1
vote
2answers
36 views

Every euclidean space ($\mathbb {R}^n$) is complete.

To prove this, I would like to use induction. For $n=1$ it is easy to prove that $\mathbb{R}$ is complete. For $n=k$ we assume it is true. For $n=k+1$, we have to show that $\mathbb {R}^{k+1}$ is ...
0
votes
0answers
2 views

Definite integral involving root of polynomial

While studying models for fluid dynamics I stumbled upon the following integral: $$\int_0^R2r (1-(\frac{r}{R})^2)^\frac{1}{n}\,dr=R^2\frac{n}{n+1}$$ I would like to prove this relation but am having ...
1
vote
0answers
13 views

Suppose that there is a finite $c$ such that $\int_0^1 | f(a+t) - f(b+t) | dt \le c$ for all $a$ and $b$. Show that $f \in L(0, 1)$.

Please help me understand the following proof. Q) Let $f$ be measurable and periodic with period $1$, that is, $f(t+1)=f(t)$. Suppose that there is a finite $c$ such that $$\int_0^1 | f(a+t) - ...
-2
votes
0answers
22 views

Sum of digits of all positive integers from 1 to 2016

How to find the sum of digits of all positive integers from 1 to 2016?
0
votes
0answers
5 views

p and q be the roots of the polynomial mx^2 + x(2-m) + 3.

Let p and q be the roots of the polynomial mx^2 + x(2-m) + 3. Let m1 ,m2 be two values of satisfying p/q + q/p= 2/3. determine numerical value of m1/(m2)^2
0
votes
0answers
4 views

Problem with change of basis of an polynomial.

Good morning, i have a problem solving this: Express $a_{0}+a_{1}x+a_{2}x^{2}$ in terms of basis: $1,x-1,x^{2}-1$ I make this: ...
2
votes
2answers
20 views

Symbol to show that implication is one-sided

Sometimes A => B is true, but B => A is not true and this fact is important and not obvious. Is there a short symbol to write it in order not to write "A => B and not B => A"
0
votes
0answers
3 views

How to solve a nonlinear recursion relation?

Given the following recursion relation \begin{equation} E^{(n)}=(E^{(n-1)}-\alpha_1)\,e^{-\alpha_2\,(\alpha_3E^{(n-1)}+b)} \end{equation} where $\alpha_i$'s and $b$ are some constants. I am trying ...
0
votes
1answer
18 views

How does $f: X\rightarrow W$ in the Theorem 6 satisfy Def. 8(b) “If $(x, y) \in f$ and $(x, z) \in f$, then y=z”?

Definition 8. Let X and Y be sets. A function from X to Y is a triple (f, X, Y), where f is a relation from X to Y satisfying (a) Dom(f) = X. (b) If (x, y)$\in f$ and (x, z) $\in f$, then y=z. ...
2
votes
2answers
38 views

Find the last two digits of $2^{2156789}$

Find the last two digits of $2^{2156789}$. My try: As $2156789 = 4* 539197 + 1$ The unit digit of $2^{2156789}$ is similar to the unit digit of $2^{4n+1}$ which is equal to 2. But I'm unable to find ...
2
votes
1answer
35 views
+50

Creating arithmetic expression equal to 1000 using exactly eight 8's and parentheses

I would like to find all the expressions that can be created using nothing but arithmetic operators, exactly eight $8$'s, and parentheses. Here are the seven solutions I've found (on the Internet) so ...
3
votes
3answers
13 views

Fixed points and cardinal exponentiation

Let the function $F: On \rightarrow On$ be defined by the following recursion: $F(0) = \aleph_0$ $F(\alpha+1) = 2^{F(\alpha)}$ (cardinal exponentiation) $F(\lambda) = \sup\{F(\alpha):\alpha \in ...
5
votes
1answer
37 views

The complete solution to a system of polynomials over $\mathbb{R}$

If I am solving a positive-dimensional system of polynomials over $\mathbb{R}$, and specifically am searching only for real solutions, how do I know that my solution is complete and there are no other ...
2
votes
1answer
30 views

Understanding algebraic operations on vector bundles

Let $X$ be a smooth manifold with vector bundles $V$ and $W$. I'm trying to understand how we can construct in general a vector bundle $V \otimes W$. In particular what manifold structure do we give ...
0
votes
0answers
23 views

Determining whether the extremal problem has a weak minimum or strong minimum or both

The extremal of the functional $\int_{0}^{\alpha}{\left((y')^2 - y^2\right)dx}$ that passes through (0,0) and (${\alpha}$,0) has a weak minimum if ${\alpha}$ < $\pi$ strong minimum if ${\alpha}$ ...
6
votes
1answer
22 views

$\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ - Theory of distribution

An exercise asks to find the wave trace $w(t)=\operatorname{tr} \left(e^{it \sqrt\Delta}\right)=\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ as a distribution (or generalized function) of the Laplacian ...
1
vote
1answer
72 views
+50

Where am i going wrong in solving this equation?

