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How To Write The Less Than Symbol In LaTeX?

How To Write The Less Than Symbol In LaTeX?

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This article aims to show you the simplest and easiest way to write the less than symbol in LaTeX.

The less than symbol is part of the inequalities symbols you are familiar with, used in everyday life. That is why today you will learn more about this particular one, and how to write this symbol in your LaTeX document.

Inequality symbols

In mathematics, by definition an inequality is a relation that makes a non-equal comparison between two values, numbers, or expressions. To represent an inequality we used the inequality symbols, there are  different kinds of inequalities, for example,

Today we center our attention on the less than a symbol. This symbol or sign denoted by “<”, states that the first number is less than the second one.

The less than inequality is known as a strict inequality, meaning that 1 is strictly less than 3.

Less than symbol in LaTeX

If you know your keyboard you know where the key “<” is, you could be tempted to use it, but sooner you would see that the output does not look like what you expected. Therefore this symbol cannot be used in text mode.

In LaTeX to write the “<” symbol, you only can use it inside math mode -denoted by a pair of dollar signs- in the latex code. The complete command would look like $<$, let’s see an example

\documentclass{article}
\begin{document}
Less than command \verb|$<$|: $<$ \\
$100 < 101$ \\
\end{document}

Going the extra mile, in case you are wondering how to write the less than or equal symbol -this is known as a not strictly inequality- you use the command \leq inside math mode, alongside the amssymb package.

\documentclass{article}
\begin{document}
Less than or equal \verb|\leq|: $\leq$ \\
$a \leq b$
\end{document}

There you have it, the LaTeX commands for the less than, and the less than or equal symbols.

Another thing to notice is that there could be more combinations that include both of these symbols but they are not so common.

I hope this tutorial was helpful, and as always keep writing in LaTeX

All images were created in LaTeX by the Author.


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# Tangent spaces at different points and the concept of connection

If $M$ is a smooth manifold and $TM$ is the tangent bundle clearly $T_pM\cong T_qM$ (as vector spaces) for every $p,q\in M$. Nobody ensures that the previous vector spaces isomorphism is natural (or canonical). In $\mathbb R^n$ we have that $T_p\mathbb R^n$ and $T_q\mathbb R^n$ are naturally isomorphic to $\mathbb R^n$ so we can differentiate a vector field along a direction, in the usual way so taking the directional derivatives of each component.

If the isomorphism between tangent spaces in different point isn't natural, why can't we differentiate a vector field in the usual way? The problem is comparing vectors belonging in different (isomorphic) vector spaces; but we can send the two vectors, with an isomorphism, in a common vector space and then subtract them. Where is the importance of a natural isomorphism?

-
If you change what you mean by "with an isomorphism in a common vector space" then your result pulled back to the original tangent spaces will change. That is, what you are proposing will depend on how you make your two tangent spaces isomorphic to a common vector space. There are lots of isomorphisms between two vector spaces of the same dimension. – KCd Nov 19 '12 at 22:35
It is true that there are lots of isomorphisms between vector spaces, but there are also lots of natural isomorphisms. For example canonical automorphisms of $V$ are the elements of $Z(GL(V))$, so homothetic transformations. – Galoisfan Nov 20 '12 at 19:24
So even if I choose a natural isomorphism, the directional derivative will not depends on the basis imposed on tangent spaces, but it depends on the choice of the natural isomorphism. – Galoisfan Nov 20 '12 at 19:28
What Kofi says is not quite right (e.g., a finite-dimensional vector space and its double dual are naturally isomorphic), but I suppose what Kofi means is that two vector spaces of the same dimension are generally not isomorphic in a canonical way. Galoisfan, do you think tangent spaces at different points of a general (abstract) manifold are naturally isomorphic in some way? What does the term "natural isomorphism" mean to you? – KCd Nov 20 '12 at 21:13
There is no such thing as a canonical isomorphism in general between two vector spaces and it is hopeless to prove there are "lots" of them between two fixed vector spaces. As an analogy, there is no such thing as a natural inner product on a (finite-dim.) real vector space. Let $V$ be the vector space of real polynomials $a+bx+cx^2$ that vanish at 45. There are many inner products on $V$, but none is "natural" or "canonical", right? – KCd Nov 21 '12 at 16:01
show 5 more comments

## 2 Answers

If a manifold has curvature, then parallel transport of vectors depends on the path. In other words, any map between tangent spaces at different points is dependent on an arbitrary choice which is particular for each pair of points and each particular manifold. It is the same problem as when relating a finite-dimensional vector space $V$ with its dual $V^*$. In order to do so, we need to fix a basis for $V$.

In regards to the importance of naturality, informally a transformation is called natural if it is "independent of its source" in a sense. For example, the isomorphism between $V$ and $V^{**}$, the double algebraic dual, is natural. We send $v\in V$ to $v^{**}\in V^{**}$ such that $v^{**}(f)=f(v)$ for all $f\in V^*$. Note that this is independent not only of the basis of $V$, but also does not reference anything particular about $V$, such as a basis, and so this isomorphism can be carried out consistently over all finite-dimensional vector spaces over a common field. Quoting Wikipedia, a transformation is not natural if it cannot be extended consistently over the entire category in question.

-
 many books say, without explaining the meaning of the adjective "natural", that the parallel transport give a way to naturally identify tangent spaces on points which can be connected by a path. – Galoisfan Nov 25 '12 at 11:54 I will explain better my doubt: If on a smooth manifold we haven't a natural identification beetwen tangent spaces, than we can't define a derivative of a vector field as "a limit of a quotient". Once we define a parallel transport, then a covariant derivative of a vector field $V$ along a curve $\gamma$ is $$lim_{t\to t_0}\frac{P^{-1}_{t_0,t}V(t)- V(t_0)}{t-t_0}$$ So i guess that $P_{t_0,t}$ is a natural isomorphism because allows to write the covariant derivative as a limit of a quotient. – Galoisfan Nov 25 '12 at 12:11 just quoting the book "Manifold and Differential Geometry - Jeffrey M. Lee" on page 502, talking about the Koszul connection: "...The definition of this connection takes advantage of the natural identification of tangent spaces..." – Galoisfan Nov 25 '12 at 13:11 I haven't read Lee's book, and I am not too familiar this this construction, but this isomorphism seems to depend on the choice of metric for the manifold and so cannot be natural. Maybe he means that for a manifold with this extra structure, the tangent spaces have natural isomorphisms between them (I cannot see why this would be true), but then we are no longer talking about general differentiable manifolds, and there is in general several connections on a given manifold. – espen180 Nov 25 '12 at 14:12

Why don't you let $M = S^2 \subset \mathbb{R}^3$ ? The equation for the Tangent plane is $$T_{(a,b,c)}S^2 = \{ (x-a)a+(y-b)b+(c-z)c = 0 \}$$ This is a collection of planes parameterized by points $(a,b,c): a^2 + b^2 + c^2 = 1$.

We can define an initial vector $\vec{x}=(x,y,z) \in T_{(a,b,c)}S^2$. Then we can ask $$\vec{x}(t + \delta t) = A(t) \vec{x}(t) \in T_{(a,b,c) + \vec{v}\delta t}S^2$$ For any time-paramterized matrix $A$, we should get a connection on the sphere that necessarily Levi-Civita connection, since this flow is not the geodesic flow.

-

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# Provision of Tier-2 computer hardware and system support from February 2010 to March 2011.

Lead Research Organisation: University of Liverpool
Department Name: Physics

### Abstract

The GRID is the world-wide computing system being constructed to provide the computing power needed to analyse the data being taken with the new large hadron collider (LHC) based at CERN that started running in 2009. This computing power is spread over large central clusters (Tier-1) and based in individual University departments (Tier-2). Hundreds of thousands of computers will be needed. Everything is connected together using the Internet and jobs are run automatically in such a way that the computers are used in the most efficient manner.

### ORCID iD

David Hutchcroft (Principal Investigator)

### Publications

10 25 50
Aaij R (2014) Search for CP violation in D ± ? K S 0 K ± and D s ± ? K S 0 p ± decays in Journal of High Energy Physics

Aaij R (2015) Measurement of forward Z ? e+e- production at s = 8 $$\sqrt{s}=8$$ TeV in Journal of High Energy Physics

Aaij R (2014) Observation of Z production in proton-lead collisions at LHCb in Journal of High Energy Physics

Aaij R (2014) Observation of the B s 0 ? J/?K S 0 K ± p ± decay in Journal of High Energy Physics

Aaij R (2014) Observation of the ? b 0 ? J/? pp - decay in Journal of High Energy Physics

Aaij R (2014) Search for Majorana neutrinos in B- ? p+ µ- µ- decays. in Physical review letters

Aaij R (2015) Observation of the B(s)(0)??'?' Decay. in Physical review letters

Aaij R (2014) First observation of a baryonic Bc+ decay. in Physical review letters

Aaij R (2014) Evidence for CP violation in B+ ? ppK+ decays. in Physical review letters

Aaij R (2014) Measurement of the ? b - and O b - baryon lifetimes in Physics Letters B

Aaij R (2015) Measurement of CP Violation in B0?J/?K{S}^{0} Decays. in Physical review letters

Aaij R (2014) Dalitz plot analysis of B s 0 ? D ¯ 0 K - p + decays in Physical Review D

Description The findings have improved the knowledge of the fundamental parameters of the universe. Including all of the particle physics experimental results from the LHC and other particle physics projects for the last seven years. Including the discovery of the Higgs and associated cultural impact.
First Year Of Impact 2010
Sector Education
Impact Types Cultural


