text stringlengths 83 79.5k |
|---|
H: Coercive/(weakly) semicontinuous function: extreme values
Consider functionals of the form
$$\phi : X \rightarrow \mathbb{R} \cup\{+\infty\},$$
where $X$ is an arbitrary, normed vector space. In particular, $X$ may be of infinite dimension. I would be fine with restrictions like Banach-spaces or separabel/reflexive... |
H: Not knowing the $\mathrm{gcd}$ and $\mathrm{lcm}$ and knowing $\mathrm{gcd+lcm}$, how to find $a$ and $b$ in $\mathrm{gcd}(a,b)$?
Here's what we have:
$\mathrm{gcd}(a,b)=d$ ; $\mathrm{lcm}(a,b)=m$ ; $a+b=30$ ; $m+d=42$ ; $b>a$.
What I tried:
if $d$ divides $a$ and $b$ so it divides $a+b$ so $d$ divides $30$.
And wi... |
H: Upper bound for the nth derivative of $\Gamma(x)^n$
I was trying to find an upper bound for
$$ \frac{d^n}{ds^n} \Gamma(s)^{n}|_{s=1}$$
yet, I only get the bound for the nth derivative of gamma, as follow:
First, the integral of the nth derivative of gamma is
$$\frac{d^n}{ds^n} \Gamma(s) = \int_0^{\infty} t^{s-1} e^... |
H: Is a locally compact Hausdorff quotient of a locally compact $\sigma$-compact first countable Hausdorff space always first countable?
Let $Y$ be a locally compact, $\sigma$-compact, first countable Hausdorff space
and $q: Y\to X$ a quotient map with $X$ Hausdorff. Suppose that $X$ is locally compact. Is $X$ first c... |
H: Given orthogonal matrix $P$, is $P \circ P$ invertible?
Let $P$ be a orthogonal matrix, i.e., $P^T P = P P^T =I.$ Then can we say that $P \circ P$ is invertible?
P.S: $A \circ B$ is the elementwise product of matrices $A$ and $B$.
AI: No.
My gut reaction to reading any question like this is skepticism: elementwis... |
H: Uniform convergence of $\varphi_n(x)=\int_{-\infty}^{\infty}\frac{\sin(n(y-x))}{n(y-x)}f(y)dy$
Let $f\in L^1(\mathbb{R})$ and $n\in\mathbb{Z}_{+}$, we define
$$\varphi_n(x):=\int_{-\infty}^{\infty}\frac{\sin(n(y-x))}{n(y-x)}f(y)dy\ .\ \ (x\in\mathbb{R})$$
(However, if $y=x$, it is interpreted as $\frac{\sin(n(y-x))... |
H: Evaluating $\lim_{n\to \infty} \prod_{k=1}^{n}\frac{1+2\cos(2x/3^k)}{3}$
$f(x)$ is defined as follows and $g(x)=|f(x)|$. I have to find the number of points of non-differentiability of $g(x)$, which will be much easier once I have dealt with this product and simplified $f(x)$.
$$f(x)=\lim_{n\to \infty}\prod_{k=1}^{... |
H: Compute the matrix of norms of $A=\begin{bmatrix}3&4\\1&-3\end{bmatrix}$
My work so far
Using the following
$\hspace{30px} L^1\ =\displaystyle \max_{\small 1\le j\le m}(\displaystyle \sum_{i=1}^n |a_{ij}|)\\ \hspace{30px} L^2\ =\sigma_{max}(A)\\ \hspace{30px} L^F\ =\sqrt{\displaystyle \sum_{i} \displaystyle \sum_{j... |
H: Why must $\int_\gamma f(z)\;d z = 0$ for *any* contour $γ$ to define antiderivative of $f$?
Whilst I was reading the following proposition from Dexter Chua's lecture notes on Complex Analysis:
Let $U \subseteq \mathbb{C}$ be a domain (i.e. path-connected non-empty open set), and $f: U \to \mathbb{C}$ be continuous... |
H: Proving the existence of a non-measurable set
I'm asking the same question as raised in this one. There hasn't been an answer yet, so can someone please help?
I did read the comments. One of the comments says the following
I see. Then I guess the point of not having $0 \in H$ is so that the union
is $(0,1]$ rather... |
H: Understanding the chain rule for differentiation operators
Suppose I want to transform a partial derivative operator from spherical to Cartesian coordinates.
