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My Math Forum Inequality problem

 Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

 January 2nd, 2019, 01:50 PM #1 Newbie   Joined: Jan 2019 From: gd Posts: 1 Thanks: 0 Inequality problem Let triangle with side a,b,c such that a+b+c=10. Prove this in.png I need help with this problem by use of triangle inequality, AG or similar inequality that I know. So I ask for hint. Last edited by skipjack; January 3rd, 2019 at 07:42 AM.
January 2nd, 2019, 02:15 PM   #2
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Quote:
 Originally Posted by jsf Let triangle with side a,b,c such that a+b+c=10. Prove this Attachment 10106 I need help with this problem by use of triangle inequality, AG or similar inequality that I know. So I ask for hint.
I can't see the image. (I can manually type it in and view it on the web but that's all.)

So:
$\displaystyle a + b + c = 10$

Here's the inequality in the image:
$\displaystyle a^2 + b^2 + c^2 + \frac{2}{5} abc < 50$

Can we take it that you have to derive the inequality from the first equation?

-Dan

Last edited by skipjack; January 3rd, 2019 at 07:44 AM.

 January 2nd, 2019, 07:04 PM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,900 Thanks: 1094 Math Focus: Elementary mathematics and beyond I've posted the image in-line. Thanks from topsquark
 January 2nd, 2019, 07:23 PM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,900 Thanks: 1094 Math Focus: Elementary mathematics and beyond Because the variables in the inequality can be interchanged with respect to value and position the equation has cyclic symmetry and an extremum is reached when all three sides are equal. After you come up with a value for that all you have to do is devise another triangle to show that the LHS of the inequality is less than the previously calculated value. Thanks from topsquark
January 3rd, 2019, 09:03 AM   #5
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Quote:
 Originally Posted by jsf I ask for hint.
Hint: as $a + b + c = 10$ and $b + c > a$, $a < 5$.
Similarly, $b < 5$ and $c < 5$, so $0 < (5 - a)(5 - b)(5 - c)$.
Multiply that inequality by 2/5 and use the fact that $(a + b + c)^2 = 100$.

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# Probability Surveys

Czasopismo Open Access z projektu Euclid

## Characterizations of GIG laws: A survey

Angelo Efoévi Koudou, Christophe Ley. Source: Probability Surveys, Volume 11, 161--176.Abstract:

Several characterizations of the Generalized Inverse Gaussian (GIG) distribution on the positive real line have been proposed in the literature, especially over the past two decades. These characterization theorems are surveyed, and two new characterizations are established, one based on maximum likelihood estimation and the other is a Stein characterization.

2014/07/17 - 16:58

## Statistical properties of zeta functions’ zeros

Vladislav Kargin. Source: Probability Surveys, Volume 11, 121--160.Abstract:

The paper reviews existing results about the statistical distribution of zeros for three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of the main results.

2014/07/04 - 22:54

## On the notion(s) of duality for Markov processes

Sabine Jansen, Noemi Kurt. Source: Probability Surveys, Volume 11, 59--120.Abstract:
We provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in Markov process theory and potential theory and give functional analytic results including existence and uniqueness criteria and a comparison of the spectra of dual semi-groups. The analytic framework builds on the notion of dual pairs, convex geometry, and Hilbert spaces. In addition, we formalize the notion of pathwise duality as it appears in population genetics and interacting particle systems. We discuss the relation of duality with rescalings, stochastic monotonicity, intertwining, symmetries, and quantum many-body theory, reviewing known results and establishing some new connections.

2014/04/30 - 16:39

## Integrable probability: From representation theory to Macdonald processes

Alexei Borodin, Leonid Petrov. Source: Probability Surveys, Volume 11, , 1--58.Abstract:

These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the $(q,t)$-deformation of those leading to the Macdonald processes, nearest neighbor dynamics on Macdonald processes, their limit to semi-discrete Brownian polymers, and large time asymptotic analysis of polymer’s partition function.

2014/03/18 - 12:10

## Integrable probability: From representation theory to Macdonald processes

Alexei Borodin, Leonid Petrov. Source: Probability Surveys, Volume 11, , 1--58.Abstract:

These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the $(q,t)$-deformation of those leading to the Macdonald processes, nearest neighbor dynamics on Macdonald processes, their limit to semi-discrete Brownian polymers, and large time asymptotic analysis of polymer’s partition function.

2014/03/18 - 12:10

## Self-normalized limit theorems: A survey

Qi-Man Shao, Qiying WangSource: Probab. Surveys, Volume 10, 69--93.Abstract:

Let $X_{1},X_{2},\ldots,$ be independent random variables with $EX_{i}=0$ and write $S_{n}=\sum_{i=1}^{n}X_{i}$ and $V_{n}^{2}=\sum_{i=1}^{n}X_{i}^{2}$. This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum $S_{n}/V_{n}$. Other self-normalized limit theorems are also briefly discussed.

2013/11/29 - 14:09

## On spectral methods for variance based sensitivity analysis

Alen AlexanderianSource: Probab. Surveys, Volume 10, 51--68.Abstract:

Consider a mathematical model with a finite number of random parameters. Variance based sensitivity analysis provides a framework to characterize the contribution of the individual parameters to the total variance of the model response. We consider the spectral methods for variance based sensitivity analysis which utilize representations of square integrable random variables in a generalized polynomial chaos basis. Taking a measure theoretic point of view, we provide a rigorous and at the same time intuitive perspective on the spectral methods for variance based sensitivity analysis. Moreover, we discuss approximation errors incurred by fixing inessential random parameters, when approximating functions with generalized polynomial chaos expansions.

2013/11/22 - 22:53

## Planar percolation with a glimpse of Schramm–Loewner evolution

Vincent Beffara, Hugo Duminil-CopinSource: Probab. Surveys, Volume 10, 1--50.Abstract:

In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy–Smirnov formula. This theorem, together with the introduction of Schramm–Loewner Evolution and techniques developed over the years in percolation, allow precise descriptions of the critical and near-critical regimes of the model. This survey aims to describe the different steps leading to the proof that the infinite-cluster density $\theta(p)$ for site percolation on the triangular lattice behaves like $(p-p_{c})^{5/36+o(1)}$ as $p\searrow p_{c}=1/2$.

2013/09/21 - 17:00

## Quantile coupling inequalities and their applications

David M. Mason, Harrison H. ZhouSource: Probab. Surveys, Volume 9, 439--479.Abstract:

This is partly an expository paper. We prove and highlight a quantile inequality that is implicit in the fundamental paper by Komlós, Major, and Tusnády [31] on Brownian motion strong approximations to partial sums of independent and identically distributed random variables. We also derive a number of refinements of this inequality, which hold when more assumptions are added. A number of examples are detailed that will likely be of separate interest. We especially call attention to applications to the asymptotic equivalence theory of nonparametric statistical models and nonparametric function estimation.

2012/11/29 - 10:15

## Erratum: Three theorems in discrete random geometry

Geoffrey GrimmettSource: Probab. Surveys, Volume 9, 438--438.Abstract:

An error is identified and corrected in the survey entitled ‘Three theorems in discrete random geometry’, published in Probability Surveys 8 (2011) 403–441.

2012/11/14 - 21:06

## Bougerol’s identity in law and extensions

Stavros VakeroudisSource: Probab. Surveys, Volume 9, 411--437.Abstract:

We present a list of equivalent expressions and extensions of Bougerol’s celebrated identity in law, obtained by several authors. We recall well-known results and the latest progress of the research associated with this celebrated identity in many directions, we give some new results and possible extensions and we try to point out open questions.

2012/11/09 - 16:08

## Quasi-stationary distributions and population processes

Sylvie Méléard, Denis VillemonaisSource: Probab. Surveys, Volume 9, 340--410.Abstract:

This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of different stochastic population size processes when 0 is an absorbing point almost surely attained by the process. The hitting time of this point, namely the extinction time, can be large compared to the physical time and the population size can fluctuate for large amount of time before extinction actually occurs. This phenomenon can be understood by the study of quasi-limiting distributions. In this paper, general results on quasi-stationarity are given and examples developed in detail. One shows in particular how this notion is related to the spectral properties of the semi-group of the process killed at 0. Then we study different stochastic population models including nonlinear terms modeling the regulation of the population. These models will take values in countable sets (as birth and death processes) or in continuous spaces (as logistic Feller diffusion processes or stochastic Lotka-Volterra processes). In all these situations we study in detail the quasi-stationarity properties. We also develop an algorithm based on Fleming-Viot particle systems and show a lot of numerical pictures.

2012/10/13 - 01:37

## Szegö’s theorem and its probabilistic descendants

N.H. BinghamSource: Probab. Surveys, Volume 9, 287--324.Abstract:

The theory of orthogonal polynomials on the unit circle (OPUC) dates back to Szegö’s work of 1915-21, and has been given a great impetus by the recent work of Simon, in particular his survey paper and three recent books; we allude to the title of the third of these, Szegö’s theorem and its descendants , in ours. Simon’s motivation comes from spectral theory and analysis. Another major area of application of OPUC comes from probability, statistics, time series and prediction theory; see for instance the classic book by Grenander and Szegö, Toeplitz forms and their applications . Coming to the subject from this background, our aim here is to complement this recent work by giving some probabilistically motivated results. We also advocate a new definition of long-range dependence.

2012/07/23 - 17:39

## Multivariate prediction and matrix Szegö theory

N.H. BinghamSource: Probab. Surveys, Volume 9, 325--339.Abstract:

Following the recent survey by the same author of Szegö’s theorem and orthogonal polynomials on the unit circle (OPUC) in the scalar case, we survey the corresponding multivariate prediction theory and matrix OPUC (MOPUC).

2012/07/23 - 17:39

## On temporally completely monotone functions for Markov processes

Francis Hirsch, Marc YorSource: Probab. Surveys, Volume 9, 253--286.Abstract:

Any negative moment of an increasing Lamperti process( Y t , t ≥ 0) is a completely monotone function of t . This property enticed us to study systematically, for a given Markov process ( Y t , t ≥ 0), the functions f such that the expectation of f ( Y t ) is a completely monotone function of t . We call these functions temporally completely monotone (for Y ). Our description of these functions is deduced from the analysis made by Ben Saad and Janßen, in a general framework, of a dual notion, that of completely excessive measures. Finally, we illustrate our general description in the cases when Y is a Lévy process, a Bessel process, or an increasing Lamperti process.

