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While working on a project, I encountered [a paper that includes this sum][1]:
$$
r(ξ, \bar{ξ}) = \frac{1}{\#} \sum_{g ∈ } r_0\bigl(g(ξ), g(\bar{ξ})\bigr)
$$
where
- $$ is "the tetrahedral group", "a discrete subgroup of isometries $ ⊂ O(3)$";
- $r_0(R) = -\frac{a + b R^2 + (3 - b) R^4 + (1 - a) R^6}{(1 + R... |
What would be some good, practical references for some of algebraic topology/geometry arguments in McDuff Salamon, *Introduction to Symplectic Topology* (3rd edition)?
Specifically, a lot of arguments seem to revolve around looking at the algebraic topology of bundles, ranging from (co)homology to characteristic cla... |
Suppose $(A\sqcup B,i_A:A\to A\sqcup B,i_B:B\to A\sqcup B)$ is a co-product of $A$ and $B$ in an arbitrary abelian category $\mathcal{A}$.
For any object $T\in \mathcal{A}$, any map $f:A\sqcup B\to T$ is fully characterized (by the universal property of co-products) by the maps $fi_A$ and $fi_B$. I also know that a... |
Wikipedia gives the definition of a bundle map as the arrow $\left( \xrightarrow{\quad \varphi \quad } \right)$ in the commuting diagram below:
[![enter image description here][1]][1]
------------
Hussemoller gives the definition of a bundle map as the arrow (note the ordered pair) $\left( \xrightarrow{\q... |
B-Isomorphic? (Husemoller, Fiber Bundles). Illustrated diagrams included. |
Suppose you given three points with known coordinates $P_1, P_2, P_3$, and also another set of three points $P'_1, P'_2, P'_3$. Now does there always exist an affine transformation
$ f( r ) = A r + b $
where $A \in \mathbb{R}^{ 3 \times 3}$ and $b \in \mathbb{R}^3 $, that will map $P_1$ to $P'_1$, $P_2$ to $P'_... |
Let $f$ and $g$ be real nonzero polynomials with *nonnegative* coefficients such that $g(0) \neq 0$.
Let
$$
h(x) = \frac{xf(x)}{(x^2+x)f(x) + g(x)}.
$$
In particular, it follows that $h$ is well defined on the interval $[0,+\infty)$, and
$$
0 \le h(x) \le 1, \qquad x \ge 0.
$$
Even more, using the partial fr... |
Let $e_1 = \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}$, and so on, where $e_i$ is a vector of all zeros, except for a $1$ in the $i^{\mathrm{th}}$ place. Since $A$ is positive definite, then $x^T A x > 0$ for any non-zero vector $x \in \Bbb R^n$. Then, $e_1^T A e_1 > 0$, and likewise for $e_2, e_3$ and so on.
... |
I got a little confused about Cartesian multiplication. Until now, I mainly used Cartesian multiplication over relations, and today my professor started combinatorics.
He showed us that performing Cartesian multiplication 'n' times over the same group for example group A results in (a1, a2, a3....,an) where each ele... |
So I'm studying number theory, and my book said that <br>
$$ x^2 + 5x + 7 \equiv 0 \pmod n $$
had at least four solutions for 1<n<20. I found four (n=3,7,13,19) but n=19 was tricky: I had to replace 7 with -50, which I assume was legit since $ 7 \equiv -50 \pmod {19} $ . Is this the only way to find the solution, by... |
Quadratic congruence equation: must I use trial & error? |
Informally, a category internal to a category $\mathbb{E}$ is a category where the collection of morphisms and the collection of objects are objects in $\mathbb{E}$, and the operations "assigning the domain", "assigning the codomain", "picking out the identity morphism", and composition are all morphisms in $\mathbb{E}... |
I’m 6th grade student (self-learning linear algebra, etc.) and this is another question I have (6th grade teacher literally didn’t know!):
I recently came across exponentiating matrices [here][1], which we basically define by plugging in a matrix $A$ into the series; $\exp(A) = e^A = \sum_{n = 0}^{\infty} \frac{1}{n... |
How to visualize matrix functions? |
I’m a middle school (6th grade) student who is self-learning [these topics][1], including linear algebra, and this is another question I have (teacher didn’t know again):
I recently came across exponentiating matrices [here][2], which we basically define by plugging in a matrix $A$ into the series; $\exp(A) = e^A = ... |
Matrix functions are just functions on multiple variables. You can visualize them in the same way (or can't, as usually matrix functions are at least 8 dimensional, which is pretty hard to visualize as a graph). But you can view Matirces as just another vector and apply higher dimensional functions to them.