I have found the following relation based on the chain rule here:
$$
\frac{\partial }{\partial \theta } = \frac{\partial x}{\partial \theta} \frac{\partial... |
H: Interpreting almost sure convergence
I'm reading: https://en.wikipedia.org/wiki/Convergence_of_random_variables#Almost_sure_convergence and here it says that
Given a probability space $(\Omega,\mathcal{F},P)$ and a random variable $X:\Omega \rightarrow \mathbb{R}$ almost sure convergence stands for $$P\left(\omega... |
H: Finding value of $\bigg(\frac{\partial u}{\partial y}\bigg)_{x}$ at point $(5,1,-3,1)$
$\displaystyle \bigg(\frac{\partial u}{\partial y}\bigg)_{x}$ at point $(u,x,y,z)=(5,1-3,1)$ . If it is given $u=x^2y^2+yz-z^3$ and $x^2+y^2+z^2=11$
What i try ::
$\displaystyle \frac{\partial u}{\partial y}=\frac{\partial }{\p... |
H: Show that the conditional variance of a Gaussian random vector is equal to a constant almost surely.
Let $(X,Y)$ be a 2-dimensional Gaussian random variable.
(a) Prove that there are constant $a$ and $b$ such that
\begin{align*}
E[Y\,|\,X]=aX+b
\end{align*}
(b) Prove that the conditional variance defined as
\b... |
H: Proving that something is a vector bundle
Let $f:X \to Y$ be a surjective morphism of irreducible varieties (over an algebraically closed field) such that, for each $y \in Y$, $f^{-1}(y)$ is a vector space of dimension $r$. Is these informations enough to say that this triple is a vector bundle?
I am tempted to sa... |
H: Length of $A/\mathfrak{m}^2$ as an $A$-module
Let $A$ be a commutative noetherian ring and $\mathfrak{m}\subset A$ a maximal ideal generated by $1$ element $f\in A$.
In that case, $\mathfrak{m}/\mathfrak{m}^2=\langle\overline{f}\rangle_A$ and $\ell(\mathfrak{m}/\mathfrak{m}^2)=1$. Now consider the exact sequence of... |
H: prove that a function $f$ is uniformly continuous if and only if there exists a modulus of continuity for $f$
Consider two metric spaces $(X,d_X)$, $(Y,d_Y)$, and a function $f: X\to Y$, $f$ is uniformly continuous.
A function $w: [0,\infty)\to [0,\infty]$ is called a modulus of continuity for $f$, if:
$w(0)=0$
$li... |
H: Evaluate the given limit by recognizing it as a Riemann sum; question regarding interval of integration
Problem Find the limit :
$\lim_{n\to \infty}\sqrt[n]{(1+1/n)(1+2/n)\cdot...\cdot(1+1/n)}$
which is same as problem solved here Evaluate the limit by first recognizing the sum as a Riemann Sum for a function defin... |
H: If $A\subset X$ is a deformation retract, then do we have $\pi_k(X,A)=0$?
Let $A$ be a deformation retract of the topological space $X$; the example in my mind is for example: $X=(\mathbb C^*)^n$ and $A=(S^1)^n$.
Notice that if we consider the homology instead, then we know $H_k(X,A)=0$ by using the long exact sequ... |
H: Proving $\lim\limits_{x\to a} |g(x)| = 0 \implies$ $\lim\limits_{x\to a} g(x) = 0$
I am trying to prove:
Let $g$ be a real valued function for all $x \in \mathbb{R}$. If $a \ne x$ is such that $\lim\limits_{x\to a} |g(x)| = 0$, then $\lim\limits_{x\to a} g(x) = 0$.
Proof:
Let $x \in \mathbb{R}$, $g(x)$ be a real ... |
H: Ratio of polynomials, how to prove $f(t) \ge 0$ for $t > 0$?
I am looking at the following function $$
f(t) = \frac{t+1}{t^2} - \frac{16}{(t+1)^3}
$$
and am struggling to prove that $f(t) \ge 0$ whenever $t>0$. The statement appears true from plotting and inspecting the graph, but this is far from a proof.
AI: Comb... |
H: Why is it ok to put all the elements to the exact power if they had different power before that?
The question could sound messy, so will demonstrate what I mean. I'm going through Algorithm course from Stanford (full screen), when the teacher makes the proof for $O(n^k)$ notation:
$$T(n) = a_kn^k + ... + a_1n+a_0$$... |
H: Combinatoric question on preimage of a function
Got stuck on the following combinatoric question. Will be glad for any suggestions.
Find the number of functions $f:\{1,2,3,4\} \rightarrow \{1,2,3,4\}$ so that for all $1\le i\le4$, $f^{-1}(\{i\})≠\{i\}$ .
(i.e. Find the number of these functions in which the pre-i... |
H: Let $y = f(x)$ be the particular solution to the differential equation $ \frac{dy}{dx} $=$y^2$ with the initial condition $f(3) = 1$
Let $y = f(x)$ be the particular solution to the differential equation $ \frac{dy}{dx} $=$y^2$ with the initial condition $f(3) = 1$. Which of the following gives an expression for $f... |
H: Solving a system of a system of equations, numerically.
