0
votes
0answers
2 views

Rings and fields: any applications to science and technology?

While there are applications of Group Theory in Particle Physics and Cryptography in security systems, what about another very important part of Abstract Algebra: Rings and Fields? Are there any ...
0
votes
0answers
6 views

Solving linear equations graphically

I have a pair of linear equations.i need to find two points from each of the equation.i have found points which are difficult to plot in the graph.help me to find two points from each of the equation ...
0
votes
0answers
8 views

How many methods are available for finding this volume?

I wonder how many methods are available for finding the volume required by the question. Two spheres (of radii $r$ and $a$, with $r \lt 2a$) meet in such a way that the centre of the one of radius ...
0
votes
0answers
5 views

A question regarding Grothendiek , topos and (adelic??) points

I am having a look at this conference: https://www.youtube.com/watch?v=yNgvvNx_P9w I am particularly interested in getting your feedback on 1:14:30 and the seconds thereafter. Could anyone expain me ...
1
vote
1answer
34 views

First 10 digits after decimal point in the number $(1+\sqrt{3})^{2015}$

The question is how to find first 10 digits after decimal point in the number $(1+\sqrt{3})^{2015}$. I keep running into this kind of problems in a context of symmetric polynomials.
0
votes
1answer
9 views

Proof structure validity: assume (a), (b), show (c). Then Permute.

I am given a collection of sets $\mathcal{E}$ and am trying to prove it is an elementary family. To show $\mathcal{E}$ is an elementary family I must show that it satisfies the following properties: ...
0
votes
0answers
13 views

Real analysis using some key constraints

Let $\alpha>1$ and $M \geq 0$. Suppose $f:\mathbb{R}→\mathbb{R}$ satisfies $|f(x)-f(y)| \leq M|x-y|^\alpha$. Prove that $f$ is a constant function. I tried taking different values of $M$ and ...
1
vote
1answer
30 views

10th derivarive of a function

I want to find $f^{(10)}(0)$ where $f(x)=ln(2+x^2)$. I know that it can be done "by hand", but I believe there is a smarter way. I think I should use Taylor series and the fact that ...
0
votes
0answers
10 views

Stable Marriage algorithms other than Gale-Shapely?

I've looked around lot and I haven't been able to find any algorithms for to the traditional stable marriage problem (I'm not talking about any of its variants like the roommate problem) besides the ...
-2
votes
0answers
6 views

Proving that Changing the Index of the Lower Bound of a Convergent Infinite Series Does Not Affect the Convergence

Would someone help me in proving that the following theorem is true? Let $j$ be a positive integer. Show that $$\sum\limits_{k=0}^\infty a_{k} \quad\textrm{ converges iff }\quad ...
0
votes
1answer
21 views

Engineering Mathematic problem with proving an equation

This is problem 20, further problems in Engineering Mathematics book by K.A.Stroud. It states: Show that the equation \begin{equation} 4\frac{d^2x}{dt^2} + 4\mu\frac{dx}{dt} + m^2x = 0 \end{equation} ...
-8
votes
0answers
8 views

How to use Best SEO tools?

Efficient SEO marketing needs an extensive content marketing funnel and the help content marketers to do the same. One can implement every aspect of the content marketing plan with the highly ...
0
votes
0answers
10 views

Graph Laplacians - self-study

I am self-studying graph laplacians in Kevin Murphy’s book “A probabilistic perspective on machine learning”. I understand that we introduce the vector f to proof that the matrix is positive ...
0
votes
1answer
11 views

Fisher Information: While calculating the expectation of score function, why do we integrate with dx?

Let $$S=\frac{d}{dy} \left[\log f(x|y)\right]$$ where $y$ is the parameter and $S$ is score function Now in text books, ...
1
vote
1answer
23 views

Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in ...
0
votes
0answers
14 views

Values of $x$ for which $\sum [(n^3+1)^\frac{1}{3}-n]x^n$ converges.

For what values of $x$ the infinite series $\sum [(n^3+1)^\frac{1}{3}-n]x^n$ converges?
0
votes
0answers
9 views

How can I use diophantine approximation to find a real number?

I have been told that the following question can be solved using Diophantine approximation, but I cannot find a way to solve it. I have no prior knowledge of Diophantine approximation and so I ...
1
vote
0answers
9 views

Conditions on $f$ to have $ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{f(x)}{(x-y)^2 (y-z)} dz dy dx $ finite?

