text: Basics of Using the TI-84 Graphing Calculator 2nd Key To access a function written in blue above the calculator keys, you must press to access the function. For anyone that is unsure this is the calculator that is. If you have a chance to buy yourself the TI-84 Plus CE (assuming that your professors prefer that you use any of the 84's), you will gain a few extra features that the Silver Edition and the older TI-84's do not have. The Download. ProfThacker 37,058 TI 83 tutorial combination permutation factorial - Duration: 9:23. Thr calculator is in perfect working condition, it has barely been used ever since it was bought although it does have a few marks and missing rubber feet because it was in a pencil case most of the time. , the 0-factorial or 5-factorial we will also show you how to use the exclamation point in maths, provide information about the n-factorial formula and the applications it can have. The TI-84 Calculator is one of the most versatile graphing calculators that a student may use, and excels in Algebra, Trigonometry, and Calculus with numerous functions to help the student calculate answers quickly and easily. Combinations Calculator. Great to add to. How to use factorial (!) in Excel - posted in Formulas and Functions: Hello, I am trying to use Excel to create a factorial formula, for example !4 = 24 Excel doesnt recognize the exclamation point so Im not sure what the best way to do this is. Factorization, the process of decomposing a number, matrix, or polynomial into a product, is one of the most common mathematical tasks graphing calculator programs are built to handle. Learners examine how to raise a number to a power using the TI-83 Plus calculator. What Power Options Are Available on Scientific Calculators? Most calculators operate using battery power for reliable performance. Experience in our production industry produced Ti-smartview Emulator Software for TI-84 Plus Family, the latest amazing item from Texas Instruments for your home. com is 100% scalable and made for the various resolutions associated with mobile devices and printers, we came up with this idea as a solution new to their TI-83 or TI-84 calculator to learn. Simplify the following:. Then, you can access the ! symbol by either hitting the number 4 or using your down arrow key until you get to the symbol. Here are six fun things you didn't know you could do with your graphing calculator. Bonjour, Il y a des choses à améliorer sur les fonctions CAS de la Prime. image/svg+xml. Unfortunately, the 30X Pro had many bugs and as a result Texas Instruments quickly pulled the calculator off the market. Standard calculators won't have this, but you can calculate small factorials simply by multiplying numbers. There are many ways to write the factorial program in C++ language. Need to know how to use your Texas Instruments graphing calculator for your college math or statistics class? You're in luck watch this video tutorial to see how to use ANOVA with a TI-83 graphing calculator. Enter the number of objects taken at a time. The online factorial calculator has the factorial function which allows to calculate online factorial of an integer. So 4! (pronounced four factorial) is You will learn more about applications of factorials in your textbook, but for now use the TI-83 to calculate 4! The factorial command is located in the MATH menu's PRB sub-menu. 32335097$when I enter$2. 1-20-10-2019-02-release for Android. Best Answer: Well, you really can't do that unless you have a special app on your calculator. Add a Free Factorial Calculator Widget to Your Site!. For anyone that is unsure this is the calculator that is. Jump to navigation Jump to search. TI-84 Plus and TI-83 Plus graphing calculator program for high school and college algebra. Since tistats. 1 Basics:Press the ON key to begin using your TI-83 calculator. We keep a huge amount of good quality reference materials on topics varying from solving systems of equations to algebra i. About Outlier Calculator. The TI 83 is a popular hand held calculator that's most often recommended for stats students. Part I: Texas Instruments TI-83, TI-83 Plus Graphics Calculator I. C/C++, PHP, BASIC, assembly, and much more. (the ! in r! means 'factorial' and is a multiplication of all numbers up to and including r, so 4! =4*3*2*1=24) How do you calculate permutations on the TI-84. It is located under the ALPHA button. Why does this happen? I thought factorials were only defined for positive integers and $0$, so what is my calculator doing to get the answer $3. The two-line display calculator offers a recall button so users can review previous work as well. The fourth root of a number is the number that would have to be multiplied by itself 4 times to get the original number. The TI-83 and TI-84 are the most commonly used graphing calculators in algebra and calculus classes. TI-30Xa Solar Calculator pdf manual download. com online calculator provides basic and advanced mathematical functions useful for school or college. The manual is available as a pdf document at www. Function Key Combination Display. Performing a Linear Regression on the TI-83+ or TI-84+ Changing the Number of Decimals Displayed on the TI BA II Plus and HP 12c; How to Change the Batteries on Your TI-83 Plus, TI-84 Plus or TI-89; Business and Finance Math #2: Calculating the Effective Annual Rate (EAR) on Your TI BA II Plus or HP 12c; Calculators. *Familiar TI-84 Plus functionality * Pre-loaded Apps and Images *3. The calculator will factor any number less than 1 million. Popular Pages. Come to Polymathlove. Type your algebra problem into the text box. When entering numbers, please use only the digits 0 through 9, and avoid decimal points, commas, letters, or other5 symbols. The TI-89 and TI-92 are significantly different, and this guide probably won't be much use for them. TI 83/84 Calculator - The Basics of Statistical Functions What you want to do >>> Put Data in Lists Get Descriptive Statistics Create a histogram, boxplot, scatterplot, etc. TI-84 Plus and TI-83 Plus graphing calculator Program includes 32 algebra functions: Roots of equations, simplifyed radicals, factors, logarithms, prime factorization and more. But I can tell you the factorial of half (½) is half of the square root of pi. Binomial Probability Calculator. Online calculators for everything. Part II: Texas Instruments TI-83, TI-83 Plus, TI-84 Plus Graphics Calculator II. 0 MB FLASH ROM memory for data archive and storage of Apps *15KB of available RAM memory *MathPrint™ functionality *USB port for computer connectivity, unit-to-unit communication with TI-84 Plus, TI-84 Plus Silver Edition, and other TI-84 Plus C Silver Edition graphing calculators. HOW TO USE YOUR CALCULATOR FOR FACTORIALS (!) please consult your TI-84 or TI-83+ graphing calculator manual or the worksheet provided. From factorial worksheet to a quadratic, we have all of it included. From decimal factorial in ti 84 to rational functions, we have every part covered. As an example, if you want the prime factorization calculator to determine how many primes are before the number 11, just type in the number 11, then select and the calculator will promptly create a list of the numbers 2, 3, 5 and 7. For the number of successes x, the calculator will return P(Xx), and P(X≥x). Texas Instruments TI-84 Plus CE Calculator. The fourth root of 104,976 is 18, as 18 x 18 x 18 x 18 is 104,976. Solve algebra questions, calculator picture equations, mixed fraction to decimal, reciprocal of a negative fraction, how to calculate scale factor. Using the factorial function on the TI-84 Plus CE graphing calculator can be useful when calculating permutations or other such statistical calculations. 0 MB FLASH ROM memory for data archive and storage of Apps *15KB of available RAM memory *MathPrint™ functionality *USB port for computer connectivity, unit-to-unit communication with TI-84 Plus, TI-84 Plus Silver Edition, and other TI-84 Plus C Silver Edition graphing calculators. ProfThacker 37,058 TI 83 tutorial combination permutation factorial - Duration: 9:23. To use the five number summary calculator, input a list of numbers in the List: area. For algebraic factoring, use the Expression Factoring Calculator for expressions such as 'x^2 + 4x + 4', and the Equation Factoring Calculator for equations such. Download Graphing calculator ti 84 - simulate for es-991 fx apk 4. The factorial option is simple to utilize. Can you use a graphing calculator as a scientific calculator? How do you do factorial on TI 84? What is the value of E on a calculator? Is a TI 84 plus a CAS calculator? What is M and M+ on a calculator? What do the MS mean on a calculator? What does it mean when there is an E on your calculator?. So 4! (pronounced four factorial) is You will learn more about applications of factorials in your textbook, but for now use the TI-83 to calculate 4! The factorial command is located in the MATH menu's PRB sub-menu. addition to your calculator. Return value : Returns the factorial of desired number. Students usually learn about factorials in introductory algebra courses for their use in probability, but will often forget what they are, as they aren’t used a whole lot until later courses. For example, on the TI-84 Plus, you need to enter the math probability menu by pressing "[math]" followed by the left key twice, and finally press "4" to enter the factorial symbol. Any time you need help on inverse or even dividing rational expressions, Mathmusic. The TI-84 is a much newer version and has more functionality and memory. 送料無料 組立設置付 通気性抜群_ガス圧式大容量跳ね上げベッド No-Mos ノーモス 薄型スタンダードポケットコイルマットレス付き 横開き セミダブル 深さラージ,【送料無料】マイクロファイバー布団カバー！. What if you were asked to find the fourth term in the binomial expansion of (2x+1) 7? Now that is more difficult. The Texas Instruments TI-84 Plus Graphing Calculator features USB on-the-go technology for file sharing with other calculators and connecting to PCs,handling calculus, engineering, trigonometric, and financial functions, 12 apps preloaded, and displays graphs and tables on split screen to trace graph while scrolling through table values. The TI-84 Plus offers 3 times the memory for Handheld Software Applications (Apps), a faster processor for getting the most out of Apps such as Cabri® Jr. Basically, it shows how many different possible subsets can be made from the larger set. The polynomial 0. Select PRB. Because it has C type internal implementation, it is fast. Binomial Coefficient. Combinations (nCr) Permutation (nPr) Number. The main purpose of finding coefficient of variance (often abbreviated as CV) is used to study of quality assurance by measuring the dispersion of the population data of a probability or frequency distribution, or by determining the content or quality of the sample data of substances. Caution: invNorm works from area to left. If you need to have advice on real numbers as well as solving equations, Polymathlove. Right from online ti 84 calculator program to algebraic expressions, we have all kinds of things covered. Examples assume all default settings, as shown in the Modes. Why does this happen? I thought factorials were only defined for positive integers and$0$, so what is my calculator doing to get the answer$3. Description : Calculating the factorial of a number. Calculations on the TI-30XIIS The TI-30XIIS calculator costs less than $20. Help is available for the following types of Binomial Probability Distribution problems:. Wabbit calculator & wabbitemu calculator, grafncalc83, graphncalc83. Actual calculator not included! • (Also suited for TI-83 Plus, TI-84 Plus Silver Edition, TI-84 Plus C and TI-84 Plus CE. Instead of calculating a factorial one digit at a time, use this calculator to calculate the factorial n! of a number n. Graphing calculator ti 84 - simulate for es-991 fx v4. For example, on the TI-84 Plus, you need to enter the math probability menu by pressing “[math]” followed by the left key twice, and finally press “4” to enter the factorial symbol. Engage your community with learning and career services for patrons of all ages. Teaching graphing calculator skills to help students with: • Speed• Mak. What about 20! or 100!? Most calculators including the TI 's series will only calculate factorials up to 69!. If you type in a number less than one and then press Enter, the number will now be automatically. The TI-83, 83+, 84, and 84+ are "graphing. Come to Graph-inequality. Text by Rachel Webb Graphics from TI. You might also like to read the more advanced topic Partial Sums. Half Factorial. Come to Mathisradical. HP 11C Features This model comes with all the features in 10C model but also has hyperbolic and inverse hyperbolic trig functions, probability (combinations and permutations), factorial, absolute value, and % change as well as backspace and a random number generator. Page 1 of 11 Getting Started with your TI-89 for Statistics This is a first draft of these TI-89 basic instructions for statistics. is in the iOS app store, and how that’s changed over time. The step-by-step way of your software really cleared my concepts and now I look forward to solve other types of algebra problems. Factorial 5 = 5*4*3*2*1 = 120 It is expressed as n! = factorial of n To implement it in. 2 comments ti 84 calculator, ti calculator app, How Many Zeros Are There in n factorial;. factorial c | factorial in c | factorial calculator | recursion c factorial | recursive factorial c | factorial c language | factorial calculator using c | c. If you type in 980311^(‐62053) on a TI‐ 83 or TI‐84 the answer given is 0. ! is the factorial function, where n! = n*(n-1)! and 0! = 1, n an nonnegative integer. calculate anything, anytime, anywhere Free online calculators for everything. This enhanced version of the TI-84 Plus is 30% lighter and thinner than earlier models. To define an exponential expression or function with base e, press [ex]. Check this amazing factorial calculator that you were searching for and you will get the complete list of all the prime factors with multiplication signs. Come to Mathmastersnyc. Each menu on the TI-84 Graphing Calculator has its own key or key combination. Features a large 4 line display. Almost every scientific calculator has a built in "factorial" function. Please buy How To Do Permutations On The Ti 89 Calculator album music original if you like the song you choose from the list. Factorial problems on a calculator. The calculator has an array of functions that can help you with your elementary stats or AP stats course. 