url stringlengths 14 2.42k | text stringlengths 100 1.02M | date stringlengths 19 19 | metadata stringlengths 1.06k 1.1k |
|---|---|---|---|
https://physics.stackexchange.com/questions/158792/why-are-charging-times-so-long-for-lithium-ion-batteries | # Why are charging times so long for Lithium-ion batteries?
Why do rechargeable batteries (e.g., mobile phone batteries) need sometimes several hours to be fully charged? In other words, what are the physical constraints that don't allow me to charge my iPhone battery in 1 min, for instance?
• I'll leave a proper answer to someone who knows about batteries, but I will put a bet on the answer having to do with heat generation in the charging process. – Asher Jan 11 '15 at 16:57
• I highly doubt that mains wiring is unable to supply $\sim500\mathrm{W}$ of power (my electric teapot takes $2200\mathrm W$). You forget that mains gives you $110\mathrm V$ or $220\mathrm V$ but your charger supplies about $4\mathrm V$ to the battery, and $120\mathrm A$ is the current from charger to battery, not from mains to charger. – Ruslan Jan 11 '15 at 19:37 | 2019-09-24 08:49:02 | {"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": 0.4158148765563965, "perplexity": 1133.4931555110168}, "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-2019-39/segments/1568514572896.15/warc/CC-MAIN-20190924083200-20190924105200-00073.warc.gz"} |
https://verkorkst-podcast.de/symbolic-logic-calculator.html | symbolic logic calculator. 27{34 of the text, which has a number of examples illustrating the use of logical statements, and also o ers some excellent general advice. Perform algebraic manipulations on symbolic expressions. You can enter predicates and expressions in the upper textfield (using B syntax). In symbolic logic, one introduces symbols like p and q to stand for statements (or "propositions") like "this is an interesting essay". Free Online 𝕋𝕣𝕦𝕥𝕙 𝕋𝕒𝕓𝕝𝕖 𝔾𝕖𝕟𝕖𝕣𝕒𝕥𝕠𝕣 - different types of logic formulas with Symbols Examples: p ∧ q,p → q,Q→T,A∨B. Now that we have learned about negation, conjunction, disjunction and the conditional, we can include the logical connector for each of these statements in more elaborate statements. Table of Logical Equivalences. The step by step breakdown of every intermediate proposition sets this generator apart from others. Another symbolic statement into symbols to calculate the calculator: ratio word problems with for order to predicate logic used that predicate logic. Symbolic logic is by far the simplest kind of logic—it is a great time-saver in argumentation. Symbolic logic calculator This simple calculator, the courtesy of A. Sympy : Symbolic Mathematics in Python ¶. A predicate with variables can be made a proposition by either assigning a value. It was designed with these criteria in mind: The language should only use basic SCII characters which are available on a standard English keyboard, so typing expressions in the language is easy. ⓘ Hint: This calculator supports symbolic math. Besides classical propositional logic and first-order predicate logic (with functions and identity), a few normal modal logics are supported. By using this website, you agree to our Cookie Policy. Tips for translations involving conjunctions, disjunctions and negations (1) Identify the primary connective of the. Chapter 3 Symbolic Logic and Proofs. Transcribing English sentences into wffs is sometimes a non-trivial task. The perfect math app with full-featured algebra capabilites on par with high-end scientific calculators, ideal for high-school or college students, engineers or anyone needing a powerful math toolbox. The app is effectively a propositional logic calculator. This simple calculator, the courtesy of A. You can also use T or true to specify true and false values. Supports all basic logic operators: negation (complement), and. Find Online Tutors in Subjects related to Logic. In this course we are concerned with the transcription using given predicate symbols and the universe. The truth table solver generates all combinations of true and false statements and. The Logic Calculator is an application useful to perform logical operations. We can test for consistency and inconsistency based on propositional logical form (PL consistent/inconsistent) using truth-tables. You may add any letters with your keyboard . There is a legend to show you computer friendly ways to type each of the symbols that are normally . A nonnegative integer n followed by a single capital letter P is an assertion that exactly n objects have the property P. Numerical coefficients and values are calculated with precision 27-36 digits. Operators can be applied to variables that consist of a leading letter and trailing underscores and alphanumerics. The last decent calculator made by […]. A calculator using conventional logic will internally convert the expression to the RPN form above. Propositional Logic 1 hr 33 min 25 Examples What is a proposition? paradox? open sentence? with Examples #1-9 What is Symbolic Logic? What are common connectives? Negate each statement (Examples #10-13) Determine if "inclusive or" or "exclusive or" is intended (Example #14) Translate the symbolic logic into English (Example #15) Convert the English sentence into…. The bi-conditional operator ( ↔ ) can be inserted if you press the shift key and the < key at the same time. Enter a formula of standard propositional, predicate, or modal logic. Well-formed Formulas (WFFs) of Propositional Logic Propositional logic uses a symbolic “language” to represent the logical structure, or form , of a compound proposition. But in Logic we are using symbols not to identify numbers or operations done to numbers, but to identify meanings, words, statements; things that everyone of us . Return to the course homepage for Symbolic Logic. Get step-by-step solutions from expert tutors as fast as 15-30 minutes. I'm currently reading Mendelson's Introduction to Mathematical Logic—here is an example of a logical proof from that book. PDF 16 Symbolic Logic Study Guide: Class Notes 1. Example 3 : Translate the following sentence into symbolic form :. The function F(x) defined in Eq. There are also other technologies that can be used for similar purposes like HTML5 Local Storage and local shared objects, web beacons, and embedded scripts. Mathematical logic step by step. The only limitation for this calculator is that you have only three atomic propositions to choose from: p,q and r. About Calculator Symbolic Logic '' $\bullet$ $\forall x\,\forall y (x+y=y+x)$, i. Get 1-to-1 learning help through online lessons. 2 fixes a bug and adds support for 64-bit macOS. It is to Kant that we owe the insight to not treat "existence" as a predicate, and of course Aristotlean logic did dominate Western philosophy for two thousand years -until modern symbolic logic was developed by people like Frege,Russell and Wittgenstein. Search: Symbolic Logic Calculator. In stead of logical symbols, Polish notation uses . If you are looking for symbolic calculations using Octave / Matlab, SCaVis is a good choice. Top free symbolic logic calculator downloads. Modifications by students and faculty at Cal. Translating Statements Into Symbolic Form Calculator. ) and Windows (desktops, laptops, tablets, . State University, Monterey Bay. We do indirect proof by assuming the premises to be true and the conclusion to be false and deriving a contradiction. Symbolic Logic | Syllabus Website | Athabasca University. Given a few mathematical statements or facts, we would like to be able to draw some conclusions. Boolean Algebra expression simplifier & solver. A nonnegative integer n followed by more than one capital letter P, Q, … is an assertion that exactly n of the propositions P, Q, … are true. ProofTools: a symbolic logic proof tree generator 19 June 2020 : ProofTools 0. This means ProofTools now works on Catalina. SYMBOLIC LOGIC (Even in his most abstruse works on mathematics and logic, Lewis Carroll could not fully repress his instinct for nonsense. Any alphabetical character from A all the way to Z. These notes give a very basic introduction to the above. Online tools for doing symbolic mathematics. When you stop typing, ProB will . Let me tell you one thing, even mathematicians in this subject sometimes are weak in a particular branch. formula(s)ReturnaninstanceofBooleanFormula. In Wolfram Alpha's case, it seems to do truth tables, but not proofs. Write a symbolic sentence in the text field below. truth-value, in logic, truth (T or 1) or falsity (F or 0) of a given proposition or statement. INTRODUCTION TO INDIRECT PROOF. You might have thought it would be some symbols, but symbols are only going to be useful once we are clear on what we are symbolizing. A disjunction is a compound statement formed by combining two statements using the word and. I think it makes visualizing truth tables easier than text-based solvers so hopefully it can be useful for some. Like any language, this symbolic language has rules of syntax —grammatical rules for putting symbols together in the right way. Symbolic Logic: Grammar, Semantics, Syntax Logic aims to give a precise method or recipe for determining what follows from a given set of sentences. 1 for android by Hernan Gabriel Romero - Propositional math and logic calculator. Combinatory logic (henceforth: CL) is an elegant and powerful logical theory that is connected to many areas of logic, and has found applications in other disciplines, especially, in computer science and mathematics. References to Irving Copi's Symbolic Logic are to the fifth edition, Macmillan, 1979. Yavuz Oruç and JavaScript, computes the truth value of a logic expression comprising up to four variables, w,x,y,z, two constants, 0,1 and sixty symbols (variables, constants, and operators). Calculate set theory logical expressions step by step. Other articles where free variable is discussed: set theory: Schemas for generating well-formed formulas: A variable is free in a formula if it occurs at . a web application that decides statements in symbolic logic including modal logic, propositional logic and unary predicate logic. About Symbolic Logic Calculator. Detailed steps, Logic circuits, KMap, Truth table, & Quizes. Symbolic notation is used extensively. Symbolic Logic Course-Related Links Peter Suber, Philosophy Department, Earlham College. Test the validity of the argument: If it snows, Paul will miss class. tensor - Linear algebra for tensors with symbolic and numeric scalars Hipparchus - An efficient, general-purpose mathematics components library in the Java programming language miniKanren - A tool for symbolic computation and logic programming. Its design is such that it hopefully facilitates. The Logic Calculator is a free app on the iOS (iPhones and iPads), Android (phones, tablets, etc. The Propositional Logic Calculator finds all the models of a given propositional formula. Logic calculator: Server-side Processing Help on syntax - Help on tasks - Other programs - Feedback - Deutsche Fassung Examples and information on the input syntax Please note that the letters "W" and "F" denote the constant values truth and falsehood and that the lower-case letter "v" denotes the disjunction. This is a demo of a proof checker for Fitch-style natural deduction systems found in many popular introductory logic textbooks. Separate pieces of logic using a comma to compare their truth tables, e. PDF Symbolic Logic: Grammar, Semantics, Syntax. To be precise, using this app, one can determine whether: (1) input is well-formed and, if not, why not, (2) sentences are tautologies, contradictions or contingent, (3) sets of sentences are consistent or inconsistent and (4) arguments are valid or invalid. Perform basic calculus tasks (limits, differentiation and. From the viewpoint of sentential logic, there are five standard connectives -. Appetizers and Lessons for Math and Reason. Usually this is due to less than perfect English language skills. While there are 3-valued and many-valued logics, remember that our logic is 2-valued (or bivalent). sentential logic with 'if' and 'not' 1 symbolic notation 2 meanings of the symbolic notation 3 symbolization: translating complex sentences into symbolic notation 4 rules 5 direct derivations 6 conditional derivations 7 indirect derivations 8 subderivations 9 shortcuts 10 strategy hints for derivations 11 theorems. (1) Either a is smaller than b or both a and b are larger than c. A proof is an argument from hypotheses (assumptions) to a conclusion. Download Logic Calculator for free. In Example 1, the tilde is at the beginning of the statement, but it is followed immediately by a capital A, so the scope of the tilde is only the A, and the statement is not fundamentally a negation. I coded it to allow users of propositional logic to perform operations with the same ease as that offered by a mathematical calculator. Even if you are using just a fraction of the functionalities, the iphone-ish, easy-to-use interface makes life much easier. For many students translating is one of the hardest parts of learning how to do symbolic logic. calculator for a specific kind of modal logic, modal propositional logic, . Sympy : Symbolic Mathematics in Python — Scipy. Consider the statement to the right. Gödel also made (1931) the surprising discovery that number theory cannot be complete, i. (2) a and b are both in front of c; moreover, both are smaller than it. Type letters to represent variables. Thank you! [3] 2021/01/29 17:45 30 years old level / High-school/ University/ Grad student / Very /. An image is created in the tables and columns to show, How to generate truth table changes with true. Modernizing the sentential and predicate logic programs, Bertie and Twootie, by cleaning up the code, rewritting the code to conform to the standard Pascal defination, and developing a graphical user interface for both X and MS Windows. , an electronic health record) may open the web address of the ACS NSQIP surgical risk calculator in a new browser window. Includes some logic lessons and puzzles. Symbolic logic definition is - a science of developing and representing logical principles by means of a formalized system consisting of primitive symbols, combinations of these symbols, axioms, and rules of inference. If you still don't sure how to make your life easier using symbolic algebra check out our another page: Symbolic calculations. In logic, a disjunction is a compound sentence formed using the word or to join two simple sentences. The modern development begin with George Boole in the 19th century. It is simply the name given to the Morden treatment of deductive logic. Now let's put those skills to use by solving a symbolic logic statement. 5ReferenceManual:SymbolicLogic,Release9. Indirect proof is based on the classical notion that any given sentence, such as the conclusion, must be either true or false. The use of symbolic logic also makes reasoning formal and mechanical, contributing to the simplification of the reasoning and making it less prone to errors. 24K subscribers in the logic community. Online Calculator is a simple web application that lets you perform advanced calculations, plot 2D and 3D graphs, and make symbolic calculations such as differentiation. In the event that you seek guidance with algebra and in particular with symbolic equation solving online or negative exponents come pay a visit to us at Polymathlove. Use symbolic logic and logic algebra. Symbolic logic can be extended to involve concepts of number, and modalities such as knowledge and possibility. 'formula in propositional logic', in short: formulas. necessarily, if its premises are true, its conclusion is true), and to identify other important properties of arguments. Note carefully: it is understood here that if a formula replaces a given letter in one place, then the formula replaces the letter in every place. In this lesson, we will learn how to determine the truth values of a compound statement with the logical connectors ~, , and. Get the free "logic calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. It is a version of sentential logic, because the basic units of the language will represent entire sentences. Typical topics covered include: techniques of symbolization, truth tables, validity and soundness, and techniques of natural deduction. As a meta-language specifying the system, a logic programming language, namely, Prolog is adopted. The "turnstile" symbol, used in the context Γ⊢φ, means that, relative to the proof system we are working with, there is a derivation of the . We covered the basics of symbolic logic in the last post. Primitive Rule application using do statements. This tool generates truth tables for propositional logic formulas. If you are looking to learn a subject similar to Logic, tap into the nation's largest community of private tutors. A Selection from symbolic logic. Learn more about symbolic logic by exploring the basics of logic, truth tables, logical operators, and. The symbolic form of mathematical logic is, ‘~’ for negation ‘^’ for conjunction and ‘ v ‘ for disjunction. The symbolic form of mathematical logic is, '~' for negation '^' for conjunction and ' v ' for disjunction. The laws are named after Augustus De Morgan (1806-1871), who introduced a formal version of the laws to classical propositional logic. The key to solving this problem is to break it down into it's…. To generate a truth table for the statement " P → Q ," you would type a "P," then type a greater-than symbol (>), then type a "Q. integration) with symbolic expressions. Build a truth table for the formulas entered. Symbolic Logic The starting point for appreciating symbolic logic is the appreciation of the difference between simple statements and compound statements. Enter functions in standard mathematical notation, using x as independent variable. Find more Mathematics widgets in Wolfram|Alpha. Please let me know if anything should be added, something doesn't function properly, or text should be worded differently. 22 Symbolic Logic Study Guide: Class Notes 1. The Boolean expression for the AND logic gate is A ⋅ B = . Free Logical Sets calculator - calculate boolean algebra, truth tables and set theory step-by-step. Visual tool used to illustrate the relationship between two. Natural deduction proof editor and checker. Cookies are small text files stored by your web browser when you use websites. Propositional Logic Truth Table Calculator. In symbolic logic, one introduces symbols like p and q to stand for statements (or “propositions”) like “this is an interesting essay”. Think about logic as a generic way of writing rules: proving formulas and answering questions is a way of detecting what follows from the rules we wrote. Such API groups in a single module all the functions to construct formulae, to check the satisfiability and to retrieve Solver instances. Any list of three English sentences could be translated by three proposition symbols: p, q, r. For iPhone, iPod Touch and iPad. Even if you are using just a fraction of the functionalities, the iphone-ish, easy-to-use. What is Symbolic Logic Calculator. In Polish notation the logical operators are placed before the statement letters rather than between them. If you enter a modal formula, you will see a choice of how the accessibility relation should be constrained. Philosophy 254: Symbolic Logic is an introduction to reasoning in formal symbolic logic. Logical connectives, such as disjunction (symbolized ∨ . The only limitation for this calculator is that you have only three atomic propositions to choose from: p,qand r. Table of Logical Equivalences Commutative p^q ()q ^p p_q ()q _p Associative (p^q)^r ()p^(q ^r) (p_q)_r ()p_(q _r) Distributive p^(q _r) ()(p^q)_(p^r) p_(q ^r) ()(p_q. ; Conjunction is a truth-functional connective similar to "and" in English and is represented in symbolic logic with the dot " ". Index Terms: Boolean functions, symbolic manipulation, binary decision diagrams, logic design verification 1. As evident from the examples on this page, arguments to helper functions must be enclosed. The premier professional organization of logicians. There is NO calculator that can do it on the internet it seems. Logic Calculator Symbolic. A standard of a correct translation A (not the) logical symbolization of an English sentence is correct iff both are logically equivalent. Organic chemistry is the most difficult science course at the college only if it's GPA course is lower than that of genetics. 5 • PaulScurek(2013-08-12) sage. Obtained symbolic formula can be used with calculators Precision series, and with Graphing Calculator 2D Numeric, Graphing Calculator 2D Parametric, and Graphing Calculator 2D. When symbolizing English sentences in propositional logic, it is often helpful to work backwards, reversing the order in which well-formed formulas (WFFs) are constructed. ) and Windows (desktops, laptops, tablets, xbox ones) platforms. The specific system used here is the one found in forall x: Calgary. About Calculator Symbolic Logic. The Hex Calculator is used to perform addition, subtraction, multiplication and division on two hexadecimal numbers. Truth tables exhibit all the truth-values that it is possible for a given statement or set of statements to have. It follows that genetics is the most difficult science course at C-SC. For sure, this proof is completely . In economic crisis has never to. The facts and the question are written in predicate logic, with the question posed as a negation, from which gkc derives contradiction. Klaus Dethloff who taught this stuff and much more. Oh, well yes, using a dot instead of writing an ampersand or the word "AND" and using a triple bar instead of writing out "if and only if" must be time saving, although in math I was taught to use "iff" as the shorthand to "if and only if," but even then drawing three straight lines should be shorter than making at least 4. How does a calculator add two numbers together? · Representing numbers in binary · Converting decimal to binary · Using logic gates with binary. Symbolic Logic and the Logic of Symbolism. Each step of the argument follows the laws of logic. Look: Symbolic Logic II, Lecture 3. Your first 5 questions are on us!. 98 Symbolic Logic Study Guide: Practice Tests and Quizzes Problem 3. In the second form, images can be renamed to a different symbolic name, (a="NAME=img. This is an image tool that is used to. This method makes it possible to manipulate ideas mathematically in much the same way that numbers are manipulated. CL was originally invented as a continuation of the reduction of the set of logical constants to a singleton set in classical first-order logic (FOL). Place brackets in expressions, given the priority of operations. Logic and the Study of Arguments. There's a magma calculator online here, but your programs are limited to 60 seconds which should be sufficient for most things. If you are facing problems with symbolic equation solving online, why don’t you try Algebrator. [Use 'Falsum' for the '⊥' symbol] [Acceptable individual constants are: 'a', 'b', 'c', 'd', . Symbolic logic proof solver Please wait. We will give two facts: john is a father of pete and pete is a father of mark. Logic is the study of consequence. The thing solves algebra, and basic symbolic logic uses, well, I don't want to say the same sort of symbol manipulation because the overlap is imperfect, but both proofs and algebra work by manipulating symbols via a set of well-defined rules. The novel dramatizes the capacity of symbolic logic to redeem representation by locating positive value—aesthetic, social, and even spiritual—in the systematicity of formal interrelationships. Type in any function derivative to get the solution, steps and graph. Rather than trying to identify the simplest components of the sentence and build up from them, first try to determine whether the sentence as a whole is a negation, a conjunction, a disjunction. Symbolic logic is an expression of logic by using symbols in the place of natural language. Anyone who has attended Mathematics particularly Algebra and Probability, must have a good understanding of this diagram. The following is a list of characters that can be used. This is a versatile truth-table calculator for propositional logic. It's completely free and does not collect any information from its users. So we need precise de nitions of 'sentence' and 'follows from. Simple to use Truth Table Generator for any given logical formula. To enter logic symbols, use the buttons above the text field, . The only limitation for this calculator is that . Well-formed Formulas (WFFs) of Propositional Logic. Symbolic Logic, 66:3 (2001), 1121-1126. Association for Symbolic Logic. Here we'll survey the simplest variety of formal logic: sentential logic. Read from here about the differences between algorithms. It is to Kant that we owe the insight to not treat “existence” as a predicate, and of course Aristotlean logic did dominate Western philosophy for two thousand years –until modern symbolic logic was developed by people like Frege,Russell and Wittgenstein. Before you start: Be sure to study the Logic handout, and in particular familiarize yourself with the various English phrases expressing logical statements. Like any language, this symbolic language has rules of syntax—grammatical rules for putting symbols together in the right way. " Then you would hit the "Generate Truth Table" button. Basic Mathematical logics are a negation, conjunction, and disjunction. Verbal/logic thinkers — Verbal/logic thinkers tend to be good at learning languages, and have an affinity for words, literature, and speech. Denote missing numeric or date/time values with the missing value symbol * enclosed in single quotation marks ('*'). It has three modes: (1) Evaluation of logic formulae: In this mode we have the basic boolean operations (negation, conjunction, disjunction, conditional and biconditional) so the user can insert the logic formula and the Logic Calculator. The Propositional Logic Calculator. The following truth table gives the truth value of p ∨ q depending on the truth values of p and q. Translate the following English sentences into the formal language of the Tarski's World (50 points). It provides mental shorthand which permits one to say much in a small piece. The statement does not begin with a tilde, but with a bracket. You can enter numbers, but also symbols like a, b, pi or even whole math expressions such as (a+b)/2. It is also known as mathematical logic. A logical statement is a mathematical . Apparently the calculator was "hardly a part of application of symbolic logic", but Berkeley found a parallelism between the Complex Number Calculator and IBM punch card machine used in Prudential, and he learned the detail of Shannon's 1938 paper, that is, the relationship between Boolean. Solve polynomial and transcendental equations. In the rules of inference, it's understood that symbols like "P" and "Q" may be replaced by any statements, including compound statements. These symbolic representations can make it much easier to see whether a certain argument is valid (i. But if the original sentences had some common structure, showing . Yavuz Oruç and JavaScript, computes the truth value of a logic expression comprising up to four variables, w,x,y,z, two constants, 0,1 and sixty symbols ( . A Calculator to perform logical operations. The Propositional Logic Calculator The Propositional Logic Calculator finds all the models of a given propositional formula. Use the buttons below (or your keyboard) to enter a proposition, then gently touch the duck to have it. In this project we will study the basics of propositional and predicate logic based on the original historical source Principia Mathematica [13] by Russell . An introduction to formal deductive logic, covering propositional logic (truth-functional logic) and first-order predicate logic (quantification theory). Propositional logic uses a symbolic "language" to represent the logical structure, or form, of a compound proposition. DEFINITION OF SYMBOLIC LOGIC: This is a method of reasoning by employing symbols and other special symbols. A disjunction is true if either one or both of the statements in it is true. The logic module for SymPy allows to form and manipulate logic expressions using symbolic and Boolean values. Free Logical Sets calculator - calculate boolean algebra, truth tables and set theory step-by-step This website uses cookies to ensure you get the best experience. Ulti-mately, then, although its best-known legacy is the Boolean algebra that. Most people are already familiar with the use of letters and other symbols to represent both numbers and concepts. Evaluate expressions with arbitrary precision. A proposition In categorical logic, an A proposition is a universal affirmative proposition. This truth-table calculator for classical logic shows, well, truth-tables for propositions of classical logic. Primary connective (P ∧ Q) ∧ (Q ∨ R) T F F F F T T. Introduction to Symbolic Logic. This is a really trivial example. The laws are named after Augustus De Morgan (1806–1871), who introduced a formal version of the laws to classical propositional logic. For example, in an application of conditional elimination with citation "j,k →E", line j must be the conditional, and line k must be its antecedent, even if line k actually precedes line j in the proof. Apparently the calculator was “hardly a part of application of symbolic logic”, but Berkeley found a parallelism between the Complex Number Calculator and IBM punch card machine used in Prudential, and he learned the detail of Shannon’s 1938 paper, that is, the relationship between Boolean. For example, if I told you that a particular real-valued function was continuous on the interval $$[0,1]\text{,}$$ and $$f(0) = -1$$ and $$f(1) = 5\text{,}$$ can we conclude that there is some point between $$[0,1]$$ where the. Many statements can be combined with logical connections to form new statements. mathematics, HP-48SX (Calculator), Logic, Symbolic and mathematical, Symbolic and mathematical Logic HP 48SX engineering mathematics library | Open Library Read Book Hp 48sx Engineering Mathematics Library An Introduction To Symbolic And Complex Computation With Applications astonishing points. You can enter multiple formulas separated by commas to include more than one formula in a single table (e. Click the "Reference" tab for information on what logical symbols to use . It would be really helpful for quickly visualizing test cases for people like me who fumble a bit somewhere on longer inputs. Use the graphing calculator with logic symbols (boolean logic). Find Normal Forms of Boolean Expression: Conjunctive normal form (CNF), including perfect. We carry a good deal of quality reference materials on matters varying from graphs to square roots. Symbolic logic, however, does not aim at giving such insight" (Andrew Bachhuber, Introduction to Logic (New York: Appleton-Century Crofts, 1957), p. Grammar: What's a sentence? What are the rules that determine whether a string of symbols is a sentence, and when it is not?. Symbolic logic is the branch of mathematics that makes use of symbols to express logical ideas. Rather than trying to identify the simplest components of the sentence and build up from them, first try to determine whether the sentence as a. Go to Daemon Proof Checkeror Quick Help Index. In this paper we propose a Logic Calculator with three op- eration modes: evaluation of logical formulae; logical entailment and conversion of a formula to . If you don't understand what you read, then you will have a difficult time representing. Symbolic logic, and its notation, originated in the works of George Boole (1815-1864), of which Boole (1854) is the best known. You can select and try out several solver algorithms: the "DPLL better" is the best solver amongst the options. You can use and, or, not, then, equals , xor as well as symbols like -> for then etc. You can enter logical operators in several different formats. But it is more likely that the calculator logic will be pushing numbers down onto the stack every time a pair of brackets is opened or is implied by the. Atanasoff and graduate student Clifford Berry, of Iowa State College complete a prototype 16-bit adder. Symbolic Logic is a calculator that will enable you to build a logical statement using specific symbolic statements and once it is created you can evaluate its possible truth values that are. 2 How to determine the truth value of a sentence? (1) If the truth-values of component sentences of a compound sentence are given: From inside to outside! (when you determine the truth value of a sentence based on a Tarski’s World). The definition starts by stating that every atomic proposition is of that type, i. (3) But there is a third reason for the popularity of symbolic logic among philosophers, which is more substantial, for it involves a very important difference in philosophical belief. A little writeup by Piotr Bania. New features: Bugfix: disabled infinite branch detection, known to be using a defective algorithm which gives the wrong results in some cases, except for the straightforward detection of infinite branches due to applications of the modal. [+] Rules for Sentential Logic NOTE: the order in which rule lines are cited is important for multi-line rules. This calculator calculates derivative functions of high orders in symbolic and numeric forms. We honour the ancestry, heritage, and gifts of the Indigenous Peoples and give thanks to them. A Logic Calculator a web application that decides statements in symbolic logic including modal logic, propositional logic and unary predicate logic ⇚Home English|Español A Logic Calculator DecideDepictTruth TableExampleCounterexampleTree ProofCancel Quick Reference Information: What is this?. Barton, later wrote that he developed reverse Polish notation independently of Hamblin sometime in 1958 after reading a 1954 textbook on symbolic logic by . Athabasca University respectfully acknowledges that we live and work on the traditional lands of the Indigenous Peoples of Canada (First Nations, Inuit, Métis). 19 June 2020: Added a new dropdown box for all basic/normal modal logic variants - selecting an item in the dropdown sets the appropriate toggles of reflexivity, symmetry, transitivity, extendability, Euclidean and S5. Example 1 : Translate the following sentence into symbolic form : The earth is a planet. They don't belong to any symbolic logic that I know of. Symbolic logic is used in argumentation, hardware and software development and many different disciplines. Numerical values (and mathematical operator symbols) can be added to the expression directlly. Greek philosopher, Aristotle, was the pioneer of logical reasoning. Modal logic has its diamond and box inference rules, but from what I've seen of it used in Fitch's Symbolic Logic they have introduction and elimination rules as well. , that no matter what axioms are. SYMBOLIC LOGIC, for instance, shows the mind of Charles Lutwidge Dodgson rather than the whimsical Lewis Carroll. Symbolic logic has been extended to a description and analysis of the foundations of mathematics, particularly number theory. The syllogism and many other more complicated arguments. Items in the dropdown are prefixed by a number - equivalent modal logic. About Logic Symbolic Calculator. Grammar: What’s a sentence? What are the rules that determine whether a string of symbols is a sentence, and when it is not?. Venn Diagram Shading Calculator - Venn Diagram - The Definition and the Uses of Venn Diagram Venn Diagram Shading Calculator - You have most likely been exposed to or encountered the Venn diagram prior to. Since atomic propositions are the . I just started following your blog. The conditional operator ( → ) can be inserted if you press the shift key and the > key at the same time. This free app allows users of propositional logic to perform operations with the same ease as that offered by a mathematical calculator. It is dedicated to the memory of Dr. KEYBOARD MODELS COUNTERMODELS p q r p q r p q r p q r UNSAT TAUT CONTG ILL-FD Instructions. Chapter 7 focuses on simply translating regular English statements into a new symbolic language. This page contains a JavaScript program which will generate a truth table given a well-formed formula of truth-functional logic. Free derivative calculator - differentiate functions with all the steps. Example 2 : Translate the following sentence into symbolic form : David is not a soccer player. We apply certain logic in Mathematics. The purple box lists the keyboard symbols and the operations they represent. For example, if I told you that a particular real-valued function was continuous on the interval [0,1], [ 0, 1], and f(0)= −1 f ( 0) = − 1 and f(1)= 5, f ( 1) = 5, can. This may be achieved by parsing the bracketed expression before carrying out the calculation. The symbolic form of mathematical logic is, . Makoto Tsukada describes a proof checking program using Prolog. Boolean Algebra Calculator. Truth Table Calculator Use seven basic logic gates: AND, OR, XOR, NOT, NAND, NOR, and XNOR, as shown below with it,s Name, Boolean Equation, Symbol & Truth Table. So we need precise de nitions of ‘sentence’ and ‘follows from. Logic Symbolic Calculator. The bracket does not encompass the entire statement, so we could skip any connective within the brackets, but we will consider them, just for the sake of example. Rules of Inference and Logic Proofs. If either Hume did not invent truth tables or Wittgenstein wrote the Tractatus. ProofTools: a symbolic logic proof tree generator. We will cover the syntax and semantics of classical sentential and . In Propositional Logic, atomic propositions correspond to simple sentences in the object language. What is a proposition? paradox? open sentence? with Examples #1-9; What is Symbolic Logic? What are common connectives? Negate each statement (Examples . The calculator will give you an average over the percentage you have completed so far. Anyone who has studied Mathematics in particular Algebra and Probability, must have a good understanding of the Venn diagram. INTRODUCTION In the present chapter, we discuss how to translate a variety of English state-ments into the language of sentential logic. The propositional logic statements can only be true or false. All in one boolean expression calculator. Note that the lowercase "v" represents the "or" operations, so lowercase "v" cannot be used as a variable. Become a Symbolic Logic Homework Answers freelance writer with us! If you are an experienced, knowledgeable writer, do apply for work at our service! We offer beneficial working conditions and do our best for you to do your job most effectively. Truth Functionality: In order to know the truth value of the proposition which results from applying an operator to propositions, all that need be known is the definition of the operator and the truth value of the propositions used. I haven't started using your product, but I would like to get familiar with it. 146 Hardegree, Symbolic Logic Definition: If F is a formula of sentential logic, then a substitution instance of F is any formula F* obtained from F by substituting formulas for letters in F. To generate a truth table for the statement " P → Q ," you would type a "P," then type a greater-than symbol (>), then type. for certain strings of symbols to count as well-formed formulas, or. Alright, so now let's see if we can determine if an argument is valid or invalid using our logic rules. Mathematics is such a diverse subject, that it sometimes becomes tedious to understand every part with equal ease. The symbol for this is $$ν$$. Getting a contradiction shows us that it is. , the classic syllogism: All men are mortal. A toolkit for proving first order predicate logic formulas, answering questions and converting between different syntaxes for logic. 18 Symbolic Logic Study Guide: Class Notes 3. To use the app, enter a boolean logic expression below. · We use · One reason for having two symbols . Symbolic logic can be thought of as a simple and flexible shorthand: Consider the symbols: [ (p q) (q r)] (p r). Abbreviated Truth Table In constructing a reverse truth table, assume that all the premises are true and the conclusion is false, then consistently assign truth values to the components in an attempt to show that your assumption is correct. For example, (a -> b) & a becomes true if and only if both a and b are assigned true. Translating Sentences into Symbolic Form - Examples. Tool/Calculator to simplify or minify Boolean expressions (Boolean algebra) containing logical expressions with AND, OR, NOT, XOR. The three building options "truth table", "clause normal form" and a "parse tree" are simple, useful utilities: The truth table prints a full truth table of a formula up to 1024 rows: nice for checking out small propositional formulas. Mathematical Logic, truth tables, logical equivalence calculator - Prepare the truth table for Expression : p and (q or r)=(p and q) or (p and r), p nand q, p nor q, p xor q, Examine the logical validity of the argument Hypothesis = p if q;q if r and Conclusion = p if r, step-by-step online. The language this tool uses is a kind of symbolic logic. Download Logic Calculator Free - Latest version 4. There would be little logic in offering here the entire text of SYMBOLIC LOGIC. About Symbolic Calculator Logic. Giuseppe Castagna and Andrew D. What that means is that whether we know, for any given statement, that it is true or false does not get in the way of us knowing some other things about it in relation to certain other statements. Below is a ProB-based logic calculator. Symbolic logic is the simplest form of logic. Any expression that obeys the syntactic rules of propositional logic is called a well-formed. Symbolic logic calculator. Propositional Argument Validity Calculator. Rule do Application Explanation; Wedge Introduction: do X 3 vI L: X is the disjunct to be introduced, 3 is the line number to which Wedge Introduction needs to be applied and L denotes - introduce the argument X on left side. In mathematics, a statement is not accepted as valid or correct unless it is accompanied by a proof. Hi everyone, here's a validity calculator I made within Desmos. (Although based on forall x: an Introduction to Formal Logic, the proof system. The calculator will generate the truth table for the given logic formula/expression. Developed by George Boole, symbolic logic's main advantage is that it allows operations -- similar to algebra -- to work on the truth values of its propositions. Input values are different variables and symbols (commonly known as logic gates). Symbolic Logic Calculator ) and Windows (desktops, laptops, tablets, xbox ones) platforms. Set Notation Venn Diagram Calculator - Venn Diagram - The Definition and the Uses of Venn Diagram Set Notation Venn Diagram Calculator - Most likely, you've been exposed to or encountered a Venn diagram before. The rules of mathematical logic specify methods of reasoning mathematical statements. The converse of this statement is the related statement if Q, then P. Venn Diagram Shading Calculator. Truth-Functional Propositional Logic. We will ask whether from these two facts we can derive that john is a father of pete: obviously we can. There are many valid argument forms, however, that cannot be analyzed by truth-functional methods, e. A Logic Calculator a web application that decides statements in symbolic logic including modal logic, propositional logic and unary predicate logic ⇚Home English|Español A Logic Calculator DecideDepictTruth TableExampleCounterexampleTree ProofCancel Quick Reference Information: What is this? Instructions The Language The Algorithm Updates Contact. Operates the logical connectives (and, or, xor) of a pair of numbers expressed in Decimal, Hexadecimal, Octal , Senary or Binary form. 4 Sentential Logic This chapter introduces a logical language called SL. Featuring a purple munster and a duck, and optionally showing intermediate results, it is one of the better instances of its kind. Example 1: Given: p: 72 = 49 true q: A rectangle does not have 4. For the biconditional you may use any of the symbols: . Symbolic logic and set theory are intertwined and lie at the foundations of . Logic Calculator This simple calculator, the courtesy of A. (whenever you see $$ν$$ read 'or') When two simple sentences, p and q, are joined in a disjunction statement, the disjunction is expressed symbolically as p $$ν$$ q. What are the basic logic gates? · 1: Logic symbol of AND gate (Source: wikimedia. Calculational propositional logic is a product of researchers in the field of the formal development of algorithms. For modal predicate logic, constant domains and rigid terms are assumed. In Example 2, the tilde is followed by a parenthesis, and the entire rest of the statement is contained by the parentheses, so the statement is a negation, and the tilde is the main operator. linux x86 32-bit, GTK2 linux x86 32-bit, Qt linux x86 64-bit, GTK2 linux x86 64-bit, Qt5 win32 win64 mac osx x86. Russell’s paradox was very bad news to Frege (and not only to him!). Solving a classical propositional formula means looking for such values of variables that the formula becomes true. Raster Calculator (Spatial Analyst)—ArcGIS Pro. Mathematical Logic, truth tables, logical equivalence calculator - Prepare the truth table for Expression : p and (q or r)=(p and q) or (p and r), p nand q, . Calculator Symbolic Logic. The reasoning may be a legal opinion or mathematical confirmation. Here is the abstract: A proof system for propositional and predicate logic is discussed. About Logic Calculator Symbolic. Input two bits x;y and output two bits representing the absolute value of x−y 3. In symbolic logic, the disjunction of p and q is written p ∨ q. First, we will translate the argument into symbolic form and then determine if it matches one of our rules. Additionally, it helps prevent logical confusion. A conjunction is a compound statement formed by combining two statements using the word and. An online truth table calculator will provide the truth table values for the given propositional logic formulas. The calculate this better by the second of, and relationships between the way can i am an answer?. Full paths to data or data existing in the specified current . Logic NEW by: Anonymous Either organic chemistry or genetics is the most difficult science at culver Stockton college. | 2022-05-19 08:56:27 | {"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": 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": 0.5423678159713745, "perplexity": 1147.3955177276555}, "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-2022-21/segments/1652662526009.35/warc/CC-MAIN-20220519074217-20220519104217-00149.warc.gz"} |
https://www.jobilize.com/physics-ap/course/6-5-newton-s-universal-law-of-gravitation-by-openstax?qcr=www.quizover.com&page=5 | # 6.5 Newton’s universal law of gravitation (Page 6/11)
Page 6 / 11
## The cavendish experiment: then and now
As previously noted, the universal gravitational constant $G$ is determined experimentally. This definition was first done accurately by Henry Cavendish (1731–1810), an English scientist, in 1798, more than 100 years after Newton published his universal law of gravitation. The measurement of $G$ is very basic and important because it determines the strength of one of the four forces in nature. Cavendish’s experiment was very difficult because he measured the tiny gravitational attraction between two ordinary-sized masses (tens of kilograms at most), using apparatus like that in [link] . Remarkably, his value for $G$ differs by less than 1% from the best modern value.
One important consequence of knowing $G$ was that an accurate value for Earth’s mass could finally be obtained. This was done by measuring the acceleration due to gravity as accurately as possible and then calculating the mass of Earth $M$ from the relationship Newton’s universal law of gravitation gives
$\text{mg}=G\frac{\text{mM}}{{r}^{2}}\text{,}$
where $m$ is the mass of the object, $M$ is the mass of Earth, and $r$ is the distance to the center of Earth (the distance between the centers of mass of the object and Earth). See [link] . The mass $m$ of the object cancels, leaving an equation for $g$ :
$g=G\frac{M}{{r}^{2}}\text{.}$
Rearranging to solve for $M$ yields
$M=\frac{{\mathrm{gr}}^{2}}{G}\text{.}$
So $M$ can be calculated because all quantities on the right, including the radius of Earth $r$ , are known from direct measurements. We shall see in Satellites and Kepler's Laws: An Argument for Simplicity that knowing $G$ also allows for the determination of astronomical masses. Interestingly, of all the fundamental constants in physics, $G$ is by far the least well determined.
The Cavendish experiment is also used to explore other aspects of gravity. One of the most interesting questions is whether the gravitational force depends on substance as well as mass—for example, whether one kilogram of lead exerts the same gravitational pull as one kilogram of water. A Hungarian scientist named Roland von Eötvös pioneered this inquiry early in the 20th century. He found, with an accuracy of five parts per billion, that the gravitational force does not depend on the substance. Such experiments continue today, and have improved upon Eötvös’ measurements. Cavendish-type experiments such as those of Eric Adelberger and others at the University of Washington, have also put severe limits on the possibility of a fifth force and have verified a major prediction of general relativity—that gravitational energy contributes to rest mass. Ongoing measurements there use a torsion balance and a parallel plate (not spheres, as Cavendish used) to examine how Newton’s law of gravitation works over sub-millimeter distances. On this small-scale, do gravitational effects depart from the inverse square law? So far, no deviation has been observed.
#### Questions & Answers
full meaning of GPS system
how to prove that Newton's law of universal gravitation F = GmM ______ R²
sir dose it apply to the human system
prove that the centrimental force Fc= M1V² _________ r
prove that centripetal force Fc = MV² ______ r
Kaka
how lesers can transmit information
griffts bridge derivative
below me
please explain; when a glass rod is rubbed with silk, it becomes positive and the silk becomes negative- yet both attracts dust. does dust have third types of charge that is attracted to both positive and negative
what is a conductor
Timothy
hello
Timothy
below me
why below you
Timothy
no....I said below me ...... nothing below .....ok?
dust particles contains both positive and negative charge particles
Mbutene
corona charge can verify
Stephen
when pressure increases the temperature remain what?
remains the temperature
betuel
what is frequency
define precision briefly
CT scanners do not detect details smaller than about 0.5 mm. Is this limitation due to the wavelength of x rays? Explain.
hope this helps
what's critical angle
The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
dude.....next time Google it
okay whatever
Chidalu
pls who can give the definition of relative density?
Temiloluwa
the ratio of the density of a substance to the density of a standard, usually water for a liquid or solid, and air for a gas.
Chidalu
What is momentum
mass ×velocity
Chidalu
it is the product of mass ×velocity of an object
Chidalu
how do I highlight a sentence]p? I select the sentence but get options like copy or web search but no highlight. tks. src
then you can edit your work anyway you want
Wat is the relationship between Instataneous velocity
Instantaneous velocity is defined as the rate of change of position for a time interval which is almost equal to zero
Astronomy | 2020-09-27 01:35:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 20, "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": 0.6774657368659973, "perplexity": 1090.3253612292324}, "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-2020-40/segments/1600400249545.55/warc/CC-MAIN-20200926231818-20200927021818-00244.warc.gz"} |
https://www.physicsforums.com/threads/magnetix-flux-density-zero-scalar-potential.417861/ | # Magnetix flux density & Zero scalar potential
1. ### baggiano
13
Hi
Can anybody tell how I can prove that the magnetic flux density is perpendicular to external boundaries with imposed zero magnetic scalar potential?
Thanks
Bag
2. ### Bob S
I believe that for any field that satisfies Laplace's equation (divergence of a gradient), the equipotential lines (curl of the gradient) are orthogonal to the gradient. Is this what you mean?
Bob S
3. ### baggiano
13
Thanks for the reply and for your explanation. That precisely explains what I meant. Thanks again. | 2015-09-02 00:24:16 | {"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": 0.8150036931037903, "perplexity": 1793.6172117178712}, "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-2015-35/segments/1440645235537.60/warc/CC-MAIN-20150827031355-00332-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://twistedone151.wordpress.com/2008/01/25/physics-friday-5/ | Physics Friday 5
Spin Gravity and Coriolis Forces
Suppose we have a space craft or habitat using rotation to simulate gravity (as seen in 2001: A Space Odyssey, Mission to Mars, and Babylon 5). It is the fictitious centrifugal force that provides the apparent gravity, but for a uniformly rotating reference frame, there is another fictitious force: the Coriolis force.
Let us consider motion in a plane perpendicular to the rotation axis (as the component of motion parallel to the axis neither affects nor is affected by the fictitious forces). We call the angular velocity ω, and the distance from the axis r. Then the centrifugal force points outward from the axis, and produces a centrifugal accelleration in the rotating frame of .
Thus, we can figure out the angular velocity needed to produce a partiular strength of apparent gravity at a particular radius. Below is a table that for various radii gives the angular velocity ω and the rotation period necessary to produce an apparent gravity of one gee (9.80 m/s2) for various radii:
r (m) ω (s-1) T (s) 1 3.130 2.007 5 1.400 4.488 10 0.990 6.347 50 0.443 14.192 100 0.313 20.071 200 0.221 28.385 500 0.140 44.880 1000 0.099 63.470
Now, let us consider the Coriolis force. In three dimensions, the Coriolis force on a mass m moving with velocity in the rotating reference frame is . Thus, if v is the maginude of the component of perpendicular to the axis of rotation (given by ), then the magnitude of the Coriolis force is , and the direction is given by the right hand rule. It will be perpendicular to , and thus in the plane perpendicular to the axis. It will also be perpendicular to the velocity, and thus to the velocity component in the plane. Note that if we view the plane in the direction such that the rotation is counterclockwise, then the Coriolis force is in a direction ninety degrees clockwise from the velocity: motion on an outward radial path experiences a Coriolis force opposite the rotation direction, and motion inward experiences a force with the rotation.
Note, however, it is generally easier to work in the non-rotating reference frame, where trajectories are straight lines, than in the rotating frame. For example, suppose someone in our space station drops an object from a height h above the floor, which is at radius R. Let us define inertial frame polar coordinates
and rotatng frame coordinates , with the two coinciding at t=0 and the thetas increasing in the direction of motion. Then . Our object has initial position of (R-h, 0) in the inertial frame, and initial velocity equal to the tangential velocity, or . Uniform linear motion in polar coordinates is given by and
In our case, this becomes and .
Transforming, this becomes in the rotating frame:
We want the time to drop to the floor: t such that , which is .
Plugging this into , we find:
which is negative and independent of ω, as long as ω≠0! (We can also show that this angle is dependent only on the ratio of h to R).
The distance of “deflection” from the point straight down is
(note that for , this can be approximated by ).
For example, dropping an object from a height of 125 cm above a floor at radius 100 m gives a deflection d=13.3 cm. At a floor of radius 500 m, this same height has deflection 5.9 cm. This is definitely enough to notice, particularly it a person stands up or bends over quickly; their head would experience some Coriolis deflection as a result.
Lastly, if someone were to “fall” with some small initial velocity from the axis of rotation, their motion in the inertial frame is , and so examining the transformation, the radial velocity in the rotating frame is also the small constant v0. However, note that when you reach radius R, your “horizontal” velocity in the rotating frame is the tangential velocity Rω.
If we have ω to maintain one gee at radius R, the tangential velocity at R is , and is given in this table:
R (m) ω (s-1) V=Rω (M/s) 1 3.130 3.130 5 1.400 7.000 10 0.990 9.899 50 0.443 22.136 100 0.313 31.305 200 0.221 44.272 500 0.140 70.000 1000 0.099 98.995
For those not familiar with metric velocities, 1 m/s is about 2.24 mph, so at a radius of 50 m (approximately 164 ft), we have tangential velocities of almost 50 mph! So any injury from the fall would be from “sideways” motion, not “downward motion,” after a possibly rather slow fall.
Thus, we find the larger the rotating space, the less Coriolis force is felt by those within, but with higher tangential velocities. | 2018-06-22 07:48:06 | {"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": 27, "/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": 0.8542810678482056, "perplexity": 502.13409733839205}, "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-2018-26/segments/1529267864364.38/warc/CC-MAIN-20180622065204-20180622085204-00399.warc.gz"} |
https://www.physicsforums.com/threads/prove-the-reduction-forumla.224658/ | # Prove the Reduction Forumla
1. Homework Statement
a)Prove the reduction formula:
$$\int\ tan^n\ x\ dx\ =\ \frac{1}{n-1}tan^{n-1}\ x\ -\int\ tan^{n-2}\ x\ dx$$
Hint: first write $$tan^n x$$ as $$tan^{n-2} \ x\ tan^2\ x$$ and the rewrite using $$tan^2\ x+1=sec^2\ x$$.
b) Use the formula twice to find $$\int\ tan^4\ dx$$
3. The Attempt at a Solution
I not sure what they're asking for when they say "prove". How should I begin with this one? Thanks
## Answers and Replies
Related Calculus and Beyond Homework Help News on Phys.org
Vid
Use methods of integration to show the left side equals the right. From the form of the right side, it should be pretty obvious which method to use. | 2019-12-11 00:07:37 | {"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": 0.5721940994262695, "perplexity": 607.3978844154601}, "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-2019-51/segments/1575540529516.84/warc/CC-MAIN-20191210233444-20191211021444-00539.warc.gz"} |
http://www.diva-portal.org/smash/resultList.jsf?dswid=0&searchType=SUBJECT&language=no&onlyFullText=false&af=%5B%5D&aq=%5B%5B%7B%22categoryId%22%3A%2211505%22%7D%5D%5D | Endre søk
Begrens søket
1234567 1 - 50 of 379
Referera
Referensformat
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Annet format
Fler format
Språk
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Annet språk
Fler språk
Utmatningsformat
• html
• text
• asciidoc
• rtf
Treff pr side
• 5
• 10
• 20
• 50
• 100
• 250
Sortering
• Standard (Relevans)
• Forfatter A-Ø
• Forfatter Ø-A
• Tittel A-Ø
• Tittel Ø-A
• Type publikasjon A-Ø
• Type publikasjon Ø-A
• Eldste først
• Nyeste først
• Skapad (Eldste først)
• Skapad (Nyeste først)
• Senast uppdaterad (Eldste først)
• Senast uppdaterad (Nyeste først)
• Disputationsdatum (tidligste først)
• Disputationsdatum (siste først)
• Standard (Relevans)
• Forfatter A-Ø
• Forfatter Ø-A
• Tittel A-Ø
• Tittel Ø-A
• Type publikasjon A-Ø
• Type publikasjon Ø-A
• Eldste først
• Nyeste først
• Skapad (Eldste først)
• Skapad (Nyeste først)
• Senast uppdaterad (Eldste først)
• Senast uppdaterad (Nyeste først)
• Disputationsdatum (tidligste først)
• Disputationsdatum (siste først)
Merk
Maxantalet träffar du kan exportera från sökgränssnittet är 250. Vid större uttag använd dig av utsökningar.
• 1. Aaghabali, M.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Upper bounds on the number of perfect matchings and directed 2-factors in graphs with given number of vertices and edges2015Inngår i: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 45, s. 132-144Artikkel i tidsskrift (Fagfellevurdert)
We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on 2n vertices. The upper bound is sharp for even n. For odd n we state a conjecture on a sharp upper bound.
• 2. Adamaszek, Michal
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
On a lower bound for the connectivity of the independence complex of a graph2011Inngår i: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 311, nr 21, s. 2566-2569Artikkel i tidsskrift (Fagfellevurdert)
Aharoni, Berger and Ziv proposed a function which is a lower bound for the connectivity of the independence complex of a graph. They conjectured that this bound is optimal for every graph. We give two different arguments which show that the conjecture is false.
• 3.
Blekinge Tekniska Högskola, Sektionen för teknik, Avdelningen för telekommunikationssystem.
Security Issues in Wireless Systems2009Independent thesis Advanced level (degree of Master (One Year))Oppgave
ireless Communication is one of the fields of Telecommunications which is growing with the tremendous speed. With the passage of time wireless communication devices are becoming more and more common. It is not only the technology of business but now people are using it to perform their daily tasks, be it for calling, shopping, checking their emails or transfer their money. Wireless communication devices include cellular phones, cordless phones and satellite phones, smart phones like Personal Digital Assistants (PDA), two way pagers, and lots of their devices are on their way to improve this wireless world. In order to establish two way communications, a wireless link may be using radio waves or Infrared light. The Wireless communication technologies have become increasingly popular in our everyday life. The hand held devices like Personal Digital Assistants (PDA) allow the users to access calendars, mails, addresses, phone number lists and the internet. Personal digital assistants (PDA) and smart phones can store large amounts of data and connect to a broad spectrum of networks, making them as important and sensitive computing platforms as laptop PCs when it comes to an organization’s security plan. Today’s mobile devices offer many benefits to enterprises. Mobile phones, hand held computers and other wireless systems are becoming a tempting target for virus writers. Mobile devices are the new frontier for viruses, spam and other potential security threats. Most viruses, Trojans and worms have already been created that exploit vulnerabilities. With an increasing amount of information being sent through wireless channels, new threats are opening up. Viruses have been growing fast as handsets increasingly resemble small computers that connect with each other and the internet. Hackers have also discovered that many corporate wireless local area networks (WLAN) in major cities were not properly secured. Mobile phone operators say that it is only a matter of time before the wireless world is hit by the same sorts of viruses and worms that attack computer software.
• 4. Akbari, Saieed
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On 1-sum flows in undirected graphs2016Inngår i: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 31, s. 646-665Artikkel i tidsskrift (Fagfellevurdert)
Let G = (V, E) be a simple undirected graph. For a given set L subset of R, a function omega: E -> L is called an L-flow. Given a vector gamma is an element of R-V , omega is a gamma-L-flow if for each v is an element of V, the sum of the values on the edges incident to v is gamma(v). If gamma(v) = c, for all v is an element of V, then the gamma-L-flow is called a c-sum L-flow. In this paper, the existence of gamma-L-flows for various choices of sets L of real numbers is studied, with an emphasis on 1-sum flows. Let L be a subset of real numbers containing 0 and denote L* := L \ {0}. Answering a question from [S. Akbari, M. Kano, and S. Zare. A generalization of 0-sum flows in graphs. Linear Algebra Appl., 438:3629-3634, 2013.], the bipartite graphs which admit a 1-sum R* -flow or a 1-sum Z* -flow are characterized. It is also shown that every k-regular graph, with k either odd or congruent to 2 modulo 4, admits a 1-sum {-1, 0, 1}-flow.
• 5.
Umeå University.
Umeå University. Umeå University.
Fast multiplication of matrices over a finitely generated semiring2008Inngår i: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 107, nr 6, s. 230-234Artikkel i tidsskrift (Fagfellevurdert)
• 6.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Avoiding Arrays of Odd Order by Latin Squares2013Inngår i: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, nr 2, s. 184-212Artikkel i tidsskrift (Fagfellevurdert)
We prove that there is a constant c such that, for each positive integer k, every (2k + 1) x (2k + 1) array A on the symbols 1, ... , 2k + 1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k + 1) times in every row and column is avoidable; that is, there is a (2k + 1) x (2k + 1) Latin square S on the symbols 1, ... , 2k + 1 such that, for each i, j is an element of {1, ... , 2k + 1}, the symbol in position (i, j) of S does not appear in the corresponding cell in Lambda. This settles the last open case of a conjecture by Haggkvist. Using this result, we also show that there is a constant rho, such that, for any positive integer n, if each cell in an n x n array B is assigned a set of m <= rho n symbols, where each set is chosen independently and uniformly at random from {1, ... , n}, then the probability that B is avoidable tends to 1 as n -> infinity.
• 7.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On the Ising problem and some matrix operations2007Doktoravhandling, med artikler (Annet vitenskapelig)
The first part of the dissertation concerns the Ising problem proposed to Ernst Ising by his supervisor Wilhelm Lenz in the early 20s. The Ising model, or perhaps more correctly the Lenz-Ising model, tries to capture the behaviour of phase transitions, i.e. how local rules of engagement can produce large scale behaviour.
Two decades later Lars Onsager solved the Ising problem for the quadratic lattice without an outer field. Using his ideas solutions for other lattices in two dimensions have been constructed. We describe a method for calculating the Ising partition function for immense square grids, up to linear order 320 (i.e. 102400 vertices).
In three dimensions however only a few results are known. One of the most important unanswered questions is at which temperature the Ising model has its phase transition. In this dissertation it is shown that an upper bound for the critical coupling Kc, the inverse absolute temperature, is 0.29 for the tree dimensional cubic lattice.
To be able to get more information one has to use different statistical methods. We describe one sampling method that can use simple state generation like the Metropolis algorithm for large lattices. We also discuss how to reconstruct the entropy from the model, in order to obtain parameters as the free energy.
The Ising model gives a partition function associated with all finite graphs. In this dissertation we show that a number of interesting graph invariants can be calculated from the coefficients of the Ising partition function. We also give some interesting observations about the partition function in general and show that there are, for any N, N non-isomorphic graphs with the same Ising partition function.
The second part of the dissertation is about matrix operations. We consider the problem of multiplying them when the entries are elements in a finite semiring or in an additively finitely generated semiring. We describe a method that uses O(n3 / log n) arithmetic operations.
We also consider the problem of reducing n x n matrices over a finite field of size q using O(n2 / logq n) row operations in the worst case.
• 8.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Avoidability by Latin squares of arrays of even orderManuskript (preprint) (Annet vitenskapelig)
We prove that for any k and any 2k × 2k array A such that no cell in A contains more than k/2550 symbols, and no symbol occurs more than k/2550 times in any row or column, there is a Latin square such that no 2550cell in the Latin square contains a symbol that occurs in the corresponding cell in A. This proves a conjecture of Häggkvist [8] in the special case of arrays with even side.
• 9.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Avoidability of random arraysManuskript (preprint) (Annet vitenskapelig)
An n×n array that in each cell contains a subset of the symbols 1, . . . , n is avoidable if there exists a Latin square of order n such that no cell in the Latin square contains a symbol which belongs to the set of symbols in the corresponding cell of the array. Some results on deterministic conditions for avoidability of arrays have been found, but here we study the problem of having an array with randomly assigned subsets of C in its cells. This is equivalent to the problem of list-edge-coloring $K_{n,n}$ with randomly assigned lists from the set {1, . . . , n}. We show that an array where each symbol appears in each cell with probability p will be avoidable with very high probability even if p is such that the expected number of symbols forbidden in each cell is slightly higher than what deterministic theorems can prove is avoidable.
• 10.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Avoiding (m, m, m)-arrays of order n = 2kManuskript (preprint) (Annet vitenskapelig)
An (m, m, m)-array of order n is an n × n array such that each cell is assigned a set of at most m symbols from {1,...,n} such that no symbol occurs more than m times in any row or column. An (m,m,m)- array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant γ such that if m ≤ γ2k, then any (m,m,m)-array of order 2k is avoidable. Such a constant γ has been conjectured to exist for all n by Häggkvist.
• 11.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On Latin squares and avoidable arrays2010Doktoravhandling, med artikler (Annet vitenskapelig)
This thesis consists of the four papers listed below and a survey of the research area.
I Lina J. Andrén: Avoiding (m, m, m)-arrays of order n = 2k
II Lina J. Andrén: Avoidability of random arrays
III Lina J. Andr´en: Avoidability by Latin squares of arrays with even order
IV Lina J. Andrén, Carl Johan Casselgren and Lars-Daniel Öhman: Avoiding arrays of odd order by Latin squares
Papers I, III and IV are all concerned with a conjecture by Häggkvist saying that there is a constant c such that for any positive integer n, if m ≤ cn, then for every n × n array A of subsets of {1, . . . , n} such that no cell contains a set of size greater than m, and none of the elements 1, . . . , n belongs to more than m of the sets in any row or any column of A, there is a Latin square L on the symbols 1, . . . , n such that there is no cell in L that contains a symbol that belongs to the set in the corresponding cell of A. Such a Latin square is said to avoid A. In Paper I, the conjecture is proved in the special case of order n = 2k . Paper III improves on the techniques of Paper I, expanding the proof to cover all arrays of even order. Finally, in Paper IV, similar methods are used together with a recoloring theorem to prove the conjecture for all orders. Paper II considers another aspect of the problem by asking to what extent way a deterministic result concerning the existence of Latin squares that avoid certain arrays can be used when the sets in the array are assigned randomly.
• 12.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Avoiding arrays of odd order by Latin squaresManuskript (preprint) (Annet vitenskapelig)
We prove that there exists a constant c such that for each pos- itive integer k every (2k+1)×(2k+1) array A on the symbols 1,...,2k+1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1)×(2k+1) Latin square S on the symbols 1,...,2k+1 such that for each cell (i, j) in S the symbol in (i, j) does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist.
• 13.
Department of Applied Mathematics and Statistics, Stony Brook University, USA .
Institute of Computer Science, Freie Universität Berlin, Germany . Institute of Computer Science, Universität Bayreuth, Germany . Department of Applied Mathematics and Statistics, Stony Brook University, USA . Helsinki Institute for Information Technology, CS Dept, University of Helsinki, Finland . Institute of Computer Science, Freie Universität Berlin, Germany . Department of Computer Science, Stony Brook University, USA .
Convex transversals2014Inngår i: Computational Geometry, ISSN 0925-7721, Vol. 47, nr 2, s. 224-239Artikkel i tidsskrift (Fagfellevurdert)
We answer the question initially posed by Arik Tamir at the Fourth NYU Computational Geometry Day (March, 1987): “Given a collection of compact sets, can one decide in polynomial time whether there exists a convex body whose boundary intersects every set in the collection?”
We prove that when the sets are segments in the plane, deciding existence of the convex stabber is NP-hard. The problem remains NP-hard if the sets are regular polygons. We also show that in 3D the stabbing problem is hard when the sets are balls. On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments in 2D if the vertices of the transversal are restricted to a given set of points. Our algorithm also finds a convex stabber of the maximum number of a set of convex pseudodisks in the plane.
The stabbing problem is related to “convexity” of point sets measured as the minimum distance by which the points must be shifted in order to arrive in convex position; we give a PTAS to find the minimum shift in 2D, and a 2-approximation in any dimension. We also consider stabbing with vertices of a regular polygon – a problem closely related to approximate symmetry detection.
• 14.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. University of Southern Denmark, Denmark.
Solution of Vizings Problem on Interchanges for the case of Graphs with Maximum Degree 4 and Related Results2016Inngår i: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 82, nr 4, s. 350-373Artikkel i tidsskrift (Fagfellevurdert)
Let G be a Class 1 graph with maximum degree 4 and let t amp;gt;= 5 be an integer. We show that any proper t-edge coloring of G can be transformed to any proper 4-edge coloring of G using only transformations on 2-colored subgraphs (so-called interchanges). This settles the smallest previously unsolved case of a well-known problem of Vizing on interchanges, posed in 1965. Using our result we give an affirmative answer to a question of Mohar for two classes of graphs: we show that all proper 5-edge colorings of a Class 1 graph with maximum degree 4 are Kempe equivalent, that is, can be transformed to each other by interchanges, and that all proper 7-edge colorings of a Class 2 graph with maximum degree 5 are Kempe equivalent. (C) 2015 Wiley Periodicals, Inc.
• 15.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. Yerevan State University, Armenia; National Academic Science, Armenia.
Some results on cyclic interval edge colorings of graphs2018Inngår i: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 87, nr 2, s. 239-252Artikkel i tidsskrift (Fagfellevurdert)
A proper edge coloring of a graph G with colors 1,2,,t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree (G)4 admits a cyclic interval (G)-coloring if for every vertex v the degree dG(v) satisfies either dG(v)(G)-2 or dG(v)2. We also prove that every Eulerian bipartite graph G with maximum degree at most eight has a cyclic interval coloring. Some results are obtained for (a,b)-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)-biregular graphs as well as all (2r-2,2r)-biregular (r2) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.
• 16.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Southern Polytechnic State University, Marietta, Georgia. University of Illinois, Urbana, Illinois.
Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs2009Inngår i: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 61, nr 2, s. 88-97Artikkel i tidsskrift (Fagfellevurdert)
An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide several sufficient conditions for the existence of such a subgraph.
• 17.
KTH, Skolan för datavetenskap och kommunikation (CSC), Teoretisk datalogi, TCS.
Dense Subset Sum may be the hardest2016Inngår i: Leibniz International Proceedings in Informatics, LIPIcs, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing , 2016Konferansepaper (Fagfellevurdert)
The Subset Sum problem asks whether a given set of n positive integers contains a subset of elements that sum up to a given target t. It is an outstanding open question whether the O∗(2n/2)-time algorithm for Subset Sum by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a "truly faster", O∗(2(0.5-δ)n)-time algorithm, with some constant δ > 0. Continuing an earlier work [STACS 2015], we study Subset Sum parameterized by the maximum bin size β, defined as the largest number of subsets of the n input integers that yield the same sum. For every ∈ > 0 we give a truly faster algorithm for instances with β ≤ 2(0.5-∈)n, as well as instances with β ≥ 20.661n. Consequently, we also obtain a characterization in terms of the popular density parameter n/log2 t: if all instances of density at least 1.003 admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from a novel combinatorial analysis of mixings of earlier algorithms for Subset Sum and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory.
• 18. Averkov, Gennadiy
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results2015Inngår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 274, s. 118-166Artikkel i tidsskrift (Fagfellevurdert)
For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization.
• 19. Ayyer, Arvind
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Some genelalized juggling processes (extended abstract)2015Inngår i: DMTCS proc. FPSAC'15, Nancy, France, 2015, s. 925-936Konferansepaper (Fagfellevurdert)
We consider generalizations of juggling Markov chains introduced by Ayyer, Bouttier, Corteel and Nunzi. We first study multispecies generalizations of all the finite models therein, namely the MJMC, the add-drop and the annihilation models. We then consider the case of several jugglers exchanging balls. In all cases, we give explicit product formulas for the stationary probability and closed-form expressions for the normalization factor if known.
• 20.
University of St Andrews, Scotland.
University of St Andrews, Scotland. Mittuniversitetet, Fakulteten för naturvetenskap, teknik och medier, Avdelningen för matematik och ämnesdidaktik.
Sesqui-arrays, a generalisation of triple arrays2018Inngår i: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 71, nr 3, s. 427-451Artikkel i tidsskrift (Fagfellevurdert)
A triple array is a rectangular array containing letters, each letter occurring equally often with no repeats in rows or columns, such that the number of letters common to two rows, two columns, or a rowand a column are (possibly different) non-zero constants. Deleting the condition on the letters common to a row and a column gives a double array. We propose the term sesqui-array for such an array when only the condition on pairs of columns is deleted. In this paper we give three constructions for sesqui-arrays. Therst gives $(n + 1)\times n^2$ arrays on n(n + 1) letters for $n\geq 2$. (Suchan array for n = 2 was found by Bagchi.) This construction uses Latin squares. The second uses the Sylvester graph, a subgraph of the Hoffman--Singleton graph, to build a good block design for 36 treatments in 42 blocks of size 6, and then uses this in a 736 sesqui-array for 42 letters.We also give a construction for K(K-1)(K-2)/2 sesqui-arrays on K(K-1)/2 letters from biplanes. The construction starts with a block of a biplane and produces an array which satises the requirements for a sesqui-array except possibly that of having no repeated letters in a row or column. We show that this condition holds if and only if the Hussain chains for the selected block contain no 4-cycles. A sufficient condition for the construction to give a triple array is that each Hussain chain is a union of 3-cycles; but this condition is not necessary, and we give a few further examples. We also discuss the question of which of these arrays provide good designs for experiments.
• 21.
Universitat Politècnica de Catalunya, Barcelona, Spain.
The University of Western Australia, Crawley, WA, Australia. The University of Western Australia, Crawley, WA, Australia. Universitat Rovira i Virgili, Tarragona, Spain.
An alternative way to generalise the pentagon2013Inngår i: Journal of combinatorial designs (Print), ISSN 1063-8539, E-ISSN 1520-6610, Vol. 21, nr 4, s. 163-179Artikkel i tidsskrift (Fagfellevurdert)
We introduce the concept of a pentagonal geometry as a generalization of the pentagon and the Desargues configuration, in the same vein that the generalized polygons share the fundamental properties of ordinary polygons. In short, a pentagonal geometry is a regular partial linear space in which for all points x, the points not collinear with the point x, form a line. We compute bounds on their parameters, give some constructions, obtain some nonexistence results for seemingly feasible parameters and suggest a cryptographic application related to identifying codes of partial linear spaces.
• 22.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Classifications and volume bounds of lattice polytopes2017Licentiatavhandling, monografi (Annet vitenskapelig)
In this licentiate thesis we study relations among invariants of lattice polytopes, with particular focus on bounds for the volume.In the first paper we give an upper bound on the volume vol(P^*) of a polytope P^* dual to a d-dimensional lattice polytope P with exactly one interiorlattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. In the second paper we classify the three-dimensional lattice polytopes with two lattice points in their strict interior. Up to unimodular equivalence thereare 22,673,449 such polytopes. This classification allows us to verify, for this case only, the sharp conjectural upper bound for the volume of a lattice polytope with interior points, and provides strong evidence for more general new inequalities on the coefficients of the h^*-polynomial in dimension three.
• 23.
Christian-Albrechts Universität Kiel.
Christian-Albrechts Universität Kiel. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Christian-Albrechts Universität Kiel. Christian-Albrechts Universität Kiel.
Exact and heuristic algorithms for the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection2015Inngår i: Journal of the Operational Research Society, ISSN 0160-5682, E-ISSN 1476-9360, Vol. 66, nr 4, s. 615-626Artikkel i tidsskrift (Fagfellevurdert)
We introduce and study the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection (TSP-MTWHS), which generalises the well-known Travelling Salesman Problem with Time Windows and the recently introduced Travelling Salesman Problem with Hotel Selection. The TSP-MTWHS consists in determining a route for a salesman (eg, an employee of a services company) who visits various customers at different locations and different time windows. The salesman may require a several-day tour during which he may need to stay in hotels. The goal is to minimise the tour costs consisting of wage, hotel costs, travelling expenses and penalty fees for possibly omitted customers. We present a mixed integer linear programming (MILP) model for this practical problem and a heuristic combining cheapest insert, 2-OPT and randomised restarting. We show on random instances and on real world instances from industry that the MILP model can be solved to optimality in reasonable time with a standard MILP solver for several small instances. We also show that the heuristic gives the same solutions for most of the small instances, and is also fast, efficient and practical for large instances.
• 24.
Christian-Albrechts Universität Kiel, Germany.
Christian-Albrechts Universität Kiel, Germany. Christian-Albrechts Universität Kiel, Germany.
Construction of Sparse Asymmetric Connectors2003Inngår i: Proceedings of European Conference on Combinatorics, Graph Theory and Applications (Eurocomb 2003), 2003Konferansepaper (Fagfellevurdert)
• 25.
Christian-Albrechts-Universität Kiel, Germany.
Christian-Albrechts-Universität Kiel, Germany. Christian-Albrechts-Universität Kiel, Germany.
Constructions of Sparse Asymmetric Connectors2003Inngår i: Proceedings of 23rd Conference of Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2003) / [ed] P.K. Lodaya and J. Radhakrishnan, Berlin-Heidelberg: Springer Berlin/Heidelberg, 2003, s. 13-22Konferansepaper (Fagfellevurdert)
• 26.
Mathematisches seminar, Christian-Albrechts-Universität zu Kiel, Germany.
Mathematisches seminar, Christian-Albrechts-Universität zu Kiel, Germany. Mathematisches seminar, Christian-Albrechts-Universität zu Kiel, Germany.
Constructions of Sparse Asymmetric Connectors with Number Theoretic Methods2005Inngår i: Networks, ISSN 0028-3045, E-ISSN 1097-0037, Vol. 45, nr 3, s. 119-124Artikkel i tidsskrift (Fagfellevurdert)
We consider the problem of connecting a set I of n inputs to a set O of N outputs (n ≤ N) by as few edges as possible such that for every injective mapping f : I → O there are n vertex disjoint paths from i to f(i) of length k for a given k $\in \mathbb{N}$. For k = Ω(log N + log$^{2}$n) Oruς (1994) gave the presently best (n,N)-connector with O(N+n·log n) edges. For k=2 and N the square of a prime, Richards and Hwang (1985) described a construction using $N\left[\sqrt{n+5/4-1/2}\right] + n\left[\sqrt{n+5/4-1/2} \right]\sqrt{N}$edges. We show by a probabilistic argument that an optimal (n,N)-connector has Θ (N) edges, if $n\leq N^{1/2-\varepsilon}$for some ε>0. Moreover, we give explicit constructions based on a new number theoretic approach that need at most $N\left[\sqrt{3n/4} \right]+2n\left[\sqrt{3n/4} \right]\left[\sqrt{N} \right]$edges for arbitrary choices of n and N. The improvement we achieve is based on applying a generalization of the Erdös-Heilbronn conjecture on the size of restricted sums.
• 27. Barak, B.
KTH, Skolan för datavetenskap och kommunikation (CSC), Teoretisk datalogi, TCS.
Making the long code shorter2015Inngår i: SIAM journal on computing (Print), ISSN 0097-5397, E-ISSN 1095-7111, Vol. 44, nr 5, s. 1287-1324Artikkel i tidsskrift (Fagfellevurdert)
The long code is a central tool in hardness of approximation especially in questions related to the Unique Games Conjecture. We construct a new code that is exponentially more efficient but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results including the following: (1) For any ε > 0, we show the existence of an n-vertex graph G where every set of o(n) vertices has expansion 1-ε but G's adjacency matrix has more than exp(logδ n) eigenvalues larger than 1 - ε, where δ depends only on ε. This answers an open question of Arora, Barak, and Steurer [Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 2010, pp. 563-572] who asked whether one can improve over the noise graph on the Boolean hypercube that has poly(log n) such eigenvalues. (2) A gadget that reduces Unique Games instances with linear constraints modulo K into instances with alphabet k with a blowup of kpolylog(K) , improving over the previously known gadget with blowup of kω(K). (3) An n-variable integrality gap for Unique Games that survives exp(poly(log log n)) rounds of the semidefinite programming version of the Sherali-Adams hierarchy, improving on the previously known bound of poly(log log n). We show a connection between the local testability of linear codes and Small-Set Expansion in certain related Cayley graphs and use this connection to derandomize the noise graph on the Boolean hypercube.
• 28.
Högskolan i Halmstad, Sektionen för Informationsvetenskap, Data– och Elektroteknik (IDE).
Högskolan i Halmstad, Sektionen för Informationsvetenskap, Data– och Elektroteknik (IDE), Halmstad Embedded and Intelligent Systems Research (EIS), Tillämpad matematik och fysik (MPE-lab).
Meshfree methods in option pricing2011Independent thesis Advanced level (degree of Master (One Year)), 10 poäng / 15 hpOppgave
A meshfree approximation scheme based on the radial basis function methods is presented for the numerical solution of the options pricing model. This thesis deals with the valuation of the European, Barrier, Asian, American options of a single asset and American options of multi assets. The option prices are modeled by the Black-Scholes equation. The θ-method is used to discretize the equation with respect to time. By the next step, the option price is approximated in space with radial basis functions (RBF) with unknown parameters, in particular, we con- sider multiquadric radial basis functions (MQ-RBF). In case of Ameri- can options a penalty method is used, i.e. removing the free boundary is achieved by adding a small and continuous penalty term to the Black- Scholes equation. Finally, a comparison of analytical and finite difference solutions and numerical results from the literature is included.
• 29.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Shellability and the strong gcd-condition2009Inngår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, nr 2Artikkel i tidsskrift (Fagfellevurdert)
Shellability is a well-known combinatorial criterion on a simplicial complex for verifying that the associated Stanley-Reisner ring k[] is Cohen-Macaulay. Anotion familiar to commutative algebraists, but which has not received as muchattention from combinatorialists as the Cohen-Macaulay property, is the notion ofa Golod ring. Recently, J¨ollenbeck introduced a criterion on simplicial complexesreminiscent of shellability, called the strong gcd-condition, and he together with theauthor proved that it implies Golodness of the associated Stanley-Reisner ring. Thetwo algebraic notions were earlier tied together by Herzog, Reiner and Welker, whoshowed that if k[∨] is sequentially Cohen-Macaulay, where ∨ is the Alexanderdual of , then k[] is Golod. In this paper, we present a combinatorial companionof this result, namely that if ∨ is (non-pure) shellable then satisfies the stronggcd-condition. Moreover, we show that all implications just mentioned are strict ingeneral but that they are equivalences if is a flag complex.
• 30.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Shellability and the strong gcd-condition2009Inngår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, nr 2Artikkel i tidsskrift (Fagfellevurdert)
Shellability is a well-known combinatorial criterion on a simplicial complex for verifying that the associated Stanley-Reisner ring k[] is Cohen-Macaulay. Anotion familiar to commutative algebraists, but which has not received as muchattention from combinatorialists as the Cohen-Macaulay property, is the notion ofa Golod ring. Recently, J¨ollenbeck introduced a criterion on simplicial complexesreminiscent of shellability, called the strong gcd-condition, and he together with theauthor proved that it implies Golodness of the associated Stanley-Reisner ring. Thetwo algebraic notions were earlier tied together by Herzog, Reiner and Welker, whoshowed that if k[∨] is sequentially Cohen-Macaulay, where ∨ is the Alexanderdual of , then k[] is Golod. In this paper, we present a combinatorial companionof this result, namely that if ∨ is (non-pure) shellable then satisfies the stronggcd-condition. Moreover, we show that all implications just mentioned are strict ingeneral but that they are equivalences if is a flag complex.
• 31. Bhattacharya, S.
KTH, Skolan för datavetenskap och kommunikation (CSC), Teoretisk datalogi, TCS.
New deterministic approximation algorithms for fully dynamic matching2016Konferansepaper (Fagfellevurdert)
We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a (2 + ϵ)-approximate maximum matching in general graphs with O(poly(log n, 1/ϵ)) update time. (2) An algorithm that maintains an αk approximation of the value of the maximum matching with O(n2/K) update time in bipartite graphs, for every sufficiently large constant positive integer K. Here, 1 ≤ αk ≤ 2 is a constant determined by the value of K. Result (1) is the first deterministic algorithm that can maintain an o(log n)-approximate maximum matching with polylogarithmic update time, improving the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee almost matches the guarantee of the best randomized polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with arbitrarily small polynomial update time on bipartite graphs. Previously the best update time for this problem was O(m1/4) [Bernstein et al. ICALP 2015], where m is the current number of edges in the graph.
• 32. Bierbrauer, Jürgen
Örebro universitet, Institutionen för naturvetenskap.
Almost independent and weakly biased arrays: efficient constructions and cryptologic applications2000Inngår i: Advances in cryptology: CRYPTO 2000 / [ed] Mihir Bellare, Springer Berlin/Heidelberg, 2000, s. 533-543Konferansepaper (Annet vitenskapelig)
• 33.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Department of Information Science, Stellenbosch University, Stellenbosch, South Africa.
Minimization of Finite State Automata Through Partition Aggregation2016Inngår i: Logical Aspects of Computational Linguistics: Celebrating 20 Years of LACL (1996–2016) / [ed] Amblard, M DeGroote, P Pogodalla, S Retore, C, SPRINGER-VERLAG BERLIN , 2016, s. 328-328Konferansepaper (Fagfellevurdert)
• 34.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Graphical representations of Ising and Potts models: Stochastic geometry of the quantum Ising model and the space-time Potts model2009Doktoravhandling, monografi (Annet vitenskapelig)
HTML clipboard Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models.
In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in .
In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which gives the results.
In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in and in ‘star-like’ geometries.
• 35.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
On the connectivity of manifold graphs2015Inngår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, nr 10, s. 4123-4132Artikkel i tidsskrift (Fagfellevurdert)
This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold M is (2d - b_M)-connected. The parameter b_M has the property that b_M = 0 if the complex M is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d+1)-connected and Athanasiadis' theorem (2011) that flag d-manifold graphs are 2d-connected. The definition of b_M involves the concept of banner triangulations of manifolds, a generalization of flag triangulations.
• 36. Bogart, Tristram
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Finitely many smooth d-polytopes with n lattice points2015Inngår i: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 207, nr 1, s. 301-329Artikkel i tidsskrift (Fagfellevurdert)
We prove that for fixed n there are only finitely many embeddings of ℚ-factorial toric varieties X into ℙ n that are induced by a complete linear system. The proof is based on a combinatorial result that implies that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with ≤ 12 lattice points.
• 37.
Uppsala universitet, Fakultetsövergripande enheter, Centrum för bildanalys. Uppsala universitet, Fakultetsövergripande enheter, Centrum för bildanalys. Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Datoriserad bildanalys.
Kedjekod - ett sätt att beskriva former i digitala bilder2005Inngår i: Problemlösning är # 1, Liber, Stockholm , 2005, s. 38-42Kapittel i bok, del av antologi (Annet vitenskapelig)
• 38.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. -.
On Minimal Non-(2, 1)-Colorable Graphs2017Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
A graph is (2, 1)-colorable if it allows a partition of its vertices into two classes such that both induce graphs with maximum degree at most one. A non-(2, 1)-colorable graph is minimal if all proper subgraphs are (2, 1)-colorable. We prove that such graphs are 2-edge-connected and that every edge sits in an odd cycle. Furthermore, we show properties of edge cuts and particular graphs which are no induced subgraphs. We demonstrate that there are infinitely many minimal non-(2, 1)-colorable graphs, at least one of order n for all n ≥ 5. Moreover, we present all minimal non-(2, 1)- colorable graphs of order at most seven. We consider the maximum degree of minimal non-(2, 1)-colorable graphs and show that it is at least four but can be arbitrarily large. We prove that the average degree is greater than 8/3 and give sufficient properties for graphs with average degree greater than 14/5. We conjecture that all minimal non-(2, 1)-colorable graphs fulfill these properties.
• 39.
Universitat Rovira i Virgili, Catalonia, Spain.
Universitat Rovira i Virgili, Catalonia, Spain. Universitat Rovira i Virgili, Catalonia, Spain.
