# All Questions

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### Finding limit including binomial coefficients

I recently came across a question from my calculus problem book and encountered a pretty good looking problem. It goes as Find $$\lim_{n \to \infty}\sqrt[1/n]\frac{3n\choose n}{2n\choose n}$$ I tried ...
18 views

### Determine p so that the line q does not have any point in common with the circle

I am preparing to take an entrance exam for a university in my country, which will happen soon. As part of my preparation, I have been practicing with some sample math tests provided by the university....
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### To find the sum of all possible perimeters of a triangle

Here's the exact question: Consider a triangle $ACD$. Point $B$ is on $AC$ with $AB = 9$ and $BC = 21$. Also $AD = CD$ , and $AD$ and $BD$ are integers. Let $S$ be the sum of all possible perimeters ...
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### show that $V = \operatorname{im} f + \ker f$ if and only if $\operatorname{im}f =\operatorname{ im}f ^2$

Let $V$ be a vector space of finite dimension $n$ and $f: V \to V$ is linear. Show that $V = \operatorname{im}f + \ker f$ if and only if $\operatorname{im}f = \operatorname{im}f ^2.$
1 vote
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### Meaning of weak solutions in the class $C^{1,\alpha}$

There are quite a few papers who, e.g. for the p-Laplace equation, show that weak solutions under some assumptions are not quite in $C^2$ but at least in $C^{1,\alpha}$ (locally). But I am wondering ...
1 vote
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### Distance between unbounded closed convex subsets of Hilbert space

Suppose that $A$ and $B$ are nonempty closed convex subsets of the Hilbert space $H$. Case I: If $A$ or $B$ is bounded, I can prove that there exist $a\in A$ and $b\in B$ such that $$d(A,B) = d(a,b),$$...
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### Trigonometric function with a single root

Considering the following trigonometric function: $f(x) = a \sin(2x+\phi_0) + b \sin(x) + c$ I need to find an explicit condition for the parameters $a, b, c, \phi_0$ such that this function has ...
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1 vote
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### Transition Probability of Orstein-Uhlenbeck Process using Girsanov's Theorem.

How I can find the weak solution and transition distribution of Orstein-Uhlenbeck Process using Girsanov's Theorem. More specifically, suppose that $$dx_t = -\lambda \:x_t\:dt + \sigma\:d\beta_t,$$ ...
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### Trying to show a claim about fibrations between smooth maps

I am trying to show that: Any locally trivial fibration is a quotient map. (Definitions below). My attempt: First, let $U ⊂ Y$ be an open set. We need to show that $p^{-1}(U)$ is open in $X$. By the ...
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### Quotient of an abelian group by torsion subgroup provides a characteristic free abelian subgroup?

I have a follow up to this question: Rank of the quotient of an Abelian group by its torsion part?. So my understanding is given a finitely generated abelian group $G$ and its torsion subgroup $T_G$ ...
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### Given a tensor $T$, find a symmetric and an anti-symmetric tensor such that $T = T_a + T_s$

I am writing this because I think my professor made a mistake, but I do not feel confident enough in what I know to tell him. Yesterday we took a multilinear algebra test, and we were given a 2-...
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### Algebra generated by a countable set is also countable

Let $M\subseteq\Omega$ be a countable set. I now want to show, that the Algebra $\mathcal{A}$ generated by $M$ is also countable. Properties of an Algebra: \begin{equation} \mathcal{A}\neq\emptyset \...
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### Conformal transformation of entire complex plane without real axis

I am trying to find conformal transformation of entire complex plane without real axis ( C/{Im z =0}) to upper half plane (Im w >0). Any help would be appreciated.
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### How to show the dual space of a normed space constructed by a sigma-finite measure space.

My question is as follows: Given a $\sigma$-finite measure space $(\Omega,\Sigma,\mu)$, let $$X=\{\{f_n\}_{n\in\omega}: f_n\in L^p(\Omega,\mu),\sum_{n=1}^\infty \|f_n\|_p^p<\infty\}$$And let the ...
1 vote
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### Solution of $u_{xx} + c^{2}u = 0$ in terms of sinh.

Let $c \in \mathbb{C}$ and $$u_{xx} + c^{2}u = 0 \ \ \text{in} \ \ (a,b)$$ with $u(b) = 0$. Why $$u(x) = u(a)\dfrac{\text{sinh}(c(x-b))}{\text{sinh}(c(a-b))} ?$$ I know the above equation can be ...
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### Equivalent definition of geometric vector bundles over schemes

The following is Hartshonre's definition of geometric vector bundles: Let $Y$ be a scheme. A (geometric) vector bundle of rank $n$ over $Y$ is a scheme $X$ and a morphism $f:X\to Y$ together with ...
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### How to manually infer that integral result?

