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H: How to find if a series is convergent given the following data?
How do I solve these type of questions and what are the correct answers for this one?
Sorry I'm not familiar with formatting math symbols so I attached an example question as an image.
Example question
AI: Try to either show that the given series are t... |
H: Measuring the Angle of a Triangle with a Protractor (Question Illustrated by Image)
Forgive my ignorance, and teach me the correct way to read an angle when I am using a protractor. From the image below, would any of the two statements below be correct? If yes, which one? If neither, what would be the correct way t... |
H: Ratio of two segments of a side of equilateral triangle
A line goes through the center (the point in which the 3 medians, bisectors and heights intersect) of an equilateral triangle and is parallel to one of the sides and splits the other two sides. What is the ratio of the two new segments of one of the sides?
Cou... |
H: Maximum volume using Lagrange multipliers
I need to determine the maximum volume of a rectangular box with these side conditions: its surface has 2m² and the sum of all its edges = 8 m of length.
How do I do that ?
AI: Hints: The volume of rectangular box is: $\space V=V(x,y,z) = x \space y \space z$.
Side conditio... |
H: What can be the value of the determinant of $A$?
Suppose that the $ 3 × 3 $ matrix $A$ has an inverse and each entry of both $A$ and $A^{-1}$
is an integer. What can be the value of the determinant of $A$ ?
I did not understand that what it means that both $A$ and $A^{−1}$
have each entry an integer?
AI: I think th... |
H: Show that $\Bbb{F}^{m×n}$ is isomorphic to $\Bbb{F}^{mn}$.
Problem-Show that $\Bbb{F}^{m×n}$ is isomorphic to $\Bbb{F}^{mn}$.
Attempt-Deffine a mapping $T:\Bbb{F}^{m×n}\rightarrow \Bbb{F}^{mn}$ by $T(E_{ij})=e_{n(i-1)+j}$ ,where $1≤i≤m$ and $1≤j≤n$.
I assume $E_{ij}$ ,$1≤i≤m$ and $1≤j≤n$ whose non-zero entry $(E_{i... |
H: How to show $P(\inf_{t\geq0}\int_0^t e^{-s}\mathrm d B_s\geq -1)>0$?
Let $(B_t)_{t\geq 0}$ be a brownian motion, $P$ the measure of the probability space which satisfys the usual conditions and $\mathbb E$ the expected value. I like to show
$$P\left(\inf_{t\geq0}\int_0^t e^{-s}\mathrm d B_s\geq -1\right)>0$$ or
$$P... |
H: $x \in W_k$ if, and only if, $\gamma(x) \leq 2^{-k}$
Let $E$ be a Hausdorff TVS and let $(V_n)_{n \in \mathbb{N}}$ be a fundamental system of neighborhoods of $0$ in $E$ such that $\bigcap_{n \in \mathbb{N}} V_n=\{0\}$. Let us set $W_1=V_1$ and define by induction the sequence $(W_n)_{n \in \mathbb{N}}$ os balanced... |
H: Does inequality for subset imply inequality for set? (Ordered numbers)
Suppose I have two ordered (decreasingly), finite sets of numbers ($\mathbb{R}_{\geq 0}$) $A$ and $B$, each split into some $n$ disjoint subsets $A = A_1 \cup A_2 \cup ... \cup A_n$ and $B = B_1 \cup B_2 \cup ... \cup B_n$. (Ordered means for in... |
H: In a metrizable TVS $E$ a point $x$ is an accumulation point of a sequence $S$ if and only if $S$ contains a subsequence which converges to $x$.
Let $E$ be a metrizable space, that is, $E$ is a Hausdorff space and if there is a countable basis of neighborhoods of $0 \in E$ in E. I want to prove: a point $x \in E$ i... |
H: Derangement of selective letters in a string.
I got to know about the derangemnt formula which can be found here:
These are links to some excellent answers to this very topic.
Link
A link to my previous question where some links related to this topic were also given.
So my problem is how to solve question when on... |
H: Show that the matrix $B$ is positive definite without using the positive pivot and determinant test.
I'm a little confused about definite matrices. We learned about $4$ different types of definite matrices: positive semidefinite, positive definite, negative semidefinite and negative semidefinite. In a positive semi... |
H: Filling up Seats
Part (a) Now suppose that not only must Sir Lancelot and Sir Gawain be diametrically opposite, but Sir Galahad and Sir Percival also demand to be diametrically opposite. How many seatings of the $10$ knights are possible?
Part (b) Suppose for this problem (though it may not be accurate in real li... |
H: Functions with multidimensional codomain
Consider a function $f: \mathbb{R}^K\rightarrow \mathbb{R}^L$, where $L>1$. For each $x\in \mathbb{R}^K$, let $x_i$ denote the $i$-th element of $x$.
