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I've been trying to solve the following differential equation:
$$y + \sqrt{(y')^2 + 1} = k$$
where $ k $ is a constant. Despite my efforts, I haven't been able to find a closed-form solution for it. It seems to be a rather tricky equation due to its nonlinearity and the presence of the square root term. I've attempte... |
Let $G$ is the proper non-trivial subgroup of $\mathbb Z\oplus \mathbb Z$, whether $G$ must be infinite cyclic ?
What I try: $\mathbb Z\oplus \mathbb Z$ can be treated as integral vector of $\mathbb R^2$. And the proper subspace of $R^2$ are the lines through the origin. The integral vector on a line is an inf... |
Non-trivial abelian subgroup of $\mathbb Z\oplus \mathbb Z$ must be infinite cyclic? |
I was given the following question to solve for homework. The solution I got was x = {x : x ≥ 6 or x≤3}. I checked my solution with the answers provided and it was correct. My working out can be seen below:
|x-5| + |x-4| - 3 = 2x-12 if x>5
-2 if 4<x≤5
-2x+6 if x≤4
... |
In Complex Variables and Applications (Churchill, Brown), Chapter 3 Exercise 13.5, we are asked to show that:
> The set of values of $\log(i^\frac{1}{2})$ is
$$
(n + \frac{1}{4})\pi i \quad (n = 0, \pm 1, \pm 2, \dots)
$$
I know the definitions of $\log$ and $\exp$. Particularly, $\log(z) = \ln\lvert z\rvert +... |
I was given the following question to solve for homework. The solution $S$ I got was $$S = \{x : x ≥ 6 \text{ or } x\leq3\}$$ I checked my solution with the answers provided and it was correct. My working out can be seen below:
$|x-5| + |x-4| - 3 =
\begin{cases} 2x-12 , & \text{if } x>5\\ -2, & \text{if }4<x≤5\\ -... |
I have a question about the proof of Proposition 2 on page 26 of [Fulton's algebraic curves. ][1]
Let $V\subset \mathbb A^n$ and $W\subset \mathbb A^m$ be affine varieties, and let $\Gamma(V)$ and $\Gamma(W)$ be their coordinate rings.
Suppose that we have a homomorphism $\alpha\colon \Gamma(V)\longrightarrow\Gam... |
> Let $X$ be a topological space. Prove that if
> $X=\bigcup\limits_{k=1}^{\infty}X_k$, $X_k$ is connected subset of $X$ and
> $X_k\cap X_{k+1}\neq\varnothing\quad\forall k\geq 1$, then $X$ is
> connected.
I can prove that if every pair $x,y$ of $X$ belongs to a connected subset of $A_{xy}$ then $X$ is connected.... |
>Solve $x+2=\sqrt{4+x\sqrt{8-x}}$
$\Rightarrow (x+2)^2=(\sqrt{4+x\sqrt{8-x}})^2 \tag{1}$
$\Rightarrow x^2+4x+4=4+x\sqrt{8-x} \tag{2}$
$\Rightarrow x^2+4x=x\sqrt{8-x} \tag{3}$
$\Rightarrow (x^2+4x)^2=(x\sqrt{8-x})^2 \tag{4}$
$\Rightarrow x^4+8x^3+16x^2=8x^2-x^3 \tag{5}$
$\Rightarrow x^4+9x^3+8x^2=0 \ta... |
> Let $X$ be a topological space. Prove that if
> $X=\bigcup\limits_{k=1}^{\infty}X_k$, $X_k$ is connected subset of $X$ and
> $X_k\cap X_{k+1}\neq\varnothing\quad\forall k\geq 1$, then $X$ is
> connected.
I can prove that if every pair $x,y$ of $X$ belongs to a connected subset of $A_{xy}$ then $X$ is connected.... |
I recently learnt about solving partial differential equations using method of separation of variables.
The problem: Find a solution of the equation $\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial x} + 2u$ in the form $u = f(x)g(y)$. Solve the equation subject to the conditions $u = 0$ and $\frac{du... |
Let $F(n,p)=\sum_{m=p}^{\infty} \frac{(n+m)!}{n!\ m!\ 2^m}$. What is the upper bound for $F(n,p)$, where $p\geq n+1$ and $n\geq 0$? Does there exist a tight bound where the value of $p$ could be easily estimated under a constraint $F(n,p)\leq 2^{-k}$, where $k>1$? |
Is there any pair of integers $a$ and $b$ such that $e^a = b$ for $a \neq 0$? I'm not quite sure how to even begin analyzing this equality.Has this problem been solved before? |
Is there any pair of integers $a$ and $b$ such that $e^a = b$ for $a \neq 0$? I'm not quite sure how to even begin analyzing this equality. Has this problem been solved before? |
Let $G$ be a finite group, and define $a(n)=\#\{g\in G\mid o(g)=n\}$.