2nd Functions that need a ) and precede the argument, such as sin, log, and all R³´P menu items. Welcome to the calculatorti. We're going to assume that you already know how to determine whether or not a probability experiment is binomial and instead just focus on how to use the calculator itself. download trigonometry identities for ti-84 examples completing the square worksheet factoring cubed variables gre permutation combination trig calculator algebra equation solver + free using TI-83 graphic calculator parabola pythagoras theorem "root locus" powerpoint adding and subtracting fractions. Alat sederhana dan efisien untuk siswa. Mathmastersnyc. An exponent tells a student how many times the base number is multiplied by itself. Public Libraries. Note that the list inputted does not have to be sorted. Active 3 years, 11 months ago. Graphing calculator ti 84 - simulate for es-991 fx v4. A blog is that is all about mathematics and calculators, two of my passions in life. Solve algebra questions, calculator picture equations, mixed fraction to decimal, reciprocal of a negative fraction, how to calculate scale factor. To define an exponential expression or function with base e, press [ex]. Math Plane - TI nspire cx cas Calculator tutorial 3. I use it in my University of Washington Calculus class, and it is also extremely useful in Algebra II & PreCal. Factorial online. Bonjour, Il y a des choses à améliorer sur les fonctions CAS de la Prime. The Outlier Calculator is used to calculate the outliers of a set of numbers. It supports computing mean, median, harmonic mean, geometric mean, minimum, maximum, range, variance, corrected variance, standard deviation, corrected standard deviation, relative standard deviation, mean deviation, median deviation and skewness. Do you need to calculate the number of ways you can arrange six people at a table or the number of ways you can select four people from a. Texas Instruments TI-84 PLUS Graphic Calculator Description: The TI-84 Plus calculator offers a built-in USB port, an improved display and preloaded applications designed to maximize student performance in numerous subjects including geometry, statistics, algebra, physics, chemistry, biology and others. If you type in 980311^(‐62053) on a TI‐ 83 or TI‐84 the answer given is 0. This document assumes that you are using a browser capable of displaying superscripts. 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I thought factorials were only defined for positive integers and$0$, so what is my calculator doing to get the answer$3. Getting Started. This factorial calculator helps you calculate any factorial operation which is a function that makes the product of all positive integers less than or equal to a given number. To the left is a picture of a TI-83 calculator. The function also works for arguments that are half an odd integer and greater than -1/2: $(-\frac1{2})!$ is defined as $\sqrt{\pi}$ and the rest are defined recursively. This calculator will compute cumulative probabilities for a binomial outcome, given the number of successes, the number of trials, and the probability of a successful outcome occurring. The Best Calculator Apps on iOS. , our interactive Geometry App. Because it has C type internal implementation, it is fast. 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Skip navigation TI-84 Calculator - 28 - Permutations, Combinations, Factorials. The factorial is normally used in Combinations and Permutations (mathematics). This isn't too bad if the binomial is (2x+1) 2 = (2x+1)(2x+1) = 4x 2 + 4x + 1. Once you get to a certain level of math classes, a regular calculator just isn't good enough anymore, you need a high-functioning scientific calculator that can perform algebraic, trigonometric, and statistical problems in addition to general mathematical tasks. Students usually learn about factorials in introductory algebra courses for their use in probability, but will often forget what they are, as they aren’t used a whole lot until later courses. The factorial symbol is the fourth one from the top. Caution: invNorm works from area to left. TI-Basic 84 Programming/Loops. The TI-84 is a much newer version and has more functionality and memory. TI-83+/84 Calculator for A. Head on over to the Android Market from your Android Phone and download a program called Andie Graph. SOURCE: on my TI 83 plus my sine cosine and tangent Alot of people seem to be having this problem. Choose VIEW menu and Enable skin from the pull down menu. Permutations and binations on TI 84 from factorial on ti83 , source:youtube. The Algebrator Software is marvelous. tpdf( is the Student's t probability density function. To protect the integrity of its exams, NCEES limits the types of calculators examinees may bring to exam sites. And yes, this includes factoring polynomials. Discrete Distributions Calculators HomePage. This calculator will generate a complete one-way analysis of variance (ANOVA) table for up to 10 groups, including sums of squares, degrees of freedom, mean squares, and F and p-values, given the mean, standard deviation, and number of subjects in each group. PRO Scientific Calculator. Texas Instruments Calculator TI-34 II - N-0503F. 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How Does My TI-84 Do That A guide to using the TI-84 for statistics Table of Contents Understanding the Instructions in this Guide 2 Basic Operations 3 Clearing the memory 3 Arithmetic operations 3 Negative numbers 4 Absolute value 4 Powers and roots 5 Adjusting the window for graphs 5 Probability and Statistics 7 Generating random integers 7. Once you get to a certain level of math classes, a regular calculator just isn't good enough anymore, you need a high-functioning scientific calculator that can perform algebraic, trigonometric, and statistical problems in addition to general mathematical tasks. If the expression x 2 does not appear as x-squared, then you need to upgrade your browser. fullscientificcalculator - A complete, extremely powerful full scientific calculator — packed into an app. Switching the TI-30XS MultiView calculator on and off & turns on the TI-30XS MultiView calculator. Note that the list inputted does not have to be sorted. 4 =√16/x, algebra solver download, solve x^4-2x^3+7x^2+6x-30. Construct, analyze and transform mathematical. Type your algebra problem into the text box. Head on over to the Android Market from your Android Phone and download a program called Andie Graph. Then, you can access the ! symbol by either hitting the number 4 or using your down arrow key until you get to the symbol. USING A TI-83 OR TI-84 SERIES GRAPHING CALCULATOR IN AN INTRODUCTORY STATISTICS CLASS W. TI-NSPIRE Calculator pdf manual download. every hour of every day across different countries, categories and devices. Can you use a graphing calculator as a scientific calculator? How do you do factorial on TI 84? What is the value of E on a calculator? Is a TI 84 plus a CAS calculator? What is M and M+ on a calculator? What do the MS mean on a calculator? What does it mean when there is an E on your calculator?. For example 0! is a special case factorial. Do you need to calculate the number of ways you can arrange six people at a table or the number of ways you can select four people from a. @JeffMcManus3 Never taught SHM to my AP Phys kids better than I did today, using a mass, spring, and @desmos @RobLiebhart #AlgII solving linear inequalities using @desmos today, while #PreAlg tackles numerical and variable expressions, plus order of operations! @TTcatalano Desmoswhat an amazing. 711224524E98. com Texas Instruments Graphing Calculator questions and answers page and calculator help forum, Help build our members by asking and answering questions on TI-83, TI-83 Plus, TI-84 Plus, TI-89, TI-89 Titanium or any other Texas Instruments graphing calculator model. TI-84 Plus is a versatile, powerful graphing calculator. % ' turns it off. NumWorks has become my go-to scientific calculator, as nearly every operation you want to do for high school math is available with a single key press (I get so tired of hitting the "2nd" key on my TI-84). Come to Graph-inequality. Factorial This code gets factorial of a number and for negative numbers returns 1. Note: Negative numbers are also included, as multiplying two negatives makes a positive. Capital budgeting will help you determine cash flows for given investments for a certain number of years in the future, thus helping you determine if the investment is worthwhile. , a binomial probability), given the number of successes, the number of trials, and the probability of a successful outcome occurring. That means you can’t use the TI-Nspire CX as a TI-eighty four like older Nspire models. 8 fx-cg500 graphing calculator 83 graphing calculator 84 graphing calculator 84 plus ce 21% Off High-resolution, full-color backlit display. Easy Steps To Success: A Graphing Calculator Guide For The TI-84 Plus, TI-83, TI-83 Plus, and TI-82 Graphing Calculators gives step-by-step keystrokes and instructions for these calculators, along with examples using these keystrokes to solve problems. You will then see the ! as one of. Enter your expressions in the same order as you would write them on paper. Asked in Calculators, TI-83 & TI-84 Graphing Calculators What is the difference between a ti-84 calculator then a ti-83 calculator ? Well, ti83s don't have archive memory(ti83 plus does though. 32335097$when I enter$2. It not only gives you the result — it gives you the working out too! Permutations and Combinations in Excel. Performs basic math operations, as well as square root and percentages. Because it has C type internal implementation, it is fast. Symbolab: equation search and math solver - solves algebra, trigonometry and calculus problems step by step. The factorial symbol, !, which is the fourth option in the PRB submenu of the MATH menu, is accessed by pressing. When entering numbers, please use only the digits 0 through 9, and avoid decimal points, commas, letters, or other5 symbols. View and Download Texas Instruments TI-85 manual book online. Note: Negative numbers are also included, as multiplying two negatives makes a positive. Check this amazing factorial calculator that you were searching for and you will get the complete list of all the prime factors with multiplication signs. I especially like this next built in function. Calculate Factorials on the Ti 84 Plus CE Color Graphing Calculator. About Binomial Coefficient Calculator. Standard Deviation Calculator - Find standard deviation, variance and range of a data set. View and Download Texas Instruments TI-30Xa Solar user manual online. com is undoubtedly the ideal site to visit!. Once the equation is solved the results are displayed on the second line and you can use the four arrow keys on the front of the calculator to edit the original equation in any way you like for a recalculation. Make sure that you do the quartiles by hand. Press the blue ‘2ND’ key (top-left) 4. The calculator will ZZZ SG[ HGX WXWRULQJ. The Algebrator Software is marvelous. Note: Negative numbers are also included, as multiplying two negatives makes a positive. Many of these values are more well described in other calculators also available on this website. This enhanced version of the TI-84 Plus is 30% lighter and thinner than earlier models. Your calculator is currently doing everything in terms of radians. As an example, if you want the prime factorization calculator to determine how many primes are before the number 11, just type in the number 11, then select and the calculator will promptly create a list of the numbers 2, 3, 5 and 7. ex) √68 = 8. Math Formulas, Calculators and Work with Steps Free online math calculators and work with steps, converters, formulas, step by step calculations, real world problems, practice problems, times tables and more to make learning K-12 pre-algebra, algebra, geometry, matrix, probability & statistics more interesting. Unfortunately, the 30X Pro had many bugs and as a result Texas Instruments quickly pulled the calculator off the market. Download Full Scientific Calculator APK latest version 1. The special case zero factorial is defined to have value 0!=1, consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects. Solve algebra questions, calculator picture equations, mixed fraction to decimal, reciprocal of a negative fraction, how to calculate scale factor. From factorial worksheet to a quadratic, we have all of it included. I know this is outside of the normal topics discussed on this site but I felt a serious need inform others about hidden functionality on the Texas Instruments TI-89. com makes available simple info on boolean algebra calculator free, denominator and completing the square and other algebra subjects. Enter the values separated by a comma in this coefficient of variation calculator to know the relative standard deviation. The notation n factorial (n!) was introduced by Christian Kramp in 1808.


