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Is there a general method to express the resolvent of an operator $A$ with $A$ appearing in the numerator? For instance, the Neumann series expansion is one approach, but it requires the operator norm of $A$ to be less than one for convergence. I am seeking a more versatile method that does not rely on such a restricti... |
*** Error corrected ***
With
$g(x) = {\large\int} \ \sin(x) \cos(x) f(x) \ dx$
known, is there a general form for
${\large\int} \ \frac{\sin(x) \cos(x)}{f(x) \ + \ a} dx$
that may contain $g(x)$ and possibly $g'(x)$? |
**Background**
In the <a href="https://math.stackexchange.com/questions/1647409/can-an-infinite-sum-of-irrational-numbers-be-rational">following question</a>, User1 asks whether an infinite sum of irrational numbers can be rational. Multiple answers<sup>1</sup> indicate the answer to this question is 'yes'. For inst... |
Can an infinite sum of non-computable numbers be computable? |
In p.76 of this book, an example 4 says "Take a strictly decreasing sequence $F_1 \supset F_2 \supset \cdots $ of closed sets in $X$ ".
I don't understand if there exists such the sequence.
I think the following statement may be true in general topology.
> Let $(X, \mathcal{O})$ be an infinite Hausdorff topologi... |
**Background**
In the <a href="https://math.stackexchange.com/questions/1647409/can-an-infinite-sum-of-irrational-numbers-be-rational">following question</a>, User1 asks whether an infinite sum of irrational numbers can be rational. Multiple answers<sup>1</sup> indicate the answer to this question is 'yes'. For inst... |
Let $A\geq 0$ be a positive semi-definite complex matrix in $M_d(\mathbb{C})$. Let $T:M_d(\mathbb{C})\to M_d(\mathbb{C})$ be a positive linear map between $d\times d$ complex matrices, i.e., $A\geq 0\Rightarrow T(A)\geq 0$.
I want to prove that $\ker(A+T(A))\subseteq \ker(A)$. Intuitively this should be easy in the... |
$\renewcommand{\al}{\alpha}
\newcommand{\bra}[1]{\left<#1\right>}
\newcommand{\brc}[1]{\left(#1\right)}
\newcommand{\Aut}[2]{\operatorname{Aut}_{#1}\brc{#2}}
\renewcommand{\C}{\mathbb{C}}
\newcommand{\si}{\sigma}
\newcommand{\brr}[1]{\left[#1\right]}$
Let $L$ be a complex semisimple finite dimensional Lie algebr... |
> People are arriving at a party one at a time. While waiting for more people to arrive they entertain themselves by comparing their birthdays. Let X be the number of people needed to obtain a birthday match, i.e., before person X arrives there are no two people with the same birthday, but when person X arrives there i... |
*** Error corrected ***
With
$g(x) = {\large\int} \ \sin(x) \cos(x) f(x) \ dx$
known, is there a general form for
${\large\int} \ \frac{\sin(x) \cos(x)}{f(x) \ + \ a} dx$
that may contain $g(x)$ and possibly $g'(x)$?
The natural place where I would look for an answer is a pair of $2F2$ or $2F1$ ... |
The problem comes from Volumn I, (7.4.3), Grothendieck's EGA. For a scheme $X$, denote the sheaf of rational functions by $\mathscr{R}(X)$.
**Corollary (7.4.3)**. Over an integral scheme $X$, any torsion-free, quasi-coherent and rank-$1$ $O_X$-module sheaf (denote this sheaf by $\mathscr{F}$) is isomorphic to a sub... |
*** Error corrected ***
With
$g(x) = {\large\int} \ \sin(x) \cos(x) f(x) \ dx$
known, is there a general form for
${\large\int} \ \frac{\sin(x) \cos(x)}{f(x) \ + \ a} dx$
that may contain $g(x)$ and possibly $g'(x)$?
The natural place where I would look for an answer is a pair of $2F2$ or $2F1$ ... |
Why does there exist such $V$? You've not tried to justify this, which of course would be one of the most important steps. You've also not linked that to actually obtaining, say, $F_{1000}$ from $F_{999}$. For help with justifying the claim $\mathfrak{a}_n\subsetneq\mathfrak{a}_{n+1}$ in the rest of the example, see [h... |
### Question:
Suppose we have three uniform $[0,1]$ random variables $X,Y,Z$. What is $P(|X-Z|\leq r_1 | |X-Y| = r_2)$?
