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H: show there is a number that occurs infinitely often in the digital representation of a polynomial
I would appreciate some help with the following problem:
Let $p(x)$ be a polynomial with integral coefficients (possibly negative).
Let $a_n$ be the digital sum in the decimal representation of $p(n)$ for $n\in \mathb... |
H: Neighborhood of a point (real number)
How can we prove that every real number has infinitely many neighbourhoods?
I know that it is true because we can consider a symmetric epsilon neighborhood of a point and there are infinitely many number of such epsilons.
But how can we prove it rigorously?
AI: You can prove it... |
H: If G is a simple graph with at least two vertices, prove that G must contain two or more vertices of the same degree.
I will use the pigeon-hole principle.
For a simple graph of n vertices the maximum degree a vertex can have is (n-1).
Let there be (n-1) boxes corresponding to degrees of 1 to (n-1). We are going to... |
H: Expected value question, with states!
Suppose an light can emit 4 different colors of light. The light starts off by emitting blue light. Each second, the color of the light changes according to the following probabilities-
When the light is blue, the next second, the light is red.(Named blue light state 0)
When th... |
H: When defining a vector space is the scalar part of the field or always a real number
I've stumbled upon an exercise that takest the set of integers $\Bbb{Z}$, defines addition and multiplication as usual but scalar multiplication as $\lfloor{\alpha}\rfloor * k$, where $\alpha$ is the scalar and $k$ the element of t... |
H: How is this proposition false?
I have a proposition:
((x v y) <=> (~x => ~y))
When I solve this, I end up getting True but the answer is False. Here's how I solved it:
when we have a <=> b, we can write it as ~a.~b + a.b and a => b becomes ~a+b
So the above equation becomes:
=> ~(x + y).~(~x => ~y) + (x + y).(~x =>... |
H: Show $(Y_{n}-a)_{+}\leq (Y_{n})_{+}+\lvert a\rvert$
In a proof, I saw the use of the following inequality
$(Y_{n}-a)_{+}\leq (Y_{n})_{+}+\lvert a\rvert(*)$
without any explanation, where $Y_{n}$ is some random variable and $a$ a constant. Note the definition
$(X)_{+}:=\max\{0,X\}$.
I am aware that $(\cdot)_{+}$ as... |
H: ***For any*** $f: B \to A$ with $(B \ne \emptyset)$, can a function $h:A \to B$ be constructed in such a way that $fhf = f$?
I've been stuck on this exercise for a while now, any help would be greatly appreciated.
Here's my try. I constructed $h$ as a left inverse of $f$.
For any function $f: B \to A$ with $B \ne ... |
H: S/pS is uncountable
From Page 163 of Rotman's Homological Algebra book:
The direct product of countably many Z is not free.
In the proof, a subgroup S is defined where S = {tuples with countably many components: each positive power of p divides almost all components}. He was trying to show that S is not free.
Then ... |
H: A deduction in Hoffman Kunze whose explanation is not given
I am self studying linear Algebra from Hoffman and Kunze and Couldn't think about how a deduction mist be true.
On page 15 authors wrote
Why if a solution $x_{1}$ ,..., $x_{n}$ belongs to F then the system of equations must have a solution with $x_{1} $,... |
H: Contrapositive arguments
I came across the following problem:-
Let $S$= {${u_{1},u_{2},...u_{n}}$} $\subseteq \Bbb C^{n}\ $and $T$= {${Au_{1},Au_{2},...Au_{n}}$}, for some matrix square matrix A $\in$$\Bbb M_{n}($C$)$.
If $S$ is Linearly independent prove that T is linearly independent for every invertible matri... |
H: Prove domain and whether the function - f(x) = 1/(1 + |x|) is one to one.
I am a first year Computer Science student studying Calculus and Linear Algebra this semester.
I have got two questions:
how to find domain regarding finding the domain of the following function.
Is this function one to one?
Question - 1 - ... |
H: Question on tensor product with field
Let $A$ be a finitely generated $K$-algebra which has no zero divisors. Here $K$ is a field of characteristic $0$. Let $K\subset L$ an algebraic field extension. Now let $f: L\to E$ and $g: \textrm{Quot}(A)\to E$ be two homomorphisms to another field $E$. The universal property... |
H: Proving $\lim_{n \to\ \infty} \sum_{k=0}^{n} \frac{1}{(k+3)k!} = e-2$
$$ \lim_{n \to \infty} \sum_{k=0}^{n} \frac{1}{(k+2)!} \leq \lim_{n \to\ \infty} \sum_{k=0}^{n} \frac{1}{(k+3)k!} \leq \lim_{n \to\ \infty} \sum_{k=0}^{n} \frac{1}{k!} -2 $$
This is what I came up with but the problem is that the upper bound ... |
H: Sum of measurable extended real-valued functions is measurable?
