All Questions
1,570,605
questions
-3
votes
0
answers
39
views
Solve $f(\lfloor x \rfloor)=\lfloor f(x) \rfloor$
I've been trying to solve the equation $f(\lfloor x \rfloor)=\lfloor f(x) \rfloor$ for real $x$ and polynomials $f$.
Of course all integer polynomials work, but I have no proof of this being the only ...
1
vote
0
answers
9
views
Find the maximum value of a linear system c = A*b (matrix multiplication)
Assuming there is a square matrix A and vector b about to be multiplied.
[A]{b}={c}
However I am interested only in the maximum value within the resulted vector c.
Is there a way to find that without ...
0
votes
0
answers
12
views
Finding $\dfrac{\sigma(u)}{u}=\alpha$ for local field $K$, $L$ unramified over $K$, $\sigma$ being Frobenius of $L/K$ and $\alpha$ having norm 1.
Let $K$ be a nonarchimedean local field, $L$ be the unique degree $n$ unramified extension of $K$. Let $\sigma$ be the Frobenius element of $L/K$, that is, $\sigma\in\mathrm{Gal}(L/K)$ such that $\...
- 805
0
votes
0
answers
25
views
If $a, b, c, d \in \Bbb R$ and $ab+cd, ad+bc \neq 0$, prove $\vert(a^2+b^2-c^2-d^2)/(ab+cd)\vert \lt 2 ⇒\vert(a^2-b^2-c^2+d^2)/(ad+bc)\vert \lt 2$
If $\vert (a^2+b^2-c^2-d^2)/(ab+cd)\vert \lt 2$ then there exists an angle $\theta$ between $0$ and $\pi$ such that $2\cos\theta=(a^2+b^2-c^2-d^2)/(ab+cd)$.
Then $a^2+b^2-2ab\cos\theta=c^2+d^2+2cd\cos\...
0
votes
0
answers
13
views
Convergence of the gauss hypergeometric function in 'Generalized hypergeometric functions', L.J. Slater
the Gauss hypergeometric function is defined by:
\begin{equation}
\label{e:pFq}
{}_{2}F_1(a;b;z)=\sum_{n=0}^{\infty}\frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}=\sum_{n=0}^{\infty}u_n,
\end{equation}
...
- 45
6
votes
1
answer
45
views
Find the limiting distribution of $\bar{X}=\frac{\sum X_i}{n}$.
Let $X_1,\dots, X_n$ be identically distributed with mean $E[X_1]=\mu$ and $\operatorname{Var}[X_1]=\sigma^2$. Assume that $\operatorname{Cov}(X_k, X_{k+1})\neq 0$ for $k=1,\dots, n$ but $\...
- 1,030
4
votes
0
answers
63
views
Is there a simpler method to compute $\int_0^{\infty} \frac{1}{(1+x)\left(\pi+\ln ^{2n } x\right)} d x$
When I encountered the integral $\int_0^{\infty} \frac{1}{(1+x)\left(\pi+\ln ^{2 } x\right)} d x $, I tried the substitution $x\mapsto \frac{1}{x} $ and found a wonderful result.
$$I=\int_0^{\infty} \...
- 12.4k
0
votes
1
answer
18
views
Inverse image of a point under a continuous surjective closed map :
Let $f:X\rightarrow Y$ be a continuous surjective closed map and $X$ is a normal space. Let there exist an open set $U\subset X$ such that $f^{-1}\{ y \}\subset U$ then show that there exist an open ...
- 612
0
votes
0
answers
21
views
a coin toss and a die roll simultaneously
Suppose that we toss a fair coin and, then toss a fair four-sided die.
(a) Defne the sample space for this experiment.
(b) Let $X$ be the random variable defined as (number of heads) + (die score ...
- 12.7k
1
vote
1
answer
29
views
Probability theory, determine the distribution of $X+Y$
Recently I tried to solve a question in probability theory but did not get quite the same answer as the conclusion and would like to understand where I am wrong somewhere. The question reads:
Suppose ...
- 391
0
votes
0
answers
12
views
Transformation between space partitions
We have a space S, being partitioned into a set of polygons P containing $n$ polygons $P_1, P_2,..., P_n$. Given $n$ constants $k_1,k_2,...,k_n $. Apply a transformation $T$ from partition $P$ to ...
