text stringlengths 83 79.5k |
|---|
H: Let $H$ a Complex Hilbert space, $T, T_n \in B(H)$. Prove that $T_n x \to Tx$.
I need help for the next exercise:
Let $H$ a complex Hilbert space and $T, T_n \in B(H)$ for $n \geq 1$ and $\|T_{n} x\| \to \|T x\|$ and $\langle T_n x,x \rangle \to \langle Tx,x \rangle$ for all $x \in H$. Prove that $T_n x \to Tx$ for... |
H: Is there a generic criterion for a function to be the gradient of another function?
I am interested to know whether there is a generic criterion in which a function $f:\mathbb{R}^n \to \mathbb{R}^n$ satisfies $f(x) = \nabla d(x)$ for some function $d:\mathbb{R}^n \to \mathbb{R}$.
Many thanks!
AI: Provided your func... |
H: Existence of Hausdorff space comprised of disjoint dense subsets
Problem $13H$ of Willard states -
For any set $X$, there is a Hausdorff space $Y$ which is the union of a collection $\{Y_x:x\in X\}$ of disjoint subsets dense in $Y$.
I have no idea how to do this. Any help is appreciated!
AI: Here’s a sketch of on... |
H: Integrate $\frac{\theta \sin \theta}{1+\cos^2 \theta}$ with respect to $\theta$
Integrate :
$$\int_0^\pi \frac{\theta \sin \theta}{1+\cos^2 \theta} d\theta$$
I tried to do a substitution by letting : $u=\cos \theta \implies du=-\sin\theta\ d\theta$
But I have a problem with that $\theta$, I don't know how to get b... |
H: Proof of the First Isomorphism Theorem
Here's what I'm trying to prove;
Let $T: V \to W$ be a vector space homomorphism. Let $N = \ker(T)$. Then, the quotient vector space $V/N$ is isomorphic to $T(V)$.
Proof Attempt:
Define the following map:
$$\forall x \in V: f(x+N) = T(x)$$
I claim that this is bijective and l... |
H: Let $H$ a complex Hilbert space, $T, T_n \in B(H)$. Prove that $T_n x \to Tx$.
I need help for the next exercise:
Let $H$ a complex Hilbert space and $T, T_n \in B(H)$ for $n \geq 1$ and $\|T_{n} x\| \to \|T x\|$ and $\langle T_n x,x \rangle \to \langle Tx,x \rangle$ for all $x \in H$. Prove that $T_n x \to Tx$ for... |
H: Using Wilson's Theorem to find solution to $n^2 \equiv -1 \pmod{p}$
I am currently studying number theory and its basic topics, and I came across this problem.
Wilson's Theorem states that if $p$ is prime, then $(p-1)!\equiv -1 \pmod{p}$.
If $p \equiv 1 \pmod{4}$ is prime, then use Wilson's Theorem to find a number... |
H: How to rewrite $R^TAR = V$ to find $A$
How to write the following equation to find $V$, the eigenvalue matrix. Note that $R$ is the orthogonal eigenvector matrix, and $A$ is one of the symmetric matrices.
AI: I assume that we have $R^TAR = V$, and $R$ is an orthogonal matrix. It follows that
$$
R^TAR = V \implies\\... |
H: Prove $\frac{tf(t)}{\int_0^t f(s)ds}\ge\frac{tg(t)}{\int_0^t g(s)ds}$ given $f(0)=g(0)=0, f'(0)\ge g'(0)\ge 0, f''(s)\ge g''(s)> 0$ for $s\in[0,t]$.
I thought about a statement that is intuitively right, but doesn't know how to prove it. So could anyone help me with this: Say two second order differentiable functio... |
H: If $A$ and $B$ are invertible matrices, then $A^2B^{-1}$ is invertible.
Let $A$ be an $n \times n$ matrix and $B$ be an $n \times n$ matrix. I know:
$$
\begin{align*}
AA^{-1}=A^{-1}A=I_n \tag{1}\\
BB^{-1}=B^{-1}B=I_n \tag{2}
\end{align*}
$$
Starting from $B^{-1}B=I_n$, I have the following series of equalities:
$$
... |
H: Is "each player always defects" a Nash equilibrium in Iterated Prisoners Dilemma
For the iterated prisoners dilemma with random ending time, is it the case that "both players defects each round" is a Nash equilibrium?
I understood a Nash equilibrium as a set of strategies for which it holds that no player benefits ... |
H: Proof of the Second Isomorphism Theorem
Here's what I'm trying to prove:
Let $V$ be a vector space. Let $M$ and $N$ be linear subspaces of $V$. Then, it is the case that $M/(M \cap N)$ is isomorphic to $(M+N)/N$.
Proof Attempt:
Define the relation $Q: M/(M \cap N) \to (M+N)/N$ as follows:
$$\forall x \in M: Q(x+M ... |
H: If $zw$ is real, must $w$ be a multiple of the conjugate of $z$?