In problem 5.18 of Isaacs' *Character Theory of Finite Groups*, the following polynomial is defined:
$$F_G(X)=\frac{1}{|G|}\sum_ma(m)X^{|G|/m}$$
**Theorem:** $F_G(k)$ is an integer for $k\in\mathbb{Z}$.
Setting $G$ to be the cyclic group ... |
I have a robot simulation model which is in terms of local coordinate system $x, y, z$. so every joint here will be with respect to a parent.
I have some data about an action and these actions are given in terms of the global coordinate system.
Now since I am feeding this data into the model and the axis are rota... |
I'm using JAX to produce a convolution
def gaussian_kernel(size: int, std: float):
"""Generates a 2D Gaussian kernel."""
x, y = jnp.mgrid[-size:size+1, -size:size+1]
g = jnp.exp(-(x**2 + y**2) / (2 * std**2))
return g / g.sum()
def gaussian_blur(image, kernel_size=5... |
Does tensor product contain Cartesian product? |
Let $f: \Bbb R^2 \setminus \{0\} \to \Bbb R $, such that:
$f$ a homogeneous function of degree $0$ (so that $\forall t>0: f(x,y)=f(tx,ty)$),
$f$ is differentiable and positive in $\Bbb R^2 \setminus \{0\}$.
I would like to prove it is constant (this may not be true, but if it is it would help me prove some oth... |
Proving a degree $0$ homogeneous function: $f: \Bbb R^2 \setminus \{0\} \to \Bbb R , f$ is constant? (notice the domain) |
Let $K$ be a number field and $K_v$ be the completion of $K$ at the place $v$. Consider an elliptic curve $E/K$ over $K$. The short exact sequence $0 \to E[2] \to E \to E \to 0$ induces a famous exact sequence $0 \to E(K)/2E(K) \to H^1(G_K, E[2]) \stackrel{\delta}{\to} H^1(G_{K_v}, E)[2] \to 0$.
My question is, why ... |
Considering https://math.stackexchange.com/questions/2405676/are-all-linear-programs-convex. Does that imply that whenever I can reformulate a real world problem into linear programming setting that I can get to the optimal solution with Dantzig's algorithm since the problem is convex? Or are there any further restrict... |
i need to find a function f(r,t) such that
f(r,r)=r & f(r,r/2)=2
the question really is straightforward but i don't have any mathematical approach to this so in reality I just wasted an hour manually trying out different possible functions. can someone tell me how I can conclusively find the function or disprove t... |
Let $V=\{a_2x^2 + a_1x + 2 : a_1, a_2 \in \mathbb R\}$. How to prove whether this is a vector space or not? |
i need to find a function f(r,t) such that
f(r,r)=r
f(r,r/2)=2
given, r>0 and t<r
the question really is straightforward but i don't have any mathematical approach to this so in reality I just wasted an hour manually trying out different possible functions. can someone te... |
i need to find a function f(r,t) such that
f(r,r)=r
f(r,r/2)=2
given, r>0 and t<r
the question really is straightforward but i don't have any mathematical approach to this so in reality I just wasted an hour manually trying out different possible functions. can someone tell m... |
Here is my question:
> Let $(R,\mathfrak m)$ be a Noetherian local ring with ideal $I\subset\mathfrak m$. Do we have $$\mathrm{depth}_IR+\dim R/I\geq\mathrm{depth}R?$$
If we have, then this is stronger than $\mathrm{depth}R\leq\dim R$ since we have $$\mathrm{depth}_IR+\dim R/I\leq\mathrm{height}(I)+\dim R/I\leq\d... |
I need to find a function $f(r,t)$ such that
$f(r,r)=r \space and \space f(r,\frac{r}{2})=2 $
where $r > 0 $ and $t < r$.