Fing the least value of $a$ for which $f(x)$ is increasing, where $$f(x)=2e^x-ae^{-x}+(2a+1)x-3$$ What i tried for increasing $f'(x)\ge 0, \forall x\in \mathbb R$. So ...
0
votes
1answer
15 views

Characterization of linear normality

A projective variety $X\subseteq\mathbb{P}^{n}$ is said to be linearly normal if the restriction map $$ H^{0}(\mathbb{P}^{n}, \mathcal{O}_{\mathbb{P}^{n}}(1))\rightarrow H^{0}(X,\mathcal{O}_{X}(1)) ...
0
votes
0answers
10 views

Reference Request-Essential Extension

Let $R$ be a commutative ring with unit. Assume $R$ is an essential extension of each of its non-zero ideals. I feel that there should be something in the literature about this, but I could not find ...
0
votes
0answers
5 views

Explicit heat kernels

For quite general domains, the Dirichlet heat kernel has a representation via the eigenfunctions of the corresponding Dirichlet problem. This form is usually not easy to analyse so I was wondering - ...
2
votes
3answers
54 views

Solving for $x$ in $\tan(3x) \tan (2x)= 1$

If $$\tan(3x) \tan(2x)= 1$$ Then $x$ is equal to Attempt: I used the '$\tan$' identity but it showed no results. The identity: $$\frac{\tan(2x)+\tan(3x)}{1-\tan(2x)\tan(3x)}$$
0
votes
0answers
4 views

Identifying some properties of a set

Let $S \subset R^2$ be defined by $S$ = {$(m+ \frac{1}{4^{|p|}} , n+ \frac{1}{4^{|q|}}): m,n,p,q \in Z$} Then, $S$ is discrete in $R^2$. The set of limit points of $S$ is the set {$(m,n) : m,n \in ...
0
votes
1answer
16 views

example positive linear operator range not closed

I'm trying to understand what closed range means for linear operators. Let $\mathcal{H}$ be a real Hilbert space. Consider a linear operator $L\in\mathscr{B}(\mathcal{H},\mathcal{H})$ ($L$ is not ...
1
vote
1answer
32 views

Number of solutions to $x+y+z = n$

If $\beta(n)$ is the number of triples $(x, y, z)$ such that $x + y + z = n$ and $0 \le z \le y \le x$, find $\beta(n)$. Attempt: I think there are many cases to look at to find $\beta(n)$. We ...
1
vote
0answers
7 views

Non-smooth curve in $\mathbb{A}^2$

In one of my exercises on Algebraic Geometry, I showed that the curve $X \subset \mathbb{A}^2$ defined by $x^3-y^2$ is irreducible but not smooth. Furthermore, they ask the following question that I ...
1
vote
1answer
566 views

Binomial Distribution for defects

I'm stuck on the following problem: A batch of components has arrived at a distributor. The batch can be characterized only if the proportion of defective components is at most 0.10. ...
1
vote
1answer
38 views

How is this possible?

Today, at a physics lecture there was a problem: $a = 20x + 5$ where a was the accelerationas a function of position($x$) and we were to solve for velocity. The professor did the following: $dv/dt = ...
10
votes
1answer
55 views

How to evaluate $\int_0^1\frac{\ln(1-2t+2t^2)}{t}dt$?

The question starts with: $$\int_0^1\frac{-2t^2+t}{-t^2+t}\ln(1-2t+2t^2)dt\text{ = ?}$$ My attempt is as follows: $$\int_0^1\frac{-2t^2+t}{-t^2+t}\ln(1-2t+2t^2)dt$$ ...
3
votes
0answers
17 views

Can we somehow use the functor $\mathbf{Set}(\mathbb{N},-)$ to define $\mathbb{N}$?

Hom functors can be used define coproducts in terms of products. In particular: $$\mathbf{Set}(A \sqcup B,X) \cong \mathbf{Set}(A,X) \times \mathbf{Set}(B,X)$$ To oversimplify a little: "a function ...
0
votes
0answers
7 views

Describe all extension groups of a given subgroup $H \trianglelefteq$ Aff$\mathbb{(F_q)}$ by Aff$\mathbb{(F_q)}/H$

Let $\mathbb{F_q}$ be a finite field. Consider the group Aff$\mathbb{(F_q)}$ Aff$\mathbb{(F_q)} := $ $ \ \begin{Bmatrix} \begin{pmatrix} a&b\\ 0&1\\ \end{pmatrix} \colon a, b \in ...
1
vote
1answer
24 views

Pack balls with maximum sum of radii

We pack $8$ balls into a cube of side length $1$ so that no two balls share an interior point. what is the maximum sum of the radii of the balls? It is possible to pack $8$ balls or radius $1/4$, ...
1
vote
3answers
59 views
+50

What is the differential of the quotient map?