I have some terminology/definitional questions:
Is $f$ a function or has it another more formal name?
Suppose that $L=3$ an... |
H: Contitional expectation contraction inequality two sub-sigma-algebras
Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{H}\subset\mathcal{G}\subset\mathcal{F}$ two $\sigma$-algebras. We know from Jensen's inequality, that for $X\in L^2(\Omega,\mathcal{F},P)$
$$
\mathbb{E}[|\mathbb{E}[X|\mathcal{H}]|... |
H: Putting balls into boxes where balls and/or boxes are distinguishable
Part (a) How many ways are there to put $4$ balls into $3$ boxes, given that the balls are not distinguished and neither are the boxes?
Part (b) How many ways are there to put $2$ white balls and $2$ black balls into $3$ boxes, given that balls o... |
H: Is the $S_4\times G$ solvable group?
We have the following claim : The group $G$ is solvable iff $S_4\times G$ is solvable.
If we consider that $S_4\times G$ is solvable we have that $1\times G\leq S_4\times G$ is solvable as a subgroup of solvable group.We consider the isomorphism
$$f:\ 1\times G \to G ,\ (1,g)\ma... |
H: Integral of $(z^2 + x^2)^{-\frac{3}{2}}$
I am studying Griffiths Introduction to Electrodynamics, in which the following integral appears:
$$\int_{-L}^L\left({z^2+x^2}\right)^{-\frac{3}{2}}\,dx$$
where $z$ denotes a constant, and $z\in \mathbb{R}$
The integration is done without much fuzz about it, as if it was sim... |
H: Does getting the same value for getting near a point in any direction in a multivariable function means the limit is that value too?
Consider the situation where we have:
$$ f:\mathbb{R}^n \rightarrow \mathbb{R}^m$$
and
$$ \vec{V} \in \mathbb{R}^n $$
and
$$ |\vec{V}| = 1 $$
and for each $\vec{V}$ we have:
$$ \lim_{... |
H: When is $v^T A w > 0$?
Consider two vectors $v,w \in \mathbb{R}^n$ and a matrix $A \in \mathbb{R}^{n \times n}$. What conditions need to be imposed on the matrix $A$ such that $v^T A w > 0$?
I understand that if $A$ is positive-definite then $v^T A v > 0, \ \forall$ non-zero $v$. However not having the same vector ... |
H: Show that $\sum_{j=1}^{\infty}\dfrac{\zeta(6j-3)-1}{2j-1} =\frac12\ln(3/2) $
Show that
$\sum_{j=1}^{\infty}\dfrac{\zeta(6j-3)-1}{2j-1}
=\frac12\ln(3/2)
$.
This came out of
some work I did recently.
Any other context would be too much.
AI: Try switching the order of summation:
$$
\sum_{j=1}^{\infty}\frac{\zeta(6j-3)... |
H: Covariance matrix in multivariate standard normal density
I am looking at the derivation of $f_{\vec{Y}}(\vec{y})$ where $\vec{Y}=A \vec{X}$ and $\vec{X}$ is a vector of i.i.d standard normal random variables. $A$ is an $n \times n$ non-singular matrix.
The multivariate standard normal density is given by :
$$f_{... |
H: Let $H\triangleleft G$. Prove that $G/H$ is abelian iff $ [G, G] \subseteq H$
The commutator of two elements $a, b \in G$ is defined as
$[a, b] = aba^{−1}b^{−1}$.
Let $[G, G] =\langle [a, b] | a, b \in G\rangle $ be the generated subgroup of all commutators of the elements of $G$.
Let $H\triangleleft G$. Pro... |
H: Solving $\int_0^1 xe^{(\log(x))^7} dx$
How do you integrate $\int_0^1 xe^{(\log(x))^k}dx ~?$ (for $k=7$).
For $k=3$ Wolfram alpha says the closed form is in terms of the generalized hypergeometric function and the Bi-airy function. For $k=5$ Wolfram alpha says the closed form is in terms of the gamma function and... |
H: How do column transformations on $n×n$ matrix affect the final inverse matrix?
$\mathbb{A}$ is $n × n$ invertible matrix, and $\mathbb{A}^{-1}$ is its inverse. $\mathbb{B}$ is a matrix which we got by applying several row transformations on $\mathbb{A}$.
$\space \mathbb{B}^{-1}$ is the inverse matrix of $\mathbb{B}... |
H: Let $f:[a,b]\to\mathbb{R}$ be integrable on $[a,b]$. Show that if $ F(x)=\int_a^x f(t)\,dt$ then $F(x)$ is continuous on $[a,b]$.
