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12 views

number of ways, n can be written as a product of k integers

For $k\ge2$, let $d_k(n)$ denote the number of ways of writing $n$ as a product of $k$ positive integers (so that $d_2(n) = d(n)$ where $d(n)$ counts the number of positive integral divisors of $n$). ...
0
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1answer
53 views

Crisp subgroup extracted from a $\mathbb Z_n$ considered as a fuzzy set

*Suppose we have a finite crisp set, but elements is fuzzy numbers which of a fuzzy set. So I make definition of the set. A = {$\tilde{0}$,$\tilde{1}$,$\tilde{2}$,...,$\tilde{n}$-$\tilde{1}$} For ...
0
votes
0answers
6 views

The case where adding a point to the data set does not change the interpolating polynomial

Given a set of points $(x_0,f(x_0)),(x_1,f(x_1)),...,(x_n,f(x_n))$, the interpolating polynomial of the function $f(x)$ over the given points is defined as $$L\left(x\right)=\sum_{i=0}^{n}f(x_{i})l_{i}...
1
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1answer
14 views

Theorem 5.11 in Hungerford Algebra $f^{-1}f(K) = K$ if and only if $\ker f < K$.

Let $f \colon G \to H$ be an homomorphism, in the proof of the theorem 5.11, Hungerford states that $f^{-1}(f(K)) = K$ if and only if $\ker f < K$ for $K$ is a subgroup of $G$. I proved the forward ...
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2answers
38 views

Find Two Rank 1 Matrices

I'm looking for rank one matrices B and C such that A = B + C and BC = 0. Where $$A = \begin{bmatrix} 0 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 0 \end{bmatrix}$$ with eigenvalues 0, -2, ...
2
votes
1answer
66 views

Bounded monotone convergence for functions

Let $f:[a,b]\rightarrow\mathbb{R}$ be a bounded, increasing function. Prove that $\;\lim\limits_{x\to b^-} f(x)=\sup\limits_{x\in\left[a,b\right[}f(x)\;.$ This looks a lot like the principle of ...
0
votes
1answer
36 views

Why is “over-parameterized linear regression” non-convex?

Consider the real matrices $V, W, X, Y$. Define the function $L(W, V) := \frac 1 2 \| Y - V W X \|_2^2$. How can it be shown that $L$ non-convex, as claimed here? Tnx. I have tried to look first at ...
4
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3answers
67 views

Would these 2 infinite sets be equal, if so why?

My friends and I were asking the following question: If Minecraft worlds were to be infinite, does that mean that every Minecraft world is identical? My friends and I are adding this constraint to say ...
1
vote
1answer
37 views

Drawing a 3d cylinder, how to find where the surface intersects the ellipse.

I want to draw a cylinder in Tikz. There are a lot of easier ways to do it on tex.stackexchange.com, but I want to do it with the ...
1
vote
3answers
80 views

Is it true that $\emptyset\subseteq \lbrace\emptyset\rbrace$?

I'd think it's true because $\emptyset\subseteq A, \forall A$. Yet I am not sure.
4
votes
2answers
60k views

How do I find percentiles of data sets (Even vs odd)?

Given the following data set with an even number of values: $100, 100, 105, 113, 129, 132, 146, 152, 176, 200$ The value representing the 30th percentile, using the formula n(p/100) where n = sample ...
-1
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4answers
26 views

Invertible non-negative matrices [closed]

Is this statement true or false? If $A$ and $B$ are $2\times2$ invertible matrices with no negative entries, $A + B$ is invertible.
0
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1answer
11 views

Describe all continuous functions on lower limit topology

Describe all continuous functions $f: (\mathbb{R}, \tau_\rightarrow) \rightarrow (\mathbb{R}, |\cdot|)$, where $(\mathbb{R}, \tau_\rightarrow)$ is lower limit topology. We know that the bases of this ...
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4answers
26 views

