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I have two difficulties on understanding the solution to an example in a course I took this semester on optimization. This example is given to illustrate the usage of Lagrange multiplier method (please see Example 1 in the image below:
$${\Large ?}\;\left\{\begin{align*}(x_1-x_2)(x_2-x_3)(x_3-x_1)&=0\\x_1^2+x_2^2+x_3^2... |
The way that this is usually done in QED is deeply related to the Ward Identities and QED's LSZ reduction formula. This discussion is taken from Chapter 67-8 in Srednicki's book "Ward Identities in QED I-II".
In particular it is related to the
amputated correlators of $\langle 0 |T\{ A^\mu(x) A^\nu (0)\} |0\rangle$, wh... |
10 True of False Problems about Nonsingular / Invertible Matrices Problem 500
10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.
The quiz is designed to test your understanding of the basic properties of these topics.
You can take the quiz as many times as you like.
The solu... |
Proving The Existence of Limits Examples 4
We will now look at some more examples of proving the existence of limits using the definition of a limit, that is $\lim_{x \to a} f(x) = L$ says that $\forall \epsilon > 0 \: \exists \delta > 0 \: \mathrm{s.t.} \: \forall x : 0 < \mid x - a \mid < \delta, \mid f(x) - L \mid <... |
First, you should note that the set of isolated points of $E$ is countable. This is in fact a general property of $\mathbb{R}$
Theorem: Let $E$ be a subset of $\mathbb{R}$ and let $F$ be the set of isolated points of $\mathbb{R}$. Then $F$ is at most countable.
Proof: Suppose otherwise, that is, that $F$ is uncountable... |
Working out the non relativistic limit of the Dirac equation, we encounter this quantity: $(\vec{\sigma} \cdot \vec{p})$ and in my notes it says that $$ (\vec{\sigma} \cdot \vec{p})^2 = p^i p^j\sigma^i\sigma^j=\vec p^{\,2} \tag{1} $$
When we couple the Dirac equation and we write $$\vec{p} \rightarrow \vec{p} - \frac{e... |
Difference between revisions of "Main Page"
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== The Problem ==
== The Problem ==
−
Let <math>[3]^n</math> be the set of all length <math>n</math> strings over the alphabet <math>1, 2, 3</math>. A ''combinatorial line'' is a set of three points in <math>[3]^n</math>, formed by taking... |
A boring method is to carefully apply the (partially) extended Euclidean algorithm.
But in the question, the modulus is a power of two (specifically $2^6$), and we can use that$$a\,x\equiv1\pmod{2^k}\implies a\,x\,(2-a\,x)\equiv1\pmod{2^{2k}}$$from which it follows this fact:
if the modular inverse of $a$ modulo $2^k$ ... |
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J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed in... |
Limits of Products of Complex-Valued Functions
Recall from the Limits of Sums and Differences of Complex-Valued Functions page that if $(S, d_S)$ and $(\mathbb{C}, d)$ are metric spaces where $d$ is the usual metric on $\mathbb{C}$ defined for all $x, y \in \mathbb{C}$ by $d(x, y) = \mid x - y \mid$ then if $A \subsete... |
Open and Closed Set Differences in Metric Spaces
Open and Closed Set Differences in Metric Spaces
Suppose that $(M, d)$ is a metric space and that $A, B \subseteq M$. Suppose that we know that $A$ is an open subset and $B$ is a closed subset. What can we say about the differences $A \setminus B$ and $B \setminus A$? Ar... |
Darboux's theorem says that any symplectic manifold $(M^{2n}, \omega)$ is locally symplectomorphic to the "trivial" symplectic manifold $( \Bbb R^{2n}, \omega_0)$, where$$\omega_0 = \sum_{j = 1}^n dx^j \wedge dy^j$$is the standard symplectic form on $\Bbb R^{2n}$. What this means is the following. Given any point $p \i... |
Dihedral Group and Rotation of the Plane Problem 52
Let $n$ be a positive integer. Let $D_{2n}$ be the dihedral group of order $2n$. Using the generators and the relations, the dihedral group $D_{2n}$ is given by
\[D_{2n}=\langle r,s \mid r^n=s^2=1, sr=r^{-1}s\rangle.\] Put $\theta=2 \pi/n$. (a)Prove that the matrix $\... |
If ice is "all around the sun" I fail to see how it can be moving at a velocity of 1000 m/s inwards. The mass of the sun is $2\cdot 10^{30}\mathrm{\;kg}$ and the radius $7\cdot 10^{8}\mathrm{\;m}$.
