3
votes
4answers
154 views

Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$

Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$ $\bf{My\; Try}::$ Let $\displaystyle I = \displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$, Now let $\displaystyle \tan ...
0
votes
1answer
24 views

evaluating the limit $\lim_{x\to 0^+} tan (2x + π/2)$.

Find : $\lim_{x\to 0^+} tan (2x + π/2)$ By looking at the graph of the function : $f$ goes to $-∞$ if x approaches 0 from right and $f$ goes to $+∞$ if x approaches 0 from left But I am ...
0
votes
1answer
36 views

Question about logic simplification

If I had to simplify ~(p^q)v~(~p^q) so first I would distribute and get: (~pV~q) v (pv~q) After distributing can I drop the parentheses and combine? What would be the next step in simplifying?
1
vote
1answer
42 views

$\operatorname{left.fin.dim}(A)=0$ if and only if $\operatorname{soc}(A_A)$ contains an isomorphic copy of every simple right $A$-module

I've been trying to find an (easy) example to show that there exists an Artin algebra $A$ such that $\operatorname{right.fin.dim}(A)\neq\operatorname{left.fin.dim}(A)$, where ...
0
votes
1answer
38 views

Having trouble proving this integral is infinite

I am working on an assignment and I have to prove that $$\frac{x^2-y^2}{(x^2+y^2)^2} \notin L_1\mathbb{R}^2)$$ to justify why Fubini's Theorem does not apply. I figured that the best way to do this ...
0
votes
1answer
108 views

Finding different eigenvectors

When I'm in a hurry (such as on an exam), I let MATLAB or my TI-89 compute eigenvectors from a square matrix. This is all good and well, except for the times I want to use the eigenvectors for forming ...
1
vote
1answer
112 views

Making Integral Stationary

We want to write and solve the Euler equation to make the integral stationary. let $ I = \int_{x_1}^{x_2} e^x\sqrt {1+y'^2} $. so $ F(x,y,y') = e^x\sqrt {1+y'^2} $. we find $ F_{y'} = ...
1
vote
1answer
49 views

Find a relationship between $f(f^{-1}(f(A)))$ and $f(A)$. Prove it in the general.

I assert that the relationship between $f(f^{-1}(f(A)))$ and $f(A)$ is $f(f^{-1}(f(A))) = f(A)$. Here is my proof: RHS: Let $y \in f(A)$. Then there exists an $x$ in $A$ such that $f(x) = y$. $f(x) ...
0
votes
1answer
522 views

Absolute Value Equivalence relation inequality Question

I'm having trouble understanding what exactly to do to see if the following relation is symmetric and transitive. I've already determined that it is reflexive. Could someone please help me? For $a, b ...
2
votes
2answers
102 views

Integration Question: Completing the Square/Trig Sub yields a different answer than integral table.

After the completing the square, $$\int \frac{dx}{x^2 + 2x - 3}$$ becomes, $$ \int \frac{dx}{(x+1)^2 - 4}$$ The integral table in my book says the antiderivative is, $$\frac{1}{2a} ln \, ...
1
vote
1answer
53 views

Galois group of reducible polynomial

I want to find Gaolois group of $(x^3-x+1)(x^2+1)$ over $ \mathbb Q$. The polynomial of degree third is irreducible and has discriminant $-23$ so it's Galois group is $S_3$. Galois group of the other ...
1
vote
3answers
211 views

Prove or disprove that $H = \{n \in \mathbb{Z}\,|\, n\, \text{is divisible by $8$ and $10$}\}$ is a subgroup of $\mathbb{Z}$.

Is my reasoning correct: since the integers divisible by $8$ and $10$ are only $1$ and $2$, can I say that $H=\{n \in \mathbb{Z}\,|\, n = 1, 2\}$. Now if $H$ is going to be a subgroup then it must be ...
1
vote
2answers
115 views

Sequence of Measurable Functions (Unsigned and Complex-Valued)

I am having several difficulties in solving the following problem about measurable functions: Let $(X,\mathcal{B})$ be a measurable space. If $f_n : X \to [0,+\infty]$ are a sequence of measurable ...
1
vote
2answers
84 views

Finding minimum on graph for given domain

So I want to find what is the minimum value of a graph on a certain domain. For example, for $y=x^2+1$ between $x=-3$ and $2$, the minimum value is at 1 at x=0. I think I know how to find minimums ...
2
votes
2answers
291 views

Prove that there exist two integers such that i - j is divisible by n.

Here's the full question: Prove that, for any $n + 1$ integers, $\{x_0, x_1, x_2, . . . , x_n\}$, there exist two integers $x_i$ and $x_j$ with $i \neq j$ such that $x_i − x_j$ is divisible by $n$. ...
0
votes
3answers
26 views

The relation between hyperbolic sine and hyperbolic cotangent

I was wondering if someone can verify (or not) the correctness of the following function? $$\frac{1}{\sinh^2X}=\coth^2X-1$$ I saw it in a paper but I am weak in math, so I am unsure if it is correct ...
1
vote
1answer
46 views

Integral Inequality: Does multiplying integrand by $t$ maintain positivity?

