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**Starting out with the wrong characterization of the formula you present seems to have set you on the wrong path.** Your first formula gives the *probability* that a random set of five distinct cards from a 52-card Poker deck will meet the criteria for "two pairs" but not for any better hand. That is, that there wil... |
I don't understand how to find the inverse element to 7 in the residue ring modulo 449. According to Euclid's algorithm, I get -64, although the answer should be the number 385. But at the same time -64 + 449 = 385. But how do I get 385 initially???? Tell me please |
I don't understand how to find the inverse element to $7$ in the residue ring modulo $449$. According to Euclid's algorithm, I get $-64$, although the answer should be the number $385$. But at the same time $-64 + 449 = 385$. But how do I get $385$ initially? |
Consider the usual Lebesgue spaces. Amid one of my studies, I started wondering if it is possible to find an example that satisfies the following problem:
**Problem.** Consider arbitrary elements $1 \leqslant p \leqslant q < \infty$ and $0 \leqslant \lambda < n\left( 1 - \frac{p}{q}\right)$, where $n$ represents the... |
I'm trying to find whether
- $\frac{\sqrt{z^2+2z+2}}{z^2-4z+3}$
is analytic in $|z| < 2$ and / or $|z| > 2$.
My attempt:
So $z^2-4z+3 = 0$ when $z=1,3$ but doesn't this exclude both of the regions? Because $1<2$ and $3>2$ unless I'm missing something big
This is where I'm stuck
Edit: I'm not sure... |
Problem 2-2 in Lee's *Introduction to Topological Manifolds* reads:
> Let $X = \{1, 2, 3\}$. Give a list of topologies on $X$ such that
> every topology on $X$ is homeomorphic to exactly one on your list.
Below is my attempted solution. Is it correct, or am I missing something? Also, is there a more efficient w... |
My question is similar to this one (https://math.stackexchange.com/questions/3642307/boundary-conditions-magnetic-field) in that it is related to the boundary conditions of the magnetic field (B-field). However, my question focuses on mathematically characterizing physicists' claims regarding the boundary conditions of... |
I am currently studying the various characterizations of amenability, and I am in particular working with Følner sets: a group $G$ is amenable if and only if for every finite subset $S \subseteq G$ and every $\varepsilon >0$ there exists a non-empty finite subset $F \subseteq G$ such that $| sF \Delta F|\leq \varepsilo... |
> For instance, how do you prove $P$(ℎ)=$P$()?
I don't think we can ever prove that mathematically, without resorting to circular reasoning at some level. But do we need a proof?
In problems involving probability, "fairness" is assumed unless there's a reason not to. And for a *fair* coin (dice or something simil... |
Suppose I have a PDE
$$ \frac\partial{\partial t}f(x,t) = F\left(f(x,t), \frac\partial{\partial x} f(x,t)\right). $$
I would like to discretize $f(x,t)$ as
$$f^\epsilon(x,t) = \sum_{n=0}^{N_\epsilon} f^\epsilon_n(t) \chi_{[n\epsilon, (n+1)\epsilon)}(x), $$
where $t$ is kept continuous, and replace partial derivativ... |
I'm asked to prove that $ \int_{-\infty}^{\infty}\frac{1}{(x^2+a^2)^2}dx = \frac{\pi}{2a^3}$
Using partial fraction decomposition on $ \frac{1}{(z^2+a^2)^2} $, and defining a contour $ \gamma_R $ to be the line segment between −R and R, together with a semicircle around $0$ of radius R. Using Cauchy's integral formu... |
In "The Book of Numbers" by John H. Conway, pg. 299, he discusses the application of surreal numbers to quantifying the growth rate of functions.