Matrix f... |
Question:
> [![enter image description here][1]][1]
>
> This figure is made up of 6 unit cells. Can we form a rectangle with integral lengths using an odd number of copies of this domino? Rotating and flipping of the figure is allowed.
This question is a past year question from a school competition.
After tryi... |
Can we form a rectangle with integral lengths using an odd number of copies of this domino? |
The definitions are equivalent: You can always recover $f$ from $\varphi$ by noting that the definition implies that $\varphi$ takes fibres to fibres, so if $\varphi(E_x) \subseteq F_y$ where $E_{-}, F_{-}$ are the fibres over the respective points, then we must have $f(x) = y$.
Who says that $(u, f)$ is a functor? ... |
Suppose $(A\sqcup B,i_A:A\to A\sqcup B,i_B:B\to A\sqcup B)$ is a co-product of $A$ and $B$ in an arbitrary abelian category $\mathcal{A}$.
For any object $T\in \mathcal{A}$, any map $f:A\sqcup B\to T$ is fully characterized (by the universal property of co-products) by the maps $fi_A$ and $fi_B$. I also know that a... |
Consider the following Lyapunov-like function:
$$ V(x,y) = \frac{1}{2}x^2 + \frac{1}{2}y^2 + \sum_{i=1}^{n}\ln\big(\cosh(q_{i})\big) \ \ \ \ \ (1)$$
where $q_{i} = \sqrt{(x-x_{i})^2 + (y-y_{i})^2} - L, \ L>0 \ (\text{constant})$, $n\geq1$ is an arbitrary positive integer number. Suppose $x_{i}, y_{i}$ are known p... |
I got a little confused about Cartesian multiplication. Until now, I mainly used Cartesian multiplication over relations, and today my professor started combinatorics.
He showed us that performing Cartesian multiplication 'n' times over the same group for example group A results in (a1, a2, a3....,an) where each ele... |
In $\mathbb{R}^d$
### Question:
a) Prove that $ ||x||_2 = Sup\left \{ x \cdot y \; s.t. \; ||y||_2 = 1 \right \}$
b) Prove that $ x \cdot y = ||x|| $ and $ ||y||=1$ iff $x = ||x|| y$
### Answer:
a) First by definition $x \cdot y = ||x|| ||y|| cos( \widehat{ x ; y} )$. Now if $||y||=1 \Rightarrow x... |
Let $R$ be a DVR with fraction field $K$ and residue field $k$ of characteristic zero.
The following result about the existence of regular models for varieties over $K$ seems to be well-known:
**Theorem:** Let $X$ be a smooth projective variety over $K$. Then there exists a regular model $\pi: \mathcal{X} \right... |
I come from the applied math and statistics world, but I was talking to my friend who comes from the pure math and number theory world—in particular Galois representation theory.
I mentioned something about the confusing definition of "matrices" in textbooks. Many textbooks talk about a matrix as the solution ... |
Consider a maximin problem with $\lambda=[\lambda_1,\lambda_2,...,\lambda_n]$
$$\max_{\lambda}\min_{i\in\{1,2,3,...,M\}}{f_i(\lambda)}$$
subject to $\lambda$ is any vector locating in a standard probability simplex, that is $\sum_{i=1}^{n}\lambda_i=1$ and $\lambda_i\geq0, \forall i\in\{1,2,...,k\}$. Note that each fu... |
(1) "Given the success of ChatGpt in dealing with human natural language" ??
Maybe Chatgpt can itself answer this question , no humans needed here !!
(2) Currently , Chatgpt is a little toy with no understanding of logic & mathematics.
What output it gives is not mathematically valid beyond what inputs it ... |
If for two strings a+b=b+a holds true, what properties can we derive about such strings? |
Let $F\subset E$ be a finite Galois extension. Prove that $\mathcal L_F(E,E)$, the vector space of all $F$-linear maps over $E$ has dimension $[E:F]$, where the scalar multiplication is defined by multiplication in $E$ pointwisely.
My attempt is to take the Galois automorphisms in $\operatorname{Gal}(E/F)$ and it su... |
In $\dfrac{365 \choose \#people}{365^{\#people}}$
this counts no repeats for the probability of none assuming Jan, Feb, March is the same thing as Feb, March Jan.