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## Microscript

I'm designing a Turing complete utility microlanguage that has a minimal set of commands. Chosen commands should on the one side have ability to compile to fast assembler code, and on the other side sould be enough descriptive so one can directly write sane code with them.

The microlanguage would be a part of my bigger project metalanguage that compiles any other languages to the microlanguage which at the end (jit or cached) compiles either to browser-executable (html + javascript), either to native code. Because it wouldn't really be used for high level programming, but only as underneath intermediate language (analogous to some bytecode language), it shouldn't have all heavy weight addons like objects, structs or reactive constructs, but these addons should be able to compile from a guest language to the host microlanguage. It should be also easy to port the microlanguage to platforms like OSX, Windows, Linux or Android.

So I'm thinking about a minimal set of commands. Somehow I think it should be imperative language, because microprocessors work that way at low-level understanding. I was thinking of supporting only these:

• variable assignment and arithmetic
• pointers and arrays like in C
• if branching
• for and while loops with labels
• defining functions with stack support
• assembler passing engine like in C for system programming

Do these commands sound cool or there is something that can be changed for better speed/usability performance?

## A2: Analog Malicious Hardware

Nothing much to do with language theory but too cool not to share.

While the move to smaller transistors has been a boon for performance it has dramatically increased the cost to fabricate chips using those smaller transistors. This forces the vast majority of chip design companies to trust a third party —often overseas—to fabricate their design. To guard against shipping chips with errors (intentional or otherwise) chip design
companies rely on post-fabrication testing. Unfortunately, this type of testing leaves the door open to malicious modifications since attackers can craft attack triggers requiring a sequence of unlikely events, which will never be encountered by even the most diligent tester.

In this paper, we show how a fabrication-time attacker can leverage analog circuits to create a hardware attack that is small (i.e., requires as little as one gate) and stealthy (i.e.,
requires an unlikely trigger sequence before effecting a chip’s functionality). In the open spaces of an already placed and routed design, we construct a circuit that uses capacitors to siphon charge from nearby wires as they transition between digital values. When the capacitors fully charge, they deploy an attack that forces a victim flip-flop to a desired value. We weaponize this attack into a remotely-controllable privilege escalation by attaching the capacitor to a wire controllable and by selecting a victim flip-flop that holds the privilege bit for our processor. We implement this attack in an OR1200 processor and fabricate a chip. Experimental results show that our attacks work, show that our attacks elude activation by a diverse set of benchmarks, and suggest that our attacks evade known defenses

A2: Analog Malicious Hardware. K. Yang, M. Hicks, Q. Dong, T. Austin, D. Sylvester, Department of Electrical Engineering and Computer Science, University of Michigan, USA.

Comment by Yonatan Zunger, Head of Infrastructure for the Google Assistant:

"This is the most demonically clever computer security attack I've seen in years. It's a fabrication-time attack: that is, it's an attack which can be performed by someone who has access to the microchip fabrication facility, and it lets them insert a nearly undetectable backdoor into the chips themselves. (If you're wondering who might want to do such a thing, think "state-level actors")

The attack starts with a chip design which has already been routed -- i.e., it's gone from a high-level design in terms of registers and data, to a low-level design in terms of gates and transistors, all the way to a physical layout of how the wires and silicon will be laid out. But instead of adding a chunk of new circuitry (which would take up space), or modifying existing circuitry significantly (which could be detected), it adds nothing more than a single logic gate in a piece of empty space.

When a wire next to this booby-trap gate flips from off to on, the electromagnetic fields it emits add a little bit of charge to a capacitor inside the gate. If it just happens once, that charge bleeds off, and nothing happens. But if that wire is flipped on and off rapidly, it accumulates in the capacitor until it passes a threshold -- at which point it triggers that gate, which flips a target flip-flop (switch) inside the chip from off to on.

If you pick a wire which normally doesn't flip on and off rapidly, and you target a vulnerable switch -- say, the switch between user and supervisor mode -- then you have a modification to the chip which is too tiny to notice, which is invisible to all known forms of detection, and if you know the correct magic incantation (in software) to flip that wire rapidly, will suddenly give you supervisor-mode access to the chip. (Supervisor mode is the mode the heart of the operating system runs in; in this mode, you have access to all the computer's memory, rather than just to your own application's)

The authors of this paper came up with the idea and built an actual microchip with such a backdoor in it, using the open-source OR1200 chip as their target. I don't know if I want to guess how many three-letter agencies have already had the same idea, or what fraction of chips in the wild already have such a backdoor in them."

## Programming with a Differentiable Forth Interpreter

On arxiv. Abstract:

There are families of neural networks that can learn to compute any function, provided sufficient training data. However, given that in practice training data is scarce for all but a small set of problems, a core question is how to incorporate prior knowledge into a model. Here we consider the case of prior procedural knowledge such as knowing the overall recursive structure of a sequence transduction program or the fact that a program will likely use arithmetic operations on real numbers to solve a task. To this end we present a differentiable interpreter for the programming language Forth. Through a neural implementation of the dual stack machine that underlies Forth, programmers can write program sketches with slots that can be filled with learnable behaviour. As the program interpreter is end-to-end differentiable, we can optimize this behaviour directly through gradient descent techniques on user specified objectives, and also integrate the program into any larger neural computation graph. We show empirically that our interpreter is able to effectively leverage different levels of prior program structure and learn complex transduction tasks such as sequence sorting or addition with substantially less data and better generalisation over problem sizes. In addition, we introduce neural program optimisations based on symbolic computation and parallel branching that lead to significant speed improvements.

