1
vote
1answer
28 views

Change of variable integral

Consider the following integral \begin{equation} \int_{\Omega} f(y,My-z_2)\, g(z_1,z_2,y) ~ dz_1\, dz_2\, dy \end{equation} $f(y,My-z_2) = 1$ (a constant function for each value of $y$ and $z_2$) $...
2
votes
0answers
357 views

Suggest solutions book

Does somebody know solutions manual for book "An Introduction to Mathematical Cryptography" by Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman?
2
votes
1answer
79 views

Euclid's lemma for non-prime numbers.

I was trying to prove that $\sqrt{6}$ irrational as: Let $$\sqrt{6}=\dfrac ab$$ $$\implies a^2=6b^2$$ $$6|a^2 \implies 6|a$$. I should not be able to do the step because 6 is not a ...
1
vote
1answer
43 views

How prove this usefull identities with equations

let $f(x):R\to R$ be $C^{k+1}$, show that $$\left(\dfrac{d^2}{dr^2}\right)\left(\dfrac{1}{r}\dfrac{d}{dr}\right)^{k-1}\left(r^{2k-1}f(r)\right)=\left(\dfrac{1}{r}\dfrac{d}{dr}\right)^k\left(r^{2k}\...
14
votes
1answer
481 views

Did Einstein introduce anything new to mathematics? [duplicate]

Newton introduced calculus, so I am wondering, did Einstein introduce anything important to mathematics?
2
votes
0answers
133 views

Can “tit for tat” strategy be defined in monadic second-order logic?

Prisoner's dilema game can be represented as a game tree, which could be infinite game with corresponding infinite game (binary) tree in common case. There is well-known tit for tat strategy, which ...
1
vote
1answer
73 views

Can someone explain me the sentence about ideals?

Can someone explain me the sentence: "If $R=K[x]$ the prime ideals are $\langle f(x)\rangle $ where $f(x)$ is an irreducible polynomial in $K[x]$ and $\langle 0\rangle $, and again $\langle f(x)\...
2
votes
1answer
35 views

Symplectic basis

Let $\omega$ be a holomorphic one-form on a Riemann surface $X$ of genus $g$. Can one find a symplectic basis $(a_i,b_i)$ for $H_1(X,\mathbb{R})$ such that $\int_{a_i}\omega=\delta_{i1}$? Thanks.
0
votes
1answer
77 views

How to show that $(\operatorname{Im}{A})^⊥ = \ker(A^⊤)$, $\operatorname{Im}{(A^⊤)}= (\ker A)^⊥$? [duplicate]

Let now $A ∈ L(V)$, where $V$ be an Euclidean vector space. I need to show that: $$(\operatorname{Im}{A})^⊥ = \ker(A^⊤)$$ $$\operatorname{Im}{(A^⊤)}= (\ker A)^⊥$$
2
votes
1answer
85 views

Convexity of circle in neutral geometry

I am trying to prove that a circle is convex in neutral geometry. i.e. If $A$ and $B$ are inside a circle $C$, than any point in $AB$ is also in $C$. But I have difficulty in proving it. The case $...
3
votes
1answer
85 views

Is the product eigenvalues less than or equal to the product of singular values?

On a numerical math course I recently saw the following statement without a proof. How would one prove it? Let $A$ be an $n$ x $n$ real matrix with singular values $s_1 \geq s_2 \geq \ldots \geq ...
0
votes
2answers
71 views

Simplifying boolean algebra expression $(AB+AC)'+A'B'C$

$$\eqalign{(AB+AC)'+A'B'C&=\overline{(AB+AC)}+\overline A \,\overline BC\\&=(\overline A+\overline B)(\overline A+\overline C)+\overline A\,\overline BC\\&=\overline A+\overline B\,\...
0
votes
1answer
69 views

Showing compactness of complete metric space

I need to show that for $K>0$, $$X=\{f:[0,1]\rightarrow [0,1]\mid |f(x)-f(y)|\leq K|x-y|\ \forall x,y \in [0,1]\}$$ with the metric $d(f,g)=\max|f(x)-g(x)|$ , (supremum metric), is a compact ...
1
vote
0answers
67 views

Zeroes of polynomials with several variables

Just consider a polynomial with multiple variables on the field of real number, A is its zeroes set. Is A a zero measure set ? How to prove that?
1
vote
1answer
153 views

Sectional Curvature of Paraboloid

I seem to have made a mistake while doing the simple exercises of calculating 2D sectional curvature of paraboloid $z=\frac a2 (x^2+y^2)$. I used polar coordinates to do this; $(r,\theta)=(\sqrt{x^2+y^...
3
votes
2answers
44 views

Distributions with finite number of moments

Is it easy to provide an example of a distribution that has, say, finite moments of order one and two, but such that $\mathbb{E}[X^k]=\infty$ or does not exist for all $k>2$ (where $k$ is not ...
0
votes
2answers
48 views

How to show that $(U^{\bot})^{\bot}=U$, if $U$ is a linear subspace of $V$ and $V$ is finite-dimensional?