I'm talking about this:
$$c(a+bi)(a-bi) = c(a^2+b^2)$$ where $c$ is real.
Is it true that $a+bi$ can only be multiplied with $c(a-bi)$ in order to get a real number or are there other complex numbers that can multiply to get a real number? If this is... |
H: Find a counter example that if $X$ banach space is not reflexive , the the operator doesn't have to be compact
Let $X, Y$ be a banach spaces, $A\in \mathcal{L} (X,Y)$ , there is a proposition that affirms that if $X$ is reflexive , and for any $ x_n\rightarrow x$ weakly in x we have that $ Ax_n\rightarrow x$ when ... |
H: Factor $3+2\sqrt{3}i$ as a product of irreducible elements in the ring $\mathbb{Z}\left [ \sqrt{-3} \right ]$
The question I am having trouble with is:
Factor $3+2\sqrt{3}i$ as a product of irreducible elements in the ring $\mathbb{Z}\left [ \sqrt{-3} \right ]$
I don't really understand how to go about this besid... |
H: Integrating $\int_0^1 \frac{\ln(1+x)\ln^3 x}{1+x}\,dx$ with restricted techniques
How does one calculate these four following integrals?
$$ \int_0^1\frac{\ln(1\pm\varepsilon x)\ln(x)^3}{1\pm \varepsilon x}\,dx,\;\forall\varepsilon\in\{-1,1\}.
$$
CONTEXT:
Our teacher asks us to to calculate these four integra... |
H: The tropical integers
Let
\begin{align}
\oplus_\mathbb{N} &= + \\
0_\mathbb{N} &= 0 \\
\odot_\mathbb{N} &= \cdot \\
1_\mathbb{N} &= 1
\end{align}
Then $(\mathbb{N}, \oplus_\mathbb{N}, 0_\mathbb{N}, \odot_\mathbb{N}, 1_\mathbb{N})$ is the ordinary rig of natural numbers. Let $\mathbb{Z} = \mathbb{N}^... |
H: Show that $P:\mathbb{R}^k\to\mathbb{R}$ by $P(x_1,...,x_k)=\sum_{(i_1,...,i_k)\in I}c(i_1,...,i_k)x_1^{i_1}\cdot\cdot\cdot x_k^{i_k}$ is continuous.
Let $k\geq 1$, let $I$ be a finite subset of $\mathbb{N}^k$ and let $c:I\to \mathbb{R}$ be a function. Show that $P:\mathbb{R}^k\to\mathbb{R}$ by $P(x_1,...,x_k)=\sum... |
H: What is the relationship between $p$-norm and 2-norm for any $p$-norm?
I know that there are some general inequalities between the $2$-norm and $1$-norm or $\infty$-norm
Suppose I am given an arbitrary $p$-norm, obviously $p \geq 1$
What can we say about the inequality between $\|x\|_2$ and $\|x\|_p$?
AI: The answe... |
H: Not sure where my argument breaks down: Let $A \subset \mathbb{R}$ be a countable set. Prove that $\mathbb{R} \setminus A$ is uncountable.
$\textit{Proof.}$ Since A is countable, we know it has the same cardinality as $\mathbb{N}$, or $\mid A \mid = \mid \mathbb{N} \mid$. Additionally, since $\mathbb{N} \subset \ma... |
H: Show $\forall c>0$ have $\int^b_af(x)dx=c\int^{b/c}_{a/c}f(cx)dx$ from the definition of integral
Let $f[a,b]\to\mathbb{R}$ be an integrable function. Prove the following, using only the definition of the integral
$$\text{For any}~c>0,\int^b_af(x)dx=c\int^{b/c}_{a/c}f(cx)dx$$
Hint: A careful choice of notation is ... |
H: Two types of proofs conditional expectation.
I am reviewing the proof about the conditional expectation of $e$ (Conditional Expectation Function Error) given $X$ is zero. This is:
$$e= Y-m(X)$$
$$E(e|X)=E(Y|X)-E(m(X)|X)$$
$$E(e|X)=E(Y|X)-E(Y|X)$$
$$E(e|X)=0$$
where $m(x)$ is $E(Y|X)$. X and Y are random variables.
... |
H: $q-p$ is a projection when $pq = p$
Consider the following theorem in the book "$C^*$-algebras and operator theory" written by Murphy.
Questions: (1) Is the theorem talking about orthogonal projections? (as defined in the text above)? Or simply projections?
(2) How to prove $(2) \implies (6)?$ I assume that the th... |
H: Question on Hölder's inequality when it's equal
I'm reading this proof where they are trying to show that the Holders inequality becomes equal iff $|f|^p$ and $|g|^q$ are multiples of each other.