I gave a two liner proof on a test for this as follows:
Proof: By the Fundamental Theorem of Calculus (First Form) $F'(x)=f(x)$, thus $F$ is differentiable. By an earlier Theorem, $F$ is... |
H: Solving $333.443,35 = 30.000\frac{1,02^n - 1}{1,02^n \cdot 0,02}$
I can't reproduce the intermediate steps when it comes to solving $n$ in:
$$333.443,35 = 30.000\frac{1,02^n - 1}{1,02^n \cdot 0,02}$$
It says
$$n = - \frac{\log (1-\frac{333.443,35}{30.000}) \cdot 0.02}{\log(1,02)}$$
$$= 12,7$$
Looks like $333.443,35... |
H: uniform integrability of all conditional expectations of a fixed $L^1$ function
Let $Z$ be a real $L^1$ random variable on a probability space $(\Omega, \mathscr{A}.\mu).$ Why is the family of all $E[Z| \mathscr{B}]$ uniformly integrable when $ \mathscr{B}$ ranges over the sub-sigma-algebras of $ \mathscr{A}$? Th... |
H: $A-mI$ as positive semidefinite matrix
Let $A$ be a symmetric real matrix. Let $m\in \mathbb{R}$ and consider $$M := A - m I$$ where $I$ denotes the identy matrix. We require that matrix $M$ be positive semidefinite. Why do we need that the minimum eigenvalue of $A$ be at least $m$?
Any help will be appreciated.
AI... |
H: How do I choose couples who are not from the same family?
Question:
How many ways can a $5-$member commission consisting of $6$ couples (sister-brother) be chosen so that there are no couples from the same family in the commission?
I need a detailed explanation to understand this problem. I know I will use combi... |
H: What is the kernel of this linear transformation?
What is the kernel of the linear transformation: $T:\mathbb{R}^3\rightarrow\mathbb{R}^2$ defined by $T(x,y,z)=(x-y,z)$?
Select one:
a. $\mathbb{R}^3$
b. $\mbox{Span}\{(1,1,0)\}$
c. $\mbox{Span}\{(1,-1,0)\}$
d. $\{(1,-1,0)\}$
Here's what I did:
I know that kernel mea... |
H: Constructing the inverse of a surjective homomorphism $g\otimes \operatorname{id}\colon B\otimes G \to C\otimes G$
Given an exact sequence of group homomorphisms on abelian groups $$A\xrightarrow{f} B \xrightarrow{g} C\to 0$$ I want to prove that the induced sequence $$A\otimes G \xrightarrow{f\otimes \operatorname... |
H: Cardinality of Set of All Sequences on $\omega$.
Synopsis
I have two short and simple questions. One is about proof-verification (kinda), and the other is about proof-understanding. Thank you for your time.
Note the following definition:
For any set $A$, Define a sequence in $A$ to be a function from some natural ... |
H: If a finite group $G$ acts transitively on a set of order $p^m$, then so does any $p$-Sylow subgroup
Here is an algebra qualifying exam problem:
Let $G$ be a finite group acting transitively on a set $X$ with cardinality $p^m$ for
some prime $p$ and nonnegative integer m. Show that any $p$-Sylow subgroup of $G$
ac... |
H: Riemann integration of a step function
From "An Introduction to Lebesgue Integration and Fourier Series" by Howard J. Wilcox and David L. Myers:
1.1 Definition: A partition $P$ of a closed interval $[a, b]$ is a finite sequence $(x_{0}, x_{1}, \ldots, x_{n})$ such that $a = x_{0} < x_{1} < \ldots < x_{n} = b$. The... |
H: What is wrong with $x=x^{2/2}=\sqrt {x^2}=\lvert x\rvert$
The title says it all, $x=x^{2/2}=\sqrt {x^2}=\lvert x\rvert$ cannot possibly be true, so what am I missing?
AI: I think there is nothing wrong with $x=x^{2/2}$. The order of operatins say you should evaluate $2/2$ first. So this is the same as $x=x^1$.
The ... |
H: Does there exist a monotonically decreasing function that is its own derivative?
I know that $f(x) = e^x$ is its own derivative. It is a monotonically increasing function. It seems intuitively plausible to me that there might be a monotonically decreasing function with the same property. Does one exist?
AI: $f(x) =... |
H: Give a function that satisfies the following criteria.
The problem I am trying to solve is: Give an example of a function whose graph is increasing on $(0,\infty)$ and concave down on $(0,\infty)$ and which passes through the points $(1,1)$ and $(2,3)$.