I have a system of 4 systems of equations.
$$\begin{align*}
C - 0 &= 1.02 \\
C - F &= 0.45 \\
C - N &= 0.24 \\
C - I &= -0.21 \\
\end{align*}$$
$$\begin{align*}
F - 0 &= 0.59 \\
F - C &= -0.45 \\
F - N &= -0.20 \\
F - I &= -0.68 \\
\end{align*}... |
H: Why is this map a unitary?
Consider following theorem from Murphy's '$C^*$-algebras and operator theory'
The proof says that if $p=1$, then the assertion that the map $H \to \bigoplus_\lambda p_\lambda(H)$ is a unitary is clear.
I don't see why this is true though. To show it is a unitary, it suffices to show that... |
H: Change of integration variable
I have the following relation for a function $\phi_A(\textbf{r})$ and it is known that $\rho_B(\lambda \textbf{r}) = \rho_A(\textbf{r})$
$\phi_A(\textbf{r}) = \int{d^3r' \frac{\rho_A(\textbf{r}')} {|\textbf{r - r}'|}} = \int{d^3r' \frac{\rho_B(\lambda \textbf{r}')} {|\textbf{r - r}'|}... |
H: have to find cartesian-coordinates from the given diagram
Honestly, I don't know where to start in this question
The cartesian co-ordinates of the point $Q$ in the figure is :
(a) $(\sqrt{3}, 1)$
(b) $(-\sqrt{3}, 1)$
(c) $(-\sqrt{3},-1)$
(d) $(\sqrt{3},-1)$
AI: Well, by the Theorem of angles opposed by the vertex,... |
H: For which real number $\alpha$ is there a value $c$ for which $\int^c_0 \frac{1}{1+x^\alpha}dx=\int^\infty_c\frac{1}{1+x^\alpha}dx$
For which real number $\alpha$ is there a value $c$ for which $\displaystyle\int^c_0 \frac{1}{1+x^\alpha}\mathrm{d}x=\int^\infty_c\frac{1}{1+x^\alpha}\mathrm{d}x$.
What I have tried:
S... |
H: Uniform continuity on different intervals
We know that $f_n:(0,1)\rightarrow\mathbb{R}$ is a sequence of nondecreasing functions on interval $(0,1)$ and that $f:(0,1)\rightarrow \mathbb{R}$ is a continuous function. Let $A\subset (0,1)$ be a dense subset of $(0,1)$, which fulfills the condition
$$ \forall_{x\in A}... |
H: Inequality involving factorial of sum
I noticed the following inequality involving factorials as a consequence of a statistics exercise:
$$
(x_1+\cdots+x_n)!\leq n^{x_1+\cdots +x_n}\,x_1!\,\cdot\cdots\cdot\,x_n!\,,
$$
where $x_1,\ldots,x_n$ are nonnegative integers. I thought such a clean inequality would have a na... |
H: Is this Lie algebra decomposition always true: $\mathcal{G} = \text{Ker}(ad_{a}) \oplus \text{Im}(ad_{a})$
I am reading: https://i.stack.imgur.com/QPS4m.png and am not understanding their decomposition $$\mathcal{G} = \text{Ker}(ad_{a}) \oplus \text{Im}(ad_{a})$$ where $\mathcal{G}$ is a Lie algebra and $ad_{a} := ... |
H: Prove that $f$ is Lebesgue integrable on $E$ if and only if $\sum_{n=0}^\infty 2^nm(\{x\in E:f(x)\geq2^n\})<\infty$
Question: Let $E$ be a finite measure space and let $f$ be a nonnegative function on $E$. Prove that $f$ is Lebesgue integrable on $E$ if and only if $\sum_{n=0}^\infty 2^nm(\{x\in E:f(x)\geq2^n\})<\... |
H: Galois Group of $x^4 - 7$ over $\mathbb{F}_5$
I am asked to find the Galois Group of the polynomial $x^4 - 7$ over $\mathbb{F}_5$. I am wondering if the following is correct:
The splitting field of $x^4 - 7 = x^4 - 2$ over $\mathbb{F}_5$ is $\mathbb{F}_5(\sqrt[4]{2},i)$ where $i,\sqrt[2]{2}$ lie in a fixed algebrai... |
H: Kernel of rational matrix has rational elements arbitrarily close to real elements
I am trying to understand this:
Let $\mathscr{S}$ be a finite system of homogeneous linear equations with rational coefficients, i.e. $Ax = 0$ for some rational matrix $A$. If $\hat{x} \in \mathbb{R}^n$ is a real solution to $\mathsc... |
H: grid "proof" for commutativity of multiplication
In this write-up by Tim Gowers on why multiplication is commutative,
https://www.dpmms.cam.ac.uk/~wtg10/commutative.html
he gives a physical grid model to which multiplication corresponds and says - "This argument, compelling as it is, doesn't quite qualify as a math... |
H: Is the finite intersection of prime ideals radical?
