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I'm confusing now about the continuity of inf-convolution.
I understand that the inf-convolution of lower semicontinuous functions is semiconcave and so it's locally Lipschitz continuous (in particular, continuous.)
However, if one consider a jump function like
$$
f(x)=1\quad(\text{for $x<0$}),\quad=0\quad(\text{for $x... |
Assume $(\Omega,\mathcal{F}, \mathbb{P})$ is a probability space, $\{F_t\}_{t\leq T}$ is an adapted filteration and $M_t$ is a martingale with respect to that with $M_0=1$. We can define another probability measure on $F_T$ using the following radon nikodym derivative:
\begin{align}
\frac{d\mathbb{Q}}{d\mathbb{P}}=M_T
... |
Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ordering make $A^*$ well-quasi-ordered as well (reference to Corollary $1.7$ of [2]).
References:
http://www.math.harvard.edu/... |
Can someone please explain how they get from the step outlined in red to the one in blue? I tried using partial fractions to break it up but that didn't work, and I'm not sure how else I could rearrange it.
|
Question: Show that if the square of a rational number is an integer, than the rational number itself is an integer. Suppose the rational number is of the form $(\frac{m}{n})$, I have deduced that if $n^2$ divides $m^2$ then $n$ divides $m^2$. The hint in my textbook says to use the prime factorization theorem but I a... |
So I am just learning about conditional expectation in the modern probability sense. I understand that it is a random variable.
What I am having a hard time understanding is how you calculate the outputs of the conditional expectation in specific examples.
For instance I'm reading through this example and the author do... |
The question asks:
Parametrize the intersection of the surfaces using trigonometric functions.
$$y^2-z^2=x-6$$
$$y^2+z^2=81$$
$\mathbf{r}(t)=$ ____
My first step was recognizing $y^2+z^2=81$ as a circle in the $yz$-plane, which could be defined as $y=9\cos(t)$ and $z=9\sin(t)$.
I then thought that I could substi... |
At each integer $n$, $B_r(n)=\{n\}$ for small $r$, so $B_r(n)=\{n\} \subset \mathbb{Z}$. Since any subset is a union of some integers, does this imply that all subsets are open?
Also, since there is no limit point in a subset of $\mathbb{Z}$ because there's no sequence of integers converging to an integer, does this me... |
I've read many definitions in different books, and some of them specifically point out in their definition the existence of left inverse, left identity, and associativity. But the grand majority does not make this distinction, why would this be?
Plus, reading Kripke's Introduction to Analysis, there's an example that r... |
Evaluate
$$\iiint _{D}\vec{\nabla} \cdot\vec{F}\,dV$$
with
$$\vec{F}=\left \langle x^{2},y,z \right \rangle$$
$$D=\left \{ \left ( x,y,z \right )|x^{2}+y^{2}+1\leq z\leq 5 \right \}$$
$$\vec{\nabla}\cdot\vec{F}=2x+2$$
$$\int_{y=-2}^{2}\int_{-\sqrt{4-y^{2}}}^{+\sqrt{4-y^{2}}}\int_{x^{2}+y^{2}+1}^{5}2x+2\,dx\... |
I'm trying to prove that when $p ≡ 1 \pmod3$ is a prime, $p$ is reducible over the Eisenstein integers, and I've gotten to the point where, provided $p\,|\,u^2 - u + 1$ for some integer $u$, then $p$ is reducible, and in trying to prove the existence of such a $u$, I arrived at this problem. I may be going in the wrong... |
Let $p$ be a prime number and $A,B,C\in M_n(\mathbb{Z}_p)$ be nonsingular circulant matrices. How can I prove that this matrix $$\begin{bmatrix} A &B\\
B &C \end{bmatrix}$$ is nonsingular?
I have proved that $\det\begin{bmatrix} A &B\\
B &C \end{bmatrix}=\det(A)\det(C-BA^{-1}B)$. So the rest is to show that $\det(C-BA^... |
Consider the following infinite series:
$\sum_{i = 1}^{\infty} 2^i x^{(2^i)} = 2x^2 + 4x^4 + 8x^8 + 16x^{16} + ...$
I know that it converges for values of x between 0 and 1- can anyone help find a closed form in terms of x?
|
It is well known that the natural map $\phi :SL_2(Z) \to SL_2(Z/NZ)$ is surjective. So that the kernel, i.e. the principal congruence subgroup is of finite index.
