1
vote
0answers
25 views

Integrals over subset of measure space

Let $(X, \mathcal{M}, \mu)$ be a measure space. Suppose $E \in \mathcal{M}$ and $f \in L^+$ where $L^+$ is a space of measurable functions from $X$ to $[0, \infty]$. $\int_E f$ is defined by $\int_X f\...
0
votes
2answers
20 views

For each ideal $I$ of $B$, there seems to be a corresponding morphism $f/I :A/f^{-1}I \rightarrow B/I$. Is this right?

(All my rings are commutative with $1$.) Suppose $f : A \rightarrow B$ is a morphism of rings. Then for each ideal $I$ of $B$, there seems to be a corresponding morphism $$f/I :A/f^{-1}I \rightarrow ...
2
votes
1answer
53 views

Is this integral continuous? (with respect to $z$)

Consider the integral $$\int_0^\infty f(t)e^{tz}\,dt,$$ where $f$ is an integrable function. Is this integral continuous with respect to $z$ (complex variable) on the domain $\{z=x+yi:x<0,y\in\...
1
vote
0answers
22 views

A bound for a solution of a PDE

Let $u(t,x):\mathbb{R_+}\times\mathbb{R}\rightarrow\mathbb{R}$ be a very smooth function, which satisfies the equation: $$\dfrac{\partial u}{\partial t}+f(x,u)\dfrac{\partial u}{\partial x}=g(x,u),$$ ...
0
votes
1answer
28 views

Let $X$ be a metric space and $Y$ be a open set where $Y=\cup_{x \in Y} B(x, r(x)) $ The union in this theorem will have to be infinite. Why?

Let $X$ be a metric space and $Y$ be a open set($Y \subseteq X$). Now, according to a theorem, $Y=\cup_{x \in Y} B(x, r(x)) $ i.e. $Y$ as a union of open balls. The union in this theorem will have to ...
0
votes
1answer
18 views

To find the centre of the inner circle that is tangent to the unit circle and the x-axis

We have a unit circle $C:x^2+y^2=1$. Let $l:y=m(x+1)$. We consider a circle $C'$ at a centre on $l$ that is inscribed to an upper semi-circle, i.e., a circle that is tangent to the circle $C$ and the ...
0
votes
0answers
10 views

Compact résolvant inequality

I want to prove that if an operator $A$ with domain $D(A)=\left\{u\in L^2\;\text{such that}\; Au\in L^2(\mathbb{R}^n) \right\}$ has a compact resolvante then there existe a constante $c>0$ such ...
0
votes
1answer
17 views

Partial Derivative of a quadratic form

I want to derive, w.r.t $x$, this: $x'Ax+2y'B'x+y'Cy$ The reference says: "Assuming $A$ positive definite, then the partial derivative is: $2(Ax+By)$." Why the transpose $x'$ it's not in the ...
1
vote
1answer
48 views

Proving $(f \circ g)_{*}= f_{*} \circ g_{*}$

I have two continuous functions $f: X \to Y$ and $g: Y \to X$ such that $f \circ g =id_Y$ and $g \circ f =id_X$. If I induce maps $f_{*}$ and $g_{*}$ everywhere I search tells me that it's clear that $...
0
votes
0answers
13 views

Best open approximations of the diagonal

Suppose $A$ is a topological space and $\Delta_A\subseteq |A|\times |A|$ is the diagonal relation. $\Delta_A$ is rarely open (indeed it is open if an only if the topology on $A$ is discrete), but it ...
1
vote
0answers
15 views

Applied Rational Choice Theory

I am a programmer researching the application of Rational Choice Theory. I have found many links to the philosophical nature of it. And fewer documents formalizing it's mathematical principles. ( ...
3
votes
0answers
28 views

Proof verification: Mantel's theorem

if a graph $G=(V,E)$ on $n$ vertices contains no triangles than it contains at most $n^2/4$ edges. Proof: Let v$\in$V be a vertex of maximum degree k. since G contains no triangles, there are no ...
0
votes
0answers
9 views

Can every precedence table be represented as an “activity on arc” activity network

I have the precedence table Activity | depends on A | - B | - C | - E | A,B F | B,C G | A,C And I want to ...
1
vote
1answer
24 views

Problem about parallelogram

In the diagram, $ABCD$ is a parallelogram. The points $E$ and $F$ are the midpoints of $BC$ and $AD$ respectively. The lines $DE$ and $BF$ intersect the diagonal $AC$ at $Q$ and $P$ respectively. ...
0
votes
0answers
18 views

Gradient of 3d delta-function

I need to evaluate the following expression: $\int \mathrm{d}\boldsymbol{r} \left[\nabla_{\boldsymbol{R}_\alpha}\delta(\boldsymbol{r}-\boldsymbol{R}_\alpha)\right]v(\boldsymbol{r})$ and I want to ...
4
votes
0answers
41 views

Most general space on which we can do calculus

I have two somewhat related questions: Question 1: What is the most general space (set of objects) on which we can do calculus? Is it a metric space, or can we relax the conditions a bit further? ...
0
votes
3answers
19 views

find the locus of point of intersection of such tangents to an ellipse which make constant intercept on y axis.