I would like to simplify the formula manually as follows: $$\int_0^a{\sin \left( \frac{n\pi}{a}x \right) \frac{1}{x}\sin \left( \frac{m\pi}{a}x \right) dx}$$ Discuss the cases where (1) $n=m$ and (2)...
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### Arrangements of two sets of $n$ objects such that at any point, at least as many of the first set has appeared as the second set

Twenty children are queueing for ice cream that is sold at R5 per cone. Ten children have an R5 coin, the others want to pay with an R10 bill. At the beginning, the ice cream does not have any change. ...
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### Relationship between steering angle and turning angle from centre of mass of a car

I'm aware of the Ackermann Steering Angle, however, I still struggle to come up with an equation that model the relationship between $\dot{x}, \dot{y}$ (the horizontal and vertical velocity of the ...
28 views

Through testing, I have convinced myself that these two shapes are identical: $$(x,y) = \frac{1}{\sqrt{a\cos^2\theta + b\sin^2\theta}}( a\cos\theta, b\sin\theta)$$ $$r(\phi) = \sqrt{\frac{ab}{a\sin^2\... • 327 3 votes 0 answers 31 views ### Converse of Dynkin's formula It is a well known (see of theorem 19.21 of https://link.springer.com/book/10.1007/978-1-4757-4015-8 ) that any Feller stochastic process X with infinitesimal generator (\mathcal{L},\mathcal{D}) ... • 71 -2 votes 0 answers 26 views ### For what values of 𝑝 p is the series convergent? [closed] question is in this link$$\sum_{n=1}^∞ \frac{(-1)^n(ln(n))^p}{n^2}$$I have tried alternating series test and its conditions which the an part should be bigger than 0 and an+1 <= an and that it ... • 1 -2 votes 0 answers 14 views ### Euclidean geometry + affine transformations [closed] Good afternoon, I don't understand this exercise can anyone help me please? Show the following are equivalent for any map F:E^n → E^n: F is an affine transformation in some coordinate system, F(ax + ... • 1 0 votes 2 answers 38 views ### Open subset in \Bbb R defintion question Definition: A set A \subseteq \Bbb R is said to be open if \forall a \in A, \exists \epsilon >0 such that V_\epsilon(a) \subseteq A. Question: why there is V_\epsilon(a) \subseteq A in ... 0 votes 1 answer 26 views ### Homeomorphisms of unit squares satisfying properties We were given the following problems in our topology course as fun, challenging problems (they are not homework, and we are incouraged to discuss with others) Given I=[0,1], Problem 1: Find a ... 0 votes 0 answers 26 views ### Finding Burton's elementery number theory tough, any lower level book suggestion I am studying number theory from David Burton's book elementary analysis and second chapter on primes, in that exercise I had no idea how to start attempting questions on primes. The first chapter was ... • 103 0 votes 0 answers 10 views ### Prove if S \subseteq Ass_N(M) then their exist a submodule N such that Ass_R(N)=S. R is a commutative ring and M is a R-module. I am trying to prove, If S \subseteq Ass_R(M) then their exist a submodule N such that Ass_R(N)=S. I want use Zorn lemma to prove this problem. ... • 137 0 votes 1 answer 39 views ### Prove that \operatorname{adj}(AB-BA) = \operatorname{adj}(AB) - \operatorname{adj}(BA) then A, B \in M_{22} Prove or Disprove :$$\operatorname{adj}(AB-BA) = \operatorname{adj}(AB) - \operatorname{adj}(BA)$$where A and B are arbitrary n \times n matrices, and \operatorname{adj} is the adjugate(... • 1,234 0 votes 0 answers 22 views ### Bad reduction of normalization of singular curve 5y^2=x^4-17 Let C be a projective curve defined by 5y^2=x^4-17 over \Bbb{Q} and C' be its normalization(N.B. C is singular at infinity). I want to determine in which prime p, C' has bad reduction. ... • 4,275 1 vote 0 answers 25 views ### Quick exponentiation of bit matrices Is there a method for quickly rising to a power a matrix with only 0s and 1s? I am aware of the diagonalization method. However, it is general and requires a lot of work. Due to the constraint I ... 