All Questions

0
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0answers
3 views

Given two functions, how to find range of values for which first function has values greater than second?

Given two functions $ f(x)$ and $ g(x)$, how do I find the range $ [a,b]$, such that $ \forall{x} \in [a,b], f(x) > g(x)$. Is/are there any standard way of solving such problems. Do they apply to ...
0
votes
0answers
4 views

f(x) = x sin(1/x) uniformly continuous on (0,1) ??

I see how the function $f(x) = x sin(1/x)$ is continuous at the point 0 by the inequality of $|sin(\alpha)| \leq 1$. But for uniformly continuous, I try to produce the two sequences in $(0,1)$ which ...
0
votes
0answers
8 views

$\vert f'(a) \vert < k$,prove that there exists $I$, $a\in I$ that $\vert f'(x)\vert<k$, $f \in C^1$

so here's the question: $\vert f'(a) \vert < k$,prove that there exists $I$, $a\in I$ that $\vert f'(x)\vert<k$, $f \in C^1$, since the first derivative is continuous, I tried using the ...
0
votes
0answers
4 views

Hydrostatic force against side walls of a 38'6“L x 6'6”W x 2'H

I'm no good a calc and need some help. I need to figure out the amount of force that will be exerted on the walls of a tank of salt water (sg. 1.026) that is 40'long x 6'6"wide x 2'tall. All help will ...
0
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0answers
9 views

Why does the binomial formula use multiplication?

This is Problem 2.2 from Tsitsiklis, Bertsekas, Introduction to Probability, 2nd edition. You go to a party with 500 guests. What is the probability that exactly one other guest has the sam ...
0
votes
1answer
10 views

Dimension of a Subspace (with respect to span)

Just have a general question regarding the dimension of a subspace when it is shown like this: $W = span${$x, x_1, x_2,...$} How does the 'span' affect the dimension of a subspace?
0
votes
0answers
3 views

Perfectly Repeating Square Grid on a Hexagonal Tile Base

I am attempting to discover what the sizing ratios of squares and hexagons are. What I want to do with this information is determine (if possible) what size squares in a grid I should place over a ...
0
votes
1answer
14 views

How can we solve x=8sin(x)?

Been thinking about this for quite a while, I know 0 is one of the answers but I just can't figure out how can I find the others (not by ploting the graphs but working it out steps by steps), please ...
0
votes
0answers
11 views

Analogue of deMoivre theorem for real numbers

If $\phi_1, \phi_2, ... $ is a series of positive acute angles so that $\tan \phi_{m+1} = \tan \phi_m \sec \phi_1 + \sec \phi_m \tan \phi_1$ then prove that- $$\tan \phi_{m+n} = \tan \phi_m \sec \...
0
votes
0answers
6 views

distribution of fractions of partial sums of exponential random variables

Let $X_1, X_2, ...$ are iid exponential random variables, $S_k=\sum_{i=1}^{k}X_i$. I want to find the distributions of $S_k/S_n$ for $k=1,...,n-1$. i first used transformation $Y_i=S_k/S_n , (k=1,......
1
vote
4answers
20 views

I was told that I couldn't “pull the limit in”. Tell me exactly how I'm messing up, please!

So, the problem that we were solving was The limit as n approaches infinity of (n/(n+1))^n (picture for clarity) To figure out whether the series converged or diverged, after simplification, I asked ...
0
votes
2answers
13 views

There are 3 sections in a question paper with 5 questions each.

There are 3 sections in a question paper each containing 5 questions. A candidate has to solve only 5 questions, choosing at least one question from each section. In how many ways can he make his ...
-1
votes
1answer
26 views

Find 1^1 + 2^2 + … + 99^99 mod 3

Calculate the following numbers in modular arithmetic. Justify your answers (a) What is $1^1 + 2^2 + · · · + 99^{99}\ (mod\ 3)$ ? I know that $1^1 + 2^2 + · · · + 99^{99} = \sum_{n=0}^{32} (3n + 1)^{...
0
votes
1answer
15 views

Ideals and Field [duplicate]

Let $A \neq0$ be a ring. Then the following are equivalent i) A is a field ii) the only ideals in A are $0$ and $(1)$ iii) every homomorphism of A into a non-zero ring B is injective. I have ...
1
vote
0answers
8 views

Continuity of supremum of polynomial

Prove: Let $A \subseteq \mathbb{R}$ be a compact set. Prove that the function $f \colon\mathbb{R^{n+1}} \to \mathbb{R}$ $\qquad f(x_0,..., x_n) = \sup_{x\in A} \prod_{j=0}^{n} (x-x_j)$ is continuous....
0
votes
0answers
9 views

Solve the Distribution equation $xT= 1$

My Problem is to find all solutions of the distribution equation $xT =1$. I don't know how to solve it. The solutions should be $pv\frac1x+c$ , $c \in \mathbb{R}$. My idea was to create a function ...
0
votes
1answer
18 views

What is the logic behind this puzzle's answer?