## Theory of syntax extensions: does it exist?

I've become interested in the question of syntax extensions: not the general theory of extending syntax, nor the mechanisms for extending syntax (there are many, from the C preprocessor to Racket's extended syntax-case), but some basis for deciding what syntaxes should be provided and how they ought to be organized. Most languages have fixed syntax, either because their creators took it for granted or as a matter of principle. Those that do allow syntax extension usually have cultural constraints against overusing it, and the extended syntaxes provided by the standard library are a rag-bag whose members depend on history or the creator's whims or both. We seem to be at the same state where syntax libraries are concerned that we were in in the 1960s for procedure libraries.

Is there state of the art about this that I haven't heard about yet? Please inform and/or enlighten me.

## For the record...

This google-oracle trial (##googacle) is hysterical. sarah jeong seems to be the person to follow.

Are there any PL experts/LTU-folk as expert witnesses?

## STEPS Toward the Reinvention of Programming, 2012 Final Report

It appears that the final report from the STEPS program has been recently released at long last:

STEPS Toward the Reinvention of Programming, 2012 Final Report

If computing is important -- for daily life, learning, business, national defense, jobs, and more -- then qualitatively advancing computing is extremely important. Fro example, many software systems today are made from millions to hundreds of millions of lines of program code that is too large, complex and fragile to be improved, fixed, or integrated. (One hundred million lines of code at 50 lines per page is 5000 books of 400 pages each! This is beyond humane scale.)

What if this could be made literally 1000 times smaller -- or more? And made more powerful, clear, simple, and robust? This would bring one of the most important technologies of our time from a state that is almost out of human reach -- and dangerously close to being out of control -- back into human scale.

An analogy from daily life is to compare teh great pyramid of Giza, which is mostly solid bricks piled on top of each other with very little usable space inside, to a structure of similar size made from the same materials, but using the later invention of the arch. The result would be mostly usable space, and require roughly 1/1000 the number of bricks. In other words, as size and complexity increase, architectural design dominates materials.

The "STEPS Toward Expressive Programming Systems" project is taking the familiar world of personal computing used by more than a billion people every day -- currently reqiring hundreds of millions of lines of code to make and sustain -- and substantially recreating it using new programming techniques and "architectures" in dramatically smaller amounts of program code. This i made possible by new advances in design, programming, programming languages, systems organization, and the use of science to analyze and create models of software artifacts.

..and there seems to be a new research activity with Alan Kay starting up (HARC which might at least inspired by STEPS (if not used a jumping off point).

## A Farewell to FRP in Elm

Making signals unnecessary with The Elm Architecture

...the big benefit is that Elm is now significantly easier to learn and use. As the design of subscriptions emerged, we saw that all the toughest concepts in Elm (signals, addresses, and ports) could collapse into simpler concepts in this new world. Elm is designed for ease-of-use, so I was delighted to stumble upon a path that would take us farther with fewer concepts. To put this in more alarmist terms, everything related to signals has been replaced with something simpler and nicer.

## Algebra Of Pointers

I have been thinking a little about pointers, and what kind of mathematical structure they form. They seem to be like a one-dimensional vector (as opposed to a simple scalar). If we use the terms location for a point in n-dimensional space, and distance for a vector, then we get the following properties, adding a distance to a location results in a location, subtracting a distance from a location results in a location, subtracting a location from a location results in a distance.

Where is gets more tricky is that adding a location to a location would seem to be invalid (I don't know what type of thing this is), and yet to average two locations, to get the mid-point seems perfectly reasonable. Topologically what is going on here? What is the type of a location added to a location?

If we treat all locations as distances with an origin at zero, then everything makes sense again in terms of types, except for zero itself which seems to still be a location and not a distance (or it all gets horribly self-referential).

Any thoughts on how to make sense of this, what the type of the sum of two locations might be, etc?

## Session Types in a Linearly Typed Multi-Threaded Lambda-Calculus

The paper Session Types in a Linearly Typed Multi-Threaded Lambda-Calculus, Hongwei Xi, Zhiqiang Ren, Hanwen Wu, William Blair, 2016.

We present a formalization of session types in a multi-threaded lambda-calculus (MTLC) equipped with a linear type system, establishing for the MTLC both type preservation and global progress. The latter (global progress) implies that the evaluation of a well-typed program in the MTLC can never reach a deadlock. As this formulated MTLC can be readily embedded into ATS, a full-fledged language with a functional programming core that supports both dependent types (of DML-style) and linear types, we obtain a direct implementation of session types in ATS. In addition, we gain immediate support for a form of dependent session types based on this embedding into ATS. Compared to various existing formalizations of session types, we see the one given in this paper is unique in its closeness to concrete implementation. In particular, we report such an implementation ready for practical use that generates Erlang code from well-typed ATS source (making use of session types), thus taking great advantage of the infrastructural support for distributed computing in Erlang.

## Programming by poking: why MIT stopped teaching SICP

A blog post; excerpt:

In this talk at the NYC Lisp meetup, Gerry Sussman was asked why MIT stopped teaching the legendary 6.001 course, which was based on Sussman and Abelson’s classic text The Structure and Interpretation of Computer Programs (SICP). Sussman’s answer was that: (1) he and Hal Abelson got tired of teaching it (having done it since the 1980s). So in 1997, they walked into the department head’s office and said: “We quit. Figure out what to do.” And more importantly, (2) that they felt that the SICP curriculum no longer prepared engineers for what engineering is like today. Sussman said that in the 80s and 90s, engineers built complex systems by combining simple and well-understood parts. The goal of SICP was to provide the abstraction language for reasoning about such systems.

Today, this is no longer the case. Sussman pointed out that engineers now routinely write code for complicated hardware that they don’t fully understand (and often can’t understand because of trade secrecy.) The same is true at the software level, since programming environments consist of gigantic libraries with enormous functionality. According to Sussman, his students spend most of their time reading manuals for these libraries to figure out how to stitch them together to get a job done. He said that programming today is “More like science. You grab this piece of library and you poke at it. You write programs that poke it and see what it does. And you say, ‘Can I tweak it to do the thing I want?'”. The “analysis-by-synthesis” view of SICP — where you build a larger system out of smaller, simple parts — became irrelevant. Nowadays, we do programming by poking.

Also, see the Hacker news thread. I thought this comment was useful:

What should we consider fundamental?

A fair question, and a full answer would be too long for a comment (though it would fit in a blog post, which I'll go ahead and write now since this seems to be an issue). But I'll take a whack at the TL;DR version here.

AI, ML, and NLP and web design are application areas, not fundamentals. (You didn't list computer graphics, computer vision, robotics, embedded systems -- all applications, not fundamentals.) You can cover all the set theory and graph theory you need in a day. Most people get this in high school. The stuff you need is just not that complicated. You can safely skip category theory. What you do need is some amount of time spent on the idea that computer programs are mathematical objects which can be reasoned about mathematically. This is the part that the vast majority of people are missing nowadays, and it can be a little tricky to wrap your brain around at first. You need to understand what a fixed point is and why it matters. You need automata theory, but again, the basics are really not that complicated. You need to know about Turing-completeness, and that in addition to Turing machines there are PDAs and FSAs. You need to know that TMs can do things that PDAs can't (like parse context-sensitive grammars), and that PDAs can to things that FSAs can't (like parse context-free grammars) and that FSAs can parse regular expressions, and that's all they can do.

You need some programming language theory. You need to know what a binding is, and that there are two types of bindings that matter: lexical and dynamic. You need to know what an environment is (a mapping between names and bindings) and how environments are chained. You need to know how evaluation and compilation are related, and the role that environments play in both processes. You need to know the difference between static and dynamic typing. You need to know how to compile a high-level language down to an RTL. For operating systems, you need to know what a process is, what a thread is, some of the ways in which parallel processes lead to problems, and some of the mechanisms for dealing with those problems, including the fact that some of those mechanisms require hardware support (e.g. atomic test-and-set instructions). You need a few basic data structures. Mainly what you need is to understand that what data structures are really all about is making associative maps that are more efficient for certain operations under certain circumstances.

You need a little bit of database knowledge. Again, what you really need to know is that what databases are really all about is dealing with the architectural differences between RAM and non-volatile storage, and that a lot of these are going away now that these architectural differences are starting to disappear. That's really about it.

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# Show that the two solutions of the SDE are equivalent

I have a process: $$dr_t = (W_t^1 - ar_t)dt +\sigma dW_t^2$$ where $W_t^1$ and $W_t^2$ are brownian motions with instantaneous correlation coefficient $\rho$. I want to show that the solution of this process can be written in a form: $$r_t = r_0 e^{-at}+ \int_0^tk(t,s)dW_s^1 + \sigma\int_0^th(t,s)dW_s^2$$. I started with the substitution $\tilde{r}_t = e^{at}r_t$ which led me to the solution. $$r_t = r_0 e^{-at}+ \int_0^t e^{a(s-t)}W_s^1ds + \sigma\int_0^te^{a(s-t)}dW_s^2$$ which differs from the asked solution in the first integrand ($ds$ instead of $dW_s^1$). I thought I could use Fubini theorem but I am not quite sure how. So essentially what I would like to know is how to show that these two solutions are equivalent. Thanks for any suggestions.