Let $f, g: X \to \overline{\mathbb{R}}$ be measurable (here $\overline{\mathbb{R}}$ denotes the extended real numbers).
Let $E_1 = \{ x \in X: f(x) = - \infty, g(x) = +\infty \}, E_2 = \{ x \in X: f(x) = + \infty, g(x) = -\infty \}$. Define
$$
h(x) =
\... |
H: Prove that ,$\left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2} \geq 8$
For any positive a, b prove that $$\left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2} \geq 8$$
My approach:
Using the well known inequality,
$ \boxed{\mathrm{AM} \geq \mathrm{GM}}$
$\left(a+\frac{1}{a}\right)^{2}+\left(b+... |
H: Proving $\sum_{n=0}^\infty\frac{(-1)^n\Gamma(2n+a+1)}{\Gamma(2n+2)}=2^{-a/2}\Gamma(a)\sin(\frac{\pi}{4}a)$
Mathematica gives
$$\sum_{n=0}^\infty\frac{(-1)^n\Gamma(2n+a+1)}{\Gamma(2n+2)}=2^{-a/2}\Gamma(a)\sin(\frac{\pi}{4}a),\quad 0<a<1$$
All I did is reindexing then using the series property $\sum_{n=1}^\infty (-1)... |
H: My answer is 36225. Please verify if it is correct
A computer program prints out all integers from 0 to ten thousand in base 6 using the numerals 0,1,2,3,4 and 5. How many numerals it would have printed ?
AI: The numbers from $0$ to $5$ have $1$ digit in base $6$.
The numbers from $6$ to $35$ have $2$ digits in bas... |
H: Does midpoint convexity at a single point imply full convexity at that point?
Let $f:[a,b] \to \mathbb [0,\infty)$ be a continuous function, and let $c \in (a,b)$ be a fixed point.
Suppose that $f$ is midpoint-convex at the point $c$, i.e.
$$
f((x+y)/2) \le (f(x) + f(y))/2,
$$
whenever $(x+y)/2=c$, $x,y \in [a,b]$.... |
H: Is there a metric on a finite-dimensional (non-trivial) vector space $V$ over $\mathbb{R}$ which makes it compact?
I know that a finite-dimensional (non-trivial) vector space $V$ over $\mathbb{R}$ which is normed isn't compact, but what about when it has a metric in general?
AI: Take any bijection $f: V \to [0,1] ... |
H: Smallest subset of integers that you can use to produce 1,2,....,40
Find the smallest subset of integers that you can use to produce $1,2,...,40$ by only using $"+"$ or $"-"$, and each number in the subset can be used at most one time.
There is a hint that $0$ must be in the set, but I cannot see any justification ... |
H: The interval in which the function $ f(x)=\sin(e^x)+\cos(e^x)$ is increasing is/are?
question
options and answers
The interval in which the function $f(x)=\sin(e^x)+\cos(e^x)$ is increasing is/are?
I don't understand how to approach such problems. it would be helpful if you could kindly guide me through the process... |
H: Why $P(\limsup A_n) = 1$ when $P(\cup A_n) = 1$?
Let us have a sequence of independent sets like $\{A_n\}$, suppose for any $n$ belongs to $N$, we have $P(A_n)< 1$. If $P(\cup A_n) = 1$ why $P(A_n \ \ i.o) = 1$?
AI: This is not true. If $A_n$'s form a partition of the sample space then $P (\cup A_n)=1$ but $A_n$'s... |
H: Why is the set of all bases of a vector space a manifold?
Let $V$ be an $\mathbb{R}$ vector space. Then the collection of all ordered bases of $V$ denoted by $P(V)$ seemingly is manifold. How does one see this ? I suppose that second-countability is inhereted from some $\mathbb{R}^n$ diffeomorphic to $V$. But how t... |
H: p is prime and $p=4t+3$, I am wondering how to prove
when $p ≡ 1\pmod7$
$$(-1)^\frac{(p+1)}{4} = 1$$
when $p ≡ 5 \pmod3$
$$(-1)^\frac{(p+1)}{4} = -1$$
My attempt: For question 1, $p=4t+3$,hence $$\frac{(p+1)}{4} = t+1$$
$4t+3≡1\pmod7$, hence $$t+1≡4\pmod7$$
For question 2, $p=4t+3$,hence $$\frac{(p+1)}{4} = t+1... |
H: Application of Identity Theorem
Let {$a_n$} and {$b_n$} be sequences of complex numbers such that each $a_n$ is non zero,
$$\lim_{n\to\infty}a_n = \lim_{n\to\infty}b_n=0$$ and such that for every natural number k, $$\lim_{n\to\infty}\frac{b_n}{a_{n}^k}=0$$
Suppose f is an analytic function on a connected open subse... |
H: Questions about differentiability
Which of these following statements is true?