Keep a number between $1$ and $60$, including $1$ and $60$. The aim is to find kept the number. $1,3,5,7,9,\\ 11,13,15,17,19,\\ 21,23,25,27,29,\\ 31,33,35,37,39,\\ 41,43,45,47,49,\\ 51,53,55,57,59\...
0
votes
0answers
8 views

Convolution of 3 functions

Assume that I have an equation like this: $f(t)=\int{g(\tau)h(2t-\tau)k(t-\tau)d\tau}$, in which g(t), h(t) and k(t) are three arbitrary functions. It can be seen that it is similar to convolution ...
0
votes
0answers
5 views

commutant of a non unital $C^*$ algebra

If $A$ is a non-unital $C^*$ algebra,is the commutant of $A$ empty? Does there exist a theorem which states that every $C^*$ algebra has commutant(centralizer)?
0
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0answers
5 views

Equivalent definitions of Kähler metric on a Riemannian manifold.

In the book Compact Manifolds with Special Holonomy by Dominic Joyce, Ch. 4.4 there is a proposition 4.4.2 Let $M$ be a manifold of dimension $2m$, $J$ an almost complex structure on $M$, and $g$...
-1
votes
3answers
36 views

How do I prove that √5+√7 irrational?

I got stuck at : a^2/b^2 = 12+2√35 I understand that 12 is rational and now I need to prove that √35 is irrational. so i defined ∀c,d∈R while d isn't 0 that: c^2/d^2 = √35 so - c^2=(d^2)*√35 It ...
1
vote
2answers
19 views

Distance of a plane

I have posted this question in Stack Overflow programming forum. Someone there feels it might be more suited to Mathematics. I have to warn you I am rusty in math and was terrible in Algebra. ...
1
vote
0answers
11 views

Gradient of the Lie exponential map on SO(n)

I am interested in computing the gradient of $f(e^A)$ when $A$ is a skew-symmetric matrix. If we write $e^A = B$ and we denote the gradient of the function $f$ on the ambient space evaluated at $B$ ...
1
vote
2answers
13 views

Show that $2\cos(mp-nq)$ is one of the values of $\left( \frac{x^m}{y^n}+\frac{y^n}{x^m} \right)$

Q:If $2\cos p=x+\frac{1}{x}$ and $2\cos q=y+\frac{1}{y}$ then show that $2\cos(mp-nq)$ is one of the values of $\left( \frac{x^m}{y^n}+\frac{y^n}{x^m} \right)$My Approach:$2\cos p=x+\frac{1}{x}\...
1
vote
0answers
15 views

how to get rid of the orders of zeroes of analytic function?

I was reading complex variable theory by fisher.In ch 3 i faced the following problem. let me draw the background. Then i will ask the problem,show my attempt. Suppose that h is an analytic function ...
4
votes
1answer
19 views

What is the intersection of all $L^p(\mathbb{R}^n)$ spaces?

I wondered this, and tried to find an answer online, but the only thing I could find was a statement that the set of functions which are in all $L^p(\mathbb{R}^n)$ is well-studied. But what functions ...
0
votes
0answers
11 views

Consider the constrained optimization $\max f(.)=g_1(.)f_1(.)+g_2(.)f_2(.)$, can we optimize $g_1(.),f_1(.),g_2(.),f_2(.)$ indep. to optimize f(.)?

Consider the following optimization problem \begin{equation*} \begin{aligned} & \underset{P_1,P_2,\lambda_1,\lambda_2}{\text{maximize}} & & C(P_1,P_2,\lambda_1,\lambda_2) \\ & \text{...
0
votes
0answers
12 views

Complex Factorisation

From my maths textbook I have noted a general formula - $$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,...
0
votes
0answers
6 views

Finding time factor in 3d wave function usin seperation of variables

$$\nabla^2\varphi=\dfrac{1}{c^2}\dfrac{\partial^2\varphi}{\partial t^2}$$ By using seperation of variable I assume a function $\varphi(r,t)=\psi(x)T(t)$. So I end up with two equations $$\dfrac{d^2T}{...
0
votes
2answers
19 views

Minkowski Set Addition

What is the Minkowski sum of Q to itself, where Q is the set of all rational numbers? I can't find a way to solve this with freshman knowledge. Thank you in advance.
0
votes
0answers
9 views

Compute the Fourier series (example)

Determine the Fourier series of the function $f\colon [-\pi,\pi]\to\mathbb{R}$ given by $f(x):=\lvert x^3\rvert+12$. First of all, this surely means to find the Fourier series of the function which ...
0
votes
0answers
14 views

If there is an $r>0$ such that $\frac{f^{(n)}(c)}{n!}(x-c)^n \rightarrow 0$ when $|x-c|<r$, then $f$ is real analytic at $c$?

I am working on a problem $f \in C^{(\infty)}$ show that $f$ can be represented as a power series in an interval about $c$ if and only if there is an $r>0$ such that $\frac{f^{(n)}(c)}{n!}(...
1
vote
0answers
9 views

How to solve $I = \int_0^T (1-\alpha \beta(t)) \beta(t)^{n-1} dt $ where $\beta(t) = \frac{1-e^{-rt}}{1-\alpha e^{-rt}}$

How to solve the integral: $I = \int_0^T (1-\alpha \beta(t)) \beta(t)^{n-1} dt $ where $\beta(t) = \frac{1-e^{-rt}}{1-\alpha e^{-rt}}$ with the assumptions that $0<\alpha<1$, $n\in \Bbb{Z}$ ...
0
votes
0answers
10 views

Find how many payements will I need to do and what is going to be the final amount.