He gives the following correspondences:
$$\begin{aligned}
\frac{1}{ω}&&\cdots&&&\ln x \\
n&&\cdots&&&x^n&&(n\text{ real}) \\
ω&&\cdots&&&e^x \\
ω + n &&\cdots&&&x^n e... |
Using Surreal Numbers to measure function growth rate - Tetration? |
How can I find the sum of sum of series $\sum_{n=0}^{\infty}(2n^2+3)/(2\cdot 3^n)$? I proved that it is convergent, but I am stuck. I tried putting $x$ instead $3$ and studied it as a power series, but it didn't help. I also operated and reduced the problem to find the sum of $n^2 x^n$, but I couln't conclude anything.... |
How can I find the sum of sum of series $\sum_{n=0}^{\infty}\frac{(2n^2+3)}{(2\cdot 3^n)}$? I proved that it is convergent, but I am stuck. I tried putting $x$ instead $3$ and studied it as a power series, but it didn't help. I also operated and reduced the problem to find the sum of $n^2 x^n$, but I couln't conclude a... |
I have a box full with $n$ balls of different color but only one of each color e.g. 1 red, 1 green, 1 yellow, 1 blue etc.
We pull out balls without laying them back.
Here if we want to compute all possibilities (order doesn't matter) after $k$ pulls, the answer is $\binom{n}{k}$. This is clear.
**But how d... |
As a hint : separation is the clue
$$\sum_{n=0}^{\infty}\frac{(2n^2+3)}{(2\cdot 3^n)}
\\\frac 12\sum_{n=0}^{\infty}\frac{(2n^2+3)}{3^n}
\\\frac 12\sum_{n=0}^{\infty}\frac{2n^2}{3^n}+\frac 12\sum_{n=0}^{\infty}\frac{3}{3^n}
$$
the second part is the geometric progression but for the first, you can start with $f(x)... |
In "The Book of Numbers" by John H. Conway, pg. 299, he discusses the application of surreal numbers to quantifying the growth rate of functions.
He gives the following correspondences:
$$\begin{aligned}
\frac{1}{ω}&&\cdots&&&\ln x \\
n&&\cdots&&&x^n&&(n\text{ real}) \\
ω&&\cdots&&&e^x \\
ω + n &&\cdots&&&x^n e... |
How can I find the sum of sum of series $\sum_{n=0}^{\infty}\frac{(2n^2+3)}{(2\cdot 3^n)}$? I proved that it is convergent, but I am stuck. I tried putting $x$ instead $3$ and studied it as a power series, but it didn't help. I also operated and reduced the problem to find the sum of $n^2 x^n$, but I couln't conclude a... |
today I got my midterm test result of Discrete math (It can only raise the grade). I got 63, which is above the average, but still I don't feel comfortable with that.
I have 7 courses in the semester, and the time management is pretty hard, because everything is time consuming.
I understand that I need to change t... |
My question is similar to this one (https://math.stackexchange.com/questions/3642307/boundary-conditions-magnetic-field) in that it is related to the boundary conditions of the magnetic field (B-field). However, my question focuses on mathematically characterizing physicists' claims regarding the boundary conditions of... |
I have a stochastic differential equation of the type:
$$ dX(t) = \mu(t) X(t)dt + \sigma(t) X(t) dW(t) \tag{1}
$$
However, my $\mu(t)$ is a complicated function of $t$ as well as $W(t)$, somewhat like:
$$ \mu(t) = a(t) + b(t) + \sigma W(t) \tag{2}
$$
I feel like $W(t)dt$ that I get by substituting $(2)$ into ... |
How to prove unique solvability of an SDE? |
I'll detail the approach in J.G.'s comment. Let's work in the complex plane in order to deal with both coordinates at the same time, as suggested by J.G. From now on, I'll refer to the linked animation in the same comment, with the same initial situation.
Let $R$ and $r$ be the radii of the static big circle (cente... |
Since:
$$pmi(x,y)\quad =\quad \log(\frac{P(X=x,Y=y)}{P(X=x)P(Y=y)})\quad =\quad -t$$
iff
$$ 2^{t}P(Y=y | X=x) \quad =\quad P(Y=y)$$
the interpretation of a negative PMI seems very clear to me. So I don't really understand the following comment in the Wikipedia article on PMI (section Positive PMI) explaining why it... |
I know that topological space is first countable if each point has a countable neighborhood basis. A neighborhood basis at a point. Consider the following topological space, X=R with t={A⊂R:0 not in A} U {A⊂R:0∈A and R∖A is finite}.
