0
votes
0answers
5 views

Time complexity of a recursive function on a given set

I am computing a function $fun$ which is defined as follows. $fun(m,s)=\sum_{\sigma_{p}\subset s;|\sigma_p|=m}\left [\prod_{i}i\in \sigma_p \sum_{j=1}^{|s-\sigma_p|}\sum_{\gamma_p\subset ...
0
votes
1answer
24 views

Continued Fraction

I am working on the following question "Use the continued fraction $[1;0,1,1,2,1,1,4,1,1,6,1,1]$ to get an estimate for $e$." But I got stuck when I tried to compute $q_i$, since $a_1=0$ , $q_1 =0$. ...
0
votes
0answers
7 views

Fourier Series of Complex Valued Functions

Write the Fourier series of functions in the space of complex valued functions L2([0; 1]), which we view as periodic functions on R. Specify the coefficients of the expansion and also express them as ...
1
vote
1answer
12 views

If $\lim t_n=0$, when does $\lim s_nt_n \neq 0$?

I am looking for a real-valued sequence $s_n$ such that $\lim s_nt_n \neq 0$, given that $\lim t_n=0$. Any hints?
3
votes
1answer
33 views

Prove a family of function is equi-continuos.

Let $f_n(x)=\sin \sqrt{x+4n^2x^2}$ on $[0,\infty)$. (1) Prove that $f_n$ is equi-continuous on $[0,\infty)$. (2) $f_n$ is uniformly bounded. (3) $f_n \to 0$ pointwise on $[0,\infty)$ ...
0
votes
1answer
12 views

Solving a system of ODEs with variable in matrix A

I'm looking at a system x'=Ax as follows: $$\left[ \begin{matrix}x'(t) \\y'(t)\end{matrix} \right] = \left[\begin{matrix}0 & 1\\4/t^2 & -1/t\end{matrix}\right]\left[ \begin{matrix}x(t) ...
3
votes
2answers
374 views

Reference book for “Dynamical Systems”

I want to do my thesis about oscillations. I am a math student so I enjoy rigorous texts and hate sketchy ones. I am looking for a textbook or a good source that could help me with dynamical systems. ...
12
votes
3answers
3k views

Recommendation for a book and other material on dynamical systems

I currently have the book Dynamical Systems with Applications Using Mathematica by Stephen Lynch. I used it in an undergrad introductory course for dynamical systems, but it's extremely terse. As an ...
3
votes
3answers
292 views

Textbook Recommendation: Topological Dynamics

I need to take credits satisfying a topology requirement, and can structure it myself. My field of study is dynamical systems, can someone recommend a textbook that handles differential ...
2
votes
1answer
20 views

Recommended second textbook for dynamical systems?

I recently finished a course on dynamical systems supplemented by Strogatz's textbook. There are a few parts of the book that we didn't cover (in particular, the material on fractals), but the ...
2
votes
1answer
13 views

Is the co-limit of a chain of normal subspaces necessarily normal?

Suppose $ X_0 \subset X_1 \subset X_2 \subset \dots$ is a chain of normal subspaces of $X$ such that $X= \cup_{i=1}^{\infty} X_i$. Assume that $X$ has the colimit topology w.r.t. these subspaces. Can ...
2
votes
1answer
28 views

Algebraic subfield of transcendental extension

I was recently thinking about whether it is possible to generate an infinite dimensional algebraic extension over a base field using just finitely many transcendental elements. Specifically, given a ...
0
votes
0answers
12 views

Determining Counts of Discrete Objects Using Linear Algebra

I'm teaching myself linear algebra and was able to solve the following question using trial and error, but--how would one setup and solve a question like this using Linear Algebra? I have 32 bills ...
0
votes
1answer
31 views

A proof about boundedness for continuous functions

Let $I := [a,b]$ and let $f : I \rightarrow \mathbb{R}$ be a continuous function such that $f(x) > 0$ for each $x$ in $I$. Prove that there exists a number $a > 0$ such that $f(x) \geq a$ for ...
0
votes
0answers
11 views

Point in a spherical triangle test

Given three latitude/longitude coordinates on a sphere forming a triangle, how do I test if a point p is inside that triangle? I know latitude and longitude implies Earth and Earth is not perfectly ...
0
votes
1answer
16 views

Determining if a point is inside an infinite 3d elliptical tilted cone

I have an infinite 3d elliptical, tilted cone that is defined by a vertex point P(x,y,z) and by 4 angles: the first pair of angles represent the spatial orientation of the cone: θ is the polar angle ...
-1
votes
0answers
11 views

