# All Questions

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I am studying a paper where it is mentioned that Newton iteration may be used to compute the inverse of $n \times n$, well- conditioned matrix in parallel time $o(\log^2n)$ and that this computation ...
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### Consider a Group $G$ of order $20$ such that $G=Aut$($Z_{25},+) \cong (U_{25}, \cdot)$. Analyze the Sylow subgroups in G.

$G=Aut$($Z_{25},+) \cong (U_{25}, \cdot)$ I know that there is one 5-Sylow subgroup and number of $2-Sylow$ subgroups is either $1$ or $5$. (a) How do I decide whether the number of distinct 2-Sylow ...
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### $(x-a)^2(x-a-1) + p \equiv 0 \bmod{p^3}$

Let p be a prime and a belong to Z. Find all solutions to the equation $$(x-a)^2(x-a-1) + p \equiv 0 \bmod{p^3}$$ I'm having a hard time working with this as such few variables are given. We know p ...
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### Can flux be proportional to $r^2$ in divergence theorem?

The motivation for the divergence being interpreted as the flux of stuff used the following: $$\text{div} F(a) = \lim_{r\to0}\frac{3}{4\pi r^3}\int_{|x-a|=r} F\cdot n dA$$ Without the $r^3$ in the ...
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### Units on a ring

Let $A$ be a ring such that there exists units $u_1, \dots , u_p : u_1 + \dots + u_p=0$ and $p$ a prime. Did $A$ have a subring isomorphic to $\mathbb{Z}/p \mathbb{Z}$ in general Thanks in advance.
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### Proving an Integral domain is a field. [duplicate]