The thickness of a shell of ice with that inner radius and mass would be (assuming the usual density of ice of about 0.9x ... |
, and to attain this field in specific regions of the brain, the electric current should pass through different head layers via skin, fat, skull, meninges, and cortex (part of the brain). In order to model the brain, different layers should be considered, including gray and white matters.The meninges, three layers of p... |
Consider a problem with three variables: $u$, $\sigma_l$, and $\sigma_w$ where $\sigma_w > \sigma_l$. I want to represent the following relationship using integer programming. \begin{equation} u = \begin{cases} \sigma_w - x & x < \sigma_l \\ 0 & x > \sigma_l \end{cases} \end{equation}
Using simple either-or constraints... |
In the mathtools documentation, page 27, a command
\Set* (along with
\Set) is defined to stretch according to the large input inside:
\providecommand\given{}\newcommand\SetSymbol[1][]{ \nonscript\:#1\vert \allowbreak \nonscript\: \mathopen{}}\DeclarePairedDelimiterX\Set[1]\{\}{ \renewcommand\given{\SetSymbol[\delimsize... |
Directional Derivatives
We will now look at a new type of derivative known as a
directional derivative. Directional Derivatives of Two Variable Functions
Let $z = f(x, y)$ be a two variable real-valued function. Recall from the Partial Derivatives page that the partial derivative $\frac{\partial z}{\partial x} = \lim_{... |
Identity Matrices
Identity Matrices
Definition: A square $n \times n$ matrix $I$ is considered an Identity Matrix if all entries along the main diagonal are 1 and all other entries are 0. Alternatively we can define identity matrices such that if $(I)_{ij} = 1$ if $i =j$ and $(I)_{ij} = 0$ otherwise.
We note that all i... |
Measurable Functions
So far we have looked at three major classes of functions.
On the Step Functions on General Intervals page we said that a function $f$ on a general interval $I$ is a step function if there exists a closed and bounded interval $[a, b] \subseteq I$ such that there exists a partition $P = \{ a = x_0, ... |
Sometimes it is more appropriate to utilize what is known as the
vector form of the equation of plane. Vector Form Equation of a Plane
Let $\vec{n} = (a, b, c)$ be a normal vector to our plane $\Pi$, that is $\Pi \perp \vec{n}$. Instead of using just a single point from the plane, we will instead take a vector that is ... |
The BBC article Event Horizon Telescope ready to image black hole describes the Event Horizon Telescope, a coordinated observing technique with several radio telescope arrays across the globe forming a synthetic aperture with an Earth-sized baseline.
$$\frac{\lambda}{r_{Earth}} \sim \frac{r_{Sag A*}}{D_{Sag A*}} \sim 1... |
The Annals of Statistics Ann. Statist. Volume 10, Number 1 (1982), 297-301. An Inequality Comparing Sums and Maxima with Application to Behrens-Fisher Type Problem Abstract
A sharp inequality comparing the probability content of the $\ell_1$ ball and that of $\ell_\infty$ ball of the same volume is proved. The result i... |
Question
Find the resistance that must be placed in parallel with a $10.0 \Omega$ galvanometer having a $100 \mu \textrm{A}$ sensitivity to allow it to be used as an ammeter with: (a) a 20.0-A full-scale reading, and (b) a 100-mA full-scale reading.
