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The differential equation for the Legendre polynomials
$P_n(x)$ is given by:
$(1 - x^2) \frac{d^2P_n}{dx^2} - 2x \frac{dP_n}{dx} + n(n + 1)P_n = 0$. Now I want to prove that $\begin{equation} \int_{-1}^{1} P_l(x) P_m (x) = \dfrac{2}{2l+1} \delta_{lm} \end{equation} $.
It's possible to prove the orthogonal rela... |
Is it possible to prove orthogonal form of integral of legendre polynomial solely from legendre's differential equation without using anything? |
I remember loosely that I saw somewhere a space notation $C_{\to0}^0(\mathbb R^n)$ for the space of all continuous function that vanishes at infinity. But I can't find a reference. Does anyone know a textbook that use this notation? |
I'll assume you already know or have established that $\sim$ is in fact an equivalence relation.
Assume $\phi(t)=\phi(s)$. To prove injectivity, we need to prove $t=s$.
By the definition of $\phi$, we have that $[(t, 1)]=[(s, 1)]$. By the definition of an equivalence class, this tells us that $(t, 1) \sim (s, 1... |
I'll assume you already know or have established that $\sim$ is in fact an equivalence relation.
Assume $\phi(t)=\phi(s)$. To prove injectivity, we need to prove $t=s$.
By the definition of $\phi$, we have that $[(t, 1)]=[(s, 1)]$. By the definition of an equivalence class, this tells us that $(t, 1) \sim (s, 1... |
We are all familiar with the regular Pythagorean theorem of $a^2+b^2=c^2$ where $a$ and $b$ are legs of a right triangle and $c$ is the hypotenuse. This formula can be visualized as turning each side of the triangle into a square and then saying the sum of the squares corresponding to the legs is equal to the square co... |
Generalizing Pythagorean Theorem to regular n-gon / semicircles extension? |
This is what i got so far.
Case 1: one person get 3 jobs and everybody else (4 people left) get 1 job. And there are 5 ways to pick a person to assign 3 jobs
$\binom{7}{3} \cdot 4!\cdot 5 = 4200$
Case 2: two people get 2 jobs and everybody else (3 people left) get 1 job. And there are $ \binom{5}{2} $ way ... |
How many ways to assign 7 different jobs to 4 different people assuming one person must have at least one job? |
I have a function f(n) defined over the set of natural numbers, where each input n corresponds to a unique output f(n), ensuring that the function's values are distinct. I am trying to prove that the set of outputs of this function must contain a power of 2 somewhere.
Intuitively, it seems plausible, as the set of n... |
This is what i got so far.
Case 1: one person get 3 jobs and everybody else (4 people left) get 1 job. And there are 5 ways to pick a person to assign 3 jobs
$\binom{7}{3} \cdot 4!\cdot 5 = 4200$
Case 2: two people get 2 jobs and everybody else (3 people left) get 1 job. And there are $ \binom{5}{2} $ wa... |
How many ways to assign 7 different jobs to 5 different people assuming one person must have at least one job? |
This problem is in my book:
>Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point $P$ on
the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the center of the circle and ... |
How to prove: The stereographic projection is orientation reversing (Complex Analysis by Theodore W. Gamelin 1.3.3)? |
Almost every programs consider $\frac{1}{0}$ as $+\infty$ and $-\frac{1}{0}$ as $-\infty$ according to the IEEE 754 floating-point standard. Thus $\pm\infty\cdot 0=\pm\frac{1}{0}\cdot 0=\pm\frac{0}{0}=\text{NaN} $.
Some programs also consider $\log(0)$ as $-\infty$ according to:
$$ \lim_{x\to 0^+}\log(x)=-\infty $$... |
> A group of ten students includes triplets Alex, Alison and Alice
> wearing black, white and yellow t-shirts respectively. They are to be
> seated in groups of five around two tables: one green and one red. The
> triplets insist on sitting together. Determine the number of seating
> arrangements possible.
**Cho... |
Almost every programs consider $\frac{1}{0}$ as $+\infty$ and $-\frac{1}{0}$ as $-\infty$ according to the IEEE 754 floating-point standard. Thus $\pm\infty\cdot 0=\pm\frac{1}{0}\cdot 0=\pm\frac{0}{0}=\text{NaN} $.