Let $R$ be an integral domain containing a field $F$ as a subring. Show that if $R$ is a finite dimensional vector space over $F$, then $R$ is a field. This is a Ph.D. entrance question, I recently ...
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I am working on some exercises in the book Geometry: A Metric Approach with Models by R.S. Millman. He defines the following map: $$d_S(P,Q):\mathbb R^2\times\mathbb R^2\to\mathbb R\\\ ... 4answers 111 views ### Closed form of a recurrence relation using generating functions It's been awhile since I have done this. The sequence is \displaystyle a_n = a_{n-1} + 5~a_{n-2} with a_{0}=0 and a_{1}=1. I found the generating function to be \displaystyle G(x) = ... 1answer 185 views ### Prove that if Q^tQ = I and A = QR, then \|Ax - b\| = \|Rx - Q^tb\| I have a linear algebra final tomorrow and was practicing a few proofs. I want to make sure this proof is correct. Prove that: If Q^tQ = I and A = QR, then \|Ax - b\| = \|Rx - Q^tb\| ... 2answers 91 views ### Question about derivation in Jordan algebra Let (G,\circ) be a Jordan algebra, then \sigma:G\to G given by$$c\mapsto a\circ(b\circ c)-b\circ(a\circ c),\quad\forall c\in G,$$is a derivation, where a and b are two fixed elements of G. ... 4answers 2k views ### Find \lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n} Find$$\lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}.This is a recent exam question, which I couldn't figure out in the exam. My guess is it doesn't ... 2answers 274 views ### Is it an abuse of notation to omit the leading zero in a decimal less than 1? Is it acceptable to write .001 rather than 0.001 when using decimal notation? Are there contexts in which omitting the leading zero is acceptable, and other situations in which it is not? 1answer 114 views ### Finding Partial Derivative (n-dimensional) using implicit differentiation vs explicitly solving This is a book example (not a homework question) about implicit differentiation on a composite of functions in n-dimensional space. But my book explains this example in a very unclear manner. So I ... 1answer 113 views ### Niceness of the projection of a closed subscheme of affine space? Let k be an algebraically closed field, and suppose C\subseteq \mathbb{A}^{n+m}_k is a closed subscheme. What can we say about the image under the projection \pi: \mathbb{A}^{n+m}_k\rightarrow ... 0answers 68 views ### How to show that \exists~f\in C(X,\mathbb R) such that f is unbounded? [duplicate] Let X be a non-compact metric space. How to show that \exists~f\in C(X,\mathbb R) such that f is unbounded? 2answers 145 views ### Trees with vertex set I am having hard time understanding and solving the following question: There are exactly three trees with vertex set {1,2,3}. Note that all these trees are paths; the only difference is which ... 1answer 125 views ### Total no. of ordered pairs (x,y) in x^2-y^2=2013 Total no. of ordered pairs (x,y) which satisfy x^2-y^2=2013 My try:: (x-y).(x+y) = 3 \times 11 \times 61 If we Calculate for positive integers Then (x-y).(x+y)=1.2013 = 3 .671=11.183=61.33 ... 1answer 107 views ### Block matrix notation Given that A is a real, rectangular matrix of dimension m \times n and \begin{align} A = \left[\begin{array}{c} I \\ e^{\intercal} \\ -e^{\intercal}\end{array}\right] \end{align} is represented ... 3answers 412 views ### Relationship between matrix equations and determinants? What is the relationship between a matrix equation and the determinants of the matrix? Say if you have the matrix equation A^3 - 4A = 0, what can I learn about the matrix A's determinants? Or ... 1answer 158 views ### About measure theoretic interior and boundary Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given ... 2answers 120 views ### Number Theory: x^2+y^2=a^2 Is there a coprime triple (x,y,z) such that x^2+y^2=a^2, x^2+z^2=b^2, y^2+z^2=c^2, where a,b,c are integers P.S. such solution doesn't exist for a,b,c<1000, as the computer says P.P.S. ... 1answer 534 views ### Has this phenomenon been discovered and named? Ifx-\frac{x}{2}=\frac{x}{2},$$and$$\frac{x}{\sqrt{x}}=\sqrt{x},$$and$$x-\uparrow(x-\uparrow^22)=x-\uparrow^22$$when (x\uparrow^n-[A])\uparrow^nA=x, where A is some constant, and one uses ... 2answers 401 views ### What are some reasonable things to prove about the Collatz Conjecture? I am writing an undergraduate paper on the 3n+1 problem, and I am looking for some theorems related to the problem that would be reasonable for someone with my mathematical background to prove. I'm ... 1answer 101 views ### Conditional Random Variable and Iterated Expectation First, I can't seem to find a definition of a "conditional random variable"; I understand the concepts of conditional expectation and conditional probability, but is there a notion for the random ... 2answers 149 views ### Find a lattice with exactly three congruence relations A lattice L is a partially ordered set such that any two elements a and b have a least upper bound a\lor b and a greatest lower bound a\land b. A congruence relation on L is an ... 4answers 33k views ### Definite Integral of square root of polynomial I need to learn how to find the definite integral of the square root of a polynomial such as:$$\sqrt{36x + 1}$$or$$\sqrt{2x^2 + 3x + 7} $$EDIT: It's not guaranteed to be of the same form. ... 0answers 112 views ### Free basis and free group How can I prove the following result? Let G be a group, X\subseteq G and let F_a(X) be the free group on X. Then the subgroup of G generated by X is isomorphic with F_a(X) if and only ... 1answer 266 views ### area of square between tangent(externally) circles Two externally tangent circles,have a square between them,standing on the same base as the two circles.The circles have a radius of 1 unit each.The top two vertices of the square are touching one ... 