But what if we replace Z with some ring of integers? Is this map still surjective?
|
The output system is:
$x(t)$ -->(S)--> $y(t) = \int_{-\infty}^{t}x(\tau) d\tau$
Recall that the system is causal if the output at $t$ depends only on input before $t$, or if the impulse response $h(t)=0,\forall t <0$
I guess the system is not causal, but I am not sure how to obtain the impulse response if we are give... |
I have a matrix of size nxm which consists of 0s and 1s..so i have to place 2x2 squares in matrix where there is 0. You cant place square where 1 is present.
The question is maximum 2x2 squares that can be placed in given 3x5 matrix eg
0 0 1 0 0
0 0 1 0 0
0 0 1 1 1..
So in above example we can place max of two 2x2 in t... |
I was asked by my teacher to prove that $7<e^2<8$ using only algebraic methods and knowing that $2<e<3$.
I don't know how to do this, where to start from, but I guess that I would need some kind of a function where: $f'(\xi )=\frac{f(b)-f(a)}{b-a}$ so as to exploit monotonicity of $f'$. Any hint?
|
For a function $F(x,y,z)$ with $(a,b,c)$ on the level surface $F(x,y,z)=k$, where $F(x,y,z)=k$ defines $z$ implicitly as a function of $x$ and $y$. Using the chain rule, assuming $F_z(a,b,c)\neq0$ show that at the point $(a,b,c)$,
$\frac{\partial z}{\partial x}=g_x(a,b)=-\frac{F_x(a,b,c)}{F_z (a,b,c)}$
I start by using... |
I have two questions. First, given a finite dimensional complex vector space $V$ and a finite group representation $\rho:G \to GL(V)$, Maschke's theorem tells us that we may decompose $V$ into a direct sum
$$V= \bigoplus_{\lambda}m_{\lambda}V^{\lambda}$$
of irreducible representations $V^{\lambda}$ of multiplicity $m_{... |
I should be able to get this problem...I'm studying for a qualifying exam and the question is to show that if $\mu(X) < \infty$, $f_n \to f$ a.e., and $\int f_n^2 \leq C$, then $f_n \to f$ in $L^1$.
I know $f \in L^1$ since this follows if $f^2 \in L^1$ since $\mu(X) < \infty$; then $\int |f|^2 = \int \liminf |f|^2 \... |
I'm looking for a quadratic form of the form $q(x,y)=ax^2 + bxy + cy^2 \in F[x,y]$, where $F$ has characteristic 2, and $q(x,y)$ has no roots besides the obvious one, $x=y=0$. I've proved the case of finite fields with characteristic $>2$ by group-theoretic arguments, but this case seems like it will require much more ... |
In statistics one aspect of the I.I.D. concept that bothers is when I think about it in the context of ordered statistics. As most of you already know, $Y_1,Y_2,Y_3,...,Y_n$ are I.I.D. when the parameters are the same.
Now, here are two things I'm confused about.
In the context of ordered statistics, each of $Y_{(1)... |
$$
\begin{align*}
x_{1} &= 1 + x_{2}\\
x_{2} &= 1 + \frac{1}{2} x_{3} + \frac{1}{2} x_{1}\\
&\vdots\\
x_{i} &= 1 + \frac{1}{2} x_{i+1} + \frac{1}{2} x_{1}\\
&\vdots\\
x_{n-2} &= 1 + \frac{1}{2} x_{n-1} + \frac{1}{2} x_{1}\\
x_{n-1} &= 1 + \frac{1}{2} x_{n} + \frac{1}{2} x_1 \\
x_{n} &= 0 \\
\end{align*}
$$
EDIT: I foun... |
Confused about multiplying conditional probabilities.
Does $$\Bbb P (A | B ) = \Bbb P(A | C) \Bbb P(C|B)$$ hold true ever?
|
I know a set $B$ is dense in $\mathbb{R}$ if an element of $B$ can be found between any two real numbers $a,b$ $s.t.$ $a<b$.