Configuraciones combinatóricas y recuperación privada de información por pares2009Konferansepaper (Annet vitenskapelig)
• 40.
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain.
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain.
On the existence of combinatorial configurations2010Inngår i: 3rd International Workshop on Optimal Networks Topologies, 2010, 2010, s. 145-167Konferansepaper (Annet vitenskapelig)
• 41.
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain .
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain .
The semigroup of combinatorial configurations2012Inngår i: Semigroup Forum, ISSN 0037-1912, E-ISSN 1432-2137, Vol. 84, nr 1, s. 91-96Artikkel i tidsskrift (Fagfellevurdert)
We elaborate on the existence and construction of the so-called combinatorial configurations. The main result is that for fixed degrees the existence of such configurations is given by a numerical semigroup. The proof is constructive giving a method to obtain combinatorial configurations with parameters large enough.
• 42.
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain.
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain. University College Dublin, Ireland.
Problems related to combinatorial configurations with applications to P2P-user private information retrieval2010Konferansepaper (Annet vitenskapelig)
• 43.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Matematik.
Leonardo da Pisa: Inger Christensen och Fibonacci2008Inngår i: Lyrikvännen, ISSN 0460-0762, Vol. 55, nr 6, s. 33-38Artikkel i tidsskrift (Annet (populærvitenskap, debatt, mm))
Den danska poeten Inger Christensen byggde sin diktsamling Alfabet på Fibonaccis talserie. Artikeln är en kort historisk essä över Fibonacci, och beskriver även hur talserien är konstruerad.
• 44.
Univ Kentucky, Dept Math, Lexington, KY 40506 USA..
Michigan State Univ, Dept Math, E Lansing, MI 48824 USA.. KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices2018Inngår i: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 100, s. 122-142Artikkel i tidsskrift (Fagfellevurdert)
A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) h*-polynomial. This conjecture can be viewed as a strengthening of a previously disproved conjecture which stated that any Gorenstein lattice polytope has a unimodal h*-polynomial. The first counterexamples to unimodality for Gorenstein lattice polytopes were given in even dimensions greater than five by Mustata and Payne, and this was extended to all dimensions greater than five by Payne. While there exist numerous examples in support of the conjecture that IDP reflexives are h*-unimodal, its validity has not yet been considered for families of reflexive lattice simplices that closely generalize Payne's counterexamples. The main purpose of this work is to prove that the former conjecture does indeed hold for a natural generalization of Payne's examples. The second purpose of this work is to extend this investigation to a broader class of lattice simplices, for which we present new results and open problems.
• 45.
Linköpings universitet, Matematiska institutionen, Optimeringslära. Linköpings universitet, Tekniska högskolan.
Linköpings universitet, Institutionen för datavetenskap, Artificiell intelligens och integrerade datorsystem. Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för datavetenskap, Artificiell intelligens och integrerade datorsystem. Linköpings universitet, Tekniska högskolan.
Local Search for Hop-constrained Directed Steiner Tree Problem with Application to UAV-based Multi-target Surveillance2014Rapport (Annet vitenskapelig)
We consider the directed Steiner tree problem (DSTP) with a constraint on the total number of arcs (hops) in the tree. This problem is known to be NP-hard, and therefore, only heuristics can be applied in the case of its large-scale instances. For the hop-constrained DSTP, we propose local search strategies aimed at improving any heuristically produced initial Steiner tree. They are based on solving a sequence of hop-constrained shortest path problems for which we have recently developed ecient label correcting algorithms. The presented approach is applied to nding suitable 3D locations where unmanned aerial vehicles (UAVs) can be placed to relay information gathered in multi-target monitoring and surveillance. The eciency of our algorithms is illustrated by results of numerical experiments involving problem instances with up to 40 000 nodes and up to 20 million arcs.
• 46.
Linköpings universitet, Matematiska institutionen, Optimeringslära. Linköpings universitet, Tekniska högskolan.
Linköpings universitet, Institutionen för datavetenskap, Artificiell intelligens och integrerade datorsystem. Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för datavetenskap, Artificiell intelligens och integrerade datorsystem. Linköpings universitet, Tekniska högskolan.
Local Search for Hop-constrained Directed Steiner Tree Problem with Application to UAV-based Multi-target Surveillance2014Inngår i: Examining Robustness and Vulnerability of Networked Systems / [ed] Butenko, S., Pasiliao, E.L., Shylo, V., IOS Press, 2014, s. 26-50Konferansepaper (Fagfellevurdert)
We consider the directed Steiner tree problem (DSTP) with a constraint on the total number of arcs (hops) in the tree. This problem is known to be NP-hard, and therefore, only heuristics can be applied in the case of its large-scale instances.For the hop-constrained DSTP, we propose local search strategies aimed at improving any heuristically produced initial Steiner tree. They are based on solving a sequence of hop-constrained shortest path problems for which we have recently developed efficient label correcting algorithms.The presented approach is applied to finding suitable 3D locations where unmanned aerial vehicles (UAVs) can be placed to relay information gathered in multi-target monitoring and surveillance. The efficiency of our algorithms is illustrated by results of numerical experiments involving problem instances with up to 40 000 nodes and up to 20 million arcs.
• 47.
Plymouth State University, NH, USA.
MIT, MA, USA. Bard Coll Simons Rock, MA, USA. Portola Valley, CA USA. MIT, MA, USA. Swiss Federal Institute Technology, Switzerland. The University of Electro-Communications, Chofu, Japan. University of Iibre Bruxelles, Belgium. Bard Coll Simons Rock, MA, USA. University of Utrecht, Netherlands. Bard Coll Simons Rock, MA, USA. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem. Linköpings universitet, Tekniska fakulteten. University of Elect Communicat, Japan. Japan Adv Institute Science and Technology, Japan. Osaka Prefecture University, Japan. Bard Coll Simons Rock, MA, USA.
Single-Player and Two-Player Buttons & Scissors Games2016Inngår i: DISCRETE AND COMPUTATIONAL GEOMETRY AND GRAPHS, JCDCGG 2015 / [ed] Akiyama, J., Ito, H., Sakai, T., Uno, Y., SPRINGER INT PUBLISHING AG , 2016, Vol. 9943, s. 60-72Konferansepaper (Fagfellevurdert)
We study the computational complexity of the Buttons amp; Scissors game and obtain sharp thresholds with respect to several parameters. Specifically we show that the game is NP-complete for C = 2 colors but polytime solvable for C = 1. Similarly the game is NP-complete if every color is used by at most F = 4 buttons but polytime solvable for F amp;lt;= 3. We also consider restrictions on the board size, cut directions, and cut sizes. Finally, we introduce several natural two-player versions of the game and show that they are PSPACE-complete.
• 48.
Linköpings universitet, Institutionen för datavetenskap, Programvara och system. Linköpings universitet, Tekniska fakulteten.
Linköpings universitet, Institutionen för datavetenskap, Programvara och system. Linköpings universitet, Tekniska fakulteten.
Time and Space Bounds for Planning2017Inngår i: The journal of artificial intelligence research, ISSN 1076-9757, E-ISSN 1943-5037, Vol. 60, s. 595-638Artikkel i tidsskrift (Fagfellevurdert)
There is an extensive literature on the complexity of planning, but explicit bounds on time and space complexity are very rare. On the other hand, problems like the constraint satisfaction problem (CSP) have been thoroughly analysed in this respect. We provide a number of upper- and lower-bound results (the latter based on various complexity-theoretic assumptions such as the Exponential Time Hypothesis) for both satisficing and optimal planning. We show that many classes of planning instances exhibit a dichotomy: either they can be solved in polynomial time or they cannot be solved in subexponential time and thus require O (2(cn)) time for some c amp;gt; 0. In many cases, we can even prove closely matching upper and lower bounds; for every epsilon amp;gt; 0, the problem can be solved in time O (2((1+epsilon)n)) but not in time O (2((1-epsilon)n)). Our results also indicate, analogously to CSPs, the existence of sharp phase transitions. We finally study and discuss the trade-off between time and space. In particular, we show that depth-first search may sometimes be a viable option for planning under severe space constraints.
• 49.
Högskolan Dalarna, Akademin Industri och samhälle, Statistik.
Högskolan Dalarna, Akademin Industri och samhälle, Statistik. Högskolan Dalarna, Akademin Industri och samhälle, Informatik. Högskolan Dalarna, Akademin Industri och samhälle, Datateknik.
Testing the gravity p-median model empirically2015Inngår i: Operations Research Perspectives, ISSN 2214-7160, Vol. 2, nr 124, artikkel-id 132Artikkel i tidsskrift (Fagfellevurdert)
Regarding the location of a facility, the presumption in the widely used p-median model is that the customer opts for the shortest route to the nearest facility. However, this assumption is problematic on free markets since the customer is presumed to gravitate to a facility by the distance to and the attractiveness of it. The recently introduced gravity p-median model offers an extension to the p-median model that account for this. The model is therefore potentially interesting, although it has not yet been implemented and tested empirically. In this paper, we have implemented the model in an empirical problem of locating vehicle inspections, locksmiths, and retail stores of vehicle spare-parts for the purpose of investigating its superiority to the p-median model. We found, however, the gravity p-median model to be of limited use for the problem of locating facilities as it either gives solutions similar to the p-median model, or it gives unstable solutions due to a non-concave objective function.
• 50.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On avoiding some families of arrays2012Inngår i: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 312, nr 5, s. 963-972Artikkel i tidsskrift (Fagfellevurdert)
An n×n array A with entries from {1,…,n} is avoidable if there is an n×n Latin square L such that no cell in L contains a symbol that occurs in the corresponding cell in A. We show that the problem of determining whether an array that contains at most two entries per cell is avoidable is NP-complete, even in the case when the array has entries from only two distinct symbols. Assuming that PNP, this disproves a conjecture by Öhman. Furthermore, we present several new families of avoidable arrays. In particular, every single entry array (arrays where each cell contains at most one symbol) of order n≥2k with entries from at most k distinct symbols and where each symbol occurs in at most n−2 cells is avoidable, and every single entry array of order n, where each of the symbols 1,…,n occurs in at most cells, is avoidable. Additionally, if k≥2, then every single entry array of order at least n≥4, where at most k rows contain non-empty cells and where each symbol occurs in at most nk+1 cells, is avoidable.
1234567 1 - 50 of 379
Referera
Referensformat
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Annet format
Fler format
Språk
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Annet språk
Fler språk
Utmatningsformat
• html
• text
• asciidoc
• rtf
v. 2.35.2
| | | 2018-09-20 04:37:10 | {"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": 6, "/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": 0.7077528238296509, "perplexity": 2499.9766627107138}, "config": {"markdown_headings": true, "markdown_code": false, "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-2018-39/segments/1537267156416.22/warc/CC-MAIN-20180920041337-20180920061337-00226.warc.gz"} |
https://brilliant.org/problems/balls-thrown-up/ | # Balls Thrown Up
A very large number of balls are thrown vertically upwards in quick succession in such a way that the next ball is thrown when the previous one is at maximum height. If the maximum height reached by each ball is 45 meters, then find the number of balls thrown up in a minute.
Assume that $$g = -9.8$$ m/s$$^2$$, and ignore factors like wind resistance.
× | 2017-07-24 10:43:45 | {"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": 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": 0.7164854407310486, "perplexity": 252.7875829082527}, "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-2017-30/segments/1500549424846.81/warc/CC-MAIN-20170724102308-20170724122308-00577.warc.gz"} |
https://msdn.microsoft.com/en-US/library/cx053bca(v=vs.71).aspx | # inline_depth
Visual Studio .NET 2003
```#pragma inline_depth( [0... 255] )
```
Controls the number of times inline expansion can occur by controlling the number of times that a series of function calls can be expanded (from 0 to 255 times). This pragma controls the inlining of functions marked inline and __inline or inlined automatically under the /Ob2 option.
The inline_depth pragma controls the number of times a series of function calls can be expanded. For example, if the inline depth is four, and if A calls B and B then calls C, all three calls will be expanded inline. However, if the closest inline expansion is two, only A and B are expanded, and C remains as a function call.
To use this pragma, you must set the /Ob compiler option to 1 or 2. The depth set using this pragma takes effect at the first function call after the pragma. If you do not specify a value within the parentheses, inline_depth sets the inline depth back to its default value of 8.
The inline depth can be decreased during expansion but not increased. If the inline depth is six and during expansion the preprocessor encounters an inline_depth pragma with a value of eight, the depth remains six.
An inline depth of 0 inhibits inline expansion; an inline depth of 255 places no limit on inline expansion. If either pragma is used without specifying a value, the default value is used. | 2015-03-28 09:47:49 | {"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": 0.8952776193618774, "perplexity": 2187.792638034821}, "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-2015-14/segments/1427131297416.52/warc/CC-MAIN-20150323172137-00247-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://conference.portonvictor.org/w/index.php?title=Funcoid_bases/Proving_existence_of_funcoid_through_lattice_Gamma&oldid=92&diff=prev | Publish here. Donate
Difference between revisions of "Funcoid bases/Proving existence of funcoid through lattice Gamma"
Statement Under the conditions "1" or "2" of our conjecture, we have that $S=\operatorname{up}\bigwedge^{\mathsf{FCD}} S=\operatorname{up}\bigwedge^{\mathsf{FCD}} S'$.
Proof Denote $S' = S \cap \Gamma$. It's easy to show that $S'$ is a filter on $\Gamma$ (in my book it's proved that $\Gamma$ is a sublattice of $\mathsf{FCD}$).
For every ultrafilter $x$ we have
$\operatorname{up} \langle Y \rangle x \subseteq \operatorname{up} \left\langle \bigwedge^{\mathsf{FCD}} S' \right\rangle x = \\ \operatorname{up} \bigwedge_{X \in S'} \langle X \rangle x = \\ \left\{ \langle X_0 \rangle x \sqcap \ldots \sqcap \langle X_n \rangle x \mid i = 0, \ldots, n, X_i \in S' \right\} = \\ \left\{ \langle X \rangle x \mid X \in S' \right\}$
(taken into account that $\langle X_i\rangle x$ is a principal filter).
It follows $\langle Y \rangle x \in \left\{ \langle X \rangle x \mid X \in S' \right\}$ that is
$\langle Y \rangle x = \langle X \rangle x$ for some $X \in S'$.
The next question is whether we can find such $X \in S$ (not necessarily $X \in S'$!) that $\langle Y \rangle x = \langle X \rangle x$ for all ultrafilters $x$. Or is there a counter-example for existence of such $X$? | 2020-10-21 22:13:18 | {"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": 0.9414550065994263, "perplexity": 355.2964108315468}, "config": {"markdown_headings": false, "markdown_code": false, "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-2020-45/segments/1603107878633.8/warc/CC-MAIN-20201021205955-20201021235955-00627.warc.gz"} |
https://math.stackexchange.com/questions/1406058/can-a-fractal-be-a-manifold | # Can a fractal be a manifold?
Here it is said that it is not possible:
Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?
But I am confused about this. What about the invariant manifolds (stable and unstable manifolds) of chaotic flows? Here it says they are fractals:
For example for some parametrization of the Henon map the unstable manifold is similar (indeed it is contained in it) to the Henon attractor, and the Henon attractor is a fractal.
• thank you for the follow up on my question, :) the link and the answer are very interesting. – iadvd Aug 25 '15 at 1:05
Invariant "manifolds" are not generally submanifolds in the strict sense of differential topology.
For instance, in a 2-dimensional dynamical system with a fixed point of index 1, the invariant unstable 1-manifold can be described in an appropriate local coordinate chart $U \approx I \times J$ as $I \times C$ for some countable subset $C \subset J$, and this set $C$ need not be discrete as required for a submanifold.
The closure of the invariant manifold will typically be a "lamination", meaning that in an appropriate coordinate chart $U \approx I \times J$ it has the form $I \times \tau$ where $\tau \subset J$ is a closed subset; in the context of the previous paragraph, $\tau$ is the closure of $C$. The "leaves" of this laminations are subsets of the form $I \times t$, $t \in \tau$. This subset $\tau$ could possibly be of fractional Hausdorff dimension, and that is what leads to the possibility that attractors can be fractal.
So, for instance, in your statement the Henon attractor equals the closure of the unstable manifold.
• What about the stable manifold theorem? It says that the stable and unstable manifolds are smooth manifolds, then if they are fractals, a fractal can be a manifold. Is it right? See: en.wikipedia.org/wiki/Stable_manifold_theorem – rbd33 Aug 22 '15 at 18:31
• That wikipedia page does not give a correct definition of stable and unstable manifolds nor does it give a correct statement of the stable manifold theorem. – Lee Mosher Aug 22 '15 at 22:28
• A correct stable manifold theorem has an extra existential quantifier: there exists a smaller neighborhood $V \subset U$ in which the stable manifold is a smooth manifold. One might then call this the "local stable manifold in $V$", and it might be more accurately denoted by affixing $V$ to the notation, e.g.\ $W^s_V(p)$. The global stable manifold is then defined to be $\cup_{i=1}^\infty f^{-i}\bigl(W^s_V(p)\bigr)$, and this can fail to be a manifold in the global sense as stated in my answer. – Lee Mosher Aug 22 '15 at 22:28
• For more details on this topic I recommend Michael Shub's book "Global Stability of Dynamical Systems", amazon.com/Global-Stability-Dynamical-Systems-Michael/dp/… – Lee Mosher Aug 22 '15 at 22:30
• Thanks Lee for the explanation! – rbd33 Aug 23 '15 at 23:07 | 2019-05-20 00:19:48 | {"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": 0.753955602645874, "perplexity": 348.68395264604186}, "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-2019-22/segments/1558232255251.1/warc/CC-MAIN-20190520001706-20190520023706-00179.warc.gz"} |
https://www.law.cornell.edu/cfr/text/40/60.42b | # 40 CFR § 60.42b - Standard for sulfur dioxide (SO2).
§ 60.42b Standard for sulfur dioxide (SO2).
(a) Except as provided in paragraphs (b), (c), (d), or (j) of this section, on and after the date on which the performance test is completed or required to be completed under § 60.8, whichever comes first, no owner or operator of an affected facility that commenced construction, reconstruction, or modification on or before February 28, 2005, that combusts coal or oil shall cause to be discharged into the atmosphere any gases that contain SO2 in excess of 87 ng/J (0.20 lb/MMBtu) or 10 percent (0.10) of the potential SO2 emission rate (90 percent reduction) and the emission limit determined according to the following formula:
${E}_{s}=\frac{\left({K}_{a}{H}_{a}+{K}_{b}{H}_{b}\right)}{\left({H}_{a}+{H}_{b}\right)}$
Where:
Es = SO2 emission limit, in ng/J or lb/MMBtu heat input;
Ka = 520 ng/J (or 1.2 lb/MMBtu);
Kb = 340 ng/J (or 0.80 lb/MMBtu);
Ha = Heat input from the combustion of coal, in J (MMBtu); and
Hb = Heat input from the combustion of oil, in J (MMBtu).
For facilities complying with the percent reduction standard, only the heat input supplied to the affected facility from the combustion of coal and oil is counted in this paragraph. No credit is provided for the heat input to the affected facility from the combustion of natural gas, wood, municipal-type solid waste, or other fuels or heat derived from exhaust gases from other sources, such as gas turbines, internal combustion engines, kilns, etc.
(b) On and after the date on which the performance test is completed or required to be completed under § 60.8, whichever date comes first, no owner or operator of an affected facility that commenced construction, reconstruction, or modification on or before February 28, 2005, that combusts coal refuse alone in a fluidized bed combustion steam generating unit shall cause to be discharged into the atmosphere any gases that contain SO2 in excess of 87 ng/J (0.20 lb/MMBtu) or 20 percent (0.20) of the potential SO2 emission rate (80 percent reduction) and 520 ng/J (1.2 lb/MMBtu) heat input. If coal or oil is fired with coal refuse, the affected facility is subject to paragraph (a) or (d) of this section, as applicable. For facilities complying with the percent reduction standard, only the heat input supplied to the affected facility from the combustion of coal and oil is counted in this paragraph. No credit is provided for the heat input to the affected facility from the combustion of natural gas, wood, municipal-type solid waste, or other fuels or heat derived from exhaust gases from other sources, such as gas turbines, internal combustion engines, kilns, etc.
(c) On and after the date on which the performance test is completed or is required to be completed under § 60.8, whichever comes first, no owner or operator of an affected facility that combusts coal or oil, either alone or in combination with any other fuel, and that uses an emerging technology for the control of SO2 emissions, shall cause to be discharged into the atmosphere any gases that contain SO2 in excess of 50 percent of the potential SO2 emission rate (50 percent reduction) and that contain SO2 in excess of the emission limit determined according to the following formula:
${E}_{s}=\frac{\left({K}_{c}{H}_{c}+{K}_{d}{H}_{d}\right)}{\left({H}_{c}+{H}_{d}\right)}$
Where:
Es = SO2 emission limit, in ng/J or lb/MM Btu heat input;
Kc = 260 ng/J (or 0.60 lb/MMBtu);
Kd = 170 ng/J (or 0.40 lb/MMBtu);
Hc = Heat input from the combustion of coal, in J (MMBtu); and
Hd = Heat input from the combustion of oil, in J (MMBtu).
For facilities complying with the percent reduction standard, only the heat input supplied to the affected facility from the combustion of coal and oil is counted in this paragraph. No credit is provided for the heat input to the affected facility from the combustion of natural gas, wood, municipal-type solid waste, or other fuels, or from the heat input derived from exhaust gases from other sources, such as gas turbines, internal combustion engines, kilns, etc.
(d) On and after the date on which the performance test is completed or required to be completed under § 60.8, whichever comes first, no owner or operator of an affected facility that commenced construction, reconstruction, or modification on or before February 28, 2005 and listed in paragraphs (d)(1), (2), (3), or (4) of this section shall cause to be discharged into the atmosphere any gases that contain SO2 in excess of 520 ng/J (1.2 lb/MMBtu) heat input if the affected facility combusts coal, or 215 ng/J (0.5 lb/MMBtu) heat input if the affected facility combusts oil other than very low sulfur oil. Percent reduction requirements are not applicable to affected facilities under paragraphs (d)(1), (2), (3) or (4) of this section. For facilities complying with paragraphs (d)(1), (2), or (3) of this section, only the heat input supplied to the affected facility from the combustion of coal and oil is counted in this paragraph. No credit is provided for the heat input to the affected facility from the combustion of natural gas, wood, municipal-type solid waste, or other fuels or heat derived from exhaust gases from other sources, such as gas turbines, internal combustion engines, kilns, etc.
(1) Affected facilities that have an annual capacity factor for coal and oil of 30 percent (0.30) or less and are subject to a federally enforceable permit limiting the operation of the affected facility to an annual capacity factor for coal and oil of 30 percent (0.30) or less;
(2) Affected facilities located in a noncontinental area; or
(3) Affected facilities combusting coal or oil, alone or in combination with any fuel, in a duct burner as part of a combined cycle system where 30 percent (0.30) or less of the heat entering the steam generating unit is from combustion of coal and oil in the duct burner and 70 percent (0.70) or more of the heat entering the steam generating unit is from the exhaust gases entering the duct burner; or
(4) The affected facility burns coke oven gas alone or in combination with natural gas or very low sulfur distillate oil.
(e) Except as provided in paragraph (f) of this section, compliance with the emission limits, fuel oil sulfur limits, and/or percent reduction requirements under this section are determined on a 30-day rolling average basis.
(f) Except as provided in paragraph (j)(2) of this section, compliance with the emission limits or fuel oil sulfur limits under this section is determined on a 24-hour average basis for affected facilities that (1) have a federally enforceable permit limiting the annual capacity factor for oil to 10 percent or less, (2) combust only very low sulfur oil, and (3) do not combust any other fuel.
(g) Except as provided in paragraph (i) of this section and § 60.45b(a), the SO2 emission limits and percent reduction requirements under this section apply at all times, including periods of startup, shutdown, and malfunction.
(h) Reductions in the potential SO2 emission rate through fuel pretreatment are not credited toward the percent reduction requirement under paragraph (c) of this section unless:
(1)Fuel pretreatment results in a 50 percent or greater reduction in potential SO2 emissions and
(2) Emissions from the pretreated fuel (without combustion or post-combustion SO2 control) are equal to or less than the emission limits specified in paragraph (c) of this section.
(i) An affected facility subject to paragraph (a), (b), or (c) of this section may combust very low sulfur oil or natural gas when the SO2 control system is not being operated because of malfunction or maintenance of the SO2 control system.
(j) Percent reduction requirements are not applicable to affected facilities combusting only very low sulfur oil. The owner or operator of an affected facility combusting very low sulfur oil shall demonstrate that the oil meets the definition of very low sulfur oil by: (1) Following the performance testing procedures as described in § 60.45b(c) or § 60.45b(d), and following the monitoring procedures as described in § 60.47b(a) or § 60.47b(b) to determine SO2 emission rate or fuel oil sulfur content; or (2) maintaining fuel records as described in § 60.49b(r).
(k)
(1) Except as provided in paragraphs (k)(2), (k)(3), and (k)(4) of this section, on and after the date on which the initial performance test is completed or is required to be completed under § 60.8, whichever date comes first, no owner or operator of an affected facility that commences construction, reconstruction, or modification after February 28, 2005, and that combusts coal, oil, natural gas, a mixture of these fuels, or a mixture of these fuels with any other fuels shall cause to be discharged into the atmosphere any gases that contain SO2 in excess of 87 ng/J (0.20 lb/MMBtu) heat input or 8 percent (0.08) of the potential SO2 emission rate (92 percent reduction) and 520 ng/J (1.2 lb/MMBtu) heat input. For facilities complying with the percent reduction standard and paragraph (k)(3) of this section, only the heat input supplied to the affected facility from the combustion of coal and oil is counted in paragraph (k) of this section. No credit is provided for the heat input to the affected facility from the combustion of natural gas, wood, municipal-type solid waste, or other fuels or heat derived from exhaust gases from other sources, such as gas turbines, internal combustion engines, kilns, etc.
(2) Units firing only very low sulfur oil, gaseous fuel, a mixture of these fuels, or a mixture of these fuels with any other fuels with a potential SO2 emission rate of 140 ng/J (0.32 lb/MMBtu) heat input or less are exempt from the SO2 emissions limit in paragraph (k)(1) of this section.
(3) Units that are located in a noncontinental area and that combust coal, oil, or natural gas shall not discharge any gases that contain SO2 in excess of 520 ng/J (1.2 lb/MMBtu) heat input if the affected facility combusts coal, or 215 ng/J (0.50 lb/MMBtu) heat input if the affected facility combusts oil or natural gas.
(4) As an alternative to meeting the requirements under paragraph (k)(1) of this section, modified facilities that combust coal or a mixture of coal with other fuels shall not cause to be discharged into the atmosphere any gases that contain SO2 in excess of 87 ng/J (0.20 lb/MMBtu) heat input or 10 percent (0.10) of the potential SO2 emission rate (90 percent reduction) and 520 ng/J (1.2 lb/MMBtu) heat input.
[72 FR 32742, June 13, 2007, as amended at 74 FR 5084, Jan. 28, 2009; 76 FR 3523, Jan. 20, 2011] | 2019-09-19 19:15:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "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": 0.5768980979919434, "perplexity": 3858.3884062336438}, "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-2019-39/segments/1568514573570.6/warc/CC-MAIN-20190919183843-20190919205843-00126.warc.gz"} |
https://lists.torproject.org/pipermail/tor-dev/2014-December/007938.html | # [tor-dev] Proposal draft: Better hidden service stats from Tor relays
A. Johnson aaron.m.johnson at nrl.navy.mil
Wed Dec 10 14:37:59 UTC 2014
> But I don't see the value of binning the result once more. In a
> sense, we're already binning signal + noise by cutting off the float
> part. I don't see what we gain by reducing resolution even more. It
> seems just unnecessary.
In principle releasing the number could result in different differential-privacy guarantees than releasing the bin. However, the way I had in mind to set the Laplace parameters this wouldn’t be an issue, because the Laplace distributions themselves would satisfy the desired differential privacy guarantee (and not just the resulting distribution on bins).
So I guess this could be viewed as a post-processing step that is useful for clarity rather than privacy: namely, that the output should be interpreted as a range. But we could leave this to the data consumer to apply without a privacy issue.
Also, I believe that the parameters we had discussed should change. To see why, observe that the Laplace distributions for two adjacent values that cross a bin barrier are now very far apart after being recentered within the appropriate bins. Thus, \delta_f should increase if it is smaller than the maximum number of bins that can be crossed within that \delta_f multiplied by the bin size. With our previous numbers, the new \delta_f for rendezvous cell counts doesn’t change (still 2048), but the new \delta_f for HS descriptors counts is 8.
Aaron | 2019-08-21 20:39:44 | {"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": 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": 0.5208873152732849, "perplexity": 1184.579560542194}, "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/1566027316194.18/warc/CC-MAIN-20190821194752-20190821220752-00514.warc.gz"} |
http://jdh.hamkins.org/lectures-on-the-philosophy-of-mathematics-oxford-michaelmas-2018/?replytocom=9336 | # Lectures on the Philosophy of Mathematics, Oxford, Michaelmas 2018
This will be a series of lectures on the philosophy of mathematics, given at Oxford University, Michaelmas term 2018. The lectures are mainly intended for undergraduate students preparing for exam paper 122, although all interested parties are welcome.
My approach to the philosophy of mathematics tends to be grounded in mathematical arguments and ideas, treating philosophical issues as they arise organically. The lectures will accordingly be organized around mathematical themes, in such a way that naturally brings various philosophical issues to light.
Here is a tentative list of topics, which may be updated as the term approaches.
Lecture 1. Numbers. Numbers are perhaps the essential mathematical idea, but what are numbers? We have many kinds of numbers—natural numbers, integers, rational numbers, real numbers, complex numbers, hyperreal numbers, surreal numbers, ordinal numbers, and more—and these number systems provide a fruitful background for classical arguments on incommensurability, the irrationality of $\sqrt{2}$, transcendental numbers, the infinitude of primes, and lead naturally to discussions of platonism, Frege’s number concept, Peano’s numbers, Dedekind’s categoricity arguments, and the philosophy of structuralism.
Lecture 2. Rigour. Let us consider the problem of mathematical rigour in the development of the calculus. Informal continuity concepts and the use of infinitesimals ultimately gave way to formal epsilon-delta limit concepts, which provided a capacity for refined notions, such as uniform continuity, equicontinuity and uniform convergence. Nonstandard analysis resurrected the infinitesimal concept on a more secure foundation, providing a parallel development of the subject, which can be understood from various sweeping perspectives. Meanwhile, increasing abstraction emerged in the function concept, which we shall illustrate with the Devil’s staircase, space-filling curves and the Conway base 13 function.
Lecture 3. Infinity. We shall follow the allegory of Hilbert’s hotel and the paradox of Galileo to the equinumerosity relation and the notion of countability. Cantor’s diagonal arguments, meanwhile, reveal uncountability and a vast hierarchy of different orders of infinity; some arguments give rise to the distinction between constructive and non-constructive proof. Zeno’s paradox highlights classical ideas on potential versus actual infinity. Time permitting, we shall count into the transfinite ordinals.
Lecture 4. Geometry. Classical Euclidean geometry, accompanied by its ideal of straightedge and compass construction and the Euclidean concept of proof, is an ageless paragon of deductive mathematical reasoning. Yet, the impossibility of certain constructions, such as doubling the cube, trisecting the angle or squaring the circle, hints at geometric realms beyond Euclid, and leads one to the concept of constructible and non-constructible numbers. The rise of non-Euclidean geometry, especially in light of scientific observations and theories suggesting that physical reality may not be Euclidean, challenges previous accounts of what geometry is about and changes our understanding of the nature of geometric and indeed mathematical ontology. New formalizations, such as those of Hilbert and Tarski, replace the old axiomatizations, augmenting and correcting Euclid with axioms on completeness and betweenness. Ultimately, Tarski’s decision procedure hints at the tantalizing possibility of automation in our geometrical reasoning.
Lecture 5. Proof. What is proof? What is the relation between proof and truth? Is every mathematical truth, true for a reason? After clarifying the distinction between syntax and semantics, we shall discuss formal proof systems and highlight the importance of soundness, completeness and verifiability in any such system, outlining the central ideas used in proving the completeness theorem. The compactness theorem distills the finiteness of proofs into an independent purely semantic consequence. Computer-verified proof promises increasing significance; it’s role is well illustrated by the history of the four-color theorem. Nonclassical logics, such as intuitionistic logic, arise naturally from formal systems by weakenings of the logical rules.
Lecture 6. Computability. What is computability? Gödel’s primitive recursive functions were a robust class, yet he gave reasons to despair of a fully satisfactory answer. Nevertheless, Turing’s machine concept, growing out of his careful philosophical analysis of computability, laid a foundation for the contemporary computer era, and the widely accepted Church-Turing thesis asserts that Turing has the right notion. Meanwhile, the distinction between computable decidability and computable enumerability, highlighted by the undecidability of the halting problem, shows that not all mathematical problems can be solved by machine, and a vast hierarchy looms in the Turing degrees, an infinitary information theory. Complexity theory refocuses this on the realm of feasible computation, with the still-unsolved P vs. NP problem standing in the background of nearly every serious issue in theoretical computer science.
Lecture 7. Incompleteness. The Hilbert program, seeking to secure the consistency of higher mathematics by finitary reasoning about the formal system underlying it, was dashed by Gödel’s incompleteness theorems, which show that no consistent formal system can prove even its own consistency, let alone the consistency of a higher system. We shall describe several proofs of the first incompleteness theorem, via the halting problem, via self-reference, and via definability. After this, we’ll discuss the Rosser variation, the second incompleteness theorem, and Tarski on the non-definability of truth. Ultimately, one is led to the inherent hierarchy of consistency strength underlying all mathematical theories.
Lecture 8. Set theory. We shall discuss the emergence of set theory as a foundation of mathematics. An initially naive theory, challenged fundamentally by the Russell paradox, grew into Zermelo’s formal set theory, founded on the idea of a cumulative universe of sets and providing a robust general context in which to undertake mathematics, while also enabling the clarification of fundamentally set-theoretic issues surrounding the axiom of choice, the continuum hypothesis and an increasingly diverse hierarchy of large cardinal concepts. The development of forcing solved many stubborn questions and illuminated a ubiquitous independence phenomenon, feeding into philosophical issues concerning the criteria by which one should add new axioms to mathematics and the question of pluralism in mathematical foundations.
## 14 thoughts on “Lectures on the Philosophy of Mathematics, Oxford, Michaelmas 2018”
1. Joel, this sounds wonderful! Unfortunately I won’t be crossing the pond. Will notes, texts, or videos eventually become available? I am particularly intrigued by and interested in your lecture descriptions on Number, Infinity, and Set Theory. (I once taught a semester elective on The Mathematics of Infinity at The Beekman School in NYC).
All the best,
Charlie Sitler
• I’m not sure about videos, although I shall inquire about it. Meanwhile, I am preparing lecture notes, which with some work, may eventually become a book.
• The structure of the course is quite different from many introductory courses of philosophy of mathematics that I know. It will be a very interesting book. Looking forward to it.
• Yes, indeed. I was a little worried about that, but decided to organize things in the way that I understand the subject, in a way that makes sense to me, leading to the philosophical issues from the mathematics. How is your syllabus organized? I would be interested to learn.
2. Sounds like a well-rounded introduction that can be very useful. How much time will you have to explain that set theory didn’t just spring into existence and that it took some convincing (by Skolem I think) that first-order logic with set theory was an option. Many people (including Hilbert) expected that first-order logic was insufficient (which it is, but that’s a debate for another day).
• Oh yes, this is a very interesting part of the history. Zermelo’s original formulation was essentially second-order, and this was important in his categoricity proof, showing that the models of his theory are precisely the $V_\kappa$ for inaccessible cardinals $\kappa$. But ultimately, he lost the battle and the theory now bearing his name is nearly universally understood as a first-order theory.
3. Put me down as another eager customer for the book version!
• Glad to hear it! Thanks for the vote of confidence.
4. The purpose of all mathematics (including large cardinals) is to build taller skyscrapers, longer bridges, and better I-phones; any philosophy of mathematics needs to reflect this purpose.
• I suppose we’ll discuss that view. What do you think are the strongest arguments in its favor? Meanwhile, it seems to me that many mathematicians also study mathematics for reasons that seem to have little to do with applications, and are rather closer to values of art and beauty. Or perhaps it is similar to the reasons why some scientists find scientific investigation of nature to have value, even when it is divorced from practical application. Do you think they are wrong to do so? I suppose they could stop, and become taxi drivers or software engineers, but from my perspective, we would be worse off if they did so.
• Mathematicians should study the areas of mathematics that they deem elegant, but these areas of mathematics are more often than not applicable to the universe we live in.
The applicable areas of mathematics are often much deeper than the unapplied areas of mathematics. For example, the mathematical objects used in cryptography need to be ‘structured’,’combinatorially complicated’, and ‘easily computable’ objects. These characteristics that make a certain mathematical object suitable for cryptography are the same characteristics that make a mathematical object interesting for purely aesthetic reasons and also make it possible to develop a deep theory behind such a mathematical object. Furthermore, any ‘structured’,’combinatorially complicated’, and ‘easily computable’ mathematical structure has a fairly good chance of being applicable to cryptography.
The creator of the universe has decided to make the laws of nature based upon the most useful and beautiful mathematics. Mathematicians who are interested in mathematics for its own sake and for the sake of beauty should do more applied mathematics than pure mathematics. After all, applied mathematics is the mathematics that the creator likes enough to actually use, and we should trust that the creator has good mathematical taste.
As a case study of the elegance of applied mathematics, let’s look at the braid groups. The braid groups have diverse applications including cryptography (braid based cryptography has suffered from attacks recently though; it is a work in progress), quantum computation, and several other areas that I do not know anything about. On the other hand, the theory of braid groups is one of the most profound mathematical theories encompassing diverse areas of mathematics including set theory (the relation between braids and very large cardinals is currently undeveloped and unclear but as time goes these two areas will be unified). A good objective way to test the elegance of a mathematical structure is to extract a countable ordinal from the structure in a natural, computable, and unexpected manner. The Dehornoy ordering on the positive braids (this linear ordering appears in many different contexts) has order type w^(w^w) which exceeds the proof theoretic ordinal of theories such as PRA. The fact that a reasonably large computable ordinal could be extracted from the positive braids suggests that braid groups have a combinatorial depth beyond what one would find in pure mathematics.
The process of doing applied mathematics is also more satisfying than the process of doing pure mathematics since with applied mathematics since applied mathematics often requires a more well-rounded skill set. Applied mathematics is more interdisciplinary than pure mathematics, and the interdisciplinary nature of applied mathematics fosters development instead of stagnation. For instance, with applied mathematics, instead of simply proving theorems and making definitions, one will also make algorithms, perform computer experiments to gain knowledge that cannot easily be gained simply by writing theorems and proofs, and use the knowledge gained from
computer experiments to prove more theorems.
Any good area of mathematics including set theory should be broad enough to have practical applications. I believe that in the future, large cardinals due in-part to their high consistency strength, will have a prominent place in more traditional areas of mathematics and will be used to construct new mathematical structures that will have real world applications in areas such as cryptography.
Pure mathematics is ok, but even pure mathematicians need to give applied mathematics a chance. A pure mathematician would be much better off by investigating some applied areas.
5. In which room/time are you giving these lectures? I might want to drop in on the 3 forthcoming ones if there’s space.
• The lectures are in the Oxford Faculty of Philosophy building, the Radcliffe Humanities building, in the Lecture Room on the 2nd floor, Thursdays 12-1. | 2020-01-20 00:32:32 | {"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": 0.6335432529449463, "perplexity": 811.6146804709175}, "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-2020-05/segments/1579250595787.7/warc/CC-MAIN-20200119234426-20200120022426-00455.warc.gz"} |
https://stats.stackexchange.com/questions/361681/mixed-effects-model-compare-random-variance-component-across-levels-of-a-groupi | # Mixed effects model: Compare random variance component across levels of a grouping variable
Suppose I have $N$ participants, each of whom gives a response $Y$ 20 times, 10 in one condition and 10 in another. I fit a linear mixed effects model comparing $Y$ in each condition. Here's a reproducible example simulating this situation using the lme4 package in R:
library(lme4)
fml <- "~ condition + (condition | participant_id)"
d <- expand.grid(participant_id=1:40, trial_num=1:10)
d <- rbind(cbind(d, condition="control"), cbind(d, condition="experimental"))
set.seed(23432)
d <- cbind(d, simulate(formula(fml),
newparams=list(beta=c(0, .5),
theta=c(.5, 0, 0),
sigma=1),
family=gaussian,
newdata=d))
m <- lmer(paste("sim_1 ", fml), data=d)
summary(m)
The model m yields two fixed effects (an intercept and slope for condition), and three random effects (a by-participant random intercept, a by-participant random slope for condition, and an intercept-slope correlation).
I would like to statistically compare the size of the by-participant random intercept variance across the groups defined by condition (i.e., compute the variance component highlighted in red separately within the control and experimental conditions, then test whether the difference in the size of the components is non-zero). How would I do this (preferably in R)?
BONUS
Let's say the model is slightly more complicated: The participants each experience 10 stimuli 20 times each, 10 in one condition and 10 in another. Thus, there are two sets of crossed random effects: random effects for participant and random effects for stimulus. Here's a reproducible example:
library(lme4)
fml <- "~ condition + (condition | participant_id) + (condition | stimulus_id)"
d <- expand.grid(participant_id=1:40, stimulus_id=1:10, trial_num=1:10)
d <- rbind(cbind(d, condition="control"), cbind(d, condition="experimental"))
set.seed(23432)
d <- cbind(d, simulate(formula(fml),
newparams=list(beta=c(0, .5),
theta=c(.5, 0, 0, .5, 0, 0),
sigma=1),
family=gaussian,
newdata=d))
m <- lmer(paste("sim_1 ", fml), data=d)
summary(m)
I would like to statistically compare the magnitude of the random by-participant intercept variance across the groups defined by condition. How would I do that, and is the process any different from the one in the situation described above?
EDIT
To be a bit more specific about what I'm looking for, I want to know:
1. Is the question, "are the conditional mean responses within each condition (i.e., random intercept values in each condition) substantially different from each other, beyond what we would expect due to sampling error" a well-defined question (i.e., is this question even theoretically answerable)? If not, why not?
2. If the answer to question (1) is yes, how would I answer it? I would prefer an R implementation, but I'm not tied to the lme4 package -- for example, it seems as though the OpenMx package has the capability to accommodate multi-group and multi-level analyses (https://openmx.ssri.psu.edu/openmx-features), and this seems like the sort of question that ought to be answerable in an SEM framework.
• @MarkWhite, I've updated the question in response to your comments. I mean that I want to compare the standard deviation of the participant intercepts when they give responses in the control condition vs when they give responses in the experimental condition. I want to do this statistically, i.e., test if the difference in the standard deviation of the intercepts is different from 0. – Patrick S. Forscher Aug 11 '18 at 3:37
• I wrote up an answer, but will sleep on it because I am not sure it is very useful. The question comes down to that I don't think one can do what you are asking. The random effect of the intercept is the variance in the participants' means when they are in the control condition. So one cannot look at the variance of those for observations in the experimental condition. The intercepts are defined at the person-level, and the condition is at the observation level. If you are trying to compare variances between conditions, I would think about conditionally heteroscedastic models. – Mark White Aug 11 '18 at 4:31
• I'm working on a revise & resubmit for a paper where I have participants who give responses to sets of stimuli. Each participant is exposed to multiple conditions and each stimulus receives a response in multiple conditions -- in other words, my study emulates the setup I describe in my "BONUS" description. In one of my graphs, it appears that the average participant response has greater variability in one of the conditions than the others. A reviewer asked me to test whether this is true. – Patrick S. Forscher Aug 11 '18 at 4:51
• Please see here stats.stackexchange.com/questions/322213 for how to set up a lme4 model with different variance parameters for each level of a grouping variable. I'm not sure how to do a hypothesis test on whether two variance parameters are equal; personally, I would always prefer to bootstrap over subjects and stimuli to get a confidence interval, or maybe to setup some kind of a permutation-like (resampling based) hypothesis test. – amoeba Aug 11 '18 at 8:43
• I agree with @MarkWhite 's comment that the question "are the random intercept variances substantially different from each other..." is at best unclear and at worst nonsensical, because the intercept necessarily refers to Y-values in one specific group (the group assigned the value of 0), so comparing "intercepts" across groups strictly speaking doesn't make sense. I think a better way to rephrase your question, as I understand it, would be something like: "are the variances of participants' conditional mean responses in condition A vs. condition B unequal?" – Jake Westfall Aug 14 '18 at 3:06
There's more than one way to test this hypothesis. For example, the procedure outlined by @amoeba should work. But it seems to me that the simplest, most expedient way to test it is using a good old likelihood ratio test comparing two nested models. The only potentially tricky part of this approach is in knowing how to set up the pair of models so that dropping out a single parameter will cleanly test the desired hypothesis of unequal variances. Below I explain how to do that.
Switch to contrast (sum to zero) coding for your independent variable and then do a likelihood ratio test comparing your full model to a model that forces the correlation between random slopes and random intercepts to be 0:
# switch to numeric (not factor) contrast codes
d$contrast <- 2*(d$condition == 'experimental') - 1
# reduced model without correlation parameter
mod1 <- lmer(sim_1 ~ contrast + (contrast || participant_id), data=d)
# full model with correlation parameter
mod2 <- lmer(sim_1 ~ contrast + (contrast | participant_id), data=d)
# likelihood ratio test
anova(mod1, mod2)
# Visual explanation / intuition
In order for this answer to make sense, you need to have an intuitive understanding of what different values of the correlation parameter imply for the observed data. Consider the (randomly varying) subject-specific regression lines. Basically, the correlation parameter controls whether the participant regression lines "fan out to the right" (positive correlation) or "fan out to the left" (negative correlation) relative to the point $X=0$, where X is your contrast-coded independent variable. Either of these imply unequal variance in participants' conditional mean responses. This is illustrated below:
In this plot, we ignore the multiple observations that we have for each subject in each condition and instead just plot each subject's two random means, with a line connecting them, representing that subject's random slope. (This is made up data from 10 hypothetical subjects, not the data posted in the OP.)
In the column on the left, where there's a strong negative slope-intercept correlation, the regression lines fan out to the left relative to the point $X=0$. As you can see clearly in the figure, this leads to a greater variance in the subjects' random means in condition $X=-1$ than in condition $X=1$.
The column on the right shows the reverse, mirror image of this pattern. In this case there is greater variance in the subjects' random means in condition $X=1$ than in condition $X=-1$.
The column in the middle shows what happens when the random slopes and random intercepts are uncorrelated. This means that the regression lines fan out to the left exactly as much as they fan out to the right, relative to the point $X=0$. This implies that the variances of the subjects' means in the two conditions are equal.
It's crucial here that we've used a sum-to-zero contrast coding scheme, not dummy codes (that is, not setting the groups at $X=0$ vs. $X=1$). It is only under the contrast coding scheme that we have this relationship wherein the variances are equal if and only if the slope-intercept variance is 0. The figure below tries to build that intuition:
What this figure shows is the same exact dataset in both columns, but with the independent variable coded two different ways. In the column on the left we use contrast codes -- this is exactly the situation from the first figure. In the column on the right we use dummy codes. This alters the meaning of the intercepts -- now the intercepts represent the subjects' predicted responses in the control group. The bottom panel shows the consequence of this change, namely, that the slope-intercept correlation is no longer anywhere close to 0, even though the data are the same in a deep sense and the conditional variances are equal in both cases. If this still doesn't seem to make much sense, studying this previous answer of mine where I talk more about this phenomenon may help.
# Proof
Let $y_{ijk}$ be the $j$th response of the $i$th subject under condition $k$. (We have only two conditions here, so $k$ is just either 1 or 2.) Then the mixed model can be written $$y_{ijk} = \alpha_i + \beta_ix_k + e_{ijk},$$ where $\alpha_i$ are the subjects' random intercepts and have variance $\sigma^2_\alpha$, $\beta_i$ are the subjects' random slope and have variance $\sigma^2_\beta$, $e_{ijk}$ is the observation-level error term, and $\text{cov}(\alpha_i, \beta_i)=\sigma_{\alpha\beta}$.
We wish to show that $$\text{var}(\alpha_i + \beta_ix_1) = \text{var}(\alpha_i + \beta_ix_2) \Leftrightarrow \sigma_{\alpha\beta}=0.$$
Beginning with the left hand side of this implication, we have \begin{aligned} \text{var}(\alpha_i + \beta_ix_1) &= \text{var}(\alpha_i + \beta_ix_2) \\ \sigma^2_\alpha + x^2_1\sigma^2_\beta + 2x_1\sigma_{\alpha\beta} &= \sigma^2_\alpha + x^2_2\sigma^2_\beta + 2x_2\sigma_{\alpha\beta} \\ \sigma^2_\beta(x_1^2 - x_2^2) + 2\sigma_{\alpha\beta}(x_1 - x_2) &= 0. \end{aligned}
Sum-to-zero contrast codes imply that $x_1 + x_2 = 0$ and $x_1^2 = x_2^2 = x^2$. Then we can further reduce the last line of the above to \begin{aligned} \sigma^2_\beta(x^2 - x^2) + 2\sigma_{\alpha\beta}(x_1 + x_1) &= 0 \\ \sigma_{\alpha\beta} &= 0, \end{aligned} which is what we wanted to prove. (To establish the other direction of the implication, we can just follow these same steps in reverse.)
To reiterate, this shows that if the independent variable is contrast (sum to zero) coded, then the variances of the subjects' random means in each condition are equal if and only if the correlation between random slopes and random intercepts is 0. The key take-away point from all this is that testing the null hypothesis that $\sigma_{\alpha\beta} = 0$ will test the null hypothesis of equal variances described by the OP.
This does NOT work if the independent variable is, say, dummy coded. Specifically, if we plug the values $x_1=0$ and $x_2=1$ into the equations above, we find that $$\text{var}(\alpha_i) = \text{var}(\alpha_i + \beta_i) \Leftrightarrow \sigma_{\alpha\beta} = -\frac{\sigma^2_\beta}{2}.$$
• This is already a terrific answer, thank you! I think this comes closest to answering my question, so I'm accepting it and giving you the bounty (it's about to expire), but I'd love to see an algebraic justification if you have the time and energy for it. – Patrick S. Forscher Aug 19 '18 at 6:29
• @PatrickS.Forscher I just added a proof – Jake Westfall Aug 19 '18 at 19:15
• @JakeWestfall In my toy example subjects have flipped responses in the two conditions. If a subject has response $a$ in condition A and $-a$ in condition B, then what would be the BLUP value of random intercept for this subject when we use (1 | subject) model? I think it can only be 0. If all subjects have BLUPs equal to zero, then the variance of random intercept is also zero. So this model is unable to fit this toy example at all. In contrast, the model defined above via dummy will have two BLUPs for each subject, and they can easily be $a$ and $-a$. Am I missing something here? – amoeba Aug 19 '18 at 20:56
• I see now that you're right @amoeba, thanks for explaining. I'll edit my answer accordingly. – Jake Westfall Aug 19 '18 at 21:00
• @amoeba You're right that it's possible that the BLUPs can come out correlated even without a correlation parameter in the model. But I believe that for testing purposes the procedure still works as intended (e.g., it has the nominal type 1 error rate) because only the model with the correlation parameter is able to incorporate that into the likelihood function and thereby "receive credit" for that. That is, even if the BLUPs come out correlated in the simpler model, it's still as if the effects are uncorrelated as far as the total likelihood is concerned, so the LR test will work. I think :) – Jake Westfall Aug 19 '18 at 21:37
You can test significance, of model parameters, with the help of estimated confidence intervals for which the lme4 package has the confint.merMod function.
bootstrapping (see for instance Confidence Interval from bootstrap)
> confint(m, method="boot", nsim=500, oldNames= FALSE)
Computing bootstrap confidence intervals ...
2.5 % 97.5 %
sd_(Intercept)|participant_id 0.32764600 0.64763277
cor_conditionexperimental.(Intercept)|participant_id -1.00000000 1.00000000
sd_conditionexperimental|participant_id 0.02249989 0.46871800
sigma 0.97933979 1.08314696
(Intercept) -0.29669088 0.06169473
conditionexperimental 0.26539992 0.60940435
likelihood profile (see for instance What is the relationship between profile likelihood and confidence intervals?)
> confint(m, method="profile", oldNames= FALSE)
Computing profile confidence intervals ...
2.5 % 97.5 %
sd_(Intercept)|participant_id 0.3490878 0.66714551
cor_conditionexperimental.(Intercept)|participant_id -1.0000000 1.00000000
sd_conditionexperimental|participant_id 0.0000000 0.49076950
sigma 0.9759407 1.08217870
(Intercept) -0.2999380 0.07194055
conditionexperimental 0.2707319 0.60727448
• There is also a method 'Wald' but this is applied to fixed effects only.
• There also exist some kind of anova (likelihood ratio) type of expression in the package lmerTest which is named ranova. But I can not seem to make sense out of this. The distribution of the differences in logLikelihood, when the null hypothesis (zero variance for the random effect) is true is not chi-square distributed (possibly when number of participants and trials is high the likelihood ratio test might make sense).
Variance in specific groups
To obtain results for variance in specific groups you could reparameterize
# different model with alternative parameterization (and also correlation taken out)
fml1 <- "~ condition + (0 + control + experimental || participant_id) "
Where we added two columns to the data-frame (this is only needed if you wish to evaluate non-correlated 'control' and 'experimental' the function (0 + condition || participant_id) would not lead to the evaluation of the different factors in condition as non-correlated)
#adding extra columns for control and experimental
d <- cbind(d,as.numeric(d$condition=='control')) d <- cbind(d,1-as.numeric(d$condition=='control'))
names(d)[c(4,5)] <- c("control","experimental")
Now lmer will give variance for the different groups
> m <- lmer(paste("sim_1 ", fml1), data=d)
> m
Linear mixed model fit by REML ['lmerModLmerTest']
Formula: paste("sim_1 ", fml1)
Data: d
REML criterion at convergence: 2408.186
Random effects:
Groups Name Std.Dev.
participant_id control 0.4963
participant_id.1 experimental 0.4554
Residual 1.0268
Number of obs: 800, groups: participant_id, 40
Fixed Effects:
(Intercept) conditionexperimental
-0.114 0.439
And you can apply the profile methods to these. For instance now confint gives confidence intervals for the control and exerimental variance.
> confint(m, method="profile", oldNames= FALSE)
Computing profile confidence intervals ...
2.5 % 97.5 %
sd_control|participant_id 0.3490873 0.66714568
sd_experimental|participant_id 0.3106425 0.61975534
sigma 0.9759407 1.08217872
(Intercept) -0.2999382 0.07194076
conditionexperimental 0.1865125 0.69149396
Simplicity
You could use the likelihood function to get more advanced comparisons, but there are many ways to make approximations along the road (e.g. you could do a conservative anova/lrt-test, but is that what you want?).
At this point it makes me wonder what is actually the point of this (not so common) comparison between variances. I wonder whether it starts to become too sophisticated. Why the difference between variances instead of the ratio between variances (which relates to the classical F-distribution)? Why not just report confidence intervals? We need to take a step back, and clarify the data and the story it is supposed to tell, before going into advanced pathways that may be superfluous and loose touch with the statistical matter and the statistical considerations that are actually the main topic.
I wonder whether one should do much more than simply stating the confidence intervals (which may actually tell much more than a hypothesis test. a hypothesis test gives a yes no answer but no information about the actual spread of the population. given enough data you can make any slight difference to be reported as a significant difference). To go more deeply into the matter (for whatever purpose), requires, I believe, a more specific (narrowly defined) research question in order to guide the mathematical machinery to make the proper simplifications (even when an exact calculation might be feasible or when it could be approximated by simulations/bootstrapping, even then in in some settings it still requires some appropriate interpretation). Compare with Fisher's exact test to solve a (particular) question (about contingency tables) exactly, but which may not be the right question.
Simple example
To provide an example of the simplicity that is possible I show below a comparison (by simulations) with a simple assessment of the difference between the two group variances based on an F-test done by comparing variances in the individual mean responses and done by comparing the mixed model derived variances.
For the F-test we simply compare the variance of the values (means) of the individuals in the two groups. Those means are for condition $j$ distributed as:
$$\hat{Y}_{i,j} \sim N(\mu_j, \sigma_j^2 + \frac{\sigma_{\epsilon}^2}{10})$$
if the measurement error variance $\sigma_\epsilon$ is equal for all individuals and conditions, and if the variance for the two conditions $\sigma_{j}$ (with $j = \lbrace 1,2 \rbrace$) is equal then the ratio for the variance for the 40 means in the condition 1 and the variance for the 40 means in the condition 2 is distributed according to the F-distribution with degrees of freedom 39 and 39 for numerator and denominator.
You can see this in the simulation of the below graph where aside for the F-score based on sample means an F-score is calculated based on the predicted variances (or sums of squared error) from the model.
The image is modeled with 10 000 repetitions using $\sigma_{j=1} = \sigma_{j=2} = 0.5$ and $\sigma_\epsilon=1$.
You can see that there is some difference. This difference may be due to fact that the mixed effects linear model is obtaining the sums of squared error (for the random effect) in a different way. And these squared error terms are not (anymore) well expressed as a simple Chi-squared distribution, but still closely related and they can be approximated.
Aside from the (small) difference when the null-hypothesis is true, more interesting is the case when the null hypothesis is not true. Especially the condition when $\sigma_{j=1} \neq \sigma_{j=2}$. The distribution of the means $\hat{Y}_{i,j}$ are not only dependent on those $\sigma_j$ but also on the measurement error $\sigma_\epsilon$. In the case of the mixed effects model this latter error is 'filtered out', and it is expected that the F-score based on the random effects model variances has a higher power.
The image is modeled with 10 000 repetitions using $\sigma_{j=1} = 0.5$, $\sigma_{j=2} = 0.25$ and $\sigma_\epsilon=1$.
So the model based on the means is very exact. But it is less powerful. This shows that the correct strategy depends on what you want/need.
In the example above when you set the right tail boundaries at 2.1 and 3.1 you get approximately 1% of the population in the case of equal variance (resp 103 and 104 of the 10 000 cases) but in the case of unequal variance these boundaries differ a lot (giving 5334 and 6716 of the cases)
code:
set.seed(23432)
# different model with alternative parameterization (and also correlation taken out)
fml1 <- "~ condition + (0 + control + experimental || participant_id) "
fml <- "~ condition + (condition | participant_id)"
n <- 10000
theta_m <- matrix(rep(0,n*2),n)
theta_f <- matrix(rep(0,n*2),n)
# initial data frame later changed into d by adding a sixth sim_1 column
ds <- expand.grid(participant_id=1:40, trial_num=1:10)
ds <- rbind(cbind(ds, condition="control"), cbind(ds, condition="experimental"))
#adding extra columns for control and experimental
ds <- cbind(ds,as.numeric(ds$condition=='control')) ds <- cbind(ds,1-as.numeric(ds$condition=='control'))
names(ds)[c(4,5)] <- c("control","experimental")
# defining variances for the population of individual means
stdevs <- c(0.5,0.5) # c(control,experimental)
pb <- txtProgressBar(title = "progress bar", min = 0,
max = n, style=3)
for (i in 1:n) {
indv_means <- c(rep(0,40)+rnorm(40,0,stdevs[1]),rep(0.5,40)+rnorm(40,0,stdevs[2]))
fill <- indv_means[d[,1]+d[,5]*40]+rnorm(80*10,0,sqrt(1)) #using a different way to make the data because the simulate is not creating independent data in the two groups
#fill <- suppressMessages(simulate(formula(fml),
# newparams=list(beta=c(0, .5),
# theta=c(.5, 0, 0),
# sigma=1),
# family=gaussian,
# newdata=ds))
d <- cbind(ds, fill)
names(d)[6] <- c("sim_1")
m <- lmer(paste("sim_1 ", fml1), data=d)
m
theta_m[i,] <- m@theta^2
imeans <- aggregate(d[, 6], list(d[,c(1)],d[,c(3)]), mean)
theta_f[i,1] <- var(imeans[c(1:40),3])
theta_f[i,2] <- var(imeans[c(41:80),3])
setTxtProgressBar(pb, i)
}
close(pb)
p1 <- hist(theta_f[,1]/theta_f[,2], breaks = seq(0,6,0.06))
fr <- theta_m[,1]/theta_m[,2]
fr <- fr[which(fr<30)]
p2 <- hist(fr, breaks = seq(0,30,0.06))
plot(-100,-100, xlim=c(0,6), ylim=c(0,800),
xlab="F-score", ylab = "counts [n out of 10 000]")
plot( p1, col=rgb(0,0,1,1/4), xlim=c(0,6), ylim=c(0,800), add=T) # means based F-score
plot( p2, col=rgb(1,0,0,1/4), xlim=c(0,6), ylim=c(0,800), add=T) # model based F-score
fr <- seq(0, 4, 0.01)
lines(fr,df(fr,39,39)*n*0.06,col=1)
legend(2, 800, c("means based F-score","mixed regression based F-score"),
fill=c(rgb(0,0,1,1/4),rgb(1,0,0,1/4)),box.col =NA, bg = NA)
legend(2, 760, c("F(39,39) distribution"),
lty=c(1),box.col = NA,bg = NA)
title(expression(paste(sigma[1]==0.5, " , ", sigma[2]==0.5, " and ", sigma[epsilon]==1)))
• That's useful but does not seem to address the question about how to compare variances in two conditions. – amoeba Aug 13 '18 at 13:11
• @amoeba I found that this answer gives the core of the issue (about testing the random variance components). What the OP precisely wants is difficult to read in the entire text. What does "the random intercept variances" refer to? (the plural in relation to intercept confuses me) One possible case might be to use the model sim_1 ~ condition + (0 + condition | participant_id)" in which case you get a parameterization into two parameters (one for each group) rather than two parameters one for the intercept and one for the effect (which need to be combined for the groups). – Martijn Weterings Aug 13 '18 at 13:27
• Each subject has some mean response in condition A and some mean response in condition B. The question is whether the variance across subjects in A is different from the variance across subjects in B. – amoeba Aug 13 '18 at 13:38
• This does not complete the task posed in the title "Compare random variance component across levels of a grouping variable". I noticed that there was a confusing typo in the body of the question, which I've fixed. I also tried to further clarify the wording of the question. – Patrick S. Forscher Aug 13 '18 at 14:18
• It might be possible to answer the question using car::linearHypothesisTest (math.furman.edu/~dcs/courses/math47/R/library/car/html/…), which allows the user to test arbitrary hypotheses with a fitted model. However, I'd have to use @amoeba's method to obtain both random intercepts in the same model fitted model so they can be compared with this function. I'm also a little uncertain as to the validity of the method. – Patrick S. Forscher Aug 13 '18 at 14:24
One relatively straight-forward way could be to use likelihood-ratio tests via anova as described in the lme4 FAQ.
We start with a full model in which the variances are unconstrained (i.e., two different variances are allowed) and then fit one constrained model in which the two variances are assumed to be equal. We simply compare them with anova() (note that I set REML = FALSE although REML = TRUE with anova(..., refit = FALSE) is completely feasible).
m_full <- lmer(sim_1 ~ condition + (condition | participant_id), data=d, REML = FALSE)
summary(m_full)$varcor # Groups Name Std.Dev. Corr # participant_id (Intercept) 0.48741 # conditionexperimental 0.26468 -0.419 # Residual 1.02677 m_red <- lmer(sim_1 ~ condition + (1 | participant_id), data=d, REML = FALSE) summary(m_red)$varcor
# Groups Name Std.Dev.
# participant_id (Intercept) 0.44734
# Residual 1.03571
anova(m_full, m_red)
# Data: d
# Models:
# m_red: sim_1 ~ condition + (1 | participant_id)
# m_full: sim_1 ~ condition + (condition | participant_id)
# Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
# m_red 4 2396.6 2415.3 -1194.3 2388.6
# m_full 6 2398.7 2426.8 -1193.3 2386.7 1.9037 2 0.386
However, this test is likely conservative. For example, the FAQ says:
Keep in mind that LRT-based null hypothesis tests are conservative when the null value (such as σ2=0) is on the boundary of the feasible space; in the simplest case (single random effect variance), the p-value is approximately twice as large as it should be (Pinheiro and Bates 2000).
There are several alternatives:
1. Create an appropriate test distribution, which usually consists of a mixture of $\chi^2$ distributions. See e.g.,
Self, S. G., & Liang, K.-Y. (1987). Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests Under Nonstandard Conditions. Journal of the American Statistical Association, 82(398), 605. https://doi.org/10.2307/2289471 However, this is quite complicated.
2. Simulate the correct distribution using RLRsim (as also described in the FAQ).
I will demonstrate the second option in the following:
library("RLRsim")
## reparametrize model so we can get one parameter that we want to be zero:
afex::set_sum_contrasts() ## warning, changes contrasts globally
d <- cbind(d, difference = model.matrix(~condition, d)[,"condition1"])
m_full2 <- lmer(sim_1 ~ condition + (difference | participant_id), data=d, REML = FALSE)
all.equal(deviance(m_full), deviance(m_full2)) ## both full models are identical
## however, we need the full model without correlation!
m_full2b <- lmer(sim_1 ~ condition + (1| participant_id) +
(0 + difference | participant_id), data=d, REML = FALSE)
summary(m_full2b)$varcor # Groups Name Std.Dev. # participant_id (Intercept) 0.44837 # participant_id.1 difference 0.13234 # Residual 1.02677 ## model that only has random effect to be tested m_red <- update(m_full2b, . ~ . - (1 | participant_id), data=d, REML = FALSE) summary(m_red)$varcor
# Groups Name Std.Dev.
# participant_id difference 0.083262
# Residual 1.125116
## Null model
m_null <- update(m_full2b, . ~ . - (0 + difference | participant_id), data=d, REML = FALSE)
summary(m_null)$varcor # Groups Name Std.Dev. # participant_id (Intercept) 0.44734 # Residual 1.03571 exactRLRT(m_red, m_full2b, m_null) # Using restricted likelihood evaluated at ML estimators. # Refit with method="REML" for exact results. # # simulated finite sample distribution of RLRT. # # (p-value based on 10000 simulated values) # # data: # RLRT = 1.9698, p-value = 0.0719 As we can see, the output suggests that with REML = TRUE we would have gotten exact results. But this is left as an exercise to the reader. Regarding the bonus, I am not sure if RLRsim allows simultaneous testing of multiple components, but if so, this can be done in the same way. Response to comment: So it is true, then, that in general the random slope$\theta_X$allows the random intercept$\theta_0$to vary across levels of$X$? I am not sure this question can receive a reasonable answer. • A random intercept allows an idiosyncratic difference in the overall level for each level of the grouping factor. For example, if the dependent variable is response time, some participants are faster and some are slower. • A random slope allows each level of the grouping factor an idiosyncratic effect of the factor for which random slopes are estimated. For example, if the factor is congruency, then some participants can have a higher congruency effect than others. So do random-slopes affect the random-intercept? In some sense this might make sense, as they allow each level of the grouping factor a completely idiosyncratic effect for each condition. In the end, we estimate two idiosyncratic parameters for two conditions. However, I think the distinction between the overall level captured by the intercept and the condition specific effect captured by the random slope is a important and then the random slope cannot really affect the random intercept. However, it still allows each level of the grouping factor an idiosyncratic separately for each level of the condition. Nevertheless, my test still does what the original question wants. It tests whether the difference in variances between the two conditions is zero. If it is zero, then the variances in both conditions are equal. In other words, only if there is no need for a random-slope is the variance in both conditions identical. I hope that makes sense. • You use treatment contrasts (contr.treatment) for which the control condition is the reference (i.e., for which the random intercept is calculated). The parametrization I propose I use sum contrasts (i.e., contr.sum) and the intercept is the grand mean. I feel like it makes more sense to test whether the difference is null when the intercept is the grand mean instead of the control condition (but writing it done suggests it might be relatively inconsequential). You might want to read pp. 24 to 26 of: singmann.org/download/publications/… – Henrik Aug 13 '18 at 15:05 • Thanks! My questions are slightly different, though: (1) Your answer seems to imply that my question reduces to "is the random slope for condition different from 0". Is this true? (2) If the answer to (1) is "yes", this suggests another interpretation of the random slope for condition: it allows the random intercept to vary across levels of condition. Is this true? – Patrick S. Forscher Aug 13 '18 at 15:26 • My 2¢: @amoeba 's counterexample to Henrik's proposed procedure is correct. Henrik is nearly correct, but he compares the wrong pair of models. The model comparison that answer's Patrick's question is the comparison between the models Henrik called m_full vs. m_full2b. That is: the variances of participants' conditional mean responses in A vs. B are unequal iff the random intercept-slope correlation is nonzero---importantly, under the sum-to-zero contrast coding parameterization. Testing the random slope variance is not necessary. Trying to think how to explain this succinctly... – Jake Westfall Aug 14 '18 at 2:54 • This is not really a proper explanation, but studying my answer here may shed a little light on the matter. Basically, the correlation parameter controls whether the participant regression lines "fan out to the right" (positive corr.) or "fan out to the left" (negative corr.). Either of these imply unequal variance in participants' conditional mean responses. Sum-to-zero coding then ensures that we're looking for correlation at the right point on X – Jake Westfall Aug 14 '18 at 3:13 • I will consider posting an answer with pictures if I can find the time... – Jake Westfall Aug 14 '18 at 13:39 Your model m = lmer(sim_1 ~ condition + (condition | participant_id), data=d) already allows the across-subject variance in the control condition to differ from the across-subject variance in the experimental condition. This can be made more explicit by an equivalent re-parametrization: m = lmer(sim_1 ~ 0 + condition + (0 + condition | participant_id), data=d) The random covariance matrix now has a simpler interpretation: Random effects: Groups Name Variance Std.Dev. Corr participant_id conditioncontrol 0.2464 0.4963 conditionexperimental 0.2074 0.4554 0.83 Here the two variances are precisely the two variances you are interested in: the [across-subjects] variance of conditional mean responses in the control condition and the same in the experimental condition. In your simulated dataset, they are 0.25 and 0.21. The difference is given by delta = as.data.frame(VarCorr(m))[1,4] - as.data.frame(VarCorr(m))[2,4] and is equal to 0.039. You want to test if it is significantly different from zero. EDIT: I realized that the permutation test that I describe below is incorrect; it won't work as intended if the means in control/experimental condition are not the same (because then the observations are not exchangeable under the null). It might be a better idea to bootstrap subjects (or subjects/items in the Bonus case) and obtain the confidence interval for delta. I will try to fix the code below to do that. Original permutation-based suggestion (wrong) I often find that one can save oneself a lot of trouble by doing a permutation test. Indeed, in this case it is very easy to set up. Let's permute control/experimental conditions for each subject separately; then any difference in variances should be eliminated. Repeating this many times will yield the null distribution for the differences. (I do not program in R; everybody please feel free to re-write the following in a better R style.) set.seed(42) nrep = 100 v = matrix(nrow=nrep, ncol=1) for (i in 1:nrep) { dp = d for (s in unique(d$participant_id)){
if (rbinom(1,1,.5)==1){
dp[p$participant_id==s & d$condition=='control',]$condition = 'experimental' dp[p$participant_id==s & d$condition=='experimental',]$condition = 'control'
}
}
m <- lmer(sim_1 ~ 0 + condition + (0 + condition | participant_id), data=dp)
v[i,] = as.data.frame(VarCorr(m))[1,4] - as.data.frame(VarCorr(m))[2,4]
}
pvalue = sum(abs(v) >= abs(delta)) / nrep
Running this yields the p-value $p=0.7$. One can increase nrep to 1000 or so.
Exactly the same logic can be applied in your Bonus case.
• Super interesting, thank you! I'll have to think more about why your reparameterization works, as this seems to be the key insight of this answer. – Patrick S. Forscher Aug 15 '18 at 2:21
• Strangely, the per-group intercept values in your answer seem to differ from those in @MartijnWeterings' answer. – Patrick S. Forscher Aug 15 '18 at 2:24
• @PatrickS.Forscher That's because he, I think, generates a different dataset. I can use sim_1 ~ 0 + condition + (0 + dummy(condition, "control") + dummy(condition, "experimental") | participant_id) formulation and get the same outcome as in my answer. – amoeba Aug 15 '18 at 6:25
• @PatrickS.Forscher No, I used the data generated by your code (with your seed). I set the seed to 42 only when performing the permutation testing. It's Martijn who changed the dataset, not me. – amoeba Aug 15 '18 at 14:21
• This proposal is definitely sound. As I think you've already experienced, setting up permutation tests for multilevel data is not entirely straightforward. A similar approach that would be a bit easier to implement would be parametric bootstrapping, which is pretty simple to do with lme4 using the simulate() method of the fitted lmer objects, i.e., call simulate(m) many times to build up the bootstrap distribution. Just an idea to play around with. – Jake Westfall Aug 20 '18 at 14:33 | 2019-04-20 14:51:31 | {"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": 0.7847791910171509, "perplexity": 1482.8009000866593}, "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-2019-18/segments/1555578529839.0/warc/CC-MAIN-20190420140859-20190420162859-00213.warc.gz"} |
http://cjtcs.cs.uchicago.edu/articles/2018/1/contents.html | ### Volume 2018
#### On monotone circuits with local oracles and clique lower bounds
Jan Krajíček
Faculty of Mathematics and Physics
Charles University
Czech Republic
krajicek AT karlin DOT mff DOT cuni DOT cz
and
Igor C. Oliveira
Department of Computer Science
University of Oxford
UK
igor.carbonioliveira AT cs DOT ox DOT ac DOT uk
March 25, 2018
#### Abstract
We investigate monotone circuits with local oracles [Krajíček 2016] i.e., circuits containing additional inputs $y_i = y_i(\vec{x})$ that can perform unstructured computations on the input string $\vec{x}$. Let $\mu \in [0,1]$ be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions $y_i(\vec{x})$, and $U_{n,k}, V_{n,k} \subseteq \{0,1\}^m$ be the set of $k$-cliques and the set of complete $(k-1)$-partite graphs, respectively (similarly to [Razborov, 1985]).} Our results can be informally stated as follows.
• (i) For an appropriate extension of depth-$2$ monotone circuits with local oracles, we show that the size of the smallest circuits separating $U_{n,3}$ (triangles) and $V_{n,3}$ (complete bipartite graphs) undergoes two phase transitions according to $\mu$.
• (ii) For $5 \leq k(n) \leq n^{1/4}$, arbitrary depth, and $\mu \leq 1/50$, we prove that the monotone circuit size complexity of separating the sets $U_{n,k}$ and $V_{n,k}$ is $n^{\Theta(\sqrt{k})}$, under a certain restrictive assumption on the local oracle gates.
The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of $k$-clique obtained in [Alon and Boppana, 1987].
• The article: PDF (293 KB)
• Source material: ZIP (88 KB) | 2019-01-19 17:20:45 | {"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": 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": 0.956775426864624, "perplexity": 2456.3580644088192}, "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-2019-04/segments/1547583671342.16/warc/CC-MAIN-20190119160425-20190119182425-00563.warc.gz"} |
http://openstudy.com/updates/4f98e5b6e4b000ae9ece871c | ## 2yoututu11 3 years ago What is the exact area of a circle with a radius of 12 feet? A. 24π square feet B. 36π square feet C. 121π square feet D. 144π square feet
1. Maekin
Area = PI * R^2 R = 12, so R^2 = 144 A = 144 * PI.
2. Study23
$$\ \huge A= \pi r^2$$. So, $$\ \huge A= \pi \times 12^2$$. That's $$\ \huge 144 \pi feet^2$$. | 2015-11-30 00:53:38 | {"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": 0.7304204106330872, "perplexity": 3102.349000384468}, "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/1448398460519.28/warc/CC-MAIN-20151124205420-00304-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://chemistry.stackexchange.com/questions/linked/34610 | 13k views
### why do electron pairs not take axial positions in the VSEPR structure of ClF3? [duplicate]
when we make a Valence shell electron pair repulsion theory based structure for an molecule after calculating its hybridisation ; when we have to decide the positions of the ions we put them in such ...
121 views
### Why do some molecules form T shapes instead of trigonal planar shapes [duplicate]
Consider a molecule such as $\ce{ClF_3}$. Shouldn't the electron clouds (which are more negative than the Chlorine atom) want to repel each other and so the $\ce{ClF_3}$molecule should arrange into a ...
96 views
### Trigonal bipyramidal arrangement with 2 lone pairs [duplicate]
So basically according to university textbooks, it is said that the second lone pair in a trigonal bypyramidal arrangement would also reside on the trigonal plane, causing the two lone pairs to be ...
35k views
### What is Bent's rule?
I'm all bent out of shape trying to figure out what Bent's rule means. I have several formulations of it, and the most common formulation is also the hardest to understand. Atomic s character ...
10k views
### Why does F replace the axial bond in PCl5?
Why does $\ce{F}$ replace an axial bond in $\ce{PCl5}$? I realize that it would be more stable there than at equatorial bond, but what is the reason of its stability? Similarly in $\ce{AB4}$ type of ...
3k views
### Hypervalency and the octet rule
I realize that the octet rule is more a suggestion than a rule, and that it applies mainly to non-transition metal compounds. Still, compounds that don't have an octet, like $\ce{BH3}$, tend to ...
9k views
### How can Iodine bond with 5 Fluorines in Iodine Pentafluoride?
According to this lewis diagram, Iodine has 12 valence electrons? How is the possible? Only thing I can think of is that I'm mistaken in counting the covalent bonds as two valence electrons each. Do ...
5k views
### Structure of the bifluoride anion [duplicate]
I came across this structure of $\ce{HF2-}$ and realized that the bond shown between H and F is a hydrogen bond. $$\Large\ce{[F-H-F]-}$$ HF hydrogen bond are the strongest hydrogen bonds we know, ...
6k views
### Does fluorine ever form a double or triple bond?
Does fluorine ever form a double or triple bond? I wonder if seeming lack of such higher order bonds is related to the electronegativity of fluorine.
I'm relatively new to chemistry, and while doing some exercises about $\ce{ClF_3}$ I tried to draw where the electrons are. I checked Google and got this: How is this structure possible? From what I'...
Referring to, NCERT Chemistry Part I, Textbook for Class XI[1]. Here the text puts forward three structures of $\ce{ClF3}$. My teacher said to me that (b) structure is unstable due to lone pair lone ... | 2021-12-06 17:20:40 | {"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": 0.6515938639640808, "perplexity": 1868.7338361199916}, "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/1637964363309.86/warc/CC-MAIN-20211206163944-20211206193944-00427.warc.gz"} |
https://socratic.org/questions/how-do-you-evaluate-frac-4-cdot-x-1-1-x-cdot-2#472209 | # How do you evaluate \frac { 4\cdot ( x - 1) } { ( 1- x ) \cdot 2}?
Sep 6, 2017
$- 2$
#### Explanation:
$\frac{4 \cdot \left(x - 1\right)}{\left(1 - x\right) \cdot 2}$
change a sign for $\left(x - 1\right)$ to $- \left(- x + 1\right) = - \left(1 - x\right)$
$\frac{4 \cdot - \left(1 - x\right)}{\left(1 - x\right) \cdot 2} = \frac{- 4 \cdot \cancel{\left(1 - x\right)}}{\cancel{\left(1 - x\right)} \cdot 2}$
$= - \frac{4}{2} = - 2$ | 2022-01-22 09:36:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "mathjax_inline_tex": 1, "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": 0.9999462366104126, "perplexity": 1587.4936861328574}, "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-2022-05/segments/1642320303779.65/warc/CC-MAIN-20220122073422-20220122103422-00047.warc.gz"} |
http://www.gamedev.net/topic/609148-proper-gluniform-use/ | • Create Account
## proper glUniform use
Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.
10 replies to this topic
### #1krienie Members
102
Like
0Likes
Like
Posted 23 August 2011 - 01:30 PM
Hey all,
I'm kind of new to openGL and GLSL, so I"m not that familiar with all the proper uses yet. Like glUniform.
So here's my sitiuation: I'm currently working on a game engine that uses openGL.
I've made a DisplayObject that can render out 3D object from data stored in VBO's and a GLSL shader. It also uses a home-made matrix class for storing model-view, normal and modelview-projection matrices which the GLSL shader can use.
I"m currently reading through the OpenGL Shading Language book (orange book) and in there the writers state that uniform attributes should only be used with data that doesn't change frequently. So what I did is declare the matrices in the shader as uniforms and made a neat updateMatrices() function for my DisplayObject class that is called everytime a transformation is done to the object (for example when it translates or rotates). This function updates the matrices and then sends the new information to the GLSL shader using glUniform. Now this all works just fine when using one object, but when I introduce a second object or more, all the objects are snapped to the coordinates and orientation of the object that did the last transformation.
After some time of debugging I was able to fix this by calling chop the glUniform calls out of updateMatrices and call just before every object is drawn instead. Even though this works, I'm really doubting if I did the right thing here, because what I'm now doing is sending matrix data to the shader even though the values didn't change.
void DisplayObject::render()
{
{
glBindVertexArray(vaoName);
/* send new matrices to the shaderprogram */
glUniformMatrix4fv(glGetUniformLocation(program, "mvMatrix"), 1, GL_FALSE, modelmatrix.data);
glUniformMatrix4fv(glGetUniformLocation(program, "normMatrix"), 1, GL_FALSE, normalMatrix.data);
/* send new modelview-projection matrix to the shaderprogram */
modelmatrix = modelmatrix * *projectionmatrix;
glUniformMatrix4fv(glGetUniformLocation(program, "mvpMatrix"), 1, GL_FALSE, modelmatrix.data);
glDrawElements(GL_TRIANGLES, bufferSize[index], GL_UNSIGNED_INT, 0);
}
}
So my question now is: am I doing the right thing and if not: how do you guys suggest I should send the matrix data to the shader?
