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I'm trying to do an integral which arises from another integral that is simple in polar coordinates (so I can work out the answer that way). However I am using cartesian coordinates and trying to see if I get the same answer. The integral (with $G>0$) is,
$$I(G)=4\sqrt{1+G^2}\int_{0}^{G}\text{d} x\,\frac{x \tan^{-1}... |
**Expression:**
[![enter image description here][1]][1]
**My attempt:**
1. $(xy)' + (yz)$
2. $(x'+y') + (yz) \quad \textit{After applying de Morgan's Axiom}$
3. $x' + (y' + yz) \quad \textit{After applying 1st Distributive Axiom}$
4. $x'+ (y' + y) + (y'z) \quad \textit{Rewriting}$
5. $x' + 1 + (y'z... |
It is well known that every column sum in the character table of a finite group is an integer.
Nevertheless, it is not easy to find an example where a column sum is negative. The smallest example has order 96.
For example, one can take SmallGroup(96,3) in the notation of GAP, whose character table is presented under ... |
Character table of groups of order 96, coincidence or there is a reason? |
I got this integral $\int_{1/2}^1t^{{1\over f(x)}-3} \sin t^2dt$ (note: $t$ is raised to the ${1\over f(x)}-3$ in case its hard to see.) I am really struggling to find the answer to this, I'm not even really sure how to approach it
----------
The original question was $\lim_{x\to 0^+} \int_{1/2}^1t^{\frac{1}{f(x... |
The goal is to find $m$ swaps $s_1, s_2, \dots, s_m$ such that any $n$-permutation can be encoded as a binary sequence of length $m$, $x_1, x_2, \dots, x_m$, where $x_i$ indicates whether to perform the $s_i$ swap (if $x_i=1$) or skip it (if $x_i=0$).
For example, if we list the swaps as $s_1=(1,2),\; s_2=(2,3),\; s... |
I've been trying to solve the following question:
>Show that the polynomial ring $\mathbb{Z}[x,y,z]$ is a finitely generated module over its subring generated by the following three elements:$$x+y+z,\ xy+xz+yz,\ xyz.$$
Could you give me a hint? Say $R$ is the ring generated by the above three elements. My idea w... |
Consider a function on R defined by f(x) = $\frac{x}{1+\left|x\right|}$
We can prove through substitution that f(x1)= f(x2) ⇒ x1 = x2 and hence, the function is injective.
However, I am confused as to how this can be so.
The domain of the above function is R, the set of real numbers (which is an infinite set).... |
How can a function from an infinite set to a subset be one-to-one? |
If we agree to define $S=\lim_{n\to \infty}\sum_0^na_n$,then, only convergent series can have meaningful $S$. But if we decide to define $S$, to be the Caesaro mean, now we can also assign a real number to $S$ for some divergent series as well. My question is such relaxation of the definition frown upon in the mathemat... |
Are Caesaro summation and Abelian summation frown upon in mathematical community? |
A planar graph is *triangulated* if its faces are bounded by three edges. A triangle of a planar graph is a *separating triangle* if it does not form the boundary of a face. That is, a separating triangle has vertices both inside it and outside it; therefore its removal separates the graph.
Given planar graphs $H_1$... |
Does the operation stitch preserve non-Hamiltonicity? |
Let $f(x,y)=(x^2-y^2,2xy),$ where $x>0,y>0.$ Let $g$ be the inverse of $f$ in a neighbourhood of $f(2,1)$. Then the determinant of the Jacobian matrix of $g$ at $f(2,1)$ is equal to...............?
**Solution:**
Let $f(x,y)=(f_1(x,y),f_2(x,y))$,where $f_1(x,y)=x^2-y^2$ and $f_2(x,y)=2xy$,then the Jocobian of the... |
Is the determinant of the Jacobian matrix of $g$ at $f(2,1)$ correct? |
I've learned about differentiability of several variables. I know that the requirement of differentiable is that its partial derivatives are continuous. As for counterexample,
$xy\sin(1/(x^2+y^2))$ is differentiable but its partial derivatives are not continuous.