2012/05/12 - 07:23

## Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation

Svante JansonSource: Probab. Surveys, Volume 9, 103--252.Abstract:

We give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case of critical Galton–Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton–Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton–Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree.

The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding theorems for this model.

This survey paper contains many known results from many different sources, together with some new results.

2012/03/09 - 00:34

## A lecture on the averaging process

David Aldous, Daniel LanoueSource: Probab. Surveys, Volume 9, 90--102.Abstract:

To interpret interacting particle system style models as social dynamics, suppose each pair { i , j } of individuals in a finite population meet at random times of arbitrary specified rates ν ij , and update their states according to some specified rule. The averaging process has real-valued states and the rule: upon meeting, the values X i ( t − ), X j ( t − ) are replaced by ½( X i ( t − )+ X j ( t − )),½( X i ( t − )+ X j ( t − )). It is curious this simple process has not been studied very systematically. We provide an expository account of basic facts and open problems.

2012/01/24 - 09:24

## Around the circular law

Charles Bordenave, Djalil ChafaïSource: Probab. Surveys, Volume 9, 1--89.Abstract:

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a n × n random matrix with i.i.d. entries of variance 1/ n tends to the uniform law on the unit disc of the complex plane as the dimension n tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.

2012/01/04 - 05:38

## Recent progress on the Random Conductance Model

Marek BiskupSource: Probab. Surveys, Volume 8, 294--373.Abstract:

Recent progress on the understanding of the Random Conductance Model is reviewed and commented. A particular emphasis is on the results on the scaling limit of the random walk among random conductances for almost every realization of the environment, observations on the behavior of the effective resistance as well as the scaling limit of certain models of gradient fields with non-convex interactions. The text is an expanded version of the lecture notes for a course delivered at the 2011 Cornell Summer School on Probability.

2011/12/31 - 08:06

## Topics on abelian spin models and related problems

Julien DubédatSource: Probab. Surveys, Volume 8, 374--402.Abstract:

In these notes, we discuss a selection of topics on several models of planar statistical mechanics. We consider the Ising, Potts, and more generally abelian spin models; the discrete Gaussian free field; the random cluster model; and the six-vertex model. Emphasis is put on duality, order, disorder and spinor variables, and on mappings between these models.

2011/12/31 - 08:06

## Three theorems in discrete random geometry

Geoffrey GrimmettSource: Probab. Surveys, Volume 8, 403--441.Abstract:

These notes are focused on three recent results in discrete random geometry, namely: the proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice is $\sqrt{2+\sqrt{2}}$ ; the proof by the author and Manolescu of the universality of inhomogeneous bond percolation on the square, triangular, and hexagonal lattices; the proof by Beffara and Duminil-Copin that the critical point of the random-cluster model on ℤ 2 is $\sqrt{q}/(1+\sqrt{q})$ . Background information on the relevant random processes is presented on route to these theorems. The emphasis is upon the communication of ideas and connections as well as upon the detailed proofs.

2011/12/31 - 08:06

## Scaling limits and the Schramm-Loewner evolution

Gregory F. LawlerSource: Probab. Surveys, Volume 8, 442--495.Abstract:

These notes are from my mini-courses given at the PIMS summer school in 2010 at the University of Washington and at the Cornell probability summer school in 2011. The goal was to give an introduction to the Schramm-Loewner evolution to graduate students with background in probability. This is not intended to be a comprehensive survey of SLE.

2011/12/31 - 08:06

## Conformally invariant scaling limits in planar critical percolation

Nike SunSource: Probab. Surveys, Volume 8, 155--209.Abstract:

This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov’s theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE κ ), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path converges in the scaling limit to chordal SLE 6 . No prior knowledge is assumed beyond some general complex analysis and probability theory.

2011/10/30 - 07:32

## Fundamentals of Stein’s method

Nathan RossSource: Probab. Surveys, Volume 8, 210--293.Abstract:

This survey article discusses the main concepts and techniques of Stein’s method for distributional approximation by the normal, Poisson, exponential, and geometric distributions, and also its relation to concentration of measure inequalities. The material is presented at a level accessible to beginning graduate students studying probability with the main emphasis on the themes that are common to these topics and also to much of the Stein’s method literature.

2011/10/30 - 07:32

## A basic theory of Benford’s Law

Arno Berger, Theodore P. HillSource: Probab. Surveys, Volume 8, 1--126.Abstract:

Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford’s Law (BL) or, in a special case, as the First Digit Law . The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.

2011/07/29 - 09:17

## Reviewing alternative characterizations of Meixner process

E. Mazzola, P. MuliereSource: Probab. Surveys, Volume 8, 127--154.Abstract:

Based on the first author’s recent PhD thesis entitled “Profiling processes of Meixner type”, [50] a review of the main characteristics and characterizations of such particular Lévy processes is extracted, emphasizing the motivations for their introduction in literature as reliable financial models. An insight on orthogonal polynomials is also provided, together with an alternative path for defining the same processes. Also, an attempt of simulation of their trajectories is introduced by means of an original R simulation routine.

2011/07/29 - 09:17

## Moments of Gamma type and the Brownian supremum process area

Svante JansonSource: Probab. Surveys, Volume 7, 1--52.Abstract:

We study positive random variables whose moments can be expressed by products and quotients of Gamma functions; this includes many standard distributions. General results are given on existence, series expansion and asymptotics of density functions. It is shown that the integral of the supremum process of Brownian motion has moments of this type, as well as a related random variable occurring in the study of hashing with linear displacement, and the general results are applied to these variables.

2010/12/30 - 03:10

## Limit theorems for discrete-time metapopulation models

F.M. Buckley, P.K. PollettSource: Probab. Surveys, Volume 7, 53--83.Abstract:

We describe a class of one-dimensional chain binomial models of use in studying metapopulations (population networks). Limit theorems are established for time-inhomogeneous Markov chains that share the salient features of these models. We prove a law of large numbers, which can be used to identify an approximating deterministic trajectory, and a central limit theorem, which establishes that the scaled fluctuations about this trajectory have an approximating autoregressive structure.

2010/12/30 - 03:10

## Symbolic extensions of smooth interval maps

Tomasz DownarowiczSource: Probab. Surveys, Volume 7, 84--104.Abstract:

In this course we will present the full proof of the fact that every smooth dynamical system on the interval or circle X , constituted by the forward iterates of a function f : X → X which is of class C r with r > 1, admits a symbolic extension, i.e., there exists a bilateral subshift ( Y , S ) with Y a closed shift-invariant subset of Λ ℤ , where Λ is a finite alphabet, and a continuous surjection π : Y → X which intertwines the action of f (on X ) with that of the shift map S (on Y ). Moreover, we give a precise estimate (from above) on the entropy of each invariant measure ν supported by Y in an optimized symbolic extension. This estimate depends on the entropy of the underlying measure μ on X , the “Lyapunov exponent” of μ (the genuine Lyapunov exponent for ergodic μ , otherwise its analog), and the smoothness parameter r . This estimate agrees with a conjecture formulated in [15] around 2003 for smooth dynamical systems on manifolds.

2010/12/30 - 03:10

## Regeneration in random combinatorial structures

Alexander V. GnedinSource: Probab. Surveys, Volume 7, 105--156.Abstract:

Kingman’s theory of partition structures relates, via a natural sampling procedure, finite partitions to hypothetical infinite populations. Explicit formulas for distributions of such partitions are rare, the most notable exception being the Ewens sampling formula, and its two-parameter extension by Pitman. When one adds an extra structure to the partitions like a linear order on the set of blocks and regenerative properties, some representation theorems allow to get more precise information on the distribution. In these notes we survey recent developments of the theory of regenerative partitions and compositions. In particular, we discuss connection between ordered and unordered structures, regenerative properties of the Ewens-Pitman partitions, and asymptotics of the number of components.

2010/12/30 - 03:10

## Combinatorics and cluster expansions

William G. FarisSource: Probab. Surveys, Volume 7, 157--206.Abstract:

This article is about the connection between enumerative combinatorics and equilibrium statistical mechanics. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. The passage from species to generating functions is a combinatorial analog of the Fourier transform. Indeed, there is a convolution multiplication on species that is mapped to a pointwise multiplication of the exponential generating functions. The statistical mechanics side deals with a probability model of an equilibrium gas. The cluster expansion that gives the density of the gas is the exponential generating function for the species of rooted connected graphs. The main results of the theory are simple criteria that guarantee the convergence of this expansion. It turns out that other problems in combinatorics and statistical mechanics can be translated to this gas setting, so it is a universal prescription for dealing with systems of high dimension.

2010/12/30 - 03:10

## Addendum to Moments of Gamma type and the Brownian supremum process area

Svante JansonSource: Probab. Surveys, Volume 7, 207--208.Abstract:

Supplementary references and material are provided to the paper entitled ‘Moments of Gamma type and the Brownian supremum process area’, published in Probability Surveys 7 (2010) 1–52.