Let $V$ be an Euclidean vector space with scalar product $(.|.)$. If $S ⊂ V$ is any subset of $V$ , define the orthogonal complement of $S$ by $$S^{\bot}=\left\{v\in V| \forall s\in S:\left(s|v\right)=...
5
votes
1answer
173 views

Filtered colimits commute with forgetful functors

In many cases, filtered colimits commute with forgetful functors to $\mathbf{Set}$, for example with $\mathbf{CRing} \to \mathbf{Set}$ or $R-\mathbf{Mod} \to \mathbf{Set}$. Is there a slick way of ...
1
vote
3answers
152 views

Hypergeometric function integral representation

How to prove the following relation? $$ \, _2{F}_1(K,K;K+1;1-m) = \frac{\Gamma (K+1)}{\Gamma (K)} \int_0^{\infty } \frac{1}{(1+x) (m+x)^K} \, dx $$ where $_2{F}_1(.,.;.;.)$ is the hypergeometric ...
2
votes
1answer
100 views

If a family $\{f_n\}$, is uniformly integrable, then also $\{|f_n|\}$ is.

I want to show that if a set of functions $\{f_n\}_n$ is uniformly integrable, then also $\{|f_n|\}_n$ is also uniformly integrable. How can I show this? My guess is to use separate $f_n$ into real ...
1
vote
0answers
215 views

About Intersection of two convex polytope?

the intersection of two convex hull of two polytope P and Q , is it the convex hull of the intersection of P&Q ? Conv(P) ∩ Conv(Q) = conv(P∩Q) ???.
0
votes
1answer
57 views

Multiple hypothesis testing

Let's suppose I have 10 independent measurements with results close to zero. How can I claim that they are in agreement with the theory, them being zero? The errors of these 10 results are not equal, ...
1
vote
1answer
74 views

Pullback of the skyscraper sheaf

Let $\phi:X\longrightarrow Y$ be a morphism of schemes, and let $y\in Y$. Let $k(y)$ be the constant sheaf $k(y)$ on the closed subset $\{\bar{y}\}$. Then what is $\phi^*(k(y))$? By definition, $\phi^...
1
vote
1answer
21 views

an injective map can not take several intersecting arcs onto line segment

I read a result in the theory of harmonic mappings, and i think it might be true in general setting as well. But i am unable to get a proof of this. Can anyone help me with proving it. The statement ...
0
votes
0answers
36 views

About the birational maps between surfaces

Let $S^{'}$ and $S$ two complex algebraic projective surfaces. Suppose that there exist an étale cover $p:S^{'}\rightarrow S$ with connected fibers. Now suppose that $S^{'}$ is ruled. is it true that $...
5
votes
1answer
688 views

What comes before precalculus [closed]

I studied in Europe and followed mostly European academic ways. I'm tutoring a young person , and I will to know what math class comes before pre calculus. Because soon we will be starting some ...
1
vote
6answers
90 views

Solving $\int \frac{1}{\sqrt{x^2 - c}} dx$

I want to solve $$\int \frac{1}{\sqrt{x^2 - c}} dx\quad\quad\text{c is a constant}$$ How do I do this? It looks like it is close to being an $\operatorname{arcsin}$? I would have thought I could ...
0
votes
1answer
48 views

Which are matrices 2×2 that commute with the matrix

Which are matrices $2\times 2$ that commute with the matrix $$\left[\begin{array}{cc}1&1\\1&1\end{array}\right]?$$
2
votes
2answers
1k views

Solve for x in tanx-2x=0

I know homework questions are generally frowned upon here, but I've run into the following equation, which I've tried to solve and am having a genuinely hard time with: $$\tan(x)-2x=0,x\in(-\pi/2, \...
3
votes
2answers
171 views

Solving a 2 independent variables (2nd degree) recurrence relation

Changes to the recurrences and definition are changed! See here: $f(n, 1) = 2n^2 $ and $f (n, k) = 0$ for $k \geq 2n$ and for $k < 0$ and $f(n, 2n-1) = 1$ for all $n$. Question: Is it possible ...
2
votes
1answer
92 views

Equivalence of induced representation

Let $H$ be a subgroup of $G$. In Wiki, it gives an algebraic construction of induced representation. And it is equivalent to the vector space $Hom_{H}(\mathbb{C}[G],V)$ i.e. $ \{ f:G \rightarrow V ...
1
vote
1answer
54 views

For positive random variables, $X_n \stackrel{p}{\to} X$ and $E(X_n) \to E(X)$ implies $E|X_n-X|\to 0$

Let $X,X_1,X_2\ldots$ be positive random variables. Suppose $X_n \stackrel{p}{\to} X$ and $E(X_n) \to E(X)$. Show that $E|X_n-X|\to 0$. I observed certain things. For example, it is enough to prove ...
0
votes
1answer
62 views

How to write approximations of a sequence $x_n = {1/3^n}$

Write three approximations of the sequence ${x_n} = {1/ 3^n}$, using the following scheme - $P_0= 1, P_1 = 0.33332$ and $P_n = (6/5)P_{n-1} - (1/5)P_{n-2}$ for $n = 2, 3,\dots$ Further, make a ...
0
votes
0answers
33 views