If
>We interpret the question as asking for the probability that **precisely two** people share a birthday.
>Imagine lining up our $n\ge ... |
Matrix functions are just functions on multiple variables. You can visualize them in the same way (or can't, as usually matrix functions are at least 8 dimensional, which is pretty hard to visualize as a graph). But you can view Matirces as just another vector and apply higher dimensional functions to them.
Matrix f... |
I have a product of the form $e^{-tx}\times G_{p,q}^{m,n}\left(x\left|\begin{smallmatrix}\mathbf{a}_n, \mathbf{a}_{p-n}\\
\mathbf{b}_m, \mathbf{b}_{q-m}\end{smallmatrix}\right.\right)$. I need to write the product as Meijer's-G function. I know the shifting property $z^{\mu}G_{p,q}^{m,n}\left(x\left|\begin{smallmatrix... |
I have come across an interesting problem whilst looking to estimate how many cylinders of equal radius I can fit on a 3D printer build platform.
As I was writing out this post I think I've managed to answer my own question. It's only taken me all day learning about floor functions...however, I would like to know if... |
How do you rearrange an equation with the product of floor functions? |
Let $f(x) = 2x$ for $x<0$ and $f(x)=2x+1$ for $x>0$. Why is this function not differentiable? I understand that if a function is differentiable, then it has to be continuous which this function is not. That being said, I am trying to understand it in a different way such as by using the Epsilon-Delta definition.
Fo... |
Why Is This Jump Discontinuity Not Differentiable? |
Let $F\subset E$ be a finite Galois extension. Prove that $\mathcal L_F(E,E)$, the vector space of all $F$-linear maps $E\to E$ over $E$ has dimension $[E:F]$, where the scalar multiplication is defined by multiplication in $E$ pointwisely.
My attempt is to take the Galois automorphisms in $\operatorname{Gal}(E/F)$ ... |
**Context**: I'm currently an engineering college student taking the equivalent of a calculus/real analysis class (more rigorous than a standard calculus course but we didn't construct the reals etc.).
Starting from the least upper bound (taken for granted), we proved the monotone sequence theorem, then the adjacen... |
Let's take PA1 for example. From godel's 2nd incompleteness theorem, PA1 can't prove its own consistency, more specifically it can't prove that the largest consistent subset of the theory is PA1 itself.
But PA1 has infinite axioms, so can PA1 prove atleast for a given finite set of axioms ( of PA1 ) that they are ... |
is arithmetic finitely consistent? |
How to evaluate
$$
I=\int_{0}^{1} \int_{0}^{1} \int_{0}^{1}
\frac{1}{\sqrt{3-x^2-y^2-z^2} }\text{d}x
\text{d}y\text{d}z?
$$
Some simple calculation shows that
$$
I=\frac{\sqrt{2} -1}{4}\pi+\frac{\pi^2}{4}
-\frac{3\pi}{4}\arctan\left ( \sqrt{2} \right )+\frac12\int_{0}^{1}\arctan\left ( \frac{1}{\sqrt{x} } ... |
A differentiable and also convex function has the following property.
$$
(\nabla f (x) - \nabla f (y))^T(x-y) \geq 0
$$
I can derive it from the first order condition for a convex function,
but I don't know meanings of the inequality in a graph.
Does it mean that a tangent line is below the graph? but how... |
In $\mathbb{R}^d$
### Question:
a) Prove that $ ||x||_2 = Sup\left \{ x \cdot y \; s.t. \; ||y||_2 = 1 \right \}$
b) Prove that $ x \cdot y = ||x|| $ and $ ||y||=1$ iff $x = ||x|| y$
### Answer:
a) First by definition $x \cdot y = ||x|| ||y|| cos( \widehat{ x ; y} )$. Now if $||y||=1 \Rightarrow x... |
Let $G$ be a finite group acting freely on a smooth projective complex variety $X$. This induces an action on $\mathrm{Coh}(X)$ by pullback. Let $\pi\colon X \rightarrow Y=X/G$ denote the quotient.
How do I prove the following claim?