Does there exist a ring $R$ and finitely many prime ideals $P_i$ such that $\cap_{i = 1}^n P_i$ is not radical ideal?
In other words, is the finite intersection of prime ideals a radical ideal?
AI: Consider a commutative unital ring $R$ and some prime ideals $P_1,... |
H: Sonin's identity
The following identity attributed to N.Y. Sonin states the following:
Suppose $f\in C^2[a,b]$. Let $\rho(x)=\frac12-\{x\}$, where $\{x\}$ is the fractional part of $x$, and $\sigma(x)=\int^x_0\rho(t)\,dt$. Then
$$
\sum_{a< n\leq b}f(n)=\int^b_a f(t)\,dt +\rho(b-)f(b)-\rho(a)f(a)-\big(\sigma(b)f'(b)... |
H: Irreducible polynomial for infinitely many values
I want to prove that there are infinitely many values of k such that the polynomial $x^{9}+12x^{5}-21x+k$ is irreducible. I sense that I have to use Eisenstein and the number 3 but I don't see exactly how. Any help would be appreciated.
AI: General idea: What does E... |
H: Trouble understanding QCQP
Using a graphical method, indicate the feasible region and solve the minimization problem. $$\begin{array}{ll} \text{minimize} & f := x_1^2 + x_2 + 4\\ \text{subject to} & c_1 := -x_1^2-(x_2+4)^2 +16 \ge 0\\ & c_2 := x_1 - x_2 - 6 \ge 0\end{array}$$
I draw the problem as such:
And I d... |
H: If $a^2 + b^2 + c^2 = 1$, what is the the minimum value of $\frac {ab}{c} + \frac {bc}{a} + \frac {ca}{b}$?
Suppose that $a^2 + b^2 + c^2 = 1$ for real positive numbers $a$, $b$, $c$. Find the minimum possible value of $\frac {ab}{c} + \frac {bc}{a} + \frac {ca}{b}$.
So far I've got a minimum of $\sqrt {3}$. Can ... |
H: Does this lead to a contradiction within ZF?
Let the following be an axiom (which I will denote P):
If $x,y$ are sets and $f:y\to x$ is a surjection, then the existence of an injection $f:x\to y$ guarantees a choice function exists.
Does P lead to contradiction within ZF in any obvious Russell's paradox like sens... |
H: Show that $u(x)=0$ for all $x\in\Omega$
Suppose $\Omega\subset R^n$ is a bounded open domain and $u(x)$ is a smooth function that satisfies
$$\left\{\begin{matrix}
\Delta u+x_{1}u^{2}u_{x_1}=0 \text{ for all } u\in\Omega\\
u(x)=0 \text{ for all } x\in\partial\Omega
\end{matrix}\right.$$
Show that $u(x)=0$ for all ... |
H: Showing $f_j(0)$ converges to $f(0)$ where $f_j$ and $f$ are rational functions.
Let $A=\{z \in \mathbb{C}: \frac{1}{2} < |z| < 1 \}$. Let $\{f_j\}_{j=1}^\infty$ be a sequence of rational functions and let $f$ be a rational function. Suppose none of these functions have poles on $A \cup \{0 \}$, $f_j$ converges uni... |
H: CW complexes are T$_1$
Given the constructive definition of CW-complexes (i.e. the one Hatcher gives in his Algebraic Topology book) how would one prove that every singleton in closed. He states in page 522 that every point pulls back to closed subsets of the closed discs $D_\alpha^n$ under every characteristic map... |
H: Total derivative of vector function
Let's asume that I have a (vector) function $f \space (s,t,u,v):\mathbb{R}^4 \rightarrow \mathbb{R}^n.$ I would like to calculate: $\frac{d}{dt}\biggr|_{t=t_0} f(t,t,t,t).$
Intuitively:
This should just be given by:
$$\frac{d}{dt}\biggr|_{t=t_0} f(t,t,t,t)=\frac{\partial}{\part... |
H: Given 6 distinct points in $3$-$D$ space, can the distances between $3$ of the points be determined if all other distances between points are known?
