2
votes
1answer
63 views

Model a function with specific shape

I need a function with a specific shape: Quadratic/gaussian concave shape ($-x^2$ like) Centered in $\frac{1}{2}$ where it reaches the max value 1 On 0 and ...
10
votes
3answers
194 views

Series of iterates of a power series

Let $f = \sum_{n \ge 2} a_{n} x^{n}$ be a formal power series of order higher/equal two. By $$ f^{\circ n} := \underbrace{f \circ \dots \circ f}_{\text{ n times}}$$ we denote the $n^{th}$ iterate of ...
4
votes
3answers
91 views

Determine the limit of $\frac{1}{n}\sum_{i =2}^n \frac{1}{\ln i}$ as $n \to \infty$

For the sequence $a_n =\frac{1}{n}\sum_{i =2}^n \frac{1}{\ln i}$ (with $n \ge 2$), I would like to determine if the limit exists, and, if so, find its value. Some observations I have made so far: ...
1
vote
2answers
201 views

A problem on continuous i.i.d random variable

Let $X_1, X_2, X_3, X_4$ be i.i.d continuous random variables with a common distribution function $F$. How to prove that "all the 4! possible orderings of $X_1, \dots, X_4$ are equally likely" without ...
2
votes
0answers
92 views

van der pol and liapunov

i have attempted this question and done as much as i possibly could, any help regarding this question would be very helpful and appreciated. a) show that the second-order differential equation for ...
5
votes
1answer
135 views

What points of affine space can be mapped to zero by an étale morphism?

Let $K$ be a field and $n$ a positive integer. For what points $x\in\mathbb{A}^n_K$ can I find an étale morphism $f_x:\mathbb{A}^n_K\to \mathbb{A}^n_K$ mapping $x$ to zero and how does such a ...
3
votes
1answer
179 views

Is $\sum_{n=2}^\infty\log\left(1+\frac{(-1)^n}{\sqrt{n}}\right)$ convergent?

The question is motivated by the following exercise: Find an example of a sequence of complex numbers $\{a_n\}$ such that $\sum a_n$ converges but $\prod(1+a_n)$ diverges. A necessary condition ...
1
vote
1answer
355 views

Center of Mass double integral

A lamina occupies the region which is the intersection of $x^2+y^2-2y \leq 0$ and the first quadrant of the $xy$-plane. Find the center of mass if the density at a point of the lamina is twice the ...
3
votes
3answers
748 views

Given one primitive root, how do you find all the others?

For example: if $5$ is a primitive root of $p = 23$. Since $p$ is a prime there are $\phi(p - 1)$ primitive roots. Is this correct? If so, $\phi(p - 1) = \phi(22) = \phi(2) \phi(11) = 10$. So ...
2
votes
1answer
49 views

number of elements in vector space

Given that $k$ is a finite field with $q$ elements and $V$ is a $n$-dimensional $k$-vector space, then by basis representation, we know that for $v \in V, v=a_1v_1+a_2v_2+\cdots+a_nv_n$ uniquely. ...
2
votes
1answer
117 views

What's the lowest real $x$ such that $\zeta(x)$ converges?

It's easy to prove that$$\zeta(1)=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...$$ diverges, and $$\zeta(2)=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...$$ converges to $\frac{\pi^2}{6}$. Intuiting the ...
4
votes
3answers
1k views

Determine all subgroups of $\mathbb{R}^*$ (nonzero reals under multiplication) of index $2$.

Determine all subgroups of $\mathbb{R}^*$ (nonzero reals under multiplication) of index $2$. how can I able to solve this problem
1
vote
1answer
87 views

Continuous dependence of zeros on a parameter

Let $F:I\times J\to\mathbb{R}$ be a $C^k$ (or analytic) function, with $I,J$ real open intervals. Set $f_\lambda(x):=F(\lambda,x)$ and consider the parametric equation $$f_\lambda(x)=0\,.$$ Assume its ...
2
votes
1answer
144 views

Let $ f:\Delta \mapsto \Delta $ be an analytic and bijective mapping.

Let $ f:\Delta \mapsto \Delta $ be an analytic and bijective mapping. My question is whether this implies $ f (z)=kz $ for some $ k \in\mathbb {C} $ such that $| k|=1 $. Here, $\Delta :=\{z\in \mathbb ...
3
votes
1answer
2k views

what's the difference between Syntactic consequence ⊢ and Semantic consequence ⊨ [duplicate]

Can you help me to differentiate the Syntactic consequence $\vdash$ and Semantic consequence $\vDash$ ? I think $A \vdash B$ means, "$A$ proves $B$" and $A \vDash B$ means , if $A$ is true, then ...
1
vote
1answer
115 views

How can I prove this two formula $| M^{n} | = | M |^{n}$

I have no idea, how to prove this. $\left| M \cdot N \right| = \left| M \right| \cdot \left| N \right|$ $\left| M^{n} \right| = \left| M \right|^{n} $ please help me. it would be so nice to ...
2
votes
1answer
259 views

