# All Questions

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### How many 4 digits prime numbers can be formed from 0,1,…,9 without repeated digits?

I'm just curious about the prime numbers in combinatorics. Can we use the combinatorics rule to find the number of prime number from given number, for example from the above condition? My attempt: I ...
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### Showing $\phi(f \cdot g) = \phi(f) + \phi(g)$

For $\phi \in C_n(X; G)$ a cocycle being thought of as a function from paths in X to G, I want to show: $\phi(f \cdot g) = \phi(f) \cdot \phi(g)$. What I'm not sure is how I'm supposed to relate a ...
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### Integration $\int \frac{x}{x^2-5x+6}dx$

Evaluate the Integral: $\int \frac{x}{x^2-5x+6}dx$ I solved twice and once I got $3log\left|x-3\right|-2log\left|x-2\right|+C$ and I tried again and changed one step and I got ...
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### Nature of the series $\sum_{n=3}^{\infty} \dfrac{1}{(\log\log n)^{\log n}}$

Does $\sum_{n=3}^{\infty} \dfrac{1}{(\log\log n)^{\log n}}$ converge or diverge ? I tried some tests , but nothing conclusive is coming . Pleas help
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I have the following definitions which may be true or false but for me it seems like both true A) If ($a_{2n} - a_n$) converges to $0$ then $a_n$ converges. B) If $a_n$ converges then ($a_{2n} - ... 0answers 3 views ### Why we always chose minimum Ratio in Simplex method? Why we always chose minimum ratio in simplex method? (linear programming problem) 0answers 15 views ### Pronunciation of Rng - the non-unital Ring I chuckled the first time I heard that a Ring without a multiplicative identity (Ring without the i) is called a ... 1answer 8 views ### first and second projection i have read that in the ordered pair$z=(x_1,x_2)$, an element of a direct product$Z=X_1 \times X_2$of sets$X_1$and$X_2$, the element$x_1$is called the first projection and$x_2$is called the ... 1answer 8 views ### what will be the PDF of the magnitude of this random variable x+j y? if we have a complex random variable [x+j*y] where (j :sqrt(-1)) and x,y both have Gaussian distribution and statistically dependent , so what will be the distribution (PDF) of the magnitude of this ... 2answers 16 views ### To show following function is discontinous Given$f(x) = [x + 1] (\sin(1/x))$, where[.] denotes greatest integer function ; when$x\in (-1,0) \cup (0,1)$$$f(x) = 0 , \text{ otherwise}$$ Question is to show f has discontinuity of second ... 0answers 8 views ### Summing the sequence$a(n) = \sin(n x) \exp(-nt)$Consider the sequence$a(n)$defined by$a(n) = \sin(n x) \exp(-nt)$, where$n = 0, 1, 2, 3, 4, \ldots$. The parameter$x$is a real number. Parameter$t$is a positive real number. It is clear that ... 1answer 5 views ### Formula of regular 2m-gon inscribed in a unit cirlce From pg 80 of Introduction to Calculus and Analysis I by R. Courant: If we let$f_m$denote the area of the regular$m$-gon inscribed in a unit circle, the area of the inscribed$2m$-gon is given by ... 1answer 12 views ### Combinations of fruits and their “nutrients” As a computer scientist and not a mathematician, I know not some of the formal language to describe my problem, so I'll present it in a word problem form. Maybe someone can help me hone my search and ... 0answers 6 views ### Question on weak star convergence in subspace Let$X$be some normed linear space and let$X^\ast$denote its dual space endowed with the weak star topology. Let$U^\ast$be some subspace of$X^\ast$. If I want to show that$\varphi_n$... 0answers 5 views ### Minimal polynomial in$T$-invariant subspace I am stuck on the following problem. Problem: Let$V$be a finite dimensional vector space over field$F$and$T$a linear transformation from$V$to$V$.$W$is an invariant subspace. Let$h_1$be ... 0answers 10 views ### Proving 2 random variables differ with positive probability Suppose that conditional on$x$,$y$is normal with mean$x'\beta_0$and variance$\sigma_0^2$. The log of the conditional density is then $$... 