All Questions

0
votes
1answer
70 views

Proof wanted that there is no positive integer matrix with positive integer eigenvalues u,v,w, if $0<u<v$ and $1\le w-v\le 2$

I have the following conjecture : If u,v,w are integers with $0<u<v<w$, then there is a POSITIVE INTEGER 3x3 - matrix A with eigenvalues u,v,w if and only if $w-v\ge 3$. I approved the ...
1
vote
3answers
95 views

Question regarding permutations and combinations?

Hi, I was just wondering on how you are supposed to approach this question. I keep getting 114 as an answer, but the answers say it is 174. How would anyone do this question, because I feel like I'm ...
4
votes
0answers
352 views

How to solve a non-homogeneous second-order linear difference equation with both a forward and a backward difference?

Quite a long title for this: I'm looking for the general solution of the following difference equation: $$ax_{t+1} -bx_t + x_{t-1} = c + u_t$$ where $a,b,c$ are real constants and $u_t$ is a bounded ...
1
vote
2answers
516 views

Double integral of a rational function

Consider the region $D$ given by $1\leq x^2+y^2\leq2\land0\leq y\leq x$. Compute $$\iint_D\frac{xy(x-y)}{x^3+y^3}dxdy$$ Attempt: The region $D$ is part of a ring in the first quadrant below the line $...
7
votes
1answer
218 views

Solutions for $ \frac{dy}{dx}=y $?

Al-right, this may be a very basic question but I'm confused about this. We all know that one differential equation can only have one solution. Consider: $$ \frac{dy}{dx}=y $$ The solution is: $$ y=...
0
votes
2answers
66 views

Contour intergals of rational fuction

Consider $$F=\frac {x}{x^3+y^3}dx+\frac{y}{x^3+y^3}dy$$ 1) Show that $\int_GF=0$, where $G$ is the arc of a circle or radius $r$ in the first quadrant. 2) Compute the integral of $F$ along the ...
3
votes
1answer
253 views

Learning Fibre Bundle from “Topology and Geometry” by Bredon

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if ...
1
vote
2answers
217 views

What is Cumulative Binomial probabilities?

I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a Cumulative Binomial probability is. So my question is, What is ...
1
vote
0answers
190 views

What is $\operatorname{Ass}\operatorname{Ext}^i(M,N)$?

This is exercise 1.2.27 of Bruns-Herzog: Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $N$ an arbitrary $R$-module. Deduce that $\operatorname{Ass}(\operatorname{Hom}_R(M,N)) = \...
1
vote
2answers
153 views

Solving ln/exponent question

How do I change the subject of the equation from x to y in the following equation: $$x=[4.105-\ln(\sqrt{y})]^2$$
2
votes
1answer
94 views

Show solvability of ODE without explicitly calculating solution

Show that $$ u + u^{(4)} - u^{(2)} = f $$ has a solution $u \in H^4(\mathbb R)$ (without explicitly calculuting it) for every $f \in L^2(\mathbb R)$! What criteria for solvability for such ODE's ...
0
votes
1answer
65 views

Partial derivative of a vector

I'm trying to show: $\displaystyle \frac{\partial} {\partial t}( \nabla(\phi))= \nabla\frac{\partial \phi} {\partial t} $ Am I allowed to write: $\displaystyle \frac{\partial} {\partial t}\begin{...
3
votes
0answers
672 views

Maximum modulus principle, 3 questions

I have several questions regarding the maximum modulus principle, but first let me interpret my understanding of this theorem: Assuming we have some analytic, non-constant function $\;f:\Omega\...
1
vote
2answers
149 views

I need to solve $\dfrac{dx}{dt}= 2x(1-0.0001x)-0.01xy, \dfrac{dy}{dt} = -0.5y+0.0001xy$

I need to solve $$ \begin{align} \frac{dx}{dt} &= 2\,x\,(1-0.0001\,x)-0.01\,x\,y \\ \frac{dy}{dt} &= -0.5\,y+0.0001\,x\,y \end{align} $$ Can anyone tell how do we solve such problems, if ...
1
vote
0answers
23 views

Possible known ways to improve Back Substitution

I am quite new to this field and just implement algorithms. I am currently using back-substitution as a way to invert a lower triangular matrix. I would like to ask if there are known ways to improve ...
17
votes
10answers
7k views

“I have found a dead body on my car.”