I have an inkling that, as $n \rightarrow \infty$, $q=2^n \rightarrow \infty$, and we can play with the values of $\dfrac {p} {q}$ such that for any $a,b \in \mathbb{R}$, $0<b-a< \dfrac {p} {q}$... |
I am trying to find $a$ and $b$ such that $\sqrt{9+6\sqrt{2}}=a+b$, obviously this can just be evaluated by a computer to find $\sqrt{3}$ and $\sqrt{6}$ but I'm wondering if there's a relatively simple algebraic way of calculating it other than using the above equation and $9+6\sqrt{2}=a^2+2ab+b^2$
|
Let $A$ be a non-empty set and let $X$ be a subset of $S_A$
Now let $F(X) = \{a \in A : \sigma(a) = a, \forall \sigma \in X\}$, $M(X) = A\backslash F(X)$, and $D = \{ \sigma \in S_A : \mid M(\sigma)\mid < \infty \}$
Prove $D$ is a normal subgroup. The straight forward one-step or two-step subgroup test with added norma... |
Angle of sine is in degrees, can anyone show me an easy soln to this? This was question was given to us for 1minute without calcu.
I know that $\sin4^\circ=\sin176^\circ$, $\sin8^\circ=\sin172^\circ$.....
But dont know how to proceed
|
Can I do universal instantiation on the following predicate :
$ \forall x\;S(x)\; \lor\; \forall x\;L(x)$ become $S(c)\lor L(c)$
or is it has to be $\forall x\; ((S(x) \lor L(x))$ to be able to do the instantiation ?
|
Given $a, b \in \mathbb{R}$, such that $b-a>1$, there is at least one $k\in \mathbb{Z}$ such that $a<k<b$.
My attempt:
Consider $E:=(a,b)\cap \mathbb{N}$. We need to show that $E$ is not empty. Set $b:=a+1+\varepsilon$. Then $(a,b)=(a, a+1+\varepsilon)$ for some $\varepsilon > 0$. But $(a, a+1+\varepsilon)$ contains th... |
Hi i have weird problem with calculate eigenvector from simplest matrices. So have something like this:
$A = \begin{bmatrix} \frac{1}{2} & 0 \\ 2 & \frac{1}{2} \end{bmatrix}$
Eigenvalues are : $\lambda_{1} = \lambda_{2} = \frac{1}{2}$
So i need calculate eigenvector next:
$\begin{bmatrix} 0 & 0 \\ 2 & 0 \end{bmatrix}\b... |
I have a question, how was the geometric series collapsed to be in the form of $2^{k+1}$?
|
It's the definition of partial derivative from Rudin's PMA.
Why he consider $(25)$ for real functions $f_i$? What about if $f_i$ in $(25)$ replaced by vector-valued function $\mathbf{f}=(f_1,f_2,\dots,f_m)$? I guess that it could be more general.
Any answer is greatly appreciated.
|
Suppose I've got a sequence of vectors $\{x_n\}_{n∈N}$ in $\mathbb{R}^k$. Why is it that exactly one of the following three facts must hold:
$\lim
_{n→∞}
{||x_n||} = ∞$,
$\lim
_{n→∞}
{x_n} = 0$, or there exists a convergent subsequence with a nonzero limit?
My progress is that if I have a sequence, then it must either... |
Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$
$H$ is the set of fixed points on $A$
$A$ : set, $H \le S_A$, $F(H) = \{ a \in A : \sigma (a) = a, \forall \sigma \in H \}$
Let $\tau \in N_{S_A}(H), \sigma \in H$
$\tau[F(\sigma)] = [F(\tau\sigma\tau^{-1})] = [F(\sigma... |
How to find $\lim_{x\to 0}\sin^2(\frac{1}{x})\sin^2 x$ ? I tried using taylor expansion:
$$((x-\frac{x^3}{6}+\frac{x^5}{120}+O(x^5))(\frac{1}{x}-\frac{1}{6x^3}+\frac{1}{120x^5}+O(x^{-5})))^2$$
but it gets very complex. I am looking for simple evalutation. Another option that I tried was using $\sin A\sin B =\frac{1}{2... |
The question I'm stuck on asks: Find the matrix representation of the differential operator D acting on the space of
polynomials of degree at most $3$ with basis $(3, 1 + x, x − x^
2
, 1 + x^
3)$, onto the space of
polynomials of degree at most $3$ with basis $(1 + x, 1 − x, 1 + x^
2)$
I have the answer but I have no i... |
I read that the function $f(x)$ has a vertical tangent at $x=a$ in the domain of $f$ if
$$f'(a^-) \to +\infty$$ and $$f'(a^+) \to +\infty$$ Or both approach to $-\infty$.