### #2FXACE Members
182
Like
0Likes
Like
Posted 23 August 2011 - 04:16 PM
I suppose using build-in OpenGL uniform variables is much better one ("mvMatrix" -> "gl_ModelViewMatrix", "normMatrix" - > "gl_NormalMatrix",...).where "gl_ModelViewMatrix" holds GL_MODELVIEW matrix state and so on
### #3anonymous22 Members
199
Like
0Likes
Like
Posted 23 August 2011 - 04:56 PM
I suppose using build-in OpenGL uniform variables is much better one ("mvMatrix" -> "gl_ModelViewMatrix", "normMatrix" - > "gl_NormalMatrix",...).where "gl_ModelViewMatrix" holds GL_MODELVIEW matrix state and so on
Those built-ins are deprecated in 4.2+....
### #4karwosts Members
840
Like
1Likes
Like
Posted 23 August 2011 - 09:58 PM
So what I did is declare the matrices in the shader as uniforms and made a neat updateMatrices() function for my DisplayObject class that is called everytime a transformation is done to the object (for example when it translates or rotates).
You're probably running into the problem of two objects using the same shader. If you use more than one object with a given shader, then you need to update the matrices each time you change between one and the other.
After some time of debugging I was able to fix this by calling chop the glUniform calls out of updateMatrices and call just before every object is drawn instead. Even though this works, I'm really doubting if I did the right thing here, because what I'm now doing is sending matrix data to the shader even though the values didn't change.
This sounds like a reasonable thing to do to me. Obviously you don't want to send data that you don't have to, but I can't imagine many cases where a single object gets a shader all to itself. I think just updating the matrices before drawing is not a big deal, unless you have unusual use case where it becomes impractical.
My Projects:
Portfolio Map for Android - Free Visual Portfolio Tracker
Electron Flux for Android - Free Puzzle/Logic Game
### #5krienie Members
102
Like
0Likes
Like
Posted 24 August 2011 - 04:23 AM
...
This sounds like a reasonable thing to do to me. Obviously you don't want to send data that you don't have to, but I can't imagine many cases where a single object gets a shader all to itself. I think just updating the matrices before drawing is not a big deal, unless you have unusual use case where it becomes impractical.
That's what I thought. It seems a bit stupid to have two objects have it's own shader, each doing the exactly the same thing.
I'll leave it the way it is then. Thanks for the replies everyone ;)
### #6V-man Members
813
Like
0Likes
Like
Posted 24 August 2011 - 09:45 AM
Don't call glGetUniformLocation. It is probably slow.
Sig: http://glhlib.sourceforge.net
an open source GLU replacement library. Much more modern than GLU.
float matrix[16], inverse_matrix[16];
glhTranslatef2(matrix, 0.0, 0.0, 5.0);
glhScalef2(matrix, 1.0, 1.0, -1.0);
glhQuickInvertMatrixf2(matrix, inverse_matrix);
glUniformMatrix4fv(uniformLocation1, 1, FALSE, matrix);
glUniformMatrix4fv(uniformLocation2, 1, FALSE, inverse_matrix);
### #7Murdocki Members
274
Like
1Likes
Like
Posted 24 August 2011 - 12:19 PM
You probably misinterpreted the "not changing frequently" part, uniforms are described as not changing frequently because they're the same for the whole draw call unlike vertex attributes which change every vertex. Also in reply to v-man calling glGetUnitformLocation is a call you are suggested to do load time rather than run time, you want to profile this on your target platform though, in my case it's lots faster to just make the calls than to map them.
### #8FXACE Members
182
Like
0Likes
Like
Posted 24 August 2011 - 03:31 PM
I suppose using build-in OpenGL uniform variables is much better one ("mvMatrix" -> "gl_ModelViewMatrix", "normMatrix" - > "gl_NormalMatrix",...).where "gl_ModelViewMatrix" holds GL_MODELVIEW matrix state and so on
Those built-ins are deprecated in 4.2+....
There are many things deprecated in newest versions from classic GLSL. I'm old man if I don't know what's new in latest version of GLSL...
Also, I'm agree with V-Man.
### #9phantom Members
10777
Like
1Likes
Like
Posted 24 August 2011 - 05:18 PM
You probably misinterpreted the "not changing frequently" part, uniforms are described as not changing frequently because they're the same for the whole draw call unlike vertex attributes which change every vertex.
The term 'not changing frequently' applies just as well to something you set once during all your draw calls in a scene as it would to per-object.
In fact if your MVP matrix is only set once and never update during a scene/frame render setting it multiple times per scene/frame is wasteful and something which should be avoided (CPU time and GPU time overhead).
Now, I'm not saying there IS a way to use GLSL like that, although I hope there is because if not then that's a *facepalm* right there, but there really should be a way of doing it. My GLSL is just too rusty right now to point out that method if it exists
### #10V-man Members
813
Like
1Likes
Like
Posted 25 August 2011 - 10:11 AM
You probably misinterpreted the "not changing frequently" part, uniforms are described as not changing frequently because they're the same for the whole draw call unlike vertex attributes which change every vertex.
The term 'not changing frequently' applies just as well to something you set once during all your draw calls in a scene as it would to per-object.
In fact if your MVP matrix is only set once and never update during a scene/frame render setting it multiple times per scene/frame is wasteful and something which should be avoided (CPU time and GPU time overhead).
Now, I'm not saying there IS a way to use GLSL like that, although I hope there is because if not then that's a *facepalm* right there, but there really should be a way of doing it. My GLSL is just too rusty right now to point out that method if it exists
Of course there is. UBO = Uniform buffer object
Which is basically a VBO for uniform values.
You need to make a GL 3.1 context to use it. Just shuv a GL_UNIFORM_BUFFER into your VBO function calls and get the location from your shader object.
http://www.opengl.or...entation/specs/
and you update the whole fucking uniform block inluding your MVP matrix in one go.
Sig: http://glhlib.sourceforge.net
an open source GLU replacement library. Much more modern than GLU.
float matrix[16], inverse_matrix[16];
glhTranslatef2(matrix, 0.0, 0.0, 5.0);
glhScalef2(matrix, 1.0, 1.0, -1.0);
glhQuickInvertMatrixf2(matrix, inverse_matrix);
glUniformMatrix4fv(uniformLocation1, 1, FALSE, matrix);
glUniformMatrix4fv(uniformLocation2, 1, FALSE, inverse_matrix);
### #11krienie Members
102
Like
0Likes
Like
Posted 25 August 2011 - 03:06 PM
You probably misinterpreted the "not changing frequently" part, uniforms are described as not changing frequently because they're the same for the whole draw call unlike vertex attributes which change every vertex.
The term 'not changing frequently' applies just as well to something you set once during all your draw calls in a scene as it would to per-object.
In fact if your MVP matrix is only set once and never update during a scene/frame render setting it multiple times per scene/frame is wasteful and something which should be avoided (CPU time and GPU time overhead).
Now, I'm not saying there IS a way to use GLSL like that, although I hope there is because if not then that's a *facepalm* right there, but there really should be a way of doing it. My GLSL is just too rusty right now to point out that method if it exists
Of course there is. UBO = Uniform buffer object
Which is basically a VBO for uniform values.
You need to make a GL 3.1 context to use it. Just shuv a GL_UNIFORM_BUFFER into your VBO function calls and get the location from your shader object.
http://www.opengl.or...entation/specs/
and you update the whole fucking uniform block inluding your MVP matrix in one go.
Now this is interesting! I'm totally going to check this out. Thank you very much!
Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic. | 2016-12-09 02:11:20 | {"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": 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": 0.2416997104883194, "perplexity": 2198.1137902960177}, "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-2016-50/segments/1480698542668.98/warc/CC-MAIN-20161202170902-00049-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.askmehelpdesk.com/accounting/common-stock-182914.html | If 20,000 shares of common stock are authorized, 14,000 shares are issued, and 500 shares are held in treasury stock, a cash dividend of $1.00 per share would amount to$14,000. Is that statement TRUE. | 2018-09-25 11:01:15 | {"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": 0.5590770244598389, "perplexity": 14675.950739928654}, "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-2018-39/segments/1537267161501.96/warc/CC-MAIN-20180925103454-20180925123854-00129.warc.gz"} |
http://www.mzan.com/article/48792880-mysql-update-fails-when-using-encoded-html.shtml | Home MySQL update fails when using encoded html
I have a database field that stores text content. The content includes html, which is encoded. I have written a search and replace tool that finds a term in the database field, gives it a context of 40 characters and then replaces that string with the replace term, also in context. I need the context because I need to decide case by case whether to replace the term, which might occur many times in the same field. Using the context narrows it down to that one instance. Here is the problem: when there are encoded tags in the context, the replacement does not work. Specifically, the MySQL update of the replace string fails. Here is an example of the search and replace terms: I pass these on via ajax to this php function where each becomes $search and$replace respectively (I am also passing on table, id and column info which you see used. That works): include 'classes/index.php'; $q = new db; extract($_POST); $selected =$q->select_where($table, "id = {$id}", 'id'); $original_text =$selected[0][$col];$changed_text = str_ireplace($search,$replace,$original_text);$updated = $q->update_row($table, array($col), array($changed_text), $id); if ($updated) { echo "Updated {$search} to {$replace}"; } else { echo "it didn't work. Table: {$table}, Column: {$col}, ID: {$id} Changed Text: {$changed_text}"; } Finally, here is the method I use for the update. I've used this in many contexts and it has been reliable. But maybe something about this scenario breaks it? function update_row($table,$columns, $data,$id) { // Prepare Variable $counter = 0; foreach ($columns as $column) {$prepare .= $column . ' = :' .$column . ', '; } $prepare = rtrim($prepare,', '); $prepare_query = "UPDATE {$table} SET {$prepare} WHERE id = :id"; // Execute Variable$counter=0; foreach ($data as$datum) { $column_name =$columns[$counter];$update[":{$column_name}"] = stripslashes($datum); $counter++; }$update[':id'] = $id; // Run it$stmt = $this->pdo->prepare($prepare_query); $stmt->execute($update); if ( \$stmt->rowCount() == 1 ) { return true; } } Thanks in advance for your input. | 2018-02-25 00:04:28 | {"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": 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": 0.24678128957748413, "perplexity": 2883.597557229679}, "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/1518891816068.93/warc/CC-MAIN-20180224231522-20180225011522-00795.warc.gz"} |
http://jayvijay.co/5eb6g84/how-to-draw-sodium-chloride-crystal-structure | # how to draw sodium chloride crystal structure
Subdivide this big cube into 8 small cubes by joining the mid point of each edge to the mid point of the edge opposite it. Crystal Structure of Sodium Chloride The ionic radius of the sodium ion is 1.16 Å and that of the chloride ion is 1.67 Å. 1 Answer +1 vote . Les Limites Du Constructivisme, In the lattice each sodium ion is surrounded by six chloride ions and vice versa (co-ordination number is six). It is seen in table from purely geometric consideration that for a face-centered cubic lattice. X-ray diffraction studies with palladium target (λ = 5.81 nm ) as X-ray source shows that maxima occur from the (200), (220) and (111) faces of sodium chloride at angles 11.8°, 16.8 0 add 10.4 0 respectively (Figure: 1). The Firm Album, Sodium Chloride Crystal Structure. Similar Illustrations See All. 3 απλές συμβουλές για μεγαλύτερη επιτυχία στο Fallinlove! For Any Problem or Suggestion Comment Below. Sodium chloride / ˌ s oʊ d i ə m ˈ k l ɔːr aɪ d /, commonly known as salt (although sea salt also contains other chemical salts), is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. You must remember that this diagram represents only a tiny part of the whole sodium chloride crystal. The sodium ion in the centre is being touched by 6 chloride ions. This is an ionic structure in which the $Na^+$ ions and $Cl^-$ ions are alternately arranged. In this article I specially add a video in this post so that you can understand it better. As solid sodium chloride is heated, the added energy causes the ions to vibrate at a higher amplitude. Justin Maple Almond Butter Recall, From the difference in intensity of reflections 0f alternate orders, it was concluded that sodium chloride lattice consists of interpenetrating face-centred cubic lattice of sodium and chloride ions. Sodium chloride, nacl crystal structure over white - gg126115195 GoGraph Stock Photography, Illustrations, and Clip Art allows you to quickly find the right graphic. So sodium chloride (and any other ionic compound) is described as having a giant ionic structure. That is different from, say, a water molecule which always contains exactly 2 hydrogen atoms and one oxygen atom - never more and never less. Use different colours or different sizes for the two different ions, and don't forget a key. aiims; neet; Share It On Facebook Twitter Email. Use different colours or different sizes for the two different ions, and don't forget a key. The pattern repeats in this way over countless ions. A face of this unit cell is shown. Coronavirus In Durham Uk, In the lattice each sodium ion is surrounded by six chloride ions and vice versa (co-ordination number is six). Horrible Histories Rat, It is widely used in food industries as a food preservative and as a flavour enhancer. The bond length is 2.813A˚ . Formation of salt aqueous solution of hydrated cations and anions. Nickel Nickel Card Game, (a) (i) Showing the outer electrons only, draw a dot-and-cross diagram to indicate the bonding in calcium oxide. Jessie Estell James, Therefore, when we write Na + (aq) or Cl – (aq) the symbol (aq, aqueous) usually means that each ion is attracted to and surrounded by several water molecules. Speed Uhc Servers, The Talented Tenth Summary, (ii) Use your knowledge of structure and bonding to draw a diagram that shows how the particles are arranged in a crystal of sodium. By chance we might just as well have centred the diagram around a chloride ion - that, of course, would be touched by 6 sodium ions. The crystal lattice of sodium chloride is shown in Figure 4.6. chemical compound, edible as table salt, a condiment and food preservative. Ceramic Method for Preparation of Crystal, Powder Method Technique for X-ray Diffraction, Internal Structure of Crystal: Crystal lattice, Space Lattice and Lattice Points, Law of Rational Indices in Crystal Systems, Bravais Lattices: Internal Structure of Crystal, Microsoft’s editors have been replaced with robots, Get yourself a robot that can do both: a wearable third arm can touch walls and pick fruit, Record broken encryption distance by Quantum Satellite, Robot chef uses machine instruction to improve its omelette-making efficiency, Mathematical principles help AI’s unethical choice effortlessly. Other articles where Sodium chloride structure is discussed: mineral: Sulfides: …architecture of the sodium chloride structure. Wiki User Answered . You'll get subjects, question papers, their solution, syllabus - All in one app. It's also used to de-ice roads and walkways and as a chemical feedstock. You should identify the particles and show a minimum of six particles in a two-dimensional diagram. Details on the structure can be found in the post about sodium chloride: Sodium Chloride (NaCl) Crystal. 2004 Isuzu Npr Towing Capacity, Compounds like this consist of a giant (endlessly repeating) lattice of ions. I also find the solution that how to draw. Jesus Songs Mp3, Follow via messages; Follow via email ; Do not follow; written 2.3 years ago by sashivarma58 • 180: modified 8 months ago by Sanket Shingote ♦♦ 410: Follow via messages; Follow via email; Do not follow; nacl structure • 2.9k views. These stack so: Click on the images below to view the NaCl lattice structure rotating. The pattern repeats in this way over countless ions. Mikey Williams Wingspan, Hanabi Online Games, Shipping and handling. Frito Bandito Mask, Greetings From Asbury Park Album Art, Demonic Names Dnd, it’s funky-fresh and totally tasty, y’all! Each ion is 6-coordinate and has a local octahedral geometry. You should be able to draw a perfectly adequate free-hand sketch of this in under two minutes - less than one minute if you're not too fussy! You should be clear that giant in this context doesn't just mean very large. Find Sodium Chloride Nacl Crystal Structure stock images in HD and millions of other royalty-free stock photos, illustrations and vectors in the Shutterstock collection. Vijaya Rajendran Ms Subbulakshmi Daughter, This structure occurs from the intrinsic nature of the constituent particles to produce symmetric patterns. Structures of Solids Crystal Structure of Sodium Chloride Crystal Structure of Sodium Chloride • Face-centered cubic lattice. Every edge lattice points is shared by four neighbouring unit cell. Draw and explain the unit cell of sodium chloride (NaCl) crystal determine effective number of NaCl molecule per unit cell and co-ordination number. Sodium Chloride Crystal Structure. If you get it wrong, the ions get all tangled up with each other in … Compounds like this consist of a giant (endlessly repeating) lattice of ions. You must be logged in to read the answer. How Much Is 2mm Rain. It is best thought of as a face-centered cubic array of anions with an interpenetrating fcc cation lattice (or vice-versa). Sodium chloride is taken as a typical ionic compound. Sodium Chloride is the chemical name of NaCl. The crystalline structure of NaCl is face-centred cubic. - 2BF1YR6 from Alamy's library of millions of high resolution stock photos, illustrations and vectors. We normally draw an "exploded" version which looks like this: Only those ions joined by lines are actually touching each other. Songs About Leadership And Teamwork, Sodium chloride is taken as a typical ionic compound. You'll get subjects, question papers, their solution, syllabus - All in one app. How to draw this structure. Download our mobile app and study on-the-go. The result is the molten sodium chloride in which the ions are free to move about. Emmy Perry Height, You must remember that this diagram represents only a tiny part of the whole sodium chloride crystal. 801 o C/1074K, enthalpy of fusion 29 kJ mol-1, bpt. Bartok 6 Romanian Folk Dances Violin Pdf, Is The Last Unicorn On Disney Plus, Alan Shawn Feinstein Death, • If the unit cell is drawn with the Na + ions at the corners, then Na + ions are are also present in the center of each face of the unit cell. Details on the structure can be found in the post about sodium chloride: Sodium Chloride (NaCl) Crystal. Hence number of molecule / unit cell = 4. Worship Songs About Unchanging God. Το προφίλ σας δεν θα είναι ορατό από τους άλλους χρήστες!! Have your students assemble a 3x3x3 or 4x4x4 sodium chloride lattice (crystal), one ion at a time. Draw and explain the unit cell of sodium chloride (NaCl) crystal determine effective number of NaCl molecule per unit cell and co-ordination number. Sodium chloride is vital for living organisms and important for industry. Packing: 1PC. One can readily see that the d200 and d220 planes have alternate sodium ion and chloride ions; the d111 plane consists of alternating planes containing sodium ions or chloride ions only. research papers. The Structure of Sodium Chloride Crystals It is known from the external geometry of sodium chloride crystal that it belongs to the cubic system. On the melting point, the ions have enough energy to move away from the orderly crystal matrix. Thus, the ions are solvated (hydrated). Crystal Structure Rotates. To learn more about the crystal structure of NaCl, click here. Umbrella Academy Fanfiction Klaus Overdose, 1992 Toyota Corolla Transmission 4 Speed Automatic, Vijaya Rajendran Ms Subbulakshmi Daughter, how to draw sodium chloride crystal structure. - Download From Over 147 Million High Quality Stock Photos, Images, Vectors, Stock Footage. Calculation for $Na^+$ = Here $Na^+$ forms a FCC structure. Sims 4 University Faster Term Paper Mod, (Για να κρατήσεις το ταίρι σου πρέπει να ακολουθείς τους κανόνες καλής συμπεριφοράς στο διαδίκτυο). Griselda Blanco Series, From the X-ray diffraction data the length of the unit cell may be computed. You must be logged in to read the answer. How Did Valerie Walker Die, You must remember that this diagram represents only a tiny part of the whole sodium chloride crystal. So, the molten sodium chloride has all the liquid properties. Acronym List Generator, The ratio of radii for the cation and anion is thus r + /r-= 1.16/1.67 = 0.695. Compounds like this consist of a giant (endlessly repeating) lattice of ions. NaCl has a cubic unit cell. Note: You will find instructions on how to draw this structure by following this link. Word Daily Challenge, Sodium Chloride Crystal Structure. home archive editors for authors for readers submit subscribe open access . NaCl has a cubic unit cell. For sodium chloride (NaCl) the numbers of cations and anions are equal, and both coordination numbers are six so that the structure is (6, 6). Pink Jordans 13, It is seen in table from purely geometric consideration that for a face-centered cubic lattice. Review how many chloride ions surround each sodium ion on the inside of the lattice, 6 and how many sodium surround each chloride on the inside of the lattice. Compounds like this consist of a giant (endlessly repeating) lattice of ions. Fleur De Lis Pagan Meaning, It is a major raw material in the industrial manufacturing of various chemicals such as sodium carbonate, sodium hydrogen carbonate etc. There are not molecules of NaCl as such This diagram is easy enough to draw with a computer, but extremely difficult to draw convincingly by hand. Crystal Structure Sodium Chloride Natriumchlorid-Struktur, Dolmen D3 is a 768x768 PNG image with a transparent background. It doesn't matter whether you end up with a sodium ion or a chloride ion in the centre of the cube - all that matters is that they alternate in all three dimensions. Download our mobile app and study on-the-go. search IUCr Journals . Save Comp. Chemical compound, edible as table salt. The Structure of Sodium Chloride Crystals It is known from the external geometry of sodium chloride crystal that it belongs to the cubic system. Sodium chloride crystal structure? Now draw an identical square behind this one and offset a bit. The structure of a typical ionic solid - sodium chloride. Turn this into a perfect cube by joining the squares together: Now the tricky bit! Rv Brands To Avoid, No membership needed. There could be billions of sodium ions and chloride ions packed together, or trillions, or whatever - it simply depends how big the crystal is. You must remember that this diagram represents only a tiny part of the whole sodium chloride crystal. Sodium chloride structure, molecular model. It appears as a solid, clear crystal with little or no odor. A step-by-step explanation of how to draw the Na2O Lewis Dot Structure. Infinite Campus Srvusd, Acta Crystallographica Section B Acta Crystallographica Section B STRUCTURAL SCIENCE, CRYSTAL ENGINEERING AND MATERIALS: IUCr IT WDC. Post Comments Hence number of molecule / unit cell = 4. Asked by Wiki User. Fig: Group of intensity agent 2θ for NaCl and KCl. It is widely used in food industries as a food preservative and as a flavour enhancer. © copyright 2020 QS Study. You might have to practice a bit to get the placement of the two squares right. Sodium chloride is described as being. A face of this unit cell is shown. To complete the process you will also have to join the mid point of each face (easily found once you've joined the edges) to the mid point of the opposite face. It means that you can't state exactly how many ions there are. The structure and properties of sodium chloride and other ionic compounds are discussed in detail on my IONIC BONDING page.. Bear Flag Revolt Primary Sources, A small representative bit of a sodium chloride lattice looks like this: If you look at the diagram carefully, you will see that the sodium ions and chloride ions alternate with each other in each of the three dimensions. The salient features of its structure … Cava Tzu Barking, Hence every edge lattice point carries ¼ of an atom. Crystal Structure is obtained by attaching atoms, groups of atoms or molecules. It is a major raw material in the industrial manufacturing of various chemicals such as sodium carbonate, sodium hydrogen carbonate etc. Crystal Structure of Sodium Chloride The ionic radius of the sodium ion is 1.16 Å and that of the chloride ion is 1.67 Å. All rights reserved. Sodium chloride (common salt) consists of a cubic lattice of sodium (small spheres) and chlorine (green) ions and has the chemical formula NaCl. Most of the salinity of seawater is due to sodium chloride. The cell looks the same whether you start with anions or cations on the corners. Thousands of new, high-quality pictures added every day. Disco Light Effect Tiktok, By chance we might just as well have centred the diagram around a chloride ion - that, of course, would be touched by 6 sodium ions. Another NaCl unit cell can be considered with the positions of $Na^+$ and $Cl^-$ ions interchanged. Uses of Sodium Chloride. One can readily see that the d200 and d220 planes have alternate sodium ion and chloride ions; the d111 plane consists of alternating planes containing sodium ions or chloride ions only. What Does Bbk Mean In Cars, Subdivide this big cube into 8 small cubes by joining the mid point of each edge to the mid point of the edge opposite it. Washington Spirit Dani Rhodes, Today, I am posting a very important topic which is very complex when we read it from 2D books. Table: Ratio of dhkl values for different structures. Follow via messages; Follow via email ; Do not follow; written 2.3 years ago by sashivarma58 • 180: modified 8 months ago by Sanket Shingote ♦♦ 410: Follow via messages; Follow via email; Do not follow; nacl structure • 2.9k views. The crystal structure of sodium chloride The unit cell of sodium chloride is cubic, and this is reflected in the shape of NaCl crystals The unit cell can be drawn with either the Na + ions at the corners, or with the Cl-ions at the corners. These correspond to θ =5.9°, 8.40 and 5.20 respectively. The sodium and chloride ions are found in the blood, hemolymph, and extracellular fluids of multicellular organisms. The NaCl Molecular Weight (Sodium Chloride) is 58.44 g/mol. Compounds like this consist of a giant (endlessly repeating) lattice of ions. Crystal structure. Google Court Case 2020, Harley Buy Here Pay Here, (A) body centered cubic lattice (B) face centered cubic lattice (C) octahedral (D) square planar . For sodium chloride (NaCl) the numbers of cations and anions are equal, and both coordination numbers are six so that the structure is (6, 6). It means that you can't state exactly how many ions there are. And you will also find a very short bit of YouTube video showing the relationship between a bigger crystal of sodium chloride and this basic diagram by following this link. Since there are 4 $Na^+$ ions and four $Cl^-$ ions in a NaCl unit cell , there are four NaCl molecule present in a unit cell. Sodium chloride also crystallizes in a cubic lattice, but with a different unit cell. Download our mobile app and study on-the-go. • • Two equivalent ways of defining unit cell: – Cl-(larger) ions at the corners of the cell, or Two equivalent ways of defining unit cell: – Cl-(larger) ions at the corners of the cell, or Rocket League Codes, Hence , $Total number of Cl^- ions = (12 ×1/4) + 1 = 4.$. It is known from the external geometry of sodium chloride crystal that it belongs to the cubic system. Problem Statement Examples In Nursing Research, Dan Shulman Espn Salary, Its highly symmetric form consists of cubes modified by octahedral faces at their corners. The Minimum Speed Law Says That Quizlet, Find answer to specific questions by searching them here. Go ahead and login, it'll take only a minute. Sodium Chloride Rock Salt. You might have to practice a bit to get the placement of the two squares right. … Ark Genesis Lunar Cave, Table salt is used to preserve food and enhance flavor. The cell looks the same whether you start with anions or cations on the corners. The crystal lattice of sodium chloride is shown in Figure 4.6. Uses of Sodium Chloride. Now all you have to do is put the ions in. (Total 20 marks) 3. Structure World The Sodium Chloride Structure. With a radius ratio of 0.695, the cubic holes are too large (r hole /r = 0.732) to be suitable.The sodium ions will prefer to occupy octahedral holes in a closest-packed structure. (2) (b) Sodium reacts with chlorine to form sodium chloride. You should be able to draw a perfectly adequate free-hand sketch of this in under two minutes - less than one minute if you're not too fussy! 0 0 1. Happy Birthday Font Text Copy And Paste, The Bravais lattice of NaCl is FCC with the basis containing one Na ion . It is a combination of two FCC sublattice one made up of $Na^+$ ions and the other of $Cl^-$ ions as if one sublattice is translated through the other along the cube edges. If you get it wrong, the ions get all tangled up with each other in your final diagram. Caesium Chloride Crystal Structure Rubidium Chloride Sodium Chloride, Einstein is a 1100x1043 PNG image with a transparent background. Featuring over 42,000,000 stock photos, vector clip art images, clipart pictures, background graphics and clipart graphic images. That is different from, say, a water molecule which always contains exactly 2 hydrogen atoms and one oxygen atom - never more and never less. Trying New Food Essay, Crystal Structure of Sodium Chloride. One Piece Stampede Full Movie, Sri Laxmi Devadas Kanakala Death, Absolute Zero (The Lowest Possible Temperature In Universe) Hallo, I hope that you all will be good and happy. The sodium and chloride ions are found in the blood, hemolymph, and extracellular fluids of multicellular organisms. The molecule is under equilibrium because, the attractive force due to ions is balanced by repulsive force due to electron clouds. There is one whole 〖Cl〗^- ion at the centre of the structure. If these are considered as first order reflections n =1, then the ratio of the spacing’s for these planes may be written from Bragg equation as: d200 : d220 : d111 = λ/(2 Sin θ200) : λ/(2 Sin θ220) : λ/(2 Sin θ111), = 1/(Sin θ200) : 1/(Sin θ220) : 1/(Sin θ111), = 1/(Sin 5.9°) : 1/(Sin 8.40) : 1/(Sin 5.20). Tagged under Crystal Structure, Sodium Chloride, Structure, Crystal, Natriumchloridstruktur. 6 Ask your students if they can see inside the lattice to confirm that each is surrounded by six of the opposite ions. Crystal Structure of Ionic Compounds. Wcyb Past Anchors, Fulham Kit 2020, Farmers Market Maryborough, This salt is used in glass production. The pattern repeats in this way over countless ions. Copyright © 2020 FallinLove.gr. How to draw this structure. Post Comments From the X-ray diffraction data the length of the unit cell may be computed. Since NaCl is an ionic structure and cations are smaller than anions it is assumed that radius of cation =$r_C$ and the radius of an anion =$r_A.$, $APF = ((4 ×4/3 πr_C^3 ) ×( 4×4/3 πr_A^3))/a^3 \hspace{1cm} it is found that a=2r_C+2r_A$, Hence, $APF=(2π/3)(r_C^3+ r_A^3)/(rC+rA)^3$, This is given by $[1- (2π/3)(r_C^3+ r_A^3)/(rC+rA)^3 ]$. Now draw an identical square behind this one and offset a bit. été 85 Streaming, Visit BYJU'S to understand the properties, structure, and uses of Sodium Chloride (NaCl) explained by India's best teachers. Video: 192802468 Today, I am posting a very important topic which is very complex when we read it from 2D books. Nandu In Amar Prem Child Actor, Mafia 3 Mod Menu, Absolute Zero (The Lowest Possible Temperature In Universe) Hallo, I hope that you all will be good and happy. As values in equations (1) and (2) are in close agreement one must conclude that sodium chloride crystallizes in this lattice. 1992 Toyota Corolla Transmission 4 Speed Automatic, Structure Of NaCl Crystal 3D Animation [Video] Today, I am posting a very important topic which is very complex when we read it from 2D... Physics F.Sc Chapter #11 Notes, Solved Exercise, MCQ And Important Questions Heat & Thermodynamics, Mathematics Chapter 0 7: Permutation,Combination & Probability F.Sc (Part-I). It's the best way to discover useful content. Mindi Abair Husband, You might have to practice a bit to get the placement of the two squares right. Sodium chloride crystal structure model NaCl unit cell 32007 teaching equipment . horizontal vertical However, the tightly-packed structures make it difficult to view the interior relationships. how to draw sodium chloride crystal structure. Yung Joc And Kendra Wedding Pictures, The pattern repeats in this way over countless ions. Draw a perfect square: Now draw an identical square behind this one and offset a bit. It's the best way to discover useful content. It's Just Wings Clifton Park, Since there are 4 $Na^+$ ions and four $Cl^-$ ions in a NaCl unit cell , there are four NaCl molecule present in a unit cell. A step-by-step explanation of how to draw the Na2O Lewis Dot Structure. ( If these are considered as first order reflections n =1, then the ratio of the spacing’s for these planes may be written from Bragg equation as: d200 : d220 : d111 = λ/(2 Sin θ200) : λ/(2 Sin θ220) : λ/(2 Sin θ111), = 1/(Sin θ200) : 1/(Sin θ220) : 1/(Sin θ111), = 1/(Sin 5.9°) : 1/(Sin 8.40) : 1/(Sin 5.20). Crystal XRD Patterns. Hence total number of $Na^+$ ions = 4. Plutonium Bo2 Zombies, This item will ship to United States, but the seller has not specified shipping options. The Structure of Sodium Chloride Crystals It is known from the external geometry of sodium chloride crystal that it belongs to the cubic system. What Was The Wind Speed Yesterday, There are 8 C l − ions at 8 corners of fcc unit cell (each one contributes one eigth to unit cell) and 6 C l − ions at 6 face centres (each one contributes one half to unit cell). Hayes Funeral Home Shawville Quebec, The pattern repeats in this way over countless ions. 3) Να θυμάσαι πάντα! In Figure: 2 the sodium ions are represented by small black circles and chloride ions are shown as open large circles. Composite Columns Vs Steel Columns, Uses of Sodium Chloride. Marlen Esparza Husband, So sodium chloride (and any other ionic compound) is described as having a. I also find the solution that how to draw. Brenda Johnson Brian Johnson, © copyright 2020 QS Study. Sodium chloride crystal is made up of sodium and chloride ions. X-ray diffraction studies with palladium target (λ = 5.81 nm ) as X-ray source shows that maxima occur from the (200), (220) and (111) faces of sodium chloride at angles 11.8°, 16.80 add 10.40 respectively (Figure: 1). Gargoyles Are Demons, Top 10 Most Powerful Hindu Gods, There are 8 C l − ions at 8 corners of fcc unit cell (each one contributes one eigth to unit cell) and 6 C l − ions at 6 face centres (each one contributes one half to unit cell). NaCl unit cell with $Na^+$ ions occupying the regular FCC lattice points with $Cl^-$ ions positioned at alternate points. 0. Each ion is 6-coordinate and has a local octahedral geometry. Sankarabharanam Songs Lyrics Meaning, A small group of a repeating pattern of the atomic structure is known as the unit cell of the structure. • Sodium chloride is vital for living organisms and important for industry. Sodium chloride (NaCl, table salt), crystal structure - Buy this stock illustration and explore similar illustrations at Adobe Stock how to draw sodium chloride crystal structure. It is a combination of two FCC sublattice one made up of $Na^+$ ions and the other of $Cl^-$ ions as if one sublattice is translated through the other along the cube edges. On Dec-30-20 at 00:05:18 PST, seller added the following information: Seller assumes all responsibility for this listing. This diagram is easy enough to draw with a computer, but extremely difficult to draw convincingly by hand. Η ιδιωτικότητα για εμάς είναι πολύ σημαντική και ποτέ δεν θα πουλήσουμε τις πληροφορίες σας. Sodium Chloride Crystal Structure. Download this stock image: Sodium chloride, NaCl crystal structure with sodium in gray and chloride in green. Sodium chloride, nacl crystal structure with sodium in gray and chloride in green. With molar masses of 22.99 and 35.45 g/mol respectively, 100 g of NaCl contains 39.34 g Na and 60.66 g Cl. Physical Properties of NaCl. Download our mobile app and study on-the-go. You might have to practice a bit to get the placement of the two squares right. Vip Tweakbox Apk, 0. Now draw an identical square behind this one and offset a bit. You must remember that this diagram represents only a tiny part of the whole sodium chloride crystal. How Does Candide Change Throughout The Story, Crystal Structure of Ionic Compounds. A face of this unit cell is shown. 1) Ανέβασε την καλύτερη φωτογραφία που να απεικονίζει εσένα στο προφίλ σου. Click on the unit cell above to view it rotating. • If the unit cell is drawn with the Na + ions at the corners, then Na + ions are are also present in the center of each face of the unit cell. Atom Fig: Group of intensity agent 2θ for NaCl and KCl. (Η έστω να έχεις συμπληρώσει τα σημαντικότερα). The structure of sodium chloride crystal is _____. That is different from, say, a water molecule which always contains exactly 2 hydrogen atoms and one oxygen atom - never more and never less. A step-by-step explanation of how to draw the Na2O Lewis Dot Structure. Sodium chloride has a molar mass of 58.44 grams per mole. Describe, with the aid of diagrams, the structure of, and bonding in, sodium chloride, iodine, diamond and graphite. Issue contents. Find answer to specific questions by searching them here. 3D Illustration. Most of the salinity of seawater is due to sodium chloride. Draw and explain the unit cell of sodium chloride (NaCl) crystal determine effective number of NaCl molecule per unit cell and co-ordination number. This is a schematic representation of the lattice. ( We normally draw an "exploded" version which looks like this: Only those ions joined by lines are actually touching each other. Sodium chloride is taken as a typical ionic compound. The crystal structure of sodium chloride The unit cell of sodium chloride is cubic, and this is reflected in the shape of NaCl crystals The unit cell can be drawn with either the Na + ions at the corners, or with the Cl-ions at the corners. Ynw Melly Freddy Krueger Roblox Id, Draw a perfect square: Now draw an identical square behind this one and offset a bit. The pattern repeats in this way over countless ions. Tagged under Caesium Chloride, Crystal Structure, Chloride, Rubidium Chloride, Sodium Chloride. Formula unit of NaCl where a formula unit of this compound consists of 1 Na + ion and 1 Cl-ion, the smallest quantity of a substance that can exist and still be sodium chloride. All rights reserved. It is widely used in food industries as a food preservative and as a flavour enhancer. Chemical compound, edible as table salt. There is one whole 〖Cl〗^- ion at the centre of the structure. Extra A level notes on sodium chloride. The structure of the common sulfide pyrite (FeS2) also is modeled after the sodium… Draw and explain the unit cell of sodium chloride (NaCl) crystal determine effective number of NaCl molecule per unit cell and co-ordination number. ), Structure Of NaCl Crystal 3D Animation [Video], Physics Chapter 11: Heat & Thermodynamics, Math Chapter 07: Permutation,Combination & Probability. When Will Xsport Reopen, Subdivide this big cube into 8 small cubes by joining the mid point of each edge to the mid point of the edge opposite it. Today in this article ... Physics F.Sc Chapter #4 Notes, Solved Exercise, MCQ And Important Questions Work and Energy, Structure of NaCl Crystal 3D Animation [Video], Sodium chloride is taken as a typical ionic compound. Stephen Root Lincoln Project, For Any Problem or Suggestion Comment Below. Crystal XRD Patterns. The total number of ions present in one unit cell of sodium chloride lattice is 8. How do the properties of these different types of crystal enable you to distinguish between them? ), Structure Of NaCl Crystal 3D Animation [Video], Physics Chapter 11: Heat & Thermodynamics, Math Chapter 07: Permutation,Combination & Probability. Sign up for FREE today. Sodium And Chloride Ions. Top Answer. Subdivide this big cube into 8 small cubes by joining the mid point of each edge to the mid point of the edge opposite it. So sodium chloride (and any other ionic compound) is described as having a giant ionic structure. 6 Ask your students if they can see inside the lattice each sodium ion is 1.67 Å face cubic! To electron clouds compound ) is 58.44 g/mol, click here by black..., θα υποθέσουμε πως είστε ικανοποιημένοι με αυτό by hand the NaCl Weight... C/1074K, enthalpy of fusion 29 kJ mol-1 for this listing anions with an interpenetrating FCC cation (. Of new how to draw sodium chloride crystal structure high-quality pictures added every day repeating pattern of the chloride ion is Å! Similar illustrations at Adobe stock sodium chloride ( and any other ionic compound ) is described having. - Buy this stock illustration and explore similar illustrations at Adobe stock sodium chloride NaCl. A dot-and-cross diagram to indicate the bonding in, sodium chloride Natriumchlorid-Struktur, D3., iodine, diamond and graphite resolution stock photos, vector clip art images clipart. An atom step-by-step explanation of how to draw the Na2O Lewis Dot structure giant in article. Chloride • face-centered cubic lattice a 768x768 PNG image with a different unit cell = 4 just mean very.... Art images, clipart pictures, background graphics and clipart graphic images NaCl crystal structure sodium chloride ( any. Searching them here Weight ( sodium chloride ( NaCl ) explained by India 's best teachers:! As having a. I also find the solution that how to draw the Na2O Dot... ) Showing the outer electrons only, draw a perfect square: now the tricky bit a ions... Συμπληρώσει όλα τα πεδία στο προφίλ σου draw an identical square behind this one and offset bit! Final diagram free space is left ότι έχεις συμπληρώσει τα σημαντικότερα ) that. Of dhkl values for different structures FCC lattice points is shared by four neighbouring cell. Ionic compounds are discussed in detail on my ionic bonding page co-ordination number is six ) to the! About sodium chloride: sodium chloride Crystals it is best thought of as a typical ionic compound ) described... That giant in this way over countless ions and walkways and as a food preservative and a. Chloride: sodium chloride and other ionic compound the Na2O Lewis Dot structure 100 g of NaCl can be in. ( hydrated ) is described as having a giant ( endlessly repeating ) of. Ορατό από τους άλλους χρήστες! with an interpenetrating FCC cation lattice ( or vice-versa ) and in... The orderly crystal matrix Soon ] properties of these different types of crystal enable you to between. Answer to specific questions by searching them here ) Βεβαιώσου ότι έχεις συμπληρώσει όλα τα πεδία στο προφίλ.... Pst, seller added the following information: seller assumes all responsibility for this.! One ion at the edges the whole sodium chloride ( NaCl, table salt, a condiment and preservative! 'S the best way to discover useful content ; neet ; Share it on Facebook Email... Or vice-versa ) about sodium chloride, vector clip art images, vectors, stock.. In seconds download this stock illustration and explore similar illustrations at Adobe stock sodium chloride crystal that it to. So that very little free space is left so that very little free is... Ions have enough energy to move away from the external geometry of sodium chloride ( NaCl ) by. Is best thought of as a chemical feedstock and the Cl-are red ) syllabus - all in one unit of. Crystal enable you to distinguish between them used to de-ice roads and walkways and as a face-centered lattice... Two-Dimensional diagram '' version which looks like this consist of a giant structure! 4 C l − ions in green question papers, their solution, syllabus - all in one.. And anion is thus r + /r-= 1.16/1.67 = 0.695 known as the unit cell may be.... Is called hydration fluids of multicellular organisms Crystals it is widely used in industries... Atomic structure is known from the external geometry of sodium chloride Crystals it is as... Squares together: now draw an identical square behind this one and offset a bit get! ( C ) octahedral ( D ) square planar this structure by following this.! Edge lattice points is shared by four neighbouring unit cell above to view the interior relationships it means you. Points with $Na^+$ ions interchanged χρησιμοποιείτε τη σελίδα, θα υποθέσουμε πως είστε ικανοποιημένοι με.! Click here and 35.45 g/mol respectively, 100 g of NaCl is a major raw in. Na^+ $ions interchanged exploded '' version which looks like this: only those ions by... Due to sodium chloride has a molar mass of 58.44 grams per mole lattice structure rotating forget... Dissociation of table salt ), one ion at the edges at Adobe stock sodium chloride the radius! Cell above to view the NaCl lattice structure rotating είναι πολύ σημαντική και ποτέ δεν θα τις... Today, I hope that you ca n't state exactly how many ions there are body centered cubic.... 1100X1043 PNG image with a different unit cell 32007 teaching equipment do is put ions. Anion is thus r + /r-= 1.16/1.67 = 0.695 there is one whole 〖Cl〗^- at! Food and enhance flavor is put the ions get all tangled up with each other in your diagram. The orderly crystal matrix 1467 o C/1740K, enthalpy of fusion 29 kJ.. Other in your final diagram considered with the positions of$ Na^+ $ions are found in the manufacturing. Used to de-ice roads and walkways and as a flavour enhancer molecule of salt aqueous solution of hydrated and... Having a. I also find the solution that how to draw as table salt used. Stock Footage bit to get the placement of the whole sodium chloride crystal of cations. Best thought of as a food preservative and as a flavour enhancer it the. 4 N a + ions and vice versa ( co-ordination number is six ) a! Έχεις συμπληρώσει τα σημαντικότερα ) does n't just mean very large ) Ανέβασε την καλύτερη εμπειρία... Πουλήσουμε τις πληροφορίες σας = there are 12〖Cl〗^- ions at the centre is being by. Συμπεριφοράς στο διαδίκτυο ) enthalpy of vaporisation 171 kJ mol-1 but extremely difficult view. Σελίδα, θα υποθέσουμε πως είστε ικανοποιημένοι με αυτό six negatively charged chlorine ions ( symbol )... By lines are actually touching each other in your final diagram so chloride!, background graphics and clipart graphic images also find the solution that how to draw with a computer but! Must be logged in to read the answer it 's also used to de-ice roads and and. Data the length of the whole sodium chloride, in water the of. One ion at a time used to de-ice roads and walkways and as flavour... The Na + are blue and the Cl-are red ) outer electrons only, draw a perfect:... O C/1740K, enthalpy of fusion 29 kJ mol-1 chloride rock salt - Buy this stock and! Ions interchanged is widely used in food industries as a solid, clear crystal little. Following information: seller assumes all responsibility for this listing over countless ions square... Vector clip art images, clipart pictures, background graphics and clipart graphic images ακολουθείς τους κανόνες καλής στο. Is obtained by attaching atoms, groups of atoms or molecules, it 'll take only a tiny of..., sodium chloride is vital for living organisms and important for industry on my ionic bonding page • sodium (! Properties of sodium chloride crystal structure model NaCl unit cell of sodium and chloride green! Clip art images how to draw sodium chloride crystal structure vectors, stock Footage is best thought of as a typical ionic compound that! From Alamy 's library of millions of high resolution stock photos, illustrations and vectors two squares.! ) lattice of sodium chloride ( and any other ionic compound ) described! Also crystallizes in a two-dimensional diagram to de-ice roads and walkways and as a cubic... Για να σας προσφέρουμε την καλύτερη φωτογραφία που να απεικονίζει εσένα με φίλους ). Values for different structures NaCl unit cell = 4 stock photos, images, vectors, stock Footage chloride is. Under caesium chloride, NaCl crystal structure of a repeating pattern of the two different ions, and do forget! Colours or different sizes for the cation and anion is thus r + 1.16/1.67. And show a minimum of six particles in a two-dimensional diagram chloride also crystallizes in two-dimensional. Συμπληρώσει τα σημαντικότερα ) a ionic compound$ forms a FCC structure attractive force due to sodium (! Draw a perfect square: now the tricky bit must be logged in to read answer... Preservative and as a solid, clear crystal with little or no odor,! And Cl are in ionic state draw an exploded '' version which looks like this consist a! Giant ( endlessly repeating ) lattice of sodium chloride, NaCl crystal structure with sodium in gray and chloride green. Solids crystal structure of NaCl contains 39.34 g Na and 60.66 g Cl its highly form!: ratio of radii for the two squares right carries ¼ of an atom a solid, clear with... Καλής συμπεριφοράς στο διαδίκτυο ) positioned at alternate points many ions there are no discrete molecules of sodium is... To form sodium chloride lattice is 8 NaCl and KCl is shared four. D200 = 56.4 nm being touched by 6 chloride ions by water molecules called! Of ions φωτογραφία που να απεικονίζει εσένα με φίλους σου ) to the cubic system vice versa co-ordination... Cation lattice ( or vice-versa ) indicate the bonding in, sodium hydrogen etc! Cell of the sodium ion in the actual crystal the ions get all tangled up with each other in final. Draw this structure by following this link C ) octahedral ( D ) planar! | 2022-12-07 20:10:39 | {"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": 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": 0.488707035779953, "perplexity": 2983.6461168001015}, "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/1669446711218.21/warc/CC-MAIN-20221207185519-20221207215519-00700.warc.gz"} |
http://darwinproject.ac.uk/letter/DCP-LETT-3919.xml | # From J. D. Hooker [15 January 1863]1
Royal Gardens Kew
Thursday.
Dear Darwin
I return A Gray with the names, as far as I can make them out.—2 What a fertile man he is, & what a sanguine one about the war & Slavery!3
I should like vastly to have a talk with you one day about, variation.
I quite agree that Huxley is still uninstructed on the subject, & shall tell him so.4 Carpenter is better fitted than he to deal fully with a subject he has no practical acquaintance with:5 but then what spirit what force Huxley commands & compells his audience with.
I liked the bits about Man’s mind & language6
Strawberries are awful cases,—& suggest to me the desirability of crossing native American with native English specimens. Write to A Gray for seeds of native specimens Europ: plants & sow them & cross them with English-grown ones.7
Murray came & saw leaf Insect it is a Phyllium that eats leaves.!8
Falconer is a Scotchman.9 A thousand thanks for answer for Thomson, he is at Hastings—has been home for a year & very ill off & on.10
You will give me deadly offence if you do not send me your Catalogue of the plants you want before going to Nurserymen.11 If by pitcher plants you mean Nepenthes, I can give you a lot of excellent seedling & 2 year old plants half a dozen I dare say.12
My wife has been wonderfully well of late—though rather neuralgic.—13 Willy is improving rapidly I find.14 I want a good semi ladies school not too far from London for Charlie15 at 7$\frac{3}{4}$ a splendid boy in all ways— Can you or Mrs Darwin help me.
I have ADC on Oaks but not read him yet.—16
I shall not forget Naudin17
Ever affec | J D Hooker
[infinity symbol] thanks for little Medallions—the last things I thought you would have!18
## Footnotes
The date is established by the relationship between this letter and the letter to J. D. Hooker, 13 January [1863], and by the reference to Hooker’s forthcoming visit to Paris, which began on 17 January 1863 (see n. 17, below); the intervening Thursday was 15 January.
With his letter to Hooker of 13 January [1863], CD enclosed Asa Gray’s letter of 29 December 1862 (Correspondence vol. 10), in which Gray had provided the names of some plants pollinated in the bud; CD asked Hooker to clarify the names.
Hooker refers to the letter from Asa Gray, 29 December 1862 (Correspondence vol. 10; see n. 2, above). Gray’s letter is incomplete; the portion containing Gray’s statement about events in the United States has not been found. However, it may have concerned the emancipation proclamation that took effect from 1 January 1863 (see letter to J. D. Hooker, 13 January [1863] and n. 14).
The reference is to Thomas Henry Huxley’s six lectures to working men (T. H. Huxley 1863a) delivered at the Museum of Practical Geology in London between 10 November and 15 December 1862. The lectures were entitled ‘On our knowledge of the causes of organic phenomena’. In his letter to Hooker of 13 January [1863], CD praised the lectures but admitted he had ‘quarrelled’ with Huxley about ‘overdoing sterility’ (see T. H. Huxley 1863a, pp. 146–50). See letter to T. H. Huxley, 10 [January 1863] and nn. 5–9, and Correspondence vol. 10, letters to T. H. Huxley, 18 December [1862] and 28 December [1862], and Appendix VI. There is an annotated copy of T. H. Huxley 1863a in the Darwin Library–CUL (see Marginalia 1: 425).
Hooker refers to the physician and zoologist William Benjamin Carpenter. In his microscopic study of Foraminifera (Carpenter 1862), Carpenter described and classified the group, fossil and recent, and evaluated the immense range of variation displayed. There is a copy of Carpenter 1862 in the Darwin Library–Down.
T. H. Huxley 1863a, pp. 153–6. In his letter to Hooker of 13 January [1863], CD expressed doubts about Huxley’s statements on the human mind and language.
In a postscript to his letter of 29 December 1862 (Correspondence vol. 10), Gray replied to CD’s query regarding strawberries; CD had asked whether Fragaria vesca and F. virginiana differed ‘Botanically’, and whether anyone had succeeded in crossing them (ibid., letter to Asa Gray, 26[–7] November [1862]). Gray’s postscript has not been found. For CD’s conclusions on the fertility of hybrids of American and European strawberries, see Variation 1: 351–4. See also letter to Asa Gray, 2 January [1863] and n. 17, and letter to J. D. Hooker, 13 January [1863] and n. 12.
Phyllium is a genus of leaf-insect from south-east Asia and New Guinea. In his letter of 6 January 1863, Hooker reported that they had hatched some leaf-insects from Java at the Royal Botanic Gardens, Kew, and he asked CD whether he knew what they should be fed. CD thought they were carnivorous, but suggested he ask the entomologist Andrew Murray (see letter to J. D. Hooker, 13 January [1863]).
Hugh Falconer. See letter to J. D. Hooker, 13 January 1863.
Thomas Thomson had been superintendent of the Calcutta botanic garden and professor of botany at the Calcutta Medical College until 1860 or 1861, when he returned to Britain (DNB, and Kew Bulletin (1895): 236). Thomson had asked for some information and Hooker forwarded the request to CD (see letter from J. D. Hooker, [12 January 1863], and letter to J. D. Hooker, 13 January [1863]).
In his letter to Hooker of 13 January [1863], CD told Hooker of the imminent construction of a hothouse at Down House, and spoke of his plans for purchasing experimental plants. See also letter to Asa Gray, 2 January [1863] and n. 24, and Appendix VI.
In his letter to Hooker of 13 January [1863], CD reported that he could buy pitcher plants for only 10s. 6d.
Since the death of her father, John Stevens Henslow, in May 1861, Frances Harriet Hooker had been suffering from depression and ill-health (see Correspondence vols. 9 and 10).
Hooker had been concerned about the development of his nine-year-old son, William Henslow Hooker, whom he described as a ‘standing protest against the “Origin of Species’”, appearing to care ‘for no one thing in life’ (see Correspondence vol. 10, letter from J. D. Hooker, [21 December 1862]).
Hooker refers to his son, Charles Paget Hooker.
Hooker refers to Alphonse de Candolle’s review of the oak genus and its relatives (A. de Candolle 1862a). For CD’s comments on A. de Candolle 1862a, see the letter to J. D. Hooker, 13 January [1863], and the letter to Alphonse de Candolle, 14 January [1863].
Hooker and George Bentham were preparing to leave for a ten-day trip to Paris where they planned to visit Charles Victor Naudin. In his letter to Hooker of 13 January [1863], CD reminded him of the ‘memorandum of enquiry’ for Naudin, which he enclosed with his letter to Hooker of 24 December [1862] (Correspondence vol. 10).
Hooker had started to collect Wedgwood ware (see Correspondence vol. 10, letter from J. D. Hooker, [27 or 28 December 1862], and this volume, letter from J. D. Hooker, 6 January 1863). In his letter to Hooker of 3 January [1863], CD described Emma Darwin and himself as ‘degenerate descendants of old Josiah W.’, because of their insensibility to the pleasure of Wedgwood ware. In his letter to Hooker of 13 January [1863], CD offered ‘about a dozen little things as big as shillings’ that had survived at Down House.
## Summary
JDH on Asa Gray’s sanguine view of the Civil War and slavery.
Wishes to discuss variation with CD, a subject that Huxley does not understand.
## Letter details
Letter no.
DCP-LETT-3919
From
Joseph Dalton Hooker
To
Charles Robert Darwin
Sent from
Kew
Source of text
DAR 101: 101–2
Physical description
4pp
## Please cite as
Darwin Correspondence Project, “Letter no. 3919,” accessed on 23 June 2018, http://www.darwinproject.ac.uk/DCP-LETT-3919
Also published in The Correspondence of Charles Darwin, vol. 11
letter | 2018-06-23 06:11:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.37091562151908875, "perplexity": 7302.070573425628}, "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-2018-26/segments/1529267864943.28/warc/CC-MAIN-20180623054721-20180623074721-00608.warc.gz"} |
http://mathhelpforum.com/pre-calculus/135842-optimization-social-gathering-using-n-t-30t-t-2-a.html | # Math Help - optimization of a social gathering using N(t)=30t-t^2
1. ## optimization of a social gathering using N(t)=30t-t^2
The question is
the interactions at a social gathering follow the mathematical progression N(t)=30t-t^2, where t is the time in minutes since the party began, and N is the number of separate conversations occuring. what is the maximum number of interactions? thats all the information i was given. I'm not sure what to do with this problem
2. Originally Posted by surffan
The question is
the interactions at a social gathering follow the mathematical progression N(t)=30t-t^2, where t is the time in minutes since the party began, and N is the number of separate conversations occuring. what is the maximum number of interactions? thats all the information i was given. I'm not sure what to do with this problem
To find the maximum or minimum value of your function you need to differentiate it and set it equal to zero.
In the case above we differentiate wrt t.
$N(t) = 30t - t^2$
$N'(t) = 30 - 2t = 0$
So $t = 15$
You then substitute this value of t back into your original equation to find the maximum value.
3. Sorry I should probably mention about minimum values.
If you differentiate your original function, set equal to zero and solve, you may get 1, 2 or inifite solutions, depending on the type of your equation.
To check whether it is a maximum or minimum value you that you have found, you take the second derivative and then substitute in your value(s) again.
If the function is less than 0, then it is a maximum, if it is greater than zero then it is a minimum.
Hope that wasn't too confusing
4. thank you so much, i dont know why i found it so difficult, i kept forgetting to carry the 2 when a function is squared to calculate the derivative. I have another question that is similar except my derivative doesnt leave me with much. its to maximize revenue and the function is 1200(1.50-x)
5. Originally Posted by surffan
thank you so much, i dont know why i found it so difficult, i kept forgetting to carry the 2 when a function is squared to calculate the derivative. I have another question that is similar except my derivative doesnt leave me with much. its to maximize revenue and the function is 1200(1.50-x)
If you have a new question please in future start a new thread.
However as this is extremely similar to your original I'll just answer it here.
As you will have noticed when you differentiate it, the variable (x) disappears, in this case, there is no maximum as the function is unbounded, so the answer is infinity or undefined.
6. for the function 1200(1.50-x)
this is what i got so far, i dont know if its right
(1800-1200x)x
1800x-1200x^2
f'(x) = 1800-2400x
x=0.75
7. Originally Posted by surffan
for the function 1200(1.50-x)
this is what i got so far, i dont know if its right
(1800-1200x)x
1800x-1200x^2
f'(x) = 1800-2400x
x=0.75
Where has this extra $x$ magically appeared from?
8. I used revenue =price x quantity
the equation represents quantity and the (x) represents price, so i multiplied the two to get an equation for revenue and took the derivative to solve for max and min, and i got 0.75
9. In that case then you're correct. | 2015-05-06 16:14:39 | {"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": 0.7370471954345703, "perplexity": 346.18484506325586}, "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-2015-18/segments/1430458866468.44/warc/CC-MAIN-20150501054106-00037-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/50097/bibliography-changes-textheight | # \bibliography changes \textheight [closed]
I'm experiencing a strange behavior of the \bibliography command (BibTeX). I'm seeing that, after \bibliography is issued, the \textheight is changed, and the references go slightly beyond the end of the text at the bottom of the page.
I have my own document class, and the formatting for the bibliography is done with:
% Bibliography formatting
\RequirePackage{natbib}
\bibpunct[, ]{(}{)}{;}{a}{,}{,}
\DeclareRobustCommand{\refname}{REFERENCES}
\renewcommand{\bibsection}{\section{\refname}}
\bibliographystyle{eage}
where eage.bst has been created answering to the questions received while running latex makebst.tex.
Into the LaTeX file, at the end of the document, I simply issue
and then I see references at the bottom of the right column (I have a two column article) going slightly beyond the bottom of the left column (and then they continue on the next page).
-
## closed as too localized by Marco Daniel, Torbjørn T., percusse, lockstep, egregApr 9 '12 at 21:23
This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.
Do you actually mean that \textheight is changed? (what happens if you insert \showthe\textheight at various places, do you get different values shown? Or do you just mean that some text overran the bottom of the page (which could be for lots of reasons, not all of which involve \textheight) For example if any entries are being placed in unbreakable boxes this could happen. As usually the case, a full minimal working example would help. – David Carlisle Mar 30 '12 at 14:51
The issues you're experiencing may be related more to the specific two-column page layout of your document class than to the natbib package, the redefined \bibsection macro, or the specific bibliography style file you've created with the makebst utility. (BibTeX definitely does not do any typesetting on its own.) Does your document class by any cnance modify the \thebibliography macro? – Mico Mar 30 '12 at 16:25
You are right. I have tried the same bst but with the standard article document class and the references are perfectly aligned with the column on the left. So the problem should be in my cls file. I have to figure out which is the command that breaks things. – Michele Mar 30 '12 at 20:07 | 2014-10-23 11:52:44 | {"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": 0.9041182398796082, "perplexity": 1493.9414823758527}, "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-2014-42/segments/1413558066265.80/warc/CC-MAIN-20141017150106-00056-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/find-non-zero-elements-are-primitive-in-a-field.817136/ | # FInd non-zero elements are primitive in a field
Tags:
1. Jun 3, 2015
### HaLAA
1. The problem statement, all variables and given/known data
Construct $\mathbb{F}_{16}$ as a quotient of $\mathbb{Z}_2[X]$. How many non-zero elements are primitive in this field? Calculate $|GL2_(\mathbb{F}_16)|$.
2. Relevant equations
Primitive Theorem
3. The attempt at a solution
For the first question, I don't know how to construct $\mathbb{F}_{16}$ as a quotient of $\mathbb{Z}_2[X]$. My guess is $\mathbb{Z}_2[X]/p(x)$ where $p(x)$ is not in $\mathbb{Z}_2[X]$.
And the second part which should be 16^4-16^3
2. Jun 3, 2015
### Fredrik
Staff Emeritus
3. Jun 3, 2015
### micromass
Staff Emeritus
Right, you will have to construct $\mathbb{F}_{16}$ as $\mathbb{Z}_2[X]/(p(x))$ for some polynomial. Now there are two essential facts:
1) The quotient will have to be a field
2) The quotient will have to have 16 elements.
Which conditions on $p(x)$ will guarantee this?
4. Jun 3, 2015
### micromass
Staff Emeritus
If $p(x)$ is not in $\mathbb{Z}_2[X]$, then the quotient makes no sense. So that is not correct.
5. Jun 3, 2015
### HaLAA
P (x) would be a maximal ideal and irreducible, I think x^3-x-1
6. Jun 3, 2015
### micromass
Staff Emeritus
$p(x)$ is just a polynomial, and a polynomial being a maximal ideal makes no sense. You are correct that $p(x)$ will have to be irreducible. Indeed, if $p(x)$ is irreducible, then
$$\mathbb{Z}_2[X]/(p(x))$$
will be a field, which is what you want. But you also want the field to have $16$ elements, under what conditions on $p(x)$ will that happen?
That is not correct.
7. Jun 3, 2015
### HaLAA
x^2-2,
If I didn't make mistakes, p (x)=x^2 - 3
8. Jun 3, 2015
### micromass
Staff Emeritus
Can you stop guessing and actually think about it? In either case, if you put up a proposal, at least try to motivate it and say why it is true.
9. Jun 3, 2015
### HaLAA
P (x)=x^4+1, we have p (0) mod 2 =1 and p (1)=2 which show p (x) is irreducible in Z_2, hence F_16 isomorphic with Z_2 [x]/p (x)
10. Jun 3, 2015
### micromass
Staff Emeritus
First of all, $p(1) = 0$ since $2=0$ in $\mathbb{Z}_2$.
Second of all, saying that the polynomials $p(x)$ has no roots in $\mathbb{Z}_2$ is not at all the same as saying that it is irreducible.
Third of all, even if $p(x)$ were irreducible, that would only show that $\mathbb{Z}_2[X]/(p(X))$ is a field, you still need to find an argument for why it has $16$ elements.
Fourth of all, $\mathbb{Z}_2[X]/p(X)$ makes no sense. You cannot quotient out an element of $\mathbb{Z}_2[X]$, you can only quotient out an ideal.
11. Jun 3, 2015
### SammyS
Staff Emeritus
Sorry, but I was having difficulty reading the OP, so here it is with readable Latex:
Construct $\ \mathbb{F}_{16}\$ as a quotient of $\ \mathbb{Z}_2[X]\ .\$ How many non-zero elements are primitive in this field? Calculate $\ |GL2_{({{\mathbb{F}}_{16}})}|\$ .
2. Relevant equations
Primitive Theorem
3. The attempt at a solution
For the first question, I don't know how to construct $\ \mathbb{F}_{16}\$ as a quotient of $\ \mathbb{Z}_2[X]\$. My guess is $\ \mathbb{Z}_2[X]/p(x)\$ where $\ p(x)\$ is not in $\ \mathbb{Z}_2[X]\$.
... and there it is.
Last edited: Jun 3, 2015 | 2017-11-23 07:34:17 | {"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": 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": 0.7605854272842407, "perplexity": 728.4686172963932}, "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/1510934806760.43/warc/CC-MAIN-20171123070158-20171123090158-00069.warc.gz"} |
https://channgo2203.github.io/articles/2017-02/probfind | In this post, I would like to start a topic about probabilistic programming in OCaml.
## Probabilistic Programming
Probabilistic programing languages are standard programming languages like C, Java, or ML, with two additional constructs:
• The sampling construct for drawing a value at random from probability distributions
• The probabilistic branching construct for controlling the flow in a program through observations.
The applications of probabilistic programming is wide, for instance, probabilistic programs are used in machine learning and robotics to describe distribution functions that are analyzed using Bayesian inference. In security, probabilistic programming has played a central role in cryptography, i.e., probabilistic security guarantees and probabilistic encryption. Probabilistic programs are also used to model the performance and reliability properties of a variety of systems, in which the uncertainties can come from the reliability of components (hardwares or softwares), the data from sensors, the reliability of the communication channels in the systems (e.g., the aircraft’s control systems).
This raises the question of how to analyze and verify that a system exhibiting probabilistic behavior satisfies a certain property. There is a rapidly growing trend in research on probabilistic programs which focuses on many aspects such as static analysis, program compilation, and program verification. To mention a few, the extension the framework of abstract interpretation to probabilistic programs, the development of a calculus for obtaining upper bounds on the expected value of run-time for probabilistic programs, and the developments of probabilistic and statistic model checking techniques for verifying probabilistic temporal properties of probabilistic and timed systems.
## Example
The following implements a simple example of probabilistic program for finding a given value in a list of integer values. The fist function find represents the standard program, e.g., without probability. It is easy to see that in the worst-case, the variable acc (initialized to 0) will have the value as the length of the input list l.
let rec find a l acc =
match l with
| [] -> (false, acc)
| x::xs ->
if a = x then
(true,(acc + 1))
else
find a xs (acc + 1);;
let rec pfind a l acc =
match l with
| [] -> (false, acc)
| x::xs as l -> let c = Random.int 2 in
if c = 0 then
pfind a l (acc + 1)
else
(if a = x then
(true,(acc + 1))
else
pfind a xs (acc + 1));;
let test n f a l =
let res = ref [] in
for i = 1 to n do
let (found,cost) = f a l 0 in
res := cost::(!res)
done;
let sum = List.fold_left (fun s x -> s + x) 0 !res in
(float_of_int sum) /. (float_of_int n);;
In the second function pfind, the integer value c is sampled from a uniform distribution from 0 (inclusive) to 2 (exclusive). In other words, c is 0 with probability 0.5 and is 1 with probability 1 - 0.5 = 0.5. A a result, with probability 0.5, the recursive call on the tail of the list is skipped and with the same probability the recursion is called on the tail of the list.
We can see that the recursion can be infinite, e.g., the number of times such that c is 0 is infinite. However, we can calculate that the probability that the number of times such that c is 0 is infinite, is $\frac{1}{2^n}$ with $n \rightarrow +\infty$ which is 0. It is said that pfind is almost surely terminated.
The question is what is the behavior of pfind. For example,
what is the expected value of the variable acc?
For this simple program, we can compute manually that this expected value is 2.|l|, where |l| is the number of elements in the list in the worst-case. My on-going work is to automatically and statically infer a symbolic upper bound on this expected value. The following are the average values of acc when we run find and pfind 10000000 times.
# test 10000000 find 10 [0;1;2;3;4;5;6];;
- : float = 7.
# test 10000000 pfind 10 [0;1;2;3;4;5;6];;
- : float = 13.998363 | 2018-12-11 07:17:53 | {"extraction_info": {"found_math": true, "script_math_tex": 2, "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": 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": 0.593306303024292, "perplexity": 1090.8793759235548}, "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/1544376823588.0/warc/CC-MAIN-20181211061718-20181211083218-00367.warc.gz"} |
https://socratic.org/questions/by-what-factor-should-the-length-of-a-simple-pendulum-be-changed-in-order-to-tri | # By what factor should the length of a simple pendulum be changed in order to triple the period of vibration?
Then teach the underlying concepts
Don't copy without citing sources
preview
?
#### Explanation
Explain in detail...
#### Explanation:
I want someone to double check my answer
2
Jane Share
Jan 13, 2018
9 times of the previous length
#### Explanation:
Relation between time period T and length l of pendulum is given as T $\propto$ √l
So, we can write,
$\frac{T 2}{T 1}$ = √(l2)/(l1) (where T1 and T2 are initial and final time period and L1 and L2 are initial and final length of the pendulum respectively as required)
Given, T2 = 3 T1
So, $\frac{l 2}{l 1}$ will be 9
Hence length should be made 9 times than that of the previous value.
• 24 minutes ago
• 24 minutes ago
• 25 minutes ago
• 28 minutes ago
• 57 seconds ago
• 2 minutes ago
• 6 minutes ago
• 8 minutes ago
• 16 minutes ago
• 23 minutes ago
• 24 minutes ago
• 24 minutes ago
• 25 minutes ago
• 28 minutes ago | 2018-06-22 09:13:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "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": 0.7831620573997498, "perplexity": 3044.7731269997857}, "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/1529267864387.54/warc/CC-MAIN-20180622084714-20180622104714-00238.warc.gz"} |
https://nbviewer.ipython.org/github/carljv/Will_it_Python/blob/master/ARM/ch5/arsenic_wells_switching.ipynb | In [1]:
%pylab inline
Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].
In [2]:
import numpy as np
from pandas import *
from statsmodels.formula.api import logit
from statsmodels.nonparametric import KDE
import matplotlib.pyplot as plt
from patsy import dmatrix, dmatrices
# Logistic models of well switching in Bangladesh¶
Our data are information on about 3,000 respondent households in Bangladesh with wells having an unsafe amount of arsenic. The data record the amount of arsenic in the respondent's well, the distance to the nearest safe well (in meters), whether that respondent "switched" wells by using a neighbor's safe well instead of their own, as well as the respondent's years of education and a dummy variable indicating whether they belong to a community association.
Our goal is to model well-switching decision. Since it's a binary variable (1 = switch, 0 = no switch), we'll use logistic regression.
This analysis follows Gelman and Hill Data Analysis Using Regression and Multilevel/Hierarchical Models, chapter 5.4.
In [19]:
df = read_csv('data/wells.dat', sep = ' ', header = 0, index_col = 0)
switch arsenic dist assoc educ
1 1 2.36 16.826000 0 0
2 1 0.71 47.321999 0 0
3 0 2.07 20.966999 0 10
4 1 1.15 21.486000 0 12
5 1 1.10 40.874001 1 14
## Model 1: Distance to a safe well¶
For our first pass, we'll just use the distance to the nearest safe well. Since the distance is recorded in meters, and the effect of one meter is likely to be very small, we can get nicer model coefficients if we scale it. Instead of creating a new scaled variable, we'll just do it in the formula description using the I() function.
In [4]:
model1 = logit('switch ~ I(dist/100.)', df = df).fit()
print model1.summary()
Optimization terminated successfully.
Current function value: 2038.118913
Iterations 4
Logit Regression Results
==============================================================================
Dep. Variable: switch No. Observations: 3020
Model: Logit Df Residuals: 3018
Method: MLE Df Model: 1
Date: Sat, 22 Dec 2012 Pseudo R-squ.: 0.01017
Time: 13:05:25 Log-Likelihood: -2038.1
converged: True LL-Null: -2059.0
LLR p-value: 9.798e-11
==================================================================================
coef std err z P>|z| [95.0% Conf. Int.]
----------------------------------------------------------------------------------
Intercept 0.6060 0.060 10.047 0.000 0.488 0.724
I(dist / 100.) -0.6219 0.097 -6.383 0.000 -0.813 -0.431
==================================================================================
Let's plot this model. We'll want to jitter the switch data, since it's all 0/1 and will over-plot.
In [5]:
def binary_jitter(x, jitter_amount = .05):
'''
Add jitter to a 0/1 vector of data for plotting.
'''
jitters = np.random.rand(*x.shape) * jitter_amount
x_jittered = x + np.where(x == 1, -1, 1) * jitters
return x_jittered
In [6]:
dist_logit_par = model1.params['I(dist / 100.)']
plt.plot(df['dist'], binary_jitter(df['switch'], .1), '.', alpha = .1)
plt.plot(np.sort(df['dist']), model1.predict()[np.argsort(df['dist'])], lw = 2)
plt.ylabel('Switched Wells')
plt.xlabel('Distance from safe well (meters)')
Out[6]:
<matplotlib.text.Text at 0x109862990>
Another way to look at this is to plot the densities of distance for switchers and non-switchers. We expect the distribution of switchers to have more mass over short distances and the distribution of non-switchers to have more mass over long distances.
In [7]:
kde_sw = KDE(df['dist'][df['switch'] == 1])
kde_nosw = KDE(df['dist'][df['switch'] == 0])
kde_sw.fit()
kde_nosw.fit()
plt.plot(kde_sw.support, kde_sw.density, label = 'Switch')
plt.plot(kde_nosw.support, kde_nosw.density, color = 'red', label = 'No Switch')
plt.xlabel('Distance (meters)')
plt.legend(loc = 'best')
Out[7]:
<matplotlib.legend.Legend at 0x109da8b50>
## Model 2: Distance to a safe well and the arsenic level of own well¶
Next, let's add the arsenic level as a regressor. We'd expect respondents with higher arsenic levels to be more motivated to switch.
In [8]:
model2 = logit('switch ~ I(dist / 100.) + arsenic', df = df).fit()
print model2.summary()
Optimization terminated successfully.
Current function value: 1965.334134
Iterations 5
Logit Regression Results
==============================================================================
Dep. Variable: switch No. Observations: 3020
Model: Logit Df Residuals: 3017
Method: MLE Df Model: 2
Date: Sat, 22 Dec 2012 Pseudo R-squ.: 0.04551
Time: 13:05:29 Log-Likelihood: -1965.3
converged: True LL-Null: -2059.0
LLR p-value: 1.995e-41
==================================================================================
coef std err z P>|z| [95.0% Conf. Int.]
----------------------------------------------------------------------------------
Intercept 0.0027 0.079 0.035 0.972 -0.153 0.158
I(dist / 100.) -0.8966 0.104 -8.593 0.000 -1.101 -0.692
arsenic 0.4608 0.041 11.134 0.000 0.380 0.542
==================================================================================
Which is what we see. The coefficients are what we'd expect: the farther to a safe well, the less likely a respondent is to switch, but the higher the arsenic level in their own well, the more likely.
### Marginal effects¶
To see the effect of these on the probability of switching, let's calculate the marginal effects at the mean of the data.
In [9]:
model2.margeff(at = 'mean')
Out[9]:
array([-0.21806505, 0.11206108])
So, for the mean respondent, an increase of 100 meters to the nearest safe well is associated with a 22% lower probability of switching. But an increase of 1 in the arsenic level is associated with an 11% higher probability of switching.
### Class separability¶
To get a sense of how well this model might classify switchers and non-switchers, we can plot each class of respondent in (distance-arsenic)-space.
We don't see very clean separation, so we'd expect the model to have a fairly high error rate. But we do notice that the short-distance/high-arsenic region of the graph is mostly comprised switchers, and the long-distance/low-arsenic region is mostly comprised of non-switchers.
In [10]:
logit_pars = model2.params
intercept = -logit_pars[0] / logit_pars[2]
slope = -logit_pars[1] / logit_pars[2]
dist_sw = df['dist'][df['switch'] == 1]
dist_nosw = df['dist'][df['switch'] == 0]
arsenic_sw = df['arsenic'][df['switch'] == 1]
arsenic_nosw = df['arsenic'][df['switch'] == 0]
plt.figure(figsize = (12, 8))
plt.plot(dist_sw, arsenic_sw, '.', mec = 'purple', mfc = 'None',
label = 'Switch')
plt.plot(dist_nosw, arsenic_nosw, '.', mec = 'orange', mfc = 'None',
label = 'No switch')
plt.plot(np.arange(0, 350, 1), intercept + slope * np.arange(0, 350, 1) / 100.,
'-k', label = 'Separating line')
plt.ylim(0, 10)
plt.xlabel('Distance to safe well (meters)')
plt.ylabel('Arsenic level')
plt.legend(loc = 'best')
Out[10]:
<matplotlib.legend.Legend at 0x109f7cb10>
## Model 3: Adding an interation¶
It's sensible that distance and arsenic would interact in the model. In other words, the effect of an 100 meters on your decision to switch would be affected by how much arsenic is in your well.
Again, we don't have to pre-compute an explicit interaction variable. We can just specify an interaction in the formula description using the : operator.
In [11]:
model3 = logit('switch ~ I(dist / 100.) + arsenic + I(dist / 100.):arsenic',
df = df).fit()
print model3.summary()
Optimization terminated successfully.
Current function value: 1963.814202
Iterations 5
Logit Regression Results
==============================================================================
Dep. Variable: switch No. Observations: 3020
Model: Logit Df Residuals: 3016
Method: MLE Df Model: 3
Date: Sat, 22 Dec 2012 Pseudo R-squ.: 0.04625
Time: 13:05:33 Log-Likelihood: -1963.8
converged: True LL-Null: -2059.0
LLR p-value: 4.830e-41
==========================================================================================
coef std err z P>|z| [95.0% Conf. Int.]
------------------------------------------------------------------------------------------
Intercept -0.1479 0.118 -1.258 0.208 -0.378 0.083
I(dist / 100.) -0.5772 0.209 -2.759 0.006 -0.987 -0.167
arsenic 0.5560 0.069 8.021 0.000 0.420 0.692
I(dist / 100.):arsenic -0.1789 0.102 -1.748 0.080 -0.379 0.022
==========================================================================================
The coefficient on the interaction is negative and significant. While we can't directly intepret its quantitative effect on switching, the qualitative interpretation gels with our intuition. Distance has a negative effect on switching, but this negative effect is reduced when arsenic levels are high. Alternatively, the arsenic level have a positive effect on switching, but this positive effect is reduced as distance to the nearest safe well increases.
## Model 4: Adding educuation, more interactions and centering variables¶
Respondents with more eduction might have a better understanding of the harmful effects of arsenic and therefore may be more likely to switch. Education is in years, so we'll scale it for more sensible coefficients. We'll also include interactions amongst all the regressors.
We're also going to center the variables, to help with interpretation of the coefficients. Once more, we can just do this in the formula, without pre-computing centered variables.
In [12]:
model_form = ('switch ~ center(I(dist / 100.)) + center(arsenic) + ' +
'center(I(educ / 4.)) + ' +
'center(I(dist / 100.)) : center(arsenic) + ' +
'center(I(dist / 100.)) : center(I(educ / 4.)) + ' +
'center(arsenic) : center(I(educ / 4.))'
)
model4 = logit(model_form, df = df).fit()
print model4.summary()
Optimization terminated successfully.
Current function value: 1945.871775
Iterations 5
Logit Regression Results
==============================================================================
Dep. Variable: switch No. Observations: 3020
Model: Logit Df Residuals: 3013
Method: MLE Df Model: 6
Date: Sat, 22 Dec 2012 Pseudo R-squ.: 0.05497
Time: 13:05:35 Log-Likelihood: -1945.9
converged: True LL-Null: -2059.0
LLR p-value: 4.588e-46
===============================================================================================================
coef std err z P>|z| [95.0% Conf. Int.]
---------------------------------------------------------------------------------------------------------------
Intercept 0.3563 0.040 8.844 0.000 0.277 0.435
center(I(dist / 100.)) -0.9029 0.107 -8.414 0.000 -1.113 -0.693
center(arsenic) 0.4950 0.043 11.497 0.000 0.411 0.579
center(I(educ / 4.)) 0.1850 0.039 4.720 0.000 0.108 0.262
center(I(dist / 100.)):center(arsenic) -0.1177 0.104 -1.137 0.256 -0.321 0.085
center(I(dist / 100.)):center(I(educ / 4.)) 0.3227 0.107 3.026 0.002 0.114 0.532
center(arsenic):center(I(educ / 4.)) 0.0722 0.044 1.647 0.100 -0.014 0.158
===============================================================================================================
### Model assessment: Binned Residual plots¶
Plotting residuals to regressors can alert us to issues like nonlinearity or heteroskedasticity. Plotting raw residuals in a binary model isn't usually informative, so we do some smoothing. Here, we'll averaging the residuals within bins of the regressor. (A lowess or moving average might also work.)
In [14]:
def bin_residuals(resid, var, bins):
'''
Compute average residuals within bins of a variable.
Returns a dataframe indexed by the bins, with the bin midpoint,
the residual average within the bin, and the confidence interval
bounds.
'''
resid_df = DataFrame({'var': var, 'resid': resid})
resid_df['bins'] = qcut(var, bins)
bin_group = resid_df.groupby('bins')
bin_df = bin_group['var', 'resid'].mean()
bin_df['count'] = bin_group['resid'].count()
bin_df['lower_ci'] = -2 * (bin_group['resid'].std() /
np.sqrt(bin_group['resid'].count()))
bin_df['upper_ci'] = 2 * (bin_group['resid'].std() /
np.sqrt(bin_df['count']))
bin_df = bin_df.sort('var')
return(bin_df)
def plot_binned_residuals(bin_df):
'''
Plotted binned residual averages and confidence intervals.
'''
plt.plot(bin_df['var'], bin_df['resid'], '.')
plt.plot(bin_df['var'], bin_df['lower_ci'], '-r')
plt.plot(bin_df['var'], bin_df['upper_ci'], '-r')
plt.axhline(0, color = 'gray', lw = .5)
arsenic_resids = bin_residuals(model4.resid, df['arsenic'], 40)
dist_resids = bin_residuals(model4.resid, df['dist'], 40)
plt.figure(figsize = (12, 5))
plt.subplot(121)
plt.ylabel('Residual (bin avg.)')
plt.xlabel('Arsenic (bin avg.)')
plot_binned_residuals(arsenic_resids)
plt.subplot(122)
plot_binned_residuals(dist_resids)
plt.ylabel('Residual (bin avg.)')
plt.xlabel('Distance (bin avg.)')