Final Answer
$5.00 \times 10^{-5} \Omega$ $10.0 \textrm{ m}\Omega$ Cal... |
This example is an implementation of the assessment of a new total hip replacement (THR) technology described in chapter 3.5 of Decision Modelling for Health Economic Evaluation. A more detailed report is available at this location. This reports goes a bit further in the analysis. For the sake of simplicity we will not... |
I am stuck in this exercise from my textbook:
Consider a one-period market model with $N+1$ assets: a bond, a stock and $N-1$ call options. The prices of the bond are $B_0=1$ and $B_1 = 1+r$, where $r$ is a constant. The prices of the stock are given by a constant $S_0$ and a random variable $S_1$ taking values in $\{0... |
Preprints (rote Reihe) des Fachbereich Mathematik Refine Year of publication 1993 (13) (remove) Has Fulltext yes (13) (remove)
240
244
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Efficient algorithms and structural results are presented for median problems with 2 new facilities including the classical 2-Median problem, the 2-Median problem with forbidden reg... |
I am trying to prove the following relation
$$ \langle\Phi|\hat{\bar{\psi}}(x)A\hat\psi(x)|\Phi\rangle=\bar\varphi(x)A\varphi(x)+\langle0|\hat{\bar{\psi}}(x)A\hat\psi(x)|0\rangle\,, $$ where $|\Phi\rangle$ is a state that contains a single electron and it is defined in this way
$$ |\Phi\rangle=\sum_{r=1}^2\int d^3\!q\;... |
Perceptron is the most basic and primary implementation of a biological neuron in machine intelligence. Moreover the concept of perceptron can be leveraged to build more complex neural networks which we will see later. These are used mainly for supervised learning and can be modified to work with unsupervised learning ... |
Difference between revisions of "Probability Seminar"
(→Probability related talk in PDE Geometric Analysis seminar: Monday, 3:30pm to 4:30pm, Van Vleck 901)
(→February 21, Diane Holcomb, KTH)
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Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polyn... |
Problem 533
Consider the complex matrix
\[A=\begin{bmatrix} \sqrt{2}\cos x & i \sin x & 0 \\ i \sin x &0 &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x \end{bmatrix},\] where $x$ is a real number between $0$ and $2\pi$.
Determine for which values of $x$ the matrix $A$ is diagonalizable.
When $A$ is diagonalizable, find... |
I want to see if a propeller is bad or good for generating static thrust. This is what I know about the propeller and the engine used to turn it during a test:
Engine shaft power, $P$ = 0.58 hp
Propeller diameter, $D$ = 8.5 ft
Revolutions per minute, $RPM$ = 245
Static thrust, $T$ = 18.75 lbf
Can I calculate the effici... |
What is meant by a local Lagrangian density?
How will a non-local Lagrangian look like?
What is the problem that we do not consider such Lagrangian densities?
What is a local Lagrangian density?
A classical field theory on Minkowski space $\mathbb R^{d,1}$ is specified by a space $\mathcal C$ of field configurations $\... |
Matrices
Definition: A Matrix $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$ is a rectangular array of numbers or expressions known as entries to the matrix. If $A$ has $m$ rows and $n$ col... |
Any longtime reader cannot help but notice that we get many
abstract duplicate questions, e.g. this recent question on partial fraction computation, which is not essentially different from many other questions of the same shape, e.g. this question. Once you know how to solve one of these problems you can solve them all... |
The vector $\vec X = [X_1,\dots,X_N]^T$ has a rotationally invariant distribution. That is, if $A$ is any orthogonal matrix, then the distribution of $\vec X$ and $A \vec X$ are the same. Hence by letting $A$ be an orthogonal matrix that takes $[1,\dots,1]^T$ to $[\sqrt N,0,\dots,0]^T$, your problem is the same as comp... |
Tagged: rank of a matrix Problem 643
For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix.
(a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$. (b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$. (c) $C = \begin{b... |
Null Space of a Linear Map
Definition: If $T \in \mathcal L (V, W)$ then the Null Space or Kernel of the linear transformation $T$ is the subset of $V$ defined as $\mathrm{null} (T) = \{ v \in V : T(v) = 0 \}$, that is, the null space of $T$ is the set of vectors from $V$ that are mapped to the zero vector in $W$ under... |
Solar core is 34% of the Sun mass, so sun will continue to be a star, at some point after the event. It will implode and probably intensify the processes compared to previous conditions and to a comparable star with 66% of the Sun mass. Dynamics of collapsing processes, may lead to ejection of plasma, so it definitely ... |
this is a mystery to me, despite having changed computers several times, despite the website rejecting the application, the very first sequence of numbers I entered into it's search window which returned the same prompt to submit them for publication appear every time, I mean ive got hundreds of them now, and it's stil... |
It is easier to talk about these concepts by moving past Knight a bit. There has been a lot of discussion since then. The easiest way to start talking about this is in discussing the three main schools of thought in probability and statistics. The schools are, in order of discovery, the Bayesian, the Likelihoodist and ... |
Table of Contents
Riemann Stieltjes-Integrals with Integrators of Bounded Variation
Recall from the Functions of Bounded Variation page that if $f$ is a function defined on the closed interval $[a, b]$ and if $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$, then the variation of $f$ associated with $P$ is:... |
Evolution equations and subdifferentials in Banach spaces
1.
Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555
2.
Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Tokyo, 169-8555
Studies for th... |
The classical Brown Representability Theorem states: Denote $hCW_*$ the homotopy category of pointed CW-complexes. Let $F : hCW_* \to Set_*$ be a contravariant functor. Then $F$ is representable if and only if
$F$ respects coproducts, i.e. $F(\vee_{i \in I} X_i) = \prod_{i \in I} F(X_i)$ for all families $X_i$ of point... |
Consider the MWE:
\documentclass{memoir}\usepackage{graphicx}\captiontitlefont{\slshape} % All captions slanted\begin{document}\begin{figure} \centering \includegraphics[width=\textwidth]{example-image} \caption{Flowchart of a gradient-based optimization algorithm. $ \Delta \alpha_i^{(k)} $ is the change of $ \alpha_i ... |
The First and Second Arens Products on A** The First Arens Product
Let $\mathfrak{A}$ be a Banach algebra. Consider the second dual, $\mathfrak{A}^{**}$, which is clearly a Banach space. We would like to make $\mathfrak{A}^{**}$ a Banach algebra too, but it is not entirely obvious what the multiplication on $\mathfrak{... |
Thank you for using the timer!We noticed you are actually not timing your practice. Click the START button first next time you use the timer.There are many benefits to timing your practice, including:
Does GMAT RC seem like an uphill battle? e-GMAT is conducting a free webinar to help you learn reading strategies that ... |
In Hardy-Littlewood's 1923 paper
"Some problems of 'Partitio Numerorum' III" it is proven, assuming a weak version of GRH (namely that there is $\varepsilon>0$ s.t. all zeroes of $L(s,\chi)$ have $\Re(s)<3/4-\varepsilon$), that for all $k\geqslant 3$, when $n\to\infty$ through the integers with same parity than $k$:$$ ... |
I think that the question is sufficiently precise if we think at a
realistic meaning of the word “inconsistent”. Also nowadays, for non logicians the adjective “inconsistent” doesn't really mean “free of contradictions” (this is only the obvious meaning given by modern Mathematical Logic), but rather it means not accep... |
Tagged: field Problem 283
Let $F$ be a field and let
\[H(F)=\left\{\, \begin{bmatrix} 1 & a & b \\ 0 &1 &c \\ 0 & 0 & 1 \end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\] be the Heisenberg group over $F$. (The group operation of the Heisenberg group is matrix multiplication.)
Determine which mat... |
The Root Test for Positive Series
The Root Test for Positive Series
We will now look at a test that is analogous to the ratio test. This test can be useful in certain cases, specifically when there are terms raised to the $n^{\mathrm{th}}$ power.
Theorem 1 (The Root Test for Positive Series): If the sequence of terms $... |
Difference between revisions of "De Bruijn-Newman constant"
(→Writeup)
(→Threads)
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* [https://terrytao.wordpress.com/2018/05/04/polymath15-ninth-thread-going-below-0-22/ Polymath15, ninth thread: going below 0.22?], Terence Tao, May 4, 2018.
* [https://terrytao.wordpress.com/2018/05/04/polymath15-ninth... |
Why do people use Quadratic Programming techniques (such as SMO) when dealing with kernelized SVMs? What is wrong with Gradient Descent? Is it impossible to use with kernels or is it just too slow (and why?).
Here is a little more context: trying to understand SVMs a bit better, I used Gradient Descent to train a linea... |
Invertibility of a Linear Map Examples 1
Let $V$ and $W$ be vector spaces. Recall from the Invertibility of a Linear Map page that a linear map $T \in \mathcal L (V, W)$ is said to be invertible if there exists a linear map $S \in \mathcal L (W, V)$ such that $ST = I_V$ and $TS = I_W$ where $I_V$ is the identity map on... |
Square Lebesgue Integrable Functions
We will now discuss another class of functions known as square Lebesgue integrable functions which we define below.
Definition: A function $f$ is said to be Square Lebesgue Integrable on an interval $I$ if $f$ is a measurable function on $I$ and $f^2$ is Lebesgue integrable on $I$. ... |
Tagged: linearly dependent Problem 603
Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the functions \[f(x)=\sin^2(x) \text{ and } g(x)=\cos^2(x)\] in $C[-2\pi, 2\pi]$.