**Claim** $\pi_\ast \mathcal{O}_X$ splits as a direct sum according to the irred... |
While working on a project, I came across [a paper that includes this sum on page 15][1] as the definition of a support function $r$ for a surface with tetrahedral symmetry:
$$
r(ξ, \bar{ξ}) = \frac{1}{\#} \sum_{g ∈ } r_0\bigl(g(ξ), g(\bar{ξ})\bigr)
$$
where
- $$ is "the tetrahedral group", "a discrete subgro... |
Is this single-variable function being called with two variables (and if so, how do I handle that), or am I misreading the notation? |
Let $f : [0,\infty) \to \mathbb{R}$ be monotone-increasing. Then, it is well-known that the derivative $f'$ exists and takes nonnegative values almost everywhere on $[0,\infty)$.
Then, is it true that
\begin{equation}
f(b) - f(a) \geq \int_a^b f'(t) dt \text{ for "all" } 0 \leq a < b <\infty
\end{equation}
holds... |
Why is $\overline{z^{n}} = (\bar{z})^{n}$ true only for $n \in \mathbb{Z}$? What about a real or complex exponent in general?
With $k \in \mathbb{Z}$ (I am not choosing a branch, everything goes) I have
$$
\overline{\log z} = \overline{\ln \rho + i (\theta + 2k\pi)} = \ln \rho - i (\theta + 2k\pi) = \ln \rho + i (... |
This is probably going to be a silly question, but here it goes: assume you have a complicated derivation where some function $f(x,\lambda)$ is manipulated in various ways (for example, into something like $\int_a^b dx \ |\partial_\lambda f(x,\lambda)|^2$ ).
Suppose one is interested in the case where $\lambda$ is ... |
Let's take PA1 for example. From godel's 2nd incompleteness theorem, PA1 can't prove its own consistency, more specifically it can't prove that the largest consistent subset of the theory is PA1 itself.
But PA1 has infinite axioms, so can PA1 prove atleast for a given finite set of axioms ( of PA1 ) that they are ... |
Let's take PA1( First order axioms of peano arithmetic ) for example. From godel's 2nd incompleteness theorem, PA1 can't prove its own consistency, more specifically it can't prove that the largest consistent subset of the theory is PA1 itself.
But PA1 has infinite axioms, so can PA1 prove atleast for a given finit... |
Consider a maximin problem with $\lambda=[\lambda_1,\lambda_2,...,\lambda_n]$
$$\max_{\lambda}\min_{i\in\{1,2,3,...,M\}}{f_i(\lambda)}$$
subject to $\lambda$ is any vector locating in a standard probability simplex, that is $\sum_{i=1}^{n}\lambda_i=1$ and $\lambda_i\geq0, \forall i\in\{1,2,...,k\}$. Note that each fu... |
How to show the continuity of the following map? |
I have learned the component formula for the scalar product for quite a while, but really, it doesn't seem intuitive at all. I have 2 questions that confuse me while imagining what the scalar product really is, which are:
1- Why is the scalar product defined as $\vec{u} . \vec{v} = \sum u_{i}v_{i}$ (It doesn't make ... |
What is the correct MSD function for a biased random walk on a triangular grid? |
I've been researching some operator spaces and have stumbled upon the following problem. Suppose $x,y\in\mathcal{B}(\mathcal{H})$ (for $\mathcal{H}$ separable) such that $0\leq x\leq y\leq 1$. Further let $t\in[0,1]$ and $\rho$ be a density matrix (i.e. $\rho\geq 0$, $\mathrm{Tr}(\rho)=1$). Is it true that
$$
\mathrm... |
I have written the following algorithm pseudocode to depth-first traverse an undirected connected graph $G = (V, E)$:
```
DFS(G, v):
T = (V' = {v}, E' = empty, v)
s = [] // stack with push and pop
visited = [] // array with append and size
all_visited = [] // array
for u in {u | uv in E}:
s.push(u... |
If a point equally distant of the vertices of a regular tetrahedron, does this mean it is the center of the sphere circumscribed in that tetrahedron?