References:[1] B. L. J. Braaksma, Asymptotic expansions and analytic continuations for a class of Barnes-integrals. Compositio Math. 15 (1964), 239–341.[2] B. D. Carter and M. D. Springer, The distribution of products, quotients and powers of independent H-function variates. SIAM J. Appl. Math. 33 (1977), no. 4, 542–558.[3] J.-F. Chamayou and G. Letac, Additive properties of the Dufresne laws and their multivariate extension. J. Theoret. Probab. 12 (1999), no. 4, 1045–1066.[4] D. Dufresne, The beta product distribution with complex parameters. Comm. Statistics – Theory and Methods 39 (2010), no. 5, 837–854.[5] D. Dufresne, G distributions and the beta-gamma algebra. Preprint, University of Melbourne, 2009.[6] C. Fox, The G and H functions as symmetrical Fourier kernels. Trans. Amer. Math. Soc. 98 (1961) 395–429.[7] M. Kaluszka and W. Krysicki, On decompositions of some random variables. Metrika 46 (1997), no. 2, 159–175.[8] A. M. Mathai and R. K. Saxena, On the linear combinations of stochastic variables. 20 (1973), 160–169.[9] A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Lecture Notes in Mathematics, Vol. 348, Springer-Verlag, Berlin-New York, 1973.[10] A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function. Theory and Applications. Springer, New York, 2010. xiv+268 pp. ISBN: 978-1-4419-0915-2[11] C. S. Meijer, On the G-function. I–VIII. Nederl. Akad. Wetensch., Proc. 49, (1946) 227–237, 344–356, 457–469, 632–641, 765–772, 936–943, 1063–1072, 1165–1175 = Indagationes Math. 8 (1946), 124–134, 213–225, 312–324, 391–400, 468–475, 595–602, 661–670, 713–723.[12] E. W. Weisstein, Fox H-Function. MathWorld. http://mathworld.wolfram.com/FoxH-Function.html[13] E. W. Weisstein, Meijer G-Function. MathWorld. http://mathworld.wolfram.com/MeijerG-Function.html

2010/12/30 - 03:10

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### D-2, r. 12 - Decree respecting the installation of petroleum equipment

9.02. Pay sheet: The employer must remit to the employee, together with his wages, a pay sheet containing sufficient information to enable the employee to verify the computation of his wages. That pay sheet must include, in particular, the following information, where applicable:
(1)  the name of the employer;
(2)  the name of the employee;
(3)  the identification of the employee’s occupation;
(4)  the date of the payment and the work period corresponding to the payment;
(5)  the number of hours paid at the employee’s base rate;
(6)  the number of hours of overtime paid with the applicable premium;
(7)  the nature and amount of the bonuses, indemnities, allowances or commissions that are being paid;
(8)  the employee’s base rate;
(9)  the amount of wages before deductions;
(10)  the nature and amount of the deductions effected;
(11)  the amount of the net wages paid to the employee.
R.R.Q., 1981, c. D-2, r. 33, s. 9.02; O.C. 769-92, s. 18; O.C. 1152-99, s. 10.


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# Abstract Completeness

 Title: Abstract Completeness Authors: Jasmin Christian Blanchette (j /dot/ c /dot/ blanchette /at/ vu /dot/ nl), Andrei Popescu and Dmitriy Traytel Submission date: 2014-04-16 Abstract: A formalization of an abstract property of possibly infinite derivation trees (modeled by a codatatype), representing the core of a proof (in Beth/Hintikka style) of the first-order logic completeness theorem, independent of the concrete syntax or inference rules. This work is described in detail in the IJCAR 2014 publication by the authors. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems as well as various flavors of FOL---e.g., with or without predicates, equality, or sorts. Here, we give only a toy example instantiation with classical propositional logic. A more serious instance---many-sorted FOL with equality---is described elsewhere [Blanchette and Popescu, FroCoS 2013]. BibTeX: @article{Abstract_Completeness-AFP, author = {Jasmin Christian Blanchette and Andrei Popescu and Dmitriy Traytel}, title = {Abstract Completeness}, journal = {Archive of Formal Proofs}, month = apr, year = 2014, note = {\url{https://isa-afp.org/entries/Abstract_Completeness.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Depends on: Collections Used by: Abstract_Soundness, Incredible_Proof_Machine Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


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List the various data abstraction concepts and the corresponding modeling concepts in the EER model.

List the various data abstraction concepts and the corresponding modeling concepts in the EER model.
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doplovif
Here is a list of data abstraction concepts and their variants in EER model:
* Classification and instantiation - The EER variant of classification is represented by classifying entities into entity types based on their attributes and relationships. Relationship instances becomerelationship types. Forthe need offurther classification, we introduce superclass /subclass concepts and sometimes even categories. In EER model, there is no handling of class exceptions and no showing of examples (instances).
* Identification - EER model implements identification hy sticking to the rule of name uniqueness. Every entity type, relationship type, attribute, subclass, superclass and category has to have unique name that no other object in database has.
* Specialization and Generalization - The concept of specialization and generalization in the EER model is represented by subclasses. They are more specific objects than their respective superclasses and arerelated to them by IS-A-SUBCLASS-OF (or shortly IS-A) relationship.
* Aggregation and Association - EER schema uses aggregation abstraction in form of creating new compositeobjects. We can aggregate attributes to createa composite attribute, entity type or even a relationship type if there’s need to. Association in EER modeling is simply a relationship type between two or more independent entity types.


text:
Consider the function $f(x) = 7 (x - 2)^{2/3}$. For this function there are two important intervals: $(-\infty, A)$ and $(A,\infty)$ where $A$ is a critical value.
Find $A$
For each of the following intervals, tell whether $f(x)$ is increasing (type in INC) or decreasing (type in DEC).
$(-\infty, A)$:
$(A,\infty)$:
For each of the following intervals, tell whether $f(x)$ is concave up (type in CU) or concave down (type in CD).
$(-\infty, A)$:
$(A,\infty)$:


text:
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# What is the units digit of 13^4*17^2*29^3?

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Intern
Joined: 06 Oct 2010
Posts: 40
What is the units digit of 13^4*17^2*29^3? [#permalink]

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17 Oct 2010, 12:50
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15% (low)

Question Stats:

67% (01:36) correct 33% (00:38) wrong based on 230 sessions

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What is the units digit of $$13^4*17^2*29^3$$?

(A) 9
(B) 7
(C) 5
(D) 3
(E) 1
[Reveal] Spoiler: OA

Last edited by stonecold on 24 Jan 2017, 12:43, edited 2 times in total.
Edited.
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Re: What is the units digit of 13^4*17^2*29^3? [#permalink]

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17 Oct 2010, 14:32
4
KUDOS
I think that the fastest and more important, the safest way to solve this is:
First brake the problem to three multiplication problems:
1. 13*13*13*13
2. 17*17
3. 29*29*29
In all of the problems we don't have to multiply by the whole number, but only by the last digit (because that is what we are asked about).
Second, simply multiply: 13*13*13*13. It is actually 3*3*3*3 (we are interested in the last digit). Thus - 3*3=9, 9*3=7(the last digit), 7*3=1 (the last digit).
17*17 is actually 7*7 = 9 (the last digit).
29*29*29 is actually -9*9*9- that can be interpreted into 9*9=1 (the last digit) and 1*9=9.
So eventually we will have- 9 (29^3) * 9 (17^2) * 1 (13^4). Therefore - 9*9=1(the last digit), 1*1=1

So the correct answer is E -1.
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Re: What is the units digit of 13^4*17^2*29^3? [#permalink]

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17 Oct 2010, 17:03
Awesome! I think that explanation will help me solve a lot of "units digit" questions. Thanks!
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Re: What is the units digit of 13^4*17^2*29^3? [#permalink]

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17 Oct 2010, 17:25
1
KUDOS
General rule :

To find the unit's digit- find the remainder when divided by 10.

to find the last two digits - find the remainder when divided by 100.

....and so on.

For advance theory of remainders check the Math book of gmat Club.
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Re: What is the units digit of 13^4*17^2*29^3? [#permalink]

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04 May 2015, 00:24
1
KUDOS
just need unit digit of every component and multipy them

3^1=3
3^2=9
3^3=7
3^4=1, it is first digit

7^2=9, it is second digit

9^3=9, it is third digit

1*9*9=1

E
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Re: What is the units digit of 13^4*17^2*29^3? [#permalink]

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12 May 2016, 10:49
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Re: What is the units digit of 13^4*17^2*29^3? [#permalink]

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12 May 2016, 12:22
niheil wrote:
What is the units digit of 13^4*17^2*29^3?

(A) 9
(B) 7
(C) 5
(D) 3
(E) 1

Units digit of $$13^4$$ will be 1
Units digit of $$17^2$$ will be 9
Units digit of $$29^3$$will be 9

So, Units digit of $$13^4*17^2*29^3$$ will be 1 * 9 * 9 => 1

Answer will be option (E) 1

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Re: What is the units digit of 13^4*17^2*29^3?   [#permalink] 12 May 2016, 12:22
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## Geometry: Common Core (15th Edition)

Published by Prentice Hall

# Chapter 3 - Parallel and Perpendicular Lines - 3-2 Properties of Parallel Lines - Practice and Problem-Solving Exercises: 18

p=32

#### Work Step by Step

The labeled angle forms a corresponding angle pair with the angle marked as a right angle. The corresponding angle theorem tells us the pair is congruent, so both are right angles. $3p-6=90\longrightarrow$ definition of right angle $3p=96\longrightarrow$ addition property of equality $p=32$ division property of equality

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
CryptoDB

Paper: Universal Proxy Re-Encryption

Authors: Nico Dottling Ryo Nishimaki Search ePrint Search Google We put forward the notion of universal proxy re-encryption (UPRE). A UPRE scheme enables a proxy to convert a ciphertext under a (delegator) public key of any existing public-key encryption (PKE) scheme into another ciphertext under a (delegatee) public key of any existing PKE scheme (possibly different from the delegator one). The proxy has a re-encryption key generated from the delegator's secret key and the delegatee public key. Thus UPRE generalizes proxy re-encryption by supporting arbitrary PKE schemes and allowing to convert ciphertexts into ones of possibly different PKE schemes. In this work, we - provide syntax and definitions for both UPRE and a variant we call relaxed UPRE. The relaxed variant means that decryption algorithms for re-encrypted ciphertexts are slightly modified but still only use the original delegatee secret keys for decryption. - construct a UPRE based on probabilistic indistinguishability obfuscation (PIO). It allows us to re-encrypt ciphertexts polynomially many times. - construct relaxed UPRE from garbled circuits (GCs). We provide two variants of this construction, one which allows us to re-encrypt ciphertexts polynomially many times, and a second one which satisfies a stronger security requirement but only allows us to re-encrypt ciphertexts a constant number of times.
BibTeX
@article{pkc-2021-30981,
title={Universal Proxy Re-Encryption},
booktitle={Public-Key Cryptography - PKC 2021},
publisher={Springer},
author={Nico Dottling and Ryo Nishimaki},
year=2021
}


text:
Sales Toll Free No: 1-855-666-7446

# Moment of Inertia Calculator

Top
The inertia of a particle or larger object is a measure of its resistance to move. When a body can be modeled as a particle, its mass determines its inertia. For a larger body which is rotating about a fixed axis the particle model is no longer appropriate and its reluctance to move is determined by a quantity called its moment of inertia. It is denoted as I and SI unit is kg$m^{2}$.