I'm stuck in this step:
$$\int\vert fg \vert\leq \Vert f \Vert_p \Vert g \Vert_q \int\left( \frac{\vert f \vert^p}{p\Ve... |
H: Find formula for linear transformation given matrix and bases
Let T: $\mathbb P_2\to \mathbb R^3$ be the linear transformation with matrix $[T]_{B,A}=\begin{bmatrix}1&1&-1\cr 0&-1&-1\cr -1&0&1 \end{bmatrix}$ relative to the bases $A = \{1,2-3x.1+x^2\}$ and $B = \{(1,1,1),(1,1,0),(1,0,0)\}$ find the formula for the ... |
H: Number of chains with size $k$ in a symmetric chain paritition of subsets of $\{1,2,…,n\}$.
In a partition of the subsets of $\{1,2,…,n\}$ into symmetric chains, how many chains have only $k$ subset in them?
For a chain $A_1\subseteq A_2\subseteq ...A_l$ in the partition, we have $|A_1|+|A_l|=n$ and $|A_i|+1=|A_{i+... |
H: Is the Random Variable's Expectation the optimal Solution for the Mean Squared Error?
Lets assume we compute the Mean Squared Error between an estimated, but fixed variable $\hat{x}$ and a dataset ${x_1, x_2,...,x_n}$, sampled from a non-gaussian distributed random variable $\mathcal{X}$:
$$
\frac{1}{N} \sum^N_i (\... |
H: Are Semiregularity and Complete Hausdorff Properties preserved by Products
By preserved by products I mean - $\prod X_{\alpha}$ has property $P$ iff $X_{\alpha}$ has property $P$ for all $\alpha$ in index set
Also, $X$ is Completely Hausdorff if for $x\neq y$ in $X$, $\exists$ continuous function $f:X\to I$ with $... |
H: Is $K_a$ subspace of vector space $V$?
Let $\text{Hom}(V,V) = H$ be the set of linear transformations from $V$ to $V$. Let $a \in V, K_a \subset H$ such that for all $T \in K_a, T(a)=0.$ Is $K_a$ a subspace? Does there exist an $A$ such that $K_a=H$? Let $a_1, a_2, …, a_n$ be a basis $B$ for $V$. What is $\cap_{a_i... |
H: Conventional to write the derivative of $|x|$ wrt $x$ as $\frac{x}{|x|}$?
This might be a naive question, but it sometimes confuses me.
It's known that the derivative of $|x|=\frac{x}{|x|}$. Is it conventional that the absolute value appears in the denominator and not $\frac{|x|}{x}$ ?
AI: Yes, it is simply by conv... |
H: For integral domains $R$ and $S$, if $S \simeq R^n$ as $R$-modules, does $\mathrm{Quot}(S) / \mathrm{Quot}(R)$ have degree $n$?
Let $\phi : R \rightarrow S$ be an injective morphism of integral domains. Then, $\phi$ extends to a morphism $\mathrm{Quot}(R) \rightarrow \mathrm{Quot}(S)$, so corresponds to a field ext... |
H: family of pairwise disjoint sets in the complement of meager. set
Let $M$ be a meager subset of $\mathbb R$. I want to construct the following family in $\mathbb R\setminus M $
$$F:= \{A_{r}^\xi\colon r\in\mathbb R \ \&\ \xi<c\}$$ Such that all elemnets of $F$ are pairwise disjoint dense sets and $\bigcup F$does no... |
H: Showing a Result of the Logisitic Equation
The logisitc equation is given by $$\frac{d N}{dt}=rN\left(1-\frac{N}{K}\right)\tag{1},$$ where $K$ is the carrying capacity and $r$ is the intrinsic growth rate. I am trying to show for the logisitc equation that $$r=-\frac{d}{dt}\left(\ln\left(\frac{K-N}{N}\right)\right)... |
H: Question about function of bounded variation's derivative
$\text { Let } u_{n}(x)=\frac{1}{n^{2}} e^{-n^{2} x^{2}}, x \in \mathbb{R}$
(i) Calculate Var $u_{n}$
(ii) Let $u(x):=\sum_{n=1}^{\infty} u_{n}(x), x \in \mathbb{R} .$ Prove that $u \in B P V(\mathbb{R})$
(iii) Prove that $\sum_{n=1}^{\infty} u_{n}^{\prime}(... |
H: Linear Algebra problem involving $v^{\perp}$
Let $v,w$ be non-zero vectors in $\mathbb R^3$. Assume that the set of vectors orthogonal to them is a plane through the origin. Then each of them is a scalar multiple of the other other.
As usual, set $v,w=(\alpha_1,\beta_1,\gamma_1),(\alpha_2,\beta_2,\gamma_2)$ respe... |
H: What is the maximum integral value of $\frac{\beta-\alpha}{\tan^{-1}\beta-\tan^{-1}\alpha}$
If $0<\alpha<\beta<\sqrt3$ and if
$$k=\frac{\beta-\alpha}{\tan^{-1}\beta-\tan^{-1}\alpha}$$
then find maximum value of $\lfloor k\rfloor$.