Prove or disprove that the functions $f(x)$ and $g(x)$ are linearly independent.... |
I believe you are confusing the wing angle of attack with the pitch of the aircraft. Aircraft moving at a slow, near-stall speed, despite pointing the nose up, will still be traveling more or less horizontally. Their VSI instrument will read near zero. Whereas, if you take an aircraft moving quickly and pull the nose u... |
show that this integral:
$$\dfrac{1}{2\pi h}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{\dfrac{-i(p-p')x}{h}}x^n\varphi{(p')}dxdp'=\left(ih\dfrac{\partial }{\partial p}\right)^n\varphi{(p)}$$
where $i^2=-1$
maybe this use integration by parts？ But I fell very hard,and I can't prove it.
I think first we must this $... |
Firstly, I wasn't sure exactly where to put this. It's a typesetting query but the scope is greater than $\TeX$; however it's specific also to physics and even more specific to this site.
I've recently been reading a style guide for scientific publications (based on ISO 31-11), however there was no mention of quantum m... |
Matrix Multiplication
Definition: Given matrix $A$ of size $m \times r$ and matrix $B$ of size $r \times n$, their product denoted $AB$ is the $m \times n$ matrix whose $ij^{th}$ entries result from taking row $i$ in matrix $A$ and multiplying corresponding entries of column $j$ in matrix $B$ and summing their products... |
Because the need for color manipulation comes up fairly often in computer graphics, particularly transformations of hue, saturation, and value, and because some of this math is a bit tricky, here’s how to do HSV color transforms on RGB data using simple matrix operations.
Note: This isn’t about converting between RGB a... |
The Radius of Curvature at a Point on a Curve
Definition: Let $\vec{r}(t) = (x(t), y(t), z(t))$ be a vector-valued function that traces out the smooth curve $C$, and let $P$ be the point on $C$ at $t$. The Radius of Curvature at $P$ is $\rho (t) = \frac{1}{\kappa (t)}$ provided that $\kappa (t) \neq 0$. For functions $... |
To send content items to your account,please confirm that you agree to abide by our usage policies.If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.Find out more about sending content to .
To send content items to your Kindle, first ensure no-rep... |
Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded ... |
I'm conducting a X-ray diffraction experiment for lab, but am new to solid-state physics and crystallography. I have to find the Debye-Waller factor (DWF) of Al at room temperature using X-ray powder diffraction. The value given in the International Tables for X-Ray Crystallography Vol. 3 is approximately $0.78 \mathri... |
Answer
A=36$\pi$$\approx$113
Work Step by Step
The diameter of the inscribed circle is equal to the height of the isosceles trapezoid. h$^2$=13$^2$-($\frac{1}{2}$(18-8))$^2$ h$^2$=169-25=144 h=12 r=.5h r=6 A=$\pi$r$^2$ A=36$\pi$$\approx$113
You can help us out by revising, improving and updating this answer.Update this... |
Bounded Subsets in Euclidean Space
Definition: Let $S \subseteq \mathbb{R}^n$. The set $S$ is said to be Bounded if there exists a $\mathbf{x} \in \mathbb{R}^n$ and a positive real number $r > 0$ such that $S \subseteq B(\mathbf{x}, r)$ and $S$ is said to be Unbounded otherwise. In other words, a subset $S$ of $\mathbb... |
I am using the align command to align equations :
\begin{align*} \psi \colon & SU(2)\otimes SU(2) \otimes SU(2) \to \mathbb{C}\\ &(U_{l_1} \otimes U_{l_2} \otimes U_{l_3}) \mapsto \psi(U_{l_1} \otimes U_{l_2} \otimes U_{l_3}).\end{align*}
As I understand it, the "&" tells to latex where the object need to be vertically... |
The Dehn invariant of a polyhedron is a vector in $\mathbb{R}\otimes_{\mathbb{Z}}\mathbb{R}/2\pi\mathbb{Z}$ defined as the sum over the edges of the polyhedron of the terms $\sum\ell_i\otimes\theta_i$ where $\ell_i$ is the length of edge $i$ and $\theta_i$ is its dihedral angle.
Are all vectors in this space realizable... |
(a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ is invertible, then $ABA^{-1}=B$. (d) If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix. (e) If... |
Table of Contents
Examples of Closed Unit Balls That are NOT Compact
Recall from the Closed Unit Ball Criterion for Finite Dimensional Normed Linear Spaces page that if $(X, \| \cdot \|_X)$ is a normed linear space then the closed unit ball of $X$ is compact if and only if $X$ is finite-dimensional.
