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### Question:
Let $(A_k)_{k \in \mathbb{N}}$ a sequence of events. Prove that if $P(A_k)$ does not converge to $0$ then $\exists$ an event belonging to an infinity of $A_k$
### My answer:
1-Let writte $A'_1=A_1 \cup A_2 ..., A'_2=A_2 \cup A_3,..., A'_n=A'_n \cup A_{n+1} \cup ...$
Now we ca note that $A'_1 \cap A... |
Claim: If for any $\epsilon >0$, there exists a $\delta >0$, such that for any |x-x_0| < $\delta$, it holds
$$
(L-\epsilon)^k \leq f(x) \leq (L+\epsilon)^k,
$$
where $k$ is a positive number. Then we have
$$
\lim_{x \rightarrow x_0} f(x) =L^k.
$$
-------------------------------------
The following is ... |
Question:
> As shown in the diagram, in isosceles $\triangle ABC,\ AB=AC.$ $H$ is a point on $AC$, take points $E$ and $F$ in turn on the extension line of $BC$, and take point $BD$ on the extension line of $CB$, so that $EF=DB$, make $EG // AC$ across point $E$ to cross the extension line of $DH$ at point $G$, co... |
This paragraph is at the end of Section 6.3 in the book *Testing Statistical Hypotheses* by Lehmann and Romano:
>In most applications, $M(x)$ is a measurable function taking on values in a
Euclidean space and it is convenient to take $\mathcal B$ as the class of Borel sets. If $\phi(x) = \psi[M(x)]$ is then an arbi... |
Let $X$ be a (not reduced) connected curve (over a base-field $k$), and $X_{red}$ be the corresponding reduced curve (in the sense of Liu, 2.4.2).\
It if known (Liu, 3.3.21), that $\mathcal{O}_{X_{red}}(X_{red})$ is a finite field extension of $k$.
Furthermore (Liu, 2.4.2), $\mathcal{O}_X(X) / \sqrt{0_{\mathcal{O}_X(... |
Here, $R$ is a commutative ring with unity. I realize there are many proofs of this online, but all of them seem to assume a definition of zero divisors that Dummit and Foote does not. Here is what the definition is in Dummit and Foote:
If $a \in P$ and $b \in P$, then $ab = 0$ implies either $a = 0$ or $b = 0$. The... |
Claim: If for any $\epsilon >0$, there exists a $\delta >0$, such that for any $|x-x_0| < \delta$, it holds
$$
(L-\epsilon)^k \leq f(x) \leq (L+\epsilon)^k,
$$
where $k$ is a positive number. Then we have
$$
\lim_{x \rightarrow x_0} f(x) =L^k.
$$
-------------------------------------
The following is ... |
Let $\mathcal{C}$ be a symmetric monoidal category and let $A$ and $A^*$ be dual in the sense of [Definition 2.1][1] in nLab.
[Dold & Puppe (1984)][2] show (Thm 1.3)
that the map
$$ \text{Hom}(X, Y \otimes A^{\ast}) \xrightarrow{(-) \otimes A} \text{Hom}(X \otimes A, Y \otimes {A}^{\ast} \otimes A) \xrightarrow{... |
Why does duality of objects $A$, $A^\ast$ in a symmetric monoidal category imply an adjunction $(-) \otimes A \dashv (-) \otimes A^\ast$? |
If I sample $N+1$ integers $x, x_1, \ldots, x_N$ uniformly and independently from $\{1, \ldots, M\}$, what is the probability that $x$ contains a prime divisor that does not divide any of the $\{x_i\}$? Upper/lower bounds and/or asymptotic approximations would also be of interest.