I could not recall a general approach to a question like this,... |
H: Let $f$ be a bounded linear operator $X \to X$ s.t that $\|f(x)\|\geq m\|x\|$ for some $m$, $\forall x \in V$. Prove that $f$ cannot be compact.
Let $V$ is an infinite dimensional subpace of a Banach space and let $f$ be a bounded linear operator $X \to X$ s.t that $\|f(x)\|\geq m\|x\|$ for some $m$, $\forall x \in... |
H: What are the operations in quaternions as a division ring?
When I studied quaternions in group theory only the product was defined
Now studying rings, my notes say quaternions are a division ring, But this means that we must have 2 operations: sum and product. How are the operations defined then?
AI: Unfortunately,... |
H: Show that a meromorphic continuation exists
I am preparing for the complex analysis qualifying exam, and I recently came across this problem:
Show that $$F(z)=\int_1^\infty\frac{t^z}{\sqrt{1+t^3}}\,dt$$ defines an analytic function on $\{z\in\mathbb{C}|\text{Re }z<\frac{1}{2}\}$ that has a meromorphic extension to ... |
H: Prove that $p(x)=d(x,0)$ is not a seminorm
Let $E$ be a topological vector space and $\{p_1,p_2,\dots\}$ be a nondecreasing family of seminorms on $E$. Define $$d(x,y)=\sum_{i=1}^{\infty} a_i \frac{p_i(x-y)}{1+p_i(x-y)},$$
where $\sum_{i=1}^{\infty} a_i<\infty$.
My question: How to prove that $p(x)=d(x,0)$ is not ... |
H: EGC and waves $p-s$ for an earthquake: functions examples using Taylor's expansion
We know that the electrocardiogram (ECG) is a graphical representation of the electrical activity of the heart and in medicine plays an indispensable role. ECG is one of the indicators of the total, as well as the current state of th... |
H: Proving that $\frac{2\pi i}{f'(z_0)}=\oint_\gamma \frac{dz}{f(z) - f(z_0)}$
I have the following exercise that I'm having some trouble solving:
Let $f:U\to\mathbb C$ be an holomorphic function, and let $f'(z_0) \neq 0,z_0 \in U$. Prove that, if $\gamma=z_0+re^{it}$ with $t \in [0,2\pi]$, then, for a sufficiently s... |
H: Solving $2^n - 2\times n = a $, where $a$ is a known constant
Solving $2^n - 2\times n = a $, where $a$ is a known constant.
This is my first question.
I am having trouble solving the equation in the title...moreover I do not even know the name of that kind of equations. Any help would be veeeery appreciated.
AI: \... |
H: Proof Verification: $|\bigcup \mathscr{A}| \leq | \mathscr{A}| \cdot \kappa$
Synopsis
Please verify my proof. I would also appreciate any tips on how I might improve my mathematical writing. Thank you.
If my proof is without major issues, please note that I might delete the question soon afterwards.
Exercise
If ev... |
H: Creation of Cauchy density from $N(0, \sigma_1^2)$ and $N(0, \sigma_2^2)$ random variables
I searched for "cauchy density derivation" but I didn't find any relevant results in the first couple pages.
Problem:
Given $X \sim N(0, \sigma_1^2)$ and $Y \sim N(0, \sigma_2^2)$, $X \perp Y$, I am supposed to derive the den... |
H: General Solution for $\cos(\frac{x}{2}-1) =\cos^2(1-\frac{x}{2})$
I'm looking for an algebraic solution to : $\cos(\frac{x}{2}-1) = \cos^2(1-\frac{x}{2})$. So I simplified the equation: first off, $\cos(\frac{x}{2}-1) = \cos(1-\frac{x}{2})$. Then I divided both sides by that. and so I'm left with two things to solv... |
H: Symbolizing "There are exactly three..."
Working on the book: P.D. Magnus. "forall x: Calgary. An Introduction to Formal Logic" (p. 233)
24.4 There are exactly...
15. There are exactly three apples.
$A(x): x$ is an apple.
$$
\exists x\exists y\exists z(A(x) \land A(y) \land A(z) \land \lnot(x=y) \land \lnot(x=z) ... |
H: For all $a,b,n \in \mathbb N$, $0 \leq n(a+b+1)-n^2 +b$
I claim the following:
For all $a,b,n \in \mathbb N$, we have that $0 \leq n(a+b+1)-n^2 +b$.
This seems true for me... Although how do I really check if this is true? I was thinking about doing it by induction, but the fact that we have to deal with three va... |
H: Prove that the adhesion of $ E_{\rho} $ is equal to $ F_{\rho} $.