If $\frac{x}{y}=\frac{3}{2},$ find the value of $\frac{x^3+y^3}{2\sqrt{x^4-y^4}}.\frac{3y(x-y)}{x^3-x^2y+xy^2}$

If $\dfrac{x}{y}=\dfrac{3}{2},$ find the value of $$\dfrac{x^3+y^3}{2\sqrt{x^4-y^4}}.\dfrac{3y(x-y)}{x^3-x^2y+xy^2}. $$ I think it's a good idea to simplify the expression at first: $$\dfrac{(x+y)(x^2-...
0
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1answer
11 views

I want to model a function. Input can be natural numbers, The function should give output as zero when all of the input variables are different.

I need a non linear preferably continuous function which can detect whether all the input variables are different. The input variables are constrained to positive integers. eg input [4,3,2,1] f(x) ...
6
votes
5answers
204 views

How is this a function? - Analysis.

Let $X = \{1, 2, 3\}, Y = \{4, 5, 6\}$. Define $F \subseteq X \times Y$ as $F = \{(1, 4),(2, 5),(3, 5)\}$. Then $F$ is a function. I simply do not see how this could be a function, as there is ...
32
votes
1answer
800 views

The probability that a linear Brownian motion will hit a curve

Summary I am trying to estimate the probability that a standard linear Brownian motion will hit some curve. To make things a bit simple, I can assume that the curve is a graph of a function, that is ...
0
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0answers
23 views

Where is my mistake in calculating the Fresnel integral?

I want to prove the equation: $$ \int_0^\infty \sin x^2dx=\frac{1}{2}\sqrt\frac{\pi}2{} $$ This was my calculating process: $$ ||e^{iz^2}||=||e^{iR^2e^{i\theta}}||=e^{-R^2\sin \theta} $$ Using the ...
0
votes
2answers
37 views

Normalizing a joint PDF $f(x,y)$ and finding the marginal PDF's $f_X(x) $ and $f_Y(y)$

I have the following joint PDF: $$f_{UV}(u,v):= C \cdot \unicode[STIXGeneral]{x1D7D9}_{\{ (u,v): 0\le u \le 1, u \le v \le u+1 \}}(u,v)$$ (In case the subscript of the indicator function is a bit hard ...
0
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0answers
27 views

Evaluating number of collisions between particles in a closed container

This question is regarding whether the following method of calculating number of collisions in a closed container is correct or not. So, it goes as follows:- First of all we make a table about the ...
2
votes
3answers
27 views

How to show sequence converges

What I want to show is that the following sequence converges to 1. $a_n = \frac{\sum_{i=0}^n i!}{n!} $ My initial strategy was to use the monotone sequence theorem. It's obvious that each term is ...
-1
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0answers
14 views

Put our own formula to use matlab quadprog

Good day everyone, I have following formula $$\underset{\hat\lambda_2}{min} \left [ (\frac{1}{2} )\hat\lambda_{2}^T(\frac{1}{2}\textbf{z}^Tyy^T\textbf{z})\hat\lambda_{2} - \textbf{1}^T\hat\lambda_2 ...
0
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0answers
19 views

Let $\phi(x)$ be a formula. What does $\forall z\forall y((\phi(x)\land\phi(y))\to z=y)$ assert? (“Set Theory: A First Course” by Daniel Cunningham)

I am reading "Set Theory: A First Course" by Daniel Cunningham. There is the following exercise in this book (Exercise 1.4.6 on p.23): Let $\phi(x)$ be a formula. What does $\forall z\...
7
votes
1answer
2k views

Prove that the action of a Lie group on its Lie algebra via the adjoint representation reads $\mathrm{ad}(g)(X)=g^{-1}Xg$

I am a physics undergrad. The adjoint action of a group on itself is $\operatorname{Ad}: G \times G \to G$ is defined to be $\operatorname{Ad}:(g,h) \to g^{-1}hg$. The adjoint representation of the ...
0
votes
1answer
62 views

Question in Proof of Every Tychonoff space X can be embedded as a subspace of a cube