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Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
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Search Results: 1 - 10 of 100 matches for " "
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 Physics , 2000, DOI: 10.1046/j.1365-8711.2002.05103.x Abstract: The correlation between source galaxies and lensing potentials causes a systematic effect on measurements of cosmic shear statistics, known as the source-lens clustering (SLC) effect. The SLC effect on the skewness of lensing convergence, $S_3$, is examined using a nonlinear semi-analytic approach and is checked against numerical simulations. The semi-analytic calculations have been performed in a wide variety of generic models for the redshift distribution of source galaxies and power-law models for the bias parameter between the galaxy and dark matter distributions. The semi-analytic predictions are tested successfully against numerical simulations. We find the relative amplitude of the SLC effect on $S_3$ to be of the order of five to forty per cent. It depends significantly on the redshift distribution of sources and on the way the bias parameter evolves. We discuss possible measurement strategies to minimize the SLC effects.
 Uros Seljak Physics , 2000, DOI: 10.1046/j.1365-8711.2000.03715.x Abstract: We investigate an analytic model to compute nonlinear power spectrum of dark matter, galaxies and their cross-correlation. The model is based on Press-Schechter halos, which cluster and have realistic dark matter profiles. The total power spectrum is a sum of two contributions, one from correlations betwen the halos and one from correlations within the same halo. We show that such a model can give dark matter power spectra which match well with the results of N-body simulations, provided that concentration parameter decreases with the halo mass. Galaxy power spectrum differs from dark matter power spectrum because pair weighted number of galaxies increases less rapidly than the halo mass, as predicted by theoretical models and observed in clusters. In this case the resulting power spectrum becomes a power law with the slope closed to the observed. Such a model also predicts a later onset of nonlinear clustering compared to the dark matter, which is needed to reconcile the CDM models with the data. Generic prediction of this model is that bias is scale dependent and nonmonotonic. For red or elliptical galaxies bias in power spectrum may be scale dependent even on very large scales. Our predictions for galaxy-dark matter correlations, which can be observed through the galaxy-galaxy lensing, show that these cannot be interpreted simply as an average halo profile of a typical galaxy, because different halo masses dominate at different scales and because larger halos host more than one galaxy. We discuss the prospects of using cross-correlations in combination with galaxy clustering to determine the dark matter power spectrum (ABRIDGED).
 Physics , 2012, DOI: 10.1111/j.1365-2966.2012.21582.x Abstract: We examine scatter and bias in weak lensing selected clusters, employing both an analytic model of dark matter haloes and numerical mock data of weak lensing cluster surveys. We pay special attention to effects of the diversity of dark matter distributions within clusters. We find that peak heights of the lensing convergence map correlates rather poorly with the virial mass of haloes. The correlation is tighter for the spherical overdensity mass with a higher mean interior density. We examine the dependence of the halo shape on the peak heights, and find that the rms scatter caused by the halo diversity scales linearly with the peak heights with the proportionality factor of 0.1-0.2. The noise originated from the halo shape is found to be comparable to the source galaxy shape noise and the cosmic shear noise. We find the significant halo orientation bias, i.e., weak lensing selected clusters on average have their major axes aligned with the line-of-sight direction. We compute the orientation bias using an analytic triaxial halo model and obtain results quite consistent with the ray-tracing results. We develop a prescription to analytically compute the number count of weak lensing peaks taking into account all the main sources of scatters in peak heights. We find that the improved analytic predictions agree well with the simulation results for high S/N peaks. We also compare the expected number count with our weak lensing analysis results for 4 sq deg of Subaru/Suprime-Cam observations and find a good agreement.
 Physics , 2013, DOI: 10.1093/mnras/stt971 Abstract: We present a quantitative analysis of the largest contiguous maps of projected mass density obtained from gravitational lensing shear. We use data from the 154 deg2 covered by the Canada-France-Hawaii Telescope Lensing Survey. Our study is the first attempt to quantitatively characterize the scientific value of lensing maps, which could serve in the future as a complementary approach to the study of the dark universe with gravitational lensing. We show that mass maps contain unique cosmological information beyond that of traditional two-points statistical analysis techniques. Using a series of numerical simulations, we first show how, reproducing the CFHTLenS observing conditions, gravitational lensing inversion provides a reliable estimate of the projected matter distribution of large scale structure. We validate our analysis by quantifying the robustness of the maps with various statistical estimators. We then apply the same process to the CFHTLenS data. We find that the 2-points correlation function of the projected mass is consistent with the cosmological analysis performed on the shear correlation function discussed in the CFHTLenS companion papers. The maps also lead to a significant measurement of the third order moment of the projected mass, which is in agreement with analytic predictions, and to a marginal detection of the fourth order moment. Tests for residual systematics are found to be consistent with zero for the statistical estimators we used. A new approach for the comparison of the reconstructed mass map to that predicted from the galaxy distribution reveals the existence of giant voids in the dark matter maps as large as 3 degrees on the sky. Our analysis shows that lensing mass maps can be used for new techniques such as peak statistics and the morphological analysis of the projected dark matter distribution.
 Physics , 2009, DOI: 10.1111/j.1365-2966.2009.16117.x Abstract: We present a set of predictions for weak lensing correlation functions in the context of modified gravity models, including a prescription for the impact of the nonlinear power spectrum regime in these models. We consider the DGP and f(R) models, together with dark energy models with the same expansion history. We use the requirement that gravity is close to GR on small scales to estimate the non-linear power for these models. We then calculate weak lensing statistics, showing their behaviour as a function of scale and redshift, and present predictions for measurement accuracy with future lensing surveys, taking into account cosmic variance and galaxy shape noise. We demonstrate the improved discriminatory power of weak lensing for testing modified gravities once the nonlinear power spectrum contribution has been included. We also examine the ability of future lensing surveys to constrain a parameterisation of the non-linear power spectrum, including sensitivity to the growth factor.
 Charles R. Keeton Physics , 2002, DOI: 10.1086/345717 Abstract: The magnifications of the images in a strong gravitational lens system are sensitive to small mass clumps in the lens potential; this effect has been used to infer the amount of substructure in galaxy dark matter halos. I study the theory of substructure lensing to identify important general features, and to compute analytic cross sections that will facilitate further theoretical studies. I show that the problem of a clump anywhere along the line of sight to a lens can be mapped onto an equivalent problem of a clump in a simple convergence and shear field; clumps at arbitrary redshifts are therefore not hard to handle in calculations. For clumps modeled as singular isothermal spheres (SIS), I derive simple analytic estimates of the cross section for magnification perturbations of a given strength. The results yield two interesting conceptual points. First, lensed images with positive parity are always made brighter by SIS clumps; images with negative parity can be brightened but are much more likely to be dimmed. Second, the clumps need not lie within the lens galaxy; they can be moved in redshift by several tenths and still have a significant lensing effect. Isolated small halos are expected to be common in hierarchical structure formation models, but it is not yet known whether they are abundant enough compared with clumps inside lens galaxies to affect the interpretation of substructure lensing.
 Physics , 2015, DOI: 10.1093/mnras/stv2852 Abstract: We investigate the galaxy population in simulated proto-cluster regions using a semi-analytic model of galaxy formation, coupled to merger trees extracted from N-body simulations. We select the most massive clusters at redshift $z=0$ from our set of simulations, and follow their main progenitors back in time. The analysis shows that proto-cluster regions are dominated by central galaxies and their number decreases with time as many become satellites, clustering around the central object. In agreement with observations, we find an increasing velocity dispersion with cosmic time, the increase being faster for satellites. The analysis shows that proto-clusters are very extended regions, $\gtrsim 20 \, Mpc$ at $z \gtrsim 1$. The fraction of galaxies in proto-cluster regions that are not progenitor of cluster galaxies varies with redshift, stellar mass and area considered. It is about 20-30 per cent for galaxies with stellar mass $\sim 10^9\,{\rm M}_{\sun}$, while negligible for the most massive galaxies considered. Nevertheless, these objects have properties similar to those of progenitors. We investigate the building-up of the passive-sequence in clusters, and find that their progenitors are on average always active at any redshift of interest of proto-clusters. The main mechanism which quenches their star formation is the removal of the hot gas reservoir at the time of accretion. The later galaxies are accreted (become satellite), and the more the cold gas available, the longer the time spent as active. Central galaxies are active over all redshift range considered, although a non-negligible fraction of them become passive at redshift $z<1$, due to strong feedback from Active Galactic Nuclei.
 Physics , 2001, DOI: 10.1071/AS01017 Abstract: Gravitational lensing is most often used as a tool to investigate the distribution of (dark) matter in the universe, but, if the mass distribution is known a priori, it becomes, at least in principle, a powerful probe of gravity itself. Lensing observations are a more powerful tool than dynamical measurements because they allow measurements of the gravitational field far away from visible matter. For example, modified Newtonian dynamics (MOND) has no relativistic extension, and so makes no firm lensing predictions, but galaxy-galaxy lensing data can be used to empirically the deflection law of a point-mass. MONDian lensing is consistent with general relativity, in so far as the deflection experienced by a photon is twice that experienced by a massive particle moving at the speed of light. With the deflection law in place and no invisible matter, MOND can be tested wherever lensing is observed. The implications are that either MONDian lensing is completely non-linear or that MOND is not an accurate description of the universe.
 Physics , 2011, DOI: 10.1088/0004-637X/736/1/43 Abstract: Measuring dark matter substructure within galaxy cluster haloes is a fundamental probe of the Lambda-CDM model of structure formation. Gravitational lensing is a technique for measuring the total mass distribution which is independent of the nature of the gravitating matter, making it a vital tool for studying these dark-matter dominated objects. We present a new method for measuring weak gravitational lensing flexions, the gradients of the lensing shear field, to measure mass distributions on small angular scales. While previously published methods for measuring flexions focus on measuring derived properties of the lensed images, such as shapelet coefficients or surface brightness moments, our method instead fits a mass-sheet-transformation-invariant Analytic Image Model (AIM) to the each galaxy image. This simple parametric model traces the distortion of lensed image isophotes and constrains the flexion fields. We test the AIM method using simulated data images with realistic noise and a variety of unlensed image properties, and show that it successfully reproduces the input flexion fields. We also apply the AIM method for flexion measurement to Hubble Space Telescope observations of Abell 1689, and detect mass structure in the cluster using flexions measured with the AIM method.
 Physics , 2003, DOI: 10.1086/375851 Abstract: We investigate the gravitational lensing properties of dark matter halos with Burkert profiles. We derive an analytic expression for the lens equation and use it to compute the magnification, impact parameter and image separations for strong lensing. For the scaling relation that provides the best fits to spiral-galaxy rotation curve data, Burkert halos will not produce strong lensing, even if this scaling relation extends up to masses of galaxy clusters. Tests of a simple model of an exponential stellar disk superimposed on a Burkert-profile halo demonstrate that strong lensing is unlikely without an additional concentration of mass in the galaxy center (e.g. a bulge). The fact that most strong lenses on galactic scales are elliptical galaxies suggests that a strong central concentration of baryons is required to produce image splitting. This solution is less attractive for clusters of galaxies, which are generally considered to be dark-matter dominated even at small radii. There are three possible implications of these results: (1) dark halos may have a variety of inner profiles (2) dark matter halos may not follow a single scaling relation from galaxy scale up to cluster scale and/or (3) the splitting of images (even by clusters of galaxies) may in general be due to the central concentration of baryonic material in halos rather than dark matter.
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# A committee of three must be formed from 5 women and 5 men.