Let $ f $ be a non-constant integer function. Given a number $ \rho> 0 $, let's define:
$ E_{\rho} = \{z \in \mathbb {C}: |f(z)| <\rho \} $, $ F_{\rho} = \{z \in \mathbb {C}: |f(x)| \leq \rho \} $.
(a) Prove that the adhesion of $ E_{\rho} $ is equal... |
H: If $S$ is a subspace then is it true that $S=\bigoplus_{i=1}^{k}(S\cap \left)$?
I'm trying to solve this problem: Suppose that $V$ is a vector space with basis $\mathcal{B}=\{b_i:i\in I\}$ and $S$ is a subspace of $V$. Let $\{B_1,\ldots,B_k\}$ be a partition of $\mathcal{B}$. Then is it true that $S=\bigoplus_{i=1... |
H: Dimension of $k[x_1, \ldots, x_n]/I$
I'm tasked with the following question:
Let $R = k[x_1, \ldots, x_n]/I$, where $I$ is an ideal of $R$. Show that $\dim (R) = 0$ if and only if $R$ is a $k-$vector space of finite dimension.
From Hilbert's Base Theorem, I know that $R$ is Noetherian and with $\dim(R) = 0$ I know ... |
H: Is the sum of polynomials solvable by radicals if both polynomials are solvable by radicals?
Is there a specific theorem? I saw it as an extension of a polynomial solvable by radicals plus a constant (as this is solvable via radicals with a simple substitution of y).
AI: The answer is no. An explicit example is as ... |
H: Countability of Non-intersecting Disks, Circles, Figure Eights
Synopsis
Alas, I'm confused again.
I'm working through a problem set and I've come across the following exercise.
(a) Let $A$ be a collection of circular disks in the plane, no two of
which intersect. Show that $A$ is countable.
(b) Let $B$ be a collec... |
H: Doubt in buiding a bump function in a manifold
This definition of a bump function is given in "Introduduction to Manifolds" by Loring W. Tu: Given a point $ p $ in a manifold $ M^n$, a bump function in $p$ supported in $V$ is any non-negative function $ \rho: M \rightarrow \mathbb{R} $ which is identically $ \mathb... |
H: Does the quotient map induce a continuous map
Is the following true?
Let $X$ be a topological space and $\rho:X\to Y$ a quotient map onto a topological space $Y$. Suppose further that there exist a continuous function $f$ mapping $X$ onto itself with the property that whenever $C\subset X$ equals the preimage unde... |
H: Understanding Irrationality Measure
It is well known that real numbers are either rational or irrational.
However, one can ask whether some irrational numbers are in some sense harder to approximate by rational numbers than others. One way to make this notion precise is the Irrationality Measure, which assigns a po... |
H: Assume that $\overline{A}$ convex, is $A$ convex?
Let $A$ be bounded subset of $\mathbb{C}^d$.
It is clear that if $A$ is convex then so is $\overline{A}$. Here $\overline{A}$ denotes the closure of $A$.
Assume that $\overline{A}$ convex, is $A$ convex?
AI: No. In fact $\overline{A}$ convex doesn't even force $A$... |
H: Urn problem with two ways of expectation computation
Let $X$ denote the number of white balls selected when $k$ balls are chosen randomly from an urn containing $n$ white balls and $m$ blacks.
For $i＝1....k; j＝1....n$,
$$\begin{align*}
X_i&＝\begin{cases}
1,&\text{if }i\text{-th ball selected is white}\\
0,&\text{ot... |
H: How to understand conditional statements
$p \rightarrow q$ is read as ${\rm{if}}\:p\:{\rm{then}}\:q$.
It is clear that the result of the statement when $p$ is true is $q$. So, when $p$ is true, the truth value of the statement is the same truth value of $q$.
But how should I understand the cases when $p$ is false? ... |
H: If $R = \{(1,2),(1,4),(3,3),(4,1)\}$, then is $(1,2) \in R^2$? (Powers of Relation)
I basically got this:
$R^2 =\{(4,4),(1,1),(3,3),(4,2)\}$
But I'm not sure if I should include (1,2) as well since 2 maps to nothing?
Thanks
AI: Rather, $R$ maps $1$ to $2$, and maps $1$ to $4$, but does not map either $2$ or $4$ to ... |
H: Can I apply Cauchy-Goursat to $\int_0^\pi \frac{dz}{3+z^2} \leq \frac\pi3$
I have to prove that $\int_0^\pi \frac{dz}{3+z^2} \leq \frac\pi3$ I wanted to prove it by directly getting the integral value
I know $$\int_0^\pi \frac{dz}{3+z^2} = \int_0^\pi \frac{dz}{z^2-(\sqrt{3i})^2} = \int_0^\pi \frac{dz}{(z-\sqrt{3... |
H: If 1 is the identity of the multiplicative (semi)group what is the term for 0?