A question about the proof of Rademacher theorem

I'm referring to the proof of Rademacher theorem due to C.B.Morrey (i'm reading it on Simon: 'Lectures on geometric measure theory').\ The proof can be summarized in the following steps:\ 1)For every ...
1
vote
0answers
376 views

Turing Machine question, this is NOT HW

I was having a hard time understanding and solving this question that wants me to show the final tape and figuring out if whether or not the turning machine accepts it or not. I have a list of 20 ...
1
vote
1answer
59 views

Simple modules preserved, if exact sequences preserved by functor

I have the following question: If a functor between two categories sends exact sequences to exact sequences, how does it follow that it preserves simple modules as well? Thanks for the help.
2
votes
1answer
99 views

Prove that $f: (a,b)→ℂ$ cannot have infinitely many zeros in $(a,b)$

I have the following nonzero analytic function: $f:ℂ→ℂ$. We will consider only the restriction $f: (a,b)→ℂ$, $a,b∈ℝ$ and $a<b$. My question is: Prove that $f: (a,b)→ℂ$ cannot have infinitely many ...
0
votes
4answers
115 views

A question about an algebraic notation

I'm dealing with this notation and I can not interpret it Let $f(X)$ $\in$ $F[X]$ be a monic polynomial of degree $m$, and let $(f)$ be the ideal generated by $f$. Consider the qotient ring ...
5
votes
1answer
564 views

Show $\sup{A}-\inf{B}=\sup\{a-b:a\in A, b\in B\}$

Let $A, B \subset\mathbb{R}$ be bounded sets. Show $$\sup{A}-\inf{B}=\sup\{a-b:a\in A, b\in B\}$$
0
votes
1answer
156 views

Nonlinear Second-order ODE BVP with 4 boundary conditions

My Lagrangian comes out in this form when I impose spherical symmetry: $$ φ''(ρ)+{3\overρ} φ'(ρ)+{4μ^4\over M^2} φ(ρ)-{4μ^4\over M^4} φ^{3}(ρ)-{μ^4\over2M} ϵ=0 $$ The following boundary conditions ...
1
vote
2answers
141 views

$\cot(x)\,$ in the large $x$ limit?

I couldn't find asymptotic forms of trigonometric functions in any Math Table. In particular, I am trying to find $\;\cot(a x)\;$ in large $x$ limit. thanks,
1
vote
1answer
230 views

Partial Derivative of Modulus

Let $f(z) = u(x,y) + i v(x,y)$. Show that $\frac{\partial}{\partial z} |f(z)| = \frac{1}{2} |f(z)| \frac{f'(z)}{f(z)}$. Any tip/s how can I solve this item. So far, what I was to use the ...
0
votes
2answers
97 views

Integro-differential equation in one dimensional linear thermo-elasticity

I have this system of coupled pde's: \begin{equation} \frac{\partial^2\theta}{\partial x^2}=\frac{\partial \theta}{\partial t}+\sqrt{a}\frac{\partial^2 u}{\partial x\, \partial t} \end{equation} ...
6
votes
2answers
796 views

Pushforward of Lie Bracket

I am trying to figure out why the following equality is true : $$f_*[X,Y]=[f_*X,f_*Y]$$ where $f:M\rightarrow N$ is a diffeomorphism, $M$, $N$ are smooth manifolds, $X$, $Y$ are smooth vector fields ...
2
votes
0answers
31 views

how the number of steps needed depends on the number of nodes and depends on the transmission range?

I run the consensus algorithm, and for each round k, we record the norm of the disagreement vector(|(|δ(k)|)|>〖10〗^(-6)). We stop, at a predefined value|(|δ(k)|)|>〖10〗^(-6) and we call this ...
3
votes
2answers
887 views

Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?

I'm looking for an intuitive understanding instead of a formal proof. Thanks for the help.
2
votes
4answers
77 views

Binomial Coefficients for $(x+1)^4$

Find $(x + 1)^4$ using binomial coefficients. I'm confused as to how to start this, as I thought binomial coefficients were things like $9 \choose 2$.
0
votes
2answers
209 views

Finding Integers With Certain Properties.

How many positive integers between 100 and 999 inclusive e) are divisible by 3 or 4? For this problem, I understand that one has to employ the inclusion-exclusion principle. Those integers ...
1
vote
1answer
58 views

I don't understand this trigonometry question.