In the figure below of points in $3$-$D$ space, suppose the lengths of the blue segments are all known. Is it possible to determine the lengths of the red segments? Ea... |
H: Finding gradient of line using graph equation and given point
Consider a parabola y = x$^2$. The line that goes through the point (0, $\frac{3}{2}$) and is orthogonal to a tangent line to the part of the parabola with x > 0 is y = Ax + $\frac{3}{2}$. Find the value of A
The answer is supposedly $\frac{-1}{2}$ but I... |
H: Proving that $X^n-a$ is irreducible if $a$ is no $p$-th power for any prime $p$ dividing the degree diving the degree
Let $F$ be a field containing a primitive $n$-th root of unity (for $n \geq 2$) and let $E = F(\alpha)$ where $\alpha \in E$ is an element whose $n$-th power (but no smaller power) is in $F$. Let $\... |
H: Finding expectation of a random variable
Let $X$ be a random variable whose distribution function is given by
$$
\mathrm{F}\left(x\right) =
\left\{\begin{array}{lcl}
{\displaystyle 0} & \mbox{if} & {\displaystyle x < 2}
\\[1mm]
{\displaystyle{1 \over 3}\,x} & \mbox{if} &
{\displaystyle 2 \leq x \leq 3}
\\[1mm]
{\di... |
H: How to build a strong foundation for university mathematics?
My objective: To study pure mathematics at university level next year. I find Abstract Algebra, Number Theory and Foundations of Maths(Set Theory, FOL, etc.) to be intriguing yet mostly inaccessible with my current mathematical maturity, but I would love ... |
H: What is the idea behind the derivation of $z$-score formula?
I can't fully wrap my head around the reasoning behind the $z$-score formula.
$z = (x - \mu)/\sigma.$ Can someone explain how this formula was derived or how it works?
Edit: I understand why and how the formula and concept is used. I just don't get how th... |
H: natural isomorphism by right exactness
R is a local ring with maximal ideal $\mathfrak{m}$ and residue field k.
M is a finitely generated R-module with a projective cover:
$0 \to$ N $\to$ F $\to$ M $\to 0$.
Tensor $0\to$ $\mathfrak{m}$ $\to$ R $\to $ k $\to 0$ by N; right exactness gives a natural isomophism
$\tau_... |
H: Determinant as a polynomial
I am trying to understand the notion of a determinant of an $n \times n$ matrix as a polynomial of degree $n$ in the entries of a matrix. If I wrote a matrix of the form
$$\begin{bmatrix} a & b \\ c & d\end{bmatrix},$$
then its determinant is $ad - bc$, which is a function of four variab... |
H: Let $\frac{1}{2}<\cos2A<1$ and $6\tan A-6\tan^3A=\tan^4A+2\tan^2A+1$, find $\tan 2A$
Let $\dfrac{1}{2}<\cos2A<1$ and $6\tan A-6\tan^3A=\tan^4A+2\tan^2A+1$, find $\tan 2A$
My attempt:
\begin{align*}
6\tan A(1-\tan^2A)&=\tan^4A+2\tan^2A+1\\
12\tan^2A&=\tan2A\tan^4A+2\tan2A\tan^2A+\tan2A\\
0&=\tan2A(\tan^4A)+(2\tan2A-... |
H: What did I do wrong on this inequality problem?
The problem is:
If $b < a$, what is the range for $x$ if $ax+a < bx+b$?
Below is my work:
$$ax+a < bx + b$$
$$(a+1)x < (b+1)x$$
Divide both sides by $x$ and you get: $$a+1 < b+1$$
Which can be simplified to $a<b$ if $x$ is positive.
But $a<b$ contradicts to the sta... |
H: Test the convergence of the series with alternating signs
The series is given by ($a>0$)
$$\frac{1}{a} +\frac{1}{a+1} - \frac{1}{a+2} +\frac{1}{a+3} +\frac{1}{a+4} -\frac{1}{a+5} +\cdots$$
So how can I move forward? I can not find the general term of the series. If someone can give the general term then I can go ... |
H: Convergence of series via ratio test
I am attempting to show that a sequence which I defined as $\{a_n\}$ where $a_n =\sum_{p=0}^{n}\frac{x^p}{p!}$ converges, in order to do so I have attempted the ratio test at which point I arrive at the term $a_n =\sum_{p=0}^{n}\frac{x^p}{p!}$ and $a_{n+1} =\sum_{p=0}^{n+1}\frac... |
H: Is $\phi =\angle A"OB" = \measuredangle(AB,A"B")=\measuredangle(A'B',A"B")$? [Doubt]
Can someone clarify this doubt ?
We denote the spiral similarity by $S$, the rotation centered at $O$ with
angle $\phi$ by $\rho _O ,\phi$ , and the homothety centered at $O$ with ratio $k$ by $\chi _{ O, k}$ , then
$S _{O, k, \ph... |
H: Show that $\frac{X-\mu}{\sigma}\sim N(0,1)$ using moment functions
Let $X\sim N(\mu,\sigma^2)$. Show that $Z=\frac{X-\mu}{\sigma}\sim N(0,1)$ using moment generating functions.
\begin{align*}
M_Z(t)&=M_{\frac{X-\mu}{\sigma}}(t)\\
&=M_{X-\mu}\left(\frac t\sigma\right)\\
&=e^{-\mu t}M_X\left(\frac t\sigma\right)\\
&=... |
H: Does a Bounded linear transformation from a Banach space to Real numbers map a bounded closed set to a closed set?