The formula to calculate the moment of inertia is,
$I = MD^{2}$

Where M is the mass
D is the distance from the axis of rotation

## Steps

Step 1: Note down the given parameters from the question.
Step 2: Plug these given quantity in $I = MD^{2}$
Step 3: Calculate the unknown quantity

## Problems

The problems of Moment of Inertia are given below:

### Solved Examples

Question 1: The mass of a bob is 10g and the distance between bob and axis of rotation is 35cm. Calculate its moment of inertia?

Solution:

Given measures are,
m = 10g = 0.01kg
r = 35cm = 0.35m
The formula to calculate the moment of inertia is,
$I = MD^{2}$

I = 0.01$\times0.35^{2}$ = 0.001225 kg$m^{2}$

Question 2: Mass and distance between an object and axis of rotation are given as 5kg and 10m respectively. Calculate its moment of inertia?
Solution:

Given measures are,
m = 5kg
r = 10m
The formula to calculate the moment of inertia is,
$I = MD^{2}$

I = 5$\times10^{2}$ = 500 kg$m^{2}$


text:
IIT JAM Follow
June 11, 2022 8:19 am 25 pts
28. Let fa,} be the sequence of consecutive positive solutions of the equation tanx= x and let fb,j be the sequence of consecutive positive solutions of equation tanyx =x. Then converges bu n=l b diverges n= b converges (b) 00 diverges a, n Converges - and an (c) Both 2n= n=h 1 diverges (d) Both and n-l b, n n VnEN. Then which of the following(5) is/are 29. Let fr,} be a sequence in R such that+ (b) {x,} is Cauchy but not convergent (d) fx,} is divergent sequence correct? (a)xf is covergent sequence (C) is Oscillatory
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# Strangest language feature

What is, in your opinion, the most surprising, weird, strange or really "WTF" language feature you have encountered?

## locked by WillMay 1 '13 at 21:13

This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here. This question and its answers are frozen and cannot be changed. More info: help center.

• @gablin I think if you combined LISP delimiters with PERL regex using javascript parsing you would cover 90% of the WTF... – Talvi Watia Sep 19 '10 at 23:41

In Java you might expect

byte b = 0;
b++;

to be equal to

byte b = 0;
b = b + 1;

But it is not. In fact you get a compiler error, as the result of the addition is of type int and therefore not assignable to the byte variable b. When using the compound operator ++ The compiler automatically inserts a cast here. So

b++;

becomes

b = (byte) b + 1;

All it would take would be ONE extra line of code on Microsoft's part:

public void AddRange<S>(
IEnumerable<S> collection
) where S : T

• Not really a language feature insomuch as it is a framework feature. Pedantic, I know... – Erik Forbes Jan 5 '10 at 16:48
• I would call it a framework design oversight - but it's definitely a WTF – BlueRaja - Danny Pflughoeft Jan 6 '10 at 2:25
• It works with C# 4.0 due to new co- and contravariance features. – Michael Damatov Jan 6 '10 at 9:17
• @Denis: Try it yourself. That single line of code does solve this particular example because it is basically mimicking covariance (or contravariance, I never know which one is which). – R. Martinho Fernandes Jan 7 '10 at 2:39
• @Allon: no, that doesn’t use type inference and hence already worked pre-3.5. – Konrad Rudolph Oct 1 '10 at 14:10

In C#, this should at least generate a compiler warning, but it doesn't:

public int Something
{
get { return Something; }
set { Something = value; }
}

When called, it causes your app to crash, and you don't get a good stack trace, since it's a StackOverflowException.

• It's simple recursion, and one could write a perfectly valid recursively evaluated property. While the compiler could potentially be hard coded to catch some very simple cases of infinite recursion, to enforce it as a rule in the language would require a solution to the Halting Problem. Are you a bad enough dude to solve the Halting Problem? – David Jan 6 '10 at 13:24
• This particular case is simple, though. If the property does nothing but get or set itself (you'd need more than that for a valid use of a recursive property), generate a warning. I've seen too many developers and even whole teams stuck on a "mysterious crash" because of this issue, which is very hard to spot in a large code base. – Eric Z Beard Jan 7 '10 at 11:45
• I think this issue is considered fixed in the current version of the language. You should be using auto properties if your property is trivial. – Mehrdad Afshari Jan 7 '10 at 13:01
• Mehrdad: Since auto properties can't have an initial value, the typical solution is to implement the property yourself. – Gabe Jun 9 '10 at 16:05
• VB .NET will raise an ERROR in this case. – Joshua Aug 13 '10 at 20:43

Variable/function declarations in Javascript:

var x = 1;
function weird(){
return x;
var x = 2;
}

weird() returns undefined... x is 'taken' even though the assignment never happened.

Similarly, but not so unexpectedly

function weird2(){
var x;
return x();
function x(){ return 2 };
}

returns 2.

• this happens because of hoisting of the variable & function declarations – gion_13 Nov 22 '11 at 8:36
• This is simply an example of hoisting. Adequately Good gives a very in-depth article about hoisting. – kevinji Dec 7 '11 at 2:17

VBScript's date/time literals (why is this still so rare?):

mydate = #1/2/2010 5:23 PM#

If mydate > #1/1/2010 17:00# Then ' ...

Edit: Date literals are relative (are they technically literals, then?):

mydate = #Jan 3# ' Jan 3 of the current year

VB.NET, since it is compiled, does not support relative date literals. Date only or time only literals are supported, but the missing time or date are assumed to be zero.

Edit[2]: Of course, there are some bizarre corner cases that come up with relative dates...

mydate = #Feb 29# ' executed on 2010-01-05, yields 2/1/2029

• Scala has XML literal notation. def xmlString = <xml>foo bar <b>baz</b></xml> – fennec Jan 5 '10 at 1:00
• why is this still so rare? because literal date is not needed often -- time span literals would be more useful. – Andrey Shchekin Jan 5 '10 at 1:52
• My issue with DateTime Literals is that they can be ambiguous - mydate = #10/9/2009 18:35# - October 9 or September 10? Nowadays I'm guessing it's always mm/dd/yyyy, but for non-US users it's always very odd to have a date not in dd/mm/yyyy format. In languages that have a Date constructor, you at least can always refer to the signature, but I guess it's not that different from just looking it up in the help or memorizing it. On the other hand: A whole language construct for Dates seems very "heavy". I can understand why Visual Basic has it (Office VBA), but I wouldn't see much use i.e. in C# – Michael Stum Jan 5 '10 at 3:18
• @fennec: So does VB.NET 9, now that you mention it. – brianary Jan 5 '10 at 16:13
• @Michael Stum: Agreed. ISO 8601 should probably be used, just to keep things clear. I guess the relative weight of language features depends a great deal on how it impacts the programmer personally. – brianary Jan 5 '10 at 16:18

I've written a programming language for a client (used for experimentally driving custom hardware) with some custom types (Curl, Circuit, ...) that each have only 2 values. They are implicitly convertible to boolean, but (at the request of the client) the exact boolean value of a constant of such a type can be changed at runtime.

E.g.: The type Curl has 2 possible values: CW and CCW (clockwise and counterclockwise). At runtime, you could change the boolean value by a simple assignment statement:

ccw := true

So you could change the boolean meaning of all values of those types.

• This would seem like a good idea to me (it allows things like ccw to be defined based on what is in actual data), but only if all such assignments are the same. Differences should be flagged as conflicts, unless there's some awesome way for the compiler to translate between modules. It does seem pretty out there at first glance, I'll agree, but it's a neat idea if you're dealing with input data whose sense you have no control over. – Mike D. Jan 4 '10 at 3:37

ActionScript 3:

When an object is used by its interface, the compiler doesn't recognize the methods inherited from Object, hence:

IInterface interface = getInterface();
interface.toString();

gives a compilation error. The workaround is casting to Object

Object(interface).toString();

PHP:

. and + operators. It has its reasonable explanation, but still "a" + "5" = 5 seems awkward.

Java (and any implementation of IEEE754):

System.out.println(0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1);

Outputs 0.9999999999999999

• It's great that PHP didn't do what Javascript did with overloading the + operator for both string concatenation as well as addition. In PHP, if you see + you know you're talking about adding numbers. ...(unless you've got arrays...) – nickf Jan 4 '10 at 8:08
• The first one isn't a WTF - it's a gotcha that catches most people sooner or later regarding the finite-precision floating point representation on computers. – Joris Timmermans Jan 4 '10 at 13:28
• And it's not a Java WTF, it's WTF for any correct(!) implementation of IEEE754. – R. Martinho Fernandes Jan 4 '10 at 14:14
• C# just gives me a 1 :) – Snake Jan 7 '10 at 11:44
• That's because in C# the default type for a number with a floating point is Double. Having 0.1f instead of 0.1 would give the same result. – Allon Guralnek Jan 7 '10 at 15:29

When I was in college, I did a little bit of work in a language called SNOBOL. The entire language, while cool, is one big WTF.

It has the weirdest syntax I've ever seen. Instead of GoTo, you use :(label). And who needs if's when you have :S(label) (goto label on success/true) and :F(label) (goto label on failure/false) and you use those functions on the line checking some condition or reading a file. So the statement:

H = INPUT :F(end)

will read the next line from a file or the console and will go to the label "end" if the read fails (because EOF is reached or any other reason).