Given a statement "I have found a dead body on my car", and considering the fact that I do not own any car, is this statement true? If so, is this a special case of false implies anything?
-1
votes
1answer
266 views

The Picard-Lindelöf theorem on Wikipedia

On the Wikipedia entry of Picard-Lindelöf theorem for the local existence and uniqueness of ODE's, there is a section on the optimization of the solution's interval. There is a lemma used in this ...
0
votes
0answers
178 views

Solving system of differential equations

I have a system of differential equation to solve. Any suggestions regarding closed form or numerical method is welcome with great respect. This equation is from dynamic equation of a curve. Let us ...
1
vote
0answers
71 views

How to prove $\gamma$ is continuous?

In the paper A remark on least energy solutions in $\mathbb{R}^N$, page 2407, it said, if $u_0\in H^1(\mathbb{R}^2)$, set $\gamma(t)=t^{-1/4}u_0(x/t)$. Then $\gamma(t)$ is a continuous path in $L^2(\...
9
votes
3answers
2k views

What is golden ratio doing in this computer code?

In this file (related to random number generation), there is following line: private const int MSEED = 161803398; which resembles golden ratio. Why does the ...
8
votes
2answers
514 views

Number of Primes in Ring of Integers of a Number Field

In the ring of integers of an algebraic number field, we say that an algebraic integer $p$ is prime if, whenever $p$ divides product of two algebraic integers, $p$ divides one of them. An algebraic ...
3
votes
3answers
712 views

Find the maximum value of $xy^2z^3$ given that $x^2 + {y}^2 + {z}^2 = 1$, using AM-GM

I've been struggling with this equation and how to find the maximum value it can take: Maximise $xy^2z^3$ given that $x^2+y^2+z^2 = 1$ The question is from the book Introduction to Inequalities - ...
0
votes
1answer
31 views

How to express a contour

How would I express the contour which is the portion of the unit circle in the left hand plane going from i to -i. I though the contour would be $y(t)=e^{it}$ $t {\in} [-{\pi}/2,{\pi}/2]$ but this ...
0
votes
1answer
178 views

Passing thresholds with uniform random variables

I have encountered a challenging task: I have a bunch of uniform random variables "trying" to pass a certain threshold, and another bunch trying to pass a different threshold, and I need to estimate ...
2
votes
1answer
11k views

Does negative infinity squared = positive infinity?

I googled this question and saw this answer but I wasn't satisfied.
0
votes
1answer
82 views

Contour integrals and Cauchy's theorem

1) Let $C$ be a contour beginning and ending at 1. Suppose that $f(z)$ is analytic on $C$. Then is it true that the contour integral of $f$ around $C$ is 0? This looks to be true by Cauchy's theorem ...
1
vote
1answer
53 views

Is this matrix definite positive?

I would like to know if and why this matrix is definite positive: $$A = \sum_{i=1}^n y_i . y_i^T$$ where the $y_i$ are vectors.
8
votes
3answers
8k views

Geometric series of matrices

I am currently reading 'Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach' by J. Hubbard and B. Hubbard. In the first chapter, there is the proposition: Let A be a square ...
1
vote
1answer
43 views

Degree of a single extension of a field. [closed]

Let $K$ be a field. Let $L:=K(b)$ where $b^m\in K$ but $b^i\not\in K$ for all $0<i<m$. Then is $[L:K]=m$?
0
votes
1answer
35 views

Need some help understanding why this function is well defined and why it maps into $S^1$

Let $A$ be a unital $C^\ast$ algebra. Let $\varphi$ be the Gelfand transform. Let $u$ be unitary and $f = \varphi (u)$. Let $\ln: \mathbb C \setminus [0,\infty) \to \mathbb C$ denote the natural ...
3
votes
1answer
724 views

What is the count of the strict partitions of n in k parts not exceeding m?

Lets say we had a $k,m,n \in \mathbb{N}$ where $k < m \le n$. How many different sets $X_1,..,X_m$ with $|X_i|=k$ for $i=1,..,m$, where the sets do not include duplicates, for which the sum of ...
1
vote
1answer
391 views

Solve Burgers' Equation with side condition.