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A committee of three must be formed from 5 women and 5 men. [#permalink]  09 Oct 2017, 23:07
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A committee of three must be formed from 5 women and 5 men. What is the probability that the committee will be exclusive to one gender?

(A) 1/60

(B) 1/120

(C) 1/8

(D) 1/6

(E) 1/3

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Re: A committee of three must be formed from 5 women and 5 men. [#permalink]  10 Oct 2017, 03:55
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A committee of three must be formed from 5 women and 5 men. What is the probability that the committee will be exclusive to one gender?

(A) 1/60

(B) 1/120

(C) 1/8

(D) 1/6

(E) 1/3

Kudos for correct solution.

Here total number of members (men and women) = 10

NOw we need to choose 3 out of these. Again favourable events to one gender will be 5C3 (since both men and women are 5 in numbers)

Since $$probability = \frac{No. of favourable events}{Total possible outcomes}$$

i.e $$probability = \frac{5C3}{10C3} + \frac{5C3}{10C3}$$ (we have taken two times 5C3 because we need to have exclusive to one gender i.e either men or women, so combining them will give us total number of favorable events)

$$= \frac{1}{12} +\frac{1}{12} =\frac{1}{6}$$
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Re: A committee of three must be formed from 5 women and 5 men.   [#permalink] 10 Oct 2017, 03:55
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# Showing that the euler lagrange equations are coordinate independent

1. Jul 29, 2011

### demonelite123

so i know for example that d/dt (∂L/∂x*i) = ∂L/∂xi for cartesian coordinates, where xi is the ith coordinate in Rn and x*i is the derivative of the ith coordinate xi with respect to time. L represents the lagrangian.

so using an arbitrary change of coordinates, qi = qi(x1, x2, ..., xn)

i have that ∂L/∂x_i = ∂L/∂qi * ∂q_i/∂x_i and that ∂L/∂x*i = ∂L/∂q*i * ∂q*i/∂x*i and substituting i get:

d/dt(∂L/∂q*i * ∂q*i/∂x*i) = ∂L/∂qi * ∂qi/∂xi and using the product rule i get:

d/dt(∂L/∂q*i) * ∂q*i/∂x*i+ ∂L/∂q*i * d/dt(∂q*i/∂x*i) = ∂L/∂qi* ∂qi/∂xi

using the fact that ∂q*i/∂x*i = ∂qi/∂xi,

d/dt(∂L/∂q*i) * ∂qi/∂xi + ∂L/∂q*i * d/dt(∂qi/∂xi) = d/dt(∂L/∂q*i) * ∂qi/∂xi + ∂L/∂q*i * (∂q*i/∂xi) = d/dt(∂L/∂q*i) * ∂qi/∂xi+ ∂L/∂xi = ∂L/∂qi * ∂qi/∂xi.

if i didn't have that ∂L/∂xi on the left side then i could just cancel out the ∂qi/∂xi on both sides and i would be done. have i made a mistake somewhere or am i missing something? thanks.

Last edited: Jul 29, 2011
2. Jul 29, 2011

### vanhees71

Why don't you use the nice LaTeX functionality of this forum? It's really hard to read, what you have written. So I don't want to decipher it.

Here is my derivation of the form invariance of the EL equations under coordinate changes (diffeomorphisms). Let $x^j(q)$ denote the transformation with $j \in \{1,\ldots,f \}$. Further $q=(q^k)_{k \in \{1,\ldots, f\}}$.

First of all we need an auxilliary equation for the time derivatives. From the chain rule we have

$$\dot{x}^j=\frac{\partial x^j}{\partial q^k} \dot{q}^k.$$

Here and in the following I use Einstein's summation convention, according to which one has to sum over any pair of repeated indices. From this equation we see

$$\frac{\partial \dot{x}^j}{\partial \dot{q}^k}=\frac{\partial x^j}{\partial q^k}. \quad (1)$$

Now we have

$$\frac{\partial L}{\partial q^k}=\frac{\partial L}{\partial x^j} \frac{\partial x_j}{\partial q_k} + \frac{\partial L}{\partial \dot{x}^j} \frac{\partial \dot{x}^j}{\partial q^j} \quad (2)$$

and

$$\frac{\partial L}{\partial \dot{q}^k}=\frac{\partial L}{\partial \dot{x}^j} \frac{\partial \dot{x}^j}{\partial \dot{q}^k} \stackrel{(1)}{=}\frac{\partial L}{\partial \dot{x}^j} \frac{\partial x^j}{\partial q^k}.$$

Taking the time derivative of this, we find

$$\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q}^k} = \frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{x}^j} \cdot \frac{\partial x^j}{\partial q^k} + \frac{\partial L}{\partial \dot{x}^j} \frac{\mathrm{d}}{\mathrm{d} t} \left (\frac{\partial x^j}{\partial q^k} \right). \quad (3)$$

Now we obviously have

$$\frac{\mathrm{d}}{\mathrm{d} t} \left (\frac{\partial x^j}{\partial q^k} \right) = \frac{\partial \dot{x}^k}{\partial q^k}, \quad (4)$$

and thus, if the EL equation are valid wrt. the coordinates, $x^k$ they are also valid wrt. to the coordinates, $q^j$, which can be seen by comparing (2) and (3) and making use of (4) and the EL equation wrt. $x^k$.

3. Jul 29, 2011

### Petr Mugver

This is one of the mistakes. Instead, write:

$\frac{\partial L}{\partial x_i}=\sum_j \frac{\partial L}{\partial q_j}\frac{\partial q_j}{\partial x_i}+\sum_j \frac{\partial L}{\partial \dot{q}_j}\frac{\partial \dot{q}_j}{\partial x_i}$

You fergot the second piece. Remember:

$q_j=q_j(x_1,\dots,x_n)$
$\dot{q}_j=\sum_k\frac{\partial q_j}{\partial x_k}\dot{x}_k$

so

$\frac{\partial\dot{q}_j}{\partial x_i}=\frac{\partial^2 q_j}{\partial x_i\partial x_k}\dot{x}_k$

Last edited: Jul 29, 2011


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## Tokyo Journal of Mathematics

### Norm Conditions on Maps between Certain Subspaces of Continuous Functions

#### Abstract

For a locally compact Hausdorff space $X$, let $C_0(X)$ be the Banach space of continuous complex-valued functions on $X$ vanishing at infinity endowed with the supremum norm $\|\cdot\|_X$. We show that for locally compact Hausdorff spaces $X$ and $Y$ and certain (not necessarily closed) subspaces $A$ and $B$ of $C_0(X)$ and $C_0(Y)$, respectively, if $T:A \longrightarrow B$ is a surjective map satisfying one of the norm conditions i) $\|(Tf)^s (Tg)^t\|_Y=\|f^s g^t\|_X$, or ii) $\|\, |Tf|^s+|Tg|^t\,\|_Y=\|\, |f|^s+|g|^t\, \|_X$, \noindent for some $s,t\in \mathbb{N}$ and all $f,g\in A$, then there exists a homeomorphism $\varphi: \mathrm{ch}(B) \longrightarrow \mathrm{ch}(A)$ between the Choquet boundaries of $A$ and $B$ such that $|Tf(y)|=|f(\varphi(y))|$ for all $f\in A$ and $y\in \mathrm{ch}(B)$. We also give a result for the case where $A$ is closed (or, in general, satisfies a special property called Bishop's property) and $T:A \longrightarrow B$ is a surjective map satisfying the inclusion $R_\pi((Tf)^s (Tg)^t) \subseteq R_\pi(f^s g^t)$ of peripheral ranges. As an application, we characterize such maps between subspaces of the form $A_1f_1+A_2f_2+\cdots+A_nf_n$, where for each $1\le i \le n$, $A_i$ is a uniform algebra on a compact Hausdorff space $X$ and $f_i$ is a strictly positive continuous function on $X$. Our results in case (ii) improve similar results in~[30], for subspaces rather than uniform algebras, without the additional assumption that $T$ is $\mathbb{R}^+$-homogeneous.

