# All Questions

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### Why?$\int_0^1\int_{u(t)}^{u(t)+w(t)} f(t,v(t)) dv dt = \int_0^1 f(t,u(t)+\theta w(t))w(t) dt; ~~\theta\in[0,1]$

why: $$\int_0^1\int_{u(t)}^{u(t)+w(t)} f(t,v(t)) dv dt = \int_0^1 f(t,u(t)+\theta w(t))w(t) dt; ~~\theta\in[0,1]$$ how to get this ? Please help me Thank you.
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### Is $dxdy$ really a multiplication of $dx$ and $dy$?

In this link Is $dy/dx$ not a ratio? it was told that $\frac{dy}{dx}$ cannot be seen as a quotient, even though it looks like a fraction. My question is: does $dxdy$ in the double integral represent a ...
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### Cramér's Model - “The Prime Numbers and Their Distribution”

I was reading "The Prime Numbers and Their Distribution" by Gérald Tenenbaum and Michel Mendès France, the section about Cramér's Model, and I couldn't prove a couple of results. I would like to start ...
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### Derivative of sum of two functions is the sum of their derivatives.

Suppose $x_0 \in U \subseteq \mathbb{R}^d$, $U$ open, and $f,g : U \to \mathbb{R}^m$ differentiable at $x_0$, then $D_{f + g} (x_0) = D_f(x_0) + D_g(x_0)$. MY ATTEMPT Put $r(x) = f(x) + g(x)$. We ...
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Is ... 1answer 52 views +50 ### Estimating the number of connected components of a curve contained in a given set Let X be a metric space and \Omega\subset X an open set. Take x\in\Omega and choose r>0 such that the open ball B(x,r)\subset B(x,2r)\subset \Omega. Let \gamma:[0,1]\to X be a Lipschitz ... 3answers 115 views +50 ### Self-study: what fractions of problems to solve? I am self-studying measure-theoretic probability out of Billingsley's Probability and Measure. So far I have been trying to solve all the exercises. While the exercises are wonderful and I can ... 1answer 60 views +100 ### Confused about a version of Schauder's fixed point theorem I have read this: We have a map S:W_0 \to W_0. Moreover W_0 is not empty, convex, and weakly compact in W. Thus we can apply Schauder's fixed point theorem: Schauder's fixed point ... 2answers 334 views +350 ### Integral \int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx I'm struggling with this integralI=\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx.\tag1$$Mathematica could not evaluate it in a closed form. Its numeric value is ... 0answers 33 views +50 ### if the function f such 2\sum_{i=1}^{m}g(a_{i},f(a_{i}))>m,How many number f such this condition. give the two sets$$A=\{a_{1},a_{2},\cdots,a_{m}\},B=\{b_{1},b_{2},\cdots,b_{m}\}$$where$$a_{i}<b_{i}<a_{i+1}<b_{i+1},i=1,2,\cdots,m-1$$and define$$g(a,b)=\begin{cases} 1&a>b\\ ... 0answers 174 views +50 ### Is a field determined by its family of general linear groups? Assume thatK,L$are fields such that there is an isomorphism of groups$\mathrm{GL}_n(K) \cong \mathrm{GL}_n(L)$for all$n \in \mathbb{N}$. Does it follow that$K \cong L$? I am also interested in ... 0answers 42 views +50 ### Establishing the n-th order weak derivative. Consider $$AC^{n}:=\{f\in{AC}:f^{(k)}\in{AC}, 1\leq k \leq n-1\}$$ where$AC$stands for the space of absolutely continuous functions. Now, let$f,g\in{L_{loc}^{1}(a,b)}$and ... 0answers 63 views +100 ### A little bit of Intuition for Corepresentations from Representations Hi folks I am trying to prove what I think should be a straightforward enough result but I am having to make a somewhat unnatural definition to do it. This unnatural definition is hinted at in a paper ... 