a. Let $f:\mathbb {R}^n \to \mathbb {R}^k$ and $g:\mathbb {R}^k \to \mathbb {R}^m$. If $g\circ f$ is differentiable, then $g$ and $f$ are differentiable.
b. Let $f:\mathbb {R}^m \to \mathbb {R}^n$ with $df(p)=0$ for $p \in \mathbb{R}^m$.... |
H: taylor expansion for $\frac1{\cos(x)-\sin(x)}$
I am trying to find a closed expression for the taylor series of $\frac1{\cos(x)-\sin(x)}$ in a neighborhood of $x = 0$. One can obtain the coefficients from: $$(a_o+a_1x+a_2x^2+\cdots)(1-x-\frac{x^2}2+\frac{x^3}{3!}+\frac{x^4}{4!}--++\cdots) = 1$$
thus $a_0= 1, a_1=1,... |
H: Individual percentage of a set of numbers
I have a set of four numbers. 7.5, 18.5, 424 and 0.
Certain percentage is associated with each of it, viz 1.66%, 4.11%, 94.22% and 0.1% resp, which in total is 100%.
I want to know how these percentages were calculated? Can a generic formula be applied ?
AI: Simply put: The... |
H: Question in proof of Multipilcation of matrices in Linear Algebra
While self studying linear Algebra from Hoffman Kunze I have a question in proof of a theorem whose image I am adding :
This theorem would be useful->
( Due to some glitch both images automatically appeared at last not here).
Question:How in proof... |
H: How do I compute the derivative of this inverse function?
Let $$f(x)=\frac{1}{16}(e^{\arctan(\frac{x}{7})} + \frac{x}{7})$$
You are given that $f$ is a one-to-one function and its inverse function $f^{-1}$ is a differentiable function on $\mathbb{R}$. Also $f(0)=\frac{1}{16}$. What is the value of $(f^{-1})'(1/16)$... |
H: Finding plane equation by using parametric equation
Find the plane equation passing through $(4,-2,6)$
$x=3-2t$, $y=t$ and $z=5+2t$
How was the solution of these type of questions..Thanks
AI: Let $P(4,-2,6)$ and line $AB$: $X_\ell=A+t\,(B-A)=(3,0,5)+t\,(-2,1,2)$, then the plane equation will be $n\cdot (X-A)=0$ whe... |
H: Given a finite set of points. How can I make a ball around this points, so that the intersection of any two balls are empty
I'm looking for an algorithm or a theoretical result for the following problem:
Given the finite set of points $X = \{x_{1},\ldots,x_{d}\} \subseteq \mathbb{R}^{n}$. For $x \in X$ we define $B... |
H: Formula for the number of elements in $S_{10}$ of order $10$.
I'm trying to find a formula (mostly in terms of factorials) for the number of elements in $S_{10}$ of order $10$.
We first we count how many elements of $S_{10}$ have the same cycle structure.
So,since $lcm(a,b)=10⟺a=10,b=0$ or $a=5,b=2
$, there are onl... |
H: Measurability on subsets | tower property of conditional expectation
I have some understanding issues following the proof of the tower property of conditional expectation.
The Theorem is the following:
Let $F_0, F_1$ be $\sigma$-fields with $F_0 \subseteq F_1 \subseteq F$ and $X$ a random variable with $X \geq 0$. ... |
H: Suppose $\{A_i | i ∈ I\}$ is an indexed family of sets and $I \neq \emptyset$. Prove that $\bigcap_{i\in I}A_i\in\bigcap_{i\in I}\mathscr P(A_i)$.
Not a duplicate of
Prove that if $I ≠ \emptyset$ then $\bigcap_{i \in I}A_{i} \in \bigcap_{i \in I} \mathscr P (A_{i})$
To Prove $ \bigcap_{i \in I} A_i \in \bigcap_{i \... |
H: Proving A is a group over unusual operation
Let $A= R^2 - {(0,0)}$ and operation $*$ over A is defined by.
$$(a, b)*(a′, b′) = (aa′−bb′, ab′+a′b)$$
Question: Is $(A,*)$ a group ?
My attempt: First of all, as for all $a$,$b$ in $A$ $$(1,0)*(a,b)=(a.1-0.b,b.1-a.0)=(a,b)=(a,b)*(1,0)$$
$(1,0)$ is identity element.
Seco... |
H: I don't understand this derivation
I try to unterstand a derivation and need help.
There are given two functions
$$
s=-cos(j\pi/n),s\in[-1,1]
$$
and the nonlinear transformation
$$
y(s)=C\tan[\frac{\pi(s+1)}{4}+\frac{s-1}{2}\arctan\frac{y^*}{C}]+y^*,y\in[0,\infty)
$$
$y*$ and $C$ are constant parameters.