Then there is the $sign operator. That will use the value in a variable as a variable name. So: ANIMAL = 'DOG' DOG = 'BARK' output =$ANIMAL

will put the value 'BARK' on teh console. And because that isn't weird enough:

$DOG = 'SOUND'  will create variable named BARK (see the value assigned to DOG above) and give it a value of 'SOUND'. The more you look at it, the worse it gets. The best statement I ever found about SNOBOL (from link text) is "the power of the language and its rather idiomatic control flow features make SNOBOL4 code almost impossible to read and understand after writing it. " • the name of the language alone deserves an upvote here – kristof Jan 5 '10 at 10:52 • Perl has all those features... – Andrew McGregor Jan 5 '10 at 13:18 • PHP allows that aswell,$animal = "dog"; $dog = "bark"; echo$$animal; – Kristoffer Sall-Storgaard Jan 5 '10 at 13:38 • I should've also mentioned that my university didn't have a SNOBOL compiler. Instead, we had a SPITBOL compiler. – Jeff Siver Jan 5 '10 at 14:40 • The one time I wrote a class assignment in SNOBOL, it wasn't any fun at all. That language desperately needs better control structures, and that's far more important than having only one sort of statement (including label, main variable/value, pattern matching part or all of the former, equal sign, value to assign, and labels to jump to). – David Thornley Jan 6 '10 at 16:12 In Haskell: let 2 + 2 = 5 in 2 + 2  yields 5. • Can I get an explanation to this? – Bobby Aug 17 '10 at 10:17 • The reason is very straightforward: In haskell, (like in most other languages) you can simply redefine a function locally (haskell treats operators in the exact same way as "normal" functions), this lets the outer operator be hidden. After this, he just pattern matches against the values 2 and 2, like it's common in haskell. For instance, let 2 + 2 = 5 in 2 + 3 would yield a pattern matching failure. – fuz Aug 17 '10 at 10:54 • this is the same as let (+) = \2 2 -> 5 in (+) 2 2 – user954298 Aug 17 '10 at 12:14 • It also works in F#: let (+) 2 2 = 5 in 2 + 2 – Ming-Tang Sep 4 '10 at 17:52 Perl is full of odd but neat features. if may be used before or after the statement like this: print "Hello World" if$a > 1;
if ($a > 1) { print "Hello World"; }  The same is true for foreach: print "Hello$_!\n" foreach qw(world Dolly nurse);

• Ruby allows the same if-structures. It's one of my favourites from that language. – Matt Grande Jan 4 '10 at 18:06
• Ruby allows this, but personally I think that consistently putting the "if" first makes code easier to read. It's just as grammatically pleasing, anyway, and you don't have to reorder if you need to add an else. – Nathan Long Jan 4 '10 at 21:03
• I actually like the if modifier as it makes the programs more expressive and readable. it lets you place the more important part of the statement (the condition or the action) before the other so it is prominent. Like any feature, it is useful if used judiciously and not abused. – Prakash K Jan 5 '10 at 14:22
• fyi, this is called mutator syntax, and is syntactically awesome: last unless defined $row – Evan Carroll Jan 5 '10 at 18:24 • Good thing I can control myself, or I would start to rant on Perl... odd, sure. Neat? Not by a... okay, I shut up. – Jürgen A. Erhard Jan 14 '10 at 3:34 In PHP "true", "false" and "null" are constants which normally cannot be overridden. However, with the introduction of namespaces in PHP >=5.3, one can now redefine these constants within any namespace but the global namespace. Which can lead to the following behaviour : namespace { define('test\true', 42); define('test\false', 42); define('test\null', 42); } namespace test { var_dump(true === false && false === null); // is (bool) true }  Of course if you want your trues to be true, you can always import true from the global namespace namespace test { var_dump(\true === \false); // is (bool) false }  • If it was global, define("TRUE",false,false);define("true",true,false); would be echo(TRUE==false);//echos true and echo(true==false);//echos false... TRUE would then mean not really true, unless its lowercase. – Talvi Watia Jun 11 '10 at 2:39 LOLCODE! The whole language itself. While not exactly a WTF thing, I've never come across a language which plays out in my head in a squeeky cartoony voice. Nor have I ever looked at code before and want to exclaim "aaaawwww cuuute!" This program displays the numbers 1–10 and terminates HAI CAN HAS STDIO? IM IN YR LOOP UPPIN YR VAR TIL BOTHSAEM VAR AN 10 VISIBLE SUM OF VAR AN 1 IM OUTTA YR LOOP KTHXBYE  • Well, the language is ment as a joke, just as INTERCAL. I don't see such a great "weird feature in there"... – polemon Aug 29 '10 at 17:32 • I am sure WTF is a preserved keyword in Lolcode... – Shimmy Dec 30 '10 at 22:58 In PHP: echo 'foo' == 0; // echos '1' echo 'foo' == true; // echos '1' echo 0 == true; // echos '0'$foo = 'foo';
echo $foo['bar'] // echos 'f'  PHP has some of the most annoying type coercion... • If it's not clear above, true and false are equal to 1 and 0, respectively (and their names aren't case sensitive. TRUE or FALSE or True or False also work.) – pib Jan 16 '10 at 20:04 • == != = ....sigh – Talvi Watia Jun 11 '10 at 1:38 • few semicolons missing – Tom Pažourek Jul 7 '10 at 9:44 In C or C++ you can have a lot of fun with Macros. Such as #define FOO(a,b) (a+b)/(1-a)  if FOO(bar++,4) is passed in it'll increment a twice. • or #define false true – RCIX Jan 3 '10 at 15:49 • #define while if (who needs loops?) #define void int ("Why is the compiler complaining about no explicit return from my void functions?") #define main(argv, argc) (main)(argc, argv) (switch argv and argc for no apparent reason) – Chris Lutz Jan 4 '10 at 7:24 • While there are dumb things you can do with #define, there ARE perfectly valid reasons for defining it the way it was defined. As opposed to some of the other entries here... – Brian Postow Jan 4 '10 at 20:49 • I've seen someone on stackoverflow mention #define private public. – luiscubal Jan 4 '10 at 21:50 • @wheaties: A long time ago, in a computer lab far away, some people came up with the C language. And C was without void at first; void only really appeared with standard C. That meant there were a heck of a lot of people with old compilers who wanted to run modern C code, and #define void int worked well enough to run some C90 code in K&R compilers. – David Thornley Jan 4 '10 at 22:37 Perl filehandle-style operator calls. In the beginning, there was print "foo", "bar", "baz"; # to stdout print STDERR "foo", "bar", "baz";  Notice the ostentatious lack of a comma so that you know that's a filehandle to print-to, not a filehandle to print in a stringified manner. It's a dirty hack. Language upgrade rolls around, they make proper OO filehandles and turn x FOO y, z, abc into FOO->x(y, z, abc). Kinda cute. The same print statement effectively runs STDERR->print("foo", "bar", "baz");  Mostly you notice this when you miss a comma, or try to run something like hashof$a, $b,$c (subroutine call without parentheses) and forget to import the hashof function into your namespace from its utility package, and you get a weird error message about "Can't call method 'hashof' via package 'contents of string $a'". • Dealing with filehandles in Perl is inconceivably awkward. They are not scalar values or objects, but a 3rd category that lacks decent syntactical support. So, a bunch of Perl modules like IO::Handle were created to make them more sensible -- except these objects don't work everywhere a real filehandle would. It's a good thing programmers don't need to work with files much. – j_random_hacker Jan 31 '10 at 7:42 In Python: >>> a[0] = "hello" NameError: name 'a' is not defined >>> a[0:] = "hello" NameError: name 'a' is not defined >>> a = [] >>> a[0] = "hello" IndexError: list assignment index out of range >>> a[0:] = "hello" >>> a ['h', 'e', 'l', 'l', 'o']  These slice assignments also give the same results: a[:] = "hello" a[42:] = "hello" a[:33] = "hello"  • I agree that the a[0:] and a[42:] should raise an IndexError - but both the a[:] and a[:33] style make sense and can be usefull. – jsbueno Jan 7 '10 at 12:51 • If a[:33] can be useful, why not a[42:]? Don't see the big difference. (And a[:] is of course incredibly useful) – Jürgen A. Erhard Jan 12 '10 at 6:35 Easy pickins, Erlang is full of them. For example, 3 forms of punctuation, a_function(SomeVariable) -> statements_end_with_commas(), case PatternMatching of 0 -> now_we_end_with_semicolon; true -> except_the_last_one end. %% Function definitions end with periods!  • I guess I should add that this isn't a language feature, per se, more of an odd choice in syntax. If I had to pick an odd language feature of Erlang, it would be the representation of strings as lists of integers. Meaning [80,97,117,108]. => "Paul". I wonder what implications this had for Facebook Chat, if any? – pablo.meier Jan 6 '10 at 21:26 • A C string is also just a list of integers, sort of. – Tor Valamo Jan 10 '10 at 19:55 More of a platform feature than a language feature: on the iPhone, create an infinite loop with a few computations inside and run your program. Your phone will heat up and you can use it as a hand-warmer when it's cold outside. • I don't think that that is very good for the phone, though... – Maximilian Mayerl Jan 7 '10 at 10:10 • +1 for best use of iPhone other than paperweight or thrown missile – iandisme Jan 15 '10 at 21:56 • I have an idea for an app… :) – user142019 Aug 8 '11 at 21:09 C/C++: The Fast Inverse Square Root algorithm takes advantage of the IEEE floating-point representation (code copied from Wikipedia): float InvSqrt(float x) { union { float f; int i; } tmp; tmp.f = x; tmp.i = 0x5f3759df - (tmp.i >> 1); float y = tmp.f; return y * (1.5f - 0.5f * x * y * y); }  • Ah, this is the one that came out of Quake code, right? Love it, though it's not quite a language feature... – Rei Miyasaka Oct 11 '10 at 20:26 • Yes, it's from Quake. You're right, this has more to do with the IEEE 32-bit floating-point representation than C/C++. – Jaime Soto Oct 11 '10 at 20:55 • What's the relation to Cs language features? – Nils Dec 6 '10 at 6:22 ## Bracket identifiers in VBScript VBScript has so-called bracket identifiers, which are identifiers defined enclosed in square backets, like this: [Foo]  They're quite handy, actually, as they allow you to name variables and routines after reserved words, call methods of third-party objects whose names are reserved words and also use almost any Unicode characters (including whitespace and special characters) in identifiers. But this also means that you can have some fun with them: [2*2] = 5 [Здравствуй, мир!] = [Hello, world!] [] = "Looks like my name is an empty string, isn't that cool?" For[For[i=0]=[0]To[To[To[0] [Next[To]([For[i=0])=[For[i=0] Next  On the other hand, bracket identifiers can be a gotcha in case you forget the quotes in a statement like this: If MyString = "[Something]" Then  because If MyString = [Something] Then is a perfectly legal syntax. (And that's why an IDE with syntax highlighting is a must!) More info on bracket identifiers in Eric Lippert's blog: • Hm... In pure mathematics, [x] is sometimes used to denote a very "formal" interpretation of x, as in [0/0] and [∞/∞] (two interesting cases of limits of fractions). Although I am aware of no precise definition of [], I would say something like "I mean exactly what I write, but do not try to interpret it literally". Related? – Andreas Rejbrand Jul 6 '10 at 13:46 In C or C++, parentheses are optional for the argument to sizeof ... provided the argument isn't a type: void foo() { int int_inst; // usual style - brackets ... size_t a = sizeof(int); size_t b = sizeof(int_inst); size_t c = sizeof(99); // but ... size_t d = sizeof int_inst; // this is ok // size_t e = sizeof int; // this is NOT ok size_t f = sizeof 99; // this is also ok }  I've never understood why this is! • That's because sizeof is not a function, it's a macro. – Nick Retallack Jan 7 '10 at 12:17 • It's not a macro, it's an operator. But that doesn't explain the discrepancy. – tragomaskhalos Jan 7 '10 at 13:10 In JavaScript (and Java I think) you can escape funny characters like this: var mystring = "hello \"world\"";  If you want to put a carriage return into a string though, that's not possible. You have to use \n like so: var mystring = "hello, \nworld";  That's all normal and expected- for a programming language anyway. The weird part is that you can also escape an actual carriage return like this: var mystring = "hello, \ world";  • The same as good old C syntax. But remember that standard HTML expects CR+LF ("\r\n") newlines. – David R Tribble Jan 7 '10 at 3:28 • @Loadmaster: sure you didn't mean HTTP? afaik HTML is line-separator-agnostic – Christoph Jan 10 '10 at 14:01 • It's quite handy, actually, when you need to define a veeeeery long string without messing with string concatenation operators. – Helen Feb 8 '10 at 19:26 • @Helen except that it kind of breaks down if you want to minify or do any other sort of automated whitespace manipulation. – Breton Feb 8 '10 at 21:57 • Doesn't work in Java, though – abahgat Dec 15 '11 at 17:36 In earlier version of Visual Basic, functions without a "Return" statement just "Return None", without any kind of compiler warning (or error). This lead to the most crazy debugging sessions back when I had to deal with this language on a daily basis. • Oh man this sucked. I was so glad they pulled that out in later versions of VB.NET. They really should have just created VB.Sharp and left the compatibility argument on the table. – Jeff Atwood Jan 4 '10 at 20:21 • I'd say the return statement that lets you return a value is one of my favorite improvements of VB.NET over VB6. (My #1 favorite is exceptions vs. On Error Goto.) – Nate C-K Jan 7 '10 at 5:35 • This is also one of python's features. In a strongly typed language it wouldn't be so bad, and one could actually argue that it's good because it makes things more uniform (all functions have a return value), but with dynamic typing it always causes me headache. – Joseph Garvin Jan 7 '10 at 15:47 • Python is both strongly and dynamically typed. Just sayin'... – Jürgen A. Erhard Jan 12 '10 at 6:50 String math in Perl is pretty weird. $ perl -E '$string = "a";$string++; say $string' b$ perl -E '$string = "abc";$string++; say $string' abd$ perl -E '$string = "money";$string++; say $string' monez$ perl -E '$string = "money";$string--; say $string' -1  • Very well-known, but still cool. – Andreas Rejbrand Jul 6 '10 at 13:42 • Also, incrementing "z" gives "aa". – dan04 Jul 16 '10 at 10:10 In PowerShell, you can rename variables: >$a = "some value"
> $b = "a" >$c = "d"
> Rename-Item variable:$b$c
> $d some value  Indirect indirection! Take that, PHP! Literals work, too: > Rename-Item variable:d e >$e
some value