I write equation of two tangent and find their point of intersection. I also substitute $x=0$ in them to find $y$ intercept. But I can't eliminate the variables, which are coordinates of point of ...
1
vote
0answers
68 views

Which book would you recommended as help (assistance) for reading the so-called “Tohoku Paper”?

Recently I thought that maybe is a good time to try, read Grothendieck's "Tohoku paper" as a sort of inspiration for the future and to read some of the ideas of this great mathematician, which (among ...
0
votes
1answer
36 views

Decomposing $\ln(x)$ into sum of even and odd function.

Can somebody help me break $\ln(x)$ into sum of even and odd function. As far as I know every function can be broken in such manner. Not being able to do this as $\ln(-x)$ and $\ln(x)$ cannot exist ...
1
vote
0answers
19 views

Vector spaces as bimodules

The usual definition of a vector space $V$ over $K$ is as an abelian group, on which $(K\setminus\{0\},\cdot)$ acts on the left, such that furthermore the operation of $K$ on $V$ is compatible with ...
0
votes
0answers
10 views

Question about motion.

I don't know how to solve this problem . Please anyone help. Question:- An object start moving from immobility with a acceleration of " f m/s^2". At the end of every t seconds it increases it ...
0
votes
0answers
14 views

Total Variation distance between two Poisson distributed measures.

Let $\mu$ have the Poi(1) and $\nu$ have the Poi(2) distrbutions. What is the total variation of these two? So the total variation is definetd as : $|\mu-\nu|_{TV}=\max_{A\subset S}|\mu(A)-\nu(A)|=1/...
0
votes
0answers
5 views

Characterization of the feasible set of an optimization problem

Let $V \in \mathbb{R}^{n\times n}$ be a positive definite matrix and $D \in \mathbb{R}^{k\times k}$ a diagonal matrix with strictly positive elements on its diagonal. We also have a matrix $X \in\...
8
votes
3answers
259 views

Limit involving the Sine integral function

$$ \mbox{Prove that}\qquad \lim_{x \to \infty}\left[\vphantom{\large A}% x\,\mathrm{si}\left(x\right)+ \cos\left(x\right)\right] = 0 $$ where we define $$\mathrm{si}\left(x\right) = - \int^{\infty}_{...
1
vote
0answers
19 views

What are “boundaries” (as defined here) really called and where can I learn more?

My guess is that boundaries (perhaps under a different name) in graph theory are probably defined like this: Definition 0. Let $G$ denote a graph, $A$ denote a subset of $G$. Then a candidate ...
0
votes
1answer
11 views

Linear congruential generator

Given a linear congruential random number generator x_{n+1} = (a*x_n + c) mod m. We are given x0 and x50, what can we say about a,c and m?
0
votes
1answer
38 views

I want to prove that $B=S^2 \cap \{(x_1, x_2, x_3) : x_i \geq 0 \}$ is homeomorphic to the disk $B^2= \{ x \in \mathbb{R^2} : ||x|| \leq 1 \}$

I claim that the map $f:B \to B^2$ defined by $f(x_1, x_2, x_3)=(x_1, x_2, 0)$ is a continious, bijection with $g: B^2 \to B$, $g(x_1, x_2)=(x_1, x_2, \sqrt{1-(x_1^2+x_2^2)})$ it's inverse. So i ...
0
votes
0answers
17 views

Percentage of Sum of 2 Continuous distributions

In a factory, there are Independents 2 pipe-cutter machines. The length of the pipes from the first machine is $X_{1}$. The length of the pipes from the second machine is $X_{2}$. I know that $X1\...
1
vote
1answer
14 views

Does the law of double pseudo-complements hold for subgraphs?

As for the categories of subgraphs of a given graph $X$ with their inclusions, we have a co-Heyting algebra (and Heyting algebra). In the co-Heyting algebra, we define $\sim$$A$ as the pseudo-...
0
votes
1answer
29 views

How to prove that $h''(x)$ has at most one zero on $(0,1)$.

$h(x)=1-\sum_{i=1}^{k-1}x^i+a_kx^k+\sum_{i=k+1}^\infty x^i$, where $|a_k|\le1$, is the power series of an analytic function. Prove that $h''(x)$ has at most one zero on $(0,1)$.
2
votes
1answer
34 views

Convergence of a cosine series

Given that $$\sum_{n=1}^\infty a_n<\infty,$$ and that $$\lim_{n\to \infty}b_n=0$$ Is the series $$\sum_{n=0}^\infty a_nb_n^{-2}(1-cos(b_n))$$ necessarily convergent?
0
votes
0answers
18 views

what does $(y_{n})_0$ mean? [on hold]

In a successive differentiation math, I was told to proof something and find the value of $(y_{n})_0$.But I didn't understand that what does it mean?
0
votes
1answer
12 views

Strongly measurable implies Borel measurable in separable space

Let $(M,\mu)$ be a measure space, $X$ be a Banach space, $f: M \to X$ be a function. $f$ is said to be strongly measurable if there is a sequence of simple functions $\{f_n\}\to f$ pointwisely a.e.. $...
2
votes
0answers
18 views

Do automorphisms generate any specific equivalence?