We will now look at... |
LHCb Collaboration,; Aaij, R; Adeva, B; Adinolfi, M; Bernet, R; Bowen, E; Bursche, A; Chiapolini, N; Chrzaszcz, M; Dey, B; Elsasser, C; Graverini, E; Lionetto, F; Lowdon, P; Mauri, A; Müller, K; Serra, N; Steinkamp, O; Storaci, B; Straumann, U; Tresch, M; Vollhardt, A; Weiden, A; et al, (2015).
Differential branching f... |
I am currently going through electromagnetic form factor. I came across the fact that since the proton is not an elementary particle its scattering(elastic) with electron can be modeled using general general electromagnetic form factor of form
$ \Gamma_{\mu} = \gamma_{\mu}F_1(q^2) + i\sigma_{\mu\lambda}q^{\lambda}F_2(q... |
Topological Methods in Nonlinear Analysis Topol. Methods Nonlinear Anal. Volume 13, Number 2 (1999), 181-190. Degree and Sobolev spaces Abstract
Let $u$ belong (for example) to $W^{1,n+1}(S^n\times \Lambda, S^n)_{\lambda\in\Lambda}$ where $\Lambda$ is a connected open set in ${\mathbb R}^k$. For a.e. $\lambda\in\Lambda... |
To form a product, you give me $n$ objects, $A_1,\dots,A_n$, and I give you back an object $A_1\times\dots\times A_n$, together with $n$ maps $\pi_i\colon A_1\times\dots\times A_n\to A_i$ (one to each of the $A_i$) satisfying the universal property of the product.
So what happens if $n=0$? Then you give me $0$ objects,... |
The Normed Space Induced by an Inner Product
Theorem 1: Let $H$ be an inner produce space. Then the function $\| \cdot \| : H \to \mathbb{R}$ defined for all $x \in H$ by $\| x \| = \langle x, x \rangle^{1/2}$ is a norm on $H$. Proof:We show that $\| \cdot \|$ has all of the properties of a norm. First, suppose that $x... |
One thing to note about relativistic cosmology is its solutions evolve in time. So if at one moment the universe has a specific value of spatial curvature the next moment it would be different. The value of curvature specified in OP is quite large and thus it correspond to just a specific moment near the Big Bang (or B... |
Limits of Polynomials and Rational Functions
Before we look at some theorems regarding the limits of polynomials and rational functions, we should first formally define what each is.
Definition: A function in the form $p(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$ where $a_0, a_1, ..., a_n \in \mathbb{R}$ is said to be a ... |
Directional Derivatives Examples 2
Recall from the Directional Derivatives page that for a two variable real-valued function $z = f(x, y)$, the directional derivative of $f$ at a point $(x, y) \in D(f)$ in the direction of the unit vector $\vec{u} = (a, b)$ is given by the formula:(1)
For a three variable real-valued f... |
What are the open big problems in algebraic geometry and vector bundles?
More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over projective varieties/curves.
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute t... |
The Cauchy-Davenport Theorem says that if $A_1, \ldots, A_k$ are subsets of ${\mathbb Z}_p$, $p$ prime, then $| \sum_i A_i | \geq \min (p, \sum_i |A_i| -k +1)$.
I am looking for a generalization that bounds the number of ways each element $a \in \sum_i A_i$ can be represented as $a=\sum_i a_i$ with $a_i \in A_i$.
Speci... |
Current browse context:
nlin
Change to browse by: References & Citations Bookmark(what is this?) Condensed Matter > Disordered Systems and Neural Networks Title: Chaotic wave packet spreading in two-dimensional disordered nonlinear lattices
(Submitted on 20 Aug 2019)
Abstract: We reveal the generic characteristics of w... |
TL;DR: there is no mathematical certainty that every output value of common cryptographic hash functions is reachable, but for most that's overwhelmingly likely. A notable exception is double-SHA-256 (SHA256d) used in Bitcoin mining, where overwhelmingly likely there are some unreachable outputs.
