0
votes
0answers
5 views

square monotonic numbers

monotonic number is a number, which digits are in non-decreasing order. I found by computer that most of these numbers are squares of this numbers 3..34,3..35,3...37,3...367,3...36...7,16...67 My ...
0
votes
0answers
2 views

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, does $\sigma(n^2)/q$ divide $2n^2 - \sigma(n^2)$?

Let $\sigma(x)$ be the sum of the divisors of $x$. A number $X$ is called perfect if $\sigma(X) = 2X$. If $N$ is odd and perfect, then $N$ can be written in the Eulerian form $N = {q^k}{n^2}$, where ...
0
votes
0answers
14 views

How to integrate a function like $(x(1-x))^{-1/3}$?

How to integrate a function like $(x(1-x))^{-1/3}$ within certain limit like $0$ to $1$ ? This question must be having duplicates here but I can't find.Feel free to close if there is a duplicate.
0
votes
0answers
7 views

Intuitive meaning of the concept “computable”

My question is a follow-up question to this one: How to show that a function is computable? The original question was: Is the following function $$g(x) = \begin{cases} 1 & \mbox{if } ...
1
vote
0answers
3 views

Simplifying 4-term Boolean Expression

I am given a pretty lengthy Boolean expression: $$(¬A ∧ ¬B ∧ ¬C ∧ ¬D) ∨ (¬A ∧ B ∧ ¬C ∧ D) ∨ (A ∧ ¬B ∧ C ∧ ¬D) ∨ (A ∧ B ∧ C ∧ D)$$ which I am asked to simplify. The solution should be: $$((¬D ∨ B) ∧ ...
0
votes
0answers
6 views

“Mean-fields results” in Probability theory

I'm studying a paper on (biological) Neural Networks, and the paper studies some stability properties of an $N$-sized network, and then, as $N$ tends to infinity, it is proven that a "mean-field ...
1
vote
2answers
11 views

no of solutions of the initial value problem?

$x \dfrac{dy}{dx} = y , y (0) = 0, x \geq 0 .$ My Approach : $\dfrac{dy}{y} = \dfrac{dx}{x},$ by variable separable method, we get $lny = ln x +c $ and then raising e to both sides will get $ ...
0
votes
0answers
21 views

Real Analysis Sets, Uncountable, Interval Question

$1)$ Let $C\subseteq [0,1]$ be uncountable. Show there exists $a$ in $(0,1)$ such that $C \cap [a,1]$ is uncountable. $2)$ Let $A$ be the set of all $a$ in $(0,1)$ such that $C \cap [a,1]$ is ...
1
vote
1answer
14 views

There are 6 white balls and 9 black balls. Probability of drawing two white, then two black?

From A First Course in Probability (9th Edition): 3.5 An urn contains 6 white and 9 black balls. If 4 balls are to be randomly selected without replacement, what is the probability that the ...
0
votes
0answers
13 views

show that $f_{\epsilon} \in D(\Omega)$; moreover, $f_{\epsilon} \to f$ uniformly as $\epsilon \to 0$.

Let $K$ be a compact subset of $\Omega \subset \mathbb{R^m}$, $\Omega$ is open and nonempty and let $f \in C(\Omega)$ have support contained in $K$. For $\epsilon \gt 0$, let ...
3
votes
1answer
18 views

$A \in SO(3,\mathbb R)\setminus\{I\}$ , then there are exactly two points in $S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$ which are fixed by $A$?

Let $A \in SO(3,\mathbb R)\setminus\{I\}$ , then is it true that there exist exactly two points in $$S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$$ which are fixed by $A$? Or equivalently we ...
1
vote
3answers
37 views

proving that a function is well defined

Please, can someone help me? I have the following problem: Let $X$ be a normed space, $Y \subset X$ a linear subspace of $X$ and the function $$d_{Y}(x)=inf\{||x-y||:y \in Y\}$$;I have to prove that ...
0
votes
0answers
6 views

Use structural induction to prove that for every f ∈ G, there exists f′ ∈ G such that f and f′ are logically equivalent

Let G be a set defined as follows: • if x is a propositional variable, then x ∈ G; • if $f_1, f_2$ ∈ G,then $¬f1$ ∈ G, $(f1∨f2$) $∈$ G, and ($f1∧f2$) ∈ G; • nothing else belongs to G. ...
-7
votes
0answers
30 views

Why are there twin primes? [on hold]

Speculation encouraged. Isn't it strange that there are probably infinitely many despite the size of the numbers? Why is that?
1
vote
0answers
8 views

Primal/Dual Simplex methods clarification

I have several questions regarding these methods. Primal Simplex Method Does the pivot element always have to be a positive entry in the table? Does the RHS always have to be positive in the pivot ...
0
votes
0answers
2 views

mutual information and data processing inequality for $X\to Y\to Z$ where $Y=f(X)$