My Attempt:
Let $\beta=\tan B$ and $\alpha=\tan A$
$$k=\frac{\beta-\alpha}{\tan^{-1... |
H: Find $\lim_{x\to 0} \frac{\sqrt{ax+b}-1}{x}=1$
My answer
Let $\sqrt{ax+b}=y$
Then
$$\lim_{y\to \sqrt b} \frac{(y-1)a}{y^2-b}$$
Let $b=1$
Then $$\lim _{y\to 1} \frac{a}{\frac{y^2-1}{y-1}}$$
$$=\frac a2 =1$$
$$a=2$$
The answer is correct, but this relies on assuming $b=1$, and that doesn’t seem appropriate. What is t... |
H: When trying to find the number of unique pairs in $n$ elements why does the combination formula output a different value from $n(n-1)/2$
I was recently doing a problem on LeetCode that involved counting the number of pairs of dominoes that could be formed from a set. And I have a question about the math behind a ce... |
H: Law of cosines, Ptolemy's, angle chasing on an isosceles triangle inscribed in a circle
From HMMT:
Triangle $\triangle PQR$, with $PQ=PR=5$ and $QR=6$, is inscribed in circle $\omega$. Compute the radius of the circle with center on $QR$ which is tangent to both $\omega$ and $PQ$.
I haven't made much progress. I'... |
H: yes/ No :Is $A \cong B ?$
Let $A$ and $B$ be subspaces of $\mathbb{R}$ given by $A = (0, 1) \cup (3, 4)$ and $B = (0, 1) \cup (1, 2)$. Is $A \cong B ?$
My attempt : yes because both $A$ and $B$ are disconnected set and open interval
Is its true ??
Note :$\cong$ mean homeomorphism
AI: You're right, but you should g... |
H: Ideals $(X^2+1)$ and $(X^2+1, 7)$ of polynomial ring $\mathbb{Z}[X]$
How can I show that generated ideals $(X^2+1)$ and $(X^2+1, 7)$ of polynomial ring $\mathbb{Z}[X]$ are a prime ideal and a maximal ideal, respectively?
AI: To show $(X^2+1)$ is a prime ideal, imagine you took two polynomials with integer coeffici... |
H: How could be given two ratios are equivalent?
I'm currently learning Maths and got interested in Ratios. Currently, I'm going through equivalent ratio lesson and found that to be magical somehow.
I am impressed that given two ratios can have same value, but I don't know how is it possible and I know the rule and ca... |
H: Degree of a determinant
Let $p$ be a prime number and $n\in\mathbb N$. Consider the determinant
$$M_n = \begin{vmatrix}\frac1{x^{p^{n+1}}-x}&\frac1{x^{p^{n+1}}-x^p}\\ \frac1{x^{p^{n+2}}-x}&\frac1{x^{p^{n+2}}-x^p}\end{vmatrix} \in \mathbb F_p(x)$$
Numerical computations suggest that
$$\deg(M_n)=p-(p+2)p^{n+1}$$
Is i... |
H: Continuity of a Map on a Quotient
I am trying to understand the continuity of a map on a quotient from Loring Tu's An Introduction to Manifolds (Second Edition, page no. 72). It starts as follows.
My Question
What does it mean that $f:S \to Y$ is constant on each equivalence class? In other words, the definition o... |
H: $18$ mice were placed in $3$ groups, with all groups equally large. In how many ways can the mice be placed into $3$ groups?
In my textbook, the given answer is $18!/(6!)^3$.
But my teacher's answer is $18!/(3!)(6!)^3$.
He solved like -
Please review the attached answer and let me know which one answer is correct?... |
H: I want to know why is my idea wrong in theorem
In theorem : Let $~\mathbb{F}~$ be a field and $f(x)$ a nonconstant polynomial of degree $n$ in $\mathbb{F}(x)$. Then there exists a splitting field $\mathbb{K}$ of $f(x)$ over $\mathbb{F}$ s.t. $~[\mathbb{K}:\mathbb{F}] \leqslant n!~$ where $~[\mathbb{K}:\mathbb{F}]~$... |
H: Is $\sup$ and $\inf$ of a set single?
Lets say we have a set & its $\sup$ is $30$ whether $\max$ exists or no, is it safe to say $30$ is the ONLY $\sup$?
AI: The supremum and infimum of a set is unique. This can be proven.
Take a set $S$, with supremum $\alpha$. Suppose the supremum is not unique, and $S$ has a sec... |
H: Must an epsilon neighbourhood be small?
Is there any requirement or restriction that an epsilon neighbourhood $V_{\epsilon}(a)$ be small? Could I use $\epsilon = 1$ if my needs demand it?
AI: As long as $\epsilon > 0$, it can be whatever you need it to be. It doesn't mean that your proof of a given theorem will wor... |
H: What does it mean for an expression to be "finite but not infinitesimal"?
