All Questions

0
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0answers
4 views

Prove the following statement for differentiability at $x_0$

"The function f is differentiable at $x_0$ if and only if $f_-'(x_0)$ and $f'_+(x_0) $ exist and equal." The forward argument is rather easy but the backward argument, if we say $f_-'(x_0)$ and $f'_+(...
0
votes
2answers
16 views

If $f+g$ is measurable, does $f$ and $g$ are measurable?

If $f$ and $g$ are measurable, then so is $f+g$. But if $f+g$ is measurable, does $f$ and $g$ are measurable ? I think it is but I can't prove it. I know that $$\{f(x)+g(x)<\alpha \}=\bigcap_{r\in \...
1
vote
0answers
8 views

Taylor polynomials respect derivatives

I want to prove that the derivative of the $n$th order Taylor polynomial is the $n-1$th order Taylor polynomial of the derivative. More specifically: Suppose $f: \mathbb{R} \to \mathbb{R}$ is $n$ ...
0
votes
0answers
5 views

nCatLab notation

At this nCatLab page about the Reedy Model structure, a piece of notation is used which I do not understand (and appears not to be explained there). Suppose that we have a Reedy diagram, whose ...
0
votes
0answers
2 views

Pushfoward of smooth vector field is smooth

My books are Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An ...
0
votes
0answers
4 views

Minimal primary decomposition of ideal in $k[x,y,z]$

Let $R= k[x,y,z]$ and $\mathfrak p_1 = (x,y), \mathfrak p_2 = (x,z), \mathfrak m = (x,y,z) \triangleleft R$. Then $$\mathfrak p_1 \mathfrak p_2 = \mathfrak p_1 \cap \mathfrak p_2 \cap \mathfrak m^2$$ ...
0
votes
0answers
8 views

Why the solution to the heat equation in Fourier series always have $n\geq0$?

I think there must be a reason behind this. For every Fourier series solution of a heat equation (the most of them I have encountered) the solution even if it entirely consists of cosine terms has $+$...
0
votes
0answers
5 views

Does the Riemannian metric induced by a diffeomorphism $F$ exist because of the existence of vector field pushforwards?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...
0
votes
1answer
4 views

Is KL divergence defined for mixed joint densities?

KL divergence is generally written for continuous or discrete densities. I am interested in the case where we have a joint density with both continuous and discrete variables. If KL divergence can be ...
0
votes
0answers
24 views

Value of $\cot12^{\circ} \cdot \cot24^{\circ} \cdot \cot28^{\circ}\cdot \cot32^{\circ} \cdot \cot48^{\circ}\cdot \cot88^{\circ}$

I need a solution without a calculator. I tried concerting it into an expression of tangent but that was of no good value of $$\cot12^{\circ} \cdot \cot24^{\circ} \cdot \cot28^{\circ}\cdot \cot32^{\...
0
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0answers
4 views

Generalisation of linear connection to power series connection

In the original paper "Iterated Integrals of Differential Forms and Loop Space Homology", K.T. Chen mentioned that the concept of power series connection is a 'generalization' of linear connection on ...
1
vote
1answer
7 views

Markov process and non-deterministic random variables

How do I show the following: If $Z_1$ and $Z_2$ are non-deterministic random variables and we define the process $(X_t)_{t≥0}$ by $X_t = Z_1 cos(t)+ Z_2 sin(t)$. I want to show that this is not a ...
0
votes
0answers
15 views

Range Space of T-$\lambda$I

A theorem from Axler's Linear Algebra Done Right says that if 𝑇 is a linear operator on a complex finite dimensional vector space 𝑉, then there exists a basis 𝐵 for 𝑉 such that the matrix of 𝑇 ...
0
votes
0answers
7 views

Determine $M^\mathcal{A}_\mathcal{A}(f)$ and $M_\mathcal{B}^\mathcal{A}(f)$; $f:V\to V$

Given are the following bases of the vector space $V = \{p \in \mathbb{R}[t] \mid \operatorname{deg}(p) ≤ 3\}$ of all polynomials of degree less than or equal to $3$: $\mathcal{A} = (t^3, t^2, t, ...
-1
votes
2answers
19 views

Derivative of an integral with parameter

I want to know how to calculate the derivative of this integral, assuming that all our functions are bounded and smouths on some interval of $\mathbb{R}$: $$ F(x)=\int_{\phi(x)}^{\psi(x)} f(x,t) dt $$
0
votes
1answer
12 views

stieltjes-integral with sgn(sinx) How integrate?

$\int_{-\pi}^{\pi} (x+2) d(sign\sin x)$ how to calculate the integral Stieltjes? I know what $\int f(x) dg(x) = \int f(x)g(x)'dx$ But derivative sign(sinx)'=0? But then how to decide?
0
votes
0answers
71 views

Proof of $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$

Prove that $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$ I have actually got the HINT for this Proof from a very nice book: The HINT i started with is: $$\frac{1}{\sin^2 x}=\frac{1}{4\sin^2\...
1
vote
1answer
23 views

Why do we need noncommutative local rings?

