All Questions

0
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0answers
14 views

General solution methods for third-order linear differential equation with variable coefficients

I'm trying to solve the following IVP: $y''' + ay''+by'+cy=0 $, with initial conditions $y(x_{0})=y_{0},y'(x_{0})=y_{1},y''(x_{0})=y_{2}$, where $y,a,b$, and $c$ are all functions of the ...
0
votes
1answer
9 views

Difference between Compactness Proof Structure Creation and Creation of a Forcing Extension Structure

Having just started to learn Forcing and having looked though the large number of Forcing questions on StackExchange, I apologize if I overlooked the following 'overview' novice question: Enderton ...
0
votes
1answer
14 views

Residue of $m$-order pole Computation

I'm having trouble with the following residue computation: Find the order of the pole and corresponding residue for $$\bigg(\frac{z}{2z+1}\bigg)^3$$ The solution is given as: order = $3$ and ...
0
votes
0answers
12 views

largest known twin semiprimes and consecutive semiprimes

What are the largest known semiprimes differing by 2? What is the largest known pair of consecutive numbers that are both semiprimes? Has anyone computed this?
0
votes
0answers
35 views

What are the elements of a fractional summation?

For example, what are the elements in: $$\sum_{i=0}^{2.5} i$$ Background: As part of a program I wanted to generate a sequence along $i^2$ in a range $[a,b]$ with a given sum $c$. So, \begin{align}...
0
votes
0answers
17 views

Cauchy-Riemann equations with partial derivative being continuous

I am trying to understand a proof in complex analysis. And there is a step that I am not able to see why this ''re writing'' can be done. define $f(z) = f(x+iy) = u(x,y) + iv(x,y)$ two small ...
1
vote
0answers
13 views

Maximization of the function when some parameters are unknown.

I would like to know if my understanding about how to find a maximum of the function when some parameters are unknown is correct. Consider the following maximization problem. $\max_{x}V=\int_0^{a(x)}...
0
votes
0answers
16 views

biholomorphic map from rectangle to disk

I know that there exists a biholomorphic map from the open rectangle to the open disk. However, I wanted to see an explicit/closed form of the map (or its inverse). I tried to compose multiple ...
0
votes
2answers
22 views

Uniform continuity of the function $f(x)= \frac{\sqrt{x^2-1}}{x+\log x}$ in the set $E=[1,+ \infty)$

I have the function $f(x)= \frac{\sqrt{x^2-1}}{x+\log x}$ in the set $E=[1,+ \infty)$and I have to discuss the uniform continuity of f in E. I've calculated the derivative $y'$ and it tends to $0$ ...
2
votes
0answers
12 views

Optimise allocation to minimise variance

Background I am trying to allocate customers $C_i$ to financial advisers $P_j$. Each customer has a policy value $x_i$. I'm assuming that the number of customers ($n$) allocated to each adviser is ...
0
votes
0answers
29 views

Strong induction

I have this question, Prove, $7 + 77 + 777 +7777 + 77...$n digits..$77 = 7/81[(10^n × 10) - 9n - 10]$ By induction. Now since this question was given in the exercise that involves proving various ...
-4
votes
1answer
48 views

New way of finding primes - Can it be proven wrong?

So, I believe I may have discovered a new way for finding primes, but I'm not sure if there is a definite proof; If there is then I might have solved the Goldbach Conjecture, and if not, then I have ...
1
vote
2answers
21 views

Circular permutation with constraints

If four boys and four girls play tricks, how many ways can they join hands, provided that at least two girls are together? My plan is to determine the circular permutation of the eight (boys + girls),...
0
votes
1answer
25 views

Prove or disprove: $\lvert\mathcal{R}\rvert=\lvert\mathcal{R}^{-1}\rvert$

Prove or disprove: If $\mathcal{R}$ is a relation then $\lvert\mathcal{R}\rvert=\lvert\mathcal{R}^{-1}\rvert$. I think it is true but I do not know how to prove it. Facts: $\mathcal{R}^{-1}=\{(...
0
votes
0answers
14 views

Homeomorphism inducing auto-homeomorphism on function space

Suppose that $(X,d_X)$ and $(Y,d_Y)$ are complete metric spaces and let $\phi:X\rightarrow Y$ be a homeomorphism (resp. bi-Lipschitz embedding). Let $C(Z,Z)$ be the set of continuous functions from $...
4
votes
1answer
37 views

Value of a binomial series

Some time ago a question was asked here regarding the value of the sum $$\sum_{i=0}^k \frac{{2i \choose i}}{4^i}$$. But it was deleted later by the OP.I went around it but didn't find a solution. ...
0
votes
1answer
18 views

a function must have to consider about its domain and codomain?