But for $f(x)=\sin^{-1}x$ $f'(x)=\frac{1}{\sqrt{1-x^2}}$ and we have
$$f'(-1^+) \to +\infty$$ and $$f'(1^-) \to +\infty$$ but $f'(-1^-)$ and $f'(1+)$... |
Let $$t_{n}=\frac{1}{n}\left(1+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n}}\right),\ n=1,2,\dots$$ then I want to know if $\sum_{n=1}^{\infty}t_{n}$ converges/diverges and the sequence$\{t_{n}\}$ converges and diverges
for it I thought of finding $\lim_{n\to\infty}t_{n}\\=\lim_{n\to\infty}\frac{1}{n}\sum_{r=1}^{n}\fra... |
Let $M$ be a W*-algebra and consider the following map:
$$\gamma: M\times M^*\to M^*: (a,f)\to af$$
where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under the weak-star topology $\sigma(M^*,M)$.
Q: Is $\gamma$ weakly-weak-star jointly continuous? I mean, assume that $\{a_i\}$... |
Is there a method to approximate the inverse of a sine function in ones head?
I know one can approximate the inverse of a cosine with the following equation:
$\cos^{-1}(x) = \sqrt{7(1000-1000x)}-\frac{1}{2}$
Does a similar equation exist for inverse of sines?
|
I know how to come up with $F(n, k) = \sum \limits_{p=1}^n p^k$ recursively knowing $F(n, k-1)$.
But what if I want to find it in a very short time?
I know how to find fast $f_n = \sum\limits_{i=1}^{n-1} a_i f_i$, where $(a_1, ..., a_{n-1})$ are constants. It can be achieved using matrix multiplication in logarithmic t... |
I know that the maximum size of an independent vertex set, also known as the independence number and denoted by $\alpha$, plus the the size of the minimum vertex cover, $\tau$, is equal to the number of vertices in the graph, $n$.
$$\alpha(G) + \tau(G) = |G|$$
However, for a complete graph I don't understand how this i... |
Task:
Find the cardinality of all such functions $f: P(\mathbb N) \rightarrow P(\mathbb N)$ that $f(\bigcup S) = \bigcup \lbrace f(Z) \mid Z \in S \rbrace$
The answer is: $\mathfrak c$
My approach:
Define $R(A)$ as the set of all equivalence relations on $A$. Let's denote the set of desired function by $A$. There ar... |
The conjecture is as follow: In $\mathbb{C}^{n}$, there exists $\{v_1,\cdots,v_{n^2}\}$ such that the following holds:
$$ \left| \left \langle v_i, v_j \right \rangle \right| = \begin{cases} 1 & i = j\\ \frac{1}{n+1} & i \ne j\end{cases}$$
I have a prove for when $n = 2$, basically what I did is just assuming withou... |
In the definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$, it is said that : objects remain the same and arrows' directions are changed (that is, and arrow $f:A \to B$ in $\mathcal C$ is and arrow $f:B \to A$ in $\mathcal C^{op}$, and vice versa. Now I think this can not be a safe definiti... |
As far as I know, mathematics is written based on a language which can be for example set theory or category theory. My concern is about the power of these languages. How can we realize which language is more powerful and we can construct more structures with them? Maybe, there is an assumption which implies all langua... |
I'm trying to count the frequency of a flush hand in $7$-card poker. Since a flush hand could be thought of as having $5$ cards with the same suit while the other $2$ doesn't matter, I wrote down as follows. $$4\ _{13}C_{5}\ _{47}C_{2}=5564988$$ But this clearly is too much than the actual frequency of flush. What did ... |
Could you please help me proving rigorously the following fact from Mayer's book:
(a) if $X_t$ is a process adapted with respect to filtration $\{\mathcal{F}_t\}_{t\ge 0}$ and for every $\varepsilon>0$ is progressively measurable with respect to $\{\mathcal{F}_{t+\varepsilon}\}_{t\ge 0}$. Then $X_t$ is progressively ... |
Blue and red rectangles are drawn on a blackboard. Exactly 7 of the rectangles are squares. There are 3
red rectangles more than blue squares. There are 2 red squares more than blue rectangles. How many blue rectangles are there?