The question really is straightforward but I don't have any mathematical approach to this so in reality I just wasted an hour manually trying out different possible functions. can someone tell... |
I need to find a function $f(r,t)$ such that
$f(r,r)=r$ and $f(r,\frac{r}{2})=2$,
where $r > 0 $ and $t < r$.
The question really is straightforward, but I don't have any mathematical approach to this, so I just wasted an hour trying out different possible functions. Can someone tell me how I can conclusively fin... |
How to find asymptotic bound for $ T(n) = n + \sum_{i=1}^{\lceil \log_3 n \rceil} T(\lfloor \frac {n}{3^i} \rfloor), n\ge 1 $? |
Why is the polar representation for lines $x cos\theta + y sin \theta = d$? |
I want to find the intersections of two hyperbolas in polar coordinates. One of their foci coincides, we use this as the pole. (The right focus of the left hyperbola is the same as the left focus of the right hyperbola, as shown on the [picture][1]).
I'm trying to solve the following equation to get their intesectio... |
Some context beforehand:- I was writing an expression for force between two charges with two mediums placed in between them. I noticed that the deviation of the force I calculated to be varying by a constant factor at particular distances from the actual value of force.
I know the expression for the actual force bu... |
I'm gonna guess that it's true for any $m$.
Assume your hypothesis. Then $a^a=b^b \pmod 2\implies a=b.$ And let $g$ be a primitive $\pmod 5.$ Write $a=g^k, b=g^l.$ Then $a^a\equiv b^b\implies g^{ak-bl}=1.$ So $\varphi (5)=4\mid (ak-bl).$ But $1\le a,b\lt5.$ Thus $a\equiv b\pmod 5.$
So $x^x$ is injective $\p... |
Let $G$ be a group of Order 42 which has an element of order 6. I want to classify all groups like this up to isomorphism.
Let $g\in G$ be the element of order 6. Then $C_6\cong\langle g\rangle\leq G$. As $42=6\cdot 7$ and $6<7$ there is only one 7-Sylow Subgroup, which thus is normal and ismorphic to $C_7$. Let $P$... |
Fubini's theorem for differential forms? Why does $\int_{t_0}^{t_1}(\oint_{\partial\Omega}j)dt=\int\limits_{[t_0,t_1]\times\partial\Omega}dt\wedge j$? |
I have read in Book An introduction to Banach space theory by Robert Megginson
Theorem
Goldstine's Theorem
Let $X$ be a normed space and let $Q$ be the natural map from $X$ into $X^{**}.$ Then $Q(B_X)$ is weakly* dense in $B_{X^{**}}.$
Corollary
Let $X$ be a normed space and let $Q$ be the natural map from $... |
I have read in Book An introduction to Banach space theory by Robert Megginson
**Theorem**
Goldstine's Theorem
Let $X$ be a normed space and let $Q$ be the natural map from $X$ into $X^{**}.$ Then $Q(B_X)$ is weakly* dense in $B_{X^{**}}.$
**Corollary**
Let $X$ be a normed space and let $Q$ be the natural ma... |
I recently learnt about solving partial differential equations using method of separation of variables.
The problem: Find a solution of the equation $\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial x} + 2u$ in the form $u = f(x)g(y)$. Solve the equation subject to the conditions $u = 0$ and $\frac{du... |
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$\color{blue}{H(X)=-\frac{1}{2} \log\left ((2\pi e)^n\det{\Sigma} \right)}.$$ Then, one can see that
$$\log f_{X}(x) = \underbrace{\frac{1}{2} \log((2 \pi)^n\det{\Sigma})}_{\color{blue}{A:=-H(X)+\frac{1}{2} \log(e^n)}} -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu). \tag{... |
**Expression:**
$$(x\times y)'+(y\times z)$$
**My attempt:**
1. $(xy)' + (yz)$
2. $(x'+y') + (yz) \quad \textit{After applying de Morgan's Axiom}$
3. $x' + (y' + yz) \quad \textit{After applying 1st Distributive Axiom}$
4. $x'+ (y' + y) + (y'z) \quad \textit{Rewriting}$
5. $x' + 1 + (y'z) \quad \te... |
How to simplify the boolean expression $(x\times y)'+(y\times z)$? |
Why is the polar representation for lines $x \cos\theta + y \sin \theta = d$? |
I'm gonna guess that it's true for any $m$.