Hope one of you can help me! Thank you! |
If a point equally distant of the vertices of a regular tetrahedron, does this mean it is the center of the sphere circumscribed in that tetrahedron? |
Given some vector $x = (x_1,...,x_d)$ in $\mathbb{R}^d$ we may embed it in the tensor algebra $T(\mathbb{R}):= \prod_{n=0}^\infty (\mathbb{R}^d)^{\otimes n}$ via the tensor exponential map $\exp_\otimes$. Specifically, we have that
$$\exp_\otimes(x) := \sum_{n=0}^\infty \frac{x^{\otimes n}}{n!}:= \sum_{n=0}^\infty \f... |
How is the tensor exponential map defined over a generic Banach space? |
I'm studying modal logic and on Chapter 3 of [this book][1] one of the proposed exercises (3.40) is to prove that necessity distributes over implication, that is,
\begin{equation}
\square(\varphi \rightarrow \psi) \; \rightarrow \; (\square \varphi \rightarrow \square \psi) \;\; \; \; \;\text{axiom } (\mathbf{K})
\... |
The symmetric group is a factor of the braid group (see e.g. https://math.stackexchange.com/q/1165136/169085, https://math.stackexchange.com/q/1816878/169085). Consequently one can visualize an element in the symmetry group like a braid with no information regarding which strand is on top at a crossing (e.g. as done he... |
I'm over all a little confused about base change in general, and also a little confused about what it means to be stable under base change. Let me highlight my confusion in the context of affine morphisms.
So, let $f:X\rightarrow Y$ be an affine morphism, and $g:Z\rightarrow Y$ be another morphism between schemes $X... |
While finding eigenvalues of a particular matrix, I end up with the following: $\sqrt{p^{2\alpha}-4p^{\alpha}+8p^{\alpha-1}+4}$, where $p$ is an odd prime and $\alpha \geq 1$. The next step is to check if this eigenvalue is an integer or not. How do I go about checking the integrality of such an expression? |
I know that if $a \mid b$ and $a \mid c$ then $a \mid b+c$.
Which is easy to see since
$a \mid b \Leftrightarrow \exists k \in \mathbb{N} : a \cdot k=b$ and
$a \mid c \Leftrightarrow \exists n \in \mathbb{N}: a \cdot n=c$.
Thus $b+c=ak+an=a(n+k)$, which implies $a \mid c+b$.
But does $a \mid b$ and $a \nm... |
I’m not a mathematician and I’m working with some transforms in physical chemistry.
I use a transform to pass from the time domain of phase domain in a process that use a square wave to perturb and modulate a stimulus on a system.
In this case, the transform is
$$
A(\phi)= \int_{0}^{T} A(t)\sin(kwt + \phi)\,dt,
... |
Suppose instead of defining the derivative as
$$f'(x) = \lim_{h\to 0}\frac{f(x + h) - f(x)}{h}$$
we define it as
$$f_q'(x) = \lim_{q\to 1}\frac{f(qx) - f(x)}{qx - x}.$$
Does this alternative definition still satisfy the linear approximation characterization of the derivative, i.e. do we have
$$f(x + h) = f(x) + ... |
A differentiable and also convex function has the following property.
$$
(\nabla f (x) - \nabla f (y))^T(x-y) \geq 0
$$
I can derive it from the first order condition for a convex function,
$$
f(y) \geq f(x)+\nabla f(x)^T(y-x)
\\
f(x) \geq f(y)+\nabla f(y)^T(x-y)
$$
by adding the two inequalities and ... |
Let ABC be a triangle with angle B = 90 degrees. Let D, E be points on AB such that BD = BC and AE = AC. Let F, G be the feet of perpendicular from E and F onto BC and AB respectively. If EF = 12 And DG = 11, what is DE?
While there are a ton of similar triangles, using their ratios did not lead to the nicest of equ... |
Suppose instead of defining the derivative as
$$f'(x) = \lim_{h\to 0}\frac{f(x + h) - f(x)}{h}$$
we define it as
$$f_q'(x) = \lim_{q\to 1}\frac{f(qx) - f(x)}{qx - x}.$$
Does this alternative definition still satisfy the linear approximation characterization of the derivative, i.e. do we have
$$f(x + h) = f(x) + ... |
The question is :$u,v\in\mathbb{R}^2$, prove $f(u,t)=\int_{\mathbb{R}^2}\left[\nabla^\perp\ln||u-v||\right]p(v)dv$ (Here $\nabla$ is gradient operator and the vector $\mathbf{a}^\perp$ is obtained by rotating vector $\mathbf{a}$ by 90 degrees.) satisfies the condition
$\exists C>0, \forall u_1,u_2\in\mathbb{R}^2,t\g... |
How to prove this function is Lipschitz? |
**Background**
Let $\zeta(\cdot)$ be the <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>. I'm looking for a visual proof of the infinite series identity $$\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1. \tag{1}\label{1}$$
This identity arises in the study of <a href="htt... |
$u,v\in\mathbb{R}^2$, $p(v)$ is bounded and non-negative, and $\int_{\mathbb{R}^2}p(v)dv=1$, prove $f(u,t)=\int_{\mathbb{R}^2}\left[\nabla^\perp\ln||u-v||\right]p(v)dv$ (Here $\nabla$ is gradient operator ,the vector $\mathbf{a}^\perp$ is obtained by rotating vector $\mathbf{a}$ by 90 degrees) satisfies the condition
... |
Let $G$ be a group and $g \in G$. Assume that $g$ is conjugate to $g^2$. Prove that $g^{-1}$ is conjugate to $g^{-4}$.