1answer 8 views ### Asymmetric simple random walk? It comes from the book Probability: Theory and Example. I don't understand the part marked with red line. Why it cannot converge to an interior point of (a,b)? Can anyone help? Thanks so much! 1answer 32 views ### how can publish my log approximation formula I've successfully found out a formula which can give log value of any base till 4-5 places after decimal I want to know whether it can get published because I've seen some journals which have ... 1answer 20 views ### A simple question about free group Fix r\in \mathbb{N} and let \mathbb{F}_{r}=\langle g_{1}, ...,g_{r}\rangle be the rank-r free group. I have asked a question several days ago: Is \mathbb{F}_{2} a subgroup of \mathbb{F}_{3}? ... 0answers 21 views ### Does there exist a continuous function between the following sets: Does there exist a continuous function between the following sets: A.f:(-1,1)\rightarrow (-1,1] which is onto and one-one B.f:\{(x,y):y^2=4x\}\rightarrow \mathbb R which is one-one What ... 1answer 8 views ### Rational Canonical Form Confusion; Choosing Basis Which Gives the Rational Canonical Form. I am reading the theory of finitely generated modules over a PID. One of the applications of the the theory is that one can derive the theory of rational canonical form of a linear operator on a ... 3answers 13 views ### Limit About Complex Variable I am trying to see if I have right understanding of the limit. when the book mentions$$\lim_{z\to z_0} f(z) = L$$this just means that f(z_0) = L is this correct? 1answer 31 views ### det (AB)=det(A)det(B) is possible when A and B are ???? det (AB)=det(A)det(B) is possible when A and B are ???? what is the answer when A and B are ???? This is a Fill in Blanks that I found in my paper but I don't have this answer 1answer 9 views ### Prove existence/non-existence of a pdf given mean, std, range Given: Mean = 100, Range = [4, 10000], std = 3000 Is it possible to prove whether a pdf exists or not that satisfies these values? If it does exist, what would be approximate shape of the ... 0answers 24 views ### The convergence of the multiplication of two convergent series? If we know that $${\sum\limits_{n=1}^{\infty} }a_n$$ and $${\sum\limits_{n=1}^{\infty} }b_n$$ are convergent What about their ... 0answers 5 views ### Composite residuosity statement. Consider the following definition. A number z is said to be n-th residue modulo n^2 , if there exists a number y \in \mathbb{Z}_{n^2}^* such that$$z\equiv y^n \mod n^2$$Let us take n=6 ... 2answers 20 views ### Solve this Differential Equation [x\csc(\frac{y}{x})-y]dx+ydy=0. [x\csc(\frac{y}{x})-y]dx+ydy=0 My work: [\csc(\frac{y}{x})-\frac{y}{x}]dx+\frac{y}{x}dy=0 Let u=\frac{y}{x}\rightarrow y=ux\rightarrow dy=udx+xdu [\csc(u)-u]dx+u(udx+xdu)=0 ... 1answer 15 views ### Uniform convergence to exponential exercise Yesterday I encountered the following exercise in a tutorial sheet from the University of Lyon : define a sequence of functions (f_n) (with f_n:[0,\infty) \to {\mathbb R}) by ... 0answers 19 views ### Every map from a compact Hausdorff space is continuous Could someone help me figure out my mistake? I just proved that if X is compact Hausdorff then f: X \to Z is continuous. Here's my proof: Let U be open in Z. Let x \in f^{-1}(U). Since X ... 0answers 14 views ### If the sum of two i.i.d. random variables is normal, must the variables themselves be normal? It is well known that if two i.i.d. random variables are normally distributed, their sum is also normally distributed. Is the converse also true? That is, suppose X and Y are two i.i.d. random ... 1answer 25 views ### Equivalence classes of \mathbb R Let X be a locally compact, connected, locally connected, Hausdorff space. Considder U_1\supseteq U_2\supseteq\cdots of open and non-empty connected subsets with compact frontiers such that ... 0answers 18 views ### how to transform among each number assume there are 16 numbers in base 3 the basic construction for numbers from 0 to 3^(3^2)-1 numbers is using 0..15 0 1 2 3 ... 14 15 ... 0answers 7 views ### How to find the interior of this set? let S=\{A\in M_n(\mathbb R):tr(A)=0\} The question is to check whether S is Nowhere dense .I think the set is closed and hence the problem reduces to findind int(S).How to do that? 0answers 11 views ### Non-orthogonal basis I have a set of complex vectors (maybe 10,000 vectors, each of which has maybe 200 elements). I know that each of the complex vectors is a linear combination of a small (maybe 10) collection of ... 