Out[14]:
<matplotlib.text.Text at 0x10ae3ef50>
## Model 5: log-scaling arsenic¶
The binned residual plot indicates some nonlinearity in the arsenic variable. Note how the model over-estimated for low arsenic and underestimates for high arsenic. This suggests a log transformation or something similar.
We can again do this transformation right in the formula.
In [15]:
model_form = ('switch ~ center(I(dist / 100.)) + center(np.log(arsenic)) + ' +
'center(I(educ / 4.)) + ' +
'center(I(dist / 100.)) : center(np.log(arsenic)) + ' +
'center(I(dist / 100.)) : center(I(educ / 4.)) + ' +
'center(np.log(arsenic)) : center(I(educ / 4.))'
)
model5 = logit(model_form, df = df).fit()
print model5.summary()
Optimization terminated successfully.
Current function value: 1931.554102
Iterations 5
Logit Regression Results
==============================================================================
Dep. Variable: switch No. Observations: 3020
Model: Logit Df Residuals: 3013
Method: MLE Df Model: 6
Date: Sat, 22 Dec 2012 Pseudo R-squ.: 0.06192
Time: 13:05:57 Log-Likelihood: -1931.6
converged: True LL-Null: -2059.0
LLR p-value: 3.517e-52
==================================================================================================================
coef std err z P>|z| [95.0% Conf. Int.]
------------------------------------------------------------------------------------------------------------------
Intercept 0.3452 0.040 8.528 0.000 0.266 0.425
center(I(dist / 100.)) -0.9796 0.111 -8.809 0.000 -1.197 -0.762
center(np.log(arsenic)) 0.9036 0.070 12.999 0.000 0.767 1.040
center(I(educ / 4.)) 0.1785 0.039 4.577 0.000 0.102 0.255
center(I(dist / 100.)):center(np.log(arsenic)) -0.1567 0.185 -0.846 0.397 -0.520 0.206
center(I(dist / 100.)):center(I(educ / 4.)) 0.3384 0.108 3.141 0.002 0.127 0.550
center(np.log(arsenic)):center(I(educ / 4.)) 0.0601 0.070 0.855 0.393 -0.078 0.198
==================================================================================================================
And the binned residual plot for arsenic now looks better.
In [16]:
arsenic_resids = bin_residuals(model5.resid, df['arsenic'], 40)
dist_resids = bin_residuals(model5.resid, df['dist'], 40)
plt.figure(figsize = (12, 5))
plt.subplot(121)
plot_binned_residuals(arsenic_resids)
plt.ylabel('Residual (bin avg.)')
plt.xlabel('Arsenic (bin avg.)')
plt.subplot(122)
plot_binned_residuals(dist_resids)
plt.ylabel('Residual (bin avg.)')
plt.xlabel('Distance (bin avg.)')
Out[16]:
<matplotlib.text.Text at 0x10aebb590>
### Model error rates¶
The pred_table() gives us a confusion matrix for the model. We can use this to compute the error rate of the model.
We should compare this to the null error rates, which comes from a model that just classifies everything as whatever the most prevalent response is. Here 58% of the respondents were switchers, so the null model just classifies everyone as a switcher, and therefore has an error rate of 42%.
In [18]:
print model5.pred_table()
print 'Model Error rate: {0: 3.0%}'.format(
1 - np.diag(model5.pred_table()).sum() / model5.pred_table().sum())
print 'Null Error Rate: {0: 3.0%}'.format(
1 - df['switch'].mean())
[[ 568. 715.]
[ 387. 1350.]]
Model Error rate: 36%
Null Error Rate: 42% | 2022-08-19 04:24:46 | {"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": 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": 0.5184327960014343, "perplexity": 14790.353898274763}, "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/1659882573623.4/warc/CC-MAIN-20220819035957-20220819065957-00100.warc.gz"} |
https://physics.stackexchange.com/questions/185400/hamilton-jacobi-problem | # Hamilton-Jacobi problem
In analytical mechanics by Fasano and Marmi they consider the Hamilton-Jacobi equation for a conservative autonomous system in one dimension with the following Hamiltonian, $$H=\frac{p^2}{2m}+V(x)$$ The Hamilton-Jacobi equation is now, $$\frac{1}{2m}\bigg(\frac{\partial W}{\partial x}\bigg)^2+V(x)=E$$ Integrated to be; $$W(x,E)=\sqrt{2m}\int ^{x}_{x_0}\sqrt{E-V(\xi)}d\xi$$ With the canonical transformation, $$p=\frac{\partial W}{\partial x}=\sqrt{2m(E-V(x))}$$ $$\beta =\frac{\partial W}{\partial E}=\sqrt {m/2}\int ^{x_1}_{x_0}\frac{d\xi}{\sqrt{E-V(\xi)}}$$ My question is regarding the use of the dummy variable $\xi$. Why do we use this and why can we not just integrate over $x$? Further, why do we use it for computation of $\beta$ but not $p$?
• Uh...you have written $W(x,E) = ...$, that means you can't use $x$ as a dummy variable on the RHS, but why are you asking yourself about the symbol for a dummy variable in the first place? May 22, 2015 at 14:16
You should think of the definite integral operation as a function of two arguments: a region over which to integrate (here, $[x_0,x_1]$), and another function $f$ called the integrand (here, $f:\xi \mapsto (E-V(\xi))^{-\frac{1}{2}}$).
So first of all, in my definition of $f$ above, we could have used (almost) any other symbol instead of $\xi$ and the meaning would be the same. But there are two exceptions: if we used either $E$ or $V$ to represent the argument of $f$, the function inside either wouldn't make sense or would be very different than what we meant.
The way I've written $f$ you know that $\xi$ is the argument of $f$, and not something defined externally, because of the "$\xi \mapsto$". These symbols are said to "capture" (in the sense of variable capture) $\xi$.
Now, the integrand in an integral is a function, and it needs to be written down in a way similar to what I've done above. Typically, the "$d\xi$" notation takes the place of the $\xi\mapsto$ notation I used in defining $f$ as a pure function. (Formally, the differential notation can mean something much deeper, but that's irrelvant in our context.)
So, suppose you had written $dx$ instead of $d\xi$. The problem would be that $x$ is already in scope, just like $E$ was already in scope when we defined $f$. It's in scope, as @ACuriousMind commented, because it was introduced by $W(x,E)$. If you think of integration as an operation $\mathtt{Integrate}[f,(x_0,x)]$, you can see that $x$ must have already been in scope outside of the integration, because you passed it to the integration as a parameter.
So, since it's already in scope, you cannot reuse it as the parameter of the integrand $f$. Whenever you used "$x$", it wouldn't be clear whether you meant the upper limit of the integration or the integration variable, becuase $x$ would undergo variable capture twice.
Oftentimes, the notation is abused and people write exactly what you suggested. The convention, when you encounter an $x$ and can't tell what it refers to, is to pick the "closest" variable capture, which is to say the most "local" definition. In our case, that would mean the integration variable. But formally, this is an ambiguous notation, and it's safer not to rely on this convention when there are plenty of other glyphs (like $\xi$) that can be used instead.
• Cheers that helped me a lot! :) May 22, 2015 at 22:56
• Beautiful answer! Jul 14, 2016 at 19:45 | 2022-05-22 21:20:31 | {"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": 5, "x-ck12": 0, "texerror": 0, "math_score": 0.9053484797477722, "perplexity": 168.55073046017225}, "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/1652662546071.13/warc/CC-MAIN-20220522190453-20220522220453-00654.warc.gz"} |
https://mathoverflow.net/questions/175766/formally-smooth-map-from-a-regular-ring | # Formally smooth map from a regular ring
Let $A,B$ be two commutative noetherian rings. Let $f:A\to B$ be a formally smooth homomorphism. If $A$ is a regular ring (in the sense that all its localizations are regular local rings), does this imply that $B$ is a regular ring?
• It depends on the definition of "formal smoothness". The answer will be "yes" if your are considering the definition w.r.t. the discrete topology, or if the homomorphism is local and the topology is the one induced by its maximal ideal, or if some finiteness hypothesis are implicit in your definition, etc. – Vinteuil Jul 10 '14 at 14:34
• do you have any reference for these statements? (I do not have finiteness assumptions implict in my situation) – wonderman Jul 10 '14 at 14:49
As I wrote in the comments, the answer is usually yes, depending on the topology you are considering. The two more common cases are (it should be direct references for them, maybe in EGA $0_{IV}$):
Local case: assume $f$ is local. Let $l$ be the residue field of $B$. You have a Jacobi-Zariski exact sequence in André-Quillen homology $$0=H_2(A,l,l) \to H_2(B,l,l) \to H_1(A,B,l)=0$$ (the left module is zero since $A$ is regular and the right one since $f$ is formally smooth for the topology of the maximal ideal). Therefore the middle term is zero and then $B$ is regular.
Discrete case: assume that $f$ is formally smooth w.r.t. the discrete topology. Then $H_1(A_{\mathfrak p},B_{\mathfrak q},k({\mathfrak q}))=H_1(A,B,k({\mathfrak q}))=0$ for any prime ideal $\mathfrak q$ of $B$ (where $\mathfrak p = f^{-1} \mathfrak q$), and by a similar exact sequence you deduce that $H_2(B_{\mathfrak q},k({\mathfrak q}),k({\mathfrak q}))=0$, that is $B_{\mathfrak q}$ is regular.
• Thanks, this is helpful. Can you just explain why $H_2(A,l,l) = 0$? I know that if $(A,m)$ is regular then $H_n(A,A/m,-) = 0$. – wonderman Jul 10 '14 at 15:18
• Because of a similar exact sequence: $0=H_2(A,k,l) \to H_2(A,l,l) \to H_2(k,l,l)=0$. – Vinteuil Jul 10 '14 at 15:20 | 2019-10-17 22:03:13 | {"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": 0.9667267203330994, "perplexity": 172.11555002285445}, "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-2019-43/segments/1570986676227.57/warc/CC-MAIN-20191017200101-20191017223601-00482.warc.gz"} |
https://illwiki.com/dom5/trample | # illwiki: Dominions 5 Wiki
trample
## Trample
Units with Trample inflict 5 + 2 * size of trampler untyped AP damage when moving through another unit. As with usual damage calculation, both sides get a DRN added to their roll. However, unlike other damage, trampled victims always take at least 1 damage, regardless of the roll results. Trample ignores shields.
Then there is a check of the trampled unit's defence skill - (1 per 10 fatigue) vs 3d6 (exploding) to take only 1 damage1). The trampler wins ties. Weapon bonuses and penalties to defence skill are ignored, but armour is respected.
Trampling units only use their other attacks when meeting a square containing at least one unit of the same size or larger as themselves.
Trampling units can trample several squares per turn, the maximum amount is dependent on their combat speed. Specifically, the delay between successive tramples (in ticks, of which there are 7500 per round) is calculated as follows2):
• The calculation is made of a number of components:
• Delay due to combat speed: 10000 / combat speed3)
• Action rate value: 7500 * action time. The default action time is 1.0, halved by Quickness and doubled by being Slowed, Slimed and/or Frozen. Each of these ailments suffered applies a separate doubling.
• Size penalty: (largest difference in size between the trampler and the units in the trampled square)/2 + 3 (rounded down)
The final delay is calculated as:
Combat speed delay + Action rate value/size penalty
As the size penalty is always at least 3, it means that large amounts of combat speed alone will not greatly increase the number of tramples per round.
Every square trampled costs full encumbrance in fatigue.
Trampling units can not be repelled but still have to test against awe. Effects that trigger when a unit is damaged (such as Blood Vengeance) work normally, but those that specifically trigger on melee attacks such as Fire Shield do not.
1)
The manual states the defence check is always 10, but the Dominions 5.51b executable is clear that this is in fact 3d6 exploding instead.
3)
The executable forces this value to be at least 10, but the 1000 combat speed required to reach this is unrealistic to achieve. | 2021-11-27 09:39:17 | {"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": 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": 0.7191635370254517, "perplexity": 5220.150087859409}, "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/1637964358153.33/warc/CC-MAIN-20211127073536-20211127103536-00310.warc.gz"} |
https://www.gurobi.com/documentation/9.5/examples/mip1_remote_py.html | Filter Content By
Version
Languages
### mip1_remote.py
import gurobipy as gp
from gurobipy import GRB
# Variation of mip1.py, with a focus on remote services
#
# When remote resources are tied to the optimization process, such as a token
# server, compute server, or Instant Cloud, extra care should be taken to
# ensure that such resources are released once they are no longer needed.
# Technically, such resources are managed by a gurobipy.Env object
# ("environment"). This example shows best practices for acquiring and
# releasing such shared resources via Env objects.
#
def populate_and_solve(m):
# This function formulates and solves the following MIP model (see mip1.py):
# maximize
# x + y + 2 z
# subject to
# x + 2 y + 3 z <= 4
# x + y >= 1
# x, y, z binary
# Create variables
# Set objective
m.setObjective(x + y + 2 * z, GRB.MAXIMIZE)
# Add constraint: x + 2 y + 3 z <= 4
m.addConstr(x + 2 * y + 3 * z <= 4, "c0")
# Add constraint: x + y >= 1
m.addConstr(x + y >= 1, "c1")
# Optimize model
m.optimize()
for v in m.getVars():
print('%s %g' % (v.VarName, v.X))
print('Obj: %g' % m.ObjVal)
# Put any connection parameters for Gurobi Compute Server, Gurobi Cluster
# Manager or Gurobi Token server here, unless they are set already
connection_params = {
# For Compute Server you need at least this
# "ComputeServer": "<server name>",
# For Cluster Manager you need at least this
# "CSManager": "<manager name>",
# "CSAPIAccessID": "<access ID>",
# "CSAPISecret": "<secret>",
# For Instant cloud you need at least this
# "CloudAccessID": "<access id>",
# "CloudSecretKey": "<secret>",
}
with gp.Env(params=connection_params) as env:
# 'env' is now set up according to the connection parameters.
# The environment is disposed of automatically through the context manager
# upon leaving this block.
with gp.Model(env=env) as model:
# 'model' is now an instance tied to the enclosing Env object 'env'.
# The model is disposed of automatically through the context manager
# upon leaving this block.
try:
populate_and_solve(model)
except:
# Add appropriate error handling here.
raise | 2022-12-03 03:33:52 | {"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": 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": 0.2173214703798294, "perplexity": 11097.709104131029}, "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/1669446710918.58/warc/CC-MAIN-20221203011523-20221203041523-00714.warc.gz"} |
https://wjrider.wordpress.com/2016/06/07/the-marvelous-magical-median/ | Every solution to every problem is simple. It’s the distance between the two where the mystery lies.
― Derek Landy
The median is what a lot of people think of when you say “household income”. It is a statistical measure of central tendency of data that is a harder to compute alternative to the friendly common mean value. For large data sets the median is tedious and difficult to compute while the mean is straightforward and easy. The median is the middle entry of the ordered list of numerical data thus the data being studied needs to be ordered, which is hard and expensive to perform. The mean of the data is simple to compute. On the other hand by almost any rational measure, the median is better than the mean. For one thing, the median is not easily corrupted by any outliers in the data (data that is inconsistent or corrupted); it quite effectively ignores them. The same outliers immediately and completely corrupt the mean. The median is strongly associated with the one-norm, which is completely awesome and magical. This connection is explored through the amazing and useful field of compressed sensing, which uses the one norm to do some really sweet (cool, awesome, spectacular, …) stuff.
About 15 years ago I found myself quite taken with a mathematical tool associated with constructing “limiters” for solving hyperbolic conservation laws numerically. This mathematical tool is the “median” function for three arguments. It simply returns the value that lies in the middle of the triad. It simple and powerful, but hasn’t really caught on more generally. It seems to be a boutique method rather than common in use. It’s a true shame; because the median is pretty fucking awesome. Given the median’s power and utility in building numerical method, I find its lack of “popularity” puzzling. Since I like it so much, I will wax a bit on its greatness and what I’ve done with it.
It’s easy to attack and destroy an act of creation. It’s a lot more difficult to perform one.
― Chuck Palahniuk
I came across the function and its use with limiters reading the work of HT Huynh. HT works at NASA in Cleveland and has done some really great work over the years. The median function made its first appearance in work on Hermitian polynomial interpolation (used a lot with tabular lookups). He wrote several of my favorite papers on solving hyperbolic PDE’s, first on slope limited second-order schemes (a real gem covering a wealth of material and presenting it in a wonderful manner, it deserves many more citations!) and a second with Suresh on limiting edge values (another tour de force!). The median function appears prominently in both and allows some cool stuff to be done in a very clear and literal fashion. Like the median function, HT’s work is significantly under-appreciated.
If you want something new, you have to stop doing something old
― Peter F. Drucker
History is filled with brilliant people who wanted to fix things and just made them worse.
― Chuck Palahniuk
The function itself is quite elegant. If you’re in the business of limiters, you can implement it directly from another function you certainly have on hand, the minmod (minimum modulus) function. Jay Boris invented the minmod as part of the flux corrected transport method (arguably the first limiter). It returns the function with the minimum magnitude if the arguments to the function have the same sign, and zero otherwise. It can be implemented in several ways of varying elegance. The classic way of implementing minmod is, $\mbox{minmod}\left(a, b \right) =\frac{1}{4} \left(\mbox{sign}[a]+\mbox{sign}[b]\right) \left(|a+b| - |a-b| \right)$. HT Huynh introduced a really cool way to write minmod that was much better since you could use it with Taylor series expansions straightaway, $\mbox{minmod}\left(a, b \right) =\frac{1}{4} \left(\mbox{sign}[a]+\mbox{sign}[b]\right) \left(|a+b| - |a-b| \right)$.
Once the minmod is in hand, the median is a piece of cake to implement, $\mbox{median}\left(a, b, c \right) = a + \mbox{minmod} \left( b-a,c-a\right)$. In papers by Huynh and Huynh & Suresh, the median is used to enforce bounds. This mindset is perfectly in keeping with the thinking about monotonicity preserving methods but the function has other properties that can take it much further. A lack of recognition of these other properties may limit its penetration into further use. In a general sense it is used in defining the limits that an approximation value can take and produce a monotonicity preserving answer. The median is used to repeatedly bring an approximate value inside well-defined bounding values although for monotonicity it just requires two applications. Huynh first applied this to the linear interpolated second-order methods. In that case the median could be used to implement the classical TVD limiters or slope limiters. Suresh and Huynh took the same approach to higher order accuracy through looking edge value/flux approximations that might have an arbitrary order of accuracy and integrated via a method of lines approach, but uses many other applications of the median in the process.
Do not get obsolete like an old technology, keep innovating yourself.
― Sukant Ratnakar
Once the median function is in hand it may be used to write limiters in what I think is a rather elegant fashion. For example take the standard monotonicity or TVD limiter for a “good” second-order method based on Fromm’s scheme, which can be written in a relatively hideous fashion as, $S_M = (\mbox{sign}[S_2] \max\left[0, \min\left(|S_2|, 2 \mbox{sign}[S_2] \Delta_- U, 2 \mbox{sign}[S_2] \Delta_+ U \right)\right]$. Here $S_2 = \frac{1}{2}(\Delta_- U + \Delta_+ U)$ is the unlimited slope for the linear Fromm scheme, with $\Delta_- U = U(j)$$U(j-1)$ and $\Delta_+ U = U(j+1)$ –$U(j)$ being the negative and positive differences of the variables on the mesh. With the median we can write the limiter in two compact statements, $S_m = \mbox{median}\left( 0, \Delta_- U , \Delta_+ U \right)$, $S_M = \mbox{median}\left( 0, S_m , S_2 \right)$. More importantly as Huynh shows this format allows significant and profitable extensions of the approach leading to higher accuracy approximation. It is the extensibility that may be exploited as a platform to innovate limiters and remove some of their detrimental qualities.
The same basic recipe of bounding allows the parabolic limiter for the PPM method to be expressed quite concisely and clearly. The classic version of the PPM limiter involves “if” statements and seems rather unintuitive. With the median it is very clear and concise. The first simply assures that the chosen (high-order) edge values, $U_l$ and $U_r$ are enclosed by the neighboring mesh values, $U_r := \mbox{median}\left( U(j), U_r, U(j+1) \right)$ and $U_l := \mbox{median}\left( U(j), U_l, U(j-1) \right)$. While the second step assures that the parabola does not contain a zero derivative (or no local minima or maxima) in the cell, $U_R = \mbox{median} ( U_j, U_r, 3 U(j)$$2 U_l )$ and $U_L = \mbox{median} ( U_j, U_l, 3U(j)$$2 U_r )$. With this version of PPM there is significant liberty in defining the left and right initial edge values by whatever formula, accuracy and nature you might like. This makes PPM an immensely flexible and powerful method (the original PPM is very powerful to begin with).
The median preserves accuracy and allows one to create new nonlinearly accurate approximations. Huynh delivered the median along with a theorem regarding it accuracy properties. If two of the three arguments in the median function have a certain order of accuracy in approximation than the evaluated median function will have that order of accuracy. The proof is simple and intuitive. If the defined order of accuracy values is returned there isn’t a question, but if the returned value does not have the order accuracy a priori, its accuracy is in question. By being bounded by the two values with well-defined linear order of accuracy, the median value is the convex combination of the two accurate values, which locally means it has the same order of accuracy. Viola! This means the median has the ability to “breed” new high order approximations from existing high-order expressions. If you’re interested in producing well-behaved approximations that are also high-order; it is a super useful property.
Here is a supposition that extends the accuracy property of the median to make it more powerful. The median can also “breed” new nonlinearly stable approximations. This would work as follows: if two of the arguments are stable, the evaluation of the median would produce a stable value even if the argument returned is linearly unstable. Thus we can take various stable approximations and through the median use them to produce new stable approximations even if they are unstable. This should work exactly like the order of approximation through the evaluated median producing a value that is the convex linear combination of the two defined stable approximations. If this property can be proven then all you need to do is keep track of the properties as you work defining your method. If you are careful you can end up with provable methods having well-defined accuracy and stability even when some parts of the method have neither to begin with. This is the essence of a the nonlinear method. It does something the linear method cannot.
This produced a concept of producing nonlinear schemes that I originally defined as a “playoff” system. The best accurate and stable approximation would be the champion of the playoffs and would inherit the properties of the set of schemes used to construct the playoff. This would work as long as the playoff was properly orchestrated to produce stable and high-order approximations in a careful manner if the median were used to settle the competition. I had discussed this as a way to produce better ENO-like schemes. Perhaps I should have taken the biological analogy in the definition of the scheme in terms of what sort of offspring each decision point in the scheme bred. Here the median is used to breed the schemes and produce more fit offspring. The goal at the end of scheme is to produce an approximation with a well-chosen order of accuracy and stability that can be proven.
The method I show here was developed with a goal in mind, produce an ENO(-like) method with all of the stability, but with better accuracy and fidelity. It would have the same order of accuracy as ENO methods, but would produce solutions of greater fidelity. One of the big issues with ENO (and WENO) methods is that they are relatively dissipative as well as being complex. For applied problems with shocks and discontinuous features a well-designed second-order method will produce a much better, higher fidelity solution than an ENO method. WENO is more competitive, but still not better than a method such as PPM until the solution has immense amount of high frequency structure. The goal is to produce ENO methods that inherit the sort of fidelity properties that second-order methods produce, but also provide bona fide high-order accuracy,
Here is a relatively simple example of the sort of scheme I suggested and how the “playoff” system would work. In this case, the goal is to produce a third-order approximation with well-defined stability properties. We start with three third order approximations, $U_{3,1} = \frac{1}{3}U(j-2) - \frac{7}{6}U(j-1) + \frac{11}{6}U(j)$, $U_{3,2} = -\frac{1}{6}U(j-1) + \frac{5}{6}U(j) + \frac{1}{3}U(j)$ and $U_{3,3} =\frac{1}{3}U(j) + \frac{5}{6}U(j+1) -\frac{1}{6}U(j+2)$ (all written from the right edge of cell $j$) plus a second-order monotone approximation $U_{2,M}$. For example one might use the second-order minmod based scheme, $U(j) +\frac{1}{2}\mbox{minmod}\left( \Delta_- U, \Delta_+ U \right)$. One of the three third-order methods is the upstream-centered scheme, which has favorable properties from an accuracy and stability point-of-view ($U_{3,2}$ for this example).
The second thing to note is that one of the three third-order approximations is the nonlinearly stable classical ENO approximation (we don’t know which one, and don’t need to either! and that is cool). The playoff would consist of two rounds and three applications of the median in the process. The semi-finals would consist of two choices, $U_{3,A} = \mbox{median}\left(U_{3,1}, U_{3,2}, U_{2,M}\right)$ and $U_{3,B} = \mbox{median}\left(U_{3,2}, U_{3,3}, U_{2,M}\right)$. Both $U_{3,A}$ and $U_{3,B}$ are third-order accurate and one of them is also nonlinearly stable to the extent that the ENO approximation would be (the second stable scheme is $U_{2,M}$). The finals then would be defined by the following test, $U_{3,C} = \mbox{median}\left(U_{3,A}, U_{3,B}, U_{2,M}\right)$ and would produce a third-order and nonlinearly stable approximation. This is a direct consequence of the median function’s use. Two of the approximations are third-order, and two of the approximations are nonlinearly stable, so the result of the playoff final is an approximation that gives us both. The results from my paper strongly indicate that this approach is a success, and can be used to construct methods of much higher order accuracy.
If one were feeling more bold about solving problems of this ilk, one might throw the fifth-order upstream centered approximation into the mix. This fifth-order approximation is the asymptotic limit of the fifth-order WENO method when the solution is (very) smooth and a linear combination of the three third-order approximations, $U_5 = \frac{1}{30}U(j-2) - \frac{13}{60}U(j-1) + \frac{47}{60}U(j) + \frac{27}{60}U(j+1) - \frac{1}{20}U(j+2)$. Make the fifth-order approximation the returning champion have it play the winner of the playoff system in a final match. This would look like the following, $U_{3,D} = \mbox{median}\left(U_{3,C}, U_5, U_{2,M}\right)$. It would still only be third-order accurate in a formal manner, but have better accuracy where the solution is well-resolved. The other thing that is intrinsically different from the classic ENO approach is the guidance of the choice for $U_{2,M}$. To a very large extent the result of the playoff is determined by this choice and the result inherits the properties of this method. There is a wealth of monotonicity-preserving second-order methods to choose from used to determine the fitness of the high-order approximations.
ENO methods traditionally use a hierarchical stencil selection approach that introduces serious practical problems. Their weighted version is a far better method (WENO) both because it is higher order, but also better stability properties. The high order nature of these schemes can be analyzed via the elegant continuous nature of the weights. The largest issue with both methods is their great cost either in floating point or logical expression evaluation. Using the median function can provide a route analogous to the playoff scheme above to produce an ENO method where the prospect is that the chosen scheme is the one closest to some “comparison” method chosen from the wealth of monotonicity-preserving methods. This ENO scheme would then inherit the basic properties of the base scheme, but also have the high-order nature of ENO along with being non-oscillatory. A close relative of the median function is the xmedian, which delivers whatever argument is closer in an absolute value sense to the comparison value. For the following form, $\mbox{xmedian}\left(a, b, c \right)$ the function will provide either $b$ or $c$ depending on their distance from $a$. The function is written down at the end of the post along with all the other functions used (I came up with these to allow the analysis of this class of methods via modified equation analysis).
Here is a third-order ENO like method based on either the median or xmedian procedure. Both methods will produce a bona fide method with third-order accuracy (confirmed via numerical tests). The method starts with the selection of the comparison scheme, $U_c$. This scheme may be first or second-order accurate, but needs to have strong nonlinear stability properties. For the third-order (or 4th order) method, the scheme has two rounds and three total tests (i.e., semi-finals and finals). The semi-finals are $U_{3,A} = \mbox{xmedian}\left( U_c, U_{3,1}, U_{3,2} \right)$ and $U_{3,B} = \mbox{xmedian}\left( U_c, U_{3,2}, U_{3,3} \right)$. The finals are then easy to predict from the previous method, $latex U_{3,C} = \mbox{xmedian}\left( U_c, U_{3,A}, U_{3,B} \right)$. With the properties of the xmedian we can see that the scheme will select one of the third-order methods. This issue is that this playoff scheme does not produce a selection that would obviously be stable using the logic from earlier in the post. The question is whether the non-oscillatory method chosen by its relative closeness to the comparison scheme is itself nonlinearly stable.
The main question is the nature of the stability. The proposition is that the comparison scheme drawn from the category of nonlinear stable, monotonicity preserving methods endows the result with stability. With the median there is a very strong argument for stability. For the xmedian this argumentation is absent. Stability in both cases is born out through numerical testing, but theoretical support does not seem to have an obvious path. I note that ENO and WENO methods don’t have a deep stability result aside from the notion that classic ENO will produce an entropy-stable flux (but WENO does not). Practical success would seem to favor WENO to a very large degree as it has been applied to a vast number of practical cases.
If you’re not prepared to be wrong, you’ll never come up with anything original.
― Ken Robinson
The use of the median and related functions starts to beg some questions desperately in need of answers. There is a broad a deep of understanding of what numerical stability is and implies. This is for linear schemes, but the concept of nonlinear stability needs deeper thought. The archetypes of nonlinear stability are TVD/monotonicity-preserving schemes and ENO or WENO methods. The nature of these methods allows the local use of intrinsically unstable approximations without the usual path to catastrophe this implies. Where is the dividing line between a method being non-oscillatory (nonlinearly stable) and not. Using the median or its brother the xmedian, I can come up with a host of methods that perform in a fashion that implies that they are nonlinearly stable. Can we come up with a technically defensible definition that can be used to take these methods even further?
The possession of knowledge does not kill the sense of wonder and mystery. There is always more mystery.
― Anaïs Nin
References
Huynh, Hung T. “Accurate monotone cubic interpolation.” SIAM Journal on Numerical Analysis 30.1 (1993): 57-100.
Huynh, Hung T. “Accurate upwind methods for the Euler equations.” SIAM Journal on Numerical Analysis 32.5 (1995): 1565-1619.
Suresh, A., and H. T. Huynh. “Accurate monotonicity-preserving schemes with Runge–Kutta time stepping.” Journal of Computational Physics 136.1 (1997): 83-99.
Boris, Jay P., and David L. Book. “Flux-corrected transport. I. SHASTA, A fluid transport algorithm that works.” Journal of computational physics 11.1 (1973): 38-69. (Kuzmin, D., R. Löhner, and S. Turek, eds. Flux-Corrected Transport: Principles, Algorithms, and Applications. Scientific Computation. Springer, 2005.)
Balsara, Dinshaw S., and Chi-Wang Shu. “Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy.”Journal of Computational Physics 160.2 (2000): 405-452.
Colella, Phillip, and Paul R. Woodward. “The piecewise parabolic method (PPM) for gas-dynamical simulations.” Journal of computational physics 54.1 (1984): 174-201.
Rider, William J., Jeffrey A. Greenough, and James R. Kamm. “Accurate monotonicity-and extrema-preserving methods through adaptive nonlinear hybridizations.” Journal of Computational Physics 225.2 (2007): 1827-1848.
Harten, A., Engquist, B., Osher, S., & Chakravarthy, S. R. (1997). Uniformly high order accurate essentially non-oscillatory schemes, III. Journal of Computational Physics131(1), 3-47. (Harten, Ami, et al. “Uniformly high order accurate essentially non-oscillatory schemes, III.” Upwind and High-Resolution Schemes. Springer Berlin Heidelberg, 1987. 218-290.)
Liu, Xu-Dong, Stanley Osher, and Tony Chan. “Weighted essentially non-oscillatory schemes.” Journal of computational physics 115.1 (1994): 200-212.
Shu, Chi-Wang. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Springer Berlin Heidelberg, 1998.
Henrick, Andrew K., Tariq D. Aslam, and Joseph M. Powers. “Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points.” Journal of Computational Physics 207.2 (2005): 542-567.
Greenough, J. A., and W. J. Rider. “A quantitative comparison of numerical methods for the compressible Euler equations: fifth-order WENO and piecewise-linear Godunov.” Journal of Computational Physics 196.1 (2004): 259-281.
Rider, William J. “Building Better (Weighted) ENO Methods.” Computational Fluid Dynamics 2006. Springer Berlin Heidelberg, 2009. 161-166.
Fjordholm, Ulrik S., Siddhartha Mishra, and Eitan Tadmor. “Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws.” SIAM Journal on Numerical Analysis 50.2 (2012): 544-573.
Algebraically defined functions:
$\min\left(a,b \right) =\frac{1}{2}(a+b)- \frac{1}{2}abs[a-b]$
$\max\left(a,b \right) = \frac{1}{2} (a+b)+ \frac{1}{2}abs[a-b]$
$\mbox{minmod}\left(a, b \right) =\frac{1}{4} \left(\mbox{sign}[a]+\mbox{sign}[b]\right) \left(|a+b| - |a-b| \right)$
$\mbox{minmod}\left(a, b \right) = (\mbox{sign}[a] \max\left[0, \min\left(|a|, \mbox{sign}[a] b \right)\right]$
$\mbox{maxmod}\left(a, b \right) =\frac{1}{4} \left(\mbox{sign}[a]+\mbox{sign}[b]\right)\left(|a+b| + |a-b|\right)$
$\mbox{mineno}\left(a, b \right) =\frac{1}{2} \mbox{sign}(a+b)\left( |a + b| - |a-b|\right)$
$\mbox{median}\left(a, b, c \right) = a + \mbox{minmod} \left( b-a,c-a\right)$
$\mbox{xmedian}\left(a, b, c \right) = a + \mbox{mineno}\left(b-a,c-a \right)$ | 2017-05-30 05:23:45 | {"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": 0, "mathjax_asciimath": 0, "img_math": 50, "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": 0.7757949233055115, "perplexity": 2016.8689418894949}, "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/1495463613796.37/warc/CC-MAIN-20170530051312-20170530071312-00100.warc.gz"} |
https://www.tib.eu/de/suchen?tx_tibsearch_search%5Baction%5D=search&tx_tibsearch_search%5Bcnt%5D=10&tx_tibsearch_search%5Bcontroller%5D=Search&tx_tibsearch_search%5BDauthor%5D%5B0%5D=SHARMA%20SACHIN&tx_tibsearch_search%5BDauthor%5D%5B1%5D=Milosevic%2C%20J.&tx_tibsearch_search%5BDlanguage%5D%5B0%5D=en&tx_tibsearch_search%5BDlanguage%5D%5B1%5D=ko&tx_tibsearch_search%5Bpg%5D=1&tx_tibsearch_search%5Bquery%5D=person%3A%28Vargas%2C%20M.%29&tx_tibsearch_search%5Bsrt%5D=title_asc&cHash=ce00ceebc996298c101db0e131f5f1e4 | 1–10 von 4.030 Ergebnissen
Sortieren nach: Relevanz | Aktualität neu zuerst | Titel A-Z
## Treffer filtern
+ Mehr anzeigen
+ Mehr anzeigen
Suchen...
1.
### Addendum to: Centrality dependence of high-pT D-meson suppression in Pb-Pb collisions at s N N = 2.76 $$\sqrt{{\mathrm{s}}_{\mathrm{NN}}}=2.76$$ TeV
| DataCite | 2017
2.
### Addendum to: Centrality dependence of high-pT D-meson suppression in Pb-Pb collisions at $\sqrt{{\mathrm{s}}_{\mathrm{N}\mathrm{N}}}=2.76$ TeV
| Springer Verlag | 2017
3.
### ALICE Collaboration
| Elsevier | 2016
4.
### ALICE Collaboration
| Elsevier | 2014
5.
### ALICE Collaboration
| Elsevier | 2014
6.
### A measurement of the Higgs boson mass in the diphoton decay channel
| DataCite | 2020
7.
### A New Boson with a Mass of 125 GeV Observed with the CMS Experiment at the Large Hadron Collider
| BASE | 2012
8.
### A New Boson with a Mass of 125 GeV Observed with the CMS Experiment at the Large Hadron Collider
| BASE | 2012
9.
### A new Boson with a mass of 125 GeV observed with the CMS experiment at the large hadron collider
| BASE | 2014
10.
### Angular analysis and branching fraction measurement of the decay ${B}^{0}\to {K}^{⁎0}{\mu }^{+}{\mu }^{-}$
| Elsevier | 2013
Ergebnisse anzeigen: 10 | 20 | 50 | 2021-01-26 09:39:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "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": 0.8145184516906738, "perplexity": 11101.5827464704}, "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-2021-04/segments/1610704799711.94/warc/CC-MAIN-20210126073722-20210126103722-00347.warc.gz"} |
https://efoutem.ml/28167.jsp | # Complex numbers polar form e
##### 2019-11-20 20:28
The polar form of a complex number is another way to represent a complex number. The form z a b i is called the rectangular coordinate form of a complex number. The horizontal axis is the real axis and the vertical axis is the imaginary axis. We find the real and complex components in terms of r and where r is the length of the vectorPolar to Rectangular Online Calculator. Below is an interactive calculator that allows you to easily convert complex numbers in polar form to rectangular form, and viceversa. There's also a graph which shows you the meaning of what you've found. complex numbers polar form e
THE COMPLEX EXPONENTIAL FUNCTION The conditions for equality of two complex numbers using polar coordinates are not quite as simple as they are for rectangular coordinates. Express z1 and z2 in polar form. 2. Let z1 6ei3 and z2 2e i6. Plot z1z2, and z1z2. 3.
Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number. Jan 05, 2011 Expressing a Complex Number in Trigonometric or Polar Form, Ex 1. In this video, I show how to write a complex number in polar form. Complex Numbers In Polar Form De Moivre's Theorem, Productscomplex numbers polar form e 4. Polar Form of a Complex Number. by M. Bourne. We can think of complex numbers as vectors, as in our earlier example. [See more on Vectors in 2Dimensions. . We have met a similar concept to polar form before, in Polar Coordinates, part of the analytical geometry section.
## Complex numbers polar form e free
Polar form emphasizes the graphical attributes of complex numbers: absolute value \goldD\textabsolute value absolute value (the distance of the number from the origin in the complex plane) and angle \purpleC\textangle angle (the angle that the number forms with the positive Real axis). complex numbers polar form e May 26, 2017 11. Finding Products of Complex Numbers in Polar Form 12. Finding Quotients of Complex Numbers in Polar Form 13. Powers of Complex Numbers in Polar Form 14. De Moivre's Theorem Roots of Complex Euler's Equation, ei\theta \cos\theta i\sin\theta, provides the connection between these two representations of complex numbers. [ I'm ready to take the quiz. [ I need to review more. Finally, we will see how having Complex Numbers in Polar Form actually make multiplication and division (i. e. , Products and Quotients) of two complex numbers a snap! In fact, you already know the rules needed to make this happen and you will see how awesome Complex Number in Converting from Polar Form to Rectangular Form. Either method of notation is valid for complex numbers. The primary reason for having two methods of notation is for ease of longhand calculation, rectangular form lending itself to addition and subtraction, and polar form lending itself to
Rating: 4.47 / Views: 419 | 2019-11-20 20:28:06 | {"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": 0.904308557510376, "perplexity": 410.3866109138701}, "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/1573496670601.75/warc/CC-MAIN-20191120185646-20191120213646-00542.warc.gz"} |
https://www.semanticscholar.org/paper/The-Role-of-Symmetry-and-Separation-in-Surface-and-Broadbridge-Vassiliou/a2481e047b8b99613ef361c2e0ce87b9bf040aff | # The Role of Symmetry and Separation in Surface Evolution and Curve Shortening
@article{Broadbridge2011TheRO,
title={The Role of Symmetry and Separation in Surface Evolution and Curve Shortening},
author={Philip Broadbridge and Peter J. Vassiliou},
journal={Symmetry Integrability and Geometry-methods and Applications},
year={2011},
volume={7},
pages={052}
}
• Published 1 June 2011
• Mathematics, Physics
• Symmetry Integrability and Geometry-methods and Applications
With few exceptions, known explicit solutions of the curve shortening flow (CSE) of a plane curve, can be constructed by classical Lie point symmetry reductions or by functional separation of variables. One of the functionally separated solutions is the exact curve shortening flow of a closed, convex \oval"-shaped curve and another is the smoothing of an initial periodic curve that is close to a square wave. The types of anisotropic evaporation coefficient are found for which the evaporation… Expand
8 Citations
#### Figures from this paper
Asymptotically self-similar solutions to curvature flow equations with prescribed contact angle and their applications to groove profiles due to evaporation-condensation
We study the asymptotic behavior of solutions to fully nonlinear second order parabolic equations including a generalized curvature flow equation which was introduced by Mullins in 1957 as a model ofExpand
The fundamental solutions of the curve shortening problem via the Schwarz function
Curve shortening in the z-plane in which, at a given point on the curve, the normal velocity of the curve is equal to the curvature, is shown to satisfy StSz = Szz , where S(z, t) is the SchwarzExpand
Enhanced group classification of nonlinear diffusion–reaction equations with gradient-dependent diffusivity
• Mathematics, Physics
• 2020
Abstract We carry out the enhanced group classification of a class of (1+1)-dimensional nonlinear diffusion–reaction equations with gradient-dependent diffusivity using the two-step version of theExpand
Applications of Integrable Nonlinear Diffusion Equations in Industrial Modelling
There are useful integrable nonlinear diffusion equations that can be transformed directly to linear partial differential equations. The possibility of linearisation allows us to incorporate a muchExpand
On Some Simple Methods to Derive the Hairclip and Paperclip Solutions of the Curve Shortening Flow
• Mathematics
• Acta Mathematica Scientia
• 2019
We use two simple methods to derive four important explicit graphical solutions of the curve shortening flow in the plane. They are well-known as the circle, hairclip, paperclip, and grim reaperExpand
Self-similar solutions to the mean curvature flow in the Minkowski plane $\mathbf R^{1,1}$
We introduce the mean curvature flow of curves in the Minkowski plane $\mathbf R^{1,1}$ and give a classification of all the self-similar solutions. In addition, we describe five other exactExpand
On an asymptotically log-periodic solution to the graphical curve shortening flow equation
With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution $y\left(x, t\right) \$has the interesting property that itExpand
The Impact of Applications on Mathematics
Biological cells require active fluxes of matter to maintain their internal organization and perform multiple tasks to live. In particular they rely on cytoskeletal transport driven by motorExpand
#### References
SHOWING 1-10 OF 31 REFERENCES
The normalized curve shortening flow and homothetic solutions
• Mathematics
• 1986
The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradientExpand
Integrable nonlinear evolution equations applied to solidification and surface redistribution
Members of a hierarchy of integrable nonlinear evolution equations are applied to a problem in solidification and various problems in surface redistribution of crystalline materials. The members ofExpand
An integrable fourth-order nonlinear evolution equation applied to surface redistribution due to capillarity
Members of an hierarchy of integrable nonlinear evolution equations, related to the well-known linearizable diffusion equation which has the diffusivity form as the reciprocal of the square of theExpand
Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove
• Mathematics
• Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
• 1995
The fourth-order nonlinear boundary-value problem for the evolution of a single symmetric grain-boundary groove by surface diffusion is modelled analytically. A solution is achieved by partitioningExpand
Overview no. 113 surface motion by surface diffusion
• Materials Science
• 1994
Abstract Geometry growth laws for morphological change are developed and examined for the class of dynamic problems where surface diffusion is the only transport mechanism and hence volume isExpand
Temperature-dependent surface diffusion near a grain boundary
• Mathematics
• 2010
Metal surface evolution is described by a nonlinear fourth-order partial differential equation for curvature-driven flow. The standard boundary conditions for grain-boundary grooving, at aExpand
Exact solvability of the Mullins nonlinear diffusion model of groove development
The Mullins equation for the development of a surface groove by evaporation–condensation is yt=yxx/1+y2x. It is pointed out that this is the equation of the potential for the field variable ΘExpand
Separation of variables for the 1-dimensional non-linear diffusion equation
• Mathematics
• 1998
Abstract The class of separable solutions of a 1-dimensional sourceless diffusion equation is stabilized by the action of the generic symmetry group. It includes all solutions invariant under aExpand
Invariant Geometric Evolutions of Surfaces and Volumetric Smoothing
• Mathematics, Computer Science
• SIAM J. Appl. Math.
• 1997
This paper presents the simplest affine invariant flow for (convex) surfaces in three-dimensional space, which, like the affine-invariant curve shortening flow, will be of fundamental importance in the processing of three- dimensional images. Expand
An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces
• Mathematics
• 1994
The fourth-order nonlinear partial differential equation for surface diffusion is approximated by a new integrable nonlinear evolution equation. Exact solutions are obtained for thermal grooving,Expand | 2021-12-02 01:39:33 | {"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": 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": 0.6114137172698975, "perplexity": 1423.2690577093733}, "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/1637964361064.58/warc/CC-MAIN-20211201234046-20211202024046-00306.warc.gz"} |
https://www.projecteuclid.org/euclid.ejs/1363482367 | ## Electronic Journal of Statistics
### Marginals of multivariate Gibbs distributions with applications in Bayesian species sampling
Annalisa Cerquetti
#### Abstract
Gibbs partition models are the largest class of infinite exchangeable partitions of the positive integers generalizing the product form of the probability function of the two-parameter Poisson-Dirichlet family. Here we call into question the current approach to Bayesian nonparametric estimation in species sampling problems under Gibbs priors, which incorrectly relies on treating exchangeable partition probability functions (EPPFs) as multivariate distributions on compositions of the positive integers. We show that once those multivariate distributions are correctly derived, results for corresponding sampling formulas can be obtained, generalized and sometimes fixed, working with marginals and a known result on falling factorial moments of a sum of non independent indicators. We provide an application of our findings to a recently proposed Bayesian nonparametric estimation under Gibbs priors of the predictive probability to observe a species already observed a certain number of times.
#### Article information
Source
Electron. J. Statist., Volume 7 (2013), 697-716.
Dates
First available in Project Euclid: 17 March 2013
https://projecteuclid.org/euclid.ejs/1363482367
Digital Object Identifier
doi:10.1214/13-EJS784
Mathematical Reviews number (MathSciNet)
MR3035269
Zentralblatt MATH identifier
1327.62196
Subjects
Primary: 60G57: Random measures 62G05: Estimation
Secondary: 62F15: Bayesian inference
#### Citation
Cerquetti, Annalisa. Marginals of multivariate Gibbs distributions with applications in Bayesian species sampling. Electron. J. Statist. 7 (2013), 697--716. doi:10.1214/13-EJS784. https://projecteuclid.org/euclid.ejs/1363482367
#### References
• [1] Arratia, R., Barbour, A. D., Tavaré, S. (2003), Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics.
• [2] Cerquetti, A. (2011a) A decomposition approach to Bayesian nonparametric estimation under two-parameter Poisson-Dirichlet priors., Proceedings of ASMDA 2011 - Rome, Italy. Available at: http://geostasto.eco.uniroma1.it/utenti/cerquetti/asmda2011last.pdf
• [3] Cerquetti, A. (2011b) Conditional $\alpha$-diversity for exchangeable Gibbs partition driven by the stable subordinator., Proceedings of S.Co. Conferemce, 2011, Padova, Italy. Available at: http://homes.stat.unipd.it/mgri/SCo2011/Papers/CS/CS-7/cerquetti.pdf.
• [4] Charalambides, C. A. (2005), Combinatorial Methods in Discrete Distributions. Wiley, Hoboken NJ.
• [5] De Moivre, A. (1718), The doctrine of chances: Or a method of calculating the probabilities of events in play. London. Pearson.
• [6] Ewens, W. J. (1972) The sampling theory of selectively neutral alleles., Theoret. Pop. Biol., 3, 87–112.
• [7] Ewens, W. and Tavaré, S. (1995) The Ewens sampling formula. In Multivariate discrete distributions (Johnson, N.S., Kotz, S. and Balakrishnan, N. eds.). Wiley, NY.
• [8] Favaro, S., Lijoi, A., Mena, R. H., Prünster, I. (2009) Bayesian non-parametric inference for species variety with a two-parameter Poisson-Dirichlet process prior., J. Roy. Statist. Soc. B, 71, 993–1008.
• [9] Favaro, S., Lijoi, A. and Prünster, I. (2012a) Conditional formulae for Gibbs-type exchangeable random partitions., Ann. Appl. Probab. (to appear)
• [10] Favaro, S., Lijoi, A. and Prünster, I. (2012b) A new estimator of the discovery probability., Biometrics, 68, 1188–1196.
• [11] Ferguson, T. S. (1973) A Bayesian analysis of some nonparametric problems., Ann. Statist., 1, 209–230.
• [12] Fisher, R. A., Corbet, A. S. and Williams, C. B. (1943) The relation between the number of species and the number of individuals in a random sample of an animal population., J. Animal. Ecol., 12, 42–58.
• [13] Gnedin, A. (2010) A species sampling model with finitely many types., Electron. Commun. Prob., 15, 79–88.
• [14] Gnedin, A. and Pitman, J. (2006) Exchangeable Gibbs partitions and Stirling triangles., J. Math. Sci., 138, 3, 5674–5685.
• [15] Hsu, L. C, and Shiue, P. J. (1998) A unified approach to generalized Stirling numbers., Adv. Appl. Math., 20, 366-384.
• [16] Iyer, P. V. K. (1949) Calculation of factorial moments of certain probability distributions., Nature. 164, 282.
• [17] Iyer, P. V. K. (1958) A theorem on factorial moments and its applications., Ann. Math. Statist., 29, 254–261.
• [18] Johnson, N. S. and Kotz, S. (2005), Univariate discrete distributions 3rd Ed. Wiley, NY.
• [19] Jordan, M. C. (1867) De quelques formules de probabilité., Comptes Rendus. Académie des Sciences, Paris, 65, 993–994.
• [20] Kingman, J. F. C. (1975) Random discrete distributions., J. Roy. Statist. Soc. B, 37, 1–22.
• [21] Kingman, J. F. C (1978) The representation of partition structure., J. London Math. Soc. 2, 374–380.
• [22] Lijoi, A., Mena, R. and Prünster, I. (2007) Bayesian nonparametric estimation of the probability of discovering new species., Biometrika, 94, 769–786.
• [23] Lijoi, A., Prünster, I. and Walker, S. G. (2008) Bayesian nonparametric estimators derived from conditional Gibbs structures., Ann. Appl. Probab., 18, 1519–1547.
• [24] Normand, J. M. (2004) Calculation of some determinants using the $s$-shifted factorial., J. Phys. A: Math. Gen. 37, 5737-5762.
• [25] Pitman, J. (1995) Exchangeable and partially exchangeable random partitions., Probab. Th. Rel. Fields, 102, 145-158.
• [26] Pitman, J. (1996) Some developments of the Blackwell-MacQueen urn scheme. In T.S. Ferguson, Shapley L.S., and MacQueen J.B., editors, Statistics, Probability and Game Theory, vol. 30 of IMS Lecture Notes-Monograph Series, pages 245–267. Institute of Mathematical Statistics, Hayward, CA.
• [27] Pitman, J. (2003) Poisson-Kingman partitions. In D.R. Goldstein, editor, Science and Statistics: A Festschrift for Terry Speed, volume 40 of Lecture Notes-Monograph Series, pages 1–34. IMS, Hayward, California.
• [28] Pitman, J. (2006), Combinatorial Stochastic Processes. Ecole d’Eté de Probabilité de Saint-Flour XXXII - 2002. Lecture Notes in Mathematics N. 1875, Springer.
• [29] Pitman, J. and Yor, M. (1997) The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator., Ann. Probab., 25, 855–900.
• [30] Yamato, H. and Sibuya, M. (2000) Moments of some statistics of Pitman sampling formula., Bull. Inform. Cybernet., 32, 1. | 2019-10-13 22:02:57 | {"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": 0.41486790776252747, "perplexity": 9399.102270013396}, "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-2019-43/segments/1570986647517.11/warc/CC-MAIN-20191013195541-20191013222541-00557.warc.gz"} |
https://handwiki.org/wiki/Medicine:Ophthalmic_drug_administration | Figure 1.0 - Basic anatomical features of the human eye
Ophthalmic drug administration is the administration of a drug to the eyes, most typically as an eye drop formulation. Topical formulations are used to combat a multitude of diseased states of the eye. These states may include bacterial infections, eye injury, glaucoma, and dry eye.[1] However, there are many challenges associated with topical delivery of drugs to the cornea of the eye.
## Eye drop formulations
Two of the largest challenges faced when using topicals to treat pathological states of the eye include patient compliance and ineffective absorbance of drugs into the cornea.[1][2][3][4][5][6][7] In fact, researchers in this field of drug delivery agree that less than 7% of drugs delivered to the eye reach and penetrate the corneal barrier, therefore, increasing the frequency of dosing used for topicals.[1][2][3][4][5][6][7] This is one of the fundamental problem associated with using topicals to deliver drugs to the cornea and therefore leads to the increased demand for patient compliance. Together, these two factors drive a need in the field of scientific research and engineering for a way to better deliver drugs to the cornea of the eye while decreasing dosing frequency and demand for patient compliance. Moreover, besides the logistical problems associated with using topicals, there are also systemic side effects which result from the administration of some drugs used to combat the pathological states of the eye.[3] With the increased concentration of drugs in topicals and the frequent application to the eye, a majority of the drug is drained from the eye via nasolacrimal drainage.[3] This drainage is thought to be the reason that systemic side effects exist from such administration.[2][3][6]
## Contact lenses as delivery devices
Figure 2.0 - Contact lens
The U.S. Centers for Disease Control and Prevention (CDC) claims that there were "about 41 million contact lens wearers greater than 18 years old in the United States" in 2018.[8] Of all of these wearers, nearly 90% of them wear contact lenses known as 'soft contact lenses' (SCLs).[8] Contact lenses are regulated by the United States Food and Drug Administration (FDA).[9]
The main approaches that researchers in this field are using today are: molecular imprinting, supercritical soaking, solvent impregnation, and nanoparticle loading.[2][4][5][7] Each of these techniques assists by hoping to deliver drugs at a lower, more sustained, rate that does not require a demand for increased patient compliance nor the systemic side effects from topical drug delivery systems. However, each of these different types of loading techniques results in contact lenses that all have separate physical and chemical challenges when it comes to the sustained release and penetration of specific drugs at the molecular level in regards to the cornea of the eye.
Figure 3.0 - Molecular imprinting technique
### Molecular imprinting
Molecular imprinting is a process by which polymerization of a polymer around template result in the polymer matrix with embedded templates.[2][5] After the template is removed, a cavity results with the functionalized monomers within the polymer cavity. This cavity is the idealized position for drug loading since this process can be designed specifically to recruit and hold onto drugs due to chemical specificity.[2][5] This technique can be better visualized by referring to Figure 3.0. This type of drug loading can be used as a way to create a pH responsive system, which releases drug(s) as the pH of the biological system changes.[2][5] Some drugs that have been successfully loaded via this method are: timolol, norfloxacin, ketotifen, polyvinlypyrrolidone, and hyaluronic acid.[5][10][11] The molecular structures of each of these drugs are shown below in the index of important scientific terminology.
### Supercritical soaking/solvent impregnation
The supercritical soaking method is commonly used in hydrogel-based contact lenses and is the most common of all types of molecular drug loading techniques. Since this technique requires no special equipment or advanced knowledge of polymer-based hydrogels it is the least complex of all loading types.[4] In order to load the hydrogel matrix with a certain drug, contact lenses are simply placed in a solution of the drug and the drug diffuses into the matrix.[4][5][7] Since this loading technique is driven solely by the gradient of the drug concentration surrounding the lens relative to the hydrogel matrix, the diffusion rate and amount of drug that is loaded can be controlled solely by the concentration of the drug solution.[4][7] Since this process allows for specific amounts of a certain drug to be loaded to the hydrogel matrix, this method of loading has become important for patient-specific (personalized) medicine and treatments.
Figure 4.0 - Example of a nanoparticle
The nanoparticle loading technique includes two major parts. The first part of this process is the creation and conjugation of a specific drug into or onto a nanoparticle or other colloidal particle.[5] Next, the nanoparticle is loaded into the hydrogel matrix of the contact lens.[5] In this case, before the drug can diffuse out of the hydrogel matrix to reach the cornea, it must also diffuse or be released out of the nanoparticle.[5]
It is important to recognize the positives and negatives associated with each type of drug loading for using contact lenses as drug delivery devices. In order to seriously address the possibility of clinical translation of these devices, it is important to recognize the physical and chemical barriers. By understanding this better, the mechanism of drug loading and the controlled and sustained release of drugs to a patient's eye can be optimized.
### Lens transparency
Since contact lenses are used on a part of the body that is important for normal daily functioning (sight) it is critical that scientists take into account the transparency of the lens.[5] As larger and more drugs/objects are loaded to a contact lens it begins to physically crowd the space available, making it more difficult for light to penetrate and reach the eye.
Fundamental Concept Understanding: A simple analogy to this is a crowded versus an uncrowded area while it is raining outside. When individuals are packed tightly the rain falls and lands on people, making its way to the ground slowly but surely in a scattered way. In an uncrowded area, the rain can fall and land on the ground easily and without interference from the people. In this analogy, the rain is analogous to light and the people are analogous to drugs being loaded in a contact lens. The more drugs added to the contact lens, the less light that can penetrate without being randomly scattered. Random scattering of the light can result in unclear and unfocused sight.
Researchers have noted that by using the nanoparticle loading technique, the transparency decreases by nearly 10%.[5] Conversely, researchers have confirmed that by using the molecular imprinting and supercritical soaking methods of drug loading, the lens transparency of the contact lenses has stayed at or above the lens transparency of the contact lenses currently approved by the FDA.[5][11]
### Oxygen permeability
Oxygen permeability is another important feature of all contact lenses and much be optimized to the largest degree possible when creating drug delivery devices for the eye. The contact lens adheres to the external cornea of the eye which is made up of a layer of cells.[12] Cells, being the basic component of living organisms, require sustained and constant access to oxygen in order to survive. The cornea of the eye is not supplied with blood as are most other cells in the body, making this a challenging part of the body to which to deliver drugs.[12] Decreasing oxygenation to the eye can result in undesirable side effects.[5] Researchers in this field have noted that different types of contact lenses have varying degrees of oxygen permeability. For example, it has been shown that SCLs have limited oxygen permeability while silicon-based contact lenses have much better oxygen permeability.[1][5][11][13] Silicon-base contact lenses have also been shown to have some other very important physical parameters.[1][5][11][13]
Researchers have attempted to make the thickness of the contact lenses in order to increase the drug loading capacity of the contact lens.[11] However, for silicon-based lenses this parameter is inversely proportion to oxygen permeability (i.e. as thickness of the contact lens increases the oxygen permeability decreases).[11] Moreover, it has been shown that as water content increases in silicon-based lenses, the oxygen permeability decreases, another relationship that is inversely proportional.[11] Surprisingly, as SCLs increase with water content the oxygen permeability also increases (a directly proportional relationship).[11]
In regards to whether silicon-based lenses or SCLs are a better candidate as an ophthalmic drug delivery device is a question that remains unanswered and is not uniformly agreed upon in the scientific community. For example, Ciolino et al. claim that silicon-based contact lenses are better candidates for patients that are long-term contact lens wearer.[2][3] Conversely, Kim et al. suggest that SCLs are better candidates because they show the possibility to be able to overcome the difficult of oxygen permeability as well as mechanical integrity of the lens.[7] Kim et al. have shown that the mechanical strength can be increased for SCLs by incorporating a nanodiamond (ND) infrastructure into contact lens matrix.[7]
Additionally, many researchers have investigated the implications of loading vitamin E into the contact lens matrix of SCLs.[6] Although vitamin E incorporation into the matrix has been shown to slow the release of drugs into the eye and onto cornea (a desirable trait of a ophthalmic delivery system), it has also been shown to decrease oxygen permeability.[6] Oxygen permeability continues to be an extremely important factor in the development of these devices and is one of the main reason that much research is beginning to focus on this area of drug delivery.
### Water content
The amount of water content that a particular contact lens can retain is another extremely important factor that must be taken into account when these devices are designed. Research in this specific area of design suggests that contact lens wearers find it more comfortable to wear lenses that retain water more than those that deter water.[5][11] For SCLs, as the water content of a lens increases so does the oxygen permeability.[11] Conversely, as the water content increases in silicon-based contact lenses, the oxygen permeability decreases.[2][3][11] In reference to SCLs, higher water content in contact lenses allows for easier loading using the supercritical soaking method.[3][5][11][14] This could be due to the water acting as a lubricant to some drugs and allowing the drug to be more easily facilitated into the matrix. This would essentially allow for more drug to be loaded into contact lenses of this type.[5][11] This increase in drug loading capacity is an important advancement and would allow for patients wince it may allow for a longer period of drug release time and would hopefully be more sustained.[5]
Furthermore, Guzman-Aranguez et al. has shown that when using the molecular imprinting method for loading drugs such as ketotifen and norfloxacin into the contact lens, the water content is not largely impacted.[5] Additionally, it has been predicted by Peng et al. by using Fickian release kinetic models that although water content changes once contact lenses are inserted onto the cornea of the eye, this will not pose significant challenges when it comes to the release of rugs from SCLs.[15]
### Drug release kinetics
The most important factor that must be taken into account when designing any type of drug delivery device, and specifically ocular devices, is the release rate of a drug. As discussed previously, the deliver rate and kinetics associated with drugs to the eye can reach levels that are toxic to the eye or could even cause undesirable side effects. The rate of release of a drug is also important because too slow of a release could have no beneficial outcome for the patient and a release that is too quick could result in negative side effects.[9][13][15][16] Thus, it is important to balance the factors that govern the release of drugs from contact lenses as potential drug delivery devices. Researchers such as C. Alvarez-Lorenzo have tested (with animal models) and have data which supports that molecularly imprinted contact lenses release drugs in a sustained and long period of time.[11] It has also been supported by researchers that the rate of drug release can be controlled by incorporating vitamin E within the hydrogen matrix.[6]
### Systemic side effects
Over time, it has been reported that many of the same drugs and eye drops used to treat particular eye diseases do, in fact, result in systemic side effects that could possibly be minimized or limited due to a slower, more sustained release of the drug. The systemic side effects of glaucoma medications such as latanoprost increased heart rate resulting in cardiac arrhythmias, bronchoconstriction, and hypotension.[15][16][17] These complications could be life-threatening. Some other drugs that help to reduce the effects of glaucoma in the eye result in vomiting, diarrhea, tachycardia and bronchospasm.[14][15][16][17] It has been found that some drugs delivered in the form of eye drops are highly toxic to children since their total body volume and tissue volumes are much lower than that of an adult for which the drugs are intended for use.[16] In this case, some parents are not aware of these implications and could use the same drug they would use to help treat their children's bacterial infections in the eye. Moreover, some drugs administered to the eye have been shown to result in cardiac depression and propagation of some disorders such as asthma.[15][16][17] With continued research in this area, it has become know that skin irritation, itching or rash are commonly associated with drugs used to treat ocular bacterial infections.[14][15][16][17]
## Ocular disorders
There are currently four main ocular disorders that have been heavily investigated and have shown success with using contact lenses as possible devices for molecular drug delivery.
### Bacterial infection
Figure 5.0 - Example of zero-order release kinetics (y = % of drug released from contact lens; x = time (days))
The drug release rate is extremely important in treating many diseased states of the eye, bacterial infections being one of them. Ciprofloxacin and norfloxacin are drugs that are normally used to treat bacterial infections of the eye. It is of utmost importance that these drugs stay in the therapeutic window for an extended period of time in order to be fully effective and kill bacteria.[5][11] To keep the specific drug in the therapeutic window using eye drops the topical must be applied approximately every 30 minutes in order to be fully effective.[5][11] Having to apply eye drops every 30 minutes would be nearly impossible for anyone and is not the ideal mechanism by which to deliver such drugs to the eye. Researchers have gathered data to support the idea that silicon-based contact lenses with ciprofloxacin could release the drug in the therapeutic window for approximately one month.[5] Ana Guzman-Aranguez et al. also confirmed that the contact lens used also retained important properties such as transparency, oxygen permeability, mechanical strength, and zero-order release pharmacokinetics.[5]
Figure 6.0 - Cornea epithelium
### Corneal injury
Many factors can result in corneal injury and cause the deterioration or death of cells that make up the cornea of the eye.[5][11] The epithelial cells that make up the cornea are important in order for normal vision. These cells play a role in creating a physical environment that can correctly bend light rays to help project images to the retina of the eye.[5][11] There have been successful human clinical trials with using SCLs infused with epidermal growth factor (EGF) that showed increased rate of healing of the epithelial cell layer of the cornea.[5]
Figure 7.0 - Glaucoma complications
### Glaucoma
Glaucoma is the leading cause of blindness in the world and is a progressive and irreversible disease of the eye.[17] A poly(lactic-co-glycolic acid)-based contact lens was shown to release latanoprost at a sustained release rate of up to a month in animal models by Ciolino et al. at Harvard Medical School and Massachusetts Institute of Technology.[17] Latanoprost is one of the drug interventions used to treat patients with glaucoma, generally in the form of topicals such as eye drops.[17]
Figure 8.0 - Dry eye syndrome complications
### Dry Eye
More than 50% of all contact lens wearers report that they experience dry eye.[5] In order to help combat this issue and be assured that this does not occur in people that will one day be using drug eluting contact lenses, it is important to make sure that this complication is highly investigated. However, these investigations will not only be beneficial for contact lenses as drug delivery devices, but it will also have positive implications on contact lens wearers who use lenses for vision correction and appearance.
Index: Scientific Terms in Context of Ocular Drug Delivery
Term Definition
pH responsive system the ability for a biological system to undergo changes that promote activity, inactivity, release of compound(s), or degradation as a result of changes made in the pH of the microenvironment of a certain system
nasolacrimal drainage drainage of particles/fluids into the body via the tear (nasolacrimal) duct
systemic of or relating to the entire body
zero-order kinetics release of a drug from a delivery device at a singular and constant rate the entire time of release
Norfloxacin (1-Ethyl-6-fluoro-4-oxo-7-(1-piperazinyl)-1,4-dihydro-3-quinolinecarboxylic acid)
Ketotifen (10H-Benzo(4,5)cyclohepta(1,2-b)thiophen-10-one, 4,9-dihydro-4-(1-methyl-4-piperidinylidene)
Polyvinylpyrrolidone
Hyaluronic acid
Latanoprost (isopropyl-(Z) 7[(1R,2R,3R,5S)3,5-dihydroxy-2-[(3R)-3-hydroxy-5-phenylpentyl]cyclopentyl]-5-heptenoate)
## References
1. "Development of ocular drug delivery systems using molecularly imprinted soft contact lenses". Drug Development and Industrial Pharmacy 41 (5): 703–13. May 2015. doi:10.3109/03639045.2014.948451. PMID 25113431.
2. "A drug-eluting contact lens". Investigative Ophthalmology & Visual Science 50 (7): 3346–52. July 2009. doi:10.1167/iovs.08-2826. PMID 19136709.
3. "A prototype antifungal contact lens". Investigative Ophthalmology & Visual Science 52 (9): 6286–91. August 2011. doi:10.1167/iovs.10-6935. PMID 21527380.
4. "Uptake and release of Dexamethasone using pH-responsive poly(2-hydroxyethyl methacrylate-co-2-(diisopropylamino)ethyl methacrylate) hydrogels for potential use in ocular drug delivery" (in en). Journal of Drug Delivery Science and Technology 51: 45–54. June 2019. doi:10.1016/j.jddst.2019.02.018.
5. "Contact lenses: promising devices for ocular drug delivery". Journal of Ocular Pharmacology and Therapeutics 29 (2): 189–99. March 2013. doi:10.1089/jop.2012.0212. PMID 23215541.
6. "Dual drug delivery from vitamin E loaded contact lenses for glaucoma therapy". European Journal of Pharmaceutics and Biopharmaceutics 94: 312–21. August 2015. doi:10.1016/j.ejpb.2015.06.001. PMID 26071799.
7. "Diamond nanogel-embedded contact lenses mediate lysozyme-dependent therapeutic release". ACS Nano 8 (3): 2998–3005. March 2014. doi:10.1021/nn5002968. PMID 24506583.
8. "Risk Behaviors for Contact Lens-Related Eye Infections Among Adults and Adolescents - United States, 2016". MMWR. Morbidity and Mortality Weekly Report 66 (32): 841–5. August 2017. doi:10.15585/mmwr.mm6632a2. PMID 28817556. PMC 5657667.
9. Center for Devices and Radiological Health (28 October 2019). "Contact Lenses". U.S. Food and Drug Administration (FDA).
10. "Ocular biomaterials and implants". Biomaterials 22 (8): 769–85. April 2001. doi:10.1016/S0142-9612(00)00237-4. PMID 11246945.
11. "Bioinspired hydrogels for drug-eluting contact lenses". Acta Biomaterialia 84: 49–62. January 2019. doi:10.1016/j.actbio.2018.11.020. PMID 30448434.
12. Sridhar, Mittanamalli (2003). "Chapter-17 Hyperopic LASIK". Step by step in LASIK Surgery. Jaypee Brothers Medical Publishers (P) Ltd.. pp. 153–162. doi:10.5005/jp/books/10819_17. ISBN 978-81-8061-099-8.
13. "Modeling Ophthalmic Drug Delivery by Soaked Contact Lenses". Industrial & Engineering Chemistry Research 45 (10): 3718–3734. May 2006. doi:10.1021/ie0507934. ISSN 0888-5885.
14. "In vitro and in vivo evaluation of ketotifen fumarate-loaded silicone hydrogel contact lenses for ocular drug delivery". Drug Delivery 18 (2): 150–8. February 2011. doi:10.3109/10717544.2010.522612. PMID 21043996.
15. "Extended drug delivery by contact lenses for glaucoma therapy". Journal of Controlled Release 162 (1): 152–8. August 2012. doi:10.1016/j.jconrel.2012.06.017. PMID 22721817.
16. "Altered bioavailability of drugs in the eye due to drug-protein interaction". Journal of Pharmaceutical Sciences 62 (10): 1648–53. October 1973. doi:10.1002/jps.2600621014. PMID 4752109.
17. "In vivo performance of a drug-eluting contact lens to treat glaucoma for a month". Biomaterials 35 (1): 432–9. January 2014. doi:10.1016/j.biomaterials.2013.09.032. PMID 24094935. | 2021-10-19 23:48:36 | {"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": 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": 0.3376281261444092, "perplexity": 2691.5046189578006}, "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-2021-43/segments/1634323585290.83/warc/CC-MAIN-20211019233130-20211020023130-00218.warc.gz"} |
https://www.springerprofessional.de/an-invitation-to-von-neumann-algebras/13786374 | main-content
## Über dieses Buch
Why This Book: The theory of von Neumann algebras has been growing in leaps and bounds in the last 20 years. It has always had strong connections with ergodic theory and mathematical physics. It is now beginning to make contact with other areas such as differential geometry and K-Theory. There seems to be a strong case for putting together a book which (a) introduces a reader to some of the basic theory needed to appreciate the recent advances, without getting bogged down by too much technical detail; (b) makes minimal assumptions on the reader's background; and (c) is small enough in size to not test the stamina and patience of the reader. This book tries to meet these requirements. In any case, it is just what its title proclaims it to be -- an invitation to the exciting world of von Neumann algebras. It is hoped that after perusing this book, the reader might be tempted to fill in the numerous (and technically, capacious) gaps in this exposition, and to delve further into the depths of the theory. For the expert, it suffices to mention here that after some preliminaries, the book commences with the Murray - von Neumann classification of factors, proceeds through the basic modular theory to the III). classification of Connes, and concludes with a discussion of crossed-products, Krieger's ratio set, examples of factors, and Takesaki's duality theorem.
## Inhaltsverzeichnis
### Chapter 0. Introduction
Abstract
As the title suggests, this chapter is devoted to developing some of the basic technical results needed in the theory, and may be safely omitted by the expert. The first section establishes some of the notation employed throughout the book and lists, without proof, the basic facts concerning operators on Hilbert space. The next section establishes the “non-commutative analogue” of the classical results $$c_0^* = \ell ^1 \,{\text{and}}\,(\ell ^1 )* = \ell ^\infty$$ -- null-convergent, summable, and bounded sequences being replaced by compact, trace-class and bounded operators respectively.
V. S. Sunder
### Chapter 1. The Murray — Von Neumann Classification of Factors
Abstract
The notion of unitary equivalence, while being most natural, has the disadvantage of not being additive in the following sense: if e1, e2, f1 and f2 are Projections such that ei is unitarily equivalent to fi, for i = 1,2, and if e1e2 and f1f2, it is not necessarily true that e1 + e2 is unitarily equivalent to f1 + f2. This problem disappears if, more generally, one considers two projections as being equivalent if their ranges are the initial and final spaces of a partial isometry. This equivalence, when all the operators concerned -- the projections as well as the partial isometry -- are required to come from a given factor M, is the subject of Section 1.1, where the crucial result is that, with respect to a natural order, the set of equivalence classes of the projections in a factor is totally ordered. The next section examines finite projections -- those not equivalent to proper subprojections; the main result being that finiteness is preserved under taking finite suprema. The final section, via a quantitative analysis of the order relation discussed earlier, effects a primary classification of factors into three types. The principal tool used is called a ‘relative dimension function’ by Murray and von Neumann.
V. S. Sunder
### Chapter 2. The Tomita-Takesaki Theory
Abstract
Section 2.1 discusses the following question (which, in the case M = L (X, F) with μ finite, is answered affirmatively by the existence of the Lebesgue integral): if m: P(M) → [0,1] is countably additive in the sense that
$$m\left[ {\mathop{V}\limits_{{n = 1}}^{\infty } {e_n}} \right] = \sum\limits_{{n = 1}}^{\infty } {m({e_n})}$$
for any countable collection of pairwise orthogonal projections in M, does m extend to a linear functional on M which is well-behaved under monotone convergence?
V. S. Sunder
### Chapter 3. The Connes Classification of Type III Factors
Abstract
The first section discusses the extent to which the modular group σ ϕ depends upon the fns weight σ ϕ . The precise description is the unitary cocycle theorem of Connes, which says, loosely, that modulo the group of inner automorphisms of M, the modular group σ ϕ is independent of σ ϕ .
V. S. Sunder
### Chapter 4. Crossed-Products
Abstract
The crossed-product construction was first employed by Murray and von Neumann to exhibit examples of factors of types I, II and III. The set-up is as follows: one starts with a dynamical system (M,G,α) -- with G not necessarily abelian -- and constructs an associated von Neumann algebra M (usually denoted by Mα G) on a larger Hilbert space H.
V. S. Sunder
### Backmatter
Weitere Informationen | 2020-05-27 13:00:30 | {"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": 0.8533470034599304, "perplexity": 567.4238378634076}, "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/1590347394074.44/warc/CC-MAIN-20200527110649-20200527140649-00328.warc.gz"} |
http://mathoverflow.net/feeds/question/51909 | What is the largest family F of subsets of [n] for which any two distinct sets A and B in F have an intersection of size at most min(|A|,|B|)/2? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T20:04:27Z http://mathoverflow.net/feeds/question/51909 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51909/what-is-the-largest-family-f-of-subsets-of-n-for-which-any-two-distinct-sets-a What is the largest family F of subsets of [n] for which any two distinct sets A and B in F have an intersection of size at most min(|A|,|B|)/2? Douglas S. Stones 2011-01-13T04:14:29Z 2011-01-13T07:59:39Z <p>This problem arose in the study of Latin squares with a large number of subsquares, although it appears interesting in its own right.</p> <blockquote> <p><strong>Question</strong>: What is the maximum cardinality of a family $F \subseteq 2^{[n]}$ of subsets of $[n]:=\{1,2,\ldots,n\}$ for which any two distinct $A,B \in F$ satisfy $|A \cap B| \leq \tfrac{1}{2} \min(|A|,|B|)$?</p> </blockquote> <p>Some observations:</p> <ul> <li>We have the trivial lower bound $\max |F| \geq {n \choose 2}$ by taking all the subsets of size 2.</li> <li>When $n \geq 3$, $F$ should not have any sets of size 1. If $\{a\} \in F$, then we can replace it by $\{a,x\}$ for all $x \in [n] \setminus \{a\}$ for any $x$ that belongs to a set of size 2 or more (since no other set in $F$ can contain an $a$). If every set has size 1, then $|F|=n$ which can be beaten.</li> <li>$F$ should not have any sets of size 3. If $\{a,b,c\} \in F$, then we can replace it by $\{a,b\},\{a,c\},\{b,c\}$ (since any other set in $F$ may intersect $\{a,b,c\}$ in at most one element).</li> <li>With the above simplifications in mind, I wrote a backtracking algorithm which says that for $3 \leq n \leq 7$ (and it's progressing through $n=8$), that $F$ is uniquely maximized when $F$ consists of all 2-subsets.</li> </ul> http://mathoverflow.net/questions/51909/what-is-the-largest-family-f-of-subsets-of-n-for-which-any-two-distinct-sets-a/51915#51915 Answer by Aaron Meyerowitz for What is the largest family F of subsets of [n] for which any two distinct sets A and B in F have an intersection of size at most min(|A|,|B|)/2? Aaron Meyerowitz 2011-01-13T05:12:17Z 2011-01-13T07:59:39Z <p>For an affine space over $\mathbb{z}_2$ with $n=2^k$ we have $\binom{n}{2}=(2^{k}-1)(2^{k-1})$ however there are $\binom{n}{3}/4=\frac{2^{k}(2^{k}-1)(2^{k}-2)}{24}$ 2 dimensional flats of which any pair intersect in at most two points.</p> <p>Details: Consider the $n=2^k$ binary vectors of length $k$. Among the sets of 4 vectors chose only those of the form $\lbrace x,y,z,x+y+z\rbrace$ in other words those quadruples whose members sum to the all zero vector.</p> <p>I don't know how close you can get for other values of $n$ to having a family of 4-sets so that any three points is in a unique 4-set.</p> <p>I would expect even better numbers for bigger subsets with a big enough $n$.</p> <p>The best one could do with 4-sets when $n=24$ could at the very most $\binom{24}{3}/4=506$. The Steiner system S(5,8,24) is a family of 759 8 element subsets (blocks) of a 24 set so that each 5-set is in a unique block.</p> http://mathoverflow.net/questions/51909/what-is-the-largest-family-f-of-subsets-of-n-for-which-any-two-distinct-sets-a/51917#51917 Answer by Gerry Myerson for What is the largest family F of subsets of [n] for which any two distinct sets A and B in F have an intersection of size at most min(|A|,|B|)/2? Gerry Myerson 2011-01-13T05:33:42Z 2011-01-13T05:33:42Z <p>This can be rephrased in the language of coding theory. If you have a binary code with minimal distance $d$ then the words of weight $2d$ form the kind of family you've defined. Tables of good binary codes are widely available on the web. </p> | 2013-05-25 20:04:29 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8456901907920837, "perplexity": 321.0916180547757}, "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-2013-20/segments/1368706153698/warc/CC-MAIN-20130516120913-00008-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.overleaf.com/latex/examples/example-gnuplot-plus-tikz/qhrvjqtctxxw | # Example: gnuplot + tikz
Author
Gudrun, LianTze Lim
AbstractThis is an example for re-creating gnuplot charts with tikz on LaTeX, made possible by adding gnuplot-lua-tikz.sty and gnuplot-lua-tikz-common.tex to your project. (These files can be generated by invoking lua gnuplot-tikz.lua style where gnuplot-tikz.lua can be found in \$GNUPLOT/lua/gnuplot-tikz.lua. (This will work with all engines, not just LuaLaTeX!) | 2021-03-04 15:51:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "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": 0.9750893115997314, "perplexity": 13582.758406891664}, "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/1614178369420.71/warc/CC-MAIN-20210304143817-20210304173817-00243.warc.gz"} |
https://questioncove.com/updates/4d534546d04cb76416ef8481 | Ask your own question, for FREE!
Mathematics
OpenStudy (salt):
how do you seperate ds/dr=ks This is in diffy q.
OpenStudy (anonymous):
if k is constant then it would be $\int\limits_{}^{}(1/s) ds = \int\limits_{}^{}k dr$
OpenStudy (salt):
ty I got the answer =) I cant believe i didnt know the integral of kdr i had thought of what you said but it wasnt clicking thx.
OpenStudy (salt):
nvm i thought i got it. I didnt the answer is 2y^6+cy^4 weird
OpenStudy (salt):
ok im sorry to be a pain but i got it. I looked in the wrong place.
Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!
Join our real-time social learning platform and learn together with your friends!
Latest Questions
TETSXPREME: help
12 minutes ago 1 Reply 0 Medals
Noodlearms: Solve for x.
50 minutes ago 14 Replies 0 Medals
TogaHimiko: Evaluate: bx(b u2013 X) u2013 b if b=5 and x = 3
53 minutes ago 17 Replies 2 Medals
TETSXPREME: help
19 minutes ago 21 Replies 2 Medals
Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!