|
Let $X$ be a topological space and $A \subseteq X$ a subset.
$A$ is called sequentially compact iff every sequence in $A$ has a convergent subsequence with limit in $A$. A point $x \in X$ is an $\omega$-accumulation point of $A$ iff every open neighborhood of $x$ contains infinitely many points of $A$.
Is the following... |
As the titles suggests, I am trying to see if we can upper bound
$\mathbb{E}[X \text{ }| \text{ }X > c] - c$
For now, I am assuming bounded mean on both sides: $0 < m \leq \mathbb{E}[X] \leq M < \infty$, and bounded second moment: $\mathbb{E}[X^2] \leq V \leq \infty$. Additionally, X is a random variable with unbound... |
I try to get numerical solution of ordinary harmonic oscillator with symplectic integrator. The problem is that what I obtain doesn't conserve energy (but symplectic integration should do).
I consider 2 simplectic integrators - symplectic Euler method and symplectic 4th order method.
The equations of motion are:
$$ \d... |
Let $p$ be a prime. Let $n$ be a positive integer dividing $p-1$.
Suppose that $a^{(p-1)/n}=1$ in the finite field $\mathbb F_p$ with exactly $p$ elements. Then does there exist $b$ with $a=b^n$?
Of course if there exists such an element $b$ then $a^{(p-1)/n}=1$ by Fermat little theorem.
|
I want to define for which $a,b,c \in \mathbb{R}$ $$(X,Y) = X^T
\begin{pmatrix}
b+6 & 0 & 11 \\
0 & 6 & a \\
11 & -2 & 1
\end{pmatrix}
Y + cy_1^2$$
is an inner product of $\mathbb{R}^3$.
($y_1$ first coord. of $Y$).
I have shown that $a = -2, c = 0$ by just controlling symmetry and bilinearity of $( \cdot, \cdot)$.... |
Consider the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$.
Why would it suffice to suffices to assume that this vector lies in the unit sphere in $\mathbb{R}^{d+1}$? How would I show this to be true?
|
Suppose that $(x_n)_{n\in\mathbb N}$ is a sequence of positive real numbers satisfying
$x_n\to 0$; and
$n^{-1}(\log x_n)\to c<0$
as $n\to\infty$. I want to show that $x_n=O(\exp(n c))$; that is, the sequence $\{x_n\exp(-n c)\}_{n\in\mathbb N}$ remains bounded. I keep running in circles and I have been unable to come ... |
Shown is the graph of $y=f(x)$,a polynomial function of degree $10$ whose domain is restricted to $[1,5]$.Function $f$ is symmetric about $x=3$.Compute the number of solutions to the equation $f(x)=f(f(x))$.
My effort
Considering the cases where $f(a)=f(b)$ where $a \ge 1 $ and $b \le 5 $ ,I must have that $a=b$ or $b... |
$R$ is a commutative ring. We need to prove that $R[x]$ has a subring isomorphic to $R$.
Let $S$ be that subring of $R[x]$ which has polynomials of even degree. Now I consider a mapping from $S$ to $R$ as : $f(x) \to f(0)$ where $f(x)$ is a polynomial of even degree and $f(0)$ is clearly in $R$.
The mapping defined a... |
Given that an event occurs at some rate $\lambda$ per unit time, I know that the probability that n such events occur in unit time is given by the Poisson distribution.