Suppose that $f$ is a $\mathcal{C}^\infty$ function. $$ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{f(x)}{(x-y)^2 (y-z)} dz dy dx $$ Which are the conditions on $f$ that makes this integral finite ? ...
0
votes
0answers
7 views

calculation using relatedness formula for document relation in ontology?

While I am studying on text mining, I have found the formula below for measuring document relation in ontology, but it looks a bit conceptual to me, if anyone knows how to apply to the formula below, ...
1
vote
1answer
16 views

Induced map on the homology

Although there are good articles about this theme like induced map homology example, I would like to get a more explicit answer. I know that one way to find such a map is the following: $ f:X\to Y ...
6
votes
1answer
52 views

Two numbers that cannot both be squares

I was wondering where to start with the following question: Show for $a,b \in \mathbb{N}$ that $a+b^2$ and $a^2+b$ cannot be both squares. Here $\mathbb{N}$ is the positive integers ($0$ not ...
0
votes
4answers
45 views

how to solve $2^x \bmod 53 ≡ 1$?

I was writing network security exam, one of the question is $2^x \bmod 53 ≡ 1$. This they asked because most of the encryption and decryption algorithms involves modulus calculation
0
votes
1answer
43 views

A conditional probability question

Let A and B two events and if $P(A)=0.5$ and $P(B)=0.4$ what is the $P(B\mid A^C)$?
2
votes
3answers
50 views

Determining if function odd or even

This exercise on the Khan Academy requires you to determine whether the following function is odd or even f(x) = $-5x^5 - 2x - 2x^3$ To answer the question, the instructor goes through the following ...
1
vote
1answer
40 views

Show $\int_E {(f_1 + f_2)d\mu } = \int_E {f_1 d\mu } + \int_E {f_2 d\mu } $

In my textbook, given a measure space $(\Omega,F,\mu)$, the integration for a non-negative $F$ measurable function $f$ on $E$ is defined as $$\int_E f\ \mathsf d\mu = \sup_{0 \le h \le f} I_E\left( h ...
0
votes
0answers
21 views

Why do we need a Borel function in order to use this lemma?

Im trying to understand a proof for differentiably a.e for functions $F$ given by $$F(x)= \int_{-\infty}^{x}f\ \mathsf dt$$ for $f$ Lebesgue measurable and $L^{1}$. He defines a finite Borel measure ...
0
votes
1answer
17 views

Two dimensional Lie Algebra - what do we know without knowing the Bracket?

I am having trouble understanding how Lie algebras act. I.e. if I am trying to work with a two dimensional Lie algebra, there isn't much I can do without knowing the Lie Bracket that is defined on the ...
1
vote
3answers
24 views

Column Space/Row Space

I just have a small question. I was wondering if someone could explain to me the difference between "column space" and "basis for column space" as well as "row space" and "basis for row space". I've ...
0
votes
0answers
6 views

Conditions for the existence of moments of the supremum of a random variable

let $\{x_i\}_{i=1}^\infty$ be a random variable with finite first $n$ moments. Under what conditions (if at all) do the first $n$ moments of the random variable $\sup_i x_i$ (i.e., the supremum of ...
6
votes
4answers
68 views

When $\overline{(a,b)}$ does not equal $[a,b]$

Lee topological manifolds 2.13 c) says For any pair of points a,b in X show that $\overline{(a,b)}\subset[a,b]$. I have done that, but next: Give an example to show that equality need not ...
0
votes
0answers
31 views

What is a good way to Google and obtain Math resources?

I'm having a great deal of difficulty finding math resources on Google and stackoverflow. Is there a scheme to search for MathJax equations? Any good ways to search the web via Wolfram Alpha?
0
votes
0answers
21 views

SVD for square matrix

I already know the concept of SVD applyed on an mxn matrix. Eigen vectors can't exist for a non-square matrix, but singular-vectors can. My question is: does SVD on a square matrix relate to ...
0
votes
0answers
7 views

a problem in gauss lemma

I was reading the Gauss Lemma from the do carmos Rienmannian geometry book which says that Let $p \in M$ and let $v \in T_pM$ such that $\exp _p v$ is defined Let $w\in T_pM$ is identified with ...
3
votes
1answer
19 views

Cancellation law of equal in distribution

I came across this gem while discussing with my friends, If $X$ and $Y$ are two real valued random variables (not necessarily independent) that satisfy $$X =^d X+Y$$ (where $=^d$ means equal in ...
0
votes
0answers
6 views

Prove $K_4-Cover$ is NP-Complete

I'm studying for a computational theory exam, and as part of my studying I'm trying to solve previous years' exams. I have come across this problem and I'm having some difficulty with it: Let $ G ...
0
votes
0answers
11 views

factor graphs - example

I am self-studying graphs - and stumbled upon factor graphs - e.g. as described on https://en.wikipedia.org/wiki/Factor_graph. I have trouble concretizing what the factor vertices represent. Would ...
1
vote
1answer
34 views

are there closed form solution to $n \cdot y + \log(y) = x$?

