(x) = \frac{2x}{||x||_2}$
$f$ is certainly injective, because it has a well-defined inverse. It is not surjective because not every point in $\mathbb{Q}$ is hit. It's not bijective because both aren't satisfied.
$g$ is not injective because the points $(2,3,4)$ and $(2,4,3)$ (for example) yield the same value of $g$. It is, however, surjective, as every point in $[1,\infty)$ is hit. It's not bijective because both aren't satisfied.
$h$ isn't injective, because, for example, $h(1,2,3)$ and $h(2,4,6)$ both yield the same value. It isn't surjective either, because the only points in $\mathbb{R}^3$ that are hit are vectors of length $2$, which certainly don't compose all of 3-space.
PS: Make sure that the way I interpreted your functions is correct. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 18, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9422638416290283, "perplexity": 446.99198853667417}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00494-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://brilliant.org/problems/it-has-some-capacity-i-guess/ | # it has some capacity I guess
In the following problem, the switch is closed at t=0, then at t=infinity what will be the charge store in capacitor C_{2}, take C=1F?
× | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9495102763175964, "perplexity": 2148.3900821897932}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423320.19/warc/CC-MAIN-20170720181829-20170720201829-00226.warc.gz"} |
https://www.knowpia.com/knowpedia/Rule_of_inference | BREAKING NEWS
Rule of inference
## Summary
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inference called modus ponens takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion.
Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursive are important; i.e. rules such that there is an effective procedure for determining whether any given formula is the conclusion of a given set of formulae according to the rule. An example of a rule that is not effective in this sense is the infinitary ω-rule.[1]
Popular rules of inference in propositional logic include modus ponens, modus tollens, and contraposition. First-order predicate logic uses rules of inference to deal with logical quantifiers.
## Standard form
In formal logic (and many related areas), rules of inference are usually given in the following standard form:
Premise#1
Premise#2
...
Premise#n
Conclusion
This expression states that whenever in the course of some logical derivation the given premises have been obtained, the specified conclusion can be taken for granted as well. The exact formal language that is used to describe both premises and conclusions depends on the actual context of the derivations. In a simple case, one may use logical formulae, such as in:
${\displaystyle A\to B}$
${\displaystyle {\underline {A\quad \quad \quad }}\,\!}$
${\displaystyle B\!}$
This is the modus ponens rule of propositional logic. Rules of inference are often formulated as schemata employing metavariables.[2] In the rule (schema) above, the metavariables A and B can be instantiated to any element of the universe (or sometimes, by convention, a restricted subset such as propositions) to form an infinite set of inference rules.
A proof system is formed from a set of rules chained together to form proofs, also called derivations. Any derivation has only one final conclusion, which is the statement proved or derived. If premises are left unsatisfied in the derivation, then the derivation is a proof of a hypothetical statement: "if the premises hold, then the conclusion holds."
## Example: Hilbert systems for two propositional logics
In a Hilbert system, the premises and conclusion of the inference rules are simply formulae of some language, usually employing metavariables. For graphical compactness of the presentation and to emphasize the distinction between axioms and rules of inference, this section uses the sequent notation (${\displaystyle \vdash }$ ) instead of a vertical presentation of rules. In this notation,
${\displaystyle {\begin{array}{c}{\text{Premise }}1\\{\text{Premise }}2\\\hline {\text{Conclusion}}\end{array}}}$
is written as ${\displaystyle ({\text{Premise }}1),({\text{Premise }}2)\vdash ({\text{Conclusion}})}$ .
The formal language for classical propositional logic can be expressed using just negation (¬), implication (→) and propositional symbols. A well-known axiomatization, comprising three axiom schemata and one inference rule (modus ponens), is:
(CA1) ⊢ A → (B → A)
(CA2) ⊢ (A → (B → C)) → ((A → B) → (A → C))
(CA3) ⊢ (¬A → ¬B) → (B → A)
(MP) A, A → B ⊢ B
It may seem redundant to have two notions of inference in this case, ⊢ and →. In classical propositional logic, they indeed coincide; the deduction theorem states that AB if and only if ⊢ AB. There is however a distinction worth emphasizing even in this case: the first notation describes a deduction, that is an activity of passing from sentences to sentences, whereas AB is simply a formula made with a logical connective, implication in this case. Without an inference rule (like modus ponens in this case), there is no deduction or inference. This point is illustrated in Lewis Carroll's dialogue called "What the Tortoise Said to Achilles",[3] as well as later attempts by Bertrand Russell and Peter Winch to resolve the paradox introduced in the dialogue.
For some non-classical logics, the deduction theorem does not hold. For example, the three-valued logic of Łukasiewicz can be axiomatized as:[4]
(CA1) ⊢ A → (B → A)
(LA2) ⊢ (A → B) → ((B → C) → (A → C))
(CA3) ⊢ (¬A → ¬B) → (B → A)
(LA4) ⊢ ((A → ¬A) → A) → A
(MP) A, A → B ⊢ B
This sequence differs from classical logic by the change in axiom 2 and the addition of axiom 4. The classical deduction theorem does not hold for this logic, however a modified form does hold, namely AB if and only if ⊢ A → (AB).[5]
In a set of rules, an inference rule could be redundant in the sense that it is admissible or derivable. A derivable rule is one whose conclusion can be derived from its premises using the other rules. An admissible rule is one whose conclusion holds whenever the premises hold. All derivable rules are admissible. To appreciate the difference, consider the following set of rules for defining the natural numbers (the judgment ${\displaystyle n\,\,{\mathsf {nat}}}$ asserts the fact that ${\displaystyle n}$ is a natural number):
${\displaystyle {\begin{matrix}{\begin{array}{c}\\\hline {\mathbf {0} \,\,{\mathsf {nat}}}\end{array}}&{\begin{array}{c}{n\,\,{\mathsf {nat}}}\\\hline {\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}\end{array}}\end{matrix}}}$
The first rule states that 0 is a natural number, and the second states that s(n) is a natural number if n is. In this proof system, the following rule, demonstrating that the second successor of a natural number is also a natural number, is derivable:
${\displaystyle {\begin{array}{c}{n\,\,{\mathsf {nat}}}\\\hline {\mathbf {s(s(} n\mathbf {))} \,\,{\mathsf {nat}}}\end{array}}}$
Its derivation is the composition of two uses of the successor rule above. The following rule for asserting the existence of a predecessor for any nonzero number is merely admissible:
${\displaystyle {\begin{array}{c}{\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}\\\hline {n\,\,{\mathsf {nat}}}\end{array}}}$
This is a true fact of natural numbers, as can be proven by induction. (To prove that this rule is admissible, assume a derivation of the premise and induct on it to produce a derivation of ${\displaystyle n\,\,{\mathsf {nat}}}$ .) However, it is not derivable, because it depends on the structure of the derivation of the premise. Because of this, derivability is stable under additions to the proof system, whereas admissibility is not. To see the difference, suppose the following nonsense rule were added to the proof system:
${\displaystyle {\begin{array}{c}\\\hline {\mathbf {s(-3)} \,\,{\mathsf {nat}}}\end{array}}}$
In this new system, the double-successor rule is still derivable. However, the rule for finding the predecessor is no longer admissible, because there is no way to derive ${\displaystyle \mathbf {-3} \,\,{\mathsf {nat}}}$ . The brittleness of admissibility comes from the way it is proved: since the proof can induct on the structure of the derivations of the premises, extensions to the system add new cases to this proof, which may no longer hold.
Admissible rules can be thought of as theorems of a proof system. For instance, in a sequent calculus where cut elimination holds, the cut rule is admissible. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 14, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.947434663772583, "perplexity": 504.2750188744473}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662625600.87/warc/CC-MAIN-20220526193923-20220526223923-00283.warc.gz"} |
http://www.aorinevo.com/stevens/mat-121-calculus-ia/differentiation-rules-reference-page/ | # Differentiation Rules Reference Page
The derivative of $$f(x)$$ with respect to $$x$$ is denoted by $$f'(x)$$ where
$$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}.$$
Equivalent notation for $$f'(x)$$ include:
$$y’, \frac{df}{dx}, \frac{dy}{dx}, \frac{d}{dx}[f(x)]$$
Basic Differentiation Rules: Let $$c$$ be a constant.
• $$\displaystyle \frac{d}{dx}[c] = 0$$
• $$\displaystyle \frac{d}{dx}[x^n] = nx^{n-1}$$
• $$\displaystyle \frac{d}{dx}[e^x] = e^x$$
• $$\displaystyle \frac{d}{dx}[a^x] = a^x \ln a$$
• $$\displaystyle \frac{d}{dx}[\sin x] = \cos x$$
• $$\displaystyle \frac{d}{dx}[\csc x] = -\csc x \cot x$$
• $$\displaystyle \frac{d}{dx}[\cos x] = -\sin x$$
• $$\displaystyle \frac{d}{dx}[\sec x] = \sec x \tan x$$
• $$\displaystyle \frac{d}{dx}[\tan x] = \sec^2 x$$
• $$\displaystyle \frac{d}{dx}[\cot x] = -\csc^2 x$$
Derivative of a General Inverse Function:
$$\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}$$
More Basic Differentiation Rules:
• $$\displaystyle \frac{d}{dx}[\ln x] = \frac{1}{x}$$
• $$\displaystyle \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$$
• $$\displaystyle \frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1-x^2}}$$
• $$\displaystyle \frac{d}{dx}[\sec^{-1} x] = \frac{1}{|x|\sqrt{x^2-1}}$$
• $$\displaystyle \frac{d}{dx}[\cos^{-1} x] = \frac{-1}{\sqrt{1-x^2}}$$
• $$\displaystyle \frac{d}{dx}[\csc^{-1} x] = \frac{-1}{|x|\sqrt{x^2-1}}$$
• $$\displaystyle \frac{d}{dx}[\tan^{-1} x] = \frac{1}{1+x^2}$$
• $$\displaystyle \frac{d}{dx}[\cot^{-1} x] = \frac{-1}{1+x^2}$$
Properties of Derivatives:If $$f(x)$$ and $$g(x)$$ are differentiable at $$x$$ and $$c$$ is a constant, then
• Sum/Difference Rule:
$$\frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)]$$
• Product Rule:
$$\frac{d}{dx}[f(x) \cdot g(x)] = \frac{d}{dx}[f(x)] \cdot g(x) + \frac{d}{dx}[g(x)] \cdot f(x)$$
$$\frac{d}{dx}[f(x) \cdot g(x)] = \frac{d}{dx}[f(x)] \cdot g(x) + \frac{d}{dx}[g(x)] \cdot f(x)$$
\begin{aligned} &\frac{d}{dx}[f(x) \cdot g(x)] \\ & \qquad = \frac{d}{dx}[f(x)] \cdot g(x) + \frac{d}{dx}[g(x)] \cdot f(x) \end{aligned}
• Quotient Rule:
$$\frac{d}{dx}\left[ \frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot\frac{d}{dx}[f(x)] – \frac{d}{dx}[g(x)] \cdot f(x)}{(g(x))^2}$$
$$\frac{d}{dx}\left[ \frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot\frac{d}{dx}[f(x)] – \frac{d}{dx}[g(x)] \cdot f(x)}{(g(x))^2}$$
\begin{aligned} & \frac{d}{dx}\left[ \frac{f(x)}{g(x)}\right] \\ & \qquad = \frac{g(x) \cdot\frac{d}{dx}[f(x)] – \frac{d}{dx}[g(x)] \cdot f(x)}{(g(x))^2} \end{aligned}
• Constant Multiplier Rule:
$$\frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)]$$
• Chain Rule: If, in addition, $$f(x)$$ is differentiable at $$g(x)$$ then
$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot \frac{d}{dx}[g(x)]$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000098943710327, "perplexity": 1893.9277411638657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886116921.7/warc/CC-MAIN-20170822221214-20170823001214-00300.warc.gz"} |
https://math.stackexchange.com/questions/2516263/difference-between-cantors-ordinals-and-von-neumann-ordinals | # Difference Between Cantor's Ordinals and Von Neumann Ordinals?
Cantor's Naive Set Theory allows the construction of the set of all ordinals, which contains itself, which triggers the Burali-Forti Paradox. ZFC both disallows a set of the size of all ordinals and typically uses Von Neumann's definition of ordinals. Under Von Neumann's definition, a set $\alpha$ is an ordinal number iff
1. If $\beta$ is a member of $\alpha$, then $\beta$ is a proper subset of $\alpha$;
2. If $\beta$ and $\gamma$ are members of $\alpha$ then one of the following is true: $\beta=\gamma$, $\beta$ is a member of $\gamma$, or $\gamma$ is a member of $\beta$;
and 3. If $\beta$ is a nonempty proper subset of $\alpha$, then there exists a $\gamma$ member of $\alpha$ such that the intersection $\gamma \cap \beta$ is empty. (Definition from Wolfram Alpha).
The first rule implies that no ordinal is an element of itself, hence even if the axiom of foundation and axiom of replacement did not exist, the set of all ordinals could not be an ordinal, since it would violate the first rule, and the Burali-Forti Paradox would therefore still be resolved. So... there seems like their must be a significant difference between Cantor's definition of ordinals and Von Neumann's, since Cantor's Naive Set Theory still allows the set of all ordinals to be an ordinal. Cantor's ordinals can be elements of themselves. Why can Cantor's ordinals be elements of themselves? What is it about his definition of ordinals that allows them to be elements of themselves?
• There is a class of all ordinals, which is itself well-ordered as a class. Could you give a source for "Cantor's naive set theory allows the construction of the set of all ordinals"? – Patrick Stevens Nov 12 '17 at 8:40
• @PatrickStevens Here's a link to Wolfram's rather terse explanation of the Burali-Forti Paradox, which recognizes the 'set of all ordinals'--but if you're asking why the set of all ordinals exists in Cantor's naive set theory, it's because the Unrestricted Comprehension Principle (which is used in naive set theory, as opposed to the replacement axiom of ZFC) allows the construction of sets of the form $(x | P(x))$, and hence allows the existence of the set $(x|x$is an ordinal$)$. – J.P. Escarcega Nov 12 '17 at 9:18
• About "Why can Cantor's ordinals be elements of themselves?" see Limitation of size: Canot has an "informal" principle "that identifies certain "inconsistent multiplicities" that cannot be sets because they are "too large". In modern terminology these are called proper classes." – Mauro ALLEGRANZA Nov 12 '17 at 9:39
## 1 Answer
"By the numbering [Anzahl] or the ordinal number of a well-ordered set $\frak M$ I mean the general concept or universal [Allegemeinbegriff, Gattungsbegriff] which one obtains by abstracting the character of its elements and by reflecting upon nothing but the order in which they occur."
(J.W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite pp. 127–128; the quote is attributed to G. Cantor "Mitteilungen zur Lehre vom Transfiniten" on p.388 in the 1932 collection "Gesammelte Abhandulngen mathematischen und philosophischen Inhalts".)
This is not a rigorous definition, but to me, the only modern way of interpreting this is to consider ordinals either as an abstract category, or more likely at the time, as a collection of equivalence classes of well-ordered sets under the order-isomorphism equivalence relation.
The latter interpretation seems to be more in the spirit of the time, so I will stick with it.
This means that an ordinal is not a set, in modern perspective, since for any non-empty well-ordered set, there is a proper class of well-ordered sets which are isomorphic to this given order.
Therefore the main difference between the von Neumann ordinals and Cantor's ordinals, is that the former are sets, and the latter are not. Moreover, the collection of Cantorian ordinals is not even a proper class, in modern terms, since its "elements" are not sets.
Cantor's ordinals were not elements of themselves. Even if you take the naive set theoretic approach, just because a set can be a member of itself, doesn't mean that every set will be a member of itself.
Let me also make a tangential remark, that a complete name of the von Neumann ordinals should be "the von Neumann ordinal assignment". Because we simply show that there is a canonical choice from every Cantorian ordinal: a transitive set well ordered by $\in$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8657187819480896, "perplexity": 352.2541794445279}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257259.71/warc/CC-MAIN-20190523143923-20190523165923-00193.warc.gz"} |
http://blogannath.blogspot.com/2009/11/vedic-mathematics-lesson-27-solving.html | ## Wednesday, November 18, 2009
### Vedic Mathematics Lesson 27: Solving Equations 2
In the previous lesson, we examined the derivation of different formulae for solving different types of equations so that we don't have to go through the process of solving for the unknown variable from first principles. But Vedic Mathematics provides another method of solving some equations on sight that we will examine in this lesson.
You can find all the previous posts about Vedic Mathematics below:
Multiplication Special Case 3
Vertically And Crosswise I
Vertically And Crosswise II
Squaring, Cubing, Etc.
Subtraction
Division By The Nikhilam Method I
Division By The Nikhilam Method II
Division By The Nikhilam Method III
Division By The Paravartya Method
Digital Roots
Straight Division I
Straight Division II
Vinculums
Divisibility Rules
Simple Osculation
Multiplex Osculation
Solving Equations 1
The Vedic method that we are going to examine in this lesson is based on a sutra that reads "Sunyam Samyasamuccaye". The literal translation of this sutra is that "if the Samuccaya is the same, then that Samuccaya must be zero".
What exactly does Samuccaya mean though? It turns out that the term means different things under different contexts. This may sound bad at first, but it actually turns out to be a good thing because the more contextual meanings Samuccaya has, the more contexts under which the sutra can be used. And the more contexts under which the sutra can be used, the more types of equations we may be able to solve on sight without expending any labor!
The first, and simplest, meaning of Samuccaya is an unknown quantity that occurs as a common factor throughout an equation. Take for instance, the equation 4x + 5x = 3x + x. Basic algebra skills would suffice for one to conclude that x = 0 is the solution to this equation. That is precisely what the first application of the sutra also says: since x is an unknown common factor throughout the equation, it is the sumuccaya and equating it to zero gives us the solution x = 0 right away.
However, the method can be applied in equations like the one below also:
7(x - 2) = 3(x - 2)
In this case, (x - 2) is the sumuccaya, and setting it to zero, gives us the solution x = 2. It is possible to expand the terms, collect like terms, transpose and adjust, and reach the same conclusion, but you can solve the equation on sight using this sutra.
The second meaning of Samuccaya is the product of independent terms in equations of the form (x + a)(x + b) = (x + c)(x + d). Suppose you are confronted with the equation (x + 2)(x + 3) = (x + 1)(x + 6). We notice right away that the product of the independent terms on the left hand side is 2*3 = 6, and the product of the independent terms on the right hand side is 1*6 = 6 also. Since they are the same, we can conclude that x = 0 is the solution to that equation!
The astute reader may have noticed that this type of equation was dealt with in the previous lesson as an equation of type 2. The formula for the solution of a type 2 equation has (cd - ab) as its numerator. The two products, cd and ab, represent the products of the independent terms on the right hand side and left hand side of the equation respectively. If they are equal, then the numerator becomes zero, thus making x equal to zero. So you could derive the same solution using the formula in that lesson instead of this sutra.
The third meaning and application of this sutra is in the solution of equations of the form m/(ax + b) + m/(cx + d) = 0. Notice that the numerators of the two terms are the same (m). The sutra says that the sum of the denominators is then the sumuccaya, which can be set to zero for the solution of the equation.
Let us take 1/(x - 1) + 1/(x - 2) = 0 as an example. Since the numerators are the same, calculate the sum of the denominators. It is x - 1 + (x - 2) = 2x - 3. Equating it to zero gives us x = 3/2 right away. We don't have to cross-multiply, expand terms, collect like terms, transpose or adjust. Simply by recognizing that the sum of the denominators is a samuccaya when the numerators are identical enables us to solve the equation on sight!
An extension of this meaning is that if the numerators of the two fractional expressions are not the same numerical constant, but the same unknown quantity, that unknown quantity is a samuccaya too (this can be considered an extension of the first meaning of samuccaya since the numerator becomes a common factor throughout the equation). Consider the following equation:
(x - 1)/(2x + 1) + (x - 1)/(3x - 4) = 0
The sum of the denominators is 2x + 1 + 3x - 4 = 5x - 3. Setting this to zero gives us x = 3/5 right away. But to solve the equation fully, we also need to set x - 1 (the common numerator) to zero, thus giving us x = 1 as another solution to the equation. By cross-multiplying, creating a quadratic equation and using the quadratic formula to solve it, you can verify that x = 1 and x = 3/5 are indeed the two solutions to the equation above.
The fourth meaning of samuccaya can be used to solve some equations that seemingly look very difficult to solve. They are equations of the form (ax + b)/(cx + d) = (ex + f)/(gx + h). In this context, the samuccaya means the sum of the numerators and the sum of the denominators. The sutra says that if the sum of the numerators is the same as the sum of the denominators, then that sum should be equated to zero.
Let us apply to an example to see how to use this interpretation of samuccaya. Suppose one needs to solve (3x + 1)/(x - 2) = (2x - 5)/(4x - 2). We immediately notice that the sum of the numerators is 3x + 1 + 2x - 5 = 5x - 4. The sum of the denominators is x - 2 + 4x - 2 = 5x - 4. Since the sum of the numerators is the same as the sum of the denominators, the sutra can be applied and 5x - 4 can be equated to zero. That immediately gives us one of the solutions to the equation, x = 4/5.
Obviously, once the cross-multiplication and other massaging of the equation is done, it will turn out to be a quadratic equation, so we still need to solve for the other solution, but being able to identify one solution with hardly any calculations is still a step in the right direction.
This application of the sutra is particularly useful when you have equations of the sort (ax + b)/(cx + d) = (cx + d)/(ax + b). These types of equations are easy to spot, and you can also tell right away that this sutra is applicable since the sum of the numerators is ax + b + cx + d, which is also the sum of the denominators. By setting this sum to zero as the sutra tells us to do for solving this equation, we get x = (-b - d)/(a + c).
Now, if a = c in the above equation, then it turns out to be a simple linear equation without any quadratic terms when expanded, and transposed and adjusted. Then, we can conclude that x = (-b - d)/2a is the only solution to the equation. But if a is not equal to c, then the equation will expand out to a quadratic equation that will require us to use a different method to obtain the other solution.
The next meaning of samuccaya may be able to help in that regard though. In this context, samuccaya also means the difference between the numerator and denominator on both sides of the equation. Consider the equation (5x + 3)/(3x + 2) = (3x + 6)/(x + 5). On the left hand side, the difference between the numerator and denominator is 5x + 3 - 3x - 2 = 2x + 1. The difference between the numerator and denominator on the right hand side is 3x + 6 - x - 5 = 2x + 1. The sutra then says that this common difference, being the samuccaya, can be equated to zero for a solution to the equation. Thus, we get x = -1/2.
The important thing to note in this context is that by "difference", I literally mean difference (without any particular order of terms in the subtraction). I am not talking about any order between the numerator and denominator when I do the subtraction on each side of the equation. If the equation were N1/D1 = N2/D2, I am calculating either N1 - D1 or D1 - N1 (depending on my convenience), and I am either calculating N2 - D2 or D2 - N2 (again depending on my convenience). I can compare N1 - D1 against N2 - D2 or against D2 - N2. Similarly, I can compare D1 - N1 against either D2 - N2 or N2 - D2.
As an aside, this is actually quite easy to understand since D2 - N2 is simply -1*(N2 - D2). Since -1 can never be equal to zero, it is the difference that has to be equal to zero.
But, why is this important? There are several reasons why this could be important: First of all, it expands the applicability of this sutra. If N1 - D1 ≠ N2 - D2, you can still try and see whether N1 - D1 = D2 - N2, and if it is, the sutra still applies, with its attendant labor-savings in solving the equation.
More importantly, let us now revisit equations of the sort (ax + b)/(cx + d) = (cx + d)/(ax + b). We already saw that we could use the previous meaning of sumuccaya to get one of the solutions to this equation as (-b -d)/(a + c). Now, consider the difference between the numerators and denominators. On the left hand side, we get ax + b - cx - d when we do N1 - D1. On the right hand side, we are guaranteed to get the same difference when we do D2 - N2 because the actual terms on the numerator and denominator are the same on both sides of the equation, only their positions have been changed.
Thus, according to this sutra, ax + b - cx - d should be set equal to zero for another solution to this equation. This gives us x = (d - b)/(a - c). Notice that if a = c, then the equation is not really a quadratic equation and will expand out to be a linear equation. This second formula is undefined in that circumstance, as it should be!
Consider the equation (5x + 1)/(4x - 2) = (4x - 2)/(5x + 1). Using the last two meanings of the sutra, we can say that the two solutions to this quadratic equation are x = 1/9 and -3. We did not do any cross-multiplication, expanding of terms, collecting of terms, application of the quadratic formula or anything of the sort. We simply looked at the equation, recognized it as something that can be tackled with this sutra, applied the appropriate meaning of the sutra, and got both solutions of the equation within seconds! That is the power of this sutra, especially the last two meanings we have been examining!!
Now, notice that this makes it trivial to solve equations such as (4x + 5)^2 = (3x + 2)^2. All we have to do is rewrite it in a slightly different form as follows, and we are pretty much done: (4x + 5)/(3x + 2) = (3x + 2)/(4x + 5). The solutions are then x = -1 and x = -3! Thus, if you have a quadratic equation that is of the form (ax + b)^2 = (cx + d)^2, the two solutions to it are x = (-b - d)/(a + c) and x = (d - b)/(a - c). Obviously, if a = c in the equation above, it is not a quadratic equation, so we can ignore the second solution. No squares, square roots, discriminants and all the rest of the quadratic paraphernalia! I don't know of any other system anywhere that formalizes this property of quadratic equations of the above sort!!
This brings up an even more powerful property of this sutra to solve quadratic equations that are not expressed readily in the form (ax + b)^2 = (cx + d)^2. But we will examine that in a later lesson.
As you may have guessed by now, this word samuccaya has some more meanings also. Correspondingly, this sutra has a lot more applications also. We will examine them in future lessons. In the meantime, here is a table summarizing the results we have covered in this lesson.
Type Of Equation At-sight solution technique Unknown quantity is a common factor throughout the equation Set the unknown quantity equal to zero (x + a)(x + b) = (x + c)(x +d) and ab = cd x = 0 m/(ax + b) + m/(cx + d) = 0 Set the sum of the denominators equal to zero. Thus, x = (–b – d)/(a + c) Instead of m above, numerator is an unknown quantity In addition to above solution, set the unknown quantity equal to zero for another solution (ax + b)/(cx + d) = (ex + f)/(gx + h) and ax + b + ex + f = cx + d + gx + h (sum of the numerators = sum of denominators) Equate the sum to zero for one solution (ax + b)/(cx + d) = (ex + f)/(gx + h) and |ax + b – cx – d| = |ex + f – gx – h| (difference between numerator and denominator is the same on both sides of the equation) Equate the difference to zero for one solution (ax + b)^2 = (cx + d)^2 Combine the two solution strategies above to derive x = (–b – d)/(a + c) and (if a ≠ c), x = (d – b)/(a – c)
As always, practice is key to success in the application of these methods to the solution of equations. Identifying the samuccaya and applying the right interpretation of the sutra to any given equation takes time and practice. But the rewards can be quite substantial. Good luck, and happy computing!
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http://oak.snr.missouri.edu/nr3110/topics/ci.php | ## Confidence Limits for Single Sample Mean
### To calculate confidence limits for single sample mean
Confidence limits for single sample mean
You can calculate this with:
$$\bar{x} \pm t_{\alpha (2), df} \sqrt{\frac{s^2}{n}}$$
$$\bar{x} \pm t_{\alpha (2), df} s_\bar{x}$$
where Xi is the mean of each sample, tα(2),v is the t value for the confidence range, with a(2) significance and v degrees of freedom. The n is the number of samples taken. Note the last part of the equation is the standard error.
Also See:
Chapter 9 - One-Sample Hypotheses pages 100-102 in:
Zar, J. H. 2007. Biostatistical Analysis. Prentice-Hall, Inc. Englewood Cliffs, New Jersey. 718 pp. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9763166308403015, "perplexity": 3093.2823334445457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608058.57/warc/CC-MAIN-20170525102240-20170525122240-00284.warc.gz"} |
http://internetdo.com/2023/01/solve-lesson-4-page-67-math-10-learning-topic-kite/ | ## Solve lesson 4, page 67, Math 10 learning topic – Kite>
Problem In the coordinate plane Oxy, for the line $$\Delta 😡 = – 5$$ and the point $$F\left( { – 4;0} \right)$$. Take 3 points $$A\left( { – 3;1} \right),B\left( {2;8} \right),C\left( {0;3} \right)$$ a) Calculate the the following ratio: $$\frac{{AF}}{{d\left( {A,\Delta } \right)}},\frac{{BF}}{{d\left( {B,\Delta } \right)}},\frac{{CF}}{{d\left( {C,\Delta } \right)}}$$ b) Ask each point […] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.902331531047821, "perplexity": 1242.3449214595241}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499524.28/warc/CC-MAIN-20230128054815-20230128084815-00397.warc.gz"} |
https://www.intechopen.com/books/ionic-liquids-new-aspects-for-the-future/hydrodynamics-of-ionic-liquids-in-bubble-columns | Open access peer-reviewed chapter
# Hydrodynamics of Ionic Liquids in Bubble Columns
By Vicky Lange, Barry J. Azzopardi and Pete Licence
Submitted: April 11th 2012Reviewed: July 17th 2012Published: January 23rd 2013
DOI: 10.5772/51658
## 1. Introduction
Over the last ten years, research into ionic liquids (ILs) and their physical/chemical and thermodynamic properties has intensified such that significant progress has been achieved in their application to a wide range of chemical processes. If one considers the patent and secondary literature however, it can be seen that there are very few IL based applications that have successfully reached commercialization. The first IL based process on a pilot scale, called Difasol, was the dimerization of olefins with a biphasic, homogeneous catalyst developed by the Institue Française du Petrole (IFP) [1]. Other examples of applications on pilot scale or even industrial scale include acid scavenging (BASIL, BASF) [2], extractive distillation (BASF) [3], compatibilizers in pigment pastes (Degussa/Evonik) [4], cooling agent (BASF) [5] and storage of gases (Air Products) [6]. It may be a fair assumption that one of the major challenges that restrict their application on an industrial scale is a lack of engineering data which are needed for the optimum scale-up, design and operation of industrial units used with ILs. Thus for the successful transition of ILs from academic labs to industry, not only must the fundamental physical/thermodynamic properties be investigated but we must also better understand the hydrodynamic or flow behaviour that govern reactions on a larger scale under real process conditions. This chapter focuses on IL based multiphase flow, a topic of specific interest to a variety of industrial platforms including biotechnology, biphasic catalysis and gas extraction involving for example H2, CO, CO2, H2S and SO2.
To apply ILs under multiphasic conditions, the simplest equipment to contact the liquid and gas is the bubble column. In these devices, the gas phase is bubbled through a column of liquid to promote close contact between the two phases. Due to their ease of construction and large applicability, these reactors find widespread use in industry. However despite its simple construction, the hydrodynamics involved inside these units is quite complex because of the very deformable nature of the gas-liquid interface. In bubble column operation, the specific interfacial area is an important criterion since it determines the rate of heat and mass transfer across the interface. The specific interfacial area is defined as the surface area of the bubbles per unit volume of the reactor space occupied by the two-phase flow mixture. To maximise this area and improve the efficiency of transport processes, small bubbles and a uniform distribution across the column cross section are desired.
This chapter examines how the physical properties of ILs, in particular its viscosity hinder achieving optimally high values of specific interfacial area. Available gas holdup data as well as bubble characteristics are analysed. Firstly factors which can influence the specific interfacial area such as the flow regime and gas holdup are discussed in Section 2. Section 3 describes the experimental approach and measurement techniques employed to study ILs in bubble columns. Bubble formation and coalescence in ILs is examined in Section 4, whilst the effect of operating conditions on the gas holdup is considered in Section 5. Lastly in Section 6, results from statistical analysis of the gas holdup data are presented to describe the frequency and velocity of the flow structures formed.
## 2. Influencing the specific interfacial area
For a given gas-liquid system, the transport coefficients and the interfacial area are highly dependent on the prevailing operating regime of the bubble column as this is directly related to the bubble size distribution. Because of the very deformable gas/liquid interface, there are a large number of ways in which the different phases may distribute in the column. To simplify this problem, the flow is conventionally distinguished by two main flow regimes, i.e., the homogeneous bubble regime and the heterogeneous (churn-turbulent) regime. The homogeneous regime normally exists at very low gas flow rates and is characterised by smaller, more uniformly sized bubbles which rise with similar velocity. At higher gas flow rates, a wide distribution of bubble sizes is observed due to bubble coalescence and the flow is described as heterogeneous. In this regime, the bubble swarm consists mostly of large fast-ascending bubbles and a few smaller bubbles which are trapped in the liquid recirculation flow. In cases where the column diameter is small (≤100 mm), the larger bubbles of heterogeneous flow are stabilised by the surrounding walls and can occupy the entire column cross section, forming gas plugs (called slug flow). As a result of the larger, fast-ascending bubbles, the mean gas phase residence time and the specific interfacial area in the heterogeneous regime is lower in comparison to the homogeneous regime. In reality, due to high gas throughputs the heterogeneous regime is often observed in industrial bubble columns. Although, the homogeneous regime which offers greater interfacial contacting and a low gas shear environment can be more desirable, in particular for those applications which involve sensitive media, for example biotechnology.
To estimate the specific interfacial area, the bubble size distribution is used in conjunction with the gas holdup or void fraction. The gas holdup is defined as the fraction of the gas present in the two-phase mixture in the reactor. It depends strongly on the operating conditions, physiochemical properties of the two phases, the gas distributor design (the number and size of the holes) and the column geometry (height to diameter ratio). To better understand the effect of these parameters on the flow behaviour in the bubble column, their effect on the gas holdup and the gas flow rate has been investigated by several researchers through extensive experimentation as well as theoretical analysis in the last few decades. An extensive review was given by [7], and there have been continuous publications since.
Understanding the effect of system pressure (or gas density) has been important since most industrial applications of bubble columns operate under pressurized conditions. This is because the increased solubility of a gas with pressure is expected to enhance the mass transfer and reaction rates. Based on the findings by several workers [8-15], it has been commonly established that gas holdup increases with operating pressure. The increase of gas holdup at the elevated system pressures has been attributed to the smaller bubbler size which is caused by a reduction in the coalescence rate, the enhancement of the bubble breakup or the decreased size of bubbles formed at the gas disperser [16-18].