I am thinking about a structure (in terms of predicate logic), where we have a carrier set A and some relations over A (no functions). I am thinking about all the automorphisms for that structure. I ...
0
votes
1answer
36 views

An example of subset $A$ such that $A \cap K$ is open in $K$ for each compact set $K$, but $A$ is not open.

Let $X$ be a topological space. For any $A \subseteq X$, consider two possible conditions on $A$: 1) $A$ is open in $X$; 2) $A \cap K$ is open in $K$, for each compact set $K \subseteq X$. Then $(...
5
votes
0answers
31 views

Number of primitive roots mod $p$ that are not primitive roots mod $p^2$

Consider the primitive roots of a prime $p$ in the range $1...p$ which are not primitive roots mod $p^2$. Let $n(p)$ be this number. While looking for an answer to this question, it seems that the ...
-4
votes
0answers
28 views

Determinant of a Matrix and Determinant of Its Transpose Question [on hold]

Why does the determinant of a transposed matrix have a matching inverse cycle for each permutation of the non-transposed matrix?
0
votes
1answer
16 views

How to prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference?

how can i prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference? i mean supose that $A=(a_1,...,a_m)$ and $B=(b_1,...,b_m)$ both ...
1
vote
0answers
28 views

Reference Request: Sequence of Positive Integers Excluding a Quadratic Integer Sequence

I remember that I have seen a similar problem to this one here: The set of natural numbers that don't belong to a set. If my memory serves me right, that problem is about a sequence excluding ...
1
vote
1answer
33 views

Determining maxima and minima of the function $f(x)=\frac{\cos x}{1+2\sqrt{x}}$

I have the next function $f(x)=\dfrac{\cos x}{1+2\sqrt{x}}$. I want to find the maxima and minima of the function. The derivative of this function is $$f'(x)= \frac{-(\sqrt{x}\cdot \sin x+2x\cdot \...
0
votes
3answers
19 views

Show that $f$ is continuous in the relative topology

Let $(X, T)$ be the subspace of $\mathbb R$ given by $X = [0,1]\cup [2,4]$. Define $f: (X,T) \to \mathbb R$ by $$f(x)= \begin{cases}1 & x \in [0,1] \\ 2 & x \in [2,4]\end{cases}$$ Show that $f$...
0
votes
1answer
21 views

The dimension of $:W_1\cap W_2$

$$W_1=\operatorname{span}\left(\begin{pmatrix}1&1\\ 0&0\end{pmatrix},\begin{pmatrix}3&1\\ -1&0\end{pmatrix}\right)$$ $$W_2=\operatorname{span}\left(\begin{pmatrix}1&1\\ 1&0\...
-4
votes
2answers
32 views

Use the laws of logic to simplify the expression [on hold]

Use the laws of logic to simplify the expression P∨¬(¬P→Q)
2
votes
0answers
29 views

Inequality of absolute values of complex sums

Let $c_1,\dots, c_n\geq 0$ and $x,\dots,x_n\in\mathbb R$. Then $$\left\lvert \sum_k \frac{c_k x}{(x_k-i x)^2}\right\rvert\leq \left\lvert \sum_k \frac{c_k }{x_k-i x}\right\rvert$$ seems to hold for ...
0
votes
0answers
18 views

Logical combinations of exponential random variables

I have a scenario where events can be modeled as exponentially distributed random variables, however, I need to consider logical combinations of these variables. For example, say I have three timers ...
1
vote
0answers
11 views

Common Estimates Suggestions

Consider two Markov Chains $X-Y_1-Z_1$ and $X-Y_2-Z_2$ defined on same alphabet space $\mathcal{X}$, such that $Z_1= g_1(Y_1)$ and $Z_2=g_2(Y_2)$ for some functions $g_1,g_2$. Assume further that ...
0
votes
2answers
37 views

Formula for all terms in a sequence.

I am looking to find an algorithm for a repeating pattern, in order to solve a programming problem. I need a formula for all terms of the following sequence: ...
0
votes
2answers
25 views

evaluation of double integral using change of order of integration

How to evaluate the following double integral $\int\limits_s^t\int\limits_s^u e^{-\lambda(t-v)}(u-v)^{-\beta-1}dvdu $ where $\lambda$ and $\beta$ are positve constants.
-1
votes
0answers
24 views

Find all integers $p>q\ge 0$, such that for all real $x,y \in [0;1]$ following inequality holds $[px]+[py]\ge [qx+y]+[x+qy]$

Find all integers $p>q\ge 0$, such that for all real $x,y \in [0;1]$ following inequality holds $$\lfloor px \rfloor + \lfloor py\rfloor \ge \lfloor qx+y\rfloor+\lfloor x+qy \rfloor$$ I have no ...
0
votes
1answer
24 views

Finitely presented subgroup of finitely presented group

If I am given a group $G$, which is finitely presented by $\langle S \mid R \rangle$, and I am given a finitely presented subgroup $H$ of $G$. Is it true that $H$ takes the form $\langle T \mid R' \...

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