1
vote
0answers
7 views

Comparing PDE solutions for different Riemannian metrics

I'm looking for the approach to compare PDE solutions on the Remannian manifolds when those solutions are obtained under two different metrics. To be more specific, suppose we have two Riemannian ...
0
votes
0answers
4 views

Prove that $l^{\infty}(\mathbb{Z^+})$ is not separable.

Let $l^{\infty}(\mathbb{Z^+})$ be the set of all bounded complex functions on $\mathbb{Z^+}$.Then prove that $l^{\infty}(\mathbb{Z^+})$ is not separable. My attempt:Suppose $E\subset ...
0
votes
0answers
4 views

Does the Schur complement Theorem establish in following cases?

Let $M$ be an $(n+m)\times(n+m)$ real non-symmetric positive semidefinite (PSD) matrix partitioned as \begin{eqnarray*} M=\left(% \begin{array}{cc} A~~B\\ C~~D\\ \end{array}% \right), ...
0
votes
0answers
3 views

What is pseudodiagonality in matrix/tensor?

What is the difference between diagonality and pseudodiagonality? Does this apply to tensor too? https://www.math.uzh.ch/fileadmin/math/preprints/06_11.pdf
0
votes
0answers
5 views

Etymology of Multipole Expansion

Can anyone tell me why multipole expansions are given the name they are based on their definition as being expansions about two angles? I know in Physics you get multipole expansions that are relevant ...
0
votes
0answers
6 views

Solving the Telegraph Equation using Partial Differential Equations and Sturm-Liouville theory

I've been asked to do the following question, and I've got through the brunt of it (so this is going to be a rather long question...), but I'm just having a bit of trouble applying Sturm-Liouville ...
0
votes
0answers
10 views

how to add shipping cost to unit cost

If I have x number of units at DIFFERENT unit cost from supplier, after shippings costs, how much shipping expenses will each unit carry? Basically I want to know how much each unit will jump after ...
0
votes
0answers
10 views

Interpretation of the derivative of the delta function

I've seen in several places the derivative of the delta function regarded as a mathematical object I understand that it can be realized mathematicly as a generalized function which returns the value ...
1
vote
2answers
22 views

If a linear transformation T is nilpotent, show that $\alpha_0+\alpha_1T+…+\alpha_kT^k$ is invertible

If a linear transformation T is nilpotent, show that $\alpha_0+\alpha_1T+......+\alpha_kT^k$ is invertible provided that $0\ne\alpha_0\in F,$ for some field F. I am in the mid way, and am stuck at the ...
0
votes
1answer
15 views

Query regarding an alegbraic inequality

Do we have any inequality of the type $(a-b)^q \geq C_q(a^q-b^q),$ where $q>0$ and $a,b$ are real numbers?
0
votes
2answers
14 views

System of linear equation

Determine the value for k for which the system of linear equation has infinitely many solution. 2x - y = 2 4x + ky = 4
0
votes
0answers
18 views

Is it possible to study information theory while studying a first course on probability?

I'm currently taking a course on intro to probability. The course is not mathematically rigorous and does not invoke theorems from real analysis, etc. The course covers all the way from basic ...
2
votes
2answers
14 views

how to calculate |exp(-ia)+exp(-ia')|^2

What is the correct way to calculate something like $|\exp(-ia)+\exp(-ia')|^2$ ? I have tried simply multiplying the term inside the absolute value by its complex conjugate, ...
-2
votes
1answer
22 views

Recurring Decimals [duplicate]

If: x = 0.999... 10x = 9.999... [Multiply by 10 on both sides] 9x = 9 [Subtract x, or 0.999... from both sides] x = 1 [Divide by 9 on both sides] Therefore, 0.999... = 1 Why is this? How is it ...
1
vote
0answers
10 views

Problematic Initial Condition of a Recurrence Relation

I encountered this equation, and tried to solve it: $T(n) = T(\sqrt{n})+log(n)$ Under the initial condition T(1)=1. Can someone tell me why is this initial condition helpful? I mean, of course ...
0
votes
3answers
21 views

Prove $n^n\prod_{i=1}^{n}(x_i^{n}+1) \ge \sum_{i=1}^{n} x_i +\sum_{i=1}^{n}\frac{1}{x_i}$

if $x_i$ is positive real number that $\prod_{i=1}^{n} x_i=1$,Prove:$$n^n\prod_{i=1}^{n}(x_i^{n}+1) \ge \sum_{i=1}^{n} x_i +\sum_{i=1}^{n}\frac{1}{x_i}$$ Additional info:I'm looking for solutions ...
0
votes
1answer
19 views

Help explain linear algebra/differential calculus theorem in simpler terms.