I am trying to find a closed-form solution to $n \cdot y + \log(y) = x$ How do we deal with the fact that there aren't, if there aren't? Is it possible to rewrite this in a better way as a ...
0
votes
1answer
10 views

The relationship between function space embeddings and their respective inequalities

Let $L^{p,\infty}$ be the weak $L^p$ space consisting of measurable functions $f$ satisfying \begin{equation*} ||f||_{p,\infty}:=\sup_{\rho}\rho\lambda (|f|>\rho)^{\frac{1}{p}}<\infty . ...
0
votes
0answers
4 views

Does $ 10 \otimes 10 = 45_a \oplus 54_s \oplus 1_s ,$ tell us that the elements of $\mathfrak{so}(10)$ acting on $54_s$ are symmetric?

I'm currently not sure what is meant by a symmetric Lie algebra representation. On the one hand it could mean that if we write the basis vectors of the $54_s$ representation as $10 \times 10$ matrices ...
5
votes
0answers
22 views

Closed-forms of the integrals $\int_0^1 K(\sqrt{k})^2 \, dk$, $\int_0^1 E(\sqrt{k})^2 \, dk$ and $\int_0^1 K(\sqrt{k}) E(\sqrt{k}) \, dk$

Let denote $K$ and $E$ the complete elliptic integral of the first and second kind. The integrand $K(\sqrt{k})$ and $E(\sqrt{k})$ has a closed-form antiderivative in term of $K(\sqrt{k})$ and ...
-3
votes
0answers
22 views

How to choose starting points in bisection method?

How to choose starting point in bisection method for example given this question $$f(x) = x^3 - x - 1$$ find root by bisection method... In this question initial points not given, how to do I find ...
-1
votes
2answers
32 views

An equation involving fractional expressions

(3/x + 5/x+2) = 2 How come when I solve this equation by setting the RHS of the equation to zero I receive x = 1, -3 AND when I ...
5
votes
1answer
39 views

Problem in Banach Fixed Point Theorem for a functional equation

I was recently presented this within the context of topological spaces: I am asked to show that there exists a unique continuous function $ f\colon \left[0,\frac{1}{2}\right] \rightarrow \Bbb R $ ...
0
votes
0answers
8 views

size of input of algorithm according to Moore's law

I am reading about algorithms in a book tilted Algorithms by Sanjay DasGupta. Here author mentione as below In 1965, computer chip pioneer Gordon E. Moore noticed that transistor density in ...
1
vote
2answers
53 views

Determinant of M [on hold]

How to find the determinant of the $n\times n$ matrix $M$, whose all the entries are zero except 1st row, 1st column and diagonal entries: $$M= \begin{bmatrix} -x & a_2 & a_3 & \cdots ...
0
votes
1answer
7 views

Approximation by $\mbox{Im }(t-z)^{-1}$ with $\mbox{Im } z > \epsilon$

It is a standard fact of harmonic analysis that the span of the functions $$g_z(t) = \mbox{Im } (t-z)^{-1},$$ ranging over all $z \in \mathbb{C}$ with $\mbox{Im } z > 0$, is dense in ...
0
votes
0answers
26 views

how to construct system matrix A , given only eigen values [on hold]

how to construct system matrix A if only eigenvalues are given as follows 0,2 and 4 ?
0
votes
0answers
11 views

Hyperbolic equations with time dependent coefficients associated with the time derivatives

I'm concerned with evolution equations (of second order) and am hoping for some literature hints regarding a special situation. The equations I'm working with basically look like (complemented with ...
0
votes
4answers
103 views

Which number is bigger?

Which number is bigger? $1.01^{101}$ or $2$? and how about $e^{\pi}$ or $\pi^e$? Tried some algebraic manipulations to no end, so would love some suggestions or some different ways to approach those ...
1
vote
0answers
9 views

Diffusion equation in polar coordinates with non-zero boundary conditions (BC)

I'm trying to solve the diffusion equation in polar coordinates: $$c_t = \frac{D}{r^2}[2r\,c_r + r^2\,c_{rr}] = \frac{D}{r}[2\,c_r + r\,c_{rr}] \tag{1}$$ with the following BC: $$c(0,t)=0, \quad ...

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