Why are we need noncommutative local rings, is there any geometric meaning?
0
votes
0answers
6 views

Averaging with respect to predictive distribution for predicting positive and real-valued ratings

I have given a rating matrix $R$. In this setting, ratings are positive and real-valued scores. $R_{ui}$ is the rating score from user u to item i. My task is to predict a missing rating score $R_{...
2
votes
1answer
28 views

Proof that $\int_{0}^{1} \frac{f(x)}{f(x) + f(1-x)}dx = 1/2$, where $f:[0,1]\rightarrow \mathbb{R}$ is strictly positive and continuous?

I'm trying to prove this for Riemann integration (not very rigorously, just finding an outline for how the proof should go). The substitution $y=1-x$ seemed handy, as I got $$\int_0^1 \frac{f(x) - f(...
3
votes
1answer
28 views

Exercise 3.22 from W. Lawvere. Sets For Mathematics.

I can't solve Exercise 3.22 from W. Lawvere - Sets For Mathematics book. Exercise 3.22. Show that for any $A$, $1 \times A \simeq A$. The exercise has the following hint. Hint: To show one of ...
2
votes
0answers
22 views

Compute $\oint_{|z|=r}z^2 \sin(\bar z)dz$

As mentioned above I am interested in the value of $$\oint_{|z|=r}z^2 \sin(\bar z)dz$$ where $r>0$ although I'm somewhat helpless at the moment. What I got so far: $$z^2 \sin(\bar z)=z^2 \sum_{n=0}^...
0
votes
0answers
25 views

Solve Pell's equation

Bonjour, How can I solve a pell's equation $x^2-181y^2=180$? I try continued fraction method but it is too long.
1
vote
3answers
39 views

Is $e^x$ the only non-trivial function for which the differential operator is the identity operator?

Given that the derivative of a function is defined as: $$\frac{d}{dx}f(x)=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\tag{1}\label{1}$$ and: $$\frac{d}{dx}e^x=e^x\tag{2}\label{2}...
0
votes
0answers
42 views

When the sum of two permutations is a permutation?

I have to work with encrypted data, a problem arises as follows: Assume that we have $\sigma_1, \sigma_2 \in S_X$, where we consider the set $X$ as the abelian group $Z_n$. Each element of $X$ is an ...
-3
votes
2answers
22 views

If A is path connected then intA is connected - is it true? Explain [on hold]

My views :if A is path connected then intA is also path connected. So intA is connected.
0
votes
0answers
11 views

If $K$ and $F$ are monotone, when is $I+KF$ monotone?

It is known that if $K$ and $F$ are monotone, that $I+KF$ may not be monotone. For example, if $F(x,y)=(x+y, y-x) $ and $K(u,v)=(u+2v, v-2u)$ then $F$ and $K$ are monotone. However, $I+KF$ is not ...
0
votes
0answers
5 views

Canonical Polyadic (CP) Tensor decompositon

I am trying to solve cp decomposition for 3 and 4 way tensor. For 3 way tensor: $Y=Q\times _1 A\times _2B\times _3\times C$ For 4 way tensor: $Y=Q\times _1 A\times _2B\times _3\times C\times _4 D$ ...
2
votes
0answers
10 views

What sorts of models are out there for cooperative games with strategic elements?

I want to look at games where there are coalitions as well as individuals who are trying to maximize personal utility. What papers are out there for these sorts of games?
0
votes
1answer
21 views
0
votes
1answer
30 views

How can I find the function $f(t,x)$ from the given integral equation?

$$ \int_0^\infty f(t, x) = \frac{2}{B+tC} $$ given, initial condition : $f(0,x)=\frac{N_0}{x_0}\exp(-x/x_0)$ here, $N_0,x_0,B,C$ are constants
2
votes
2answers
30 views

Find $x>0$ for which the integral value $\int_0^{\sqrt {x}} \sin (\frac{2\pi t}{t+2}) dt$ is the largest

Find $x>0$ for which the integral value of $$\int_0^{\sqrt {x}} \sin \left(\frac{2\pi t}{t+2}\right) dt$$ is the largest. My try: Let: $$f(x)=\int_0^{\sqrt {x}} \sin \left(\frac{2\pi t}{t+2}\...
0
votes
0answers
14 views

Replacing Constants with Unary Functions

Suppose I have a first order theory over some signature $\Sigma$ with constant symbols. Is there a name for the theory I obtain by replacing the constant symbols with unary function symbols along ...
1
vote
0answers
34 views

Collatz conjecture, Tao-Collatz remainder and mod n.