### Things I know already / have tried:
I have that the CDF of $|X-Y|$ is $P(|X-Y|<r) = 2r-r^2$.
Also,
$$
P(|X-Y|=r | X=x) = \begin{cases}
2 & 0\leq r \leq \max\{x, 1-x\} \\
1 & x < r \le... |
If that's the definition, then how do we know *double* covers of $SO(n)$ exist? |
how to show all spin groups are double covers? |
Numerical LA question: Is there an algorithm that takes an upper triangular matrix $T$ over the complex numbers as input, and spits out a bidiagonal matrix $B$ which it's similar to?
### Remarks
By $T$ being similar to $B$, I mean that there exists an invertible matrix $P$ such that $B = PTP^{-1}$.
Such a matr... |
Suppose there is an $n\times n$ matrix $M$ whose entries are 0 and 1, and exactly $2n-1$ many entries are 1's.
I am wondering if we can get a lower-triangular matrix from $M$ by only allowing to swap rows and columns?
|
Let's say I have $f: \mathbb{R}^{2} \to \mathbb{R}$, and I want to know if $$\lim_{(x,y) \to (0,0)} f(x,y)$$ exists.
Suppose I substitute $x$ and $y$ for their polar forms $r\cos(\theta )$, $r\sin(\theta )$. Then, when analysing $\lim_{r \to 0} f(r, \theta)$, I get some $\theta$-dependent trigonometric function (e.... |
Is there an algorithm that takes an upper triangular matrix $T$ over the complex numbers as input, and spits out a bidiagonal matrix $B$ which it's similar to?
### Remarks
By $T$ being similar to $B$, I mean that there exists an invertible matrix $P$ such that $B = PTP^{-1}$.
Such a matrix certainly exists: Fo... |
Why does there exist such $V$? You've not tried to justify this, which of course would be one of the most important steps, and don't seem to have used assumptions on $X$. You've also not linked that to actually obtaining, say, $F_{1000}$ from $F_{999}$. For help with justifying the claim $\mathfrak{a}_n\subsetneq\mathf... |
> Is there an algorithm that takes an upper triangular matrix $T$ over the complex numbers as input, and outputs a bidiagonal matrix $B$ which $T$ is similar to?
### Remarks
By $T$ being similar to $B$, I mean that there exists an invertible matrix $P$ such that $B = PTP^{-1}$.
Such a matrix certainly exists: ... |
I am currently studying Cubic residue characters from Kenneth Ireland and Michael Rosen's "A Classical Introduction to Modern Number Theory", and this is the definition given in the book:
If $N(\pi) \neq 3$, the cubic residue character of $\alpha$ modulo $\pi$ is given by
$(a)$ $ (\frac{\alpha}{\pi})_3 = 0$ if $\p... |
**Question**
Given a sequence $a_n$, expressed in closed form, when it is possible to express such series as a linear recurrence relation?
I'm looking for a characterization for such sequences, but also some necessary condition will be fine.
___
**My work**
I managed to prove some sufficient condition w... |
When a sequence satisfy a linear recurrence relation? |
> Let $Y_t=t\sin W_t$ where $W_t$ - Wiener process. Calculate $\left< Y \right>_t$.
The correct answer is: $\left< Y \right>_t=\int_0^t(2s\sin W_s+s^2 \cos (2W_s))ds$
I received something different and I don't know where I made a mistake:
$$\begin{align} Y_t &= t\sin W_t \\ dY_t &= td(\sin W_t)+\sin(W_t)dt \\ \s... |
> Is there an algorithm that takes an upper triangular matrix $T$ over the complex numbers as input, and outputs a bidiagonal matrix $B$ which $T$ is similar to?
>
> We may assume that $T$ is invertible.
### Remarks
By $T$ being similar to $B$, I mean that there exists an invertible matrix $P$ such that $B = PT... |
In the proof Theorem 5.2.1 (equality (5.2.5)), Hörmander claims that for any smooth function $\psi \in C^{\infty} (\mathbb{R}^n)$ the following equality is true:
$ \frac{\partial}{\partial \epsilon}(\epsilon^{-n} \psi (\frac{x}{\epsilon})) = \sum_{j=1}^n \frac{\partial}{\partial x_j}(\epsilon^{-n}\psi_j(\frac{x}{\ep... |
> Is there an algorithm that takes an [upper triangular](https://en.wikipedia.org/wiki/Triangular_matrix) matrix $T$ over the complex numbers as input, and outputs a [bidiagonal matrix](https://en.wikipedia.org/wiki/Bidiagonal_matrix) $B$ which $T$ is similar to?