This is a financial mathematics problem: On time $0$ we have to pay $297505.48$ of a loan. Assuming a capitalization interest of $8\%$ and a fixed annual payment of $49623.55$, how many payments will ...
1
vote
1answer
21 views

Theory of equations problem

$f(x)$ is a cubic polynomial $x^3 + ax^2 + bx + c$ such that $f(x) = 0$ has three distinct integral roots and $f(g(x)) = 0$ does not have real roots, where $g(x) = x^2 + 2x - 5$ What is the minimum ...
2
votes
2answers
21 views

Prove/disprove, that the relation is reflexive, symmetric, antisymmetric and transitive

I want to prove or disprove, if the relation $R$ with \begin{align} iRj:\Longleftrightarrow (\forall k \in \mathbb{N} \text{ with } k \text{ is a prime number}:k \mid i \Longrightarrow k \mid j) \end{...
0
votes
0answers
8 views

Explain recurrence and Dynamic Programming methods

Well during competitive programming, Dynamic Programming and Recursion is one of the most favorite topics. It kind of draws the line between an average and a good coder. Now my question is, is there ...
0
votes
0answers
12 views

how to show a sequence of finite-rank operators converges uniformly

Let $G\in L^2(0,1)^2$. Define a linear mapping $A:L^2(0,1)\mapsto L^2(0,1)$ such that $$ Af(x):=\int_0^1G(x,\xi)f(\xi)d\xi,\quad \forall f\in L^2(0,1). $$ Since $L^2(0,1)$ is a separable Hilbert space,...
-2
votes
2answers
39 views

How to solve $(x^2-4x+4)^2=0$? [on hold]

How to solve this equation ? $$(x^2-4x+4)^2=0$$
0
votes
2answers
19 views

Does there exist any function f(x,y) which satisfies the following condition?

I would like to construct a function $f(x,y)$ such that $f(x,y)\leq 255$ for all integers $x,y\in [0,255]$ and conversely, i.e., if (x,y) is given then I can get back $x,y$ from this given $f(x,y)$. ...
0
votes
0answers
5 views

Not normal state

Does anybody know an example of a state on a von Neumann algebra that is not normal? If it has relevance to physics it would be nice.
2
votes
1answer
20 views

Showing that $f: \mathbb{R}^3 \to \mathbb{R}, f(x,y,z):=3z+3xy^2-x^3$ is a manifold

$M \subseteq \mathbb{R}^3$ $$M=\{(x,y,z) \in \mathbb{R}^3:f(x,y,z)=0\}$$ with $$f(x,y,z):=3z+3xy^2-x^3$$ I want to show that $M$ is a manifold and determine its dimension. $f:\mathbb{R}^3 \to \...
0
votes
0answers
18 views

How to determine the dimension of the set of matrices with rank $r$?

There is a proposition saying that all $m\times n$ matrices form a smooth manifold of dimension $(m-r)(n-r).$ But I cannot see how this can be true? Since when $m=2, n=2, r=1$, it seems that the ...
1
vote
1answer
25 views

How can we quickly tell if $\mathbb{Z}_n$ is decomposable or not?

If we are given a group, say $\mathbb{Z}_n$ (for some integer $n\ge 2$) what is a quick way to tell if the group is decomposable or not? My book merely states the definition of a decomposable group ...
0
votes
1answer
8 views

Measurability of the sum of measurable functions $(X_t)_{t \in I}$ ranging over a random index set $N$.

Assume I have a collection of real-valued measurable functions $(X_i)_{i \in I}$ on the measurable space $(\Omega,\mathcal{F})$. Let $N:\Omega \rightarrow 2^\Omega$ such that for every $\omega \in \...
0
votes
0answers
8 views

Conic sections and polynomials over C

Let K be a field with $char(K) \neq 2$. Every polynomial $f \in K[X,Y] $ has a unique representation $$ a_{11}x^2+a_{22}y^2+2a_{12}xy+ 2a_{13}x + 2a_{23}y + a_{33}$$ that can be identified by a ...
0
votes
1answer
25 views

Riemann Zeta function, nontrivial zeroes

How can we prove what are, say the first 4 non-trivial zeroes of the Riemann $\zeta$ on the critical line $Re(z_j)=\frac{1}{2}$, $j=1,2,3,4$ the first two with negative imaginary part and the second ...
-2
votes
0answers
13 views
0
votes
0answers
17 views

Difference tables and vector spaces

At the linear algebra course our professor was introducing the notion of vector spaces and we worked on some examples. We proved that different kind of sets are vector spaces with regards to addition(...
-1
votes
0answers
6 views

Question in Henkin Semantics (First order logic)

I need to prove the following: First of all let L* be a language obtained from adding constant symbols to L and let Tm be a Henkin theory, then If $\Gamma$, $\phi[\overline{c}_{\phi}/x]\vDash\psi$ ...

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