0
votes
0answers
3 views

Negative of arbitrary lower bound is arbitrary upper bound?

Statement: Let $A \subset \mathbb{R}$ be bounded above. Prove that $inf\{x\in\mathbb{R}: -x\in A\} = -sup(A)$. Proof: Since $A$ is (implicitly nonempty and) bounded above, $sup(A)$ exists. Denote $S ...
0
votes
0answers
2 views

Operations on two normal distributions using order statistics

$G(x)$ is a Normal distribution with mean $\mu$ and standard deviation $\sigma$. I observe realization of $X$ which are a function of $s$. The distribution $F(s)$ is found as the root (between 0 and ...
0
votes
0answers
12 views

Urysohn's Lemma, Stone-Weierstrass

Let $X$ be a compact space. Show that the following statements are equivalent: a) $X$ is homeomorph to a compact subset of $\mathbb{R}^n$ b) There are functions $f_1,\dotso, f_n\in ...
0
votes
1answer
6 views

Inequality in recurrence relation

I'm having a mental block understanding what is probably a simple inequality in a guess and check example for a recurrence relation. Would someone please explain to me how they obtain the inequalities ...
0
votes
0answers
6 views

Proving theorems using the Compactness theorem

We say an infinite set $S$ is closed under $\wedge$ if for all $a,b$ $\in S$ so $a\wedge b \in S$. I need to prove that if S is closed under $\wedge$ and for all $a \in S$ we know is that $a$ is ...
1
vote
0answers
15 views

If $f$ is differentiable in $(a,b)$ then $\frac{1}{f}$ is differentiable at $(a,b)$, provided $f(a,b)\neq0$

"Suppose that $f$ is a differentiable function at $(a,b)$. Prove that $\frac{1}{f}$ is differentiable in $(a,b)$, provided $f(a,b)\neq0$" We were given the following definition of differentiability: ...
0
votes
0answers
14 views

Mathematics help please

Suppose that the dollar cost of producing q appliances is c(q)=1000-0.05q+0.3q^2 And the demand function is given by p=20-0.025q Q1: compute the marginal cost when the quantity is equal to 1 Q2: ...
1
vote
0answers
4 views

Subtracting scaled projection matrix from identity matrix

I am trying to understand what the following operation signifies. $$ \rm W_n=I-2u_n u_n^H/u_n^Hu_n $$ where I and $u_n$ is described in section 6.3.4.2.3 of this document. My question is, what does ...
0
votes
0answers
9 views

Image: proof re fractional exponents

Can someone help me prove that $(a^m)^{1/n} = (a^{1/n})^m$ per the textbook excerpt captured in the image?
0
votes
0answers
8 views

About сonvergence of partial sums of basis of Banach space

Let $B$ is some separable Banach space with Schauder basis $\{e_i\}_{i=1}^\infty \subset B$. Let $\{\alpha_i\}_{i=1}^\infty$ - is some sequence of complex numbers. Let $p_n = \sum_{i=1}^n \alpha_i ...
2
votes
0answers
5 views

Intuition Homology and homology groups

Could someone give an intuition on the concepts of homology and homology groups? I have been reading the definition of these, but don't have a clear understanding of them. Thanks!
0
votes
0answers
5 views

Using GPS coordinates in trillateration

for a project we need to find a certain position. The info we have : 3 surrounding positions and the distance between those positions and the point we are looking for. We've got a setup like this ...
0
votes
1answer
11 views

show that $\int_{0}^{1}\frac{1}{x^2}\ln\left[\frac{(1+x^2)^2}{1-x^2}\right]dx=\pi$

Most integrals involved $\ln(x)$ seem to produced results of $\pi^2$, $\sqrt\pi$, $\pi\ln(2)$ etc, but rarely $\pi$ on its own. Here is one (1) ...
0
votes
0answers
5 views

Spivak's Calculus on Manifolds - Statement of Lemma 2-10 is incorrect?

In Spivak's Calculus on Manifolds, there is a Lemma 2-10 that is later used to prove the Inverse Function Theorem. Lemma 2-10 : Let $A \subset \mathbb{R}^n$ be a rectangle and let $f : A \to ...
1
vote
1answer
9 views

Continuity of Lipchitz constant of local lipschitz function

Suppose $f:\mathbb{R}\to \mathbb{R}$ be local lipschitz, which is equivalent to Lipschitz on compact sets. That is, for any $R>0$, there exists some $L >0$ such that $$\sup_{|x|,|y|\le ...
1
vote
0answers
11 views

Compute $\int_{0}^{\infty} \frac{x^{1/3}}{(x^2 + 1)^2} dx$

So I want to compute $\int_{0}^{\infty} \frac{x^{1/3}}{(x^2 + 1)^2} dx$ using complex analysis, Cauchys theorem and the residue theorem. What I did was the following: define $g(z) = e^{1/3(\ln|z| + ...
0
votes
0answers
8 views

Assumptions that led to Wigner's surmise for probability density of spacing between eigenvalues of real symmetric random matrices.