Also according to the definition of a positive infinitesimal:
a hyperreal number b is positive infinitesimal if b is positive but less than every positive real number.
So how are real numbers other than 0 able to have infinitesimals around... |
H: What is the definition of value?
Imagine a number line. Each number has a value, but how is value defined? For rational numbers this is self evident, but what about real numbers? For example, we know that $\pi>e$, because $\pi>3$ while $e<3$. So here we have a rational number between them. This however doesn't tell... |
H: Proof for Cauchy's Definition of Centre of Curvature.
Cauchy defined the center of curvature as the intersection point of normals drawn to two infinitely close points on a curve. Is there any way to prove this?
I am unable to get a starting point for this.
AI: I'll assume you are dealing with biregular plane curve ... |
H: How I can write a negligible compact set?
My problem:
Suppose $E$ is a negligible compact set, i.e. $E \subset \mathbb{R}$ is compact and $|E|=0$ where $|.|$ denote the Lebesgue measure. Can I write $E=\bigcap_{k \in \mathbb{N}} U_k$ where $U_k$ is open and $|U_k| \leq 2^{-k}$?
My attempt:
I tried taking for every ... |
H: Is it possible to show, $S_4$ has no subgroup isomorphic to $Q_8$ using homomorphisms?
Question: Prove that $S_4$ has no subgroup isomorphic to $Q_8$?
My attempt: If $Q_8$ isomorphic to subgroup of $S_4$ then there exists a homomorphism from $f: S_4\rightarrow Q_8$ such that,
$\frac{S_4}{\ker f}≈Q_8$ (Am I correc... |
H: Is knowing a sheaf equivalent to knowing all the stalks?
We have a topological space $(X, \tau)$, and a sheaf of functions $F: \tau \rightarrow \textbf{Set}$ on $X$. Now I build a new sheaf of stalks of F, defined as $G(U) \equiv \prod_{x \in X} F_x$. I feel that this $G(U)$ ought to carry the same information as $... |
H: Relation between covariance and uncorrelation/independence
I need to have an explanation... If I have a covariance equal to zero, the random variables are only uncorrelated, or are uncorrelated and independent at the same time?
Thanks
AI: They are uncorrelated but not independent, in general.
Conversely, if the var... |
H: Expectation value of $Y= \lfloor(\frac{x}{2})\rfloor+1$
I need help to solve this question:
Let us define $X \sim \text{Geo}(2/3)$, $Y=\Big\lfloor \frac{X}{2} \Big\rfloor + 1$.
Find $\mathbb{E}[Y]$=?
I cant seem to get the infinite sum correctly, if someone can give me any tip or guidance would be greatly appreciat... |
H: Multivariate Taylor series with Hessian evaluated at a linear combination of $x$ and $\Delta x$
I found the following theorem, but I don't understand it and was unable to prove it. Is it true? Is there a proof for it?
Theorem: Let $f : R^d → R$ be such that $f$ is twice-differentiable and
has continuous derivatives... |
H: Bounded sequence in $L^{\infty}$ has weakly convergent subsequence
It would be great if someone could name me a reference where I can find a proof for the following statement, thank you! :)
Given is a bounded sequence $(f_{n})_{n\in\mathbb{N}}\in L^{\infty}$. Then there exists a $f\in L^{\infty}$ and a subsequence... |
H: Open convex subset of compact-open topology
Let $X$ be a locally-compact metric space, $E$ be a Banach space, and equip $C(X,E)$ with the compact-open topology. Suppose that $Z\subseteq C(X,E)$ is such that $\operatorname{span}(Z)$ is dense in $C(X,E)$. Then, is $co(Z)\triangleq \left\{\sum_{i=1}^n k_iz_i:\,k_i \... |
H: $\int_0^1 \frac{x^p}{1-x^q}\; dx＝∞$
When $p>-1, \;q>0$, I want to prove
$$\int_0^1 \frac{x^p}{1-x^q}\; dx = \infty.$$
Any help would be appreciated. I observed by graph soft this is true, but I cannot prove. I want to find some function $\varphi(x)$ which satisfies
$$\int_0^1 \varphi(x)dx = \infty,$$
such that
$ ... |
H: The characteristic of a ring with unity
Let $R$ be a ring with unity. Suppose $n\cdot 1\neq 0$ for all $n\in\mathbb N$. Here $n\cdot 1$ means the sum of $n$ unities. I would like to prove that $R$ has characteristic $0$. To this end, I want to show that we cannot have $n\cdot a=0$ for all $a\in R$ for some $n\in\ma... |
H: Existence of open convex set separating a closed, compact convex set and a point
If $(V, ||\cdot||)$ is a normed vector space and $A \subseteq C$ is compact (and therefore closed since the topology on $V$ induced by $||\cdot||$ is Hausdorff) and convex, and $x \in V \backslash A$, then does there exist an open, con... |
H: Calculate: $\lim _{x\to \infty }\left(\frac{\sqrt{x^3+4}+\sin \left(x\right)}{\sqrt{x^3+2x^2+7x+11}}\right)$.