And if it is differentiable then we can use chain r... |
What is the regular language for L = {w | w has even length, and starts and ends with the same symbol}? |
For $x^2+y^2=4$,
$$z=2\sqrt{4}\Rightarrow z=4$$
Since the radius of the basis is $2$, then the volume of the cone is
$$V=\frac{\pi\cdot2^2\cdot4}{3}\Rightarrow V=\frac{16\pi}{3}$$
However, using double integral and polar coordinates, I get
$$V=\int_{0}^{2\pi}\int_{0}^{2} 2r^2 dr d\theta=\frac{32\pi}{3}$$
Wha... |
Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. Let $x$ be a real and let
$$
f(x) = P\left(\frac{1}{a-b+x} + \frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right)
$$
The plot of $x$ vs $f(x)$ is show below along ... |
In a triangle if $P\left(\frac{1}{a-b+x}+\frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right)=\frac{1}{2}$ then $x=\pm\frac{3}{4}, \pm\frac{3}{5}$ or $0$? |
Let $f$ be an increasing and continuous real function with exactly one root. Let $\tau=\min\left\{k \in \mathbb{Z}, f(k) \geq 0\right\}$ be the first integer such that f is nonnegative (i.e. the first integer above the root). Now, let $f_n$ be a sequence of increasing, continuous real functions that converge pointwise ... |
I am looking for some geometrical insights into the Reynolds transport theorem. Firstly, the case when looking at the transport of scalar quantities is rather trivial, let $\rho \omega$ be a spatial k-form on $M^k$, the integral over time-dependent region $\Omega(t) = \psi_{t,t_0} (\Omega(t_0)) \subset M$ then reads
... |
New to optimization, trying to understand the characteristics of a problem I am in the process of modelling, and looking to solve in Python. Given is a decision variable **k**,the sum of **N** other decision variables **c**, and two constants, **a** and **T**.
I want to maximize **k** subject to **(a + c) * k <= T*... |
Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$.
**Direct problem** - If we know the PDF $p_X$, then the PDF $p_Y$ is for... |
The linear system: $Ax=b$
I know that if the system is underdetermined, there will be closed-form solutions. But if $b\sim any~distribution$, are there closed-form solutions?
More detailedly, this is what I am trying to solve:
$$
\arg \min_x \lVert x - X\rVert_2, X\sim any~distribution, X, x \in \mathbb{R}^{n\t... |
If the volume scales V (km^3) with the area A (km^2) by a relationship V = 0.67*A^(1.262). Then how would the volume change if an area of 1161 km^2 decreases by 210.76 km^2? |
I try to understand Dedekind's cuts and I'm stuck where he proves next statement:
Let $\lambda \in \mathbb{Z}_+$, $\sqrt{D} \notin \mathbb{Z}$:
$$
\lambda^2 < D < (\lambda+1)^2.
$$
Let's take $r \in \mathbb{R}_+$ and if $r^2 > D$ then $r$ belongs to $A_2$ and if $r^2 < D$ then it belongs to $A_1$.
Dede... |
I try to understand Dedekind's cuts and I'm stuck where he proves next statement:
Let $\lambda \in \mathbb{Z}_+$, $\sqrt{D} \notin \mathbb{Z}$:
$$
\lambda^2 < D < (\lambda+1)^2.
$$
Let's take $r \in \mathbb{R}_+$ and if $r^2 > D$ then $r$ belongs to $A_2$ and if $r^2 < D$ then it belongs to $A_1$.
Dede... |
A couple of remarks, that hopefully are able to shed some light on your question(s)
- I would strongly advise against using the sign of the Lagrange multiplier to check if its a minimum or maximum. Lets say you have the constraint $x=y+2$. Would you use $g=y-x+2$ or $g=x-y-2$?
- You made a mistake with your seco... |
I try to understand Dedekind's cuts and I'm stuck where he proves next statement:
Let $\lambda \in \mathbb{Z}_+$, $\sqrt{D} \notin \mathbb{Z}$:
$$
\lambda^2 < D < (\lambda+1)^2.
$$
Let's take $r \in \mathbb{R}_+$ and if $r^2 > D$ then $r$ belongs to $A_2$ and if $r^2 < D$ then it belongs to $A_1$.