Some programs also consider $\log(0)$ as $-\infty$ according to:
$$ \lim_{x\to 0^+}\log(x)=-\infty $$... |
> A group of ten students includes triplets Alex, Alison and Alice
> wearing black, white and yellow t-shirts respectively. They are to be
> seated in groups of five around two tables: one green and one red. The
> triplets insist on sitting together. Determine the number of seating
> arrangements possible.
**Cho... |
> A group of ten students includes triplets Alex, Alison and Alice
> wearing black, white and yellow t-shirts respectively. They are to be
> seated in groups of five around two tables: one green and one red. The
> triplets insist on sitting together. Determine the number of seating
> arrangements possible.
**Cho... |
I remember loosely that I saw somewhere a space notation $C_{\to0}^0(\mathbb R^n)$ for the space of all continuous function that vanishes at infinity. But I can't find a reference. Does anyone know a textbook that use this notation?
For books which doesn't use $C_0(\mathbb R^n)$ for function vanishing at infinity, s... |
I did an analytic exercise: f is a polynomial with degree n, s.t.$\int_0^1 f(x)x^k=0$, (k=1,2,...,n). Prove: $\int_0^1 f^2(x)dx=(n+1)^2(\int_0^1 f(x)dx)^2$. This implies that $$\begin{vmatrix}
1-\frac{1}{(n+1)^2}& \frac{1}{2}& \frac{1}{3} \cdots & \frac{1}{n+1} \\
\frac{1}{2}& \frac{1}{3} & \frac{1}{4}& \cdots & \fr... |
How to prove this determinant is zero? |
I did an analytic exercise: f is a polynomial with degree n, s.t.$\int_0^1 f(x)x^k=0$, (k=1,2,...,n). Prove: $\int_0^1 f^2(x)dx=(n+1)^2(\int_0^1 f(x)dx)^2$. This implies that $$\begin{vmatrix}
1-\frac{1}{(n+1)^2}& \frac{1}{2}& \frac{1}{3} & \cdots & \frac{1}{n+1} \\
\frac{1}{2}& \frac{1}{3} & \frac{1}{4}& \cdots & \f... |
I did an analytic exercise: If f is a polynomial with degree n, s.t.$\int_0^1 f(x)x^kdx=0$, (k=1,2,...,n), then $\int_0^1 f^2(x)dx=(n+1)^2(\int_0^1 f(x)dx)^2$. This implies that $$\begin{vmatrix}
1-\frac{1}{(n+1)^2}& \frac{1}{2}& \frac{1}{3} & \cdots & \frac{1}{n+1} \\
\frac{1}{2}& \frac{1}{3} & \frac{1}{4}& \cdots &... |
I did an analytic exercise:
> If $f$ is a polynomial of degree $n$ such that
$$\int_0^1 f(x)x^kdx=0$$ for all $k=1,2,\dots,n$, then
$$\int_0^1 f^2(x)dx=(n+1)^2\left(\int_0^1 f(x)dx\right)^2.$$
This implies that
$$\begin{vmatrix}
1-\frac{1}{(n+1)^2}& \frac{1}{2}& \frac{1}{3} & \cdots & \frac{1}{n+1} \\
... |
For two (say continuous) function $f,g\in C(\mathbb R^n)$, let us define
\begin{equation}
f\ast g(x):=\lim_{R\to+\infty}\int_{B^n(0,R)}f(x-y)g(y)dy,
\end{equation}
whenever the integral converges.
It is clear that under such definition the convolution can be non-associative (e.g. [this post][1]). Indeed by taki... |
Can convolution fails to be commutative and distributive in the sense of (generalized) Riemann integral? |
This is Exercise 3.1.13 [here][1].
Let $\gamma:[a,b] \to U \subset\mathbb{C}$ be a piecewise $C^1$, injective (or can also be closed) curve. Show that there exists a continuous function $f$ such that $\int_\gamma f(z)dz \neq 0$.
**My idea:** Use a bump function $f$ near a point $\gamma(t_0)$ to show $\int_\gamma ... |
For two (say continuous) function $f,g\in C(\mathbb R^n)$, let us define
\begin{equation}
f\ast g(x):=\lim_{R\to+\infty}\int_{B^n(0,R)}f(x-y)g(y)dy,
\end{equation}
whenever the integral converges.
It is partially clear that under such definition the convolution can be non-associative (e.g. [this post][1]). Inde... |
Let $P = \mathbb{R}^2 \setminus \{(0,0)\}$. We define a relation on $P$ as follows: $(x,y) \sim (z,w)$ if and only if there exist $\lambda \in \mathbb{R}\setminus \{0\}$ such that $x = \lambda z$ and $y = \lambda w$.