Specifically I need to know that whether a bounded linear transformation from the sequence space l_1 to Real numbers map the closed unit sphere to a closed set?
I'm thinking of this problem for two day... |
H: How can I find the distance between the centres of two circles?
I'm a senior year maths student and I stumbled upon a question from a maths competition from a previous year. I seem to be on the cusp of solving it but I am unable to solve for the radius (to give me the answer).
The question reads as follows:
A recta... |
H: Prove that if $U$ is a linear operator on $V$, then $UT=TU$ if and only if $U=g(T)$ for some $g(T)$.
Let $T$ be a linear operator on a vector space $V$, and suppose that $V$ is a $T-$cycle subspace of itself. Prove that if $U$ is a linear operator on $V$, then $UT=TU$ if and only if $U=g(T)$ for some polynomial $g(... |
H: Evaluating $\int_0^\infty \frac{\tan^{-1}(t)}{e^{2\pi t}-1}\,\mathrm{d}t$
As stated in the title, I want to evaluate the integral
$$\int_0^\infty \frac{\tan^{-1}(t)}{e^{2\pi t}-1}\,\mathrm{d}t$$
Because of the $e^{2\pi t}$, it seems that complex analysis techniques are required for this one, which I am not so famil... |
H: Why is the following method of finding out a conserved quantity wrong?
Let a system be defined by $\dot{x}=y; \dot{y}=f(x).$ And let $E(x,y)$ be a conserved quantity of the system. Then
$$
\frac{\partial{E}}{\partial{x}}\dot{x}+\frac{\partial{E}}{\partial{y}}\dot{y}=0.
$$
My question is why cannot I just rewrite th... |
H: A question based on location of roots using information about derivative at some points
I am trying some questions asked in masters of mathematics exam of my university and I was unable to solve this particular problem.
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be twice continuous function with $f(0) = f(1) = f'(0... |
H: Does there exist a second category set that is not a Baire space?
Can someone give an example of a second-category set $Y$ in a metric space $X$ but $Y$ is not a Baire space.
We know that Baire space$\implies $ Second Category.
But I am trying to show that the converse in not true.
Clearly ,$Y$ must not be complete... |
H: A characterization of identity operator on Hilbert space
Let $H$ be a Hilbert space and $T\in B(H)$ be a bounded linear operator on $H$, then $T=I$ $\Longleftrightarrow$ $\langle\psi,T\psi\rangle=1$ for every $\|\psi\|=1$.
It is easy to examine the "$\Longrightarrow$". But how to show the opposite implication?
AI: ... |
H: Why point is a circle with radius zero?
I was reading this What is a point circle, a real circle and an imaginary circle? and i get confused with the statement that is written in the accepted answer , i.e
A point "circle" is just a point; it's a circle with a radius of zero
But point itself is a circle and when we ... |
H: How to prove that the induced topology is the coarsest and identification topology is the finest topology that keeps the map continuous?
I am reading maps between topological space from Isham, Chris J. Modern differential geometry for physicists. Vol. 61. World Scientific, 1999.. Here he defines the induced topolog... |
H: Largest number at singleton
I have a funny question - can I call one and only element of singleton the largest number?
For example: I have a singleton : $\{1\}$. Can I call 1 the largest number?
Here is what I think: largest number $n$ from set $L$ is a number for which is true $n - k$ (also from $L$) not equal $0$... |
H: How many unique "$\phi$-nary" expansions are there for $1$?
I was playing around the expansions of numbers in irrational bases, namely base $\phi=\frac{1+\sqrt5}{2}$. Of course, I should immediately define what it means to symbolize digits in a non-integer base.
At least in my case, the expansions consist of $\lcei... |
H: What is actual difference between transitivity and quasitransitivity?
I have been trying to construct q.t. relation, but always get transitive relation.
It seems to me, that transitivity includes q.t. Ok, but how would look like pure q.t. relation?