Assume your hypothesis. Then $a^a=b^b \pmod 2\implies a=b.$ And let $g$ be a primitive $\pmod 5.$ Write $a=g^k, b=g^l.$ Then $a^a\equiv b^b\implies g^{ak-bl}=1.$ So $\varphi (5)=4\mid (ak-bl).$ But $1\le a,b\lt5.$ Thus $a\equiv b\pmod 5.$
So $x^x$ is injective $\p... |
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$\color{blue}{H(X)=-\frac{1}{2} \log\left ((2\pi e)^n\det{\Sigma} \right)}.$$ Then, one can see that
$$\log f_{X}(x) = \underbrace{\frac{1}{2} \log((2 \pi)^n\det{\Sigma})}_{\color{blue}{A:=-H(X)+\frac{1}{2} \log(e^n)}} -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu). \tag{... |
Some context beforehand:- I was writing an expression for force between two charges with two mediums placed in between them. I noticed that the deviation of the force I calculated to be varying by a constant factor at particular distances from the actual value of force.
I know the expression for the actual force bu... |
Assuming you mean that all points in the plane are equidistant from $A$ and $B$, then notice that the vector $$v:= v_2-v_1= ( x_2-x_1,y_2-y_1,z_2-z_1 )^T$$ is normal to the desired plane. Furthermore, the midpoint of the line segment between $A$ and $B$ must lie in this plane. The midpoint is the point $m=\left(\frac{x... |
Let $K$ be a number field and $K_v$ be the completion of $K$ at the place $v$. Consider an elliptic curve $E/K$ over $K$. The short exact sequence $0 \to E[2] \to E \to E \to 0$ induces a famous exact sequence $0 \to E(K)/2E(K) \to H^1(G_K, E[2]) \stackrel{\delta}{\to} H^1(G_{K_v}, E)[2] \to 0$.
For $M$: discrete p-... |
In Principles of Mathematical Analysis Rudin states that:
> Theorem 3.33: Given $\sum a_n$, put $\alpha = \lim_{n \to \infty}\sqrt[n]{n}$. Then $$$$ ... $$$$ (c) if $\alpha = 1$, the test gives no information
For proving (c) he states:
> We consider the series $$\Sigma\frac{1}{n}, \Sigma\frac{1}{n^2}$$ For eac... |
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$\color{blue}{H(X)=-\frac{1}{2} \log\left ((2\pi e)^n\det{\Sigma} \right)}.$$ Then, one can see that
$$\log f_{X}(x) = \underbrace{\frac{1}{2} \log((2 \pi)^n\det{\Sigma})}_{\color{blue}{A:=-H(X)+\frac{1}{2} \log(e^n)}} -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu). \tag{... |
I'm working on a problem where I want to find the supremum over the orthogonal group $O_n(\mathbb{R})$ of the sum of the upper triangular elements of matrices in this group, specifically we want to compute
$$
m_n=\sup_{M \in O_n(\mathbb{R})} \sum_{1 \leq i \leq j \leq n} m_{ij}
$$
I have shown that this is equi... |
In Principles of Mathematical Analysis Rudin states that:
> Theorem 3.33: Given $\sum a_n$, put $\alpha = \lim_{n \to \infty}\sqrt[n]{\lvert a_n \rvert}$. Then $$$$ ... $$$$ (c) if $\alpha = 1$, the test gives no information
For proving (c) he states:
> We consider the series $$\Sigma\frac{1}{n}, \Sigma\frac{1... |
There is a gun with 5 chambers and 1 bullet. There is also a row of x people, with x > 2. Starting from Person 1, each person takes turns shooting the gun (spinning the chamber with every attempt) at Person 1, except for Person 1 who shoots the gun (also spinning the chamber with every attempt) at a random person (that... |
Suppose $A,B \in M(n \times n, \mathbb{C})$ or $ A,B \in M(n \times n, \mathbb{R}) $. Under which hypothesis can I state that:
$\rho(AB) \leq \rho(A)\rho(B)$ ?