Define the function $\phi : \mathbb{R} \to {^{P}\big/_{\sim}}$ by $\phi(t) = [(t,1)]$ for $t \in \ma... |
We have, for $z = e^{\frac{2 \pi i}{17}}$, $$(z + z^2 + z^4 + z^8 + z^{16} + z^{15} + z^{13} + z^9) (z^3 + z^5 + z^6 + z^7 + z^{10} + z^{11} + z^{12} + z^{14}) \in \mathbb Z.$$
This happens because the Galois group $\operatorname{Gal}(\mathbb Q(z)/\mathbb Q)$ acts on the partition of roots by either swapping them or i... |
When can non-trivial roots of unity be partitioned such that product of their sums is an integer? |
A prime number $\overline{a_1a_2\ldots a_n}$ is called a permutable prime number, if for any $\sigma\in S_n$, the number $\overline{a_{\sigma{(1)}}a_{\sigma{(2)}} \ldots a_{\sigma{(n)}}}$ is prime. For example 13 is a permutable prime number. Are there infinity many permutable prime numbers? |
I am reading these notes Derived categories, resolutions, and Brown representability
Henning Krause (https://arxiv.org/pdf/math/0511047.pdf) about derived and triangulated categories
I am having trouble understanding the proof of condition MS3 in this lemma of section 3:
[![enter image description here][1]][1]
... |
How to prove that the shortest path between two points in sphere is a part of the great circle? |
This comes straight from problem 1.5 (p. 12) of *Handbook of Product Graphs, Second Edition* by Hammack, Imrich, and Klavžar.
It even comes with a hint in the appendix:
> **Exercise 1.5** Hint: Show that the homomorphic image of an odd cycle is a closed walk that contains an odd cycle.
But I think maybe there'... |
Let $A\subseteq \Bbb R^n$ an open set and let $\vec x\in \Bbb R^n$ a fixed vector. Show that the set $A+\vec x={\{\vec a+\vec x:\vec a\in A\}}$ is an open set in $\Bbb R^n$.
Dem:
An element of the set $A+\vec x$ is of the form $\vec a+\vec x$ for some $\vec a\in A$ and because $A$ is an open set:
$\implies \fo... |
We need to find the range of $x$ for the following problem:
$$\frac{(x+6)(x-4)^2}{(x-6)(x+4)^2} + \frac{(x-6)(x+9)^2}{(x+6)(x-9)^2} < \frac{2x^2 + 72}{x^2 - 36}$$
They invlove the use of squares, which we have to avoid while solving inequalities.....And sadly, I cannot think of ways for trying this question. Can ... |
To my understanding, rigorously speaking, probability for real-valued random variables or random vectors ($n$-tuples of real numbers) is defined only for $\mathcal{B}(\mathbb{R}^{n})$ (the Borel subsets of $\mathbb{R}^{n}$), i.e. probability (for random vectors) is, by definition, strictly a function $\mathsf{P} : \mat... |
Does "definition" of probability in terms of densities imply probability for non-Borel subsets of $\mathbb{R}^{n}?$ |
Suppose I have a sequence of i.i.d. random variables $X_1, X_2, \dots$ with positive mean $E[X_1] = \mu$. Is the function
$$f(n) = \Pr( X_1 + X_2 + \dots + X_n \leq A)
$$
monotonically non-increasing in $n$ (perhaps for $n$ large enough)?
Intuitively, I think this should be the case. By the law of large numbers,... |
This comes straight from problem 1.5 (p. 12) of *Handbook of Product Graphs, Second Edition* by Hammack, Imrich, and Klavžar.
It even comes with a hint in the appendix:
> **Exercise 1.5** Hint: Show that the homomorphic image of an odd cycle is a closed walk that contains an odd cycle.
But I think maybe there'... |
I have a doubt regarding the use of the Bolzano-Weierstrass theorem in a dynamic optimization problem. Exactly, the problem is the neoclassical growth model (or Ramsey model). In the website that I post below, in page 14, they invoke the aforementioned theorem so as to prove the existence of a solution. This is not the... |
I am unsure if this belongs to math or philosophy.
Let's say there's 0-dimensional space, however multiple objects exist within in, occupying the same "spot". If multiple objects exist, is the space still zero-dimensional?