Broadly given an operator $*$ the term identity is used for an element $e$ such that $x * e = x$ for all elements. However is there a term for a value $ x * O = O$ for all values? This was brought to mind by this question What is the id... |
H: Covariance of two mixed binomial distributions with geometric distribution
Let X1, X2... be iid Bernoulli random variables with parameter 1/4, let Y1, Y2... be another sequence of iid Bernoulli random variables with parameter 3/4 and let N be a geometric random variable with parameter 1/2. Assume the Xi's, Yj's and... |
H: Explanation for behaviour of graph of $y=x^2e^{-x^2}$ (Maxwell-Boltzmann distribution)
Consider the function
$$y=x^2e^{-x^2}$$
The graph initially behaves as a parabola then in later part exponential part of it dominates; i.e., the graph looks exponential after maximum of the curve.
Actually this graph is related t... |
H: Equivalent definitions of the support of a measure
It's not a homework, actually I was reading an article where the following was stated. Let $\Omega$ be a toplogical space and $\mathcal{F}$ its Borel $\sigma$-algebra, i.e. the $\sigma$-algebra generated by its open sets. Let $\mu$ be a probability measure on $\Ome... |
H: If $11+11=4$ and $22+22=16$, then $33+33=\text{???}$ (Facebook math quiz)
From a Facebook math quiz:
$$\begin{align}
11 + 11 &= 4 \\
22 + 22 &= 16 \\
33 + 33 &= \text{???}
\end{align}$$
Maybe I'm just stupid but the answer to this is $36$, however I think it should be zero.
The reason I think this is because if you... |
H: What is the boundary point of the set of all convergent sequences
I'm not sure as to how to define this.
As in let $C$ be the set of all convergent sequence of real numbers.
Then what would a boundary point of $C$ be in $(l_\infty, d_\infty)$.?
AI: $C$ is its own boundary.
Any convergent sequence $(a_n)$ is in the ... |
H: Suppose $S$ is a subring $R$ and R is free of finite rank as a module over $S$. Is there a Ring homomorphism from $R$ to $S$?
Suppose $S$ is a subring $R$ and R is free of finite rank as a module over $S$. Is there a Ring homomorphism from $R$ to $S$? The reason I ask is to prove that if S has invariable basis numb... |
H: Is a holomorphic function with nonvanishing derivative almost injective?
Let $\Omega \subseteq \mathbb C$, be an open, bounded, connected, contractible subset with smooth boundary.
Let $f:\Omega \to \mathbb C$ be holomorphic, and suppose that its derivative $f'$ is everywhere non-vanishing.
Is true that for almost... |
H: Number of permutations $(p_1,\dots,p_6)$ of $\{1,\dots,6\}$ such that for any $1\le k\le5,(p_1,\dots,p_k)$ is not a permutation of $\{1,\dots,k\}$
Problem (INMO 1992 problem #4)
Find the number of permutations $(p_1,p_2,p_3,p_4,p_5,p_6)$ of $\{1,2,3,4,5,6\}$ such that for any $k$ such that $1 \le k \le 5,$ $(p_1,..... |
H: Proof regarding extreme point of a convex set
I´m having trouble proving the following:
Let $C\subset\mathbb{R}^n$. Prove $x \in C$ is an extreme point of $C$ if and only if $C\setminus {x}$ is convex.
For the sufficiency, I figured if $x$ is an extreme point then the result follows since any other point can be e... |
H: Looking for a math expression to fit these series of inputs/graphs
I require a math formula that lets me move smoothly between these graphs:
It must always intersect x=0 and x=1 as seen, and its peak must always be at 1. It can be a parametric equation. How might I generate something like this.
For example that mi... |
H: Gradient fields with holes and the fundamental theorem of line integrals
I'm learning vector calculus for the GRE, so this is my first time encountering these concepts. To my knowledge, a gradient field $\mathbf{F}(x,y) : D\to \mathbb{R}^2$ is one where there exists a scalar function $f(x,y) : D\to \mathbb{R}$ such... |
H: Implicit function the right approach
Given $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
what is $y’_x$?
I’ve gone back and forth on this and I thought I could perhaps use the implicit function theorem, but then again is there a need to? I have the answer options but can’t seem to get to them so I must be doing something wro... |
H: $\{ \sqrt n x^n\}$ is divergent in $(C[0,1],d_2)$
Let $f_n \in C[0,1]$ such that $f_n(x)=\sqrt nx^n, n \in \Bbb Z^+$. I want to show that $\{f_n\}$ is divergent in $(C[0,1],d_2)$.