#### Article information

Source
Tokyo J. Math., Volume 40, Number 2 (2017), 421-437.

Dates
First available in Project Euclid: 9 January 2018

https://projecteuclid.org/euclid.tjm/1515466834

Digital Object Identifier
doi:10.3836/tjm/1502179236

Mathematical Reviews number (MathSciNet)
MR3743727

Zentralblatt MATH identifier
06855943

#### Citation

GHODRAT, Razieh Sadat; SADY, Fereshteh; JAMSHIDI, Arya. Norm Conditions on Maps between Certain Subspaces of Continuous Functions. Tokyo J. Math. 40 (2017), no. 2, 421--437. doi:10.3836/tjm/1502179236. https://projecteuclid.org/euclid.tjm/1515466834

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text:
Mathematics, philosophy, code, travel and everything in between. More about me…

English | Czech

# All The Numbers In The World

We know that the natural numbers 1,2,3,… go off to infinity. But what happens when we consider negative numbers -1,-2,-3,… as well? How many numbers do we get? And what about fractions? Do we have multiple infinities?

Welcome to another part of the “Maths For Normal People” series. You will find all the previous parts in this article.

So far, we have been talking only about $\mathbb{N}$, the set of natural numbers. As the name implies, these are the numbers that come to mind most naturally and so they were invented first. As mathematical thought developed, new numbers had to be invented.

To facilitate subtracting larger numbers from smaller ones, humans developed the notion of whole numbers, whose set we denote $\mathbb{Z} = \{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$. The dots “in both ways” mean that there is an infinity of negative numbers and an infinity of positive numbers.

To facilitate division of numbers that are not commensurable (i.e. can’t be divided without a remainder), we devised the idea of rational numbers, or fractions. Their set is denoted $\mathbb{Q}$ and can no longer be easily described by listing a few elements and adding dots. Instead, we say that

$$\mathbb{Q} = \{\text{all numbers }\frac{p}{q}\text{ such that }p\text{ is whole and }q\text{ is natural}\}.$$

And finally, to make it possible to take square roots of all positive numbers, we invented real numbers, whose set is denoted $\mathbb{R}$ and contains, roughly speaking, all numbers with all possible combinations of decimal digits. The genesis of number sets is a deep and interesting topic that I am going to cover in an article of its own. For now, let’s settle for this short introduction.

Now, the obvious question is: how many numbers are there in the respective sets $\mathbb{Z},\mathbb{Q},\mathbb{R}$? We have already established that there are $\aleph_0$ numbers in $\mathbb{N}$, and that this $\aleph_0,$ or “Aleph-0” is a name we give to infinity.

## Negativity Brings Nothing New

Perhaps this won’t come off as a surprise: there are just as many whole numbers as there are natural numbers. If you take the set of natural numbers and give a minus sign to each of them, you get all the negative whole numbers. Merging this set with the original natural numbers and adding zero, you arrive at $\mathbb{Z}$. But concerning set cardinalities, all you’ve done is you multiplied $\aleph_0$ by two and added one. But according to the rules we have derived, $2\cdot\aleph_0 + 1 = \aleph_0$. Therefore,

$$|\mathbb{N}| = |\mathbb{Z}|.$$

If you don’t believe me, I’ve got a better proof. We can easily find a bijection between $\mathbb{N}$ and $\mathbb{Z}$. One way to do it is to assign odd natural numbers to positive whole numbers and even natural numbers to negative whole numbers. (You might want to refresh your infinity counting skills in the “Infinity Plus Three” article I wrote back in March.)

## Rationally More

The question of rational numbers is not all that obvious. Fractions are created by making all possible combinations of whole and natural numbers, and so there are many, many more fractions than whole numbers.

Right?

No.

There are no more fractions than natural numbers, and we can prove it by numbering all the fractions by natural numbers, which really is nothing else than creating a bijection between fractions and natural numbers. (Whenever you are counting things, you are establishing a bijection between those things and natural numbers.)

Let’s write all the positive fractions out in a table and number them diagonally in the direction of the arrows. Well, you’ll forgive me if I don’t write out all the positive fractions, right? :-)

We see that it is indeed possible to number all positive fractions in this manner.

Having established that there are as many positive fractions as there are natural numbers, we can infer that this is valid for all fractions. The negative ones can be obtained simply by putting a minus sign in front of the positive ones, so their number is obviously the same. And putting the positive and negative ones together creates a set with cardinality $2\cdot\aleph_0$, which obviously equals $\aleph_0$. So we have

$$|\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}|.$$

Is $|\mathbb{R}| = |\mathbb{N}|$ or not? And why? We’ll see next week!


text:
## asynchronous – Syntax for TypeScript Arrow Functions With Generics

I don’t know if this is done properly or not, but I posted an answer in StackOverflow back in Sept. 2020 about implementing generics in TypeScript. I got downvoted 3 times, and I was wondering if the following piece of code was problematic:

This is how I defined an async generic function expression in TypeScript

const request = async <T>(param1: string, param2: number) => {
const res = await func();
return res.response() as T;
}

And a more complex pattern, in case you’d like to wrap your function inside a generic counterpart, such as memoization (Example uses fast-memoize):

const request = memoize(
async <T>(
url: string,
token?: string
) => {
}
);

See how you define the generic after the memoizing function.

This function has worked well for me, so I don’t see any issues.

This is the original SO question.

## docker – Cannot gather stats and Invalid template file after installing Porto theme

After setting up Magento 2 in developer mode, I am getting a couple of errors:

(2021-04-19 21:39:09) report.INFO: Broken reference: the 'yotpo_bottomline' tries to reorder itself towards 'product.info.addto', but their parents are different: 'product.info.main' and '' respectively. () ()
(2021-04-19 21:39:09) report.CRITICAL: Cannot gather stats! Warning!stat(): stat failed for /app/pub/static/frontend/Magento/luma/en_US/mage/requirejs/mixins.js {"exception":"(object) (Magento\Framework\Exception\FileSystemException(code: 0): Cannot gather stats! Warning!stat(): stat failed for /app/pub/static/frontend/Magento/luma/en_US/mage/requirejs/mixins.js at /app/vendor/magento/framework/Filesystem/Driver/File.php:95)"} ()
(2021-04-19 21:39:10) report.CRITICAL: Invalid template file: 'owl_list.phtml' in module: 'Smartwave_Filterproducts' block's name: 'homefeaturedlist_0' {"exception":"(object) (Magento\Framework\Exception\ValidatorException(code: 0): Invalid template file: 'owl_list.phtml' in module: 'Smartwave_Filterproducts' block's name: 'home\featuredlist_0' at /app/vendor/magento/framework/View/Element/Template.php:280)"} ()
(2021-04-19 21:39:10) report.CRITICAL: Invalid template file: 'grid.phtml' in module: 'Smartwave_Filterproducts' block's name: 'homefeaturedlist_1' {"exception":"(object) (Magento\Framework\Exception\ValidatorException(code: 0): Invalid template file: 'grid.phtml' in module: 'Smartwave_Filterproducts' block's name: 'home\featuredlist_1' at /app/vendor/magento/framework/View/Element/Template.php:280)"} ()

I followed the Magento 2 documentation to install Magento in docker. Then, in an attempt to follow the Porto docs, I copied the Porto theme files, the theme patch for Magento 2.2.x, theme patch for Magento 2.3.x, and the theme patch for Magento 2.4.1.

Any help on resolving the above errors would be appreciated.

## android – Erro com Nav Controller

Estou enfrentando problemas ao rodar meu app pois ele nao abre a main, o problema esta no meu nav controler mas nao sei como solucioná-lo.

Log:

E/AndroidRuntime: FATAL EXCEPTION: main
Process: com.example.pokedex, PID: 19404
java.lang.RuntimeException: Unable to start activity ComponentInfo{com.example.pokedex/com.example.pokedex.ui.main.MainActivity}: java.lang.IllegalStateException: Activity com.example.pokedex.ui.main.MainActivity@17e9680 does not have a NavController set on 2131230964
at android.os.Handler.dispatchMessage(Handler.java:106)
at android.os.Looper.loop(Looper.java:169)
at java.lang.reflect.Method.invoke(Native Method)
at com.android.internal.os.RuntimeInit\$MethodAndArgsCaller.run(RuntimeInit.java:438)
at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:807)
Caused by: java.lang.IllegalStateException: Activity com.example.pokedex.ui.main.MainActivity@17e9680 does not have a NavController set on 2131230964
at com.example.pokedex.ui.main.MainActivity.configNavDrawer(MainActivity.kt:69)
at com.example.pokedex.ui.main.MainActivity.onCreate(MainActivity.kt:46)
at android.app.Activity.performCreate(Activity.java:7051)
at android.app.Activity.performCreate(Activity.java:7042)
at android.app.Instrumentation.callActivityOnCreate(Instrumentation.java:1215)
at android.os.Handler.dispatchMessage(Handler.java:106)
at android.os.Looper.loop(Looper.java:169)
at java.lang.reflect.Method.invoke(Native Method)
at com.android.internal.os.RuntimeInit\$MethodAndArgsCaller.run(RuntimeInit.java:438)
at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:807)

Main Activity:

class MainActivity : AppCompatActivity() {

private lateinit var toggle: ActionBarDrawerToggle
var currentPath: String? = null
val TAKE_PICTURE = 1
val SELECT_PICTURE = 2

override fun onCreate(savedInstanceState: Bundle?) {
super.onCreate(savedInstanceState)
setContentView(R.layout.main_activity)
configBottomNavegacao()
configNavDrawer()