The effect of the liquid phase properties has also been investigated, although to a lesser degree. For convenience, most studies of bubble columns have used water as the liquid phase, even though in reality the physical properties of many of the liquids used industrially differ significantly from those of water. In particular, the ILs considered here have viscosities considerably higher than water. Several studies have looked at the effect of liquid viscosity during bubble column operation [19-26]. These show that at a given gas flow rate, the gas holdup decreases with increasing viscosity. Furthermore in a larger diameter column, the transition point from homogeneous to heterogeneous flow has been found to shift to lower gas velocities with increasing liquid viscosity.
## 3. Determination of gas holdup
When a gas is bubbled through a column filled with liquid, the bed of liquid begins to expand or swell as soon as gas is introduced. If the gas holdup is expressed in terms of a global or total voidage for a bubble column, it can be calculated by measuring the difference between the gassed and ungassed height of the liquid since this represents the total gas volume present in the column. The total gas holdup can also be determined from pressure gradient measurements using pressure sensors. However, in cases where more local information on the gas phase is required, the gas holdup can be measured by a selection of invasive or non-invasive techniques. A comprehensive review of methods which can be applied for closed systems (pipes, columns etc.) was presented in [27], wherein all pertinent literature until roughly 1977 was surveyed. More recent reviews have been presented by workers such as [28] and [29]. Examples of some of the options available include using needle probes, quick-closing valves (QCVs), flush mounted conductance probes, capacitance probes, wire mesh sensors, neutron and radioactive absorption, resistive and ultrasonic tomography.
To obtain the local gas holdup for an IL, the conductance probe is an appropriate choice since the IL is a very good electrical conductor, while the conductivity of the gas phase is infinitely low. This measurement technique follows the approach developed by researchers such as [30], whereby the electrical impedance of the gas-liquid region close to a system of electrodes is measured and the phases distinguished due to the difference in the electrical conductivity of the phases. The conductivity is measured between two metallic rings (ring-type conductance probe) which are mounted flush with the bubble column walls. For the experiments presented here, the metallic rings were constructed by placing two stainless steel plates between three alternating plates of an acrylic resin and machining out a cylinder through them which had an internal diameter equal to the column diameter. Figure 1(a) shows a picture of these probes, while a schematic of their configuration is presented in Figure 1(b). The configuration is characterised by the thickness of the rings s and the spacing between them D e . The column diameter is represented by D t . The dimensions D e /D t and s/D t were 0.357and 0.075 respectively.
Each probe pair was supplied with an a.c. carrier voltage of -1 to 1 volt peak to peak, at a frequency of 20 kHz. An instrumentation amplifier, a full wave rectifier and a band pass filter were installed before the signal was sent to the data acquisition board. A cut-off frequency of 100 Hz was applied in order to eliminate the high frequency noise generated from the power supply. To account for any differences in the test fluid conductivity during experiments due to any temperature variations, all conductivities were normalized to produce a dimensionless conductance by measuring the conductivity at the start of each experimental run with the column full of liquid.
To obtain values for gas holdup from the conductance measurements, it is necessary to determine the relationship between the liquid phase conductance and gas holdup. This relationship is obtained via careful calibration. The simplest approach involves artificially creating instantaneous gas fractions between the probes, for example using plastic non-conducting beads to simulate bubbly flow inside the column. A detailed description for the calibration procedure is available in [31].
For the IL 1-ethyl-3-methylimidazolium ethylsulfate, [C2C1Im][EtOSO3], the time-varying gas holdup in a cylindrical bubble column (inner diameter 0.038 m; height 1.1 m) has been measured by utilising two pairs of conductance probes installed flush in the column walls. The experimental setup is illustrated in Figure 2. For comparison, water, a glycerol/water solution and solution of glycerol/water with sodium chloride dissolved in it were also investigated. To prepare the glycerol solutions, a total of 15% of weight of water was mixed with glycerol to produce a liquid which has a viscosity similar to that of [C2C1Im][EtOSO3]. A total of 1.3% by weight of sodium chloride was also added to one of the glycerol/water mixtures to ensure sufficient conductivity for use with the conductance probes. The physical properties of the liquid phases used in all experiments at 20˚C are summarised in Table 1. To determine the viscosities and conductivities of the liquids a Brookfield viscometer and WTW KF 340 conductivity meter were used respectively. The viscosity of water and surface tensions were obtained from physical property tables. The properties of [C2C1Im][EtOSO3] were obtained from the paper by [32].
Prior to experiments, [C2C1Im][EtOSO3] was dried and degassed under vacuum at 60°C for 24 hours since it is known that ILs will absorb a couple weight percent of water when left open to the atmosphere. Here, the main concern is the effect of the absorbed water on the physical properties of [C2C1Im][EtOSO3], in particular its viscosity as well as ion mobility which relates to its electrical conductivity. The estimated water content of [C2C1Im][EtOSO3] after drying was approximately 0.76 wt% water, as measured by Karl Fischer test.
Property [C2C1Im][EtOSO3] Glycerol (15% water) Glycerol (15% water + 1.3% NaCl) Water Density (kg/m3) 1241 1212 1224 1000 Viscosity (mPa s) 91.5 93.1 107.9 1.09 Surface tension (N/m) 0.047 0.062 0.062 0.072 Conductance (S/cm) 4160 1.4 485 42
### Table 1.
Physical properties of the liquid phases (T = 20˚C).
Measurements for gas holdup at a given operating pressure were performed by increasing the gas superficial velocity to a maximum value of 60 mm/s, while the operating pressure was systematically varied in the range of 0.1-0.8 MPa. The column was first filled with the liquid under test at room temperature (20˚C) and compressed nitrogen gas from a gas cylinder was then fed into the bottom of the column through a single nozzle gas distributor which had an inner diameter of 0.635 mm. Glass beads were also employed at the gas inlet to the column for improved phase distribution. A pressure regulating valve at the gas cylinder fixed the maximum gas inlet pressure and a pressure relief valve set at 15 MPa protected the facility against overpressure. Flow was created in the system by using a needle valve which was installed at the inlet to a gas rotameter and had an outlet open to the atmosphere. To minimise liquid carry-over from the column into the downstream instrumentation, several pieces of wire mesh were installed at the top of the column to catch stray droplets.
## 4. Bubble Formation and Coalescence
When a gas is passed through an orifice into a pool of liquid, certain forces act on the gas which breaks it up to form individual bubbles. If the bubble column is transparent, photographic techniques involving a high-speed camera can be quite helpful to analyse the shape and the size of the bubbles formed. Stills taken from high-speed videos at atmospheric pressure of [C2C1Im][EtOSO3], glycerol solutions and water are shown in Figure 3. These reveal that even at the lowest gas superficial velocity (u gs = 3 mm/s), the homogeneous flow regime is essentially absent for the IL and the glycerol solutions. In a distinct contrast to water, it is seen that the bubbles formed in these high viscous liquids are bigger and less uniform in size.
In [C2C1Im][EtOSO3] three different sized bubbles are formed. The largest are bullet-shaped Taylor bubbles which characterize slug flow. These only occur for the largest gas flow rates studied; instead clearly defined spherical-cap shaped bubbles are formed at velocities ≥ 15 mm/s. The second, intermediate sized bubbles in the IL are essentially spherical of 0.5 – 2.5 mm, while the smallest are also spherical however with diameters ≤ 250 µm. It is believed that the smallest bubbles are formed by the bursting of Taylor bubbles at the gas/liquid interface. These tiny bubbles formed have very low rise velocities and tend to flow with the liquid. As a result, they can be distributed throughout the entire column due to backmixing of the liquid.
As the bubbles rise in the column, their size can decrease or increase through various breakup or coalesce mechanisms respectively. In both [C2C1Im][EtOSO3] and glycerol solutions coalescence is observed immediately at the lower gas velocities. In contrast, due to the reduced viscosity in water coalescence is only seen at velocities > 30 mm/s. The smaller viscous or drag forces in water also leads to the formation of spherical-cap bubbles which in comparison to the IL and other viscous liquids, are rather irregular and distorted in shape (Figure 3(D)).
In the IL, coalescence of Taylor or spherical-cap bubbles is frequently observed, however coalescence between spherical-cap and intermediate bubbles is less common. In Figure 4, the coalescence between two spherical-cap bubbles is shown as an example. In the first frame the lower bubble appears to be distorted by being in the wake of the upper spherical-cap bubble. It can also be seen that the intermediate-sized bubbles in the wake of the leading spherical-cap bubble do not coalescence with the upper bubble but instead they are forced to one side by the arrival of the second spherical-cap bubble. Thereafter the intermediate-sized bubbles eventually reposition themselves in the wake of the combined spherical-cap bubble.
Bubble coalescence in [C2C1Im][EtOSO3] was also observed in high-speed videos taken of a larger diameter column (125 mm) with a gas distributor consisting of a plate with 25, 1 mm diameter holes. Trailing bubbles were seen to travel considerable distances across the column, move into the wake of a preceding one and then coalescence. A still at a gas superficial velocity of 10 mm/s is provided in Figure 5, which shows the formation of small spherical-cap bubbles even in the larger sized column.
The behaviour of single bubbles rising through liquids has been classified on the basis of three dimensionless groups, Morton, Eötvös and Reynolds numbers, defined as µ 4 gΔρ/ρ2σ3, gΔρD 2 3 and ρuD/µ respectively. Here µ is the liquid viscosity, g the acceleration due to gravity, Δρ is the difference between the gas and the liquid, ρ is the liquid density, σ is the surface tension, D is the column diameter and u is its rise velocity. The ranges of the dimensionless groups for which different types of bubbles exist and in which the effects of the channel walls become important has been identified by [33]. For [C2C1Im][EtOSO3] the Morton number is 0.014 and the Eötvös number on the order of 400. It is found that the characterization map of [33] accurately predicts the formation of spherical-cap bubbles at smaller bubble sizes.
The motion and deformation of a single bubble rising in ILs ([C4C1Im][BF4], [C8C1Im][BF4]and [C4C1Im][PF6]) were also studied by [34] through image analysis of high-speed videos. Two new empirical correlations were proposed to correlate the drag coefficient as a function of Reynolds number and the aspect ratio as a function of a new dimensionless parameter which could group experimental data for bubbles in ILs. The predicted drag coefficients agreed well with the experimental data however, further experiments are needed to verify these new correlations for a wider range of ILs with different ion pairs and physical properties.
## 5. Gas holdup at low and elevated pressures
To achieve a better understanding of the flow phenomenon present time traces of the gas holdup obtained from the conductance signals can be analysed by using statistical functions. Time-averaged values are usually obtained first and thereafter, the variations in the amplitude and in the frequency space can be explored. In each experiment presented here, the gas holdup was measured by sampling the data at 1000 Hz over a time period of 60 s. Examples of time traces at the same gas superficial velocity (u gs = 50 mm/s) are given in Figure 6. Figure 6(a) shows the effect of increasing pressure in the IL system. At 0.1 MPa, the time series shows periods of very low gas holdup values which alternate with significant periodic increases to the gas holdup. This confirms the presence of alternating aerated liquid slugs with large gas pockets and even Taylor bubbles which were seen in the photographs. In contrast, at 0.8 MPa the time trace shows that the flow is no longer intermittent. The peaks are noticeably lower and non-uniform which indicates a bubbly heterogeneous flow of smaller sized bubbles.
In Figure 6(b), it is seen that similar time traces are obtained for [C2C1Im][EtOSO3] and the glycerol solutions at 0.8 MPa suggesting that similar sized bubbles were present in both viscous systems. However when compared with water at 0.8 MPa (Figure 6(c)) a more uniform time trace of smaller gas holdup values is observed for water which suggests that the ‘larger’ bubbles present in the heterogeneous flow in the viscous systems did not form in water.
Time-averaged gas holdup values have also been calculated and plotted against gas superficial velocity for the liquids. Figures 7(a) and 7(b) show examples at atmospheric pressure and at a pressure typical of industrial conditions (0.8 MPa) respectively. These increase monotonically with gas superficial velocity. In comparison to water, the gas holdup in [C2C1Im][EtOSO3] is significantly lower while similar results are obtained for [C2C1Im][EtOSO3] and the viscous glycerol solutions. It is therefore reasonable to conclude that the observed reduction in the gas holdup in [C2C1Im][EtOSO3] is due to its higher viscosity. For high viscous media in bubble columns, the reduced gas holdup can be attributed to increases in the magnitude of viscous or drag forces exerted during bubble formation so that a stable bubble diameter is attained before its detachment at the gas distributor. By stabilizing the bubble interface, bubble coalescence is promoted and bubble breakup is suppressed in the gas distributor region. It is probable that this led to the formation of larger fast-rising bubbles in the IL and glycerol solutions. These faster bubbles spend a shorter time in the column and consequently the overall gas holdup is decreased.
For the estimation of the gas holdup, there exists a vast number of equations in the open literature, most of which are empirically based. In the review by [35], 37 published equations for gas holdup are listed. In selecting equations to predict gas holdup overall correlations based on experimental data is often used, despite the fact that it is generally better to use more physically based methods since empirical correlations are usually most suited for the experimental conditions which they were developed from. However, due to the extreme sensitivity of the gas holdup to other factors for example the gas distributor design, purity of the continuous phase and the physical properties of the phases, developing a generalised model can be complex.
The time-averaged gas holdup data for [C2C1Im][EtOSO3] was tested against three available models in the literature: Urseanu et al. [26], Wilkinson et al. [20] and Krishna et al. [36]. It was found that the prediction by [26] gives the best agreement at both low and elevated pressures (Figure 8). This model was based on their work with high viscous liquids (0.05-0.55 Pa s) at elevated pressures (0.1-1 MPa). The model by [20] which was developed specifically for the industrial scale-up of pressurized bubble column reactors is presented in Figure 9. The model is shown to be reliable at lower gas superficial velocities while the correlation by [36] significantly overestimates the gas holdup in the homogenous regime as well as the transition point.
The effect of system pressure (or gas density) on the gas holdup can be significant in certain circumstances. Figure 10 shows how increasing the pressure increases the time-averaged gas holdup in [C2C1Im][EtOSO3], water and glycerol solutions. It is believed that increased system pressure leads to enhanced local turbulence. This destabilises the larger bubbles that would otherwise form in the heterogeneous regime at atmospheric pressure. As a result bubble breakup occurs and small bubbles which have lower rise velocities are created. These bubbles have longer residence times in the column and the gas holdup is consequently increased. However it is clear that for the range of gas superficial velocities investigated, the change in gas holdup with pressure is not very significant for [C2C1Im][EtOSO3]. Similar findings were reported by [26] in their work with high viscous oils at elevated system pressures.
## 6. Detailed Behaviour of the Flow
### 6.1. Flow patterns
To examine the flow in more detail, the variations in amplitude of the gas holdup time traces can be considered using the probability density functions (PDFs), i.e., the fraction or how often particular gas holdup values occur. In the approach made popular by [37], PDF plots of gas holdup time series are often used to identify different flow patterns in gas-liquid flows, based on the distinctive shape that exists for each regime. For example, the signature PDF for homogeneous bubbly flow is characterised by a narrow single peak at low gas holdup, whereas the PDF for slug flow is double-peaked. PDFs for [C2C1Im][EtOSO3] and glycerol are presented in Figures 11 and 12 respectively. It is seen that the shapes of the PDFs for both liquids are quite similar. At atmospheric pressure, there is a single peak at low gas holdup with a broadening tail which extends to a smaller second peak at higher gas holdup values. According to [37] this signature shape is indicative of spherical-cap bubbly flow or even slug flow. The small second peak for the spherical-cap bubble is marked with arrows. This is validated by what was seen in the high-speed videos (Section 3).
At higher pressures, the probabilities of higher gas holdup values decrease for both [C2C1Im][EtOSO3] and glycerol. This provides evidence that Taylor or larger spherical-cap bubbles are not formed in the viscous systems at system pressures exceeding atmospheric. Instead the PDF traces are single-peaked, although there is some spread in the distributions which suggests that the flow consists of unequally sized bubbles, i.e., heterogeneous. The Morton number and the Eötvös number for [C2C1Im][EtOSO3] were on the order of 50 and 30 respectively for the system at higher pressures. The characterization map of [33] predicts that ellipsoidal bubbles instead of spherical-cap bubbles are expected. It also indicates that the terminal velocities of the bubbles will be strongly influenced by the walls.
Generally for gas-liquid flows, the most common way of identifying which flow pattern occurs for a given set of flow rates is to use a flow pattern map. For bubble columns, these flow pattern maps are often plots of the gas superficial velocity against the column diameter. If the gas holdup data for [C2C1Im][EtOSO3] is plotted on such a graph, conditions at which heterogeneous flow were observed would be in the homogenous region of the map. The inability of these maps to accurately predict the flow regimes in high viscous media such as [C2C1Im][EtOSO3] is probably due to the fact that they have been developed empirically using data obtained from air-water experiments.
### 6.2. Structure frequencies
Much can also be learned about the flow structures in [C2C1Im][EtOSO3] by examining their frequencies. This can be obtained by power spectrum analysis of the gas holdup time traces. Here, power spectrum densities (PSDs) have been obtained by using the Fourier transform of the auto-covariance functions. Essentially the Fourier transform is used to transform the time series from a time domain into a frequency spectrum. Examples of PSDs for [C2C1Im][EtOSO3] at 0.8 MPa are shown in Figure 13. At each gas flow rate, a clear peak in the range of 2 – 2.5 Hz is seen. These peak values are the frequencies of recurrence of any periodic structures. The frequencies obtained using this method have been compared with those determined from manually counting the peaks in the time traces. Values obtained from the two methods agree within 10%.
The effect of increasing pressure on the structure frequencies in [C2C1Im][EtOSO3] is shown in Figure 14. It is seen that for the same gas flow rates, the frequencies generally increase with increasing system pressure. The increasing number of structures at higher pressures further suggests that at higher pressures bubble breakup or coalescence suppression occurs in [C2C1Im][EtOSO3].
The frequencies obtained for [C2C1Im][EtOSO3] at 0.1 MPa and 0.8 MPa can also be compared to the equivalent data for glycerol-NaCl. These are shown in the subplots presented in Figure 15. At atmospheric pressure it is seen that there are only small differences between the frequencies for the different liquids and they decrease with increasing gas flow rates. However at a higher system pressure, larger variation is observed particularly at the highest gas flow rates. To analyse this further aspects of the axial variations in the flow have also been examined. The flow development in [C2C1Im][EtOSO3] and glycerol-NaCl based on their frequency and PDF data obtained from both conductance probes are shown in Figures 16 and 17 respectively. It is seen that frequencies obtained for the probe downstream (B) in [C2C1Im][EtOSO3] is slightly higher than probe upstream (A) while the PDF downstream displays a taller peak indicating a larger probability of small gas holdup values there. The opposite is observed in the results for glycerol-NaCl. A possible explanation for this could be that the difference in molarity between the two liquids had an effect on the coalescence of bubbles. The glycerol-NaCl solution has a molarity of 0.17 M, while that of [C2C1Im][EtOSO3] is essentially infinite. In the literature, a critical value of 0.2 M has been cited for the suppression of coalescence in liquids [38-40]. Thus, it is possible that coalescence is much reduced for the IL at higher pressures.
### 6.3. Structure velocities
The velocities of bubbles can be determined from image analysis of high-speed videos or more objectively from cross-correlating the signals of probes placed slightly apart. For the latter, a time lag which corresponds to a peak in the cross-correlation function represents the average time required for flow structures to travel between the two probes. Since the distance between the two probes are known, the structure velocity is then easily calculated. This method of velocity measurement has been used to determine the structure velocities in [C2C1Im][EtOSO3]. In Figure 18, the flow velocities in [C2C1Im][EtOSO3] at atmospheric pressure and 0.8 MPa are shown respectively. For comparison, the equivalent data for glycerol is also presented. It can be seen that there is very good agreement between both viscous systems.
The effect of increasing system pressure on the bubble rise velocities in [C2C1Im][EtOSO3] is illustrated in Figure 19. It is seen that in general the structure velocities decrease with increasing pressure. This is expected given the smaller bubble sizes in the system at elevated pressures. For gas-liquid flows the correlation by Nicklin et al. [41] is often used to predict the velocities of flow structures including void fraction waves and slugs. This correlation has been tested against the present IL data. It is found that the correlation predicts the correct linear trend, although with higher absolute values.
## 7. Conclusion and outlook
From the perspective of industrialization, research on the hydrodynamics of ionic liquids in bubble columns has been presented in this chapter. Despite being in its beginning stages, it is evident that this work is important for future industrial scale-up and process design. The experimental results reveal that the flow characteristics of the ionic liquid are similar to those of the other viscous media studied, both at atmospheric and elevated system pressures. Due to its high viscosity, the bubbles formed in the ionic liquid are larger compared to water which results in a significant reduction to the gas holdup. This creates smaller specific interfacial areas and less effective gas-liquid contacting. Thus to successfully achieve intimate contact between an ionic liquid and a gas stream, the viscosity of the ionic liquid is an important factor to be considered in the choice and design of industrial equipment.
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https://www.physicsforums.com/threads/entropy-changes.319295/ | # Entropy changes
1. Jun 11, 2009
### CatWoman
1. The problem statement, all variables and given/known data
(a) A 0.5kg mass of copper (specific heat 385 J kg-1K-1) at 600K is plunged into a litre of water at 20C. What is the equilibrium temperature, T2, of the system? What is the change in entropy of the water?
(b) A litre of water is heated slowly at a constant rate from 20C to T2. What is the change in entropy?
How do you account for the difference between (a) and (b)?
2. Relevant equations
not totally sure
3. The attempt at a solution
(a) ∆T_water=T_2-T_1=T_2-293 so ∆Q_water=C ∆T=〖4200 (T〗_2-293)
∆T_copper=T_2-T_1=T_2-600 so ∆Q_copper=C ∆T=〖385/2 (T〗_2-600)
Heat lost by copper is heat gained by water so ∆Q_water= ∆Q_copper so
〖4200 (T〗_2-293)=〖385/2 (T〗_2-600) => T_2=336K=63 degrees Centigrade
This is not an isothermal process so cannot use S=Q/T to calculate entropy before and after to calculate entropy difference.
Instead, use ∆S=C ln〖T_2/T_1 〗 so ∆S=4200 ln〖336/293=575 J K^(-1) 〗
However, if I treat it like an isothermal process, where
∆Q=4200 (336-293)=4200×43=180600 Joules
Then ∆S=∆Q/T=180600/293=616 J K^(-1)
The extra entropy gained could be because some of the water has turned to steam.
Please could somebody tell me what I am doing right, how I should calculate the entropy differently for part a and part b, and why, as I am very confused!!!
Many thanks :)
2. Jun 11, 2009
### Mapes
Entropy is a state variable, dependent only pressure, temperature, and amount of material. I'm having a hard time seeing why the entropy change of water between 20°C and some arbitrary temperature $T_2$ could be path dependent, even if some quantity has boiled and recondensed.
3. Jun 11, 2009
### CatWoman
Many thanks - yuor logic makes sense to me so I don't know what the question is trying to get at.
Similar Discussions: Entropy changes | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9140852093696594, "perplexity": 2356.9002173064587}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890874.84/warc/CC-MAIN-20180121195145-20180121215145-00101.warc.gz"} |
https://physics.stackexchange.com/questions/143790/the-hamiltonian-and-energy | # The Hamiltonian and Energy
if anyone can give assistance on this question it is much appreciated! Suppose I have a Hamiltonian $$H=\frac{p^{2}}{2m}+V(r)+F(r,t)$$ where $$F(r,t) = Q(r)\Re{(e^{{iT(t)}})}:\quad Q(r), T(t),T'(t) \neq 0$$ is seperable.
Now since $$\frac{\partial H}{\partial t}=Q(r)\Re (ie^{iT(t)}\cdot \frac{d T}{dt}) \neq 0$$ does this mean the Hamiltonian (being time dependent) is not conserved (this is my guess, since it does not equal zero)
Also since the Hamiltonian is time dependent and we have that $$T=\frac{p^2}{2m}, \quad V=V(r),\quad E=T+V \Longrightarrow H-E \neq 0 \Longleftrightarrow H \neq E$$ i.e. because we have that extra time dependent term, I'm guessing that signifies the Hamiltonian cannot be the total system energy. Any information/places is apppreciated :)
Your Hamiltonian has the "potential" $U = V + F$, just because $F$ is time-dependent doesn't make it not part of the potential. You are correct that the Hamiltonian is not conserved.
The Hamiltonian is the total energy of the system if the Lagrangian does not contain terms linear in the velocity or when the generalized coordinates in the Lagrangian do not depend explicitly on time. In that case, the Hamiltonian is T+V and is automatically the total energy.
However, if these conditions are not met, it does not mean that $H$ is not the total energy. The obvious example here is the vector potential. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9886848330497742, "perplexity": 206.55321723002203}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570871.10/warc/CC-MAIN-20220808183040-20220808213040-00468.warc.gz"} |
http://mathhelpforum.com/differential-geometry/90948-bounded-variation.html | 1. Bounded variation
what we mean when we say this function is of bounded variation please help me
Thanks
2. Originally Posted by Amer
what we mean when we say this function is of bounded variation please help me
Thanks
Given $f:[a,b] \to \mathbb{R}$ and a partition $P=\{a=t_0 < t_1 < ... < t_n=b \}$ then the vatiatin of fo over P is $V(f,P)=\sum_{n=0}^{n}|f(t_i)-f(t_{i-1}|$
Notice that this is very similar to a Riemann sum.
Now if we take the supreemum of all partitions of [a,b] This is called the total variation if $\sup_{P}=V(f,p) < \infty$
3. it is clear now thanks | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9980471730232239, "perplexity": 394.7782565679717}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721008.78/warc/CC-MAIN-20161020183841-00363-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://en.wikipedia.org/wiki/Talk:Poisson's_ratio | # Talk:Poisson's ratio
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## Large Deformation Formula
Can someone tell me what strain measure is used in the graph showing small vs. large deformation lines for the large deformation formula? - EndingPop 18:18, 18 March 2007 (UTC)
small deformations assumes that strain is linear (the first derivative is a constant number), large deformation assumes that the strain is not linear (depending on the precision is it not constant up to second or larger "n-th" derivative). If you are talking about "Engineering strain", "true strain", "logarithmic strain" or "lagrange strain" then it is of no importance here: this plot could apply to any of them. The point is that whether you represent strain deformation using a linear or non-liear (more precise) function. Janek Kozicki 20:40, 19 March 2007 (UTC)
## arnt the eqns under orthotropic material true for isotropic?
this should be mentioned. maybe
---
Orthotropic means it is different in different directions. Isotropic means it is the same in all directions. Isotropic equations would be found under the generalize section.--thegreatco (talk) 17:48, 14 May 2008 (UTC)
## Area?
The definition given is in terms of strain in the transverse direction. But is that the average strain in the plane orthogonal to the stress, or what? —BenFrantzDale 07:14, 20 October 2005 (UTC)
how about a nice little equation to go along with the description...? --alex
## Moved from article
G is actually the Shear Modulus, a materials resistance to a change in shape without a change in volume. G is equal to shear stress, tao, over shear strain, gamma. Tom Harrison Talk 19:50, 24 May 2006 (UTC)
## References
References would be good for the values listed for various materials. — DIV (128.250.204.118 05:59, 14 May 2007 (UTC))
## A bit of humor
The Poisson Conspiracy! http://home.messiah.edu/~tvandyke/podcast5.mp3 153.42.211.95 18:44, 5 October 2007 (UTC)
## Merge
The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section.
The result was merge POISSON'S EFFECT into POISSON'S RATIO. The merge was proposed almost two months ago (on 19th March). All opinions (the four expressed below and the one of the user who initially proposed it) are in favor of the merge. -- Federico Grigio, alias Nahraana (talk) 17:54, 25 April 2008 (UTC)
Poisson's effect really should get merged in. That page has some good stuff, so it'll take a little thought, but not much. —Ben FrantzDale (talk) 02:10, 29 April 2008 (UTC)
You say the page has good stuff and that you want it to be merged. Now, what is the reason why it should be merged? Federico Grigio, alias Nahraana (talk) 08:08, 1 May 2008 (UTC)
They should be merged because they cover the same topic. Together they contain more details than either one, but the effect and the ratio go hand in hand; the ratio is just a measure of the effect. —Ben FrantzDale (talk) 13:08, 1 May 2008 (UTC)
I see. I agree to merge the articles, because of the same reasons as above. Federico Grigio, alias Nahraana (talk) 08:34, 5 May 2008 (UTC)
I agree, they should be merged. They are taught as one in all my engineering courses. The ratio is the measure of the effect. thegreatco (talk) 17:22, 6 May 2008 (UTC)
Merge it. Greg Locock (talk) 01:58, 12 May 2008 (UTC)
I'll be honest, I've never merged a page and I don't think Poisson's Ratio is the place to start. --thegreatco (talk) 17:44, 14 May 2008 (UTC)
The above discussion is closed. Please do not modify it. Subsequent comments should be made in a new section.
Hi. I started merging (deleted some repeated information and copied-pasted two sections).
The section "Calculation of Poisson’s effect" is still left in Poisson's effect. Maybe, someone who likes equations can carefully implant its non-repeated bit into this article.
Federico Grigio, alias Nahraana (talk) 11:31, 15 May 2008 (UTC)
After we finish the merge (when all content has been removed from Poisson's effect) we can place #REDIRECT [[Poisson's ratio]] on it. Federico Grigio, alias Nahraana (talk) 11:35, 15 May 2008 (UTC)
## Added distinction between pre-and post-yield behavior
I made a slight modification to distinguish between pre-yield and post-yield deformation. This is strongly related to mods I made earlier today on the Necking(engineering) page. Also made it slightly less metals-centric. Crosslink (talk) 22:37, 16 February 2009 (UTC)
## Definition of Poissons Ratio
The definition given in the main article refers only to tensional forces (ie: stretching). George Davis (Structural geology) defines it as "the degree to which a core of rock bulges as it shortens" under compression.Iain —Preceding unsigned comment added by 41.0.29.98 (talk) 10:45, 23 October 2009 (UTC)
## Greater than 0.5 or not?
I was wondering when I read "ν cannot be less than −1.0 nor greater than 0.5." And later on there is this honeycomb thing with ν up to 3.88. How does that fit together??? (I was wondering even before I read the second thing cause I could easily design a grid which would have ν out of the upper range. Obviously a kind of honeycomb.) So does the first statement have to be relativated somehow? Like "for bulk specimens..." or so? Peterthewall (talk) 17:41, 18 January 2010 (UTC)
Wouldn't be isotropic. Greglocock (talk) 02:52, 19 January 2010 (UTC)
Yes, thanks. I came to that conclusion as well now ;-) But it took a while. When I first read it there it seemed to be so general. I added in the intro that there are as well materials exhibiting values > 0.5. Peterthewall (talk) 11:33, 19 January 2010 (UTC)
I read the reference to the paper by Robert (Bob) Park and couldn't where the paper stated Possion's ratio of 0.5 for mild steel in the post-yield range. If anyone can point me towards some information for the value once mild steel has yielded I would be very happy. — Preceding unsigned comment added by 132.181.5.142 (talk) 23:07, 27 November 2014 (UTC)
These guys certainly think so : "For isotropic elastic continuum analysis, a single value of Poisson’s ratio is sufficient to fully characterize the material response since standard plastic flow rules assume incompressibility, and a Poisson’s ratio of 0.5 in the post-yield regime. F"
Cheers Greglocock (talk) 00:23, 28 November 2014 (UTC)
## Changing the equations for orthotropic and transversely isotropic materials
Some users have been changing the equations for orthotropic and transversely isotropic materials without understanding the significance of the quantities. These equations can be written in many ways. However, the definitions of the quantities in the equations must be consistent throughout the article. They are consistent as of now. For example, changing $\nu_{xy}/E_x$ to $\nu_{xy}/E_y$ changes the meaning of $\nu_{xy}$. The new meaning is non-standard and a explanation has to be given on why such a menaing has been chosen. Also, that change in meaning has to be reflected throughout the article, including in places where various Poisson's ratios of composites have been listed. Please discuss your reasons for changing these equations before going ahead with the changes. Bbanerje (talk) 22:04, 15 February 2010 (UTC)
To be consistent with the matrix for transverse isotropy on the Hooke's Law page, it might be best to change Gyx to Gxy or explicitly state they are equivalent as you have done for other relations. —Preceding unsigned comment added by 18.95.5.74 (talk) 01:55, 1 March 2010 (UTC)
## But what does it mean?
transverse shear deformation blah blah blah. I read the whole article and still don't know what it means. This article is obviously written by intelligent, articulate, educated people, but it carries a prerequisite of maybe two semesters of physics or engineering or something.
Could someone please just throw in a statement (preferably in the first paragraph), to the effect of, "a poisson ratio of [?] means it's very strong, or it will break easily when twisted, or whatever it means. Meanwhile, I'm off to find out if a polycarbonate sheet with a poisson ratio of 0.37-0.38 is suitable for my project.
thanks, —Preceding unsigned comment added by Dusty.crockett (talkcontribs) 02:47, 20 June 2010 (UTC)
The second paragraph says:
When a sample cube of a material is stretched in one direction, it tends to contract (or occasionally, expand) in the other two directions perpendicular to the direction of stretch. Conversely, when a sample of material is compressed in one direction, it tends to expand (or rarely, contract) in the other two directions. This phenomenon is called the Poisson effect. Poisson's ratio ν (nu) is a measure of the Poisson effect.