The deriva... |
H: What does $\exists^{=1}$ stands for?
In this paper, they use $\exists^{=1}$, I know that $\exists$ stands for there exists. But what does $\exists^{=1}$ stand for? My guess is "Only one existential quantifier". But can someone confirm it?
AI: For any constant $k\in\mathbb N$, $\exists^{=k}x\,\phi(x)$ is a standard ... |
H: Integration problem where denominator of the integral variable is fractional powered
I am a bit stuck on this integration problem
\begin{equation}
\int \frac{1}{a+(x-b)^c} dx
\end{equation}
Is there a name on this type of integral that I can look up to?
AI: \begin{equation}
I=\int \frac{dx}{a+(x-b)^c} =\int \frac{d... |
H: How to simplify the following matrix problem?
Taken from the lecture notes: Introduction to the principles and methods of data assimilation in the geosciences - Marc Bocquet, where I am currently working on page 12.
Given that:
$$\mathrm{P^a=(I-KH)B+[KR-(I-KH)BH^T]K^T} \tag{1.27}$$
the expression in brackets in th... |
H: Finding the basis of a product topology
Let $X = \{1, 2, 3\}$, $T = \{\varnothing, \{1\}, \{1, 2\}, X\}$, $Y = \{4, 5\}$, and $U = \{\varnothing, \{4\}, Y\}$. How do I find the basis $B$ for the product topology on $X \times Y$?
Definition: A topological space $(X, T)$ is a Hausdorff space provided
that if $x$ and ... |
H: Random vector in a circle of unity radius
The random variable $(X,Y)$ is uniformly distributed in the circle $x^2+y^2\leq 1$.
Find the joint density $f_{XY}(x,y)$ and the marginal density of $X$ and $Y$.
$$
\begin{split}
f_{XY}(x,y) &= \frac{1}{\pi} \\
f_X(x) &= \frac{2}{\pi}\sqrt{1-x^2} \\
f_Y(y) &= \f... |
H: Is there any easy way to calculate the value of this determinant?
\begin{vmatrix}
0 & 3 & 1 & 2 & 10! & e^{-7}\\
1 & 2 & -1 & 2 & \sqrt{2} & 2 \\
-1 & -2 & 3 & -3 & 1 & -\frac{1}{5} \\
-2 & -1 & 3 & 2 & -2 & -9 \\
0 & 0 & 0 & 0 & 4 & 2 \\
0 & 0 & 0 & 0 & 1 & 1 \\
\end{vmatrix}
I still can't see any easy way to c... |
H: How to find 3D point of a triangle in a 3D space
I have a triangle in $3D$ space, with $2$ points defined (lets call them $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$) and distances to the $3^{rd}$ point known (lets call it $C(x_3, y_3, z_3$)) as well as the $z_3$ known.
I need to make a universal formula to find $C$,... |
H: Why does an a local analytic isomorphism imply a mapping onto a disk?
in one of the answers to my questions on StackExchange (Open Mapping Theorem Serge Lang Proof) the person answering the question states that "The map u:z↦(z−a)g1(z) is local analytic isomorphism by Theorem 6.1(c) above, so we can take an open nei... |
H: Suppose every element of $\mathcal F$ is a subset of every element of $\mathcal G$. Prove that $\bigcup \mathcal F\subseteq \bigcap\mathcal G$.
Not a duplicate of
Prove that if F and G are nonempty families of sets, then $\bigcup \mathcal F \subseteq \bigcap \mathcal G$
Validity of this proof: Prove that $\cup \mat... |
H: Maximize $\boxed{\mathbf{x}+\mathbf{y}}$ subject to the condition that $2 x^{2}+3 y^{2} \leq 1$
Maximize $\mathbf{x}+\mathbf{y}$ subject to the condition that $2 x^{2}+3 y^{2} \leq 1$
My approach
$\frac{x^{2}}{1 / 2}+\frac{y^{2}}{1 / 3} \leq 1$
Let $z=x+y$
$\mathrm{Now}, 4 \mathrm{x}+6 \mathrm{y} \frac{d y}{d x}=0... |
H: Is this conclusion on orders of magnitude correct?
Let $f(n_1,n_2) = \mathcal{O}\left(\frac{n_1n_2^2}{(n_1-n_2)^3}\right)$, where $n_1$ and $n_2$ are natural numbers. If $n_1\propto n_2$, that is, if the two variables grow proportionally, is it true that $f(n_1,n_2) =\mathcal{O}(1)$?
AI: It depends on the coefficie... |
H: Reference request: SLLN for correlated/exchangeable random variables
Is there any strong law of large number for correlated but identically distributed random variables or exchangeable random variables?