Let $X\to Y\to Z$ be three random variables. The data processing inequality states $I(X;Y)\geq I(X;Z)$. Further assume $Y=f(X)$ where $f:\mathcal{X}\to\mathcal{Y}$ is an arbitrary function. What ...
0
votes
0answers
5 views

total variation for closed set zero if measure is zero on closed subsets

Let $\mu$ be a complex borel measure on $\Omega$, $|\mu|$ its total variation and $A \subseteq \Omega$ a closed set s.t. for each closed set $B\subseteq A$ we have $\mu(A)=0$. Now does it hold that ...
0
votes
0answers
13 views

Use convolution theorem to evaluate $\int_0^\infty e^{-((|a-su|)/c)^b}e^{-(u/k)^p}du$

$$\int_0^\infty e^{-((|a-su|)/c)^b}e^{-(u/k)^p}du$$ I cannot figure out what to do to solve a case like this, where the variable $u$ is only supported from $0$ to $\infty$.
-1
votes
0answers
10 views

question about weierstrass approximation theorem true or false justify

Is the following assertion true or false? There exists a nonzero function $f \in C([0,1])$ such that $$\int_0^1f(x)x^ndx=0 (\forall n \in \mathbb N)$$ holds. (Hint: use the weierstrass approximation ...
0
votes
0answers
6 views

Showing Bernoulli function is constant on streamlines

An incompressible inviscid fluid, under the influence of gravity, has the velocity field $$\textbf u = (− \cos(x)\sin(y), \, \, \sin(x)\cos(y), \, \, 0)$$ with the $z$-axis vertically upwards, where $g$ ...
3
votes
2answers
36 views

Find minimal integer $n>1$ for which $2^n > n^{1000}$.

Find the minimum integer $n>1$ for which $2^n > n^{1000}$. I have taken the $log$ on both sides, but not reached any result. I would appreciate if anybody will solve it accurately. Thanks in ...
-1
votes
0answers
5 views

Product Spaces and Integration

I have a True/False Question which I am not sure about. Background: Let (Ω1,Σ1,µ1) and (Ω2,Σ2,µ2) be two probability spaces and define (Ω,Σ,µ) by Ω = Ω1 × Ω2, Σ = Σ1 × Σ2 and µ = µ1 × µ2. Let ...
0
votes
1answer
25 views

How do I show that the probability of the union of events is not larger than the sum of the individual probabilities?

In numeric analysis class, we are supposed to show that $$P\Bigl(\bigcup_{n\in\mathbb N}A_n\Bigr)\le\sum_{n\in\mathbb N}P(A_n).$$ This is easy to show using induction for a union of finitely many ...
0
votes
1answer
20 views

Combinatorial argument

Think of a set with $m+n$ elements as composed of two parts, one with $m$ elements and the other with $n$ elements. Give a combinatorial argument to show that $\dbinom{m+n}{r}$ = ...
1
vote
2answers
16 views

Condition for roots to lie in certain intervals

The set of values of $p$ such that both the roots of the equation $$f(x)=(p−5)x^2−2px+(p−4)=0$$ are positive and one of the roots is less than $2$ and the other root lies between $2$ & $3$ ...
1
vote
1answer
23 views

Topological spaces that remain non-metrizable, if the definition of metric space allows $d(x,y) = 0$ where not necessarily $x = y$?

In the definition of metric space, only one thing strikes me as unnatural: the requirement that $d(x,y) = 0$ implies $x = y$. As a programmer, I don't find it uncommon to deal with equivalence ...
1
vote
1answer
23 views

Integer (or whole) numbers in arbitrary fields.

Given an arbitrary field $K$, may I define an integer in $K$? I have found how to define an algebraic number in $K$ and how to define an integer algebraic number in $K$. For instance, let ...
7
votes
2answers
60 views

$p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$; $\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}} = ? $

Let $p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$ The sum $$\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}}$$ Can be expressed as $\frac{a}{b}$. Where $\lfloor \cdot \rceil$ denotes the ...
3
votes
2answers
51 views

Is there any partial sums of harmonic series that is integer?

is there any partial sums of harmonic series that add up to an integer? partial sums not as trivial as the first term only i.e. 1, or the powers of 2 i.e the infinite geometric series for 2. This ...
0
votes
0answers
5 views

Generating space representation of all elements of a point group using generators

A well known set of groups are the 32 three-dimensional crystallographic point groups which elements represent the transformation matrices of the symmetry elements (rotation, reflection etc.). If one ...
0
votes
4answers
28 views

Equation $\log(x^2+2ax)=\log(4x-4a-13)$ has only one solution; then exhaustive set of values of $a$ is

Equation: $$\log(x^2+2ax)=\log(4x-4a-13)$$ It has only one solution; then exhaustive set of values of $a$ is ?? I don't even know where to begin The answer is : $$(-13/4,-13/12) \cup [-1]$$
0
votes
0answers
3 views

Which open projections are support of closed left ideals in a C*-algebra

Let $A$ be a C*-algebra. Let $\phi$ be a positive linear functional on $A$. The support of $\phi$ is defined by the smallest projection $q\in A^{**}$ with $\phi(q)=||\phi||$ and in the literature ...
1
vote
4answers
47 views

Minimum distance between the curves $f(x) =e^x$ and $g(x) =\ln x$

What is the minimum distance between the curves $f(x) =e^x$ and $g(x) = \ln x$? I didn't understand how to solve the problem. Please help me.
2
votes
4answers
66 views

Prove that $\frac{7}{12}<\ln 2<\frac{5}{6}$ using real analysis

I studying in Real Analysis 2, but I have no idea how to solve this problem. My guess is to use Mean Value Theorem or a similar theorem? Could any one help me? Thanks.
0
votes
0answers
6 views

What are disadvantages of solving difference equations backwards?