The problem here is that multiple objects existing at the same spot introduces dimensional... |
Consider two vector spaces $\mathbb{C}^n$ and $\mathbb{C}^m$, where $0<n<m<\infty$. Now, I'd like to define matrices $A\in\mathbb{C}^{n\times n}$ and $B\in\mathbb{C}^{m\times m}$ in the following way.
With $A_i\in\mathbb{C}^{n\times n}$ and $B_i\in\mathbb{C}^{m\times m}$, decide a common set of coefficients $s_i$ t... |
Is the Eigen spectrum of a matrix completely defined by the algebra of its parts? |
I am working with the book "Abstract Algebra - Theory and Applications" by Thomas W. Judson. On page 205 it lists the following exercise: "A ring R is a Boolean ring if for every $a ∈ R, a^2 = a$. Show that every Boolean ring is a commutative ring.".
Trying to work on this exercise, I came up with the non-commutativ... |
For the multi-variable Riemann integral (specifically, integrating functions $f : \mathbb{R}^n \rightarrow \mathbb{R}$ over subsets of $\mathbb{R}^n$), the most common treatment (see footnote below) seems to be defining the integral in terms of *rectangular* partitions of the domain of integration (partition composed o... |
Given square matrices $A$, $B$ and $X$ and that
$$
AXA^T = BXB^T
$$
Does that imply $A = B$ ?
What about a specific condition, e.g. when $A, B$ and $X$ are symmetric?
Also $X$ is not the zero matrix, but the question is not for a specific $X$, as @lisyarus correctly pointed out in the comments.
Sizes of $A,... |
We know that that given $X_1,...,X_n$ independent random variables which are distributed exponentially, their minimum is also distributed exponentially. Is the converse true? Given $X = \min\{X_1, ..., X_n\}$ distributed exponentially, and $X_1, ..., X_n$ are independent, is it true that $X_i$ is distributed exponentia... |
If minimum of independent random variable is exponential, is each one exponential? |
Why do various SDE books consider only equations with the following form:
$$\displaystyle \mathrm {d} X_{t}=\mu (X_{t},t)\,\mathrm {d} t+\sigma (X_{t},t)\,\mathrm {d} B_{t}$$
I understand that it is analytical and computational convenient, and also many time-varying phenomena in various fields in science and engi... |
I am working with the book "Abstract Algebra - Theory and Applications" by Thomas W. Judson. On page 205 it lists the following exercise: "A ring R is a Boolean ring if for every $a ∈ R, a^2 = a$. Show that every Boolean ring is a commutative ring.".
Trying to work on this exercise, I came up with the non-commutativ... |
Given square matrices $A$, $B$ and $X$ and that
$$
AXA^T = BXB^T
$$
Does that imply $A = B$ ?
What about a specific condition, e.g. when $A, B$ and $X$ are symmetric?
Also $X$ is not the zero matrix, but the question is not for a specific $X$, as @lisyarus correctly pointed out in the comments.
Specifical... |
Put 60 identical balls into 10 identical boxes such that each box has at least one ball while all boxes contain a different number of balls. How many ways can you do so? |
How many ways can you do so? |
Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$ with spectra $\sigma(A), \sigma(B)$.
We know that : $\sigma(A \otimes 1 + 1 \otimes B) = \overline{\sigma(A+B)}$ and $A \otimes 1 + 1 \otimes B$ defines an unbounded self-adjoint operator on $\mathcal{H_1} \otimes \mathc... |
We have, for $z = e^{\frac{2 \pi i}{17}}$, $$(z + z^2 + z^4 + z^8 + z^{16} + z^{15} + z^{13} + z^9) (z^3 + z^5 + z^6 + z^7 + z^{10} + z^{11} + z^{12} + z^{14}) \in \mathbb Z.$$
This happens because the Galois group $\operatorname{Gal}(\mathbb Q(z)/\mathbb Q)$ acts on the partition of roots by either swapping them or i... |
I have a question about the definition of Cauchy sequence in $\sigma$-finite measure, e.g. Lebesgue measure. The reason why I have this question is because I was taught that
for a Cauchy sequence $(f_n)$ we can assume its limit $f$ for all $x\in A$, and zero otherwise where each $f_n$ is in $L^{\infty}$.
My quest... |
So I've been trying to understand this theorem for the past week and there is a proof (Not rigorous) that was given to me that I just cannot understand. I'm not gonna lie I'm not so good at mathematics but, I've read it like 50 times+ and it just doesn't make sense to me.