Examples and definitions seem to me ( in my opinion, which might be... |
H: differentials and tangent space of a fibre
The setup I have is as follows: Let $f: X \to Y$ be a morphism of non-singular $n$-dimensional varieties (separated reduced irreducible scheme of finite type over $k$) over $k$ an algebraically closed field. For all closed points $y \in Y$, the fibre $f^{-1}(y)$ is a finit... |
H: Show that the inequality $\left|\int_{0}^{1} f(x)\,dx\right| \leq \frac{1}{12}$ holds for certain initial conditions
Given that a function $f$ has a continuous second derivative on the interval $[0,1]$, $f(0)=f(1)=0$, and $|f''(x)|\leq 1$, show that $$\left|\int_{0}^{1}f(x)\,dx\right|\leq \frac{1}{12}\,.$$
My at... |
H: Binary number and measure
Consider I express all the number between $[0,1]$ into binary number. Define the set $X:=\{x\in[0,1]|(0.1x_{1}1x_{2}1x_{3}...),x_{i}\in\{0,1\}\}$. Now I believe this set is uncountable, totally disconnected. But what about the measure of $X$? Can it be measure zero?
My intuition tell me th... |
H: Singular values of matrices which preserve the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1$
Let $0<a<b$, $ab=1$, and let
$$
D_{a,b}=\biggl\{(x,y) \,\biggm | \, \frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1 \biggr\}
$$
be the ellipse with diameters $a,b$.
Let $A \in \operatorname{SL}_2(\mathbb R) \setminus \operatorn... |
H: Question about proof that sub $*$-algebra of $B(H)$ is strongly dense in bicommutant.
Consider following fragment from Murphy's "$C^*$-algebras and operator theory":
In the proof of lemma 4.1.4, why does $u(x) \in K$ follow from $pu = up?$
AI: First, we will show that $u(K) \subseteq K$. Let $y \in K$, then $p(y) ... |
H: Confusion on proof of limit laws epsilon delta
at https://www.youtube.com/watch?v=9tYUmwvLyIA from 35:33 to 39:33, Herb Gross says:
\begin{align}
f(x) &= L+ [f(x)-L] \\
g(x) &= M+ [g(x)-M]
\end{align}
Multiplying these 2, we get:
\begin{align}
f(x)g(x) &= LM + L[g(x) -M] + M[f(x) -L] +[f(x) -L][g(x)-M] \\
f(x)g(x) ... |
H: $\mathbb{R}$ is not isomorphic to a proper subfield of itself
let $\mathbb{R}$ be the field of real numbers. I found stated in this pretty work On Groups that Are Isomorphic to a Proper Subgroup, that there is no proper subfield $K$ of $\mathbb{R}$ which is isomorphic to $\mathbb{R}$ itself.
Does someone have a pro... |
H: Approximating an expression for $\rho$ tends to zero
I need to approximate the following expression within $-M\leq z \leq M$ for $\rho$ tends to zero,
$\frac{\sqrt{\rho^2+(z+M)^2}-(z+M)}{\sqrt{\rho^2+(z-M)^2}-(z-M)}$,
where $M$ is a constant and $z$ is a variable
My lecturer insists that a Taylor expansion about $\... |
H: Example of number field with certain conditions on ramification index and degree
I am looking for a number field with degree $n$ over $\mathbb{Q}$ and with a ramified prime $p$ with ramification index $e$ such that $\textrm{gcd}(n, p-1) = 1$ and $\textrm{gcd}(e, p-1)>1$. I would also be interested in a slightly str... |
H: Conditions for interchanging order of limits and summations
Let $f: \mathbb{N} \times \mathbb{N} \to \overline{\mathbb{R}}$. Then under which conditions is the expression $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$ valid?
Would anyone have a rig... |
H: Can we imply $\exists x\phi(x)\rightarrow \exists x\psi(x)$ from $\exists x(\phi(x)→\psi(x))$
let $\Sigma$ be a consistent set of formulas in first-order logic and implies $\exists x(\phi (x)→\psi(x))$. which one of these statements is logical implication from $\Sigma$?
a) $\forall x\phi(x)→\forall y\psi(y) $
b) ... |
H: If $R$ is an Euclidean domain that is not a field, is $R[X]$ a PID?
I have a question in ring theory whose answer I am looking for.
Consider R to be a Euclidean Domain such that R is not a field. Then is polynomial ring R[X] is always a PID or not.
Attempt : R is ED implies R[X] is ED and R is ED implies R is PID... |
H: Given $\gamma(t)$ a $\mathcal{C}^1$ path on $\mathbb{C}^{\times}$, why is it true that $|\gamma(t)|$ is also $\mathcal{C}^1$?
Suppose $\gamma:[0,1]\to\mathbb{C}^{\times}$ is continuously differentiable, there is a proof I saw involved using the fact that $s(t)=|\gamma(t)|$ must also continuously differentiable but ... |
H: Coordinate Geometry Question using matrices
Let on the x-y plane, the distance between the points $A(x_1,y_1)$ and $B(x_2,y_2)$ be $d$. Another point $P(a,b)$ satisfies the equations $x_1a+y_1b=1$ and $x_2a+y_2b=1$ and the distance between point $P$ from the origin,$O(0,0)$ is $p$. Find the area of triangle $\Delt... |
H: Complement of family of binary sequence
Consider set of the binary sequence, we choose an element $(0,0,0,...)$,then construct a family of sequence by changing each entry, so that each element is different from zero vector by one entry. Denote the elements by $x_{n}$. Now we proceed the similar way, from starting o... |
H: bilinear form of positive matrix
If $A$ is a positive-definite matrix and $x$,$y$ are non negative vectors, is $f(A)=x^TAy$ positive ?