On a previous question, I got something related to linear algebra and linear algebra, but having no background in linear algebra and a little background in vector calculus(mainly from physics), I ...
0
votes
0answers
6 views

if f(t) in F[t] is separable and E/F is an algebraic extension of F, then how can I be sure that f(t) is separable as an element of E[t]

if f(t) in F[t] is separable and E/F is an algebraic extension of F, then how can I be sure that f(t) is separable as an element of E[t] I thought it is a trivial question...but now I think it is ...
1
vote
0answers
13 views

how to extract frequency from a set of numbers

Given the numbers 4,1,0,4,0,0,4,1,0,4 it is obvious there's a dominating frequency of 4 appearing every four numbers. Given 5,1,1,3,0,0,6,1,0,4 again it looks that there's a spike of about 4 (4.5 to ...
1
vote
3answers
28 views

Calculus (integration [multiplication])

Determine $\int e^{2x} \sqrt{e^x+1}dx$ Is there a multiplication rule for integration or something?
3
votes
1answer
18 views

Characterization of entire functions to be a polynomial

I need some help with this proposition: If $f:\mathbb{C}\longrightarrow \mathbb{C}$ is an entire function such that $\{z\in \mathbb{C}:f(z)=w\}$ is finite for all $w\in \mathrm{Im} (f)$ then $f$ is a ...
0
votes
0answers
18 views

Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
0
votes
3answers
29 views

Direct evaluation of a series from Euler's identity.

Is there a direct way to evaluate: $$ \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!}=-1 $$ Note that this follows from Euler's identity.
0
votes
0answers
5 views

The invariance of the Ricci tensor under diffeomorphisms and its non-ellipticity.

Consider $(M,g)$ a compact Riemannian manifold. When viewed as a second order (non-linear) differential operator $$ \text{Ric} : C^{\infty}(\text{Sym}^2_+T^*M) \to C^{\infty}(\text{Sym}^2T^*M), $$ the ...
0
votes
4answers
46 views

Why $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1^n)$?

I am working out a Fourier Series problem and I saw that the suggested solution used $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1^n)$ to simply the expressions while finding the Fourier Coefficients ...
0
votes
1answer
18 views

Simplified Galois proof?

I have learned about Galois epoch-making proof that any polynomial of the fifth degree has no solution representable in terms of its coefficients. Can his proof be simplified and clarified in modern ...
0
votes
4answers
22 views

Simplifying/solving a logarithm

Need help with simplifying this logarithm. $$log_24^{2n}$$ Would I just pull the 2n to the front: $$2n*log_24$$ So it would simplify to $$4n$$ Is this correct or am I completely wrong?
0
votes
2answers
16 views

Two circles with the same radius r intersects each other and one passes to the centre of the other.

Two circles with the same radius r intersects each other and one passes to the centre of the other.Then the length of the common chord is can someone help me I think its answer is r because passes ...
0
votes
3answers
31 views

Calculus (what is y when x is?)

Given $y>0$ and $$dy/dx = (3x^2+4x)/y$$ If the point $(1,rad10)$ is on the graph relating x and y, then what is $y$ when $x=0$? I'm not sure whether or not to integrate, or just plug in the ...
0
votes
0answers
27 views

How to prove cyclic here? [duplicate]

Let $G$ be a finite group with identity $e$. Suppose that for every positive integer $d$, the number of elements $x\in G$ with $x^d = e$ is at most $d$. Show that $G$ must be cyclic.
1
vote
1answer
23 views

Explain rings and is [S, /, -] a ring?

Okay, so we are going to use the base set of numbers [i], which contains all possible cases of ai, where a is any real number. Here are 4 possible groups on this set --> [i,*]... [i,+]... [i,/] ...
1
vote
2answers
19 views

How do I convert a fraction in base 10 to a quad fraction (base 4)?

I am totally confused when it comes to converting fractions or floating point numbers to a different base. I have no problem converting whole numbers to any base but when it comes to fractions or ...
0
votes
1answer
14 views

Problem book for abstract linear algebra

Kindly suggest a good book for abstract linear algebra other than finite dimensional vector space by P R Halmos
2
votes
0answers
42 views

Are there infinite many primes p such that 2p-1 is also prime?