In 1957, Wigner surmised (guessed) that the probability density of the spacing between adjacent eigenvalues of real symmetric matrices is given by $$ P(s) = \frac{\pi}{2}s e^{\frac{-\pi s^2}{4}} $$ ...
1
vote
1answer
18 views

If two finite algebraic structures are surjective images of each other, then they are isomorphic

Let $A$ be some algebraic structure (a monoid, a group, a vector space etc) and suppose $\varphi$ is some homomorphism. Suppose $|A| = |\varphi(A)|$ as sets. If $|A|$ is finite, then $\varphi$ must be ...
0
votes
0answers
10 views

Let $S$ be a set of vectors. If each finite subset of $S$ is linearly independent, then $S$ is linearly independent

Here's a similar question which doesn't answer mine: If every subset of $S$ is linearly independent, then $S$ is independent Let $S$ be finite. Let $S_1 \cup S_2 \cup \ldots \cup S_n = S$ such that ...
1
vote
1answer
7 views

Process for solving this system of equations

I have this system of equations for which I'd like to solve for $x$,$y$, and $r$ where $a$,$b$, and $t$ are constants: 1: $0 = (x-a)^2 + (y - b)^2 - t^2$ 2: $y = \dfrac{bx-rx+ar}{a}$ 3: $r = ...
0
votes
0answers
10 views

Prove $\int_{0}^{1}\frac{1}{x^2}\ln\left[\frac{(1+x^2)^2}{1-x^2}\right]dx=\pi$

Most integrals involved $\ln(x)$ seem to produced results of $\pi^2$, $\sqrt\pi$, $\pi\ln(2)$ etc, but rarely $\pi$ on its own. Here is one (1) ...
0
votes
0answers
5 views

Intersection of a variety with its tangent space

If you have a complete intersection of quadrics, what can be said about the dimension of its intersection with a general tangent space? Do you get the expected dimension? In my particular situation, I ...
0
votes
0answers
8 views

Eigenvectors of “weighted” Hermitian matrix?

Consider two real matrices $\boldsymbol{H}$ and $\boldsymbol{D}$ with the following properties: $\boldsymbol{H}$ is a symmetric matrix (since it is a real matrix this is equivalent to being ...
0
votes
0answers
3 views

Knot sequence for a natural cubic (B-)spline interpolant

say I am given $n+1$ data points $(x_i,y_i)$ with $0\leq i \leq n$ and $x_0 < x_1 < \dots < x_n$. I want to interpolate these with a natural cubic spline $s(x)$ ($C_2$ continuous at knots ...
0
votes
0answers
10 views

Converting a Parametric with trig and inverse trig functions to Rectangular form

I came up with a parametric equation for rotating a function $f(t)$ on a graph in three dimensions $$y=\sqrt{f(t)^2+t^2}\sin{\left(\beta+\arctan{\frac{f(t)}{t}}\right)}\cos{\alpha}$$ ...
0
votes
1answer
8 views

Find volume of cube with the help of eqn of plane

The volume of cube whose two faces lie on the plane 6x-3y+2z+1=0 and 6x-3y+2z+4=0?
0
votes
1answer
6 views

Fourier series of the first bernoulli polynomial

I have read somewhere, that states that, for all $x\notin\mathbb{Z}$, $$B_1(x-\left \lfloor x \right \rfloor)=-\sum_{k=1}^{\infty}\frac{\sin(2\pi kx )}{k\pi},$$ where $B_1(x-\left \lfloor x \right ...
0
votes
0answers
11 views

Necessary condition for a subgroup to be a Hall subgroup

Let $G$ be a finite group and $H \le G$ of order $m$. Assume that we have the following property: P: For all $g \in G$, $o(g) | m$ if and only if $g \in H^x$ for some $x \in G$. Under this ...
1
vote
1answer
22 views

Can a elementary row operation change the size of a matrix?

My question is very simple - Can an elementary row operation change the size (eg: $2\times2$ or $3\times 2$) of a matrix? I think the answer should be no, but while reading Linear Algebra by Hoffman ...
0
votes
1answer
14 views

Cardinality of set of functions with coefficients from a set with cardinality omega

Let $A_1$, and $A_2$ be subsets of $\mathbb{R}$ of cardinality $\aleph_0$, $\aleph_1$ respectively. Let $P_1$ be the set of polynomials of the form $a_n(x^n) + a_{n−1}(x^{n−1}) +··· a_1x + a_0$ where ...
0
votes
1answer
12 views

Construct vector field along a curve

Let $M$ be a smooth manifold. I am trying to construct a (piecewise smooth) vector field $V$ along a curve $c$ that takes on prescribed values $K_{i}$ at times $t_{i}$. Say we construct V along ...
0
votes
3answers
36 views

Whether the function $f(x,y)$ is continuous at $(0,0)$

QUESTION: $$f(x,y)=\begin{cases}x \sin \frac{1}{y} + y \sin \frac{1}{x} & \text{if } xy \not = 0 \\ 0 & \text{if } xy = 0\end{cases}$$ Show that $f(x,y)$ is continuous at ...
1
vote
2answers
18 views

Simplest solution for differential equation

Find the simplest solution: $y' + 2y = z' + 2z$ I think proper notation is not sure, y' means first derivate of y. ($\frac{dy}{dt}+ 2y = \frac{dz}{dt} + 2z$) $y(0)=1$ I got kind of confused, is ...
0
votes
1answer
23 views

Number of ways to put n labeled balls distributed among k unlabeled boxes. All boxes should be non-empty.