I am learning Topology from Dolciaini Expository text in Topology by S.G Krantz. I have question in proof of following theorem: Here $I^{C}$ denote the collection of all functions from C to I. I am ...
0
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1answer
14 views

Prove that projection operator is continuous

[Ciarlet (2.2-4)] Let $K$ be a compact subset of a normed vector space $(X, \Vert \cdot\Vert)$. (1) Show that, given any $x \in X$, there exists $y \in K$ such that $\Vert x - y\Vert = \inf_{z \in K}\...
1
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1answer
22 views

Does it matter how you find total lateral area of a rectangular prism

Let's consider a rectangular prism with length 3, height 6 and width 12 . To find lateral area, does it matter if we do $3(12) + 3(12) + 6(12) + 6 (12)$ giving us 216 then + 2(3)(6) for total surface ...
1
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0answers
40 views

How many Chia plots can I fit on a hard drive, when attempting to fill it completely and maximizing the overall number of plots?

I'm making plots for Chia, a cryptocurrency which isn't "mined" in the strict sense of the word, but instead requires that one have many files known as "plots" on a storage drive (...
2
votes
5answers
74 views

Find the center of the circle given two tangent lines and one point of tangency

I'm attempting to find the center of the purple circle (and/or the radius) given the following information: A point of tangency and the slope of the line (orange line) A point on a line that is ...
0
votes
1answer
18 views

Ring homomorphism from Wiki

From the page of Wiki: $f: \mathbb Z_6\rightarrow Z_6$ defined by $f([a_6])=[4a_6]$ is a ring homomorphism. $f(a)=\{a_n\}=a \pmod n$. The kernel is $3\cdot \mathbb Z_6$ and image is $2\cdot\mathbb Z_6$...
0
votes
1answer
17 views

Prove that in the conjugate gradient method, ${d^{(k)}}^\top Qd^{(k)} = - {d^{(k)}}^\top Qg^{(k)}$

Let $f:\mathbb{R^n} \rightarrow\mathbb{R},f(\mathbf{x})=\mathbf{x^TQx-x^Tb}$, where $\mathbf{b \in \mathbb{R^n}}$ and $\mathbf{Q}$ is a real symmetric positive definte n$\times$n matrix. How to show ...
0
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0answers
22 views

Having 4 equations, find $x$ in terms of known variables.

Having these 4 equations, which are having 4 unknown variables ( x, y, α, and β ) as follows: $$ \left\{ \matrix{ x^{\,2} - y^{\,2}= A^{\,2} + B ^{\,2} - 2AB \cos \left( {\alpha - Δ } \right) \...
1
vote
2answers
23 views

Why is $Av_j=\lambda_jv_j$?

I don't quite understand the solution give to this exercise, so I would like some clarification on that: Let $v_1,...,v_p \in \mathbb{R}^p$ be orthonormal vectors, and for some $$-1< \lambda_p<\...
1
vote
0answers
7 views

How to choose the points to triangulate a domain

Choosing Mathematica to expose the problem, defined the points: ...
0
votes
1answer
12 views

Proving divergence of an alternating series

Consider following series $$\sum_{n=0}^\infty \frac{(3-(-1)^n)\cos((n-1)\pi)}{2n}.$$ According to answers in my textbook, this is a divergent series. The problem is, I don't know how to prove it. I ...
1
vote
1answer
18 views

Evaluating a Double Integral $\int\int \sqrt{x(2a-x)+y(2b-y)}$

Please help me in evaluating the double integral $\int\int \sqrt{x(2a-x)+y(2b-y)}$ over the region bounded by the circle $x^2+y^2-2ax-2by=0$. My thought was to change the variable by substituting $x=r(...
0
votes
0answers
7 views

Volume integral of a scalar times a scalar gradient

I have a volume integral problem that I would like to express as a surface integral: $$\int_V \psi \frac{\partial \phi}{\partial y} \partial V. $$ If the $\psi$ wasn't there I think this holds: $$\...
0
votes
1answer
26 views

Morphism of tangent space induce from a morphism of varieties.