Collatz conjecture is equivalent to $n\times 3^{k} = 2^{ak+1} - TCR$ where, for me, $k$=odd steps, and $ak+1 $=even steps. Note that total steps = k +( ak+1) steps. Some numbers have the same total ...
0
votes
0answers
16 views

Show that $\{\sigma_i,\sigma_j\}=0, \text{if } i \neq j$

The following should be shown $\{\sigma_i,\sigma_j\}=0, \text{if } i \neq j$ This statements are given: $$\{A,B\}=AB+BA$$ $$\sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\sigma_2=\begin{...
0
votes
1answer
12 views

Laplacian in space with non uniform step

I am trying to find the laplacian of a point in 3D but the major issue is that the distances between my points not constant. For example in 1D : Illustration, the problem is the same for each axis. ...
1
vote
1answer
25 views

Differential equation of the form $x^2y''+(1-2p)xy'+(p^2-q^2)y=x^n$

$$x^2y''+(1-2p)xy'+(p^2-q^2)y=x^n $$ can be written in the form $$ x^a(x^b(x^cy)')'=x^n, $$ where $$ a=1+2p\pm q, b=1\mp2q, c=-p\pm q. $$ As you can see there are two possible sets of values of $a,b,...
0
votes
0answers
11 views

Substitution method for definite integrals

If I am trying to solve a definite integral by means of a substitution, my teacher told me that the substitution function cannot have any extrema in the interval of integration. Is there any way to ...
0
votes
1answer
22 views

When is square root of transpose and transpose of square root of a matrix are equal?

Square root of transpose and transpose of square root of a matrix. $B^{\frac{1}{2}^T}=B^T{^{\frac{1}{2}}} $ (This is not true in general.) where $B$ is symmetric. But what are the instances where ...
0
votes
0answers
15 views

Understanding SVM geometrically

I have a question about SVM. I learned that the dot product of w*x is the projection of x on w. Therefore, as I understand it, wx-b=1 is the line which consists of all the points with a projection of ...
0
votes
0answers
10 views

How to update slope of linear regression line using Bayesian statistics

Suppose we have the following table of measurements: $X = (0.5, 1, 1.5, 2, 2.5, 3)$ with given outcomes ($Y(x)$): $Y = (0.0619, 0.0888, 0.1564, 0.1940, 0.2555, 0.2890)$. One group of scientists ...
0
votes
0answers
8 views

Conditional independence equivalent definition

Consider a probability space $(\Omega, \mathbb{F}, P)$. Let $\mathbb{F}_1$, $\mathbb{F}_2$ and $\mathbb{F}_3$ be sub-$\sigma$-algebras of $\mathbb{F}$. Assume that $\mathbb{F}_1$ and $\mathbb{F}_2$ ...
1
vote
0answers
18 views

Cohen Set Theory and the Continuum Hypothesis p44 Partial Truth Formulae

In Cohen, Set Theory and the Continuum Hypothesis, page 44 the ability to form Partial Truth Formulae is described : "We leave as an exercise for the reader the proof of the following fact: For each ...
0
votes
0answers
12 views

Dual of 0-1 integer program

I have worked with LP & IP with solvers like Gurobi and CPLEX. To play around the processes of encoding some practical problem, I am learning how to construct a dual LP from https://en.wikipedia....
1
vote
0answers
7 views

Dominant Weyl chamber and co-adjoint orbit for Riemann symmetric pair

Let $K$ be a compact Lie group with Lie algebra $\mathfrak{k}_0$. Suppose that $\sigma$ is an involutive automorphism of $K$ which defines a symmetric pair $(\mathfrak{k}_0,\mathfrak{k}_0^\sigma)$. ...
0
votes
1answer
15 views

Find the points in which a map is a submersion

Let $F: \mathbb{R^3} \mapsto \mathbb{R^2}, F: (x,y,z) \mapsto (x^2+y^2+z^2-1, ax+by+cz), a,b,c \in \mathbb{R}$ such that $a^2+b^2+c^2=1$. Find the points in which the map $F$ is a submersion. My idea ...
-1
votes
1answer
21 views

Give an estimate for the approximation $a^n=b^n$

Given $|a-b|<\epsilon$ and $|a|\le K$; $|b|\le K$ give an estimate for the approximation $a^n \approx b^n$, where $n$ is a positive integer. We have that $|a^n-b^n|<\epsilon_0$; $|a|^n\le K^n$; ...
0
votes
0answers
6 views

xyz - dq transform

I'd like to ask you guys about the xyz-dq transform. So I found a lot of useful information about it online already, but I have a problem I hope you can help me with. I want to transform 3 sinusoidal ...
0
votes
1answer
38 views

Prove that $0 \le xy+yz+xz - 2xyz \le 7/27, \:\:\:x,y,z \ge 0$ with $x+y+z=1$.

Prove that $$0 \le xy+yz+xz - 2xyz \le 7/27, \:\:\:x,y,z \ge 0$$ with $x+y+z=1$. Attempt: We must prove $xy + yz + xz \le 2xyz + 7/27$ Assume $x,y,z>0$, since the case when all equal $0$ is ...
0
votes
1answer
12 views

Reference request with examples, finite difference method for $1D$ heat equation ,with mixed boundary conditions.

I am looking for book or online material on numerical (finite difference method) solution of $1D$ heat equation, given initial distribution of temperature, with mixed boundary conditions: Dirichlet ...

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