For prime $p$ and $x, x_1, \ldots, ... |
This paragraph is at the end of Section 6.3 in the book *Testing Statistical Hypotheses* by Lehmann and Romano:
>In most applications, $M(x)$ is a measurable function taking on values in a
Euclidean space and it is convenient to take $\mathcal B$ as the class of Borel sets. If $\phi(x) = \psi[M(x)]$ is then an arbi... |
I know that:
* If $f$ is continuous and monotonous on $[a, b]$, then $$f([a, b]) = [\min(f(a), f(b)), \max(f(a), f(b))]$$
* If $f$ is continuous and unimodal on $[a, b]$ with an extremum at $x_0 \in [a, b]$, then $$f([a, b]) = [\min\{f(a), f(x_0), f(b)\}, \max\{f(a), f(x_0), f(b)\}]$$
* If $f=\cos$ or $f=\sin$, th... |
Under what assumptions on f can I calculate f([a, b])? |
Assume that $GL(\mathcal{H})$ is the group of invertible linear transformations of Hilbert space $\mathcal{H}$ and $U(\mathcal{H})$ is the group of unitary operators on $\mathcal{H}$. We know that $\mathbb{C}^*$ and $U(1)$ act respectively on $GL(\mathcal{H})$ and $U(\mathcal{H})$.
Is $GL(\mathcal{H})/\mathbb{C}^* ... |
Is $GL(\mathcal{H})/\mathbb{C}^* \cong U(\mathcal{H})/U(1)$? |
Does the epsilon delta definition of a limit always hold? I was thinking about the function $$f(x)=\begin{cases}
1 & x= 0 \\
0 & x\neq 0
\end{cases}
$$
and it seems that $$\lim_{x \to 0} f(x) = 0$$, however unless I'm mistaken this would not converge to $0$ (or even at all) via the epsilon delta ... |
Consider $F(x) = \int_0^{x^2} \frac{t^2+1}{t^4+2t^2+4}dx$. I want to study
$$
\lim_{x\to\infty} F(x), \quad \lim_{x\to\infty} \frac{F(x)}{x}
$$
Ideas: First, we can observe than $f(t) = \frac{t^2+1}{t^4+2t^2+4}$ is a continuous function over $\mathbb{R}$, so $G(x) = \int_0^{x} \frac{t^2+1}{t^4+2t^2+4}dx $ is wel... |
Does every partially ordered commutative ring admit an order preserving homomorphism to a real closed field? |
The number of r-element subsets of a set of order n is given by $_nC_r$, with is equivalent to the factorial expression $$\frac{n!}{r!(n-r)!}$$ Well, whenever I see a factorial I get the urge to analytically continue it with the gamma function, which would make the expression above $$\frac{Γ(n+1)}{Γ(r+1)Γ(n-r+1)}$$ Whi... |
Analytic continuation of the order of a set: -1/2 element sets? Sets with an imaginary number of elements? |
Consider $F(x) = \int_0^{x^2} \frac{t^2+1}{t^4+2t^2+4}dt$. I want to study
$$
\lim_{x\to\infty} F(x), \quad \lim_{x\to\infty} \frac{F(x)}{x}
$$
Ideas: First, we can observe than $f(t) = \frac{t^2+1}{t^4+2t^2+4}$ is a continuous function over $\mathbb{R}$, so $G(x) = \int_0^{x} \frac{t^2+1}{t^4+2t^2+4}dt $ is wel... |
An algebra has one of four unification types - unitary, finitary, infinitary or nullary. A variety $\mathbf{V}$ has type unary, if every member has unary type, finitary, if every member has finitary type, infinitary, if every member has infinitary type, and nullary, if there exists an algebra in $\mathbf{V}$ with nulla... |
Does every variety of locally finite closure algebras have a unification type? |
The number of r-element subsets of a set of cardinality n is given by $_nC_r$, with is equivalent to the factorial expression $$\frac{n!}{r!(n-r)!}$$ Well, whenever I see a factorial I get the urge to analytically continue it with the gamma function, which would make the expression above $$\frac{Γ(n+1)}{Γ(r+1)Γ(n-r+1)}... |
Analytic continuation of the cardinality of a set: -1/2 element sets? Sets with an imaginary number of elements? |
# Question: Calculating Average Variance in a Gaussian Mixture Model
I am working with a Gaussian Mixture (GM) model within the framework of the Expectation-Maximization (EM) algorithm. My goal is to estimate a set of variables, and I am currently facing a challenge with calculating the average variance of the poste... |
How to Calculate the Average Variance of P_{S|XZ} with Respect to P_{Z|X}? |
I have a logical satement of the form:
For every number $a≥0$, there exist two prime numbers $p$ and $p+2$ such that $$a+4<p<2^{a+4}$$ and $$a+4<p+2<2^{a+4}$$
Then how one can negate this statment?