|
Under which conditions is the spectral radius submultiplicative? |
according to different sources in the internet, homogeneous function is defined as if f(tx,ty)=t^m*f(x,y) for some integer m, but my question is why integer, why not a real number(assuming the function is defined over reals to reals), consider this function f(x,y)=(x^2+y^2)^(1/3), this intuitively seems to be a homogen... |
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$\color{blue}{H(X)=-\frac{1}{2} \log\left ((2\pi e)^n\det{\Sigma} \right)}.$$ Then, one can see that
$$\log f_{X}(x) = \underbrace{\frac{1}{2} \log((2 \pi)^n\det{\Sigma})}_{\color{blue}{A:=-H(X)+\frac{1}{2} \log(e^n)}} -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu). \tag{... |
We are given a function $f\colon R\to R$ that satisfies $f(x+f(y))=f(x)+f(y)$ for any real $x,y$, and also $f(0)\neq0$. The equation seems to be entirely dependent on the values of $f$. It seems impossible to have anything to say about $f(x+y)$. Clearly, functions of the form $x+b$ (with $b\neq 0$) satisfy this equatio... |
For simplicity lets consider finite probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and two random vatiables $\xi, \eta$ on it. I wonder is there any connection between statistical and functional dependence of random variables? For example, if I have $\xi = f\circ \eta$ for some $f$, does it imply that $\xi$ and ... |
How to prove this proposition about collection of continuous functions $\{f_i\}_{i \in I}$? |
Fix a $V\in\mathbb{N}$ with $V\ge 4$. Randomly pick $V$ points on a sphere (independently and uniformly with respect to the surface area measure). You may think of the convex hull of these $V$ points. With probability 1, this constitutes a nondegenerate convex polyhedron. In fact, all faces would be triangles (probabil... |
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$\color{blue}{H(X)=-\frac{1}{2} \log\left ((2\pi e)^n\det{\Sigma} \right)}.$$ Then, one can see that
$$\log f_{X}(x) = \underbrace{\frac{1}{2} \log((2 \pi)^n\det{\Sigma})}_{\color{blue}{A:=-H(X)+\frac{1}{2} \log(e^n)}} -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu). \tag{... |
**Joachimsthal's notation**
This "trick" to find the equation of the tangent to a conic section (ellipse, parabola, hyperbola) can be understood using the notation of [Ferdinand Joachimsthal](https://mathshistory.st-andrews.ac.uk/Biographies/Joachimsthal/) (1818-1861), a German geometer who had been a student of Kum... |
I have a problem with understanding what is the difference between:
$\psi (y) := \int_{\mathbb{R}^m} x \cdot f_{X|Y}(x | y) \ dx$ ,therefore $\psi (Y)$ = $E(X \mid Y)$
and
$\phi (y) := \int_{\mathbb{R}^m} h(x) \cdot f_{X|Y}(x | y) \ dx $ ,therefore $ \phi (Y)$ = $E(h(X) \mid Y)$
Also what is the difference... |
There is a gun with $5$ chambers and $1$ bullet. There is also a row of $x$ people, with $x > 2$. Starting from Person $1$, each person takes turns shooting the gun (spinning the chamber with every attempt) at Person 1, except for Person 1 who shoots the gun (also spinning the chamber with every attempt) at a random pe... |
according to different sources in the internet, homogeneous function is defined as if $f(tx,ty)=t^mf(x,y)$ for some integer $m$, but my question is why integer, why not a real number(assuming the function is defined over reals to reals), consider this function $f(x,y)=(x^2+y^2)^{1/3}$, this intuitively seems to be a ho... |
according to different sources in the internet, homogeneous function is defined as if $f(tx,ty)=t^mf(x,y)$ for some integer $m$, but my question is why integer, why not a real number(assuming the function is defined over reals to reals), consider this function $f(x,y)=(x^2+y^2)^{1/3}$, this intuitively seems to be a ho... |
For simplicity lets consider finite probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and two random variables $\xi, \eta$ on it. I wonder is there any connection between statistical and functional dependence of random variables? For example, if I have $\xi = f\circ \eta$ for some $f$, does it imply that $\xi$ and ... |
What is the formula that provides the relationship between *θ* angle and *r* length? 0 ≤ θ < π/2
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/Pvaar.png |
How to formulate the line segment drawn from the left vertex point of the ellipse? |
I'm working on a problem,
> Let $\mathcal{E}:y^2=x^3-2x+1$ be an elliptic curve over the field $\mathbb{F}_5$, and let $P=(0,1)$ be among the points on $\mathcal{E}$. Find the equation of the line on which $P$, $2P$ and $4P$ all lie. Deduce that $7P=O$, the neutral element of the group law (point at infinity).