Say you have the initial value of 100th and 99th term and you want to see how much error is in 1st term instead of picking values from the table for 1st and 2nd term and propagating to 99th and 100th ...
1
vote
0answers
4 views

When is the Stieltjes integral of bounded variations?

I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ...
1
vote
1answer
14 views

Existance of an (in)finite theory having infinite model

Please help me to study the following simple cases: Let $P$ be a binary predicate symbol. I am trying to find out, if there exists a satisfiable $T$ having infinite models only, for the following ...
0
votes
1answer
15 views

Translation of arguments into symbolic logic

If all men were good, there would be no wars. Some men are not good. Therefore, there will be wars. *I am confused on what sign to use on the some part
0
votes
0answers
8 views

Degrees of vertices in a circuit must be even

Let $G$ be a graph with a circuit. Let $C$ denote the subgraph of $G$ consisting of vertices and edges of the circuit. Then for every vertex in $C$, $\deg (v)$ considered in $C$ is even. I would ...
0
votes
0answers
10 views

Suppose that $G$ is the disjoint union $G=\cup_{i=1}^n Sg_iT$.Prove that $[G:T]=\sum_{i=1}^n[S:S\cap g_iTg_i^{-1}]$

Let $S,T\leq G$,where G is a finite group, and suppose that $G$ is the disjoint union $$G=\cup_{i=1}^n Sg_iT$$ Prove that $[G:T]=\sum_{i=1}^n[S:S\cap g_iTg_i^{-1}]$ I don't have any idea how to start ...
0
votes
2answers
20 views

Solve analytically a nonlinear first order ODE

How can one possibly find the general solution to the following nonlinear ODE? $\frac{dy(x)}{dx}=e^{y(x)/2}$ I tried Mathematica, which gives the solution $y(x)=-2 ln[1/2 (-x - c)]$ However I ...
-1
votes
0answers
18 views

Problem with limits sinus in exp.

I have limits $ \lim_{x\to \infty} (3^{sinx \pi}-1).ln(x^2+2)$ I do with left bracket to $e^{sin x \pi ln 3}$ But i don't know next step. Anyone help?
1
vote
0answers
19 views

Optimal strategy for a moment to take an exam

In my problem set there was an exercise involving optimal stopping theory. Here is the problem: There is an exam, a list of $n$ questions and $n$ students. Student $A$ knows answers to $k$ of them. He ...
0
votes
0answers
16 views

Show that $\sigma(\mathcal{H})$ is equal to $\mathcal{P}(\mathbb{N})$.

Let $\mathbb{N} = \{1,2,3,4,\dots \}$ and define the sets $A_k \subset \mathbb{N}$ by $$ A_k = \{k,2k,3k,\dots \} $$ for $k = 1,2,\dots$. We denote by $\mathcal{H}$ the collection $\{A_1, A_2, ...
0
votes
0answers
8 views

Fourier transform calculus of tempered distributions

For example I wanted to ask confirmation of this calculation, if $u \in \mathcal{S}'(\mathbb{R}^n)$ then $\widehat{D^\alpha u} =(2\pi i \xi)^\alpha \widehat{u}$. By definition $\langle \varphi , u ...
0
votes
0answers
5 views

Sufficient conditions for finite covariance

Consider two real valued random variables defined on the same probability space $(\Omega, \mathcal{F}, P)$. Consider the covariance $Cov(X,Y)$. Under which sufficient conditions the covariance is ...
0
votes
1answer
12 views

Inclusion-exclusion error clarification

Suppose you pick a number between $1$ and $30$ uniformly at random. Let $A$ be the event that the number is even. Let $B$ be the event that the number is divisible by $3$. Let $C$ be the event that ...
2
votes
4answers
34 views

Analyticity of $\overline {f(\bar z)}$ given $f(z)$ is analytic

Suppose $f$ is an analytic function on a domain $D$. Then I need to show that $\overline {f(\bar z)}$ is also analytic. Here is what I did - Suppose $f(z) = u(x,y) + iv(x,y)$ where $u$ and $v$ are ...
3
votes
2answers
24 views

What is the meaning of the notation [A|B] in Linear Algebra.

I am going through Linear Algebra right now, we are using the book Elementary Linear Algebra by Andrilli. In one of the theorems he uses this notation without really introducing it. Here is the ...
0
votes
0answers
12 views

Co-ordinate Parabola Circle Contained in it; Difference in maximum and minimum possible radius

If the Difference of radii of larget and smallest Circle passing through the focus of Parabola $$Y^2=4x$$ and toughing parabola in at least one point is My Approach Let Circle be $$C: ...

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