I know it is not first countable. Is this because at the local base x=0, R\A is finit... |
If a topological space X is not first countable, is it second countable? |
I know that topological space is first countable if each point has a countable neighborhood basis. A neighborhood basis at a point. Consider the following topological space, X=R with t={A⊂R:0 not in A} U {A⊂R:0∈A and R∖A is finite}.
I know it is not first countable. Is this because at the local base x=0, R\A is finit... |
Problem 2-2 in Lee's *Introduction to Topological Manifolds* reads:
> Let $X = \{1, 2, 3\}$. Give a list of topologies on $X$ such that
> every topology on $X$ is homeomorphic to exactly one on your list.
Below is my attempted solution. Is it correct, or am I missing something? Also, is there a more efficient w... |
I know that topological space is first countable if each point has a countable neighborhood basis. A neighborhood basis at a point. Consider the following topological space, X=R with t={A⊂R:0 not in A} U {A⊂R:0∈A and R∖A is finite}.
I know it is not first countable. Is this because at the local base x=0, R\A is finit... |
Let $A$ be a $k$-algebra for a commutative ring $k$. Denote by $A^e$ the $k$-algebra $A\otimes A^{op}$. The center of a bimodule yields a functor $C$ from the category of right $A^e$-modules $Mod_{A^e}$ to the category of $k$-modules $Mod_k$.
There are forgetful functors $U_1:{}_{A^e}Mod_{A^e\otimes A^e}\rightarrow Mo... |
Suppose there is a function $f : [a,b] \rightarrow \mathbb{R}$ $(a, b\in \mathbb{R},$ with $a<b)$ which is discontinuous everywhere. This means that for any $x \in [a, b], \exists \epsilon_{x}> 0$ such that for any $\delta >0, \exists x_{\delta} \in [a, b]$ such that $|x-x_{\delta}|< \delta $, but $|f(x) - f(x_{\delta}... |
Is there some notion of "Uniformly Discontinuous" function? |
As part of a personal project, I found the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1], which discusses deriving equations for a 3D surface I want to parameterize. However, despite my efforts to decipher the paper, I can't follow everything it is describing. The ... |
How do I handle this single-variable function being called with two variables? |
Let $W\subseteq X\times Y$ where $\forall x\in X , \exists ! y\in Y , (x,y) \in W $.
How to proof that always $X \neq X\times Y$ ?
I try to proof that $x = (x,y)$ cannot exists - however I realize that maybe my assumption that if $X = X\times Y$ then $x = (x,y)$ is wrong - at least maybe I should first proof that... |
Two slightly different polynomial expansions for $\Xi(0)$. Could a connection between these two be derived? |
My problem is simplified as follows: I'm estimating a binary choice probit model where the error term follows a mixture of two normal distributions. This is:
$y_i = \mathbf{1} \left( \beta_0 + \beta_1 x_i + \epsilon_i > 0 \right)$
Where $\epsilon_i \sim 0.3 \mathcal{N}(-1,1) + 0.7 \mathcal{N}(1.2,0.2)$. I follow... |
What structure do smooth maps and diffeomorphisms preserve? |
Let $W\subseteq X\times Y$ where $\forall x\in X , \exists ! y\in Y , (x,y) \in W $.
How to proof that if $X\neq \emptyset$ then always $X \neq X\times Y$ ?
I try to proof that $x = (x,y)$ cannot exists - however I realize that maybe my assumption that if $X = X\times Y$ then $x = (x,y)$ is wrong - at least maybe... |
For a convex function $f$, I am interested in bounds of the form $$\frac{f(y)-f(x)}{y-x}\geq (1-\lambda)f'(y)+\lambda f'(x),\quad y<x.$$
For $\lambda=1$, this becomes true of any convex function $f$. On the other hand, for $\lambda= 0$, this only holds for a line. So I'm wondering which functions satisfy the above f... |
Suppose there is a function $f : I \rightarrow \mathbb{R}$ (I is some non-empty interval of real numbers) which is discontinuous everywhere. This means that for any $x \in I, \exists \epsilon_{x}> 0$ such that for any $\delta >0, \exists x_{\delta} \in I$ such that $|x-x_{\delta}|< \delta $, but $|f(x) - f(x_{\delta})|... |
Let $W\subseteq X\times Y$ where $\forall x\in X , \exists ! y\in Y , (x,y) \in W $.