From the book principle of mathematical analysis define 2.1: Consider two sets $A$ and $B$, whose elements may be any objects whatsoever, and suppose that with each element $x$ of $A$ there is ...
0
votes
1answer
22 views

Is $\{ 0 \}$ a basis of the free module $\{ 0 \}$?

I'm studying modules by reading Dummit and Foote, and I'm having a problem understanding the definition of a free module. I read this stackexchange question, but I couldn't figure it out. The ...
0
votes
1answer
18 views

Finding a joint CDF F

Given the joint density of two random variables $X$ and $Y$, $f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$ How do I find the joint CDF ? I know it'll be: $F_{XY}(x,y)=\int\int_R f_{XY}(x,y)=\int\...
0
votes
0answers
42 views

$x_{0}=?$ such that limit of $x_{n}$ is 2.

I have the following sequence $(x_n)_{n\geq0}; x_{n+1}=2^{\frac{x_n}{2}}; x_0$ is a real number.I need to find $x_0$ such that the limit of this sequence is $2$. Some hints?
0
votes
0answers
51 views

How to integrate n-th tetration?

How can the following indefinite integral be computed ? $ \int{x↑↑n} dx $ where $n$ = {$x$ $\in$ $N^+$ : ${x > 2}$} Here ${x↑↑n}$ refers to $n$th tetration of $x$. I tried searching over the ...
1
vote
0answers
12 views

Prouhet-Thue-Morse sequence and Arithmetic Progressions

This question is a fragment from a question posted by @Mathphile for which the link will be provided below. Let $(x_i)$ be an arithmetic progression of length $M$. Let $(t_i)$ be the $i$th element in ...
2
votes
0answers
14 views

Show that $\Omega \setminus N_A \ni x \mapsto 1_A(x) \cdot \int\limits_{\Upsilon} k(x,t)f(t)d\nu(t)$ is measurable

Assume $(\Omega,\mathcal{A},\mu)$ and $(\Upsilon,\mathcal{B},\nu)$ are measure spaces with $\sigma$-finite measures $\mu, \nu$ and $k \in L^2(\Omega \times \Upsilon, \mathcal{A} \otimes \mathcal{B}, \...
2
votes
1answer
18 views

Proving this sequence converges in $L^2(\mathbb{P})$

We have some IID sequence, $\left\{ {{X_n}} \right\}_{n = 1}^\infty $, of standard normal random variable on the probability space $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$. Also $\left\{ {{\...
0
votes
1answer
6 views

Is a function of admissible heuristics in A* search admissible?

I don’t understand how to approach this problem. $h_1, h_2, h_3$ are three admissible heuristics for an optimisation problem to be solved using A* search. Is the heuristic defined by $$h(n) = \frac{...
0
votes
0answers
17 views

Oddity with comparison of norms on finite dimensional vector space in Multigrid Methods by Bramble

For the context of my question, we are assuming that $M_J$ is a finite dimensional vector space equipped with inner product $(\cdot, \cdot)$, and that we have two symmetric positive definite operators ...
0
votes
1answer
20 views

maximizing entropy -> solving the dual problem analytically

I have to maximize entropy and therefore formulated the dual function, calculated its derivatives, set them equal to zero and now I have to solve the following system analytically: $$ e^{-\mu_1 - \...
0
votes
0answers
10 views

Reference request: networks of morphisms where each morphism goes from a product to a product

I am studying networks of morphisms where each morphism goes from a product to a product. Does anyone know if there is literature or references on this? In particular I study the following structure:...
0
votes
0answers
35 views

How to solve this integral with exponential?

$$\int_\mathbb R \frac{e^{ixy}}{y^2+1} dy - \int_\mathbb R \frac{e^{ixy}}{y^2+4} dy$$ I am wondering if it is feasible by hand or if my examinator does not want that I calculate it explicitly
0
votes
1answer
40 views

compute double integral$ \iint_{|x-y|\leq r}{\rm d}x{\rm d}y$?

I want to compute the following double intergal but I do not know how to do $$ \iint_{|x-y|\leq r}{\rm d}x{\rm d}y $$ where $(x,y)\in \mathbb R^3 \times \mathbb R^3$ and $r$ is a positive constant.
0
votes
1answer
21 views

Polynomial rings and congruence classes

Let's consider the polynomial $m(x)$ over a field $\mathbb{Z}_{3}$. We know that $[m(x)]_{m(x)}=m(a)=0$. Now $m(x)=x^{3}+1$; in my lectures slide, it's said at this point that: $m(a)=0$ implies that ...
-1
votes
1answer
32 views

Calculation of digits

Suppose that an array of numbers from $1$ to $10^{23}$ is given. Let's calculate the sum of digits of each number. What kind of numbers among them are more with a $3$-digit sum or a $2$-digit sum? ...
0
votes
1answer
20 views

11 x [27] = 297 & 792 ÷ 11 = [72] …why is that?