Dede... |
Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. Let $x$ be a real and let
$$
f(x) = P\left(\frac{1}{a-b+x} + \frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right)
$$
The plot of $x$ vs $f(x)$ is show below along ... |
When a spherical cap is cut by a plane y=y1 that is perpendicular on the plane that made the sperical cap z=z1, I managed to calculate the volumes of the resulting two pieces by the following integral:
\int\limits_y1^{y2} {\int\limits_{-\sqrt {1 - {\y^2}-{\z1^2}} }^{\sqrt {1 - {\y^2}-{\z1^2}} }{\sqrt {1 - {\x^2}-{\y... |
I am trying to find a way to randomly spawn objects within the FOV (field of view) of a player camera inside a video game. All the objects will be spawned at the same z-height so I will only be looking at a 2D coordinate system with a x- and y-axis.
The first case I looked at is when the player FOV is 90 degrees. I... |
Will all points inside the FOV be reachable by this vector? |
I am trying to find a way to randomly spawn objects within the FOV (field of view) of a player camera inside a video game. All the objects will be spawned at the same z-height so I will only be looking at a 2D coordinate system with a x- and y-axis.
The first case I looked at is when the player FOV is 90 degrees. I... |
Let $\mathbb{Z}_p$ be the ring of $p$-adic integers and $\overline{\mathbb{Q}}_p$ a fixed algebraic closure of the field $\mathbb{Q}_p$ of $p$-adic numbers. Let $R$ be a complete Noetherian local $\mathbb{Z}_p$-algebra. It is not hard to see that if $R$ is a finite $\mathbb{Z}_p$-algebra, then the set $\operatorname{Ho... |
can you guys help me check the fleshed out logic of 'my' proof of the cancellation law for the natural numbers? It's in Peano's system of the natural numbers with the recursive definitions of addition and multiplication. I was trying to prove it without trichotomy and stumbled across a proof given by Mauro Allegranza [... |
> Let $f$ be holomorphic, bounded on $|z|>1$ and continuous on $|z|\geq 1$. Prove that $|f|$ attains maximum on $|z|=1$, i.e
> $$\sup\limits_{|z|\geq 1}|f(z)|=\max\limits_{|z|=1}|f(z)|.$$
My attempt:\
Consider $g(z)=f\left(\dfrac{1}{z}\right)$. From hypothesis, we know that $g$ is holomorphic, bounded on $0<|z|<1$... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
> If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.
*Proof.* We use the already proven fact that $b^{2a}-4a^b$ is a perfect square iff there are positive integers $c,d... |
I see two definitions of “Goppa codes”. The first is through polynomials, as in [this thesis](http://infosec.seu.edu.cn/space/kangwei/senior_thesis/Goppa.pdf). This is also true in some of the core textbooks in the field, such as van Lint 3rd edition (1999), and MacWilliams and Sloane (1977).
However, another defini... |
A planar graph is *triangulated* if its faces are bounded by three edges. A triangle of a planar graph is a *separating triangle* if it does not form the boundary of a face. That is, a separating triangle has vertices both inside it and outside it; therefore its removal separates the graph.
Given planar graphs $H_1$... |
For a $k-cycle$ $(1, 2, 3, 4, 5)$ and a transposition $(1, 2)$.Calculation of "$(1, 2, 3, 4, 5) \cdot (1, 2)$". We all know that groups satisfy the associative property, i.e. $(a \cdot b) \cdot c=a \cdot (b \cdot c)$.
I will get $ \left \{ 1, 2, 3, 4, 5 \right \} \longmapsto \left \{ 2, 3, 4, 5, 1 \right \} $, wh... |
How to calculate "$(1, 2, 3, 4, 5) \cdot (1, 2)$"? |
I considered this problem, but, I cannot solve.
$a,b,c$ are a real number.
$k$ is defined as follows.
$$k=\frac{(a^2+b^2+c^2)(a^3+b^3+c^3)}{(a^5+b^5+c^5)}$$
if $ab+bc+ca=0$, How many possible values of $k$ are there?