(pokemons_activity_nav_host.findNavController())

toggle = ActionBarDrawerToggle(this, drawerLayout, R.string.open, R.string.close )
toggle.syncState()

supportActionBar?.setDisplayHomeAsUpEnabled(true)

when(it.itemId){
R.id.drawer_camera -> dispatchCameraIntent()
R.id.drawer_fav -> dispatchFavoritosIntent()
}
true
}
}

private fun dispatchFavoritosIntent() {
}

private fun configNavDrawer() {
.setupWithNavController(navController)

// verifica onde vai ou nao mostrar o drawer
navController.addOnDestinationChangedListener { controller, destination, arguments ->
when(destination.id){
R.id.listaPokemons -> mostraNavDrawer()
R.id.listaFavPokemons -> mostraNavDrawer()
R.id.search -> mostraNavDrawer()
R.id.detalhesPokemons -> mostraNavDrawer()
else -> ocultaNavDrawer()
}
}
}

private fun mostraNavDrawer() {
navView.visibility = View.VISIBLE
}
private fun ocultaNavDrawer(){
navView.visibility = View.GONE
}

private fun configBottomNavegacao() {
val navHostFragment = supportFragmentManager
.findFragmentById(R.id.pokemons_activity_nav_host) as NavHostFragment
val navController = navHostFragment.navController

// verifica onde vai ou nao mostrar o bottom navigation
navController.addOnDestinationChangedListener { controller, destination, arguments ->
when(destination.id){
R.id.listaPokemons -> mostraBottomNav()
R.id.listaFavPokemons -> mostraBottomNav()
R.id.search -> mostraBottomNav()
R.id.detalhesPokemons -> mostraBottomNav()
}

}

}

//oculta o botao de navegaçao
}

// exibe o botao de navegaçao
private fun mostraBottomNav() {
}

activity main:

<androidx.drawerlayout.widget.DrawerLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
xmlns:tools="http://schemas.android.com/tools"
android:id="@+id/drawerLayout"
android:layout_width="match_parent"
android:layout_height="match_parent"
tools:context=".ui.main.MainActivity">

<androidx.constraintlayout.widget.ConstraintLayout
android:layout_width="match_parent"
android:layout_height="match_parent">

</androidx.constraintlayout.widget.ConstraintLayout>

<androidx.fragment.app.FragmentContainerView
android:id="@+id/pokemons_activity_nav_host"
android:layout_width="match_parent"
android:layout_height="match_parent"
app:defaultNavHost="true"

<androidx.constraintlayout.widget.ConstraintLayout
android:layout_width="match_parent"
android:layout_height="match_parent">

android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_marginStart="0dp"
android:layout_marginEnd="0dp"
app:layout_constraintBottom_toBottomOf="parent"
app:layout_constraintLeft_toLeftOf="parent"
app:layout_constraintRight_toRightOf="parent"

</androidx.constraintlayout.widget.ConstraintLayout>

android:layout_width="wrap_content"
android:layout_height="match_parent"
android:id="@+id/navView"
android:layout_gravity="start"
android:fitsSystemWindows="true"/>

</androidx.drawerlayout.widget.DrawerLayout>

## dnd 5e – As a Warlock, can I cast Vampiric Touch on myself to grant my pact weapon an extra 3d6 necrotic damage?

### No, you can’t improve an existing melee attack with the Vampiric Touch

The Vampiric Touch spell states:

The touch of your shadow-wreathed hand can siphon life force from others to heal your wounds. Make a melee spell attack against a creature within your reach. On a hit, the target takes 3d6 necrotic damage, and you regain hit points equal to half the amount of necrotic damage dealt. Until the spell ends, you can make the attack again on each of your turns as an action.

So, firstly you take the Cast A Spell action and expend a 3rd level spell slot. Then you deal 3d6 necrotic damage, if you succeed with the melee spell attack. This melee spell attack is a part of the spell — you literally have to touch the target with “your shadow-wreathed hand” to apply the spell effect.

Within 1 minute, you can repeat this action without expending more spell slots, providing you maintain the concentration. So you have to choose — either you make a weapon attack with your pact weapon, or you make a spell attack with the Vampiric Touch.

See also Can you apply the Vampiric Touch and Shocking Grasp damage together in one attack?

## design – elementor no-repeat css value works randomly on unique single post template

I’m working on our WP website with elementor pro template builder. I have one single post template that I’m using for all the posts. The background “hero” image of the header section is the featured image of the post.

In the following post (with firefox/Chrome Mac) no-repeat value doesn’t work:
https://dev.mkorke.net/patagonie/centolla/

In the following post (with firefox/Chrome Mac) no-repeat value does work:

Both examples supposedly use the same single post template with background values as:
…background-image: url(“photograph.jpg”);
min-height: 60%;
background-position: center;
background-repeat: no-repeat;
background-size: cover;…

In the single post template, if I change the value to default then back to no-repeat, it works again, … until it doesn’t.

What am I missing. Thanks for the help

## D&D 3.5e — “Mettle”

In D&D 3.5e, evasion allowed someone to take no damage on a successful Reflex save against something that usually had halved damage on a successful save. This feature was originally available to monks and rogues, and became available to other classes via supplemental content.

Later in the same edition, additional classes were introduced that had a feature called “mettle,” which worked the same way, except instead of Reflex saves, it worked on both Fortitude and Will saves.

Note that D&D 3.5e only had these three saving throw types—Fortitude based on Constitution, Reflex based on Dexterity, and Will based on Wisdom. There were no saving throws associated with Strength, Intelligence, or Charisma. (These associations were only defaults, however, so special features could and occasionally did change the ability score associated with a given type of save.)

Mettle covered both Fortitude and Will because it was (and, I believe, continues to be, in 5e) relatively rare for those types of saving throws to still have some reduced effect on a successful save. Usually, those kinds of saves just negated whatever the effect was entirely.

To my knowledge, there was no “half-mettle” feature that covered only Fortitude saving throws.

## D&D 4e — “Mettle” again

In D&D 4e, we still have Fortitude, Reflex, and Will, but now they are no longer saving throws, they are “defenses” and work the same way that AC does—certain powers had you roll your attack against these defenses instead of AC. (Saving throws were an entirely separate mechanic and there was only the one type, “saving throw.”) Also, Fortitude was based on Strength or Constitution, Reflex was based on Dexterity or Intelligence, and Will was based on Wisdom or Charisma, whichever was higher out of each pair.

But evasion and mettle remained effectively the same—evasion caused attacks that targeted your Reflex to do nothing on a miss, even if they usually did something on a miss, and mettle did likewise for attacks that targeted Fortitude or Will.

## D&D 5e — none, I believe

As far as I can tell, D&D 5e doesn’t have mettle, or anything similar to it. It might be hard to make in 5e, since Dexterity remains the saving throw associated most strongly with partial or reduced effect on a successful save, but “everything but Dexterity” is now five separate saving throws, instead of just two, which makes having mettle cover all of those seem a fair bit more imbalanced. At any rate, Wizards of the Coast has yet to cover it in their publications.

## Mathematica Assist. \$yx=qxy\$

In Mathematica program, How to define a product of two elements $$x$$ and $$y$$ such that $$yx=qxy$$ with $$q$$ is a number commuting with both $$x$$ and $$y$$.

## soft fork – Is there network split risk for Taproot activation with two releases (Bitcoin Core and Bitcoin Taproot)?

Every soft fork or consensus change involves a (very small) non-zero risk of a network split. That risk is considerably lower for a soft fork than say a hard fork (where all nodes need to upgrade). That’s why soft forks aren’t attempted every month or year. All you can do is minimize that risk.

Aaron lays out some scenarios that are theoretically possible. Any incompatibility between “Bitcoin Core” and “Bitcoin Taproot” during the Speedy Trial deployment is in my view highly unlikely. If Speedy Trial fails to activate and we reach November 2022 (please note 2022 not 2021) without miners activating then we are in a similar scenario to the UASF in 2017 where it depends on what the economic majority is running. I can’t predict what the economic majority would be running in November 2022 but I highly suspect the delaying of Taproot activation would be at the top of everyone’s minds.

You do have to weigh up these risks of a network split with miners deliberately blocking Taproot activation potentially forever. If we were to say no more UASFs ever again because we don’t want to take any network split risk that would be handing miners a permanent veto to block the activations of soft forks that have community consensus. So you have to weigh up the risk of the latter which would be just as concerning (if not more concerning) to people.

So in summary these are subtle trade-offs. A number of developers have worked hard to minimize the risk of a network split. But it doesn’t get to zero unless you literally never try a soft fork again. And that would mean that Bitcoin would never seriously improve again.

## c# – Why does CreateProcess ignore STARTUPINFO window size values?

I am trying to use CreateProcess to start Notepad at a specific window size to avoid the flashing that occurs when it appears somewhere for a split second before I can resize it and move it.

CreateProcess starts Notepad but CreateProcess and Notepad both ignore the window dimensions in the STARTUPINFO structure. Notepad appears in the normal place at the normal size where it was last closed.

No combination of dwFlags that I tried worked. Notepad either does not appear at all or ignores my settings and appears in a place and size determined by the operating system.

Why is CreateProcess ignoring the values I set in STARTUPINFO? Am I missing something?