I use the definition and got this
$$||f||_2 = \sqrt{\int^1_0 (\sqrt nx^n)^2}$$
Which eventually gave me
$$\sqrt {\cfrac {n} {2n+1}}$$
Ta... |
H: Conditions on a matrix having 1 as eigenvalue
Suppose $A\in\mathbb{R}^{m \times n}$, $m\geq n$, and assume $A (A^TA)^{-1}A^T$ has 1 as an eigenvalue.
Is it possible to say anything about the structure of A?
AI: $P := A(A^\top A)^{-1} A^\top$ is the orthogonal projection onto the column space of $A$. Orthogonal proj... |
H: Changing norm between $\ell^2$ and $\ell^\infty$.
For two finite sets $A$ and $B$, let $x_{i,j}$ be in both $\ell^2(A)$ and $\ell^\infty(B)$. Then is it possible that
$$
\| \|x_{i,j} \|_{\ell^\infty(B)}\|_{\ell^2(A)} = \| \|x_{i,j} \|_{\ell^2(A)}\|_{\ell^\infty(B)}
$$
or inequality
$$
\| \|x_{i,j} \|_{\ell^\infty(B... |
H: Finding angle between two hexagonal planes
I am looking to get angle between two hexagonal planes. I have coordinates of all 12 vertices (2 hexagons). Is there anyway I can find the angle between the planes. And also I have the information about the centers of each hexagonal plane.
You can see the two hexagons, I n... |
H: Quick question about antisymmetric relationship.
Here we go,
It is a really yes or no question.
If aRb is a|b then is this antisymmetric? a, b belongs to integers including 0*
AI: Is it? Consider if $a \mid b$ and $b \mid a$. Can this happen for integers $a,b$ and them not be equal?
Hint: Consider $b=-a$.
Solution:... |
H: Perfect Matching on Bipartite Graph
So I was trying to solve this problem
Let $H$ be a bipartite graph with bipartition $A,B$ such that $|A| = |B| = k$. Prove that the graph contains a perfect matching when every vertex has degree of at least $\frac{k}{2}$
And this is what I tried:
We know that each vertex in $A... |
H: Find $\mathbb{P}(A\cap B^c)$ where $A=\{X_1+X_2<1\}$ and $B=\{X_1+X_2+X_3<1\}$
Let $(X_1,X_2,X_3)$ three independent random variables with uniform distribution $[0,1]$.
Let $A=(X_1+X_2<1)$. Find $\mathbb{P}(A)$.
$\rightarrow \mathbb{P}(A)=\int_{0}^{1}[\int_{1-x_2}^{1}dx_1]dx_2=\frac{1}{2}$.
Let $B=(X_1+X_2+X_3<1... |
H: Pointwise convergence in probability and maximum of finite random variables
Let $\{G_n(\theta)\}_{n=1}^\infty$ be a sequence of random variables indexed by $\theta\in\Theta$.
Suppose that
$$
\forall\theta\in\Theta, G_n(\theta) \to_p 0 \quad(\text{pointwise convergence}).
$$
Pick any finite indices $\{\theta \}_{... |
H: Cauchy-Schwarz applied multiple times with difficulty on second application
Below, $a_i$ is a column vector with matching dimension as the square matrix $G$ and $b_{ij}$ is a scalar. I apply CS once over $(i,j)$ to obtain the first inequality but I'd like to apply CS once more to the first term on the right hand si... |
H: Convergence radius of $\sum_{n=1}^\infty \frac{(4-x)^n}{\sqrt{n^4+5}}$
Find the convergence radius of $$\sum_{n=1}^\infty \frac{(4-x)^n}{\sqrt{n^4+5}}$$
I've recently started self-learning about series, so I'm having a little trouble. Looking at this example, I tried the $n^{th}$-root test:
$$\sqrt[n]{\bigg|\frac... |
H: $1999$ Iberoamerican Number theory problem
Let $n$ be an integer greater than 10 such that everyone of its digits belongs to the set $S$=$\{1,3,7,9\}$. Show that $n$ has a prime divisor greater than or equal to 11.
Obviously n cannot have prime divisor 2 or 5 , now I have to show that n cannot equal to number of ... |
H: Logarithm power rule does not provide a complete solution. Have the logarithm rules failed me?
I am solving this question:
$log_3(m-7)^2 = 4$
There are two ways to solve it.