2 votes 0 answers 33 views ### Use cases for L^p and l^p spaces where p\neq 1,2,\infty Soft question: L^1,L^2, and L^\infty spaces all have many practical uses and an easy intuition behind them (Along with the l^1, etc. versions). Just for visualization, I was playing around ... • 16.2k -1 votes 0 answers 27 views ### Proof related to Chebyshev polynomials Consider x_j (j=0,1,....,n) the roots of T_n+1(x)=2xT_n(x)-T_n-1(x). Prove that for x belonging to [-1;1], the maximum of the absolute value of the multiplicand of (x-xj), j=0,...n equals to 1/2^n. 1 vote 1 answer 16 views ### Given a \sigma-algebra \mathcal{S}, is \mu:\mathcal{S}\to 0 a valid measure? Definition A real-valued function \mu defined on a \sigma-algebra \mathcal{S} is a measure if: \mu(\emptyset)=0; \mu(A)\geq0 for all A\in\mathcal{S}; \mu(\cup_k A_k)=\Sigma_k\mu(A_k) if ... • 508 0 votes 1 answer 7 views ### CGAL: rotate Nef Polyhedron CGAL gives an example of how to rotate a Nef_polyhedron 90 degrees around the x-axis (https://doc.cgal.org/latest/Nef_3/Nef_3_2transformation_8cpp-example.html). I am not exactly sure what the ... 4 votes 0 answers 73 views ### What is elementary mathematics? In Tao's exposition What Is Good Mathematics, he refers to the proof of Szemerédi's Theorem as "elementary" but at the same time "remarkably intricate and not easily comprehended". ... • 41 0 votes 0 answers 7 views ### Techniques to analyze locally ranked choices Suppose that we have, say, 100 menus M_1,\ldots,M_{100}, with each menu having 4 choices C_{i1},\ldots,C_{i4}. Each choice is a vector of length k and we can think of the components of the ... • 10.3k 1 vote 0 answers 27 views ### What is the density of integers whose prime factor exponents are in order? Let n = p_1^{a_1} p_1^{a_2} \cdots p_{\omega(n)}^{a_{\omega(n)}} be the prime factorization of n for some prime p_1, p_2, \ldots p_{\omega(n)} and exponents a_1, a_2, \ldots, a_{\omega(n)}. If ... • 16.4k 0 votes 0 answers 29 views ### How do we solve this two variable equation (involving periodic functions)? Is there a way to solve these sort of two variable system of equations:$$x^2((\sin(Ax))^2+(\sin(Bx))^2) + My((\sin(Cy))^2+(\sin(Dy))^2) - E=0$$where A, B, C, D, E, M \in \mathbb{R}. I could not ... • 29 -1 votes 1 answer 25 views ### Can an unbounded series of functions converge pointwise to a bounded function? I’m trying to find an example of a sequence of unbounded functions f_n: [0,1] \rightarrow \mathbb{R} which converge pointwise to a bounded function f: [0,1] \rightarrow \mathbb{R} . I can find ... • 37 -2 votes 0 answers 15 views ### Curve 25519 and prime-order Ristretto group: given different group elements is it possible to get the same curve point? [closed] Given two random elements x_1, x_2 from the prime-order Ristretto group and a Montgomery point y from Curve 25519, is it possible that values s_1 = y^{x_1} and s_2 = y^{x_2} will end up the ... • 669 1 vote 1 answer 21 views ### Prove that any orientation preserving isometry preserving a symmetric, positive definite bilinear form belongs to SO(n) Let V = \mathbb R^n, and Q : V\times V\to \mathbb R be a symmetric and positive definite bilinear form. I define SO(n) to be$$ SO(n) := \left\{ A \in SL_n(\mathbb R) ~|~ A^T A = I_n \right\}. $$... • 153 0 votes 1 answer 20 views ### For a distributive lattice, show x ~\hat{\land}~ a = y ~\hat{\land}~ a and x ~\hat{\lor}~ a = y ~\hat{\lor}~ a imply x = y. Let \hat{\land}, \hat{\lor} be binary operators denoting the infimum and supremum of two elements in a poset. I was given the following problem. For a distributive lattice \langle ~L, ~\hat{\lor}~,... • 2,650 0 votes 0 answers 19 views ### Terminology for collection of subsets of G = \cup_i G_i (a partitioning) of subsets of G which contain exactly one element of G_i for any i Let G_1,\ldots,G_k form a partitonining of a finite set G, and let \mathfrak S be the collection of subsets of G which contain exactly one element from each G_i, i.e$$ \mathfrak S := \{S \...
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Suppose the matrix representation of $T$ with respect to the ordered bases $B_1=\{(−2,1,0),(−2,0,1)\}$ for $V_1$ and $B2=\{(1,0,−2),(0,1,−2)\}$ for $V_2$ is given by $[(1,0),(0,0)]$. What's the ...