**My solution**:
**There exist** a number $a≥0$ such that **for every** natural number $d$ with $$a+4<d<2^{a... |
How one can negate this statement? |
The number of r-element subsets of a set of cardinality n is given by $_nC_r$, with is equivalent to the factorial expression $$\frac{n!}{r!(n-r)!}$$ Well, whenever I see a factorial I get the urge to analytically continue it with the gamma function, which would make the expression above $$\frac{Γ(n+1)}{Γ(r+1)Γ(n-r+1)}... |
The population proportion parameter is an estimator? |
I read in one russian book about complex vectors. There was formula for product of two complex vectors. I was confused because I didn't find what this formula is.
[![enter image description here][1]][1]
$a$ and $b$ are vectors not numbers.
$(a_1,a_2)$ and so on is scalar product of two vectors.
What this form... |
I have a logical satement of the form:
For every number $a≥0$, there exist two prime numbers $p$ and $p+2$ such that $$a+4<p<2^{a+4}$$ and $$a+4<p+2<2^{a+4}$$
Then how one can negate this statment?
**My solution**:
**There exist** a number $a≥0$ such that **for every** natural number $d$ with $$a+4<d,d+2<... |
Can you explain how to convert this notation to a function to someone unfamiliar with group theory? |
While I was reading *Norman Biggs' Algebraic Graph Theory* I came across the following (Page 12 **2g**) exercise/additional result:
> Let $g_{ij}(r)$ denote the number of walks of length $r$ in $\Gamma$ from $v_i$ to $v_j$. If we write $\mathbf{G}(z)$ for the matrix
$$
\mathbf{G}(z)_{ij}=\sum_{r=1}^{\infty} g_{ij... |
I'm studying first order ODEs, and I have the following theorem: "Let $(1)\left\{
\begin{array}{c}
y'=f(y, t) \\
y(t_0)=y_0
\end{array}
\right.$ 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 the solution is granted in a ... |
While working on a project, I came across a paper that includes this definition: $g ∈ $, where $$ is the "tetrahedral group" ("a discrete subgroup of isometries $ ⊂ O(3)$"). The paper then uses $g(z)$ as a function with a complex input, where $z$ is "the local complex coordinate on the unit 2-sphere in $^3$ obtained by... |
While I was reading *Norman Biggs' Algebraic Graph Theory* I came across the following (Page 12 **2g**) exercise/additional result:
> Let $g_{ij}(r)$ denote the number of walks of length $r$ in $\Gamma$ from $v_i$ to $v_j$. If we write $\mathbf{G}(z)$ for the matrix
$$
\mathbf{G}(z)_{ij}=\sum_{r=1}^{\infty} g_{ij... |
At the end of [this question](https://math.stackexchange.com/a/1281982/114910) a user states
> A function is recursive primitive iff the order needed to prove the induction is at most $\omega$.
Intuitively this makes sense, but is it true? Ackerman is not primitive recursive, which implies the order needed is n... |
Let $K$ be a local field and let $\widehat{K^*}$ and $\widehat{\mathbb{Z}}$ denote the profinite completions of $K$ and $\mathbb{Z}$. As the title suggests I'm having difficulties understanding the isomorphism ${\mathcal{O}^*_K} \times \widehat{\mathbb{Z}} \cong \widehat{K^*}$ that is given by $(u, z) \mapsto u \pi^z$ ... |
If I sample $N+1$ integers $x, x_1, \ldots, x_N$ uniformly and independently from $\{1, \ldots, M=2^k\}$, what is the probability that $x$ contains a prime divisor that does not divide any of the $\{x_i\}$? Upper/lower bounds and/or asymptotic approximations would also be of interest.