There are $n$ labeled balls and $k$ unlabeled boxes. The balls should be distributed among the $k$ boxes. All boxes should contain at least one ball. Question: In how many different ways the balls ...
1
vote
2answers
26 views

Infinite differentiability of a function with a removable discontinuity

How would I prove that $\frac x{e^x-1}$ is infinitely differentiable? (This question came up since the No 1 answer in Maclaurin series for $\frac{x}{e^x-1}$ states that the function is infinitely ...
0
votes
0answers
10 views

Finding the Tangent space that makes the tetrahedral of minimal volume with coordinate axis

I was asked to find the tangent space of $x^{2}\over4$+$y²\over9$+$z²\over16$=1 with $x>0,y>0,z>0$ that forms with the coordinate axis the tetrahedral of minimum volume. My attempt: By ...
5
votes
1answer
46 views

How does one integrate $x^2 \frac{e^x}{(e^x+1)^2}$?

How can I show this? $$ \int_{-\infty}^{\infty} x^2 \frac{e^x}{(e^x+1)^2} dx = \pi^2/3$$ I tried applying residuals, but the pole is of infinite(?) order.
0
votes
0answers
12 views

evaluating limit of two functions

Let $P(n) = a^{P(n-1)}-1$ such that for all $n = 2 ,3 ,$ and so on. And let $P(1) = a^x -1$ where a belongs to all real positive numbers, then we have to evaluate $\lim\limits_{x\to0} P(n)/ x$ ...
1
vote
0answers
5 views

$\mathcal{T_B}$ is the intersection of all topologies containing $\mathcal{B}$

Let $\mathcal{B}$ be a basis on a set X, and let $\mathcal{T_B}$ be the topology it generates. Show that $\mathcal{T_B} =\bigcap \{ \mathcal{T} \subseteq P(X) \mathcal{T}$ is a topology on X and ...
1
vote
1answer
27 views

Prove/disprove converge series

Can you help me or give me a hint with this, I don't know from where to start: prove/disprove this: $$\sum_{n=0}^\infty \frac{1}{(2n)!}=\frac{e^1+e^{-1}}{2}$$ Thanks!
1
vote
0answers
3 views

R $\subseteq \omega$ iff R is the fibre of some formula in two free variables.

I'm trying to show $R \subseteq \omega$ is recursive iff it is a fibre of some formula of two free variables in the language $(+ , . ,$ Succ $,1, 0)$. The implication $\rightarrow$ is trivial.However ...
0
votes
1answer
12 views

Voronoi edges example

I have 4 line segments: 0 0 2 0 // 1st line segment 2 0 2 1 // 2nd line segment 2 1 0 1 0 1 0 0 and I wrote some CGAL code to print the Voronoi edges. However, ...
0
votes
0answers
20 views

For every $x$ and $y$ there exists $z$ such that $x-y=z$

If I have the statement. For every $x$ and $y$ there exists $z$ such that $x-y=z$ What would the predicate be for that statement? And how would it be written in symbolic notation? I can't seem ...
0
votes
0answers
10 views

Let t(n) be the number of strings of n letters that can be produced by concatenating copies of the string “a”, “bc”, “cb” find t(3) and t(4)

For each integer n>= 1, let $t_n$ be the number of strings of n letters that can be produced by concatenating (running together) copies of the strings "a", "bc", and "cb" For example, $t_1$ = 1("a" ...
1
vote
0answers
15 views

Log normal simulation.

I want to calculate numerically the expectation of a lognormal random variable $Y=e^X$, where $X$ is normally distributed with mean $m$ and variance $V$. The expectation is known as ...
0
votes
0answers
7 views

Distributions of components to distribution of vector

Suppose that I have independent variables $x_1,\ldots,x_n$ with tractable (not necessarily identical) distributions. I'm interested in the distribution of $\boldsymbol{x}=(x_1,\ldots,x_n)'$ and, if ...
0
votes
1answer
15 views

Doubts in Volume, Hypervolume in $R^4$

Recently I was reading about triple integrals and I came across the statement - "We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out ...
0
votes
1answer
17 views

Is the sequence of functions $g_n=ng_1(nx)$ a cauchy sequence?

Given a function $g_1(x) \in \mathcal L^2(\Bbb R)$ that satisfies: $$\int_{-\infty}^{\infty}dx \space g_1(x)=1$$ one can define a sequence of functions $g_n=ng_1(nx)$. Does $g_n(x)$ define a ...
0
votes
0answers
19 views

Any power of $x$ from $1$ to $n$, inclusive, starts with a $9$

For any positive integer $n$, prove that we may choose a sufficiently long string of $9$s for a positive integer $x$ so that any power of $x$ from $1$ to $n$, inclusive, starts with a $9$. I was ...
-1
votes
2answers
41 views

Proof that the area of a rectangle is $\ell\times b$ [on hold]

Can anybody prove that the area of a rectangle is length * width

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