0
votes
0answers
2 views

Differential geometry qsn

Find the curvature and torsion of the curves given by i. r=(a(3u-u^3),3au^2,a(3u+u^2)) ii. r=a(1+cosu), a sin u,2asin(1/2)u)
1
vote
0answers
7 views

How does u^Tv = p dot ||u|| follow from the projection onto line?

Before anybody asks, this is not a homework question. I just saw the formula given in Andrew Ng's Coursera course in the SVM section. For reference: the projection formula is proj_w_(p) = p((p dot w) ...
2
votes
0answers
7 views

Help in step of the proof of Burnside's $p^aq^b$ theorem (Doerk-Hawkes book)

I'm reading the proof of the $p^aq^b$ Burnside's theorem from the book Finite soluble groups by Doerk and Hawkes. The fifth step of the proof says 2.5. Let $M$ and $H$ be maximal subgroups of $G$ ...
1
vote
0answers
5 views

How to show the sum of the images of such $m$ projections is direct and is the whole space?

There are $m$ projections (whose square are themselves) $\phi_1,\cdots,\phi_m$ acting on a finite-dimensional vector space $V$ such that $$\phi_i\phi_j=0\quad i\ne j\tag{1}$$ where $0$ denotes the ...
1
vote
0answers
9 views

Property of hermitian matrices (eigen values)

In a paper I have read the following ${\bf G}$ is a Hermitian matrix, that 1) ${\bf G}$ is diagonalizable 2) the singluar values are same as the eigen values Is number 2 correct? I cant seem to ...
1
vote
0answers
7 views

Ratio between subsegments of space diagonal of a cube

Let $ABCDEFGH$ be a cube where $ABCD$ is the ground face and $E$ is about $A$, $F$ above $B$, and so on. Consider the space diagonal $AG$, and call $P$ the orthogonal projection of $B$ on $AG$. The ...
1
vote
1answer
16 views

Difference between convergence in measure and convergence almost everywhere

This question is an extension of a question asked earlier. Let $(X,\mathcal{M},\mu)$ be a measure space and let $f_{n}: X \to Y$, where $\{f_{n}\}$ is a sequence of functions. The proof wiki ...
1
vote
1answer
28 views

How to prove the sum of squares larger than 1/n without induction? [duplicate]

known that: $1\geq R_1 \geq R_2 \geq \dots \geq R_n \geq 0$ and $\sum_{i=1}^n R_i=1$ To prove: $\sum_{i=1}^n R_i^2 \geq \frac{1}{n}$ Using induction, the problem can be easily proved. I'd like to ...
1
vote
2answers
31 views

Error in proving inequality $1 - x \leq e^{-x}$

Fact states as following, $$1 - x \leq e^{-x}$$ This is how I try to prove it: \begin{align*} \ln (1 - x) &\leq \ln (e^{-x})\\ \ln 1/ \ln x &\leq -x\\ \ln 1 &\leq -x \times \ln x ...
-1
votes
1answer
10 views

Floor and Ceiling Series (I)

Prove or disprove: 1) $$\sum_{n=2}^{\infty}{\frac{1}{\lfloor n^2/2 \rfloor}} = \frac{1+\zeta(2)}{2}$$ 2) $$\sum_{n=1}^{\infty}{\frac{1}{\lceil\ n^2/2 \rceil}} = \frac{\zeta(2)}{2} + \frac{\pi}{2} ...
2
votes
1answer
18 views

Does the set of all piecewise constant functions form a subspace of the vector space $\mathbb{R}^\mathbb{R}$ over $\mathbb{R}$?

