# All Questions

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### My proof: Frobenius Map generates $\mbox{Gal}(\mathbb F_{p^n}/\mathbb F_p)$

I would like to ask, whether anyone can confirm or correct the following version of the proof that the Frobenius map generates the Galois Group of a finite field. Proof First note that ...
It's been many years since I studied maths, and I'm trying to figure out the half factorials $(7/2)!$ without a calculator. I did $(7/2) \times (5/2) \times (3/2) \times (1/2) = (105/16) ^ \pi = ... 1answer 52 views ### Finding the types of singularities of$f(z)=\frac{1}{z\cdot (e^z -1 )}$I am getting trouble to find the types of singularities of $$f(z)=\frac{1}{z\cdot (e^z -1 )}$$ What I tried to do is:$z=0z=2\pi k i$for$z=2\pi k i$its in order 1, but for the first one I ... 2answers 189 views ### A question about tableau method for first-order logic I have a doubt about tableau method for f-o logic. In Smullyan's book (First-Order Logic, 1968, Dover reprint) the method is defined (pag.53) for formulae but - if I'm not wrong - all examples that ... 3answers 326 views ### How to show$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$) $$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$ I tried to solve but came up with$\pi(2-e^{-a}) $. Could you tell me where did I do the mistake? if$x=z$then ... 1answer 139 views ### Gödel Incompletness theorem I am not familiar with model theory. As a matter of fact, I only had my first Logic and Set theory courses last semester. But still, there is a question that bothers me, and It could be nice if ... 2answers 118 views ### Computing distance from line to point in geodetic environment Supposing to be in a cartesian plan and that I have the following point: $$A(x_{1},y_{1}), B(x_{2},y_{2}), C(x_{3},y_{3}), D(x_{4},y_{4})$$ $$P(x_{0},y_{0})$$ Now immagine two lines, the fist one ... 1answer 465 views ### How to solve this fraction within minute? (or trick) I want to solve this question within minute but bcz of fraction it take more than a minute. does any one know a trick to solve this type question.plz share thanks 2answers 259 views ### Find the number of irreducible factors of$x^{63} - 1$I have to find the number of irreducible factors of$x^{63} - 1$over$\mathbb{F}_2$using the$2$-cyclotomic cosets modulo$63$. Is there a way to see how many the cyclotomic cosets are and what is ... 2answers 119 views ### Are these two 2-manifolds homeomorphic? I have a 2-Sphere with a finite number$k$of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ... 1answer 87 views ### Determine whether the given binary relation is equivalence relation$D$is the binary relation defined on$R$as follows: For all$x,y\in R,xDy\Leftrightarrow xy >0$. Determine whether the given binary relation is equivalence relation and if it is, give the ... 4answers 100 views ### How to understand that the$\lim_{x\to 0}\frac1x\cos(\frac1x)$does not exist. I'm studying the$\lim_{x\to 0}\frac1x\cos(\frac1x)$. Can I understand that the limit does not exist simply splitting it into two parts? That is, $$\lim_{x\to 0}\frac1x\cos(\frac1x)=\lim_{x\to 0}\frac ... 0answers 67 views ### Continuity in a physical context I'm currently trying to solve an exercise for my quantum mechanics class and have run into a bit of a jam: Suppose we have the following potential : V(x) = 0 if x > |a/2| but V(x) = V_0 if ... 1answer 167 views ### Biholomorphic map between the upper half plane with a slit and the unit disk Let \mathbb{H} = \{z\in \mathbb{C}| \ Im(z) > 0 \}. I want to find a biholomorphic mapping between \Omega_{} = \mathbb{H}-\{it \ | \ t \leq 1 \} and D(0,1). Any hint ? 0answers 156 views ### How to use Chebyshev's inequality or the law of large numbers to a probability question Let x be a random bit string that takes values \{1,0\}^n. Let r be the value of the most significant (MSB) bit of x (and r is a r.v. 1 or 0 that are equally likely). Let g be our guess for the MSB ... 2answers 165 views ### Prove that set contains least element. Let A\not=\emptyset ,A\subset \mathbb{Z} and if (\exists d\in \mathbb{Z})(\forall a\in A):d\le a then set A contains least element. How do I prove this? I understand I can use WOP principle. What ... 1answer 137 views ### Composition of orthogonal projection Given \gamma: \mathbb{R}^2 \rightarrow \mathbb{R}^2 (rotation around o) and \sigma: \mathbb{R}^2 \rightarrow \mathbb{R}^2 (reflection in one of the lines through the origin), I have to show that ... 