The first way (expand the brackets):
$log_3(m^2 -14m + 49) = 4$
$m^2 - 14m + 49 = 3^4$
$m^2 - 14m - 32 = 0$
$m = 16,-2$
And both of these are ... |
H: Solve $\int_3^4 \frac{\ln \left(x\right)+x}{e^x+x^x}\:dx$
Solve: $$\int_3^4 \frac{\ln \left(x\right)+x}{e^x+x^x}\:dx$$
I am not really sure how to attempt this. Although at first glance I thought it would be quite easy, Symbolab says that there are no steps to solve such an integral. However, WolframAlpha gives t... |
H: Compare two numbers without comparative operators
Is it possible to have a function which compares two numbers without comparative operators so that it returns 1 if they are equal and 0 if they are not? (No <, >, ==, etc.)
e.g.
$f(x, y) = ?$
$f(20, 20) = 1$
$f(15, 20) = 0$
It's possible to get 1 if they are equal -... |
H: How to calculate $\max\left\{\,\dfrac{1}{x}+\max\left\{\,\dfrac{2x}{x+y}, 1\right\} \right\} $
I am doing a case analysis for coming up with the worst-case performance of an algorithm. At some points, I need to calculate the maximum value of the following:
$$\dfrac{1}{x}+\max\left\{\,\dfrac{2x}{x+y}, 1\right\}$$
wh... |
H: Proving a question on connectivity of topological subspaces
I have been trying to solve the following question:
Let $X$ be a topological space and $Y$ a subspace of $X$, that is connected (in the relative topology). Show that if $Z$ is any subspace of $X$, such that $Y \subseteq Z \subseteq \bar{Y}$, ($\bar{Y}$ is ... |
H: Is there any closed form for $\displaystyle \prod_{n=1}^{\infty} \left(1-\frac{1}{\zeta(n)}\right)$
How do i evalulate the following infinite product? $$\displaystyle \prod_{n=2}^{\infty} \left(1-\frac{1}{\zeta(n)}\right)$$
Notation: $\zeta(n)$ is Riemann zeta function.
I'm interested to evaluate the above produ... |
H: The $\pm$ symbol in the square root of fractions, as well as in the quadratic formula
If I had $ \sqrt {a^2}= \frac{1+b}{4c^2}$ and wanted to solve for $a$, an easy thing to do would be to take the square root of both sides, giving $a=\pm\sqrt{\frac{1+b}{4c^2}}$
And then I would proceed to simplify the fraction by ... |
H: Joint probability distribution with dependant domains
Given a joint distribution function with this specific domain:
$$f_{X,Y}(x,y) = g(x,y)*1_{x,y\ge 0,\space x+y\le1}$$
I'd like to find the expected values of $X,Y$, i'm just unsure about the integral bounds:
$$f_X(x)=\int_0^{1-x}f_{X,Y}(x,y)dy$$
$$f_Y(y)=\int_0^{... |
H: If $\operatorname{lcm}(m, m + k) = \operatorname{lcm}(n, n + k)$, then $m = n$
Let $m, \ n, \ k \in \Bbb N $ be such that $ \operatorname{lcm}[m , m + k] = \operatorname{lcm}[n , n + k],$ then prove that $ m = n.$
Though I wasn't able to proceed much, but here is a sketch of what I tried.
First let $l = \operator... |
H: Upper bound of a set
A non-empty subset $S ⊆ \mathbb{R}$ is bounded above by $k ∈ \mathbb{R}$ if
$s ≤ k$ for all $s ∈ S$. The number $k$ is called an upper bound for $S$.
Could by this definition we say that $S$ can have the greatest element and that element is the upper bound?
AI: By that definition, if $S$ has a... |
H: $\frac{1}{x^2}\ = (-5+\sqrt{3})$
How can I solve the equation: $$\frac{1}{x^2}\ = (-5+\sqrt{3})$$
I tried this:
$$
x^2\cdot\frac{1}{x^2} = x^2\cdot(-5+\sqrt{3})
$$
$$
1=-5x^2+\sqrt{3}x^2
$$
$$
1= x^2(-5+\sqrt{3})
$$
$$
x^2=\frac{1}{(-5+\sqrt{3})}
$$
$$
x=\pm \sqrt{\frac{1}{-5+\sqrt{3}}}
$$
Not sure if this is corr... |
H: Finding all entire functions such that for all $|z| \geq 1$ we have $|f(z)| \leq \frac{1}{|z|}$
I am studying for my complex analysis exam and this question popped up few years ago on the "mini quiz":
Find all $f \in H(\mathbb{C})$ such that for all $|z| \geq 1$ we have $|f(z)| \leq \frac{1}{|z|}$.
The hint was t... |
H: "Surjections have Right Inverse" to the "Axiom of Choice"
I have learned that the statements
"Every surjective function has right inverse" and the "Axiom of Choice" are equivalent each other. I could easily prove the $\Longleftarrow$ direction, but it's little tricky to do the reverse direction. The problematic par... |
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