What part of that don't you understand? How can it be improved?PAR (talk) 03:10, 20 June 2010 (UTC)
@Dusty.crockett: The Poisson's ratio of a material is a measure of stiffness, not a measure of strength. Someone who knows a bit of undergraduate level mechanics (usually an engineering student) can help in choosing a material with required properties. However, such a task is best left to professionals. Bbanerje (talk) 04:30, 20 June 2010 (UTC)
Bbanerje, I don't think it's proper to call it a "stiffness" property; it's a unique intensive material property. Wizard191 (talk) 15:47, 20 June 2010 (UTC)
Wizard191, the Poisson's ratio has a precisely defined meaning in linear elasticity as a scaling factor for terms in the stiffness tensor. Therefore it is a measure of stiffness though it is dimensionless. The Young's modulus, which is the other stiffness measure in linear elasticity, is also an intensive property. The Poisson effect, which is a qualitative description of the scaling phenomenon, describes a larger range of situations than does the Poisson's ratio which is well-defined only for infinitesimal strains. Bbanerje (talk) 22:56, 20 June 2010 (UTC)
Honestly, I'm not familiar with the stiffness tensor at all. However, I can say that this article doesn't mention stiffness once and I've never heard of Poisson's ratio being related to stiffness. However, if you have a reference for your definition, I would like to see it. Wizard191 (talk) 16:50, 21 June 2010 (UTC)
The stiffness tensor is also known as the elasticity tensor. It is the "tensor of proportionality" in the tensor expression of Hooke's law which states that $\sigma_{ij}=C_{ijmn}\varepsilon_{mn}$, $C_{ijmn}$ being the stiffness or elasticity tensor. It corresponds to k in the scalar statement of Hooke's law: F=k x. We should probably make its name consistent across all articles. I am in favor of calling it the stiffness tensor, since as it becomes larger, the force (stress) needed to produce a given displacement (strain) increases. The stiffness tensor will be a function of two of the elastic moduli, one of which may or may not be the Poisson ratio, so you cannot really say the Poisson ratio alone is a measure of stiffness, I think. PAR (talk) 20:02, 21 June 2010 (UTC)
"What part of that don't you understand? How can it be improved?PAR"
Thanks for asking. I see that my question might be a bit obtuse; the definition of the ratio itself is pretty clear, but I was hoping to find a more concise "bottom line" explanation. I've worked it out (only took a little bit of effort) -- a material with a ratio close to 0 retains its original proportions when stretched better than one with a larger (positive or negative) ratio. If it expands in the direction perpendicular to the stretch, the ratio is negative; if it contracts, the ratio is positive. I realize that might be too simplistic, there might be other factors that would render such a statement inaccurate. BTW, The table of ratios for various materials -- that's a nice touch. Dusty.crockett (talk) 00:45, 23 June 2010 (UTC)
Now I've gotten around to reading some of this discussion, I see the above quote from George Davis, "the degree to which a core of rock bulges as it shortens" under compression. Although limited in scope, it seems a very clear illustration of exactly what is being measured. For clarification, my only reason for posting here is to provide the perspective of a non-engineer who see the term "Poisson's Ratio" on some product specs, and just wants to fit it into context. I'm basically not qualified to add content to a technical subject like this. Thanks. Dusty.crockett (talk) 13:53, 23 June 2010 (UTC)
## bonkers
The way the equation is defined won't give you a poisson's ratio of 0.5 for a perfectly incompressible material. It gives a ratio of 2 as defined in the article. Draw a quick before and after square diagram to see what I mean. I understand the mechanics, but the definition of strain directions in the equation is misleading and gives you the inverse ratio.
Just looking at the figure provided, strain (as elongation) in an x direction is larger than strain in the y direction, which gives a value higher than 1.
It seems more intuitive to just define it as the ratio of contraction to associated longitudinal extension —Preceding unsigned comment added by Georesidue (talkcontribs) 19:30, 25 June 2010 (UTC)
This definitely needs to be clarified. Strain is not a dimension (s) of an element, it is a dimensionless quantity ds/s, where ds is the change in a dimension when stressed, so taking a ratio of width to height as the Poisson ratio of a distorted cube is not correct. Also, we have to work in three dimensions, not two.
There are three concepts, the position of an infinitesimal element inside the material $[x,y,z]$, the displacement of that element $u=[u_x,u_y,u_z]$ and the strain tensor $\varepsilon\,$ which is a measure of how much the element is distorted. For the simple example of a cube compressed on its x faces, only the diagonal elements of strain are not zero. The diagonal elements of strain are $\varepsilon=[du_x/dx,du_y/dy,du_z/dz]$. In the simple example, if you have a cube, one unit of length on a side, and slightly compress it on its +x and -x faces, it will change from having dimensions $[1,1,1]\,$ to $[(1-b), (1+a), (1+a)]\,$ where a and b are small, (usually) positive values. In other words, an infinitesimal element inside the cube originally at $[x,y,z]$ will be moved to $[(1-b)x,(1+a)y,(1+a)z]\,$ so that its displacement from its original position will be $u=[-bx,ay,az]\,$ and the diagonal elements of the strain will be $\varepsilon=[-b,a,a]\,$. Poisson's ratio will then be $\nu=-\varepsilon_{yy}/\varepsilon_{xx}=-\varepsilon_{zz}/\varepsilon_{xx}=a/b\,$. If the material maintains constant volume, we must have $(1-b)(1+a)^2=1\,$. Solving for Poisson's ratio in terms of a gives $\nu=(1+a)^2/(2+a)\,$. In the limit of small values of a, that's $\nu=1/2\,$.
Now we just have to make that clear in the article.PAR (talk) 12:33, 26 June 2010 (UTC)
Right, that all makes total sense. I think the misleading part in the article (at least for me) comes from the inexact way of defining how the principal stresses are oriented for the example shown. It currently states:
"Assuming that the material is stretched or compressed along the axial direction (the y axis in the diagram):"
But for the equation specified, it needs to be the former, not the latter, which also makes the picture backwards. Either the picture is backwards or the equation is inverted. OR just remove any mention of x and y axes and specify directions of compression and tension. I think...
Thanks for the response —Preceding unsigned comment added by Georesidue (talkcontribs) 19:35, 26 June 2010 (UTC)
## Incremental Poisson's ratio
I notice that the definition of Poisson's ratio has been replaced with an incremental form
$\nu = -\frac{d\varepsilon_\mathrm{trans}}{d\varepsilon_\mathrm{axial}} = -\frac{d\varepsilon_\mathrm{y}}{d\varepsilon_\mathrm{x}}= -\frac{d\varepsilon_\mathrm{z}}{d\varepsilon_\mathrm{x}}$
Though not incorrect in the strict sense, this definition is does not accurately represent the quantity used by engineers. The reason is that the constant ratio $\nu$ is applicable only for linear elastic materials in which case
$\frac{d\varepsilon_\mathrm{trans}}{d\varepsilon_\mathrm{axial}} = \frac{\varepsilon_\mathrm{trans}}{\varepsilon_\mathrm{axial}} \,.$
The definition should agree with recognized standards such as the ASTM standrard "ASTM E132 - 04 Standard Test Method for Poisson's Ratio at Room Temperature" which clearly states that
"When uniaxial force is applied to a solid, it deforms in the direction of the applied force, but also expands or contracts laterally depending on whether the force is tensile or compressive. If the solid is homogeneous and isotropic, and the material remains elastic under the action of the applied force, the lateral strain bears a constant relationship to the axial strain. This constant, called Poisson’ratio, is an intrinsic material property just like Young’modulus and Shear modulus."
The standard relations between the Poisson's ratio and the other moduli of elasticity are not generally valid for incrementally linear elasticity. I'd suggest that you revert back to the standard definition and add a caveat saying that incremental Poisson's ratios are sometimes used in nonlinear elasticity. Bbanerje (talk) 02:26, 26 July 2010 (UTC)
The standard relations between the Poisson's ratio and the other moduli of elasticity ARE generally valid for incrementally linear elasticity, as long as the other moduli are also defined incrementally. The real question is - what is the (incremental) Poisson's ratio for a stressed material? If it is a constant, independent of the amount of deformation, then the "integrated" Poisson's ratio is a function of the amount of deformation, as presently described in the article. Conversely, if the "integrated" Poisson's ratio is constant, which is what you are talking about, then the incremental Poisson's ratio must vary with the amount of deformation. Only when the deformations are very small with respect to the dimensions of the unstressed test object do the two become very nearly the same. I do not have access to ASTM E132-04, but I cannot imagine that it does not contain caveats about carrying the application of the test stress or strain too far. In other words, the quotation you give cannot be giving the complete picture. If you do have access to this publication, can you investigate what these caveats are? Does it give specific limits on the amount of relative deformation (or strain) under which the measurement is valid? If it does not, it is incomplete, and if it does, are those limits large or small with respect to the unstressed dimensions of the test object? PAR (talk) 13:46, 26 July 2010 (UTC)
"Only when the deformations are very small with respect to the dimensions of the unstressed test object do the two become very nearly the same". I don't have the full ASTM standard but it's intended primarily for metals in which the elastic regime is small and relatively linear. The popular definition of Poisson's ratio is valid only for infinitesimal strains and linear elasticity. The Poisson effect may be observed beyond that regime. But the Poisson's ratio is neither constant nor does it have any fixed relationship with other moduli beyond the small strain regime. That can be verified by exploring any of a number of hyperelastic material models. My point is that using the term "Poisson's ratio" when we mean a more general "Poisson's effect" confuses rather than clarifies. You can see an example of the problem in P. Gretener, 2003, "AVO and Poisson's ratio", The Leading Edge; January 2003; v. 22; no. 1; p. 70-72, Society of Exploration Geophysicists. Bbanerje (talk) 22:54, 26 July 2010 (UTC)
──────────────────────────────────────────────────────────────────────────────────────────────────── I'm still having trouble understanding your objection. Maybe it is trouble with definitions. Considering a cube of side L subject to normal stress on the x-faces only, not necessarily small, with its stretched x or axial length being $L+\Delta L$ and its contracted transverse length being $L-\Delta L'$ we can define the "finite" strains as $\varepsilon_x=\Delta L/L$ and $\varepsilon_y=-\Delta L'/L$. If we then subject the cube to a further infinitesimal stress, the axial or x faces will move to $L+\Delta L+\delta L$ and the transverse faces will move to $L-\Delta L'-\delta L'$, where the $\delta$'s are infinitesimally small. The infinitesimal strains will be $d\varepsilon_x=\delta L/(L+\Delta L)$ and $d\varepsilon_y=-\delta L'/(L-\Delta L')$. Now we can define a finite Poisson's ratio as
$\nu_{finite}(\varepsilon_x)=\Delta L'/\Delta L$
where $\varepsilon_x$ is chosen as the measure of the pre-existing strain on the cube. An infinitesimal Poisson's ratio can be defined as:
$\nu_{inf}(\varepsilon_x)=\delta L'/\delta L$
And the "usual" Poisson's ratio will then be $\nu_{inf}(0)$ which will be equal to the limit of $\nu_{finite}(\varepsilon_x)$ as the $\varepsilon_x$ approaches zero. This will be a constant of the material, independent of its dimensions. These Poisson's ratios are not necessarily equal to each other except when $\varepsilon_x$ approaches zero. The finite Poisson's ratio can be found by integrating the infinitesimal Poisson's ratio from zero out to the final $\varepsilon_x$. Linear elasticity assumes infinitesimally small strain variations about $\varepsilon_x$. Usually, one considers the case of an initially unstressed material, i.e. infinitesimally small strain variations about $\varepsilon_x=0$. The present article, when calculating the large scale deformations, assumes that $\nu_{inf}(\varepsilon_x)$ is independent of $\varepsilon_x$, allowing the integration to be easily performed.
I am in favor of saying that Poisson's ratio is $\nu_{inf}(0)$, i.e. the ratio of the transverse to axial changes in distance (or strain) for very small values of those distances (or strains) for an unstressed material. This is given by the "incremental" definition provided. If we say Poisson's ratio is the ratio of the strains for a linear material, we are saying essentially the same thing, since linear implies infinitesimally small variations. If we simply say its the ratio of strains, large or small, then we are giving a definition which is not a constant of the material except under very wierd conditions. PAR (talk) 00:29, 27 July 2010 (UTC)
It's a question of definition, i.e., what is the accepted definition of Poisson's ratio and which source should we use to determine that definition? For instance, the article uses a concept called strain. What is the definition of strain that is used in the definition of Poisson's ratio? Is it the infinitesimal strain, the true strain, the Lagrangian Green strain, or some other strain measure? My take is that we should keep it simple and define the Poisson's ratio as a ratio of infinitesimal strains in linear elasticity and we can call similar ratios in nonlinear or finite strain elasticity something else. Bbanerje (talk) 23:01, 27 July 2010 (UTC)
I agree, but isn't the ratio of infinitesimal strains $d\varepsilon_t/d\varepsilon_a$ rather than $\varepsilon_t/\varepsilon_a$? PAR (talk) 22:10, 28 July 2010 (UTC)
We may be talking about two different things. The infinitesimal strain tensor is defined as $\varepsilon = \tfrac{1}{2}\left((\nabla\mathbf u)^T + \nabla\mathbf u\right)$ where $\|\mathbf u\| \ll 1 \,\!$ and $\|\nabla \mathbf u\| \ll 1$. So a ratio of such strains is $\varepsilon_1/\varepsilon_2$, for example. People usually find that ratio easier to understand than ratios of differentials or derivatives. For the benefit of others here's why I think the simple ratio is good enough. For a linear material with constant $\nu$ (ignoring signs)
$\varepsilon_1 = \nu\varepsilon_2 \implies d\varepsilon_1/d\varepsilon_2 = \nu$. Alternatively, we can do
$d\varepsilon_1/d\varepsilon_2 = \nu \implies \varepsilon_1 = \int \nu d\varepsilon_2 + C = \nu\varepsilon_2 + C \implies \nu = \varepsilon_1/\varepsilon_2 - C/\varepsilon_2$
If $\varepsilon_1 = 0 \implies \varepsilon_2 = 0$ we have $C=0$ and we get the standard definition of Poisson's ratio.Bbanerje (talk) 01:10, 30 July 2010 (UTC)
## Allowable range of ν
Try again from above: Allowable range of ν
It would be useful to have a short section that identifies the allowable range of ν, which is -1 to 1/2, based on the equation 3B(1-2ν) = 2G(1+ν)
132.250.22.8 (talk) 20:26, 6 October 2010 (UTC) goldfish
## Cause of the Poisson effect
Is it me, or does the "cause" part simply restate the effect, without explaining the underlying mechanism? To me it reads as "Materials exhibit the Poisson effect because their molecular structure exhibits the Poisson effect." Could someone give a clearer explanation of why this happens? TheCat5001 (talk) 10:16, 3 February 2011 (UTC) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 55, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9420896768569946, "perplexity": 867.7206869162687}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207932182.89/warc/CC-MAIN-20150521113212-00241-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.cuemath.com/ncert-solutions/q-1-exercise-2-2-fractions-and-decimals-class-7-maths/ | # Ex 2.2 Q1 Fractions-and-Decimals-Solutions NCERT Maths Class 7
Go back to 'Ex.2.2'
## Question
Which of the drawing $$(a)$$ to $$(d)$$ show:
i) \begin{align}2 \times \frac{1}{5}\end{align}
ii) \begin{align}2 \times \frac{1}{2}\end{align}
iii) \begin{align}3 \times \frac{2}{3}\end{align}
iv) \begin{align}3 \times \frac{1}{4}\end{align}
## Text Solution
What is Known?
Fractions and Drawings
What is unknown?
Matching of fractions with shaded part of the drawings.
Reasoning:
Matching can be easily done by comparing the fractions and shaded areas.
Steps:
i) \begin{align}2 \times \frac{1}{5}\end{align} matches with $$(d)$$ since two circles are divided in to five parts and one part of both the circles is shaded.
\begin{align} 2 \times \frac{1}{5} = \frac{1}{5} + \frac{1}{5} = \frac{2}{5} \end{align}
ii) \begin{align}2 \times \frac{1}{2}\end{align}matches with $$(b)$$ as one half of both the drawings is shaded.
\begin{align} 2 \times \frac{1}{2} = \frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1 \end{align}
iii) \begin{align}3 \times \frac{2}{3}\end{align}matches with $$(a)$$ since two third of the three circles is shaded.
\begin{align} 3 \times \frac{2}{3} = \frac{2}{3} + \frac{2}{3} + \frac{2}{3} = 3 \times \frac{2}{3} = 2 \end{align}
iv) \begin{align}3 \times \frac{1}{4}\end{align}matches with $$(c)$$ since one fourth of three squares is shaded.
\begin{align} 3 \times \frac{1}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4} \end{align}
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• Completely personalized curriculum | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 24, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000100135803223, "perplexity": 3209.3950581503427}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496666229.84/warc/CC-MAIN-20191113063049-20191113091049-00205.warc.gz"} |
https://www.clutchprep.com/physics/practice-problems/137712/consider-the-circuit-shown-in-the-figure-below-let-r-36-0-a-find-the-current-in- | Solving Resistor Circuits Video Lessons
Concept
# Problem: Consider the circuit shown in the figure below. (Let R = 36.0 Ω.)(a) Find the current in the 36.0-Ω resistor.A(b) Find the potential difference between points a and bV
###### FREE Expert Solution
95% (219 ratings)
###### Problem Details
Consider the circuit shown in the figure below. (Let R = 36.0 Ω.)
(a) Find the current in the 36.0-Ω resistor.
A
(b) Find the potential difference between points a and b
V | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9361240863800049, "perplexity": 1745.7662127689775}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363465.47/warc/CC-MAIN-20211208083545-20211208113545-00217.warc.gz"} |
https://www.matchfishtank.org/curriculum/math/algebra-1/linear-equations-inequalities-and-systems/lesson-10/ | # Linear Equations, Inequalities and Systems
## Objective
Identify solutions to systems of inequalities graphically. Write systems of inequalities from graphs and word problems.
## Common Core Standards
### Core Standards
?
• A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
• A.REI.D.12 — Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
?
• 8.EE.C.8
## Criteria for Success
?
### Problem 1
Some treasure has been buried at a point ${(x,y)}$ on the grid, where $x$ and $y$ are whole numbers.
Clue 1: $x> 2$
Clue 2: $x+y< 8$
Clue 3: $2y-x\geq 0$
Which of the following points could be a possible location for the treasure?
(3,2) (2,3) (5,3) (3,5) (4,3) (5,2)
### Problem 2
Write a system of linear inequalities that only has the region named as part of the solution set.
## Problem Set
?
The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.
• Include problems where there is no solution for the system of inequalities as well as all solutions for the system of inequalities. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8559577465057373, "perplexity": 1227.5851816290883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402128649.98/warc/CC-MAIN-20200930204041-20200930234041-00173.warc.gz"} |
https://cris.tau.ac.il/en/publications/a-model-explaining-the-anomalous-fading-effect-in-thermoluminesce | # A model explaining the anomalous fading effect in thermoluminescence (TL)
J. L. Lawless, R. Chen*, V. Pagonis
*Corresponding author for this work
Research output: Contribution to journalArticlepeer-review
## Abstract
An energy-level model consisting of an electron trap, a hole trap and a hole recombination center is proposed to explain the anomalous-fading effect of thermoluminescence which has been observed in several materials. The present model is related to a thermoluminescence glow-curve consisting of two peaks. The relevant set of coupled differential equations is considered and a set of the relevant parameters is chosen. The equations are solved both by making plausible analytical approximations and numerically by using a Matlab solver; the results of the two approaches are in very good agreement. The simulated sample is excited at 100K and then held at room temperature or lower for different lengths of time before simulating the heating stage. It is found that as expected, the low-temperature peak at ∼400K decays very quickly and quite anomalously, the high-temperature peak at ∼570K is also fading much faster than expected for a peak occurring at this temperature. The dependence on the fading temperature, an effect which has been found in some materials is also demonstrated in the simulations. The numerical simulations and the analytical approximations can explain these results and show why this decoupling between the peak temperature and the fading takes place.
Original language English 106881 Radiation Measurements 160 https://doi.org/10.1016/j.radmeas.2022.106881 Published - Jan 2023 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9265281558036804, "perplexity": 1218.3282266841438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949958.54/warc/CC-MAIN-20230401094611-20230401124611-00301.warc.gz"} |
http://mathhelpforum.com/pre-calculus/62380-quick-question-about-fractions.html | # Math Help - Quick question about fractions
1. ## Quick question about fractions
If I have the fraction (x+2)/(x^2-9), where do I find the zeros -- the numerator or the denominator?
Would the zeros of the equation be just -2, or would they be 3 and -3 (x+3, x-3 from the denominator)?
Thanks for your help!
2. Originally Posted by live_laugh_luv27
If I have the fraction (x+2)/(x^2-9), where do I find the zeros -- the numerator or the denominator?
Would the zeros of the equation be just -2, or would they be 3 and -3 (x+3, x-3 from the denominator)?
Thanks for your help!
I assume you have a rational function here.
$f(x)=\frac{x+2}{(x+3)(x-3)}$
The zero in this case is -2. You have two vertical asymptotes at x=3 and x=-3
3. Originally Posted by masters
I assume you have a rational function here.
$f(x)=\frac{x+2}{(x+3)(x-3)}$
The zero in this case is -2. You have two vertical asymptotes at x=3 and x=-3
That's what I thought...thanks for confirming | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9578162431716919, "perplexity": 1127.218895182127}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663135.34/warc/CC-MAIN-20140930004103-00469-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://owenduffy.net/blog/?p=12578 | FT82-43 matching transformer for an EFHW
A published design for an EFHW matching device from 80-10m uses the following circuit.
Like almost all such ‘designs’, they are published without supporting measurements or simulations.
The transformer is intended to be used with a load such that the input impedance Zin is approximately 50+j0Ω, Gin=0.02S.
Analysis of a simple model of the transformer with a load such that input impedance is 50+j0Ω gives insight into likely core losses.
Let us calculate the magnetising admittance of the 3t primary at 3.6MHz. The core is a FT82-43 ferrite toroid.
Gcore is the real part of Y, 0.00691S.
If Yin of the loaded transformer is 0.02S, we can calculate the core efficiency as 1-Gcore/Gin=1-0.00691/0.02=65.4%, core loss is 1.87dB.
Now as to whether 65% efficiency is acceptable is a question for the user. This is intended for QRP use, so 5W SSB telephony input is not like to damage it, and you could think that an inefficient antenna system doubles the benefit of QRP, QRP^2 if you like.
Can it be improved?
If an efficiency target for the transformer is set at say 90% or better, it takes an 6t primary to achieve that on 3.6MHz.
Gcore is the real part of Y, 0.00173S.
If Yin of the loaded transformer is 0.02S, we can calculate the core efficiency as 1-Gcore/Gin=1-0.00173/0.02=91.3%, core loss is 0.39dB.
Of course twice the turns are needed on the other winding, and the compensation capacitor will need review.
Where do these designs come from?
It seems 2t or 3t primary windings on #43 cores are very common which might suggest ‘designers’ have simply changed the core dimensions.
The geometry of the core varies from size to size, so just as inductance and impedance are very dependent on magnetic properties of the material (complex permeability), the also depend on cross section area and path length. The calculator shots above show a metric, ΣA/l, which captures the geometry, the larger it is, the fewer turns for same inductance / impedance. ΣA/l for and FT240-43 core is 0.00106 whereas for the FT82-43 core discussed in this article, it is less than half of that at 0.000468 and so drives a need for more turns.
Note that the next size core up or down in a series could have greater or lesser ΣA/l (m), there is no general rule that going to a smaller or a larger core requires more or less turns. Some datasheets show ΣA/l or the inverse Σl/A .
Does it matter?
Well, in ham radio, everything works. But systems that work better increases the prospects of contacts. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8060762286186218, "perplexity": 2544.1419943964174}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583684033.26/warc/CC-MAIN-20190119221320-20190120003320-00061.warc.gz"} |
http://hal.in2p3.fr/in2p3-00837672 | # IIB supergravity on manifolds with SU(4) structure and generalized geometry
1 Théorie
IP2I Lyon - Institut de Physique des 2 Infinis de Lyon
Abstract : We consider N=(2,0) backgrounds of IIB supergravity on eight-manifolds M_8 with strict SU(4) structure. We give the explicit solution to the Killing spinor equations as a set of algebraic relations between irreducible su(4) modules of the fluxes and the torsion classes of M_8. One consequence of supersymmetry is that M_8 must be complex. We show that the conjecture of arxiv:1010.5789 concerning the correspondence between background supersymmetry equations in terms of generalized pure spinors and generalized calibrations for admissible static, magnetic D-branes, does not capture the full set of supersymmetry equations. We identify the missing constraints and express them in the form of a single pure-spinor equation which is well defined for generic SU(4)\times SU(4) backgrounds. This additional equation is given in terms of a certain analytic continuation of the generalized calibration form for codimension-2 static, magnetic D-branes.
Document type :
Journal articles
http://hal.in2p3.fr/in2p3-00837672
Contributor : Sylvie Flores <>
Submitted on : Monday, June 24, 2013 - 9:47:47 AM
Last modification on : Thursday, February 6, 2020 - 4:28:10 PM
### Citation
D. Prins, D. Tsimpis. IIB supergravity on manifolds with SU(4) structure and generalized geometry. Journal of High Energy Physics, Springer, 2013, 07(2013), pp.180. ⟨10.1007/JHEP07(2013)180⟩. ⟨in2p3-00837672⟩
Record views | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8919479846954346, "perplexity": 2117.2520174866577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400197946.27/warc/CC-MAIN-20200920094130-20200920124130-00523.warc.gz"} |
http://cms.math.ca/cmb/kw/comparability%20graph | A Note on Conjectures of F. Galvin and R. Rado In 1968, Galvin conjectured that an uncountable poset $P$ is the union of countably many chains if and only if this is true for every subposet $Q \subseteq P$ with size $\aleph_1$. In 1981, Rado formulated a similar conjecture that an uncountable interval graph $G$ is countably chromatic if and only if this is true for every induced subgraph $H \subseteq G$ with size $\aleph_1$. TodorÄeviÄ has shown that Rado's Conjecture is consistent relative to the existence of a supercompact cardinal, while the consistency of Galvin's Conjecture remains open. In this paper, we survey and collect a variety of results related to these two conjectures. We also show that the extension of Rado's conjecture to the class of all chordal graphs is relatively consistent with the existence of a supercompact cardinal. Keywords:Galvin conjecture, Rado conjecture, perfect graph, comparability graph, chordal graph, clique-cover number, chromatic numberCategories:03E05, 03E35, 03E55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9358601570129395, "perplexity": 300.77405323524994}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375094491.62/warc/CC-MAIN-20150627031814-00132-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/mass-conservation-in-radially-symmetric-parabolic-pde-problems.390544/ | # Mass conservation in radially symmetric parabolic PDE problems
1. Mar 29, 2010
### ndalchau
Dear all,
I'm trying to solve the 2d heat equation in a radially symmetric domain, numerically using the Crank-Nicolson method. i.e.
$$\dfrac{\partial u}{\partial t} = D\left( \dfrac{\partial^2u}{\partial r^2}+\dfrac{1}{r}\dfrac{\partial u}{\partial r}\right)$$
Applying the Crank-Nicolson method basically results in a recurrence relation:
$$(-q+\frac{z}{r})u_{i-1,j+1} + (1+2q)u_{i,j+1} - (q+\frac{z}{r})u_{i+1,j+1} = (q-\frac{z}{r})u_{i-1,j} + (1-2q)u_{i,j} + (q+\frac{z}{r})u_{i+1,j}$$
where $$q=\dfrac{\delta t}{2(\delta r)^2}$$, $$z=\dfrac{\delta t}{4\delta r}$$ and $$u_{i,j}$$ is the solution at $$r=i\delta r, t=j\delta t$$.
You can write this into a matrix equation of the form $$Au\{j+1\}=Bu\{j\}$$, which basically enables you to solve the problem. This all works very nicely for the 1d heat equation (the same differential equation but without the first spatial derivative). However, I now have the problem that the solution doesn't conserve mass. That is, given some initial condition across $$r\in (0,R)$$, the sum of the solution points decreases over time.
Clearly, if the column sums of A and B are the row vector of 1s, then I am guaranteed mass conservation. However, this is not the case for my Crank-Nicolson implementation here. This results from the $$\frac{z}{r}$$ changing values and signs between neighbouring rows, which means they don't cancel.
Anyone got any comments? Is there a way of solving these parabolic PDEs in radially symmetric domains that preserves the mass conservation law? Or is my implementation incorrect maybe?
Your assistance would be greatly appreciated,
Neil | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9402201175689697, "perplexity": 300.9556053175793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267865995.86/warc/CC-MAIN-20180624005242-20180624025242-00048.warc.gz"} |
https://www.nag.com/numeric/nl/nagdoc_latest/cpphtml/e04/e04rhf.html | # NAG CPP Interfacenagcpp::opt::handle_set_simplebounds (e04rh)
Settings help
CPP Name Style:
## 1Purpose
handle_set_simplebounds is a part of the NAG optimization modelling suite and sets bounds on the variables of the problem.
## 2Specification
```#include "e04/nagcpp_e04rh.hpp"
#include "e04/nagcpp_class_CommE04RA.hpp"
```
```template <typename COMM, typename BL, typename BU>
void function handle_set_simplebounds(COMM &comm, const BL &bl, const BU &bu, OptionalE04RH opt)```
```template <typename COMM, typename BL, typename BU>
void function handle_set_simplebounds(COMM &comm, const BL &bl, const BU &bu)```
## 3Description
After the handle has been initialized (e.g., handle_init has been called), handle_set_simplebounds may be used to define the variable bounds ${l}_{x}\le x\le {u}_{x}$ of the problem. If the bounds have already been defined, they will be overwritten. Individual bounds may also be set by e04tdf (no CPP interface).
This will typically be used for problems, such as:
Linear Programming (LP)
$minimize x∈ℝn cTx (a) subject to lB≤Bx≤uB, (b) lx≤x≤ux , (c)$ (1)
$minimize x∈ℝn 12 xTHx + cTx (a) subject to lB≤Bx≤uB, (b) lx≤x≤ux, (c)$ (2)
Nonlinear Programming (NLP)
$minimize x∈ℝn f(x) (a) subject to lg≤g(x)≤ug, (b) lB≤Bx≤uB, (c) lx≤x≤ux, (d)$ (3)
or linear Semidefinite Programming (SDP)
$minimize x∈ℝn cTx (a) subject to ∑ i=1 n xi Aik - A0k ⪰ 0 , k=1,…,mA , (b) lB≤Bx≤uB, (c) lx≤x≤ux, (d)$ (4)
where ${l}_{x}$ and ${u}_{x}$ are $n$-dimensional vectors. Note that upper and lower bounds are specified for all the variables. This form allows full generality in specifying various types of constraint. In particular, the $j$th variable may be fixed by setting ${l}_{j}={u}_{j}$. If certain bounds are not present, the associated elements of ${l}_{x}$ or ${u}_{x}$ may be set to special values that are treated as $-\infty$ or $+\infty$. See the description of the optional parameter Infinite Bound Size which is common among all solvers in the suite. Its value is denoted as $\mathit{bigbnd}$ further in this text. Note that the bounds are interpreted based on its value at the time of calling this function and any later alterations to Infinite Bound Size will not affect these constraints.
See Section 3.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.
## 4References
Candes E and Recht B (2009) Exact matrix completion via convex optimization Foundations of Computation Mathematics (Volume 9) 717–772
## 5Arguments
1: $\mathbf{comm}$CommE04RA Input/Output
Communication structure. An object of either the derived class CommE04RA or its base class NoneCopyableComm can be supplied. It is recommended that the derived class is used. If the base class is supplied it must first be initialized via a call to opt::handle_init (e04ra).
2: $\mathbf{bl}\left({\mathbf{nvar}}\right)$double array Input
On entry: ${l}_{x}$, bl and ${u}_{x}$, bu define lower and upper bounds on the variables, respectively. To fix the $j$th variable, set ${\mathbf{bl}}\left(j-1\right)={\mathbf{bu}}\left(j-1\right)=\beta$, where $|\beta |<\mathit{bigbnd}$. To specify a nonexistent lower bound (i.e., ${l}_{j}=-\infty$), set ${\mathbf{bl}}\left(j-1\right)\le -\mathit{bigbnd}$; to specify a nonexistent upper bound (i.e., ${u}_{j}=\infty$), set ${\mathbf{bu}}\left(j-1\right)\ge \mathit{bigbnd}$.
Constraints:
• ${\mathbf{bl}}\left(\mathit{j}-1\right)\le {\mathbf{bu}}\left(\mathit{j}-1\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{nvar}}$;
• ${\mathbf{bl}}\left(\mathit{j}-1\right)<\mathit{bigbnd}$, for $\mathit{j}=1,2,\dots ,{\mathbf{nvar}}$;
• ${\mathbf{bu}}\left(\mathit{j}-1\right)>-\mathit{bigbnd}$, for $\mathit{j}=1,2,\dots ,{\mathbf{nvar}}$.
3: $\mathbf{bu}\left({\mathbf{nvar}}\right)$double array Input
On entry: ${l}_{x}$, bl and ${u}_{x}$, bu define lower and upper bounds on the variables, respectively. To fix the $j$th variable, set ${\mathbf{bl}}\left(j-1\right)={\mathbf{bu}}\left(j-1\right)=\beta$, where $|\beta |<\mathit{bigbnd}$. To specify a nonexistent lower bound (i.e., ${l}_{j}=-\infty$), set ${\mathbf{bl}}\left(j-1\right)\le -\mathit{bigbnd}$; to specify a nonexistent upper bound (i.e., ${u}_{j}=\infty$), set ${\mathbf{bu}}\left(j-1\right)\ge \mathit{bigbnd}$.
Constraints:
• ${\mathbf{bl}}\left(\mathit{j}-1\right)\le {\mathbf{bu}}\left(\mathit{j}-1\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{nvar}}$;
• ${\mathbf{bl}}\left(\mathit{j}-1\right)<\mathit{bigbnd}$, for $\mathit{j}=1,2,\dots ,{\mathbf{nvar}}$;
• ${\mathbf{bu}}\left(\mathit{j}-1\right)>-\mathit{bigbnd}$, for $\mathit{j}=1,2,\dots ,{\mathbf{nvar}}$.
4: $\mathbf{opt}$OptionalE04RH Input/Output
Optional parameter container, derived from Optional.
1: $\mathbf{nvar}$
$n$, the current number of decision variables $x$ in the model.