I ge... |
Why do 3 collinear points on an elliptic curve sum to $O$? |
Unfortunately, there is an ambiguity over the definition of positive definiteness, and probably there is no agreement among the authors. I also stumbled upon the Gilbert Strang's definition and asked to myself "wait, does positive definiteness imply symmetry?".
The ambiguity is that:
*definition 1*: positive defi... |
**Polar form relative to center**
In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate θ measured from the major axis, the ellipse's equation is
$r(\theta)=\dfrac{b}{\sqrt{1-(e\cos\theta)^2}}$
[![enter image description here][1]][1]
**Polar form relative to fo... |
My approach:
$\lim_{n\rightarrow\infty} \sum_{k=1}^n \frac{1}{k}\tan(\frac{k\pi}{4n + 4}) = \lim_{n\rightarrow\infty} \sum_{k=1}^{n+1} \frac{1}{k}\frac{n+1}{n+1}\tan(\frac{k\pi}{4n + 4}) = \lim_{n\rightarrow\infty} \sum_{k=1}^{n+1} \frac{1}{k}\frac{n+1}{n+1}\tan(\frac{\pi k}{4(n + 1)}) = \lim_{n\rightarrow\infty} \s... |
Here, I **first** provide a proof with required details based on relative entropy, and **then** I discuss how your **main issue** on dropping the expectation constraint when using the KKT method can be solved (without the expectation constraint, there are infinitely many other feasible KKT points, and the entropy tends... |
**Polar form relative to center**
In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate θ measured from the major axis, the ellipse's equation is
$r(\theta)=\dfrac{b}{\sqrt{1-(e\cos\theta)^2}}$
[![enter image description here][1]][1]
**Polar form relative to fo... |
$x$ and $y$ are variables in the set $\mathbb R$. With bijective functions $f$ and $g$, $x$ and $y$ respectively goes to $a$ and $b$ which is in the set $[-1, 1]$. So, $f(x)=a$ and $g(y)=b$.For example, $f$ and $g$ can be hyperbolic tangent function.
Is there $f$ and $g$ that satisfies the followings?
1. The set ... |
My only problem is this step:
From making this:
$\sum_{n\in\mathbb Z}(-1)^nz^nq^{n^2}\sum_{n\in\mathbb Z}z^nq^{n^2}={A^2(q)\over A(q^2)}\sum_{n\in\mathbb Z}(-1)^nz^{2n}q^{2n^2}\tag1$
To this:
$\sum_{n\in\mathbb Z}(-1)^nq^{2n^2}={A^2(q)\over A(q^2)}\tag2$
in this proof:
https://math.stackexchange.com/que... |
**Polar form relative to center**
In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate θ measured from the major axis, the ellipse's equation is
$r(\theta)=\dfrac{b}{\sqrt{1-(e\cos\theta)^2}}$
[![enter image description here][1]][1]
**Polar form relative to fo... |
Here, I **first** provide a proof with required details based on relative entropy, and **then** I discuss how your **main issue** on dropping the expectation constraint when using the KKT method can be solved.
--------------
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$\color{blue}{H(X)=\frac{1}{2} \log\l... |
What is the importance of the density of $C_c^\infty(\mathbb R^n)$ and $L_c^\infty(\mathbb R^n)$ on $L^p(\mathbb R^n)$, for every $1 \leq p < \infty$? |
I'm currently on a calculus book. It starts with inequalities and the rules of it. So there is my question:
1. If **a < b** and **b < c**, then **a < c**.
2. If c is any number and **a < b**, then it is also true that **a + c < b + c** and **a - c < b - c**.
3. If **a < b** and **c < d** then **a + c < b + d**.... |
I'm currently on a calculus book. It starts with inequalities and the rules of it. So there is my question:
1. If **a < b** and **b < c**, then **a < c**.
2. If c is any number and **a < b**, then it is also true that **a + c < b + c** and **a - c < b - c**.
3. If **a < b** and **c < d** then **a + c < b + d**.... |
I'm currently on a calculus book. It starts with inequalities and the rules of it. So there is my question:
1. If **a < b** and **b < c**, then **a < c**.
2. If **c** is any number and **a < b**, then it is also true that **a + c < b + c** and **a - c < b - c**.
3. If **a < b** and **c < d** then **a + c < b + ... |
A simple proof that there are prime gaps of size $n$ for every $n$ can be seen in the first answer to [this question][1]. I consider these examples of prime gaps of length $n$ to be "trivial".