- 1
2
votes
0
answers
26
views
Determining whether or not the alternating series test can be applied to the following infinite series
I was working on some series calculus questions and am struggling with this particular one:
This is what I answered:
CONV – The graph of $y=\frac{x^3+1}{x^4+1}$ is positive but decreasing for $x\geq ...
- 341
1
vote
2
answers
47
views
Question about arranging $6$ identical balls into $3$ identical boxes
The problem description is as follows:
$6$ Identical balls are randomly placed in $3$ identical boxes. What is the probability that each box will have $2$ balls?
My attempt:
The total number of ...
- 11
0
votes
0
answers
8
views
Expectation of network modularity
Given an uncertain undirected graph $\mathcal{G} = (V, E, p:E \to [0, 1])$ with $|V| = n$ and $E \subseteq \binom{V}{2}$,
and a partition $P: V \to [k]$ for some positive integer $k$, we want to ...
- 1,154
0
votes
0
answers
15
views
manipulating boolean algebra expression to prove xy+ x'z=(x+z)(x'+y)
I've managed to prove how $(x+z)(x'+y)= x'y+ xz$, but I fail to see how it works for a similar yet slightly different form : $x'y+ xz = (x+z)(x'+y)$ from left hand side.
for reference
\begin{equation}
...
0
votes
1
answer
23
views
Are there nontrivial distributions completely determined by finite moments
We know that the Gaussian distribution is completely determined by its first 2 moments, i.e., $\mathbb{E}X,\mathbb{E}X^2$. Are there some well-studied examples of distributions determined by its first ...
- 2,575
2
votes
1
answer
44
views
Steady State Temperature Distribution in a Rectangular Plate
We need to solve the following :
$$
\nabla^2𝑢=0, 0\leq 𝑥\leq𝑎, 0\leq y\leq b
$$
satisfying the boundary conditions
$$
𝑢(0,y)=0, 0\leq y\leq b \\
𝑢(𝑥,0)=𝑢(𝑥,𝑏)=0, 0\leq 𝑥\leq 𝑎 \\
𝑢_x(a,𝑦)=...
- 21
1
vote
0
answers
32
views
Follow Up Question: Witten's explanation of Feynman diagrams
This is a follow-up to a recent question of mine Witten's proof of Wick Formula of QFT. The background of this question can be found there if needed, but I feel my question is simple enough ...
- 637
4
votes
1
answer
55
views
Non contractible subspace of $\mathbb{R}^2$
I'm having trouble proving that the subspace $X$ of $\mathbb{R}^2$ such that $X$ is the union of $[-1,1] \times \{ 0 \}$ and the line segments that join the points $(0,\frac{1}{n})$ with the point $(1,...
- 174
0
votes
0
answers
26
views
Angle Subtraction as a corollary to Angle Addition
I'm working with Greenberg's Euclidean and Non-Euclidean Geometries: Development and History 4th Edition. For a copy of the axioms used, see https://www.ms.uky.edu/~droyster/courses/fall11/ma341/...
- 635
0
votes
0
answers
17
views
Defining surreal addition on signed ordinals
Consider surreal numbers as signed ordinals $\alpha\rightarrow\{-,+\}$. Suppose we already have $x<y$ defined for any two surreals, as well as $F|G$ as the simplest surreal $z$ strictly between the ...
- 2,980
0
votes
0
answers
22
views
Prove that Mobius map on the upper half plane is analytic.
Let $\mathbb{H}^2 = \{x \in \mathbb{C} \mid \Im (z) >0\}$ be the upper half plane. Then prove that $T: \mathbb{H}^2 \rightarrow \mathbb{H}^2$ defined by
$T(z) = \displaystyle{\frac{az+b}{cz+d}}$ ...
- 139
0
votes
1
answer
33
views
Finding directional angle of vector in $\mathbb{R}^2$
I'm not sure if this is correct or not and need someone to check.
I have a vector $\vec{v} = 4\left(\frac{-1}{2}, 1\right) - \frac{1}{2}(4, 8)$
I simplified it to $(-4, 0)$
So the directional angle ...
- 23
-1
votes
1
answer
65
views
Find $\lim_{z\to 1+i}\left(\frac{z+2-i}{3}\right)^\frac{z-i}{z-1-i}$
I am trying to find the limit
$$ \lim_{z\to 1+i}\left(\frac{z+2-i}{3}\right)^\frac{z-i}{z-1-i} $$
If we plug in $1+i$ for $z$, we get the following
$$ \lim_{z\to 1+i}\left(\frac{z+2-i}{3}\right)^\frac{...