Can anyone for the love of god help dumb th... |
Collapsing either one of the circles in the bouquet of two circles to the basepoint, how can I describe this by a map (in the free product) and how is this related to that $\mathbb{R}^2\backslash\{p,q\}$ is **not** homeomorphic to a bouquet of two circles! However there is a **deformation retraction** of that space to ... |
We know that that given $X_1,...,X_n$ independent random variables which are distributed exponentially, their minimum is also distributed exponentially. Is the converse true? Given $X = \min\{X_1, ..., X_n\}$ distributed exponentially, and $X_1, ..., X_n$ are independent, is it true that $X_i$ is distributed exponentia... |
For example if we have 22 people and we want to distribute them into two groups there are $\binom{22}{11}/2$ possibilities.
But I'm wondering, can we generalise this if we say how many possibilities are there to distribute $n$ people into $m$ equally large groups? (Even more difficult, what if each group is of a di... |
I have a question about the definition of Cauchy sequence in $\sigma$-finite measure, e.g. Lebesgue measure. The reason why I have this question is because I was taught that
for a Cauchy sequence $(f_n)$ we can assume its limit $f$ for all $x\in A$, and zero otherwise where each $f_n$ is in $L^{\infty}$.
My quest... |
I'm a beginner of algebraic geometry and I find some questions as follows.
Consider the projective bundle (in sense of Grothendieck) $\mathbb{P}_{\mathbb{P}^n}(\mathscr{O}(a_1)\oplus\cdots\cdots\mathscr{O}(a_r))$ on $\mathbb{P}^n$. I have the following questions:
[Q1] We know that $\mathrm{Bl}_{\mathbb{P}^a}\math... |
Projective bundles on projective spaces and birational contractions? |
In [this](https://github.com/mitmath/18303/blob/master/supp_material/poissonFD.ipynb) notebook from MIT's Intro to Linear PDEs course, it is unclear to me why the Kronecker product is used to formulate the coefficients matrix $A$ for solving the linear system of equations $A u = b$. Given Poisson's equation on the unit... |
Is it true: $\int ( \int f(u,z) d\mu(u))^2 dz \leq \|\mu \|_{TV} \int ( \int f(u,z) du)^2 dz$? |
Let $G = \mathrm{GL}_n$, let $B$ be a Borel subgroup of $G$.
The set $\mathcal{B}$ of $G$-conjugates of $B$ can be given an algebraic variety structure and is known as the *variety of Borel subgroups* or the *complete flag variety*. It is isomorphic to $G/N_G(B) = G/B$.
A classic result is that $\bigcup_{g\in G} ... |
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability sapce and let $X: \Omega \to C^+(\mathbb{R})$ be a random variable taking values in the space of real valued non-negative continuous functions on the real line. I want to verify that the map $(\omega, x) \to X(\omega)(x)$ is measurable with respect to the product... |
In [this](https://github.com/mitmath/18303/blob/master/supp_material/poissonFD.ipynb) notebook from MIT's [Intro to Linear PDEs](https://github.com/mitmath/18303/tree/master) course, it is unclear to me why the Kronecker product is used to formulate the coefficients matrix $A$ for solving the linear system of equations... |
Let $\mathfrak{g}$ be a finite dimensonnal semisimple Lie algebra, let $e \in \mathfrak{g}$ be a regular nilpotent element ie. an element whose adjoint endomorphism is nilpotent and of maximal rank. Denote by $\mathfrak{g}^e$ the centralizer of $e$
in $\mathfrak{g}$ ie. the set of $y\in\mathfrak{g}$ such that $[e,y]=... |
There is a challenging question assigned regarding Counting. here is the question:
>we call every path from point A to B a **good** path, if in every step it has moved up, down, or right and not crossed the same line more than once. we write down the number of squares beneath every possible *good* path. what would b... |
> Question
>
>
>
> 1. Is it true that $(A\cap B)^\perp=A^\perp+B^\perp$?
> 2. Does this require the space to be a Hilbert space or is it valid for prehilbert spaces too?