A function $f\in \mathbb{R}^\mathbb{R}$ is piecewise constant if and only if it is a constant function $x\to c$ or there exist $a_1<a_2<\cdots<a_n$ and $c_0,...,c_n$ in $\mathbb{R}$ such that ...
0
votes
2answers
17 views

Understanding the connection of the roots of an irreducible polynomial and a basis for field extensions

Let $\alpha,\beta,\gamma \in E$ be the roots of an IRREDUCIBLE polynomial $p(x)\in Q[x]$ (where E/Q is an extension field. Can I use these roots to construct a basis for E over Q? Why?
3
votes
0answers
18 views

For finitely generated free abelian groups $A,B$ if there is an onto homomorphism $A \to B$, then $\operatorname{rank}(A) \geq \operatorname{rank}(B)$

$\newcommand{\rank}{\operatorname{rank}}$For two finitely generated, free abelian groups $A,B$ prove that if there is an onto homomorphism $A \rightarrow B$, then $\rank(A) \geq \rank(B)$ Assume that ...
0
votes
0answers
9 views

Solving for a 3D point in a 5D graph given 3 pairs of 2D points.

I am attempting to solve the values $C$, $D$, and $S$, given three pairs of $[M,R]$. $$R = \frac {M}{C - MDC + DC\left(MS\right)^2}$$ I have been able to solve for a related equation (or rather, ...
0
votes
0answers
9 views

A generalization of simple random walk

Suppose $S_n, n\geq 0$ is a martingale on $\mathbb{R}$ such that $S_0=0$ and $|S_{n+1}-S_{n}|\in [\frac{1}{2}, 1]$. Prove that there exists $c,C>0$ s.t. $$ \frac{c}{\sqrt{n}} \leq P( S_1\geq ...
1
vote
0answers
7 views

Find the linear maps that preserve this bilinear form.

This time I want to know which linear maps are the ones that preserve the following bilinear form: $$\beta(x,y)=2(x_1y_1+x_2y_2)$$ and they give as a hint the matrix: $$\begin{pmatrix} ...
1
vote
1answer
17 views

Determining if a sum of trig function is periodic

$$2 + \sin(2\pi\cdot t) + 3\cos(3\pi\cdot t) - 5\sin(7t-\frac{\pi}{4})$$ Is there any manual, easy, way of knowing such a function is not periodic? I'd love to know if there's any method which ...
2
votes
1answer
22 views

Proving that the sum of this series is even…

The question is- Let $n$ be any natural number. Now, it is required to prove $$\left[\frac n1\right]+ \left[\frac n2\right] + \left[\frac n3\right]+\cdots+\left[\frac nn\right]+\left[\sqrt n\right]$$ ...
0
votes
0answers
12 views

Boolean Algebra Karnaugh Maps

I'm having trouble solving this: Simplify the expression F = W'X'Y'Z' + W'X'YZ' + WX'Y'Z' + WX'YZ' + WXYZ + W'XYZ using a Karnaugh Map. The book I have very poorly describes how to do Karnaugh Maps. ...
2
votes
3answers
16 views

Linear independence for a set of real valued continuous functions

Let $V$ be the vector space of all real valued continuous functions. Is the following set $\{\cos t, \sin t, \mathrm{e}^t\}$ linearly independent? I usually understand what and how to determine ...
2
votes
0answers
8 views

Can this be proved using the MCT instead of the DCT?

I've seen various version of the DCT prove that if f is a real valued, or extended real valued, or complex, integrable function, and if $\{E_n\}_n$ is a sequence of disjoint measureable subsets, then: ...
2
votes
0answers
9 views

Smallest witness for checking the primality of a number

In this link https://primes.utm.edu/prove/prove2_3.html it is stated that the smallest witness for a composite number is always less than $2ln(n)^2$ , assuming the extended Riemann-hypothesis. ...
1
vote
0answers
9 views

Pyramid with unit sides inside a cube

Let $ABCDEFGH$ be a unit cube with base $ABCD$. Let $P$ be the top of the pyramid with base $ABCD$ and all edges of length $1$. One has a standard 2-dimensional projection of this cube on the back ...
0
votes
0answers
4 views

Find a recursive definition for inorder: binary Tree(T) → list(T ) where inorder(T ) is the list of nodes from an inorder traversal of T .