- 1
0
votes
1
answer
18
views
Detail in proof of uniqueness of Brownian motion maximum
I am reading Morter's book on Brownian motion, and am confusing about a portion of Theorem 2.11. The lemma being proved is that for two non-overlapping intervals, the maxima are almost surely distinct....
- 379
0
votes
0
answers
13
views
Calculating a sub percentage when other weighted sub percentages and the total percentage are known.
I am working on retirement planning and having a problem working out an equation. I have a good model with all retirement funds and projections including deferred growth annuity equations.
I am trying ...
- 9
1
vote
0
answers
22
views
Universal covering of manifold is regular covering?
From Wiki, the regular cover is that its action is transitive on some fiber. From this definition, I feel the universal cover of manifold (local homeomorphism to Euclidean open sets) should be ...
- 5,507
-1
votes
4
answers
71
views
Is “height” an area?
If the height of a function is $f(x)$ and the area of a rectangle of height $f(x)$ and width $1$ is $f(x) \times 1$ does this mean the height of line or a function is exactly an area of a rectangle ...
- 11
-2
votes
1
answer
17
views
Discretemaths graph theory
A particular tree with $9$ vertices has precisely $5$ vertices of degree $1$, and precisely $2$ vertices of degree $2$. The remaining $2$ vertices have degrees of $a$ and $b$. Find $a$ and $b$, given $...
- 1
0
votes
1
answer
29
views
Composition of functions where f(g(x^2)), how do you handle the g(x^2) function?
Plugging the question into symbolab, it only applies the square to the x within the g(x) function. Example: f(x) = x^2 - 2, g(x) = x - 7. The g(x^2) = x^2-7. f(g(x^2)) becomes x^4-14x^2+47. The ...
- 3
0
votes
0
answers
5
views
Probabilistic upper bound for $E_{X_{1:m}} ( \frac{1}{m} \sum_{i}^{} [X_i > \tau(X_{1:m})] ) - E_{Y} \left[ Y > \tau(X'_{1:m}) \right] > \epsilon$.
Let $\tau: \mathbb{R}^n \times \ldots \times \mathbb{R}^n \to \mathbb{R}$, which takes samples $X_{1:m}$ as an input and returns a real value. Is there any tight upper bound for
$$ \mathbb{P}_{X'_{1:m}...
- 311
-1
votes
1
answer
14
views
Does a sequence of finite integrals with non-negative integrands which converges on a measure space also converge on any measurable set?
Let $(X, \mathcal{A}, \mu)$ denote a measure space, and let $\{f_n\}_{n=1}^{\infty}$ be a sequence of non-negative, real-valued functions on $X$ for which each integral in (1) is finite, and suppose ...
- 59
0
votes
0
answers
19
views
How to generalize the triangles condition in Morera's Theorem to translates and dilates of any toy contour?
I saw in COMPLEX ANALYSIS written by Stein the generalization of Morera's Theorem, but the hint made me confused. Why the fe(x) defined there satisfy the condition(16), namely that integral of fe(x) ...
- 1
1
vote
1
answer
24
views
Existence of others homomorphisms in $(\ell_\infty)'$
Let $\ell_\infty$ the space of bounded sequences in $\mathbb{C}$. Knowing that $\ell_\infty$ is a $C^*$-algebra with coordinate-to-coordinate multiplication, show that exist a homomorphism $\tau:\ell_\...
- 408
0
votes
0
answers
12
views
Calculating recovery time for constant harvesting
We are given a continuous growth model with constant harvesting as:
$\frac{dN}{dt} = rN(1-\frac{N}{K})-Y_0$
The question asks to calculate for
$\frac {T_R(Y_0)}{T_R(0)}$
where $T_R(Y)$ denotes the ...
0
votes
0
answers
16
views
Queue number above 2 for planar graphs?
Are there any known planar graphs with queue number greater than $2$?
The queue number of a graph counts the minimum number of subsets that the edges must be divided into to avoid all nested pairs of ...
- 868
0
votes
0
answers
8
views
Find a "smaller" positive semidefinite rank 1 matrix
Given a positive semidefinite matrix $A\in \mathbf{C}^{n\times n}$ and a vector $x\in \mathbf{C}^{n\times 1}$, I would like to find a positive semidefinite matrix $B\in \mathbf{C}^{n\times n}$ such ...