$P(n) = e^{-\lambda}\frac{\lambda^n}{n!}$
However, let's say I measure the time intervals at which the next event occurs. I obtain a histogram since th... |
In PDE-constrained optimization problems, the distributed control problem
$$ \begin{array}{ll}
\displaystyle \min_{y,u} & J(y,u) = \frac{1}{2}\|y-y_d\|_{L^2(\Omega)}^2 + \frac{\alpha}{2}\|u\|_{L^2(\Omega)}^2 \\
\text{subject to} & \mathcal{D}y = u, \quad \text{on}\ \Omega, \\
& \phantom{\mathcal{D}}y = g, \quad \text{... |
Prove $\displaystyle\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}\,dx$ exists.
I want to use Dominated convergence theorem to show the limit, but I'm stuck on finding the bound $g(x)$ here. Once I get $g(x)$, I will just exchange the limit and integral.
|
My problem is the following:
I have a dataset in particular have $4$ dimensions, for didactic reasons I need to represent this dataset as a $m\times n$ matrix array such that the ($i$-th, $j$-th) cell containing a $4$-dimensional data. So far I have not found as repsentar this mathematically.
|
I have to prove $$P[x_1 < X \le x_2,y_1 < Y \le y_2] = P[x_1 < X \le x_2]P[y_1 < Y \le y_2]$$ using the property $F(x,y)=F(x)F(y)$, which is the cumulative distribution function, where $X$ and $Y$ are independent. I know to prove this I need to start from the definition of a cumulative distribution function. I did $$F(... |
Suppose we have three random variables $X, Y$ and $Z$, it is known that $$P(X >
Y ) = P(Y > Z) = \dfrac23$$ Then,
$P(Z > X) > 0.5$.
$X-Z$ has positive mean.
The median of $X$ is larger than the median of $Z$.
The median of $Y$ is smaller than that of $X$.
None of the above is correct.
I guess the answer is $2.$ since... |
While learning calculus I stumbled upon this concept of convergence. Is this some general concept or just related to sequence and series. What is its importance?
|
I understand how to use the compound angle formula when solving $\sin(\pi/12)$.
However I dont understand how I can use a compound angle formula to show
$$\arctan(3)-\arctan(1/2)=\pi/4$$
Thankyou
Any references would be highly appreciated, I looked online but couldn't find any that targeted the arc-tan/sin/cos.
|
I have a problem trying solving this ODE:
$$(x+y)^2dx+(2xy+x^2-1)dy=0$$
I tried the following s=teps:
$$M=(x+y)^2dx, N=(2xy+x^2-1)dy$$
$$\frac{\partial M}{\partial y}=2(x+y)=2x+2y$$
$$\frac{\partial N}{\partial x}=2y+2x=\frac{\partial M}{\partial y}\rightarrow Exact $$
$$\varnothing_1=\int Mdx=\int (x+y)^2dx=\frac{(x+... |
I'm interesting to know how do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{{(-1)}^{n²}}{{(i\pi)}^{n}}$$, I have tried to evaluate it using two partial sum for odd integer $n$ and even integer $n$ ,but i can't since it's alternating series ,and i would like to know if it's well know series also what about it's value... |
I'm pretty sure that any point in $\mathbb{R}^n$ can be written as a sum of finitely many unit vectors (in $\mathbb{R}^n$, of course). However, I have no idea how to go about proving this.
Any ideas?
|
$(X_1,\dots, X_n)$ is a random sample extracted from a uniform distribution on the interval $$(\alpha-\beta, \alpha+\beta) \ \ \ \ \alpha \in \mathbb{R}, \beta \in \mathbb{R}^{+}$$
Demonstrate that $E[X_1]$ depends only from $\alpha$ and $Var[X_1]$ depends only from $\beta$
Calculate the estimators $T_1$ of $E[X_1](... |
Let me ask a very simple question: Let $(A_n)$ be a sequence of integers defined by $A_0 = 1$ and
$$\forall n \geq 1 : A_n = \sum_{k=1}^{n} 2^{k+1} \cdot A_{n-k}.$$
There is an explicit formula for $A_n$, and it is quite simple. I can find it using the formalism of generating functions. But I wonder, is there another w... |
Let $f\colon X \to Y$ be a uniformly continuous function. Then I think it is "well-known" that it may be approximated by a Lipschitz function, and how well one can do this depends on the modulus of continuity of $f$.