Find a recursive definition for inorder: binary Tree(T) → list(T ) where inorder(T ) is the list of nodes from an inorder traversal of T . I have no idea what this question is even asking me. What ...
1
vote
0answers
11 views

Infinite Bessel function sum

Let $$f(x)=\sum_{p=0}^\infty J_p(x)$$ We have that $f(0)=1$ and a Bessel function like curve. Can $f(x)$ be expressed in terms of known functions? $$f(x)=1-\sum_{n=0}^\infty ...
2
votes
1answer
27 views

Why does the a*c cheat work when factoring trinomials?

When factoring a trinomial, in the form $ax^2 + bx + c$, I am told that one can multiply $a$ and $c$ which gives a product whose factors add to $b$. So if I have $2x^2 + 5x -3$ that gives me $-6$. ...
1
vote
0answers
12 views

Let $V_1,V_2$ be subspaces of $V$. If $\dim(V_1+V_2)=\dim(V_1 \cap V_2) + 1$ then prove that $ V1 \subseteq V2 $ or $V2 \subseteq V1$.

We know that $\dim(V_1+V_2)=\dim(V_1)+\dim(V_2)-\dim(V_1 \cap V_2)$, But $\dim(V_1+V2)=\dim(V1 \cap V2)+ 1$ (given), Now if we assume that $V_1 \subseteq V_2$ , $V_1 \cap V_2= V_1$, Then at one side ...
0
votes
0answers
4 views

How to create a ring in MAGMA with relations?

I'm using MAGMA221 and would like to create a ring over $GF(2)$ with respect to a list of relations. Here's what I have so far: $\mathtt{Z:=GF(2);} \\\mathtt{P<x,y,z>:=PolynomialRing(Z,3);}$ ...
0
votes
0answers
9 views

Compact set in the interior of a cone

Suppose compact set $S \subseteq R^n$ is in the interior of $x_0+C$, where $C$ denotes a solid convex cone in $R^n$ with the vertex at $0$. I am trying to prove that $\exists r>0$ such that $$S ...
-1
votes
0answers
12 views

Moving object position

An object is moving with 20 points p/s. Currently the object is at position x: 30, y: 50, z: 90. The object is moving to x: 4^6, y: 4^8 z: 9. What are the coordinates after 25 minutes? This is what ...
0
votes
0answers
9 views

A variation on a problem of Polya and Szego

Among the various propositions on real series and sequences in "Problems and Theorems od Analysis I" Pt. I Chap. 4 by Polya and Szego, I noted n.178 at page 39 which implies what follows. Let ...
1
vote
3answers
12 views

Finding a coordinate over a right angle in a triangle where the other two coordinates are known

a B ------- C \ | \ | \ | c \ | b \ | \ | \| A Alright, this is a triangle I have, and ...
0
votes
2answers
22 views

Number 9 and age of mother when child is born.

If a mother's age is divisible by 9 when a child is born then once you go to the next decade,n every 11 years the child's age and mother's age are always the same two numbers in reverse order. For ...
3
votes
0answers
7 views

Infinitely iterated square roots in groups

Let $G$ be a group. What are possible conditions on $G$ to ensure that there is no sequence $\{g_i\}_{i\in\mathbb Z}\subset G\backslash\{1\}$ such that $g_{i+1}=g_i^2$ for all $i\in\mathbb Z$? Does ...
-4
votes
1answer
30 views

Find all natural numbers *a*, that satisfy the following:

Find all natural numbers a for which $$ \frac{a^4+4}{17} $$ is prime.
0
votes
0answers
12 views

Conditional distribution of mixed process

$$N(t)\sim (1-p) \operatorname{Poiss}(\lambda_0 t)+p\operatorname{Poiss}(\lambda_1 t)$$ is a mixture of two Poisson processes. As we know that Poisson mixtures don't have the independent increments ...
-4
votes
0answers
21 views

If $F(x) = \frac{1}{2}x(x+1)$, evaluate the following

If $F(x)=\frac{1}{2}x(x+1)$, evaluate the following: $F(1)$ $F(2)$ $F(3)$ $F(x-1)$ $F(5)-F(4)$ $F(7)-F(6)$ $F(x+1)$ $F(x)-F(x-1)$ $F(x^2)$
1
vote
1answer
11 views