Calculate:
$$\lim _{x\to \infty }\left(\frac{\sqrt{x^3+4}+\sin \left(x\right)}{\sqrt{x^3+2x^2+7x+11}}\right)$$
Here's my attempt:
I first tried to the statement up. I first let $f(x)=\left(\frac{\sqrt{x^3... |
H: Projecting 3D points onto 2D coordinate system of a plane
This is a rather basic question though I could not find a post with the answer.
I have a set of points in 3D. Let us define one of them $s$ as an observation point. we calculate the 3D Euclidean distances and choose the furthest point from $s$, denoted $r$. ... |
H: Ideals in a UFD
Consider the ideal $I=(ux,uy,vx,uv)$ in the polynomial Ring $\mathbb Q[u,v,x,y]$, where $u,v,x,y$ are indeterminates. Prove that every prime Ideal containing I contains the Ideal $(x,y)$ or the Ideal $(u,v)$.
I am not able to choose the correct combinations of products of the four indeterminates to ... |
H: Given a convergent sum $\sum_{n=1}^{\infty}a_n$, prove/disprove: $\sum_{n=1}^{\infty}a_n(1-a_n)$
Given a convergent sum $\sum_{n=1}^{\infty}a_n \ $, prove/disprove: $\sum_{n=1}^{\infty}a_n(1-a_n)$ is convergent
My Attempt:
By dividing the question into cases, as for the first case; $\sum_{n=1}^{\infty}a_n$ is defi... |
H: Prove or disprove: $\ker (TS)=\{0\}$ $\implies$ $\ker (S)=\{0\}$
Prove or disprove: $\ker (TS)=\{0\}$ $\implies$ $\ker (S)=\{0\}$.
$\bullet~$ $\textbf{My attempt:}$
(not necessarily)
we will define
$$S:\mathbb{R}^2 \rightarrow \mathbb{R}^2 ~\text{ and }~ T:\mathbb{R}^2 \rightarrow\mathbb{R}^2$$
from the given
$$\... |
H: How is the MSE calculated?
I have trouble with the following problem:
Consider the independent random variables $[X_1,\ldots,X_{19}]$ For each
$i$ their probability mass function is given by $p(X_i=-1)$ = $p(X_i= 1) = p$ and $p(X_i= 0) = 1-2p$. We use $T = \frac{1}{38}\sum_{i=1}^{19}Xi^2 $ as an estimator for p. C... |
H: Probability with a diagnostics tests
Question: A virus has been spread around a population. The prevalence of this virus is 84%. A diagnostic test, with a specificity of 94% and sensitivity of 15%, has been introduced. If a patient is drawn randomly from the population, what is the probability that:
a) a person has... |
H: Prove $\int_{a} ^{b} f(x) \, dx=\lim_{x\to b^-} g(x) - \lim_{x\to a^+} g(x) $
Can anyone tell me how to prove the following theorem.
Let $f:[a, b] \to\mathbb {R} $ be Riemann integrable on $[a, b] $ and let $g:(a, b) \to\mathbb {R} $ be such that $g'(x) =f(x) $ for all $x\in(a, b) $. Then the limits $$\lim_{x\to a^... |
H: What is the difference between $A^TA$ and $A^2$?
What is the difference between squaring a matrix by multiplying it by its transpose and squaring it by multiplying it by itself. When I am asked to square a matrix, which method is preferred?
AI: In general we have $A^TA \ne A^2.$
Example: $A=\begin{bmatrix} 0 & 1 \\... |
H: Is orthogonal complement in a Hilbert space unique?
Let $H$ be a Hilbert space and let $X \leq H$ where $X$ is a closed subspace of $H$. Suppose that there is another (closed?) subspace $Y$ with $X \oplus Y = H$ and $X \perp Y$. Is it true that $Y = X^\perp?$
Attempt:
Let $y \in Y$. Then $\langle y,x \rangle = ... |
H: How to solve this equation involving natural log
I have $2$ related questions:
I know value of $\ln(x) / \ln(y)$, say it is $v$, how can I find value of $x/y$?
If $\ln(x) = v_1$ and $\ln(y) = v_2$ , what is $x/y$ ?
Thanks for your help. Apologies if these are very basic questions.
AI: Note $ln(x) =v_1\implies... |
H: Motivation of the definition of topology
In general topology the the definition of topology is the following:
Let X be a non empty set. A set $\tau$ of subsets of $X$ is said to be a topology on $X$ if
$X \in \tau$ and $\emptyset \in \tau$
The union of any (finite or infinite) number of sets in $\tau$ belongs to ... |
H: "bracelet type" Combinatorics
This question seems ok but I'm having real difficultly working out the answer using the method they provided. It's so hard to keep track of all the options. Does anyone know of a better more algebraic method?