How to proof that if $W\neq \emptyset$ then always $W \neq X\times Y$ ?
I try to proof that $x = (x,y)$ cannot exists - however I realize that maybe my assumption that if $W = X\times Y$ then $x = (x,y)$ is wrong - at least maybe... |
So, in my textbook, I came across a theorem/axiom (?) that states:
> Given a polynomial $P(x)$ with integer coefficients, its roots, if they exist, are divisors of the independent/constant term of the polynomial
I've wrapped my head around this and I can't figure out why. There's a explanation in my book, but I c... |
Why is it that: "If polynomials have integer coefficients, the roots of those polynomials will be a divisor (factor) of the constant term"? |
Theorem:
If $L/K$ is Galois and $E$ is an intermediate field, then the subspace topology on $\text{Gal}(L/E)$ as a subset of $\text{Gal}(L/K)$ equals the Krull topology on $\text{Gal}(L/E)$.
I want to prove that the open set of $\text{Gal}(L/E)$ under krull topology is also open set of subspace topology of $\text{G... |
Let $W\subseteq X\times Y$ where $\forall x\in X , \exists ! y\in Y , (x,y) \in W $.
How to prove that if $W\neq \emptyset$ then always $X \neq X\times Y$ ?
I try to prove that $x = (x,y)$ cannot exist - however I realize that maybe my assumption that if $W = X\times Y$ then $x = (x,y)$ is wrong - at least maybe ... |
Let $W\subseteq X\times Y$ where $\forall x\in X , \exists ! y\in Y , (x,y) \in W $.
How to prove that if $W\neq \emptyset$ then always $W \neq X\times Y$ ?
I try to prove that $x = (x,y)$ cannot exist - however I realize that maybe my assumption that if $W = X\times Y$ then $x = (x,y)$ is wrong - at least maybe ... |
I have a fairly simple discrete Markov Chain with a transition probability matrix that looks like this:
\begin{bmatrix}0&{n_1}&0&0&0&...\\
{n_2}&{n_3}&0&0&0&...\\
0&{n_4}&0&{n_5}&0&...\\
...&...&...&...&...&...\\
...&0&0&0&n_k&0\end{bmatrix}
Basically, just a long line of states, all of which can go into a ne... |
How to determine the stationary distribution of a Markov Chain with N states? |
Let $W\subseteq X\times Y$ where $\forall x\in X , \exists ! y\in Y , (x,y) \in W $.
How to prove that if $W\neq \emptyset$ then always $W \neq X$ ?
I try to prove that $x = (x,y)$ cannot exist - however I realize that maybe my assumption that if $W = X$ then $x = (x,y)$ is wrong - at least maybe I should first p... |
I am struggling a little bit on probability and I was hoping someone could clear things up in a slightly informal way. I am new to probability so I am in an awkward position where I need to learn how to solve problems + have an informal understanding of why things work without getting into nitty gritty. I was hoping so... |
Is my informal understanding of probability definitions correct? |
I have a function `$f$` that is continuous over the set of variables, `$f: \ K \times M$`, where `$K$` is a compact and `$M$` is a metrical space. And I want to prove that the function `$g(y)=sup_{x\in K} f(x,y)$` is continues.
As `$K$` is a compact, `$f$` is uniformly continuous with respect to its first argument.... |
How to prove continuity of g(y)=max f(x,y)? |
Let $W\subseteq X\times Y$ where $\forall x\in X , \exists ! y\in Y , (x,y) \in W $.
How to prove that if $W\neq \emptyset$ then always $W \neq X$ ?
I try to prove that $x = (x,y)$ cannot exist - however I realize that maybe my assumption that if $W = X$ then $x = (x,y)$ is wrong - at least maybe I should first p... |
I have a function $f$ that is continuous over the set of variables, $f: \ K \times M$, where $K$ is a compact and $M$ is a metrical space. And I want to prove that the function $g(y)=\sup_{x\in K} f(x,y)$ is continues.