1answer 718 views ### Summation of logarithmic series I am solving a recurrence relation and it requires me to sum the following series upto \log{n} terms - 1/\log(n) + 1/\log(n/2) + 1/\log(n/4)..... The base in each term is 2. Any help on ... 1answer 64 views ### What is a codomain of diagonal functor? I'm reading a "Graph Transformations. An Introduction to the Categorical Approach" by H.J.Schneider. In a example 6.3.3 Graph Category constructed as a comma category of a identity functor id_{Set} : ... 0answers 73 views ### Resolvable spaces a space X is called a resolvable space if it is expressible as a union of two disjoint dense subsets. I want to find a resolvable but not lindelof space? Is there any example such a space? 0answers 118 views ### \mathfrak b \leq \mathfrak r. Is this proof correct? I am trying to prove that \mathfrak b \leq \mathfrak r where \mathfrak r is the minimal cardinality of a reaping family of sets. A family \mathcal R \subseteq [\omega]^\omega of sets \mathcal ... 1answer 75 views ### A problem of urns. Suppose there are N balls of different colors and K urns. For each ball i=1,...,N it is extracted a flat integer random number k_i between 1 and K and the ball i is randomly assigned ... 2answers 401 views ### Can't read integral method [closed] I type this : fun = @(x) exp(-x.^2).*log(x).^2; q = integral(fun,2,4); q; when I run the above code, I get an error message Undefined function or method ... 1answer 87 views ### Philosophical side of MATH. knowing the path then walk it. [closed] Can I find a book that gives me the purpose of theorems and definitions without going deep into proofs. It's just like knowing the path then walk it. That's will me the understanding reach the next ... 4answers 243 views ### how to evaluate \int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin x}}\text{d}x I was solving a physics problem and eventually the problem boiled down to solving the following integral:$$\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin x}}\text{d}x$$I have already tried ... 1answer 62 views ### Continuous extension of positive functions on a C-embedded set. If A is a discrete, closed and C-embedded subset of a completely regular Hausdorff space X. Then how can we prove that for every continuous function f:A\rightarrow (0,\infty), there exists a ... 5answers 191 views ### The Limit of \frac{\cos x } {x e^{x}}- \frac{1}{x} as x \to 0 I need to evaluate$$\lim_{x \to 0} (\frac{\cos x } {x e^{x}}- \frac{1}{x})$$Using neither L'Hôspitale rule, nor Taylor series... My try:$$\frac{\cos x}{xe^x}-\frac{1}{x}=\frac{\cos x - ... 1answer 98 views ### How much time will the pipe take? There are four outlet pipes of the same capacity fixed one above the other to a water tank. The first pipe is at the bottom level and the fourth pipe is at three-fourths of the height of the tank. The ... 1answer 83 views ### Show that$(\Bbb{N}, |)$is a distributive lattice. Show that the set of Natural numbers with divisibility form a distributive Lattice where for any$x, y\in\mathbb{N}$we have$x$meet$y = \operatorname{gcd}(x,y)$and$x$joint ... 1answer 44 views ### Find the constants in a 2D flow (incompressible, newtonian) $$u_1=x_1^2x_2$$ $$u_2=A+Bx_1x_2^2$$ $$p=cosx_1$$ The fluid is an incompressible Newtonian fluid$\impliesu_i,i=0$and$tor_ij=-p\delta_ij+\mu u_ij$Fluid bounded by a stationary rigid plate at ... 3answers 76 views ### Representing Complex Exponentials with Real and Imaginary Parts My confusion lies with this : http://www.wolframalpha.com/input/?i=modulus+%28cos%282+pi+r_1%29%2Bcos%282+pi+r_2%29%2Bi+%28sin%282+pi+r_1%29%2Bsin%282+pi+r_2%29%29%29+squared I was looking at ... 1answer 65 views ### If two powers of permutations are equal and have no common symbols, they're the identity. - Mulholland p. 44 Proof to Theorem 4.2 Theorem 4.2 (Order of a Permutation): The order of a permutation written in disjoint cycle form is the least common multiple of the lengths of the cycles. Proof: One cycle: As we noted above, a ... 1answer 45 views ### Mean of absolute difference series of two random series, uniformy distributed? Suppose we have two series of 100 (or more) random numbers between 0 and 1. Naturally, the average of series 1 is close to 1/2, and the average of series 2 is also close to 1/2. (screenshot attached.) ... 