A further question, what about random variables taking values in arbitrary Banach spaces?
AI: The SLLN when the v... |
H: volume of solid obtained by rotating the graph about $x=2$ line
Find the volume of solid generated by revolving the regin bounded by the graph of the equation $y=2x^2,y=0,x=2$ about $x=2$ line is
What i try
Put $x=2$ in $y=2x^2$ . We get $y=8$
So Volume of solid is
$$\pi\int^{8}_{0}r^2dy$$
I did not understand H... |
H: $f:\mathbb {R}^2 \to \mathbb {R}$, $f(x,y) = \lvert x \rvert y$. Does $df(1,0)$ exist?
Let $f:\mathbb {R}^2 \to \mathbb {R}$, $f(x,y) = \lvert x \rvert y$
I know that $\frac {\partial }{\partial y} f$ exists in $\mathbb {R}^2$ and that $\frac {\partial }{\partial x} f$ doesn't exist in $\mathbb{R}^2$. Does $df(1,0)... |
H: Find Remainder when $(x+1)^n$ divided by $x^2+1$
I put $(x+1)^n=p(x)(x^2+1)+bx+c$ for some $p(x)$ as the other exercise where we asked to find the remainder when one polynomial is divided by another polynomial. But to make $p(x)(x^2+1)$ go so I could find $b,c$ I have to put $x=i$ which is something I shouldn't put... |
H: Show that $\{v_1,v_2,\dots,v_n\}$ is a basis of a vector space iff a chain of subspaces is complete.
Let $V$ be a vector space over a field $F$.
A chain $\{0\}=V_0\subseteq V_1\subseteq\dots\subseteq V_{n-1}\subseteq V_n=V$ of subspaces $V_1,V_2,\dots,V_{n-1}$ of $V$ is said to be complete if there is no subspace ... |
H: The projective space $\Bbb RP^{2n}$ cannot be the total space of a nontrivial covering map
Why is the projective space $\Bbb RP^{2n}$ not the total space of a nontrivial covering map? I've heard this in class but I can't see why it holds.
AI: You've already worked out the answer in the comments, but just so that th... |
H: show that $e^{f(x)}$ is integrable if $f(x)$ is integrable
Let $f(x)$ be integrable in the path $[a,b]$, I need to prove that $e^{f(x)}$ is also integrable in $[a,b]$
My attempt is to argue that if $f(x)$ is integrable so $F=\int{f(x)}$ is continous
and than I thought about a way to get relation between $e^F$ to $... |
H: Prove that if $A^*+A=AA^*$ then $A$ is normal
Prove that if $A^*+A=AA^*$ then $A$ is normal.
I've tried some basic algebraic operations with no luck.
AI: \begin{align*}
&A^*+A=AA^*\\
\implies &(A-I)(A^*-I)=I\\
\implies &A-I=(A^*-I)^{-1}\\
\implies &(A^*-I)(A-I)=I\\
\implies &A^*A=A+A^*=AA^*\tag*{$\blacksquare$}
\en... |
H: exp(A+B) = exp(A)exp(B) for matrices proof
In this thread, On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?, it was mentioned that absolute convergence is required for swapping sums. What theorem is used precisely?
AI: I think actually absolute convergence is only mentioned in the origi... |
H: Stochastic order
I need help with the following exercise:
$X$ and $Y$ are random variables in $\mathbb{R}$ with the distribution functions $F_X$ and $F_Y$, so $X$ is stochastically smaller or equal to $Y$, i.e. $(X\leq_{st}Y)$ if $F_X(t)\geq F_Y(t)$ holds for all $t\in\mathbb{R}$.
How can I show the following state... |
H: find the solutions of $y^{\prime \prime}-4 y^{\prime}+3 y=8 e^{-x}+9$ s.t $\lim _{x \rightarrow \infty} e^{-x} y(x)=7$
I have the ODE $$y^{\prime \prime}-4 y^{\prime}+3 y=8 e^{-x}+9$$
I am asked to find a solution such that: $$\lim _{x \rightarrow \infty} e^{-x} y(x)=7$$
This question feels a liitle bit tricky, how... |
H: A curious property of exponential sums for rational polynomials?
An article led me to generate some graphs of exponential sums of the form $S(N)=\sum_{n=0}^Ne^{2\pi i f(n)}$, where $f(n)= {n\over a}+{n^2\over b}+{n^3\over c}$ with $a,b,c\in\mathbb{N}_{>0},\,$ leaving me amazed at their great variety. Here are some ... |
H: Similarity transformation and representation of matrices
I'm trying to understand this passage of a book:
Why this last expressions shows that the $i$th column of $\bar{A}$ is the representation of $Aq_i$ with respect to the basis $\{q_1,\ldots q_n\}$? I can't understand how the author reached this conclusion.