4answers 1k views +50 ### An exotic sequence Let$a=\frac{1+i\sqrt 7}{2}$and$u_n=\Re(a^n)$show that$(|u_n|)\to +\infty$I think basics method does not works here. Any ideas ? 2answers 61 views +50 ### help with showing completeness Let$\left\{H_n\right\}_{n=1}^\infty$be a sequence of Hilbert spaces and let$H=\left\{\left\{x_n\right\}:x_n\in H_n, \sum ||x_n||^2<\infty \right\}$. Define the inner product as ... 0answers 55 views +50 ### How find this sequence such this three condition (AMM Problem 2012) Prove that for all$n\ge 4$, there exsit integer$x_{1},x_{2},\cdots,x_{n}$such following conditions, (1):$$x_{1}=1,x_{k-1}<x_{k}<3x_{k-1},2\le k\le n-2$$ (2): ... 5answers 413 views +100 ### Proving a certain map on the closed unit disc must be the identity Re-bountied! Prove: Let$f$be a continuous function on the closed unit disc with two properties: 1.$f$is the identity on the boundary, i.e., on the unit circle. That is, if$|z| = 1$, then ... 1answer 51 views +100 ### Locally complete intersection in a fiber Let Y be an affine noetherian scheme,$Z = V_+(F_1, \ldots, F_r)$a closed subscheme of$\mathbb{P}^d_Y$that is flat over Y. Let$y_0 \in Y$be a point such that$Z_{y_0}$is a complete intersection ... 0answers 177 views +150 ### Probability that two halves of a partition have the same sum Consider the first$n \leq m $rows of a circulant matrix with$m$columns and call this matrix$M$. Let us say that$M_{i,j} \in \{0,1\}$. If$m>1$we can partition the columns of$M$into two ... 0answers 52 views +50 ### Allocating observations to test whether two expected values are equal$\mu_{1}-\mu_{2}=\Delta\neq 0?$Question: Suppose that you wish to test the hypothesis$H_{0}:\mu_{1}=\mu_{2}$versus$H_{1}:\mu_{1}\neq \mu_{2}$,where both variances$\sigma^2_{1}$and$\sigma^2_{2}$are know ,A tatal of ... 0answers 72 views +100 ### Characterization of Volumes of Lattice Cubes Let$C$be a lattice cube in$\mathbb{R}^n$. Characterize all possible volumes for$C$. A cube is called a lattice cube if and only if every vertex has integer coordinates. I broke this proof into ... 1answer 88 views +50 ### How prove$\sin{b}\sin{c}\sin{(b-c)}(\sin^2{b}+\sin^2{c}+\sin^2{(b-c)}+\sin{c}\sin{a}\sin{(c-a)}(\sin^2{c}+\sin^2{a}+\cdots=0$maybe have this condition: if $$a,b,c\in (0,\pi),a+b+c=\pi$$ show that ... 1answer 32 views +50 ### Equidistribution of lattice points on spheres in dimensions$d \ge 4$(Pommerenke's theorem) I am looking for either an English translation of: Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden. Acta Arith. 5, 227-257 (1959) Or a reference in ... 0answers 36 views +50 ### Proof of L'Hôpital's Rule for$x \to$finite$a^{+}$(J Stewart pp A-46) Reduce all cases of indeterminate form to$x \to 0^{+}$and$f(x),g(x) \to 0$. But Stewart proves it for$x \to finite \; a^{+}$. I don't know why? Ergo in place of$x \to 0^{+}$, I'll chagrin about ... 1answer 41 views +50 ### Solve:$\int_{0}^{2\pi}g \psi e^{i n \theta}\,\text{d}\theta = n/(n-i\alpha) \int_{0}^{2\pi}\psi e^{i n \theta}\,\text{d}\theta$For$\alpha>0$, I want to find a$g(\alpha, \theta)$such that $$\int_{0}^{2\pi}g(\alpha, \theta)\psi(\theta)e^{i n \theta}\,\text{d}\theta = \frac{n}{n-i\alpha} \int_{0}^{2\pi}\psi(\theta)e^{i n ... 1answer 86 views +50 ### find a chance that all N points lie on the half circle. We are given a circle with N randomly allocated points on it. Task is to find a chance that all N points lie on the one half of circle. I have drafted some solution: 1. Since there are no way to put ... 1answer 122 views +50 ### Air friction question Suppose I throw a ball up in the air (with air friction b>0) at time t=0 and it lands at time T_1. So we have the equation:$$m\frac{\mathrm{d}v}{\mathrm{d}t} = -bv + mg$$And suppose I ... 