- 27
0
votes
1
answer
91
views
How is the Law of Large Numbers not a Lie?
I'm sure something is wrong somewhere in my premise, but I never understood how the Law of Large Numbers can be accurate in a mathematical world where a previous outcome of a statistical event doesn't ...
- 109
2
votes
2
answers
77
views
The biggest number $N$ that satisfies certain requirements.
The problem is as such:
Say a natural number $n$ as special if it does not have the digit $0$, has $2021$ as the sum of its digits, and the sum of the digits of $2n$ does not exceed 1202. Let $N$ be ...
- 329
0
votes
1
answer
30
views
Discontinuity of f and g
Let $f:g:X\to \mathbb R$ be functions where $X$ is a subset of $\mathbb R$. If f is continuous at $a\in X$ and $f(a)\neq 0$, while $g$ is discontinuous at $a$. Prove or disprove that $fg$ is ...
- 11
2
votes
1
answer
45
views
Kernel of homomorphism is equal to normal closure
Suppose we have two groups $G = \langle g_1, \cdots, g_r, \cdots g_n\rangle$, $H = \langle g_1, \cdots, g_r\rangle$ and a homomorphism $f: G \to H$ such that $f(g_i) = g_i$ if $i\leq r$ and $f(g_i) = ...
- 349
2
votes
0
answers
40
views
Area enclosed by $\sin^2(\pi x)+\sin^2(\pi y)>1$ and $x,y\in[-1,1]$
The area of regin traced by the
point in the cartesian plane which satisfy the equation $\sin^2(\pi x)+\sin^2(\pi y)>1$ , where $x,y\in[-1,1]$
My Try:
We can write
$\sin^2(\pi x)+\sin^2(\pi y)>...
- 5,442
-1
votes
0
answers
16
views
Expectation of a random matrix smaller than expectation of difference between it and its independent copy
Supoose $X$ is $m\times n$ matrix, and $X'$ is its an independent copy. How to show that
$$
\mathbb{E}(||X||) \leq \mathbb{E}(||X-X'||),
$$
where $||\cdot||$ is the spectral norm.
- 63
-3
votes
1
answer
36
views
If X ~ Poisson(λ), then find E(X(X+1)^2) [closed]
Having some difficulty with this problem. Any help would be appreciated.
- 5
1
vote
2
answers
31
views
Deformation retract on subspace of $S^3$
Let $M = \{(x, y, 0, 0) \in \mathbb{R}^4 ~|~ x^2 + y^2 = 1\}$ and $N = \{(0, 0, z, w) \in \mathbb{R}^4 ~|~ z^2 + w^2 = 1\}$ be subspaces of $S^3$. Construct a deformation retract of $S^3 \setminus M$ ...
- 103
1
vote
0
answers
17
views
Casting shadows of parametric convex surfaces to arbitrary planes
Given a smooth function $f: \mathbb R^3 \rightarrow \mathbb R$, a surface $S = \{f=0\}$ casts a (orthogonally projected) shadow to a plane with unit normal $\mathbb n$ that, provided $S$ is convex, ...
- 355
0
votes
1
answer
19
views
How to find the reduced cost of a variable in a large data set
I have been given a large data set with a list of starting nodes, their destination nodes, and the length to each destination node from each starting node. Using Dijkstra's algorithm, I coded a ...
- 420
2
votes
1
answer
31
views
Noetherian ring - difficulty in understanding the union of infinite ideals
I was reading the proof that a commutative unital ring is Noetherian if and only if every ideal of it is finitely generated. It said that given a strictly ascending chain $$I_0\subsetneq I_1\subsetneq\...
- 81
0
votes
0
answers
65
views
Explaining $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{ak+b}=+\infty$, for positive real $a$ and $b$
Can somebody please explain a proof for this limit Calculus claim? The notes I was reading only used it but did not give a proof
If $a,b>0$, then
$$\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{ak+b}=+\...
-2
votes
0
answers
25
views
What is the total number of stickers in 3bags,9 books and 6 single stickers
What is the total number of stickers in $3$ bags,$9$ books and $6$ single stickers
$1$ sticker $=1$;
$1$ page $= 10$;
$1$ book $=100$;
$1$ bag $= 1,000$
Need answer to be thousands hundreds tens ones.
- 1