I am trying to show that a morphism of varieties $f: X \to Y$ induces a linear map between the tangent spaces $\tilde{f}: T_aX \to T_{b} Y$, for $a \in X$ and $b = f(a) \in Y$. My idea was to use the ...
0
votes
0answers
7 views

partial derivative of a definite integral of products of two functions

why is it that the derivative of the the following definite (from 0 to v) integral ∫C(i)*P(i)di is P(i) and not ∫P(i)di. Thanks
1
vote
2answers
25 views

How to prove $\frac{e^{jx} - je^{-jx}}{je^{jx}-e^{-jx}} = \frac{\tan x-1}{\tan x+1} $

How do I prove this equation: $$ \frac{e^{jx} - je^{-jx}}{je^{jx}-e^{-jx}} = \frac{\tan x-1}{\tan x+1} \tag{1} $$ where $j=\sqrt{-1}$. I've tried to prove it from the right hand $$ \tan x = \frac{e^{...
1
vote
1answer
20 views

Is $RP^2$ homeomorphic to Mobius Band quotient its boundary circle?

If $M = I^2/[(0,x)\sim(1,1-x)]$ is Mobius Band and $C$ = it's boundary circle, then it can be easily shown (by taking CW-structure) that $M/C$ is homotopy-equivalent to $RP^2$ = $S^2/(\text{andipodal ...
0
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0answers
8 views

how to construct a map of a continent having n different countries in such a way that four colors are needed to color bordering countries.

(a) Let n be your special number. Construct a map of a continent having n different countries in such a way that four colors are needed to color bordering countries in different colors. Using blue for ...
1
vote
0answers
20 views

Two students, one after another without replacement, were chosen randomly from the above sample of $600$ students. What is the probability

I am sorry I don't know how to create in here so I have attached the picture of it, since this table contains the Data of the problem. So I have been asked to answer the following two questions. $a)$...
0
votes
0answers
4 views

Question on equivalent ways of describing homotopy extension property.

I am following a lecture note in algebraic topology. There I came across a property which is known as Homotopy Extension Property or in short HEP. The definition is given as follows $:$ Definition $:$ ...
2
votes
1answer
47 views

What do you call a two-dimensional ray?

A ray looks like: -------> A 2D analoge would look like: ^ | | | -----------> Is there a name for this in higher than ...
0
votes
1answer
22 views

Property of logarithms

In a longer proof, I read the following lines: For a deterministic sequence $(x_n)_{n \geq 1}$ it holds $$\lim_{n \to \infty} \left( \prod_{i = 1}^{n} x_i \right)^{1/n} = a \iff \lim_{n \to \infty} 1/...
0
votes
0answers
5 views

Property of Bernstein polynomials that $\sum_{k=0}^n\frac{k}{n}B_k^n(t) = t, n > 0$

I see this identity pop up a, but I have not seen a proof for it, nor can I prove it myself. The only progress I have made is that $\frac{k}{n}{n \choose k} = {n - 1 \choose k - 1}$, so that $\sum_{k=...
0
votes
0answers
10 views

Tate twist in Poincaré duality for étale cohomology of varieties with $\mathbb F_q$-structure

Let $q$ be the power of some prime number, let $\mathbb F_q$ be the field with $q$ elements and let $\overline{\mathbb F}$ be an algebraic closure of $\mathbb F_q$. Let $X$ be a smooth quasi-...
1
vote
4answers
9k views

$c\mid a,b\iff c\mid\gcd(a,b)$ [GCD Universal Property]

If $c$ is a common divisor of $a$ and $b$ then $c$ divides the greatest common divisor of $a$ and $b$. What can we use to prove this?
0
votes
1answer
13 views

How to convert the given ODE into standard Bessel form?

The given ODE is $$x^2y''-xy'+(x^2-4)y=0$$. How do we convert this into standard bessel equation?

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