(TestMethod())
public void CreateProcessTest() {
const uint NORMAL_PRIORITY_CLASS = 0x0020;
const uint STARTF_USESHOWWINDOW = 0x0001;

// create structures needed by CreateProcess
var pInfo = new Kernel32.PROCESS_INFORMATION();
var pSec = new Kernel32.SECURITY_ATTRIBUTES();
var tSec = new Kernel32.SECURITY_ATTRIBUTES();
pSec.nLength = Marshal.SizeOf(pSec);
tSec.nLength = Marshal.SizeOf(tSec);

// set the app and a file to open
var app = Environment.GetEnvironmentVariable("windir") + @"notepad.exe";
var arguments = @" C:somefile.txt";

// the started app window does not use these values
var sInfo = new Kernel32.STARTUPINFO();
sInfo.dwX = 800;  // desired x-y position of the window
sInfo.dwY = 400;
sInfo.dwXSize = 200; // desired size of the window
sInfo.dwYSize = 400;

// no combination of these flags that I tried makes any difference
// Notepad always appears as normal and ignores the size settings above
sInfo.dwFlags = STARTF_USESHOWWINDOW;
sInfo.wShowWindow = (short) Win32.SW_SHOW;

// create the process
var result = Kernel32.CreateProcess(app, arguments,
ref pSec, ref tSec, false, NORMAL_PRIORITY_CLASS,
IntPtr.Zero, null, ref sInfo, out pInfo);
}


text:
# General formula for normal moments

Yesterday I wrote about how to find moments of the normal distribution using Sage. Then GlennF left a comment saying it’s not too hard to work out the moments analytically and outlined a proof. I’ll fill in a few details here.

First, start with a standard normal distribution Z. That is, Z has mean 0 and variance 1. By symmetry, the odd moments of Z are 0. For the even moments, integration by parts shows that E(Z2m) = (2m – 1) E(Z2m – 2). Apply this relation recursively until you get E(Z2m) = (2m – 1)!!. (See this post if you’re unfamiliar with double factorial. Note that (-1)!! is defined to be 1.)

For a general normal random variable X with mean μ and variance σ2, define Z = (X – μ)/σ. Then Z is a standard normal and X = σZ + μ. Apply the binomial theorem and note that the odd terms are zero.

## 9 thoughts on “General formula for normal moments”

1. GlennF

I like your statement of it better than mine. For some reason I can’t explain today, I chose to re-express the double factorial in terms of ordinary factorials. It looks much cleaner *not* doing that, though.

2. SteveBrooklineMA

Very nice. Is there an application? I thought that high moments were interesting for theoretical questions, but not really useful when applied to “real” data.

3. I’ve haven’t had much need for higher moments. But I’ve been working with Edgeworth expansions lately, and higher moments pop up. In this tech report I needed up to 5th moments of a normal. With more terms in my Edgeworth approximations, I’d need higher moments.

4. How are the odd terms zero? For the generalized version?

5. The normal density is even, so when you multiply it by an odd power you get an odd function. And the integral of an odd function over the real line is zero by symmetry: the left side is the negative of the right side.

6. Timothy Crack

Sir, I derived this myself and then went searching for confirmation. I found your formula, but my answer differs slightly from yours. Where you have (2j-1)!!, I have [2max(j,1)-1]!!, because the case j=0 in the summation would yield (-1)!! here. Alternatively, I stripped out the j=0 term from the summation and had
$E(X^n)= \begin{cases} \mu,& n=1 \\ \mu^n+\sum_{j=1}^{floor(\frac{n}{2})} {n \choose 2j} \sigma^{2j} (2j-1)!! \mu^{n-2j},& n\ge 2 \end{cases}$
Am I missing something? Have you defined -1!! to be unity?

7. A slight variation on the analytical proof, without using integration by parts explicitly (although it’s the same trick), would be, if N(0,1) is the density,, and omitting the integration limits to +- infinity,
\int{x N(0,1)dx}=0
\int{\frac{d}{dx} xN(0,1)dx}=[xN(0,1)]_{-\infty}^{+\infty} = 0
and
\int{\frac{d}{dx}xN(0,1)s}=\int{[N(0,1) – x^2N(0,1)]dx}
so \int{x^2N(0,1)dx}=\int{[N(0,1)}=1
The same trick works connecting the 2n+1 moment with the 2n and 2n+2 moments, so we get the full result by forward or backward induction, whichever we like best.


text:
## Tuesday, September 02, 2014 ... /////

### Gavin Schmidt's analogy ceasefire

...I agree with him...

Shockingly enough, the climate fearmonger Gavin Schmidt of RealClimate.ORG wrote something that I agree with:

On arguing by analogy
Climate skeptics often like to say that the climate alarmists are analogous to the defenders of Lysenkoism, phlogiston, phrenology, and eugenics, while the climate alarmists themselves love to compare themselves to Galileo Galilei, Isaac Newton, Albert Einstein, Charles Darwin, and maybe even Paul Ehrlich who has predicted that hundreds of millions of Americans would starve to death by the year 2000.

Schmidt's main point that I agree with is that these analogies are pretty much worthless for the debate about the substance because these analogies are only relevant for someone who already accepts the basic points about the substance. If one already knows (or at least assumes) that the climate panic is hogwash, he will know that the skeptics' favorite analogies are approximately right while the alarmists' analogies are misleading. If she still believes that the climate is dangerously changing, she will refuse the skeptics' favorite analogies and buy the alarmists' ones.

None of these analogies can really settle the starting point, namely the substance of the climate panic or the non-existence of it. All people who are rational know that. (The previous proposition isn't supported by empirical data but it is backed by a logical argument.) If we compare a group of people to some generally accepted big winners (or losers) of the history of science, we are making an assumption that we haven't proven by this analogy itself. And the analogy is only as relevant as its assumptions are valid.

In the previous paragraph, I mentioned "people who are rational". It's because many people are not rational and if they hear someone comparing a group (e.g. climate skeptics) to someone who is a loser, they automatically deduce that the group (e.g. climate skeptics) is wrong and bad even though they have actually been given no evidence whatever.

The analogies and other demagogic arguments may have a great impact on the reasoning and beliefs of such gullible people.

It is not hard to see that most of the devoted believers in the climate change orthodoxy belong among these gullible brainwashed impressionable morons. The previous sentence isn't an analogy; it is a sociological comment about this particular situation. You may check that it's probably right when you open the comments under Schmidt's article. Most of the commenters disagree with Schmidt. They think it's important to compare climate skeptics with unpopular groups. No doubt about it, these critics love to elaborate upon such analogies.

While I agree with Schmidt about his main points that these sociological analogies have nothing to say about the substance, I disagree with many other things he believes or writes. Of course, I disagree with the substance of his climate hysteria. But I disagree with some of Schmidt's other sociological claims, too. For example, he insanely believes in the "consensus science":
Given the rarity of a consensus-overturning event, the only sensible prior is to assume that [the majority of gullible sheep and multiply counted would-be experts living in a deep and mindless group think is always right].
But this logical implication is completely wrong. Consensus-overturning events may be "rare" if we use a certain measure, but that doesn't imply that the majority is right about controversial questions.

In fact, every discovery in science is a consensus-overturning event and almost all of progress in science is composed of these events. So while they may be "rare" in the sense of "percentage of the time" that they occupy on the timeline, they represent almost 100% of the events that the history of science is composed of. To discover something means to find something that others don't know or believe to be non-existent (or to prove the non-existence of something that is believed to exist, like the luminiferous aether). The more important the discovery is, the more likely it is for a significant opposition to emerge. The more important the discovery is, the more visibly it overturns the existing knowledge (or it changes the opinions about more important questions).

Otherwise, Schmidt realizes and acknowledges that analogies between two situations are helpful and independent of the assumptions about the scientific substance if they show that a particular method or logic would lead (or would have led or actually has led) to completely wrong or undesirable outcomes in other situations. For example, we may see that the political intimidation of scientists would have distorted lots of other scientific research as well and many discoveries could have become impossible, or impossible to spread.

The undesirability of the non-experts' political intervention into the scientific research isn't an analogy. It is a general principle that may be seen theoretically as well as by looking at many examples from the history of science. Of course that the intervening political forces may sometimes happen to be right, too. That's why the existence of such undesirable influences doesn't prove and doesn't disprove the climate change psychopathy.

#### snail feedback (10) :

If A argues that the theory of global warming must be true because it is supported by a 97% consensus, and B returns by citing the case of Galileo, B may be putting forward an analogy (I am like Galileo) which is almost certainly a very weak argument, but he may be using Galileo as a counterexample, which could be quite a strong argument.

Gavin recently lashed out at someone trying to justify calling skeptics deniers, in this short video clip:

http://tinypic.com/r/2lsehp2/5

One gauge of rationality is whether someone is alarmed about "climate change" or "global warming." If the first, they are most likely a herd follower, because the real controversy is about global warming, not change, and the terminology change is only a rhetorical device. If the latter, there is at least a chance of rationality.

Of course, some may claim to be alarmed about "climate change" but really be worrying about global warming, just because they recognize the value of the change term in convincing the gullible.

A pretty nice surprise, indeed!

I agree but you may also have other examples where the minority side didn't do as well as Galileo did, like the case of the rejected relationship between AIDS and HIV (Schmidt mentions it as well), not to mention "staged moonlanding" and other conspiracy theories. If you pick those analogies instead of Galileo, the minority side suddenly looks very different and its humiliation looks more understandable.

I was intrigued that a question about free will came up from the audience at the end. Anton's response was that the universe does not know what will happen next, due to objective uncertainty, so free will is not excluded. Of course, that is a very different statement than saying free will exists. Plunging in to the deep end, I would say that the main problem with free will is that if it depends on something then it is not free, but if it does not depend on something then we can't say anything about it mathematically or causally. If it operates but we can't say anything about it and it doesn't appear to be causally due to anything, it will look random. So how can we tell free will from randomness? I guess it might take an extraordinarily sensitive experiment which showed a skewed result, but it is hard to imagine what that setup might be. On the whole it seems to me that QM's randomness, along with our subjective experience of free will go together quite nicely. As you say, Libet's experiments are still debated as to what they really imply, and still allow "free won't" (nice paper! - I'm digesting it now).