Checking for a function to be homeomorphism

Let $h=(h_1,h_2):U\to \mathbb{R}^2$ be one-one and continuous in a neighborhood $U$ of the origin in $\mathbb{R}^2$ with $h(0)=0$ and $h_1$ harmonic. Then $h:U\to h(U)$ is bijective and continuous. If ...
1
vote
4answers
357 views

Find the derivative of y with respect to the given independent variable

Find the derivative of y with respect to the given independent variable: $y = 3^{-x} \stackrel{D}{\longrightarrow} y' = 3^{-x} \cdot (-1) \cdot \ln 3 $ This is my teacher's solution. I don't ...
1
vote
1answer
47 views

Unitary transformation of Fubini-Study metric

I am trying to solve a problem in Introduction to Complex Geometry by D. Huybrechts, question 3.1.6 which is the following: let $A\in GL(n+1, \mathbb{C})$ be a $\mathbb{C}$-linear transformation $\...
2
votes
3answers
62 views

In statistics what does mutually exclusive mean?

For homework a question related to Venn diagrams is 'Are the probabilities of having not used a spinner and not tossed a coin in the game mutually exclusive' Don't know what is meant by it so can't ...
1
vote
2answers
47 views

Proving a theorem about vector spaces.

This is one of the exercise problems I found in Halmos' 'Finite Dimensional Vector Spaces': Let $V$ be a finite-dimensional vector space and $M$ and $N$ be two of its subspaces. Then, if $$M \...
0
votes
2answers
41 views

Distance from a proper subspace of a metric space.

Let $(X,d)$ be a metric space. And $A \subset X$ be a proper closed subspace. Proper here implies every closed and bounded ball is compact. Define $d_A(x)= inf_{a \in A} d(x,a)$. Show that this ...
0
votes
1answer
72 views

Evaluating the limit $\lim_{x\to-1}\frac{\sqrt{x}-1}{x-1}$

How would you solve $\displaystyle\lim_{x\rightarrow-1}\left(\dfrac{\sqrt{x}-1}{x-1}\right)$ ? I tried multiplying it by the conjugate. I don't know how to get rid of the square root.
1
vote
1answer
131 views

Inequality about the euler characteristic of surfaces

Let $S$ a complex algebraic projective surfaces. Suppose that there exist a surjective morphism $p:S \rightarrow B$ such that $p$ has connected fibers and $B$ is a smooth curve. Call $F_{\eta}$ the ...
2
votes
2answers
78 views

Finding the area of the surface of a planar region

Find the area of the surface bounded by the region $x+y+z=3a/2$, within the bounds of $0≤x,y,z≤a$ What i tried I first take the unit normal vector $n=1/3<1,1,1>$.I believe i must do an ...
1
vote
1answer
710 views

Ideal contained in a finite union of prime ideals

Let $I \subset R$ be an ideal and $P_i$ $(i=\{1,...,n\})$ prime ideals with $I\subseteq\bigcup_{i=1}^nP_i$. Prove that then $I$ is contained in one $P_i$. I don't know how to show this because I ...
0
votes
1answer
45 views

How to show that $S^{\bot}$ is a linear subspace of $V$?

Let $V$ be an Euclidean vector space with scalar product $(.|.)$. If $S ⊂ V$ is any subset of $V$ , define the orthogonal complement of $S$ by $$S^{\bot}=\left\{v\in V| \forall s\in S:\left(s|v\right)=...
0
votes
1answer
89 views

a question about central simple algebras

I have a question about double centralizer therorem. Thanks for any help. " If $A$ is a finite-dimensional central simple algebra over a field $F$ and $B$ is a simple subalgebra of $A$, then $C_A(C_A(...
1
vote
1answer
174 views

On the largest and smallest topology on a given set.

Let $\{ \mathscr{T}_{\alpha} \}_{\alpha \in \Sigma }$ be a family of topologies on a given set $X$. Question: How can I find the unique smallest topology on $X$ containing all the collections $\...
0
votes
1answer
115 views

Proof of Hensel's Lemma (particular version)

Let $k\in \mathbb{N}$ and $p$ an odd prime number which do not divide $a$. We suppose that there is a solution $u_k$ to the equation $x^{2} \equiv a [p^{k}]$. Then, there is only one solution $u_{k+1}$...
1
vote
2answers
64 views

Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.

Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$. A hint is given that for $0 \le \alpha \le \beta$ there exist a number $\xi \in (\beta, \alpha + \beta)$ s.t. $$(\...
0
votes
1answer
47 views

Basis of a tensor

I'm new to tensors, and I need to understand what a certain basis actually is, how to visualise it. Say we have the $r$-dimensional vector space $T_p M$ and $n$-dimensional dual space $T^{*}_p M$. ...
-3
votes
3answers
89 views

Is (-1)^n a Cauchy sequence?

I proved $(-1)^n$ is a Cauchy sequence using this method. I just wanted to check if it is correct. Let $n,m \gt N$. $$ |(-1)^n-(-1)^m|= |(-1)^n+(-1)^{m+1}| \le |(-1)^n|+ |(-1)^m+1| \le 1+1= 2 $$ ...

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