Firstly, I substitute $c=-\frac{ab}{a+b}$ into the equation of $k$, when the value of a+b ... |
How many possible values of $k$ are there? |
A Sturm-Liouville's problem is often presented with the following boundary values:
\begin{align}
\alpha_1\,y(a) + \beta_1\, y'(a) &= 0 \\
\alpha_2\,y(b) + \beta_2\, y'(b) &= 0
\end{align}
Would it have sense to use higher derivatives ? for instance:
\begin{align}
\alpha_1\,y(a) + \beta_1\, y'(a) + \gamma_1\, y... |
I am trying to solve for X the following: $XA + B(X \otimes Y)Q=C$ is there a closed form solution to this? Thank you in advance. |
Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ be a continuous function. Take $x,y \in \mathbb{R}^{n}$ with $f(x) < f(y)$ and let $L = \{\alpha x + (1-\alpha)y$ : $\alpha \in [0,1]\}$ be the straight line between these two points. Is it true that the continuity of $f$ implies that $f(L) \subset [f(x), f(y)]$? |
Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ be a continuous function. Take $x,y \in \mathbb{R}^{n}$ with $f(x) < f(y)$ and let $L = \{\alpha x + (1-\alpha)y$ : $\alpha \in [0,1]\}$ be the straight line between $x$ and $y$. Is it true that the continuity of $f$ implies that $f(L) \subset [f(x), f(y)]$? |
When a spherical cap is cut by a plane $y=y_1$ that is perpendicular on the plane that made the sperical cap $z=z_1$, I managed to calculate the volumes of the resulting two pieces by the following integral:
$$\int\limits_{y_1}^{y_2} {\int\limits_{-\sqrt {1 - {y^2}-{z_1^2}} }^{\sqrt {1 - {y^2}-{z_1^2}} }{\sqrt {1 - ... |
I am trying to solve for X the following: $XA + B(X \otimes Y)Q=C$ is there a closed form solution to this? Thank you in advance.
Note: This problem iteratively is part of optimization problem-solving the proposed equation is the result of the derivation of the previous problem, what I need now is to isolate the v... |
I am trying to solve for X the following: $XA + B(X \otimes Y)Q=C$ is there a closed form solution to this? Thank you in advance.
Note: This problem is part of optimization problem-solving iteratively the proposed equation is the result of the derivation of the previous problem, What I need now is to isolate the ... |
Let $C(0,0,\sqrt{2})$ and $F(0,\sqrt{2},0)$ be two points in $\Bbb R^3$.\
$AF,BF$ are arbitrary perpendicular lines through $F$ on the plane $y=\sqrt{2}$.\
These lines project to the lines $DF,EF$ under the perspective projection centered at $C$ onto $xy$-plane.\
Why are the lines $DF,EF$ perpendicular?
[![enter ... |
When a spherical cap is cut by a plane $y=y_1$ that is perpendicular on the plane that made the sperical cap $z=z_1$, I managed to calculate the volumes of the resulting two pieces by the following integral:
$$\int\limits_{y_1}^{y_2} {\int\limits_{-\sqrt {1 - {y^2}-{z_1^2}} }^{\sqrt {1 - {y^2}-{z_1^2}} }{\sqrt {1 - ... |
A complex Hadamard matrix of order $N$ is a square matrix $A = [a_{ij}]$ of size $N$ with entries $|a_{ij}| = 1$ for all $i,j$; and further that $A^\dagger A= N I$ (or $A/\sqrt{N} $ is a unitary matrix).
Further, a complex Hadamard matrix of Buston-type is such that $a_{ij}$ is the $q$-th root of unity for some posi... |
A very important example of this is the incredible expressive power of first-order arithmetic. For instance, in the structure $(\mathbb{N},+,\cdot)$ it is possible to express the fundamental theorem of arithmetic as a first-order sentence. At first glance this seems very unlikely: to state that for all $n\in\mathbb{N... |
To solve this, I think we need to use combinatorial reasoning.=
Consider a bit string of length \( n \). There are \( 2^n \) possible bit strings of this length because each bit can independently be either 0 or 1.