Let $g: [0, 1]\to \mathbb R$ be a continuous function such that, for all $x\in [0, 1]$, $\displaystyle\int_x^1 g(t) \ dt\geq 0$. I'd like to show that $\displaystyle\int_0^1 tg(t)\ dt\geq 0$. Well, ...
0
votes
1answer
32 views

Vectors - theory on cross product

If $X$ is a point on a line through $P$ and $Q$, $X=OX, P=OP, Q=OQ$ (all are vectors but $X$) then: $$X \times (Q-P) = P \times Q$$ I subbed in the $OX$, etc and simplified, but did not get each ...
0
votes
1answer
34 views

If $\lim_{x\to 6} f(x) = -216$, find $\lim_{x \to 6} [f(x)]^{1/3}$

If $\lim_{x\to 6} f(x) = -216$, find $\lim_{x \to 6} [f(x)]^{1/3}$. It says to use the fractional limit law, I have no idea what this means. The textbook example is: If $\lim_{ x \to 6} ...
35
votes
1answer
976 views

Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $

I've found here the following integral. $$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$ I've never seen it before and I also ...
2
votes
2answers
54 views

Integrate $\int_{-\infty}^{\infty} \frac{1}{x\sqrt{1+x}}dx$

The following is the method that I want to use. By letting $$u=(1+x)^{-\frac{1}{2}}$$ $$dx = -2u^{-3}du$$ and $$x=u^{-2}-1$$ So ignoring the limits of integration so far, I get $$ -2\int{ ...
3
votes
0answers
113 views

Is multiplication in mixed radix numeral systems complicated?

The wikipedia article on mixed radix numeral systems says Mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. This sounds like "naive ...
4
votes
1answer
114 views

Definite integral inequality

Let $f:[a,b] \to R$ be a differentiable function so that $f(b)=0$.Prove that there is a $c \in (a,b)$ which satisfies: $$ f'(c) \int_a^c f(t) dt + c \ge \frac{a+b}{2} $$ Any useful ideas that can ...
0
votes
1answer
53 views

Algebraic element

$K \leq L, a \in L$ I am looking at the proof that if $a$ is algebraic over $K$, then $K(a)=K[a]$ : We show that $K[a]$ is a field, then we have that $K \subseteq K[a] \subseteq K(a) \subseteq L$. ...
1
vote
1answer
50 views

Matrix Problem about commutative multiplication [duplicate]

The problem is: "Find all $2\times 2$ matrices A that have the property that for any $2\times 2$ matrix B, AB = BA." Given hint: "The given equation must hold for all B. Try matrices B that have ...
0
votes
3answers
51 views

Integrating $\cos(x)^3dx$

My attempt at integrating $\cos(x)^3dx$: $$\begin{align}\;\int \cos^3x\mathrm{d}x &= \int \cos^2x \cos x \mathrm{d}x \\&= \int(1 - \sin^2 x) \cos x \mathrm{d}x \\&= \int \cos x dx - \int ...
0
votes
1answer
23 views

Integration by substitution - determine support

I was wondering if anyone may be able to direct me to the correct materials to understand how the support of a function changes with substitution. For example, if I have a simple integral with a ...
0
votes
2answers
86 views

Prove that $G - F$ is open and $F - G$ is closed given $F$ closed and $G$ open in $M$.

proof: Since $G$ is open, $\exists B_{r}(x) \subset G$ for any $x$ and for any $r > 0$, $B_{r}(x) \cap F = \emptyset.$ So $B_{r}(x) = B_{r}(x) - F \subset G - F.$ (I am not sure how to ...
1
vote
1answer
1k views

How many nonnegative integer solutions are there to the pair of equations $x_1+x_2+…+x_6=20$ and $x_1+x_2+x_3=7$?

How many nonnegative integer solutions are there to the pair of equations \begin{align}x_1+x_2+\dots +x_6&=20 \\ x_1+x_2+x_3&=7\end{align} How do you find non-negative integer solutions?
4
votes
1answer
135 views

Every matrix is a limit of a sequence of invertible matrices.

I'm trying to prove that every matrix $n\times n$ is a limit of a sequence of invertible matrices $n\times n$. I know this intuitively, but I couldn't prove formally. I'm trying to prove this using ...
1
vote
3answers
32 views

Show that $(f_n)^2$ doesn't converge uniformly to $p^2$".