My attempt is to use the definition and manipulate the given equation.
Given that $g$ is conjugate to $g^2$, there exists an $x \in G$ such that: $g=xg^{2}x^{-1}$.\
I take the inverse:\
$g^{-1}=(... |
Let ABC be a triangle with angle B = 90 degrees. Let D, E be points on AB such that BD = BC and AE = AC (they lie in the order B-E-D-C). Let F, G be the feet of perpendicular from E and F onto BC and AB respectively. If EF = 12 And DG = 11, what is DE?
While there are a ton of similar triangles, using their ratios d... |
**Background**
Let $\zeta(\cdot)$ be the <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>. I'm looking for a visual proof of the infinite series identity $$\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1. \tag{1}\label{1}$$
This identity arises in the study of <a href="htt... |
Let ABC be a triangle with angle B = 90 degrees. Let D, E be points on AB such that BD = BC and AE = AC (they lie in the order B-E-D-C). Let F, G be the feet of perpendicular from E and D onto BC and AB respectively. If EF = 12 And DG = 11, what is DE?
While there are a ton of similar triangles, using their ratios d... |
Let $G$ be a group and $g \in G$. Assume that $g$ is conjugate to $g^2$. Prove that $g^{-1}$ is conjugate to $g^{-4}$.
My attempt is to use the definition and manipulate the given equation.
Given that $g$ is conjugate to $g^2$, there exists an $x \in G$ such that: $g=xg^{2}x^{-1}$.\
I take the inverse:\
$g^{-1}=(... |
**Background**
Let $\zeta(\cdot)$ be the <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>. I'm looking for a visual proof of the infinite series identity $$\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1. \tag{1}\label{1}$$
This identity arises in the study of <a href="htt... |
While working on a project, I came across [a paper that includes this sum on page 15][1] as the definition of a support function $r$ for a surface with tetrahedral symmetry:
$$
r(ξ, \bar{ξ}) = \frac{1}{\#} \sum_{g ∈ } r_0\bigl(g(ξ), g(\bar{ξ})\bigr)
$$
where
- $$ is "the tetrahedral group", "a discrete subgro... |
Let ABC be a triangle with angle B = 90 degrees. Let D, E be points on AB such that BD = BC and AE = AC (they lie in the order B-E-D-A). Let F, G be the feet of perpendicular from E and D onto BC and AB respectively. If EF = 12 And DG = 11, what is DE?
While there are a ton of similar triangles, using their ratios d... |
Let $f$ be a $\rho$-lipschitz smooth function, meaning that
$$
||\nabla f(x) - \nabla f(y)|| \leq \rho ||x-y||, \forall x,y \in dom(f)
$$
Is following inequality true?
$$
\nabla f(x) \leq \nabla f(y) + \rho ||x-y||
$$
How do I remove norm from the inequation and show that it is true?
It comes to me whe... |
Find three non equivalent topologis on infinite set.
For case infinite countable set we can use [this][1] answer for rational numbers.
Also for example in $ \mathbb {R}$ we can theree non equivalent metric : discrete metric, euclidean metric and $p-adic $ metric.
But two infinite uncountable set don't hav... |
Let $G$ be a group and $g \in G$. Assume that $g$ is conjugate to $g^2$. Prove that $g^{-1}$ is conjugate to $g^{-4}$.
My attempt is to use the definition and manipulate the given equation.
Given that $g$ is conjugate to $g^2$, there exists an $x \in G$ such that: $g=xg^{2}x^{-1}$.\
I take the inverse:\
$g^{-1}=(... |
I encountered this problem studying the local period map and I'm wondering how to solve it.
I would like to prove that, given $V$ a complex vector space and $W \subseteq V$ a one-dimensional subspace, the tangent space of $\mathbb{P}(V)$ at $W$ is
$$T_W\mathbb{P}(V)=\text{Hom}(W,V/W).$$
Where do I start and what... |
Find three non equivalent metrics on infinite set.