## 6Exceptions and Warnings
Errors or warnings detected by the function:
All errors and warnings have an associated numeric error code field, errorid, stored either as a member of the thrown exception object (see errorid), or as a member of opt.ifail, depending on how errors and warnings are being handled (see Error Handling for more details).
Raises: ErrorException
$\mathbf{errorid}=1$
comm::handle has not been initialized.
$\mathbf{errorid}=1$
comm::handle does not belong to the NAG optimization modelling suite,
has not been initialized properly or is corrupted.
$\mathbf{errorid}=1$
comm::handle has not been initialized properly or is corrupted.
$\mathbf{errorid}=2$
The problem cannot be modified right now, the solver is running.
$\mathbf{errorid}=4$
On entry, ${\mathbf{nvar}}=⟨\mathit{value}⟩$,
expected $\mathrm{value}=⟨\mathit{value}⟩$.
Constraint: ${\mathbf{nvar}}$ must match the current number of variables
of the model in the comm::handle.
$\mathbf{errorid}=10$
On entry, $j=⟨\mathit{value}⟩$, ${\mathbf{bl}}\left[j-1\right]=⟨\mathit{value}⟩$ and
${\mathbf{bu}}\left[j-1\right]=⟨\mathit{value}⟩$.
Constraint: ${\mathbf{bl}}\left[j-1\right]\le {\mathbf{bu}}\left[j-1\right]$.
$\mathbf{errorid}=10$
On entry, $j=⟨\mathit{value}⟩$, ${\mathbf{bl}}\left[j-1\right]=⟨\mathit{value}⟩$,
$\mathit{bigbnd}=⟨\mathit{value}⟩$.
Constraint: ${\mathbf{bl}}\left[j-1\right]<\mathit{bigbnd}$.
$\mathbf{errorid}=10$
On entry, $j=⟨\mathit{value}⟩$, ${\mathbf{bu}}\left[j-1\right]=⟨\mathit{value}⟩$,
$\mathit{bigbnd}=⟨\mathit{value}⟩$.
Constraint: ${\mathbf{bu}}\left[j-1\right]>-\mathit{bigbnd}$.
$\mathbf{errorid}=10601$
On entry, argument $⟨\mathit{\text{value}}⟩$ must be a vector of size $⟨\mathit{\text{value}}⟩$ array.
Supplied argument has $⟨\mathit{\text{value}}⟩$ dimensions.
$\mathbf{errorid}=10601$
On entry, argument $⟨\mathit{\text{value}}⟩$ must be a vector of size $⟨\mathit{\text{value}}⟩$ array.
Supplied argument was a vector of size $⟨\mathit{\text{value}}⟩$.
$\mathbf{errorid}=10601$
On entry, argument $⟨\mathit{\text{value}}⟩$ must be a vector of size $⟨\mathit{\text{value}}⟩$ array.
The size for the supplied array could not be ascertained.
$\mathbf{errorid}=10602$
On entry, the raw data component of $⟨\mathit{\text{value}}⟩$ is null.
$\mathbf{errorid}=10603$
On entry, unable to ascertain a value for $⟨\mathit{\text{value}}⟩$.
$\mathbf{errorid}=10605$
On entry, the communication class $⟨\mathit{\text{value}}⟩$ has not been initialized correctly.
$\mathbf{errorid}=-99$
An unexpected error has been triggered by this routine.
$\mathbf{errorid}=-399$
Your licence key may have expired or may not have been installed correctly.
$\mathbf{errorid}=-999$
Dynamic memory allocation failed.
Not applicable.
## 8Parallelism and Performance
Please see the description for the underlying computational routine in this section of the FL Interface documentation.
## 10Example
Examples of the use of this method may be found in the examples for: handle_solve_dfls_rcomm, handle_solve_bounds_foas, handle_solve_lp_ipm, handle_set_group and handle_solve_ipopt. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 95, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9747532606124878, "perplexity": 1883.448545960558}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104215805.66/warc/CC-MAIN-20220703073750-20220703103750-00350.warc.gz"} |
https://www.physicsforums.com/threads/finite-sum-formula-for-tangent-trigonometry.679625/ | # Finite sum formula for tangent (trigonometry)
1. Mar 20, 2013
### Vahsek
Hi everyone, I've been looking for the finite sum formulae of trig functions. I've found the easiest ones (sine and cosine). But the one for the tangent seems to be very hard. No mathematical tricks work. Plus I've looked it up on the internet. Nothing. I will greatly appreciate your help. Thanks in advance.
tan x + tan (2x) + tan (3x) + ... + tan (nx) = ???
Last edited: Mar 20, 2013
2. Mar 20, 2013
### mathman
My guess: You are out of luck. I checked Gradshteyn & Ryzhik. They have the sums for sin and cos, as well as sinh and cosh, but nothing for tan.
3. Mar 20, 2013
### Vahsek
ok, thx for your consideration though. I'll wait a bit more; maybe someone's got a way to do it.
4. Mar 20, 2013
### h6ss
I found in a textbook that $tan(x)$ can be written as an indefinite sum:
$\sum_x \tan ax = i x-\frac1a \psi _{e^{2 i a}}\left(x-\frac{\pi }{2 a}\right) + C \,,\,\,a\ne \frac{n\pi}2$ where $\psi_q(x)$ is the q-digamma function.
Computing "sum k from 1 to n of tan(k*x)" in WolframAlpha results into something much more complicated, but an answer is given.
5. Mar 20, 2013
### Vahsek
Wow. I had no idea it was that complicated. I'm in high school right now. These functions in real/complex analysis is way beyond me. Anyway, thank you everyone though. At least now I know which direction I must be heading to learn more about it.
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https://www.physicsforums.com/threads/electro-weak-vertices.591581/ | # Electro weak vertices
1. Mar 29, 2012
### Jodahr
I have a question about Feynman Diagrams:
let's say we have a process: up antidown -> W+ -> up antidown...
the first vertex is like V_CKM G PL ( mixing, gamma, projector)
the second is the same..only with the complex conjugate CKM matrix...
but why?...
If I compute the M* I have to bar the vertices..and there I got the same vertex..with the same flow..but there I would change PL to PR and interchange PL and Gamma..why is that the case?
2. Mar 30, 2012
### francesco85
Hello, in my opinion the answer is the following: the Feynman diagram you are considering is composed of two vertices: in the first an up is destroyed, an antidown is destroyed and a W+ is created; in the second vertex an up is created, an antidown is created and a W+ is destroyed; so, roughly speaking, the first is associated with a term in the lagrangian like (u dbar W-), while the second with (ubar d W+), that is its hermitian conjugate (of course I have forgot all the contraction matrices...); this is the origin of the conjugation of the CKM matrix paramters (and, of course, one should be careful with the imaginary units!)
Best,
Francesco
Last edited: Mar 30, 2012
3. Mar 30, 2012
I think another way of looking at it is like this.
That Feynman diagram also describes the processes
u → W+ + d
W+ + d → u
as all I have done here is replace the incoming anti-d with an outgoing d, and the outgoing anti-d with an incoming d.
The CKM matrix, as defined, is the factor for 'converting' down-type quarks to up-type, eg
|u> = Vud |d>
Provided only the three known generations of quarks exist, the CKM matrix must be unitary, and hence
V-1 = V
so
|d> = V*ud |u>
Last edited: Mar 30, 2012
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https://en.academic.ru/dic.nsf/enwiki/302219 | # Current mirror
Current mirror
A current mirror is a circuit designed to copy a current through one active device by controlling the current in another active device of a circuit, keeping the output current constant regardless of loading. The current being 'copied' can be, and sometimes is, a varying signal current. Conceptually, an ideal current mirror is simply an ideal inverting current amplifier that reverses the current direction as well or it is a current-controlled current source (CCCS). The current mirror is used to provide bias currents and active loads to circuits.
## Mirror characteristics
There are three main specifications that characterize a current mirror. The first is the transfer ratio (in the case of a current amplifier) or the output current magnitude (in the case of a constant current source CCS). The second is its AC output resistance, which determines how much the output current varies with the voltage applied to the mirror. The third specification is the minimum voltage drop across the output part of the mirror necessary to make it work properly. This minimum voltage is dictated by the need to keep the output transistor of the mirror in active mode. The range of voltages where the mirror works is called the compliance range and the voltage marking the boundary between good and bad behavior is called the compliance voltage. There are also a number of secondary performance issues with mirrors, for example, temperature stability.
## Practical approximations
For small-signal analysis the current mirror can be approximated by its equivalent Norton impedance .
In large-signal hand analysis, a current mirror is usually and simply approximated by an ideal current source. However, an ideal current source is unrealistic in several respects:
• it has infinite AC impedance, while a practical mirror has finite impedance
• it provides the same current regardless of voltage, that is, there are no compliance range requirements
• it has no frequency limitations, while a real mirror has limitations due to the parasitic capacitances of the transistors
• the ideal source has no sensitivity to real-world effects like noise, power-supply voltage variations and component tolerances.
## Circuit realizations of current mirrors
### Basic idea
A current mirror consists of two cascaded inverse converters with mirrored transfer characteristic.
A bipolar transistor can be used as the simplest current-to-current converter but its transfer ratio would highly depend on temperature variations, β tolerances, etc. To eliminate these undesired disturbances, a current mirror is composed of two cascaded current-to-voltage and voltage-to-current converters placed at the same conditions and having reverse characteristics. They have not to be obligatory linear; the only requirement is their characteristics to be mirrorlike (for example, in the BJT current mirror below, they are logarithmic and exponential). Usually, two identical converters are used but the characteristic of the first one is reversed by applying a negative feedback. Thus a current mirror consists of two cascaded equal converters (the first - reversed and the second - direct).
Figure 1: A current mirror implemented with npn bipolar transistors using a resistor to set the reference current IREF; VCC = supply voltage
### Basic BJT current mirror
If a voltage is applied to the BJT base-emitter junction as an input quantity and the collector current is taken as an output quantity, the transistor will act as an exponential voltage-to-current converter. By applying a negative feedback (simply joining the base and collector) the transistor can be "reversed" and it will begin acting as the opposite logarithmic current-to-voltage converter; now it will adjust the "output" base-emitter voltage so as to pass the applied "input" collector current.
The simplest bipolar current mirror (shown in Figure 1) implements this idea. It consists of two cascaded transistor stages acting accordingly as a reversed and direct voltage-to-current converters. Transistor Q1 is connected to ground. Its collector-base voltage is zero as shown. Consequently, the voltage drop across Q1 is VBE, that is, this voltage is set by the diode law and Q1 is said to be diode connected. (See also Ebers-Moll model.) It is important to have Q1 in the circuit instead of a simple diode, because Q1 sets VBE for transistor Q2. If Q1 and Q2 are matched, that is, have substantially the same device properties, and if the mirror output voltage is chosen so the collector-base voltage of Q2 is also zero, then the VBE-value set by Q1 results in an emitter current in the matched Q2 that is the same as the emitter current in Q1. Because Q1 and Q2 are matched, their β0-values also agree, making the mirror output current the same as the collector current of Q1. The current delivered by the mirror for arbitrary collector-base reverse bias VCB of the output transistor is given by (see bipolar transistor):
$I_\mathrm{C} = I_\mathrm{S} \left( e^{\frac{V_\mathrm{BE}}{V_\mathrm{T}}}-1 \right) \left(1 + \begin{matrix} \frac{V_\mathrm{CB}}{V_\mathrm{A}} \end{matrix} \right)$,
where IS = reverse saturation current or scale current, VT = thermal voltage and VA = Early voltage. This current is related to the reference current IREF when the output transistor VCB = 0 V by:
$I_{REF} = I_C \left( 1+ \frac {2} {\beta_0} \right) \ ,$
as found using Kirchhoff's current law at the collector node of Q1:
IREF = IC + IB1 + IB2.
The reference current supplies the collector current to Q1 and the base currents to both transistors — when both transistors have zero base-collector bias, the two base currents are equal, IB1=IB2=IB.
$I_{REF} = I_C + I_B + I_B = I_C + 2 I_B = I_C \left(1+ \frac {2} {\beta_0} \right) \ ,$
Parameter β0 is the transistor β-value for VCB = 0 V.
#### Output resistance
If VCB is greater than zero in output transistor Q2, the collector current in Q2 will be somewhat larger than for Q1 due to the Early effect. In other words, the mirror has a finite output (or Norton) resistance given by the rO of the output transistor, namely (see Early effect):
$R_N =r_O = \begin{matrix} \frac {V_A + V_{CB}} {I_C} \end{matrix}$,
where VA = Early voltage and VCB = collector-to-base bias.
#### Compliance voltage
To keep the output transistor active, VCB ≥ 0 V. That means the lowest output voltage that results in correct mirror behavior, the compliance voltage, is VOUT = VCV = VBE under bias conditions with the output transistor at the output current level IC and with VCB = 0 V or, inverting the I-V relation above:
$\ V_{CV}= {V_T}$ $\ \mathrm {ln}$ $\left(\begin{matrix}\frac {I_C}{I_S}\end{matrix}+1\right) \ ,$
where VT = thermal voltage and IS = reverse saturation current or scale current.
#### Extensions and complications
When Q2 has VCB > 0 V, the transistors no longer are matched. In particular, their β-values differ due to the Early effect, with
${\beta}_1 = {\beta}_{0} \ \operatorname{and} \ {\beta}_2 = {\beta}_{0}\ (1 + \frac{V_{CB}}{V_A})$
where VA is the Early voltage and β0 = transistor β for VCB = 0 V. Besides the difference due to the Early effect, the transistor β-values will differ because the β0-values depend on current, and the two transistors now carry different currents (see Gummel-Poon model).
Further, Q2 may get substantially hotter than Q1 due to the associated higher power dissipation. To maintain matching, the temperature of the transistors must be nearly the same. In integrated circuits and transistor arrays where both transistors are on the same die, this is easy to achieve. But if the two transistors are widely separated, the precision of the current mirror is compromised.
Additional matched transistors can be connected to the same base and will supply the same collector current. In other words, the right half of the circuit can be duplicated several times with various resistor values replacing R2 on each. Note, however, that each additional right-half transistor "steals" a bit of collector current from Q1 due to the non-zero base currents of the right-half transistors. This will result in a small reduction in the programmed current.
An example of a mirror with emitter degeneration to increase mirror resistance is found in two-port networks.
For the simple mirror shown in the diagram, typical values of β will yield a current match of 1% or better.
Figure 2: An n-channel MOSFET current mirror with a resistor to set the reference current IREF; VDD is the supply voltage
### Basic MOSFET current mirror
The basic current mirror can also be implemented using MOSFET transistors, as shown in Figure 2. Transistor M1 is operating in the saturation or active mode, and so is M2. In this setup, the output current IOUT is directly related to IREF, as discussed next.
The drain current of a MOSFET ID is a function of both the gate-source voltage and the drain-to-gate voltage of the MOSFET given by ID = f (VGS, VDG), a relationship derived from the functionality of the MOSFET device. In the case of transistor M1 of the mirror, ID = IREF. Reference current IREF is a known current, and can be provided by a resistor as shown, or by a "threshold-referenced" or "self-biased" current source to ensure that it is constant, independent of voltage supply variations.[1]
Using VDG=0 for transistor M1, the drain current in M1 is ID = f (VGS,VDG=0), so we find: f (VGS, 0) = IREF, implicitly determining the value of VGS. Thus IREF sets the value of VGS. The circuit in the diagram forces the same VGS to apply to transistor M2. If M2 is also biased with zero VDG and provided transistors M1 and M2 have good matching of their properties, such as channel length, width, threshold voltage etc., the relationship IOUT = f (VGS,VDG=0 ) applies, thus setting IOUT = IREF; that is, the output current is the same as the reference current when VDG=0 for the output transistor, and both transistors are matched.
The drain-to-source voltage can be expressed as VDS=VDG +VGS. With this substitution, the Shichman-Hodges model provides an approximate form for function f (VGS,VDG):[2]
\begin{alignat}{2} I_{d} & = f\ (V_{GS},V_{DG}) = \begin{matrix} \frac{1}{2}K_{p}\left(\frac{W}{L}\right)\end{matrix}(V_{GS} - V_{th})^2 (1 + \lambda V_{DS}) \\ & =\begin{matrix} \frac{1}{2}K_{p}\left(\frac{W}{L}\right)\end{matrix}(V_{GS} - V_{th})^2 \left( 1 + \lambda (V_{DG}+V_{GS}) \right) \\ \end{alignat}
where, Kp is a technology related constant associated with the transistor, W/L is the width to length ratio of the transistor, VGS is the gate-source voltage, Vth is the threshold voltage, λ is the channel length modulation constant, and VDS is the drain source voltage.
#### Output resistance
Because of channel-length modulation, the mirror has a finite output (or Norton) resistance given by the ro of the output transistor, namely (see channel length modulation):
$R_N =r_o = \begin{matrix} \frac {1/\lambda + V_{DS}} {I_D} \end{matrix} = \begin{matrix} \frac {V_{A} + V_{DS}} {I_D} \end{matrix}$,
where λ = channel-length modulation parameter and VDS = drain-to-source bias.
#### Compliance voltage
To keep the output transistor resistance high, VDG ≥ 0 V.[nb 1] (see Baker).[3] That means the lowest output voltage that results in correct mirror behavior, the compliance voltage, is VOUT = VCV = VGS for the output transistor at the output current level with VDG = 0 V, or using the inverse of the f-function, f −1:
$V_{CV}= V_{GS} (\mathrm{for}\ I_D\ \mathrm{at} \ V_{DG}=0V) = f ^{-1} (I_D) \ \mathrm{with}\ V_{DG}=0$.
For Shichman-Hodges model, f -1 is approximately a square-root function.
#### Extensions and reservations
A useful feature of this mirror is the linear dependence of f upon device width W, a proportionality approximately satisfied even for models more accurate than the Shichman-Hodges model. Thus, by adjusting the ratio of widths of the two transistors, multiples of the reference current can be generated.
It must be recognized that the Shichman-Hodges model[4] is accurate only for rather dated technology, although it often is used simply for convenience even today. Any quantitative design based upon new technology uses computer models for the devices that account for the changed current-voltage characteristics. Among the differences that must be accounted for in an accurate design is the failure of the square law in Vgs for voltage dependence and the very poor modeling of Vds drain voltage dependence provided by λVds. Another failure of the equations that proves very significant is the inaccurate dependence upon the channel length L. A significant source of L-dependence stems from λ, as noted by Gray and Meyer, who also note that λ usually must be taken from experimental data.[5]
### Feedback assisted current mirror
Figure 3: Gain-boosted current mirror with op amp feedback to increase output resistance
Figure 4: MOSFET version of wide-swing current mirror; M1 and M2 are in active mode, while M3 and M4 are in Ohmic mode and act like resistors
Figure 3 shows a mirror using negative feedback to increase output resistance. Because of the op amp, these circuits are sometimes called gain-boosted current mirrors. Because they have relatively low compliance voltages, they also are called wide-swing current mirrors. A variety of circuits based upon this idea are in use,[6][7][8] particularly for MOSFET mirrors because MOSFETs have rather low intrinsic output resistance values. A MOSFET version of Figure 3 is shown in Figure 4 where MOSFETs M3 and M4 operate in Ohmic mode to play the same role as emitter resistors RE in Figure 3, and MOSFETs M1 and M2 operate in active mode in the same roles as mirror transistors Q1 and Q2 in Figure 3. An explanation follows of how the circuit in Figure 3 works.
The operational amplifier is fed the difference in voltages V1 - V2 at the top of the two emitter-leg resistors of value RE. This difference is amplified by the op amp and fed to the base of output transistor Q2. If the collector base reverse bias on Q2 is increased by increasing the applied voltage VA, the current in Q2 increases, increasing V2 and decreasing the difference V1 - V2 entering the op amp. Consequently, the base voltage of Q2 is decreased, and VBE of Q2 decreases, counteracting the increase in output current.
If the op amp gain Av is large, only a very small difference V1 - V2 is sufficient to generate the needed base voltage VB for Q2, namely
$V_1-V_2 = \frac {V_B}{A_v} \ .$
Consequently, the currents in the two leg resistors are held nearly the same, and the output current of the mirror is very nearly the same as the collector current IC1 in Q1, which in turn is set by the reference current as
$I_{ref} = I_{C1} (1 + 1/ { \beta}_1) \ ,$
where β1 for transistor Q1 and β2 for Q2 differ due to the Early effect if the reverse bias across the collector-base of Q2 is non-zero.
Figure 5: Small-signal circuit to determine output resistance of mirror; transistor Q2 is replaced with its hybrid-pi model; a test current IX at the output generates a voltage VX, and the output resistance is Rout = VX / IX.
#### Output resistance
An idealized treatment of output resistance is given in the footnote.[nb 2] A small-signal analysis for an op amp with finite gain Av but otherwise ideal is based upon Figure 5 (β, rO and rπ refer to Q2). To arrive at Figure 5, notice that the positive input of the op amp in Figure 3 is at AC ground, so the voltage input to the op amp is simply the AC emitter voltage Ve applied to its negative input, resulting in a voltage output of −Av Ve. Using Ohm's law across the input resistance rπ determines the small-signal base current Ib as:
$I_b = \frac {V_e} {r_{\pi} / ( A_v+1) } \ .$
Combining this result with Ohm's law for RE, Ve can be eliminated, to find:[nb 3]
$I_b = I_X \frac {R_E} {R_E +\frac {r_{\pi}} {A_v+1} } \ .$
Kirchhoff's voltage law from the test source IX to the ground of RE provides:
$V_X = (I_X + \beta I_b) r_O + (I_X - I_b )R_E \ .$
Substituting for Ib and collecting terms the output resistance Rout is found to be:
$R_{out} = \frac {V_X} {I_X} = r_O \left( 1+ \beta \frac{R_E} {R_E+r_{\pi}/(A_v+1)} \right) +R_E\|\frac {r_{\pi}} {A_v+1} \ .$
For a large gain Av >> rπ / RE the maximum output resistance obtained with this circuit is
$R_{out} = ( \beta +1) r_O \ ,$
a substantial improvement over the basic mirror where Rout = rO.
The small-signal analysis of the MOSFET circuit of Figure 4 is obtained from the bipolar analysis by setting β = gm rπ in the formula for Rout and then letting rπ → ∞. The result is
$R_{out} = r_O \left( 1+ g_m R_E(A_v+1) \right) +R_E \ .$
This time, RE is the resistance of the source-leg MOSFETs M3, M4. Unlike Figure 3, however, as Av is increased (holding RE fixed in value), Rout continues to increase, and does not approach a limiting value at large Av.
#### Compliance voltage
For Figure 3, a large op amp gain achieves the maximum Rout with only a small RE. A low value for RE means V2 also is small, allowing a low compliance voltage for this mirror, only a voltage V2 larger than the compliance voltage of the simple bipolar mirror. For this reason this type of mirror also is called a wide-swing current mirror, because it allows the output voltage to swing low compared to other types of mirror that achieve a large Rout only at the expense of large compliance voltages.
With the MOSFET circuit of Figure 4, like the circuit in Figure 3, the larger the op amp gain Av, the smaller RE can be made at a given Rout, and the lower the compliance voltage of the mirror.
### Other current mirrors
There are many sophisticated current mirrors that have higher output resistances than the basic mirror (more closely approach an ideal mirror with current output independent of output voltage) and produce currents less sensitive to temperature and device parameter variations and to circuit voltage fluctuations. These multi-transistor mirror circuits are used both with bipolar and MOS transistors. These circuits include:
## Notes
1. ^ Keeping the output resistance high means more than keeping the MOSFET in active mode, because the output resistance of real MOSFETs only begins to increase on entry into the active region, then rising to become close to maximum value only when VDG ≥ 0 V.
2. ^ An idealized version of the argument in the text, valid for infinite op amp gain, is as follows. If the op amp is replaced by a nullor, voltage V2 = V1, so the currents in the leg resistors are held at the same value. That means the emitter currents of the transistors are the same. If the VCB of Q2 increases, so does the output transistor β because of the Early effect: β = β0 ( 1 + VCB / VA ). Consequently the base current to Q2 given by IB = IE / (β + 1) decreases and the output current Iout = IE / (1 + 1 / β) increases slightly because β increases slightly. Doing the math,
$\frac {1} {R_{out}} = \frac {\partial I_{out} } { \partial V_{CB} } = I_E \frac {\partial } { \partial V_{CB} } \left( \frac { \beta } { \beta +1} \right) = I_E \frac {1} {(\beta + 1)^2 } \frac { \partial \beta } {\partial V_{CB}}$$= \frac {\beta I_E} { \beta +1 } \frac {1}{\beta} \frac {\beta_0} { V_A} \frac {1} {(\beta +1) } =I_{out} \frac {1} {1+V_{CB} / V_A} \frac {1} { V_A} \frac {1} {(\beta +1) } = \frac {1} { ( \beta +1 ) r_0} \ ,$
where the transistor output resistance is given by rO = ( VA + VCB ) / Iout. That is, the ideal mirror resistance for the circuit using an ideal op amp nullor is Rout = ( β + 1 ) rO, in agreement with the value given later in the text when the gain → ∞.
3. ^ Notice that as Av → ∞, Ve → 0 and IbIX.
## References
1. ^ Paul R. Gray, Paul J. Hurst, Stephen H. Lewis, Robert G. Meyer (2001). Analysis and Design of Analog Integrated Circuits (Fourth Edition ed.). New York: Wiley. p. 308–309. ISBN 0471321680.
2. ^ Gray et al.. Eq. 1.165, p. 44. ISBN 0471321680.
3. ^ R. Jacob Baker (2010). CMOS Circuit Design, Layout and Simulation (Third ed.). New York: Wiley-IEEE. pp. 297, §9.2.1 and Figure 20.28, p. 636. ISBN 978-0-470-88132-3.
4. ^ NanoDotTek Report NDT14-08-2007, 12 August 2007
5. ^ Gray et al.. p. 44. ISBN 0471321680.
6. ^ R. Jacob Baker. § 20.2.4 pp. 645–646. ISBN 978-0-470-88132-3.
7. ^ Ivanov VI and Filanovksy IM (2004). Operational amplifier speed and accuracy improvement: analog circuit design with structural methodology (The Kluwer international series in engineering and computer science, v. 763 ed.). Boston, Mass.: Kluwer Academic. p. §6.1, p. 105–108. ISBN 1-4020-7772-6.
8. ^ W. M. C. Sansen (2006). Analog design essentials. New York ; Berlin: Springer. p. §0310, p. 93. ISBN 0-387-25746-2.
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https://groupprops.subwiki.org/wiki/History_of_Sylow%27s_theorem | History of Sylow's theorem
Sylow's theorem, originally proved by Ludwig Sylow in 1872, marked an important milestone in the udnerstanding of the general structure of finite groups. This article explores the history behind this theorem and the progress made since then.
Prior results
Lagrange's theorem
Further information: Lagrange's theorem, History of Lagrange's theorem
Observations made by Lagrange in the eighteenth century during his investigation of symmetries between roots of the quintic, led to Lagrange's theorem: the order of a subgroup divides the order of the group. This result was well-known for groups of permutations.
Cauchy's theorem
Cauchy managed to prove a partial converse to Lagrange's theorem. He proved that if $p$ is a prime dividing the order of a finite group $G$, then $G$ has a subgroup of order exactly $p$. The remarkable thing about Cauchy's theorem was that it held for any finite group, and was stated in a language independent of the specific realization of the group as a group of permutations.
To prove this theorem, Cauchy proved that the symmetric group on $n$ letters contains a subgroup of order $p^m$ where $p^m$ is the largest power of $p$ dividing $n!$. This established the existence part of Sylow's theorem for symmetric groups.
The work of Sylow
Ludwig Sylow, in a paper in 1872, extended Cauchy's result. He proved that any finite group has subgroups of order $p^m$ where $p$ is a prime dividing the order of the group and $p^m$ is the largest power of it dividing the order of the group. Sylow also proved that any two such subgroups, that later came to be called $p$-Sylow subgroups, are conjugate subgroups.
The introduction of Sylow's paper
Sylow began his paper thus (translated from French):
We know that if the order of a group of substitutions is divisible by the prime number $n$, the group always contains a substitution of order $n$. This important theorem is contained in another, more general theorem, which is: "If the order of a group is divisible by $n^\alpha$, $n$ being prime, the group contains a subgroup of order $n^\alpha$." The demonstration of this theorem provides other general properties of groups of substitutions.
Sylow's first line refers to Cauchy's theorem (at the time of the writing of this paper, bibliography/referencing was not universal), and his second line states the existence part of Sylow's theorem (in fact, it states more: it states that subgroups of all possible prime powers dividing the order exist, but this follows easily from Sylow's theorem and the structure of groups of prime power order).
His first theorem: Existence and congruence condition (partial)
Theorem I of Sylow's paper reads (translated from French):
If $n^\alpha$ denotes the largest power of the prime number $n$ that divides the order of the group $G$, that group contains another $g$ of order $n^\alpha$; and moreover, if $n^\alpha\nu$ is the order of the big group contained in $G$ comprising those substitutions that permute with $g$, then the order of $G$ is of the form $n^\alpha\nu(np+1)$.
• Sylow's choice of notation is somewhat at odds with current conventions. Sylow uses $n$ for primes, and uses both small and capital letters for sets (thus, calling his subgroup $g$).
• At the time Sylow wrote his paper, terminology like normalizer was not standard; hence, Sylow gave an explicit description for what we'd today simply call $N_G(g)$, the normalizer of $g$ in $G$.
• During Sylow's time, the concept of index of a subgroup was not in wide vogue; hence his statements are all about orders of groups rather than the index of a subgroup in a group.
His second theorem: Congruence condition and conjugacy
Theorem II of Sylow's paper reads (translated from French):
With all the conditions of the preceding theorem, the group $G$ contains precisely $np + 1$ distinct groups of order $n^\alpha$, one does obtain each by transforming any other by the substitutions in $G$, each group being given by $n^\alpha\nu$ distinct transformations.
• The final clause in his sentence encapsulates the fact that starting with one Sylow subgroup, the ways of conjugating that to another Sylow subgroup correspond to one coset of its normalizer. Each such coset has size $n^\alpha \nu$ -- the size of the normalizer. The language of cosets was not familiar to Sylow; hence he needed to use this wording. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 39, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9153138995170593, "perplexity": 282.8775053898664}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347409337.38/warc/CC-MAIN-20200530133926-20200530163926-00384.warc.gz"} |
http://cogsci.stackexchange.com/questions/3600/is-there-an-accurate-online-iq-test-for-measuring-160-iqs/3601 | # Is there an accurate online IQ test for measuring 160+ IQs?
Is there some (preferably free) online and accurate IQ test?
One which does not give me an genius IQ and then tries to sell me a diploma.
I need one which measures upto 160 sd15 (or higher). Such a test would probably document its norming process somehow.
-
– Jeromy Anglim Jun 9 '13 at 7:16
There are at least two problems with measuring high intelligence:
(1) Any IQ test has a maximum difficulty. That means that all subjects above a certain intelligence answer all questions correctly and get the same maximum score. This is called the "ceiling effect".
Now you might say, that we simply need to construct a test that is difficult enough for even the most intelligent person to make some mistakes.
(2) The problem with special "high IQ" tests is that you cannot draw a large enough sample to norm this test. An IQ score is not an absolute value, like height, where you measure from a zero point to a certain length and each length has a meaning in itself in relation to that zero point. IQ does not have a zero point (there is no measurable total absence of intelligence), and the values of an IQ test are defined in relation to the average intelligence of the population that person comes from.
To norm a test, it is applied to a large sample of the population, and the mean and the shape of the distribution are calculated. The mean is defined as 100, the distribution is defined by the standard deviation, which for an IQ test is 15. The norming is done for different countries and at different times. This also means, that you cannot compare an IQ score from France to one from Spain, and that you cannot compare an IQ score from 1950 to one from 2007, at least not without some mathematical trickery similar to calculating the temperature in Centigrade from one given in Fahrenheit.
Now, since there are very few very intelligent people, even if all of them took the test your sample would be too small to reliably calculate a mean and the distribution. (And you couldn't test any of them ever again, because they already took the test and know the questions.)
Finally, you wouldn't know how the scores from this test relate to scores from other tests. In an IQ test with which you test the whole population, the average is set to be 100. Since all different IQ tests are normed for the whole population, you can set all their means at 100 and compare their scores. For a test that tests only a non-average subset of the population, like very intelligent people, the average must be somewhere else, because of course highly intelligent people are not of average intelligence. But where do you set it in relation to the 100 of the average population? You would need their scores from a normal IQ test to calculate this, but (now the circle closes) since a normal IQ test cannot measure them due to the ceiling effect, you don't know where they are in relation to the average IQ of 100.
Conclusion:
While you can create an intelligence test to measure the intelligence of highly intelligent people, this test will not give you a result that has any meaning in relation to the IQ, so in fact it is not an IQ test.
IQ is not intelligence (a trait), but a construct. IQ is what is measured by an IQ test, as the saying goes. A high intelligence test measures a different construct.
-
I dont see the problem, IQ testing in the normal range upto 140 is very successfull, for higher range, all you need is harder items and a larger sample. IQ testing has existed for over 50 years and norming has been done on millions. There exist many offline administered tests that measures upto 170 or higher. – z457731 Jun 11 '13 at 7:27
@z457731 That was covered in the reply by Jensen RCM. I'll try again: In the middle IQ ranges (average IQ around 100) there are many millions of people that you can test. From these results, you calculate the difficulty of your questions. This is important: the difficulty is not a factum that exists independent of the population. You need to test people to know how difficult a question is. To simplify the process, the difficulty is the number of people who cannot answer the question (from your test subjects). [contd.] – what Jun 11 '13 at 8:56
[contd.] For extremely difficult questions there might not be anyone among your subjects who can answer this question. So you don't know its difficulty. From your standpoint, all the questions that no-one could answer, are equally difficult (i.e. impossible) to answer. Or you might have one subject who has an IQ of 180, and he can answer all these difficult questions. Since only he can answer these questions, they appear equally difficult. [contd.] – what Jun 11 '13 at 8:56
[contd.] You don't know if there would be someone with IQ 160, who can answer only some of these very difficult questions. On top of that is the problem, that you don't know if a right or wrong answer was due to knowledge or guesswork. That person that had all those difficult questions right might just have had a lucky day. While is becomes less and less likely, the more correct answers he give, we don't know which ones he guessed, and even unlikely events do happen. That is what Jensen RCM means with "noise": the more difficult the questions, the less accurate the scaling of the difficulty. – what Jun 11 '13 at 8:57 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8269825577735901, "perplexity": 651.9855194815915}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776435842.8/warc/CC-MAIN-20140707234035-00049-ip-10-180-212-248.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3216012/pivotal-quantity-for-the-location-parameter-of-a-two-parameter-exponential-distr | # Pivotal Quantity for the location parameter of a two parameter exponential distribution
Let $$X$$ be a random variable with probability density function $$f(x,\theta, \beta)=\beta e^{-\beta(x-\theta)} \mathbb{1}_{(\theta,\infty)}$$ with $$\beta>0, \theta \in \mathbb{R}$$ (a two parameter exponential distribution) from which a random sample is taken. If $$\beta$$ is known and $$\theta$$ unknown, find an optimal confidence interval for $$\theta$$.