We can view the projective space $P(\mathbb R^n)$ as the quotient of $S^n/\sim$ where $x \sim y$ if and only if $x = -y$. The quotient map $q: S^n \to P(\mathbb R^n)$ is the map $x \mapsto [x]$ ...
1
vote
2answers
55 views

Find $\sum\limits_{r=0}^n(-1)^r\binom{n}{r}^{-1}$ for $n$ even

If $n$ is an even natural number, then find $$\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$$ I tried to solve the question using conventional method, by trying to use calculus, but I ...
0
votes
0answers
11 views

Conjecture about Rabin-Miller pseudo prime test

At oeis A014233 a method for exact testing of primes up to a level using Rabin-Miller is described. I have used this method together with squeezed prime tables to test the Rabin-Miller pseudo prime ...
5
votes
3answers
226 views

How many 6 digit numbers are possible with no digit appearing more than thrice?

How many 6 digit numbers are possible with at most three digits repeated? My attempt: The possibilities are: A)(3,2,1) One set of three repeated digit, another set of two repeated digit and ...
0
votes
2answers
62 views

Why we not check conditions while solving questions?

Note:Down ward problem is just an example to express my question(I know the both solution of problem are insufficient but the first solution is in my 10+2 book and second one is mine which is ...
2
votes
1answer
12 views

Given the splitting field of a polynomial, how can I show that there are three intermediate extensions which aren't normal?

So $f = x^3 +9x -2$, and $E$ is its splitting field. I need to show that there are exactly three intermediate field extensions $K$ such that $\mathbb{Q} \subset K$ is not normal. By Descartes' rule ...
0
votes
1answer
67 views

recursive generating functions

\begin{align} f(0) & = 1 \\ f(1) & = 1 \\ f(2) & = 2 \\ f(2n) & = f(n)+f(n+1), \;\;\;n\gt1 \\ f(2n+1) & = f(n-1)+f(n), \;\;\;n\ge1 \\ \end{align} I am trying to figure out ...
0
votes
1answer
55 views

calculus of finite differences

If D,E,$\delta,\mu$ be the operators with usual meaning and if hD=U, where h is the interval of differencing,How to prove the following relations between operators:- ...
0
votes
0answers
5 views

I want to calulate the range of an operator $S$ which maps $L^{2}$ into $H^{1}(\Omega)$?

I have an equation $s(\lambda,\mu)=l(\mu)$, where s(.,.) is a symetric positive definite bilinear form in $L^{2}(\Omega)$, and $l(.)$ is in $H^{1}(\Omega)$. I want to show that the range of the ...
1
vote
0answers
45 views

$(C_2 )^3$ is not a subgroup of $S_4$

Prove $(C_2)^3$ is not a subgroup of $S_4$. (Using group actions.) I could think of a permutation argument that $(C_2)^3$ is not a subgroup of $S_4$. But I would like to argue it by considering ...
9
votes
2answers
126 views

Prove that $\sin x \cdot \sin (2x) \cdot \sin(3x) < \tfrac{9}{16}$ for all $x$

Prove that $$ \sin (x) \cdot \sin (2x) \cdot \sin(3x) < \dfrac{9}{16} \quad \forall \ x \in \mathbb{R}$$ I thought about using derivatives, but it would be too lengthy. Any help will be ...
0
votes
0answers
11 views

Show that $K(X)$ is not a field for $X = Z(xy)$

Let $X = Z(xy) \subset \mathbb{A}^2$. I want to show that the rational fuctions on $X$ defined as $ K(X) := \{(U,f) : U ⊂ X \text{ is open and dense and } f \in \mathcal{O}_X(U)\}/ \sim$, where ...
-1
votes
1answer
31 views

Find $a$ if $\int\frac{da}{a\sqrt{s-a^2}}=\int dx$

How to solve $\int\frac{da}{a\sqrt{s-a^2}}=\int dx$ if $a$ is a function of $x$ I solved it for $s=1$, obtained $\frac{1}{\cosh(x)}$, now I need only for $s=-1$
0
votes
2answers
12 views

why this part of a statement is not in the hypothesis

The question basically is For each of the following problems, identify the hypothesis (what you can assume is true) and the conclusion (what you are trying to show is true). Let $f(x) = ...
1
vote
0answers
12 views

Calculate minimal Variance

My task is to calculate the minimal variance. I got a result, but don't know for sure if it's correct. Maybe some of you could help me out here. Let $X$ be some real-valued random variable. We know ...
2
votes
2answers
54 views

Curve sketching without a computer program

How to sketch the curve x^6 + y^6 = (x^4)*y without using a computer program ? Could someone give me the step by step ?
1
vote
0answers
11 views

An m-dimensional space with each 'point' in the space having an n-dimensional value

Say I have an $m$-dimensional space (continuous or discrete) such that every point in that space has a value, and that value is an $n$-dimensional vector (continuous or discrete). My question is how ...
0
votes
1answer
15 views

Convex set in a vector space gives a norm

Given an $\mathbb{R}$ or $\mathbb{C}$ vector space $X$ and a function $p:X\rightarrow[0,\infty)$ with $p(x)=0$ iff $x=0$ and $p(\alpha x)=|\alpha|p(x)$ for all $x,\alpha$, I want to show that $p$ is a ...
6
votes
2answers
382 views
+50

A proviso in l'Hospital's rule

L'Hospital's Rule states that $$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$ can be applied when: (1) $f$, $g$ are differentiable; (2) $g'(x) \neq 0$ for $x$ near $a$ (except ...

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