1answer 123 views +50 ### Strange closed forms for hypergeometric functions So in the process of trying to find a derivation for this answer, the following interesting equalities arose (one can check with Wolfram Alpha/Mathematica):$$\frac{8\sqrt{2}G^4}{5\pi^2} ... 0answers 50 views +50 ### Stability in first_dim piecewise linear map and cycles Consider the first_dim piecewise linear map: $$x^, = \begin{cases} \frac{1}{2}x + 1; & if\ x\le0\\ -\alpha\ x + 1; & if\ 0 < x \end{cases}$$$(i)$Find the fixed point(s) of the map and ... 0answers 36 views +50 ### Approximation of stochastic processes in Protter I'm reading Stochast integration and stochastic differential equation by Protter. In particular I have a question about Theorem 10 in chapter 2.4. Here Protter defines a simple predictable processes ... 0answers 73 views +50 ### Levy collapse gone bad Let$\kappa$be strongly inaccessible, and let$\mu<\kappa$be regular. What is the effect on$\kappa$of forcing with the following? (1) The product of$Col(\mu,\alpha)$for$\alpha<\kappa$, ... 2answers 2k views +200 ### The Hexagonal Property of Pascal's Triangle Any hexagon in Pascal's triangle, whose vertices are 6 binomial coefficients surrounding any entry, has the property that: the product of non-adjacent vertices is constant. the greatest common ... 0answers 45 views +400 ### How to denote this in game theoretic notation I'm writing a paper that demonstrates that linguists can use the concepts in game theory to infer what interlocutors naturally infer when the literal meaning of their utterances doesn't ostensibly ... 1answer 39 views +50 ### Does congruence guarantee length conversion? Suppose that a linear transformation$M:R^2 \rightarrow R^2$maps a triangle$ABC$to a congruent triangle$A'B'C'$($\{A, B, O\}, \{B, C, O\},\{C, A, O\}$are not colinear, and$A,B,C\neq O$) Is it ... 1answer 84 views +50 ### How prove this set inequality$|B|\ge 2|A|^2-1$let$A$is a finite set ,and the element are positive integers,and let $$B=\{\dfrac{a+b}{c+d}|a,b,c,d\in A\}$$ show that $$|B|\ge 2|A|^2-1$$ where$|X|$is define finite set$X$numbers This is a ... 3answers 76 views +100 ### Relation between Right Riemann sum and definite integral Let a partition$\{t_0,\ldots,t_n\}$of the interval$[a,b]$and let$f$an integrable function. (we may also assume that$f$is differentiable on$[a,b]$) We know that the Right Riemann sum is ... 0answers 66 views +50 ### How find this$aA_{m+1}=\overline{\sigma_{0}\sigma_{1}\sigma_{2}\cdots\sigma_{m}}$Question let$m$is positive numbers,and such$m\ge 5$,and $$A_{m+1}=\overline{1234\cdots m}=1\times (m+1)^m+2\times (m+1)^{m-1}+\cdots+(m-1)\times (m+1)+m$$(or see ... 0answers 76 views +50 ### How prove this there exsit positive integer$N$,such for every integer$n\ge N$, have$f(n)=n$Let$f:\mathbb Z^+\to\mathbb Z^+$be a function satisfying the following conditions: (1): For any positive integers$m$and$n$, ... 2answers 93 views +50 ### In search of Probability text recommendations The probability class I recently finished (taught at an upper-undergraduate or lower-graduate level) used the text by Grimmett and Stirzaker. I really disliked this book. I am familiar with measure ... 2answers 44 views +50 ### Question about minimizing points in a line Let$L$be defined as follows $$L = \{ (x,y,z) : x - y + 2z =4 , \; \; \; 2x+y-z = 1 \}$$ I want to find the point in$L$closest to$(0,0,0)$My approach LEt$(x,y,z)$be arbitrary point in$L$, ... 0answers 62 views +50 ### Irreducible homogeneous ideals We have the following question: Let$I$be a homogeneous ideal. Is it true that$I\$ is irreducible if and only if it can't be written as the intersection of two homogeneous ideal? So, is it ...

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