So I need help finding the pivotal quantity for this example. I thought using $$U_{i}=X_{i}-\theta \sim \operatorname{Exp}(\beta)$$ then $$\overline{U}=\overline{X}-\theta \sim \operatorname{Gamma}(n,\beta)$$ to homologate the procedure to find a confidence interval for $$\lambda$$ from $$\operatorname{Exp} \sim (\lambda)$$.
• You did find a pivotal quantity $U$ for $\theta$. Alternatively, since $X_i-\theta$ is i.i.d $\mathsf{Exp}$ with mean $1/\beta$, we have $2\beta(X_i-\theta)$ i.i.d $\chi^2_2$, thus giving the pivot $2\beta\sum (X_i-\theta)\sim \chi^2_{2n}$. Now the confidence interval can be obtained using chi-square fractiles. – StubbornAtom May 6 at 16:20
Indeed $$\overline X-\theta$$ is a valid pivot for $$\theta$$ when $$\beta$$ is known. All you have to do now is find $$a,b$$ such that $$a<\overline X-\theta with the desired confidence coefficient $$P_{\theta}\left[\theta\in(\overline X-b,\overline X-a)\right]$$ (for a two-sided interval). This is easily done using software.
Note that you have $$X_i-\theta\stackrel{\text{i.i.d}}\sim\mathsf{Exp}$$ with mean $$1/\beta$$, which implies $$X_{(1)}-\theta\sim \mathsf{Exp}$$ with mean $$1/(n\beta)$$ where $$X_{(1)}=\min\limits_{1\le j\le n}X_j$$. So yet another pivotal quantity is $$T(\mathbf X,\theta)=2n\beta(X_{(1)}-\theta)\sim \chi^2_2$$
We expect a confidence interval based on this pivot to be 'better' (in the sense of shorter length, at least for large $$n$$) than the one based on $$\sum\limits_{i=1}^n X_i$$ as $$X_{(1)}$$ is a sufficient statistic for $$\theta$$.
Now you can derive a two-sided confidence interval with confidence level $$1-\alpha$$ starting from
$$P_{\theta}(\chi^2_{1-\alpha/2,2}< T< \chi^2_{\alpha/2,2})=1-\alpha\quad\forall\,\theta$$
Here $$\chi^2_{\alpha,2}$$ is of course the $$(1-\alpha)$$th fractile of $$\chi^2_2$$, i.e. $$P(\chi^2_2>\chi^2_{\alpha,2})=\alpha$$. Notably if you have an observed sample at hand, then calculations involving chi-square fractiles for the confidence interval can be done by hand since printed chi-square tables are readily available. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 32, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9900407791137695, "perplexity": 387.52531930002624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998913.66/warc/CC-MAIN-20190619043625-20190619065625-00370.warc.gz"} |
https://www.intechopen.com/chapters/52177 | Open access peer-reviewed chapter
# Land-Atmosphere Interaction in the Southwestern Karst Region of China
By Jiangbo Gao, Wenjuan Hou, Kewei Jiao and Shaohong Wu
Submitted: November 6th 2015Reviewed: June 28th 2016Published: October 26th 2016
DOI: 10.5772/64740
## Abstract
Land-atmosphere interaction in the southwestern Karst region of China was investigated from two aspects: response of land cover to climate change and climatic effects of Karst rocky desertification. The first part focused on the temporal-spatial variation of growing-season normalized difference vegetation index (NDVI) and its relationship with climate variables. The relationships between growing-season NDVI with temperature and precipitation were both positive, indicating its limiting role on the distribution and dynamic of vegetation cover in the study area. The second part was designed to investigate whether the changed vegetation cover and land surface processes in the Karst regions was capable of modifying the summer climate simulation over East Asia. It was shown that land desertification resulted in the reduced net radiation and evaporation in the degraded areas. The East Asian summer monsoon was weakened after land degradation. Such circulation differences favored the increase in moisture flux and clouds, and thereby causing more precipitation in southeast coastal areas. Based on the above findings, it can be concluded that vegetation cover in Karst region was sensitive to climate change at larger scale, and on the other hand, there was significant feedback of vegetation cover change to regional climate by altering water and energy balance.
### Keywords
• Karst rocky desertification
• climate change
• land cover
• southwest China
• land-atmosphere interaction
## 1. Introduction
During the past decades, the vegetation-climate interaction has been a research focus of meteorology, climatology, geography, and ecology. The contents mainly include the impact of climate change on ecosystem and the feedback of vegetation cover change to atmosphere. Investigation on the correlation between vegetation variation and climate change and its influencing mechanisms are the basis for the studies on climate change adaptation and mitigation.
The response of terrestrial ecosystem to climate change, a complex issue in the field of global change, has been focused on in the last 30 years [1]. Vegetation cover has been proven to be governed by climatic factors, such as precipitation, temperature, solar radiation, and CO2 concentration. Therefore, variation in vegetation and its relationship with climatic factors reflected the sensitivity and vulnerability of the ecosystem to climate change (i.e., the responding processes) [2]. In many studies, the normalized difference vegetation index (NDVI) was selected to detect the impact of climate change on vegetation activity in Eurasia, [35]. Although the temperature increase was detected to dominate the vegetation cover and its dynamic in the northwestern China, western China, and the Tibetan Plateau, the impact of precipitation in the arid and semiarid regions may be more significant. The complicated and spatial heterogeneous effects of climate change on NDVI indicate the need to conduct further investigation at regional scales. Recently, in order to make clear the role of vegetation cover in the regional climate change, several studies on the feedback of land cover to atmosphere were conducted, especially after 1990s [6]. Land cover change (LCC) was documented as important as atmospheric circulation and solar orbit perturbations in climate change [7]. On the other hand, the feedback is regional-dependent due to the complicated climate and LCC in different regions.
The Karst region in the southwest China presents the transformation from vegetation covered landscape to exposed basement rocks, which was defined as the Karst rock desertification (KRD). In this region, the natural ecosystem is vulnerable while the human disturbance is severe. Earlier studies mainly emphasized the impact of land use change on vegetation cover [810], lacking consideration of climate change impacts at large scales. Furthermore, it is unknown the climatic effects of land cover change in the Karst region, especially land degradation. Therefore, in this chapter, the southwestern Karst region of China was selected to conduct land-atmosphere interactions research.
## 2. Study area
The southwestern Karst region of China, at 101°73'–112°44'E and 21°26'–29°25'N, and the Guizhou Karst Plateau, in the center of the southwestern Karst region (Figure 1), were selected to conduct research of climatic impacts on vegetation cover and climatic effects of vegetation degradation, respectively. They are located in the subtropical/tropical monsoon climate zone with annual precipitation of above 900 mm. The temperature and precipitation present great difference in spatial patterns, because of the typical topographical features with widely distributed mountains. Besides the Guizhou Karst Plateau, the southwestern Karst region, approximately 5.5 × 106 km2, includes Guangxi Zhuang Autonomous Region (GX) and eastern part of Yunnan Province (YN).
There are six vegetation types in the study area, including broadleaf forest, coniferous forest, shrub, grass, meadow, and cultural vegetation, with shrub covering the largest area. Because of the widely distributing bare limestone and the unsuitable land use since 1950s, KRD covers over 20% of the total area with the desertification rate of 2.5 × 104 km2 per year, and thus has become the most serious environmental problem in the study area. Rocky desertification in GKP exhibits three characteristics of severe degree, large area and high risk. However, litter research was carried out to assess the long-term vegetation dynamics and its influence on regional climate change.
## 3. Materials and methods
### 3.1. Statistical methods
#### 3.1.1. Trend analysis
The NDVI trend from 1982 to 2013 at pixel scale was estimated using the ordinary least squares (OLS) based on the ArcGIS 10.1 platform:
θslope=n×i=1ni×NDVIi=1nii=1nNDVIin×i=1ni2(i=1ni)2E1
where θis the regression slope and nrepresents the study year during the research period. The positive value of θmeans increasing NDVI.
#### 3.1.2. Mann-Kendall (MK) test
Mann-Kendall analysis, applied as a nonparametric, rank-based method for evaluating trends in time-series data [11], was used to detect the changing trend because it is known as more resilient to outliers. A rank sequence (Sk) for time series was built:
SK=i=1kri(k=2,3,,n)E2
where kis the dataset record length over years, and riis the altered data series for original dataset:
ri=(1xi>xj0xi<xj)(j=1,2,,i)E3
Under the assumption of random and independent time series, the statistic Zis defined:
Zk=[SkE(Sk)]Var(Sk)(k=1,2,,n)E4
Moreover, Z1 = 0, E(Sk) and Var (Sk) is the mathematical expectation and variance, respectively:
E(Sk)=n(n1)4E5
Var(Sk)=n(n1)(2n+5)72E6
The positive Zkvalue means the trend is increasing. Compared Zkwith Zα, the result of |Zk| > Zα(Z0.05 = 1.96) means the trend is statistically significant.
#### 3.1.3. Ordinary linear square
In order to compare the relative importance of temperature and precipitation for NDVI, the multivariate regression and the standardized coefficients were applied together. The higher standardized values mean important roles. The MATLAB 8.1 was used to establish multivariate linear model:
NDVI=b0+b1×Temperature+b2×Precipitation+εE7
where b0, b1, and b2 are the regression parameters, while εis the regression residual. Because of the different range for values of temperature and precipitation, it required normalization to compare the relative importance of climatic factors in the NDVI variations:
bi'=bi×t=1n(xtx¯)t=1n(yty¯)E8
#### 3.1.4. Geographically weighted regression (GWR)
The GWR analysis, coupled in ArcGIS 10.1, was conducted to reveal the spatial variations in relationships between NDVI and climatic variables. Both the spatial distribution and the dynamics of NDVI were considered by the GWR model. GWR extends the traditional OLS to consider the spatial heterogeneity in climate-vegetation correlations by assigning weight values [12]:
yi=β0(μi,νi)+k=1pβk(μi,νi)xik+εiE9
where yi, xik, and εi, represent the dependent variable, the independent variables, and the random error term at location i, respectively. Note that (µi, νi) expresses the coordinate location of the ith point, kdenotes the independent variable number. β0 and βkare the regression parameters at location i.
The regression coefficients were estimated by:
β(μi,νi)=(XTW(μi,νi)X)1XTW(μi,νi)YE10
βis the unbiased estimate of the regression coefficient. Wis the weighting matrix, and Xand Yare matrices for independent and dependent variables, respectively. The kernel function, used to determine the weight, was performed as the exponential distance decay:
ωij=exp(dij2b2)E11
ωijexpresses the weight of observation jfor location i, dijrepresents the Euclidean distance between points iand j, and bis the kernel bandwidth.
### 3.2. WRF climate model and experimental design
The WRF-ARW was developed as the next generation for regional climate model. It includes different parameterization schemes for longwave and shortwave radiation, cloud microphysics, cumulus, and land surface processes. The simplified simple biosphere model (SSiB), coupled with WRF model, was selected to simulate land surface energy balance. According to the SSiB model description, there are 12 types of vegetation cover, while the vegetation and soil parameters were set for every types. Defining different vegetation cover types in this study enabled investigation of the impact of land degradation and Karst rocky desertification using the WRF-SSiB model. The domain for WRF model was set as follows: dimensions of 196 × 154 horizontal grid points with center at 35°N and 110°E. In this domain, the influencing factors for East Asian summer monsoon can be included, for example, the upper level westerly jet (ULJ) and low-level jet (LLJ), the Bay of Bengal and the southeast trade wind, and so on [13]. The WRF downscaling ability was assessed by comparing the simulations with different physical schemes (Table 1), and the optimal combination was concluded from the assessment. For the execution of the WRF, we used the NCEP DOE Reanalysis-2 [14], hereafter NCEP R-2, at 6-h intervals to provide initial conditions and lateral boundary conditions.
Two experiments were done. One was the Case C, using the original SSiB vegetation map (as shown in Figure 2a), the other was Case D with the degraded land cover types (Figure 2b). The degraded types were decided based on the spatial pattern of different rocky desertification degrees [15]. For example, if the deserted areas accounted more than 30% of the corresponding counties, the SSiB vegetation was modified to bare soil (type 11 in SSiB model). The type 9 (broadleaf shrubs with bare soil) was used to replace original vegetation types in areas described as desert and potential desert areas larger than 45% of the counties and smaller than 30% of the counties, respectively. Based on the reset of vegetation cover types, two vegetation maps were used in WRF model, and was further used to conduct Case C and Case D.
1WSM 3RRTMMM5(Dudhia)Precipitation0.701.684.07
Temperature0.893.484.65
2KesslerRRTMMM5(Dudhia)Precipitation0.371.025.50
3Purdue LinRRTMMM5(Dudhia)Precipitation0.652.646.28
4WSM5RRTMMM5(Dudhia)Precipitation0.672.846.58
5FerrierRRTMMM5(Dudhia)Precipitation0.662.816.30
6WSM 3CAMCAMPrecipitation0.651.914.33
Temperature0.882.974.08
7WSM 3RRTMGRRTMGPrecipitation0.673.045.41
Temperature0.892.243.65
### Table 1.
Descriptive statistics of precipitation and temperature from WRF/SSiB with different microphysics and radiation schemes for June 2000 over 18°-52°N, 86°-136°E.
## 4. Results and discussion
### 4.1. Variations in growing-season NDVI
As shown in Figure 3, the rate of 0.0015/year during 1982–2013 was estimated for the growing-season NDVI trend in the Karst region of southwest China. The maximum value can be found in 2009 with significant variations between different years. It is indicated in Figure 3(b) that the year of 1994 was a tipping point, which means that there were two states before and after this year for the NDVI anomaly. We observed decreasing trend for some years, although the overall trend was increasing. Furthermore, the M-K trend test showed significant increasing trend, especially after the year 2004. As for the variation in NDVI of different vegetation types, the increasing rate was highest for coniferous forest, and the smallest value for meadow (Table 2).
Vegetation typeGrowing-season NDVI valueNDVI rateCorrelation coefficients
AverageMaximumMinimumTemperaturePrecipitation
Shrub0.69520.83690.48660.00150.149**0.130**
Grassland0.69460.84050.41260.00130.493**0.289**
Coniferous forest0.68710.82700.39320.00160.252**0.063
Cultural vegetation0.67060.83980.35760.00150.374**0.182**
### Table 2.
Statistical characteristics of growing-season NDVI for different vegetation types during 1982–2013.
Figure 4 shows the spatial distribution of NDVI values in the study area, ranging from 0.32 to 0.85. Due to higher temperature and more precipitation in Guangxi Zhuang Autonomous Region, there were high values of NDVI in the east part of the study area. Under the background of complex climate change, there was also spatial heterogeneity for the dynamical variation of NDVI. The higher increasing rate was observed in the northwest and the smaller values in the southeast (Figure 5).
### 4.2. Correlations between NDVI and climate factors
We observed warming rate of 0.018°C/year in the study area (Figure 6a). It fluctuated from −0.6 ∼ 0.8°C for average growing-season temperature. The year of 1995 was a tipping point for temperature and NDVI changes. Specifically, the average temperature for different months presented obvious variations with a maximum temperature (25.2°C) in July. For the changes in precipitation, Figure 6(c) shows a decrease of −1.21mm/year during 1982–2013. The dynamic processes for precipitation can be classified as falling under three stages: 1982–1992, 1993–2002, and 2003–2013 (Figure 6d). Additionally, the significant uptrend for temperature can be concluded from the Mann-Kendall test.
#### 4.2.1. Traditional linear regression for NDVI and climate variables
As shown in Figure 7(a), there was obvious synergy for NDVI and temperature, but the synergy for NDVI and precipitation was relatively weak (Figure 7b). The lower regression coefficients of precipitation indicated the weaker impact of precipitation on vegetation cover change. The reason may be that there was rich rainfall in the study area, and the annual variation cannot play significant roles. Moreover, the correlations between NDVI and climatic variables were different for different vegetation types (shown in Table 2). The largest regression coefficient was in grassland.
In most areas, the relationship between NDVI and temperature (Figure 8a) was positive due to the strengthened photosynthesis and vegetation activity by the increase in temperature. It should be pointed out that only within an appropriate range, the temperature rise can result in beneficial effects, and if the temperature is too high, it will cause negative impact on vegetation growth. Figure 8(b) shows the regression coefficient for NDVI and precipitation. Although the correlation was positive in most of the areas, there were some negative values in the northern part of the study area.
#### 4.2.2. Local regression for the spatial relationships
The later one means applying the changing rate of NDVI (Figure 5) as the dependent variable of GWR while the changing rate of climatic factors as independent variables. Figure 9 lists the GWR regression coefficients, where colors ranging from blue to red represented values from low to high. Additionally, the standard errors were analyzed by the points with different sizes.
There was positive relationships between multiyear average NDVI and temperature (Figure 9a), however, the regression coefficients for NDVI and precipitation contained both positive and negative values (Figure 9b). It was found that the positive values for NDVI and precipitation were mainly located in Yunnan Province, where the climate is more arid than other areas of the study area. The GWR regression coefficients for dynamic relationships were listed in Figure 9(c) and (d). The NDVI was lower with increasing surface temperature, which may be explained as more serious aridity due to the warming. On the other hand, the correlation between the changing rate of NDVI with precipitation were positive, meaning that the increase in NDVI during 1982–2013 could have been caused mainly by the precipitation variations.
### 4.3. The decrease in NDVI during 2009–2012 and its climatic explanation
Additional to the uptrend of NDVI from 1982 to 2013, there were some years when the NDVI decreased, that is, from 2009 to 2012. The decreasing rate during this time was −0.017/year. The significant decline was mostly in Guizhou Province where a decreased rate less than −0.02/year was observed (Figure 10). Correlation analysis between NDVI and climate change, revealed that the impact of temperature on the decreased NDVI was more profound than that from precipitation (Figure 11). Furthermore, the negative relationships between NDVI and precipitation also indicated the indirect impact of precipitation on temperature change. The increase in precipitation with more cloud could have led to the decrease in solar radiation and temperature, thus inhibiting photosynthesis.
### 4.4. Assessing the dynamic downscaling of WRF
Uncertainty on the downscaling capability of regional climate model (RCM) has in most cases led to skepticism for its use. Despite the weakness, the RCM dynamic downscaling is better than the simulations from General Circulation Model (GCM) or reanalysis datasets [13]. Furthermore, the uncertainty increases when the RCM is used to simulate the impact of land cover change on regional climate. In this section, the state-of-the-art RCM's downscaling ability was evaluated first, and was followed by analysis of the climatic effects of land degradation.
To reveal the improvement of WRF simulations over reanalysis dataset, daily rainfall, temperature, and other circulation factors from WRF and reanalysis were compared with the APHROD (Asian Precipitation-Highly-Resolved Observational Data) precipitation dataset, the GTS (Global Telecommunication System) temperature dataset, and the JRA-25 (Japanese 25-year Reanalysis) atmospheric variables dataset. The assessment was conducted from the viewpoint of correlation coefficient (R), bias and root mean square error (RMSE) over the years of 1998, 2000, and 2004 and over 18°–52°N, 86°–136°E (Table 3). The lower Bias and RMSE and the higher R values indicate better performance.
VariablesBiasRMSER
PrecipitationNCEP R-21.954.220.60
WRF/SSiB1.573.160.78
TemperatureNCEP R-2−1.933.620.86
WRF/SSiB−2.294.210.85
VQ700NCEP R-22.8911.380.65
WRF/SSiB−1.377.490.70
### Table 3.
Descriptive statistics of ensemble mean JJA daily precipitation, temperature and water vapor flux at 700 hpa from WRF/SSiB and NCEP R-2 over 18°-52°N, 86°-136°E.
We further observed that the phenomenon of most rainfall occurring in the south of China, especially in the south of Yangtze River, can be detected from both WRF simulation and APHROD dataset. From the WRF simulation, there was also an obvious increasing trend from the northwest to southeast in the south of about 38°N with the minimum temperature in Qinghai-Tibetan Plateau. The WRF simulation of precipitation out-performed NCEP R-2, and was probably caused by the improved simulations of low level water vapor flux (Table 3), a key factor influencing the atmospheric convection in East Asian summer monsoon. Although the simulated surface temperature from WRF was not improved over NCEP R-2, the clearer spatial information for temperature was presented from WRF output, which suggests that, it is also an applicable tool in downscaling temperature.
### 4.5. Influence on precipitation and temperature due to KRD
The area over 20°–34°N, 104°–124°E was chosen to investigate the impact of Karst rocky desertification on precipitation and temperature, because the significant and consistent effects were located in this region. There was spatial variation in the precipitation changes among the regions (Figure 12a). The reduced rainfall was mainly observed in the middle of Guizhou Karst Plateau. The areas with increased precipitation, mainly the middle and lower parts of Yangtze River and the surrounding areas, were of much larger magnitude and extent than that with decreased rainfall. It can be inferred that the consistent but nonsignificant reduction in rainfall with Guizhou Karst Plateau was due to high moisture influence from the Bay of Bengal. The land surface warming mainly occurred in the areas where the original vegetation types were replaced with bare soil type (Figure 12b), while the rainfall changes not only occurred within the desertification area but also beyond the area.
### 4.6. Influence of KRD on land surface energy balance
As shown in Figure 13, the substantial changes of surface energy components occurred in Guizhou Karst Plateau. In the degraded areas, the higher albedo (Figure 13a) led to more reflected shortwave radiation from the land surface (Figure 13b). Due to the higher surface skin temperature (Figure 12b), the outgoing longwave radiation increased significantly, which further caused the reduced net longwave radiation at the surface (Figure 13c). Both the reduction of the net shortwave radiation and the net longwave radiation certainly resulted in the decrease in land surface net radiation (Figure 13d). More sensible heat flux was also induced by the warmer surface (Figure 13e), however, the reduction in surface latent heat flux (Figure 13f) was much more than the sensible heat flux increase. The decrease in evaporation was probably contributed by changes in vegetation and soil properties, such as the lower LAI and roughness length, and the higher surface albedo. It can be concluded that evaporation decrease produced the most profound influence on the hydrological balance at land surface. Additionally, the above-mentioned higher temperature in the degraded areas was caused by the reduced evaporative cooling.
Consistent with the spatial changes in precipitation, there were areas with significantly changed energy budget extending beyond the degraded area. Outside the Guizhou Karst Plateau, the variations in sensible heat flux and latent heat flux were controlled by the precipitation differences. For example, in the areas between 30°–34°N, 112°– 120°E (i.e., the southeastern coastal area of China), the increased evaporation (Figure 13f) was caused by the increase in precipitation (Figure 12a), which further led to the lower temperature (Figure 12b), and the lower sensible heat flux (Figure 13e). The issue on the impact of atmospheric circulations on precipitation will be discussed in the next section.
Figure 13(g) shows the impacts of cloud albedo and land surface albedo on shortwave radiation. In the degraded areas within Guizhou Karst Plateau, the cloud fraction was reduced due to the less evaporation and moisture flux convergence after land degradation, and the reduced cloud fraction further led to more incoming shortwave radiation. However, the increase in upward shortwave radiation (Figure 13a) due to the higher land surface albedo was much more than the downward shortwave radiation, which resulted in the reduced net shortwave radiation (Figure 13b). Moreover, in the southeastern coastal areas of China, the increased cloud fractions, consistent with more rainfall, led to the decrease in incoming shortwave radiation, dominating the alteration in net shortwave radiation.
### 4.7. Effects of KRD on atmospheric circulation
The modified water and energy budget due to Karst rocky desertification was the first-order effects. Because of the different input of heat and moisture into atmospheric circulation, the large-scale circulation features were altered, resulting in climatic effects beyond the desertification area. As shown in Figure 14, the weakened 3-month mean wind vector at 700 hPa between Case D and Case C was caused by the lower surface heating in GKP (Figure 13d). The monsoon airflow from the Bay of Bengal, an important moisture source for the East Asia, was weakened from the degraded areas to the northeast. Furthermore, the weakened southwest airflow had significant impacts on the East Asian monsoon, especially, the anomaly cyclone (Figure 14) and the stronger horizontal convergence in the southeastern coastal area that led to the strengthened vertical ascending motion and the increase in precipitation.
On the other hand, the longitude-height section of the composite difference of zonal circulation along 24°–30°N between Case D and Case C was plotted to conduct further analysis (Figure 15). After the land degradation in GKP, an anomalous descending motion appeared in both the upper and middle level of troposphere over GKP and the middle and lower troposphere of the adjacent regions to the east. Such circulation modification caused the strengthened ascending motion over 114°–122°E. Moreover, the stronger lifting over the coastal areas led to the increase in the vertically integrated moisture flux convergence (VIMFC) from 1000 to 300 hPa. Consequently, the different circulation and moisture flux reduced the rainfall over GKP and promoted the formation of clouds and the positive rainfall anomalies over southeastern coastal areas of China (Figure 12a). Also, in the southeast China, the surface cooling (Figure 12b) was induced by the increased amount of clouds and further a negative net cloud radiation forcing.
## 5. Conclusions
The growing-season NDVI increased significantly during the last 30 years in the Karst region of the southwest China. There were also differences in the increase rate of vegetation types. The distribution of NDVI presented obvious spatial patterns, specifically, lower values in the western part and higher values in the east. The correlation between NDVI and climatic factors implied the limiting role of temperature for the vegetation growth and distribution in the study area, although the regression coefficients presented spatial heterogeneity. Additionally, the decreased NDVI was analyzed to detect the influencing mechanism. It was found that the increased cloud cover and rainfall led to the decrease in solar radiation and temperature, and further impeded photosynthesis.
We also observed that after the land cover change, there is need to consider its climatic effects through the impact of LCC on land surface water and energy budget. Karst rocky desertification (i.e., extensive exposure of basement rocks, serious soil erosion, drastic decrease in soil productivity and appearance of desert-like landscape) can modify the energy budget at land surface and then the regional climate. Specifically, after land degradation, the higher surface albedo and temperature caused the reduced net shortwave radiation and net longwave radiation. The sensible heat flux was increased by the higher temperature. Specifically, the substantial increase in sensible heat flux from ground offset the decrease in that from canopy. Due to higher stomatal resistance and lower LAI, the latent heat flux in KRD was reduced significantly. Less atmospheric heating from degraded land resulted in relative subsidence and less moisture flux convergence (MFC). The decrease in rainfall was probably attributed by both the reduced MFC and the reduced evaporation. A feedback loop was activated when precipitation was affected, for example, the altered soil moisture, vegetation growth, and phenology can further result in less diabatic heating rates, less moisture flux convergence, and lower rainfall. Moreover, the changed rainfall beyond the degraded areas was more significant. The modified energy and water balance due to land degradation weakened the southwest monsoon flow and affected the atmospheric circulation and moisture flux. In the southeastern coastal areas, the precipitation increased due to two reasons: (1) the weaker low-layer anticyclone causing the stronger vertical ascending motion, (2) the air mass diverging in the lower troposphere accompanying rising up over southeastern China.
## Acknowledgments
We thank the National Basic Research Program of China (Grant No. 2015CB452702), the National Natural Science Foundation of China (Grant No. 41671098, 41301089), the National Science and Technology Support Program of China (Grant No. 2012BAC19B10, Grant No. 2013BAC04B02) for supporting this work.
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Jiangbo Gao, Wenjuan Hou, Kewei Jiao and Shaohong Wu (October 26th 2016). Land-Atmosphere Interaction in the Southwestern Karst Region of China, Land Degradation and Desertification - a Global Crisis, Abiud Kaswamila, IntechOpen, DOI: 10.5772/64740. Available from:
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https://eprint.iacr.org/2018/232 | ## Cryptology ePrint Archive: Report 2018/232
Improved fully homomorphic public-key encryption with small ciphertext size
Masahiro Yagisawa
Abstract: A cryptosystem which supports both addition and multiplication (thereby preserving the ring structure of the plaintexts) is known as fully homomorphic encryption (FHE) and is very powerful. Using such a scheme, any circuit can be homomorphically evaluated, effectively allowing the construction of programs which may be run on ciphertexts of their inputs to produce a ciphertext of their output. Since such a program never decrypts its input, it can be run by an untrusted party without revealing its inputs and internal state. The existence of an efficient and fully homomorphic cryptosystem would have great practical implications in the outsourcing of private computations, for instance, in the context of cloud computing. In previous work I proposed the fully homomorphic public-key encryption scheme with the size of ciphertext which is not small enough. In this paper the size of ciphertext is one-eighth of the size in the previously proposed scheme. Because proposed scheme adopts the medium text with zero norm, it is immune from the the “p and -p attack”. As the proposed scheme is based on computational difficulty to solve the multivariate algebraic equations of high degree, it is immune from the Gröbner basis attack, the differential attack, rank attack and so on.
Category / Keywords: public-key cryptography / fully homomorphic public-key encryption, multivariate algebraic equation, Gröbner basis, non-associative ring | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8626914024353027, "perplexity": 1356.6506623710432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376829115.83/warc/CC-MAIN-20181217183905-20181217205905-00259.warc.gz"} |
http://mathhelpforum.com/statistics/227540-confidence-interval.html | ## Confidence interval
A confidence interval was used to estimate the proportion of statistics students who are female. A random sample of 72 statistics students generated the following 90% confidence interval: (0.438,0.642). Using the information above, what total size sample would be necessary if we wanted to estimate the true proportion to within 0.08 using 95% confidence? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9558549523353577, "perplexity": 609.6778146661313}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375096738.25/warc/CC-MAIN-20150627031816-00251-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/vasimr-plasma-thruster.863993/ | # VASIMR plasma thruster
Tags:
1. Mar 26, 2016
### AlkamlBan
First off, I want to apologize if this thread is in the wrong place but I am new this. Anyway, so I recently checked on the VASIMR plasma engine since I was interested in it. But one thing I didn't understand about it is if it ejects the neutral gas it uses. What I mean is that it obviously needs a neutral gas to work but does it eject it out of the thruster or can it be used "infinitely" (I mean for a long time not literally). In the case where it does eject it out the back is it possible for air to be inserted, have the neutral gases secluded and then use them for continuing the thrust?
2. Mar 27, 2016
### Staff: Mentor
3. Mar 27, 2016
### AlkamlBan
Interesting, I was considering its capabilities for flight on Earth where it could have an air intake but I am not so sure if the heat for the plasma coming out the back would burn things or not.
4. Mar 27, 2016
### Staff: Mentor
Compared to more conventional propulsion methods it would be horribly inefficient in an atmosphere. It also requires too much power to be of practical use within the timescales of atmospheric flight.
5. Mar 27, 2016
### Staff: Mentor
It's basically a very low power electric rocket motor, its energy comes from the battery (chemical or nuclear, or solar cells). The propellant is not a fuel so is not a source of energy. The motor is not for use on Earth.
6. Mar 27, 2016
### AlkamlBan
Ok got that, but one last thing I want to know is if the plasma coming out the back is hot enough to burn something at a distance or is it safe (aka hot but only at a small distance from the thruster?
7. Mar 27, 2016
### Staff: Mentor
It is very hot, but it has a low density. The effect will depend on what you do and how the environment looks like.
Draft saved Draft deleted
Similar Discussions: VASIMR plasma thruster | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.850877046585083, "perplexity": 1133.066806214756}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824899.75/warc/CC-MAIN-20171021224608-20171022004608-00388.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=2018_AIME_I_Problems/Problem_4&diff=prev&oldid=134562 | # Difference between revisions of "2018 AIME I Problems/Problem 4"
## Problem 4
In and . Point lies strictly between and on and point lies strictly between and on so that . Then can be expressed in the form , where and are relatively prime positive integers. Find .
## Solution
. Let be midpoint of , then .
## Solution 1 (No Trig)
We draw the altitude from to to get point . We notice that the triangle's height from to is 8 because it is a Right Triangle. To find the length of , we let represent and set up an equation by finding two ways to express the area. The equation is , which leaves us with . We then solve for the length , which is done through pythagorean theorm and get = . We can now see that is a Right Triangle. Thus, we set as , and yield that . Now, we can see = . Solving this equation, we yield , or . Thus, our final answer is . ~bluebacon008
## Solution 2 (Easy Similar Triangles)
We start by adding a few points to the diagram. Call the midpoint of , and the midpoint of . (Note that and are altitudes of their respective triangles). We also call . Since triangle is isosceles, , and . Since , and . Since is a right triangle, .
Since and , triangles and are similar by Angle-Angle similarity. Using similar triangle ratios, we have . and because there are triangles in the problem. Call . Then , , and . Thus . Our ratio now becomes . Solving for gives us . Since is a height of the triangle , , or . Solving the equation gives us , so our answer is .
## Solution 3 (Algebra w/ Law of Cosines)
As in the diagram, let . Consider point on the diagram shown above. Our goal is to be able to perform Pythagorean Theorem on , and . Let . Therefore, it is trivial to see that (leave everything squared so that it cancels nicely at the end). By Pythagorean Theorem on Triangle , we know that . Finally, we apply Law of Cosines on Triangle . We know that . Therefore, we get that . We can now do our final calculation: After some quick cleaning up, we get Therefore, our answer is .