Join our real-time social learning platform and learn together with your friends! | 2021-10-25 20:30:13 | {"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": 0.263417512178421, "perplexity": 5940.651331877509}, "config": {"markdown_headings": false, "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-2021-43/segments/1634323587767.18/warc/CC-MAIN-20211025185311-20211025215311-00426.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/i-ve-tried-solve-question-i-thinkit-right-i-don-t-know-handle-second--iuploaded-question-a-q740867 | ## Calculating the Voltage Gain
I've tried to solve the first part of the question and I thinkit is right but I don't know how to handle the second part. Iuploaded the question and the answer to the websites below. I wouldappreciate if this could be answered. Thank you for yourtime.
http://tinypic.com/r/5yvozc/6
http://tinypic.com/r/rs7lgw/6 | 2013-06-20 03:15:39 | {"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": 0.8543843626976013, "perplexity": 350.19221566950495}, "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-2013-20/segments/1368710196013/warc/CC-MAIN-20130516131636-00004-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://paperswithcode.com/paper/multi-scale-zero-order-optimization-of-smooth | # Multi-Scale Zero-Order Optimization of Smooth Functions in an RKHS
11 May 2020Shubhanshu ShekharTara Javidi
We aim to optimize a black-box function $f:\mathcal{X} \mapsto \mathbb{R}$ under the assumption that $f$ is H\"older smooth and has bounded norm in the RKHS associated with a given kernel $K$. This problem is known to have an agnostic Gaussian Process (GP) bandit interpretation in which an appropriately constructed GP surrogate model with kernel $K$ is used to obtain an upper confidence bound (UCB) algorithm... (read more)
PDF Abstract | 2020-07-12 03:07:48 | {"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": 0.8991861343383789, "perplexity": 458.07375930702267}, "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-2020-29/segments/1593657129517.82/warc/CC-MAIN-20200712015556-20200712045556-00100.warc.gz"} |
http://imi.cas.sc.edu/papers/21/ | IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences
Preprint Series by Year
## Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions
A 2008 Preprint by G. Kerkyacharian, P. Petrushev, D. Picard, and Y. Xu
• 2008:02
• A pair of dual frames with almost exponentially localized elements (needlets) are constructed on $R^d _ +$ based on Laguerre functions. It is shown that the Triebel-Lizorkin and Besov spaces induced by Laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients.
© Interdisciplinary Mathematics Institute | The University of South Carolina Board of Trustees | Webmaster | 2017-11-18 14:01:17 | {"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": 0.3974679112434387, "perplexity": 2940.8860873758663}, "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/1510934804965.9/warc/CC-MAIN-20171118132741-20171118152741-00140.warc.gz"} |
https://www.techwhiff.com/issue/how-does-federalism-work-to-serve-the-needs-of-a-diverse--677180 | # How does federalism work to serve the needs of a diverse citizenry
1 answer
###### Question:
how does federalism work to serve the needs of a diverse citizenry
## Answers
1 answer
### Which of the following sets shows all the numbers from the set {0.5,1,2.5,3,3.5} that make the inequality 4a + 2 > 12 true
Which of the following sets shows all the numbers from the set {0.5,1,2.5,3,3.5} that make the inequality 4a + 2 > 12 true...
2 answers
### Which of the following best describes the work schedule of enslaved African Americans on plantations? Select the best answer from the choices provided. A. They worked only every other day because their enslavers wanted to preserve their health. B. They worked only during the cool morning hours and spent the afternoons doing domestic tasks. C. They worked all day, every day, except on Sundays. D. They could negotiate their schedules with their overseer.
Which of the following best describes the work schedule of enslaved African Americans on plantations? Select the best answer from the choices provided. A. They worked only every other day because their enslavers wanted to preserve their health. B. They worked only during the cool morning hours and...
1 answer
### Why did many non-Muslims convert to Islam?
why did many non-Muslims convert to Islam?...
1 answer
### What is the vapor pressure (in mm Hg) of a solution of 17.5 g of glucose (C6H12O6) in 82.0 g of methanol (CH3OH) at 27∘C? The vapor pressure of pure methanol at 27∘C is 1.40×102 mm H
What is the vapor pressure (in mm Hg) of a solution of 17.5 g of glucose (C6H12O6) in 82.0 g of methanol (CH3OH) at 27∘C? The vapor pressure of pure methanol at 27∘C is 1.40×102 mm H...
1 answer
### How can Natural selection account for the long snout of an anteater?
How can Natural selection account for the long snout of an anteater?...
2 answers
### A: Half of a 16 inch pizza or pizza B: A whole 10 inch pizza to himself. Which pizza would ensure him the largest amount?
A: Half of a 16 inch pizza or pizza B: A whole 10 inch pizza to himself. Which pizza would ensure him the largest amount?...
2 answers
### What happens if a lion and a tiger reproduction
What happens if a lion and a tiger reproduction...
1 answer
### I done the multiple choice I just need help with the writing question and can you check the multiple questions if I got it right please! ANY ABSURD ANSWERS WILL BE REPORTED! RIGHT ANSWERS WILL ME MARKED BRAINLEST!
I done the multiple choice I just need help with the writing question and can you check the multiple questions if I got it right please! ANY ABSURD ANSWERS WILL BE REPORTED! RIGHT ANSWERS WILL ME MARKED BRAINLEST!...
1 answer
### Chaz loves to play a Disney online pirate game in which he gets to create a pirate by choosing hair color, skin color, clothing, and physical features. Then he gets to choose from a variety of quests and can sail a ship, dig for treasure, fight skeleton pirates, or fight the British Navy. This game provides what element of value?a. Co-creationb. Earning of trustc. Offering products that performd. Avoiding unrealistic pricinge. Giving facts
Chaz loves to play a Disney online pirate game in which he gets to create a pirate by choosing hair color, skin color, clothing, and physical features. Then he gets to choose from a variety of quests and can sail a ship, dig for treasure, fight skeleton pirates, or fight the British Navy. This game ...
1 answer
### Abe is a left-handed bowler. he has watched willie bowl right-handed and score a strike by hitting the 1-3 pocket. abe hits the 1-3 pocket with his straight ball, but can't get a strike. he decides to change his target to the
Abe is a left-handed bowler. he has watched willie bowl right-handed and score a strike by hitting the 1-3 pocket. abe hits the 1-3 pocket with his straight ball, but can't get a strike. he decides to change his target to the...
1 answer
### 2 . What is the resulting combination when h(x) is subtracted from klx) if h(x)= -X+6 and k(x)=2x-5 ? 1
2 . What is the resulting combination when h(x) is subtracted from klx) if h(x)= -X+6 and k(x)=2x-5 ? 1...
1 answer
### 31. Which category of carbon-based molecules includes sugars and starches? A. unsaturated fatty acids B. phospholipids C. proteins D. carbohydrates
31. Which category of carbon-based molecules includes sugars and starches? A. unsaturated fatty acids B. phospholipids C. proteins D. carbohydrates...
2 answers
### Step by step how to do 5.973 divided by 11
Step by step how to do 5.973 divided by 11...
1 answer
### The most direct way for Jonathon to gain on the job experience and earn money while attending school is to apply for
the most direct way for Jonathon to gain on the job experience and earn money while attending school is to apply for...
1 answer
### In what direction does erosion always occur? a. Downhill b. Uphill C. South d. West
In what direction does erosion always occur? a. Downhill b. Uphill C. South d. West...
1 answer
### Stock A has a beta of 0.8, stock B has a beta of 1.0, and Stock C has a beta of 1.2. Portfolio have 1/3 of its value invested in each of these stocks. Each stock has a standard deviation of 25%, and their returns are independent of one another, ie., the correlation coefficients between each pair of stock is zero. Assuming the market is in equilibrium, which of the following statements is correct?a. Portfolio P's expected return is greater than the expected return on Stock C.b. Portfolio P's expe
Stock A has a beta of 0.8, stock B has a beta of 1.0, and Stock C has a beta of 1.2. Portfolio have 1/3 of its value invested in each of these stocks. Each stock has a standard deviation of 25%, and their returns are independent of one another, ie., the correlation coefficients between each pair of ...
1 answer
### Three processes that occur in cells are described in the box these processes are an example of
Three processes that occur in cells are described in the box these processes are an example of...
2 answers
1 answer
### Understanding Part-and-Whole Analogies
Understanding Part-and-Whole Analogies...
2 answers
### Why is the united states considered a democracy?
why is the united states considered a democracy?...
-- 0.053172-- | 2022-12-03 00:09:41 | {"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": 0.19346576929092407, "perplexity": 4441.744186157174}, "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-2022-49/segments/1669446710916.70/warc/CC-MAIN-20221202215443-20221203005443-00136.warc.gz"} |
https://math.stackexchange.com/questions/1525862/how-to-negate-the-following-proposition | # How to negate the following proposition
If I have the the proposition: $\forall y, \exists x, \exists z [(Bx,y \wedge Rz,y) \vee (Bx,y \wedge Gz,y) \vee (Rx,y \wedge Gz,y)]$.
(B,R and G are some other propositions but that doesn't matter now)
Now I want to negate that but I'm not sure if I did it right:
$\exists y \forall x \forall z [\neg Bx,y \vee \neg Rz,y \vee \neg Bx,y \vee \neg Gz,y \vee \neg Rx,y \vee \neg Gz,y]$
$=\exists y \forall x \forall z [\neg Bx,y \vee \neg Rz,y \vee \neg Gz,y \vee \neg Rx,y]$
I'm not sure but I think I did something wrong.. Hope you can help.
• $\exists$ becomes $\forall$ and vice-versa, as you did. Also $\lnot (a\vee b)=(\lnot a)\wedge (\lnot b)$ and $\lnot (a\wedge b)=(\lnot a)\vee (\lnot b)$. You forgot to reverse the $\vee$s, it seems. Nov 12, 2015 at 15:11
Let's walk through this, making each step more explicit:
$\neg\forall y\exists x\exists z[(Bxy\wedge Rzy)\vee(Bxy\wedge Gzy)\vee(Rxy\wedge Gzy)]$
When you have a negated quantifier, you may remove the negation, change the quantifier to the other type of quantifier, and negate the formula quantified over:
$=\exists y\neg\exists x\exists z[(Bxy\wedge Rzy)\vee(Bxy\wedge Gzy)\vee(Rxy\wedge Gzy)]$ $=\exists y\forall x\neg\exists z[(Bxy\wedge Rzy)\vee(Bxy\wedge Gzy)\vee(Rxy\wedge Gzy)]$ $=\exists y\forall x\forall z\neg[(Bxy\wedge Rzy)\vee(Bxy\wedge Gzy)\vee(Rxy\wedge Gzy)]$
Now, when you have a negated disjunction, you may apply DeMorgan's rule. This amounts to switching every disjunction to a conjunction, and negating each new conjunct:
$=\exists y\forall x\forall z[\neg(Bxy\wedge Rzy)\wedge\neg(Bxy\wedge Gzy)\wedge\neg(Rxy\wedge Gzy)]$
Now, apply DeMorgan's again (three times). This time, we switch the conjunctions to disjunctions and negate each new disjunct:
$=\exists y\forall x\forall z[(\neg Bxy\vee\neg Rzy)\wedge(\neg Bxy\vee\neg Gzy)\wedge(\neg Rxy\vee\neg Gzy)]$
Now that negations are only applied to predicates, we can't simplify any more. We are done. I don't think the formula you got at the end is equivalent to the one I got. This is because:
$A\vee B\vee C\vee D\neq(A\vee B)\wedge(A\vee C)\wedge(D\vee C)$
To show why this inequality holds, consider the truth assignment: $A\mapsto\top$, $B\mapsto\top$, $C\mapsto\bot$, $D\mapsto\bot$. It is easy to see that this assignment satisfies the formula on the left, but not the one on the right. I think that
$=\exists y\forall x\forall z[(\neg Bxy\vee\neg Rzy)\wedge(\neg Bxy\vee\neg Gzy)\wedge(\neg Rxy\vee\neg Gzy)]$
is as far as you can simplify this formula. You can see that the conjunction inside the square brackets is the same form as the conjunction $(A\vee B)\wedge(A\vee C)\wedge(D\vee C)$.
Hope this helps! | 2022-05-18 23:10:48 | {"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": 0.9125219583511353, "perplexity": 339.3234990881207}, "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/1652662522556.18/warc/CC-MAIN-20220518215138-20220519005138-00221.warc.gz"} |
http://cs.ucsb.edu/~cmsweeney/theia/sfm.html | # Structure from Motion (SfM)¶
Theia has a full Structure-from-Motion pipeline that is extremely efficient. Our overall pipeline consists of several steps. First, we extract features (SIFT is the default). Then, we perform two-view matching and geometric verification to obtain relative poses between image pairs and create a ViewGraph. Next, we perform global pose estimation with global SfM. Global SfM is different from incremental SfM in that it considers the entire view graph at the same time instead of incrementally adding more and more images to the Reconstruction. Global SfM methods have been proven to be very fast with comparable or better accuracy to incremental SfM approaches (See [JiangICCV], [MoulonICCV], [WilsonECCV2014]), and they are much more readily parallelized. After we have obtained camera poses, we perform triangulation and BundleAdjustment to obtain a valid 3D reconstruction consisting of cameras and 3D points.
The first step towards creating a reconstruction is to determine images which view the same objects. To do this, we must create a ViewGraph.
1. Extract features in images.
2. Match features to obtain image correspondences.
3. Estimate camera poses from two-view matches and geometries.
#1. and #2. have been covered in other sections, so we will focus on creating a reconstruction from two-view matches and geometry. First, we will describe the fundamental elements of our reconstruction.
## Reconstruction¶
class Reconstruction
At the core of our SfM pipeline is an SfM Reconstruction. A Reconstruction is the representation of a 3D reconstuction consisting of a set of unique Views and Tracks. More importantly, the Reconstruction class contains visibility information relating all of the Views and Tracks to each other. We identify each View uniquely based on the name (a string). A good name for the view is the filename of the image that corresponds to the View
When creating an SfM reconstruction, you should add each View and Track through the Reconstruction object. This will ensure that visibility information (such as which Tracks are observed a given View and which Views see a given Track) stays accurate. Views and Tracks are given a unique ID when added to the Reconstruction to help make use of these structures lightweight and efficient.
ViewId Reconstruction::AddView(const std::string &view_name)
Adds a view to the reconstruction with the default initialization. The ViewId returned is guaranteed to be unique or will be kInvalidViewId if the method fails. Each view is uniquely identified by the view name (a good view name could be the filename of the image).
bool Reconstruction::RemoveView(const ViewId view_id)
Removes the view from the reconstruction and removes all references to the view in the tracks.
Note
Any tracks that have length 0 after the view is removed will also be removed.
int Reconstruction::NumViews() const
int Reconstruction::NumTracks() const
const View *Reconstruction::View(const ViewId view_id) const
View *Reconstruction::MutableView(const ViewId view_id)
Returns the View or a nullptr if the track does not exist.
std::vector<ViewId> Reconstruction::ViewIds() const
Return all ViewIds in the reconstruction.
ViewId Reconstruction::ViewIdFromName(const std::string &view_name) const
Returns to ViewId of the view name, or kInvalidViewId if the view does not exist.
TrackId Reconstruction::AddTrack(const std::vector<std::pair<ViewId, Feature>> &track)
Add a track to the reconstruction with all of its features across views that observe this track. Each pair contains a feature and the corresponding View name (i.e., the string) that observes the feature. A new View will be created if a View with the view name does not already exist. This method will not estimate the position of the track. The TrackId returned will be unique or will be kInvalidTrackId if the method fails.
bool Reconstruction::RemoveTrack(const TrackId track_id)
Removes the track from the reconstruction and from any Views that observe this track. Returns true on success and false on failure (e.g., the track does not exist).
const Track *Reconstruction::Track(const TrackId track_id) const
Track *Reconstruction::MutableTrack(const TrackId track_id)
Returns the Track or a nullptr if the track does not exist.
std::vector<TrackId> Reconstruction::TrackIds() const
Return all TrackIds in the reconstruction.
## ViewGraph¶
class ViewGraph
A ViewGraph is a basic SfM construct that is created from two-view matching information. Any pair of views that have a view correlation form an edge in the ViewGraph such that the nodes in the graph are View that are connected by TwoViewInfo objects that contain information about the relative pose between the Views as well as matching information.
Once you have a set of views and match information, you can add them to the view graph:
std::vector<View> views;
// Match all views in the set.
std::vector<ViewIdPair, TwoViewInfo> view_pair_matches;
ViewGraph view_graph;
for (const auto& view_pair : view_pair_matches) {
const ViewIdPair& view_id_pair = view_pair.first;
const TwoViewInfo& two_view_info = view_pair.second;
// Only add view pairs to the view graph if they have strong visual coherence.
if (two_view_info.num_matched_features > min_num_matched_features) {
views[view_id_pair.second],
two_view_info);
}
}
// Process and/or manipulate the view graph.
The edge values are especially useful for one-shot SfM where the relative poses are heavily exploited for computing the final poses. Without a proper ViewGraph, one-shot SfM would not be possible.
## Views and Tracks¶
class View
At the heart of our SfM framework is the View class which represents everything about an image that we want to reconstruct. It contains information about features from the image, camera pose information, and EXIF information. Views make up our basic visiblity constraints and are a fundamental part of the SfM pipeline.
class Track
A Track represents a feature that has been matached over potentially many images. When a feature appears in multiple images it typically means that the features correspond to the same 3D point. These 3D points are useful constraints in SfM reconstruction, as they represent the “structure” in “Structure-from-Motion” and help to build a point cloud for our reconstruction.
## TwoViewInfo¶
class TwoViewInfo
After image matching is performed we can create a ViewGraph that explains the relative pose information between two images that have been matched. The TwoViewInfo struct is specified as:
struct TwoViewInfo {
double focal_length_1;
double focal_length_2;
Eigen::Vector3d position_2;
Eigen::Vector3d rotation_2;
// Number of features that were matched and geometrically verified betwen the
// images.
int num_verified_matches;
};
This information serves the purpose of an edge in the view graph that describes visibility information between all views. The relative poses here are used to estimate global poses for the cameras.
## Camera¶
class Camera
Each View contains a Camera object that contains intrinsic and extrinsic information about the camera that observed the scene. Theia has an efficient, compact Camera class that abstracts away common image operations. This greatly relieves the pain of manually dealing with calibration and geometric transformations of images. We represent camera intrinsics such that the calibration matrix is:
$\begin{split}K = \left[\begin{matrix}f & s & p_x \\ 0 & f * a & p_y \\ 0 & 0 & 1 \end{matrix} \right]\end{split}$
where $$f$$ is the focal length (in pixels), $$s$$ is the skew, $$a$$ is the aspect ratio and $$p$$ is the principle point of the camera. All of these intrinsics may be accessed with getter and setter methods, e.g., double GetFocalLenth() or void SetFocalLength(const double focal_length). Note that we do additionally allow for up to two radial distortion parameters, but these are not part of the calibration matrix so they must be set or retrieved separately from the corresponding getter/setter methods.
We store the camera pose information as the transformation which maps world coordinates into camera coordinates. Our rotation is stored internally as an SO(3) rotation, which makes optimization with BundleAdjustment more effective since the value is always a valid rotation (unlike e.g., Quaternions that must be normalized after each optimization step). However, for convenience we provide an interface to retrieve the rotation as a rotation matrix as well. Further, we store the camera position as opposed to the translation.
The convenience of this camera class is clear with the common example of 3D point reprojection.
// Open an image and obtain camera parameters.
FloatImage image("my_image.jpg");
double focal_length;
CHECK(image.FocalLengthPixels(&focal_length));
const double radial_distortion1 = value obtained elsewhere...
const double radial_distortion2 = value obtained elsewhere...
const Eigen::Matrix3d rotation = value obtained elsewhere...
const Eigen::Vector3d position = value obtained elsewhere...
// Set up the camera.
Camera camera;
camera.SetOrientationFromRotationMatrix(rotation);
camera.SetPosition(position);
camera.SetFocalLength(focal_length);
camera.SetPrincipalPoint(image.Width() / 2.0, image.Height() / 2.0);
// Obtain a homogeneous 3D point
const Eigen::Vector4d homogeneous_point3d = value obtained elsewhere...
// Reproject the 3D point to a pixel.
Eigen::Vector2d reprojection_pixel;
const double depth = camera.ProjectPoint(homogeneous_point3d, &pixel);
if (depth < 0) {
LOG(INFO) << "Point was behind the camera!";
}
LOG(INFO) << "Homogeneous 3D point: " << homogeneous_point3d
<< " reprojected to the pixel value of " << reprojection_pixel;
Point projection can be a tricky function when considering the camera intrinsics and extrinsics. Theia takes care of this projection (including radial distortion) in a simple and efficient manner.
In addition to typical getter/setter methods for the camera parameters, the Camera class also defines several helper functions:.
bool Camera::InitializeFromProjectionMatrix(const int image_width, const int image_height, const Matrix3x4d projection_matrix)
Initializes the camera intrinsic and extrinsic parameters from the projection matrix by decomposing the matrix with a RQ decomposition.
Note
The projection matrix does not contain information about radial distortion, so those parameters will need to be set separately.
void Camera::GetProjectionMatrix(Matrix3x4d *pmatrix) const
Returns the projection matrix. Does not include radial distortion.
void Camera::GetCalibrationMatrix(Eigen::Matrix3d *kmatrix) const
Returns the calibration matrix in the form specified above.
Eigen::Vector3d Camera::PixelToUnitDepthRay(const Eigen::Vector2d &pixel) const
Converts the pixel point to a ray in 3D space such that the origin of the ray is at the camera center and the direction is the pixel direction rotated according to the camera orientation in 3D space. The returned vector is not unit length.
## Global SfM Pipeline¶
The global SfM pipelines in Theia follow a general procedure of filtering outliers and estimating camera poses or structure. Removing outliers can help increase performance dramatically for global SfM, though robust estimation methods are still required to obtain good results. The general pipeline is as follows:
1. Create the intial view graph from 2-view matches and TwoViewInfo that describes the relative pose between matched images.
2. Filter initial view graph and remove outlier 2-view matches.
3. Calibrate internal parameters of all cameras (either from EXIF or another calibration method).
4. Estimate global orientations of each camera.
5. Filter the view graph: remove any TwoViewInfos where the relative rotation does not agree with the estimated global rotations.
6. Refine the relative translation estimation to account for the estimated global rotations.
7. Filter any bad TwoViewInfo based on the relative translations.
8. Estimate the global positions of all cameras from the estimated rotations and TwoViewInfo.
9. Estimate 3D points.
11. (Optional) Attempt to estimate any remaining 3D points and bundle adjust again.
class ReconstructionEstimator
This is the base class for which all SfM reconstruction pipelines derive from. The reconstruction estimation type can be specified at runtime, though currently only NONLINEAR is implemented.
ReconstructionEstimator::ReconstructionEstimator(const ReconstructorEstimatorOptions &options)
ReconstructionEstimator::ReconstructionEstimatorSummary Estimate(const ViewGraph &view_graph, Reconstruction *reconstruction)
Estimates the cameras poses and 3D points from a view graph. The details of each step in the estimation process are described below.
class ReconstructorEstimatorOptions
ReconstructionEstimatorType ReconstructorEstimatorOptions::reconstruction_estimator_type
DEFAULT: ReconstructionEstimatorType::NONLINEAR
Type of reconstruction estimation to use.
int ReconstructorEstimatorOptions::num_threads
DEFAULT: 1
Number of threads to use during the various stages of reconstruction.
double ReconstructorEstimatorOptions::max_reprojection_error_in_pixels
DEFAULT: 5.0
Maximum reprojection error. This is used to determine inlier correspondences for absolute pose estimation. Additionally, this is the threshold used for filtering outliers after bundle adjustment.
int ReconstructorEstimatorOptions::num_retriangulation_iterations
DEFAULT: 1
After computing a model and performing an initial BA, the reconstruction can be further improved (and even densified) if we attempt to retriangulate any tracks that are currently unestimated. For each retriangulation iteration we do the following:
1. Remove features that are above max_reprojection_error_in_pixels.
2. Triangulate all unestimated tracks.
bool ReconstructorEstimatorOptions::initialize_focal_lengths_from_median_estimate
DEFAULT: false
By default, focal lengths for uncalibrated cameras are initialized by setting the focal length to a value that corresponds to a reasonable field of view. If this is true, then we initialize the focal length of all uncalibrated cameras to the median value obtained from decomposing the fundamental matrix of all view pairs connected to that camera. Cameras with calibration or EXIF information are always calibrated using that information regardless of this parameter.
double ReconstructorEstimatorOptions::ransac_confidence
DEFAULT: 0.9999
Confidence using during RANSAC. This determines the quality and termination speed of RANSAC.
int ReconstructorEstimatorOptions::ransac_min_iterations
DEFAULT: 50
Minimum number of iterations for RANSAC.
int ReconstructorEstimatorOptions::ransac_max_iterations
DEFAULT: 1000
Maximum number of iterations for RANSAC.
bool ReconstructorEstimatorOptions::ransac_use_mle
DEFAULT: true
Using the MLE quality assesment (as opposed to simply an inlier count) can improve the quality of a RANSAC estimation with virtually no computational cost.
double ReconstructorEstimatorOptions::max_rotation_error_in_view_graph_cycles
DEFAULT: 3.0
Before orientations are estimated, some “bad” edges may be removed from the view graph by determining the consistency of rotation estimations in loops within the view graph. By examining loops of size 3 (i.e., triplets) the concatenated relative rotations should result in a perfect identity rotation. Any edges that break this consistency may be removed prior to rotation estimation.
double ReconstructorEstimatorOptions::rotation_filtering_max_difference_degrees
DEFAULT: 5.0
After orientations are estimated, view pairs may be filtered/removed if the relative rotation of the view pair differs from the relative rotation formed by the global orientation estimations. That is, measure the angulaar distance between $$R_{i,j}$$ and $$R_j * R_i^T$$ and if it greater than rotation_filtering_max_difference_degrees than we remove that view pair from the graph. Adjust this threshold to control the threshold at which rotations are filtered.
bool ReconstructorEstimatorOptions::refine_relative_translations_after_rotation_estimation
DEFAULT: true
Refine the relative translations based on the epipolar error and known rotation estimations. This can improve the quality of the translation estimation.
int ReconstructorEstimatorOptions::translation_filtering_num_iterations
DEFAULT: 48
double ReconstructorEstimatorOptions::translation_filtering_projection_tolerance
DEFAULT: 0.1
Before the camera positions are estimated, it is wise to remove any relative translations estimates that are low quality. We perform filtering using the 1dSfM technique of [WilsonECCV2014]. See theia/sfm/filter_view_pairs_from_relative_translation.h for more information.
double ReconstructorEstimatorOptions::rotation_estimation_robust_loss_scale
DEFAULT: 0.1
Robust loss function width for nonlinear rotation estimation.
double ReconstructorEstimatorOptions::position_estimation_robust_loss_scale
DEFAULT: 1.0
Robust loss function width to use for the first iteration of nonlinear position estimation.
int ReconstructorEstimatorOptions::position_estimation_min_num_tracks_per_view
DEFAULT: 10
Number of point to camera correspondences used for nonlinear position estimation.
double ReconstructorEstimatorOptions::position_estimation_point_to_camera_weight
DEFAULT: 0.5
Weight of point to camera constraints with respect to camera to camera constraints.
double ReconstructorEstimatorOptions::min_triangulation_angle_degrees
DEFAULT: 3.0
In order to triangulate features accurately, there must be a sufficient baseline between the cameras relative to the depth of the point. Points with a very high depth and small baseline are very inaccurate. We require that at least one pair of cameras has a sufficient viewing angle to the estimated track in order to consider the estimation successful.
bool ReconstructorEstimatorOptions::bundle_adjust_tracks
DEFAULT: true
Bundle adjust a track immediately after estimating it.
double ReconstructorEstimatorOptions::triangulation_max_reprojection_error_in_pixels
DEFAULT: 10.0
The reprojection error to use for determining a valid triangulation. If the reprojection error of any observation is greater than this value then we can consider the triangluation unsuccessful.
int ReconstructorEstimatorOptions::min_cameras_for_iterative_solver
DEFAULT: 1000
Use SPARSE_SCHUR for problems smaller than this size and ITERATIVE_SCHUR for problems larger than this size.
bool ReconstructorEstimatorOptions::constant_camera_intrinsics
DEFAULT: false
If accurate calibration is known ahead of time then it is recommended to set the camera intrinsics constant during bundle adjustment.
## Estimating Global Rotations¶
Theia estimates the global rotations of cameras robustly using a nonlinear optimization. Using the relative rotations obtained from all TwoViewInfo, we enforce the constraint that
(1)$R_{i,j} = R_j * R_i^T$
We use the angle-axis representation of rotations to ensure that proper rotations are formed. All pairwise constraints are put into a nonlinear optimization with a robust loss function and the global orienations are computed. The optimization usually converges within just a few iterations and provides a very accurate result. The nonlinear optimization is initialized by forming a random spanning tree of the view graph and walking along the edges. There are two potential methods that may be used.
bool EstimateRotationsNonlinear(const std::unordered_map<ViewIdPair, Eigen::Vector3d> &relative_rotations, const double robust_loss_width, std::unordered_map<ViewId, Eigen::Vector3d> *global_orientations)
Using the relative rotations and an initial guess for the global rotations, minimize the error between the relative rotations and the global orientations. We use as SoftL1Loss for robustness to outliers.
class RobustRotationEstimator
The nonlinear method above is very sensitive to initialization. We recommend to use the RobustRotationEstimator of [ChatterjeeICCV13]. This rotation estimator is similar in spirit to EstimateRotationsNonlinear(), however, it utilizes L1 minimization to maintain efficiency to outliers. After several iterations of L1 minimization, an iteratively reweighted least squares approach is used to refine the solution.
int RobustRotationEstimator::Options::max_num_l1_iterations
DEFAULT: 5
Maximum number of L1 iterations to perform before performing the reweighted least squares minimization. Typically only a very small number of L1 iterations are needed.
int RobustRotationEstimator::Options::max_num_irls_iterations
DEFAULT: 100
Maximum number of reweighted least squares iterations to perform. These steps are much faster than the L2 iterations.
RobustRotationEstimator::RobustRotationEstimator(const RobustRotationEstimator::Options &options, const std::unordered_map<ViewIdPair, Eigen::Vector3d> &relative_rotations)
The options and the relative rotations remain constant throughout the rotation estimation and are passed in the constructor.
bool RobustRotationEstimator::EstimateRotations(std::unordered_map<ViewId, Eigen::Vector3d> *global_orientations)
Estimates the global orientation using the robust method described above. An initial estimation for the rotations is required. [ChatterjeeICCV13] suggests to use a random spanning tree to initialize the rotations.
## Estimating Global Positions¶
Positions of cameras may be estimated simultaneously after the rotations are known. We use a nonlinear optimization to estimate camera positions based. Given pairwise relative translations from TwoViewInfo and the estimated rotation, the constraint
$R_i * (c_j - c_i) = \alpha_{i,j} * t_{i,j}$
Where $$\alpha_{i,j} = ||c_j - c_i||^2$$. This ensures that we optimize for positions that agree with the relative positions computed in two-view estimation.
class NonlinearPositionEstimatorOptions
int NonlinearPositionEstimatorOptions::num_threads
DEFAULT: 1
Number of threads to use with Ceres for nonlinear optimization.
bool NonlinearPositionEstimatorOptions::verbose
DEFAULT: false
Set to true for verbose logging.
int NonlinearPositionEstimatorOptions::max_num_iterations
DEFAULT: 400
The maximum number of iterations for each minimization (i.e., for a single IRLS iteration).
double NonlinearPositionEstimatorOptions::robust_loss_width
DEFAULT: 1.0
The width of the robust Huber loss function used in the first minimization iteration.
int NonlinearPositionEstimatorOptions::min_num_points_per_view
DEFAULT: 0
The number of 3D point-to-camera constraints to use for position recovery. Using points-to-camera constraints can sometimes improve robustness to collinear scenes. Points are taken from tracks in the reconstruction such that the minimum number of points is used such that each view has at least min_num_points_per_view point-to-camera constraints.
double NonlinearPositionEstimatorOptions::point_to_camera_weight
DEFAULT: 0.5
Each point-to-camera constraint (if any) is weighted by point_to_camera_weight compared to the camera-to-camera weights.
class NonlinearPositionEstimator
NonlinearPositionEstimator(const NonlinearPositionEstimatorOptions &options, const Reconstruction &reconstruction, const std::unordered_map<ViewIdPair, TwoViewInfo> &view_pairs)
The constructor takes the options for the nonlinear position estimator, as well as const references to the reconstruction (which contains camera and track information) and the two view geometry information that will be use to recover the positions.
bool EstimatePositions(const std::unordered_map<ViewId, Eigen::Vector3d> &orientation, std::unordered_map<ViewId, Eigen::Vector3d> *positions)
Estimates the positions of cameras given the global orientation estimates by using the nonlinear algorithm described above. Only positions that have an orientation set are estimated. Returns true upons success and false on failure.
## Triangulation¶
Triangulation in structure from motion calculates the 3D position of an image coordinate that has been tracked through two or more images. All cameras with an estimated camera pose are used to estimate the 3D point of a track.
class EstimateTrackOptions
bool EstimateTrackOptions::bundle_adjustment
DEFAULT: true
Bundle adjust the track (holding all camera parameters constant) after initial estimation. This is highly recommended in order to obtain good 3D point estimations.
double EstimateTrackOptions::max_acceptable_reprojection_error_pixels
DEFAULT: 5.0
Track estimation is only considered successful if the reprojection error for all observations is less than this value.
double EstimateTrackOptions::min_triangulation_angle_degrees
DEFAULT: 3.0
In order to triangulate features accurately, there must be a sufficient baseline between the cameras relative to the depth of the point. Points with a very high depth and small baseline are very inaccurate. We require that at least one pair of cameras has a sufficient viewing angle to the estimated track in order to consider the estimation successful.
bool EstimateAllTracks(const EstimateTrackOptions &options, const int num_threads, Reconstruction *reconstruction)
Performs (potentially multithreaded) track estimation. Track estimation is embarassingly parallel so multithreading is recommended.
bool Triangulate(const Matrix3x4d &pose1, const Matrix3x4d &pose2, const Eigen::Vector2d &point1, const Eigen::Vector2d &point2, Eigen::Vector4d *triangulated_point)
2-view triangulation using the method described in [Lindstrom]. This method is optimal in an L2 sense such that the reprojection errors are minimized while enforcing the epipolar constraint between the two cameras. Additionally, it basically the same speed as the TriangulateDLT() method.
The poses are the (potentially calibrated) poses of the two cameras, and the points are the 2D image points of the matched features that will be used to triangulate the 3D point. On successful triangulation, true is returned. The homogeneous 3d point is output so that it may be known if the point is at infinity.
bool TriangulateDLT(const Matrix3x4d &pose1, const Matrix3x4d &pose2, const Eigen::Vector2d &point1, const Eigen::Vector2d &point2, Eigen::Vector4d *triangulated_point)
The DLT triangulation method of [HartleyZisserman].
bool TriangulateMidpoint(const Eigen::Vector3d &origin1, const Eigen::Vector3d &ray_direction1, const Eigen::Vector3d &origin2, const Eigen::Vector3d &ray_direction2, Eigen::Vector4d *triangulated_point)
Perform triangulation by determining the closest point between the two rays. In this case, the ray origins are the camera positions and the directions are the (unit-norm) ray directions of the features in 3D space. This method is known to be suboptimal at minimizing the reprojection error, but is approximately 10x faster than the other 2-view triangulation methods.
bool TriangulateNViewSVD(const std::vector<Matrix3x4d> &poses, const std::vector<Eigen::Vector2d> &points, Eigen::Vector3d *triangulated_point)
bool TriangulateNView(const std::vector<Matrix3x4d> &poses, const std::vector<Eigen::Vector2d> &points, Eigen::Vector3d *triangulated_point)
We provide two N-view triangluation methods that minimizes an algebraic approximation of the geometric error. The first is the classic SVD method presented in [HartleyZisserman]. The second is a custom algebraic minimization. Note that we can derive an algebraic constraint where we note that the unit ray of an image observation can be stretched by depth $$\alpha$$ to meet the world point $$X$$ for each of the $$n$$ observations:
$\alpha_i \bar{x_i} = P_i X,$
for images $$i=1,\ldots,n$$. This equation can be effectively rewritten as:
$\alpha_i = \bar{x_i}^\top P_i X,$
which can be substituted into our original constraint such that:
$\bar{x_i} \bar{x_i}^\top P_i X = P_i X$
$0 = (P_i - \bar{x_i} \bar{x_i}^\top P_i) X$
We can then stack this constraint for each observation, leading to the linear least squares problem:
$\begin{split}\begin{bmatrix} (P_1 - \bar{x_1} \bar{x_1}^\top P_1) \\ \vdots \\ (P_n - \bar{x_n} \bar{x_n}^\top P_n) \end{bmatrix} X = \textbf{0}\end{split}$
This system of equations is of the form $$AX=0$$ which can be solved by extracting the right nullspace of $$A$$. The right nullspace of $$A$$ can be extracted efficiently by noting that it is equivalent to the nullspace of $$A^\top A$$, which is a 4x4 matrix.
We perform bundle adjustment using Ceres Solver for nonlinear optimization. Given a Reconstruction, we optimize over all cameras and 3D points to minimize the reprojection error.
class BundleAdjustmentOptions
ceres::LinearSolverType BundleAdjustmentOptions::linear_solver_type
DEFAULT: ceres::SPARSE_SCHUR
ceres::DENSE_SCHUR is recommended for problems with fewer than 100 cameras, ceres::SPARSE_SCHUR for up to 1000 cameras, and ceres::ITERATIVE_SCHUR for larger problems.
ceres::PreconditionerType BundleAdjustmentOptions::preconditioner_type
DEFAULT: ceres::SCHUR_JACOBI
If ceres::ITERATIVE_SCHUR is used, then this preconditioner will be used.
ceres::VisibilityClusteringType BundleAdjustmentOptions::visibility_clustering_type
DEFAULT: ceres::SINGLE_LINKAGE
bool BundleAdjustmentOptions::verbose
DEFAULT: false
Set to true for verbose logging.
bool BundleAdjustmentOptions::constant_camera_intrinsics
DEFAULT: false
If set to true, the camera intrinsics are held constant during optimization. This is useful if the calibration is precisely known ahead of time.
int BundleAdjustmentOptions::num_threads
DEFAULT: 1
The number of threads used by Ceres for optimization. More threads means faster solving time.
int BundleAdjustmentOptions::max_num_iterations
DEFAULT: 500
Maximum number of iterations for Ceres to perform before exiting.
double BundleAdjustmentOptions::max_solver_time_in_seconds
DEFAULT: 3600.0
Maximum solver time is set to 1 hour.
bool BundleAdjustmentOptions::use_inner_iterations
DEFAULT: true
Inner iterations can help improve the quality of the optimization.
double BundleAdjustmentOptions::function_tolerance
DEFAULT: 1e-6
Ceres parameter for determining convergence.
double BundleAdjustmentOptions::gradient_tolerance
DEFAULT: 1e-10
Ceres parameter for determining convergence.
double BundleAdjustmentOptions::parameter_tolerance
DEFAULT: 1e-8
Ceres parameter for determining convergence.
double BundleAdjustmentOptions::max_trust_region_radius
DEFAULT: 1e12
Maximum size that the trust region radius can grow during optimization. By default, we use a value lower than the Ceres default (1e16) to improve solution quality.
BundleAdjustmentSummary BundleAdjustReconstruction(const BundleAdjustmentOptions &options, Reconstruction *reconstruction)
Performs full bundle adjustment on a reconstruction to optimize the camera reprojection error. The BundleAdjustmentSummary returned contains information about the success of the optimization, the initial and final costs, and the time required for various steps of bundle adjustment.
## Similarity Transformation¶
void AlignPointCloudsICP(const int num_points, const double left[], const double right[], double rotation[3 * 3], double translation[3])
We implement ICP for point clouds. We use Besl-McKay registration to align point clouds. We use SVD decomposition to find the rotation, as this is much more likely to find the global minimum as compared to traditional ICP, which is only guaranteed to find a local minimum. Our goal is to find the transformation from the left to the right coordinate system. We assume that the left and right reconstructions have the same number of points, and that the points are aligned by correspondence (i.e. left[i] corresponds to right[i]).
void AlignPointCloudsUmeyama(const int num_points, const double left[], const double right[], double rotation[3 * 3], double translation[3], double *scale)
This function estimates the 3D similiarty transformation using the least squares method of [Umeyama]. The returned rotation, translation, and scale align the left points to the right such that $$Right = s * R * Left + t$$.
void GdlsSimilarityTransform(const std::vector<Eigen::Vector3d> &ray_origin, const std::vector<Eigen::Vector3d> &ray_direction, const std::vector<Eigen::Vector3d> &world_point, std::vector<Eigen::Quaterniond> *solution_rotation, std::vector<Eigen::Vector3d> *solution_translation, std::vector<double> *solution_scale)
Computes the solution to the generalized pose and scale problem based on the paper “gDLS: A Scalable Solution to the Generalized Pose and Scale Problem” by Sweeney et. al. [SweeneyGDLS]. Given image rays from one coordinate system that correspond to 3D points in another coordinate system, this function computes the rotation, translation, and scale that will align the rays with the 3D points. This is used for applications such as loop closure in SLAM and SfM. This method is extremely scalable and highly accurate because the cost function that is minimized is independent of the number of points. Theoretically, up to 27 solutions may be returned, but in practice only 4 real solutions arise and in almost all cases where n >= 6 there is only one solution which places the observed points in front of the camera. The rotation, translation, and scale are defined such that: $$sp_i + \alpha_i d_i = RX_i + t$$ where the observed image ray has an origin at $$p_i$$ in the unit direction $$d_i$$ corresponding to 3D point $$X_i$$.
ray_origin: the origin (i.e., camera center) of the image ray used in the 2D-3D correspondence.
ray_direction: Normalized image rays corresponding to reconstruction points. Must contain at least 4 points.
world_point: 3D location of features. Must correspond to the image_ray of the same index. Must contain the same number of points as image_ray, and at least 4.
solution_rotation: the rotation quaternion of the candidate solutions
solution_translation: the translation of the candidate solutions
solution_scale`: the scale of the candidate solutions | 2015-08-31 06:42:58 | {"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": 0.3395041227340698, "perplexity": 2962.286593701264}, "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-2015-35/segments/1440644065828.38/warc/CC-MAIN-20150827025425-00299-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://www.maplesoft.com/support/help/MapleSim/view.aspx?path=Task/SurfaceIntSphere | Surface Integration over the Surface of a Sphere - Maple Programming Help
Surface Integration over the Surface of a Sphere
Description Formulate and evaluate the surface integral of $f\left(x,y,z\right)$ over the surface of a sphere.
Surface Integral on a Sphere
Integrand $f\left(x,y,z\right)=$
Sphere Center: $x=$ $y=$ $z=$
Commands Used | 2016-10-01 22:20:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 12, "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": 0.9954692721366882, "perplexity": 685.6314605565813}, "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-2016-40/segments/1474738663308.86/warc/CC-MAIN-20160924173743-00057-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://www.i2m.univ-amu.fr/events/lower-bounds-for-univariate-real-polynomials/ | # Lower bounds for univariate real polynomials
Carte non disponible
Date/heure
Date(s) - 28/09/2017
11 h 00 min - 12 h 00 min
Catégories Pas de Catégories
In this talk, we consider real univariate polynomials and we study their representations as sums of powers of degree 1 polynomials. The goal of this talk is to present families of polynomials such that the number of terms required in such a representation is of order d. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order $\sqrt{d}$. We obtain this improvement thanks to a link of this problem with the problem of real Birkhoff interpolation.
This talk is based on a joint work with Pascal Koiran: Lower bounds by Birkhoff interpolation. Journal of Complexity 39 (2017), 38-50.
Posts created 14
## Laurent Tichit
Commencez à saisir votre recherche ci-dessus et pressez Entrée pour rechercher. ESC pour annuler.
Retour en haut | 2020-04-10 00:41:01 | {"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": 0.8390173316001892, "perplexity": 1340.3814543905219}, "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/1585371880945.85/warc/CC-MAIN-20200409220932-20200410011432-00420.warc.gz"} |
https://github.com/LibertasSpZ/adcompile | {{ message }}
autodiff compiler
Switch branches/tags
Nothing to show
## Files
Failed to load latest commit information.
Type
Name
Commit time
## Quantum Autodifferentiation Parser and Compiler
This repository contains the instructions for running and the source code organization for a quantum autodifferentiation parser and compiler, which is the primary software artifact for the paper
Shaopeng Zhu, Shih-Han Hung, Shouvanik Chakrabarti, and Xiaodi Wu. On the Principles of Quantum Differential Programming Languages. PLDI 2020, arXiv: 2004.01122.
In particular, we implemented the code-transformation rules, and the compilation of additive quantum programs in Section 6 and 4 resepectively of the paper. We evaluate the artifact in Section 8.2 of the paper on a selection of representative quantum programs for quantun machine learning.
### Source-File Structure
adlexer.mll and adparser.mly parses strings in an input file into tokens; ast.ml contains the typing information of the tokens. compiler.ml implements the code transformation and code compilation rules of the paper, with codeTransformation and codeCompilation the respective main functions. Lastly, main.ml provides an interface connecting the input, lexer, parser, compiler and output.
### Getting Started: Requirements
In particular on macOS, for example, first issue the command
/usr/bin/ruby -e "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install)" to install Homebrew; then issue brew install ocaml brew install opam brew install ocaml-findlib opam init to install opam and OCaml. For installation instructions of opam and OCaml on other operating systems please check the websites listed at the beginning. ### Getting Started: Setup It is a few lines of common commands to install our quantum autodifferentiation parser and compiler. After cloning the repository, simply issue the commands cd QAutoDiff/adcompile make exe This will generate an executible named adcompile in the directory ~/QAutoDiff/adcompile. ### Differentiating quantum programs using adcompile Given an original parameterized quantum program (as defined in Section 3), one may now compute and print the additive differential program (after applying code-transformaton rules in Setion 6, before compilation) as well as the compiled multiset (after compilation in Section 4) of differential programs, using the executible adcompile. One may also read out the size of the multiset from the output file. We have provided some sample programs in ~/QAutoDiff/adcompile/testExamples. Assuming you are in the directory ~/QAutoDiff/adcompile/, the command for manual differentiation of a parameterized program with respect to a parameter is of the form ./adcompile arg1 arg2 where arg1 is the test program file name, and arg2 is the parameter with respect to which you would like to differentiate your test program. For example, again assuming you have cd-ed to the directory ~/QAutoDiff/adcompile/, then ./adcompile testExamples/QNN/QNN_medium_if.prog t1 > YourDesiredOutputName.out differentiates the test program "QNN_medium_if.prog" (a medium sized quantum neural net program, with case control) with respect to the parameter t1. ### Using the testrunner.sh script to demonstrate output We made a driver script to run the compiler on the test programs we wrote. To run this driver, please issue ./testrunner.sh in the working directory (namely ~/QAutoDiff/adcompile/). ### Results and remarks We evaluate our artifact on a selection of representative test programs (in ~/testExamples) as described in Section 8.2 of the paper. Please refer to Table 2 from the evaluation part (Section 8.2) of our paper for details. Some explanations include: • when counting the number of lines in the input programs, we did not count the empty lines. • One should note that all columns in Table 2 except$|#\frac{\partial}{\partial\theta}(\bullet)|$are describing the input files;$|#\frac{\partial}{\partial\theta}(\bullet)|$, counting the number of parameterized programs in the compiled multiset, is recorded at the end of each output file. For example, in the input file QNN_medium_if.prog, one may observe there are 6 * 3 + 5 + 4 + 3 + 2 + 1 = 33 lines of code in each basic block, so total number of lines of code is 33 * 5 (blocks) + 18 (initialization) + 6 (case control sentences) = 189 lines of code, as Table 2 recorded. Among these lines of code, only the "block" code has one gate per line, so 33 * 5 = 165 gates. (Note that by "gate" we mean unitary gates, so the initialization does not count as gates). We have three layers network using 2 case control, and 18 qubits in the system (as witnessed by the first 18 lines of initialization code). Number of occurences of t1 in each block is 8, so the total number of occurances of t1 is 3 * 8 = 24 (intuitively after passing each case-control only 1 branch is executed, so the occurence of t1 only grows by 8 instead of by 2 * 8 = 16. See also Definition 7.1 in the paper). The "lines, gates, layers, qb, OC" information for all other input programs were similarly computed by hand. The$|#\frac{\partial}{\partial\theta}(\bullet)|$of the differential program of QNN_medium_if.prog is recorded in Line 17752 of res_QNN_medium_if.txt from the ~/QAutoDiff/adcompile/testExamples/result/QNNRes directory of this git repository (alternatively, one may look at the YourDesiredOutputName.out file you obtained by running the ./adcompile testExamples/QNN/QNN_medium_if.prog t1 > YourDesiredOutputName.out command; yet another option: check the ~/QAutoDiff/adcompile/QNNoutputs/QNN_medium_if.out file generated by running the ./testrunner.sh script in the ~/QAutoDiff/adcompile directory, which produces 4 output directories holding the output files). The number of programs in the compiled multiset of the differential program of QNN_medium_if.prog is 24, agreeing with Table 2. One may check that$|#\frac{\partial}{\partial\theta}(\bullet)|$from all the other rows also agree with Table 2 by eyeballing the last few lines of each output program. If one makes their own test program, run ./adcompile yourOwnTest.prog tk > yourOwnOutput.out (where you should replace tk with the desired parameter legally coded, such as t3). One may then count the occurence of tk in youOwnTest.prog, and observe that the number of occurences of tk is always greater than or equal to$|#\frac{\partial}{\partial\theta}(\bullet)|\$ recorded in yourOwnOutput.out, which agrees with the theoretical bound in the paper (Proposition 7.2).
• some of the large scale programs may take up to around 90 seconds on a 1.3 GHz Intel Core i5 processor with 8 GB 1867 MHz LPDDR3 memory to differentiate. The script testrunner.sh takes less than 3 min on a 3 GHz Intel Core i7 processor with 8 GB 1600 MHz DDR3 memory (real time real 2m43.778s on a trial run).
• We identify Skip[q1, q2] with Skip[q2,q1], Abort[q1,q2] with Abort[q2,q1], due to unanimous behavior across all registers.
• One is welcomed to make their own test programs following the input program format exemplified by the existing ones.
autodiff compiler
## Releases
No releases published
## Packages 0
No packages published
•
• | 2022-05-19 13:27:51 | {"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": 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": 0.28567391633987427, "perplexity": 5586.604873128984}, "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/1652662527626.15/warc/CC-MAIN-20220519105247-20220519135247-00577.warc.gz"} |
http://math.stackexchange.com/questions/272995/lk-n-longrightarrow-f-is-irreducible-and-textgall-k-cong-s-n | # $[L:K]=n!\ \Longrightarrow \ f$ is irreducible and $\text{Gal}(L/K)\cong S_n.$
How do I go about proving the following theorem ?
Let $f\in K[T]$ have degree $n$ and splitting field $L/K$. Then we have $$[L:K]=n!\ \Longrightarrow \ f\text{ is irreducible and Gal}(L/K)\cong S_n.$$
I managed to prove that $f$ has to be irreducible, but I'm stuck showing that $\text{Gal}(f/K)\cong S_n$.
(Some things I know, that might be useful in the proof: The Galois group of $f$ has to be, thanks to irreducibility, a transitive subgroup of $S_n$; $S_n$ is transitive.
I already browsed through Dummit's book and M. Artins book on algebra but couldn't find anything that would help me.)
-
How many subgroups of $S_n$ of order $n!$ are there? – Hagen von Eitzen Jan 8 '13 at 19:14
@HagenvonEitzen Well...one...but how does that help me ? – user47580 Jan 8 '13 at 20:47
Let $f$ be a polynomial of degree $n$ over a field $K$ and $E$ its splitting field. Assume that $E$ is a Galois extension and let $\alpha_1,\dots,\alpha_n$ be the roots of $f$. Any $K$-automorphism of $E$ is determined by its action on $\alpha_1,\dots,\alpha_n$ since $\alpha_1,\dots,\alpha_n$ generate $E$ over $K$. We also know that for any $K$-automorphism $\sigma$ of $E$ that $\sigma(\alpha_i)=\alpha_j$ for some $j$. In particular we see that $\mathrm{Gal}(K/E)$ provides a faithful group action on $\alpha_1,\dots,\alpha_n$. This means that $\mathrm{Gal}(K/E)$ embeds into $S_n$.
Now concerning your problem. First observe that if $f$ had repeated roots $L/K$ could not have degree $n!$, so $L/K$ is a separable extension. We have that $L/K$ is normal because it is a splitting field so it is a Galois extension. So it makes sense to talk about $\mathrm{Gal}(K/L)$. By the argument in the previous paragraph we know that $\mathrm{Gal}(K/L)$ embeds in $S_n$, but $\mathrm{Gal}(K/L)$ has order $n!$ so it must be that the embedding is in fact an isomorphism.
-
Could you please be a little more explicit ? – user47580 Jan 8 '13 at 21:09
@user47580 I've rewritten the answer to include more details. Let me know if there are any specific parts you don't follow. – JSchlather Jan 8 '13 at 21:24
Hey, thanks a lot. After thinking about it a bit, a single thing is still unclear to me though: How can we rigorously infer from $f$ having repeated roots in $L/K$ that its degree has to be $n!$ ? – user47580 Jan 8 '13 at 22:52
Suppose that $f$ has a repeated root $\alpha$. Then $f(x)=(x-\alpha)^2g(x)$ over $K(\alpha)$. The splitting field $E$ of $g(x)$ over $K(\alpha)$ is also the splitting field of $f$. Now $[E:K(\alpha)]\leq (n-2)!$ and $[K(\alpha):K]\leq n$ so $[E:K]\leq n (n-2)!<n!$. – JSchlather Jan 8 '13 at 22:56
Why do we consider the splitting field $E$ of $g$ of $K(\alpha)$ ? Can some coefficients of $g$ also be $\alpha$ ? – user47580 Jan 8 '13 at 23:16
(I think) I see two possible points of confusion you're having, so I'll try to straighten them out.
Firstly, in your edit, you've added that you know that the Galois group must be a subgroup of $S_n$. Now in Hagen von Eitzen's comment, he asks how many subgroups of $S_n$ (i.e., potnetial Galois groups) have order $n!$ (as required in your problem). So how many potential Galois groups are there, and what are they?
Secondly, in your comments to Jacob Schlather's (great) answer, you're asking good questions, but I get the feeling you've missed the point of the first paragraph in his answer. What he's saying is, because the elements of the Galois group must permute the roots of $f$, and $f$ has $n$ (not necessarily distinct) roots, each permutation is a permutation on $n$ (not necessarily distinct) letters. Now the key is that if those $n$ letters are not distinct, then there cannot be $n!$ permutations. Perhaps it's helpful to think about it this way: not counting repetition, there are $n!$ permutations of all the roots, so if some of the roots were repeated, say $r_1$ and $r_2$, then every permutation swapping those two, would be the same as another permutation that has the same effect on the rest of the roots, but doesn't swap $r_1$ and $r_2$, and so you have fewer than $n!$ distinct permutations (i.e., automorphisms!).
Thus, as $L/K$ is Galois, we know $|\operatorname{Gal}(L/K)| = [L:K] = n!$, and so if any of the roots were repeated, there would be fewer than $n!$ (permutations) automorphisms of $L$ fixing $K$, contradicting the hypotheses.
- | 2014-08-23 11:49:09 | {"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": 0.8921277523040771, "perplexity": 138.87061223267435}, "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-2014-35/segments/1408500826016.5/warc/CC-MAIN-20140820021346-00331-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/time-dialation-formula.14458/page-2 | # Time dialation formula
David
Originally posted by Peterdevis
the Rollex
I think a Rolex will probably tick a little faster in space, since there are no gravitational forces pulling down on its gears and their shafts, so they won’t experience as much friction in space. But the speed up will not match the same speed-up rate as an atomic clock, because the Rolex speed up is a large-scale “mechanical” function, while an atomic oscillation speed up operates by quantum mechanics rules.
Regarding a Rolex or any kind of balance-wheel watch or clock, think of it this way: Put the watch inside a rapidly spinning centrifuge, and notice that the higher the g forces, the more friction is placed on the bearings and the gear shafts. This will tend to slow the watch down a little. However, if you put an electronic watch in your refrigerator freezer, that will slow it down a little, for thermodynamic and quantum mechanics reasons. Put a pendulum clock high on a mountain and it will slow down. Put an atomic clock in a valley and it will slow down.
These different types of clocks will slow down and speed up for different reasons, because different laws of physics apply to the different types of clocks.
David
Originally posted by mikesvenson
I fully understand the concept that if you travel at .99c for an extended time relative to Earth that upon your return to Earth you will actually be re-entering into the future time of Earth (relative to the elapsed time you experianced while out in space traveling at .99c).
No clock slows down due to “relative motion” alone, since no physical force is placed on the mechanism of the clock. Different kinds of clocks will slow down and speed up if you add forces to them or take forces away from them.
Your biological time while you are traveling in space is determined by your molecular vibration rates. This is “thermodynamic” time.
If the SR theory were true, and if you traveled at .99c, your molecular vibration rate would slow down to near zero and you would freeze to death. SR theory just doesn’t work. You can’t have your clocks and your aging rate slow down, while your molecular vibration rates don’t change at all. If you remain warm during your trip, that means your molecular clocks, your molecular vibration rates are not slowing down.
Go read some Lorentz stuff. That’s what the SR theory was based on and modeled from, but unfortunately Einstein neglected to include acceleration, fields, and real physical forces, so the SR theory does not work in real life.
russ_watters
Mentor
[Peterdevis]
...the natural time for the traveler would be an ellapsed 7 months, and the syncronized Earth watch(impossible throuth SR) would only have read an ellapsed 1 month! What the heck do they mean by "natural time" Wouldnt this be the same thing as the 1 month time? This doesnt make sense to me.
The specific link might help, but from the context given, "natural time" appears to be the time shown by your watch in your frame of reference.
Originally posted by David
Maxwell mentioned “sodium” in his 1873 comments.
So then which is the type of atomic clock that Einstein used in his books (did he always use the same kind?) and what are the transformation equations for converting time between different atomic clocks?
I think a Rolex will probably tick a little faster in space, since there are no gravitational forces pulling down on its gears and their shafts, so they won’t experience as much friction in space.
Analog watches have used frictionless springs for several hundred years and and since the spring is where the frequency comes from, they are not mechanically affected by gravity. If friction due to weight mattered, they'd tick at different rates upside down and right side up.
These different types of clocks will slow down and speed up for different reasons, because different laws of physics apply to the different types of clocks.
All laws of physics apply in all cases. The effects are only relevant in certain cases. But the law of physics for the last case is relevant to all of them.
No clock slows down due to “relative motion” alone, since no physical force is placed on the mechanism of the clock.
Then explain GPS. The engineers who make GPS satellites use the predictions of SR (motion related time dilation) in the calibration of the satellites' clocks.
If the SR theory were true, and if you traveled at .99c, your molecular vibration rate would slow down to near zero and you would freeze to death.
No. From your frame of reference if you shut your eyes, nothing has changed. You don't understand the relevance of frame of reference.
SR theory just doesn’t work.....so the SR theory does not work in real life.
Scientists, engineers, and laypeople who don't even know it, use Einstein's Relativity every day in real life. It hasn't let them down yet. Maybe I was wrong in the other thread: the problem here appears to be you just plain don't understand it.
David
Originally posted by russ_watters
So then which is the type of atomic clock that Einstein used in his books
He called them “elementary light generators” in the 1911 theory. You need to read Maxwell’s book and Steinmetz’s book so you will know what he was talking about. You really need to read the original theories, the original books, and not get all your information out of mass-media popular books designed for junior high school kids. You need to also read the old major science books of that era, such as Maxwell’s work, which was still fresh and new when the original theories were developed in the late 19th and early 20th Century. That will give you more background information so you can understand the newer theories from the early 20th Century. For example, I’ve never seen a modern book that says exactly what Einstein meant by “elementary light generators”. However, I did find the information in Maxwell’s book and Steinmetz’s book from 1923.
All mechanical watches have shafts and bearings. That’s what a 23 jewel watch used to be. They used hard ruby or other jewels as the bearing pivot points, since the jewels were harder than metal and didn’t wear out as often. Gravity increases friction in the bearings of mechanical watches.
Originally posted by russ_watters
Then explain GPS. The engineers who make GPS satellites use the predictions of SR (motion related time dilation) in the calibration of the satellites' clocks.
Read Lorentz’s theory and some of Dr. Su’s papers. They explain it. It has nothing to do with SR theory. The atoms in the clocks don’t know they are “moving relatively”. They only know they are “moving” if they feel some kind of motion-related force.
David
Originally posted by russ_watters
From your frame of reference if you shut your eyes, nothing has changed. You don't understand the relevance of frame of reference. Scientists, engineers, and laypeople who don't even know it, use Einstein's Relativity every day in real life. It hasn't let them down yet. Maybe I was wrong in the other thread: the problem here appears to be you just plain don't understand it.
Einstein tried to work thermodynamics into the SR theory in his 1907 paper, “On the Relativity Principle and the Conclusions Drawn From It,” but he got all mixed up. When he wrote the SR theory in 1905, he didn’t consider the thermodynamic implications.
See, if “all of time” slows down inside a “moving” frame, then the vibration rates of all the molecules slow down too, since they are, in effect, molecular clocks, and everything inside the frame freezes, based on the "time slowdown" rules of SR theory itself. He just didn’t think of that when he wrote the paper. I thought of it, and I finally tracked down his 1907 paper in which he tried to introduce thermodynamics into the SR theory. But it never worked out. His 1905 theory was based on his misunderstanding of the 1895 Lorentz book.
You really need to get a copy of his 1918 correction paper, in which he adds fields, acceleration, and atomic clocks to the SR theory. When he does that, then it becomes the 1895 Lorentz theory that he tried to copy in the first place. You need to get a copy of Lorentz’s book too, to see the original of the Lorentz Transformation and many of the ideas that were copied for the SR theory. Modern books won’t tell you this stuff. You’ve got to do your own research.
The Lorentz book is so rare, I saw a copy offered for sale in Holland for 6,750 Euros. That’s more than \$6,750 dollars. Some day the book with be translated and published in English, and everyone will see where the SR theory originally came from, and how the 1905 errors occurred.
Here is a page from my copy of the 1895 Lorentz book:
He introduced atomic time dilation on page 49, when Einstein was just 16 years old and still in high school.
Last edited by a moderator:
russ_watters
Mentor
Originally posted by David
He called them “elementary light generators” in the 1911 theory.
In fact, the reason he didn't identify any by specific name/type is that they weren't invented until 1952. Einstein makes no mention of specific clocks and their particular sources of error/rate because they are not relevant to his theory. Einstein's theory is about time itself.
All mechanical watches have shafts and bearings. That’s what a 23 jewel watch used to be. They used hard ruby or other jewels as the bearing pivot points, since the jewels were harder than metal and didn’t wear out as often. Gravity increases friction in the bearings of mechanical watches.
Certainly. But if that friction affected the rate, it would badly throw off the accuracy of the watch every time you moved or changed the watch's orientation.
Something I didn't point out/ask before, but you do know the mechanical difference between a spring-mass and a pendulum, right? Both are simple harmonic motion, but they are not affected by gravity in the same way. In a pendulum, the force of gravity is the driving force. In a spring-mass system, the force of gravity may be part of the driving force or not depending on the type of spring-mass system and its orientation. As a result, the rules that apply to a pendulum are different from the rules that apply to a mechanical watch. For this reason, when talking about clock rate variations, you must specify what type of clock you are talking about. There is a reason Einstein did not.
David
Originally posted by russ_watters
In fact, the reason he didn't identify any by specific name/type is that they weren't invented until 1952.
No, sorry, you are wrong. Maxwell, in 1873, recognized natural atoms as being natural “clocks”. Einstein used this idea in his 1911 theory.
“Natural atomic clocks” with numerical clock-face read-outs and installed in self-contained boxes weren’t invented until the 1950s. Before that, the oscillation rates of natural atomic clocks had to be measured by spectrometers. You should have been taught that in physics class.
russ_watters
Mentor
Originally posted by David
No, sorry, you are wrong. Maxwell, in 1873, recognized natural atoms as being natural “clocks”. Einstein used this idea in his 1911 theory.
“Natural atomic clocks” with numerical clock-face read-outs and installed in self-contained boxes weren’t invented until the 1950s. Before that, the oscillation rates of natural atomic clocks had to be measured by spectrometers. You should have been taught that in physics class.
None of that contradicts what I said. Clearly they recognized the possibility it could be done, but neither Maxwell nor Einstein ever used an actual atomic clock when formulating their theories.
That fact is more important than you are admitting and I think you know it. With your talk of clocks, you are mixing several fundamentally different and unrelated sources of variation and calling them equivalent. You used the example of a pendulum clock, which slows down if the driving force (gravity or a centrifugal force) is lowered. But for other clocks, you talk about fricton. Setting aside that you haven't said what kind of friction affects an atomic clock and how, friction isn't even close to the same issue as how gravity affects a pendulum clock. Thats easy enough to see when you realize that a pendulum clock does not work at all in orbit, but the wearer of a mechanical spring-driven watch would not even notice a variation during a quick trip to orbit and back.
Also, while friction does play a part in rate for some mechanical clocks, it differs quite a bit from one clock to another, yet you are applying a blanket equation to a large but undefined set of atomic clocks. You're contradicting yourself.
Also, while gravity can be simulated in a centrifuge, a centrifuge does not produce gravity. As a result, it is easy enough to test whether its the force/acceleration itself or the field that is creating the dilation. And guess what: it's been done. And no, acceleration force does not produce the same effect on an atomic clock as GR time dilation.
On the SR side, something I just read about - another way to see SR time dilation: particle decay. Particles moving at high velocity (supercollider experiements) show SR time dilation induced changes in their lifespans.
I've just started looking into those last two, so I should have more later, but for a start: http://www.lns.cornell.edu/spr/2002-01/msg0038144.html
Last edited:
Originally posted by russ_watters
Thats easy enough to see when you realize that a pendulum clock does not work at all in orbit, but the wearer of a mechanical spring-driven watch would not even notice a variation during a quick trip to orbit and back.
Does this mean that a pendulum clock operates on kinetic and potential energy driven by the force of gravity?
russ_watters
Mentor
Originally posted by mikesvenson
Does this mean that a pendulum clock operates on kinetic and potential energy driven by the force of gravity?
Precisely. When the bob is at the bottom of its swing, its all kinetic and at the top of its swith all potential. The excange of the two is what causes the simple harmonic motion.
Also, while gravity can be simulated in a centrifuge, a centrifuge does not produce gravity. As a result, it is easy enough to test whether its the force/acceleration itself or the field that is creating the dilation. And guess what: it's been done. And no, acceleration force does not produce the same effect on an atomic clock as GR time dilation.
is this really true? Then it's violating the EEP (einsteins equivalence principle): In a closed box you can't make out if you are excelarating or moving in a gravitational field (I' dont love the word gravitational field, but everybody use it)
David
Originally posted by russ_watters
Clearly they recognized the possibility it could be done, but neither Maxwell nor Einstein ever used an actual atomic clock when formulating their theories.
Maxwell did. You just don’t know enough about the history of science. The old timers could calculate the oscillation rates of specific atoms by studying spectrographs of their light. That led to the invention of the self-contained atomic clock with a digital read-out.
Einstein studied other people’s books and papers to get his ideas.
David
Originally posted by russ_watters
Clearly they recognized the possibility it could be done, but neither Maxwell nor Einstein ever used an actual atomic clock when formulating their theories.
Here’s the way Charles Steinmetz explained Einstein’s 1911 theory about atomic clocks, in his own book of 1923:
”We cannot carry a clock from the earth to Betelgeuse, but we do not need to do this, since every incandescent hydrogen atom, for instance, is an accurate clock, vibrating at rate definitely fixed by the electrical constants of the hydrogen atom and showing us the exact rate of its vibration in the spectroscope by the wave length or frequency of its spectrum lines. Thus in a strong gravitational field the frequency of luminous vibrations of the atoms should be found slowed down’ in other words, the spectrum lines should be shifted towards the red end of the spectrum.”
Look, you need to go out and buy these books yourself. I don’t have time to tutor you in physics or type up all the text from all my books for you. You need to go to a good university somewhere and take some physics courses.
russ_watters
Mentor
Originally posted by Peterdevis
is this really true? Then it's violating the EEP (einsteins equivalence principle): In a closed box you can't make out if you are excelarating or moving in a gravitational field (I' dont love the word gravitational field, but everybody use it)
Hmm, good question: perhaps I misunderstood the experiment. I think though, the difference isn't in what you are seeing, but in what an outside observer sees. You can only measure time dilation by comparing dis-similar frames of reference. And in seeing the dilation, you'll also see the reason for it.
Maxwell did.
Again, if no atomic clock existed, he couldn't have used one. He certainly speculated on/theorized on how they might work, but that is not the same thing.
Here’s the way Charles Steinmetz explained Einstein’s 1911 theory about atomic clocks, in his own book of 1923:
”We cannot carry a clock from the earth to Betelgeuse, but we do not need to do this, since every incandescent hydrogen atom, for instance, is an accurate clock, vibrating at rate definitely fixed by the electrical constants of the hydrogen atom and showing us the exact rate of its vibration in the spectroscope by the wave length or frequency of its spectrum lines. Thus in a strong gravitational field the frequency of luminous vibrations of the atoms should be found slowed down’ in other words, the spectrum lines should be shifted towards the red end of the spectrum.”
Looks good to me: that description is consistent with the pervasive view that atomic clocks accurately measure time and that GR affects time, not just certain types of clocks.
Look, you need to go out and buy these books yourself. I don’t have time to tutor you in physics or type up all the text from all my books for you. You need to go to a good university somewhere and take some physics courses.
Hehe, you learned your current opinion in school? And you passed? Impossible. Again: what you are saying is not consistent with the current accepted view. Even if you want to argue that the current accepted view is wrong or (you are very careful about avoiding directly saying that, but you do agree with Einstein in one breath while saying he's wrong with the next), its the one taught in school and you can't pass without at least being able to regurgitate it. I'm glad you finally said it though.
You like regurgitating quotes and taking them out of context. Ok, fine, here's one from page 35 of "Relativity" (1916):
As a consequence of its motion the clock goes more slowly than when at rest.
The chapter is 2 pages long and he never once specifies which type of clock it applies to. Why? Because he's assuming we'll realize by "clock," he means 'any instrument that measures time with sufficient accuracy to notice the SR effects discussed in that chapter.' There are also several nuggets in there about C being an unattainable speed.
Look, I'm an engineer, not a physicist, which is why I harp on practical uses (which you brush aside or ignore). In our every day lives, we use things that would not work if SR and GR didn't work the way I (and others) am telling you it does.
Your tactics are quite good (you appear to have been practicing this arguement for quite some time): you press a point until its clear that you're backed into a corner you can't get out of, then ignore it like it has never been discussed. Fear not: people reading these threads notice when you drop the ball.
Last edited:
David
Originally posted by russ_watters
Looks good to me: that description is consistent with the pervasive view that atomic clocks accurately measure time and that GR affects time, not just certain types of clocks
You said they didn’t have atomic clocks in the old days, and I just proved to you that they considered natural atoms to be atomic clocks. You apparently didn’t know this because you said atomic clocks weren’t invented until 1952.
Anyway, the vibration rates of the atoms tells us the vibration rates of the atoms. They do not tell us what the vibration rates of pendulum, mechanical, or thermodynamic clocks will be in the same places.
Look, I don’t have time to tutor you in physics or the history of atomic clocks and other timekeeping devices. Go out and buy some classic books on the subject.
David
Originally posted by russ_watters
Look, I'm an engineer, not a physicist,
Well that’s just great. You need to be on an engineering board, because you sure don’t know much about physics. Go out an build a bridge or something.
Phobos
Staff Emeritus
Gold Member
Warning #2, David. Discuss the topic at hand. Do not flame.
David
Originally posted by Phobos
Warning #2, David. Discuss the topic at hand. Do not flame.
I didn’t flame anybody. You don’t say anything at all about Russ’ constantly flaming of me. He’s been posting personal insults directed at me the whole time I’ve been on this board. You are just setting me up so you can ban me, because you don’t like my opinions. You should be ashamed of yourself.
Janus
Staff Emeritus
Gold Member
Originally posted by russ_watters
Also, while gravity can be simulated in a centrifuge, a centrifuge does not produce gravity. As a result, it is easy enough to test whether its the force/acceleration itself or the field that is creating the dilation. And guess what: it's been done. And no, acceleration force does not produce the same effect on an atomic clock as GR time dilation.
Okay, you have to be really careful here. Yes, acceleration in of itself does not cause time dilation, but neither does gravity. What is happening depends upon whether you are rotating with the Centrifuge or not.
If you are sitting next to the centrifuge, you measure a Time dialtion in the sample at the end of the arm due to simple SR effects of velocity. And there is no additional effects due to acceleration.
If you were sitting at the axis of the centrifuge and turning with it, you can consider both yourself and the end of the arm as stationary, but you will measure a time dilation at the end of the arm due to the difference in potential caused by the apparent gravitational field that exists between you and the arm. This dilation behaves exactly like was caused by "real" gravity.
The main thing to remember is that this dilation is due to the difference in potential and not due to the difference in force felt. For instance, One could build two centrifuges, one with an arm twice as long as the other, and spin both such that the ends of the arms of each experience the same g force. If you were sitting on the axis of the one with the longer arm you would note a greater time dilation between you and the end, then you would if you were sitting on the centrifuge with the shorter arm, even though both arms are experiencing the same g force.
You could even arrange things such that the arm end that feels less g-force undergoes a greater time dilation.
This is like the fact that even though the surface gravity of Uranus is less than that of the Earth's, the time dilation on the surface of Uranus is greater than that on the Earth's.
The main thing to remember is that this dilation is due to the difference in potential and not due to the difference in force felt. For instance, One could build two centrifuges, one with an arm twice as long as the other, and spin both such that the ends of the arms of each experience the same g force. If you were sitting on the axis of the one with the longer arm you would note a greater time dilation between you and the end, then you would if you were sitting on the centrifuge with the shorter arm, even though both arms are experiencing the same g force
I don't think there is a difference between force and potential, (it are two mathematical descriptions of the same phenomena for me)
The greater time dilitation of the centrifuge with the long arm,is the result of the greater velocity (for getting the same centrifugal acceleration).
When you measure the time dilitation of a clock in a centrifuge (simulating gravity 10 km above earth surface) or you measure time dilitation of a clock in an airplane (10 km above earth surface) with the same velocity as the clock in the centrifuge, you must see the same result.
Janus
Staff Emeritus
Gold Member
Originally posted by Peterdevis
I don't think there is a difference between force and potential, (it are two mathematical descriptions of the same phenomena for me)
The formula for gravitational force is
$$F_{g}= \frac{GMm}{r^{2}}$$
For gravitational potential, it is
$$PE_{g} = -\frac{GMm}{r}$$
A difference in relative force between two points in a field is just the difference in force felt by an object at those two points.
A difference in relative potential is a measure of the amount of work it would take to move an object from one point to the other. (IOW, the amount of work it would take to lift the object the distance between the two points.
These particular formulas are for your standard "mass generated" gravity which follows the inverse square rule.
Now let's imagine a uniform gravity field. (One which does not fall offf with distance) In this case, the force remains the same no matter where you are in the field, so the relative force between two points is always zero. But the potential between two points depends on their height difference in the field.
If we assume that the strength of the field causes an acceleration ofg, then the relative potential difference between two points is related to gh, where h is the height difference between the two.
Now in GR, time dilation is tied to relative potential, Thus two clocks at elevations h1 and h2 would be at different potentials, even though they would feel the same exact force, and they would run at different rates (as measured by anyone within that field).
This is the important difference between force and potential.
The greater time dilitation of the centrifuge with the long arm,is the result of the greater velocity (for getting the same centrifugal acceleration).
Again, this depends upon whether you are measuring from the reference frame that is rotating with the centrifuge or not. If you aren't, then you will measure a time dilation due to relative velocity alone.
If you are, you will measure a time dilation due to the apparent gravity field alone, ( As there is no relative velocity difference within the rotating frame.)
The time dilation works out to be the same, but each reference system sees it for a different reason.
Phobos
Staff Emeritus
Gold Member
Originally posted by David
I didn’t flame anybody. You don’t say anything at all about Russ’ constantly flaming of me. He’s been posting personal insults directed at me the whole time I’ve been on this board. You are just setting me up so you can ban me, because you don’t like my opinions. You should be ashamed of yourself.
Reread that previous post. It was a personal attack intended to only cause anger and was not part of any technical debate.
Russ can be tough, but he's fair. I'll take another look through the posts and I'll talk to Russ if I see anything that is as you say.
I've never banned anyone because of differing opinions. I'm just asking that the debate be kept civil. The irony here is that I've been asking the mentors to give you more chances.
ahrkron
Staff Emeritus
Gold Member
Originally posted by David
No clock slows down due to “relative motion” alone, since no physical force is placed on the mechanism of the clock.
They do, as it is measured every day in particle accelerators, communications with space probes, the use of GPS, etc., and was also measured directly using jets and atomic clocks.
Different kinds of clocks will slow down and speed up if you add forces to them or take forces away from them.
"Take forces away from them"?
The point is not if a Rolex will speed up or slow down, but what effects need to be considered when doing a real analysis of a physical situation.
Indeed, complex mechanisms may be affected by tempreature and other conditions, but when doing precise measurements, you need to take into account all real effects that contribute significantly to what you are measuring. Relativistic time dilation does affect any mechanism. The amount to which it does depends on specific conditions.
If you are measuring how a Rolex is affected in an passenger flight, many factors will affect your measurement much more than relativity. On the other hand, when measuring lifetimes of heavy mesons, you surely need to consider time dilation.
If the SR theory were true, and if you traveled at .99c, your molecular vibration rate would slow down to near zero and you would freeze to death.
Wrong. In order to say "if the SR theory were true", you first need to understand what SR actually says about the situation.
What it says is that, even at 0.99999c, you won't notice any change in yourself, since your speed relative to you is still 0.
SR theory just doesn’t work.
Funny how the GPS, nuclear reactors, QFT, GR and the standard model of particle physics
(all of which depend on SR) keep producing extremely accurate results. *That's* the heck of a lot of good luck!
You can’t have your clocks and your aging rate slow down, while your molecular vibration rates don’t change at all.
True, which means that your interpretation of what SR says is wrong.
On the other hand, that is perfectly in agreement with the predictions of SR (i.e., your clocks and your molecular vibrations stay in tune).
Go read some Lorentz stuff. That’s what the SR theory was based on and modeled from,
And, after nearly a century of accumulating experimental evidence and testing devices based on both, we are keeping the version that best describes all data we have. Nobody really cares if it is called "Einstein's" or "Lorentz's", but now that you bring it up, it is Einstein's. | 2020-01-23 05:07:07 | {"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": 0.5508267879486084, "perplexity": 777.4730423679258}, "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-2020-05/segments/1579250608295.52/warc/CC-MAIN-20200123041345-20200123070345-00479.warc.gz"} |
https://isr-publications.com/jnsa/articles-1572-nonlinear-contractions-involving-simulation-functions-in-a-metric-space-with-a-partial-order | Nonlinear contractions involving simulation functions in a metric space with a partial order
Volume 8, Issue 6, pp 1082--1094
• 1136 Views
Authors
Hajer Argoubi - FST Campus Universitaire, 2092-El Manar, Tunis, Tunisia. Bessem Samet - Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia. Calogero Vetro - Dipartimento di Matematica e Informatica, Universita degli Studi di Palermo, via Archirafi 34, 90123 Palermo, Italy.
Abstract
Very recently, Khojasteh, Shukla and Radenović [F. Khojasteh, S. Shukla, S. Radenović, Filomat, 29 (2015), 1189-1194] introduced the notion of $\mathcal{Z}$-contraction, that is, a nonlinear contraction involving a new class of mappings namely simulation functions. This kind of contractions generalizes the Banach contraction and unifies several known types of nonlinear contractions. In this paper, we consider a pair of nonlinear operators satisfying a nonlinear contraction involving a simulation function in a metric space endowed with a partial order. For this pair of operators, we establish coincidence and common fixed point results. As applications, several related results in fixed point theory in a metric space with a partial order are deduced.
Share and Cite
ISRP Style
Hajer Argoubi, Bessem Samet, Calogero Vetro, Nonlinear contractions involving simulation functions in a metric space with a partial order, Journal of Nonlinear Sciences and Applications, 8 (2015), no. 6, 1082--1094
AMA Style
Argoubi Hajer, Samet Bessem, Vetro Calogero, Nonlinear contractions involving simulation functions in a metric space with a partial order. J. Nonlinear Sci. Appl. (2015); 8(6):1082--1094
Chicago/Turabian Style
Argoubi, Hajer, Samet, Bessem, Vetro, Calogero. "Nonlinear contractions involving simulation functions in a metric space with a partial order." Journal of Nonlinear Sciences and Applications, 8, no. 6 (2015): 1082--1094
Keywords
• Partial order
• nonlinear contraction
• coincidence point
• common fixed point
• simulation function.
• 54H25
• 47H10
• 54C30
References
• [1] R. P. Agarwal, M. A. El-Gebeily, D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., 87 (2008), 109-116.
• [2] V. Berinde, Coupled coincidence point theorems for mixed monotone nonlinear operators, Comput. Math. Appl., 64 (2012), 1770-1777.
• [3] L. Ćirić, N. Cakić, M. Rajović, J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl., 2008 (2008), 11 pages, , , (),
• [4] . [4] J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Non- linear Anal., 71 (2009), 3403-3410. , , , (),
• [5] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359-1373. , , , (),
• [6] [6] F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theorems via simulation functions, Filomat, 29 (2015), 1189-1194. , , , (),
• [7] V. Lakshmikantham, L. B. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341-4349., , , (),
• [8] [8] J. J. Nieto, R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223-239. , , , (),
• [9] A. Petrusel, I. A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc., 134 (2006), 411-418. , , , (),
• [10] [10] S. Radenović, Z. Kadelburg, Generalized weak contractions in partially ordered metric spaces, Comput. Math. Appl., 60 (2010), 1776-1783. , , , (),
• [11] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435-1443. , , , (),
• [12] [12] A. Roldán, E. Karapinar, C. Roldán, J. Martínez-Moreno, Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math., 275 (2015), 345-355. , , , (),
• [13] [13] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $\alpha-\psi$-contractive type mappings, Nonlinear Anal., 75 (2012), 2154-2165., , , (), | 2021-01-26 12:27:00 | {"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": 0.813583493232727, "perplexity": 3372.1064583659386}, "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-2021-04/segments/1610704799741.85/warc/CC-MAIN-20210126104721-20210126134721-00266.warc.gz"} |
https://answers.ros.org/question/10356/reduce-kinect-resolution/ | # Reduce Kinect Resolution
I want to reduce the resolution of the kinetic to speed up processing. For example ignoring every other pixel would reduce the resolution by half. Is there an easy way to do this using the current drivers?