Please,check my solution to item "a" and give me a hint to solve item "b". Thanks in advance. Problem: "Let $p:\mathbb {R} \rightarrow \mathbb R$ be a polynomial whose degree is $\ge 1$. (a) Show ...
1
vote
2answers
54 views

A stricter Fermat's little theorem

By Fermat's little theorem we know that $a^{p-1} \equiv 1 \pmod{p}$ for all primes p. But it is often possible to find $x$ such that $a^{x} \equiv 1 \pmod{p}$ and x < p - 1. Is there anyway to ...
0
votes
1answer
105 views

prove a theorem about an upper bound of entropy of a random vector

There is a theorem that: if Z is any zero-mean, complex random vector with covariance $E[ZZ^H]=R_z$, then $H(Z)\leq \log|{\pi eR_z}|$, with equality holding if and only if Z has a circularly ...
2
votes
0answers
64 views

Prove circle packing solution is optimal

Background: This is a follow on from this question of how to maximise the area of two non overlapping circles of arbitrary radii packed into a rectangle of arbitrary width and height. I proposed a ...
1
vote
2answers
119 views

calculate the absolute value of a complex number

I have to calculate the absolute value of $\lvert{i(2+3i)(5-2i)\over(2-3i)^3}\rvert$ solely using properties of modulus, not actually calculating the answer. I know that I could take the absolute ...
2
votes
1answer
85 views

Derivative and Integrals of Matrix functions

I am interested in calculating quantities such as $$\frac{\operatorname{d}}{\operatorname{d}\!t} e^{A(t)},\quad \frac{\operatorname{d}}{\operatorname{d}\!t} e^{\int_{a}^{t}A(s)\operatorname{d}\! s}$$ ...
0
votes
2answers
36 views

Questions regarding ternary operations

I have a question regarding ternary operations over a set S. Is it true that for any ternary operation h in S, there are binary operations f and g such that h(x,y,z)=f(g(x,y),z)? And the second ...
2
votes
1answer
88 views

Norm on a Geometric Algebra

In the literature, for example "New Foundations for Classical Mechanics" by David Hestenes, the author introduces a function on the Geometric Algebra $$||M||^2=\langle M M^\dagger \rangle_0,$$ where ...
1
vote
1answer
49 views

Distributional derivative of a hölder function

Let $f$ be a $\alpha$-Hölder function in $\mathbb{R}^n$. Question : does it have distributional derivatives in a $L^p$ space ? (modulo a suitable relationship between $\alpha$ and $n$). I know the ...
3
votes
1answer
245 views

Complement of a set is not a set

How does one prove that for all sets S, there is no set T that contains all x not in S? This is working in ZFC, of course.
3
votes
2answers
314 views

Homomorphism problem gone wrong

Okay, so I'm working on a homework problem in abstract algebra, and I have found the solution already, what I want to know is why my initial line of reasoning didn't work - i..e, what have I done or ...
1
vote
1answer
108 views

Compute volume of tetrahedron using a triple integral

I'm trying to compute the volume of a tetrahedron with the vertices (0, 0, 0), (0, 0, 1), (2, 0, 0), (0, 2, 0). It needs to be done using a triple integral. Not allowed to use "det" or other ...
1
vote
1answer
396 views

law of iterated expectations with nested conditioning sets

What I'm given: $\bf{g_i}=\bf{x_i}$$u_i$ where $\mathbf{x_i}$ is a k-dimensional vector and $E(u_i|\mathbf{g_{i-1},...,g_1})=0$ I want to show that $E(u_i|u_{i-1},...,u_1)=0$ My work so far: ...
0
votes
1answer
69 views

Can we not apply the Hensel Lifting Lemma in this case?

Check if the equation $x^2=-1 \text{ in } \mathbb{Z}_2$ has a solution, and if it has, calculate the three first positions of the solution. So, we are looking for a solution $\pmod 2$, one solution ...
0
votes
1answer
41 views

evaluating a simple complex integral

Why is $\int_{0}^{2\pi}i dt = 2\pi i$? I'd like to use the fundamental theorem of calculus, but the version I know only works for functions $:\mathbb{R} \rightarrow \mathbb{R}$. Is it that we're just ...
2
votes
2answers
279 views

In any finite graph with at least two vertices, there must be two vertices with the same degree

Show that, in any finite graph with at least two vertices, there must be two vertices with the same degree. HINTS ONLY!
2
votes
2answers
81 views

Number of ways to enter 1,2,3,4,5,6 on the spots marked on three intersecting circles and have the sums of 14

How to solve it , i applied so many method but i could not find right answer, suggest me some method combination question gre 44
2
votes
0answers
51 views

Why do Ad(K) orbits in the $-1$ eigenspace of a Cartan decomposition intersect the Weyl chamber once?

Let $G$ be a semisimple Lie group and let $\frak p\oplus t$ be a Cartan decomposition of $\frak g$ and $K$ the connected subgroup with Lie algebra $\frak t$. Choose a maximal abelian subalgebra ...
3
votes
1answer
125 views

Solution of Dirichlet problem in a ball?

If $u\in C^2(B_{R}(0))\cap C^0(B_R(0)))$ is harmonic, then $\large u(x)=\frac{R^2-|x|^2}{nW_nR}\int_{\partial B_R(0)}\frac{u(y)}{|x-y|^n}\,ds(y)=\int_{\partial B_R(0)}k(x,y)u(y)\,ds(y)$ by the ...
1
vote
1answer
63 views

What's the total of battery if 50mA of current are provided to a load for 1 hour?

What's the total charge that moves from one terminal of a battery to the other if 50mA of current are provided to a load for 1 hour? I think I'm suppose to use the formula I = Q/t, where I is the ...

15 30 50 per page