• I would try the following: Consider $h(t) W_t$ where $h$ is a function of time only. Then using Ito, this can be written in integral form as $\int_0^t h'(s) W_s \mathrm{d}s + \int_0^t h(s) \mathrm{d}W_s$. So $h'(s) = e^{a (s - t)}$ in your case. Then $h(s) = e^{a(s - t)} / a$ and $\int_0^t e^{a (s - t)} W_s \mathrm{d}s = \int_0^t \left( 1 - e^{a (s - t)} / a \right) \mathrm{d}W_s$. Warning: typed on my phone. – LocalVolatility Mar 11 '18 at 19:32
• You lost a $d$ in front of $r_t$ in your first equation. @LocalVolatility's has given you the right answer, so I am not going to repeat it. – Hans Mar 11 '18 at 20:11
• @LocalVolatility: Would you like to check out my question regarding SABR model solution derivation? quant.stackexchange.com/q/38719/6686 – Hans Mar 11 '18 at 20:12
• excellent solution LocalVolatility..cheers – Michael Mark Mar 11 '18 at 23:20
• @LocalVolatility you could write an answer out of your comment if you have time. – Daneel Olivaw Mar 12 '18 at 12:17

This is a slightly extended and corrected (thanks @DaneelOlivaw) version of my comment.

Consider a process $h(t) W_t$, where $h$ is a function of time only. Using the Ito product rule, this can be expressed in integral form as

$$h(t) W_t = \int_0^t h'(s) W_s \mathrm{d}s + \int_0^t h(s) \mathrm{d}W_s,$$

where we used that $h(0) W_0 = 0$. Matching terms we find that

$$h'(s) = e^{a (s - t)}$$

and thus

$$h(s) = \frac{1}{a} e^{a (s - t)}.$$

Rearranging yields

\begin{eqnarray} \int_0^t e^{a (s - t)} W_s \mathrm{d}s & = & \frac{1}{a} W_t - \frac{1}{a} \int_0^t e^{a (s - t)} \mathrm{d}W_s\\ & = & \frac{1}{a} \int_0^t \left( 1 - e^{a (s - t)} \right) \mathrm{d}W_s. \end{eqnarray}

"I thought I could use Fubini theorem but I am not quite sure how."

Note that we can write:

\begin{align} \int_0^t e^{a(s-t)}W^1_s\text{d}s & = \int_0^t e^{a(s-t)}\left(\int_0^s\text{d}W^1_u\right)\text{d}s \\[6pt] & = \int_0^t \int_0^se^{a(s-t)}\text{d}W^1_u\text{d}s \end{align}

In the Appendix of Heath et al. (1992) you can find a nice statement of the stochastic Fubini theorem, which they themselves take from Ikeda and Watanabe (1981).

Let $s \leq T$ with $T \in \mathbb{R}^+$, we then define the continuous function $\phi(s)=e^{a(s-t)}$ for $s \in [0;T]$ $-$ note that in our case the function only depends on one variable thus for $u \in [0;T]$ we have $\phi(s,u)\triangleq\phi(s)$ which makes things easier. The function $\phi(s)$ is well-defined for any value of $t \in \mathbb{R}$. Therefore:

1. The function $\phi(s)$ is deterministic thus mesurable w.r.t. the filtration generated by $W^1$;
2. $\phi(s)$ is clearly square-integrable over $[0;T]$;
3. For any $\tau \in [0;T]$: $\int_0^T (\int_0^{\tau}\phi(s)\text{d}W^1_u)\text{d}s=\int_0^T \phi(s)(\int_0^{\tau}\text{d}W^1_u)\text{d}s=W_{\tau}^1\int_0^T \phi(s)\text{d}s$ $=cW_{\tau}^1$ for some $c \in \mathbb{R}$, which is continuous in $\tau$ by property of Brownian Motion.

The hypothesis of Lemma 0.1 of Heath et al. are fulfilled (p.98), thus applying Corollary 2 (p.99):

\begin{align} \int_0^t e^{a(s-t)}W^1_s\text{d}s & = \int_0^t \int_0^se^{a(s-t)}\text{d}W^1_u\text{d}s \\[6pt] & = \int_0^t \left(\int_u^te^{a(s-t)}\text{d}s\right)\text{d}W^1_u \\[6pt] & = \int_0^t \left(\frac{1-e^{a(u-t)}}{a}\right)\text{d}W^1_u \end{align}

@LocalVolatility's comment gives an alternative derivation avoiding Fubini, although I believe the $1/a$ must be outside the parenthesis.

• I am still unclear how to finish from here. Could you please explain how to justify switching the order of integration? Thanks – Michael Mark Mar 12 '18 at 20:55
• @MichaelMark I have added some additional details. – Daneel Olivaw Mar 13 '18 at 13:39


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# Is higher-order unification decidable for terms without abstractions within applications?

Consider the problem of higher order unification - that is, finding a substitution for the equation a = b, where a and b are open terms on the lambda calculus, such that a and b have the same alpha-beta normal form. That problem is undecidable in general. Now, consider the case where:

1. b is a closed term on the shape (λ t . (t Y0 Y1 Y2 ...)).
2. a is on the shape (λ t . (t (F X0) (F X1) (F X2) ... )), where F is the only variable.
3. Xn and Yn have no lambdas within applications.

For example, a valid instance of this problem is:

a = (λ t .
(F (λ a b c d . (a (c d) (a (c (c d)) (a (c (c (c d))) b)))))
(F (λ a b c d . (a (c (c d)) (a (c d) (a (c (c (c (c d)))) b)))))
(F (λ a b c d . (a (c (c d)) (a (c (c d)) b)))))

b = (λ t .
(λ a b . (a (a (a (a (a (a b)))))))
(λ a b . (a (a (a (a (a (a (a b))))))))
(λ a b . (a (a (a (a b))))))

The solution is the substitution:

F = (λ l f . (l (λ h t e . (h (t e))) (λ n . n) (λ p e . (f (p e))) (λ z . z)))

Is this problem decidable?

• I'm not sure this is an answer but If you eta-expand each instance of F into x -> F x then a and b are both higher order patterns. Higher order patterns not only have decidable unification but if a unifier exists there is a most general unifier. I'm not sure if the eta expansion will throw off beta equivalence however but my bet is that it will not. So not quite an answer but perhaps progress. – Jake Sep 29 '15 at 23:43
• It is a great progress, but I thought it was restricted to the simply typed lambda calculus? (Although that case probably has a simple type, now that I think about it.) – MaiaVictor Sep 30 '15 at 0:05
• Well I didn't think it had to be simply typed but the handbook of automated reasoning seems to assume simply typed lambda calculus for its terms. So perhaps you are right. Hopefully your specific case can fit into a simply typed setting. – Jake Sep 30 '15 at 0:50


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# zbMATH — the first resource for mathematics

On compactly supported spline wavelets and a duality principle. (English) Zbl 0759.41008
The authors consider a multiresolution analysis of $$L^ 2(\mathbb{R})$$ generated by the $$m$$-th order $$B$$-spline $$N_ m(x)$$. Then they describe the wavelet subspaces $$W_ k$$ and construct a basic spline-wavelet with compact support that generates $$W_ k$$. By Fourier transform techniques one can handle the corresponding pair of two-scale relations and a decomposition formula. Here the main idea is a duality principle which states that the pair of two-scale relations can be used as the decomposition formula, and vice versa. This approach yields a very desirable decomposition of every $$L^ 2$$-function into a direct sum of compactly supported spline wavelets.

##### MSC:
 41A15 Spline approximation 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 41A05 Interpolation in approximation theory 41A30 Approximation by other special function classes
Full Text:
##### References:
 [1] Charles K. Chui, Multivariate splines, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. With an appendix by Harvey Diamond. · Zbl 0687.41018 [2] Charles K. Chui and Jian-zhong Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113 (1991), no. 3, 785 – 793. · Zbl 0755.41008 [3] Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909 – 996. · Zbl 0644.42026 · doi:10.1002/cpa.3160410705 · doi.org [4] Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases of \?²(\?), Trans. Amer. Math. Soc. 315 (1989), no. 1, 69 – 87. · Zbl 0686.42018 [5] I. J. Schoenberg, Cardinal spline interpolation, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12. · Zbl 0264.41003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Proving Unrealizability for Syntax-Guided Synthesis

Proving Unrealizability for Syntax-Guided Synthesis We consider the problem of automatically establishing that a given syntax-guided-synthesis (SyGuS) problem is unrealizable (i.e., has no solution). Existing techniques have quite limited ability to establish unrealizability for general SyGuS instances in which the grammar describing the search space contains infinitely many programs. By encoding the synthesis problem's grammar G as a nondeterministic program P_G, we reduce the unrealizability problem to a reachability problem such that, if a standard program-analysis tool can establish that a certain assertion in P_G always holds, then the synthesis problem is unrealizable. Our method can be used to augment any existing SyGus tool so that it can establish that a successfully synthesized program q is optimal with respect to some syntactic cost -- e.g., q has the fewest possible if-then-else operators. Using known techniques, grammar G can be automatically transformed to generate exactly all programs with lower cost than q -- e.g., fewer conditional expressions. Our algorithm can then be applied to show that the resulting synthesis problem is unrealizable. We implemented the proposed technique in a tool called NOPE. NOPE can prove unrealizability for 59/134 variants of existing linear-integer-arithmetic SyGus benchmarks, whereas all existing SyGus solvers lack the ability to prove that these benchmarks are unrealizable, and time out on them.