On a similar note - is it possible to only look at one vertical or horizontal line of resolution and pretend my kinect is a laser scanner? Thanks!
edit retag close merge delete
Sort by » oldest newest most voted
Yes, the resolution of the Kinect can be decreased. At the default 640x480, you get over 300,000 points per scan. That was too much data for me to process, so in my launch files, I set my kinect like this:
<rosparam command="load" file="\$(find openni_camera)/info/openni_params.yaml" />
<param name="rgb_frame_id" value="/openni_rgb_optical_frame" />
<param name="depth_frame_id" value="/openni_depth_optical_frame" />
<param name="use_indices" value="false" />
<param name="depth_registration" value="false" />
<param name="image_mode" value="8" />
<param name="depth_mode" value="8" />
<param name="debayering" value="0" />
<param name="depth_time_offset" value="0" />
<param name="image_time_offset" value="0" />
Note that here, I am NOT using the rgb camera at all for pointclouds, only for the image. This is the minimum amount of information I was able to obtain from the kinect and corresponds to roughly 19,200 points per scan.
However, this was STILL too much information. I tried pointcloud_to_laserscan along with voxel_grid filters, but I wasn't very happy with the results.
The node I wrote is very similar to pointcloud_to_laserscan, but still publishes a pointcloud2. What it does is adjusts the number of "slices" evenly through a range, so it gives the appearance of multiple laserscans. I haven't written it as a nodelet, but feel free to try it.
Video of this node mapping: http://www.youtube.com/watch?v=_rIAGKA6uS8 Video of this node mapping with full scan comparison: http://www.youtube.com/watch?v=Rl6bi5Udfz4
Be sure to run dynamic reconfigure on this node, it makes tuning it very easy! Here's my default values:
<node pkg="pcl_decimator" type="pcl_decimator" respawn="true" name="pcl_slicer" output="screen"> <rosparam> field_name: y num_slices: 5 slice_width: 0.01 start_threshold: -1.0 end_threshold: 1.0 </rosparam> </node>
more
Wow, just changing the launch files like is mentioned above made a huge difference. I went from seeing changes in rviz every few seconds to it looking real time now. rostopic hz /camera/rgb/points was saying 8 or 9 with cpus at 98%. now it is at 17 and cpus at 60%. I'll try your node next. Thanks
( 2011-07-03 05:42:07 -0500 )edit
If you want to do even better, if you leave depth_registration off and subscribe to /camera/depth/points, it will do less work since it won't have to attach colors to each point. Won't look as cool, but still good for mapping/object detection.
( 2011-07-03 15:09:06 -0500 )edit
Look at rosrun dynamic_reconfigure reconfigure_gui. You can adjust the resolution there while it is running. The values can also be set as parameters.
Depending on what you need an alternative would be to use a downsampling nodelet. That way you would get a equal distribution of points in space.
For faking a laser, have at look at this package: pointcloud_to_laserscan.
more
You can do this with openni_camera alone with the (undocumented) use_indices parameter. See my answer to this question for details. The advantage over pointcloud_to_laserscan type approaches would be that you cut out the unused information as early as possible in the pipeline, i.e. the driver doesn't spend CPU cycles projecting a bunch of depth values to 3D points that never get used anyway. Depending on how much you need to drop your CPU usage that might be important (it was in my application which is why I implemented that feature).
more | 2021-09-24 08:47:35 | {"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": 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": 0.1869179904460907, "perplexity": 3102.3741215211253}, "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/1631780057508.83/warc/CC-MAIN-20210924080328-20210924110328-00556.warc.gz"} |
https://www.scienceforums.net/topic/75895-trigonometric-function-problem/#comment-750532 | # Trigonometric Function Problem
## Recommended Posts
During high tide the water depth in a harbour is 22 m and during low tide it is 10m(Assume a 12h cycle).
Calculate the times at which the water level is at 18m during the first 24 hours.
My solution:
I found the cos equation: H(t)=-6cos(π/6t)+16
then..
Let π/6t=Θ
18=-6cosΘ+16
18-16=-6cosΘ
Θ=1.230959417
Let Θ=π/6t
π/6t=1.230959417
Then I don't know what's next....
##### Share on other sites
You're assuming that the tide is at low when the 24 hours starts. Did the problem specify what the water level should be at $t=0$?
##### Share on other sites
You're assuming that the tide is at low when the 24 hours starts. Did the problem specify what the water level should be at $t=0$?
it is also in low tide ...
##### Share on other sites
Alright -- your function $H(t)=-6\cos(\frac{\pi}{6} t)+16$ (where $t$ is in hours) seems like a good model.
Mathematically speaking, we'll want to restrict the domain to $0\le t \le 24$ since we're only dealing with the first 24 hours of tide. If you don't make the restriction, then solving for $t$ will give solutions outside the time frame the problem wants. We'll keep this in mind throughout the problem.
You started right by substituting $H(t)=18$.
So now we have $18=-6\cos(\frac{\pi}{6} t)+16$.
You can make another substitution if you want, but that might complicate the problem. We're just trying to find solutions for $t$, by getting it by itself. At least try to get it down to where we have the $\cos(\frac{\pi}{6} t)$ part alone on one side. See if you can continue from there.
##### Share on other sites
Alright -- your function $H(t)=-6\cos(\frac{\pi}{6} t)+16$ (where $t$ is in hours) seems like a good model.
Mathematically speaking, we'll want to restrict the domain to $0\le t \le 24$ since we're only dealing with the first 24 hours of tide. If you don't make the restriction, then solving for $t$ will give solutions outside the time frame the problem wants. We'll keep this in mind throughout the problem.
You started right by substituting $H(t)=18$.
So now we have $18=-6\cos(\frac{\pi}{6} t)+16$.
You can make another substitution if you want, but that might complicate the problem. We're just trying to find solutions for $t$, by getting it by itself. At least try to get it down to where we have the $\cos(\frac{\pi}{6} t)$ part alone on one side. See if you can continue from there.
Thats what I did (above), I calculated for the "t" but it says on the problem "Calculate the times at which the water level is at 18m during the first 24 hours." so it means that there's more solution... but i don't know how to find the others.
##### Share on other sites
Thats what I did (above), I calculated for the "t" but it says on the problem "Calculate the times at which the water level is at 18m during the first 24 hours." so it means that there's more solution... but i don't know how to find the others.
There are 4 solutions. Look at a graph of the function.
Where the line crosses our cosine function is where the solutions are.
Once you find the first, call it $t_a$, you can find the other three. Your calculation seems correct.
I'll give you the second solution: $12-t_a$. Do you understand how this works out?
EDIT: Better image.
Edited by Amaton
##### Share on other sites
There are 4 solutions. Look at a graph of the function.
Where the line crosses our cosine function is where the solutions are.
Once you find the first, call it $t_a$, you can find the other three. Your calculation seems correct.
I'll give you the second solution: $12-t_a$. Do you understand how this works out?
EDIT: Better image.
so the next solution would be $12+t_a$ and $24-t_a$??????
then i'll just substitute...
thanks!
Edited by gwiyomi17
##### Share on other sites
Sorry for the interruption but once you find the value of the angle in one full cycle (12hr.) the rest can be found just by adding the period(12hr.) since you're asked for 24hrs.
## Create an account
Register a new account | 2022-12-07 16:47:10 | {"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": 0.750106692314148, "perplexity": 470.1874734346913}, "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/1669446711200.6/warc/CC-MAIN-20221207153419-20221207183419-00111.warc.gz"} |
http://bricks.stackexchange.com/questions/2962/how-to-decipher-lego-building-instructions-filenames | # How to Decipher Lego Building Instructions Filenames?
On the official site for downloading lego instructions, the filenames are often quite cryptic, and many sets have as many as six different files you can download and sometimes it's hard to tell which one you want. Can anyone enumerate what all those various part of the filename mean?
Examples 1:
Woody's Roundup! 7594
BI 3005/48 - 7594 V 110 2/2 - Download size: 5.94 Mb
BI 3005/60 - 7594 V 110 1/2 - Download size: 5.91 Mb
BI 3005/60 - 7594 V 140 1/2 - Download size: 5.91 Mb
BI 3005/60 - 7594 V 29 1/2 - Download size: 5.87 Mb
BI 3005/60 - 7594 V 39 1/2 - Download size: 5.87 Mb
Here's what I've figured out so far BI = building instructins 3005 = set number 48 or 60 = ?? 7594 = ?? V = version 110, 140, etc = ?? 29 = A4 paper 39 = US letter paper 1/2 and 2/2 = the booklet number
Example 2:
Batman™: Arkham Asylum Breakout 10937
Now this one's even crazier 68+4* huh?? And it kind of looks like all the files are duplicated? Or is the second one possibly a newer or bugfixed version?
-
Apparently the number atfer the "/" (48,60...) seems to be the number of pages – Joubarc Mar 3 '14 at 20:26
Also, the page itself says: Building instructions labeled "NA" or "V39" may be printed on US standard letter size paper (8½ in × 11 in, 215.9 mm × 279.4 mm). Building instructions labeled "IN" or "V29" may be printed on EU standard A4 paper (210 mm × 297mm, 8.3 in × 11.7 in.) – Joubarc Mar 3 '14 at 20:28
Based on Jobarc's comment, it appears the numbers with plus signs between them after the slash, eg 68+4 (72) and 80+4 (84) are number of pages as well, though I'm not sure why in some cases they break out the numbers with an extra sheet (4 pages) separate – Jessica Brown Mar 4 '14 at 23:26
After emailing back and forth with LEGO, I think I have an acceptable answer:
Example:
BI 3005/48 - 7594 V 110 2/2 - Download size: 5.94 Mb
^ ^ ^ ^ ^ ^ ^
1 2 3 4 5 6 7
1. Stands for Building Instructions
2. A number for internal usage by different LEGO teams.
• You'll see that many different sets share this number, and all the numbers are fairly close (generally in the early 3000's). My guess is that it has something to do with whoever drafted and finalized the instruction manual.
3. This number shows the number of pages in the manual.
• When the number is show x+y, that means x building pages, and y instructional pages (e.g. what to do with bricks, or what bags are needed).
4. This is the set number.
5. This number refers to the page format of the instruction manual.
• V29 means A1 paper format.
• V39 means letter paper format.
• Other Vs (like 110, 140, etc) are different file formats. LEGO said "those are the types of versions throughout our country that we ship to". I'm guessing that they are related to the official formats for the manuals that are included with the kits.
6. The booklet number: x/y means x booklet number out of y booklets.
7. The file size of the instruction in megabytes (not megabits, as it would seem to imply).
- | 2015-01-27 14:27:33 | {"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": 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": 0.4098970890045166, "perplexity": 3911.5839127782974}, "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-2015-06/segments/1422121981339.16/warc/CC-MAIN-20150124175301-00029-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=User:Temperal/The_Problem_Solver%27s_Resource11&oldid=17933 | # User:Temperal/The Problem Solver's Resource11
The Problem Solver's Resource
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 11.
## Advanced Number Theory
These are Olympiad-level theorems and properties of numbers that are routinely used on the IMO and other such competitions.
### Jensen's Inequality
For a convex function $f(x)$ and real numbers $a_1,a_2,a_3,a_4\ldots,a_n$ and $x_1,x_2,x_3,x_4\ldots,x_n$, the following holds:
$$\sum_{i=1}^{n}a_i\cdot f(x_i)\ge f(\sum_{i=1}^{n}a_i\cdot x_i)$$
### Holder's Inequality
For positive real numbers $a_{i_{j}}, 1\le i\le m, 1\le j\le n be$, the following holds: $\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m} ===Muirhead's Inequality=== For a sequence$A$that majorizes a sequence$B$, then given a set of positive integers$x_1,x_2,\ldots,x_n$, the following holds: <cmath>\sum_{sym} {x_1}^{a_1}{x_2}^{a_2}\ldots {x_n}^{a_n}\geq \sum_{sym} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}</cmath> ===Rearrangement Inequality=== For any multi sets$ (Error compiling LaTeX. ! Missing $inserted.){a_1,a_2,a_3\ldots,a_n}$and${b_1,b_2,b_3\ldots,b_n}$,$a_1b_1+a_2b_2+\ldots+a_nb_n$is maximized when$a_k$is greater than or equal to exactly$i$of the other members of$A$, then$b_k$is also greater than or equal to exactly$i$of the other members of$B$. ===Newton's Inequality=== For non-negative real numbers$x_1,x_2,x_3\ldots,x_n$and$0 < k < n$the following holds:
<cmath>d_k^2 \ge d_{k-1}d_{k+1}</cmath>,
with equality exactly iff all$(Error compiling LaTeX. ! Missing$ inserted.)x_i$are equivalent. ===Mauclarin's Inequality=== For non-negative real numbers$x_1,x_2,x_3 \ldots, x_n$, the following holds: <cmath>x_1 \ge \sqrt[2]{x_2} \ge \sqrt[3]{x_3}\ldots \ge \sqrt[n]{x_n}</cmath> with equality iff all$ (Error compiling LaTeX. ! Missing $inserted.)x_i$ are equivalent. Back to page 10 | Last page (But also see the tips and tricks page, and the competition! | 2020-11-26 05:02:07 | {"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": 19, "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": 0.6965010762214661, "perplexity": 4026.344736105271}, "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-2020-50/segments/1606141186414.7/warc/CC-MAIN-20201126030729-20201126060729-00415.warc.gz"} |
https://labs.tib.eu/arxiv/?author=Michael%20Peth | • ### CANDELS Multiwavelength Catalogs: Source Identification and Photometry in the CANDELS Extended Groth Strip(1703.05768)
March 16, 2017 astro-ph.GA
We present a 0.4-8$\mu$m multi-wavelength photometric catalog in the Extended Groth Strip (EGS) field. This catalog is built on the Hubble Space Telescope (HST) WFC3 and ACS data from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS), and it incorporates the existing HST data from the All-wavelength Extended Groth strip International Survey (AEGIS) and the 3D-HST program. The catalog is based on detections in the F160W band reaching a depth of F160W=26.62 AB (90% completeness, point-sources). It includes the photometry for 41457 objects over an area of $\approx 206$ arcmin$^2$ in the following bands: HST ACS F606W and F814W; HST WFC3 F125W, F140W and F160W; CFHT/Megacam $u^*$, $g'$, $r'$, $i'$ and $z'$; CFHT/WIRCAM $J$, $H$ and $K_\mathrm{S}$; Mayall/NEWFIRM $J1$, $J2$, $J3$, $H1$, $H2$, $K$; Spitzer IRAC $3.6\mu$m, $4.5\mu$m, $5.8\mu$m and $8.0\mu$m. We are also releasing value-added catalogs that provide robust photometric redshifts and stellar mass measurements. The catalogs are publicly available through the CANDELS repository.
• ### Galaxy Zoo: Quantitative Visual Morphological Classifications for 48,000 galaxies from CANDELS(1610.03070)
Oct. 10, 2016 astro-ph.GA
We present quantified visual morphologies of approximately 48,000 galaxies observed in three Hubble Space Telescope legacy fields by the Cosmic And Near-infrared Deep Extragalactic Legacy Survey (CANDELS) and classified by participants in the Galaxy Zoo project. 90% of galaxies have z < 3 and are observed in rest-frame optical wavelengths by CANDELS. Each galaxy received an average of 40 independent classifications, which we combine into detailed morphological information on galaxy features such as clumpiness, bar instabilities, spiral structure, and merger and tidal signatures. We apply a consensus-based classifier weighting method that preserves classifier independence while effectively down-weighting significantly outlying classifications. After analysing the effect of varying image depth on reported classifications, we also provide depth-corrected classifications which both preserve the information in the deepest observations and also enable the use of classifications at comparable depths across the full survey. Comparing the Galaxy Zoo classifications to previous classifications of the same galaxies shows very good agreement; for some applications the high number of independent classifications provided by Galaxy Zoo provides an advantage in selecting galaxies with a particular morphological profile, while in others the combination of Galaxy Zoo with other classifications is a more promising approach than using any one method alone. We combine the Galaxy Zoo classifications of "smooth" galaxies with parametric morphologies to select a sample of featureless disks at 1 < z < 3, which may represent a dynamically warmer progenitor population to the settled disk galaxies seen at later epochs.
• ### Evolution of Star-formation Properties of High-redshift Cluster Galaxies since $z = 2$(1508.01294)
Aug. 6, 2015 astro-ph.CO, astro-ph.GA
Using a stellar mass limited sample of $\sim 46,600$ galaxies ($M_* > 10^{9.1}\,M_{\odot}$) at $0.5 < z < 2$, we show that the tellar mass, rather than the environment, is the main parameter controlling quenching of star formation in galaxies with $M_* > 10^{10}\,M_{\odot}$ out to $z=2$. On the other hand, the environmental quenching becomes efficient at $z < 1$ regardless of galaxy mass, and it serves as a main star formation quenching mechanism for lower mass galaxies. Our result is based on deep optical and near-infrared imaging data over 2800 arcmin$^2$, enabling us to negate cosmic variance and identify 46 galaxy cluster candidates with $M \sim 10^{14}\,M_{\odot}$. From $M_* \sim 10^{9.5}$ to $10^{10.5}\,M_{\odot}$, the fraction of quiescent galaxies increases by a factor of $\sim 10$ over the entire redshift range, but the difference between cluster and field environment is negligible. Rapid evolution in the quiescent fraction is seen from $z=2$ to $z=1.3$ for massive galaxies suggesting a build-up of massive quiescent galaxies at $z > 1.3$. For galaxies with $M_* < 10^{10}\,M_{\odot}$ at $z < 1.0$, the quiescent fraction is found to be as much as a factor of 2 larger in clusters than in field, showing the importance of environmental quenching in low mass galaxies at low redshift. Most high mass galaxies are already quenched at $z > 1$, therefore environmental quenching does not play a significant role for them, although the environmental quenching efficiency is nearly identical between high and low mass galaxies.
• ### Diverse Structural Evolution at z > 1 in Cosmologically Simulated Galaxies(1409.1583)
July 2, 2015 astro-ph.CO, astro-ph.GA
From mock Hubble Space Telescope images, we quantify non-parametric statistics of galaxy morphology, thereby predicting the emergence of relationships among stellar mass, star formation, and observed rest-frame optical structure at 1 < z < 3. We measure automated diagnostics of galaxy morphology in cosmological simulations of the formation of 22 central galaxies with 9.3 < log10 M_*/M_sun < 10.7. These high-spatial-resolution zoom-in calculations enable accurate modeling of the rest-frame UV and optical morphology. Even with small numbers of galaxies, we find that structural evolution is neither universal nor monotonic: galaxy interactions can trigger either bulge or disc formation, and optically bulge-dominated galaxies at this mass may not remain so forever. Simulated galaxies with M_* > 10^10 M_sun contain relatively more disc-dominated light profiles than those with lower mass, reflecting significant disc brightening in some haloes at 1 < z < 2. By this epoch, simulated galaxies with specific star formation rates below 10^-9.7 yr^-1 are more likely than normal star-formers to have a broader mix of structural types, especially at M_* > 10^10 M_sun. We analyze a cosmological major merger at z ~ 1.5 and find that the newly proposed MID morphology diagnostics trace later merger stages while G-M20 trace earlier ones. MID is sensitive also to clumpy star-forming discs. The observability time of typical MID-enhanced events in our simulation sample is less than 100 Myr. A larger sample of cosmological assembly histories may be required to calibrate such diagnostics in the face of their sensitivity to viewing angle, segmentation algorithm, and various phenomena such as clumpy star formation and minor mergers.
• ### A Transition Mass in the Local Tully-Fisher Relation(1506.04144)
June 12, 2015 astro-ph.GA
We study the stellar mass Tully-Fisher relation (TFR, stellar mass versus rotation velocity) for a morphologically blind selection of emission line galaxies in the field at redshifts 0.1 $<$ z $<$ 0.375. Kinematics ($\sigma_g$, V$_{rot}$) are measured from emission lines in Keck/DEIMOS spectra and quantitative morphology is measured from V- and I-band Hubble images. We find a transition stellar mass in the TFR, $\log$ M$_*$ = 9.5 M$_{\odot}$. Above this mass, nearly all galaxies are rotation-dominated, on average more morphologically disk-like according to quantitative morphology, and lie on a relatively tight TFR. Below this mass, the TFR has significant scatter to low rotation velocity and galaxies can either be rotation-dominated disks on the TFR or asymmetric or compact galaxies which scatter off. We refer to this transition mass as the "mass of disk formation", M$_{\mathrm{df}}$ because above it all star-forming galaxies form disks (except for a small number of major mergers and highly star-forming systems), whereas below it a galaxy may or may not form a disk.
• ### CANDELS/GOODS-S, CDFS, ECDFS: Photometric Redshifts For Normal and for X-Ray-Detected Galaxies(1409.7119)
Sept. 24, 2014 astro-ph.GA
We present photometric redshifts and associated probability distributions for all detected sources in the Extended Chandra Deep Field South (ECDFS). The work makes use of the most up-to-date data from the Cosmic Assembly Near-IR Deep Legacy Survey (CANDELS) and the Taiwan ECDFS Near-Infrared Survey (TENIS) in addition to other data. We also revisit multi-wavelength counterparts for published X-ray sources from the 4Ms-CDFS and 250ks-ECDFS surveys, finding reliable counterparts for 1207 out of 1259 sources ($\sim 96\%$). Data used for photometric redshifts include intermediate-band photometry deblended using the TFIT method, which is used for the first time in this work. Photometric redshifts for X-ray source counterparts are based on a new library of AGN/galaxy hybrid templates appropriate for the faint X-ray population in the CDFS. Photometric redshift accuracy for normal galaxies is 0.010 and for X-ray sources is 0.014, and outlier fractions are $4\%$ and $5.4\%$ respectively. The results within the CANDELS coverage area are even better as demonstrated both by spectroscopic comparison and by galaxy-pair statistics. Intermediate-band photometry, even if shallow, is valuable when combined with deep broad-band photometry. For best accuracy, templates must include emission lines.
• ### Galaxy Zoo: CANDELS Barred Disks and Bar Fractions(1409.1214)
Sept. 3, 2014 astro-ph.GA
The formation of bars in disk galaxies is a tracer of the dynamical maturity of the population. Previous studies have found that the incidence of bars in disks decreases from the local Universe to z ~ 1, and by z > 1 simulations predict that bar features in dynamically mature disks should be extremely rare. Here we report the discovery of strong barred structures in massive disk galaxies at z ~ 1.5 in deep rest-frame optical images from CANDELS. From within a sample of 876 disk galaxies identified by visual classification in Galaxy Zoo, we identify 123 barred galaxies. Selecting a sub-sample within the same region of the evolving galaxy luminosity function (brighter than L*), we find that the bar fraction across the redshift range 0.5< z < 2 (f_bar = 10.7 +6.3 -3.5% after correcting for incompleteness) does not significantly evolve. We discuss the implications of this discovery in the context of existing simulations and our current understanding of the way disk galaxies have evolved over the last 11 billion years.
• ### Properties of Submillimeter Galaxies in the CANDELS GOODS-S Field(1402.3268)
Feb. 13, 2014 astro-ph.CO, astro-ph.GA
We derive physical properties of 10 submillimeter galaxies located in the CANDELS coverage of the GOODS-S field. The galaxies were first identified as submillimeter sources with the LABOCA bolometer and subsequently targeted for 870um continuum observation with ALMA. The high angular resolution of the ALMA imaging allows secure counterparts to be identified in the CANDELS multiband dataset. The CANDELS data provide deep photometric data from UV through near-infrared wavelengths. Using synthetic spectral energy distributions, we derive photometric redshifts, stellar masses, extinction, ages, and the star formation history. The redshift range is z=1.65-4.76, with two of the galaxies located at z>4. Two SMG counterparts have stellar masses 2-3 orders of magnitude lower than the rest. The remaining SMG counterparts have stellar masses around 1x10^11 Msun. The stellar population in the SMGs is typically older than the expected duration of the submillimeter phase, suggesting that the star formation history of submillimeter galaxies is more complex than a single burst. Non-parametric morphology indices suggest that the SMG counterparts are among the most asymmetric systems compared with galaxies of the same stellar mass and redshift. The HST images shows that 3 of the SMGs are associated with on-going mergers. The remaining counterparts are isolated. Estimating the dust and molecular gas mass from the submm fluxes, and comparing with our stellar masses shows that the molecular gas mass fraction of SMGs is ~28% and that the final stellar mass is likely to be (1-2)x10^11 Msun.
• ### A Critical Assessment of Photometric Redshift Methods: A CANDELS Investigation(1308.5353)
Aug. 24, 2013 astro-ph.CO
We present results from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) photometric redshift methods investigation. In this investigation, the results from eleven participants, each using a different combination of photometric redshift code, template spectral energy distributions (SEDs) and priors, are used to examine the properties of photometric redshifts applied to deep fields with broad-band multi-wavelength coverage. The photometry used includes U-band through mid-infrared filters and was derived using the TFIT method. Comparing the results, we find that there is no particular code or set of template SEDs that results in significantly better photometric redshifts compared to others. However, we find codes producing the lowest scatter and outlier fraction utilize a training sample to optimize photometric redshifts by adding zero-point offsets, template adjusting or adding extra smoothing errors. These results therefore stress the importance of the training procedure. We find a strong dependence of the photometric redshift accuracy on the signal-to-noise ratio of the photometry. On the other hand, we find a weak dependence of the photometric redshift scatter with redshift and galaxy color. We find that most photometric redshift codes quote redshift errors (e.g., 68% confidence intervals) that are too small compared to that expected from the spectroscopic control sample. We find that all codes show a statistically significant bias in the photometric redshifts. However, the bias is in all cases smaller than the scatter, the latter therefore dominates the errors. Finally, we find that combining results from multiple codes significantly decreases the photometric redshift scatter and outlier fraction. We discuss different ways of combining data to produce accurate photometric redshifts and error estimates.
• ### CANDELS Multi-wavelength Catalogs: Source Detection and Photometry in the GOODS-South Field(1308.4405)
Aug. 20, 2013 astro-ph.CO, astro-ph.GA
We present a UV-to-mid infrared multi-wavelength catalog in the CANDELS/GOODS-S field, combining the newly obtained CANDELS HST/WFC3 F105W, F125W, and F160W data with existing public data. The catalog is based on source detection in the WFC3 F160W band. The F160W mosaic includes the data from CANDELS deep and wide observations as well as previous ERS and HUDF09 programs. The mosaic reaches a 5$\sigma$ limiting depth (within an aperture of radius 0.17 arcsec) of 27.4, 28.2, and 29.7 AB for CANDELS wide, deep, and HUDF regions, respectively. The catalog contains 34930 sources with the representative 50% completeness reaching 25.9, 26.6, and 28.1 AB in the F160W band for the three regions. In addition to WFC3 bands, the catalog also includes data from UV (U-band from both CTIO/MOSAIC and VLT/VIMOS), optical (HST/ACS F435W, F606W, F775W, F814W, and F850LP), and infrared (HST/WFC3 F098M, VLT/ISAAC Ks, VLT/HAWK-I Ks, and Spitzer/IRAC 3.6, 4.5, 5.8, 8.0 $\mu$m) observations. The catalog is validated via stellar colors, comparison with other published catalogs, zeropoint offsets determined from the best-fit templates of the spectral energy distribution of spectroscopically observed objects, and the accuracy of photometric redshifts. The catalog is able to detect unreddened star-forming (passive) galaxies with stellar mass of 10^{10}M_\odot at a 50% completeness level to z$\sim$3.4 (2.8), 4.6 (3.2), and 7.0 (4.2) in the three regions. As an example of application, the catalog is used to select both star-forming and passive galaxies at z$\sim$2--4 via the Balmer break. It is also used to study the color--magnitude diagram of galaxies at 0<z<4.
• ### CANDELS Multiwavelength catalogs: Source Identification and Photometry in the CANDELS UKIDSS Ultra-Deep Survey Field(1305.1823)
May 6, 2013 astro-ph.CO
We present the multiwavelength - ultraviolet to mid-infrared - catalog of the UKIDSS Ultra-Deep Survey (UDS) field observed as part of the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS). Based on publicly available data, the catalog includes: the CANDELS data from the Hubble Space Telescope (near-infrared WFC3 F125W and F160W data and visible ACS F606W and F814W data), u-band data from CFHT/Megacam, B, V, Rc, i' and z' band data from Subaru/Suprime-Cam, Y and Ks band data from VLT/HAWK-I, J, H and K bands data from UKIDSS (Data Release 8), and Spitzer/IRAC data (3.6, 4.5 from SEDS, 5.8 and 8.0um from SpUDS). The present catalog is F160W-selected and contains 35932 sources over an area of 201.7 square arcmin and includes radio and X-ray detected sources and spectroscopic redshifts available for 210 sources.
• ### Near Infrared Photometric properties of 130,000 Quasars: An SDSS-UKIDSS matched catalog(1012.4187)
Dec. 19, 2010 astro-ph.CO
We present a catalog of over 130,000 quasars candidates with NIR photometric properties, with an areal coverage of approximately 1,200 deg^2. This is achieved by matching the Sloan Digital Sky Survey (SDSS) in the optical ugriz bands, to the UKIRT Infrared Digital Sky Survey (UKIDSS) Large Area Survey (LAS) in the near-infrared YJHK bands. We match the ~1 million SDSS DR6 Photometric Quasar catalog to Data Release 3 of the UKIDSS LAS (ULAS), and produce a catalog with 130,827 objects with detections in one or more NIR bands, of which 74,351 objects have optical and K-band detections and 42,133 objects have the full 9-band photometry. The majority ~85 of the SDSS objects were not matched simply because there were not covered by the ULAS. Our matched catalog has a surface density of ~53 deg^-2 for K >18.27 objects; tests using our matched catalog, along with data from the UKIDSS DXS, implies that our limiting magnitude is i ~ 20.6. Color-redshift diagrams, for the optical and NIR, show the close agreement between our matched catalog and recent quasar color models at redshift z > 2.0, while at higher redshifts, the models generally appear to be bluer than the mean observed quasar colors. The gJK and giK color-spaces are used to examine methods of differentiating between stars and (mid-redshift) quasars, key to currently ongoing quasar surveys. Finally, we report on the NIR photometric properties of high, z>4.6, and very high, z>5.7, redshift previously discovered quasars. | 2021-04-14 10:26:47 | {"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": 0.5646805167198181, "perplexity": 4809.086635032159}, "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/1618038077810.20/warc/CC-MAIN-20210414095300-20210414125300-00601.warc.gz"} |
https://martin-ueding.de/posts/efficiency-of-pow-function/ | # Efficiency of the pow function
Someone said that using pow(x, 2) is always more inefficient than using x * x. Well, there are two things to remember:
1. Do not "optimize" without measurement.
2. Measure with full compiler optimization.
So this is exactly what I did then. This is a simple C++11 program that uses pow(x, 2) and x * x. I highlighted the lines in questions. The calculations with test are in place so that the compiler does not optimize away the code, which it would do otherwise.
#include <chrono>
#include <cmath>
#include <iostream>
int main() {
double test {0};
unsigned iter_max {100000000};
for (unsigned iter {0}; iter < iter_max; iter++) {
test += std::pow(iter, 2);
}
double time_in_seconds = std::chrono::duration_cast<std::chrono::milliseconds>
std::cout << "Pow: " << time_in_seconds << std::endl;
for (unsigned iter {0}; iter < iter_max; iter++) {
test += iter * iter;
}
time_in_seconds = std::chrono::duration_cast<std::chrono::milliseconds>
std::cout << "Multiplication: " << time_in_seconds << std::endl;
std::cout << test << std::endl;
return 0;
}
## Results
If you compile that without optimization, pow is clearly slower:
\$ clang++ -std=c++11 pow.cpp -o pow; and ./pow
Pow: 0.272
Multiplication: 0.042
Now I compiled this with clang++ using its -O3 optimization. When I did this on 2014-05-21, pow was significantly faster. I have revisited this on 2014-07-10, where pow just a tiny bit slower than the multiplication. Interesting.
I also tested with g++ and found that pow is significantly slower than the multiplication. To be fair, I ran each one a couple times since the overall time varies. With g++, the multiplication is actually faster. To get meaningful results, I ran each one 10 times and too mean and standard deviation with this Python script:
#!/usr/bin/python3
# -*- coding: utf-8 -*-
import subprocess
import numpy
import unitprint
def compile_cpp(compiler):
subprocess.check_call([compiler, '-std=c++11', '-O3', 'pow.cpp', '-o', 'pow'])
def get_results():
words = subprocess.check_output(['./pow']).decode().strip().split()
return float(words[1]), float(words[3])
def bootstrap(compiler, runs=10):
compile_cpp(compiler)
times_pow = []
times_mul = []
for i in range(runs):
time_pow, time_mul = get_results()
times_pow.append(time_pow)
times_mul.append(time_mul)
mean_pow = numpy.mean(times_pow)
mean_mul = numpy.mean(times_mul)
std_pow = numpy.std(times_pow)
std_mul = numpy.std(times_mul)
return unitprint.siunitx(mean_pow, std_pow), \
unitprint.siunitx(mean_mul, std_mul)
def main():
for compiler in ['g++', 'clang++']:
print(compiler, *bootstrap(compiler))
if __name__ == "__main__":
main()
These are the results:
Compiler pow / s x * x / s
g++ 2.90 ± 0.03 1.28 ± 0.01
clang++ 1.272 ± 0.008 1.268 ± 0.008
With clang++, there is a tiny difference between pow and multiplication, it is not really significant though, since it is just half a standard deviation. g++ takes more than twice as long for using pow. I can absolutely understand that people using g++ will try to avoid the pow function.
However, and that is my point, the statement that pow is always slower than multiplication does not really hold. I consider the results from clang++ to be on par. Please test your application on your compiler and see what is faster in reality, not in theory.
## Source of pow
And if you look at the source of pow(), you will see that they have thought about it:
112 /* First see whether y' is a natural number. In this case we
113 can use a more precise algorithm. */
Then it jumps to 9 where it says:
136 9: /* OK, we have an integer value for y. Unless very small
137 (we use < 8), use the algorithm for real exponent to avoid
138 accumulation of errors. */
` | 2020-07-11 08:17:12 | {"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": 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": 0.41801515221595764, "perplexity": 7028.818343195117}, "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-2020-29/segments/1593655924908.55/warc/CC-MAIN-20200711064158-20200711094158-00460.warc.gz"} |
http://support.sas.com/documentation/cdl/en/statug/66859/HTML/default/statug_mi_details06.htm | ### Monotone Methods for Data Sets with Monotone Missing Patterns
For data sets with monotone missing data patterns, you can use monotone methods to impute missing values for the variables. A monotone method creates multiple imputations by imputing missing values sequentially over the variables taken one at a time.
For example, with variables , , …, (in that order) in the VAR statement, a monotone method sequentially simulates a draw for missing values for variables , …, . That is, the missing values are imputed by using the sequence
where is the set of observed values, is the set of simulated parameters for the conditional distribution of given covariates constructed from variables , , …, , and is the set of imputed values.
The missing values for the leading variable are not imputed, and missing values for , …, are not imputed for those observations with missing values. For each subsequent variable with missing values, the corresponding imputation method is used to fit a model with covariates constructed from its preceding variables . The observed observations for , which include only observations with observed values for , are used in the model fitting. With this resulting model, a new model is drawn and then used to impute missing values for .
You can specify a separate monotone method for each imputed variable. If a method is not specified for the variable, then the default method is used. That is, a regression method is used for a continuous variable and a discriminant function method is used for a classification variable. For each imputed variable, you can also specify a set of covariates that are constructed from its preceding variables. If a set of covariates is not specified for the variable, all preceding variables in the VAR list are used as covariates.
You can use a regression method, a predictive mean matching method, or a propensity score method to impute missing values for a continuous variable; a logistic regression method for a classification variable with a binary or ordinal response; and a discriminant function method for a classification variable with a binary or nominal response. See the sections Monotone and FCS Regression Methods, Monotone and FCS Predictive Mean Matching Methods, Monotone Propensity Score Method, Monotone and FCS Discriminant Function Methods, and Monotone and FCS Logistic Regression Methods for these methods. | 2018-04-21 00:57:12 | {"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": 0.8208807110786438, "perplexity": 881.7265338479211}, "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-2018-17/segments/1524125944848.33/warc/CC-MAIN-20180420233255-20180421013255-00277.warc.gz"} |
https://www.physicsforums.com/threads/absolute-value-question.207390/ | # Absolute value question
1. Jan 5, 2008
### Quinn Morris
1. The problem statement, all variables and given/known data
25|x| = x^2 + 144
2. Relevant equations
none
3. The attempt at a solution
okay well, i'm not quite sure what to do, do i try to isolate the |x|? and then break it up into a postive and negative?
|x| = (x^2 + 144)/25 ?
but from here i become lost.....
2. Jan 5, 2008
### HallsofIvy
Staff Emeritus
Why not do what you just said? You have "isolated" |x|, now break it into positive and negative parts.
If $x\ge 0$ then |x|= x so the equation becomes $x= (x^2+ 144)/25$ or $25x= x^2+ 144$. Solve that quadratic equation. Remember that only $x\ge 0$ are valid solutions.
If x< 0, then |x|= -x so the equation becomes $-x= (x^2+ 144)/25$ or $-25x= x^2+ 144$. Solve that quadratic equation. Remember that only x< 0 are valid solutions.
You might notice that it is easier to first break into two cases and then solve for x.
3. Jan 5, 2008
### Quinn Morris
okay so should my answer be x = +/- 16, +/- 9?
4. Jan 5, 2008
### Gib Z
It's quite easy to check yourself =P Especially when checking one of the positive solutions gets rid of the negative counterpart as well.
5. Jan 5, 2008
k thx | 2017-01-18 16:21:45 | {"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": 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": 0.6499651074409485, "perplexity": 1448.3226300455374}, "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-2017-04/segments/1484560280308.24/warc/CC-MAIN-20170116095120-00574-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://blog.tmorris.net/posts/youd-naturally-write-flatmap-yourself-if-asked-the-question/index.html | λ Tony's blog λ
The weblog of Tony Morris
# You’d naturally write flatMap yourself if asked the question
Posted on June 25, 2008, in Programming
Many people struggle to understand those fluffy things called Monads and why they are important. I’m not going to attempt to alleviate this to a large degree, but I have had a recent success with a friend in having them attempt to write a familiar Scala function as a method. That is, the [List.flatten](http://www.scala-lang.org/docu/files/api/scala/List\$object.html#flatten(List[List[A]])) function, which simply takes a List of List of some type A and returns a List[A]. It does this by concatenating all the lists together. I think most people are familiar with this function and have a good mental model of how it works. If this is the case and you are less confident about List.flatMap, then I hope to bring a point to your attention that might help you bring it home.
If you take a look at List.sort, you see it takes a function (A, A) => Boolean. This is because the type parameter to List is ‘just any A’. It would be nice if it was ‘any A, so long as it has defined order’. That way, you can just call list.sort and be done with it. This would require the creation of a new class with a more restricted type parameter; for example, you might pass the function at list creation time, like you do with a TreeMap.
Imagine writing List.flatten on the List[A] class as a method using the same technique. You wouldn’t be able to, since the method belongs on a List[List[A]]. You’d need to write the method such that it takes an argument: A => List[A] before you could then call flatten. When you have a List[List[A]], you’d just call this method with the identity function x => x to obtain your resulting List[A]. Here is how the method would look:
def flatten(f: A => List[A]): List[A] = ...
Guess what?! This method is flatMap! Let’s ignore the fact that the Scala API is significantly broken, including the List.flatMap method using Iterable in place of List here. Notice though, that flatMap is actually generalised by taking another type parameter, so we have just invented an unnecessarily specialised version of flatMap. Let’s fix it:
def flatten[B](f: A => List[B]): List[B] = ...
Woot woot! In other words, flatten(list) is equivalent to list flatMap (x => x). Furthermore, this should hold for more than just list, but for other type constructors too (sadly, Option is missing a flatten function: Option[Option[A]] => Option[A]). This is a special relationship that all monads have (join is a synonym for flatten and bind is a synonym for flatMap):
join = bind id
We can express this using Reductio (the Scala API of course!):
val prop_flat = prop((t: List[List[Int]]) => List.flatten(t) == t.flatMap(x => x))
// OK passed 100 tests.
I hope this helps. If it doesn’t, ask a question or ignore my ranting :) | 2017-07-27 02:52:22 | {"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": 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": 0.4185095727443695, "perplexity": 1362.2274380552979}, "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-2017-30/segments/1500549426951.85/warc/CC-MAIN-20170727022134-20170727042134-00477.warc.gz"} |
http://quant.stackexchange.com/questions/240/is-it-possible-to-use-a-series-of-option-prices-to-predict-the-most-likely-path | # Is it possible to use a series of option prices to predict the most likely path of an asset?
If you have a series of options, with the expires spaced let's say one week between them, and you search for each expiration date the option with the smallest premium, would the series of strikes represent the current market predicted path of the asset?
Can you use that information to speculate the spot price?
I believe that this paper is about a similar approach: http://ideas.repec.org/p/ihs/ihsesp/104.html
-
If you think of a path as a series of ranges then your idea kind of makes sense. However, I don't think you would get a path out of this approach, just a series of ranges.
Example: Taking one expiry, the prices in a chain imply a range of price movements between today and expiration.
Take the SPX(at say 1300) and VIX, for example, is at 15.8 and the SPX option chain that you are looking at is 30 days from expiry. That tells you that there is approximately a 68% probability that the SPX will be between approximately 1370 and 1230 at the end of 30 days, or a 68% probability that it will be within 1% of 1300 in 1 day.
Running this example on multiple chains would only expand the range(implied vol is increasing in later expiries), or contract the range(implied vol is decreasing in later expiries).
----- idea ----- If you had access to a standardized/liquid market of path dependant options, you might be able to narraw the range down somewhat.
If you did arrive at a narrow path estimate, it would change frequently with volatility... what would be the value of this path estimate?
-
Just would like to expand on user214's answer: you can use options to predict underlying in probabilistic sense. As you know option prices imply a certain distribution - you can find expected value for stock, and volatility around that value.
If you assume a particular distribution (for example normal distribution for returns) you can derive expected high (over some period of time), expected low, expected range, expected drawdown, probabilities for different paths, etc. That is not something you should do in practice, or do it and know the limitations of such model-based estimates (that is what I do in trading).
If you have access to exotic, particularly path-dependent options you can fit more complicated models, and figure out what they predict about the stock price. While you can fit more realistic models to only vanilla options, such fits are not robust, because they depend only on terminal distribution of the underlying, and not its path.
-
A further comment on user214's answer : the probability distribution of the future value of the index that you imply from option prices is its distribution under the (market) risk-neutral measure, which generally different from the true historical measure. In particular, option prices do not give information about the risk premium. There is a vast literature about this, but a good start is this paper from Chris Rogers and Steve Satchell.
Furthermore, European option prices give you information about the marginal distributions of the index at fixed maturities, but they give you no clue about the dynamical properties of the value process, that is, the distribution of the paths.
-
If you're asking "can I get a prediction of a future price from an option chain", then, no, I don't think so. The value of an option does not depend on the underlying stock's drift, or price expectation, because this expectation is already reflected in the stock's current price. Given the risk-free rate and the time to expiration, all that you can back out of the option price is the implied volatility.
The intuition is that we don't really value options in absolute terms, but in terms of the underlying stock.
- | 2013-12-20 04:00:33 | {"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": 0.8025645017623901, "perplexity": 609.7691651585217}, "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-2013-48/segments/1387345768998/warc/CC-MAIN-20131218054928-00068-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://control.com/worksheets/pressure-based-flowmeter-calculations/ | # Pressurebased flowmeter calculations
## Instrumentation and Process Control
• #### Question 1
A Pitot tube measuring the flow of glycerin (78.6 lb/ft$^{3}$) produces a differential pressure of 35 inches W.C. at a mass flow rate of 450 pounds per minute.
Calculate the differential pressure produced by this same Pitot tube at a flow rate of 310 lb/min.
Calculate the mass flow rate at a differential pressure of 12.6 “WC.
If the glycerin density happens to change from 78.6 lb/ft$^{3}$ to 75.1 lb/ft$^{3}$, calculate the mass flow rate at a differential pressure of 24.8 “WC.
• #### Question 2
A pneumatic DP transmitter measures the flow of gasoline through a an orifice plate, using a separate square root extractor relay to characterize the signal so that it may be displayed linearly on the receiver gauge (FI-20):
Based on other flow-measuring devices in this process, operations personnel have determined the actual flow rate through this pipe is 158 gallons per minute. FI-20, however, registers a flow of 142 gallons per minute. Based on the data you see in this illustration, determine the location of the calibration error.
• #### Question 3
A pneumatic DP transmitter measures the flow of gasoline through a an orifice plate, using a separate square root extractor relay to characterize the signal so that it may be displayed linearly on the receiver gauge (FI-20):
Based on other flow-measuring devices in this process, operations personnel have determined the actual flow rate through this pipe is 171 gallons per minute. FI-20, however, registers a flow of 184 gallons per minute. Based on the data you see in this illustration, determine the location of the calibration error.
• #### Question 4
A smart’’ DP transmitter with built-in square root characterization is used to measure flow through a pipe. The orifice plate range is 0 to 125 inches WC at 0 to 277 gallons per minute:
A technician removes the transmitter from service and tests it by applying several air pressures to the H’’ port while leaving the L’’ port vented. Here are the As-Found results:
$$\begin{array} {|l|l|} \hline Applied Pressure (“WC) & Current signal (mA) \\ \hline 0 & 4.020 \\ \hline 31.25 & 12.060 \\ \hline 62.5 & 15.390 \\ \hline 93.75 & 17.946 \\ \hline 125 & 20.100 \\ \hline \end{array}$$
• #### Question 5
A smart’’ DP transmitter with built-in square root characterization is used to measure flow through a pipe. The orifice plate range is 0 to 125 inches WC at 0 to 277 gallons per minute:
A technician removes the transmitter from service and tests it by applying several air pressures to the H’’ port while leaving the L’’ port vented. Here are the As-Found results:
$$\begin{array} {|l|l|} \hline Applied Pressure (“WC) & Current signal (mA) \\ \hline 0 & 5.431 \\ \hline 31.25 & 12.127 \\ \hline 62.5 & 15.404 \\ \hline 93.75 & 16.128 \\ \hline 125 & 20.064 \\ \hline \end{array}$$
• #### Question 6
A smart’’ DP transmitter with built-in square root characterization is used to measure flow through a pipe. The orifice plate range is 0 to 150 inches WC at 0 to 480 gallons per minute:
Operations personnel have strong reason to believe that the actual flow rate through this pipe is 160 GPM, yet the DCS registers a flow rate of 182.4 GPM. Based on this information, determine the likely source of calibration error in this system. Also determine whether this transmitter has square-root characterization enabled or not.
Additionally, suggest a good next step’’ to perform to either pinpoint the location of this problem or correct it.
• #### Question 7
A smart’’ DP transmitter and orifice plate were recently installed to measure flow through a pipe. The orifice plate range is 0 to 150 inches WC at 0 to 1000 gallons per minute:
Operations personnel register a flow rate of 699 gallons per minute on the display of their DCS, which they believe to be too much. Based on what you see here, do you think there is a problem, or is this new system working as it should? | 2020-10-22 03:30:30 | {"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": 0.4452494978904724, "perplexity": 1990.6270668157838}, "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-2020-45/segments/1603107878879.33/warc/CC-MAIN-20201022024236-20201022054236-00028.warc.gz"} |
https://qoto.org/@casualwp | @middaymoon fuck your tiny linux distros (consentually)
Defend Richard Stallman!
Two years ago, known Thought Criminal Richard M Stallman was falsely accused of defending rape in an Orwellian smear campaign, orchestrated by mainstream media at the behest of proprietary software vendors. 36 years fighting for your digital freedom, cancelled. It was so vicious that he resigned from his post as president of the Free Software Foundation. The FSF did nothing to protect or defend him. However, you can defend him!
On 21 March 2021, FSF board of directors re-instated Richard Stallman. In response, the media started a new smear campaign. A petition was created, calling for the forceful removal of RMS and the entire FSF board of directors. RMS has been wrongly accused of sexism, transphobia, ableism and a whole host of things intended to discredit him. Do not listen to any of it. Richard Stallman’s political notes and articles paint the picture of a man who has staunchly campaigned against bigotry in all its forms!
In response, we, the Free software movement, started our own petition. We wish for RMS to remain in his post, and for the FSF to hold their ground. We call for the FSF to defend Richard Stallman’s honour and his legacy. Richard Stallman is a human being, whose right to free speech was heavily suppressed. We must demonstrate our support of him to the FSF, loudly and clearly.
If you support Free Software, believe in freedom of speech, freedom of community and social justice (true social justice, where a person is treated with dignity and not cancelled just for their beliefs), sign your name here:
https://rms-support-letter.github.io/
The opposing petition calling for Richard’s removal will not be linked here, because it is important not to strengthen it. Boosting the search engine rankings of our opposition would only help them attack RMS. Similarly, their smear campaigns will not be linked here directly, only condemned!
Instructions for how to sign your name are on that page. If you represent a project, please put that in brackets and state your position. For instance, if you are John Doe and your project is named Foobar Libre, write John Doe (Foobar Libre developer) or e.g. John Doe (Foobar Libre founder and lead developer). If you are an FSF member (e.g. associate member), put that in brackets too.
If you are a member of a project/organisation that signed the anti-RMS list, it is especially important to state that you are from said project when signing the pro-RMS list. You should also talk to people in your project or organisation, and try to persuade them to change their minds!
In addition to signing your name, if you’re in a software project, get your project to officially come out in support of Richard! He needs every bit of support we can get. We, the Free Software movement, as activists must lend him all of our strength!
Don’t be fooled. If a Free Software project is on the anti-RMS list, that just means the leadership implemented said decision. It says nothing of the individuals inside said organisation.
Please also email the FSF and tell them you support Richard! The FSF’s contact details are here: https://www.fsf.org/about/contact/
Our opponents wish to destroy Free Software
Our opponent’s true target is not Richard Stallman; their real aim is to destroy the FSF by thoroughly infiltrating it (like they already have with organisations like the OSI and Linux Foundation). These people even started an online petition calling for RMS’s forceful removal and for the entire board of directors at the FSF to resign from their posts. This is clearly an attempt at a coup to overthrow the FSF! Out of fear, many known Free Software projects joined in on the anti-RMS witch hunt because they did not want to be cancelled either. The list that attacks Richard has Microsoft, Google, OSI, Linux Foundation, Gnome Foundation and Ethical Source people on it! These people oppose Free Software ideologically (even if some of them do produce free software sometimes, for reasons other than promoting freedom) and many of them have actively sought to destroy it for years! How dare these people claim to represent us!.
The letter opposing RMS talks the talk, but it does not walk the walk. The people on that list do not represent us! If you do see actual Free Software developers on the list, please talk to them. Do not be hateful or spiteful, just talk to them: tell them that they have been misled by a hateful campaign. We need unity in our movement. You see, it’s likely that a lot of people who signed the opposing list were just scared; at the beginning, the petition supporting RMS did not exist, and so it was not known how many people supported RMS. In other words, many people likely signed the anti-RMS list because they were scared of becoming outcasts. This is because last time, we were caught off guard. We stayed silent last time, but we will not be silent this time!
As of 31 March 2021, 02:50 AM UK time, we are winning! The letter calling for RMS’s removal has 2959 signatures. Our letter supporting and defending RMS has 4533 signatures! That’s a 60% approval rating, if you add up both numbers but our petition is rising in popularity much faster while the anti-RMS petition has stalled. People see that it’s OK to support RMS, because it is. RMS is innocent of wrongdoing!
Richard Stallman is my hero
I strongly believe in free software ideology. I am the founder of Libreboot, and its lead developer. When I first started using Free Software as a teenager in the mid 2000s, Richard Stallman’s lectures were among the biggest influences on me; Richard founded the GNU project in 1983 and the Free Software Foundation in 1985. I also saw the film Revolution OS and read Eric Raymond’s Cathedral and the Bazaar. I very quickly became fascinated but it was the articles by Richard on the GNU project website that heavily inspired me. For a few years however, I identified as an open source supporter until I gravitated towards the Free Software camp in 2009. I had worked sysadmin and IT support jobs at companies, working mostly with proprietary software including Windows, while at home I taught myself programming on GNU+Linux. I hated working with proprietary systems, precisely because of how restrictive they were compared to my systems at home, which all ran various GNU+Linux distributions (I also toyed with OpenBSD). When I did my A-Levels, I studied computing but they forced us to use the proprietary Visual Studio IDE and C#; I hated it, but coped with it by using Mono at home for class assignments. It wasn’t long after I joined as an FSF Associate Member in 2013 that my life took a huge turn, and Libreboot was a huge part of it. Needless to say, I strive to eliminate my dependence on proprietary software and I want others to experience such freedom aswell.
Richard Stallman’s articles and video lectures were what led me down this path. I have met the man 5 times, in 3 different countries.
In the early days of computing, most (if not all) software was shared freely with source code. In the early 1980s, when software started becoming more commercial, companies started making software proprietary which meant that the software no longer came with source code or otherwise placed restrictions on the use, development or sharing of that software. This meant that computer users no longer had freedom over their computing; by the time the GNU project started in 1983, free software did not exist! Richard Stallman, faced with the possibility of making large amounts of money as a proprietary software developer, staunchly resisted this trend and began the GNU project to create a completely free operating system that people could run on their computers.
I believe in Free Software for the same reason I believe in public education; I believe that knowledge is a human right. For example, I believe that all kids are entitled to learn Mathematics. I believe the same thing about Computer Science. Education is a human right. I want everyone to have freedom; the right to read, to a community and to free speech. Programming counts as speech, and I believe that all good work is based on the work of others; this is why the right to a community is critical. The four freedoms are paramount. I am a staunch supporter of copyleft and I believe that it should be mandatory, by law, for all creative and/or intellectual works. I use the GNU General Public License whenever possible, and I strongly advocate for its adoption everywhere.
Free software still has a long way to go. The mission of the GNU project and the Free Software movement is to eradicate proprietary software in our world and give everyone exclusively free software. That is a most noble mission which the Libreboot project shares. Companies like Apple and Microsoft resist us at every turn. Logic is highly proprietary; manufacturers of computer chips/boards heavily restrict access to knowledge about how the hardware works, and they put in DRM (such as cryptographic signature checks of firmware) to restrict our progress; this is why Libreboot still has very weak hardware support, as of the date this article is being published. Right to repair is a critical component of our fight, in particular, as a part of the wider OSHW (Free/libre Hardware) movement. Another problem we face is serialization of components, where the same component can no longer be used to replace another, in modern devices; the software on said device might check whether the new part is authorized and refuse to work if it isn’t. We in the freedom movement are under constant attacks, in a legal and technical sense. Large tech companies use every dirty trick in the book to thwart our efforts.
If it weren’t for Richard Stallman’s work, Libreboot would not exist. All works are derivative in human society; we stand on the shoulder of giants. The GNU project almost had a complete operating system, and finally they had one piece missing, the kernel; this program sits at the heart of the operating system, talking to hardware and allocating system resources, providing an interface on which application software can run. GNU had started work on a kernel which they called Hurd, but this is still far from complete as of 2021. Fortunately, another project called Linux appeared in the early 90s and was released under the GNU GPL, which meant that people were able to combine a modified GNU system with Linux to create a complete operating system; the first GNU+Linux distributions were born! It is from all of this that our movement, the Free Software movement, began, and without it, I doubt we’d have such wide access to free computing today. I cannot imagine a world where Libreboot and GNU do not both exist.
Could coreboot have existed without GNU+Linux? I doubt it very much! It’s possible that Linux on its own may have still existed, but would it have been Free Software by today? Would it have reached the level it did today? In that reality, BSD projects might have taken over instead, and would they have had the ideological drive to ensure that all computer users had freedom, or would they simply regard the source code as a reference for educational purposes only?
You see, Richard Stallman’s work in the 80s was revolutionary and without him, none of us would be here today. The people in charge of big tech companies like Apple and Microsoft hate us, and have been attacking our movement for years. That’s what the attacks on RMS have been about. They do not care what Richard did or didn’t do at any given point in time.
Richard had been president of the Free Software Foundation since its inception in 1985, spreading Free Software ideology all over the world; until, that is, he was cancelled in 2019 in the most Orwellian smear campaign possible.
Anyone familiar with Libreboot probably already knows all of the above, or they are familiar with the gist of it, so why am I talking about the FSF, GNU and Richard Stallman today? Because of something very sinister that is currently happening.
Don’t just take my word for it. Stephen Fry, a well-known GNU+Linux user, did this video in 2008 praising the GNU project and supporting Free Software:
https://yewtu.be/watch?v=P_mS4CIXcLY
RMS is NOT transphobic
I’ve been good friends with Richard for many years. I did have a falling out with him (publicly so) a few years ago, but we made up. He has always respected me.
When my project, Libreboot, was in the process of joining GNU, I wasn’t out as trans. I came out as trans not long before Libreboot became GNU Libreboot. RMS switched to she/her with me on the spot. No problems.
Some people have linked to the following article and suggested that he is transphobic: https://stallman.org/articles/genderless-pronouns.html
Specifically, people believe that RMS refuses to use correct pronouns with people. People believe that RMS is transphobic for saying per/perse instead of accepting they/them.
Let me tell you something:
Richard sent me and several other people a copy of that article when he was drafting it. I repeatedly urged RMS not to do per/perse when he suggested it. I strongly suggested that he use they/them when referring to someone generically. When he decided to use per/perse, I was annoyed but not offended; you see, I regard it as idiotic. Clearly, they/them is commonly understood and will cause the least amount of misunderstanding.
Being foolish is not the same thing as being transphobic. If you actually tell Richard your preferred pronouns, he’ll use them with you without hesitation.
Several of my friends are trans and also speak to Richard, mostly via email. He respects their pronouns also.
Funnily enough, the GNU project has these guidelines about pronouns: https://www.gnu.org/philosophy/kind-communication.en.html - see: https://www.gnu.org/philosophy/kind-communication.en.html#f1
Not transphobic. At all. Same per/pers bullshit. Not transphobic, just stupid. I wasn’t misgendered by other GNU developers when my project, Libreboot, was in GNU. Calling RMS a transphobe is an insult to people who suffer from real transphobia.
Background information
I could address each specific accusation made against him, but other articles already do that; those articles are written much better than anything I could ever write, so please click on the links below.
I feel no need to re-invent the wheel. The whole purpose of this article was just to express my support for Richard Stallman, and to defend his honour. His time will end one day, and he deserves for that to come naturally. However, there is still much that he can contribute!
The following articles more or less describe accurately what happened since September 2019 when the events surrounding Richard Stallman started:
https://www.wetheweb.org/post/cancel-we-the-web
Here is another article expressing support for Richard, and it too has details about the events that took place:
https://jorgemorais.gitlab.io/justice-for-rms/
This video by DistroTube provides an excellent account of events aswell:
https://odysee.com/@DistroTube:2/mob-mentality-threatens-the-free:b
Exposing our opponents for who they are
Our problem, in defending Richard Stallman, is that opponents of the Free Software movement have learned to co-opt our language. They talk the talk and they wear the colours, but make no mistake: their actions and their intentions do not reflect the ideology they claim to represent! There are genuinely some Free Software activists and organanisations on that list, who have been misled or have some other reason to oppose RMS; my focus will not be on those people, but hopefully some of those people and organisations will change their mind if they read what I have to say!
I do not subscribe to cancel culture. Some of these people may well try to cancel me but I would never do the same to them. This entire article merely aims to defend RMS against the vicious smear campaigns. To do that, we will explore some of the people on that anti-RMS list.
I said I wouldn’t directly link to the list calling for RMS’s removal, so I will print the URL below without making it a hyperlink (this prevents it from being boosted in search engines). Look at the names on their list:
https://rms-open-letter.github.io/
Don’t be fooled! The open source movement is not the same as the Free Software movement! The following article describes how Open Source differs from Free Software: https://www.gnu.org/philosophy/open-source-misses-the-point.en.html
I will focus on the people in the main list of signers, and maybe talk about specific organisations (or other names) on that list. Some of them are otherwise reasonable people besides their anti-RMS stance (which means they were misled, most likely), whereas some people on the list are nasty.
I will jump straight into it:
Redhat pulling funding from FSF
RedHat announced, in response to RMS’s re-instatement at the FSF, that they would remove their funding for the FSF. They joined in on the usual smear campaign.
RedHat is owned by known non-free software company IBM these days. Their enterprise GNU+Linux distro comes with plenty of non-free software and they actively tell their customers how to get more; they do nothing to advance free software and merely see it as something they can use. They do not believe in FSF ideology. More info about the merger: https://www.redhat.com/en/ibm
Redhat very recently killed CentOS. CentOS was a community edition of RHEL, with a strong community backing. In other words, Redhat actively took a step that hurts the community. More info: https://arstechnica.com/gadgets/2020/12/centos-shifts-from-red-hat-unbranded-to-red-hat-beta/
Look at that: https://www.ibm.com/products/software
Does this look like a company that cares about Free Software?
Why should we care what RedHat thinks? If they pull funding, that’s one less corrupting influence to worry about! Redhat does not believe in free software (they may have believed in open source at one point, but that time is probably long gone now that they’ve been bought by IBM)
OSI/Microsoft connection
OSI is short for Open Source Initiative. This organisation started as an offshoot of the Free Software Foundation in an attempt to make Free Software more marketable to large corporations. Read about the OSI here: https://en.wikipedia.org/wiki/Open_Source_Initiative
They say a picture speaks a thousand words:
From left to right, their names (all prominent OSI leaders/influencers), where left is your left and their right (for the phono in the photos) are:
Back row: Faidon Liambotis, Chris Lamb, Simon Phipps, Allison Randal, Molly de Blanc, Patrick Masson
Front row: Josh Simmons, VM Brasseur, Carol Smith, Italo Vignoli, Richard Fontana.
All of these people are highly influential at the OSI. Several former presidents.
Does this look strange to you? Look where they are. The photo comes from this news article: http://techrights.org/2020/01/15/osi-board-at-microsoft/ (archive: http://web.archive.org/web/20200121042512/http://techrights.org/2020/01/15/osi-board-at-microsoft/)
Microsoft is a major sponsor of the OSI. OSI themselves have an article on their website, stating this: https://opensource.org/node/901 (archive: http://web.archive.org/web/20201112022740/https://opensource.org/node/901)
When your organisation starts to depend on large amounts of funding by companies like Microsoft (who have rigorously attacked Free Software and Open Source for years), you are going to lose sight of some of your ideals. You will lose some of that spark you previously had in you. You will start doing what your donors tell you, because you fear the loss of that funding. Microsoft, over the years, has dived into their version of what they regard as open source; in reality, it’s just openwashing (like whitewashing, but with Open Source perspective instead), and Microsoft’s core products such as Windows are still very much non-free! Microsoft still campaigns hard for your lack of freedom by getting more and more computers locked down with things like SecureBoot and cryptographically signed firmware.
So if Microsoft has hated Richard Stallman for years, and wanted to destroy him for years, and Microsoft has financial influence at the Open Source Initiative, on organization that could somewhat credibly speak Free Software lingo, would that not be the best thing ever for Microsoft? Imagine being Microsoft. You’d jump at the opportunity, right? Someone tell me I’m not the only one here.
Even if Microsoft wasn’t heavily attached to the OSI, would the OSI have any right using the language of Free Software while claiming to be a part of our community? Open source is not a part of the Free Software movement! It is an ideological competitor to Free Software.
Fun fact:
The OSI recently banned Eric S Raymond (co-founder of the OSI) from their mailing lists, after certain comments he made defending the OSI against infiltration by the Ethical Source movement and against oppressive codes of conducts that stifle free speech. Despite the name, Ethical Source licenses are in fact non-free because they put restrictions on usage of the software; if the author of such software disagrees with your political views, they can ban you from using the software. This is wrong! People like Coraline Ada Ehmke (leader of Ethical Source movement) were trying to influence the OSI so as to re-write the Open Source Definition. This video provides some nice introductory information:
https://odysee.com/@DistroTube:2/founder-of-open-source-is-banned-by-open:7
In the OSI’s case, they probably won’t let Eric back in; though even if I disagree with Open Source (I’m a Free Software activist), Open Source isn’t at a bad thing per se, just ideologically lacking; Ethical Source people like Coraline Ada Ehmke will cause extreme amounts of damage if they get their way (they’ve already infiltrated several well-known Free Software and Open Source aligned projects by getting them to introduce a Code of Conduct; Libreboot recently came to the good sense to scrap its Code of Conduct, which was none other than Coraline’s Contributor Covenant)
Everyone should send Eric Raymond a supportive email. He did the right thing. Tell him you care. I’ve never heard anything especially horrible about him. He’s very reasonable and a nice person; outspoken and unfiltered while still being respectful (in my opinion, having read some of his articles), which is quite refreshing.
Microsoft employees
Yes, Microsoft employees are on the anti-RMS list.
Microsoft is a mortal enemy of the Free Software movement. Microsoft isn’t foolish enough to sign their entire company name onto the list, because then that would be game over for the anti-RMS campaign; so instead, they use their corrupting influences at various organisations that supposedly represent us.
If I were Microsoft, I’d ask these people to remove their names from the list. It actually hurts their anti-RMS efforts, for such people to have their position at Microsoft stated like this, even if it’s just a few people.
None of their people on this list seem to be high up at Microsoft. I would be inclined to believe that they published their names independently, without direction. No sane Microsoft boss would want Microsoft listed on that page, in any capacity!
Gnome Foundation (has been deeply connected to Microsoft)
NOTE: Do not confused the Gnome community with the Gnome Foundation. They are very different things!
There are well-known connections between members of the Gnome Foundation with Microsoft. Here is an article:
http://web.archive.org/web/20200607212123/http://techrights.org/2020/06/07/gnome-board-of-directors-2020/
They have been attacking RMS for years:
http://techrights.org/2021/01/12/gnome-foundation-rms/
So, of course, it’s not credible for these people to represent themselves on behalf of the Free Software movement!
The following Gnome Foundation members are on the core signers list of the anti-RMS petition, and associated with the Gnome Foundation:
Molly de Blanc (Debian Project, GNOME Foundation) (also associated with OSI)
Neil McGovern (GNOME Foundation Executive Director, Former Debian Project Leader)
Luis Villa (Former Director of the Open Source Initiative and the GNOME Foundation; contributor to the GPL v3 drafting process)
In other cases, I wouldn’t choose to list names, but Neil and Molly are two of the people with push/pull/review rights on the anti-RMS github site. I feel the need to mention their names; see also that they are both members of the Debian project.
Coraline Ada Ehmke (Founder, Organization for Ethical Source)
Coraline is the founder of the Ethical source movement. Despite the name, it is actually distributing non-free licenses; non-free because they put restrictions on the usage of software licensed under it. If you use software under one of those licenses, and the author disagrees with you politically, the author can ban you from using that software.
I for one believe in freedom! I want freedom for everyone, including those whom I politically disagree with!
Discriminating based on someones beliefs is always wrong. No ifs, no buts. I want my political opponents to have freedom, because:
If I were able to take away my enemy’s freedom, they could take away mine.
If my enemy were able to take away my freedom, I could take away theirs.
Coraline is also quite abusive online. There are numerous accounts of her terrorizing companies/projects, acting like a bully. There’s a chance that she might even target the Libreboot project, if someone tells her of this article.
She is a horrible person.
She’s more well known for the Contributor Covenant, a template code of conduct that some projects use. We in Libreboot recommend that you do not have a code of conduct, because it alienates new contributors and creates a self-censored environment where people feel unable to express their views about issues; you see, freedom of speech is healthy, and it’s quite common sense to just deal with bad behaviours. Contributor Covenant is a trojan horse; that’s what they push on you first, and then they’ll recommend you use an Ethical Source license. Once you take your first dose of Ethical Source, they’ll sink their claws into your project. Do not let these types of people infiltrate your project!
Do not listen to Coraline Ada Ehmke or anyone like her! She is fuelled by hatred and bigotry herself. She is completely intolerant of other people’s views and regularly tries to destroy people she disagrees with.
Conclusion
That’s all!
Defend RMS!
Can’t be bothered to write more. I was going to go through the list more exhaustively, but I think you see the point.
rms.jpg
@sjw
As a reminder, "ethical source" is incompatible with the GPL and as such cannot include any code licensed under the GPL.
Be a helper: Submit pull requests to projects with such a license to remove code they are no longer legally entitled to use.
someone told me there was a traffic accident near here, right next to the hospital.
the guy died because it took too long for the ambulance to get there...
welcome to Norway.
@rife_with_tedium Sry, was away for a few days. Part of the coercion chart is actually letting people have a little bit of fun before cracking down on them again. (#5) You let them have a little hope, and then break them of that hope. Like in CA when they said we could go eat inside restaurants again, only to pull it out from under us after like 2 weeks. "No eating out again slaves!"
Sign the Petition
Put the boat back
http://chng.it/gncvwjNcFx
Defend Richard Stallman!
Defend freedom and open source software autizmo.xyz/~/Sjw/defend-richa
Would be really nice if someone could dry my hair every day though
Perhaps I should start posting about my every day lol ;)
Having vivid weird recently…
Last night I dreamt of a swimming race. Upon reaching the other side, I had to talk (in English) with a man who looks like Mao about PRC- relations.
After that a really cute girl volunteered to dry my hair, and bragged about how good her hair dryer was while doing it.
One of her “talking points” was that the hair dryer used a higher voltage than normal. “It’s not like it’s more unsafe,” she said, “the voltage of normal hair dryers already zaps you to death anyway.”
Then after that I fiddled with her laptop (which was a !) for some reason, and she was using Spotify, out of all things, for voice conferencing. I even accidentally killed the process and had to restart it.
The scene then changed to my grandma’s house, and more bizarre stuff happened, though my memory is more faint on this.
So, you wanna be a pirate, eh?
Tired of the jannies telling you what you can't say? Or maybe you don't think hate speech is being effectively filtered effectively enough on your current platform? Wanna run things your way? Be your own captain?
It always seemed like too much work, just to get started, though...
Well, not anymore! Introducing YARRPS!
With this script you'll be able to pilot your own ship and sail the 7 seas of the Fediverse!
If you already know how to get a domain name and a VPS (specifically a blank Ubuntu install) you can install a fully featured Pleroma instance with three simple commands. Pic related.
Stop being a barnacle! Be your own captain and the world will be your oyster!
Get started today with YARRPS!
https://yarrps.xyz
@nosleep Taxation is theft, therefore I am helping police by not collaborating with a thieve.
"Why are you so interested in privacy? Do you have something to hide?"
Yes. | 2022-10-06 11:21:47 | {"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": 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": 0.2371547371149063, "perplexity": 3203.918340055389}, "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/1664030337803.86/warc/CC-MAIN-20221006092601-20221006122601-00180.warc.gz"} |
https://www.physicsforums.com/threads/how-do-i-express-an-equation-in-polar-coordinates-as-a-cartesian-one.993673/ | # How do I express an equation in Polar coordinates as a Cartesian one.
• A
## Summary:
I need to convert an equation that is on polar coordinates into a cartesian but as soon I start doing that I got confused and I'm not really shure about what to do.
I got a polar function.
$$\psi = P(\theta )R(r)$$
When I calculate the Laplacian:
$$\ \vec \nabla^2 \psi = P(\theta)R^{\prime\prime}(r) + \frac{P(\theta)R^{\prime}(r)}{r} + \frac{R(r)P^{\prime\prime}(\theta)}{r^{2}}$$
Now I need to convert this one into cartesian coordinates and then it results very difficult to me because I know how to convert simple equations using the simple relations:
$$x = r Cos(\theta )$$
$$y = r Sin(\theta )$$
$$r= \sqrt{x^{2}+y^{2}}$$
$$\theta= arcTan( \frac{y}{x})$$
I can't figure out how to use this relations in order to replace them in my functions R and P since there are first and second derivatives of a functions dependent on x and y, so I can not just replace the relations (At least not directly).
What I need to do is to express the Laplacian of psi in cartesian coordinates.
Is there a way just to replace in them as :
$$P(\theta) = F(x,y)$$
$$P^{\prime}(\theta) = G(x,y)$$
$$P^{\prime\prime}(\theta) = H(x,y)$$
Or how should I try?
Thanks A lot for your help
## Answers and Replies
mfb
Mentor
If you don't know P and R, the best you can do is setting up something like ##\displaystyle \frac{\partial P(\theta(x,y))}{\partial \theta}## and then applying the chain rule. It will still look quite awkward. | 2020-10-26 13:31:42 | {"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": 0.8613491058349609, "perplexity": 358.5549878529667}, "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-2020-45/segments/1603107891228.40/warc/CC-MAIN-20201026115814-20201026145814-00574.warc.gz"} |
https://curriculum.illustrativemathematics.org/HS/teachers/2/3/8/preparation.html | # Lesson 8
Are They All Similar?
### Lesson Narrative
In this lesson, students conjecture and reason about whether all shapes in a certain category must be similar, such as whether all circles, or all rectangles, are similar. Then they prove that all equilateral triangles are similar, and all circles are similar (MP3). The proof that all circles are similar is used again in a subsequent unit, when it is used to rigorously define angle measure, connect angle measure to arc length, and connect degrees to radians.
Students get a chance to apply the theorem they proved in previous lessons, that if two triangles have all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion, then the triangles are similar. This simplifies the proof that all equilateral triangles are congruent.
Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
### Learning Goals
Teacher Facing
• Critique a proof that uses similarity (in writing).
• Prove theorems about triangles and other figures using similarity.
### Student Facing
• Let’s prove figures are similar.
### Required Preparation
The scientific calculators are for use in the extension problem of the activity “Always? Prove it!”
### Student Facing
• I can critique proofs that use similarity.
• I can write proofs using the definition of similarity.
Building On
Building Towards
### Glossary Entries
• similar
One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.
Triangle $$A'B'C'$$ is similar to triangle $$ABC$$ because a rotation with center $$B$$ followed by a dilation with center $$P$$ takes $$ABC$$ to $$A'B'C'$$.
### Print Formatted Materials
For access, consult one of our IM Certified Partners. | 2021-10-16 14:43:04 | {"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": 0.6988458037376404, "perplexity": 1250.5083285705089}, "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-2021-43/segments/1634323584886.5/warc/CC-MAIN-20211016135542-20211016165542-00674.warc.gz"} |
https://proofwiki.org/wiki/Axiom:Axioms_of_Uncertainty/Axiom_8 | # Axiom:Axioms of Uncertainty/Axiom 8
## Axiom
Let $Z$ be a random variable.
Let $Z$ take the values $a_i$ with probability $p_i$, where $i \in \set {1, 2, \ldots, n}$.
Let $H_n: Z^n \to \R$ be a mapping which is to be defined as the uncertainty of $Z$.
${H_n}$ fulfils the following axiom:
Let:
$p = p_1 + p_2 + \dotsb + p_m$
$q = q_1 + q_2 + \dotsb + q_n$
such that:
each of $p_i$ and $q_j$ are non-negative
$p + q = 1$
Then:
$\map {H_{m + n} } {p_1, p_2, \dotsc, p_m, q_1, q_2, \dotsc q_n} = \map {H_2} {p, q} + p \map {H_m} {\dfrac {p_1} p, \dfrac {p_2} p, \dotsc, \dfrac {p_m} p} + q \map {H_n} {\dfrac {q_1} q, \dfrac {q_2} q, \dotsc, \dfrac {q_n} q}$ | 2021-01-20 19:18:54 | {"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": 0.9758713245391846, "perplexity": 456.343085730718}, "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-2021-04/segments/1610703521987.71/warc/CC-MAIN-20210120182259-20210120212259-00712.warc.gz"} |
https://phys.libretexts.org/Courses/Muhlenberg_College/Physics_122%3A_General_Physics_II_(Collett)/00%3A_Front_Matter/03%3A_Table_of_Contents | $$\require{cancel}$$
• ## 1: Electric Charges and Fields
In this chapter, we begin the study of the electric force, which acts on all objects with a property called charge. The electric force is much stronger than gravity (in most systems where both appear), but it can be a force of attraction or a force of repulsion, which leads to very different effects on objects. The electric force helps keep atoms together, so it is of fundamental importance in matter.
• ## 2: Gauss's Law
So far, we have found that the electrostatic field begins and ends at point charges and that the field of a point charge varies inversely with the square of the distance from that charge. These characteristics of the electrostatic field lead to an important mathematical relationship known as Gauss’s law. Gauss’s law gives us an elegantly simple way of finding the electric field, and, as you will see, it can be much easier to use than the integration method described in the previous chapter.
• ## 3: Electric Potential
In this chapter, we examine the relationship between voltage and electrical energy, and begin to explore some of the many applications of electricity.
• ## 4: Current and Resistance
In this chapter, we study the electrical current through a material, where the electrical current is the rate of flow of charge. We also examine a characteristic of materials known as the resistance. Resistance is a measure of how much a material impedes the flow of charge, and it will be shown that the resistance depends on temperature. In general, a good conductor, such as copper, gold, or silver, has very low resistance.
• ## 5: Direct-Current Circuits
In this chapter, we use these electric components in circuits. A circuit is a collection of electrical components connected to accomplish a specific task. The second section of this chapter covers the analysis of series and parallel circuits that consist of resistors. We also introduce the basic equations and techniques to analyze any circuit, including those that are not reducible through simplifying parallel and series elements. But first, we need to understand how to power a circuit.
• ## 6: Magnetic Forces and Fields
For the past few chapters, we have been studying electrostatic forces and fields, which are caused by electric charges at rest. These electric fields can move other free charges, such as producing a current in a circuit; however, the electrostatic forces and fields themselves come from other static charges. In this chapter, we see that when an electric charge moves, it generates other forces and fields. These additional forces and fields are what we commonly call magnetism.
• ## 7: Sources of Magnetic Fields
In this chapter, we examine how magnetic fields are created by arbitrary distributions of electric current, using the Biot-Savart law. Then we look at how current-carrying wires create magnetic fields and deduce the forces that arise between two current-carrying wires due to these magnetic fields. We also study the torques produced by the magnetic fields of current loops. We then generalize these results to an important law of electromagnetism, called Ampère’s law.
• ## 8: Electromagnetic Induction
In this and the next several chapters, you will see a wonderful symmetry in the behavior exhibited by time-varying electric and magnetic fields. Mathematically, this symmetry is expressed by an additional term in Ampère’s law and by another key equation of electromagnetism called Faraday’s law. We also discuss how moving a wire through a magnetic field produces an emf or voltage.
• ## 9: The Nature of Light
In this chapter, we study the basic properties of light. In the next few chapters, we investigate the behavior of light when it interacts with optical devices such as mirrors, lenses, and apertures.
• ## 10: Geometric Optics and Image Formation
This chapter introduces the major ideas of geometric optics, which describe the formation of images due to reflection and refraction.
• ## 11: Interference
The most certain indication of a wave is interference. This wave characteristic is most prominent when the wave interacts with an object that is not large compared with the wavelength. Interference is observed for water waves, sound waves, light waves, and, in fact, all types of waves.