When Independence $\Rightarrow$ Independence of higher moments (Prob/ Stats)

suppose {$X_n$} is iid. Then, is $X_i$ independent of $X_j^3$ for j≠i? If so, why? Secondly, is $X_i^2$ independent of $X_j^2$ for j≠i? Intuition: yes no If there's a difference, why? (note: ...
1
vote
1answer
602 views

coordinate geometry - find point in right-angled triangle

I'm making a map. I've come across a geometry problem, and I'm not so knowledgeable about maths! Let me illustrate with pictures. I am trying to plot flightpaths with a curved line, using a ...
0
votes
0answers
31 views

Diffrental equation solution [on hold]

How can I solve this equation? $$\frac{\partial f}{\partial x} =\frac{a-x}{y} \frac{\partial f}{\partial y}$$ where $a$ is a constant. So what is $f$?
3
votes
1answer
51 views

How to curve fit an unknown function?

I have data which can be described by $y=f(x,z)$ where $z$ varies from 170 ~ 154. Now values given by $ks$ are known sample values that equals value given in the table header, $uks$ are unknown ...
0
votes
1answer
22 views

how to identify the subspace of vectors? [on hold]

Which of the following subsets of $\Bbb R^3$ are subspaces of $\Bbb R^3$? A. The $3\times 3$ matrices with all zeros in the second row B. The $3\times 3$ matrices whose entries are all integers C. ...
0
votes
0answers
8 views

if $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$

if laplace transform $F(s_{0})$ for some $s_{o}$exists then it exists for all $s>s_{o}$. i need to prove this . now, ...
1
vote
1answer
2k views

Defining a Limit Point of A Set

Limit Point is defined as: Wolfram MathWorld: A number $x$ such that for all $\epsilon \gt 0$, there exists a member of the set $y$ different from $x$ such that $|y-x| \lt \epsilon$. Proof Wiki: ...
0
votes
1answer
13 views

Boundedness Theorem for continuous functions on intervals

Just want to confirm this is a suitable proof: Assume $f$ is not bounded on $I$. So, for any $n \in \mathbb{N}$, $\lvert f(x)\rvert > n$. Since $I$ is bounded, $x_n$ is also bounded. By ...
1
vote
2answers
38 views

$C$ is an uncountable set. Show that $B(x,\epsilon)\cap C$ is uncountable.

Suppose $X$ is a separable metric space, and $C$ is an uncountable subset of $X$. Prove that there is a point $x \in C$ such that for each $\epsilon>0$, $B(x;\epsilon)\cap C$ is uncountable. I ...
0
votes
1answer
13 views

Supercuspidal representations

If $(\pi,V)$ is a representation of $G=GL_n(F)$ where $F$ is a nonarchimedean local field, and $0 \subset V_2 \subset V_1 \subset V$ is a filtration of $V$ into $G$-invariant subspaces, with $V/V_1$ ...
2
votes
3answers
40 views

Show the closed form of the sum $\sum_{i=0}^{n-1} i x^i$ [duplicate]

Can anybody help me to show that when $x\neq 1$ $$\large \sum_{i=0}^{n-1} i\, x^i = \frac{1-n\, x^{n-1}+(n-1)\,x^n}{(1-x)^2}$$
7
votes
0answers
57 views

Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
0
votes
0answers
21 views

How to find eigen vector for an eigen value in generalized eigen value problem

I have a generalised eigen value problem of the form $A$x = λ$B$x. I have computed the eigen value (say λ1) I am interested in using Eigen library(C++). However, because the library does not support ...
0
votes
0answers
82 views

Help to understand the Monty Hall Problem [on hold]

I have read the Monty Hall Problem and explanations on Wikipedia several times and fail to grasp how exactly 'conditional probability' is being realistically applied. More specifically, my logical ...
0
votes
1answer
25 views

Integrability of characteristic function

I have a following questions that I am having trouble. Let $E = \{(\frac{a}{b}, \frac{c}{b}) : a,b,c \in \mathbb{Z}, a \text{ and } b \text{ are relatively prime}\}.$ For what $a \in [1,2]$ is the ...
-1
votes
0answers
25 views

a question about abstract algebra,prove that the ring is commutative. [duplicate]

(1)A ring R is a booleean ring if for every $a\in R$,$a^2=a$. Show that every Boolean ring is a commutative ring. (2)Let R be a ring,where $a^3=a$ for all $a\in R$.Prove that R must be a commutative ...
0
votes
1answer
13 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=(\frac{2xy-2xy^2}{(1+x^2)^2}+\frac{8}{13})i+(\frac{2y-1}{1+x^2}+2y)j$$ Determine $\int_cF*dr$, where $C$ is the path $C_1+C_2+C_3$ from $(2,0)$ to $(5,6)$ shown. I ...
0
votes
3answers
29 views

Modulo arithmetic proof

Show that if none of the numbers in the list 1a,2a,..(p-1)a are congruent to 0 mod p, then no two numbers in the list are congruent to each other mod p. I am not sure how to try to demonstrate this. ...
0
votes
0answers
14 views

How to calculate solid angle of a rectangular detector of 20cm x 10cm?