~awesome1st
## Solution 4 (Coordinates)
Let , , and . Then, let be in the interval and parametrically define and as and respectively. Note that , so . This means that However, since is extraneous by definition, ~ mathwiz0803
## Solution 5 (Law of Cosines)
As shown in the diagram, let denote . Let us denote the foot of the altitude of to as . Note that can be expressed as and is a triangle . Therefore, and . Before we can proceed with the Law of Cosines, we must determine . Using LOC, we can write the following statement: Thus, the desired answer is ~ blitzkrieg21
## Solution 6
In isosceles triangle, draw the altitude from onto . Let the point of intersection be . Clearly, , and hence .
Now, we recognise that the perpendicular from onto gives us two -- triangles. So, we calculate and
. And hence,
Inspecting gives us Solving the equation gives
~novus677
## Solution 7 (Fastest via Law of Cosines)
We can have 2 Law of Cosines applied on (one from and one from ),
and
Solving for in both equations, we get
and , so the answer is
-RootThreeOverTwo
## Solution 8 (Easiest way- Coordinates without bash)
Let , and . From there, we know that , so line is . Hence, for some , and so . Now, notice that by symmetry, , so . Because , we now have , which simplifies to , so , and . It follows that , and our answer is .
-Stormersyle
## Solution 9 Even Faster Law of Cosines(1 variable equation)
Doing law of cosines we know that is * Dropping the perpendicular from to we get that Solving for we get so our answer is .
-harsha12345
• It is good to remember that doubling the smallest angle of a 3-4-5 triangle gives the larger (not right) angle in a 7-24-25 triangle.
## Solution 10 (Law of Sines)
Let's label and . Using isosceles triangle properties and the triangle angle sum equation, we get Solving, we find .
Relabelling our triangle, we get . Dropping an altitude from to and using the Pythagorean theorem, we find . Using the sine area formula, we see . Plugging in our sine angle cofunction identity, , we get .
Now, using the Law of Sines on , we get After applying numerous trigonometric and algebraic tricks, identities, and simplifications, such as and , we find .
~Tiblis
## Solution 11 (Trigonometry)
We start by labelling a few angles (all of them in degrees). Let . Also let . By sine rule in we get Using sine rule in , we get . Hence we get . Hence . Therefore, our answer is | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9947811961174011, "perplexity": 719.6373693233396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141716970.77/warc/CC-MAIN-20201202205758-20201202235758-00629.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/84133-linear-transformations.html | 1. Linear Transformations
Hello everyone,
I am currently having trouble with this problem. Any help would be greatly appreciated!
(a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image.
T is the counterclockwise rotation of 45 degrees in $R^2$, v = (2,2)
2. Originally Posted by larson
Hello everyone,
I am currently having trouble with this problem. Any help would be greatly appreciated!
(a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image.
T is the counterclockwise rotation of 45 degrees in $R^2$, v = (2,2)
Notice that $T$ is a linear transformation. You can get the matrix of the transofrmation by figuring out how it acts on its standard basis. Let $\bold{i} = (1,0) \text{ and }\bold{j} = (0,1)$ (just remember to think of them as coloumns, when we deal with $\mathbb{R}^n$ we sometimes things of vectors as an $n\times 1$ matrix). Compute $T(\bold{i})$ and $T(\bold{j})$ (again as coloumns) and form the $2\times 2$ matrix $[ T(\bold{i}) ~ ~ ~ T(\bold{j})]$.
3. Originally Posted by ThePerfectHacker
Notice that $T$ is a linear transformation. You can get the matrix of the transofrmation by figuring out how it acts on its standard basis. Let $\bold{i} = (1,0) \text{ and }\bold{j} = (0,1)$ (just remember to think of them as coloumns, when we deal with $\mathbb{R}^n$ we sometimes things of vectors as an $n\times 1$ matrix). Compute $T(\bold{i})$ and $T(\bold{j})$ (again as coloumns) and form the $2\times 2$ matrix $[ T(\bold{i}) ~ ~ ~ T(\bold{j})]$.
I'm sorry... I'm still kind of confused. How do I find T?
4. hi
hi
$\left[\begin{matrix} 1 \\ 0 \end{matrix}\right]$ rotates into $\left[\begin{matrix} cos(\theta) \\ sin(\theta) \end{matrix}\right]$ and $\left[\begin{matrix} 0 \\ 1 \end{matrix}\right]$ rotates into $\left[\begin{matrix} -sin(\theta) \\ cos(\theta) \end{matrix}\right]$
Rotation matrix here will be:
A = $\left[ \begin{matrix} cos(\frac{\pi}{4}) & -sin(\frac{\pi}{4}) \\ sin(\frac{\pi}{4}) & cos(\frac{\pi}{4}) \end{matrix} \right]$
Since $A \left[ \begin{matrix} 1 \\ 0 \end{matrix}\right] = \left[\begin{matrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix}\right]$ and $A \left[ \begin{matrix} 0 \\ 1 \end{matrix}\right] = \left[\begin{matrix} -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix}\right]$
Now calculate $Av$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 26, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9364727139472961, "perplexity": 314.44091669955077}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281492.97/warc/CC-MAIN-20170116095121-00208-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/stuck-on-homework-problem.131245/ | Stuck on homework problem
1. Sep 8, 2006
drh
http://tinyurl.com/hjybb
thats a picture of the problem, anyone know how to solve it??
2. Sep 8, 2006
Staff: Mentor
Write the formula for centripetal acceleration.
From that, determine the tangential velocity, which when the ball leaves the tube, is horizontal, i.e. in +x-direction, so that gives vx.
Assuming no air resistance, what can one assume about the horizontal velocity?
Now as soon as the ball leaves the tube, it starts to fall, so there is a downward vy which increases downward with acceleration of gravity. From this one can determine the time that the ball air airborne.
Know vx and t in air, one can determine the distance traveled. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9648892283439636, "perplexity": 918.8010223699299}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171251.27/warc/CC-MAIN-20170219104611-00120-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.computer.org/csdl/trans/tp/1984/03/04767523-abs.html | Issue No. 03 - March (1984 vol. 6)
ISSN: 0162-8828
pp: 314-318
Thomas E. Flick , Naval Research Laboratory, Washington, DC 20375.
Keinosuke Fukunaga , School of Electrical Engineering, Purdue University, West Lafayette, IN 47907.
ABSTRACT
A quadratic metric dAO (X, Y) =[(X - Y)T AO(X - Y)]¿ is proposed which minimizes the mean-squared error between the nearest neighbor asymptotic risk and the finite sample risk. Under linearity assumptions, a heuristic argument is given which indicates that this metric produces lower mean-squared error than the Euclidean metric. A nonparametric estimate of Ao is developed. If samples appear to come from a Gaussian mixture, an alternative, parametrically directed distance measure is suggested for nearness decisions within a limited region of space. Examples of some two-class Gaussian mixture distributions are included.
INDEX TERMS
CITATION
Thomas E. Flick, Keinosuke Fukunaga, "An Optimal Global Nearest Neighbor Metric", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 6, no. , pp. 314-318, March 1984, doi:10.1109/TPAMI.1984.4767523 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8563805222511292, "perplexity": 2937.6947326056993}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320863.60/warc/CC-MAIN-20170626184725-20170626204725-00245.warc.gz"} |
https://einsteinathome.org/content/not-gravity-geometry | # Not Gravity, Geometry?
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Topic 194949
In GR, how does the gravitational force, F = GM1M2/r^2, become geometry?
-LD
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Mike Hewson
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### Not Gravity, Geometry?
Quote:
In GR, how does the gravitational force, F = GM1M2/r^2, become geometry?
Ah, now that is a core question. It may take more than a bit of explaining though .... and beware there is no neat/pat answer that will likely satisfy your intuition. It really is quite a paradigm shift. Go and get a cup of your favorite brew before reading. I'll give you Mike's Tour Of GR! :-)
Firstly : one has to include time in the geometry. Just talking about 3-D space alone won't cut it. Try this simple example. Fire a cannonball at some fixed angle into the air. You'll find that where it lands depends ( ignoring lots of other fiddly non-gravitational stuff ) on the velocity with which it was fired. Indeed for given initial and final points even, there are two trajectories - the high lob and the low, flat shot - if you allow both the angle and direction of velocity to change.
The same applies to orbits around a central body, like planets and moons, where the subsequent evolution of the path requires considering the magnitude as well as the direction of the velocity vector. One could imagine the Earth in it's current orbital position around the Sun, with it's usual velocity, and it'll keep circling. But suppose it was at exactly the same position but with say, five times it's usual speed - it certainly won't hang around the Sun for much longer. So there's a time ( rate of change ) required in the explanation. That's as true for GR as it is for classical explanations.
Secondly : when we say geometry it's crucial, though no doubt annoying, to have to consider the basis of measurement. As Einstein found out with Special Relativity, some 'obvious' ideas in classical physics turned out to be wrong, or at least misleading and approximate. So there are many choices of geometric description, and for Einstein the challenge was to come up with a formulation that would be physically true ( same predictions ) regardless of special choices of how/where/when the geometry is defined. Thus the Earth ought continue to orbit the Sun, say, as viewed by a whole range of observers in various positions, with different clocks and measuring sticks.
Fortunately much of this was already done, incidentally, by a chap called Riemann. He studied what geometry would be like if it wasn't according to Euclid/Pythagorus etc. His essential breakthrough was a way of describing geometry locally and within the thing being discussed. So one might look at an apple and say : it is round much like a sphere, more so near the top where the stem is, is dimpled/puckered either end and is pretty smooth overall. These are words which are really 'external' descriptors. Riemann's way of deducing this equivalently was to say : I have a point on the apple and I compare two paths diverging from that point, further down each path I find they meet each other again. He comes up with a 'value' at each point on the apple, such that if one considers the totality of all these values you could arrive at what it would look like if you did see the apple from afar.
Now the reasons for going to a local ( in time as well as space ) rather than a general description is several fold :
- 3D in space plus one in time is 4 dimensions. Hard to visualise per se.
- to split the problem into two parts. Begin by stating the geometry in terms of what distribution of matter/energy produces it, then given that, see what response some object has in that geometry. Hence 'matter tells space how to warp, and space tells matter how to move'.
- spacetime is flat whenever viewed from close enough. This means that for a short enough time and/or for a small enough distance any movement looks Euclidean ( or Gallilean or Newtonian ).
So thirdly : how do you describe movement in detail in this framework? Well those 'values' I mentioned above are really a set of values at each point in spacetime. They 'explain' how you transition from one point to the next if you are freely falling ( only subject to gravity ). This is where the 'warping' business comes in : at each point in spacetime where gravity is acting ( and gravity is everywhere acting on everything ), these 'metric tensors' are a local guide to how directions change and thus which way to go next. I've described in another thread an analogy with small villages on some undulating landscape. At the centre of each hamlet is a signpost where roads intersect that has directions indicating the way to nearby villages. The metric prescribes how these signposts ought vary from place to place. [ The full horror is calculus, infinitesimals and equations with partial derivatives ... ]
If you were describing a globally flat spacetime ( unrealistically meaning no mass or energy was about ) then the metrics will state that the signposts don't vary from village to village. If you grind through the math in this scenario then you'd wind up describing the law of inertia in free space, where things just keep going if they are already. Specifically they wont deviate from a straight line.
Fourthly : I have left light out thus far. In GR the phrase 'straight line' is replaced by 'the path that light follows' or 'null geodesic'. For ordinary life these are easily seen to be the same thing. If you can arrange matters to view three objects by eye and see that they overlap/occult one another simultaneously then we say they are in a line. Bricklayers, surveyors, shooters etc explicitly do this all the time. So one way of mapping the curvature of spacetime is to study what the light rays are up to. The time component comes out as a change in the frequency of the light radiation, and thus is a measure of how time ( ie. clocks ) varies around and about.
Add in that any non-zero rest mass can't match ( or exceed ) light speed then you have an overall rule that matter won't escape the confines of the light paths. Hence the 'cone' analogy to spacetime points, where curvature means the cones are wobbly shaped if compared to the flat case. The way the cones change their shape from here to there is encoded in the metric.
So finally back to Newton's case of a single central mass that influences another one some distance away - your originally query. The good & bad news is that we have an approximate but not exact solution to the GR equations for this. It's still pretty good and has been observationally well confirmed more than a few times though ( eclipses, Mercury .. ). It was figured out by an artillery officer on the eastern European front in WWI, and he died not long after mailing it to Einstein. It's the ( Karl ) Schwartzchild ( maybe without the 't' ? ) solution, and has been expanded upon by others to include rotation and electric charge too ( see Kerr ).
The solution has a special quality with regard to a certain distance from the central body, the Schwartzchild radius. This radius depends on the mass of the body and some fundamental universal constants. If a body happens to lie completely within it's own Schwartzchild radius it will become a black hole. I won't recount all the observational features of black holes bar the prime one - that not even light can travel from within that radius to without ( discounting quantum effects ). For me that radius is about that of a proton I think, for the Earth about an apple size, for the Sun about the width of a mountain. As Mike/Earth/Sun each are not compact or dense enough - insufficient mass within a given volume - then while not black holes, there may be some measurable deviation of light rays passing by ( as viewed from far away ). An eclipse just after WW1 discovered the effect for the Sun, later technology in the 1960's ( plus the GPS more recently ) confirmed that for the Earth. No one has yet come along to demonstrate the effect around me though .....
Personally I try to avoid the word 'curvature' or at least mentally substitute it with the phrase 'observers differ'. That way time can be 'curved' by differently situated clocks progressively disagreeing with each other.
Quote:
For the mathematically inclined : deep in the gore of GR is a 4 by 4 matrix that is used to convert/connect one spacetime vector/point/event to another. There is a row for each of the space directions plus one for time, and each of those rows has four columns - for each space direction plus one for time. This is one way of representing the metric. The metric is one thing you definitely want to discover for a given problem. If you know it then you can say how things will 'fall' or behave in the absence of non-gravitational forces.....
Cheers, Mike.
( edit ) To be more precise, I ought say by 'metric tensors' I mean a metric tensor which is evaluated at many points. A tensor is sort of a multi-functional function. So instead of a single valued function - one number in, one number out - a tensor can have many things both in and out. In a sense we bundle lots of single-valued functions together, for instance how stuff in the z-direction depends on time, or how time depends on stuff in the x-direction. But they work as a group, and reasonable ideas of symmetry ( ie. reasonable universes without surprising behaviours that we haven't yet seen ) contract 4 x 4 = 16 functions to 10 independent ones.
Another way to visualise is : at each point in spacetime ( each moment in space & for each instant ) you have this associated tensor 'gadget' or 'box' that you can crank. We don't of course have an infinite listing of boxes, but Einstein's equation that governs their character. What remains is initial/starting/boundary conditions. We may well know how things vary from ( spacetime ) point to point but that still leaves some freedom in choice of the 'baseline'. Alot of discussions I read about GR astronomical problems divulge alot of assumptions upon these conditions. In a way one might solve say two neutron stars circling each other, however they aren't really alone so you have to 'connect' their behaviour to the rest of the universe at the 'boundaries'. Boundary also applies to the time co-ordinate, thus from whence and until whence is quite relevant.
I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal
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### That's how I see it. Take a
That's how I see it. Take a four-dimensional Riemannian manifold and endow it with a pseudoeuclidean metric (that of special relativity). Then calculate the tangent space at a chosen point. You have to differentiate the manifold using differential operators. Question: do they form a Lie algebra? If yes, which one? I have been unable to answer this question. But is it a good question?
Tullio
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### RE: It's the ( Karl )
Message 98231 in response to message 98229
Quote:
It's the ( Karl ) Schwartzchild ( maybe without the 't' ? ) solution...
Indeed it's Schwarzschild (black shield, not child ;-)
Gruß,
Gundolf
Computer sind nicht alles im Leben. (Kleiner Scherz)
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### RE: RE: It's the ( Karl )
Message 98232 in response to message 98231
Quote:
Quote:
It's the ( Karl ) Schwartzchild ( maybe without the 't' ? ) solution...
Indeed it's Schwarzschild (black shield, not child ;-)
Ah, so not 'the child of Schwartz' then. Thanks, I've seen many spellings. With the 't' is probably an anglicized mangling ... :-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal
Mike Hewson
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### RE: That's how I see
Message 98233 in response to message 98230
Quote:
That's how I see it.
I'm relieved! It's a toughy to understand, much less explain! :-)
Quote:
Take a four-dimensional Riemannian manifold and endow it with a pseudoeuclidean metric (that of special relativity). Then calculate the tangent space at a chosen point. You have to differentiate the manifold using differential operators. Question: do they form a Lie algebra? If yes, which one? I have been unable to answer this question. But is it a good question?
Don't know much about Lie algebras per se, except that they mean 'smooth', 'differentiable', 'continuous' and what not. So that's tantamount to asking if we can ( or not ) quantize spacetime? Good question indeed ...
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal
LivingDog
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### RE: RE: In GR, how does
Message 98234 in response to message 98229
Quote:
Quote:
In GR, how does the gravitational force, F = GM1M2/r^2, become geometry?
Ah, now that is a core question. It may take more than a bit of explaining though .... and beware there is no neat/pat answer that will likely satisfy your intuition. It really is quite a paradigm shift. Go and get a cup of your favorite brew before reading. I'll give you Mike's Tour Of GR! :-)
Firstly : ...
Secondly : ...
So thirdly : ...
Fourthly : ...
So finally ...
Thanks for the PG Tips.
-LD
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LivingDog
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### I originally asked, "In GR,
Message 98235 in response to message 98229
I originally asked, "In GR, how does the gravitational force, F = GM1M2/r^2, become geometry?" To which Mike replied (the quotes) and then my replies below his replies. (I edited the 'hec' out of the quotes and lost the original flow. So now I have to explain the above... like a pendulum do.)
Quote:
...
These are words which are really 'external' descriptors. Riemann's way of deducing this equivalently was to say : I have a point on the apple and I compare two paths diverging from that point, further down each path I find they meet each other again. He comes up with a 'value' at each point on the apple, such that if one considers the totality of all these values you could arrive at what it would look like if you did see the apple from afar.
May I have some of the math? I have an MS in Physics, and have taken a GR course (millennium ago), and am now reading the Princeton Phone book (MTW's blerb about GR).
E.g. are you refering to the affine connection? $$\Gamma^{\alpha}_{\mu\nu}$$ the metric tensor?? $$g_{\mu\nu}$$ BOTH???
"Why no tex? ... you have no tex!?! AHHH!! HE HAS NO TEX!!!" (apologies to Mike Judge)
Quote:
...
- to split the problem into two parts. Begin by stating the geometry in terms of what distribution of matter/energy produces it,
$$T_{\mu\nu}$$?
Quote:
...
then given that, see what response some object has in that geometry. Hence 'matter tells space how to warp, and space tells matter how to move'.
OHhhh... I see. Since the mass moves in a way according to the spacetime!
Quote:
...
- spacetime is flat whenever viewed from close enough. This means that for a short enough time and/or for a small enough distance any movement looks Euclidean ( or Gallilean or Newtonian ).
Yes, b/c measurements have a certain precision. "Flat" really means "I cannot measure curvature below the precision of my equipment."
Quote:
...
So thirdly : how do you describe movement in detail in this framework? Well those 'values' I mentioned above are really a set of values at each point in spacetime. They 'explain' how you transition from one point to the next if you are freely falling ( only subject to gravity ). This is where the 'warping' business comes in : at each point in spacetime where gravity is acting ( and gravity is everywhere acting on everything ), these 'metric tensors' are a local guide to how directions change and thus which way to go next.
Yeah... so the mass distribution (typically a sphere, ala Scwarzschild) determines $$g_{\mu\nu}$$ and then that determines the proper interval. $$g_{\mu\nu}$$ tells things how two points are connected - "curved" via $$g_{\mu\nu}$$ or "flat" via $$\eta_{\mu\nu}$$.
Right, in short: $$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu}$$
Quote:
...
Fourthly : I have left light out thus far. In GR the phrase 'straight line' is replaced by 'the path that light follows' or 'null geodesic'.
Is this the role of the Killing vector? If one solves for X in the Killing equation, then one knows the Killing vector, which is the path of a photon in this metric (with given mass distribution), which is called the "geodesic". IOW, solve for X in:
$$X_{\mu};_{\nu} + X_{\nu};_{\mu} = 0$$
Quote:
...
For ordinary life these are easily seen to be the same thing. If you can arrange matters to view three objects by eye and see that they overlap/occult one another simultaneously then we say they are in a line.
Right... we say they are in a "straight" line, but in reality they are following along the geodesic! Yes?? :)
Quote:
...
The time component comes out as a change in the frequency of the light radiation, and thus is a measure of how time (i.e. clocks) varies around and about.
Ohhh, so that's how we know the spacetime is curved - by the change in the color of the light as it travels from one point to another. Yes?? :(
Yes, on the light cone analogy. I think I am understanding it now.
Quote:
...
we have an approximate but not exact solution to the GR equations for this.
I thought it was exact. Was this the problem with the cosmological term?
Quote:
...
It was figured out by an artillery officer on the eastern European front in WWI, and he died not long after mailing it to Einstein. It's the ( Karl ) Schwartzchild (maybe without the 't' ?) solution,
Oh wow... I did not know that. What a shame.
Quote:
...
Personally I try to avoid the word 'curvature' or at least mentally substitute it with the phrase 'observers differ'. That way time can be 'curved' by differently situated clocks progressively disagreeing with each other.
Well, if my replies are correct, then I can accept curvature since it is really how two points are connected - "geodesic-ally" - via the $$g_{\mu\nu}$$. Yes?
Thanks Mike!
-LD
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Mike Hewson
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### RE: May I have some of the
Message 98236 in response to message 98235
Quote:
May I have some of the math? I have an MS in Physics, and have taken a GR course (millennium ago), and am now reading the Princeton Phone book (MTW's blerb about GR).
Hadn't heard that ( phone book ) phrase, but I see what you mean! :-)
I've been keeping the description away from math specifics because (a) I'm not that knowledgeable enough to frame it correctly ( but I'm studying .... ) and (b) tensor arithmetic ( gymnastics with indices ) tends to obscure the physical meaning.
Quote:
OHhhh... I see. Since the mass moves in a way according to the spacetime!
The underlying model is a smooth manifold so that it can be differentiated as many times as needed, and is classically/continuously so ( no weird quantum/foamy bits at really small scales ).
Quote:
Yes, b/c measurements have a certain precision. "Flat" really means "I cannot measure curvature below the precision of my equipment."
Yup indeed, but I really should have said "Minkowskian" ( as per Special Relativity ) so that
ds^2 = - dt^s + dx^2 + dy^2 + dz^2
becomes the infinitesimal line element ( there are several conventions possible here ). Or that metric tensor becomes :
[pre]-1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1[/pre]
with all the "cross" partial derivatives being zero ie. not the case with more general curvature. You can draw simple spacetime diagrams so that ( say, time on vertical axis and a space dimension on the horizontal ) is easily analysed with 'flat' triangles and whatnot. Meaning that it is sufficient to know the co-ordinate differences of events to yield the lengths of non-infinitesimal separations. With GR the co-ordinate differences don't give that, you have to "integrate along" as the metric changes from point to point ( a landscape stuffed full of an infinite number of villages lying along your curve, each an infinitesimal jump away from the previous and the next ). The co-ordinate values of two particular events of interest are endpoints to that integration ( I start and end at some named villages ).
Quote:
Yeah... so the mass distribution (typically a sphere, ala Scwarzschild) determines $$g_{\mu\nu}$$ and then that determines the proper interval. $$g_{\mu\nu}$$ tells things how two points are connected - "curved" via $$g_{\mu\nu}$$ or "flat" via $$\eta_{\mu\nu}$$.
Right, in short: $$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu}$$
Yup, eta is the straight co-ordinate difference approach whereas gee is a function ( group thereof ..... ) that varies across your spacetime landscape.
Quote:
Is this the role of the Killing vector? If one solves for X in the Killing equation, then one knows the Killing vector, which is the path of a photon in this metric (with given mass distribution), which is called the "geodesic". IOW, solve for X in:
$$X_{\mu};_{\nu} + X_{\nu};_{\mu} = 0$$
Dunno. Probably :-)
All I know of that is the Killing vector tells you where to go to preserve/not-distort distances on an object.
Quote:
Right... we say they are in a "straight" line, but in reality they are following along the geodesic! Yes?? :)
Well, we say a straight line is the geodesic. By fiat. Fait a compli. Say it is so as an axiom, then move on with deductions assuming that. In a sense it is word-play, but I think there is a variational principle here : one has two endpoints ( spacetime events ) such that over all possible paths between, the one with the minimum "total distance" is that which light takes ( all 'adjacent' paths are longer ). You can't beat it.
Quote:
Ohhh, so that's how we know the spacetime is curved - by the change in the color of the light as it travels from one point to another. Yes?? :(
Well anything time-dependent, but as photons get involved sooner or later in any practical time definition then that's a good indicator of change.
There's a neat short section on page 26-27 of MTW that compares observers with 'good' and 'bad' clocks : essentially one can make acceleration appear/disappear solely by choice of clock behaviour ( position measurements unchanged ). Reversing the approach, one can say that by choosing a local spacetime metric as flat ( inertial/unaccelerated ) forces a re-definition of the time standard ie. clocks alter. So things become inertial locally by suitable choice of clock. Which clock? The one that makes acceleration go away!! :-)
Quote:
Yes, on the light cone analogy. I think I am understanding it now.
Like a Hogwart's hat, it's roughly conical but squashy. :-)
Quote:
I thought it was exact. Was this the problem with the cosmological term?
Nope, it's linear approximations on non-linear equations. That's the whole rub of GR. Essentially the field itself has energy, so that 'feeds back'. A true & exact solution must encompass it's own presence, so to speak. QCD with quarks and gluons has a similiar character. So the mass of a proton say, can be partitioned into 'bare' quark masses plus a mass due to interaction energies of the gluons. Likewise a binary neutron star system has alot of 'gravitational self energy' meaning the entire system energy is well more than the total of the separated masses ( say around 30% - ish more ?? ).
Quote:
Oh wow... I did not know that. What a shame.
Poor lad, got a crappy skin infection from the horrible mud of the trenches and died from blood poisoning ( no antibiotics then ).
Quote:
Well, if my replies are correct, then I can accept curvature since it is really how two points are connected - "geodesic-ally" - via the $$g_{\mu\nu}$$. Yes?
Yup, what is straight for one guy with his rulers/clocks is wiggly for another with different ones. This is more than SR, where for a given relative speed you can stroll around one or other frame with a metric that is constant over the entire given frame. In GR you have to live with rulers and clocks that morph even in the same frame. So as you stroll from village to village, clock in one hand and ruler in the other, they are subtly changing as you go .... remember the metric is an agglomeration of functions relating your space and time degrees of freedom, the specific evaluation of which is time/space dependent.
Quote:
Thanks Mike!
A pleasure, but beware I could be easily lacking the proper rigour here. Thanks for the W&G links too, I didn't know they did TV ads !? :-)
Cheers, Mike.
[ edit ] You could of course integrate along a path in Minkowski/SR frames, but you get a simple answer - the same as found by subtracting co-ordinates and then doing a spot of ( flat ) trigonometry.
[ edit ] Another aspect is somewhat more philosophical, but it works. Because we have yet to find any phenomena that are not subject to gravitational or inertial effects ( NB one may have to look hard though ) we say gravity/inertia is universal. This is tantamount to saying that gravity/inertia is not a property of each specific body/particle so much as a characteristic of the spacetime they exist in. We attribute 'free fall' ( no non-gravitational accelerations ) to the background and not to the particles per se. The gravitational force disappears. So in a way that is a neat compression of the thinking/algorithm, find out what spacetime is up to and then given that : all bodies will behave similiarly. { Except one trouble, the presence of a body will change the solution and the bigger the mass/energy the more the change }
[ edit ] Note with regard to the 'good' and 'bad' clocks. It's the second derivative ( of one clock reading compared to the other ) which has to be non-zero. So we're not talking of the standard SR time dilation ( constant ratio between clocks in different frames ), but that said dilatation changes as time proceeds .... this is how an astronaut waving goodbye as he/she descends towards a black hole event horizon gradually slows down and 'freezes' as viewed by a distant observer. Black holes had been called 'frozen stars' prior to Penrose et al in the 1960's . Black holes are black because the light is infinitely frequency shifted to zero frequency, or equivalently the energy barrier to surmount is always greater than what any photon can start off with ( an infinite shift beats any finite frequency ).
[ edit ] Is it 'dilation' or 'dilatation' ?? I'm never quite sure of that one !! :-)
I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal
Mike Hewson
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### A concrete but very
A concrete but very artificial 'example' as regards the time change. Strictly speaking this likely falls over as I'm attributing all changes to the time axis alone ... so this is an 'in principle' explanation for 'some universe'. :-)
Suppose I have an 'ordinary' clock marking intervals for a falling body. Say we're on the Moon, so as to exclude all that air resistance stuff. So we have 'free fall' or no non-gravitational forces. Then if I look at the distance fallen for t = 0, 1, 2, 3, 4, 5 .... seconds I'll get ( distance ) x = 0, 1, 4, 9, 16, 25 ..... units. That is : x goes like t^2.
Now let's have another clock, but fudged so that it marks time but according to the square root of the reading on the first clock ( T = t^(1/2) ) . So on this clock, at time T = 0, 1, 2, 3, 4, 5 .... seconds I'll get ( distance ) x = 0, 1, 2, 3, 4, 5 .... units !!! So : x goes like the square root of [ the square of time ] ...
[ I haven't changed the length ruler ]
In the first case my distance is quadratic with time, whereas in the second it is linear. The first case says I have an acceleration ( dx^2/dt^2 != 0 ). Is dT^2/dt^2 non-zero? You bet it is! dT^2/dt^2 = (-1/4) * t^(-3/2). And dx^2/dT^ 2 = 0, so I have un-accelerated behaviour by that choice of 'dodgy' clock.
Note that the longer you run things, the seconds of 'T' time represent shorter intervals in 't' time. So any physical process is doing less per equal 'T' interval compared with the 't' interval. Or put another way : to make the 'T' time system mark those distance increments as equal every 'T' second, thus eliminating the acceleration in the 'T' frame, then for every 't' second ( with the speed increasing in the 't' system ) I have to jump in quicker on each 't' tick.
Cheers, Mike.
[ edit ] So for when the first clock 'strikes' t = 0, 1, ~1.414, ~1.732, 2, ~ 2.236, ~2.449, ~2.646, ~2.828, 3 ...... the second clock 'strikes' T = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 .... hence t(T+1) - t(T) = 1, ~0.414, ~0.318, ~0.268, ~0.236, ~0.213, ~0.197, ~0.182, ~0.172 ...
[ edit ] This is, of course, a local ( in time as well as space ) effect/argument. I'll hit something eventually as the body falls and/or the acceleration increases as I get closer to whatever is the central gravitating body. Generally non-gravitational forces interrupt our pure GR discussion by eliminating the 'free fall' assumption. Good thing too ... else what would stop the mass/energy density rising to make black holes far more common?? It's quantum that stops opposite electric charges sitting on top of one another. ;-) :-)
[ edit ] So if I could instantly flip my metabolism, thinking etc over to the 'T' rate from the 't' then : the acceleration would go away, and other stuff happening around the place would progressively slow down by my perception. If you have a functional dependence b/w 'T' and 't' other than the above ( parabola on it's side ) the arguments still qualitatively hold. Question : could the strengthening of a gravitational field speed up time ???? If not, why not ??? What ( pretty fundamental ) change would be needed for that to occur ???? :-) :-)
I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal
Mike Hewson
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### RE: Question : could the
Message 98238 in response to message 98237
Quote:
Question : could the strengthening of a gravitational field speed up time ???? If not, why not ??? What ( pretty fundamental ) change would be needed for that to occur ???? :-) :-)
By 'speeding up' in the context as above means that for each 'T' second passing by, more than one second passes in 't' time. So instead of t(T+1) - t(T) going from unity down to zero, it goes upwards instead. That way from a 'T' time aligned persona more is physically happening per 'T' second. But I still want to get rid of any accelerations by using the 'T' clock ....
The only way ( by time adjustment alone ) I can get the same distance covered during longer 't' intervals ( ie. removing acceleration ) is to have the distance travelled per 't' second decrease. So the further the object falls the slower it goes. There are no non-gravitational forces acting. And I'm getting closer towards a central body.
So now I'm falling 'down' and going slower with time. Hmmmm .... we would call this anti-gravity, would we not?? :-) :-)
All this is tantamount to saying that with our 'usual' assumptions about the universe no clock will run faster than one which is clear of all gravitational fields. Any field increase must slow the clock ( go closer to some mass ). You can speed a clock up but only by going from a stronger field to a weaker one - like coming back up from nearby a black hole event horizon to some distant position.
[ Another way around is to find out about the position of objects earlier than otherwise. Supraluminal signals will cause us to record any distance increments 'sooner'. By that I mean the case of normal attractive gravity but with tachyons now. Essentially this is what Newton assumed, instantaneous transmission, an infinite light speed and no clock variations of any sort. The One Big Clock for Everything. Terry Pratchett has a delightful parody of this in Thief of Time. ]
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8005945682525635, "perplexity": 1178.6515015448072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540548537.21/warc/CC-MAIN-20191213020114-20191213044114-00153.warc.gz"} |
https://earthscience.stackexchange.com/questions/16377/unwarranted-claim-of-higher-degree-of-accuracy-in-zircon-geochronology | # Unwarranted claim of higher degree of accuracy in zircon geochronology
The uncertainty in the half life of uranium-238 is stated at 0.05% [1]. The same paper gives the date 251.941 myr ± 31 kyr. 251.941 $$\times$$ 0.05% = 125 kyr.
How are the authors justified in claiming an accuracy of ± 31 kyr when the uncertainty in the half life alone is ± 125 kyr? On top of that, they list two other kinds of analytical uncertainties, which would only increase the overall uncertainty.
[1] Burgess, S. et al, "High-precision timeline for Earth’s most severe extinction", Proceedings of the National Academy of Sciences USA, Volume 111, 2014. https://www.pnas.org/content/pnas/111/9/3316.full.pdf
## 1 Answer
I can't be entirely sure but I'll make an informed guess:
That value doesn't come for a single measurement. Therefore, if the error in the age of a single sample is $$\pm125$$ kyr, you just need to average 16 samples to get it down to $$\pm31$$ kyr.