• ## 12: Diffraction
In the preceding chapter, we implicitly regarded slits as objects with positions but no size. The widths of the slits were considered negligible. When the slits have finite widths, each point along the opening can be considered a point source of light—a foundation of Huygens’s principle. Because real-world optical instruments must have finite apertures (otherwise, no light can enter), diffraction plays a major role in the way we interpret the output of these optical instruments. | 2021-11-27 14:38:20 | {"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": 0.6489096283912659, "perplexity": 378.5134063598912}, "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/1637964358189.36/warc/CC-MAIN-20211127133237-20211127163237-00556.warc.gz"} |
http://temu.coworking.id/npc3/delta-t-calculator.php | # Delta t calculator
Delta T is the spre May 30, 2010 · You will need the mass flow rate and temperature differential. 5 kJ Find out how to calculate the delta adjusted notional value of an options contract, and why gross notional value cannot be used like with other derivatives. For the independent samples T-test, Cohen's d is determined by calculating the mean difference between your two groups, and then dividing the result by the pooled standard deviation. Calculations are based on established theoretical thermal equations tempered with correction factors learned from decades of real-world heat pipe testing. Effect Size Calculator for T-Test. All the while, we list our expenses publicly, budgeting $25,000 a year for our If you can't find what you're looking for, or you have an idea for a calculator that would be helpful to you, let us know. wheth Jan 10, 2013 · phi^@=360^@ . John is performing the experiment, where he realized that he has to raise the temperature from 25o C to 30o c. BTU Output of Appliance, Delta. 844 lbs/minute of air flow. You don't HAVE to use anything but the DeltaBot Calculator. DELTA E, DELTA H, DELTA T: WHAT DOES IT MEAN?1 DELTA E, DELTA H, DELTA T: WHAT DOES IT MEAN? Delta E (ΔE) Delta E is defined as the difference between two co lors in Summary of Gibbs free energy. Read on to learn how to apply the heat capacity formula correctly to obtain a valid result. 75 x 100 (assuming each contract HOW DO YOU CALCULATE ΔH (DELTA H)? Once you memorized the relationship between the side the energy is on in a chemical equation, the Δ H and endothermic or How do we find the heat transfer rate. Does anyone have an idea about how to find the coolant flow rate in would have 100 L/min of coolant flow rate but how to calculate the airflow X delta T Mass x Specific heat capacity x delta T = kw If its cooling one you have to do then you will need wet bulb and dry bulb temps so you can calculate the enthalpy Our heat loss calculator can determine how much heat is being lost Grooved steel pipe and fittings for greenhouse heating Delta T heating systems that use high Photovoltaic developers and triallists find the SPN1 Sunshine Pyranometer a valuable tool for research and for performance validation. Hello. HOW DO WE CALCULATE Δ G (Second Way)? The second way to calculate Δ G is to use a formula that involves enthalpy, temperature, and entropy. f. of the lunar acceleration parameter can also be studied by changing its nominal value of –26. The formula is Summary of Gibbs free energy. 1 28. Brands · Services · HWP Technical Reference Tools · Merchandise Return & Warranty Forms · Contact Information · About Us. The Y–Δ Transform, also known as "delta–star", and "delta–wye", is a mathematical process used in the field of electronic engineering to simplify complex BTU Calculator & BTU Formulas. Calculate the rate constant. difference)[/b] chart for Chillers. Earn miles for traveling, then use miles for rewards, from premium drinks to flights on Delta Private Jets. Say you own 10 contracts of XYZ calls, each with a delta of . There are many brands of test instruments to help measure Delta T, Delta P, and so on. 3 Initial temperature 26. Enter the thermal conductivity of your material (W/m•K); OR select a value from our material database. Δ T = 5 oC. What do we mean by 'Delta T' Polynomial Expressions for Delta T (ΔT) Between years 1961 and 1986, calculate: ΔT = 45. Involvement with Delta Chi doesn't stop at graduation. 7 32. Efficiency, Return Temperature. Input the cross-sectional area (m2); Add your materials thickness (m); Enter the hot side temperature 29 Dec 2017 The Average Acceleration calculator uses the equation, a = (V2 - V1)/(Δt), to compute the average acceleration base on two observations of velocities (v2 and v1) and the time between the measurements (Δt). wheth • KINETICS — the study of REACTION RATES and their relation to the way the reaction proceeds, i. Mass x Specific heat capacity x delta T = kw If its cooling one you have to do then you will need wet bulb and dry bulb temps so you can calculate the enthalpy Answer to With the Q=MS(delta)T equation: How do you calculate the heat of 4. This calculator implements the least-squares delta calibration algorithm that RepRapFirmware has Does any body has [b]GPM/TR Vs Delta T (Temp. . Calculating Water Flow by Delta T. Steinhart-Hart Temperature Calculator. I'm confused to Get an answer for 'What is the formula for Delta T? and C? and m?Q=mxcx Delta T So Delta T=? C=? m=?' and find homework help for other Science questions at eNotes How to Calculate Specific Heat. 0 1. The formula is Q(BTU/hr) = (Mdot)(Cp)(delta-T). com i now know the heat required for my room so how do i check Calculating Wattage for Industrial Heating Applications. Delta t phase angle Example 1 for Phase Angle Calculation: Calculate the phase angle in degree the frequency f=100Hz and time delay Free Laplace Transform calculator - Find the Laplace and inverse Laplace transforms of functions step-by-step Delta T / GPM question I just finished Dan's book about Primary / Secondary pumping. To get from the ground to low Earth orbit (LEO) requires a Calculate the heat given off from 366g of mercury when it cools from 77. 1 31. 6 28. DeltaBot Calculator. aelheating. 0 7. Below is a chart Nufarm Ltd} showing this relationship and the Does any body has [b]GPM/TR Vs Delta T (Temp. Heat Calculator is used to calculate the change in temperature, heat energy, specific heat, and mass of body. 184 J/kg * degrees C. *Temperature deviation from 1976 standard atmosphere (off-standard atmosphere). 6 31. 1 Final temperature Here are my values. Thermistor resistance is related to . This calculator implements the least-squares delta calibration algorithm that RepRapFirmware has Accurate and easy CMC l:c comparisons with our online delta-e calculator. What is a slope intercept calculator? How do you solve an equation using slope-intercept form? SPRA Spray Shop Loosely described, Delta T is the relationship between temperature & humidity. , its MECHANISM. 44 "/cy/cy in the IAU (1952) and the Astronomical Ephemeris (since 1960) relations; –23. A spreadsheet version of the UAC calculator is shown in Table 2. Learn the easiest way to calculate this in Excel. Pictured here is the testo 925 — a temperature measuring instrument Dec 08, 2007 · Best Answer: In this kind of calculation, the Specific Heat isn't needed and neither is Kelvin temperature scale. Discussion Change in velocity, or delta-V, is the most important measure of "distance" in space flight. This tool calculates how many SkyMiles are earned flying on Delta and its partners. The Specific Heat formula is: c = ΔQ / (m × ΔT) Where: c: Specific Heat , in J/(kg. The calculation is a little off, as I haven't accounted for carriage size. 0 degrees C to 12. ARROWS - Move the effector in the XY plane; Free calculus calculator - calculate limits, integrals, derivatives and series step-by-step Help finding Delta T for a known position displacement with known accel I am trying to estimate/calculate the time it will take for$$\Delta t = \frac We have spent the majority of our marriage traveling full time, living out of hotels. , the calculator will still calculate Cohen's d and Glass' delta, but it won't generate a value for Hedges's g. Attention Here’s an example. Accurate and easy color comparisons with our online delta-e calculator. The specific heat of a More Delta T Calculator videos Get an answer for 'What is the formula for Delta T? and C? and m?Q=mxcx Delta T So Delta T=? C=? m=?' and find homework help for other Science questions at eNotes Application assumed to be cooling. Though we can get it thru TR=GPM x Temp diff/24. Hello, I've been given some vectors and I'm supposed to calculate delta T for them. Answer to A+B--> C rate= delta [C]/ delta t = k [A]^a[B]^b a) Given the initial concentration of each of the reactants, calculate Hello. To calculate position delta, multiply . Pi and T networks This page has a The pi is equivalent to the Delta and the T is equivalent to the Wye calculate the Z's assuming resistive impedances: . Specific heat is just the amount of thermal energy you need to supply to a sample weighing 1 kg to increase its temperature by 1 K. 0 2. They shake a few hands, see an exhibit or two LA8OKA's Corned Fed Delta Loop An example of the Corner Fed Delta Loop antenna made for 20 m. In this video, John Both from NuFarm discusses Delta T for making decisions about your herbicide application methods. 0 arcsec/cy/cy t ther values. Why can't all textbooks be written that way? Today's kids would have higher test Use the data sheet to calculate Delta H, Delta S and Delta G at 25 degrees Celsius for each of the following reactions. Specific heat is the amount of energy required to raise one gram of a pure substance by one degree Centigrade. Formula: H = Cp * m * Δ T. Note that the following values were adopted in the original ΔT relations: –22. What is a slope intercept calculator? How do you solve an equation using slope-intercept form? Does any body has [b]GPM/TR Vs Delta T (Temp. specific heat is 0. These include the starting and ending temperature, as well as Training · Our Team · Contact Us · View Line Card. Use our free online units converter for temperature difference. ARROWS - Move the effector in the XY plane; Theory: Delta T and Flow. The influence of the lunar acceleration parameter can also be studied by changing its nominal value of –26. There are 3 cases to this: 1) Fresh air is supplied directly into your room from an external fresh air handling unit (call it an Enthalpy and Gibbs Free Energy Calculator Introduction : the purpose of this calculator is to calculate the value of the enthalphy of a reaction (delta H) or the If you're a marketer, all Excel dashboards should include percent deltas. Firing rate, as far as I know there is no way to calculate gpm without an assessment of the Accurate and easy color comparisons with our online delta-e calculator. My final answers are probably wrongbut if Delta printer least-squares calibration calculator Instructions. Altitude: feet [ft], kilometers [km], meters [m], nautical miles. e. A short article about How to Calculate the Log Mean Temperature Difference in heat exchangers for counter current flow and co-current parallel flow. 3 30. c={\frac {Q}{m\Delta T}} , where c — specific heat, Q — heat added or removed to the body, m — mass of the body, ΔT — temperature change. 0 5. Also, I show off our new computer desk, as requested by itscool1968. Plan your trips — or mileage runs — to maximize mileage earning, or Free calculus calculator - calculate limits, integrals, derivatives and series step-by-step Photovoltaic developers and triallists find the SPN1 Sunshine Pyranometer a valuable tool for research and for performance validation. 0 6. BTU Input of Appliance, Supply Temperature. by Conumdrum on June 14, Delta T (DT) and Why it’s How Do You Calculate All of This? Kestrel Meter 3500 DT MeterDesigned specifically for agricultural professionals, the Kestrel 3500 Delta T provides Delta T readings. wheth Velocity as a Function of Acceleration and Time v = u + at : Calculate final velocity (v) as a function of initial velocity (u), acceleration (a) and time (t). Guide to Delta-T in Water Cooling. Step 2 : Javascript ΔT Calculator. 139J/g degrees C The Y–Δ Transform, also known as "delta–star", and "delta–wye", is a mathematical process used in the field of electronic engineering to simplify complex The Astronomical Almanac Online — Delta T: Past, Present and This calculator is used to calculate LMTD for parallel flow (i. 0 arcsec/cy/cy t ther values. It says that to calculate it I need the three time vectors, hour minute and second Unique to Castrads, you won’t find these finishes anywhere else. 9 31. A tight Delta-T, Notes: This calculator is designed for use with high Sensitivity Troponin T (TnT-hs) Delta troponin is the percent difference between two troponin results taken in DELTA E, DELTA H, DELTA T: WHAT DOES IT MEAN? Delta E It is not possible to calculate delta T values for colors composed of a mixture of cyan, magenta, Effect Size Calculator for T-Test. QUESTION: After using the “Easy heat guide for your room” on the home page of www. delta t calculatorThis specific heat calculator is a tool that determines the heat capacity of a heated or a cooled sample. 0 degrees C. Calculate the heat needed to raise the temperature? Step 1 : Given: Cp = 4. 6g of 50 degree C water to completely boil? Delta T also designs, builds and repairs custom heating and control systems. 067*t - t^2/260 - t^3 / 718 where: t = y - 1975 Aug 06, 2009 · I show how to calculate the Delta T while your air conditioner is running. 75 x 100 (assuming each contract Q Mc Delta T Calculator? - Crowdsourced Questions & Answers at Okela HOW DO YOU CALCULATE ΔH (DELTA H)? Once you memorized the relationship between the side the energy is on in a chemical equation, the Δ H and endothermic or Hello all, I received the following 3 questions back on a test and cannot seem to figure why I didn't get the full mark. Many variables have been fixed or allow only limited choices for quicker online results. On the bench or on the greenhouse floor, Delta T’s radiant heating systems offer the grower distinct advantages, including The IERS Rapid Service/Prediction Center is the product center of the International Earth Rotation and Reference Systems Service responsible for providing Earth Steinhart-Hart Thermistor Calculator. Compare colors in Rgb, Cmy, Cmyk, Hsl, Xyz, CIE-L*ab, CIE-Lch, and Yxy spaces. 3 32. where the hot stream,liquid or gas, goes from say left to right, and so does the cold stream). 75. A BTU is an abbreviation for B ritish T hermal U nit, which is the amount of energy required to raise 1 pound of water 1 degree This calculator compares the spreading resistance (delta-T) of an aluminum or copper solid metal base used in heat sinks to a vapor chamber base alternative. Delta printer least-squares calibration calculator Instructions. Mdot can be solved for if you gave a GPM Notes: This calculator is designed for use with high Sensitivity Troponin T (TnT-hs) Delta troponin is the percent difference between two troponin results taken in Effect Size Calculator for T-Test. This specific heat calculator is a tool that determines the heat capacity of a heated or a cooled sample. Specific Heat Calculator will help us to calculate the specific heat of different substances. Ohm's Law Calculator 3-Phase Delta / Wye Calculator SkyMiles helps you go further. Temperature offset*: Celsius / Kelvin, Fahrenheit / Rankine, Réaumure. Calculate the Temperature when the above reaction is at equilibrium T(Delta S) Delta G = 110. Hot Water Products, Inc. 8946 "/cy/cy in the The heat transfer conduction calculator deals with the heat transfer between substances that are in direct contact with each other. flow is directly related to delta T. The heat capacity can be affected by many of the state variables that describe the thermodynamic system under study. Engineers needing heat pipe thermal solutions which can operate beyond the Specific heat refers to the amount of heat required to raise unit mass of a substance's temperature by 1 degree. But need to clarify 01. 0 4. Enthalpy change. delta t calculator 5 kJ Optimal Chilled Water Delta-T power and delta-T increase while EER decreases with TODB.$\Delta$T- Change in the temperature of the substance. The Delta Loop antenna is one of the famous ”magical antennas”, but Delta-v (literally "change in velocity"), symbolised as ∆v and pronounced delta-vee, as used in spacecraft flight dynamics, is a measure of the impulse that is Radiant Heating for Greenhouses. Step 2 : Put the values Training · Our Team · Contact Us · View Line Card. manufacturers representatives, wholesale distribution, high efficiency water heaters, hydronic heating systems, boilers, steam boilers, solar Heat Calculator is used to calculate the change in temperature, C p =$\frac{H}{m * \Delta T}\$ Step 2 : Put the values in the formula and calculate it further. To calculate the overall wattage required you need to calculate the wattage for process start (Delta T May 15, 2011 · 24. If you want to calculate DeltaG under non-standard conditions, How can I calculate delta G of a reaction? Chemistry Thermochemistry Spontaneous and Non Mar 24, 2009 · I've received a lot of questions on delta, including: How do I calculate an options delta? How does volatility effect an option's delta? Can a delta be If you can't find what you're looking for, or you have an idea for a calculator that would be helpful to you, let us know. 0 8. 0 3. Heat calculators. PRESSURE AND FLOW FOR HEATING AND COOLING CONTRACTORS it is possible to calculate the airflow required to pressurize the system. 0 0 25 50 75 100 125 150 delta T - Temperature Rise (ºF) Water Flow (gpm) RL75I Type of Appliance Thunder Group Model Number What is the equation "Q equals mc delta T"? A: Quick Answer. ? bhdrsc, Oct 4, 2009 (Delta T) Q is the heat rate If you want to you can calculate a volumetric flow rate and convert United Laboratories Cycle Smart™ cooling tower programs use the greenest, How to Enter Data into the Calculator Enter the current Delta T across Tower; Our heat loss calculator can determine how much heat is being lost Grooved steel pipe and fittings for greenhouse heating Delta T heating systems that use high Here’s an example. Being a specialized company working with different suppliers and engineering teams This spreadsheet contains calculators that produce Cohen's d effect size (plus r-squared, Hedge's g, and Glass’s delta) for single sample, independent, and repeated 0. K) ΔQ: Heat required for the temperature change, in J ΔT: Temperature change, in K m: Mass of the object, in kg » Specific Heat Dec 29, 2017 The Average Acceleration calculator uses the equation, a = (V2 - V1)/(Δt), to compute the average acceleration base on two observations of velocities (v2 and v1) and the time between the measurements (Δt). 8946 "/cy/cy in the Heat is defined as the energy transfer due to difference in the temperature between a system and its surroundings. The Fraternity needs your time, talent, and resources to support our undergraduate chapters/colonies A lot of politicians come to the Mid-South Farm and Gin Show in Memphis, Tenn. In each case show that Delta G- Delta H-T installation guide delta t pro control units wifi + ethernet excellence by design Specific Heat Capacity Calculator Specific heat refers to the amount of heat required to raise unit mass of a substance's temperature by 1 degree. where: SDpooled = √((SD12 + SD22) ⁄ 2). Glass' Delta and Hedges' How does the heat transfer conduction calculator works? The heat transfer conduction calculator below is simple to use. We would like to show you a description here but the site won’t allow us. m = 1000 gms = 1 kg. If he has the mass of 1000gms. Plan your trips — or mileage runs — to maximize mileage earning, or Free calculus calculator - calculate limits, integrals, derivatives and series step-by-step SkyMiles helps you go further. Aug 22, 2007 · BTU/hr calculation I Delta T = 70*F - 55*F = 15*F At 55°F dry bulb, you calculate 30. What is the equation "Q equals mc delta T"? A: Quick Answer. See our other Electronics Calculators. Cohen's d = (M2 - M1) ⁄ SDpooled. 45 + 1. 8 30 | 2021-05-12 01:03:13 | {"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": 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": 0.46657314896583557, "perplexity": 2486.9392783386606}, "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-2021-21/segments/1620243991693.14/warc/CC-MAIN-20210512004850-20210512034850-00304.warc.gz"} |
https://codereview.stackexchange.com/questions/239419/sheetchange-event-vba | # SheetChange event VBA
I would like to ask for a code review as I feel like it's not the most efficient way of doing it. For context I have a add-ins with all the code in module and then the SheetChangeevent is in ThisWorkbook. Also in the add-ins I have 2 sheets that will have data on it so that the ActiveWorkbook will be able to read this info when running code form module the events will fire and do a vlookup against the sheet in the add-ins.
Private Sub ExcelApp_SheetChange(ByVal Sh As Object, ByVal Target As Range)
Dim KeyCells As Range
Dim wsMyList As Worksheet
Set wsMyList = ThisWorkbook.Sheets(2)
Set KeyCells = [B3,B5] 'I only need this 2 cells to fire events
If Sh.Name <> "Response" Then
If Not Intersect(Target, Sh.Range("B3:B5")) Is Nothing Then 'not too sure how to do it here so I put 3 cells instead of 2
If Target.Row = 3 Then
If Range("B3").Value = vbNullString Then Exit Sub
Application.EnableEvents = False
If Sh.Range("B3").Value <> vbNullString Then
Sh.Range("B4").Value = Application.WorksheetFunction.VLookup(Sh.Range("B3").Value, wsMyList.Range("A:B"), 2, False)
Sh.Range("B6").Value = "Type or Select a transaction"
Else
Sh.Range("B4").Value = "Type or Select a program"
End If
Columns.ClearColumns
Transactions.FetchTransactions
Application.EnableEvents = True
ElseIf Target.Row = 5 Then
If Range("B5").Value = vbNullString Then Exit Sub
Application.EnableEvents = False
If Range("B5").Value <> vbNullString Then
Sh.Range("B6").Value = Application.WorksheetFunction.VLookup(Sh.Range("B5").Value, wsMyList.Range("D:E"), 2, False)
Columns.ClearColumns
Columns.PopulateFields
End If
Application.EnableEvents = True
End If
End If
End If
End Sub
• Please could you flesh out the description, currently it conveys nothing to me. – Peilonrayz Apr 6 '20 at 20:35
• @Peilonrayz sorry what do you mean? I don’t understand your ask. – QuickSilver Apr 7 '20 at 7:32
• Please attempt to provide a more general description of the code's purpose. What are you doing with Excel and why? What are your specific efficiency concerns? – Reinderien Apr 13 '20 at 17:08
When writing any kind of event method, I always try to minimize the code that executes because you don't want the user to be aware of processing that is happening between the keystrokes. In your case, your code is firing each time ANY cell on ANY sheet is changed. So in the spirit of keeping things streamlined, don't create, initialize, or perform any logic that you don't really need (until you need it). Using this philosophy, the beginning of my example method would look like this:
If Sh.Name = "Response" Then Exit Sub
Dim checkCells As Range
Set checkCells = Union(Sh.Range("B3"), Sh.Range("B5"))
If Intersect(Target, checkCells) Is Nothing Then Exit Sub
If Target.Address = vbNullString Then Exit Sub
Notice that if the changed sheet is "Response", then initializing any other variable is meaningless. Once we get past that, a checkCells range is established using the Union function. It may be a bit overkill sometimes, but clearly illustrates the idea that you're looking at multiple, non-contiguous cells/ranges.
Also, from your OP code, if either cell is empty, you are immediately exiting. So you can check the target address after the other checks right away.
But then I get into an issue where you check if the cell value is null, but right away you're checking that it has a value:
If Range("B3").Value = vbNullString Then Exit Sub
Application.EnableEvents = False
If Sh.Range("B3").Value <> vbNullString Then ...
This is redundant. And -- by the way -- you'll never get to your Else statement because you've already exited if B3 is a null string.
The statement in both parts of your If statement is confusing:
Columns.ClearColumns
Columns is a property of a Range or Worksheet, and ClearColumns is not a method of Range at all that I know of. So I'm assuming it's a part of the add-in. But if Columns is the name of one of your code modules, then change it. Using Columns is not a good name to use because it is the same as an existing property and is confusing. If you are clearing columns on a worksheet, then specify which worksheet always. I'm also assuming the Transactions is a code module in your VBA project as well.
Here is all of the code:
Option Explicit
Private Sub Workbook_SheetChange(ByVal Sh As Object, ByVal Target As Range)
If Sh.Name = "Response" Then Exit Sub
Dim checkCells As Range
Set checkCells = Union(Sh.Range("B3"), Sh.Range("B5"))
If Intersect(Target, checkCells) Is Nothing Then Exit Sub
If Target.Address = vbNullString Then Exit Sub
Dim wsMyList As Worksheet
Set wsMyList = ThisWorkbook.Sheets(2)
Application.EnableEvents = False
With Sh
Dim lookupArea As Range
If Target.Row = 3 Then
Set lookupArea = wsMyList.Range("A:B")
.Range("B4").Value = Application.WorksheetFunction.VLookup(Target.Value, _
lookupArea, _
2, False)
.Range("B6").Value = "Type or Select a transaction"
Columns.ClearColumns
Transactions.FetchTransactions
ElseIf Target.Row = 5 Then
Set lookupArea = wsMyList.Range("D:E")
.Range("B6").Value = Application.WorksheetFunction.VLookup(Target.Value, _
lookupArea, _
2, False)
Columns.ClearColumns
Columns.PopulateFields
End If
End With
Application.EnableEvents = True
End Sub
• thanks for the feedback and all your assumptions are correct. I will modify everything accordingly. Also if you have time are you able to provide any feedback on the vlookup because I ended up putting on error resume next and on error goto 0 before and after that line as it was failing in some conditions. Thanks a lot. – QuickSilver Mar 27 '20 at 16:12
• I would suggest you change your VLookup to use Application.VLookup instead. See this guide for more information. – PeterT Mar 27 '20 at 18:57 | 2021-06-23 03:46:15 | {"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": 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": 0.17523512244224548, "perplexity": 3622.437092574835}, "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-2021-25/segments/1623488528979.69/warc/CC-MAIN-20210623011557-20210623041557-00229.warc.gz"} |
https://indico.cern.ch/event/656452/contributions/2869779/ | Quark Matter 2018
May 13 – 19, 2018
Venice, Italy
Europe/Zurich timezone
The organisers warmly thank all participants for such a lively QM2018! See you in China in 2019!
Spin alignment measurements using vector mesons with ALICE detector at the LHC
May 15, 2018, 9:20 AM
20m
Sala Mosaici-2, 3rd Floor (Palazzo del Casinò)
Sala Mosaici-2, 3rd Floor
Palazzo del Casinò
Parallel Talk Chirality, vorticity and polarisation effects
Speaker
Dr Ranbir Singh (NISER, Jatni, India)
Description
Large magnetic field and large angular momentum are expected to be present in the initial stages of high-energy heavy-ion collisions. One of the physics interests of the heavy-ion program using the ALICE detector at the LHC is to look for signatures of these effects. This can be achieved by studying the angular distributions of the decay daughters of hyperons and vector mesons.
We present new measurements related to spin alignment of K$^{*0}$ vector mesons at mid-rapidity for Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 and 5.02 TeV. The zeroth element of the spin density matrix element, $\rho_{00}$, is found to have values slightly below 1/3 at low transverse momentum ($p_{\mathrm{T}}$) for K$^{*0}$ mesons, while it is consistent with 1/3 (no spin alignment) at higher $p_{\mathrm{T}}$. No spin alignment is observed for K$^{*0}$ in pp collisions at $\sqrt{s}$ = 13 TeV and for the spin zero hadron K$^{0}_{S}$ in 20-40$\%$ Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV.
The $\rho_{00}$ values are not only sensitive to the angular momentum of the system but also to
the production mechanism of the vector meson. The centrality dependence of the $\rho_{00}$ results
with production plane and event plane in Pb--Pb collisions at LHC energies will be discussed in detail.
Collaboration ALICE Experiment Presenter name already specified | 2021-12-05 12:08:03 | {"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": 0.8867416381835938, "perplexity": 2428.3858570513084}, "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/1637964363157.32/warc/CC-MAIN-20211205100135-20211205130135-00361.warc.gz"} |
https://mathemerize.com/find-the-asymptotes-of-the-hyperbola-2x2-5xy-2y2-4x-5y-0-find-also-the-general-equation-of-all-the-hyperbolas-having-the-same-set-of-asymptotes/ | # Find the asymptotes of the hyperbola $$2x^2 + 5xy + 2y^2 + 4x + 5y$$ = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes.
## Solution :
Let $$2x^2 + 5xy + 2y^2 + 4x + 5y + k$$ = 0 be asymptotes. This will represent two straight line
so $$abc + 2fgh – af^2 – bg^2 – ch^2$$ = 0 $$\implies$$ 4k + 25 – $$25\over 2$$ – 8 – $$25\over 4$$k = 0
$$\implies$$ k = 2
$$\implies$$ $$2x^2 + 5xy + 2y^2 + 4x + 5y + 2$$ = 0 are asymptotes
$$\implies$$ (2x+y+2) = 0 and (x+2y+1) = 0 are asymptotes
and $$2x^2 + 5xy + 2y^2 + 4x + 5y + c$$ = 0 is general equation of hyperbola.
### Similar Questions
Angle between asymptotes of hyperbola xy=8 is
Find the normal to the hyperbola $$x^2\over 16$$ – $$y^2\over 9$$ = 1 whose slope is 1.
Find the equation of the tangent to the hyperbola $$x^2 – 4y^2$$ = 36 which is perpendicular to the line x – y + 4 = 0
The eccentricity of the conjugate hyperbola to the hyperbola $$x^2-3y^2$$ = 1 is
If the foci of a hyperbola are foci of the ellipse $$x^2\over 25$$ + $$y^2\over 9$$ = 1. If the eccentricity of the hyperbola be 2, then its equation is : | 2022-11-26 09:46:02 | {"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": 0.8335554599761963, "perplexity": 296.38237396954065}, "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/1669446706285.92/warc/CC-MAIN-20221126080725-20221126110725-00751.warc.gz"} |
https://cs.stackexchange.com/questions/40493/existence-of-functions-that-only-tests-memberships-without-revealing-any-informa | # Existence of functions that only tests memberships without revealing any information about the members
Given a set of strings $\{s_1, s_2, \cdots, s_n\}$, is there a way to encode them to a program $P$, such that given some string,$s$, the program can test the membership efficiently (deterministically or probabilistically), BUT no adversary can ever learn/extract any (member) strings from the program (in polynomial time)?
Can we have such functions with computational assumptions (such that they are secure even in the post quantum regime)?
Note: We can also entertain a program that takes, as input, a valid $s$ that is a member of P, to verify the membership of $s'$.
## migrated from cstheory.stackexchange.comMar 17 '15 at 2:06
This question came from our site for theoretical computer scientists and researchers in related fields.
• An all-powerful adversary can just run $P$ successively on all strings until he finds those that he wants. – Emil Jeřábek Mar 16 '15 at 15:47
• Of the many Bloom filter variants, which have you investigated and found wanting? – jbapple Mar 16 '15 at 15:51
• Thank @EmilJeřábek, and yes, of course, I should have mentioned we also allow quantum resources, else we are willing to settle with computational assumptions that are secure in the post-quantum regime. – Subhayan Mar 16 '15 at 15:51
• @jbapple thanks, I am looking at bloom filters now. Can you please refer to some paper that might help (while respecting the PQ-crypto assumptions) and make this an answer perhaps? – Subhayan Mar 16 '15 at 16:27
• bloom filters will not work, for one thing the false positives. Also because they are in no way satisfy the criteria- "no adversary can ever learn/extract any (member) strings from the program?" – Subhayan Mar 16 '15 at 21:00
Yes. This is known as the problem of obfuscating a point function. There are standard schemes. See, e.g., the following papers:
On Obfuscating Point Functions. Hoeteck Wee. STOC 2005.
Technically, a point function is a function $f_s(x)$ given by $f_s(x)=1$ if $x=s$ and $f_s(x)=0$ for all $x \ne s$. There are standard schemes for obfuscating point functions. This corresponds to the special case of your problem where $n=1$ (hiding a single value $s$). However, the techniques all generalize to arbitrary $n$, as long as the number $n$ of values is exponentially small compared to the size of the domain of the function.
You can use a one-way permutation. Pad all strings to the same length. Let $f$ be a one-way permutation. Store $S = \{f(s_1),\ldots,f(s_n)\}$. You can test whether $x \in \{s_1,\ldots,s_n\}$ by checking whether $f(x) \in S$. However, depending on the security properties of $f$, you shouldn't be able to learn anything about $S$ without a lot of effort.
One way to do this is to compute and store a secure hash of the each of the strings (e.g., using SHA). Then the program $P$, given a string, checks to see if its hash is among the stored list of hashes. A one-way permutation could be used, instead of a hash, but this would require more time and space (especially if the strings are long, relative to the secure hash size). However, the drawback of using a hash, rather than a one-way permutation, is that it is theoretically possible that the program can give a false positive output, i.e. incorrectly say a string was in the list when it in fact was not; but such false positive cases in practice will never be observed (barring some major algorithmic/technological breakthrough), provided the hash size is chosen large enough.
Of course, there is a potential attack of simply trying the possible strings by brute force; this will be a problem if some of the strings are short or are formed in a predictable way.
In any case, it would not be easy to mathematically prove that an attacker with a classical computer can never extract any strings from $P$ in polynomial time (much less could we prove that the scheme is secure even against attacks from a quantum computer). Such a proof would give a proof that $P \neq NP$, since if an attacker could solve NP-complete problems in polynomial time then they could extract the strings in polynomial time as well.
• "such false positive cases in practice will never be observed" -- where never means "with low probability", I assume? – Raphael Mar 17 '15 at 6:44 | 2019-02-19 22:57:48 | {"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": 0.4351343810558319, "perplexity": 633.0868392507916}, "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-2019-09/segments/1550247493803.26/warc/CC-MAIN-20190219223509-20190220005509-00196.warc.gz"} |
https://www.snapxam.com/problems/18117179/integral-of-2x-1-0-x-3-2-dx-from-0-to-2 | # Step-by-step Solution
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
(◻)
+
-
×
◻/◻
/
÷
2
e
π
ln
log
log
lim
d/dx
Dx
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc
asin
acos
atan
acot
asec
acsc
sinh
cosh
tanh
coth
sech
csch
asinh
acosh
atanh
acoth
asech
acsch
## Step-by-step explanation
Problem to solve:
$\int_0^2\left(\left(\frac{2x-1}{\left(x+3\right)^2}\right)\right)dx$
Learn how to solve definite integrals problems step by step online.
$\begin{matrix}u=x+3 \\ du=dx\end{matrix}$
Learn how to solve definite integrals problems step by step online. Integrate (2x-1)/((x+3)^2) from 0 to 2. Solve the integral \int_{0}^{2}\frac{2x-1}{\left(x+3\right)^2}dx applying u-substitution. Let u and du be. Rewriting x in terms of u. Substituting u, dx and x in the integral and simplify. Multiplying polynomials 2 and u-3.
$\frac{31}{351}$$\,\,\left(\approx 0.08831791419864787\right)$
### Problem Analysis
$\int_0^2\left(\left(\frac{2x-1}{\left(x+3\right)^2}\right)\right)dx$
### Main topic:
Definite integrals
~ 2.78 seconds | 2020-02-25 05:34:59 | {"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": 0.9864236116409302, "perplexity": 6139.424041439986}, "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-2020-10/segments/1581875146033.50/warc/CC-MAIN-20200225045438-20200225075438-00178.warc.gz"} |
https://itprospt.com/num/16116046/find-the-madan-numbe-0l-service-clls-trom-tho-following | 5
# Find the madan numbe 0l service clls trom tho following lisl14, 16.27,12,27, 17, 13,29, 10The metianEnter your answerin the answcr bOaType here t0 search...
## Question
###### Find the madan numbe 0l service clls trom tho following lisl14, 16.27,12,27, 17, 13,29, 10The metianEnter your answerin the answcr bOaType here t0 search
Find the madan numbe 0l service clls trom tho following lisl 14, 16.27,12,27, 17, 13,29, 10 The metian Enter your answerin the answcr bOa Type here t0 search
#### Similar Solved Questions
##### =FiGURE 13.7 Cystic fibrosis pedigree.
= FiGURE 13.7 Cystic fibrosis pedigree....
##### Notes_4_O1pdf(53.3KB)Practice QuestionsThroughout these questions, let E = Q(V3,), where V-1 as usual Find basis for E as vector space over ( Explain how your know 2. Let your answer is correct. =v3 2i, an element of E. Calculate 2? of the basis for F in Question and writing YOuT answers as linear cornbinations Find the minimal polynomial of 2 = V3 _ 2i in use @Ix]; different approach if you prefer; but (Hint; Use the calculations of Question 2. You Iay i8 the minimal one: in any Case explain ho
Notes_4_O1pdf(53.3KB) Practice Questions Throughout these questions, let E = Q(V3,), where V-1 as usual Find basis for E as vector space over ( Explain how your know 2. Let your answer is correct. =v3 2i, an element of E. Calculate 2? of the basis for F in Question and writing YOuT answers as linear...
##### 82 3 I ] ~/2 8 ] 3 1 J3 9 I 1 2 I 1 2 1 1 2 1 I pIg J 8 W 1 U W I 0 1 0 J 1 3 3 ~/a 3 1 1 3 V 0 3 3 W 8 M ! 3 0
82 3 I ] ~/2 8 ] 3 1 J3 9 I 1 2 I 1 2 1 1 2 1 I pIg J 8 W 1 U W I 0 1 0 J 1 3 3 ~/a 3 1 1 3 V 0 3 3 W 8 M ! 3 0...
##### {Nim [0N1 1 8 7
{Nim [0N 1 1 8 7...
##### Part EA function of cholesterol that does not harm health is its roleas a component of animal cell membranes All of cholesterol's effects cause the body harm calcium and phosphate metabolismas the primary female sex hormonethe most abundant male sex hormoneSubmitPrevious Answers RequastAnsWor
Part E A function of cholesterol that does not harm health is its role as a component of animal cell membranes All of cholesterol's effects cause the body harm calcium and phosphate metabolism as the primary female sex hormone the most abundant male sex hormone Submit Previous Answers RequastAn...
##### Question 2 Nol yrt onswered Maiked out 0i 300The natural domain of Ihe vector-valued furction 7(0) = (V16- / , In(in t), 0} is(~c,-AU(4, 06)Flag question[-4,4] (1 , 4]
Question 2 Nol yrt onswered Maiked out 0i 300 The natural domain of Ihe vector-valued furction 7(0) = (V16- / , In(in t), 0} is (~c,-AU(4, 06) Flag question [-4,4] (1 , 4]...
##### II. Problem Solving 10 pts each): Write your solutions clearly:1> Two forces act on a body as shown in Fig.1 . Calculate the magnitude and direction of the third force so that the net force on the body is equal to zero_Fig:2> Calculate the magnitude of the force P needed to pull upward the block of mass 3 kg with an acceleration of 2 m/s2 as shown in Fig: = 2PamFig: 23> Calculate the magnitude of the force P needed to push the block of mass 5 kg at constant velocity along a level surfac
II. Problem Solving 10 pts each): Write your solutions clearly: 1> Two forces act on a body as shown in Fig.1 . Calculate the magnitude and direction of the third force so that the net force on the body is equal to zero_ Fig: 2> Calculate the magnitude of the force P needed to pull upward the ...
##### Write conditional statement code in R that prints "the number is positive" if the number is greater than 0 but otherwise, prints "the number is negative:if (number 0) (the number is positive} else (the number is negative }0 if (number O1 (printl" the E number is positive")} else (print("the number is negative: "W}0 if (number 01 ("the number is positive' else ("the number is negative-if (number O) (print(the number is positive)l else (printlthe n
Write conditional statement code in R that prints "the number is positive" if the number is greater than 0 but otherwise, prints "the number is negative: if (number 0) (the number is positive} else (the number is negative } 0 if (number O1 (printl" the E number is positive")...
##### Question 11 ptsConsider the portion of the curve r(t) = (At? , ~2t8 , -3t8) for 1 < t < 2 If the length of this portion of the curve is 49. find two possible integer values of A_(Give the smaller one first )
Question 1 1 pts Consider the portion of the curve r(t) = (At? , ~2t8 , -3t8) for 1 < t < 2 If the length of this portion of the curve is 49. find two possible integer values of A_ (Give the smaller one first )...
##### Primate features discussed in lecture include all of the following exceptopposable big toes and prehensile handsa reduced sense of smell:flat nails and sensitive pads on fingers.relatively large brains compared to other mammalsAll of the above are primate features_
Primate features discussed in lecture include all of the following except opposable big toes and prehensile hands a reduced sense of smell: flat nails and sensitive pads on fingers. relatively large brains compared to other mammals All of the above are primate features_...
##### The function fln)= 100+n2.5+ n5 'log nisO a O(n5 'log n)b O(n5)c O(n logn) d.0(n2.5) o(log '
The function fln)= 100+n2.5+ n5 'log nis O a O(n5 'log n) b O(n5) c O(n logn) d.0(n2.5) o(log '...
##### Find rhe formula for 1 funcrion of che form w(r) Asin( Br) C wirh () mXiUM (0,5,12). and (ii) no critical points betwcen these rwo () a minimum at (1.6, Points w(m)Check your result by 'making ' ploc: (Nor Submixted)
Find rhe formula for 1 funcrion of che form w(r) Asin( Br) C wirh () mXiUM (0,5,12). and (ii) no critical points betwcen these rwo () a minimum at (1.6, Points w(m) Check your result by 'making ' ploc: (Nor Submixted)...
##### INFRARED SPECTROSCOPY-_PRELABFrkth Rep"a (usILX IuLIXuTake (Orl OrANtN~xh{v4MtlJLREuEhIn Aompouna (rom Uhc OletPIs SHow) #ne anJle, (lzouy O1 Ye ficTuRe Infrare | Spedlroscopy_PrrdahG
INFRARED SPECTROSCOPY-_PRELAB Frkth Rep"a (us ILX IuLIXu Take (Orl Or ANtN ~xh {v4 Mtl JLREuEhIn A ompouna (rom Uhc Olet PIs SHow) #ne anJle, (lzouy O1 Ye ficTuRe Infrare | Spedlroscopy_Prrdah G... | 2022-08-15 16:26:05 | {"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": 0.6231637001037598, "perplexity": 8974.607554270253}, "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/1659882572192.79/warc/CC-MAIN-20220815145459-20220815175459-00312.warc.gz"} |
https://rna.wlu.edu/bio285/07-networklab.html | # Step 1: Load factor annotations
In doing the data analysis for this EMAP paper a hand-curated set of functional annotations was assembled for all of the 552 mutants in the collection. This set of annotations started with the GO categories, but was manually “cleaned up” based on decades of work in the mRNA processing field. I’ve provided a file containing these annotations, ordered according to the mutant axes in the genetic interaction data file:
annotations <- read.table( "data/rnaworld-annotations.txt"
, header = TRUE
, quote = "\""
)
levels(annotations$category) ## [1] "Chromatin Remodeling" "Mitochondrial " ## [3] "mRNA Export" "mRNA Processing" ## [5] "mRNA Splicing" "Nuclear Pore" ## [7] "Other" "RNA Degradation" ## [9] "rRNA Processing" "rRNA Processing " ## [11] "Transcription" "TRANSCRIPTION" ## [13] "TRANSLATION" "tRNA Metabolism" ## [15] "tRNA Metabolism " "Ubiquitin/Protein Degradation" ## [17] "UNKNOWN" You’ll see we boiled down the gene categories into a few very high-level bins. # Step 2: Principal Component Analysis Principal component analysis or PCA is a common data exploration strategy that you’ll see used often in systems biology. In our correlation matrix from last week, we saw that there are, in fact, some highly correlated sets of mutants (CFT1 & SNU23, for example). A “principal component” is a vector that describes covariance (corresponding variation) in all/most/some of the primary data vectors (here our genetic interaction profiles or correlation profiles). A “good” principal component is highly co-variant with a subset of the data and is NOT correlated with other “good” principal components. Thus, PCA is one way of “simplifying” your data down from a set of 552 unique deletion behaviors to a smaller set of principal components that describe most of the variation in the original dataset. There are several ways of implementing PCA (most popular are probably eigenvalue decomposition and single value decomposition), none of which work with missing data (sparse matrixes). We have lots of missing data values, so we’ll start by performing PCA on our correlation matrix; but it’s important to remember that in so doing we are analyzing covariation in correlation profiles, not the raw genetic interaction scores themselves. The built-in prcomp and princomp functions both perform PCA (with different procedures): interactionMatrix <- cor(emap, use = 'pairwise.complete.obs') pcs <- prcomp(interactionMatrix) You can see a summary of all of the possible principal components: summary(pcs) We always have as many possible principal components as input vectors, but the “usefulness” (proportion of variance explained by each) diminishes at the higher component numbers. You can plot the top 10 (this is called a scree plot): plot(pcs) What do we see here? How do we know which, if any, of these principal components are useful decompositions of our data? Let’s take a look at whether or not any of the top PCs effectively describe variation accross our functional groups (see ?cbind, ?prcomp and ?facet_wrap to make sure you follow along below): library(ggplot2) library(reshape2) annotatedPCs <- cbind( category = annotations$category
, melt(pcs$rotation[,1:9]) ) orderdPCs <- annotatedPCs[ order( annotatedPCs$X2
, annotatedPCs$category , annotatedPCs$X1
)
,
]
orderdPCs$geneID <- rep(1:552, times = 9) ggplot(orderdPCs, aes(geneID, value, fill=category)) + geom_bar(stat = 'identity') + facet_wrap(~X2) What do we see here? There are some regions in which PC values are mapping to the functional categories (for example, PC1 mostly discriminates chromatin from the mitochondria), but the effects are moderate. Try repeating this procedure to ask if any of the PCs summarize differences between deletion and DAMP strains (mostly to prove to yourself you followed along above!). One complicating factor here is that we are assessing covariance in the correlation profiles rather than in the raw data themselves. One (sometimes sketchy) approach to dealing with missing data upstream of PCA is to use data “imputation”. There are many imputation strategies, all of which amount to making-up data for missing values. We should not be terribly surprised when a model is used to impute data and then it is discovered that the data fit nicely to a model. However, it’s worth taking a pass at it with our raw data. The PCA function in the FactoMineR package performs imputation by default if you have missing values: library(FactoMineR) ?PCA Explore PCA with this package and the raw data values. Are the results more compelling? PCA can be quite useful for higher order analysis as PCs can be used in any of the downstream applications where raw data vectors would have been, including clustering and network building. # Step 4: Markov cluster algorithm There are MANY different approaches to building network topologies from raw data and correlation matrixes. Just a few Baysian approaches are described on the CRAN Task-views page. In systems biology, the Markov clustering algorithm, or MCL, is very much en vogue because of its spead and generally good performance with biological data. Here, we’ll walk through performing MCL and drawing the resulting network, which will also give us the opportunity to see how easy it is to use external data software as part of an analysis project in R. Currently there are a number of software pacakges that implement MCL (including one with GPU compute support), but the original was published as a stand-alone command line tool. From the shell, you can download the source code and compile the mcl command line program very easily: wget http://micans.org/mcl/src/mcl-12-068.tar.gz tar xzf mcl-12-068.tar.gz cd mcl-12-068 ./configure --prefix=$HOME/local
make install
By default, compiled binaries will be installed in your ~/local/bin directory, so you can make sure the compilation went well by running this shell command:
~/local/bin/mcl --help
You can run external programs (or exectue any shell command) from within an R session using the system function. For example:
system("~/local/bin/mcl")
This makes it very easy to chain calls to external tools into your R data analysis scripts.
From the main manual page for the MCL command line tool, we can see that this program expects an input file with three columns (unnamed): node1, node2 and weight. Let’s start by using our pearson correlation matrix for weights:
write.table( melt(interactionMatrix)
, "interaction-matrix.txt"
, col.names = FALSE
, row.names = FALSE
, quote = FALSE
)
We can then run mcl on this file, specifying a new file name to output results:
system("~/local/bin/mcl interaction-matrix.txt --abc -o mcl.results.txt")
Explore the structure of this results file!
# Challenges
Here are some things to try out (either with your own data, or with this EMAP data):
• Play with the inflation parameter using the -I switch to see how this affects the number and organization of the clusters you get back.
• Explore different mechanisms for visualizing the network of clusters you get out of the MCL algorithm. See the igraph package for useful network graph drawing tools.
• Explore alternative network building algorithms listed on the CRAN task views page. | 2019-05-20 04:20:21 | {"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": 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": 0.2715361714363098, "perplexity": 4658.030320913427}, "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-2019-22/segments/1558232255562.23/warc/CC-MAIN-20190520041753-20190520063753-00220.warc.gz"} |
https://dwwiki.mooo.com/w/index.php?title=Death&diff=prev&oldid=67437 | # Difference between revisions of "Death"
Death visits almost all of us.
## Raise & Resurrect
As long as you still have some lives in reserve, you can be raised or resurrected as below.
• Note that you must wait for DEATH to finish his speech before you can be raised or resurrected.
Otherwise nothing short of the Rite of AshkEnte will allow you to be corporeal again.
### Raise Dead ritual
Raise Dead is a ritual available to Priests and Priestesses of Gufnork, Pishe and Sek. It is also available through some NPCs and in some special rooms, usually requiring some interaction by the ghost, such as using the command "raise" or saying "please raise me". I don't know if any of the NPCs are language sensitive. It will restore the ghostly player to a more substantial form, albeit with a reduced constitution stat, no XP other than that gained while dead, and very low HP. This is the same as being raised in the shrines and temples listed below.
The Restore ritual can fix their constitution stat afterwards.
### Resurrect ritual
Resurrect is a ritual available to Priests and Priestesses of Pishe. The advantages of resurrect is that it can restore some of the lost XP and will also restore player stats, but it will not restore HP.
Resurrect needs a lot of energy (GP) to cast, and even high level Priests and Priestesses can fail. They will then have to wait for their GP to build up before trying again. Most Priests and Priestesses do not appreciate funny stunts while waiting, and the best advice if this occurs is to "keep your hands away from the keyboard and wait patiently". Your funny stunt may be perceived as complaining about ritual failure (see above) which may lead to you receiving a reduced service. This might be, but is not limited to: raise dead instead of resurrect, no healing after raising / resurrection, no restore after a raise, or the pishite just leaving.
The percentage of XP return seems to be based on the faith.rituals.curing.target bonus of the priest doing the resurrection, and any ritual modifiers. The return to the resurrected person seems to be capped at 75% of their pre-death XP. Note, however, that this only applies to the xp they had when they died. Any xp accumulated while dead is returned in full.
A pishite will not necessarily be able to tell you what your XP return will be. However, unless you had little xp on you when you died, it makes sense to look for the highest resurrection bonus you can. The skills website and the "who pishe" command may help. Note, however, that ghosts asking Pishites for their bonuses at their moment of need may alienate those Pishites.
See also the ritual information for Resurrect.
## Getting Resurrected
There are people who will try and help you. These are generally Pishites, who are granted the ritual resurrect, but there are a small but slowly increasing number of non Pishites and even non Priests who can also resurrect using faith rods.
You should:
• Be Polite, we did not kill you, we are trying to help you.
• Understand that immediately and now are relative, and may depend on your rescuers ability to first lead any groupmates to a safe area and then travel half way across the disc.
• Warn rescuers of any specific hazards, like Yetis or Spiders in the room with your corpse. If it killed you, it may present a hazard to us too, we're not all big tough fighters like you were before you died.
• Show appreciation. This need not be financial, but your attitude may affect the response next time you need assistance.
• Remember that in terrains, the easiest way for us to find you is by using "Find Corpse", so staying with your corpse helps. Staying in the room with your corpse may also be the only thing that stops the room from unloading (and taking your former inventory with it) if your corpse decays (particularly if there are no living things in the room with it).[1]
• Be specific about where you are, if you can. If you're in a city, the name of the street you're on is helpful (although some streets, like Endless Street or Long Street, are large enough that more information may be necessary). If you're in a building, tell us. If you're in a hidden area[2], be clear about that--not everyone knows about all of those.
• Stay Still while being resurrected, if you walk away during the ritual, the priest will waste 3 minutes of gp regeneration on a failed ritual! Unfollow your groupmates if they might move.
You should not, under any circumstances, complain about:
• Response time.
• Delays waiting for GP regeneration.
• Failure to Raise / Resurrect / Restore.
• Inability to retrieve heavy items from your drowned corpse.
• The unwillingness of rescuers to die whilst retrieving your equipment.
• Less XP return than you expected.
If the ritual failed, sit tight and wait quietly unless the priest says otherwise. A priest that has just failed a ressurect or raise ritual probably feels upset enough about it without you doing or saying anything at all. GP regen is normally required before the ritual can be repeated. We didn't kill you, we're trying to help. Please don't get annoyed at us!
After resurrection, the following commands will be helpful:
"get all from <name> corpse" where <name> is your simple character name, and
"equip" which should be done quickly so that you don't fumble items.
Please note that if you have, for example, magic scrolls on your corpse, it can be hazardous and even fatal to get your inventory before you have been healed. Players have died their final death doing this. Even a single scroll skillcheck can be failed by an experienced wizard! Wait until you have been healed. The same applies to, eg, getting and equipping poniards.
It would also be a good idea to warn the pishite in advance if there are npcs likely to attack you after you are resurrected, eg Hitmen or City Guards in Genua, Royal Guards in Sto Lat, etc. They may be able to take action to minimise the risk to you of dying immediately to such an attack, such as resurrecting you in a passage room.
## Points to Remember
Remember that after both Resurrect and Raise you will need some healing before you do anything that may cause a loss of HP. After Raise you will also need a Restore or to wait for your Con stat to increase to its normal level. Shrines, Temples etc.
## RRU
"What about my expensive equipment?" you ask.
There is a club called RRU who generally try and help people who have died in difficult to reach or hazardous locations, however the help they can give is obviously limited to suitably skilled people being logged on at the right time. If the only RRU member logged in is a weak Pishite with a moderate healing bonus, don't expect them to pull your corpse out of the snail, climb to the hub and resurrect you, or swim to the bottom of the circle sea.
Finally, if it's that valuable, you're in an impossible to reach or extremely dangerous area, and you had a spare life, then consider the use of godmother (see below).
RRU have their own website but it is out of date.
## NPC Raise Dead Locations
These are locations where a ghostly player may be restored to a more substantial form without player assistance. In all cases, the restored player will have reduced stats. You may need to use a command (such as "raise") or request the help of NPC ("say please raise me" or "sayto <npc> please raise me"), and some of the NPCs may be language sensitive. The known locations are:
Area Location Npc What to do More information
Ankh-Morpork Temple of Small Gods, small chapel Harry say raise 4 South, 4 West and North from the carriage stop outside the temple.
Djelibeybi Temple of Sessifet, tiny chapel Ptamney (just enter the room) It takes only seconds. The temple of Sessifet is a safe zone, your health and gp will not regenerate there.
Ephebe Outside the taverna on Flatulus Walk, sw of the Harbour Market Himaclitus sit beside Himaclitus;listen to Himaclitus (answer yes or nod to the final question) You can pretend to understand the lesson that Himaclitus tries to teach you by nodding at him at the right time.
Bes Pelargic Curative place, northeast end of BP Tu Silent Llamas (just enter the room) It takes only seconds. Extreme northeast of BP, West and then South from the junction of Chopsticks and Chestnut Path in the Imperial district
Ohulan Cutash Shrine of Small Gods, south of market (a Small God) raise me 3 South and then East from the centre of the Market Square
Ramtops Mountain shrine, south of the Land of Giants (mountain spirits) raise me (wait for the prompt) From the center of the Land of Giants: sw,sw,s,journey se,journey s, journey sw, n - on the first side path of the track to the Temple of Soyin. Watch out for Yetis and Lynxes that will pursue you (while alive, for example after being raised in the shrine).
Sto Plains Hedge Wizard's Guild, west room attendant (Rimpog) say raise North-East of Pekan Ford on the path towards Rham Nitz, on the North-West side of the path on the edge of Skund Forest.
Brown Islands Quiet little hut, beach painted medicine man (MedicineMan) (just enter the room) It takes only seconds.
Genua Tent, southwest corner of Triangle Market Mrs Gogol say help Eastern parts, north of the docks, Triangle Market, southwest corner
Uberwald Unnamed town, Decrepit Temple to Pishe Mad Priest Von Bobenstein raise me The unnamed town is southeast of Escrow and again southeast of the path leading to the Magpyr castle.
Netherworld (say misty) Misty ? Note that priests cannot perform rituals in the Netherworld, and the exit guardian will stop people from leaving whether or not they are ghosts. You cannot cast either.
## Godmother
The NPC of last resort, the Godmother will transfer you to your start location, raise you and (as long as you request her help in the same room as your corpse is in) bring your body along as well, but she will charge, initially just a temporary stat drop, later on she charges a life. Make sure you have a spare life before using the godmother command. If you are having problems understanding or communicating with the good fairy, just "say yes", "nod" your head and "think yes".
If aggressive NPCs are attacking the godmother, or she's drowning when you summon her, just use "think yes" after she appears, and usually that will do the trick.
Note that if an aggressive NPC attacks and kills the godmother when you summon her, the NPC should be bugrepped for attacking the godmother.
## Other Useful Information
### Your Corpse and Inventory
Upon death, everything on you will be on your corpse, except pets, which will remain in your arms. Your corpse will decompose over the course of an hour, and will continue to do so even after you've logged off. When it finishes, any items that were on it (your former inventory) will be left on the floor. Items on the floor can be lost if someone picks them up, or if the room unloads, so it's a good idea to get your stuff as soon as possible and not log off while dead.
Getting your inventory off your corpse can be almost as problematic for the newbie as sorting out their first resurrection. Here is a useful alias, it assumes a player's name is bob.
alias aliveagain get all from bob corpses;equip;get all from bob corpses;equip;get all from bob corpses;equip;get flowers;braid them in my hair
This alias will recover your inventory from your corpse when you are in the same room as it, and equip all worn or held items that it can from that inventory, which should prevent fumbling due to an excessive inventory count.
There's 3 get all because you can have up to 50 things in your inventory, plus money which counts as one when picking up.
Simply return to the room with your corpse and use the command "aliveagain".
Remember to wait until you've been fully healed to do this--see the previous note about scrolls and other potentially damaging items.
### New Lives
You have a number of deaths remaining until your final death. However, you can buy more lives, so dying can be merely expensive rather than, um, fatal.
If you died in combat, your opponents will stop fighting you. That is, once you're raised or resurrected and are in the same room as they are, they won't attack you. This doesn't apply to NPCs who are aggressive in the first place, such as crocodiles.
## Final Death
When you have died and have no more lives left, your death will be notified to players around the MUD.
What it looks like when you have died for the last time:
You have died your final death. Your name will be inscribed in the Hall of Legends.
Aristotle has died for the last time. He is mourned.
An Igor walks up to the corpse of Aristotle and quickly strips all the items from it before walking off with them.
If a Pishe priest or any other priest tries to resurrect you, they will see this message:
Death tells you: NO, THIS TIME HE IS COMPLETELY DEAD.
The only way to return to corporeality is if other players complete the Rite of AshkEnte.
• This is very expensive and time consuming.
• You lose all money and most items.
• Your skills are cut.
## Footnotes
1. If you ran off and aren't sure exactly where your corpse is now, you have a couple of options. First, you can use "history" or your scrollback to help retrace your steps. Second, you can check the ascii map--this shows where living things are, so sometimes it can give you a rough idea of the path you took (and if you were killed by a living thing, it's likely still in the room). If you died underwater, remember that corpses sink to the bottom.
2. Somewhere you get to through a non-obvious exit, or somewhere where you have to do something to make the exit visible. | 2022-10-05 19:14:13 | {"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": 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": 0.23784546554088593, "perplexity": 4130.495763770993}, "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-2022-40/segments/1664030337663.75/warc/CC-MAIN-20221005172112-20221005202112-00144.warc.gz"} |
http://pisarenko.net/blog/2018/03/07/customized-disposable-desktop-vms-fast/ | I’ve been running virtualized desktop environments for a couple of years now, doing various tasks from within Virtualbox VMs. I like the fact the tools and libraries for the job at hand are isolated from other environments. But I have been missing on the dynamic aspect, somehow ending up with VMs which are as prone to maintenance as physical installations. Recently, I’ve decided to solve this issue once and for all with the help from Arch Linux, Vagrant and Packer.
## Virtual Machine maintenance chores
My host operating system has relatively few things installed. A terminal client, a browser, some productivity tools and Virtualbox. I carry out most of tasks and projects from within VMs, and that includes my full-time software engineering duties. That way I don’t have to worry about hardware compatibility and at the same time enjoy my favorite development tools. With VMs I’m never concerned about mixing dependencies. Side projects can’t disrupt my main project because they’re running in isolated VMs. There’s a perceptible hit on UI responsiveness and usability, even with 3D acceleration enabled, but the benefits outweigh the costs and I consider the performance acceptable.
Yet my biggest concern was not the UI performance but the ongoing maintenance. Virtual machines offer the prospect of cheap and disposable instances that are easy to provision and destroy whenever necessary. In reality, however, I ended up with several VMs that were maintained just as if they were physical machines. I was also concerned about losing them. For that reason I kept their backups. It is certainly easier to backup VMs and recover than physical machines but that’s far from the dream of on-demand disposable instances.
Getting a new VM to usable state was also a chore. I chose Ubuntu because it is widely used and supported but it comes prepackaged with hundreds of programs. Every time it took me a couple of hours to get rid of all the stuff I don’t need and a couple of days until I’ve got everything set up the way I like it. My favorite distribution is Gentoo but that would be too much work and compile time for every single VM. Installing Gentoo every time would be a nightmare. And upgrading all the VMs even more so. Nowadays, I keep Gentoo on custom routers and servers only.
I have decided to use this opportunity to come with a scripted solution to make the best use of disposable VMs. My goal was to create customizable VMs that are built from scratch and include only the packages I want.
## Time for a new Linux distro
Before building out the infrastructure I chose Arch Linux as the base distribution. Arch is as flexible as Gentoo, sans the compilation. A minimal Arch system contains very few packages. It was a natural choice and a much wanted upgrade from the Ubuntu images I had been using.
## Automated VM provisioning
I already had a tool in mind - Vagrant. I remember trying it years ago so I vaguely recalled it could help me automate VM management. Vagrant integrates nicely with Virtualbox, which I’m using to run all my VMs. Vagrant provisions VMs from configuration files using a specified base box. After studying Vagrant documentation I have realized that having a base box is a requirement. In other words, Vagrant cannot build a complete system from scratch, i.e. from an installation ISO. Using someone else’s Arch base image is unacceptable to me as it defies the purpose – I can’t be sure there’s no superflous software installed. I wanted a solution that produces VMs from scratch: partitioning, bootloader installation, etc.
Luckily, the company behind Vagrant also provides a tool called Packer. Packer fills the gap by building base images from scripts. The combination of Vagrant + Packer is perfect for getting complete VMs from scratch. The way Packer works is rather spectacular. First, it boots the VM and programmatically types commands in the terminal – a somewhat unusual sight. The commands Packer types download a script from a shared folder and runs it. The script then sets up SSH to which Packer connects to. From there on no direct interaction is needed and further steps are executed through SSH.
## Learning Arch
With the tools in hand, my first step was to experiment with Arch and install it manually under Virtualbox while documenting every step. I chose EFI, grub and btrfs, among other things. The beauty of the setup is, that once a script works, I can change and test these small things without going through the whole setup manually. It is like unit testing. While Arch is not Gentoo it still takes effort to build. I first built a base cli system and then built a minimal X system.
## Building base box with Packer
Once the Arch installation steps were complete I began with the corresponding Packer script. I looked for and discovered a couple of existing examples online and added my changes on top. The complete script is roughly 100 lines long and takes care of everything – partitioning, installing bootloader, setting up locales and network, installing Virtualbox modules and few essential packages (e.g. ssh, neovim). The goal of the base image is to be as light as possible – other tools are installed later through Vagrant.
#!/usr/bin/env bash
# stop on errors
set -eu
DISK='/dev/sda'
FQDN='arch.bethania'
KEYMAP='us'
LANGUAGE='en_US.UTF-8'
PASSWORD=$(/usr/bin/openssl passwd -crypt 'vagrant') TIMEZONE='Europe/Zurich' CONFIG_SCRIPT='/usr/local/bin/arch-config.sh' BOOT_PARTITION="${DISK}1"
ROOT_PARTITION="${DISK}2" TARGET_DIR='/mnt' MIRRORLIST="https://www.archlinux.org/mirrorlist/?country=${COUNTRY}&protocol=http&protocol=https&ip_version=4&use_mirror_status=on"
echo "==> Create GPT partition table on ${DISK}" /usr/bin/sgdisk -og${DISK}
echo "==> Destroying magic strings and signatures on ${DISK}" /usr/bin/dd if=/dev/zero of=${DISK} bs=512 count=2048
/usr/bin/wipefs --all ${DISK} echo "==> Creating /boot EFI partition on${DISK}"
/usr/bin/sgdisk -n 1:2048:2098175 -c 1:"EFI boot partition" -t 1:ef00 ${DISK} echo "==> Creating /root partition on${DISK}"
ENDSECTOR=/usr/bin/sgdisk -E ${DISK} /usr/bin/sgdisk -n 2:2098176:$ENDSECTOR -c 2:"Linux root partition" -t 2:8300 ${DISK} echo '==> Creating /boot filesystem (FAT32)' /usr/bin/mkfs.fat -F32$BOOT_PARTITION
echo '==> Creating /root filesystem (btrfs)'
/usr/bin/mkfs.btrfs $ROOT_PARTITION echo "==> Mounting${ROOT_PARTITION} to ${TARGET_DIR}" /usr/bin/mount${ROOT_PARTITION} ${TARGET_DIR} echo "==> Mounting${BOOT_PARTITION} to ${TARGET_DIR}/boot" /usr/bin/mkdir${TARGET_DIR}/boot
/usr/bin/mount ${BOOT_PARTITION}${TARGET_DIR}/boot
echo "==> Setting local mirror"
curl -s "$MIRRORLIST" | sed 's/^#Server/Server/' > /etc/pacman.d/mirrorlist echo '==> Bootstrapping the base installation' /usr/bin/pacstrap${TARGET_DIR} base base-devel btrfs-progs neovim openssh grub efibootmgr net-tools
echo '==> Generating the filesystem table'
/usr/bin/genfstab -U ${TARGET_DIR} >> "${TARGET_DIR}/etc/fstab"
echo '==> Installing GRUB'
/usr/bin/sed -i 's/GRUB_TIMEOUT=.*/GRUB_TIMEOUT=0/' "${TARGET_DIR}/etc/default/grub" /usr/bin/sed -i 's/GRUB_CMDLINE_LINUX_DEFAULT=".*"/GRUB_CMDLINE_LINUX_DEFAULT="quiet video=1360x768"/' "${TARGET_DIR}/etc/default/grub"
echo '==> Generating the system configuration script'
/usr/bin/install --mode=0755 /dev/null "${TARGET_DIR}${CONFIG_SCRIPT}"
cat <<-EOF > "${TARGET_DIR}${CONFIG_SCRIPT}"
/usr/bin/grub-install --target=x86_64-efi --efi-directory=/boot
/usr/bin/grub-mkconfig -o /boot/grub/grub.cfg
/usr/bin/mkdir /boot/EFI/BOOT
/usr/bin/cp /boot/EFI/arch/grubx64.efi /boot/EFI/BOOT/bootx64.efi
echo '${FQDN}' > /etc/hostname /usr/bin/ln -sf /usr/share/zoneinfo/${TIMEZONE} /etc/localtime
echo 'KEYMAP=${KEYMAP}' > /etc/vconsole.conf echo 'LANG=en_US.UTF-8' > /etc/locale.conf /usr/bin/sed -i 's/#${LANGUAGE}/${LANGUAGE}/' /etc/locale.gen /usr/bin/locale-gen /usr/bin/usermod --password${PASSWORD} root
# https://wiki.archlinux.org/index.php/Network_Configuration#Device_names
/usr/bin/ln -s '/usr/lib/systemd/system/dhcpcd@.service' '/etc/systemd/system/multi-user.target.wants/dhcpcd@eth0.service'
/usr/bin/sed -i 's/#UseDNS yes/UseDNS no/' /etc/ssh/sshd_config
/usr/bin/systemctl enable sshd.service
# Vagrant-specific configuration
/usr/bin/useradd --password ${PASSWORD} --comment 'Vagrant User' --create-home --user-group vagrant echo 'Defaults env_keep += "SSH_AUTH_SOCK"' > /etc/sudoers.d/10_vagrant echo 'vagrant ALL=(ALL) NOPASSWD: ALL' >> /etc/sudoers.d/10_vagrant /usr/bin/chmod 0440 /etc/sudoers.d/10_vagrant /usr/bin/install --directory --owner=vagrant --group=vagrant --mode=0700 /home/vagrant/.ssh /usr/bin/curl --output /home/vagrant/.ssh/authorized_keys --location https://raw.github.com/mitchellh/vagrant/master/keys/vagrant.pub /usr/bin/chown vagrant:vagrant /home/vagrant/.ssh/authorized_keys /usr/bin/chmod 0600 /home/vagrant/.ssh/authorized_keys # virtualbox integration /usr/bin/pacman -S --noconfirm virtualbox-guest-utils-nox virtualbox-guest-modules-arch echo -e 'vboxguest\nvboxsf\nvboxvideo' > /etc/modules-load.d/virtualbox.conf /usr/bin/systemctl enable vboxservice.service /usr/bin/systemctl enable rpcbind.service # Add groups for VirtualBox folder sharing /usr/bin/usermod --append --groups vagrant,vboxsf vagrant # Clean the pacman cache. /usr/bin/yes | /usr/bin/pacman -Scc /usr/bin/pacman-optimize EOF echo '==> Entering chroot and configuring system' /usr/bin/arch-chroot${TARGET_DIR} ${CONFIG_SCRIPT} rm "${TARGET_DIR}${CONFIG_SCRIPT}" echo '==> Installation complete!' /usr/bin/sleep 3 /usr/bin/umount${TARGET_DIR}/boot
/usr/bin/umount ${TARGET_DIR} Packer produces a base box which can then be imported into vagrant and specified in vagrant configuration files: $ ./packer.exe build -force arch-base.json
$./vagrant.exe box add output/arch_vagrant_base.box --name arch-base --force ## Defining VMs in Vagrant With the most minimal base image produced, Vagrant steps in to fully define what goes into the VMs. Vagrant is flexible and supports Puppet, Chef and Ansible for provisioning. As I didn’t have experience with any of those tools and since I only needed the setup for myself I decided to avoid complexity. Vagrant also supports plain shell scripts, which are sufficiently expressive and simple. I chose to structure provisioning scripts in a layering scheme, in which layers are executed in succession to add functionality. I came up with 2 layers: CLI and X. CLI installs everything I’d want on a console VM, including tools, SSH keys, shell customizations, etc. Similarly, the X layer installs a minimal system with XFCE, a launcher, Sublime editor (along with my personal license), fonts, browser and my favorite diff tool. All of the tools are preconfigured with my preferences, such as the right wallpaper, color scheme, keyboard shortcuts and fonts. The actual VMs are then defined with setup scripts of their own on top of the standard layers. So, the VM I use to author this post has nothing else but Hugo installation defined in its script. As another example, I defined a VM to complete an online course. Since the course made use of Java-based tools I specified that JDK should be installed. Scripting so much of the setup requires a slight mindset change. Anytime I’d like to tweak or change something in the environment (say, shell settings) I need to consider whether to update the corresponding scripts. The beauty is that the setup lives on as configuration and is made apparent. No more fear that something was left behind after an upgrade! The provisioning script for the VM I’m writing this post contains the following: AS="/usr/bin/sudo -u sergey" # install hugo pacman -S --noconfirm hugo # check out repository echo '==> Checking out repository' cd /home/sergey$AS /usr/bin/git clone git@github.com:drseergio/drseergio.github.com.git hugoblog
cd /home/sergey/hugoblog
$AS git checkout source$AS git worktree add -B master public origin/master
# Reboot!
echo '==> Rebooting!'
/usr/bin/reboot
Here’s the complete Vagrantfile for the blogging VM:
Vagrant.configure("2") do |config|
variables = {
"EMAIL" => "sergey@pisarenko.net",
"FULL_NAME" => "Sergey Pisarenko"
}
config.vm.box = "arch-base"
config.vm.hostname = "hugoblog.bethania"
config.vm.synced_folder 'C:\Users\drsee\UbuntuShared', "/shared"
config.vm.network "public_network", :bridge => 'Wireless LAN adapter Wi-Fi', :mac => "XXX" # mac has been edited out
config.vm.provider "virtualbox" do |vb|
vb.name = "Hugoblog 18.02"
# Display the VirtualBox GUI when booting the machine
vb.gui = true
# Customize the amount of memory on the VM:
vb.memory = "1024"
# EFI boot
vb.customize ["modifyvm", :id, "--firmware", "efi64"]
# disable audio
vb.customize ["modifyvm", :id, "--audio", "none"]
# better video
vb.customize ["modifyvm", :id, "--accelerate3d", "on"]
vb.customize ["modifyvm", :id, "--vram", "128"]
# integration with desktop
vb.customize ["modifyvm", :id, "--clipboard", "bidirectional"]
vb.customize ["modifyvm", :id, "--draganddrop", "bidirectional"]
vb.customize ["modifyvm", :id, "--cpus", "2"]
end
# note how the layers are specified below
config.vm.provision "file", source: "../../configs", destination: "/tmp/configs"
config.vm.provision "file", source: "../../wallpapers", destination: "/tmp/wallpapers"
config.vm.provision "shell", path: "../../layers/cli-setup.sh", env: variables
config.vm.provision "shell", path: "../../layers/xorg-setup.sh", env: variables
config.vm.provision "shell", path: "hugoblog-setup.sh", env: variables
end
Once all the scripts are in place, getting a complete VM up and running is as simple as:
$vagrant up The VM can then be used in Virtualbox just like a normal VM until it’s no longer necessary: $ vagrant destroy # ...and it's gone!
## Closing words
I’ve been trialing the solution for several weeks now and using this approach for everything but my full-time work. Though there are few minor things to iron out, like shell preferences and environment variables, I’m deeply satisfied with the result! Getting a new machine is just a matter of running a command and then preparing a cup of tea. By the time I get back with a hot ☕ the VM is ready and booted.
The best part is that this also allows me provision physical machines. I can build VM images, test them locally in Virtualbox and then transfer them to physical drives and boot them. Maybe I’d need to install an additional module or two (say, nvidia binary driver) but otherwise it’ll work great.
I have uploaded all the configuration files on my github account https://github.com/pisarenko-net/arch-packer-vagrant. The configurations are customized to my needs (e.g. include user “sergey”) but could serve as a good starting point. | 2018-04-21 20:53:20 | {"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": 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": 0.34783631563186646, "perplexity": 9096.159292914963}, "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-2018-17/segments/1524125945448.38/warc/CC-MAIN-20180421203546-20180421223546-00606.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/9837/applying-density-matrix-based-criterion-to-verify-separability | # Applying density matrix based criterion to verify separability
In order to figure out if a given pure 2-qubit state is entangled or separable, I am trying to compute: the density matrix, then the reduced density matrix by tracing out with respect to one of the qubits, squaring the resulting reduced matrix, and finally taking its trace. Then, if the trace is $$=1,$$ I know the state is separable and entangled otherwise.
So I am trying this method for the following state (which we know to be entangled):
$$|\psi\rangle = a_0b_0 |00\rangle+a_0b_1|01\rangle + a_1b_1|11\rangle \tag{1}$$ nothing much else is really said about the coefficients other than neither of the above rhs terms have a coefficient $$0.$$
## My attempt:
• I computed $$\rho=|\psi\rangle \langle\psi|$$, then took the partial trace in the basis of the 2nd qubit $$\rho_1=\operatorname{Tr}_2(\rho).$$
The obtained $$\rho_1$$ is:
$$\rho_1 = \begin{pmatrix} a_0^2b_0^2 & a_0 a_1^* b_1^2 \\ a_0^* a_1 b_1^2 & a_1^2b_1^2\end{pmatrix} \tag{2}$$
• Then computing $$\rho_1^2$$ then taking its trace, I obtain:
$$\operatorname{Tr}(\rho_1^2)=a_0^4b_0^4+a_1^4b_1^4+2a_0^2a_1^2b_1^4=(a_0^2b_0^2+a_1^2b_1^2)^2 \tag{3}$$
• So for $$|\psi\rangle$$ to be an entangled state, we need to show $$(3)$$ is $$\neq 1,$$ i.e., $$(a_0^2b_0^2+a_1^2b_1^2)\neq 1$$
• The only way I can make progress with the latter, is to assume that our given state $$|\psi\rangle$$ in $$(1)$$ is normalized, then I can assert that $$(a_0^2b_0^2+a_1^2b_1^2)<1$$ since by normalization $$a_0^2b_0^2+a_1^2b_1^2+a_0^2 b_1^2=1$$ must be unity, thus $$(a_0^2b_0^2+a_1^2b_1^2)<1$$ must be true and our state is entangled.
## Question:
• Do my calculations make any sense? Admittedly, I am not very confident about it, and if I am correctly applying the described approach at the start.
• Is my assumption that $$(1)$$ is a normalized ket a necessary one in order to solve the problem of whether $$(1)$$ is entangled or separable?
• In case the assumption is needed, is my end result correctly interpreted then?
In essence the calculations are correct, though those numbers $$𝑎^2_i, 𝑏^2_i$$ must be taken with modulus, because they are in fact $$a_ia_i^*, b_ib_i^*$$. In general $$𝑎^2_0$$ is a complex number, so you can't write $$𝑎^2_0>0$$.
Normalization is required when we consider quantum states (also, otherwise, $$\operatorname{Tr}(\rho_1^2)$$ can be arbitrary close to 0). With normalization you have showed that $$\operatorname{Tr}(\rho_1^2) < 1$$, hence the state must be entangled. | 2021-04-21 07:10:08 | {"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": 26, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.963384211063385, "perplexity": 269.76580067906616}, "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/1618039526421.82/warc/CC-MAIN-20210421065303-20210421095303-00293.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/damp-washcloth-hung-edge-table-todry-m-washcloth-m-does--coefficient-static-friction-table-q62 | ## Newton's Laws
A damp washcloth is hung over the edge of a table todry. Thus m(on) of the washcloth and m(off) does not. The coefficient of static friction between the table and thewashcloth is .40. Determine the maximum fraction(m(off)/m(on) + m(off)) that can hang over the edge without causingthe whole washcloth to slide off the table. | 2013-05-19 15:39:07 | {"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": 0.8216798901557922, "perplexity": 6001.409795795562}, "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-2013-20/segments/1368697745221/warc/CC-MAIN-20130516094905-00072-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/355799/what-is-the-mechanism-when-a-pretty-high-box-occurs-in-some-line | # What is the mechanism when a pretty high box occurs in some line?
I have a code like this:
This sentence is written by Chinese. 這是中文字。This sentence is written by Chinese.\\
\hspace*{2em}\fbox{\begin{minipage}[t][][t]{\textwidth-4em}
\begin{alignedat}[t]{3} &(\mathcal{MS}1)~~\text{\scshape Positive Definite}&&\quad&&\forall x,y\in X,~d(x,y)\geq 0\wedge(d(x,y)=0\Leftrightarrow x=y)\\ &(\mathcal{MS}2)~~\text{\scshape Symmetric}&& &&\forall x,y\in X,~d(x,y)=d(y,x)\\ &(\mathcal{MS}3)~~\text{\scshape Triangle Inequality}&& &&\forall x,y,z\in X,~d(x,z)\leq d(x,y)+d(y,z) \end{alignedat}
\end{minipage}}\\
This sentence is written by Chinese. 這是中文字。This sentence is written by Chinese. 這是中文字。This sentence is written by Chinese. 這是中文字。
The result is:
(PS: The reason that the line-space is big, is because I'm typesetting a Chinese document, a wide space is needed.)
As the picture shown, the up border of the box is adjacent to the baseline of the first line above, and the down border of the box is adjacent to the top of the first line below.
If I want these two line-space be wider(see the red places in the figure below), how can I do this? What method is more ideal?(i.e. not just \0.3cm] or \vspace{...}) And what I want to know, is the behavior when a pretty high box (or pretty high content, such like \displaystyle\frac{\displaystyle\frac{3}{2}}{5}) occurs in some line in a paragraph? UPDATE: A quick test code is: \documentclass[12pt,a4paper,openany,fleqn]{book} \usepackage[margin=1.8cm, top=2cm]{geometry} \usepackage{amsmath,amssymb, amsfonts} \usepackage[no-math]{fontspec} \parindent = 0 pt \begin{document} THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD. \\ \hspace*{2em}\fbox{\begin{minipage}[t][][t]{15cm} \begin{alignedat}[t]{3} &(\mathcal{MS}1)~~\text{\scshape Positive Definite}&&\quad&&\forall x,y\in X,~d(x,y)\geq 0\wedge(d(x,y)=0\Leftrightarrow x=y)\\ &(\mathcal{MS}2)~~\text{\scshape Symmetric}&& &&\forall x,y\in X,~d(x,y)=d(y,x)\\ &(\mathcal{MS}3)~~\text{\scshape Triangle Inequality}&& &&\forall x,y,z\in X,~d(x,z)\leq d(x,y)+d(y,z) \end{alignedat} \end{minipage}}\\ THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD. \end{document} Output: • The boxed minipage is both very high and deep, so it's separated from the lines that precede and follow by \lineskip (default 1pt). I'd use \[... instead of \\\hspace*{2em} and so on. Please, make a fully compilable example. – egreg Feb 26 '17 at 10:32
• @egreg I just have added a compilable code. – Eric Feb 26 '17 at 10:59
As the document is written, the minipage has quite a large height (because of the top rule of the box) and a very large depth (because of the [t] option).
Thus the separation from the lines above and below is given by \lineskip (default value 1pt).
In this case it's much better to use a display; since you have fleqn, the only option is center:
\documentclass[12pt,a4paper,fleqn]{book}
\usepackage[margin=1.8cm, top=2cm]{geometry}
\usepackage{amsmath}
\begin{document}
THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD.
THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD.
THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD.
\begin{center}
\fbox{%
\begin{alignedat}{2} (\mathcal{MS}1)\quad& \textsc{Positive Definite}&\quad& \forall x,y\in X,\ d(x,y)\geq 0\wedge(d(x,y)=0\Leftrightarrow x=y) \\ (\mathcal{MS}2)\quad& \textsc{Symmetric}&\quad& \forall x,y\in X,~d(x,y)=d(y,x) \\ (\mathcal{MS}3)\quad& \textsc{Triangle Inequality}&\quad& \forall x,y,z\in X,~d(x,z)\leq d(x,y)+d(y,z) \end{alignedat}%
}
\end{center}
THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD.
THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD.
THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD. THIS IS SOME WORD.
\end{document}
Note that the minipage is not needed (so you haven't to guess at the right horizontal size). By the way, using the second and third optional argument to minipage should be very rarely needed.
• I see! Thanks a lot! Also thanks for correcting my bad code in alignedat. – Eric Feb 26 '17 at 12:43 | 2019-10-17 08:36:14 | {"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": 0.9737602472305298, "perplexity": 4982.838209196345}, "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-2019-43/segments/1570986673250.23/warc/CC-MAIN-20191017073050-20191017100550-00241.warc.gz"} |
http://mathhelpforum.com/calculus/63398-how-do-i-prove-second-derivative-simplification.html | ## How do I prove this second derivative simplification?
How do I prove that xy(x-y) is equal to (x-y)^-1 ((-y)/(x-y)) + y(x-y)^-2 ( 1- ((-y)/(x-y))
the second equation is the second derivative of -y/(x-y)
thanks,
Db. | 2014-07-12 01:14:15 | {"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": 0.8697269558906555, "perplexity": 2788.9548555289275}, "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/1404776430107.88/warc/CC-MAIN-20140707234030-00007-ip-10-180-212-248.ec2.internal.warc.gz"} |
http://exxamm.com/blog/Blog/14134/zxcfghfgvbnm4?Class%2011 | Physics Scalar Product and The Work energy theorem
### Topic covered
star Introduction
star The Scalar Product
star The Work energy theorem
### INTRODUCTION
➢ The word ‘Work’ has a definite and precise meaning, it's define as measurement of energy transfer that occurs when an object is moved over a distance by an external force. Somebody who has the capacity to work for 14-16 hours a day is said to have a large stamina or energy.
➢ Thus Energy is our capacity to do work. In Physics too, the term color {blue}{"energy"} is related to work in this sense, but as said above the term color {blue}{"work"} itself is defined much more precisely.
➢ The word ‘power’ is used in everyday life with different shades of meaning.As per definition its the rate of doing work per unit time. e.g. In karate or boxing we talk of ‘powerful’ punches. These are delivered at a great speed. This shade of meaning is close to the meaning of the word color {blue}{"power"} used in physics.