Let's denote:
- \( k \) as the number of 0's in the string
- \( n \) as the length of the bit stri... |
How many bit strings of length n contain more 0’s than 1’s? |
A complex Hadamard matrix of order $N$ is a square matrix $A = [a_{ij}]$ of size $N$ with entries $|a_{ij}| = 1$ for all $i,j$; and further that $A^\dagger A= N I$ (or $A/\sqrt{N} $ is a unitary matrix).
Further, a complex Hadamard matrix of Buston-type is such that $a_{ij}$ is the $q$-th root of unity for some posi... |
I am trying to calculate the following integral:
$$
\int\limits_{-\infty}^{\infty}\frac{x \arctan(x-a)}{(x-b)^2+y^2}d x
$$
where $a$, $b$ are positive constants and also y is a positive variable. (Actually the integral comes from Poisson's formula for the half-space). I have calculated a similar integral without $x... |
Let $A,B$ be self-adjoint operators on a Hilbert space $\mathcal{H}$ with $D(A) \subset D(B)$ and $A \leq B$. By the spectral theorem, we write :
$A = \int_{\mathbb{R}} \lambda \,E^A(d\lambda) \hspace{1cm} B = \int_{\mathbb{R}} \lambda \, E^B(d\lambda)$
Does one have : $E^A([n,+\infty[) \leq E^B([n, +\infty[)$ ?... |
Here we deal with the problem of Histogram Equalization.
Let $\Omega$ be the set of all pixels in an image, but you can just treat it as some arbitrary set in this case.\
Let $f: \Omega \rightarrow \{0,...,255\}$ be a map from pixels to pixel intensities.
We then have the pixel intensity distribution given by:... |
Let $(a,b,c)$ be the sides of a triangle and let its circumradius be $1$. Is it true that
$$
a^c + b^c - c^c < \pi
$$
**My progress**: In the special case where at least two of the three sides are equal, I have been able to show that the upper bound is about $3.13861$. If two sides are equal and the third side ... |
> If $f$ is integrable on $[0, 1]$, does that imply that an antiderivative of $f$ is differentiable on $[0, 1]$?
I'm sorry if this comes across as too basic of a question, but I'm struggling to accept this as obvious. The Fundamental Theorem of Calculus seems to support this statement, but perhaps I have yet to full... |
If $f$ is integrable on $[0, 1]$, does that imply that an antiderivative of $f$ is differentiable on $[0, 1]$? |
I am trying to calculate the determinant of a matrix that looks something like a Vandermonde matrix.
A = $\begin{bmatrix}a & a^2 & a^3 & a^4\\b & b^2 & b^3 & b^4 \\ c & c^2 & c^3 & c^4 \\ d& d^2 & d^3 & d^4 \end{bmatrix}$
I have currently performed row reduction by subtracting $R1$ from all other rows and then ma... |
How to prove determinant? |
When a spherical cap is cut by a plane $y=y_1$ that is perpendicular on the plane that made the sperical cap $z=z_1$, I managed to calculate the volumes of the resulting two pieces by the following integral:
$$\int\limits_{y_1}^{y_2} {\int\limits_{-\sqrt {1 - {y^2}-{z_1^2}} }^{\sqrt {1 - {y^2}-{z_1^2}} }{\sqrt {1 - ... |
When a spherical cap is cut by a plane $y=y_1$ that is perpendicular on the plane that made the sperical cap $z=z_1$, I managed to calculate the volumes of the resulting two pieces of the spherical cap by the following integral:
$$\int\limits_{y_1}^{y_2} {\int\limits_{-\sqrt {1 - {y^2}-{z_1^2}} }^{\sqrt {1 - {y^2}-{... |
I don't know if the next statement is true or false:
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $A,B$ and $C$ be events in $\mathcal{F}$ such that $P(A)>0$. If $P(B|A)>P(B)$ and $P(C|A)>P(C)$, then $P(B\cap C|A)\geq P(B\cap C)$.
Neither I haven't been able to prove it, nor I haven't been able to ... |
>Let $K$ be a imperfect field. Let $L/K$ be a finite field extension. Is $L$ imperfect?
Suppose that $L/K$ is separable. Is $L$ imperfect?
Suppose that $L/K$ is Galois. Is $L$ imperfect?