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## Authors

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• ### Exact and Approximate Methods for Proving Unrealizability of Syntax-Guided Synthesis Problems

We consider the problem of automatically establishing that a given synta...
04/02/2020 ∙ by Qinheping Hu, et al. ∙ 0

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• ### Decidable Synthesis of Programs with Uninterpreted Functions

We identify a decidable synthesis problem for a class of programs of unb...
10/22/2019 ∙ by Paul Krogmeier, et al. ∙ 0

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• ### Semantics-Guided Synthesis

This paper develops a new framework for program synthesis, called semant...
08/22/2020 ∙ by Jinwoo Kim, et al. ∙ 0

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• ### Gradient Descent over Metagrammars for Syntax-Guided Synthesis

The performance of a syntax-guided synthesis algorithm is highly depende...
07/13/2020 ∙ by Nicolas Chan, et al. ∙ 0

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• ### SynRG: Syntax Guided Synthesis of Expressions with Alternating Quantifiers

Program synthesis is the task of automatically generating expressions th...
07/20/2020 ∙ by Elizabeth Polgreen, et al. ∙ 0

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• ### Just-in-Time Learning for Bottom-Up Enumerative Synthesis

A key challenge in program synthesis is the astronomical size of the sea...
10/16/2020 ∙ by Shraddha Barke, et al. ∙ 0

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• ### SyGuS-Comp 2018: Results and Analysis

Syntax-guided synthesis (SyGuS) is the computational problem of finding ...
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## 1 Introduction

The goal of program synthesis is to find a program in some search space that meets a specification—e.g., satisfies a set of examples or a logical formula. Recently, a large family of synthesis problems has been unified into a framework called syntax-guided synthesis (SyGuS). A SyGuS problem is specified by a regular-tree grammar that describes the search space of programs, and a logical formula that constitutes the behavioral specification. Many synthesizers now support a specific format for SyGuS problems [1], and compete in annual synthesis competitions [2]. Thanks to these competitions, these solvers are now quite mature and are finding a wealth of applications [9].

Consider the SyGuS problem to synthesize a function that computes the maximum of two variables and , denoted by . The goal is to create —an expression-tree for —where is in the language of the following regular-tree grammar :

 Start ::=  % Plus(Start,Start)∣IfThenElse(BExpr,Start,Start)∣x∣y∣0∣1BExpr ::=  GreaterThan(Start,Start)∣Not(BExpr)∣And(BExpr,BExpr% )

and is valid, where denotes the meaning of , and

 ψmax2(f,x,y):=f(x,y)≥x∧f(x,y)≥y∧(f(x,y)=x∨f(x,y)=y).

SyGuS solvers can easily find a solution, such as

 e:=IfThenElse(GreaterThan(x,y),x,y).

Although many solvers can now find solutions efficiently to many SyGuS problems, there has been effectively no work on the much harder task of proving that a given SyGuS problem is unrealizable—i.e., it does not admit a solution. For example, consider the SyGuS problem , where is the more restricted grammar with if-then-else operators and conditions stripped out:

 Start ::=  % Plus(Start,Start)∣x∣y∣0∣1

This SyGuS problem does not have a solution, because no expression generated by meets the specification.111Grammar only generates terms that are equivalent to some linear function of and ; however, the maximum function cannot be described by a linear function. However, to the best of our knowledge, current SyGuS solvers cannot prove that such a SyGuS problem is unrealizable.222The synthesis problem presented above is one that is generated by a recent tool called QSyGuS, which extends SyGuS with quantitative syntactic objectives [10]. The advantage of using quantitative objectives in synthesis is that they can be used to produce higher-quality solutions—e.g., smaller, more readable, more efficient, etc. The synthesis problem arises from a QSyGuS problem in which the goal is to produce an expression that (i) satisfies the specification , and (ii) uses the smallest possible number of if-then-else operators. Existing SyGuS solvers can easily produce a solution that uses one if-then-else operator, but cannot prove that no better solution exists—i.e., is unrealizable.

A key property of the previous example is that the grammar is infinite. When such a SyGuS problem is realizable, any search technique that systematically explores the infinite search space of possible programs will eventually identify a solution to the synthesis problem. In contrast, proving that a problem is unrealizable requires showing that every program in the infinite search space fails to satisfy the specification. This problem is in general undecidable [6]. Although we cannot hope to have an algorithm for establishing unrealizability, the challenge is to find a technique that succeeds for the kinds of problems encountered in practice. Existing synthesizers can detect the absence of a solution in certain cases (e.g., because the grammar is finite, or is infinite but only generate a finite number of functionally distinct programs). However, in practice, as our experiments show, this ability is limited—no existing solver was able to show unrealizability for any of the examples considered in this paper.

In this paper, we present a technique for proving that a possibly infinite SyGuS problem is unrealizable. Our technique builds on two ideas.

1. We observe that unrealizability can often be proven using finitely many input examples. In IllustrativeExample, we show how the example discussed above can be proven to be unrealizable using four input examples—, , , and .

2. We devise a way to encode a SyGuS problem over a finite set of examples as a reachability problem in a recursive program . In particular, the program that we construct has an assertion that holds if and only the given SyGuS problem is unrealizable. Consequently, unrealizability can be proven by establishing that the assertion always holds. This property can often be established by a conventional program-analysis tool.

The encoding mentioned in EncodingOfGrammarAndExamples is non-trivial for three reasons. The following list explains each issue, and sketches how they are addressed

1) Infinitely many terms. We need to model the infinitely many terms generated by the grammar of a given synthesis problem .

To address this issue, we use non-determinism and recursion, and give an encoding such that (i) each non-deterministic path in the program corresponds to a possible expression that can generate, and (ii) for each expression that can generate, there is a path in . (There is an isomorphism between paths and the expression-trees of )

2) Nondeterminism. We need the computation performed along each path in to mimic the execution of expression . Because the program uses non-determinism, we need to make sure that, for a given path in the program , computational steps are carried out that mimic the evaluation of for each of the finitely many example inputs in .

We address this issue by threading the expression-evaluation computations associated with each example in through the same non-deterministic choices.

3) Complex Specifications. We need to handle specifications that allow for nested calls of the programs being synthesized.

For instance, consider the specification . To handle this specification, we introduce a new variable and rewrite the specification as . Because is now also used as an input to , we will thread both the computations of and through the non-deterministic recursive program.

Our work makes the following contributions:

• We reduce the SyGuS unrealizability problem to a reachability problem to which standard program-analysis tools can be applied (IllustrativeExample and verification).

• We observe that, for many SyGuS problems, unrealizability can be proven using finitely many input examples, and use this idea to apply the Counter-Example-Guided Inductive Synthesis (CEGIS) algorithm to the problem of proving unrealizability (cegis).

• We give an encoding of a SyGuS problem over a finite set of examples as a reachability problem in a nondeterministic recursive program , which has the following property: if a certain assertion in always holds, then the synthesis problem is unrealizable (verification).

• We implement our technique in a tool nope using the ESolver synthesizer [2] as the SyGuS solver and the SeaHorn tool [8] for checking reachability. nope is able to establish unrealizability for 59 out of 132 variants of benchmarks taken from the SyGuS competition. In particular, nope solves all benchmarks with no more than 15 productions in the grammar and requiring no more than 9 input examples for proving unrealizability. Existing SyGuS solvers lack the ability to prove that these benchmarks are unrealizable, and time out on them.

RelatedWork discusses related work. Some additional technical material, proofs, and full experimental results are given in AdditionalMaterialproofssupp-eval, respectively.

## 2 Illustrative Example

In this section, we illustrate the main components of our framework for establishing the unrealizability of a SyGuS problem.

Consider the SyGuS problem to synthesize a function that computes the maximum of two variables and , denoted by . The goal is to create —an expression-tree for —where is in the language of the following regular-tree grammar :

 Start ::=  % Plus(Start,Start)∣IfThenElse(BExpr,Start,Start)∣x∣y∣0∣1BExpr ::=  GreaterThan(Start,Start)∣Not(BExpr)∣And(BExpr,BExpr% )

and is valid, where denotes the meaning of , and

 ψmax2(f,x,y):=f(x,y)≥x∧f(x,y)≥y∧(f(x,y)=x∨f(x,y)=y).

SyGuS solvers can easily find a solution, such as

 e:=IfThenElse(GreaterThan(x,y),x,y).

Although many solvers can now find solutions efficiently to many SyGuS problems, there has been effectively no work on the much harder task of proving that a given SyGuS problem is unrealizable—i.e., it does not admit a solution. For example, consider the SyGuS problem , where is the more restricted grammar with if-then-else operators and conditions stripped out:

 Start ::=  % Plus(Start,Start)∣x∣y∣0∣1

This SyGuS problem does not have a solution, because no expression generated by meets the specification.333Grammar generates all linear functions of and , and hence generates an infinite number of functionally distinct programs; however, the maximum function cannot be described by a linear function. However, to the best of our knowledge, current SyGuS solvers cannot prove that such a SyGuS problem is unrealizable. As an example, we use the problem discussed in Introduction, and show how unrealizability can be proven using four input examples: , , , and .


text:
Still not getting getting correct solution

1. Sep 20, 2005

gmuniz

Two protons are aimed directly toward each other by a cyclotron accelerator with speeds of 1650 km/s,measured relative to the earth. find the maximum force that these protons will exert on each other.
KE_i + U_i = KE_f + U_f ;where U_i=0 and KE_f=0,so KE_i=U_f
if r= (q^2/(4*pi*epsilon_0))(1/KE_i) then F=(1/(4*pi*epsilon_0))(q^2/r^2) but the solution comes out as incorrect.

Last edited: Sep 20, 2005
2. Sep 20, 2005

Fermat

Can you explain what this eqn is about - where does it come from?
What is the question ?
What is KE_i ?
What is F ?


text:
# I Electron bands in solids

#### cube137

With regards to the number of states available to the electrons in these solids.

1. a mass of polycrystalline iron (plenty of free electrons)
2. quartz (has practically no free electrons).
3. semiconductors

Which of them has the most number of states available to the electrons?
Which has the most ease of getting them to change states considered against the ease with which they change states by themselves?

Related Atomic and Condensed Matter News on Phys.org

#### cube137

To rephrase it. An electron in isolation has discrete value. When it is part of solid, The formerly discrete nature of allowed energies for an electron is turned into thick energy bands, within which electrons can occupy any energy they desire (well, almost). In this sense, you have many states available to the electrons.

Which of these... iron, quartz or semiconductors have most states available to the electrons?