For prime $p$ and $x, x_1, \ldo... |
Let $a$, $b$, $c$,$d$, $e$, $f$ distinct complex number on the *same circle*. Then the chords $ab$, $cd$, $ef$ are concurrent ( possibly at infinity) if and only if
$$
\det\left( \begin{matrix}
1&a+b&ab\\
1&c+d&cd\\
1&e+f&ef
\end{matrix}\right)=0
$$
The condition is invariant under complex affine transforma... |
Let $a$, $b$, $c$,$d$, $e$, $f$ distinct complex number on the *same circle*. Then the chords $ab$, $cd$, $ef$ are concurrent ( possibly at infinity) if and only if
$$
\det\left( \begin{matrix}
1&a+b&ab\\
1&c+d&cd\\
1&e+f&ef
\end{matrix}\right)=0
$$
The condition is invariant under complex affine transforma... |
## Background
In my exploration of Gaussian Mixture Models (GMMs) within the scope of statistical learning, I have encountered the concept of entropy as a measure of uncertainty or randomness in a probability distribution. For single Gaussian distributions, the calculation of entropy is straightforward and is known ... |
### The Given Approach
The solution given in the question makes a mistake in the first step. It appears that the solution implies that
$$ 2^{\sin(x)} = \mathrm{e}^{-\sin(x)}, $$
which is not correct. Everything after this is based on the faulty assumption, which implies that the given approach is not going to w... |
### Question:
Let $A_1,A_2,...,A_n \subset \Omega $ a sequence of outcome such that $P(A_i) \geq c $ for all $i$.
Prove that it exists $i \neq j$ s.t. $P(A_i \cap A_j) \geq \frac{nc^2-c}{n-1}$
I have tried different ways including the following one but I did not succeed to conclude.
1-First note th... |
Consider an operator (not necessarily linear) $\mathcal{O}: C^{\infty} (\mathbb{R}) \to C^\infty(\mathbb{R})$ which satisfies the following property for any function $f \in C^\infty(\mathbb{R})$ and any constant function $c \in C^\infty(\mathbb{R})$
\begin{equation}
\mathcal{O} \left(f + c \right) = \mathcal{O} \left... |
Must an operator $\mathcal{O}$ satisfying $\mathcal{O} (f + c) = \mathcal{O} (f)$ for any constant function $c$ be a differential operator? |
I read in one russian book about complex vectors. There was formula for product of two complex vectors. I was confused because I didn't find what this formula is.
$$
\begin{aligned}
\boldsymbol{z}_{1} \boldsymbol{z}_2
&= (\boldsymbol{a}_1 + i\boldsymbol{b}_1) (\boldsymbol{a}_2 + i\boldsymbol{b}_2) \\
&= \bigl[ (\... |
Let $E/k$ and $E'/k$ be isogenous over $k$.
We know:
- $E$ and $E'$ are isogenous if and only if $\# E(k) = \# E'(k)$ [Ex V.5.4, Sil86].
- Any non-constant morphism of curves is surjective [Thm. II.2.3, Sil86].
- The kernel of an isogeny can have more that one points in it, which means that the map may not be ... |
While working on a project, I encountered [a paper that includes this sum][1]:
$$
r(ξ, \bar{ξ}) = \frac{1}{\#} \sum_{g ∈ } r_0(g(ξ), g(\bar{ξ}))
$$
where
- $$ 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^2)^3} + c... |
What do I do with a single-variable function that is called with two inputs? |
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’m trying to understand the concept of colimits for commutative rings, but unable to find a colimit(or at least a compliment) for a finite diagram of rings, is there a(non trivial) example for a compliment(and a colimit as a universal compliment) for a 2-3 rings diagram? |
The principle of the theory of the tangent to of a curve is that it is the limit of secant when the two intersection points are equal. I intend to prove that for every function $f$ differentiable on an interval $I$ and a number $a$ from that interval, the equation : $f’(a)(x-a)+f(a)=f(x)$ accepts $a$ as a double soluti... |
There are 5 people sitting on 5 chairs arranged in a straight line, all facing north. Everyone gets up and can do only one of the following things at a time.
(i) Sit back again in their original chair.
(ii)Turn their chair 180◦ to face south and then sit on it.
(iii) Sit on the chair next to their original ... |
Why this configuration is not possible for the Tower of Hanoi? |
After solving some exercises in algebra, basic information on field extensions, I have a question.
It's easy to prove that $\mathbb{Q}(\sqrt{2})\cap\mathbb{Q}(\sqrt{3})=\mathbb{Q}$. Also it can be proven that $\mathbb{Q}(\sqrt[n]{2})\cap\mathbb{Q}(\sqrt[n]{3})=\mathbb{Q}$ for all $n\ge 2$.