Solve Burgers' equation $$u_t + uu_x =0,$$ with $u=u(x,t)$ and the side condition $u(x,-1) = x^2$. Find the solution for $u=u(1,2)$ I can't figure out how to use the side condition in order to find ...
1
vote
1answer
115 views

Classification of separable algebras up to Morita equivalence

Is there a simple classification of separable algebras up to Morita equivalence, working over a particular field $k$? For example, over $\mathbb{C}$, every separable algebra is Morita equivalent to ...
1
vote
3answers
453 views

Incorrect proof of the infinities between 0 and 1 and 0 and 2

In reading another question (Explaining Infinite Sets and The Fault in Our Stars) it got me thinking about the way that you can prove that the number of numbers between 0 and 1 and between 0 and 2 are ...
1
vote
1answer
21 views

Do characters map into $S^1$

Let $A$ be a Banach algebra. Then a character is defined to be a non-trivial continuous homomorphism $A \to \mathbb C$. I'm not sure why I think this but: aren't characters really maps $A \to S^1$? I ...
1
vote
1answer
62 views

rearrangement of infinite sum

I would like to find a justification why it is correct to write for any non negative sequence $(a_{n,m})_{n,m} \subset \mathbb{R}$ that $$ \sum_{i=1}^\infty \sum_{j=1}^\infty a_{ij} = \sum_{j=1}^\...
0
votes
4answers
64 views

Is it correct to write gcd(a,b) if a<b?

While creating an algorithm to compute the greatest common divisor of two numbers I saw that — on various websites/books — when you have "gcd(a,b)" a is superior to b ; is it an obligation or am I ...
3
votes
0answers
81 views

Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials): (a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$; (b) the first $k$ ...
2
votes
1answer
341 views

Topological group with discrete topology

Let $G$ be a topological group. I came to know that if I can show the existence of a homeomorphism of $G$ which moves only finitely many points of $G$, then $G$ has only discrete topology. How can I ...
0
votes
1answer
88 views

Roundness in Taxicab Geometry

I was just wondering whether circles are considered "round" still in taxicab geometry. I know that "roundness" is probably not a well-defined term and I know what a circle /appears/ to look like in ...
3
votes
0answers
133 views

Developing Endurance

I realize many math problems solved by mathematicians require a tremendous perseverance over a period of years. My question is not about endurance over a long period of time, but over several hours. ...
0
votes
2answers
392 views

Two groups A and B are playing a game…

Two groups A and B are playing a game. The first group that wins 3 times is the winner. The probability that group A will win at on game is $\frac12$ and the same thing for group B. $X$ = The number ...
2
votes
2answers
38 views

minimization with many unknowns and one condition

I haven't done this in quite a while so excuse my perhaps silly question. I'm looking for a solution to a minimization problem (if there is one), that goes like this: I want to minimize (global) $f(...
4
votes
0answers
357 views

Homotopy limits

Let $\mathfrak C$ be a Grothendieck category and let ${\bf D}=\mathrm{D}(\frak C)$ be its derived category, that is, consider the injective model structure on the category $\mathrm{Ch}(\frak C)$ of ...
0
votes
0answers
260 views

Newton's method for the brachistochrone

Consider the potential $V(x,y)=-y$ and a particle at rest in the beginning of the coordinate system. We are going to examine the brachistochrone - the smooth curve of fastest descent. Assume we are ...
-1
votes
3answers
741 views

distance from a point to line segment not it 's perpendicular line's distance

how to find distance between line and point in the picture ? what is the shortest distancing point in the line ? Note: distance between line and point means line segment,(the intersecting point must ...
1
vote
1answer
179 views

Why does the Fund. Theorem of Contour Integrals Need Continuity?

Why does the Fundamental Theorem of Contour Integrals need continuity? When defining the integral in real analysis we don't require continuity of the function we are integrating, is it necessary to ...
1
vote
1answer
42 views

A question about Lang's explanation of ordered fields on pg 449

Let $K$ be a field, and $P$ the set of positive elements. We know that $P$ is closed under addition and multiplication. It is also easily seen that $1\in P$. Assume that $x\in P$. Then $xx^{-1}=1$. ...
0
votes
2answers
66 views

Help with this limit

Can you help me show that $\lim_{(x,y)\to (0,0)}x\log\sqrt{x^2+y^2}=0$ I have shown that $\lim_{x\to 0}x\log x=0$. I tried to use that $\log x\leq x-1, x>0$ or that $x\leq\sqrt{x^2+y^2} $ but I'm ...
2
votes
2answers
74 views

OEIS for Doubly Indexed Sequences

Is there an OEIS-like database for doubly indexed sequences? I feel like such a database would be extremely useful for mathematicians, and would be surprised if there wasn't one, but I can't seem to ...

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