As $K$ is a compact, $f$ is uniformly continuous with respect to its first argument. Also, for th... |
I have a function $f$ that is continuous over the set of variables, $f: \ K \times M$, where $K$ is a compact and $M$ is a metrical space. And I want to prove that the function $g(y)=\sup_{x\in K} f(x,y)$ is continues.
As $K$ is a compact, $f$ is uniformly continuous with respect to its first argument. Also, for th... |
I have a function $f$ that is continuous over the set of variables, $f: \ K \times M$, where $K$ is a compact and $M$ is a metrical space. And I want to prove that the function $g(y)=\sup_{x\in K} f(x,y)$ is continues.
As $K$ is a compact, $f$ is uniformly continuous with respect to its first argument. Also, for th... |
How to prove continuity of $g(y)=\max_{x\inK} f(x,y)? K is compact |
How to prove continuity of $g(y)=\max_{x\inK} f(x,y)? K$ is compact |
How to prove continuity of $g(y)=\max_{x\in K} f(x,y)? K$ is compact |
Let $ f: A \to B $ be a linear map between two $ C^* $ algebras.
What is a sufficient condition to guarantee that the linear map $ f $ takes the identity to the identity $ f(1_A)=1_B $?
For example, taking $ A=B=\mathbb{C}^{2 \times 2} $ the algebra of $ 2 \times 2 $ matrices. Is it sufficient for $ f $ to be p... |
How to prove continuity of $g(y)=\max_{x\in K} f(x,y)$? $K$ is compact |
I have a function $f$ that is continuous over the set of variables, $f: \ K \times M \to \mathbb{R}$, where $K$ is a compact and $M$ is a metrical space. And I want to prove that the function $g(y)=\sup_{x\in K} f(x,y)$ is continues.
As $K$ is a compact, $f$ is uniformly continuous with respect to its first argumen... |
I have a function $f$ that is continuous over the set of variables, $f: \ K \times M \to \mathbb{R}$, where $K$ is a compact and $M$ is a metric space. And I want to prove that the function $g(y)=\sup_{x\in K} f(x,y)$ is continues.
As $K$ is a compact, $f$ is uniformly continuous with respect to its first argument.... |
The argument presented in the body of your question is perfectly sound [with a missing minus sign: a typo?] and is worth digesting. If you find that *the notation* is what's holding you back, then write out the entire argument in **one special case** but with different notation until you believe in it. This will mak... |
Problem 2-2 in Lee's *Introduction to Topological Manifolds* reads:
> Let $X = \{1, 2, 3\}$. Give a list of topologies on $X$ such that
> every topology on $X$ is homeomorphic to exactly one on your list.
Below is my attempted solution. Is there a more efficient way to solve a problem like this one? My method f... |
Let $W\subseteq X\times Y$ where $\forall x\in X , \exists ! y\in Y , (x,y) \in W $.
How to prove that if $W\neq \emptyset$ then always $W \neq X$ ?
I try to prove that $x = (x,y)$ cannot exist - however I realize that maybe my assumption that if $W = X$ then $x = (x,y)$ is wrong - at least maybe I should first p... |
I know that topological space is first countable if each point has a countable neighborhood basis. A neighborhood basis at a point. Consider the following topological space, $X=R$ with $\mathcal T=\{A\subseteq R:0 \not\in A\} \cup \{A\subseteq R:0\in A\text{ and } R \setminus A\text{ is finite}\}$.
I know it is not f... |
Due to some context, I have reason to believe that S(K(SII)) and SSI are actually equivalent CL terms. This is my attempt at a proof (assuming a and b to be arbitrary CL terms):
$$\text{S(K(SII))ab = SII(ab) = ab(ab)}\\
\text{SSIab = Saab = ab(ab)}$$
Since both simplify to the same form, can we say that S(K(SII)... |
Here is the definitions I am working with
[![enter image description here][1]][1]
Define the map $$ (z-1)^{\frac{1}{2}}$$ defined as the inverse of $$\begin{align} f: \mathbb{C} & \rightarrow \mathbb{C} \\ z &\mapsto z^{2}+1 \end{align}$$.