AI: Two things can be done to a bracelet without changing it: rotating it an... |
H: Question about proof characterisation partial isometry
Consider the following fragment in the text "$C^*$-algebras and operator theory by Murphy":
Could someone explain why the marked step is true? I don't see how this follows from $\Vert u(x) \Vert^2 = \Vert u^* u(x) \Vert^2$. Thanks in advance.
AI: If $\|u(x)\|^... |
H: Kernel of homomorphism $\mathbb{K}[X,Y]\rightarrow \mathbb{K}[t^2,t^3]$, $X\mapsto t^2$, $Y\mapsto t^3$
I'm trying to show that the kernel of the homomorphism
$$\varphi: \mathbb{K}[X,Y]\rightarrow \mathbb{K}[t^2,t^3],$$
$$X\mapsto t^2,$$
$$Y\mapsto t^3,$$
is the ideal $I=\langle Y^2-X^3\rangle$ in $\mathbb{K}[X,Y]$... |
H: Let $f:\mathbb{R}\to(0,\infty)$ be a differentiable function. For all $x\in\mathbb{R}$ $f'(x)=f(f(x)).$ Then show that such function does not exists
What i have done is very small.
$$f'(x)=f(f(x))\implies f(f'(x))=f(f(f(x)))$$Now $$f(f(f(x)))=f'(f(x))$$Hence$$f(f'(x))=f'(f(x))$$Now i am blank. What to do for the pr... |
H: Show $\sigma(x) \in \{0,1\}$ if $x \in \{0,1\}$
Let $A$ be a unital $C^*$-algebra and $x \in A$ an element with $x^* = x=x^2 = x^3$. I want to show that $\sigma_A(x) \subseteq \{0,1\}$.
Attempt:
We know that $\sigma_A(x)^2=\sigma_A(x^2) = \sigma_A(x^3) = \sigma_A(x)^3$ so I guess the statement somehow follows from ... |
H: Impossible to pack Circles without gaps
It is intuitively apparent that circles cannot be packed without any gaps. I thought this is easy to prove, but it turns out not to me.
I have $2$ versions for this question, which likely to have opposite answers.
$1:$ Is it possible to pack finitely many circles(of radius la... |
H: Are there doubly (left- and right-) perfect sets with Lebesgue measure zero?
This question seems natural enough that the answer should be known, but I was unable to find a reference.
Call a subset $C$ of $\Bbb R$ left- (respectively, right-) perfect if it is perfect and if every point in $C$ is a limit point from t... |
H: Using exclusion and inclusion method to calculate number of functions
Let $A =\{1,2,3,4,5,6\}$.
I want to find the number of functions $f : A \rightarrow A$ such that
$|f^{-1}(i)|=i$ for every $i\in \{1,2,3\}$.
anyone has an idea, I tried a lot but didn't get to the answer.
Thank you
AI: You want $f$ to map one val... |
H: Infinite sum $\sum ^{\infty }_{n=1}\frac {n( n+1)^{2}}{(n-1)!}x^{n}$
I want to calculate the following.
$$\sum ^{\infty }_{n=1}\frac {n( n+1)^{2}}{(n-1)!}x^{n}=?$$
I knew.
$$\sum ^{\infty }_{n=1}\frac {x^{n}}{(n-1)!}=e^{x}x$$
By the way, answer is WolframAlpha!
Please tell me how to solve.
AI: Hint:
Write $$n(n+1)^... |
H: Can someone explain this small probability contradiction?
The question is: A service organization in a large town organizes a raffle. One thousand raffle tickets are sold for $\$1$ each. Each has an equal chance of winning. First prize is $\$300$, second prize is $\$200$, and third prize is $\$100$. What is that ch... |
H: Find the points closest to two lines using least squares method
Given are two lines $g(t)=a+bt$ and $h(s)=c+ds$ with $a,b,c,d \in \mathbb R^3$. I need to find the points where the two lines are closest using the least squares method. However I am unable to find a solution for this problem.
Intuitively those points ... |
H: Weak law of large numbers for epsilon sequence that tends to 0
Assume a $S_n=\sum_{i=1}^n X_i$ fulfills a weak law of large numbers, i.e. for every $\epsilon>0$,
$$P\left(|\tfrac{1}{n}S_n - \mu| > \epsilon\right) \rightarrow 0$$
Does there exist a sequence $(\epsilon_n)_n$ with $\epsilon_n \rightarrow 0$ such that
... |
H: How to evaluate $\int \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}} dx$?
I am trying to evaluate
$$\int \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}} dx \quad (1)$$
The typical way to confront this kind of integrals are the conjugates i.e:
$$\int \frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}} dx ... |
H: How prove $\int_{S^{n-1}}f(x\cdot \omega)\,d\omega=\int_{S^{n-1}}f(-x\cdot \omega)\,d\omega$
Let $f:\mathbb{R}^{n}\rightarrow \mathbb{C}$ be a continuous function.