The weak topology on an infinite dimensional linear space is not first-countable

I thought I needed help proving the above statement, but during typing I found a proof. Since I had already written it all down I will post it anyway, maybe in the future someone can benefit from it. ...
-1
votes
1answer
18 views

HELP with Equation using quadratic equation

$(2-y)^4=3(2-y)^2+1$ The answer is suppose to be $y=4\pm \sqrt{6+\frac{13}2}$. I have tried to work this problem out but I cannot get the answer that is in the book. Please help!!!!!
0
votes
1answer
13 views

Show $\mathcal N_x$ is neighbourhood system on X

I had following exercise : If we have a topological space $(X,\tau) ~~$and x$ \in X$ ,and suppose there is a family of sets defined as: $ \mathcal N_x=\{N_x;N_x\supset O_x \ni x,$for ...
0
votes
1answer
13 views

a bijection is an injective (one-to-one) , surjective (onto) map between sets. if S = (0, 1) and T =R, find a map from S to T which is

a bijection is an injective (one-to-one) , surjective (onto) map between sets. if S = (0, 1) and T =R, find a map from S to T which is my effort 1) (a) f(x) = x is a one to one function but it ...
-2
votes
0answers
17 views

PDF of $Y=\min(0,X)$ when PDF of $X$ is $\frac34(1-x^2)$ on $(-1,1)$

Let $X$ be a random variable with density $f(x) = (3/4) (1-x^2).$ Range is $-1 < x < 1.$ I have to find probability distribution of $Y = \min(0,X).$ I know that distribution function could be ...
2
votes
2answers
38 views

What is a primitive root?

So I'm trying to learn about RSA and have come across various subtopics, including the discrete logarithm problem. This mentions primitive roots, which I do not understand. Essentially all I want ...
0
votes
0answers
7 views

Fourier sine and cosine transform

Why e raise to power x is not defined for Fourier sine and cosine transform I read in some book that it is not defined for bounded region. But I could not understand the logic behind it.
4
votes
0answers
18 views

For all Dirichlet series, is $a_n$ unique to $f(s)$?

For any Dirichlet series, $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ is the function, $a_n$, always unique to $f(s)$? In other words, is it possible to show that $a_n$ is the only function that will ...
1
vote
2answers
20 views

Limit of an integral of a continuous real-valued function

If $f:[0,{\infty})\to\mathbb R$ continuous and $\lim_{x\to\infty} f(x)=a$. Show that: $$ \lim_{x\to\infty} \frac1x\int_{0}^{x} f(t)\ \mathsf dt = a. $$ If: $$ \lim_{x\to\infty} \frac1x ...
0
votes
0answers
8 views

Problem with initial values ODE

EQ = $y'+2xy$ Initial Value=$y(0)=-2$ $y'+2xy$ = $y'+y = \frac{1}{2}$ The solution of the Diff Equation $\frac{1}{e^x}$ $\int{\frac{1}{2}}e^xdx$ = $\frac{1}{2}+c$ I wonder how to check if this ...
0
votes
1answer
16 views

Is there any inner product on $M_{n \times n}$ inducing this norm?

The set $M_{n \times n}$ is the collection of all $n \times n$ matrices over $\mathbb{R}$. Definition: $\|A\|_2=Sup_{\|u\|_2=1} \|Au\|_2$. Is there any inner product on $M_{n \times n}$ inducing ...
1
vote
2answers
22 views

$y'=\frac{y^2}{2x(y-x)}$

I'm trying to solve the following differential equation: $$y'=\frac{y^2}{2x(y-x)}$$ It is supposed to have a relatively easy general solution, but I can't find it. I've tried several things, the ...
0
votes
1answer
9 views

Showing one-to-one and onto

$\alpha$: $\mathbb{Z} \times \mathbb{Z}^{+} \rightarrow \mathbb{Q}$ defined by $\alpha(n,m)=\frac{n}{m}$ Is this one to one? Is this onto? I know that if $\alpha$ is one to one I must show ...

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