• Yeah, PowerShell drives me mad. I wrote to them about the syntax one time. They responded, somewhat evasively: blogs.msdn.com/b/powershell/archive/2006/07/23/… – Rei Miyasaka Nov 20 '10 at 0:20
• The worse part is that, despite the WTFs, PowerShell looks clean when compared to other shells (sh, bash, csh, tcsh, comd.exe, etc.) and Perl. Ever tried to understand Bash's $(()),$(), [[]], etc.? – marcus Jun 13 '11 at 20:48
• @marcus: Yes, see my answers here and here for examples. – Dennis Williamson Jun 15 '11 at 11:25

in PHP the strings letters cannot be used like in C, you need to use ord() and chr() in order to convert from number to char and vica versa: "a" != 97, but ord("a") == 97.

Although, there is one exception:

for ($i = 'a';$i < 'z'; $i++) { print "$i ";
}

will print letters a to y. just like you would expect as if it was C style datatypes.

however if the test condition is changed to <= it will not print a to z as you would think, but a to yz! (total 676 items printed)

also if you change the 'z' to 'aa' which came out next after 'z' in the 676 items list, and change test condition to < again, you will see only "a" being printed out! not a to z as you would expect.

And if you change the incrementor to $i+=2 it will print only "a" again! only way to do that is to use $i++, $i++ in sequence, and now it works like expected. Nevertheless, this is a nice way in PHP to output combinations of letters a-z, although its very hard to actually use it. • "=UTF-8 vs '=ASCII. $i will render as an integer/byte – Talvi Watia Jun 11 '10 at 1:46

The bigest collection (today 1313) of decent and weird programming languages I know, you will find here: http://99-bottles-of-beer.net/ be prepared to see real weird stuff ;-) Everybody should make his one choice

Early FORTRAN where whitespace was not significant. (The anti-Python!)

DO 20 I = 1, 10

Meaning: loop from here to line 20 varying I from 1 to 10.

DO 20 I = 1. 10

Meaning: Assign 1.10 to the variable named DO20I.

Rumors are that this bug crashed a space probe.

Ruby

Time.parse often pretends that the parsing did not fail, returns now instead

require 'time'

Time.parse '2000-01-01 12:00:00'
# -> 2000-01-01 12:00:00 +0100

Time.parse '2000-99-01 00:00:00'
# -> ArgumentError: argument out of range ...

Time.parse 'now'
# -> 2010-08-13 21:26:13 +0200

Time.parse 'yesterday'
# -> 2010-08-13 21:26:18 +0200

Time.parse 'billion years ago'
# -> 2010-08-13 21:26:37 +0200

Java's access modifiers are a recent WTF to me (as I had to learn a bit of it).

Apparently packages are more intimate than class hierarchies. I can't define methods and attributes that are visible to sub-classes but not to other classes in the package. And why would I want to share the insides of a class to other classes?

But I can define attributes and methods that are visible to every class inside the package, but not to subclasses outside the package.

No matter how hard I think about this, I still can't see the logic. Switch over the access modifiers and make protected act like it works in C++ and keep the package private modifier as it is and it would make sense. Now it doesn't.

• The logic is pretty obvious to me: not a class, but a package is a unit of maintenance. Hiding members from subclasses effectively reserves the right for the maintainer to remove or rename them without breaking code in subclasses, which is pretty useful if those subclasses are in a different package, and were probably written by completely different people at a completely different time and place. The package maintainers may not even know the subclass exists at all. – reinierpost Jan 4 '10 at 9:18
• So only one person can update a package? Considering how large even some Java base packages are, that's a tall order. – Makis Jan 4 '10 at 10:49
• @Makis: "the maintainer" is the group that's responsible for the package (perhaps it should have been "the maintainers"). Programmers writing subclasses outside this group shouldn't have to worry about package internals. – outis Jan 5 '10 at 14:11
• Well, I still don't understand why that is better. Unless we keep packages really small, the problem is the same as in any application development: what you want to use OO for is exactly what this is countering. – Makis Jan 5 '10 at 18:53
• I always considered this strange too. The only reason I could thought up is the strict order of modifiers: public > protected > package > private – user57697 Jan 7 '10 at 11:10


text:
Label Encoding for Regression Networks

In the ICLR paper [Sha22L] the authors propose a highly creative reformulation of regression that significantly outperforms standard regression with neural networks (e.g. optimizing MSE) in their experiments. Their approach can be summarized as:

1. Quantizing the target $y$ into $N$ values (in a simple, uniform fashion)
2. Encoding the quantized targets as binary strings of size $M$. There are $2^{M \choose N}$ possible encodings
3. Instead of learning to predict the target directly, $M$ binary classifiers are trained to predict the entries of the binary code from step 2.
4. At inference time, the predicted codes are decoded back to the target. For that, either the binary predictions or the predicted confidences of the $M$ classifiers can be used.

The idea is inspired by earlier work on using classifiers for ordinal regression and on using binary encodings for phrasing multiclass classification as multi-binary-classification. The procedure described above was dubbed BEL regression. It has a lot of moving parts - there are many possible encoding and decoding schemes, also the hyperparameter $N$ is important. The authors perform a theoretical and experimental analysis of various schemes and give arguments for selecting some over the other. In the end, the best performing encoding / decoding choice depends on the dataset and the networks used. These best performing combinations outperformed standard regression by a large margin! See the figure below for their comparison on several data sets.

Thus, if you care a lot about having low-error neural network regression algorithms and don’t mind performing some hyperparameter search, this approach might be useful for you! The authors provide a reference implementation but for your own project you would likely need to do some implementing on your own.

This work does not compare the performance of BEL regression with non-network regressors like tree-based approaches which often outperform neural networks. Before implementing the proposed label encoding, I personally would first try something like LightGBM.

# References

• Label Encoding for Regression Networks, Deval Shah, Zi Yu Xue, Tor Aamodt. (2022)


text:
# Both sorted and unsorted glossaries with glossaries package?

Using the glossaries package, is it possible to have two glossaries:

1. one unsorted (i.e., in order of appearance/use) such as for an index of notation, and
2. the other sorted alphabetically such as for a glossary of terminology?

If so, how?

Note: I also require that the glossary of notation be subdivided into groups. I know how to do that with just one glossary, but will the usual method interfere if I also have the 2nd kind of glossary?

In case it matters, I'm using the memoir documentclass.