I Believe that
$$\int_{S^{n-1}}f(x\cdot \omega)\,d\omega=\int_{S^{n-1}}f(-x\cdot \omega)\,d\omega.\qquad \qquad (1)$$
The reason is $-\omega$ is the uni... |
H: A question about cardinality
Let $A$ be a subgroup of $\mathbb{R}$.
Assume $|\mathbb{R}\setminus A|=\aleph_0$.
Is it true that for every $a,b\in\mathbb{R}$, $a<b$
$(a,b] \cap A \neq \varnothing$ ?
I think it's true, but I'm not really sure. Can anyone shed light on this?
AI: Since, $|\mathbb{R}\setminus A|=\aleph_0... |
H: denominator with standard part 0
(i) Why do we have to simplify the numerator and denominator, and not just substitute the standard part st(c) right away since it is given as 4.
(ii) Also what does this mean (more specifically, what is the idea behind the bold part):
However, since c $\neq$ 4 the fraction is defi... |
H: How to get the point at which a line passes through another line perpendicular to it
I have two line segments that are perpendicular to each other and intersect. I want to know the point of intersection, given one endpoint of one segment and both endpoints of the other. I also know that the other endpoint of the fi... |
H: what is the negation of $\exists^{=1}x\phi(x)$?
I need to understand the negation of counting quantifiers, so in my understanding negation of $\exists^{=n}x \phi(x)$ should be $\exists^{<n}x \neg \phi(x) \lor \exists^{>n}x \neg \phi(x) $. Hence, the negation for $\exists^{=1}x\phi(x)$ is given as follows ?
$$\exist... |
H: if $f(\frac{x+y}{2}) =\frac{f(x)+f(y)}{2}$ then find $|f(2)|$
if the following functional equation $$f\bigg(\frac{x+y}{2}\bigg) =\frac{f(x)+f(y)}{2} \quad \text{ holds for all real }~ x ~\text{ and }~ y$$ If$f'(0)$ exists and equals to $-1$ then find $|f(2)|$.
My work
I tried to find $f'(x)$ so that i can intreg... |
H: Proof that $f\big(f(x)\big)$ exists, where $f(x)=\frac{4x+3}{6x-4}$
$$f(x)=\frac{4x+3}{6x-4}$$
$\operatorname {Dom}f= \Bbb{R}\setminus\left\{\frac23\right\}$
$\operatorname {Ran}f=\Bbb R$
Since $\operatorname {Ran} f$ is not a subset of $\operatorname {Dom} f$,
therefore, $f\circ f$ does not exist.
Where have I go... |
H: Eigenvalues of a matrix containing an unknown matrix
How to find eigenvalues of a symmetric matrix
$$B = \begin{bmatrix}
2\mathrm{I_{m}} & A^\intercal \\\\ A & 0
\end{bmatrix}$$
without knowing anything about $A$ besides that $A \in \mathbb{R}^{n \times m}$ ?
Finding the determinant of
$$B - \lambda \mathrm{I} = \b... |
H: Does Lindeberg's condition imply $s_n \to \infty$?
Lindeberg's theorem states that if we have a sequence of independent random variables $X_j: \Omega \to \mathbb{R}$ with zero mean, variance $\sigma_i^2$, and distribution $\alpha_i$, and we define $s_n^2 = \sigma_1^2 + \dots + \sigma_n^2$, then the distribution of... |
H: Let $f$ be measurable then prove a certain set is measurable.
Question: Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is measurable. Prove $\{(x,y)\in\mathbb{R^2}:f(x)\geq f(y)\}$ is measurable.
My thoughts: I was hoping that there would be a way to "pull back" into an open set, since open sets in $\mathbb{R}$ are m... |
H: If $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\le 1$, prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$.
QUESTION: Let $a,b,c$ be positive real numbers such that
$$\cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1$$
Prove that $$(1+a^2)(1+b^2)(1+c^2)\ge 125$$ When does equality hold?
MY APPROACH: Firstly, let's try to s... |
H: Tricolorations of a flag
Suppose that the there are 6 colors of the rainbow - red, orange, yellow, green, blue, and purple, in that order. (indigo is not included) Chan wants to design a vertical tricolor (a flag with equal vertical stripes of three different color) for a club. If Chan uses only uses the 6 rainbow ... |
H: prove that : $\sum_{j=1}^{n} P(A_{j}) = E\{\sum_{j=1}^{n}(I_{A_{j}}-P(A_{j})) I_{\bigcup_{i=1}^{n} A_{i}}\}+\sum_{j=1}^{n} P(A_{j})(1-\beta) $
let $A_1, A_2, \dots,A_n$ be probability events.
Set :
$$\beta=1-P\left(\bigcup_{k=1}^{n} A_{k}\right)$$
assume that $\beta > 0$ and prove that :
$$\sum_{j=1}^{n} P\left(A_{... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.