How can it be generaliz... |
Consider a fair lottery among a countably infinite number of people. The Axiom of Countable Additivity says this is impossible to construct: If all people have a positive (and equal) probability of winning, the sum of their probabilities of winning would diverge, while if all people have a $0$ probability of winning, t... |
Is the Axiom of Choice inconsistent with Countable Additivity? |
On [this article](https://www.ams.org/journals/proc/1964-015-06/S0002-9939-1964-0169038-4/) and Stein & Weiss (statement 1) and the books by Linares & Ponce and Duoandikoetxea (statement 2), I found the following statements of the Marcinkiewicz interpolation theorem:
1. If $T:L^{p_0}+L^{p_1}\to L_w^{q_0}+L_w^{q_1}$ ... |
Pardon me for the homework-like question, I'm just trying to better understand how we go about using infinite descent.
### Problem Statement ###
Suppose we have rational functions $f,g \in \mathbb C[t]$. We want to prove that these are constant if $f^4(t)+g^4(t) = 1$, and we want to do so utilizing infinite descen... |
I came across a paper describing how it builds a graph based on the Waxman model ([link to the probability function Waxman gives][1]) but I was wondering what the probability threshold is for determining if an edge exists between two nodes. Is it strictly a binary yes or no in terms of whether an edge exists, and if so... |
What is the probability threshold for including an edge using the Waxman model? |
The principle of the theory of the tangent to of a curve is that it is the limit of the secant when the two intersection points are equal, I mean in Calculus I. I intend to prove that for every function $f$ differentiable on an interval $I$ and a number $a$ from that interval, the equation : $f’(a)(x-a)+f(a)=f(x)$ acce... |
Pardon me for the homework-like question, I'm just trying to better understand how we go about using infinite descent.
### Problem Statement ###
Suppose we have rational functions $f,g \in \mathbb C[t]$. We want to prove that these have degree 0 if $f^4(t)+g^4(t) = 1$, and we want to do so utilizing infinite desce... |
It is well known that $SL_2(\mathbb{C})$ is the universal cover of $SO^+(1,3)$, see for example the Wikipedia page on the Lorentz group [1]. The map goes like this (some of this is not standard notation):
First, establish a bijection between $\mathbb{R}^{1,3}$ and the set $M_{s.a.}(2,\mathbb{C})$ of self-adjoint 2... |
Pardon me for the homework-like question, I'm just trying to better understand how we go about using infinite descent.
### Problem Statement ###
Suppose we have rational functions $f,g \in \mathbb C[t]$. We want to prove that these have degree 0 if $f^4(t)+g^4(t) = 1$, and we want to do so utilizing infinite desce... |
Pardon me for the homework-like question, I'm just trying to better understand how we go about using infinite descent.
### Problem Statement ###
Suppose we have rational functions $f,g \in \mathbb C[t]$. We want to prove that these have degree 0 if $f^4(t)+g^4(t) = 1$, and we want to do so utilizing infinite desce... |
Refer to [ Turing's 1936 paper][1], page 248, last paragraph. I present the paragraph in verbatim below :
> The expression "there is a general process for determining..." has been used throughout this section as equivalent to "there is a machine which will determine ... ". This usage can be justified if and only if ... |
In their paper [The Maximum of independent Geometric Random Variables as the Time for Genomic Evolution][1]the authors noted that if to consider the genomic word of L letters, than the measure of the time it takes to obtain a successful genomic word $R_L=max{X_1, \ldots, X_L}$. Where $X_i, i=1, \ldots, L$ are independe... |
Consider a fair lottery among a countably infinite number of people. The Axiom of Countable Additivity says this is impossible to construct: If all people have a positive (and equal) probability of winning, the sum of their probabilities of winning would diverge, while if all people have a $0$ probability of winning, t... |
Given a compact Hausdorff space $X$ we can construct its topological K-theory. On the other hand the continuous functions on $X$ give rise to a C*-algebra $C(X)$ and we can consider the operator K-theory of this C*-algebra. My question is how these two are related to each other.