I have an detector of $20cm*10cm$, how can i calculate the solid angle subtended by the detector if the detector is placed at $30$cm apart? Because directly i cannot use the formula ...
0
votes
1answer
62 views

$f(x) =\ln(2x^2 + 1)$ is continuous on $\mathbb{R}$

True or False The function $f : \Bbb R \to \Bbb R$ defined by $f(x) = \ln(2x^2 + 1)$ is continuous on $\Bbb R$. I know this condition that The function $f$ is continuous at some point $c$ of its ...
0
votes
1answer
18 views

Finding a linear map.

I have some problem with a question related to linear maps. I know the solution but I can not understand the reason behind it. For any polynomial $p∈P^2$ let: ...
0
votes
0answers
26 views

Equivalent presentation for the fundamental group of the projective plane

We know that $\langle a,b;(ab)^2=1\rangle$ and $\langle z;z^2\rangle$ are presentations of the fundamental group of the projective plane. Therefore, one is obtained from the other via Tietze ...
0
votes
1answer
34 views

Find the number of integers n such that the equation

Find the number of integers n such that the equation $xy^2+y^2-x-y=n$ has an infinite number of integer solutions $(x,y)$.
2
votes
0answers
11 views

A Fourier Transform Which Is Cartesian Separable

We say that the Fourier transform of a complex-valued function $f\in L^{1}(\mathbb{R}^{n})$ is separable if there exist single-variable functions $g_{1},\ldots,g_{n}$ such that $$g_{1}(\xi_{1})\cdots ...
1
vote
1answer
46 views

Proving that the matrix exponential map is surjective onto the general linear group

Let $M_n(\mathbb{F})$ be the set of all $n\times n$ with entries in $\mathbb{F}$ and let $\exp:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be defined by $$ \exp(A)=\sum_{k=0}^{\infty}\frac{A^k}{k!},$$ for ...
-4
votes
2answers
24 views

A question on proving of factorial [on hold]

Prove that $(n!)!$ is divisible by $(n!)^{(n-1)!}$.
4
votes
0answers
57 views

Linear Algebra Proof conformation

I am trying to prove this theorem in my book. Because it is provided without proof, please let me know what you think! $\mathbf{Theorem:}$ Let V be a finite , $n$ -dimensional vector space and let U ...
0
votes
1answer
40 views

Integral of a total derivative

I have seen the "total differential" $$ d \ln A = -d \ln B/c $$ Representing how infinitesimal changes in $A$ are related to infinitesimal changes in $B$. Someone then took the integral of this ...
1
vote
2answers
28 views

To show $f(x)$ has ONLY one Max in $x\in[0,1]$

I have function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$ I want to show that $f(x)$ has ONLY one maximum in $x\in[0,1]$. I get the second derivative as ...
0
votes
2answers
30 views

Can the following system be solved symbolically/analytically?

I have the following system of equations with variables $a,m$, and I'm wondering—can this system be solved symbolically/analytically? \begin{align} m &= 100 + \frac{ \left( 200 ...
2
votes
0answers
20 views

Finding the Laurent series given the poles and residues

I am working on the following problem, suppose that $f$ has a simple pole at $-1$ with $Res(f,-1) = 1$. A double pole at $2$ with $Res(f, 2) = 2$. Also $f(0) = 7/4$ and $f(1) = 5/2$. I am supposed ...
0
votes
1answer
28 views

Calculating the length of a tangent drawn to a circle from a named point

My book (New Tertiary Mathematics Volume 1 Part 1, by C Plumpton and P S W Macilwaine) describes a method for calculating the length of a tangent to a circle from the point $(x_{1}, y_{1})$ outside ...
0
votes
2answers
471 views

Internal/external division of a line question with unknown coordinates?

I've been having trouble with this question: "Given $K(3, - 1)$ and $L(-4, 2)$, find two positions of $A$ on $KL$ such that $KA = 2( KL)$." Any help would be appreciated. The first thing I tried was ...
2
votes
1answer
35 views

Find the closed-form for $\sum_{i=0}^n(-1)^i(\frac{1}{2})^i$

I start with simplifying: $$\sum_{i=0}^n(-1)^i(\frac{1}{2})^i=\sum_{i=0}^n(-\frac{1}{2})^i$$ then: $$S = 1 + (-\frac{1}{2}) + (-\frac{1}{2})^2 + ... +(-\frac{1}{2})^n$$ $$(-\frac{1}{2})S = ...

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