AI... |
H: Unknown asymptotic notation $(1 + O\Big(\frac{\log n}{n}\Big))\frac{2^n}{n}$
I am reading through Boolean Function Complexity by Stasys Jukna and I stumbled upon this notation for asymptotic bounds:
$$C(f) \leq (1 + \alpha_n)\frac{2^n}{n} \;where\; \alpha_n = O\Big(\frac{\log n}{n}\Big)$$
What exactly does the equa... |
H: Banach fixed point theorem, prove singular solution
I'm really having trouble understanding how to apply Banach's fixed-point theorem in this exercise.
Let $b_i$ and $c_{ik}$ be real numbers with $1 \leq i,k \leq n$ such that the following equation holds
$$
\sum_{i,k=1}^n c_{ik}^2 < 1
$$
Now I have to show the foll... |
H: Multiplying out a quadratic equation of differential factors acting on a function
I am having trouble understanding a multiplying out of a quadratic equation with differential factors:
from the following video:
$$
(D + A(x))(D + B(x))y(x) = 0\\
(D^2 + AD + AB + B^{'} + BD)y = 0 \\
$$
What I don't get is his remark... |
H: Function composition and inflection points
Considering two functions in $\mathbb{R}$ , $f$ and $g$, both having an inflection point on the same x-coordinate, does the function $h=f \circ g$ necessarily have an inflection point on that x-coordinate?
AI: Say $f’’(c)=g’’(c)=0$. Compute the second derivative of $f\circ... |
H: A convergence problem on bounded linear operators
It's, I think, essentially a question about the relation between the pointwise limit and the limit with respect to the operator norm and uniqueness of limits for convergent sequences in a metric space.
Define $\mathcal{B}\left(X,Y\right)$ to be a set of all bounded ... |
H: Problem for gradient and extremum points.
I try to compute extremum points for this funciton:
$$
f(x,y) = x^{4} - y^{4} - 4xy^{2}-2x^{2}
$$
The first step compute gradient:
$$
\nabla f(x,y) = [4 x^3-4 x-4 y^2, -8 x y-4 y^3]
$$
Next step
$$
\nabla f(x,y) = 0
$$ and its fail. It's not easy compute explicitly roots.
D... |
H: Congruence equation with binomial coefficient
Given a prime $p$ and some $k,t\in\Bbb{Z}^{+}$, when does the congruence equation $${x \choose k} \equiv t\pmod {p}$$ have an integer solution?
Is there some necessary and sufficient condition about $p,k,t$?
AI: By Lucas's theorem,
$${x \choose k} \equiv \prod_j {x_j... |
H: What is the proper notation for drawing random variables from processes?
I have a written statement of the form "Let $X_1$ and $X_2$ be independent order-$n$ matrices whose elements are i.i.d. normally-distributed random variables".
Because I frequently use such statements in my current paper, I require a compact n... |
H: Finding the angle between subspace and a vector.
Find the angle between the vector $x=(2,2,1,1)$ and the space formed by the linear combination of the vectors $a_1 = (3,4,-4,-1)$ and $a_2=(0,1,-1,2)$.
I found the vector $y=(3,1,3,1)$ that is perpendicular to both $a_1$ and $a_2$. Then found the angle between $x$... |
H: Does the probability of obtaining a sample for which a new element will be larger than the sample approaches 0 as the sample size increases?
Choose arbitrary $\epsilon > 0$ and arbitrary probability distribution $D$ over $[0, 1]$.
For a given natural number $m$, sample $S_m = (x_1, x_2, ..., x_m)$ as indepentent id... |
H: complex analysis question line
I know that the map $$z\mapsto\frac{R^2}{\overline{z}-\overline{a}}+a$$ takes a point to a symmetric point respect to the circle $|z-a|=R$ and am trying to get the line $y=x$ in some form like $z=i\bar z$ but this is for $y=x$.
I am unsure of how to approach/start this problem.
Any he... |
H: Will arithmetic on two sequences ever equal some value
I wrote a question a few days ago, but it seems to have disappeared. My question is this:
If I have 2 sequences:
Sequence1 = {1002, 996, 990... 1 } : essentially S1n-6
Sequence 2 = {2, 10, 18 .... x} : essentially S2n + 8
And if I want to see if nth term in... |
H: Finding parametrization of $xz^2=xy^2+y^3$
I am doing a problem on geometry and was stuck at one step. I want to parametrize the curve $xz^2=xy^2+y^3$ using two parameters $s,t$. But I cannot figure it out. I am wondering if this is a famous curve or if anyone has any thoughts?
AI: Let $y=s$ and $z=t$, then $x=\fra... |
H: Understanding Grassmannian as $\text{Gr}(k,n) \cong \text{GL}(k,k) \ \backslash \ \text{Mat}^*_{\mathbb{R}}(k,n)$.