While I support the general notion that consensus itself is not proof, the use of Galileo is itself an inappropriate analogy. The scientific community of the day supported the geo-centric theory, it was the political power of the theocracy that did not. In that sense, given the current theocratic aspirations of the right in American politics, the Galileo analogy (i.e. Church v. Science) is more supportive of the AGW side of the debate.

Thanks for the reply. Yes, if one is to be remembered for rejecting the consensus, it is better to be remembered as an Ignaz Semmelweis than as a Peter Duesberg (a prominent cell biologist who rejected the HIV virus theory).


text:
# Problem of the Week Problem D That Triangle

In the diagram, $$ABCD$$ is a rectangle. Point $$E$$ is outside the rectangle so that $$\triangle AED$$ is an isosceles right-angled triangle with hypotenuse $$AD$$. Point $$F$$ is the midpoint of $$AD$$, and $$EF$$ is perpendicular to $$AD$$.

If $$BC=4$$ and $$AB=3$$, determine the area of $$\triangle EBD$$.


text:
# Five Character ASCII Map

Note: In this post, the terms 'character' and 'color' mean essentially the same thing

This image:

can be represented as

....'''333
.eeee'''3e
..dddd33ee
%%%dd####e

(mapping colors to ascii characters)

The four color theorem states that "given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions." - Wikipedia (link)

This means that it should be possible to color a map using four colors so that no two parts which share an edge share a color.

The algorithm to color a map using only four colors is complicated so in this challenge your program only needs to color the map using five or less colors.

The previous map recolored could look like this:

which could be represented as

....'''333
.eeee'''3e
..dddd33ee
333dd....e

or equivalently

@@@@$$!!! @^^^^$$$!^ @@<<<<!!^^ !!!<<@@@@^  Challenge: Given a "map" made of ascii characters (where each character represents a different color), "recolor" the map (represent the map using different ascii characters) so that it only uses five or less colors. Example: Input: %%%%%%%%%%%%##########%% *****%%%####!!!!!!!%%%%%%%%%#^^^ (((((((***>>>>??????????%%%%%%%% &&&&&&&&$^^^^^^^))@@@%%%%%
^^^^^^%%%%%%%%%%%%##############

Possible output:

11111111111122222222223333333311
44444111222255555551111111112444
22222224441111444444444411111111
55555555222222255555553355511111
22222211111111111122222222222222

Clarifications:

• The input map will always use six or more characters
• You may use any five different characters in the output
• You can use less than different five characters in the output
• You may take the input in any reasonable format (including an array of arrays, or an array of strings)
• This is code-golf so the shortest answer wins.

• I see, at least in your example, that the "maps" aren't actually planar graphs, given that non-contiguous regions seem to have to be the same color. This means that you could easily create a graph that needed 6 or more colors to color. Should we treat the map 121 as 3 separate regions to avoid this problem, even though the example implies otherwise, or should we treat it as 2, and assume that no map will be given that needs more than 5 colors? – MildlyMilquetoast Oct 31 '17 at 5:44
• There isn't an error in the example, it's just that the example could work either way - its not wrong, just ambiguous. It would help to specify whether different regions labeled with the same character are the same or multiple regions in the rules. – MildlyMilquetoast Oct 31 '17 at 5:49
• Funnily enough, I'm writing an essay on the proof of the four color theorem right now. I have to say, the proof for the five color theorem is a lot less complicated – MildlyMilquetoast Oct 31 '17 at 5:50

# Python 2, 375361359357355353350 347 bytes

e=enumerate
C=set('12345')
def f(s):
s=[map(ord,l)for l in s]
for i,v in e(s):
for j,w in e(v):
if{w}-C:
F,N=g([],s,i,j,w)
for y,x in F:s[y][x]=min(C-N)
return s
def g(m,s,i,j,c):
n={s[i][j]}
if(n^{c}or(i,j)in m)<1:
m+=(i,j),
for x,y in(0,1),(0,-1),(1,0),(-1,0):
if len(s)>i+x>-1<j+y<len(s[0]):m,N=g(m,s,i+x,j+y,c);n|=N
return m,n

Try it online!

Takes input as a list of strings, and returns a list of lists

f takes the map input and colors it, g returns all connected characters and the set of their neighbors, to the area can be colored with a distinct color.

• 361 bytes – ovs Oct 31 '17 at 9:27
• @ovs Thanks :-) – TFeld Oct 31 '17 at 9:41
• 359 bytes – Felipe Nardi Batista Oct 31 '17 at 9:56
• @FelipeNardiBatista Thanks :) – TFeld Oct 31 '17 at 10:15
• if~-(n!={c}or(i,j)in m): for -2 bytes – Felipe Nardi Batista Oct 31 '17 at 10:31


text:
How short is your title?

What is the shortest journal paper title that you have come across? For me it is H = W, by Norman G Meyers and James Serrin (Proc Nat Acam Sci USA, 1964 – see http://www.pnas.org/cgi/reprintframed/51/6/1055).

The paper proves that $W^m_p(\Omega) = H^m_p(\Omega)$ where the first (Banach) space contains functions (distributions) whose norm,

$\Vert v\Vert_{W^m_p(\Omega)} := \left( \sum_{\vert\alpha\vert\le m}\Vert D^\alpha v\Vert_{L_p(\Omega)}^p \right)^{1/p}$

(with the usual $\mathrm{max}$ and $\mathrm{ess\, sup}$ adjustments when $p=\infty$), is finite while the second is defined as the closure of $C^\infty(\Omega) \cap W^m_p(\Omega)$ with respect to that norm. The result is true if $\Omega\subset\mathbb{R}^n$ is open but need not be if any part of the boundary is included.

By the way, not only is it a very short title, the paper itself is less than two sides.

The point of this entry to test LaTeX a bit more in this forum. I used Chrome to do this. IE seemed not to like \cap, and the LaTeX preview seems to get the math font sizes wrong…

Fit the First

Does $\LaTeXe$ work? If so then find $u\in V$ such that $a (u, v) = $ for all $v\in V$.

This was harder than it looks. Start the code with $latex but without the space and end it with$ but make sure the code begins and ends with a space. Snarkish… Hence the borrowed title.


text:
Free Version
Moderate

# Triangle Side Lengths: BC = 16

ACTMAT-VNJYKG

In the figure below, $BC = 16.$

What is the measure of ${\overline { AB } }$?

A

$8$

B

$8\sqrt { 2 }$

C

$8\sqrt { 3 }$

D

$8\sqrt {6 }$

E

$16$


text:
# Math Help - Limit Point Question

1. ## Limit Point Question

Suppose a series $z_n \to \alpha$ as $n \to \infty$.

(That is to say $( z_n )$ has a limit point $\alpha$ at infinity.)

Prove that
$\lim_{n \to \infty} \frac{z_1 + z_2 + ~...~ + z_n}{n} = \alpha$

I have found a number of series where I can prove it case by case, but am having some trouble generalizing. In addition, if the series $z_n = 1 - \frac{1}{n}$ I am having an embarrassing time showing the result at all.

Another question: Does this theorem have an actual use? I can't think of one.

(FYI: This problem is actually designed for the series to be complex. I figure if I can find out how to prove it for real series then I can generalize it to complex numbers myself.)

Thanks!
-Dan

2. Originally Posted by topsquark
Suppose a series $z_n \to \alpha$ as $n \to \infty$.

(That is to say $( z_n )$ has a limit point $\alpha$ at infinity.)

Prove that
$\lim_{n \to \infty} \frac{z_1 + z_2 + ~...~ + z_n}{n} = \alpha$

I have found a number of series where I can prove it case by case, but am having some trouble generalizing. In addition, if the series $z_n = 1 - \frac{1}{n}$ I am having an embarrassing time showing the result at all.

[snip]
If $z_n = 1 - \frac{1}{n}$ then

$\frac{z_1 + z_2 + ~...~ + z_n}{n} = \frac{ \left( 1 - \frac{1}{1} \right) + \left( 1 - \frac{1}{2} \right) + \left( 1 - \frac{1}{3} \right) + ~...~ + \left( 1 - \frac{1}{n} \right)}{n}$

$= \frac{n - \sum_{k=1}^{n} \frac{1}{k}}{n}$

$= 1 - \frac{\sum_{k=1}^{n} \frac{1}{k}}{n}$.

$\lim_{n \rightarrow \infty} \frac{\sum_{k=1}^{n} \frac{1}{k}}{n} = 0$ since the harmonic series diverges to infinity a heck of a lot slower than n does (I know that's not a solid technical argument but I'm sure it can be made so easily ..... is the glib 'easily' business the sticking point?)

3. Originally Posted by topsquark
Suppose a series $z_n \to \alpha$ as $n \to \infty$.

(That is to say $( z_n )$ has a limit point $\alpha$ at infinity.)

Prove that
$\lim_{n \to \infty} \frac{z_1 + z_2 + ~...~ + z_n}{n} = \alpha$

I have found a number of series where I can prove it case by case, but am having some trouble generalizing.
[snip]
This seems way too glib to be a correct proof:

If $z_n \rightarrow \alpha$ as $n \to \infty$, then $z_1 \rightarrow \alpha$, $z_2 \rightarrow \alpha$, .....

Then $\lim_{n \to \infty} \frac{z_1 + z_2 + ~...~ + z_n}{n} = \lim_{n \to \infty} \frac{\alpha + \alpha + ~...~ + \alpha}{n} = \lim_{n \to \infty} \frac{n \alpha}{n} = \alpha$ ....

But like I said ....

4. Originally Posted by topsquark
Suppose a series $z_n \to \alpha$ as $n \to \infty$.

(That is to say $( z_n )$ has a limit point $\alpha$ at infinity.)
Do you really mean series or do you mean sequence?
One does not usually say that a series has a limit point.