> 3. How do I prove it? Can you give me a counterexample if it's not true? And in that case, is there a way to express $(A∩B)^\perp$ ... |
Your question is specifically about the transfer principle, more precisely about the meaning of "first-order" when applying the transfer principle to "first-order formulas". The essential point that does not seem to have come out clearly in the *comments* is that the quantifiers have to be *bound* for the transfer pri... |
In [this](https://github.com/mitmath/18303/blob/master/supp_material/poissonFD.ipynb) notebook from MIT's [Intro to Linear PDEs](https://github.com/mitmath/18303/tree/master) course, it is unclear to me why the Kronecker product is used to formulate the coefficients matrix $A$ for solving the linear system of equations... |
I have an $n$ dimensional manifold $M$. I know that *it is* the cartesian product of two manifolds $M_1$ and $M_2$ but I *do not know* these two manifolds, not even their dimensions $n_1$ and $n_2$.
I am wondering if I can get some information about $M_1$ and $M_2$, e.g. the dimensions $n_1$ and $n_2$, by knowing th... |
I have a question about the definition of Cauchy sequence in $\sigma$-finite measure, e.g. Lebesgue measure. The reason why I have this question is because I was taught that
for a Cauchy sequence $(f_n)$ we can assume its limit $f$ for all $x\in A$, and zero otherwise where each $f_n$ is in $L^{\infty}$.
My quest... |
Given square matrices $A$, $B$, if there exists a symmetric matrix $X$ with entries in $(0,1)$ such that
$$
AXA^T = BXB^T,
$$
does that imply $A = B$ ?
What about a specific condition, e.g. when $A, B$ are symmetric?
Sizes of $A, B$ necessarily have to be the same for the equation to be valid. |
I have a question about the definition of Cauchy sequence in $\sigma$-finite measure, e.g. Lebesgue measure. The reason why I have this question is because I was taught that
for a Cauchy sequence $(f_n)$ we can assume its limit $f$ for all $x\in A$, and zero otherwise where each $f_n$ is in $L^{\infty}$.
My quest... |
I have an $n$ dimensional manifold $M$. I know that *it is* the cartesian product of two manifolds $M_1$ and $M_2$ but I *do not know* these two manifolds, not even their dimensions $n_1$ and $n_2$.
I am wondering if I can get some information about $M_1$ and $M_2$, e.g. the dimensions $n_1$ and $n_2$, by knowing th... |
Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$ with spectra $\sigma(A), \sigma(B)$.
We know that : $\sigma(A \otimes 1 + 1 \otimes B) = \overline{\sigma(A) +\sigma(B)}$ and $A \otimes 1 + 1 \otimes B$ defines an unbounded self-adjoint operator on $\mathcal{H_1} \otim... |
I have the following two equations
$$-\frac{\gamma+2P}{1+2Z}[Z^2+(\alpha Z-\Omega)^2]+2(P-\gamma z)[Z+\alpha(\alpha Z-\Omega)]=0$$ (1)
$$2Z[-2\gamma_1\frac{\gamma+2P}{1+2Z}Z+Z^2+(\alpha Z-\Omega)^2]-2\gamma_1^2 \frac{\gamma+2P}{1+2Z}[-\frac{\gamma+2P}{1+2Z}Z+P-\gamma Z-\gamma_1 2(P-\gamma Z)[-Z+\alpha(\alpha Z-\Omeg... |
How to get the solution of (\alpha Z-\ Omega) from the following two equations? |
I am stuck in understanding how to proceed to show if the following functions admit partial derivatives at those specific points. We have been taught not to use the definition but rather to, say, evaluate $f(x, 0)$ and $f(0, y)$, that is along the axis, and then take their derivatives and evaluating them at the critica... |
I am asking about the properties of rank over an inverse limit.
-------------------
Suppose we have a noetherian ring $R$ with uniformizer $\pi$ and noetherian ring of formal power series $R[[X]]$. Let $\mathscr{R}$ be the $\pi$-adic completion of $R[[X]]$.
Let $S \subseteq T$ be two $R [[X]]$-modules containe... |
**Hint**
By symmetry, you only have to show that the line joining $\ v_p\ $ to the centre of mass of the points $\ v_1,v_2,\dots,v_{p-1}\ $ passes through the centre of mass of all the points. The line joining $\ v_p\ $ to the centre of mass of the points $\ v_1,v_2,\dots,v_{p-1}\ $ has parametric equation
$$
v=t... |
How many ways to put 60 balls into 10 boxes such that each box has at least one ball and all boxes contain a different number of balls? |
I have trouble understanding the beginning sentence of section $3$ of a paper by C.Moore: <https://www.jstor.org/stable/2373052>, which I quoted below:
>Suppose now that $G$ is a (connected) semi-simple Lie group with Lie algebra $\mathfrak{g}_{0}$. Let $G^{*}$ be the adjoint group of $G$. Then one knows that $G^{*}... |
I recently noticed that my go-to reference for stuff related to Hilbert spaces uses the terms "Hilbert space" and "complex Hilbert space" synonymously. Hence I am now wondering what results do not hold in the real case (and vice versa).