What other solid configuration has the most states available to the electrons?

#### Henryk

Gold Member
This is an ill-posed question. If you look at hydrogen atoms, the allowed bound states have energies equal to $E_n = \frac {Ry} {n^2}$ and n is the main quantum number ranging from 1 to ... infinity. In other words, there is an infinite number of available electronic states in a single hydrogen atom. It is true for any atom as well. Now, when atoms form a solid, each of these atomic state is split into a band and within each band, there are as many allowed states as there are atoms in the entire solid (ok, strictly speaking, each band has as many states as there are primitive unit cells within the entire solid but number of bands gets multiplied by number of atoms within a primitive unit cell).
So, you have an infinite number of states multiplied by a very large number of atoms in a piece of solid.
Which infinity is greater?

#### ZapperZ

Staff Emeritus
2018 Award
To rephrase it. An electron in isolation has discrete value.
This is false. If you mean an electron "in isolation" as in free electron (i.e. not encumbered by any external potential), then it does not have "discrete values" in terms of the allowed energies and momentum. Try it. Solve the Schrodinger equation for a free electron.

When it is part of solid, The formerly discrete nature of allowed energies for an electron is turned into thick energy bands, within which electrons can occupy any energy they desire (well, almost). In this sense, you have many states available to the electrons.

Which of these... iron, quartz or semiconductors have most states available to the electrons?

What other solid configuration has the most states available to the electrons?
As has been stated, this is an ill-posed question. If you are talking about the density of states, then that's a different issues, because you will have to indicate the relevant bands and relevant energy and momentum range. Asking for a total number of states does not make sense.

And not only that, even for a particular solid, such as iron, changing its crystal structure can change its density of states. So just specifying an element alone, or a semiconductor alone, isn't sufficient.

Zz.

"Electron bands in solids"

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text:
location:  Publications → journals → CMB
Abstract view

# Real-Analytic Negligibility of Points and Subspaces in Banach Spaces, with Applications

We prove that every infinite-dimensional Banach space $X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an infinite-dimensional Banach space and $F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the $n$-torus on certain Banach spaces.


text:
How do you set up the domain name (I'm making use of GoDaddy) to mask the web server URL yet to add the below - web page link.

I'm assuming something like the Wikipedia en.wikipedia.org/wiki/something (or, if it would certainly call for httpd.conf accessibility, establishing it to add the default subpage link eg. ?page_id=2)

Currently I can set up the domain name to either be covered up entirely without revealing any kind of below - web page web links or to merely reroute my domain name to my internet server.

0
2019-12-02 02:58:26
Source Share

If I read appropriately you desire web links to sub.domain.com/path to be rerouted to domain.com/path and also not simply domain.com (might have the naked/sub domains the upside-down around yet it coincides concept).

Usually this is called a loved one redirect and also can be performed with an Apache regulation like :

RewriteEngine on
RewriteRule ^http://sub\.domain\.com(/.*)$http://domain.com$1 [L,R]

You do not require the L alternative, it is simply quits handling more regulations in instance another thing problems, yet you require the R (or R = 302) to reroute to an additional domain name. You might additionally desire the redirects to be cacheable by establishing cross country expiration days, etc

If you desire the subdomain component to be the course you can make use of :

RewriteEngine on
RewriteRule ^http://(.*)\.domain\.com(/.*)$http://domain.com/$1\$2 [L,R]

0
2019-12-03 00:27:51
Source


text:
# Misspecified linear model, interpretation of slope

#### fed2

##### Active Member
Suppose you fit a simple linear regression when the data actual follow some sort of sigmoid, 4 parm logisitc or similar.

To what extent is the following statement true/false:
"The estimated slope of the simple linear regression is the average slope of the sigmoid"

Seems intuitively true doesn't it?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
My brain can't even process it. It feels latent and cloaked issues, but maybe not. I picture the sigmoid shape and a single line going through it. So what does "average slope of sigmoid" even mean, if you weighted piecewise segments and created a single number? Will if linear regression can be used to fit binary data, I suppose it could be true, maybe.

#### fed2

##### Active Member
I guess the question is pretty vague. I got to thinking about it though and I realize that if you have a nonlinear function g(x) and consider the expectation of the usual linear regression slope betahat = Sxy/Sxx and you replace Y with its expected value g(x), and approximate g(x) with first order taylor, then you end up with something like
E(betahat) = slope of g at mean X + bias term

where the bias term is the slope of the linear regression relating the Remainder term in the taylor approx to X.

I swear i've seen this sort of theory before somewhere, but it is hard to google up again.

Maybe some R code explains better

C-like:
#logistic fun
g = function(x){
1/( 1 + exp(-x) )
}
#first derivative
g_dot = function(x){
g(x)*(1 - g(x) )
}

#first order approximation
tangent = function(x,a){
g(a) + g_dot(a)*(x - a)
}

#fit slope to logistic experiment;
runSim <- function(j){
X <- runif(10,-1,1)
gX = g(X) + rnorm(10,0,.1)
betaHat = coef( lm(gX ~ X) )[[2]];  #Fit linear regression to logistic;
#the bias is slope of SLR relating errs to X;
errs =  tangent(X,0) - gX;
bias = coef( lm(errs ~ X) )[[2]];
data.frame( betaHat=betaHat, bias = bias)
}

mySims =  do.call( 'rbind', lapply(1:1000, runSim)  )

print(  sprintf( 'expected slope at mean X = %f', g_dot(0) )  )

print(   sprintf( 'slope of slr %f', mean(mySims$betaHat ) ) ) print( sprintf( 'predicted bias %f', mean(mySims$bias  )  )  )

#so fitting linear reg gives g_dot(0) + a bias related
#to the erro of first order taylor approx?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Yes, i also wondered how this may play out in a sim. I look forward to reviewing your code when iget back to the office on Monday.


text:
### Theory:

Let us see the procedure to solve the quadratic equation by the method of completing the square.
Step 1: Write the given equation in standard form $$ax^2 + bx + c = 0$$.

Step 2: Make sure the coefficient of $$x^2$$ is $$a = 1$$. If not, make it by dividing the equation by $$a$$.

Step 3: Add and subtract the square of one half of the coefficient of $$x$$ on the left side. [That is, add and subtract $$\left(\frac{b}{2}\right)^2$$.]

Step 4: Make the equation a complete square by moving some constants to the right side.

Step 5: Solve for $$x$$ by taking square root on both sides.
Example:
Find the root of $$2x^2 + 7x - 15 = 0$$ by the method of completing the square.

Solution:

The given equation is $$2x^2 + 7x - 15 = 0$$.

Here, the coefficient of $$x^2$$ is $$2$$. So, divide the equation by $$2$$.

$\frac{2}{2}{x}^{2}+\frac{7}{2}x-\frac{15}{2}=\frac{0}{2}$

${x}^{2}+\frac{7}{2}x-\frac{15}{2}=0$

Add and subtract the square of one half of the coefficient of $$x$$ on the left side.

${x}^{2}+\frac{7}{2}x-\frac{15}{2}+{\left(\frac{7}{4}\right)}^{2}-{\left(\frac{7}{4}\right)}^{2}=0$

${x}^{2}+\frac{7}{2}x+{\left(\frac{7}{4}\right)}^{2}=\frac{15}{2}+{\left(\frac{7}{4}\right)}^{2}$

${x}^{2}+2×\frac{7}{4}x+{\left(\frac{7}{4}\right)}^{2}=\frac{15}{2}+\frac{49}{16}$

Left hand side equation reminds the identity $$(a + b)^2 = a^2 + 2ab + b^2$$.

${\left(x+\frac{7}{4}\right)}^{2}=\frac{120+49}{16}$

${\left(x+\frac{7}{4}\right)}^{2}=\frac{169}{16}$

Taking square root on both sides.

$x+\frac{7}{4}=±\frac{13}{4}$

$x=\frac{13}{4}-\frac{7}{4}$ or $x=-\frac{13}{4}-\frac{7}{4}$

$x=\frac{6}{4}$ or $x=\frac{-20}{4}$

$$x =$$ $\frac{3}{2}$ or $$x = -5$$

Therefore, the roots of the given equation are $\frac{3}{2}$ and $$-5$$.

2. Find the roots of the equation $$x^2 + x + 2 = 0$$ by completing the square method.

Solution:

The given equation is $$x^2 + x + 2 = 0$$

Here, the coefficient of $$x^2$$ is $$1$$.

$$x^2 + x + 2 = 0$$

Add and subtract the square of one half of coefficient of $$x$$ on the left side.

${x}^{2}+x+2+{\left(\frac{1}{2}\right)}^{2}-{\left(\frac{1}{2}\right)}^{2}=0$

${x}^{2}+2×\frac{1}{2}x+{\left(\frac{1}{2}\right)}^{2}=-2+\frac{1}{4}$

${\left(x+\frac{1}{2}\right)}^{2}=-\frac{7}{4}$ $$< 0$$

The above equation cannot be possible because the square of any number cannot be negative.