2answers 127 views ### Find the flux of the vector field out of the region T bounded by z = x^2 + y^2 and z = 4. Please help Find the flux of the vector field$$\vec{F}(x,y,z)= <x,y,3>$$out of the region T bounded by z = x^2 + y^2 and z = 4. It says the unit vector n is <0,0,1>, but how do you find ... 1answer 61 views ### Trying to define an abstract notion of a function that turns sets of sentences into the set of models satisfying those sentences. Let \mathsf{Sen} denote a Boolean algebra, thought of as a collection of sentences, and let \mathsf{Mod} denote a set without any additional structure, thought of as a collection of models. I'm ... 1answer 76 views ### If A is orthogonal, for any vector x such that Ax = b, \Vert x \Vert = \Vert b \Vert Is this statment true: For any vector x such that Ax = b, \Vert x \Vert = \Vert b \Vert, if A is orthogonal. I was working on a proof for my linear algebra class, when I noticed that the ... 1answer 49 views ### Simplex with edges of length at least s having smallest circumradius Is it true that of all k-simplices with edge lengths greater than or equal to some parameter s, the regular simplex with edge lengths s has the smallest circumradius? Please supply a proof or ... 2answers 46 views ### Complex Solutions to Polynomials I'm trying to use topology to prove that: z^{n} + a_{n-1}z^{n-1} + ... + a_{1}z + a_{0} = 0 has a solution in \mathbb{C} if and only if, for each positive real number c, the equation z^n + ... 2answers 175 views ### A^3+A=0 We need to show \mathrm{rank}(A)=2 Let A\ne 0 be a 3\times 3 matrix with real entries such that A^3+A=0. We need to show \mathrm{rank}(A)=2. \det A(A^2+I)=0\Rightarrow\det A=0\Rightarrow \mathrm{rank}(A)<3, Suppose ... 1answer 141 views ### Map Quadrant Conformally onto the Unit Disc and find |g'(1+i)|. If w = g(z) maps the quadrant \{z = x + iy \; : \; x,y >0\} conformally onto |w|<1 with g(1) = 1, g(i) = -1, and g(0) = -i, find |g'(1+i)|. My initial reaction was to directly ... 0answers 184 views ### Evaluating a polynomial at a root of unity? Let R = \mathbb{Z}[x]/(x^n+1) be the 2nth cyclotomic ring (for n a power of 2 in which case \Phi_{2n}(x) = x^n+1). Let g be an n-dimensional vector chosen at random from \mathbb{Z}^n ... 1answer 184 views ### How can I prove that the solutions of this differential equation is monotone? I'm trying to proof if x:I\to \mathbb R a maximal regular solution of x'=f(x), such that the image x(I)\subset \mathbb R is bounded and f:\mathbb R\to \mathbb R is C^1, then x is strictly ... 2answers 83 views ### Improper Integral \int_0^1\frac{dx}{x^p} Is this integral convergent only for p<1?$$\int_0^1\frac{dx}{x^p}$$1answer 152 views ### How to calculate smallest pattern size based on set of ratios So I have x number of items, each one has its own appearance ratio for making a pattern, of which I'm trying to determine the smallest possible pattern. The appearance ratios for all items total up to ... 1answer 113 views ### For any 11-vertex graph G, show that G and \overline{G} cannot both be planar Let G be a graph with 11 vertices. Prove that G or \overline{G} must be nonplanar. This question was given as extra study material but a little stuck. Any intuitive explanation would be great! 2answers 47 views ### P\left( n,\left( {{\lambda }_{1}}+{{\lambda }_{2}} \right)T \right) Disaggregating Tail of Poisson I have a Poisson tail P\left( x,\left( {{\lambda }_{1}}+{{\lambda }_{2}} \right)T \right) which is sum of two independent Poisson distribution with rate \lambda_1 and \lambda_2. I am trying to ... 2answers 135 views ### Geometric solution to classic committee problem Most people know the classic committe style problems. I read this solution to one of the basic version of the committe problem and was impressed, but not sure why it works. I was hoping someone ... 2answers 1k views ### Let G be a graph in which every vertex has degree 2. Is G necessarily a cycle? I suspect not but I'm having hard time showing this. Also, Let be a tree. Prove that the average degree of a vertex in T is less than 2. I know that the sum of degrees of ... 0answers 71 views ### an upper semi-continuous semi-strictly quasi-concave function May you help me? I was confused that why "an upper semi-continuous semi-strictly quasi-concave function is quasi-concave". If there is a picture,it may be better. I really need your help ～ Thank ... 2answers 135 views ### existence continuous function f:X\to [0,1] such that f(x_1)=0 , f(x_2)=\frac{1}{2}, f(x_3)=1 Let X be a Hausdorff normal topological space and x_1, x_2, x_3 are three distinct points.prove that there exist continuous function f:X\to [0,1] such that f(x_1)=0 , f(x_2)=\frac{1}{2}, ... 1answer 127 views ### Bounded and invertible operator on dense subspace Who can give me an operator like this or show it doesn't exist: Operator T: X-->Y, is a bijection from normed linear space X to normed linear space Y. X, Y are equipped with the same norm, and X is a ... 1answer 540 views ### Tricky Expectation Let X denote a Poisson distributed random variable and let Z = \max(X-\min(X,c),0), where c is a constant. How do I compute \mathbb E[Z]? Any help is greatly appreciated! 2answers 707 views ### Non-homogeneous linear recurrence relation I have this recurrence relation to solve:$$a_{n+1}=3a_n+2^{n-1}-1. The homogeneous part's solution is obviously $a_n=k3^n.$ Now I don't know how to solve the original equation, but I do know what ...
Can you give me a hint on the first part of the exercise? Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
I haven't seen the form of osc$_I(f_n-f)$.I expect your explanation.