I did a search online and found a similar notion called Sophie Germain prime, which by definition is a prime $p$ such that $2p+1$ is also prime. Sophie Germain primes are conjectured to be of infinite ...
0
votes
0answers
4 views

Canonical form for orthogonal similarity classes

Could someone point me to a reference re canonical forms for classes of matrices in $M_n(\mathbb{C})$ which are unitarily similar? That is, canonical representatives for the equivalence class defined ...
1
vote
0answers
6 views

Bound on variance of random process when signal is known

I am reading this paper (link to a Nature paper, may not be accessible) and I encountered the following. I have very little experience in probability theory and I could not find much helpful in ...
0
votes
2answers
17 views

Show that every row of matrix $S$ is a linear combination of its bottom row and the row (1 1 1 1 1 1 )

Couldn't solve the following three questions. $$S=\begin{pmatrix} 36 & 35 & 34 &33&32&31 \\ 25 & 26 & 27&28&29&30 \\ ...
-13
votes
1answer
38 views

sum of infinite series 1/2 + 1/4 + 1/8 + … [on hold]

goto 324 here: Visually stunning math concepts which are easy to explain Someone comment this writing the claim as such: Another way to think of this is that 1/2 + 1/4 + 1/8 = 0.111...binary = ...
0
votes
1answer
13 views

inequality of Kernels dimension

Exercise Let $U,V,W$ be $K$-finite-dimensional vector spaces, and $f \in \operatorname{Hom}_K(U,V)$, $g \in \operatorname{Hom}_K(V,W)$. Show that $\dim(\ker(g \circ f))\leq ...
2
votes
1answer
38 views

How to find the limit $\lim_{x\to 0} ( 4x/\sin 2x + x\cos2x )$?

Compute $$\lim_{x\to0}\left[-\dfrac{4x}{\sin 2x} + \dfrac{x}{\cos 2x}\right]$$ Obviously I can't plug in 0. I noticed the sin and cos are both 2x. Is there a way to combine them into tan? I don't ...
0
votes
1answer
23 views

Does every free $R$- module have a maximal proper submodule?

Let $R$ be a commutative ring with $1$. We know that every finitely generated $R$-module has a maximal proper submodule. Is it true for any free $R$-module? In particular, can we do the following: ...
1
vote
1answer
24 views

Matrix with eigenvalues no negatives: What is $\lim_{t\to\infty} e^{tA}$?

Here's a homework question I've been stuck on for a while. My question is what can you tell about $$\lim_{t\rightarrow\infty}e^{tA}$$ if A is $n\times n$ matrix and you know that every eigenvalue of A ...
0
votes
0answers
9 views

LUB property in a predicative logic

Is there a formulation of real analysis in a predicative logical system such that the LUB property is available? Here is a quote "Kleene uses the example of Least upper bound in his discussion of ...
0
votes
1answer
23 views

Proving equivalence of Axiom of Choice

I am working on the following question concerning the axiom of choice and one of its many equivalences. Advice as to whether I am on the right track would be appreciated. As a preface, I have looked ...
0
votes
2answers
21 views

How do you calculate this third eigenvector in this 3x3 matrix?

Scroll down to the bottom if you don't want to read how I arrived at my original two answers. My question is how are all the online calculators I check coming up with this third eigenvector (1, 1, ...
0
votes
2answers
21 views

Selfadjoint Operator: Empty Spectrum

Can a selfadjoint operator have empty spectrum? (As far as I remember, yes; but just to be sure.) The point is that if so then the closure of its spectrum cannot equal the convex hull of its ...
1
vote
1answer
27 views

If $(c_n)_n$ is the sum of geometric and arithmetic sequences. How to get the original sequences back?

If we have a geometric sequence $a= (a_n)_n$ and an arithmetic sequence $b=(b_m)_m$. We can find the $n$th term of $a+b$ easily. Now, suppose we have a sum of geometric sequnce and arithmetic ...
3
votes
3answers
192 views

Why are all non-prime numbers divisible by a prime number?

In Euclid's infinite prime numbers proof, the logic is as follows: Assume a set $S$ of all prime numbers in existence is finite (there are a finite amount of primes) Then there must be a greatest ...
3
votes
1answer
30 views

Consecutive Prime Problem

Consecutive primes whose quotient of their product and sum is itself a prime number. $$ 2 \times 3 \times 5 = 30 $$ $$ 30/10 = 3 $$ $$ 3 \times 5 \times 7 = 105 $$ $$ 105/15 = 7 $$ Question: ...
1
vote
0answers
6 views

Square Integrable local martingale or locally square integrable martingale?

I have a question about martingales. What is the difference between "locally square integrable martingale" and "square integrable local martingale"? In particular, which set does $M_{loc}^2$ ...

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