The uncertainty in the addition (or substraction) of two or more quantities is equal to the square root of the addition of the squares of the uncertainties of each quantity (assuming they arise from random errors). For example, if we have quantity A with uncertainty $$\sigma_a$$ and quantity B with uncertainty $$\sigma_b$$, the error in the quantity $$C=A+B$$ would be:
$$\sigma_c=\sqrt{\sigma_a^2+\sigma_b^2}$$
And if we call M to the average between A and B. The uncertainty in the average is
$$\sigma_m=\frac{\sqrt{\sigma_a^2+\sigma_b^2}}{2}$$
So if we average 16 samples with $$\sigma=125$$ kyr, the uncertainty in the average would be
$$\sigma_m=\frac{\sqrt{16 \sigma^2}}{16}=\frac{\sqrt{16 \times 125^2}}{16}=31$$ kyr
Uncertainty propagation can be seen in the abstract of the article you refer to:
The extinction occurred between 251.941 ± 0.037 and 251.880 ± 0.031 Mya, an interval of 60 ± 48 ka.
Where the ±48 ka comes from?
$$\sqrt{37^2+31^2}=48$$
This treatment of uncertainties assumes that uncertainties are independent of each other. As @Mark pointed in the comments, this won't be the case if the uncertainty comes from "the length of your measuring stick (the half-life of U-238)".
This is: if you measure something with a "yard-stick" that have the wrong size, you can't reduce the resulting error by just averaging many measurements.
I don't know enough of geochronology to understand all the different errors they report. But the simple example in the abstract cite I presented above shows that they are indeed treating those errors as independent. Otherwise the reported interval error (±48 ka) would not make sense.
If one significant source of error is indeed the uncertainty in the half-life of $$^{238}$$U. However, it would be wrong to treat these errors as independent only if there exists an exact value for this half-life. Alternatively, maybe there is no exact value of the half-life, and what they meant is that the half-life value can truly vary a 0.05%. This is something to look into if you want to figure out if the treatment of errors they do is correct. However, after a quick google search I found that radioactive half-life can indeed vary by a small fraction due to environmental conditions, this article explains pretty well the phenomena. Here a short excerpt:
...radioactive half-life of an atom can depend on how it is bonded to other atoms. Simply by changing the neighboring atoms that are bonded to a radioactive isotope, we can change its half-life. However, the change in half-life accomplished in this way is typically small. For instance, a study performed by B. Wang et al and published in the European Physical Journal A was able to measure that the electron capture half-life of beryllium-7 was made 0.9% longer by surrounding the beryllium atoms with palladium atoms.
If the 0.05% uncertainty on the half-life of $$^{238}$$U comes from random environmental factors, it would indeed be acceptable to consider them as an independent source of error for each sample.
• As far as I know, this only applies if the uncertainties are independent. If the uncertainty is in the length of your measuring stick (the half-life of U-238), the uncertainties are correlated and you can't reduce them by making more measurements. – Mark Mar 2 '19 at 20:48
• @Mark You are right. I've added something to my answer. Have a look. – Camilo Rada Mar 2 '19 at 21:47
• @Mark I just added something about the natural variation of radioactive decay half-life. I think that explain that they treat the errors as independent. – Camilo Rada Mar 2 '19 at 22:05
• The page you link to lists three potential mechanisms for changing the decay rate (and thus the half-life) of radioactive elements: time dilation, electron density change and bombardment with high-energy radiation. The first doesn't apply on Earth (well, technically it does, but the time dilation due to Earth's gravity is vanishingly small), the second only affects elements that decay via electron capture (Uranium doesn't), ... – Ilmari Karonen Mar 2 '19 at 22:24
• @IlmariKaronen Yes, but the OP doesn't seem to be content with just "a good approximation" as the whole question revolve around a 0.05% change in $^{238}$U half-life, a pretty tiny change. – Camilo Rada Mar 2 '19 at 22:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 13, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8706268072128296, "perplexity": 706.5787250469687}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400241093.64/warc/CC-MAIN-20200926102645-20200926132645-00447.warc.gz"} |
http://ozradonc.wikidot.com/electron-beam-dose-distribution | 9.2 - Electron Beam Dose Distribution
In a similar way to photon beams, electron beams may be represented with depth dose charts, beam profiles, and isodose charts.
## Central Axis Depth Dose
The typical features of an electron depth dose chart are shown below.
In general, electron beams demonstrate:
• A high surface dose relative to photon beams
• A broad ‘effective dose’ region
• A linear fall off in dose at depth
• A bremsstrahlung tail due to generation of photons from inelastic 'collisions' with nuclei
R90 is the point beyond which dose is less than 90% of the maximum. R50 is the point of 50% dose, and is useful in describing the quality of the electron beam. Rp is the potential range, the maximum range obtained by electrons incident on the surface.
### Comparison of depth dose values
Electrons lose energy constantly as they pass through a medium, and their rate of energy loss and amount of scattering is dependent on their energy. For lower energy electrons, lateral scattering happens shortly after they enter the tissue. This leads to a relatively rapid loss of energy, with a significant 'peak' of energy loss at zmax relative to the surface dose. Higher energy electron beams tend to undergo minimal scattering near the surface and continue onwards, losing their energy over a greater distance. This leads to significantly broader region of dose distribution, and zmax is not significantly greater than the surface dose. The final outcome of these interactions is that high energy electrons have a high surface dose relative to low energy electrons.
Beam 6 MeV 9 MeV 12 MeV 16 MeV 20 MeV Surface Dose Pre-max 90% zmax Post-max 90% R50 R10 75% 0.8 cm 1.5 cm 2 cm 2.5 cm 3 cm 83% 1 cm 2 cm 3 cm 3.5 cm 4.5 cm 88% 0.5 cm 3 cm 4 cm 5 cm 6 cm 94 % N/A 3.5 cm 5 cm 6.5 cm 8 cm 95% N/A 2.3 cm 6 cm 8.5 cm 10.5 cm
### Formula for determining depth dose figures
This is useful for exams.
#### Surface Dose
This only works in my department. There are five energies. In order, assign them a value of $2^{3-n}$ where n is in order of energies. This value is x.
(1)
$$x=2^{3-n}$$
(2)
\begin{align} \text{Surface Dose} = 100 - {E \times x} \end{align}
This gives a vague approximation of surface doses, eg (for 9 MeV where $x = 2^{3-2} = 2^1 = 2$
(3)
\begin{align} \text{Surface Dose} = 100 - {9 \times 2} = 82% \end{align}
#### zmax
zmax usually occurs at a depth of $\frac{E}{4}$
#### Post-max R90
R90 usually occurs at a depth of $\frac{E}{3}$
#### R50
R50 is a bit tricker; it occurs at a depth of $\frac{\frac{E}{3} + \frac{E}{2}}{2}$.
This is the average distance between R90 and R10
#### R10
R10 occurs at a depth of $\frac{E}{2}$
Beyond R10, the curve is relatively flat, the bremsstrahlung tail.
## Beam Profile
The beam profile is a chart of the off-axis ratios for the beam along a line perpendicular to the beam direction at a particular depth. It allows determination of beam symmetry and beam flatness.
## Isodose Charts
Electron isodose charts are constructed in the same way as photon charts but have significantly different features. There are several major differences with electron beams when compared to photons:
• The isodose curves for 20% of dose and below tend to bulge outwards, if beam energy is over 10 MeV. This is due to increased lateral range of electrons when they possess a higher starting energy.
• The isodose curves for 80% of dose and above in beams over 15 MeV show lateral constriction - the isodose lines trend towards the central axis due to loss of electronic equilibrium. This has a similar cause to the bulging of low energy isodose lines - there is increased lateral scatter of electrons at higher energies due to their increased range.
• There is rapid loss of dose after the R90 is reached for all beams
The penumbra is defined by the ICRU as the region that lies between the 20 and 80% isodose lines, at a depth of R85 / 2, where R85 is the depth beyond zmax where the percentage depth dose is 85%. The penumbra is typically broader than for a photon beam, mostly due to lateral scatter of high energy electrons. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9564074873924255, "perplexity": 2296.649265558074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221216333.66/warc/CC-MAIN-20180820101554-20180820121554-00603.warc.gz"} |
http://mathhelpforum.com/geometry/179886-tangent-circle.html | # Thread: Tangent to the Circle
1. ## Tangent to the Circle
Hi there, stuck on a tangent question.
a. Show that the point P(5,10) lies on the circle with equation (x+1)^2 + (y-2)^2 = 100.
b. PQ is a diameter of this circle as shown in the diagram. Find the equation of the tangent at Q.
I can do part a, and know how to find the equation of a tangent, but I don't know how to get the coordinates of Q so I can solve the equation.
2. Tangents to circles are always perpendicular to the circle's radius/diameter.
3. Originally Posted by HighlyFlammable
Hi there, stuck on a tangent question.
a. Show that the point P(5,10) lies on the circle with equation (x+1)^2 + (y-2)^2 = 100.
b. PQ is a diameter of this circle as shown in the diagram. Find the equation of the tangent at Q.
I can do part a, and know how to find the equation of a tangent, but I don't know how to get the coordinates of Q so I can solve the equation. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8798329830169678, "perplexity": 245.09352609169545}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719566.74/warc/CC-MAIN-20161020183839-00015-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://www.math.gatech.edu/seminars-colloquia/series/dynamical-systems-working-seminar/longmei-shu-20160401 | ## Generalized Eigenvectors for Isospectral Reduction
Series:
Dynamical Systems Working Seminar
Friday, April 1, 2016 - 13:05
1 hour (actually 50 minutes)
Location:
Skiles 170
,
Georgia Tech
Organizer:
Isospectral Reduction reduces a higher dimension matrix to a lower dimension one while preserving the eigenvalues. This goal is achieved by allowing rational functions of lambda to be the entries of the reduced matrix. It has been shown that isospectral reduction also preserves the eigenvectors. Here we will discuss the conditions under which the generalized eigenvectors also get preserved. We will discuss some sufficient conditions and the reconstruction of the original network. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.940552294254303, "perplexity": 805.3843319090142}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812579.21/warc/CC-MAIN-20180219091902-20180219111902-00617.warc.gz"} |
http://mathoverflow.net/questions/16991/what-are-the-connections-between-pi-and-prime-numbers/124656 | # What are the connections between pi and prime numbers?
I watched a video that said the probability for Gaussian integers to be relatively prime is an expression in $\pi$, and I also know about $\zeta(2) = \pi^2/6$ but I am wondering what are more connections between $\pi$ and prime numbers?
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You can use the fact that zeta(2)=pi^2/6 to prove the infinitude of primes. If there were finitely many, then the Euler product for zeta(2) would be a rational number, contradicting the irrationality of pi. – Ben Linowitz Mar 3 '10 at 19:54
This question should in my opinion be Community Wiki. – Grétar Amazeen Mar 3 '10 at 20:56
More generally, for all positive integers $n$, $\zeta(2n)$ is a rational multiple of $\pi^{2n}$. – Gerry Myerson Mar 3 '10 at 22:42
See also this related question - mathoverflow.net/questions/21367/… – François G. Dorais Jun 19 '10 at 14:52
It shows up when one considers the infinite prime together with the usual primes: mathoverflow.net/q/7656/733 – Peter Arndt May 29 '14 at 16:12
Well, first of all, $\pi$ is not just a random real number. Almost every real number is transcendental so how can we make the notion "$\pi$ is special" (in a number-theoretical sense) more precise?
Start by noticing that $$\pi=\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$$ This already tells us that $\pi$ has something to do with rational numbers. It can be expressed as "a complex number whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficients, over domains in $\mathbb{R}^n$ given by polynomial inequalities with rational coefficients." Such numbers are called periods. Coming back to the identity $$\zeta(2)=\frac{\pi^2}{6}$$ There is a very nice proof of this (that at first seems very unnatural) due to Calabi. It shows that $$\frac{3\zeta(2)}{4}=\int_0^1\int_0^1\frac{dx\,dy}{1-x^2y^2}$$ by expanding the corresponding geometric series, and then evaluates the integral to $\pi^2/8$. (So yes, $\pi^2$ and all other powers of $\pi$ are periods.) But the story doesn't end here as it is believed that there are truly deep connections between values of zeta functions (or L-functions) and certain evaluations involving periods, such as $\pi$. Another famous problem about primes is Sylvester's problem of which primes can be written as a sum of two rational cubes. So one studies the elliptic curve $$E_p: p=x^3+y^3$$ and one wants to know if there is one rational solution, the central value of the corresponding L-function will again involve $\pi$ up to some integer factor and some Gamma factor. Next, periods are also values of multiple zeta functions: $$\zeta(s_1,s_2,\dots,s_k)=\sum_{n_1>n_2>\cdots>n_k\geq 1}\frac{1}{n_1^{s_1}\cdots n_k^{s_k}}$$ And they also appear in other very important conjectures such as the Birch and Swinnerton-Dyer conjecture. But of course all of this is really hard to explain without using appropriate terminology, the language of motives etc. So, though, this answer doesn't mean much, it's trying to show that there is an answer to your question out there, and if you study a lot of modern number theory, it might just be satisfactory :-).
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The probability that two Gaussian integers are relatively prime is $6/(\pi^2 K) = 0.66370080461385348\cdots$, where $K= 1 - \frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots$ (Catalan's constant). There is no known simple expression for $K$ in terms of $\pi$. See http://www.springerlink.com/content/y826m64747254t87.
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Here is an example of a way to use $\pi$ to prove the infinitude of primes without calculating its value, or using the relatively deep fact that $\pi$ is irrational, but starting from the knowledge of $\zeta(2)$ and $\zeta(4).$ Suppose that there were only finitely many prime numbers $2= p_{1}, 3= p_{2}, \ldots, p_{k-1},p_{k}.$ From the formulae $\sum_{n=1}^{\infty} \frac{1}{n^{2}} = \frac{\pi^{2}}{6}$ and $\sum_{n=1}^{\infty} \frac{1}{n^{4}} = \frac{\pi^{4}}{90}$, we may conclude after the fashion of Euler that (respectively) we have: $\prod_{j=1}^{k} \frac{p_{j}^{2}}{p_{j}^{2}-1} = \frac{\pi^{2}}{6}$' and $\prod_{j=1}^{k} \frac{p_{j}^{4}}{p_{j}^{4}-1} = \frac{\pi^{4}}{90}.$ Squaring the first equation and dividing by the second leads quickly to $\prod_{j=1}^{k} \frac{p_{j}^{2}+1}{p_{j}^{2}-1} = \frac{5}{2}$, so $5\prod_{j=1}^{k} (p_{j}^{2}-1) = 2 \prod_{j=1}^{k}(p_{j}^{2}+1).$ This is a contradiction, since the product on the left is certainly divisible by $3$, whereas every term in the rightmost product except that for $j = 2$ is congruent to $-1$ (mod 3), so we obtain $0 \equiv (-1)^{k}$ (mod 3), which is absurd. (I would be grateful if anyone knows a reference for a proof like this. I can't believe that I am the first person to think of it).
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Neat. Is there an easier way to see that $\zeta(4)/\zeta(2) = 5/2$? – François G. Dorais Mar 16 '13 at 0:18
@Francois: I do not know. I think that there are quite a few instances in number theory where computations which eventually have a rational answer require $\pi$ in an apparently essential fashion along the way. – Geoff Robinson Mar 16 '13 at 1:01
@Geoff: Another way of seeing this: $\sum_{(a,b)=1} \frac{1}{a^2b^2}=\frac{5}{2}$. – i707107 Mar 16 '13 at 4:22
@Francois: A typo, should be $\zeta(4)/\zeta^2(2)$. – i707107 Mar 16 '13 at 4:24
@i707107: Are you saying that you can derive that formula without using values of $\zeta$ at all, or that you can calculate it from $\zeta(2)$ alone? – Geoff Robinson Mar 16 '13 at 14:51
There are a few formulas relating $\pi$ to arithmetic functions. For example, if $\sigma(n)$ is the sum of the divisors of $n$, then $\sum_1^n\sigma(n)=\pi^2n^2/12+O(n\log n)$. If $d(n)$ is the number of divisors of $n$, then $\sum_1^{\infty}n^{-2}d(n)=\pi^4/36$. If $\phi(n)$ is the Euler phi-function, then $\sum_1^n\phi(n)=3n^2\pi^{-2}+O(n\log n)$. These all appear in Section 3.5 of Eymard and Lafon, The Number $\pi$.
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By the way, do you happen to know what constant appears if we change $\sigma(n)$ into the sum $\sigma'$ of all Gaussian integer divisors of n with positive real parts? I.e. this sequence: oeis.org/A078930 Computations show that $\sum_1^n\sigma'(i)\approx Cn^2$, where C=1.7972... – mathreader Oct 18 '12 at 9:44
A formula that surely belongs here linking $\pi$ and the primes is $$2.3.5.7...=4\pi^2.$$ This is obtained via a zeta regularization in a similar way to the more well-known $\infty!=\sqrt{2\pi}$ (see e.g. here for a short discussion of this). However, to find the product of the primes, one uses the prime zeta function $$\sum_{p\; prime} \frac{1}{p^s}$$ which has the unfortunate property of having infinitely many singularities between 0 and 1 which breaks the standard regularization procedure. E. Muñoz García and R. Pérez Marco circumvent this problem (literally) by adding in an extra variable and taking the limit from a different direction.
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What is funny is that they consider 1 to be a prime number... (although this makes no difference for the product). – ACL Mar 15 '13 at 17:20
I certainly would not say that that divergent product has a value. Writing the equation that way causes much more harm, in my opinion, than the slight good that it communicates to people who already know exactly what is trying to be communicated. – Greg Martin Mar 16 '13 at 6:18
There is a nice story, initiated by L. Van Hamme, which relates several Ramanujan's formulas for $\pi$ to supercongruences modulo powers of primes. The simplest way to witness this route is to make a look at (my) http://arxiv.org/abs/0805.2788 .
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More elementary: the probability that two positive integers have GCD=1 is $6/\pi^2 = 1/\zeta(2)$ because the probability that a prime $p$ divides the GCD is 1/p^2 by considering each p by p block of pairs of positive integers. More generally, the probability that k positive integers have GCD 1 is $1/\zeta(k)$ by a similar argument.
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Leonard Euler discovered many years ago that
$\frac π 4 = \frac 3 4 \cdot \frac 5 4 \cdot \frac 7 8 \cdot \frac {11} {12} \cdot \frac {13} {12} \cdots$
where the numerators on the right-hand side are the odd prime numbers and each denominator (on both sides) is the multiple of 4 nearest to the corresponding numerator. Pretty fascinating if you ask me.
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So $\pi = \infty$, or what do you want to say? -- Your series diverges ... . – Stefan Kohl Jul 21 '13 at 23:00
Replace addition with multiplication, the infinte series should be an infinte product. – Dag Oskar Madsen Jul 21 '13 at 23:30
mathworld.wolfram.com/PrimeProducts.html, formula (33) – Dag Oskar Madsen Jul 21 '13 at 23:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9245423078536987, "perplexity": 267.0384911895654}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936461848.26/warc/CC-MAIN-20150226074101-00228-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://phys.libretexts.org/Bookshelves/Optics/Book%3A_Geometric_Optics_(Tatum)/01%3A_Reflection_and_Refraction/1.07%3A_The_Rainbow | $$\require{cancel}$$
# 1.7: The Rainbow
I do not know the exact shape of a raindrop, but I doubt very much if it is drop-shaped! Most raindrops will be more or less spherical, especially small drops, because of surface tension. If large, falling drops are distorted from an exact spherical shape, I imagine that they are more likely to be flattened to a sort of horizontal pancake shape rather than drop shaped. Regardless, in the analysis in this section, I shall assume drops are spherical, as I am sure small drops will be.
We wish to follow a light ray as it enters a spherical drop, is internally reflected, and finally emerges. See Figure I.15. We shall refer to the distance $$b$$ as the impact parameter.
We see a ray of light headed for the drop, which I take to have unit radius, at impact parameter $$b$$. The deviation of the direction of the emergent ray from the direction of the incident ray is
$D = \theta - \theta' + \pi -2\theta' +\theta - \theta' = \pi + 2\theta - 4\theta'. \label{eq:1.7.1}$
However, we shall be more interested in the angle $$r = \pi − D$$. A ray of light that has been deviated by $$D$$ will approach the observer from a direction that makes an angle $$r$$ from the centre of the bow, which is at the anti-solar point (Figure I.16)
We would like to find the deviation $$D$$ as a function of impact parameter. The angles of incidence and refraction are related to the impact parameter as follows:
$\sin\theta=b,\label{eq:1.7.2}$
$\cos\theta=\sqrt{1-b^2},\label{eq:1.7.3}$
$\sin\theta' = b/n,\label{eq:1.7.4}$
and
$\cos\theta = \sqrt{1-b^2/n^2}. \label{eq:1.7.5}$
These, together with Equation $$\ref{eq:1.7.1}$$, give us the deviation as a function of impact parameter. The deviation goes through a minimum – and $$r$$ goes through a maximum. The deviation for a light ray of impact parameter $$b$$ is
$D = \pi + 2\sin^{-1}b - 4\sin^{-1}(b/n)\label{eq:1.7.6}$
The angular distance $$r$$ from the centre of the bow is $$r = \pi − D$$, so that
$r = 4 \sin^{-1}(b/n) - 2\sin^{-1}b.\label{eq:1.7.7}$
This is shown in Figure I.17 for $$n$$ = 1.3439 (blue - $$\gamma$$ = 400 nm) and $$n$$ = 1.3316 (red - $$\gamma$$ = 650 nm).
Differentiation gives the maximum value, $$R$$, of $$r$$ - i.e. the radius of the bow – or the minimum deviation $$D_{\text{min}}$$. We obtain for the radius of the bow
$R = 4\sin^{-1}\sqrt{\frac{4-n^2}{3n^2}}- 2\sin^{-1}\sqrt{\frac{4-n^2}{3}}. \label{eq:1.7.8}$
For $$n$$ = 1.3439 (blue) this is 40° 31' and for $$n$$ = 1.3316 (red) this is 42° 17'. Thus the blue is on the inside of the bow, and red on the outside.
For grazing incidence (impact parameter = 1), the deviation is $$2 \pi -4 \sin^{-1}(1/n)$$, or 167° 40' for blue or 165° 18' for red. This corresponds to a distance from the centre of the bow $$r = 4 \sin^{-1}(1/n)-\pi$$, which is ' 12° 20' for blue and 14° 42' for red. It will be seen from figure I.17 that for radii less than $$R$$ (i.e. inside the rainbow) but greater than 12° 20' for blue and 14° 42' for red there are two impact parameters that result in the same deviation, i.e. in the same position inside the bow. The paths of two rays with the same deviation are shown in Figure I.18. One ray is drawn as a full line, the other as a dashed line. They start with different impact parameters, and take different paths through the drop, but finish in the same direction. The drawing is done for a deviation of 145°, or 35° from the bow centre. The two impact parameters are 0.969 and 0.636. When these two rays are recombined by being brought to a focus on the retina of the eye, they have satisfied all the conditions for interference, and the result will be brightness or darkness according as to whether the path difference is an even or an odd number of half wavelengths.
If you look just inside the inner (blue) margin of the bow, you can often clearly see the interference fringes produced by two rays with the same deviation. I haven’t tried, but if you were to look through a filter that transmits just one colour, these fringes (if they are bright enough to see) should be well defined. The optical path difference for a given deviation, or given $$r$$, depends on the radius of the drop (and on its refractive index). For a drop of radius $$a$$ it is easy to see that the optical path difference is
$$2a(\cos\theta_2 - \cos\theta_1) - 4n(\cos\theta'_2-\cos\theta'_1),$$
where $$\theta_1$$ is the larger of the two angles of incidence. Presumably, if you were to measure the fringe spacing, you could determine the size of the drops. Or, if you were to conduct a Fourier analysis of the visibility of the fringes, you could determine, at least in principle, the size distribution of the drops.
Some distance outside the primary rainbow, there is a secondary rainbow, with colours reversed – i.e. red on the inside, blue on the outside. This is formed by two internal reflections inside the drop (Figure I.19). The deviation of the final emergent ray from the direction of the incident ray is $$(\theta − \theta') + (π − 2\theta') + (π − 2\theta') + (\theta − \theta')$$, or $$2π + 2\theta − 6\theta'$$ counterclockwise, which amounts to $$D = 6\theta' − 2\theta$$ clockwise. That is,
$D = 6\sin^{-1}(b/n)-2\sin^{-1}b. \label{eq:1.7.9}$
clockwise, and, as before, this corresponds to an angular distance from the centre of the bow $$r = \pi − D$$. I show in Figure I.20 the angular distance from the centre of the bow versus the impact parameter $$b$$. Notice that $$D$$ goes through a maximum and hence $$r$$ has a minimum value. There is no light scattered outside the primary bow, and no light scattered inside the secondary bow. When the full glory of a primary bow and a secondary bow is observed, it will be seen that the space between the two bows is relatively dark, whereas it is brighter inside the primary bow and outside the secondary bow.
Differentiation shows that the least value of $$r$$, (greatest deviation) corresponding to the radius of the secondary bow is
$R = 6\sin^{-1} \sqrt{\frac{3-n^2}{2n^2}} - 2\sin^{-1}\sqrt{\frac{3-n^2}{2}}\label{eq:1.7.10}$
For $$n = 1.3439$$ (blue) this is 53° 42' and for $$n = 1.3316$$ (red) this is 50° 31'. Thus the red is on the inside of the bow, and blue on the outside.
In principle a tertiary bow is possible, involving three internal reflections. I don’t know if anyone has observed a tertiary bow, but I am told that the primary bow is blue on the inside, the secondary bow is red on the inside, and “therefore” the tertiary bow would be blue on the inside. On the contrary, I assert that the tertiary bow would be red on the inside. Why is this?
Let us return to the primary bow. The deviation is (Equation $$\ref{eq:1.7.1}$$) $$D = π + 2\theta − 4\theta'$$. Let’s take $$n = 4/3$$, which it will be for somewhere in the middle of the spectrum. According to Equation $$\ref{eq:1.7.8}$$, the radius of the bow $$(R = \pi − D_{\text{min}})$$ is then about 42° . That is, $$2\theta' − \theta$$ = 21° . If we combine this with Snell’s law, $$3\sin \theta = 4\sin \theta'$$ , we find that, at minimum deviation (i.e. where the primary bow is), $$\theta$$ = 60°.5 and $$\theta'$$ = 40°.8. Now, at the point of internal reflection, not all of the light is reflected (because $$\theta'$$ is less than the critical angle of 36°.9), and it will be seen that the angle between the reflected and refracted rays is (180 − 60.6 − 40.8) degrees = 78°.6. Those readers who are familiar with Brewster’s law will understand that when the reflected and transmitted rays are at right angles to each other, the reflected ray is completely plane polarized. The angle, as we have seen, is not 90°, but is 78°.6, but this is sufficiently close to the Brewster condition that the reflected light, while not completely plane polarized, is strongly polarized. Thus, as can be verified with a polarizing filter, the rainbow is strongly plane polarized.
I now want to address the question as to how the brightness of the bow varies from centre to circumference. It is brightest where the slope of the deviation versus impact parameter curve is least – i.e. at minimum deviation (for the primary bow) or maximum deviation (for the secondary bow). Indeed the radiance (surface brightness) at a given distance from the centre of the bow is (among other things) inversely proportional to the slope of that curve. The situation is complicated a little in that, for deviations between $$D_{\text{min}}$$ and $$2\pi - 4\sin^{-1}(1/n)$$, (this latter being the deviation for grazing incidence), there are two impact parameters giving rise to the same deviation, but for deviations greater than that (i.e. closer to the centre of the bow) only one impact parameter corresponds to a given deviation.
Let us ask ourselves, for example, how bright is the bow at 35° from the centre (deviation 145°)? The deviation is related to impact parameter by Equation $$\ref{eq:1.7.6}$$. For $$n = 4/3$$, we find that the impact parameters for deviations of 144, 145 and 146 degrees are as follows:
b
144 0.6583 and 0.9623
145 0.6366 and 0.9693
146 0.6157 and 0.9736
Figure I.21 shows a raindrop seen from the direction of the approaching photons.
Any photons with impact parameters within the two dark annuli will be deviated between 144° and 146°, and will ultimately approach the observer at angular distances between 36° and 34° from the centre. The radiance at a distance of 35° from the centre will be proportional, among other things, to the sum of the areas of these two annuli.
I have said “among other things”. Let us now think about other things. I have drawn Figure I.15 as if all of the light is transmitted as it enters the drop, and then all of it is internally reflected within the drop, and finally all of it emerges when it leaves the drop. This is not so, of course. At entrance, at internal reflection and at emergence, some of the light is reflected and some is transmitted. The fractions that are reflected or transmitted depend on the angle of incidence, but, for minimum deviation, about 94% is transmitted on entry to and again at exit from the drop, but only about 6% is internally reflected. Also, after entry, internal reflection and exit, the percentage of polarization of the ray increases. The formulas for the reflection and transmission coefficients (Fresnel’s equations) are somewhat complicated (Equations 1.5.1 and 1.5.2) are for unpolarized incident light), but I have followed them through as a function of impact parameter, and have also taken account of the sizes of the one or two annuli involved for each impact parameter, and I have consequently calculated the variation of surface brightness for one color $$(n = 4/3)$$ from the centre to the circumference of the bow. I omit the details of the calculations, since this chapter was originally planned as an elementary account of reflection and transmission, and we seem to have gone a little beyond that, but I show the results of the calculation in Figure I.22. I have not, however, taken account of the interference phenomena, which can often be clearly seen just within the primary bow. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8661563396453857, "perplexity": 403.1384371392632}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347445880.79/warc/CC-MAIN-20200604161214-20200604191214-00420.warc.gz"} |
https://www.physicsforums.com/threads/distributional-derivative-of-one-parameter-family-of-distributions.758242/ | # Distributional derivative of one-parameter family of distributions
1. Jun 16, 2014
### Only a Mirage
Suppose, for a suitable class of real-valued test functions $T(\mathbb{R}^n)$, that $\{G_x\}$ is a one-parameter family of distributions. That is, $\forall x \in \mathbb{R}^n, G_x: T(\mathbb{R}^n) \to \mathbb{R}$.
Now, suppose $L$ is a linear differential operator. That is, $\forall g \in T(\mathbb{R}^n)$ makes sense in terms of the normal definitions of derivates (assuming, of course, that $g$ is sufficiently smooth). $L$ also has meaning when acting on distributions by interpreting all derivatives as distributional derivatives. For example, the derivative of the distribution $\frac{\partial}{\partial x_i} G_{x_0}$ is defined by: $\forall g \in T(\mathbb{R}^n), \frac{\partial}{\partial x_i} G_{x_0}(g) = G_{x_0}(- \frac{\partial}{\partial x_i}g)$.
Note that smooth functions can multiply distributions to form a new distribution in the following way. Suppose $f:\mathbb{R}^n \to \mathbb{R}$ is a smooth function. Then $f G_x$ is defined by: $\forall g \in T(\mathbb{R}^n), (f G_x) (g) = G_x(f g)$
These facts give $L G_x$ meaning.