- Any "basic" theorems that only hold in the complex (or the real) case? (By "ba... |
I have an $n$ dimensional manifold $M$. I know that *it is* the cartesian product of two manifolds $M_1$ and $M_2$ but I *do not know* these two manifolds, not even their dimensions $n_1$ and $n_2$.
Note: the factorization into $M_1\times M_2$ could be non-unique. Please, assume that it is. And assume that $M_1$ and... |
I recently noticed that my go-to reference for stuff related to Hilbert spaces uses the terms "Hilbert space" and "complex Hilbert space" synonymously. Hence I am now wondering what results do not hold in the real case (and vice versa).
- Any "basic" theorems that only hold in the complex (or the real) case? (By "ba... |
I have the following two equations
$$-\frac{\gamma+2P}{1+2Z}[Z^2+(\alpha Z-\Omega)^2]+2(P-\gamma z)[Z+\alpha(\alpha Z-\Omega)]=0$$ (1)
$$2Z[-2\gamma_1\frac{\gamma+2P}{1+2Z}Z+Z^2+(\alpha Z-\Omega)^2]-2\gamma_1^2 \frac{\gamma+2P}{1+2Z}[-\frac{\gamma+2P}{1+2Z}Z+P-\gamma Z]-\gamma_1 2(P-\gamma Z)[-Z+\alpha(\alpha Z-\Ome... |
I had a seemingly simply question today, that goes as following.
What do we need for a mathematics to exist in a universe, or a system, more broadly speaking?
Is it a matter of having the ability to define axioms, or regularities and certain patterns?
How much does it hinge on us having the cognitive functiona... |
Let $M$ be a finitely generated noetherian module over a ring $R$.
Let $\phi$ be an automorphism of $M$.
Let $P = \{ a \in M \mid \exists n \in \mathbb{N}, \text{such that } \phi^{n} (a) =a \}$ be the sets of periodic points of $\phi$ in $M$.
For $a \in P$, let $m_a$ be the smallest positive integers su... |
(I've been self-learning differential geometry for some time now, and math.stackexchange has been of sooo much help, that I thought I might get my answer here.)
From Lee's "Introduction to Riemannian manifolds", the shape operator $s$ of a hypersurface is the Weingarten map, which is defined as $(\tilde{\nabla}_X N)... |
In $\mathbb{R}^{n+1}$, we consider an orthonormal basis $e_0,\dots,e_n$ and $\alpha_0,\dots,\alpha_n$ its dual basis. Then, the determinant is $\alpha_0 \wedge \cdots \wedge \alpha_n$. I am seeking a contracted formula for
$$
\sum_{\sigma \in \Sigma_{n}} |\sigma|\det(e_0,v_{\sigma(1)},\dots,v_{\sigma(n)})
$$
In ter... |
> Let $L \to M$ be a complex line bundle over a smooth manifold $M$. Let $\{U_\alpha\}$ be a trivializing open cover with transition maps $z_{\alpha\beta}:U_\alpha \cap U_\beta \to \Bbb C^* = \text{GL}(1,\Bbb C)$. The bundle of endomorphisms of $L$, $\text{End}(L) \cong L^* \otimes L$ is trivial since it can be defined... |
[Here][1], the Ramanujan Master Theorem is proven in the following way:
$$ \int_0^\infty\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} \lambda(n)x^n = \int_0^\infty\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} E^n\lambda(0)x^n = \int_0^\infty x^{s-1}e^{-Ex}dx \;\lambda(0) = \frac{\Gamma(s)}{E^s}\lambda(0) = \Gamma(s)\lambda(-s).$$
Th... |
**Problem definition**
Consider a dataset composed by $m$ bivariate measurements
\begin{equation*}
y_j \sim \mathcal{U}(\mathcal{E}(\ell_1,\ell_2,\theta)) \qquad j=1,2,\dots,m
\end{equation*}
uniformly distributed over an ellipse centered in the origin, with semilength $\ell_1$, semiwidth $\ell_2<\ell_1$ and or... |
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