<hr>
Establishing multivalues is immediate from the fact that $1$ a... |
I am working on understanding the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1]. They use a function $F$ in two complex variables $ξ$ and $\bar{ξ}$, but don't explicitly state an algebraic definition for it. All they do is state some conditions it must satisfy, and... |
How do I find an algebraic expression for the function $F(ξ, \bar{ξ})$ from this paper? |
I am struggling a little bit on probability and I was hoping someone could clear things up in a slightly informal way. I am new to probability so I am in an awkward position where I need to learn how to solve problems + have an informal understanding of why things work without getting into nitty gritty. I was hoping so... |
The argument presented in the body of your question is perfectly sound [with a missing minus sign: a typo?] and is worth digesting. If you find that *the notation* is what's holding you back, then write out the entire argument in **one special case** but with different notation until you believe in it. This will mak... |
I'm learning the maths behind error correction codes. For this purpose I made this question for myself:
Assume you transferred 2 information bits $x_0$, $x_1$ and a check bit $x_2$ (check bit: $x_2 = x_0 \mathbin{\mathsf{XOR}} x_1$) through an additive white gaussian noise channel (AWGNC). I.e. your input codeword i... |
How to combine two probabilities for the same event? Context: error correction codes / decoding |
I have a function $f$ that is continuous over the set of variables, $f: \ K \times M \to \mathbb{R}$, where $K$ is a compact and $M$ is a metric space. And I want to prove that the function $g(y)=\sup_{x\in K} f(x,y)$ is continues.
As $K$ is a compact, $f$ is uniformly continuous with respect to its first argument.... |
I'm learning the maths behind error correction codes. For this purpose I made this question for myself:
Assume you transferred 2 information bits $x_0$, $x_1$ and a check bit $x_2$ (check bit: $x_2 = x_0 \mathbin{\mathsf{XOR}} x_1$) through an additive white gaussian noise channel (AWGNC). I.e. your input codeword i... |
This question has been motivated by [original](https://math.stackexchange.com/questions/4870751/is-it-possible-to-approximate-any-nonnegative-schwartz-function-by-square-of-sch?noredirect=1#comment10384172_4870751) and comments therein.
Let $f$ be a nonnegative Schwartz function on $\mathbb{R}^n$ and let $\alpha \... |
> For instance, how do you prove $P$(ℎ)=$P$()?
I don't think we can ever prove that mathematically, without resorting to circular reasoning at some level. But do we need a proof?
In problems involving probability, "fairness" is assumed unless there's a reason not to. And for a *fair* coin (dice or something simil... |
I have a function $f$ that is continuous over the set of variables, $f: \ K \times M \to \mathbb{R}$, where $K$ is a compact domain of a metric space and $M$ is a metric space. And I want to prove that the function $g(y)=\sup_{x\in K} f(x,y)$ is continues.
As $K$ is a compact, $f$ is uniformly continuous with respe... |
Often, people are introduced to the notion of continuity by the idea that a function is continuous if and only if its graph can be drawn without lifting the pencil from the paper.
Now, after joining an undergraduate course and learning continuity formally using $\epsilon - \delta$ definition, I do not see any direc... |
I have a function $f$ that is continuous over the set of variables, $f: \ K \times M \to \mathbb{R}$, where $K$ is a compact domain of a metric space and $M$ is a metric space. And I want to prove that the function $g(y)=\sup_{x\in K} f(x,y)$ is continuous.
As $K$ is a compact, $f$ is uniformly continuous with resp... |
How to prove continuity of $g(y)=\max_{x\in K} f(x,y)$ when $K$ is compact? |
I am struggling a little bit on probability and I was hoping someone could clear things up in a slightly informal way. I am new to probability so I am in an awkward position where I need to learn how to solve problems + have an informal understanding of why things work without getting into nitty gritty. I was hoping so... |
I'm learning harmonic analysis with Grafakos' book *Classical Fourier Analysis*. I'm having a problem understanding the definition 5.2.1 of maximal singular integral operators.
Let me recall the definition. Here, $\Omega \colon \mathbb{S}^{n-1} \to \mathbb{C}$ is an integrable function with mean $0$. For $0 < \epsil... |
Often, people are introduced to the notion of continuity by the idea that a function is continuous if and only if its graph can be drawn without lifting the pencil from the paper.