• Since there are two commands, \makeglossaries and \makeindexI'm not sure if you need to try and create two different sorting schemes from \printglossary[type=XXX,style=YYY]. The glossaries package can also put the page numbers your item is used on with your sorted entry. The only other way of sorting a glossary that I am aware of offhand, is through the sort={} field in \newglossaryentry which is somewhat manual, but with a custom style that prints the page numbers, would give you what you describe, even though it's not nice and automatic. – EngBIRD Nov 16 '16 at 1:46

You need to use \makenoidxglossaries and \printnoidxglossary instead of \makeglossaries and \printglossary if you want different sorting rules. The noidx variant uses TeX to sort and collate rather than using makeindex or xindy. It would've helped to have had a minimal working example (MWE) to illustrate the type of entries you have. The following example uses dummy entries provided in the test files example-glossaries-brief.tex and example-glossaries-symbols.tex which should be installed with glossaries.sty:

\documentclass{memoir}

\usepackage[symbols]{glossaries}
\usepackage{glossary-mcols}

\makenoidxglossaries

\begin{document}

\printnoidxglossary[sort=word,style=mcolindexgroup]
\printnoidxglossary[type=symbols,sort=def,style=index]

\end{document}

The main glossary is sorted according to word order:

The symbols glossary is unsorted, although it looks sorted in this example simply because the sample entries were defined in that order. Another option is sort=use which orders the entries according to the first use in the document. There are many different predefined glossary styles to choose from.

Note that the noidx method is very inefficient at sorting alphabetically and is only designed for ASCII. The glossaries-extra extension package allows a mix and match approach so that makeindex/xindy may be used for the alphabetical sorting and \printnoidxglossary can be used for order of definition or order of use. (With sort=def or sort=use, \printnoidxglossary simply iterates through an internal list, which means no actual sorting is performed).

\documentclass{memoir}

\usepackage[symbols]{glossaries-extra}
\usepackage{glossary-mcols}

\makeglossaries[main]

If you use makeglossaries, it will detect that makeindex is only required for the main glossary.


text:
# Biggest number

1. Oct 21, 2004

### T@P

The question: You have to write the biggest number you can with a limited space, in other words each digit (including mathematical symbols) has an area of 1 unit. What is the greatest number that can be written on an area of x units? (note the area can be rearanged, so 2^5 would be two units, and your "paper" can go in any direction)

note i havent done the problem...

2. Oct 21, 2004

### NoTime

$$\infty$$

xxxxxxxxxxxxxxxx

3. Oct 22, 2004

### Pyrrhus

infinity is not a number.

4. Oct 22, 2004

### Noticibly F.A.T

9^9^9 maybe.

5. Oct 22, 2004

### T@P

possibly, and actually 9^9 is what i thought too, but there are other ways. for example taking a eally smail number, and then 1 over that number would be quite big. But what i was actually lookng for was a proof that some way is the most efficient way... thanks for your posts

6. Oct 22, 2004

### NateTG

7. Oct 22, 2004

### NoTime

I'm not so sure about that.
Mathematically any given line segment is composed of an infinite number of points.
Given the existence of a plank length, it would seem to resolve to a specific number.

8. Oct 22, 2004

### wave

Infinity is an abstract concept. Please explain how you can "resolve" a specific number for infinity.

9. Oct 22, 2004

### Rogerio

A line segment is not a real object! Plank doesn't apply!

It sounds the same as "there is a limited amount of real numbers in the [0,1] interval"...

10. Oct 22, 2004

### Gokul43201

Staff Emeritus
This makes no sense at all.

"Infinity" is simply not defined in the reals.

11. Oct 22, 2004

### NoTime

Is a line segment that I draw any less real than say a square or a triangle?
In other words -> Why does Plank not apply?

Infinity is a concept that was developed long before knowledge of QM.
Ma Nature seems to be saying that this is indeed the case.
Personally, I'm inclined to take Ma's word for it rather than the imagination of man.

I will note that Zeno's arrow unerringly hits the target in real life.
Does this mean that there comes a place where you can no longer half the distance?

12. Oct 22, 2004

### Hurkyl

Staff Emeritus
Mathematical ideas and concepts are defined by axioms and definitions -- formal statements in the language of mathematics. In particular they are not defined by "reality".

There is no "Planck length" for a mathematical line segment because it is not a logical consequence of the axioms.

As an aside, there is no evidence (experimental or theoretical) that there is a smallest unit of length in reality either; "Planck length" doesn't mean what you think it means.

13. Oct 22, 2004

### NoTime

I do know what an axiom is.
I also do not deny that they say exactly what you say they do.

Why should mathematics be exempt from reality?

To my knowledge the "Planck length" was determined from experimental data.
It is also the root of QM.
Is it a discontinuity or simply a region within which a determination cannot be made seems (at least to me) more problematical.
I connect with the idea that you do not like what I have said.
OTOH I have seen a number of arguments against the existence of infinities.

Why should I take one side or the other?

14. Oct 22, 2004

### wave

Infinity is an abstract concept, so it does not have any concrete existence. Furthermore, infinity is not a particular number by definition. It's fine if you don't agree. Create a new word to label your concept, instead of arguing against an established definition.

15. Oct 23, 2004

### Gokul43201

Staff Emeritus
The reals are a well defined field and do not contain an element known as "infinity". That is how they are defined, and that is that.

However, "infinity" appears in Cantorian Set Theory, (surreal numbers) as the cardinality of certain sets.

Mathematics is simply a set of results obtained through an established logical method, based on a certain framework of axioms. There is no need for a mathematical structure to have any physical meaning.

And for your information, the Planck Length is not determined from experimental data. It is simply a length scale obtained by a suitable manipulation of fundamental physical constants ($= \sqrt {hG/c^3} ~ or~about~10^{-35} m$ ). Presently, that is too small to "measure".

16. Oct 23, 2004

### Hurkyl

Staff Emeritus
Because mathematics does not deal with reality; it deals with the formal consequences of axioms.

An axiom is simply a logical statement; it gets the name "axiom" because of the way we use the statement.

Here are a few of the axioms of the real numbers:

a + (b + c) = (a + b) + c
a * b = b * a
a * (b + c) = a * b + a * c

There are about 10 in all, of varying complexity.

Now, it is entirely possible to define "infinity" to mean "7". If you did so, then infinity really would be a real number. However, no standard definition of infinity yields a real number. In fact, no definition of infinity yields an object that is any sort of familiar number! In particular, Infinite numbers (such as those in the Surreals, or in the Cardinals) are not called "infinity".

Any sort of connection between mathematics and "reality" falls under the purview of science. It is science that says real numbers have some sort of connection to the real world. If science determined that there was a fundamental length, then that would mean that science would no longer attempt to say that real numbers are lengths; it does not mean that the mathematical meaning of a real number should change.

17. Oct 23, 2004

### NoTime

Nice statement
It addresses the heart of the matter for me.
To the degree that attempts are being made to define "reality" using mathematics, then what is the requirement for mathematics to conform to "reality"?

The fundamental physical constants are called that because they can ONLY be determined by experiment.
Show me an equation that produces c. NOT a value for c, but c itself.

Take a simple well known case.
A bound electron has energy levels A and B.
What is the set of points in the interval AB?
I say that set is empty.
You can imagine that an electron can have any energy level between A and B.
It just is not true.

18. Oct 23, 2004

### NoTime

Good Point.

Ok. I will change my initial answer to

$$\pi$$

Nothing was said in the original question about "biggest" being a value.
The number of digits in Pi is claimed to be.... nevermind :surprised

19. Oct 23, 2004

### Hurkyl

Staff Emeritus
There is no requirement that mathematics conforms to reality. The general procedure is that you take experimental data and theoretical/philosophical reasoning to extract axioms. Those axioms then define a mathematical theory, and then the hypothesis is that this theory is what will describe reality.

For example, experiment led to Maxwell's equations, and from these, Einstein reasoned that the speed of light must be constant in all inertial reference frames. Einstein then took this as an axiom, which defined a new theory, then he went on to flesh out this theory and arrived at Special Relativity.

20. Oct 23, 2004

### wave

Please explain what you mean. The speed of light in a vacuum, denoted by c, is a value. What else can c be, if not a value?


text:
# Plotting f(x) = sin x and f(x) = cos x on pgf/tikz

I'm new to graphing on TikZ/pgf and have been trying to figure this out but can't seem to find any helpful info for what I want. I want to plot the above functions over one period, i.e. on the interval [0,2\pi] and would like the tick marks on the graph to show the values 0, pi/2 , pi, 3pi/2, and 2pi. Would appreciate some help.

With pgfplots

\documentclass[border=4mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.12}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
clip=false,
xmin=0,xmax=2*pi,
%axis lines=left,
%axis x line=middle,
%axis y line=left,
xtick={0,1.57,3.14,4.71,6.28},
xticklabels={$0$, $\frac{\pi}{2}$,$\pi$,$\frac{3}{2}\pi$,$2\pi$}
]
\addplot[domain=0:2*pi,samples=200,red]{sin(deg(x))}node[right,pos=0.9]{$f(x)=\sin x$};
\addplot[domain=0:2*pi,samples=200,blue]{cos(deg(x))}node[right,pos=0.9]{$f(x)=\cos x$};
\end{axis}
\end{tikzpicture}
\end{document}

Another variant will be:

\documentclass[border=4mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.12}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
clip=false,
xmin=0,xmax=2.5*pi,
xlabel= $x$,
ylabel=$f(x)$,
ymin=-1.5,ymax=1.5,
axis lines=middle,
%axis x line=middle,
%axis y line=left,
%     axis x line=middle,
xtick={0,1.57,3.14,4.71,6.28},
xticklabels={$0$, $\frac{\pi}{2}$,$\pi\,$,$\,\,\,\frac{3}{2}\pi$,$\,\,\,2\pi$},
%xticklabel style={anchor=north west}
]
node[right,pos=0.9,font=\footnotesize]{$f(x)=\sin x$};
node[right,pos=1,font=\footnotesize]{$f(x)=\cos x$};

• Change node[right,pos=0.9]{$f(x)=\sin x$}; to node[right]{$\scriptstyle f(x)=\sin x$}; – Maarten Dhondt Mar 3 '15 at 7:18


text:
# Why Eat Goldstone Boson?