So if we have a family of uniformly equicontinuous functions, we may approximate them equally well by L... |
$$\left(\left(\frac1N \sum z_i \right)- I \right)^2 = \frac1{N^2}\left(\sum (z_i - I)^2 \right)$$
How does this work? Or is there an error? I thought that you could not pull the sum out of the square?
Context is calculating $$Var\left(\frac1N \sum z_i - I\right)$$ for a Monte Carlo Estimator, where I is the desired val... |
Def: for every $\epsilon$ $\lim_{n}P(|X_n-X|>\epsilon)=0$
How to turn it into a statement of there is an N s.t. n>N...
Shall we make it $P(\lim_{n}|X_n-X|>\epsilon)=0$ first?
|
How do I solve this simultaneous
$$a + b + c = 2\to (1)$$
$$ax + by + cz = 0\to (2)$$
$$ax^2 + by^2 + cz^2 = \frac{2}{3}\to (3)$$
$$ax^3 + by^3 + cz^3 = 0\to (4)$$
$$ax^4 + by^4 + cz^4 = \frac{2}{5}\to (5)$$
$$ax^5 + by^5 + cz^5 = 0\to (6)$$
This simultaneous has 6 equations, 6 unknown and some variables are raised to... |
$$U=\operatorname{span}(v_1,v_2,v_3)$$ where
\begin{aligned}
v_1=(1,1,1,2) \\
v_2=(1,2,3,1) \\
v_3=(0,1,2,-1)\end{aligned}
I need to find if $u=(2,1,0,5) \in U$
If $u \in U$ it is a linear combination of $v_1,v_2,v_3$
$$ (2 ,1, 0, 5)=a(1, 1, 1, 2) + b(1, 2 ,3, 1)+ c(0 ,1 ,2, -1) $$
$$
\left\{
\begin{aligned}
a+b=2... |
How can one prove that for any positive integer $d$, there is a constant $A_d < 0$ such that
$$
\sup_{x∈[0,1]}
{\lvert\, p'(x)\rvert} ≤ A_d\sup_{x∈[0,1]}{\lvert\, p(x)\rvert},
$$
for all polynomials $p$ of degree at most $d$?
|
Context: I have a huge circuit with lots of input bits (around 300). Among these, only about 40 are free, the others are fixed by the current state. I have to find all satisfying assignments knowing there are no more than 10.
This is Circuit-SAT which is NP-complete. Currently I solve a Tseitin-transformed version, but... |
My goal is to prove the following inequality, known as the Dvoretsky-Kiefer-Wolfowitz inequality (1956) :
Let $(X_i)_{i \geqslant}$ be iid random variables. Let $\displaystyle F_n(x)= \frac{1}{n}\sum _{i=1}^n 1_{X_i \leqslant x}$ and $F$ the distribution function of $X_1$. Then there exists a constant $C>0$ such that... |
When we have to n×n matrices that can be made diagonal (maybe not in the same basis), is it true that the same works for their product?
|
Two random variables $X$ and $Y$ follow the same distribution.
Then
The distribution of $X − Y$ must be symmetric about $0$.
The median of $X − Y$ must be zero.
The median of $X + Y$ is twice of the median of $X$.
The mean of $X − Y$ , if finite, must be $0$.
None of the above is correct.
It is a very short question,... |
Evaluate the following:
[$-x^{n+1} e^{-x} ]_{x=0}^{x=\infty}$, where $n$ is any integer greater than 1
Any help?
|
I am studying Paul R. Halmos Measure theory. In the section 5 of chapter 1, theorem 5 states that :
If $E$ is a countable class of sets, then $R(E)$ is countable.
The proof uses class of all finite unions of differences of sets of class E.
Can anyone explain this is a simple manner or any other methods of proof?
|
Suppose that $a_n$ is a sequence of integers. Prove that $(a_n)$ converges if and only if there is a natural number $N$ big enough and an integer $m$ such that for any $n \geq N$, $a_n = m$.
I'm given this question to work on for my homework but I don't know how to start on it. Could someone guide me on how to start on... |
Let the security level ($p-$payoff, $M-$set of all strategies) $$B_i:=\sup_{\sigma_i \in M_i} \inf_{\sigma_{-i}\in M_{-i}}p_i(\sigma_{-i},\sigma_i)$$
Now I consider $3-$person zero sum game. The questions are:
When every player use strategy, which guarantee $B_i$ is that Nash equilibrium?