By Swan's theorem we have $K^0(X)\con... |
Exercise 14.7.4 from Dummit and Foote
>Let $K=\mathbb{Q}(\sqrt[n]{a})$, where $a\in \mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$ (i.e., $x^n-a$ is irreducible over $\mathbb Q$). Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d$. Prove that $E=\mathbb{Q}(\sqrt[d]{a})$. [Consider $ N_{K/E}(\sqrt[n]a)\in E... |
Refer to [ Turing's 1936 paper][1], page 248, last paragraph. I present the paragraph in verbatim below :
> The expression "there is a general process for determining..." has been used throughout this section as equivalent to "there is a machine which will determine ... ". This usage can be justified if and only if ... |
>Let $K=\Bbb Q(\sqrt[n]{a})$, where $a \in \Bbb Q^+$ and suppose $[K:\Bbb Q]=n$ (i.e., $x^n-a$ is irreducible over $\mathbb Q$). Let $E$ be any subfield of $K$ and let $[E:\Bbb Q]=d$. Prove that $E=\Bbb Q(\sqrt[d]{a})$.
Now $N_{K/F}(\sqrt[n]{a})=a\in \Bbb Q$. I also know that $N_{K/E}(\sqrt[n]{a})\in E$. What I hav... |
Exact wording of my question is a bit oxymoronic, since a norm by definition is a metric, and thus requires proper context. Let $H$ be a separable Hilbert space over the field $\mathbb{K}$. I am aware that a bounded linear operator $A\in\mathcal{B}(H)$ is said to be of *trace class* if $\mathrm{Tr}(|A|) < \infty$ for $... |
How can you show that the trace class norm $\|A\|_1:=\mathrm{Tr}(|A|)$ satisfies triangle inequality? |
Let $1 \le n_1 < n $ and $ n\ge 3$ be integers and let ${\bf \lambda}= \left( \lambda_j \right)_{j=1}^n \in {\mathbb R}$ such that $\lambda_j > 0 $ for $j=1,\cdots, n_1$ and $\lambda_j <0 $ for $j=n_1+1, \cdots, n$ and such that $1/{\bf \lambda}$ is ordered in descending order. In the following we will assume that $... |
On [this article](https://www.ams.org/journals/proc/1964-015-06/S0002-9939-1964-0169038-4/) and Stein & Weiss (statement 1) and the books by Linares & Ponce and Duoandikoetxea (statement 2), I found the following statements of the Marcinkiewicz interpolation theorem:
1. If $T:L^{p_0}+L^{p_1}\to L_w^{q_0}+L_w^{q_1}$ ... |
Let $f: D \subset \mathbb{R} \rightarrow \mathbb{R}$ continuous with $f(D) \subset D$ and let $x_0$ be a $n$-periodic atractor point of $f$. Show that every point of the cycle $\{ x_0, f(x_0),\dots,f^{n-1}(x_0) \}$ is also an $n$-periodic atractor point of $f$.
I tried using the definition of continuity in \mathbb{R... |
> If $R$ is a ring with identity and $A_R$, ${}_{R}B$ are unitary $R$-modules, then there are $R$-module isomorphisms $$A\otimes_R R\cong R \text{ and } R\otimes_R B\cong B$$
>
> **Sketch of proof:** Since $R$ is an $R\text{-}R$ bimodule $R \otimes_R B$ is a left $R$-module by Theorem 5.5. The assignment $(r,b) \maps... |
Exact wording of my question is a bit oxymoronic, since a norm by definition is a metric, and thus requires proper context. Let $H$ be a separable Hilbert space over the field $\mathbb{K}$. I am aware that a bounded linear operator $A\in\mathcal{B}(H)$ is said to be of *trace class* if $\mathrm{Tr}(|A|) < \infty$ for $... |
Is $x^n-a$ irreducible over $\Bbb Q(\zeta_n)$? |
Let $1 \le n_1 < n $ and $ n\ge 3$ be integers and let ${\bf \lambda}= \left( \lambda_j \right)_{j=1}^n \in {\mathbb R}$ such that $\lambda_j > 0 $ for $j=1,\cdots, n_1$ and $\lambda_j <0 $ for $j=n_1+1, \cdots, n$ and such that $1/{\bf \lambda}$ is ordered in descending order. In the following we will assume that $... |
How can you show that the trace class norm $\|A\|_1:=\mathrm{Tr}(|A|)$ satisfies the triangle inequality? |