3answers 27 views ### To show f(x) is discontinuous at every point$$f(x)=\begin{cases} 1 ,& \text {$x$is rational} \\ 0 , & \text{$x$is irrational}\\ \end{cases}$$How do I show this function is discontinuous at every point. How to think about it ... 0answers 17 views ### Expected number of rooms with at least one man and woman? here's the problem: "10 men and 10 women randomly go into 10 rooms. What is the expected number of rooms with at least one man and woman?" Here's my reasoning: So, I think that by linearity of ... 0answers 14 views ### BODMAS Order of operations 9÷3(6×4÷8) Here we all can apply BODMAS and solve the equation in parenthesis first giving us 3. But after this step shall we solve by 9÷3(3) = 9÷9 = 1 or 9÷3×3 = 3×3 = 9 . Please help 0answers 11 views ### Suppose H:= \{\sigma \in G| \sigma(1) = 1\}, if for any j \in \{1,2,…,n\} t_j\in G such that t_j(1) = j. Show that |G| = n|H|. Let G be a subgroup of the symmetric group S_n in n letters. Consider the following subset of G:$$H:= \{\sigma \in G| \sigma(1) = 1\}$$Suppose that G acts on the set \{1,2,...,n\} transitively ... 3answers 33 views ### Sum of convergent and non-convergent series, does it converge? And how to prove? Series a_n is convergent and b_n is not-convergent. Will the sum a_n + b_n converge? I think it will not converge, But how do I show it? I believe I have to use the definition. |a_n - A| < ... 1answer 22 views ### Why R^{op} used in the definition of “category of R-modules”? Earlier today, I was thinking: "Oh, an R-module is just an additive functor R \rightarrow \mathbf{Ab}." Anyway, I had a bit of a read over at nLab, and it says: For any small ... 1answer 22 views ### A question about analytic continuation of a function in real axis A function f(x) is an real function and analytic in an open interval of x-axis or the whole x-axis. Is there only unique way to analytically extend it to the whole complex plane? I know ... 2answers 41 views ### The convergence of a recurrcively defined sequence. Let a_1=\sqrt{2} and a_n=\sqrt{2+a_{n-1}} determine the convergence of the sequence and find its limit. I know the sequence converges to 2 and i can show this informally. But I don't know how ... 1answer 20 views ### Orthogonal Matrix with a specific row I have an assignment with the following question: Does an Orthogonal Matrix exist such that its first row consists of the following values: (1/\sqrt{3}, ... 3answers 26 views ### Help understanding a proof about vector spaces The exercise goes like this: -Let W= {(x,y,z)|2x+3y-z=0} Then W\subseteq\mathbb{R}^3, find the dimension of W. -Find the dimension [\mathbb{R}^3|W] This was a problem from my algebra exam, ... 0answers 25 views ### E(XY) = E(X).E(Y|X) . Is this true for mean = zero. I know that Joint Probability density function for two random functions X and Y$$P(XY) = P(X)\cdot P(Y|X)\tag{1}$$But I just read in a set of lecture notes that for E(X)=E(Y)=0$$E(XY) = ... 2answers 20 views ### limits of integration in spherical coordinates. Consider a cone centered about the positive z axis with its vertex at origin,a$90^{\circ}$angle at its vertex,topped by a sphere of radius$6$.Compute the volume of region bounded by sphere and ... 2answers 76 views ### Multiplying a infinite number with a rational number? Please do not down vote this question. It may be stupid, but I wonder. Why is it that we cannot multiply$3.99999\cdots$by$4$and write$16,....$? 0answers 18 views ### Find the eigenvalues and eigenvectors of T in V Let$\mathbf{V}$be the linear span of the functions 1, cos x, sin x. Let the operator T on V be given by the rule$T y(x)= y(x+\pi/4)$. Find the eigenvalues and eigenvectors of T in V. I'm not sure ... 0answers 13 views ### Find the volume of the 3D solid. Let$\delta$be the region bounded by the graphs of the functions$f(x) = x^2$and$g(x) = 2 x^2$. Find the volume of the solid generated by revolving$\delta$around the line$x = 1$. 1answer 26 views ###$\{(x,y)\in \mathbb R^2:xy=1\}$To check which pairs are Homeomorphic? A.$\{(x,y)\in \mathbb R^2:xy=0\}$B.$\{(x,y)\in \mathbb R^2:xy=1\}$C.$\{(x,y)\in \mathbb R^2:xy=0,x+y\geq0\}$D.$\{(x,y)\in \mathbb R^2:xy=1,x+y\geq 0\}\$ I ...

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