The Grassmannian $\text{Gr}(k,n)$ can be described as the quotient
$$\text{Gr}(k,n) \cong \text{GL}(k,k) \ \backslash \
\text{Mat}^*_{\mathbb{R}}(k,n) $$
where $\text{Mat}^*_{\mathbb{R}}(k,n)$ is th... |
H: Derivative of magnitude of complex trace (function of matrices with Gateaux derivative)
So if I define $y(U)=|Tr(U^*V)|^2$
If I do the Gateaux derivative:
$y_U[\tilde{U}] =\frac{d}{d\epsilon}|Tr((U^*+\epsilon\tilde{U})V)|^2
$,
Here is where I get confused because of how things are nested - the trace has a real and ... |
H: If $\cos (α + β) = 4 / 5$ and $\sin (α – β) = 5 / 13$, where $α$ lie between $0$ and $\pi/4$, find the value of $\tan2α$.
In my textbook, the given answer is only $\frac{56}{33}$. But I think the question has another answer $(\frac{16}{63})$ as well.
Please review the attached answer, share your thoughts and correc... |
H: If $A^3 = I_n$ then $\operatorname{tr}(A)\in\Bbb Z$
Let $A\in M_{n\times n}(\Bbb R)$ s.t $A^3 = I_n$. Show that $\operatorname{tr}(A) \in \Bbb Z$.
I know that $P(A) = 0$, where $ P(x) = x^3 - 1 = (x-1)(x^2+x+1)$, that is, $1$ is a engevalue of $A$. Also, Trace is the sums of the eingevalues and in $\Bbb C$ have $1... |
H: Prove the limit of parametric integral:
Suppose that:
1). $\varphi_{n}(x) \geq 0$, $\forall n \in \mathbb{N}$ on $[-1,1]$.
2). $\varphi_{n}(x) \rightrightarrows 0$ as $n \to \infty$ on $(0< \varepsilon \leq |x|\leq 1)$.
3). $\int_{-1}^{1} \varphi_{n}(x)dx \to 1$ as $n \to \infty$.
Prove that if $f \in C[-1,1]$, the... |
H: A square is cut into three equal area regions by two parallel lines find area of square.
A square is cut into three equal area regions by two parallel lines that are 1 cm apart, each one passing through exactly one of two diagonally opposed vertices. What is the area of the square ?
AI: Let $AE=x$. By symmetry (equ... |
H: Is the following a formula for expressing a dot product in terms of length?
I've come across the following formula and I've been told that it's expressing a dot product in terms of length, but I can't find any sources or derivations for it online.
$$⟨u,v⟩ = \frac{|u+v|^2 - |u|^2 - |v|^2 }{2}$$
I found some similar... |
H: Find basis and dimension of subspaces
If I have
$V=\{(x,y,z,w)\in \mathbb{R}^4 : x+y=z+w\}$
How I can find basis and dimension?
I'm new and I don't know how to proceed
AI: hint
A vector of $ V $ is of the form
$$(x,y,z,w)=(x,y,z,x+y-z)=$$
$$x(1,0,0,1)+y(0,1,0,1)+z(0,0,1,-1)=$$
$$x\vec{v_1}+y\vec{v_2}+z\vec{v_3}$$
N... |
H: How to prove that a set is not a finite union of intervals?
Consider the set $\mathbb{R} - \mathbb{Z}$. It is certainly a countable union of intervals. How does one prove that it is not in fact a finite union of intervals?
AI: I’m assuming that by interval you mean bounded interval, i.e., sets of the forms $(a,b)$,... |
H: Proving roots of quadratic equations
While working out, for the required quadratic equation, my result is: $x^2-\left(\frac{{\beta }^3+{\alpha }^3}{{(\alpha \beta )}^3}\right)x+\frac{1}{{\left(\alpha \beta \right)}^3}$
I am unable to move to the next part of the question. Any help will be appreciated.
AI: If $\al... |
H: G is solvable iff factors have prime order
A group $G$ is said to be solvable if, and only if, there exists a subnormal series of subgroups $\{e\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G$ such that each factor $\displaystyle\frac{G_{i+1}}{G_i}$ of the series is an abelian group. Prove tha... |
H: Compute the phase difference of sine wave at point in 3D space
Different points in time have values modulated sinusoidally (like a signal arriving at each point).
For the case as illustrated at the picture all points have the same value at the same time.