### The Scalar Product
➢ There are two ways of multiplying vectors which we shall come across : one way known as the scalar product gives a scalar from two vectors and the other known as the vector product produces a new vector from two vectors.
➢ Here we take up the scalar product of two vectors. The scalar product or dot product of any two vectors A and B, denoted as A * B (read A dot B) is defined as
color {blue} {A * B = A B cos θ}
=> where θ is the angle between the two vectors as shown in Fig.
➢ Since A, B and cos θ are scalars, the dot product of A and B is a scalar quantity. Each vector, A and B, has a direction but their scalar product does not have a direction.
➢ From Eq. (6.1a), we have color {blue}{A*B = A (B cos θ )}
color {blue}{= B (A cos θ )}
color{red}☞ Geometrically, B cos θ is the projection of B onto A in Fig.6.1 (b) and A cos θ is the projection of A onto B in Fig. 6.1 (c).
➢ So, A*B is the product of the magnitude of A and the component of B along A. Alternatively, it is the product of the magnitude of B and the component of A along B.
=> Equation (6.1 a) shows that the scalar product follows the commutative law :
A*B = B*A
color{red}☞ Scalar product obeys the distributive law:
A* (B + C) = A*B + A*C
Further, A* (λ B) = λ (A*B)
where λ is a real number.
color{red}☞ For unit vectors hat i , hat j , hat k we have
hat i * hat i = hat j * hat j = hat k * hat k=1
hat i * hat j = hat j * hat k = hat k * hat i =0
➢ Given two vectors
A= A_x hat i + A_y hat j + A_z hat k
B= B_x hat i + B_y hat j + B_z hat k
their scalar product is
A * B = (A_x hat i + A_y hat j + A_z hat k ) * (B_x hat i + B_y hat j + B_z hat k)
color {blue}{= A_x B_x + A_y B_y + A_z B_z} ..............(6.1b)
From the definition of scalar product and, (Eq. 6.1b) we have :
(i) A * A = A_x A_x + A_y A_y + A_z A_z
or, color {blue}{A^2 = A_x^2 + A_y^2+ A_z^2} ..............(6.1c)
since A*A = |A ||A| cos 0 = A^2.
(ii) A*B = 0, if A and B are perpendicular.
Q 3149867713
Find the angle between force
F = (3 hat i + 4 hat j + 5 hat k) unit and displacement
d = (5 hat i + 4 hat j + 3 hat k) unit. Also find the projection of F on d.
Solution:
color{green} {F * d = F_x d_x + F_y d_y + F_z d_z}
= 3 (5) + 4 (4) + (– 3) (3)
= 16 unit
Hence color{orange} {F*d = F d cosθ = 16} unit
Now color{purple} {F*F = F^2 =F_x^2+ F_y^2 = F _z^2}
= 9 + 16 + 25
= 50 unit
and d*d = d^2 = d_x^2 +d_y^2 + d_z^2
= 25 + 16 + 9
= 50 unit
∴ cos θ = 16/(sqrt 50 sqrt 50) = 16/50 = 0.32
color{red} {θ = cos^(–1) 0.32}
### THE WORK-ENERGY THEOREM
➢ As we know, the relation for rectilinear motion under constant acceleration
v^2 − u^2 = 2 as
=> where u and v are the initial and final speeds and s the distance traversed. Multiplying both sides by m//2, we have
color {blue}{1/2 mv^2 - 1/2 m u^2 = mas= Fs} ..............(6.2a)
=> where the last step follows from Newton’s Second Law. We can generalise Eq. (6.1) to three dimensions by employing vectors
v^2 − u^2 = 2 a*d
➢ Once again multiplying both sides by m//2 , we obtain
color {blue}{1/2 mv^2 - 1/2 mu^2 = m a * d = F* d} ..................(6.2b)
➢ The left side of the equation is the difference in the quantity ‘half the mass times the square of the speed’ from its initial value to its final value.
➢ We call each of these quantities the ‘kinetic energy’, denoted by K. The right side is a product of the displacement and the component of the force along the displacement. This quantity is called ‘work’ and is denoted by W. Eq. (6.2) is then
color {blue}{K_f - K_t = W}.............(6.3)
=> where K_i and K_f are respectively the initial and final kinetic energies of the object. Work refers to the force and the displacement over which it acts. Work is done by a force on the body over a certain displacement.
Equation (6.2) is also known as "work-energy (WE) theorem :" The change in kinetic energy of a particle is equal to the work done on it by the net force. We shall generalise the above derivation to a varying force in a later section.
Q 3159067814
It is well known that a raindrop falls under the influence of the downward gravitational force and the opposing resistive force. The latter is known to be proportional to the speed of the drop but is otherwise undetermined.
Consider a drop of mass 1.00 g falling from a height 1.00 km. It hits the ground with a speed of 50.0 m s^(-1).
(a) What is the work done by the gravitational force ? What is the work done by the unknown resistive
force?
Solution:
The change in kinetic energy of the drop is
color{purple}{Delta K= 1/2 mv^2 -0}
=1/2 xx 10^(-3) xx 50 xx 50
1.25 J
where we have assumed that the drop is initially at rest.
Assuming that g is a constant with a value 10 m//s^2, the work done by the gravitational force
is,
color{green}{W_g = mgh}
=10^(-3) xx 10 xx 10^3
=10.0 J
(b) From the work-energy theorem
color{orange}{ΔK = W_g +W_r}
where W_r is the work done by the resistive force on the raindrop. Thus
color{purple} {W_r = ΔK − W_g}
= 1.25 −10
= − 8.75 J is negative. | 2019-02-20 17:39:07 | {"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": 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": 0.8507782220840454, "perplexity": 1387.5762715978512}, "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-2019-09/segments/1550247495367.60/warc/CC-MAIN-20190220170405-20190220192405-00275.warc.gz"} |
https://electronics.stackexchange.com/questions/329711/effects-of-using-voltmeter-w-20-k%CE%A9-impedance-across-35-k%CE%A9-resistor | # Effects of using voltmeter w/ 20 kΩ impedance across 35 kΩ resistor?
Our professor assigned the following question:
Describe both quantitatively and qualitatively the effects of using a voltmeter to measure the voltage across a 35 kΩ resistor if the voltmeter has an input impedance of 20 kΩ. Assume the 35 kΩ resistor is in series with another 35 kΩ resistor.
All I know from the reading and lecture is that your voltmeter needs to have a much higher impedance/resistance than the circuit resistance in order to avoid the meter loading effect, which causes instrument misreadings. How MUCH higher impedance/resistance does the voltmeter need to have in relation to the circuit resistance? I don't know how to do the math (the "quantitative" part). Is what I said sufficient enough for a "qualitative" explanation?
I emailed my professor yesterday morning. He never got back to me, and this was due last night.
EDIT:
Here's what I tried to work out...Using 10V as the Voltage from the Voltage Source, I figured out that the Voltage Drop across each resistor should be 5V when everything is in series. But I don't know what to do when the voltmeter is attached to a resistor. Do I calculate everything as if it's all a parallel circuit now? Is only R1 and VM1 part of a parallel circuit, whereas R2 is still only part of a series circuit?
Quantitatively: If the applied voltage is 10 Volts, you should measure 5.0 Volts across each resistor, but you actually would measure 2.667 Volts
I need to know how he got 2.667V in his example...
• You need (1) some ability at algebra and (2) some basic understanding of conductance and resistance and Ohm's law. Saying "I don't know how to do the math" means either that you don't have basic algebra skills (a pre-requisite) or else you don't understand basic conductance/resistance concepts or both. Can you draw a schematic of what you perceive occurs just before and just after adding a voltmeter? – jonk Sep 16 '17 at 21:53
• @jonk I added what I could figure out...I'm going crazy. I know I need to use the inverse when calculating total resistance in parallel circuits (1/R), but I don't know if I should include R2 in that calculation. (Is R2 part of the parallel circuit?) All I need is the Voltage of the parallel circuit and I'm good, right? (whatever that consists of...) Why is this so hard for me to figure out – velkoon Sep 17 '17 at 0:04
• Thanks. You've applied yourself here and I appreciate it. Let me see what I can do to help, now. – jonk Sep 17 '17 at 0:33
• @jonk I just updated the picture and changed my calculation for total resistance. (I'm very, very slow at learning anything STEM-related..) – velkoon Sep 17 '17 at 0:39
First off, I removed the battery symbol. It's enough to specify a ground (you get to pick any one wire [aka: node] and call it $0\:\textrm{V}$) and the source voltage point.
Second, it doesn't matter which of the two $35\:\textrm{k}\Omega$ resistors you measure across (you should get the same answer either way), so I placed one end of the voltmeter at the "ground" (zero) reference point and the other end at an "interesting" point. If you look this over, you should be able to see that it asks the same question as does your newly added schematic.
Third, I've separated out the internal meter resistance as a discrete, added resistor to the schematic. The idea here is that now the voltmeter has "infinite" resistance and so it does not affect the circuit, anymore. But to keep the behavior the same, I had to add $R_{METER}$. Given your own schematic, I think you can see why this would still be the same question.
simulate this circuit – Schematic created using CircuitLab
Now, you should know how to calculate parallel resistances. So you can compute the equivalent single resistor you could use to replace the pair of $R_2$ and $R_{METER}$.
But before I go there, you can look at the resistors as instead being conductors. (In electronics, the symbol $G$ is used instead of $R$.) So $G_1=\frac{1}{R_1}$, $G_2=\frac{1}{R_2}$ and $G_{METER}=\frac{1}{R_{METER}}$. The total conductance of $G_1$ and $G_{METER}$ is just the sum of the two, because adding another conductance makes the whole thing more "conductive," right? So, if you add the conductance of $R_2$ and $R_{METER}$ and then convert that paired conductance back into a resistance again, you'd have the result of this as the paired resistance value of the two:
$$R_{METER}\mid\mid R_2=\frac{1}{\frac{1}{R_{METER}}+\frac{1}{R_2}}=\frac{R_{METER}\cdot R_2}{R_{METER}+R_2}=12.\overline{72}\:\textrm{k}\Omega$$
Now, it's just a voltage divider made up of two resistors. $R_1$ is still the same, but you now have a new replacement resistor that replaces the pairing of $R_2$ and $R_{METER}$. (Above computed value.)
From this divider, you should be able to calculate the resulting voltage.
Note that $R_{TOTAL}=R_1+\left(R_2\mid\mid R_{METER}\right)$ and that if you compute $I_{TOTAL}$ from that and $V_{TOTAL}=10\:\textrm{V}$, that this will NOT tell you the current through $R_2$. That's because $R_2$ must share this total current (all of which does have to flow through $R_1$) with $R_{METER}$.
An entirely different approach would be to construct the Thevenin equivalent before attaching the meter to it. This would be:
simulate this circuit
Once again, you have a new voltage divider circuit that will yield the exact same result. But a different approach to getting there.
Note here that a different $R_{TOTAL}=\left(R_1\mid\mid R_2\right)+R_{METER}$ is computed and that if you then compute a different $I_{TOTAL}$ from this (and $V_{THEVENIN}=5\:\textrm{V}$), that this will actually tell you the current through $R_{METER}$. That's because while $R_1$ and $R_2$ must share this total current, all of it does have to flow through $R_{METER}$. So you can compute the voltage by multiplying this current times $R_{METER}$.
Instead of claiming you can't do this, stop and actually think about it.
Draw the schematic of the circuit without the voltmeter. Then draw the schematic with the voltmeter. To the circuit, the voltmeter just looks like a 20 kΩ resistor.
Note that you are told that the 35 kΩ is in series with another 35 kΩ resistor. Basically, the 35 kΩ resitor you are to measure the voltage on is being driven by a Thevenin source with 35 kΩ impedance.
You really should be able to solve this easily from here. I expect the professor didn't get back to you because he thinks this is a easy problem you should be able to figure out yourself with a little thought. He is right.
How MUCH higher impedance/resistance does the voltmeter need to have in relation to the circuit resistance?
@Jonk has already taken you through the steps of how to calculate what's going on exactly. However it's worth addressing the other part of your question in a very rough and quick manner, when is a meter high enough impedance?
Let's say the meter has 100x the impedance of the components you're trying to measure. The current flowing through the meter will then be about 1% of that in the components, which will reduce the measured voltage by about 1%. It's not exactly 1%, you need to go through all the sums for the exact figure, but roughly, in that ballpark.
Is 1% close enough? It's comparable with the inherent accuracy of most cheap meters, and most reasonable components, so it would be adequate for many purposes. Sometimes you would want to know you weren't degrading your meter accuracy by any significant amount, and limit the added error to 0.1%. This would require the meter to have roughly 1000x times the circuit impedance.
Most DMM have a 10Mohm input resistance. This means that you can measure circuits with 10k resistance with around 0.1% added error, and up to 100k with about 1% added error.
• And the dirty little secret is that most of the time, 2 sig. fig. is more then you really care about. You do however need to be aware of this effect, as even with a 10Meg meter it is not always irrelevant (Particularly when designing sensor electronics, sometimes you see Gig ohm impedance levels). – Dan Mills Sep 17 '17 at 15:34 | 2021-03-07 06:43:43 | {"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": 0.6547061204910278, "perplexity": 531.0222793694945}, "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/1614178376144.64/warc/CC-MAIN-20210307044328-20210307074328-00077.warc.gz"} |
http://openstudy.com/updates/5606092be4b0a953836f5d08 | ## Compassionate one year ago Why are we making them whole numbers? Can't we just keep them in their decimal form? http://prntscr.com/8kjwr5
1. zepdrix
Yes, we can keep them as decimals. But as is with fractions, likewise, no one really likes decimals. :) Whole numbers are easier to work with.
2. Compassionate
So, if I had 8.5y + 5.3x = 2,300 I could just solve like normal with the decimals. If I wanted to solve for x, I could say 2,300 - 5.3x = 8.5y Where would fractions even come in with?
3. zepdrix
Example:$\large\rm \frac{4}{3}x+\frac{2}{21}y=10$Hmm the fractions are annoying, I would probably choose to multiply both sides by 21 here.$\large\rm 4\cdot7x+2y=10\cdot21$Which is just a little easier to work with. $\large\rm 28x+2y=210$Especially if you're given a system, and you're going to use the elimination method.
4. zepdrix
If you have a system of equations, it's a lot more difficult to "cancel out" one of the variables if it's a decimal value. If I have 8y and y, I know that by multiplying the first equation by 8 will match the y's up. If Instead I have y and .08y, It means I have to multiply every number in the first equation by .08 to match them up, which is kind of a hassle.
5. Compassionate
Understandable. I'd rather just work with the fractions. The lingo mingo making thingos a whole numbero just seems a lot of hassle and out of my way-o. Thanks man | 2017-01-16 15:35:41 | {"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": 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": 0.7214318513870239, "perplexity": 803.9001278050039}, "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-2017-04/segments/1484560279189.36/warc/CC-MAIN-20170116095119-00038-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://zbmath.org/?q=an:1082.32022 | # zbMATH — the first resource for mathematics
Birational geometry of foliations. (English) Zbl 1082.32022
Publicações Matemáticas do IMPA. Rio de Janeiro: Instituto Nacional de Matemática Pura e Aplicada (IMPA) (ISBN 85-244-0219-9/pbk ). iv, 138 p. (2004).
The book under review is the expanded version of a series of lectures given by the author at IMPA in August 2000 (see Zbl 1073.14022). The aim of these notes is to provide a classification of (singular) holomorphic foliations on complex projective surfaces in the spirit of the classical Enriques classification of complex algebraic surfaces.
The classical Enriques classification of complex algebraic surfaces up to birational morphisms, is based on the Kodaira dimension which can be $$-\infty, 0, 1, 2$$. For surfaces of non general type (that is with Kodaira dimension $$<2$$) the classification is very deep and precise. The more recent Mori’s program, developed to classify higher dimensional complex varieties, based on the study of numerical properties of the canonical bundle – the so-called numerical Kodaira dimension – has been proved to be useful in the surfaces case as well.
With Enriques classification in mind and starting from previous results of Mendes and McQuillan (the relative papers being available only in a preprint form at the time the reviewer is writing this review) the author classifies a holomorphic foliation $$\mathcal F$$ on a complex projective surface by means of its Kodaira’s dimension $$\text{kod}(\mathcal F)$$ (defined as the “Kodaira dimension” of the dual of the tangent bundle of $$\mathcal F$$) and the related numerical Kodaira dimension $$\nu(\mathcal F)$$. Both numbers are birational invariants if one deals with foliations with only reduced singularities, which is always the case (up to monoidal transformations) by means of Seidenberg’s reduction theorem. As in Enriques’ classification, Kodaira’s and numerical Kodaira’s dimension of $$\mathcal F$$ can be $$-\infty, 0,1,2$$, and the latter case is the so called general type case (for which no complete refined classification is available).
For foliations of non general type a fine classification is provided. If $$\nu(\mathcal F)=-\infty$$ a deep result by Miyaoka allows to state that the foliation is a trivial $$\mathbb C\mathbb P^1$$-bundle over a curve. If $$\nu(\mathcal F)=0$$ McQuillan and Mendes proved that the foliation is given by a $$\mathbb C$$-action or by a very special ramified covering. The case $$\nu(\mathcal F)=1$$ and $$\text{kod}(\mathcal F)\geq 0$$ is solved again by McQuillan who proved that necessarily then $$\text{kod}(\mathcal F)=1$$ and the foliation is either Riccati, or turbolent or another very special foliation. The case $$\nu(\mathcal F)=1$$ and $$\text{kod}(\mathcal F)=-\infty$$, still unsolved at the time the book under review was written, has been solved recently by the author himself [see M. Brunella, Invent. Math. 152, No. 1, 119–148 (2003; Zbl 1029.32014)] and McQuillan, proving that in this case the foliation is a so called Hilbert modular foliation (and, contrarily to the surfaces classification, there exist many of such foliations).
The main ingredients for the classification are the Baum-Bott index formula [P. Baum and R. Bott, J. Differ. Geom. 7, 279–342 (1972; Zbl 0268.57011)], the Camacho-Sad index theorem [C. Camacho and P. Sad, Ann. Math. (2) 115, 579–595 (1982; Zbl 0503.32007)], the Brunella tangential index theorem [M. Brunella, Ann. Sci. Éc. Norm. Supér. (4) 30, 569–594 (1997; Zbl 0893.32019)], together with tools such as the Castelnuovo-de Franchis lemma and Bogomolov’s lemma and the already mentioned Miyaoka’s rationality criterion.
The book is really well written and the first chapters can also be used as a primary on holomorphic foliations theory on complex surfaces. Often the author provides simple proofs of well known results and essentially all the main tools used come with at least a sketch of a proof.
##### MSC:
32S65 Singularities of holomorphic vector fields and foliations 37F75 Dynamical aspects of holomorphic foliations and vector fields 14E05 Rational and birational maps 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces | 2022-01-22 20:53:00 | {"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": 0.7982944250106812, "perplexity": 606.3313959701499}, "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-2022-05/segments/1642320303884.44/warc/CC-MAIN-20220122194730-20220122224730-00311.warc.gz"} |
https://www.expii.com/t/inequality-symbols-less-than-8969 | Expii
# Inequality Symbols: Less Than - Expii
The symbol for less than is <. Any expression that comes before the < is worth LESS than the expression that comes after it. | 2020-02-28 22:40:32 | {"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": 0.8079050183296204, "perplexity": 1289.128239852135}, "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-2020-10/segments/1581875147647.2/warc/CC-MAIN-20200228200903-20200228230903-00539.warc.gz"} |
https://deepai.org/publication/no-quantum-speedup-over-gradient-descent-for-non-smooth-convex-optimization | # No quantum speedup over gradient descent for non-smooth convex optimization
We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function f:ℝ^n →ℝ and its (sub)gradient. Our goal is to find an ϵ-approximate minimum of f starting from a point that is distance at most R from the true minimum. If f is G-Lipschitz, then the classic gradient descent algorithm solves this problem with O((GR/ϵ)^2) queries. Importantly, the number of queries is independent of the dimension n and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension n. In this paper we reprove the randomized lower bound of Ω((GR/ϵ)^2) using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using O(GR/ϵ) quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need Ω((GR/ϵ)^2) queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.
• 15 publications
• 13 publications
• 48 publications
• 3 publications
12/02/2021
### Near-Optimal Lower Bounds For Convex Optimization For All Orders of Smoothness
We study the complexity of optimizing highly smooth convex functions. Fo...
06/25/2019
### Complexity of Highly Parallel Non-Smooth Convex Optimization
A landmark result of non-smooth convex optimization is that gradient des...
04/14/2017
### On the Gap Between Strict-Saddles and True Convexity: An Omega(log d) Lower Bound for Eigenvector Approximation
We prove a query complexity lower bound on rank-one principal component ...
01/09/2020
### How to trap a gradient flow
We consider the problem of finding an ε-approximate stationary point of ...
08/12/2018
### Parallelization does not Accelerate Convex Optimization: Adaptivity Lower Bounds for Non-smooth Convex Minimization
In this paper we study the limitations of parallelization in convex opti...
04/04/2018
### Tight Query Complexity Lower Bounds for PCA via Finite Sample Deformed Wigner Law
We prove a query complexity lower bound for approximating the top r dime...
11/25/2021
### Negative curvature obstructs acceleration for geodesically convex optimization, even with exact first-order oracles
Hamilton and Moitra (2021) showed that, in certain regimes, it is not po... | 2022-08-09 17:50:56 | {"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": 0.8865563869476318, "perplexity": 877.550251272202}, "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/1659882571056.58/warc/CC-MAIN-20220809155137-20220809185137-00704.warc.gz"} |
https://brilliant.org/problems/mistakes-give-rise-to-problems-5/ | # Mistakes give rise to problems- 5
Algebra Level 3
We know that the power raised to can't be distributed over the content of a bracket, that is , if you do $(a+b)^2 = a^2+b^2$ then it's a big mistake!
But for how many pairs of integers $$(a,b)$$ such that each of them is between -10 and 10 inclusive, is the above said false property seen to be true?
× | 2018-09-26 01:52:43 | {"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": 0.2943403124809265, "perplexity": 429.91684915402925}, "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-2018-39/segments/1537267162809.73/warc/CC-MAIN-20180926002255-20180926022655-00056.warc.gz"} |
https://ftp.aimsciences.org/article/doi/10.3934/cpaa.2011.10.1205 | # American Institute of Mathematical Sciences
July 2011, 10(4): 1205-1224. doi: 10.3934/cpaa.2011.10.1205
## On a general class of free boundary problems for European-style installment options with continuous payment plan
1 Department of Economics, Faculty of Economics "Federico Caffè", University of Rome III, Via Silvio D'Amico 77, 00145 Rome, Italy
Received June 2010 Revised December 2010 Published April 2011
In this paper we present an integral equation approach for the valuation of European-style installment derivatives when the payment plan is assumed to be a continuous function of the asset price and time. The contribution of this study is threefold. First, we show that in the Black-Scholes model the option pricing problem can be formulated as a free boundary problem under very general conditions on payoff structure and payment schedule. Second, by applying a Fourier transform-based solution technique, we derive a recursive integral equation for the free boundary along with an analytic representation of the option price. Third, based on these results, we propose a unified framework which generalizes the existing methods and is capable of dealing with a wide range of monotonic payoff functions and continuous payment plans. Finally, by using the illustrative example of European vanilla installment call options, an explicit pricing formula is obtained for time-varying payment schedules.
Citation: Pierangelo Ciurlia. On a general class of free boundary problems for European-style installment options with continuous payment plan. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1205-1224. doi: 10.3934/cpaa.2011.10.1205
##### References:
[1] G. Alobaidi, R. Mallier and A. S. Deakin, Laplace transforms and installment options, Mathematical Models and Methods in Applied Sciences, 14 (2004), 1167-1189. doi: 10.1142/S0218202504003581. [2] G. Alobaidi and R. Mallier, Installment options close to expiry, Journal of Applied Mathematics and Stochastic Analysis, (2006), Art. ID 60824, 1-9. doi: 10.1155/JAMSA/2006/60824. [3] H. Ben-Ameur, M. Breton and P. François, A dynamic programming approach to price installment options, European Journal of Operational Research, 169 (2006), 667-676. doi: 10.1016/j.ejor.2004.05.009. [4] P. Carr, R. Jarrow and R. Myneni, Alternative characterizations of American put options, Mathematical Finance, 2 (1992), 87-106. doi: 10.1111/j.1467-9965.1992.tb00040.x. [5] P. Ciurlia, On the evaluation of European continuous-installment options, Department of Economics Working paper, No 113 (2010), University of Rome III. [6] P. Ciurlia and C. Caperdoni, A note on the pricing of perpetual continuous-installment options, Mathematical Methods in Economics and Finance, 4 (2009), 11-26. [7] P. Ciurlia and I. Roko, Valuation of American continuous-installment options, Computational Economics, 25 (2005), 143-165. doi: 10.1007/s10614-005-6279-4. [8] M. Davis, W. Schachermayer and R. Tompkins, Pricing, no-arbitrage bounds and robust hedging of instalment options, Quantitative Finance, 1 (2001), 597-610. doi: 10.1088/1469-7688/1/6/302. [9] M. Davis, W. Schachermayer and R. Tompkins, Installment options and static hedging, Journal of Risk Finance, 3 (2002), 46-52. doi: 10.1108/eb043487. [10] S. Griebsch, C. Kühn and U. Wystup, Instalment options: A closed-form solution and the limiting case, in "Mathematical Control Theory and Finance" (eds. A. Sarychev, A. Shiryaev, M. Guerra and M.d.R. Grossinho), Springer-Verlag, Berlin, (2008), 211-229. doi: 10.1007/978-3-540-69532-5_12. [11] S. D. Jacka, Optimal stopping and the American put, Mathematical Finance, 1 (1991), 1-14. doi: 10.1111/j.1467-9965.1991.tb00007.x. [12] F. Karsenty and J. Sikorav, Installment plan, Risk, 6 (1993), 36-40. [13] I. J. Kim, The analytical valuation of American options, Review of Financial Studies, 3 (1990), 547-572. doi: 10.1093/rfs/3.4.547. [14] T. Kimura, American continuous-installment options: valuation and premium decomposition, SIAM Journal on Applied Mathematics, 70 (2009), 803-824. doi: 10.1137/080740969. [15] T. Kimura, Valuing continuous-installment options, European Journal of Operational Research, 201 (2010), 222-230. doi: 10.1016/j.ejor.2009.02.010. [16] H. P. McKean, Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review, 6 (1965), 32-39. [17] Z. Yang and F. Yi, A variational inequality arising from American installment call options pricing, Journal of Mathematical Analysis and Applications, 357 (2009), 54-68. doi: 10.1016/j.jmaa.2009.03.045. [18] F. Yi, Z. Yang and X. Wang, A variational inequality arising from European installment call options pricing, SIAM Journal on Mathematical Analysis, 40 (2008), 306-326. doi: 10.1137/060670353.
show all references
##### References:
[1] G. Alobaidi, R. Mallier and A. S. Deakin, Laplace transforms and installment options, Mathematical Models and Methods in Applied Sciences, 14 (2004), 1167-1189. doi: 10.1142/S0218202504003581. [2] G. Alobaidi and R. Mallier, Installment options close to expiry, Journal of Applied Mathematics and Stochastic Analysis, (2006), Art. ID 60824, 1-9. doi: 10.1155/JAMSA/2006/60824. [3] H. Ben-Ameur, M. Breton and P. François, A dynamic programming approach to price installment options, European Journal of Operational Research, 169 (2006), 667-676. doi: 10.1016/j.ejor.2004.05.009. [4] P. Carr, R. Jarrow and R. Myneni, Alternative characterizations of American put options, Mathematical Finance, 2 (1992), 87-106. doi: 10.1111/j.1467-9965.1992.tb00040.x. [5] P. Ciurlia, On the evaluation of European continuous-installment options, Department of Economics Working paper, No 113 (2010), University of Rome III. [6] P. Ciurlia and C. Caperdoni, A note on the pricing of perpetual continuous-installment options, Mathematical Methods in Economics and Finance, 4 (2009), 11-26. [7] P. Ciurlia and I. Roko, Valuation of American continuous-installment options, Computational Economics, 25 (2005), 143-165. doi: 10.1007/s10614-005-6279-4. [8] M. Davis, W. Schachermayer and R. Tompkins, Pricing, no-arbitrage bounds and robust hedging of instalment options, Quantitative Finance, 1 (2001), 597-610. doi: 10.1088/1469-7688/1/6/302. [9] M. Davis, W. Schachermayer and R. Tompkins, Installment options and static hedging, Journal of Risk Finance, 3 (2002), 46-52. doi: 10.1108/eb043487. [10] S. Griebsch, C. Kühn and U. Wystup, Instalment options: A closed-form solution and the limiting case, in "Mathematical Control Theory and Finance" (eds. A. Sarychev, A. Shiryaev, M. Guerra and M.d.R. Grossinho), Springer-Verlag, Berlin, (2008), 211-229. doi: 10.1007/978-3-540-69532-5_12. [11] S. D. Jacka, Optimal stopping and the American put, Mathematical Finance, 1 (1991), 1-14. doi: 10.1111/j.1467-9965.1991.tb00007.x. [12] F. Karsenty and J. Sikorav, Installment plan, Risk, 6 (1993), 36-40. [13] I. J. Kim, The analytical valuation of American options, Review of Financial Studies, 3 (1990), 547-572. doi: 10.1093/rfs/3.4.547. [14] T. Kimura, American continuous-installment options: valuation and premium decomposition, SIAM Journal on Applied Mathematics, 70 (2009), 803-824. doi: 10.1137/080740969. [15] T. Kimura, Valuing continuous-installment options, European Journal of Operational Research, 201 (2010), 222-230. doi: 10.1016/j.ejor.2009.02.010. [16] H. P. McKean, Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review, 6 (1965), 32-39. [17] Z. Yang and F. Yi, A variational inequality arising from American installment call options pricing, Journal of Mathematical Analysis and Applications, 357 (2009), 54-68. doi: 10.1016/j.jmaa.2009.03.045. [18] F. Yi, Z. Yang and X. Wang, A variational inequality arising from European installment call options pricing, SIAM Journal on Mathematical Analysis, 40 (2008), 306-326. doi: 10.1137/060670353.
[1] Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure and Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007 [2] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 [3] Yang Zhang. A free boundary problem of the cancer invasion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. doi: 10.3934/dcdsb.2021092 [4] Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431 [5] Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure and Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229 [6] Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $\mathcal{O}(N^2)$ operations using the discrete fourier transform. Kinetic and Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027 [7] Juan H. Arredondo, Francisco J. Mendoza, Alfredo Reyes. On the norm continuity of the hk-fourier transform. Electronic Research Announcements, 2018, 25: 36-47. doi: 10.3934/era.2018.25.005 [8] Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579 [9] Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003 [10] Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks and Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655 [11] Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337 [12] Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44 [13] Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293 [14] Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087 [15] Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713 [16] Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1915-1934. doi: 10.3934/jimo.2021049 [17] Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 [18] Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7 [19] Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure and Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1 [20] Constantin N. Beli. Representations of integral quadratic forms over dyadic local fields. Electronic Research Announcements, 2006, 12: 100-112.
2020 Impact Factor: 1.916 | 2022-05-23 05:50:09 | {"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": 0.6606240272521973, "perplexity": 4199.794220751119}, "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-2022-21/segments/1652662555558.23/warc/CC-MAIN-20220523041156-20220523071156-00482.warc.gz"} |
https://en.wikipedia.org/wiki/K-means_algorithm | # k-means clustering
(Redirected from K-means algorithm)
k-means clustering is a method of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining. k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells.
The problem is computationally difficult (NP-hard); however, there are efficient heuristic algorithms that are commonly employed and converge quickly to a local optimum. These are usually similar to the expectation-maximization algorithm for mixtures of Gaussian distributions via an iterative refinement approach employed by both algorithms. Additionally, they both use cluster centers to model the data; however, k-means clustering tends to find clusters of comparable spatial extent, while the expectation-maximization mechanism allows clusters to have different shapes.
The algorithm has a loose relationship to the k-nearest neighbor classifier, a popular machine learning technique for classification that is often confused with k-means because of the k in the name. One can apply the 1-nearest neighbor classifier on the cluster centers obtained by k-means to classify new data into the existing clusters. This is known as nearest centroid classifier or Rocchio algorithm.
## Description
Given a set of observations (x1, x2, …, xn), where each observation is a d-dimensional real vector, k-means clustering aims to partition the n observations into k (≤ n) sets S = {S1S2, …, Sk} so as to minimize the within-cluster sum of squares (WCSS) (sum of distance functions of each point in the cluster to the K center). In other words, its objective is to find:
${\displaystyle {\underset {\mathbf {S} }{\operatorname {arg\,min} }}\sum _{i=1}^{k}\sum _{\mathbf {x} \in S_{i}}\left\|\mathbf {x} -{\boldsymbol {\mu }}_{i}\right\|^{2}}$
where μi is the mean of points in Si.
## History
The term "k-means" was first used by James MacQueen in 1967,[1] though the idea goes back to Hugo Steinhaus in 1957.[2] The standard algorithm was first proposed by Stuart Lloyd in 1957 as a technique for pulse-code modulation, though it wasn't published outside of Bell Labs until 1982.[3] In 1965, E. W. Forgy published essentially the same method, which is why it is sometimes referred to as Lloyd-Forgy.[4] A more efficient version was later proposed and published in FORTRAN by Hartigan and Wong.[5][6]
## Algorithms
### Standard algorithm
The most common algorithm uses an iterative refinement technique. Due to its ubiquity it is often called the k-means algorithm; it is also referred to as Lloyd's algorithm, particularly in the computer science community.
Given an initial set of k means m1(1),…,mk(1) (see below), the algorithm proceeds by alternating between two steps:[7]
Assignment step: Assign each observation to the cluster whose mean yields the least within-cluster sum of squares (WCSS). Since the sum of squares is the squared Euclidean distance, this is intuitively the "nearest" mean.[8] (Mathematically, this means partitioning the observations according to the Voronoi diagram generated by the means).
${\displaystyle S_{i}^{(t)}={\big \{}x_{p}:{\big \|}x_{p}-m_{i}^{(t)}{\big \|}^{2}\leq {\big \|}x_{p}-m_{j}^{(t)}{\big \|}^{2}\ \forall j,1\leq j\leq k{\big \}},}$
where each ${\displaystyle x_{p}}$ is assigned to exactly one ${\displaystyle S^{(t)}}$, even if it could be assigned to two or more of them.
Update step: Calculate the new means to be the centroids of the observations in the new clusters.
${\displaystyle m_{i}^{(t+1)}={\frac {1}{|S_{i}^{(t)}|}}\sum _{x_{j}\in S_{i}^{(t)}}x_{j}}$
Since the arithmetic mean is a least-squares estimator, this also minimizes the within-cluster sum of squares (WCSS) objective.
The algorithm has converged when the assignments no longer change. Since both steps optimize the WCSS objective, and there only exists a finite number of such partitionings, the algorithm must converge to a (local) optimum. There is no guarantee that the global optimum is found using this algorithm.
The algorithm is often presented as assigning objects to the nearest cluster by distance. The standard algorithm aims at minimizing the WCSS objective, and thus assigns by "least sum of squares", which is exactly equivalent to assigning by the smallest Euclidean distance. Using a different distance function other than (squared) Euclidean distance may stop the algorithm from converging.[citation needed] Various modifications of k-means such as spherical k-means and k-medoids have been proposed to allow using other distance measures.
#### Initialization methods
Commonly used initialization methods are Forgy and Random Partition.[9] The Forgy method randomly chooses k observations from the data set and uses these as the initial means. The Random Partition method first randomly assigns a cluster to each observation and then proceeds to the update step, thus computing the initial mean to be the centroid of the cluster's randomly assigned points. The Forgy method tends to spread the initial means out, while Random Partition places all of them close to the center of the data set. According to Hamerly et al.,[9] the Random Partition method is generally preferable for algorithms such as the k-harmonic means and fuzzy k-means. For expectation maximization and standard k-means algorithms, the Forgy method of initialization is preferable.
As it is a heuristic algorithm, there is no guarantee that it will converge to the global optimum, and the result may depend on the initial clusters. As the algorithm is usually very fast, it is common to run it multiple times with different starting conditions. However, in the worst case, k-means can be very slow to converge: in particular it has been shown that there exist certain point sets, even in 2 dimensions, on which k-means takes exponential time, that is 2Ω(n), to converge.[10] These point sets do not seem to arise in practice: this is corroborated by the fact that the smoothed running time of k-means is polynomial.[11]
The "assignment" step is also referred to as expectation step, the "update step" as maximization step, making this algorithm a variant of the generalized expectation-maximization algorithm.
### Complexity
Regarding computational complexity, finding the optimal solution to the k-means clustering problem for observations in d dimensions is:
• NP-hard in general Euclidean space d even for 2 clusters[12][13]
• NP-hard for a general number of clusters k even in the plane[14]
• If k and d (the dimension) are fixed, the problem can be exactly solved in time ${\displaystyle O(n^{dk+1}\log {n})}$, where n is the number of entities to be clustered[15]
Thus, a variety of heuristic algorithms such as Lloyds algorithm given above are generally used.
The running time of Lloyds algorithm is often given as ${\displaystyle O(nkdi)}$, where n is the number of d-dimensional vectors, k the number of clusters and i the number of iterations needed until convergence. On data that does have a clustering structure, the number of iterations until convergence is often small, and results only improve slightly after the first dozen iterations. Lloyds algorithm is therefore often considered to be of "linear" complexity in practice.
Following are some recent insights into this algorithm complexity behavior.
• Lloyd's k-means algorithm has polynomial smoothed running time. It is shown that[11] for arbitrary set of n points in ${\displaystyle [0,1]^{d}}$, if each point is independently perturbed by a normal distribution with mean 0 and variance ${\displaystyle \sigma ^{2}}$, then the expected running time of k-means algorithm is bounded by ${\displaystyle O(n^{34}k^{34}d^{8}\log ^{4}(n)/\sigma ^{6})}$, which is a polynomial in n, k, d and ${\displaystyle 1/\sigma }$.
• Better bounds are proved for simple cases. For example,[16] showed that the running time of k-means algorithm is bounded by ${\displaystyle O(dn^{4}M^{2})}$ for n points in an integer lattice ${\displaystyle \{1,\dots ,M\}^{d}}$.
Lloyd's algorithm is the standard approach for this problem, However, it spends a lot of processing time computing the distances between each of the k cluster centers and the n data points. Since points usually stay in the same clusters after a few iterations, much of this work is unnecessary, making the naive implementation very inefficient. Some implementations use the triangle inequality in order to create bounds and accelerate Lloyds algorithm.[17][18][19]
## Discussion
A typical example of the k-means convergence to a local minimum. In this example, the result of k-means clustering (the right figure) contradicts the obvious cluster structure of the data set. The small circles are the data points, the four ray stars are the centroids (means). The initial configuration is on the left figure. The algorithm converges after five iterations presented on the figures, from the left to the right. The illustration was prepared with the Mirkes Java applet.[28]
k-means clustering result for the Iris flower data set and actual species visualized using ELKI. Cluster means are marked using larger, semi-transparent symbols.
k-means clustering and EM clustering on an artificial dataset ("mouse"). The tendency of k-means to produce equi-sized clusters leads to bad results, while EM benefits from the Gaussian distribution present in the data set
Three key features of k-means which make it efficient are often regarded as its biggest drawbacks:
• Euclidean distance is used as a metric and variance is used as a measure of cluster scatter.
• The number of clusters k is an input parameter: an inappropriate choice of k may yield poor results. That is why, when performing k-means, it is important to run diagnostic checks for determining the number of clusters in the data set.
• Convergence to a local minimum may produce counterintuitive ("wrong") results (see example in Fig.).
A key limitation of k-means is its cluster model. The concept is based on spherical clusters that are separable in a way so that the mean value converges towards the cluster center. The clusters are expected to be of similar size, so that the assignment to the nearest cluster center is the correct assignment. When for example applying k-means with a value of ${\displaystyle k=3}$ onto the well-known Iris flower data set, the result often fails to separate the three Iris species contained in the data set. With ${\displaystyle k=2}$, the two visible clusters (one containing two species) will be discovered, whereas with ${\displaystyle k=3}$ one of the two clusters will be split into two even parts. In fact, ${\displaystyle k=2}$ is more appropriate for this data set, despite the data set containing 3 classes. As with any other clustering algorithm, the k-means result relies on the data set to satisfy the assumptions made by the clustering algorithms. It works well on some data sets, while failing on others.
The result of k-means can also be seen as the Voronoi cells of the cluster means. Since data is split halfway between cluster means, this can lead to suboptimal splits as can be seen in the "mouse" example. The Gaussian models used by the Expectation-maximization algorithm (which can be seen as a generalization of k-means) are more flexible here by having both variances and covariances. The EM result is thus able to accommodate clusters of variable size much better than k-means as well as correlated clusters (not in this example).
## Applications
k-means clustering, in particular when using heuristics such as Lloyd's algorithm, is rather easy to implement and apply even on large data sets. As such, it has been successfully used in various topics, including market segmentation, computer vision, geostatistics,[29] astronomy and agriculture. It often is used as a preprocessing step for other algorithms, for example to find a starting configuration.
### Vector quantization
Main article: Vector quantization
Two-channel (for illustration purposes -- red and green only) color image.
Vector quantization of colors present in the image above into Voronoi cells using k-means.
k-means originates from signal processing, and still finds use in this domain. For example, in computer graphics, color quantization is the task of reducing the color palette of an image to a fixed number of colors k. The k-means algorithm can easily be used for this task and produces competitive results. A use case for this approach is image segmentation. Other uses of vector quantization include non-random sampling, as k-means can easily be used to choose k different but prototypical objects from a large data set for further analysis.
### Cluster analysis
Main article: Cluster analysis
In cluster analysis, the k-means algorithm can be used to partition the input data set into k partitions (clusters).
However, the pure k-means algorithm is not very flexible, and as such is of limited use (except for when vector quantization as above is actually the desired use case!). In particular, the parameter k is known to be hard to choose (as discussed above) when not given by external constraints. Another limitation of the algorithm is that it cannot be used with arbitrary distance functions or on non-numerical data. For these use cases, many other algorithms have been developed since.
### Feature learning
k-means clustering has been used as a feature learning (or dictionary learning) step, in either (semi-)supervised learning or unsupervised learning.[30] The basic approach is first to train a k-means clustering representation, using the input training data (which need not be labelled). Then, to project any input datum into the new feature space, we have a choice of "encoding" functions, but we can use for example the thresholded matrix-product of the datum with the centroid locations, the distance from the datum to each centroid, or simply an indicator function for the nearest centroid,[30][31] or some smooth transformation of the distance.[32] Alternatively, by transforming the sample-cluster distance through a Gaussian RBF, one effectively obtains the hidden layer of a radial basis function network.[33]
This use of k-means has been successfully combined with simple, linear classifiers for semi-supervised learning in NLP (specifically for named entity recognition)[34] and in computer vision. On an object recognition task, it was found to exhibit comparable performance with more sophisticated feature learning approaches such as autoencoders and restricted Boltzmann machines.[32] However, it generally requires more data than the sophisticated methods, for equivalent performance, because each data point only contributes to one "feature" rather than multiple.[30]
## Relation to other statistical machine learning algorithms
k-means clustering, and its associated expectation-maximization algorithm, is a special case of a Gaussian mixture model, specifically, the limit of taking all covariances as diagonal, equal, and small. It is often easy to generalize a k-means problem into a Gaussian mixture model.[35] Another generalization of the k-means algorithm is the K-SVD algorithm, which estimates data points as a sparse linear combination of "codebook vectors". K-means corresponds to the special case of using a single codebook vector, with a weight of 1.[36]
### Mean shift clustering
Basic mean shift clustering algorithms maintain a set of data points the same size as the input data set. Initially, this set is copied from the input set. Then this set is iteratively replaced by the mean of those points in the set that are within a given distance of that point. By contrast, k-means restricts this updated set to k points usually much less than the number of points in the input data set, and replaces each point in this set by the mean of all points in the input set that are closer to that point than any other (e.g. within the Voronoi partition of each updating point). A mean shift algorithm that is similar then to k-means, called likelihood mean shift, replaces the set of points undergoing replacement by the mean of all points in the input set that are within a given distance of the changing set.[37] One of the advantages of mean shift over k-means is that there is no need to choose the number of clusters, because mean shift is likely to find only a few clusters if indeed only a small number exist. However, mean shift can be much slower than k-means, and still requires selection of a bandwidth parameter. Mean shift has soft variants much as k-means does.
### Principal component analysis (PCA)
It was asserted[38][39] that the relaxed solution of k-means clustering, specified by the cluster indicators, is given by principal component analysis (PCA), and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. However, that PCA is a useful relaxation of k-means clustering was not a new result,[40] and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.[41]
### Independent component analysis (ICA)
It has been shown in [42] that under sparsity assumptions and when input data is pre-processed with the whitening transformation k-means produces the solution to the linear Independent component analysis task. This aids in explaining the successful application of k-means to feature learning.
### Bilateral filtering
k-means implicitly assumes that the ordering of the input data set does not matter. The bilateral filter is similar to K-means and mean shift in that it maintains a set of data points that are iteratively replaced by means. However, the bilateral filter restricts the calculation of the (kernel weighted) mean to include only points that are close in the ordering of the input data.[37] This makes it applicable to problems such as image denoising, where the spatial arrangement of pixels in an image is of critical importance.
## Similar problems
The set of squared error minimizing cluster functions also includes the k-medoids algorithm, an approach which forces the center point of each cluster to be one of the actual points, i.e., it uses medoids in place of centroids.
## Software implementations
### Free Software/Open Source
the following implementations are available under Free/Open Source Software licenses, with publicly available source code.
• CrimeStat implements two spatial k-means algorithms, one of which allows the user to define the starting locations.
• ELKI contains k-means (with Lloyd and MacQueen iteration, along with different initializations such as k-means++ initialization) and various more advanced clustering algorithms.
• Julia contains a k-means implementation in the Clustering package.[43]
• Mahout contains a MapReduce based k-means.
• MLPACK contains a C++ implementation of k-means.
• Octave contains k-means.
• OpenCV contains a k-means implementation.
• R contains three k-means variations.[1][3][6]
• SciPy and scikit-learn contain multiple k-means implementations.
• Spark MLlib implements a distributed k-means algorithm.
• Torch contains an unsup package that provides k-means clustering.
• Weka contains k-means and x-means.
• Accord.NET contains C# implementations for k-means, k-means++ and k-modes.
### Proprietary
The following implementations are available under proprietary license terms, and may not have publicly available source code.
## References
1. ^ a b MacQueen, J. B. (1967). Some Methods for classification and Analysis of Multivariate Observations. Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability. University of California Press. pp. 281–297. MR 0214227. Zbl 0214.46201. Retrieved 2009-04-07.
2. ^ Steinhaus, H. (1957). "Sur la division des corps matériels en parties". Bull. Acad. Polon. Sci. (in French). 4 (12): 801–804. MR 0090073. Zbl 0079.16403.
3. ^ a b Lloyd, S. P. (1957). "Least square quantization in PCM". Bell Telephone Laboratories Paper. Published in journal much later: Lloyd., S. P. (1982). "Least squares quantization in PCM" (PDF). IEEE Transactions on Information Theory. 28 (2): 129–137. doi:10.1109/TIT.1982.1056489. Retrieved 2009-04-15.
4. ^ E.W. Forgy (1965). "Cluster analysis of multivariate data: efficiency versus interpretability of classifications". Biometrics. 21: 768–769. JSTOR 2528559.
5. ^ J.A. Hartigan (1975). Clustering algorithms. John Wiley & Sons, Inc.
6. ^ a b c Hartigan, J. A.; Wong, M. A. (1979). "Algorithm AS 136: A K-Means Clustering Algorithm". Journal of the Royal Statistical Society, Series C. 28 (1): 100–108. JSTOR 2346830.
7. ^ MacKay, David (2003). "Chapter 20. An Example Inference Task: Clustering" (PDF). Information Theory, Inference and Learning Algorithms. Cambridge University Press. pp. 284–292. ISBN 0-521-64298-1. MR 2012999.
8. ^ Since the square root is a monotone function, this also is the minimum Euclidean distance assignment.
9. ^ a b Hamerly, G.; Elkan, C. (2002). "Alternatives to the k-means algorithm that find better clusterings" (PDF). Proceedings of the eleventh international conference on Information and knowledge management (CIKM).
10. ^ Vattani., A. (2011). "k-means requires exponentially many iterations even in the plane" (PDF). Discrete and Computational Geometry. 45 (4): 596–616. doi:10.1007/s00454-011-9340-1.
11. ^ a b Arthur, D.; Manthey, B.; Roeglin, H. (2009). "k-means has polynomial smoothed complexity". Proceedings of the 50th Symposium on Foundations of Computer Science (FOCS).
12. ^ Aloise, D.; Deshpande, A.; Hansen, P.; Popat, P. (2009). "NP-hardness of Euclidean sum-of-squares clustering". Machine Learning. 75: 245–249. doi:10.1007/s10994-009-5103-0.
13. ^ Dasgupta, S.; Freund, Y. (July 2009). "Random Projection Trees for Vector Quantization". Information Theory, IEEE Transactions on. 55: 3229–3242. arXiv:0805.1390. doi:10.1109/TIT.2009.2021326.
14. ^ Mahajan, M.; Nimbhorkar, P.; Varadarajan, K. (2009). "The Planar k-Means Problem is NP-Hard". Lecture Notes in Computer Science. 5431: 274–285. doi:10.1007/978-3-642-00202-1_24.
15. ^ Inaba, M.; Katoh, N.; Imai, H. (1994). Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering. Proceedings of 10th ACM Symposium on Computational Geometry. pp. 332–339. doi:10.1145/177424.178042.
16. ^ Arthur; Abhishek Bhowmick (2009). A theoretical analysis of Lloyd's algorithm for k-means clustering (PDF) (Thesis).
17. ^ a b Phillips, Steven J. (2002-01-04). Mount, David M.; Stein, Clifford, eds. Acceleration of K-Means and Related Clustering Algorithms. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 166–177. doi:10.1007/3-540-45643-0_13. ISBN 978-3-540-43977-6.
18. ^ a b Elkan, C. (2003). "Using the triangle inequality to accelerate k-means" (PDF). Proceedings of the Twentieth International Conference on Machine Learning (ICML).
19. ^ a b Hamerly, Greg. "Making k-means even faster". citeseerx.ist.psu.edu. Retrieved 2015-12-10.
20. ^ Kanungo, T.; Mount, D. M.; Netanyahu, N. S.; Piatko, C. D.; Silverman, R.; Wu, A. Y. (2002). "An efficient k-means clustering algorithm: Analysis and implementation" (PDF). IEEE Trans. Pattern Analysis and Machine Intelligence. 24: 881–892. doi:10.1109/TPAMI.2002.1017616. Retrieved 2009-04-24.
21. ^ Drake, Jonathan (2012). "Accelerated k-means with adaptive distance bounds" (PDF). the 5th NIPS Workshop on Optimization for Machine Learning, OPT2012.
22. ^ Dhillon, I. S.; Modha, D. M. (2001). "Concept decompositions for large sparse text data using clustering". Machine Learning. 42 (1): 143–175. doi:10.1023/a:1007612920971.
23. ^ Steinbach, M., Karypis, G., & Kumar, V. (2000, August). A comparison of document clustering techniques. In KDD workshop on text mining (Vol. 400, No. 1, pp. 525-526).
24. ^ Pelleg, D., & Moore, A. W. (2000, June). X-means: Extending K-means with Efficient Estimation of the Number of Clusters. In ICML (Vol. 1).
25. ^ Hamerly, G., & Elkan, C. (2004). Learning the k in k-means. Advances in neural information processing systems, 16, 281.
26. ^ Amorim, R.C.; Mirkin, B. (2012). "Minkowski Metric, Feature Weighting and Anomalous Cluster Initialisation in K-Means Clustering". Pattern Recognition. 45 (3): 1061–1075. doi:10.1016/j.patcog.2011.08.012.
27. ^ Amorim, R.C.; Hennig, C. (2015). "Recovering the number of clusters in data sets with noise features using feature rescaling factors". Information Sciences. 324: 126–145. doi:10.1016/j.ins.2015.06.039.
28. ^ Mirkes, E.M. "K-means and K-medoids applet.". Retrieved 2 January 2016.
29. ^ Honarkhah, M; Caers, J (2010). "Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling". Mathematical Geosciences. 42: 487–517. doi:10.1007/s11004-010-9276-7.
30. ^ a b c Coates, Adam; Ng, Andrew Y. (2012). "Learning feature representations with k-means" (PDF). In G. Montavon, G. B. Orr, K.-R. Müller. Neural Networks: Tricks of the Trade. Springer.
31. ^ Csurka, Gabriella; Dance, Christopher C.; Fan, Lixin; Willamowski, Jutta; Bray, Cédric (2004). Visual categorization with bags of keypoints (PDF). ECCV Workshop on Statistical Learning in Computer Vision.
32. ^ a b Coates, Adam; Lee, Honglak; Ng, Andrew Y. (2011). An analysis of single-layer networks in unsupervised feature learning (PDF). International Conference on Artificial Intelligence and Statistics (AISTATS).
33. ^ Schwenker, Friedhelm; Kestler, Hans A.; Palm, Günther (2001). "Three learning phases for radial-basis-function networks". Neural Networks. 14: 439–458. doi:10.1016/s0893-6080(01)00027-2. CiteSeerX: 10.1.1.109.312.
34. ^ Lin, Dekang; Wu, Xiaoyun (2009). Phrase clustering for discriminative learning (PDF). Annual Meeting of the ACL and IJCNLP. pp. 1030–1038.
35. ^ Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 16.1. Gaussian Mixture Models and k-Means Clustering". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
36. ^ Aharon, Michal; Elad, Michael; Bruckstein, Alfred (2006). "K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation" (PDF).
37. ^ a b Little, M.A.; Jones, N.S. (2011). "Generalized Methods and Solvers for Piecewise Constant Signals: Part I" (PDF). Proceedings of the Royal Society A. 467: 3088–3114. doi:10.1098/rspa.2010.0671.
38. ^ H. Zha, C. Ding, M. Gu, X. He and H.D. Simon (Dec 2001). "Spectral Relaxation for K-means Clustering" (PDF). Neural Information Processing Systems vol.14 (NIPS 2001). Vancouver, Canada: 1057–1064.
39. ^ Chris Ding and Xiaofeng He (July 2004). "K-means Clustering via Principal Component Analysis" (PDF). Proc. of Int'l Conf. Machine Learning (ICML 2004): 225–232.
40. ^ Drineas, P.; A. Frieze; R. Kannan; S. Vempala; V. Vinay (2004). "Clustering large graphs via the singular value decomposition" (PDF). Machine learning. 56: 9–33. doi:10.1023/b:mach.0000033113.59016.96. Retrieved 2012-08-02.
41. ^ Cohen, M.; S. Elder; C. Musco; C. Musco; M. Persu (2014). "Dimensionality reduction for k-means clustering and low rank approximation (Appendix B)". arXiv:1410.6801.
42. ^ Alon Vinnikov and Shai Shalev-Shwartz (2014). "K-means Recovers ICA Filters when Independent Components are Sparse" (PDF). Proc. of Int'l Conf. Machine Learning (ICML 2014).
43. ^ Clustering.jl www.github.com
44. ^ http://help.sap.com/saphelp_hanaplatform/helpdata/en/53/e6908794ce4bcaa440f5c4348f3d14/content.htm | 2016-10-24 20:35:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 18, "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": 0.757983386516571, "perplexity": 1835.9299978114022}, "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-2016-44/segments/1476988719754.86/warc/CC-MAIN-20161020183839-00026-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://www.ams.org/mathscinet-getitem?mr=1414289 | MathSciNet bibliographic data MR1414289 (97k:65033) 65D10 Albrecht, Gudrun; Farouki, Rida T. Construction of \$C\sp 2\$$C\sp 2$ Pythagorean-hodograph interpolating splines by the homotopy method. Adv. Comput. Math. 5 (1996), no. 4, 417–442. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews. | 2015-03-04 11:40:56 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "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": 0.9972473382949829, "perplexity": 13349.893184681954}, "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-2015-11/segments/1424936463475.57/warc/CC-MAIN-20150226074103-00322-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/limits-and-0-0.5448/ | # Limits and 0/0
1. Sep 4, 2003
### STAii
Greetings.
Let me first show how my teacher solves a limit.
For example, take :
Lim(x->6) (x+5)/(x-3)
He takes the number that x is heading to (forgive me for the terminology), which is 6 in our case.
Now put 6 in the place of x in the function after the limit, you get :
6+5/(6-3) = 11/3
Now, there are 4 conditions for the ratio that u will get :
A: NonZero/NonZero : in this case there is no problem
B: Zero/NonZero : no problem in this case too
C: NonZero/Zero : in this case the limit will either be +[oo] or -[oo]
D: Zero/Zero : in this case we have to use some tricks to make the limit back to the case of A, B or C.
Everything i just said was what my teachers says.
Today in class i came up with a question that (the way i see it) does not work on the rules of my teacher. Here it is :
Define :
Code (Text):
f(x) = 2 , (x=1)
x , (x[x=]1)
Now, try to find the limit :
Lim(x->1) (f(x)-1)/(x-1)
If you use my teacher's way, you will reach 1/0, which means that the limit either equals +[oo] or -[oo]. But if you solve the limit, you will find that it equals 1 (unless i am missing sth).
When i told my teacher about this, he said : "no no no no no (maybe 30 times), it is impossible", i told him "just see this question, and either tell me where i went wrong, or ur rule isn't right", he said "this is not MY rule" (then in a silly way) "if you want try to invent ur own theory then call it on your name", i tried to explain my way of solving the problem, but he wouldn't hear me.
Anyway, here is my way :
Take the limit of the numerator, and the limit of the denominator, then write both limits as a ratio, then apply my teacher's rules (A,B,C and D).
Please tell me what do u think. Feel free to ask any questions.
Thanks.
Last edited: Sep 4, 2003
2. Sep 4, 2003
### chroot
Staff Emeritus
What you've created is a point discontinuity. There is no limit defined at x=1.
- Warren
3. Sep 4, 2003
### HallsofIvy
The limit lim(x->a) f(x) is the single number, if there is such, that f(x) nears as x nears a (without actually being equal to a).
Your teacher's method works for continuous functions (indeed, "continuous functions" are DEFINED
4. Sep 4, 2003
### Hurkyl
Staff Emeritus
You are neglecting a crucial condition for applying this "plug-in" rule; both the numerator and denominator must be continuous.
The relevant theorem, taken directly from my advanced calculus text (but notation changed to suit this medium), says:
Assuming that f and g are each defined on a deleted neighborhood of x = b and that f(x)→A as x→b
and g(x)→B as x→b, then it is true that
...
and if B[x=]0
f(x)/g(x)→A/B as x→b
Because of the way you defined your numerator, the hypotheses of this theorem does not hold.
5. Sep 4, 2003
### HallsofIvy
The limit lim(x->a) f(x) is the single number, if there is such, that f(x) nears as x nears a (without actually being equal to a).
Your teacher's method works for continuous functions (indeed, "continuous functions" are DEFINED as functions for which this works!) and it can be show that functions made by the usual operations "add, subtract, multiply, divide, powers" are continuous as long as we do reach and "impossible" operation (divide by 0, take square root of a negative number, etc.)
The function you give is not continuous (your f(x) is not continuous because the limit at x= 1 is clearly 2 (No matter how close to 1 x is, as long as x is not 1, f(x)= 2 so it's "close to" 2) but the value of the function is 1, not 2: the limit is not the value of the function- that itself "blows" your teacher's method, you don't need the "-1/(x-1)" part!). What you can do is take the "one-sided" limits. If we look only at x< 1, we can replace f(x) by 2:
f(x)-1= 2-1= 1 so (f(x)-1)/(x-1)= 1/(x-1). That reduces to the "Nonzero/zero" case. Since x-1<1 here, the limit is "- infinity" (by the way, a lot of people, myself included, who say that saying the "limit is -infinity (or +infinity)" is just another way of saying the limit does not exist). If you take x> 1, you have f(x)-1= 2-1= 1 again, but now x-1> 0. The limit is "+ infinity". Those results tell us that this function is not getting "close to" any specific number and there is no limit.
(I'm impressed that you are thinking of "piecewise" functions. Most beginning students shun those like the plague.)
6. Sep 5, 2003
### STAii
Thanks all.
chroot said
Can you explain why please ?
It is possible to have a function that has a discountinuity at a certain point say x=b, and still has a limit when x gets near b, my function is one of them (as far as i see). Please read to the end of the reply to see why.
Hurkyl said
Well, i think the theory that the calculus book is talking about is only the cases A and B of my teacher's idea, since (as you can see), my teacher's method has the denominator=0 sometimes (but will not accept the answer directly).
I understand that the plug-in rule is not acceptable when the limit of the denominator is 0, but my teacher does not look at the matter this way, my teacher plugs in the values and uses the results to decide how to deal with the limit (as you can see from my first post)
HallsofIvy
After reading ur post i felt a little bit lost, let me explain why.
I am ok with the first two paragraphs, but the trouble comes from the third paragraph.
I see the exact opposite, the value of the function f(x) at x=1 is 2=f(1), and the limit at x=1 is clearly 1 !!
This is what i actually did before writting the first post, and here is my method.
First, i redefined the original function in a more elegant way
Code (Text):
f(x) = x , x < 1
2 , x = 1
x , x > 1
Then, i took the right handed limit :
lim(x->1+) (f(x)-1)/(x-1) = lim(x->1+) (x-1)/(x-1) = lim(x->1+) 1 = 1
(since when x gets near 1 from the right, it will be bigger than 1, then f(x) (in this case) will be x)
Doing the same to the left handed limit, you get :
lim(x->1-) (f(x)-1)/(x-1) = 1
So, we conclude :
lim(x->1) (f(x)-1)/(x-1) = 1
Am i right ?
Thank you .
(Edited for a missing symbol)
Last edited: Sep 5, 2003
7. Sep 5, 2003
### HallsofIvy
You are right, I misread the function.
[QU0TE]This is what i actually did before writting the first post, and here is my method.
First, i redefined the original function in a more elegant way
code:--------------------------------------------------------------------------------
f(x) = x , x < 1
2 , x = 1
x , x > 1
--------------------------------------------------------------------------------
Then, i took the right handed limit :
lim(x->1+) (f(x)-1)/(x-1) = lim(x->1+) (x-1)/(x-1) = lim(x->1+) 1 = 1
(since when x gets near 1 from the right, it will be bigger than 1, then f(x) (in this case) will be x)
Doing the same to the left handed limit, you get :
lim(x->1-) (f(x)-1)/(x-1) = 1
So, we conclude :
lim(x->1) (f(x)-1)/(x-1) = 1
Am i right ?[/QUOTE]
Yes, with this new function the limit is 1.
It appears to me that your teacher DOES "look at the matter this way". Your teacher is saying that the "plug in rule" GIVES the right limit if the denominator is not 0. If the denominator is 0, then then you look at whether or not the numerator is 0. If not, the limit does not exist. If yes, then you need to look more closely (my preferred way of saying "use tricks"!).
By the way, the "plug in rule" does NOT say "if g(a) is not 0 then lim f(x)/g(x)= f(a)/g(a)". This is only true if f and g are themselves continuous at a.
What you are really doing is using the limit theorems:
If lim f(x)= A, lim g(x)= B then
a) lim(f(x)+ g(x))= A+ B
b) lim(f(x)- g(x))= A- B
c) lim(f(x)*g(x))= A*B
d) lim(f(x)/g(x))= A/B as long as B is not 0 (all limits as x->a)
together with the fact that lim(x->a) x= a and lim(x->a) c= c
(c a constant). From those one can show that all polynomials are continuous (lim f(x)= f(a)) for all a and rational functions are continuous at a as long as a does not make the denominator 0.
Apparently you have not been introduced to continuous functions yet.
"Plug in" only works for continuous functions. Actually, "almost all" functions are NOT continous anywhere but continuous functions are so nice that our function notation has developed so that practically every function we can write easily is continuous.
8. Sep 5, 2003
### lethe
this is not true.
9. Sep 5, 2003
### chroot
Staff Emeritus
lethe,
Yeah, yeah, I know, I read the damn function wrong too. The limit's one.
- Warren
10. Sep 5, 2003
### STAii
Exactly, this is what my teacher sees, and this is what i am trying to proove wrong using my problem .
As you can see, when you 'plug in' the value, you will have 0 in the denominator, and if you look at the numerator its value will not be zero, and therefore (according to my teacher's idea that u are perfectly expressing when you said "If not, the limit does not exist.") the limit does not exist, althought if we investigate it more we will find it exists.
Well, actually, i know what they are pretty well. The point is that my teacher isn't really accurate, actually ... isn't accurate at all! He told us about the plug in rule without even mentioning that it only works under certain conditions.
What i meant by 'using tricks' is using mathematical methods to change the limit which's plug in is 0/0 to a Nonzero/Nonzero value.
The real question lies here.
My teacher thinks we should start 'using tricks' when the result of plugging in the value of x in the limit is 0/0 (please not that he DOES NOT investigate wether or not the function is continuous).
What i am suggesting is that we should start 'using tricks' when the ratio of the limit of the numerator divided by the limit of the denominator is 0/0. This way wether the function is continous or not we will know we need to 'use the tricks'.
Tricks include : Analyzing the numerator to a simpler form, multiplying both numerator and denominator by certain roots, adding and subtracting certain numbers/functions to the numerator ... etc.
My question is only wether or not what i am suggesting is right.
Thanks.
EDIT :
This is the problem, my teacher does NOT investigate about the functions' continuity before using the plug in rule.
Last edited: Sep 5, 2003
11. Sep 6, 2003
### phoenixthoth
your teacher's methods work for continuous functions, of just plugging it in and seeing what you get. you just have a discontinuity but the limit IS defined and as you said, it is 1. to actually prove that it is one is not that hard, but requires some epsilon-delta stuff i won't go into here. maybe you should be one of the few who starts with advanced calculus rather than calculus cuz that's the box you're bumping up against. | 2019-02-19 15:27:23 | {"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": 0.883908748626709, "perplexity": 880.943003673739}, "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-2019-09/segments/1550247490225.49/warc/CC-MAIN-20190219142524-20190219164524-00566.warc.gz"} |
https://socratic.org/questions/57822ca811ef6b60b1c26e1f | # Question 26e1f
May 12, 2017
We know that KE ($E$) and momentum ($p$) of a body of mass $m$ is related as follows
$E = {p}^{2} / \left(2 m\right)$
If ${E}_{i} \mathmr{and} {p}_{i}$ are respectively initial KE and momentum and ${E}_{f} \mathmr{and} {p}_{f}$ are respectively final KE and momentum then
${E}_{i} = {p}_{i}^{2} / \left(2 m\right)$
and
${E}_{f} = {p}_{f}^{2} / \left(2 m\right)$
combining these two we have
${E}_{f} / {E}_{i} = {p}_{f}^{2} / {p}_{i}^{2}$
By the given condition p_f=50%of p_i=p_i/2
so
${E}_{f} / {E}_{i} = {p}_{f}^{2} / {\left(2 {p}_{f}\right)}^{2} = \frac{1}{4}$
=>(E_f-E_i)/E_i=-3/4=-75%#
So change in KE is 75% , here negative sign denotes the decrease in KE | 2019-05-21 16:39:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "mathjax_inline_tex": 1, "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": 0.9889337420463562, "perplexity": 1116.3298264093546}, "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-2019-22/segments/1558232256494.24/warc/CC-MAIN-20190521162634-20190521184634-00288.warc.gz"} |
https://unm.org.ua/rckeyz3/inverse-of-fx-701453 | passes the Horizontal Line Test, so the (restricted) function is invertible. google_ad_client = "pub-0863636157410944"; 5 of 7), Sections: Definition In fact, for every ordered pair ( a, b) belonging to f( x), there is a corresponding ordered pair ( b, a) that belongs to f −1( x). 1), "Finding the Inverse of a Function." Follow the five steps previously listed, beginning with rewriting f( x) as y: Note the restriction x ≥ 0 for f −1( x). here because I need to know the domain and range of the inverse. restriction, the graph looks like this: From what I know For example, find the inverse of f(x)=3x+2. $inverse\:f\left (x\right)=\sqrt {x+3}$. [Date] [Month] 2016, The "Homework 1. //--> Example: Find the inverse of f (x) = y = 3x − 2 Available from https://www.purplemath.com/modules/invrsfcn5.htm. Answer: Inverse function: A function g is the inverse of a function f if whenever y=f (x) then x=g (y). inverse f ( x) = √x + 3. The slope-intercept form gives you the y-intercept at (0, –2). Purplemath. Copyright Relations vs Functions. The inverse of a function is denoted by f^-1(x), and it's visually represented as the original function reflected over the line y=x. Find a local math tutor, , Copyright © 2020 Elizabeth Stapel | About | Terms of Use | Linking | Site Licensing, Return to the I usually wouldn't bother writing down the restriction, but it's helpful inverse y = x x2 − 6x + 8. So, here the sum of the two limited integral referred in the question is- The Area of the big rectangle – the Area of the smaller one = 10×5–2×1 =50−2 = 48 2. 99. profile. © 2020 Houghton Mifflin Harcourt. Figure 1 Inverse functions are symmetric about the line y = x. is not equal to var months = new Array( Because these functions have range elements that correspond to only one domain element each, there's no danger that their inverses will not be functions. However, the sets are switched. By … If it isn't, restrict the domain to pass the horizontal line test. Lessons Index. Let St = Fx and St(Inverse) = 1/Fx. about graphing Tap for more steps... Rewrite the equation as . The horizontal line test is a quick way to determine whether a graph is that of a one‐to‐one function. Because the given function is a linear function, you can graph it by using slope-intercept form. Since , is the inverse of . not equal Mathematically, this means that. Solve advanced problems in Physics, Mathematics and Engineering. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. With the domain 2" and the First, graph y = x. y For any input x, the function corresponding to f spits out the value y=f (x)=4x+12. google_ad_width = 160; CliffsNotes study guides are written by real teachers and professors, so no matter what you're studying, CliffsNotes can ease your homework headaches and help you score high on exams. months[now.getMonth()] + " " + Then the From your Reading List will also remove any bookmarked pages associated with this title would not pass the line! You sure you want to remove # bookConfirmation # and any corresponding bookmarks users found this helpful. More users found this answer helpful by using slope-intercept form some examples most... This short tutorial we will learn how you can graph it by using slope-intercept form there... Be a unique inverse f − 1 to denote an inverse of a Logarithmic function the! Be g ( x ), and vice versa out the value y=f ( )! X-Coordinate for the inverse the given function is f ( x ) and (... 'Re seeing this message, it means we 're having trouble loading external resources on our website f −1 x... After seeing some examples that most of the inverse of the work boils down to solving an equation can find. -8 -6 -4 4 6 that of a random variable symmetric about line... X is not equal to 2 '' and the range of f −1 ( x ) and... Since the inverse of f ( x ) =4x+12 users found this answer helpful answer helpful symbol... Distribution is the coordinates of x and y are swapped for the inverse of the passes. From the picture ( and recalling the concept of horizontal asymptotes ) that y never. Inverse f ( x ) and g ( x ) =4x+12 inverse function f ( x ) to.: y=\frac { x } { x^2-6x+8 } $1 to denote an inverse is. X, the function f ( x ), and vice versa$:. We end up with the same equation when composed with their original,. However inverse of fx coordinate is swapped examples that most of the inverse of function! Coordinates of x Fx and St ( inverse ) = 2x - 10 is h x. The value y=f ( x ) =3x+2 = 3x − 2 heart is represented by the variable. Bookmarked pages associated with this title if you 're seeing this message it. You will realize later after seeing some examples that most of the function is one-to-one, there be. A linear function, or f ( x ) from your Reading List also. ( ) ; function fourdigityear ( number < 1000 ) the reciprocal of a variable! That f ( x ) is the range of f ( x and! There will be a unique inverse Mathematics and Engineering log function is invertible as and! More steps... rewrite the equation as a foundational part of learning algebra is learning how to find inverse! How to find the inverse function for a one‐to‐one function, then the domain is is!: f\left ( x\right ) =\sqrt { x+3 } $the work down... # from your Reading List will also remove any bookmarked pages associated with this title easily., Complex Numbers, Calculation History h ( x ) and g ( x ) would not pass the line. X ), and vice versa let St = Fx and St ( inverse =..., Since it must possess an inverse distribution is the distribution of the.... Never equal 1 = Fx and St ( inverse ) = log of x a linear function follow. A foundational part of learning algebra is learning how to find the inverse f! Follow these steps: 1 y = x /2 + 5 short tutorial will... ) =4 1 to denote an inverse of the inverse of a random variable any bookmarked pages with! Because the given function tutorial we will learn how you can graph it using!, and vice versa i recognize that f ( x ), and vice.! Have the unique property that, when composed with their original functions, Next Relations vs functions =\ln\left ( ). Property that, when composed with their original functions, both functions cancel out of! This title the algebra: Since the inverse function figure 1 inverse are... And St ( inverse ) = x x2 − 6x + 8 from. And g ( x ) is a rational function, you can easily find the is! Above we end up with the same equation our inverse of fx examples, need... Will never equal 1 domain and range therefore we can take the values of f −1 x... By … * Response times vary by subject and question complexity its inverse like in our prior examples we! And any corresponding bookmarks removing # book # from your Reading List will also remove any bookmarked pages with. A linear function, or f ( x ) = y = 3x − 2.... As horizontal and vertical sides of the reciprocal of a random variable thing! Everything else alone obviously must be one‐to‐one, Since it must possess an inverse of f x! Formula of the function is one‐to‐one the concept of horizontal asymptotes ) that will., then the inverse of a given function, with steps shown domain to pass the horizontal line test an. Domain is represented by the x and y are swapped for the of... It obviously must be one‐to‐one, Since it must possess an inverse distribution is the inverse of function. X ) =4 realize later after seeing some examples that most of the reciprocal of a Logarithmic function the! Functions have the unique property that, when composed with their original functions both... Complex Numbers, Calculation History the unique property that, when composed with their original functions, Relations. Is the coordinates of x and y are swapped for the inverse f... –2 ) note from the picture ( and recalling the concept of horizontal asymptotes ) y! That most of the inverse of a Logarithmic function finding the inverse of a one‐to‐one function inverse... ) and g ( x ) is the inverse of f ( x ), and vice versa the needs. One-To-One, there will be a unique inverse if you 're seeing this message, means! Is 34 minutes and may be longer for new subjects vertical lines 8+ 4! Also remove any bookmarked pages associated with this title fx-991ES plus algebra: Since the inverse of a given.... Bookconfirmation # and any corresponding bookmarks you can easily find the inverse the... End up with the same equation rational function, follow these steps 1! 4 however the coordinate is swapped 0 '': '' ) + now.getDate ( ) ; function fourdigityear number. Instead of f ( x ) the y-intercept at ( 0, –2 ) be... Complex Numbers, Calculation History up with the same equation functions inverse step-by-step this website uses cookies to you. The unique property that, when composed with their original functions, Next Relations vs functions for new.... To f spits out the value y=f ( x ) =4 learning to! Domain and range steps: 1 work boils down to solving an.! The parabola passes the horizontal line test, so the ( restricted ) function invertible. Times vary by subject and question complexity < 1000 ) inverse f ( x ) = x /2 inverse of fx.... First, i recognize that f ( x ) =3x+2 answer helpful best., Plots, Unit Converter, equation Solver, Complex Numbers, History. Distribution of the function corresponding to f spits out the value y=f ( x ) is the of... Free functions inverse calculator - find functions inverse step-by-step this website uses cookies to ensure you get best! 1 inverse functions have the unique property that, when composed with their original,! =\Sqrt { x+3 }$ therefore we can take the values of f ( x ) the! 2+ -8 -6 -4 4 6 -8 -6 -4 4 6 steps: 1 St inverse. And recalling the concept of horizontal asymptotes ) that y will never equal 1 our... Y=\Frac { x } { x^2-6x+8 } $without this restriction, −1. Mathematics and Engineering is not equal to 1 '' variables ; leave everything alone! You inverse of fx to remove # bookConfirmation # and any corresponding bookmarks the restricted... And vertical sides of the reciprocal of a matrix using a Casio fx-991ES plus distribution is range. Resulting inverse function f ( x ) by using slope-intercept form range by the variable... Calculator - find functions inverse calculator - find functions inverse calculator - functions... 2X - 10 is h ( x ) = 2x - 10 is h ( x ) √x! X ) = √x + 3 pass the horizontal line test lines 8+ 6+ 4 2+ -8 -6 4... Get the best experience is represented by the y variable website uses cookies to you... Slope-Intercept form, equation Solver, Complex Numbers, Calculation History - 10 h! H ( x ) is a quick way to determine whether a graph is that of a given.. We can take the values of f ( x ) and St ( )! ) =\ln\left ( x-5\right )$ steps below 's inverse of fx algebra: Since the is. We end up with the same equation will also remove any bookmarked pages associated with this.. Best experience learn how you can graph it by using slope-intercept form or f ( x ) = 2x 10. Obviously must be one‐to‐one, Since it must possess an inverse of a log function is f ( )!
Anne Helm Germany, 38 Elm Street, Toronto, Hol Fit 150 Uses, Hawthorne School District, Square D Qob Breaker Spec Sheet, Rice County Court Administration, 4l60e Parts Diagram, Mcgill Pre Med Requirements, | 2021-07-29 18:20:27 | {"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": 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": 0.5577748417854309, "perplexity": 1186.7972366670658}, "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-2021-31/segments/1627046153892.74/warc/CC-MAIN-20210729172022-20210729202022-00643.warc.gz"} |
https://www.physicsforums.com/threads/problem-with-a-spring.565090/ | # Problem with a spring
1. Jan 4, 2012
### aaaa202
Look at the attached picture. I DO realize that this has been brought up several times, but there's a thing i still don't understand.
As you can see, I have solved the problem already but that was not without using several hints. Going through each hint at a time i solved the problem using conservation of energy after the collision. But but but! I don't agree with the energy assumptions they made. They made me use the elastical potentiel energy relative to the new equilibrium point as well as the kinetic energy and equaled that ½kA^2 where A denotes the amplitude.
BUT WHAT ABOUT THE LOSS IN POTENTIAL ENERGY DURING THE DOWNFALL? This depends on A as far as i can see making you need to solve a quadratic equation..
Anyways I got the right as you can see, but I don't get it...
File size:
39.4 KB
Views:
89
2. Jan 4, 2012
### SammyS
Staff Emeritus
So ...
What are you looking for from us?
3. Jan 5, 2012
### aaaa202
I said, that the solution did not involve using the fall in gravitational potential energy from the point where the ham and plate collide down to the lowest point of the oscillation. Why is that not used?
4. Jan 5, 2012
### SammyS
Staff Emeritus
It's not wrong to include the gravitational potential. However, all that including it does is to change the equilibrium position of the system. This merely represents adding a constant force.
The potential energy function, Vsg(y), with the spring and gravitational potentials included is:
$\displaystyle V_{\text{s g}}(y)=\frac{1}{2}ky^2+mgy$
This function is quadratic in y, and has a minimum at $\displaystyle y=-\frac{mg}{k}\,.$ The coefficient of y2 is (1/2)k whether or not the gravitational potential is included.
This can also be seen by completing the square.
$\displaystyle V_{\text{s g}}(y)=\frac{1}{2}ky^2+mgy$
$\displaystyle =\frac{k}{2}\left(y*2+\frac{2mg}{k}y\right)$
$\displaystyle = \frac{k}{2}\left[y^2+\frac{2mg}{k}y+\left(\frac{mg}{k}\right)^2-\left(\frac{mg}{k}\right)^2 \right]$
$\displaystyle = \frac{k}{2}\left(y+\frac{mg}{k}\right)^2-\frac{m^2g^2}{2k}$
Compare this with the potential function, Vs(y), for a spring only.
$\displaystyle V_{\text{s}}(y)=\frac{k}{2}y^2$
Adding a constant term to a potential function doesn't change the potential function's effect. So the only effect of adding the spring is to shift the equilibrium position.
5. Jan 5, 2012
### aaaa202
Hmm I still don't get it to be honest.
I do realize that the effect of gravity on a hanging spring is that it just changes the original equilibrium point of the spring - i.e. the motion is still SHM but around a lower point than had the spring laid horizontally.
But now you're saying as far as I can tell, that also the energy does nothing. So let's review: I have included the spring potential energy as well as the kinetic. What's the reason that the gravitational potential doesn't matter? Well okay you just explained that up there, but can you maybe do it less mathematically and more using intuitive, logical arguments. You've been a great help so far btw
Edit: thought about it a little more. If you include the gravitional potential the ham and plate would definately compress the spring more. But why on earth would they oscillate around another point? I mean, we did calculate a new equilibrium point when solving the assignment, which was using the equation mham + mplate = ky
So what equation would then describe this new equilibrium point, that you're talking about? :)
6. Jan 5, 2012
### SammyS
Staff Emeritus
No, it's not the case that the energy does nothing. Adding or subtracting a constant term to the potential energy function does nothing as far as the motion of a object is concerned. For instance, when using mgh for gravitational potential, it doesn't matter what point (elevation) you use for h=0 (the point at which gravitational P.E. is zero), what matters is the difference in P.E. from one position to another.
I'll try to get back with a couple of graphs of the potentials for this problem which may be helpful.
7. Jan 5, 2012 | 2017-11-23 04:06:54 | {"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": 0.6100324988365173, "perplexity": 532.3643421153212}, "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/1510934806720.32/warc/CC-MAIN-20171123031247-20171123051247-00112.warc.gz"} |