Also, for fixed $g \in T(\mathbb{R}^n)$, $(x \mapsto G_x (g))$ is a possibly (non-smooth?) function. Define the function $\psi_g : \mathbb{R}^n \to \mathbb{R}$ by $\psi_g (x) = G_x(g)$
Now, here is my question: When is the following equality true?
$$L (\psi_g) (x_0) = (L G_{x_0}) (g), \forall x_0 \in \mathbb{R}^n$$
2. Jul 10, 2014 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9882903099060059, "perplexity": 195.81957136704216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589455.35/warc/CC-MAIN-20180716193516-20180716213516-00228.warc.gz"} |
http://www.cfd-online.com/W/index.php?title=Discrete_Operator_Splitting&diff=12555&oldid=8852 | # Discrete Operator Splitting
(Difference between revisions)
Revision as of 13:17, 26 January 2008 (view source)← Older edit Latest revision as of 16:25, 28 December 2010 (view source)Ata (Talk | contribs) m (6 intermediate revisions not shown) Line 1: Line 1: - Different from many other pressure-velocity decoupling method, in Discrete Operator Splitting (DOS) method, the fully coupled discrete system is first obtained (with all boundary conditions taken into account). Next, the coupled system is split, sufficiently and necessarily, into two smaller (more importantly, well-natured) subsytems, which are coupled through source terms. Then, these two subsystems are iterated within each time marching step. + Different from many other pressure-velocity decoupling method, in Discrete Operator Splitting (DOS) method, the fully coupled discrete system is first obtained (with all boundary conditions taken into account). Next, the coupled system is split, sufficiently and necessarily, into two smaller (more importantly, well-natured) subsytems, which are coupled through source terms. Then, these two subsystems are iterated within each time marching step. DOS requires no numerical boundary conditions. The method is described as follows: - The method is described as follows. + Discretization of the momentum and continuity equation eventually leads to the following system + :$+ \left[ + \begin{matrix} + A & G\\ + D & 0 + \end{matrix} + \right] + \left\{ + \begin{matrix} + u \\ + p \\ + \end{matrix} + \right\} + = + \left\{ + \begin{matrix} + S_u \\ + S_p \\ + \end{matrix} + \right\} +$ + where $A$ in the momentum equation is the coefficient matrix for + the velocity, $G$ is the coefficient matrix for the pressure, $D$ + in the continuity equation is the coefficient matrix for the + velocity, and $S_u$ and $S_p$ are the right-hand-side known + vectors for the momentum and the continuity equations, + respectively. It is well known that such a system is + ill-conditioned, which causes difficulty to directly or + iteratively solve the whole large system simultaneously. We + split matrix $A$ into the diagonal part ($A^d$) and the + off-diagonal part + :$+ (A-A^d+A^d)u+Gp = S_u +$ + :$+ \Leftrightarrow A^d u+Gp = S_u-(A-A^d)u +$ + :$+ \Leftrightarrow u+A^{-d}Gp = A^{-d}[S_u-(A-A^d)u] +$ + :$+ \Rightarrow + Du+DA^{-d}Gp = DA^{-d}[S_u-(A-A^d)u], +$ + where $A^{-d}$ stands for the inverse of $A^d$. Let the continuity + equation be + incorporated into the above momentum equation and let the original + momentum equation + retained, we obtain the following system + :$+ DA^{-d}Gp = -S_p+DA^{-d}[S_u-(A-A^d)u], +$ + :$+ Au = S_u-Gp +$ + For convenience, we define + :$+ D^{*} \equiv D A^{-d}, +$ + so that the discrete forms of the momentum equation and the + continuity equation become + :$+ D^{*}Gp = -(S_p-D^*S_u)-D^*(Au-A^d u) +$ + :$+ Au = S_u-Gp +$ + Furthermore, let's define + :$+ L \equiv D^*G +$ + :$+ S^*_p \equiv S_p-D^*S_u, +$ + finally we have two well-posed subsystems + :$+ Lp = -S^*_p-D^*(Au-A^d u)\equiv b_p(u) +$ + :$+ Au = S_u-Gp\equiv b_u(p) +$ + So far it has been proved that the original indefinite system implies the two + definite subsystems. The reverse MUST be proved, so that they are equivalent to each other. The reverse can be found in the external link of this article. - + In CCPPE (Consistent Continuous Pressure Poisson Equation), the continuous pressure poisson equation is first derived and assigned with numerical boundary condtions (not physical boundary conditions), and the resulting system is involved with linear iteration only (suppose the nonlinear term is already linearized in time). In contrast, the DOS involves source-term iteration and linear iteration. However, while in actual practice a strict convergence criterion is imposed on the source-term iteration, only few prescibed number of iterations are imposed on the linear iteration inside the source-term iteration. Actual practice shows that when the solution reaches the level that the source-term iteration is nearly complete, the same solution also reaches the level that the criterion for the linear iteration is easily met. Hence, apart from offering simplicity and no requirement of numerical boundary conditions, DOS is efficient. - In CCPPE (Consistent Continuous Pressure Poisson Equation), the continuous pressure poisson equation is first derived and assigned with numerical boundary condtions (not physical boundary conditions), and the resulting system is involved with linear iteration only (suppose the nonlinear term is already linearized in time). In contrast, the DOS involves source-term iteration and linear iteration. However, while in actual practice a strict convergence criterion is imposed on the source-term iteration, only few prescibed number of iterations are imposed on the linear iteration inside the source-term iteration. And actual practice shows that when the solution reaches the level that the source-term iteration is nearly complete, the same solution also reaches the level that the criterion for the linear iteration is easily met. Hence, DOS is efficient, apart from offering simplicity and no requirement of numerical boundary conditions. +
## Latest revision as of 16:25, 28 December 2010
Different from many other pressure-velocity decoupling method, in Discrete Operator Splitting (DOS) method, the fully coupled discrete system is first obtained (with all boundary conditions taken into account). Next, the coupled system is split, sufficiently and necessarily, into two smaller (more importantly, well-natured) subsytems, which are coupled through source terms. Then, these two subsystems are iterated within each time marching step. DOS requires no numerical boundary conditions. The method is described as follows:
Discretization of the momentum and continuity equation eventually leads to the following system
$\left[ \begin{matrix} A & G\\ D & 0 \end{matrix} \right] \left\{ \begin{matrix} u \\ p \\ \end{matrix} \right\} = \left\{ \begin{matrix} S_u \\ S_p \\ \end{matrix} \right\}$
where $A$ in the momentum equation is the coefficient matrix for the velocity, $G$ is the coefficient matrix for the pressure, $D$ in the continuity equation is the coefficient matrix for the velocity, and $S_u$ and $S_p$ are the right-hand-side known vectors for the momentum and the continuity equations, respectively. It is well known that such a system is ill-conditioned, which causes difficulty to directly or iteratively solve the whole large system simultaneously. We split matrix $A$ into the diagonal part ($A^d$) and the off-diagonal part
$(A-A^d+A^d)u+Gp = S_u$
$\Leftrightarrow A^d u+Gp = S_u-(A-A^d)u$
$\Leftrightarrow u+A^{-d}Gp = A^{-d}[S_u-(A-A^d)u]$
$\Rightarrow Du+DA^{-d}Gp = DA^{-d}[S_u-(A-A^d)u],$
where $A^{-d}$ stands for the inverse of $A^d$. Let the continuity equation be incorporated into the above momentum equation and let the original momentum equation retained, we obtain the following system
$DA^{-d}Gp = -S_p+DA^{-d}[S_u-(A-A^d)u],$
$Au = S_u-Gp$
For convenience, we define
$D^{*} \equiv D A^{-d},$
so that the discrete forms of the momentum equation and the continuity equation become
$D^{*}Gp = -(S_p-D^*S_u)-D^*(Au-A^d u)$
$Au = S_u-Gp$
Furthermore, let's define
$L \equiv D^*G$
$S^*_p \equiv S_p-D^*S_u,$
finally we have two well-posed subsystems
$Lp = -S^*_p-D^*(Au-A^d u)\equiv b_p(u)$
$Au = S_u-Gp\equiv b_u(p)$
So far it has been proved that the original indefinite system implies the two definite subsystems. The reverse MUST be proved, so that they are equivalent to each other. The reverse can be found in the external link of this article.
In CCPPE (Consistent Continuous Pressure Poisson Equation), the continuous pressure poisson equation is first derived and assigned with numerical boundary condtions (not physical boundary conditions), and the resulting system is involved with linear iteration only (suppose the nonlinear term is already linearized in time). In contrast, the DOS involves source-term iteration and linear iteration. However, while in actual practice a strict convergence criterion is imposed on the source-term iteration, only few prescibed number of iterations are imposed on the linear iteration inside the source-term iteration. Actual practice shows that when the solution reaches the level that the source-term iteration is nearly complete, the same solution also reaches the level that the criterion for the linear iteration is easily met. Hence, apart from offering simplicity and no requirement of numerical boundary conditions, DOS is efficient. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 23, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9652265310287476, "perplexity": 819.9581705754233}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042987866.61/warc/CC-MAIN-20150728002307-00071-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://preprint.impa.br/visualizar?id=1722 | Preprint A373/2005
Metric stability for random walks (with applications in renormalization theory)
Daniel Smania | Moreira, Carlos G.
Keywords: random walk | Feigenbaum | universality | wild attractor | skew product | renormalization
Consider deterministic random walks F:I x Z-> I x Z, defined by F(x,n)=(f(x),k(x)+n), where f is an expanding Markov map on the interval I and k: I -> Z. We study the universality (stability) of ergodic (for instance, recurrence and transience), geometric and multifractal properties in the class of perturbations of the type G(x,n)=(f_n(x),p(x,n)+n) which are topologically conjugate with F and f_n are expanding maps exponentially close to f when |n| goes to infinity. We give applications of these results in the study of the regularity of conjugacies between (generalized) infinitely renormalizable maps of the interval and the existence of wild attractors for one-dimensional maps. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8505454063415527, "perplexity": 2070.107487725087}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057227.73/warc/CC-MAIN-20210921191451-20210921221451-00565.warc.gz"} |
https://sites.google.com/a/wagsymposium.org/current/fall-2017/abstracts | UCLA Fall 2017 >
Abstracts
Benjamin Bakker Transcendence of period maps Period domains $D$ can be described as certain analytic open sets of flag varieties; due to the presence of monodromy, however, the period map of a family of algebraic varieties lands in a quotient $D/\Gamma$ by an arithmetic group. In the very special case when $D/\Gamma$ is itself algebraic, understanding the interaction between algebraic structures on the source and target of the uniformization $D\rightarrow D/\Gamma$ is a crucial component of the modern approach to the Andr\'e-Oort conjecture. We prove a version of the Ax-Schanuel conjecture for general period maps $X\rightarrow D/\Gamma$ which says that atypical algebraic relations between $X$ and $D$ are governed by Hodge loci. We will also discuss some geometric and arithmetic applications. This is joint work with J. Tsimerman. Bhargav Bhatt Chern classes of crystals A vector bundle on a complex manifold has Chern classes in de Rham cohomology. Chern-Weil theory shows that these classes vanish if the bundle admits a flat connection. The analog of a bundle with flat connection in positive characteristic algebraic geometry is an isocrystal in the sense of Grothendieck. In my talk, I will recall what these objects are, and then explain why their Chern classes vanish. The key tool is algebraic K-theory. This is a report on joint work with Jacob Lurie. Wei Ho Revisiting the Hessian The Hessian of a plane cubic curve is classically described using partial derivatives or polars. In this talk, we explore another construction of the cubic Hessian (and variants), and relate these constructions to maps between certain moduli spaces of genus one curves with extra structure. We will explain how input from number theory and dynamics also helps to understand such classical objects. Katrina Honigs Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic There are many results characterizing when derived categories of two complex surfaces are equivalent, including theorems of Bridgeland and Maciocia showing that derived equivalent Enriques or bielliptic surfaces must be isomorphic and a theorem of Sosna that a K3 canonically covering an Enriques is not derived equivalent to any varieties other than itself, and that an abelian surface canonically covering a biellptic surface is derived equivalent only to itself and its dual. The proofs of these theorems strongly use Torelli theorems and lattice-theoretic methods which are not available in positive characteristic. In this talk I will discuss how to prove these results over algebraically closed fields of positive characteristic (excluding some low characteristic cases). This work is joint with M. Lieblich, L. Lombardi and S. Tirabassi. Michael Kemeny The Prym-Green conjecture for curves of odd genus A paracanonical curve is a curve together with a torsion bundle of fixed level. The moduli space of such objects is very interesting from several points of view, such as its relationship to Abelian varieties. I will give a complete description of the Betti numbers associated to the homogeneous coordinate ring of such a curve, provided the genus is odd. This is joint work with Gavril Farkas. Davesh Maulik Gopakumar-Vafa invariants via vanishing cycles Given a Calabi-Yau threefold X, one can count curves on X using various approaches, for example using stable maps or ideal sheaves; for any curve class on X, this produces an infinite sequence of invariants, indexed by extra discrete data (e.g. by the domain genus of a stable map). Conjecturally, however, this sequence is determined by only a finite number of integer invariants, known as Gopakumar-Vafa invariants. In this talk, I will propose a direct definition of these invariants via sheaves of vanishing cycles, building on earlier approaches of Kiem-Li and Hosono-Saito-Takahashi. Conjecturally, these should agree with the invariants as defined by stable maps. I will also explain how to prove the conjectural correspondence in various cases. This is joint work with Yukinobu Toda. Dulip Piyaratne Stability manifold under Fourier-Mukai transforms on an abelian threefold The notion of Fourier-Mukai transform for abelian varieties was introduced by Mukai in early 1980s. Since then Fourier-Mukai theory turned out to be extremely successful in studying stable sheaves and complexes of them, and also their moduli spaces. I will discuss how the Fourier-Mukai techniques are useful to construct Bridgeland stability conditions on any abelian threefold, and also to study the corresponding stability manifold. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8149734735488892, "perplexity": 410.45761643043977}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948517845.16/warc/CC-MAIN-20171212173259-20171212193259-00063.warc.gz"} |
http://mathhelpforum.com/differential-geometry/115863-separability.html | # Math Help - separability
1. ## separability
I was wondering how to prove a given space is separable or not.
take L^2 and L^\infty for example please !
Any idea or help is appreciated !
2. For $L^2$ you could use the fact that it's a Hilbert space, and that a Hilbert space is separable iff it has a countable orthonormal basis. $L^{\infty }$ is not so easy as the continous functions are not dense! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9124932289123535, "perplexity": 304.65311931580055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00121-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/71110/finite-dimensional-vector-spaces-over-a-complete-but-not-necessarily-valued-fiel | # Finite dimensional vector spaces over a complete but not-necessarily-valued field
I'm essentially reopening this old question of Ricky Demer which was never fully answered.
Essentially the original question: Suppose we have a topological field $F$ which is complete, Hausdorff, and non-discrete, and we put a Hausdorff topology on $F^n$ so as to make it a topological vector space over $F$; is is this topology necessarily the product topology (and hence complete, and hence closed in anything it embeds in)?
As the link shows, the answer is yes if $\tau$ comes from an absolute value on $F$, and it's easy to see the same argument works if it comes from a field ordering on $F$.
Note that the argument there shows that this question reduces to the following lemma:
Suppose we have a topological field $(F,\tau)$ which is Hausdorff and non-discrete, and we give $F$ a second topology $\tau'$, which is also Hausdorff, such that $(F,\tau')$ is a topological vector space over $(F,\tau)$. Does this force $\tau=\tau'$? What if we assume that $(F,\tau)$ is complete?
(I'm isolating completeness as a separate, possibly-unnecessary condition because the argument there only uses completeness in the reduction to the lemma, not in proving the lemma for valued fields.)
Related to this question in that one way to come up with a counterexample for both simultaneously would be to find a field with two (nondiscrete, Hausdorff) topologies with one strictly finer than the other.
• +1 $\hspace{.05 in}$ :-) $\hspace{.1 in}$
– user5810
Jul 24, 2011 at 8:46
Yes, it is "standard" that a finite-dimensional vector space over a complete, non-discrete, division algebra (!) has a unique topology compatible with a topological vector space structure (=Hausdorff, addition is continuous, scalar multiplication is continuous). This is probably proven (at least for real or complex scalars, but the proofs should generalize) in any source on general TVS's, e.g., section 5, my notes. Without completeness, or without discreteness, there are easy counter-examples.
• The proof there seems to be assuming that the topology comes from an absolute value. Again, I already know it's true in that case, the question is about general topological fields... Jul 24, 2011 at 20:54
So, there is a property of topological division rings that is equivalent to finite dimensional tvs's have a unique topology, and the property is implied by normability, but I can't find any examples that are not normable.
From Nachbin, "On Strictly minimal topological division rings":
Theorem 7. Let K be a given topological division ring. In order that every finite-dimensional vector space over K should have only one admissible topology it is necessary and sufficient that K be strictly minimal and complete.
"Admissible" here means "vector space operations are continuous". As for "strictly minimal":
A topological division ring K is said to be minimal if its topology $T_K$ is a minimal element in the ordered set of all admissible topologies on K; that is, if there exists no admissible topology X on K such that $T\lt T_K$. The topological division ring is said to be strictly minimal if there exists no topology T on K admissible with respect to $T_K$ such that $T\lt T_K$; it amounts to the same to say that the only topology on K admissible with respect to $T_K$ is $T_K$ itself.
In particular, "strictly minimal" is equivalent to the $n=1$ case of the question (i.e. completeness is not necessary, which was known already in the normed case: see Proposition 2 of Section 2.2 of Chapter 1 of Bourbaki's TVS).
In Warner's "Topological Fields", "strictly minimal" is replaced with all tvs $E$ over $K$ being "straight" (p. 224), which means that for every nonzero $a\in E$, the map $\lambda\rightarrow \lambda a$ is a homeomorphism from K onto the one-dimensional subspace generated by a. Warner mentions that no examples of straight division rings are known other than "locally retrobounded" division rings, which is more general than normable (see p. 158), which implies to me that there are known examples of non-normed locally retrobounded (and thus, straight) division rings. But, I didn't see any actual examples in the book (though I only have access to the Google books version).
Nachbin apparently has a book, "Espacos vetoriais topologicos", published in 1948, where the notion of strictly minimal is introduced. Maybe there'd be an example in there...?
• OK, this is helpful. But what I'm looking for seems to be easier than this -- If I understand this correctly, I'm not looking for a field which is strictly minimal (equivalently straight) but not normable, I'm just looking for a field which isn't strictly minimal (equivalently straight). Perhaps I should get a copy of that book, maybe it'll be in one of the exercises missing from Google Books. (Actually for a field which is strictly minimal but not normable, any field which is orderable but not normable should work, IINM.) Jul 25, 2011 at 2:20
• I suppose really I want complete but not straight, rather. Jul 25, 2011 at 2:27
• Oh yeah, I forgot to add that I couldn't find any examples that were not straight/minimally strict, either!
– B R
Jul 25, 2011 at 2:32
• Yes, SFAICT neither Warner nor Więsław (whose chapter on the subject seems to be just a copy of Nachbin) make any mention of whether or not there might exist non-discrete, non-straight fields. Annoying... Jul 25, 2011 at 3:36
For completeness, let's do one of paul's easy counterexamples. The field $\mathbb Q$ of rationals with the usual topology is not complete. (But comes from an absolute value and from a field ordering.) The two-dimensional vector space $\{a+b\sqrt{2}:a,b\in\mathbb Q\}$ with its usual topology (as a subset of the reals) is not the product topology.
• I think you need to read the question more carefully. I isolated completeness as possibly not necessary for the 1-dimensional case; I'm aware that it's necessary for higher dimensions... Jul 24, 2011 at 20:48
Well, now I feel silly -- on looking through Wieslaw again, I see he does give examples of non-discrete, non-straight fields, just not in that section. For instance, take two absolute values on the rationals; the topology they generate together still make the rationals a topological field, is not discrete, and is obviously finer than either of the ones you started with. Since it isn't even minimal, it can't be straight.
I'm not going to accept my own answer on this since this example presumably isn't complete, and I'd like a complete example. Which might well be in here too, if I keep looking... (Is the completion of this again a field? If so that should work, but I'd need to check if that's true.)
Edit: Nope, the completion of this isn't a field, so I'm still lacking for a complete example.
Edit again: OK, I'm now pretty sure Wieslaw gives one, so the answer is no. Wieslaw gives an example of a complete normed field which is not given by any absolute value (here "normed" means instead of |ab|=|a||b|, we only require |ab|≤|a||b| and |-a|=|a|). Furthermore, he shows given a power-multiplicative norm, you can find an absolute value that generates a coarser topology (so if a power-multiplicative norm wasn't equivalent to an absolute value, it isn't minimal). (Here power-multiplicative only means we require |an|=|an| for positive n, not for negative n.) And after a bit of staring at his example, I'm pretty sure it's power-multiplicative. So unless anyone can show that I've missed something, I'm going to consider this one closed. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9095253348350525, "perplexity": 440.77141502312656}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573630.12/warc/CC-MAIN-20220819070211-20220819100211-00607.warc.gz"} |
https://math.stackexchange.com/questions/1487293/otimes-categorical-generalization-of-lagrange | # $\otimes$-Categorical Generalization of Lagrange
I am reading M. Brandenburg's paper and came across the following result which is a generalization of Lagrange's theory in group theory:
Let $\mathcal C$ be a $\otimes$-category and $A\to B$ a morphism of algebras (in $\mathcal C$). Then if $M$ is an $A$-module isomorphic to $T\otimes B$ as a $B$-module and $B_{|A}\cong S\otimes A$ as an $A$-module, then $M_{|A}\cong (T\otimes S)\otimes A$.
The proof is a a simple one-liner. My question is how do we obtain Lagrange's theorem from it? At first, I thought we take $\mathcal C$ to be the cartesian closed category $\mathbf{Set}$ and let $A\to B$ be the inclusion of a subgroup in a finite group. But we are assuming $B_{|A}\cong S\times A$ and so there would be nothing to show since that would imply the order of $A$ divided the order of $B$. I know this should be easy, any help?
• Are you sure the algebras are commutative? I don't think they necessarily are (otherwise you would only get abelian groups). – Najib Idrissi Oct 19 '15 at 11:45
• My impression is that'll take @MartinBrandenburg himself to answer what he meant by that remark. :) – Omar Antolín-Camarena Oct 19 '15 at 21:35
• @NajibIdrissi Yes, you are correct. – Rachmaninoff Oct 20 '15 at 0:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9661137461662292, "perplexity": 189.97937716921857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000231.40/warc/CC-MAIN-20190626073946-20190626095946-00325.warc.gz"} |
https://cypenv.info/relationship-between-and/relationship-between-current-and-potential-difference-unit.php | # Relationship between current and potential difference unit
### GCSE Science/Current, voltage resistance and Ohm's law - Wikibooks, open books for an open world
Electronics Tutorial about the Relationship between Voltage Current and The Potential difference between two points is measured in Volts with the . The standard unit of measurement given for conductance is the Siemen, symbol (S). Current Electricity - Lesson 1 - Electric Potential Difference Because electric potential difference is expressed in units of volts, it is sometimes referred to as the .. of the mathematical relationship between work, potential energy, charge and . Charge, Current & Potential Difference in circuits. Conventional current flows around a circuit from the positive (+) side of the cell to the negative (-). Potential difference is the work done per unit charge. Resistance (W) – is the ratio of potential difference across a component to the current flowing through it, it is.
Electrons are flowing past you. One Ampere is a flow of one coulomb going past every second. The ampere is defined in terms of the force produced between two wires each carrying identical currents: Voltage[ edit ] Having looked at charge and current, we now need to look at what voltage means.
As you know, electrons repel each other. If you hold one electron near another electron, you have to push against it to hold it in place. If you try to bring it even closer, you have to do work force times distance to get it to that new position.
The voltage between one point and another is simply how much work per coulomb is required to move any small test charge from point A to point B. In most electronic components, it doesn't matter much which path the test charge takes in-between point A and point B. Voltage is related to the energy of the charges.
Let's go back to the peas in the petri dish. They can be pushed slowly or they can be pushed quickly. The faster they go, the more energy they have. It's a similar situation for the electrons, except the push isn't provided by a finger! It's provided by the battery. The battery gives the charges energy. This energy is given up to the various components in the circuit, e. The energy per unit charge is called the voltage or the potential difference.
Definition of the volt[ edit ] One volt means one joule of energy per coulomb of charge. More accurately it has 2 definitions: Electromotive Force is the amount of energy converted from non-electrical to electrical form when driving 1 coulomb of charge around a completed and closed circuit. Potential Difference is the amount of energy converted from electrical to non-electrical form when driving 1 coulomb of charge around a completed and closed circuit.
The potential difference between 2 points in a conductor is defined as 1 volt, if 1 Joule of energy is converted from electrical form to non-electrical form, when 1 coulomb of charge per second 1 amp flows through it. This will only occur between 2 points in a conductor, that has a resistance, defined as 1 ohm. It's just how difficult it is for the charges to flow through an electrical component or from one point to another in an electrical circuit.
Imagine a group of walkers travelling down a road.
They approach a fork in the road. To the left is a flat straight road leading to a nearby town. To the right is a huge mountain, over which a steep and winding road traverses.
This road also leads to the nearby town.
Naturally all the walkers chose the left route. Let's now suppose that there are millions of walkers. They are jam packed on the road, and they are all in a hurry to get to the town as quickly as possible.
Now when they come to the fork in the road which way should they go? Most will still go to the left, but a few might chose to go to the right, the road is more difficult but there is no traffic jam, so they might get there quicker.
It's a similar thing with moving charges. Like charges repel, so they would rather not pack in very closely together. Some routes, like wires have very low resistance, while other routes like bulbs have much higher resistance. More charges will go down a low resistance route than a high resistance one. Ohm's Law[ edit ] This law relates resistance, current, and voltage.
It's very easy to remember because it's obvious when you think about it. Let's think of a wire carrying current from a battery, to a bulb then back to the battery. The voltage of the battery provides the energy of the flowing electrons. Let's assume we want to increase the rate of flow of charge. Remember that current is the rate of flow of charge so we want to increase the current.
In the previous part of Lesson 1, the concept of electric potential was applied to a simple battery-powered electric circuit. In that discussionit was explained that work must be done on a positive test charge to move it through the cells from the negative terminal to the positive terminal.
This work would increase the potential energy of the charge and thus increase its electric potential. As the positive test charge moves through the external circuit from the positive terminal to the negative terminal, it decreases its electric potential energy and thus is at low potential by the time it returns to the negative terminal.
If a 12 volt battery is used in the circuit, then every coulomb of charge is gaining 12 joules of potential energy as it moves through the battery. And similarly, every coulomb of charge loses 12 joules of electric potential energy as it passes through the external circuit. The loss of this electric potential energy in the external circuit results in a gain in light energy, thermal energy and other forms of non-electrical energy.
With a clear understanding of electric potential difference, the role of an electrochemical cell or collection of cells i.
### BBC - GCSE Bitesize: Current and potential difference
The cells simply supply the energy to do work upon the charge to move it from the negative terminal to the positive terminal. By providing energy to the charge, the cell is capable of maintaining an electric potential difference across the two ends of the external circuit.
Once the charge has reached the high potential terminal, it will naturally flow through the wires to the low potential terminal. The movement of charge through an electric circuit is analogous to the movement of water at a water park or the movement of roller coaster cars at an amusement park.
In each analogy, work must be done on the water or the roller coaster cars to move it from a location of low gravitational potential to a location of high gravitational potential.
Once the water or the roller coaster cars reach high gravitational potential, they naturally move downward back to the low potential location. For a water ride or a roller coaster ride, the task of lifting the water or coaster cars to high potential requires energy. The energy is supplied by a motor-driven water pump or a motor-driven chain.
In a battery-powered electric circuit, the cells serve the role of the charge pump to supply energy to the charge to lift it from the low potential position through the cell to the high potential position.
It is often convenient to speak of an electric circuit such as the simple circuit discussed here as having two parts - an internal circuit and an external circuit. The internal circuit is the part of the circuit where energy is being supplied to the charge. For the simple battery-powered circuit that we have been referring to, the portion of the circuit containing the electrochemical cells is the internal circuit.
## GCSE Science/Current, voltage resistance and Ohm's law
The external circuit is the part of the circuit where charge is moving outside the cells through the wires on its path from the high potential terminal to the low potential terminal. The movement of charge through the internal circuit requires energy since it is an uphill movement in a direction that is against the electric field.
The movement of charge through the external circuit is natural since it is a movement in the direction of the electric field.
When at the positive terminal of an electrochemical cell, a positive test charge is at a high electric pressure in the same manner that water at a water park is at a high water pressure after being pumped to the top of a water slide. Being under high electric pressure, a positive test charge spontaneously and naturally moves through the external circuit to the low pressure, low potential location. As a positive test charge moves through the external circuit, it encounters a variety of types of circuit elements.
• Electric Potential Difference
• Charge, Current & Potential Difference
Each circuit element serves as an energy-transforming device. Light bulbs, motors, and heating elements such as in toasters and hair dryers are examples of energy-transforming devices.
In each of these devices, the electrical potential energy of the charge is transformed into other useful and non-useful forms. For instance, in a light bulb, the electric potential energy of the charge is transformed into light energy a useful form and thermal energy a non-useful form.
## Current and potential difference
The moving charge is doing work upon the light bulb to produce two different forms of energy. By doing so, the moving charge is losing its electric potential energy. Upon leaving the circuit element, the charge is less energized. The location just prior to entering the light bulb or any circuit element is a high electric potential location; and the location just after leaving the light bulb or any circuit element is a low electric potential location. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8116980791091919, "perplexity": 361.89403956473325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986705411.60/warc/CC-MAIN-20191020081806-20191020105306-00391.warc.gz"} |
https://phys.huji.ac.il/bsc | B.Sc
The B.Sc. is a 3-year degree (sometimes extended to 4 years) that provides a comprehensive introduction to classical and quantum physics. During the degree, students acquire the skills needed to undertake careers outside academia, or to continue to advanced degrees in physics.
For more details visit here.
Program Structure
All degree tracks require a combination of physics and other courses, and differ primarily in the balance between these components. The mandatory physics component in all tracks is roughly the same, and tracks differ mainly in the number of elective courses students must take:
• First year. Lays out the foundations and consists mostly of mandatory courses: Mechanics, Electrodynamics, and applied mathematics courses. Strong students may take additional elective courses in this year, but many prefer to focus on the required ones.
• Second year. Ventures beyond basic physics. First introduction to Waves, Optics, and Quantum Mechanics. Typically students take a combination of mandatory and elective courses on advanced topics.
• Third year. Usually devoted to electives on advanced topics and completion of other requirements.
Program Variants
Several options exist for studying physics as part of a B.Sc.
• Standard track and Extended track in physics. These degrees are for students whose main focus of study is physics. The extended option differs from the standard one by requiring that a higher percentage of course credits come from physics courses. In the standard option students must take a minor sequence ("Hativa") from another department.
• Joint degree with another department. This option is for students who wish to combine a degree in physics with a degree from another department. Many departments offer such an option, among them mathematics, computer science, cognitive science, and environmental studies.
• Physics as a minor sequence ("Hativa BePhysika"). Students studying for a degree in another department may take this shorter course sequence, which gives a comprehensive introduction to university-level physics. Note that this option does not lead to a degree in physics, and does not provide a path to a graduate degree in physics. We advice to start the minor in the second year of studies.
Further details regarding the different programs can be found here. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8213725090026855, "perplexity": 1269.2585576019203}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989616.38/warc/CC-MAIN-20210513234920-20210514024920-00316.warc.gz"} |
http://mathhelpforum.com/calculus/86241-polar-coordinates.html | 1. ## Polar coordinates.
Let C be the curve in the plane defined by $r=\sin \theta, 0 \leq \theta \leq 2 \pi$.
a). Draw a sketch of C.
My sketch looks like a figure of 8.
b). Find the co-ordinates of the four points at which the tangent to C is vertical.
Hint: The formula $\frac{ dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d \theta}}$ may be helpful.
I know that:
$x=r \cos \theta \Rightarrow \ x=\sin^2 \theta \cos \theta$
$y=r \sin \theta \Rightarrow \ y=\sin^3 \theta$
Therefore:
$\frac{dx}{d \theta}=2 \sin \theta \ cos^2 \theta -\sin^3 \theta$
$\frac{dy}{d \theta}=3 \sin^2 \theta$
Hence:
$\frac{dy}{dx}=\frac{ 3 \sin^2 \theta}{ 2 \sin \theta \cos^2 \theta- \sin^3 \theta}=\frac{3 \sin \theta}{2 \cos^2 \theta-\sin^2 \theta}$
I want when $\frac{dy}{dx}= \infty$. This will be when the denominator of my expression is equal to 0.
$2 \cos^2 \theta=1- \cos^2 \theta$
$\cos^2 \theta+2 \cos^2 \theta-1=0$
$3 \cos ^2 \theta=1 \Rightarrow \ \boxed{\cos \theta= \pm \frac{1}{\sqrt{3}}}$
Since I have an expression for $\cos \theta$, it is useful to write x and y in terms of $\cos \theta$.
$x=r \cos \theta \Rightarrow \ x=\sin^2 \theta \cos \theta \Rightarrow \ x=(1-\cos^2 \theta) \cos \theta$
$y=r \sin \theta \Rightarrow \ y=\sin^3 \theta=(1-\cos^2 \theta)\sqrt{1-\cos^2 \theta}$
Therefore when $\cos \theta=\frac{1}{\sqrt{3}}$ we have:
$x=\left( 1- \frac{1}{3} \right) \frac{1}{\sqrt{3}}=\left( \frac{2}{3} \right) \frac{1}{\sqrt{3}}=\frac{2}{3 \sqrt{3}}$
$y=\left( 1-\frac{1}{3} \right) \sqrt{1-\frac{1}{3}}=\frac{2}{3}\sqrt{\frac{2}{3}}=\pm \left( \frac{2}{3} \right)^{\frac{3}{2}}$
The first two points are $\left( \frac{2}{3 \sqrt{3}}, \left( \frac{2}{3} \right)^{\frac{3}{2}} \right)$ and $\left( \frac{2}{3 \sqrt{3}}, -\left( \frac{2}{3} \right)^{\frac{3}{2}} \right)$.
It's also apparent that the next x coordinate is $-\frac{2}{3 \sqrt{3}}$ with the same y coordinates as before.
The other two points are $\left( -\frac{2}{3 \sqrt{3}}, \left( \frac{2}{3} \right)^{\frac{3}{2}} \right)$ and $\left( -\frac{2}{3 \sqrt{3}}, -\left( \frac{2}{3} \right)^{\frac{3}{2}} \right)$.
The four points are (listed together for clarity) $\left( -\frac{2}{3 \sqrt{3}}, \left( \frac{2}{3} \right)^{\frac{3}{2}} \right)$, $\left( -\frac{2}{3 \sqrt{3}}, -\left( \frac{2}{3} \right)^{\frac{3}{2}} \right)$, $\left( \frac{2}{3 \sqrt{3}}, \left( \frac{2}{3} \right)^{\frac{3}{2}} \right)$ and $\left( \frac{2}{3 \sqrt{3}}, -\left( \frac{2}{3} \right)^{\frac{3}{2}} \right)$.
I'm pretty sure this is right. I would just like someone to confirm my answer.
2. Originally Posted by Showcase_22
$y=r \sin \theta \Rightarrow \ y=\sin^3 \theta$
Therefore:
$\frac{dy}{d \theta}=3 \sin^2 \theta$
Unfortunately not
Lucky you are : it does not change anything and your results are correct !
3. I mean $\frac{dy}{d \theta}=3\sin^2 \theta cos \theta!$
4. Your results are correct anyway !
5. yay!
MATH WIN!
Thankyou kindly running gag. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 30, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.846055269241333, "perplexity": 389.9159429353142}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805242.68/warc/CC-MAIN-20171119004302-20171119024302-00121.warc.gz"} |
https://www.physicsforums.com/threads/questions-re-absorption-spectrum.382720/ | # Questions re: absorption spectrum
1. Mar 1, 2010
### Mike_UK
I'm confused about some aspects of absorption spectroscopy, and hoping someone can de-confuse me
First, just a preliminary question; am I right in thinking that when an electron absorbs a photon, the electron will then emit a photon of the same frequency and energy as the one it has just absorbed?
If the answer to the above is yes, then why is it that a gas (such as hydrogen) can cause gaps (black lines) in the spectrum of an incident beam of white light? I understand that electrons within the atoms of the gas will absorb some wavelengths of the white light, but if the electron then emits a photon of the same frequency and energy, then all of the frequencies should come out of the gas intact, right?
Is it the case that the emitted photon is sent off in a different direction, so that it doesn't end up in the spectroscope? Is that why we see the black lines in the absorption spectrum?
2. Mar 1, 2010
### Stonebridge
Yes you've got the answer there.
The re-emitted photons come out in all (random) directions rather than all in the original direction. (Towards you) The result is the dark line. It's not totally "black" as there are some photons emitted in your direction. Just very few. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8064292073249817, "perplexity": 344.5334915058915}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864795.68/warc/CC-MAIN-20180622201448-20180622221448-00094.warc.gz"} |
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