Is that true: $B_1+B_2+B_3=0... |
Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$:
$$K(t,x,x)\sim (4\pi t)^{-n/2}\sum\limits_{k=0}^{\infty}u_k(x,x)t^k.$$
Often it is said that the functions $u_k(x.x)$ can be written a... |
This is a problem I found trying to find some properties related to exchangeable sequences. Anyway, I am not able to find a characterization of all solutions.. I know there are at least two completely different solutions, is it possible to find all of them?
Problem: Find all pairs of (finite non-negative) measures $\a... |
Suppose that a distribution $R\in D'(\Bbb R)$ satisfies the following estimation for an independent constant $c$:
$$\forall \phi\in D(\Bbb R)\quad |\langle R,\phi\rangle|\le c\|\phi, \,L^1(\Bbb R)\|.$$
It is easy to show that $R$ is a tempered distribution; this follows form the inequality
$$\int_{\Bbb R}|\phi(x)|dx\l... |
Prove or disprove: for each natural $n$ there exists an $n \times n$ matrix
with real entries such that its determinant is zero, but if one changes any single
entry one gets a matrix with non-zero determinant.
I think we may be able to construct such matrices.
|
Let $f$ be a continuous function on an open interval in $\mathbb R$ such that $|f|$ is differentiable. Can we show that $f$ is differentiable? I can get several examples of non-differentiable $f$ if $f$ is not continuous. But here I am stuck. Please help.
|
In How to prove that $k^3+3k^2+2k$ is always divisible by $3$?, I showed $k^3+3k^2+2k$ is divisible by $3$ using Euler's theorem, specifically, Fermat's little theorem. Then I thought that it is possible to find condition that given polynomial with integer coefficient is divisible by $3$.
First attempt:
Let $k$ be an ... |
A committee of $6$ people will be formed with $7$ men and $6$ women. The oldest of the $7$ men is A and the oldest of the $6$ women is B. It is described that the committee can include at most one of A and B. In how many ways can the committee be chosen?
My attempt:
$13C6- (6C1 \cdot 7C5+6C5 \cdot 7C1) = 1548$
Correc... |
Gauri Apartment housing society organised annual games,
consisting of three games: snooker, badminton and tennis.
In all, $510$ people were members in the apartments'
society and they were invited to participate in the games.
Each person participating in as many games as he/she feels like.
While viewing... |
If the perimeter of a the right-angle triangle is six times its smallest side,
find the ratios of the three sides.
I tried solving it by using the normal area and volume.
|
Given an integer $n >2$,for how many different rational numbers $x$
does there exist a polynomial $P(x)$ of degree $n-1$ with $P(0)=0$,and
with all integer coefficients,such that $54x^n+P(x)=315$ ?
My effort
From $P(0)$ I must have that $P(x)$ is a polynomial without constant term ,i.e $$P(x)=a_{n-1}x^{n-1}+\cdot... |
Suppose that
$$
T\begin{pmatrix}
x \\
y \\
\end{pmatrix} =
\begin{pmatrix}
x-y\\
x+y \\
2x
\end{pmatrix}
$$
I tried multiplying with a scalar as follows
$$
T \left\{\alpha\begin{pmatrix}
x\\
y \\
\end{pmatrix}\right\}
=T\begin{pmatrix}
\alpha x\\
\alpha y \\
\end{pmatrix}
= \begin{pmatrix}
... |
Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$.
We can then, of course, define a metric $d_p(q, q') := |q − q'|_p$, and we know thanks to the strong triangle inequality that $d_... |
Can every one variable equation be solved without graphing?
How would you solve the following without graphing:
$$3y + 4\sqrt{1-y^2} = 2$$
|
Picking randomly a number with two digits that can not be divided by $10$
$\color{gray}{\text{- A) Find the expectation value and the variance of the number}}$
$B)$ Find the expectetion value and the variance of the digits multiply
$C)$ Find the probability using Chebyshev's inequality that the number will be
$1.$ g... |
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