2answers 46 views ### Questions regarding bound variables $$x\in(\cap F)\cap(\cap G)=[\forall A\in F(x\in A)]\land[\forall A\in G(x\in A)]$$ Since the variable$A$is bounded by universal quantifier, it is regarded as bounded variable, according to the ... 1answer 101 views ### One can integrate every monotonic function I have a question related to the proof of "One can integrate every monotonic fucktion$f: [a,b] \to \mathbb R$." that I have as assignment. We are referring to Riemann integrals here. The idea I came ... 2answers 178 views ### Set theory aspects of category theory I have never learnt axiomatic set theory, but have studied it from Munkres's Topology book first chapter. So I do not understand the difference between a class and a set except in some vague sense. ... 4answers 160 views ### Why is$\operatorname{Hom}(M,N)$not necessarily an$R$module? Let$R$be a ring, and$M,N$be left$R-$modules. Then is it not true that$Hom_R(M,N)$has the structure of an$R$-module? I was reading the preface of the Homological Algebra book by Rotman and ... 5answers 142 views ### I am going to learn these Mathematics Topics. I need advice and suggestions please . I am really horrible when it comes to maths since I never had any maths background in my High school. I am fairly good at programming ( C++ and Java) but without mathematics I cant advance in any ... 1answer 35 views ### Difference equation formula$\sum a^t = \frac{a^t}{a-1}$. As I explain below, this question was originally posted by user YYG, but then deleted. I am reposting the question (from memory) and I will answer it myself below. Question: In Difference Equations ... 2answers 61 views ### Second isomorphism theorem for subspaces just like I did some days ago, I now have to show that$T/T\cap U \cong (U+T)/U $. Therefore I tried finding a surjective homomorphism and then, by using the first isomorphism theorem, I should be ... 3answers 142 views ### Ellipse to Standard Form This is the equation to the ellipse,$9x^2+4y^2-72x+40y+208 = 0$, and I need it in standard form. I can't figure this one out. Could this be the answer?$\frac{(x-4)^2}{9} + \frac{(y+5)^2}{4} = 1$... 2answers 664 views ### Irrational number and Baire space How to show that the set of irrational numbers is a Baire space ? 1answer 61 views ### Uniform Convergence of a Sequence of Summations Given:$f_1(x)=x$if$x\le1/2f_1(x)=1-x$if$1/2\le x\le1f_1(x+1)=f_1(x)\forall n\ge2,f_n(x)=(1/2)*f_{n-1}(2x)$Let$S_m(x)=\sum_{n=1}^m f_n(x)S_m$is a continuous function on ... 1answer 755 views ### Uniform Convergence verification for Sequence of functions - NBHM Following is a list of problems from an exam for admission into Ph.D program. I have just compiled all previous questions on uniform convergence of sequence of functions and i tried to work out . I ... 2answers 52 views ### Arithmetic Order of Operation What is the square root of -4 + 13 x 8 - (7 + 2(3 + 20/5)) The answer seems to be 3 but I wanted help trying to get to the answer! 3answers 189 views ### Geometry of the dual numbers A dual number is a number of the form$a+b\varepsilon$, where$a,b \in \mathbb{R}$and$\varepsilon$is a nonreal number with the property$\varepsilon^2=0$. Dual numbers are in some ways similar to ... 1answer 75 views ### Stopping time and filtrations I have a definition problem. I know that a filtration on a probability space is an increasing sequence of$\sigma$-algebras. I was now thinking on the fact that constant times are stopping times. I've ... 2answers 262 views ### Prove that :- If K is a compact subset of R with non empty interior then it is of the form [a,b] or [a,b] - U{In} The question is :- Let$K$be a compact subset of$\mathbb R$with non empty interior. Show that K is of the form$[a,b]$or$[a,b] \setminus \bigcup I_n$, where {$I_n$} is a countale disjoint ... 1answer 195 views ### Representation of integers as powers of the golden ratio How to prove that any integer$n$can be represented in the form of $$n= \phi^{z_1}+\phi^{z_2}+\phi^{z_3}+...+\phi^{z_m}$$ For$z_1$,$z_2$...$z_m\in\mathbb Z $and$\phi =\frac{ \sqrt ...
Does anyone know if the following optimization problem has an elegant solution? Let $A=\{a_1, a_2, \ldots, a_n\}$ be positive real numbers. Let $B=\{b_1, b_2, \ldots, b_n\}$ be unknown positive real ...