My question is:
> Can it be proven that there are prime gaps of length $n$ starting from
> integer $q,$ i.e. $q, q+1, q+... |
Let the set S = {0,1,2,3,4,5,6,7,8,9}. Let T be a subset family of S. Every element of T has element number 5. If two distinct elements A and B of T always satisfy |A∩B|≤3, what is the maximum number of elements in T?
Translated by DeepL |
In an Electrical Engineering Textbook I am given this function:
$$ H_C(j\omega) = \frac{10^6j\omega}{10^{10} + -\omega^2 + 10^5j\omega} $$
where $j = \sqrt{-1}, \omega = 2\pi f, f=frequency (Hz) $
The text book goes on to say 'the high frequency asymptote is
$$ H(j\omega) = \frac{10^6}{j\omega} $$'
If ... |
Merita has decided to play in The Clothing Combo Contest. First, she will randomly choose from a pair of brown, purple, blue, green, or black pants. Next, she will randomly choose from a black, brown, or green shirt.
If both the shirt and pants are a color that starts with a "B", she will win 10 dollars. If only one o... |
A simple proof that there are prime gaps of size $n$ for every $n$ can be seen in the first answer to [this question][1]. I consider these examples of prime gaps of length $n$ to be "trivial".
My question is:
> Can it be proven that for every $n,$ there are prime gaps of length $n$ starting from
> integer $q,$ i... |
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (ie $A$ dominates $R$ in the sense that it's maximal ideal contains the image of maximal ideal of $R$) map of local rings.
**#Edit:** As Giorgio Genovesi pointed out, essentially the "interesting" case appear where $... |
Here, I **first** provide a proof with required details based on relative entropy, and **then** I discuss how your **main issue** on dropping the expectation constraint when using the KKT method can be solved.
--------------
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$\color{blue}{H(X)=\frac{1}{2} \log\l... |
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (ie $A$ dominates $R$ in the sense that it's maximal ideal contains the image of maximal ideal of $R$) map of local rings.
**#Edit:** As Giorgio Genovesi pointed out, essentially the "interesting" case appear where $... |
Here, I **first** provide a proof with required details based on relative entropy, and **then** I discuss how your **main issue** on dropping the expectation constraint when using the KKT method can be solved.
--------------
Let $f_X \sim \mathcal N (\mu,\Sigma)$, for which $$\color{blue}{H(X)=\frac{1}{2} \log\l... |
Let $g:\mathbb{R}^n\to\mathbb{R}^m$ be linear defined by $g(x) = Ax$.
I'm looking for an analog of the adjoint relationship: $\langle\lambda,g(x)\rangle=\langle\lambda,Ax\rangle=\langle A^\top\lambda,x\rangle$ for the case where $g(x)$ is not linear.
For example, suppose we have that $h:\mathbb{R}^n\to\mathbb{R}^m$... |
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (ie $A$ dominates $R$ in the sense that it's maximal ideal contains the image of maximal ideal of $R$) map of local rings.
**#Edit:** As Giorgio Genovesi pointed out, essentially the "interesting" case appear where $... |
Here is the question that I want an answer to it using commutative diagrams (as small number of them as possible):
Prove that the free product of $ \langle g_1, \dots ,g_m | r_1, \dots ,r_n \rangle$ and $ \langle h_1, \dots ,h_k| s_1, \dots ,s_l \rangle$ is isomorphic to $\langle g_1, \dots ,g_m,h_1, \dots ,h_k|r_1... |
> Prove $S^1, S^2$ are not homeomorphic.
Assume there is a homeomorphism $f: S^1\to S^2.$\
I have proved that if we remove a point $p$ in $S^1$, $S^1\setminus\{p\}$ is connected.
I wonder if we remove a point $p$ in $S^2$, is $S^2\setminus\{p\}$ is connected? \
Moreover, I found this [link][1], but I don't unders... |
Let $R$ be discrete valuation ring with field of fractions $K$, and $\phi:R \to A$ a *finite, local* (ie $A$ dominates $R$ in the sense that it's maximal ideal contains the image of maximal ideal of $R$) map of local rings.
**#Edit:** As Giorgio Genovesi pointed out, essentially the "interesting" case appear where $... |
I am currently studying spin geometry and when dealing with the spin groups I came across the notion of a "Lie group double covering", but the book I am following does not define this notion and I couldn't find a reference on this topic. Could someone point me to such a reference? |
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