AI: The answer is no, we have
$$\left\langle \begin{bmatrix} 2 & -1 \\ -1 & 1\end{bmatrix}\begin{bmatrix} 0 \\ 1\end{bmatrix}, \begin{bmatrix} 1 \\ 0\end{bmatrix}\right\rangle = \le... |
H: Calculation of covariant derivative being chart-dependent
Ref. Schuller's lecture 7 on gravity and light (derivation starts at this timestamp).
I'm watching a lecture that introduces connection coefficients - specifically the part of the video showing for the first time how to calculate covariant derivative.
Releva... |
H: The cardinality of an equivalence relation over a set
I'm trying to prove the following statement:
let $A$ be an infinite set, then for any equivalence relation $E$ on $A$, $ |E| = |A|$
But I'm really stuck. Trying to show a bijection from $E$ to $A$ made sense only when dealing with natural numbers, using maybe ... |
H: Statistical Inversion Problem $F = Ku + \mathcal{E}$ derive conditional probability density $p(f | u)$
Consider the following Inversion Problem $f = Ku + \varepsilon$ where $f \in \mathbb{R}^{m}$, $u \in \mathbb{R}^{n}$, $K \in \mathbb{R}^{m,n}$ and $\varepsilon$ is an additive, Gaussian noise. In the Bayesian appr... |
H: Prove that $E(Y_i \bar{Y}) = \frac{\sigma^2}{n}+\mu^2$
Given $Y_1, Y_2,...Y_n$ are i.i.d random variable which follows a distribution of $N(\mu, \sigma^2)$, I'm trying to prove that $$E(Y_i \bar{Y}) = \frac{\sigma^2}{n}+\mu^2$$
Here's what I've tried:
$$E(Y_i \bar{Y}) = E(Y_i \frac{1}{n} \sum_{i=1}^{n}Y_i )$$
$$ =... |
H: Going-down theorem hypothesis
Something I don't get form the hypothesis of this theorem.
If $C$ is the integral closure of $A$, and $A$ is integrally closed (since $A$ is integral domain, it's integrally closed over its field of fractions), then $A=C$.
If $B$ is integral over $A$, then $B\subset C$. Since $A\subse... |
H: Operation in the notation for group homomorphisms
A group homomorphism $\varphi$ from the additive group of real numbers to the multiplicative group of non-zero real mumbers...
Could this be written as $$\varphi :\, (\mathbb{R},+)\to (\mathbb{R}\setminus \{0\},\cdot )?$$
I'm not sure, because I only encountered t... |
H: The number of integral points on the hyperbola $x^2 - y^2 = (2000)^2$ is
The number of integral points on the hyperbola $x^2 - y^2 = (2000)^2$ is ____? (An integral point is a point both of whose coordinates are integers.
My attempt: The equation can be rewritten as:
$$(x-y)(x+y) = 2000^2 = 2^8 \cdot 5^6$$
Now, b... |
H: Weaker conditions for differentiating under the integral sign
Standard theorems of real analysis give conditions under which it holds
$$\int_0^1 \partial_x f(x,y)dy = \frac{d}{dx}\int_0^1 f(x,y)\,.$$
In most of the formulations that I have found, it is required that, for almost every $y$, $f$ is everywhere differen... |
H: Is there an easier prime factorization method for the sum of a prime's powers?
I need to obtain prime factorizations of numbers of the type: $\sum_{i=0}^n p^i$, for any prime number $p$ (not the same one each time).
Do you know if there is a quicker algorithm to calculate these factorizations than those used for ot... |
H: What is the range of $x,y,z$ when $n$ is a known natural number in: $n=x^5+y^5+z^5$
I have the following question:
What is the range of the sum of three distinct natural numbers to the fifth power than are equal to a known natural number?
Mathematically speaking:
$$n=x^5+y^5+z^5\tag1$$
When $n\in\mathbb{N}$ is kn... |
H: Convergence in $p$th mean does not imply convergence in mean.
My book proves the following:
Let $(\Omega,\mathcal{A},\mu)$ be a finite measure space. Then every sequence $(f_n)$ in $\mathcal{L}^p$ which converges in $p$th mean to an $f \in \mathcal{L}^p$ for some $p\geq 1$ also converges to $f$ in mean.
and then as... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.