Therefore, the given equation has no real roots.


text:
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## Ideas for a 90 Point Score Goal

This topic has 1 expert reply and 0 member replies

### How long until I break the 700 score barrier? Vote here comment below

1 month

2 months

3 months

Joelski700 Newbie | Next Rank: 10 Posts
Joined
22 Aug 2017
Posted:
1 messages

#### Ideas for a 90 Point Score Goal

Tue Aug 22, 2017 9:25 am
I just took the GMAT and have my Enhanced Score Report (ESR) with me. I think I'm fine for verbal, I'm in the 83rd percentile, but my quant score is awful, 31st percentile. I'm trying to get a 700+ as fast as possible through test prep services. Right now I'm looking at Magoosh and Veritas courses since I already studied on my own through some old Manhattan Prep books. I would love input from those experienced with this process as to how long it will take before I can break the 700 barrier. I have some time before I apply second round, but not too long. I cancelled the score at the test center, so I don't think I can get my essay score, but I have any and all other relevant information that you might need to predict how long it would take for me to get to 700+

### GMAT/MBA Expert

Rich.C@EMPOWERgmat.com Elite Legendary Member
Joined
23 Jun 2013
Posted:
9433 messages
Followed by:
480 members
2867
GMAT Score:
800
Thu Aug 31, 2017 9:18 pm
Hi Joelski700,

If you're willing to share your full ESR, then I'll be happy to analyze it for you. You can feel free to PM me the Report or email it directly to me. Before I can offer you the specific advice that youâ€™re looking for, it would help if you could provide a bit more information on how you've been studying and your goals:

1) How long have you studied?
2) What study materials have you used so far?
3) How have you scored on each of your CATs (including the Quant and Verbal Scaled Scores for each)?
4) How did you score on the Official GMAT (including the Quant and Verbal Scaled Scores)?
5) What Schools are you planning to apply to?

GMAT assassins aren't born, they're made,
Rich

_________________
Contact Rich at Rich.C@empowergmat.com

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text:
# What's the condition for a matrix $A$ ($2N\times 2N$ dimension) to have eigenvalues in pairs $\pm\lambda$?

For a given even dimension square complex matrix $A$ ($2N\times 2N$ dimension),
what's the sufficient and necessary condition for the matrix $A$ such that:
if $\lambda_{1}$ is an eigenvalue, then $\lambda_{2}=-\lambda_{1}$ is also an eigenvalue?

I have found one sufficient condition: skew-symmetric matrix, but obviously there is a lot of matrices having eigenvalues in pairs are not skew-symmetric.

So I wonder is there a sufficient and necessary condition?

Or is there a name for this family of matrices (skew-symmetric matrix is a member)?

• One more sufficient condition is raised in another question, and it's proved by @egreg . The matrix $A$ in the form of $\{\{B,C\},\{-C^{*},-B^{*}\}\}$, where $B$ is hermitian, $C$ is symmetric, and $M^{*}$ means the complex conjugate of $M$ rather than the transpose conjugate of $M$. – phchen Apr 1 '16 at 12:53

## 1 Answer

Take any Hermitian matrix of the form $\mathfrak{B}=\left[\begin{array}{c|c}\mathbf{0}_n & A\\\hline A^* & \mathbf{0}_m\end{array}\right],$ where $A$ is $n\times m$. Then if $\lambda$ is an eigenvalue of $\mathfrak{B}$ with multiplicity $k$, then $-\lambda$ is also an eigenvalue with the same multiplicity. This is another sufficient condition.


text:
# (Electrostatics) Energy of a configuration

1. Apr 14, 2009

### LuisVela

1. The problem statement, all variables and given/known data

For a given configuration i found the scalar potential $$\phi(r)$$-->(as you can see its a function only of r)
My question is about calculating the energy of the system.

2. Relevant equations

$$W=-\dfrac{\varepsilon_0}{2}\int |\nabla \phi|^2 d^{3}x =\dfrac{1}{2}\int \phi \rho \,d^{3}x$$

3. The attempt at a solution

I just dont know if i should integrate $$\phi (r)$$ like a triple integral with limits $$(0,\infty)x(0,2\pi)x(0,\pi)$$ or should i perform the inverse substitution (from sferical coordinates to cartesian ) and then integrate $$\phi (x,y,z)$$ like a triple integral with limits (-oo,oo)x(-oo,oo)x(-oo,oo)

Moreover, if i perform the change in the variables,what happen to the Jacobian of the substitution (|J|=$$r^2 sin(\vartheta)$$ )????

Last edited: Apr 15, 2009


text:
(Analysis by Nick Wu)

In this problem, we have a circle of 52 points, each one annotated with a letter. Each letter is associated with exactly two points. We want to count the number of pairs of letters such that if we connected the points with the same letter, the connecting line segments would intersect.

If we take a circle and draw two intersecting line segments, and travel the circumference of the circle, we would alternate between the letters, either seeing $ABAB$ or $BABA$. However, if we take the same circle and instead draw two line segments that don't intersect, when we travel the circumference of the circle, we will not see the letters alternating. We would either see $AABB$, $ABBA$, $BAAB$, or $BBAA$.

Therefore, counting the number of intersections is equivalent to counting the number of pairs of letters where we see the sequence $ABAB$ or $BABA$.

In the provided solution, we take a slightly similar approach. Instead of explicitly looking for that sequence, we count the number of times a single letter appears between two other letters.

import java.io.*;
import java.util.*;
public class circlecross {
public static void main(String[] args) throws IOException {
PrintWriter pw = new PrintWriter(new BufferedWriter(new FileWriter("circlecross.out")));

// convert the string to an array of integers for simplicity
int[] crossArray = new int[crosses.length()];
for(int i = 0; i < crosses.length(); i++) {
crossArray[i] = crosses.charAt(i) - 'A';
}

int numPairs = 0;

// track the first and last occurrence of each numbers
for(int i = 0; i < 26; i++) {
int first = getFirstAppearance(crossArray, i);
int last = getLastAppearance(crossArray, i);
// count the number of single appearances within the range
numPairs += getNumberOfSingles(crossArray, i, first, last);
}

pw.println(numPairs);

pw.close();
}

/**
* Given an array of integers and some value, return the first index of the
* value. Return -1 if the value does not appear in the array.
* @param crossArray     The array to inspect.
* @param desiredValue   The value to look for in the given array.
* @return               The first index where desiredValue appears in crossArray, or -1 if it doesn't.
*/
public static int getFirstAppearance(int[] crossArray, int desiredValue) {
for(int i = 0; i < crossArray.length; i++) {
if(crossArray[i] == desiredValue) {
return i;
}
}
return -1;
}

/**
* Given an array of integers and some value, return the last index of the
* value. Return -1 if the value does not appear in the array.
* @param crossArray     The array to inspect.
* @param desiredValue   The value to look for in the given array.
* @return               The last index where desiredValue appears in crossArray, or -1 if it doesn't.
*/
public static int getLastAppearance(int[] crossArray, int desiredValue) {
for(int i = crossArray.length-1; i >= 0; i--) {
if(crossArray[i] == desiredValue) {
return i;
}
}
return -1;
}

/**
* Given an array of integers, a range of indices, and a lower bound, compute how many values
* greater than the lower bound appear exactly once inside the range. The range of indices
* is exclusive on the left and right indices.
* @param crossArray    The array to inspect.
* @param greaterThan   The lower bound of values we are interested in.
* @param leftIndex     The left index (exclusive) of the range we are querying.
* @param rightIndex    The right index (exclusive) of the range we are querying.
* @return
*/
public static int getNumberOfSingles(int[] crossArray, int greaterThan, int leftIndex, int rightIndex) {
int numSingles = 0;
// keep track of the number of times a given number has appeared
int[] appearances = new int[26];
// loop over the range and update how many times a given number has appeared
for(int i = leftIndex+1; i < rightIndex; i++) {
appearances[crossArray[i]]++;
}
// loop over each of the numbers from the lower bound (exclusive) and
// count how many have appeared exactly once
for(int i = greaterThan + 1; i < appearances.length; i++) {
if(appearances[i] == 1) {
numSingles++;
}
}
return numSingles;
}

}


text:
#### Binary quadratic forms and ray class groups

Cited 2 time in Cited 0 time in
• Hit : 156
DC FieldValueLanguage
dc.contributor.authorEum, Ick Sunko
dc.contributor.authorKoo, Ja Kyungko
dc.contributor.authorShin, Dong Hwako
dc.date.accessioned2020-04-28T09:20:08Z-
dc.date.available2020-04-28T09:20:08Z-
dc.date.created2018-08-06-
dc.date.created2018-08-06-
dc.date.created2018-08-06-
dc.date.issued2020-04-
dc.identifier.citationPROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, v.150, no.2, pp.695 - 720-
dc.identifier.issn0308-2105-
dc.identifier.urihttp://hdl.handle.net/10203/274031-
dc.description.abstractLet K be an imaginary quadratic field different from $\open{Q}(\sqrt {-1})$ and $\open{Q}(\sqrt {-3})$. For a positive integer N, let K-N be the ray class field of K modulo $N {\cal O}_K$. By using the congruence subgroup +/- Gamma(1)(N) of SL2(DOUBLE-STRUCK CAPITAL Z), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal(K-N/K). We also present an algorithm to find all distinct form classes and show how to multiply two form classes. As an application, we describe Gal(K-N(ab)/K) in terms of these extended form class groups for which K-N(ab) is the maximal abelian extension of K unramified outside prime ideals dividing NOK.-
dc.languageEnglish-
dc.publisherCAMBRIDGE UNIV PRESS-
dc.titleBinary quadratic forms and ray class groups-
dc.typeArticle-
dc.identifier.wosid000524939000006-
dc.identifier.scopusid2-s2.0-85060380850-
dc.type.rimsART-
dc.citation.volume150-
dc.citation.issue2-
dc.citation.beginningpage695-
dc.citation.endingpage720-
dc.citation.publicationnamePROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS-
dc.identifier.doi10.1017/prm.2018.163-
dc.contributor.localauthorKoo, Ja Kyung-
dc.contributor.nonIdAuthorEum, Ick Sun-
dc.contributor.nonIdAuthorShin, Dong Hwa-
dc.description.isOpenAccessN-
dc.type.journalArticleArticle-
dc.subject.keywordAuthorclass field theory-
dc.subject.keywordAuthorclass groups-
dc.subject.keywordAuthorcomplex multiplication-
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MA-Journal Papers(저널논문)
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