by waterfall
Tags: boson, goldstone
 P: 381 I've been reading this very good new book "The Infinity Puzzle" by Frank Close and it mentioned a lot about the Higgs boson eating the Goldstone Boson to have mass. I already checked out wikipedia but can't get an intuitive feel of it. Can anyone explain in a brief mathematical sense why the goldstone boson which has no mass has to be eaten by the higgs boson to make it massive... It would have been more intuitive if the goldstone boson is massive.. and the higgs boson eating it would turn it massive.. but the goldstone is massless. Thanks.
 Sci Advisor P: 5,366 Have you ever seen the mexican hat potential? http://en.wikipedia.org/wiki/Mexican..._hat_potential http://en.wikipedia.org/wiki/File:Me...tial_polar.svg Ok, let's first discuss the situation w/o gauge fields The Goldstone symmetry breaking mechanism says that the rotational symmetry is broken, i.e. that the state is not a V=0 but at V=Vo at some arbitrary angle θo. Now this angle θo breaks the rotational symmetry. OK, the Goldstone bosons are nothing alese but field fluctutaions at constant V=Vo but fluctuating angle θ(x); b/c the potential is constant in θ-direction there is no force and therefore the fluctuations correspond to massless particles. Now let's add gauge fields That means that we do have a local symmetry where at every spacetime point x a local rotational symmetry in θ is present; that means that fluctuations in θ(x) do no longer correspond to particles but to gauge transformations and are therefore unphysical; any change in θ(x) at some spacetime point x can be rotated away by a gauge transformation. That's the reason why Goldstone bosons disappear. Now back to Vo http://en.wikipedia.org/wiki/Higgs_mechanism The radial fluctuations in V still correspond to physical fluctuations, i.e. to Higgs bosons. But instead of having a fluctuation around V=0 we have one around V=Vo. This Vo corresponds to a mass term of the gauge fields (to understand that one has to look into the mathematical details). The interesting thing is the following: massless gauge fields always correspond to two physical polarizations (two transversal polarizations for el.-mag. waves i.e. two d.o.f. for the photon), but massive vector fields have three degrees of freedom. So what happens is that one degree of freedom disappears from the Higgs field (the θ fluctuation) and reappers as the third (longitudinal) poolarization of the gauge field. → the gauge field eats the massless Goldstone boson
P: 381
 Quote by tom.stoer Have you ever seen the mexican hat potential? http://en.wikipedia.org/wiki/Mexican..._hat_potential http://en.wikipedia.org/wiki/File:Me...tial_polar.svg Ok, let's first discuss the situation w/o gauge fields The Goldstone symmetry breaking mechanism says that the rotational symmetry is broken, i.e. that the state is not a V=0 but at V=Vo at some arbitrary angle θo. Now this angle θo breaks the rotational symmetry. OK, the Goldstone bosons are nothing alese but field fluctutaions at constant V=Vo but fluctuating angle θ(x); b/c the potential is constant in θ-direction there is no force and therefore the fluctuations correspond to massless particles. Now let's add gauge fields That means that we do have a local symmetry where at every spacetime point x a local rotational symmetry in θ is present; that means that fluctuations in θ(x) do no longer correspond to particles but to gauge transformations and are therefore unphysical; any change in θ(x) at some spacetime point x can be rotated away by a gauge transformation. That's the reason why Goldstone bosons disappear. Now back to Vo http://en.wikipedia.org/wiki/Higgs_mechanism The radial fluctuations in V still correspond to physical fluctuations, i.e. to Higgs bosons. But instead of having a fluctuation around V=0 we have one around V=Vo. This Vo corresponds to a mass term of the gauge fields (to understand that one has to look into the mathematical details). The interesting thing is the following: massless gauge fields always correspond to two physical polarizations (two transversal polarizations for el.-mag. waves i.e. two d.o.f. for the photon), but massive vector fields have three degrees of freedom. So what happens is that one degree of freedom disappears from the Higgs field (the θ fluctuation) and reappers as the third (longitudinal) poolarization of the gauge field. → the gauge field eats the massless Goldstone boson
Thanks for the descriptions. I thought the Mexican Hat thing was only used as example in Inflaton fields where the idea is simplier but this can also be used in how the Higgs Eat the Goldstone too. I'd go back to the Mexican Hat portion in the book as the key to the understanding may lie there.

P: 381
Why Eat Goldstone Boson?

 Quote by tom.stoer Have you ever seen the mexican hat potential? http://en.wikipedia.org/wiki/Mexican..._hat_potential http://en.wikipedia.org/wiki/File:Me...tial_polar.svg Ok, let's first discuss the situation w/o gauge fields The Goldstone symmetry breaking mechanism says that the rotational symmetry is broken, i.e. that the state is not a V=0 but at V=Vo at some arbitrary angle θo. Now this angle θo breaks the rotational symmetry. OK, the Goldstone bosons are nothing alese but field fluctutaions at constant V=Vo but fluctuating angle θ(x); b/c the potential is constant in θ-direction there is no force and therefore the fluctuations correspond to massless particles. Now let's add gauge fields That means that we do have a local symmetry where at every spacetime point x a local rotational symmetry in θ is present; that means that fluctuations in θ(x) do no longer correspond to particles but to gauge transformations and are therefore unphysical; any change in θ(x) at some spacetime point x can be rotated away by a gauge transformation. That's the reason why Goldstone bosons disappear. Now back to Vo http://en.wikipedia.org/wiki/Higgs_mechanism The radial fluctuations in V still correspond to physical fluctuations, i.e. to Higgs bosons. But instead of having a fluctuation around V=0 we have one around V=Vo. This Vo corresponds to a mass term of the gauge fields (to understand that one has to look into the mathematical details). The interesting thing is the following: massless gauge fields always correspond to two physical polarizations (two transversal polarizations for el.-mag. waves i.e. two d.o.f. for the photon), but massive vector fields have three degrees of freedom. So what happens is that one degree of freedom disappears from the Higgs field (the θ fluctuation) and reappers as the third (longitudinal) poolarization of the gauge field. → the gauge field eats the massless Goldstone boson
Reviewing the book, I found it also described what you said. But I need some further information that not even wiki is clear about.

Are you saying in the gauge U(1) in electromagnetism for example, the gauge freedom in the phase from 0 to 360 degrees have corresponding Goldstone boson? Where are they? Aren't they detectable? How do you measure it if detectable and if not.. how do you make it detectable? Let's just take the example of electromagnetism and not electroweak to make distinct the concept. Thanks.
P: 381
 Quote by waterfall Reviewing the book, I found it also described what you said. But I need some further information that not even wiki is clear about. Are you saying in the gauge U(1) in electromagnetism for example, the gauge freedom in the phase from 0 to 360 degrees have corresponding Goldstone boson? Where are they? Aren't they detectable? How do you measure it if detectable and if not.. how do you make it detectable? Let's just take the example of electromagnetism and not electroweak to make distinct the concept. Thanks.
You may say that goldstone boson only come about in spontaneous symmetry breaking. But the em gauge applies to it.. or specifically wave functions or schroedinger equations with phases from 0 to 360 degrees. The symmetry occurs when the system is or particle is not yet emitted from for example the double slit emitter. After it it emitted. The phase is random from 0 to 360 degrees.. so here it is spontaneous symmetry breaking.. so where is the goldstone boson here?
P: 5,366
 Quote by waterfall Are you saying in the gauge U(1) in electromagnetism for example, the gauge freedom in the phase from 0 to 360 degrees have corresponding Goldstone boson?
No.

A Goldstone boson is always related to the spontanerous brealdown of a global symmetry, not to a local gauge symmetry. So in U(1) gauge theory a Goldstone boson cannot exist.

In U(1) electromagnetism an abelian Higgs mechanism (w/o a Goldstone boson) could exist , but then you would have to add a charged scalar field with a Mexican hat potential; b/c we do not see such a field in nature this model is a purely theoretical contemplation. What we observe instead is the non-abelian Higgs mechanism (hopefully including the Higgs - to be discovered at the LHC) in the full el.-weak theory. This mechanism slightly more complicated but the basic idea is rather similar. In contrast to the abelian Higgs mechanism not the abelian but the non-abelian gauge symmetry is broken; that's why not the (abelian) photon but the (non.-abelian) W and Z are massive.
P: 381
 Quote by tom.stoer No. A Goldstone boson is always related to the spontanerous brealdown of a global symmetry, not to a local gauge symmetry. So in U(1) gauge theory a Goldstone boson cannot exist. In U(1) electromagnetism an abelian Higgs mechanism (w/o a Goldstone boson) could exist , but then you would have to add a charged scalar field with a Mexican hat potential; b/c we do not see such a field in nature this model is a purely theoretical contemplation. What we observe instead is the non-abelian Higgs mechanism (hopefully including the Higgs - to be discovered at the LHC) in the full el.-weak theory. This mechanism slightly more complicated but the basic idea is rather similar. In contrast to the abelian Higgs mechanism not the abelian but the non-abelian gauge symmetry is broken; that's why not the (abelian) photon but the (non.-abelian) W and Z are massive.
You are trying to distinguish between U(1) gauge theory vs U(1) electromagnetism, are they not the same? Are you saying U(1) electromagnetism is a global gauge symmetry that is why you could induce an abelian higgs mechanism by adding a charged scalar field? In this case,
How do you add a charged scalar field with a Mexican hat potential to U(1) electromagntism and induce an abelian Higgs mechanism to give the photon mass? Thanks.
 Sci Advisor P: 5,366 I am not trying to distinguish U(1) gauge theory and U(1) electromagnetism. The abelian Higgs mechanism I am describing would be an extra ingredient which is not present in U(1) electromagnetism b/c there is no scalar charged particle. Regarding the abelian Higgs mechanism please have a look here: http://en.wikipedia.org/wiki/Higgs_m...iggs_Mechanism
P: 381
 Quote by tom.stoer I am not trying to distinguish U(1) gauge theory and U(1) electromagnetism. The abelian Higgs mechanism I am describing would be an extra ingredient which is not present in U(1) electromagnetism b/c there is no scalar charged particle. Regarding the abelian Higgs mechanism please have a look here: http://en.wikipedia.org/wiki/Higgs_m...iggs_Mechanism
I've been studying the site above and others. It ends with the statement:

"Furthermore, choosing a gauge where the phase of the vacuum is fixed, the potential energy for fluctuations of the vector field is nonzero. So in the abelian Higgs model, the gauge field acquires a mass. To calculate the magnitude of the mass, consider a constant value of the vector potential A in the x direction in the gauge where the condensate has constant phase. This is the same as a sinusoidally varying condensate in the gauge where the vector potential is zero. In the gauge where A is zero, the potential energy density in the condensate is the scalar gradient energy:"

But I can't get what is the whole purpose of the section when we know a photon has no mass and there is no charged scalar. Is it trying to say what would happen if there is a charged scalar? But it appears to be solving for someting definite and not at all concluding that this makes the photon massless because there is no charged scalar particle.