>
> We may assume that $T$ is invertible.
### Remar... |
I have been working for this problem for a while:
$$\ln(x) = 3\left(1-\frac{1}{x}\right)$$
and by graphing, plugging and chugging values and rigorously doing the math, I can clearly see that one of the values that satisfy this condition is $x = 1$.
However, when I put this into wolfram alpha and Desmos, I see ... |
How to solve $\ln(x) = 3\left(1-\frac{1}{x}\right)$? |
I am currently studying Cubic residue characters from Kenneth Ireland and Michael Rosen's "A Classical Introduction to Modern Number Theory", and this is the definition given in the book:
If $N(\pi) \neq 3$, the cubic residue character of $\alpha$ modulo $\pi$ is given by
$(a)$ $ (\frac{\alpha}{\pi})_3 = 0$ if $\p... |
This question is motived by the following observation:
Consider the principal value integral
$$
P.V.\int_{-\eta_1}^{\eta_2}\frac{1}{a_1 x + a_2 x^2 + a_3 x^3 + \cdots}
$$
Here I choose $\eta_1$ and $\eta_2$ small enough to only contain the singular point at x=0. If $a_1\neq0$, the denominator behaves like $a_1 x... |
Recall that $n\in\mathbb{N}$ is $k$-*full* if $p\mid n$ implies $p^{k}\mid n$. In the paper I am currently reading it is said that there are $O(B^{1/k})$ $k$-full positive integers in the dyadic interval $(B/2,B]$ for any $B\geq 1$. How to obtain this bound?
Obviously, any $k^{\text{th}}$ power is a $k$-full number,... |
How many $k$-full numbers are there? |
I need help with this question.
Suppose that G is a group acting on a set of objects S, and that C is the set of colorings of elements of S using the colors in a set R. Let $\overline c$ denote the orbit of c in C with respect to the action of G. Let $\rho$ be a permutation acting on R. Prove that $\{ c \in C: \rho^... |
**Background**
In the <a href="https://math.stackexchange.com/questions/1647409/can-an-infinite-sum-of-irrational-numbers-be-rational">following question</a>, User1 asks whether an infinite sum of irrational numbers can be rational. Multiple answers<sup>1</sup> indicate the answer to this question is 'yes'. For inst... |
This question occured to me while trying to solve a three-body-esque problem involving three charged particles placed along a straight line (ignoring effects of electromagnetic radiation etc. etc.). Anyway, for particles of masses $m$, $2m$ and $5m$ and charges $q$, $q$ and $2q$, respectively, we obtain this system of ... |
Assuming we are working in axiomatic set theory, such as $ZFC$ or $NBG$ can we, for all ordered pairs $(x, y)$ find a monadic predicate, call it $First$ such that $First((x, y))=x$ Is such a thing possible if we are using Kuratowski's definition of the ordered pair, namely $(x, y) = \{ \{x \}, \{ x, y \} \}$.<br/>
One... |
Can A General Solution Be Obtained If A Particular Solution Is Known For A System of Differential Equations? |
This question pertains to "[A Entropy view of Fibonacci Trees](https://www.mathstat.dal.ca/FQ/Scanned/20-2/horibe.pdf)" (1982) by Yasuichi Horibe, [Zbl 0491.94009](https://zbmath.org/0491.94009).<br> Horibe defines a binary tree where a node has a left branch with probability x and a right branch with probability $1-x$... |
I just want to clarify a point that I haven't seen made explicitly made:
there is no globally defined tautological one-form or symplectic
potential on a closed symplectic manifold (i.e. a compact manifold without
boundary). If there were some $\theta$ (in local coordinates, expressable as $p_i dq^i$) such that $\ome... |
I am reading "Analysis on Manifolds" by James R. Munkres.
>Theorem 20.1. Let $A$ be an $n$ by $n$ matrix. Let $h:\mathbb{R}^n\to\mathbb{R}^n$ be the linear transformation $h(x)=A\cdot x$. Let $S$ be a rectifiable set in $\mathbb{R}^n$, and let $T=h(S)$. Then $$v(T)=|\det A|\cdot v(S).$$
>Proof.
$\cdots$
... |
I am reading "Analysis on Manifolds" by James R. Munkres.