I have encountered an expression in my studies and am trying to determine if it correctly represents the conditional probability ($ P_{Z|X}(z|x) $). The expression is as follows:
$$
\sum_{s \in S} P_S(s) P_{Z|S,X}(z|s,x)
$$
In this expression, ($ P_S(s) $) seems to denote the probability of \( s \) in set \( S ... |
Is This Summation Expression Equivalent to the Conditional Probability \( P_{Z|X} \)? |
Let $f$ be:
\begin{cases}
0 \space\space if\space\space x \in \mathbb Q \\
1\space\space if \space\space x \notin \mathbb Q \\
\end{cases}
The question wants me to find any values $a$ such that $f$ is continuous at $x = a$. This was my proof:
Suppose $f$ is continuous, so $\lim\limits_{x\to ... |
Is This Summation Expression Equivalent to the Conditional Probability ($ P_{Z|X} $)? |
[here][1] in my answer I got real part for [polylogarithm function][2] at $1+i$ for natural $n$
$$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$
where
$$ B_n=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} \frac{2\eta(2k)}{(n-2k)!}\operatorname{Re}\left[\left(\frac{\ln2}{2}-\frac{3\pi}{4}i\r... |
[![enter image description here][1]][1]
In the following electrical circuit the textbook says it is non-planar. I was curious from a mathematics perspective on how to approach this and found Kuratowski's Theorem about having a K5 or K3,3 subgraph if it is non-planar. Can someone point out where the K3,3 subgraph i... |
Ok, I made an attempt, but it might be wrong:
Existence of $v \in V / m_f=m_{f,v}$:
- For any $v \in V, m_{f, v}$ divides $m_f$, since if $m_f(f)=0$, then $m_f(f)(v)=0 \ \forall \ v \in V$, implying that $m_{f, v}$, which annihilates $v$, must divide $m_f$.
- The primary decomposition theorem states that $V$ ... |
The original proof by Meyers and Serrin does not need boundedness.
https://www.semanticscholar.org/paper/H-%3D-W.-Meyers-Serrin/e7a5f41b547afe5b8f38860436446a123c495738?utm_source=direct_link |
In section 3(Page 50) of
https://web.archive.org/web/20170922213351id_/http://homepages.mcs.vuw.ac.nz/~bdkim/Parity.pdf,
it is mentioned that the Kummer map, $\hat{E}(\mathfrak{m}_n)\rightarrow H^{1}(k_n,T)$ along with the Weil pairing induces a cup product of Galois cohomology groups. How is the Kummer map defin... |
The Lah numbers satisfy the following rising-falling factorials relation
\begin{equation}
x^{\overline{n}} = \sum_{k=0}^{n} L(n, k) x^{\underline{k}},
\end{equation}
where $x^{\overline{n}}$= $x(x+1)\times ... \times (x+n-1)$ and $x^{\underline{n}}$ = $x(x-1) \times ... \times (x-n+1)$.
What would be the p... |
I used boolean algebra to simplify an expression with 3 variables. After simplifying, it reduced to 2 variables. The first truth table has 8 rows and the second one has 4. How to prove they're logically equivalent?
Considering + as or, . as and and ~ as not.
The original expression is:
F = ~x (xy + ~(x + y) + ... |
How to prove two tables are logically equivalent if they have different numbers of variables? |
Following this related [Example of tensor (0,2)
acting on two vectors][1] I want to ask why we consider the map $(f\otimes g)\colon V\otimes W \to X\otimes Y$ and not $(f\otimes g)\colon V\times W \to X\otimes Y$
as $(f\otimes g)(a,b)=f(a)\otimes g(b)$ ?
Why we need $V\otimes W$ here ? A lot of times I see the ... |
It's easy to see that the derivative of $detA$ is $adj(A)^T$ applied to $A(t)$ we have $adj(A(t))^T$ and the derivative of $A(t)$ is simply $A'(t)$.
Now "multiplying" all this together with inner product ($tr$) gives $tr(adj(A)A'(t))$ which is what was desired.
I haven't seen this proof so far so I'm wondering if... |
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