Now, after joining an undergraduate course and learning continuity formally using $\epsilon$-$\delta$ definition, I do not see any direc... |
Rudin's a tough one. Check out this Khan Academy video on the topic of limits: https://www.youtube.com/watch?v=-ejyeII0i5c
The notion of a limit captures entirely the definition of continuity, which explains the equivalence of limits and continuity gotten later in the chapter (A function is continuous at $x$ if the ... |
This is my first question so I hope this sort of thing is okay to ask. I'm working my way through Rudin's Principles of Mathematical Analysis, and I'm up to chapter 4, which is on continuity in the context of functions between metric spaces. It introduces what I understand to be the standard epsilon-delta definition us... |
Let $a,b,c\ge 0:ab+bc+ca=3.$ Find the minimal value $$P=\sqrt{a^2b+b^2c+c^2a+abc}+\sqrt{a+b+c-1}.$$
I think $2+\sqrt{2}$ is desired result. $P$ attain this value when $a=b=c=1.$
I use $a+b+c\ge \sqrt{3(ab+bc+ca)}=3$ and we need to prove $$a^2b+b^2c+c^2a+abc\ge 4.$$
I got stuck here. Can you help me prove it? Thanks.... |
Let $E$ be an elliptic curve over a field $K$ given by a long Weierstrass equation $e \in K[X, Y]$, and let $\psi_n \in K[X, Y]$ be the [$n$-th division polynomials](https://math.stackexchange.com/questions/4675977/division-polynomials-in-characteristic-two) of $E$. Can we formally prove that $-\psi_{n + 1}\psi_{n - 1}... |
This is from Bachman's Harmonic Analysis book, exercise 8.3
>Let $\mu$ be a Haar measure in $G$ and let $E_1$ and $E_2$ be two compact subsets of $G$ such that $\mu(E_1)=\mu(E_2)=0$. Does this imply that $\mu(E_1E_2)=0$?
My attempt: $E_1E_2$ is compact. By outer regularity, consider an open neighborhood $U_2$ of $E... |
My problem is simplified as follows: I'm estimating a binary choice probit model where the error term follows a mixture of two normal distributions. This is:
$y_i = \mathbf{1} \left( \beta_0 + \beta_1 x_i + \epsilon_i > 0 \right)$
Where $\epsilon_i \sim 0.3 \mathcal{N}(-1,1) + 0.7 \mathcal{N}(1.2,0.2)$. I follow... |
Im sure this question is completely trivial, but I just want to check my understanding:
For the various explicit formulae in Analytic Number Theory involving sums over the nontrivial zeros of $\zeta$ how are the zeros evaluated?
For example, take $\psi(x)=x-\sum_{\zeta(\rho)=0}\frac{x^\rho}\rho -\log(2\pi)$. Wiki... |
Lately I have been reading about the cut elimination theorem, I think I get the idea however I have been struggling about some technical details concerning quantifiers.
Consider the following rule:
[![enter image description here][1]][1]
Now in the proof of the cut eliminition theorem we are using mixes inst... |
I was thinking in The following situation. Let $X$ be a topological space and $Y=X/\sim$ with quotient map $q:X\to Y$.
Let $f:Y\to Y$ be a function. Is there an easy way to prove that $f$ is continous provided that $\tilde{f}:X\to X$, a function that makes the same assignment as $f$ is continous $q\circ \tilde{f} = f\... |
When you know two sides and the angle between them, the so-called "side-angle-side" case, older textbooks would recommend using the [law of tangents](https://en.wikipedia.org/wiki/Law_of_tangents) to find the remaining angles, if you don't want to bother finding the remaining side. I think that's what you mean by "dir... |
Given a triangle $ABC$ with edges $a$, $b$, and $c$ and corresponding angles
$\alpha$, $\beta$ and $\gamma$. Assume, the values of $a$, $b$ and $\gamma$ are given. Does there exist a direct theorem (like the law of cosines, the law of sines or the law of tangents) which describes the relationship between these parame... |
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