>Theorem 20.1. Let $A$ be an $n$ by $n$ matrix. Let $h:\mathbb{R}^n\to\mathbb{R}^n$ be the linear transformation $h(x)=A\cdot x$. Let $S$ be a rectifiable set in $\mathbb{R}^n$, and let $T=h(S)$. Then $$v(T)=|\det A|\cdot v(S).$$
>Proof.
$\cdots$
... |
I am currently studying Cubic residue characters from Kenneth Ireland and Michael Rosen's "A Classical Introduction to Modern Number Theory", and this is the definition given in the book:
If $N(\pi) \neq 3$, the cubic residue character of $\alpha$ modulo $\pi$ is given by
$(a)$ $ (\frac{\alpha}{\pi})_3 = 0$ if $\p... |
I have the following problem: Let $f\in\mathcal{S}(\mathbb{R}^n)$ and $g\in C^\infty(\mathbb{R}^n)$ such that exists $\alpha>0$
$$
\left|g(x)\right|\leq \frac{1}{1+|x|^\alpha} \quad \forall x\in\mathbb{R}^n
$$
I have to prove that either $f\ast g\in\mathcal{S}(\mathbb{R}^n)$ or show why not.
From the hypothesi... |
Why are piecewise linear functions semismooth? |
Assuming we are working in axiomatic set theory, such as $ZFC$ or $NBG$ can we, for all ordered pairs $(x, y)$ find a monadic predicate, call it $First$ such that $First((x, y))=x$ Is such a thing possible if we are using Kuratowski's definition of the ordered pair, namely $(x, y) = \{ \{x \}, \{ x, y \} \}$.<br/>
One... |
The Problem says that :-
>If $\large \mathrm{N}$ is any positive integer, and $\delta_{n}=\large \int_{_{0}}^{^{N}} {{u} \choose {n}}{{N-u} \choose {N-n}} \textit{ du }$, then the value of
$\large \sum_{n=0}^{N}\delta_{n} = \\ \ \\ \ \\ (A). 0 \\ (B). N \\(C). N^2 \\ (D). \frac{1}{N} $ .
I was thinking about ... |
I am reading "Analysis on Manifolds" by James R. Munkres.
>Theorem 20.1. Let $A$ be an $n$ by $n$ matrix. Let $h:\mathbb{R}^n\to\mathbb{R}^n$ be the linear transformation $h(x)=A\cdot x$. Let $S$ be a rectifiable set in $\mathbb{R}^n$, and let $T=h(S)$. Then $$v(T)=|\det A|\cdot v(S).$$
>Proof.
$\cdots$
... |
Yes, the proof is correct. It is a well-known proof, essentially using **denominator descent** using the division algorithm, though that is *greatly obfuscated*. Below I clarify this viewpoint for the generalization below (the OP proof is *exactly* the special case $\, k = 2\,$ and $\,q = \lfloor \sqrt 2\rfloor = 1)$.... |
[OEIS A169945](https://oeis.org/A169945) introduces the concept of _thickness_ of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(x)$ with coefficients $0$ or $1$ and degree at most $n$, it denotes by $T(n,k)$, the numbe... |
This question occured to me while trying to solve a three-body-esque problem involving three charged particles placed along a straight line (ignoring effects of electromagnetic radiation etc. etc.). Anyway, for particles of masses $m$, $2m$ and $5m$ and charges $q$, $q$ and $2q$, respectively, we obtain this system of ... |
I'm not sure how to prove a statement about extended Weyl groups.
Let $V$ be a finite vector space over $\mathbb{R}$, with a positive definite symmetric bilinear form (·,·), R ⊂ V be a reduced
irreducible root system. $W$ is the Weyl group associated.\
We fix a decomposition of R into positive and negative roots:... |
Recall that $n\in\mathbb{N}$ is $k$-*full* if, for a prime number $p$ : $p\mid n$ implies $p^{k}\mid n$. In the paper I am currently reading it is said that there are $O(B^{1/k})$ $k$-full positive integers in the dyadic interval $(B/2,B]$ for any $B\geq 1$. How to obtain this bound?
Obviously, any $k^{\text{th}}$ p... |
A subset $X \subseteq \mathbb{R}^n$ is called *semialgebraic* if it is of the form
$$
X = \bigcup_{finite} \bigcap_{finite} \{ x \in \mathbb{R}^n \colon f_{i,j}(x) \star 0 \}
$$
where $\star$ represents any of the symbols $=, \leq , \geq, <, >$ and $f_{i,j} \in \mathbb{R}[X_{11}, \ldots , X_{nn}]$.