I have different cases were the wave is rotated along the ve... |
H: Proof of identity relating Prime-counting function and Stirling numbers of the second kind
In his famous 1973 paper Gallagher showed that the distribution of prime numbers in short intervals tends to a Poisson distribution. To do it he uses in one step:
$$\sum_{n \leq N}(\pi (n+h)-\pi(n))^{k}=\sum_{r=1}^{k} \sigma ... |
H: Evaluate the integral by interpreting it in terms of areas
Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry.
$$
\int_0^4 |5x−2|dx
$$
AI: So visually the absolute value means your function c... |
H: How to find Frechet derivate for this functional
My functional is $F(u)=\frac{1}{2}\int (|D_x u|^2 + |D_y u|^2)dxdy + \int fu dxdy$ where $f\in C(\Omega)$, $u\in C_{0}^2(\overline{\Omega})$ and $\Omega \subset R^2$. Well Im reading Elliptic problems in Nonsmooth domains (P.Grisvard) page 2, my question is i need t... |
H: How to uniformly sample multiple numbers whose product is within some bound
Suppose I have 3 positive integers: $n_1$, $n_2$, and $n_3$.
How do I uniformly sample $(n_1, n_2, n_3)$ so that $50 < n_1 n_2 n_3 < 100$.
I could sample each number independently with bounds between 1 and 100, then keep re-sampling if thei... |
H: Tangent lines of a differentiable function verifying $f(a) = f(b) = 0$ intersect the $x$ axis on all points outside of $[a, b]$.
Let $f$ be a differentiable function on $[a, b]$, $(a<b)$ such as that $f(a) = f(b) = 0$.
Prove that $\forall d \in \mathbb{R} $\ $[a,b] $, $\exists c \in [a,b]$ such as the tangent line ... |
H: If $\sum_{n=1}^{\infty} a_n$ converges and $\sum_{n=1}^{\infty} a_n^2$ diverges, then $\prod_{n=1}^{\infty} (1+a_n)$ diverges to $0$.
Prove that if $\sum_{n=1}^{\infty} a_n$ converges and $\sum_{n=1}^{\infty} a_n^2$ diverges, then $\prod_{n=1}^{\infty} (1+a_n)$ diverges to $0$.
We assume that $\{a_n\}$ is a real se... |
H: Künneth formula and the tensor product of cohomologies
The Künneth formula states that $$\ \Phi: H^*(M) \otimes H^*(F) \to H^*(M×F)$$ is an isomorphism, where $\ \Phi $ is the induced map of $$\ \omega \otimes \tau \to \pi ^* \omega \wedge\rho^*\tau, $$ where $\ \pi$ and $\ \rho$ are projections from $\ M× F $ to M... |
H: Showing $\lim_{n\to\infty}\int_{\mathbb{R}} f(x)f(x+n) dx=0$
Problem
Let $f(x)\in L^2(-\infty,\infty)$. Prove that
$$
\lim_{n\to\infty}\int_{-\infty}^\infty f(x)f(x+n) dx=0.
$$
My attempt
Let $N\in\mathbb{N}$ and $f_N(x):=f(x)\chi_{[-N,N]}(x)$.
For any $n\in\mathbb{N}$, consider
$$
\begin{split}
\int_{-\infty}^\inf... |
H: Reducing $\binom{t-2}{n-2}/\binom{t-1}{n-1}$ to $\frac{n-1}{t-1}$
How does
$$\frac{\displaystyle\binom{t-2}{n-2}}{\displaystyle\binom{t-1}{n-1}}$$
reduce to
$$\frac{n-1}{t-1}$$
I know that the formula for the nCk = $$\frac{n!}{k!(n-k)!}$$
When i unfold given the formula i get
$$\frac{(t-2)!(n-1)!((t-1)-(n-1))!}{(... |
H: Probability with cups and plates
I have some trouble solving the next problem:
There are 6 pairs of cups and plates (i.e. 6 cups and 6 plates). 2 of them are blue, another 2 pairs are red and the last 2 pairs are white. Each cup is place over a plate randomly. What's the probability that non of the plates and cups ... |
H: Suppose that $z = x + iy$, with $y > 0$. Show that there are positive real numbers $u$ and $v$ with $2u^{2} = |z| + x$ and $2v^{2} = |z| - x$
Suppose that $z = x + iy$, with $y > 0$. Show that there are positive real numbers $u$ and $v$ with $2u^{2} = |z| + x$ and $2v^{2} = |z| - x$. Calculate $(u+iv)^{2}$. Show th... |
H: Is there an homemomorphism that is not a diffeomorphism?
Is there an example of an homeomorphism from the reals onto itself that fails to be a diffeomorphism?
AI: Take $f(x)=x^3$: its derivative vanishes at $0$, so its inverse is not differentiable at $0$. |
H: Understanding the proof of: set whose decimal expansion contains only $4, 7$ is perfect
I'm trying to understand the proof of the following:
Let $E$ be the set of all $x \in [0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Prove the every point of $E$ is a limit point of $E$.
The proof that is... |
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