For an idealized 256-b... |
Feature #1994 LaTEX support in Wiki, Forums and Issues
Status: New Start date: 2008-10-06 Priority: Normal Due date: Assignee: - % Done:
0%
Category: Wiki Target version: - Resolution: Description
Latex support could be included especially in Wiki pages, so that Latex expressions could be displayed as images in Wiki. S... |
I am deeply confused.... For a thermally insulated ideal gas expanding freely, I think that $PV^{\gamma}=cnst$ must hold. Through the equation $PV=nRT$, it must be that $TV^{{\gamma}-1}=\rm constant$. Because the free expansion changes the volume of the gas, the temperature of the gas must change too. However, any book... |
Table of Contents
Sequence of Terms Divergence Criterion for Infinite Series
Recall from the Convergence and Divergence of Infinite Series page that an infinite series $\displaystyle{\sum_{k=1}^{\infty} a_k}$ is said to converge to the sum $s$ if the corresponding sequence of partial sums $(s_k)_{k=1}^{\infty}$ converg... |
I was thinking to solve this by computer programs but I prefer a solution.
How to obtain a list of 3 consecutive non square free positive integers? In general, how to obtain the same kind of list with $k$ elements? Thanks.
Mathematics Stack Exchange is a question and answer site for people studying math at any level an... |
Classical
acoustic theory
derives from fluid mechanics
, and centers on the mathematical
description of sound waves
. See acoustics
for the engineering
approach.
In approaching the description of a sound wave the mathematics never gives the whole story. The subtleties of thermodynamics are difficult enough to recommend... |
Let's say we have a multivariate distribution $D$ which generates random $n$-dimensional vectors $x$ for us ($x \in R^n$). We know that the dimensions of vector $x$ are correlated, and that each dimension of $x$ has a mean of 0 and a standard deviation of 1. Now, let's say we have another random
vector $y$ (of shape $(... |
Table of Contents
Inner Product Spaces Over the Field of Real Numbers
We will soon show that the set of all square Lebesgue integrable functions is an inner product space, but of course, we will first need to formally define an inner product space and inner product (which the reader is likely already familiar with).
De... |
Uniformly Cauchy Sequences of Functions
Recall from the Pointwise Cauchy Sequences of Functions page that a sequence of functions $(f_n(x))_{n=1}^{\infty}$ with common domain $X$ is said to be pointwise Cauchy if for all $\epsilon > 0$ and for all $x \in X$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ ... |
A
definite description is a denoting phrase in the form of "the X" where X is a noun-phrase or a singular common noun. The definite description is proper if X applies to a unique individual or object. For example: "the first person in space" and "the 42nd President of the United States of America", are proper. The defi... |
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Amsrefs is a package for preparing bibliographic lists. If, like me, you use bibtex then you may find this post informative. If you enter your bibliographic items into the tex file manually, \emph-asizing titles and consulting Chicago Manual of Style to ch... |
I know that $c_{d}=0.03 + 0.095c_{l}^2$.
What is the glide ratio?
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The glide ratio for a given angle of attack is the ratio of lift to drag. Both of thes... |
Recall that if we have a matrix that is in Row Echelon Form (REF), then we could use Gaussian Elimination, and if necessary, Back Substitution in order to solve a system of linear equations represented by an augmented matrix. We will now look at the similar method of Gauss-Jordan elimination by reducing a matrix to Red... |
Graph illustrating consumer (red) and producer (blue) surpluses on a supply and demand chart
In mainstream economics,
economic surplus, also known as total welfare or Marshallian surplus (named after Alfred Marshall), refers to two related quantities. Consumer surplus or consumers' surplus is the monetary gain obtained... |
Flavon-induced Higgs lepton flavour violations
Presented by Dr. Venus KEUS
Content
The current experimental limit on Charge Lepton Flavour Violating (CLFV) processes allows the branching ratios of $h \to \tau \mu$ and $h \to \tau e$ processes to be of order 10%. Since such CLFV processes are forbidden in the Standard M... |
The most important thing is practice. As you calculate more and more limits, you'll start to develop some form of intuition regarding what method to try first, and the correct approach will probably come faster to you. You might see a problem and be reminded of some other limit you did before, so you'll try a similar m... |
Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why they use Jacobian matrix to illustrate this theorem.
Let's use a simple example with only two variables. Assume there is ... |
To what extent may the interest rate models be applied for modeling implied volatity?
The story:I was checking different stochastic option pricing models for being able to replicate implied volatility term strucure (namely its hump shape). While doing that, it came to my mind that interest rate term structure is rought... |
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