5. I proved this for my analysis class last year . It usues real numbers, but you can easily generalize it. Here.

6. Originally Posted by Plato
Do you really mean series or do you mean sequence?
One does not usually say that a series has a limit point.
Yes, you are correct. I should have said sequence. (What can I say, it was a late night last night!)

-Dan

7. Originally Posted by mr fantastic
$\lim_{n \to \infty} \frac{z_1 + z_2 + ~...~ + z_n}{n} = \lim_{n \to \infty} \frac{\alpha + \alpha + ~...~ + \alpha}{n} = \lim_{n \to \infty} \frac{n \alpha}{n} = \alpha$ ....
Thank you for the effort, but this statement is incorrect. Perhaps the mistake in terminology Plato pointed out is a problem. So let me restate the original problem in more correct terms:

Suppose a sequence $( z_n )$ has the property $z_n \to \alpha$ as $n \to \infty$.

(That is to say $( z_n )$ has a limit point $\alpha$ at infinity.)

Prove that
$\lim_{n \to \infty} \frac{z_1 + z_2 + ~...~ + z_n}{n} = \alpha$

Since we are talking about a sequence here we cannot say that $z_n \to \alpha$ for all n, we are saying that the sequence approaches a limit point $\alpha$, as n approaches infinity. That is to say there is only one sequence, not an infinite number of sequences tending toward the same limit.

-Dan

8. Originally Posted by ThePerfectHacker
I proved this for my analysis class last year . It usues real numbers, but you can easily generalize it. Here.
I am somewhat confused almost from the beginning by the line
$| s _n - s | = \left | \frac{s_1 + ~...~ + s_n - s \cdot n}{n} \right |$

Is this again a case of my mistaken wording? If you define $s_n$ to be the nth partial sum of a series, this statement is quite understandable and falls out naturally. (Hopefully by now it is clear that this is not what I meant.)

-Dan

9. Originally Posted by topsquark
Suppose a series $z_n \to \alpha$ as $n \to \infty$.

(That is to say $( z_n )$ has a limit point $\alpha$ at infinity.)

Prove that
$\lim_{n \to \infty} \frac{z_1 + z_2 + ~...~ + z_n}{n} = \alpha$
There's a neat proof of this by CaptainBlack here for the case where α=0. You can easily deduce the general case from this by applying it to the sequence $(z_n-\alpha)$.

Originally Posted by topsquark
Another question: Does this theorem have an actual use? I can't think of one.
This is a case of Cesàro summation, which is an essential tool in Fourier theory. The main result about convergence of Fourier series (Fejér's theorem) is proved by using Cesàro summation.

So yes, the fact that ordinary convergence implies Cesàro convergence is certainly useful.

10. Topsquark try proving this similar result: Let $x_n>0$ and $\lim ~ x_n = x$. Prove that the geometric means $(x_1...x_k)^{1/k} \to x$.

11. Originally Posted by topsquark
I am somewhat confused almost from the beginning by the line
$| s _n - s | = \left | \frac{s_1 + ~...~ + s_n - s \cdot n}{n} \right |$

Is this again a case of my mistaken wording? If you define $s_n$ to be the nth partial sum of a series, this statement is quite understandable and falls out naturally. (Hopefully by now it is clear that this is not what I meant.)

-Dan
I am using $\sigma_n$ not $s_n$. Where $\sigma_n$ is the sequence of averages.


text:
# Java getClass().getResource("file") leads to NullPointerException

I am following the Snake Java games tutorial and always get this error:

ImageIcon iid = new ImageIcon(this.getClass().getResource("ball.png"));
ball = iid.getImage();

at javax.swing.ImageIcon.<init>(Unknown Source)
at snake2.Board.<init>(Board.java:52)
at snake2.Snake.<init>(Snake.java:10)
at snake2.Snake.main(Snake.java:22)

I actually just copied and pasted the code to see how it works. They are in the right packages too; but when I try to run it, I always end up with this error.

• Is there a ball.png file next to the .java file? In case you're using eclipse, did you refresh the source folder? Is the code above called from a subclasses method in a different package? Apr 24, 2011 at 7:51
• i imported it in the wrong place .i thought importing it in the project adds it in the resources. how about adding resources globally like i want to add D:\myresources\ to my project Apr 24, 2011 at 8:02

The image should be in the same package (folder in OS terms) as the compiled class. Check whether you have both .class and .png in the same folder. If not, you can use classpath-relative paths in getResource(..), by starting with /

Try this:

ImageIcon iid = new ImageIcon(this.getClass()
ball = iid.getImage();

Make sure image is in the same folder as java file.

Try using System.out.println(System.getProperty("java.class.path")); to find out location of your .class file and place the images in this folder.

It is general risky to load resources using relative paths, I'd always recommend using absolute paths, so do

 /ball.png

if the the image is at the root of your classpath, or add a path to the location.

You have to put the image file(ball.png) into your classpath. More details, please take a look at the Javadoc.

if the resource is in your classpath then you should be trying "this.getClass().getClassLoader().getResource("ball.png")". For you actual code to work, the ball.png needs to be in the location where your .class file is (i.e., inside the package).

• Thse two sentences re mutually contradictory. Using the class loader method loses the information about where the .class file is. Jan 20, 2014 at 3:32

Go to project >clean in the eclipse it would refresh the package explorer and you won't face this problem anymore.

You may need to add the file to your build resources, something like this:

<build>
<resources>
<resource>
<directory>path\to\resources</directory>
<includes>
<include>ball.png</include>
</includes>
</resource>
</resources>

ImageIcon iid = new ImageIcon("C:\\Users\\ranig\\My\\spaceinvaders\\ball.png");

Note: C:\\Users\\ranig\\My\\spaceinvaders\\ball.png is the whole path of ball.png image.

ImageIcon iid = new ImageIcon(this.getClass().getResource("ball.png"));

Note: If u want to only try snake code and only want to get output.

I will make it simple for you . Here is an example:

Icon bug = new ImageIcon(getClass().getResource("bug1.png"));

here "bug1.png" is the resource and if it is unavailable then it can cause error as you have discussed here.

Import an image to the same directory in which your program resides.

You can also give whole path to it as well

ImageIcon(getClass().getResource("C://me/file/bug1.png"));

The resource so named wasn't found. It needs to be in the same directory as the .class file you are calling it from. See the Javadoc.

• so by that you mean i should add the ball.png to the resource?how? Apr 24, 2011 at 7:53
• @kapitanluffy: You seem to be using the resources API without knowing the first thing about what it's for. I suggest you read the Javadoc. Apr 24, 2011 at 13:15


text:
Homomorphisms and Endomorphisms of Group Representations

# Homomorphisms and Endomorphisms of Group Representations

 Definition: Let $G$ be a group and let $(V_1, \rho_1)$ and $(V_2, \rho_2)$ be group representations of $G$. A Group Representation Homomorphism (or simply Morphism) between these group representations is a linear map $\varphi : V_1 \to V_2$ that is $G$-equivariant, that is, for all $g \in G$ and for all $v \in V_1$ we have that $\varphi([\rho_1(g)](v)) = [\rho_2(g)](\varphi(v))$. The space of all homomorphisms of $(V_1, \rho_1)$ to $(V_2, \rho_2)$] is denoted by [[$\mathrm{Hom}^G(V_1, V_2)$.

The following proposition tells us that if $(V_1, \rho_1)$ and $(V_2, \rho_2)$ are group representations of $G$ and if $\varphi \in \mathrm{Hom}^G(V_1, V_2)$ then $\ker \varphi$ is a subrepresentation of $V_1$ and $\mathrm{range} \varphi$ is a subrepresentation of $V_2$.

 Proposition 1: Let $G$ be a group and let $(V_1, \rho_1)$ and $(V_2, \rho_2)$ be group representations of $G$. If $\varphi \in \mathrm{Hom}^G(V_1, V_2)$ then $\ker \varphi$ is a subrepresentation of $V_1$ and $\mathrm{range} \varphi$ is a subrepresentation of $V_2$.
• Proof: Clearly $\ker \varphi$ is a subspace of $V_1$. Furthermore, for all $g \in G$ and all $v \in \ker \varphi$ we have that:
(1)
\begin{align} \quad \varphi([\rho_1(g)](v)) = [\rho_2(g)](\varphi(v)) = [\rho_2(g)](0) = 0 \end{align}
• So if $v \in \ker \varphi$ we see that $[\rho_1(g)](v) \in \ker \varphi$. So $\ker \varphi$ is $G$-invariant and thus $\ker \varphi$ is a subrepresentation of $V_1$.
• Similarly, clearly $\mathrm{range} \varphi$ is a subspace of $V_2$. Furthermore, for all $g \in G$ and all $w \in \mathrm{range} \varphi$ we have that there exists a $v \in V_1$ such tht $\varphi(v) = w$. So:
(2)
\begin{align} \quad [\rho_2(g)](w) = [\rho_2(g)](\varphi(v)) = \varphi([\rho_1(g)](v)) \end{align}
• So if $w \in \mathrm{range} (\varphi)$ we see that $\rho_2(g) \in \mathrm{range} \varphi$. So $\mathrm{range} \varphi$ is $G$-invarint and thus $\mathrm{range} \varphi$ is a subrepresentation of $V_2$. $\blacksquare$
 Definition: Let $G$ be a group and let $(V, \rho)$ be a group representation of $G$. An Group Representation Endomorphism is a group representation homomorphism from $(V, \rho)$ to $(V, \rho)$. The space of all group endomorphisms on $(V, \rho)$ is denoted by $\mathrm{End}^G(V)$.