This is the link of the picture: $AC = 4$ $BD = 20$ (Bigger than $A C \times 5$ so $5 \times 4 = 20$) Found: $BCD= 157.38^\circ$ $ABC \text{ or } ADC = 22.26^\circ$ Question: Find the ratio ...
0
votes
1answer
466 views

Two Dimension Heat Equation ADI Local Truncation Error

Given a two dimensional heat equation $\displaystyle \frac{\partial u}{\partial t}=K(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2})$ solved using ADI (the alternating-direction ...
3
votes
1answer
99 views

An identity related to Legendre polynomials

Let $m$ be a positive integer. I believe the the following identity $$1+\sum_{k=1}^m (-1)^k\frac{P(k,m)}{(2k)!}=(-1)^m\frac{2^{2m}(m!)^2}{(2m)!}$$ where $P(k,m)=\prod_{i=0}^{k-1} (2m-2i)(2m+2i+1)$, ...
4
votes
6answers
125 views

Prove $7|x^2+y^2$ iff $7|x$ and $7|y$

The question is basically in the title: Prove $7|x^2+y^2$ iff $7|x$ and $7|y$ I get how to do it from $7|x$ and $7|y$ to $7|x^2+y^2$, but not the other way around. Help is appreciated! Thanks.
0
votes
1answer
75 views

Representations over $\mathbb{Q}_p$

I would like to understand representations over the $p$-adic field $\mathbb{Q}_p$ and find simple $\mathbb{Q}_p[G]$ modules for a finite group $G$. Is there some famous literature like for ...
0
votes
1answer
407 views

Relation and the complementary relation: reflexivity and irreflexivity

How do I show that the relation R on a set A is reflexive if and only if the complementary relation R is irreflexive. Because of iff: I start with let R be a relation from a set A to B. The ...
-2
votes
1answer
142 views

compute The quotient

Iam trying to find what is the quotient of ZxZ by the commutator subgroup (ZxZ/[a,b]) and also the quotient of Z/2ZxZ/2Z by the commutator subgroup(Z/2ZxZ/2Z/[a,b])? Any idea? Thank you
2
votes
1answer
41 views

Is $\phi(x)=\frac{\operatorname{d}(x,A)}{\operatorname{d}(x,A)+\operatorname{d}(x,B)}$ locally Lipschitz?

Let $X$ be a metric space and $A,B\subset X$ satifying $\overline{A}\cap\overline{B}=\emptyset$. Define $\phi :X\to\mathbb{R}$ by ...
2
votes
1answer
80 views

Orienting curves with differential forms

Consider the circle given by the equation $x^2+y^2=1$. We can orient this curve by choosing the tangent vector field $(-y,x)^T$, which defines a direction. Supposedly we can do this with by taking an ...
1
vote
1answer
176 views

Composition of holomorphic function and constant functions

I am trying to solve a problem, that might seem easy to you, but i am really stuck with it and would be great if somebody could help me or give me a hint how to show that proof. Given are two ...
0
votes
1answer
41 views

Trouble computing gradient of $\mid f(z) \mid^2$.

I am implementing a published algorithm for optimal design of a digital filter. As part of the algorithm, I need to compute the gradient of $\mid H(z)\mid^2$. Where ...
16
votes
2answers
452 views

Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$

I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following: Let $G$ be a finite group ...
1
vote
1answer
87 views

A question on the category of sets

Why do SET, the category of sets and functions, is a locally small category? In other words, why do the collection of functions among two fixed sets is a set, and not a proper class?
3
votes
1answer
154 views

Why is this Poisson distribution incorrect?

Assume power failures occur independently of each other at a uniform rate through the months of the year, with little chance of $2$ or more occurring simultaneously. Suppose that $80\%$ of months have ...
2
votes
1answer
85 views

Determining the order of poles of functions, particularly of $\frac{z^2+1}{z^4-2}$ and singularity

Say I had $f(z) = \dfrac{z^2+1}{z^4-2}$. I have found the poles, but how do I find the order of the poles? Would they all be of order $4$ as they are $4$th roots? Also, how would i find the ...
1
vote
4answers
103 views

Infinite number of solutions to the Diophantine equation $x^2 + y^2 = z^3$.

I’m not sure how to approach this number theory problem I’ve been working on for a while. So basically I need to show that the Diophantine equation $$x^2 + y^2 = z^3$$ has an infinite number of ...
2
votes
1answer
393 views

Finding uncertainty in the slope/intercept for a non-linear least squares fit

I have the following function: $$M = a(\log_{10}W-2.5)+b$$ I also have a set of data with actual measured values of $W$ and $M$ (each have individual $\pm$ errors). Here's a small sampling of the ...
1
vote
1answer
46 views

Number of different combinations while ordering pizza.

Say i wanna order a pizza, I order exactly 3 different types of additions on the pizza, From 10 different options (Black olives, Green Olives, Pepperoni, Cheese, Tuna, Chocolate, Bananas, Mushrooms, ...
1
vote
3answers
796 views

Question about simplifying absolute value/exponents

Does $|x^n|^{\frac{1}{n}} = |x|$? Also n is a natural number. Sorry that this is such a stupid question, I'm just simplifying something and trying to make sure I'm doing it right.

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