A subgroup $G... |
Is a subgroup of $\operatorname{GL}(n,\mathbb{R})$ semialgebraic if and only if all its orbits are? |
The Problem says that :-
>If $\large \mathrm{N}$ is any positive integer, and $\delta_{n}=\large \int_{_{0}}^{^{N}} {{u} \choose {n}}{{N-u} \choose {N-n}} \textit{ du }$, then the value of
$\large \sum_{n=0}^{N}\delta_{n} = \\ \ \\ \ \\ (A). 0 \\ (B). N \\(C). N^2 \\ (D). \frac{1}{N} $ .
I was thinking about ... |
I am not a number theorist, but I know that number theorists are notorious for estimating sums including primes. So, I have the following question, which I may not have seen addressed, but I am sure has been thought of by others before.
Is it true that the difference
$$\sum_{p\leq N} 1/p \ - \sum_{ p<q \leq N} 1/(p... |
With
$g(x) = {\large\int} \ \sin(a x) \cos(b x) f(x) \ dx$
known, is there a general form for
${\large\int} \ \frac{\sin(a x) \cos(b x)}{f(x) \ + \ c} dx$
that may contain $g(x)$ and possibly $g'(x)$?
The natural place where I would look for an answer is a pair of $2F2$ or $2F1$ hypergeometric func... |
**Context**
The wave equation
$$
\partial_{tt}\psi=v^2\nabla^2 \psi
$$
describes waves that travel with frequency-independent speed $v$, ie. the waves are dispersionless. The character of solutions is different in odd vs even number of spatial dimensions, $n$. A point source in odd-$n$ creates a disturbance ... |
There is no right answer for this question. And the answer I am about to give is solely based on my personal experience as a Mathematics student.
Self studying Mathematics is not an easy task. I was lucky enough to have great professors at university. Most of them had a vast teaching experience. After teaching somet... |
I am given a problem to define the functions φ1(x, y), φ2(x, y) and φ3(x, y) for a single triangular element as a reference to do element assembly. However I'm not sure why the answer is:
```
φ1(x, y) := 1 − (1/2 x) − (1/2 y)
φ2(x, y) := (1/2 x) − (1/2 y)
φ3(x, y) := y
```
I understand that the `φ1(x, y)` is ... |
You toss a fair coin unlimited
number of times.
How many throws on average do you need to make before the sequence НTHT appears?
The correct answer is the expectation is the sum of the inverse probabilities
of each prefix in a word, which is also a suffix
of the same word. In the case of OROR, such suffixes-prefix... |
Toss a coin until you get НTHT. Expected number of tails tossed? |
I just want to clarify a point that I haven't seen made explicitly made:
there is no globally defined tautological one-form or symplectic
potential on a closed symplectic manifold (i.e. a compact manifold without
boundary). If there were such a one-form $\theta$ (in local coordinates, expressable as $p_i dq^i$) such... |
The Problem says that :-
>If $\large \mathrm{N}$ is any positive integer, and $\delta_{n}=\large \int_{_{0}}^{^{N}} {{u} \choose {n}}{{N-u} \choose {N-n}} \textit{ du }$, then the value of
$\large \sum_{n=0}^{N}\delta_{n} = \\ \ \\ \ \\ (A). 0 \\ (B). N \\(C). N^2 \\ (D). \frac{1}{N} $ .
I was thinking about ... |
When is the composition of Markov kernels well-defined? |
*** A further major error corrected (my apologies) ***
With
$g(x) = {\large\int} \ \sin(a x) \cos(b x) f(x) \ dx$
known, is there a general form for
${\large\int} \ \frac{\sin(a x) \cos(b x)}{\frac{1}{f(x)} \ + \ c} dx$
that may contain $g(x)$ and possibly $g'(x)$?
The natural place where I wo... |
I need help with this question.
Suppose that G is a group acting on a set of objects S, and that C is the set of colorings of elements of S using the colors in a set R. Let $\overline c$ denote the orbit of c in C with respect to the action of G. Let $\rho$ be a permutation acting on R. Prove that $\{ c \in C: \rho^... |
*** A further major error corrected (my apologies) ***
With
$g(x) = {\large\int} \ \sin(a x) \cos(b x) f(x) \ dx$
known, is there a general form for
${\large\int} \ \frac{\sin(a x) \cos(b x)}{\frac{1}{f(x)} \ + \ c} dx$
that may contain $g(x)$ and possibly $g'(x)$?