Another example of a reversed factor giving you a reversed multiple is 11 x [53] = 583 & 385 ÷ 11 = [35] Is that pattern unique to the number 11 ~ google wouldn't say :( & what is the name ...
1
vote
0answers
20 views

Existence of Rotation Matrix

Given vectors $v_1, v_2 ... v_n \in \mathbb{R}^n$ satisfying $||v_i||_2 = 1$ for all $i$ and $0 \leq\langle\ v_i,v_j\rangle \leq 1$ for all $i, j$. (Not all of them are 0 or 1). Does there always ...
4
votes
2answers
57 views

Identity of tan(x)

I came across the following formulas for analytical expressions of fundamental modes of asymmetric dielectric waveguide. $$ \tan(x) = x\frac{\pi^2-x^2}{\pi^2-4x^2} $$ This approximation is not present ...
2
votes
1answer
10 views

How to find a line field in a vector field

I have a vector field described by $$\mathbf{F}(x,y) = y \hat{x} -x\hat y$$ I am trying to find the field lines for it, ideally by choosing a starting point and then having a parameter t that will ...
0
votes
0answers
18 views

Formal solution diffusion equation

I'm not a mathematician, so please bear with me if I write things down in a non-rigorous manner. I read in either a mathematical finance or physics book (can't remember) that the formal solution of a ...
0
votes
2answers
53 views

Check of $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$ properties

For function defined as $$ f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2} $$ check if $f$ is continuous and differentiable function. My approach: I would like to use the connection between this sum and ...
0
votes
1answer
24 views

Solution of a nonlinear system of equations with 3 unknowns

How do I most elegantly solve the equation system? $\begin{array}{|l l} B= 2a^2-b^2+2c^2 \\[0.75em] C = 2a^2+2b^2-c^2 \\[0.75em] B = a^2+c^2+2ac\cdot G \\[0.75em] %b^2 = a^2+c^2-2ac\cdot G \end{array}...
0
votes
0answers
9 views

Convert $\int_0^{x_0} d^dx$ to $\int_0^{x_0} dr$, where $r=^{\textrm{def}}\|x\|$

For my Statistical Field Theory class (http://www.damtp.cam.ac.uk/user/tong/sft/sft.pdf), the prof converts integrals over each element of a vector $x$ into a single integral over the magnitude of the ...
-7
votes
0answers
23 views

question on Hausdorff space [duplicate]

show that every metric space is a Hausdorff space.
0
votes
2answers
39 views

Proving that limit does not exist

How can I prove that $\lim\limits_{n \to \infty} n$ does not exist? Shall I do it by proving the sequence $\{n\}$ is not a Cauchy sequence?
0
votes
3answers
16 views

Finding the positive \ negative domain of a simple expression

How can I find the right domain of a simple expression, I tried to check the positive domain of this function: $$ \frac{{900-6X} }{X+50} > 0$$ $$900-6X > 0 $$ $$900 > 6X$$ $$ 150 > X$$...
0
votes
2answers
27 views

Use the mid-point rule to approximate the area of the region bounded by a curve

My problem: Use the mid-point rule with $n = 2$ to approximate the area of the region bounded by $y=\sqrt[3]{16 - x^3}$, $y = x$, and $x = 0$.
0
votes
2answers
34 views

When to use a negative x for finding limits at negative infinity

$$\lim_{x\to-\infty}\cfrac{5x^2+6x}{\sqrt{16x^4-5x^2}}$$ My understanding is to always use a negative x when doing negative infinity limits. So the answer I got was $-\frac{5}{4}$. Why do you ...
1
vote
3answers
24 views

My book doesn't have the answer. Can anyone help me with volumes of solids by slicing?

How do I find the volume in the first octant bounded by the surfaces $x^2=y+2z$ and $x=2$ use the slice in the figure to compute the volume?
0
votes
1answer
21 views

Area formula for parametric surfaces

Assume for $\xi\in S^{n-1}$ the parametrization of a closed hypersurface is given by $x(\xi)=R(\xi)\xi\in \mathbb R^n$. Here $R: S^{n-1}\to \mathbb R$ is a positive function. Is there a reference for ...
0
votes
1answer
26 views

Completeness + Totally Bounded vs Compactness for subsets

We know that A metric space is compact if and only if it is complete and totally bounded. Does this hold for its subsets? I mean, is a totally bounded subset of complete metric space a ...
0
votes
2answers
20 views

25% chance occurring 7/11 times.

If an event has a probability of happening 25% of the time. How do you calculate the chances of this happening 7 out of 11 times. If A is 25% and B is 75%. What is the probability of A occurring 7 ...
0
votes
0answers
19 views

Poker chance calculation with unknown cards

I'm trying to recreate a poker chance calculation like they do on television or poker broadcasts, but without knowing the opponent cards. On the internet you can find probabilities of having this kind ...

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