For case infinite countable set we can use [this][1] answer for rational numbers.
Also for example in $ \mathbb {R}$ we can theree non equivalent metric : discrete metric, euclidean metric and $p-adic $ metric.
But two infinite uncountable set don't haven... |
I've been researching some operator spaces and have stumbled upon the following problem. Suppose $x,y\in\mathcal{B}(\mathcal{H})$ (for $\mathcal{H}$ separable) such that $0\leq x\leq y\leq 1$. Further let $t\in[0,1]$ and $\rho$ be a density matrix (i.e. $\rho\geq 0$, $\mathrm{Tr}(\rho)=1$). Is it true that
$$
\mathrm... |
I have two sets: $I=\{1,2,..,n\}$ and $J=\{1,2,..,m\}$ and a set of subsets as $P=\{p_1, p_2, .., p_d\}$ that each member of $P$ is a subset of $I$, for example, $p_d=\{3,4\}$.
This is my optimization problem where $x_j$ is the decision variable and $z_{ij}$, $LC_i$, $T_{p_d}$, $U_j$ and $c_j$ are the parameters (cons... |
Below, the notation $\max(t,0)\equiv \max(t)$ is used, and the function $\max$ is seen as a distribution. From the theory of distributions, we have that $\max'(t)=H(t)$ (in the distributional sense) where $H$ is the Heaviside distribution. Now, I am considering the one-parameter distribution $\max(at)$ where $a$ is the... |
I’m in 6th grade and this is one question I have about infinite function compositions:
I recently thought about function composition; things in the form $a_1(a_2(…(a_n(x))…)$ which are things like $e^{2x}$ or $\sin(\sin(x^2))$.
After that, I thought about infinite function compositions of the same function; $f(f(... |
Does an infinite composition of functions converge? Can we determine it? |
Can we find three random variables $X_1, X_2, X_3$ such that $E[X_2 \mid X_1] = X_1$ and $E[X_3 \mid X_2] = X_2$ but $E[X_3 \mid X_1, X_2] \neq X_2$?
I'm stuck on how to choose such random variables; in particular how to define $X_3$ in relation to the other two. At first, I was thinking of setting $X_n = \frac{S_n}... |
I'm studying first order ODEs, and I have the following theorem:
> Let
$$
\left\{
\begin{aligned}
y' &= f(y, t) \\
y(t_0) &= y_0
\end{aligned}
\right.
\tag{1}
$$
with $(t_0, y_0)\in D\subset\mathbb{R}$ and $D$ an open set, be a Cauchy problem for a first order ODE, for which existence and unicity of th... |
I am working with the same problem and have a follow up question to an old solution: https://math.stackexchange.com/a/1842770/1100158. Here they use Fubini's, but nowhere does it say that the random variables, $Z_1$ and $Z_2$, have finite expectation. Is there a reason why $E(Z_1I_{Z_1+Z_2\in A})$ is finite for any bor... |
Why is $\overline{z^{n}} = (\bar{z})^{n}$ true only for $n \in \mathbb{Z}$? What about a real or complex exponent in general?
Let $$\log z=\{\ln \rho + i (\theta + 2k\pi)\mid k\in\Bbb Z\}$$
then
$$
\overline{\log z} = \overline{\ln \rho + i (\theta + 2k\pi)} = \ln \rho - i (\theta + 2k\pi) = \ln \rho + i (-\the... |
I've been struggling with this question, which would probably be more appropriate to ask on Physics Stack Exchange which I've already done, but am hoping for some guidance here too.
A hemispherical bowl of inner radius $r$, density $ϕ$ and thickness $t$ is floating in a fluid of density $ρ$. The bowl is being filled... |
I want to know if a given set of finite axioms *in* PA1 ( 1st order peano arithmetic ) can prove infinite sentences in PA1 such that those proofs need no other axioms except those in the given finite set ? <br/>
Since PA1 can prove the consistency of every given finite set of its axioms, so I think there should be onl... |
Let $K$ be a p-adic field that is a finite extension of $\Bbb{Q}_p$.
Let $E/K$ be an elliptic curve over $K$.
Let $E(K)$ be Mordel Weil group of $E/K$.
Let $\widehat{E(K)}$ be a profinite completion of $E(K)$, that is, completion with respect to finite index subgroups.
Is it true that $E(K)=\widehat{E(K)}$ ?... |
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