The natural place where I wo... |
While reading through "[Restrictions on harmonic maps of surfaces][1]" by J. Eells and J. C. Wood I came across the lines:
> We know that there are holomorphic maps from a torus $T$ to the sphere $S$ of all degrees $\geq 2$. These provide harmonic maps $\phi: (T,g) \to (S,h)$ with degree $\geq 2$, whatever the metri... |
Recently, I posted the following question : https://math.stackexchange.com/questions/4865929/can-markov-chains-be-used-to-disprove-the-st-petersburg-paradox.
In one of the comments, it was written: "*a variable with infinite mean, will always produce a finite value*". I am trying to understand what this statement ... |
Proving that the St Petersburg Paradox has a 0 Probability of Infinite Reward? |
[![6X4 grid][1]][1]
[1]: https://i.stack.imgur.com/uyG5u.jpg
How many paths are there from start to end on a 6 × 4 grid as shown in the picture? Note that any such path comprises 6 horizontal unit lengths and 4 vertical unit lengths.
My work: 5^6. My logic: There are horizontal rows... |
Poisson distribution expresses the probability that a specific number of discrete independent events happen over a fixed time interval, as long as the events are sufficiently rare.
To be precise, I accept the following premises for Poisson events:
1. Probability of exactly $k$ events happening over a time interva... |
### Question:
Suppose we have three uniform $[0,1]$ random variables $X,Y,Z$. What is $P(|X-Y|\leq r_1 | |X-Z| = r_2)$?
### Things I know already / have tried:
I have that the CDF of $|X-Y|$ is $P(|X-Y|<r) = 2r-r^2$.
Also,
$$
P(|X-Y|=r | X=x) = \begin{cases}
2 & 0\leq r \leq \max\{x, 1-x\} \\
1 & x < r \le... |
Given a set of points forming a convex hull, is it possible to find a specific hyperplane? |
The problem says: Let (X,\Tau ) be a topological space so that A,B \subset X are connected sets in said space.
Prove that A \cup B is connected if and only if (cl(A) \cap B) \cup (A \cap cl(B)) \neq \emptyset
I have no idea where to start. Can someone help me? |
Can an infinite sum of non-computable numbers be computable, such that all finite sums of subsets of terms are non-computable? |
The problem says: Let $(X,T)$ be a topological space so that $A,B \subset X$ are connected sets in said space.
Prove that $A \cup B$ is connected if and only if $(cl(A) \cap B) \cup (A \cap cl(B)) \neq \emptyset$.
I have no idea where to start. Can someone help me? |
While reading through "[Restrictions on harmonic maps of surfaces][1]" by J. Eells and J. C. Wood I came across the lines:
> We know that there are holomorphic maps from a torus $T$ to the sphere $S$ of all degrees $\geq 2$. These provide harmonic maps $\phi: (T,g) \to (S,h)$ with degree $\geq 2$, whatever the metri... |
[![6X4 grid][1]][1]
[1]: https://i.stack.imgur.com/uyG5u.jpg
How many paths are there from start to end on a 6 × 4 grid as shown in the picture? Note that any such path comprises 6 horizontal unit lengths and 4 vertical unit lengths.
My work: 5^6. My logic: There are 5 horizontal ro... |
While reading through "[Restrictions on harmonic maps of surfaces][1]" by J. Eells and J. C. Wood I came across the lines:
> We know that there are holomorphic maps from a torus $T$ to the sphere $S$ of all degrees $\geq 2$. These provide harmonic maps $\phi: (T,g) \to (S,h)$ with degree $\geq 2$, whatever the metri... |
Recent linear state-space model papers like Mamba often use matrix exponential to discretize the system. They initialize the system in a continuous-time regime, and discretize it to run it like a vanilla RNN / CNN.
Consider the simplest continuous-time autonomous system $d\mathbf{h(t)}/dt = A\mathbf{h(t)}$ (assuming... |
The probleme des menages consists of the following:
How many ways can $n$ married couples si at a round table in such a way that there is one man between every two women and no man is seated next to his wife?
The solution on Titu Andreescu's book A path to combinatorics for undergraduates, is by Inclusion-Exclusi... |
I have the Hilbert space of square integrable functions on $[a, b]$, and what I would like to have is to discretize this space, i.e., find a sequence of finite-dimensional Hilbert spaces $H_K$ of dimension $K$ such that
$|\langle u, v \rangle - \langle u^{(K)}, v^{(K)} \rangle| \rightarrow 0$
as $K \rightarrow \i... |
I want to evaluate this double integral:
$$
\iint \frac{y}{x} \, dx \, dy \quad
$$
which is bounded by functions:
$$
y = 0 \quad
$$
$$
y = x \quad
$$
$$
x^2 + 4y^2 = 4 \quad
$$
And is in the first quadrant. The thing is that I don't quite get how to determine if the function is x-simple, y-simple,... |
The problem says: Let $(X,\tau)$ be a topological space so that $A,B \subset X$ are connected sets in said space.
For $M\subset X, \bar M$ refers to the closure of $M$.
Prove that $A \cup B$ is connected if and only if $(\bar A \cap B) \cup (A \cap \bar B) \neq \emptyset$.
I have no idea where to start. Can so... |
Here is the functional equation:
$\int_0^1 (f(x)+x)^a \, dx=0$
Is there a function $f(x)$ that satisfies it for any $a>0$ and $f(x)\not\equiv-x$?
What about the case where $a$ is an integer? |
Does a function satisfying this equation exist? |
The Problem says that :-
>If $\large \mathrm{N}$ is any positive integer, and $\delta_{n}=\large \int_{_{0}}^{^{N}} {{u} \choose {n}}{{N-u} \choose {N-n}} \textit{ du }$, then the value of
$\large \sum_{n=0}^{N}\delta_{n} = \\ \ \\ \ \\ (A). 0 \\ (B). N \\(C). N^2 \\ (D). \frac{1}{N} $ .
I was thinking about ... |
$B_t$ is said to follow a Brownian motion if i) $B_0=0$, ii) $B_t$ is continuous a.s. iii) $B_t$ has independent increments iv) $B_t-B_s\sim N(0,t-s)$. Then Wiener would go on to show that such a process does indeed exist.
What are some examples of simple and interesting stochastic processes, that can be proven to n... |
This is a question that is based off of the book $\textit{An Introduction to Nonharmonic Fourier Series}$ by Young.
Assume that $\{e_n\}_{n = 1}^\infty$ is the canonical basis for $\ell^2(\mathbb{N})$. Consider the sequence $\{e_n + e_{n+1}\}_{n = 1}^\infty$. I want to know when this sequence is minimal. Being mini... |
I want to evaluate this double integral:
$$
\iint \frac{y}{x} \, dx \, dy \quad
$$
which is bounded by functions:
$$
y = 0 \quad
$$
$$
y = x \quad
$$
$$
x^2 + 4y^2 = 4 \quad
$$
And is in the first quadrant. The thing is that I don't quite get how to determine if the domain is x-simple, y-simple, o... |
I read in Wiki (https://en.wikipedia.org/wiki/Bivector) that antisymmetric part of geometric product can be represent as $(a \land b)$ and $(a \cdot b)^2 - (a \land b)^2 = a^2b^2$
I have 2 questions:
1) How to derive $(a \cdot b)^2 - (a \land b)^2 = a^2b^2$ ?
2) Why $(a \land b)=|a||b||sin(\phi)|$ if $(a \land ... |
How to obtain $(a \cdot b)^2 - (a \land b)^2 = a^2b^2$ and why $(a \land b)=|a||b||sin(\phi)|$ if $(a \land b)^2=-|a|^2|b|^2sin^2(\phi)$? (GA) |
Here is the functional equation:
$$\int_0^1 (f(x)+x)^a \, dx=0$$
Is there a function $f(x)$ that satisfies it for any $a>0$ and $f(x)\not\equiv-x$?
What about the case where $a$ is an integer? |
Is there a non-trivial function satisfying $\int_0^1 (f(x)+x)^a \, dx=0$ for $a$ a positive real (or integer)? |
I know there are closed formulas for sums of the form $\sum_{k=0}^n k^sr^k$
and $\sum_{k=0}^n r^{2n}$ or $\sum_{k=0}^n r^{2n+1}$.
(See https://en.wikipedia.org/wiki/Geometric_series#Sum)
From https://math.stackexchange.com/questions/276892/sum-of-the-form-rr2r4-dotsr2k-sum-i-1k-r2i it seems that there is no closed... |
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