I'm looking for examples, counterexamples, explanations, and proofs of course.
**Note:** The converse holds. If $K$ is pe... |
Is a finite field extension of an imperfect field imperfect? |
Here we deal with the problem of Histogram Equalization.
Let $\Omega$ be the set of all pixels in an image, but you can just treat it as some arbitrary set in this case.\
Let $f: \Omega \rightarrow \{0,...,255\}$ be a map from pixels to pixel intensities.
We then have the pixel intensity distribution given by:... |
Consider two discrete binary random variables A and B. If $P(A=1,B=0) = P(A=1)P(B=0)$, can we conclusively say that A and B are independent?
My response is we cannot conclusively say they are independent because the $P(AB) = P(A)P(B)$ should hold for all possible values of A and B. |
I considered this problem, but, I cannot solve.
$a,b,c$ are a real numbers with $(a,b,c)\neq (0,0,0)$.
$k$ is defined as follows.
$$k=\frac{(a^2+b^2+c^2)(a^3+b^3+c^3)}{(a^5+b^5+c^5)}$$
if $ab+bc+ca=0$, How many possible values of $k$ are there?
Firstly, I substitute $c=-\frac{ab}{a+b}$ into the equation of... |
I'm trying to fill in some details in Do Carmo's curves and surfaces proof on p.149,
> If all points of a connected surface $S$ are umbilical points, then $S$ is either contained in a sphere or in a plane.
also discussed [here][1]. We have proven the local version, i.e., for every point in the surface $S$ there ... |
I have the following PDE:
Let $G = (a, b)$ and $J = (0, 1)$. Consider the Dirichlet problem with general coefficient functions $\alpha(x)$, $\beta(x)$, $\gamma(x)$
\begin{equation}
\begin{cases}
\partial_t u - \partial_x (\alpha(x) \partial_x u) + \beta(x) \partial_x u + \gamma(x) u = f(t, x), \text{ in $J \times G... |
In the complex space, I understand that we need to conjugate one of the vectors in the dot product to avoid getting that <**v**,**v**> ≤ 0. What I am unable to understand, is how taking the dot product between **w** as shown in *picture 1* and **v** conjugated still gives the correct dot product? How can the dot produc... |
Why does conjugating one of the complex vectors in the dot product, still give the correct dot product? |
#### Problems ####
1. There are multiple problems and one of them is that $H$ has many difrent definitions.
2. We call $\operatorname{H}$ a [semi derivative](https://mathworld.wolfram.com/Semiderivative.html) and $\operatorname{H}\left( \text{constant} \right)$ does not have to be a constant (e.g. see [Riemann-... |
I watched a YouTube video on factoring (namely, [100 trinomial factoring (Dedicated to Mr. Hill) by blackpenredpen](https://youtu.be/LmbqB_pfcDU?si=8q-XMm-x83qKkC5C&t=735 "this is where the method I talk about is first shown")) and was curious about the math behind it. Is there anything wrong with my logic or any wa... |
Let $C(0,0,\sqrt{2})$ and $F(0,\sqrt{2},0)$ be two points in $\Bbb R^3$.\
$AF,BF$ are arbitrary perpendicular lines through $F$ on the plane $y=\sqrt{2}$.\
These lines project to the lines $DF,EF$ under the perspective projection centered at $C$ onto $xy$-plane.\
Why are the lines $DF,EF$ perpendicular?
For a gen... |
I watched a YouTube video on factoring (namely, [100 trinomial factoring (Dedicated to Mr. Hill) by blackpenredpen](https://youtu.be/LmbqB_pfcDU?si=8q-XMm-x83qKkC5C&t=735 "this is where the method I talk about is first shown")) and was curious about the math behind the method he first shows around 12:15. Is there an... |
In the complex space, I understand that we need to conjugate one of the vectors in the dot product to avoid getting that <**v**,**v**> ≤ 0. What I am unable to understand, is how taking the dot product between **w** as shown in *picture 1* and **v** conjugated still gives the correct dot product? How can the dot produc... |
I need to solve a convex program of the following form, where the variables are $x_1,\ldots,x_n$:
\begin{align}
x \geq 0
\\
A x \geq b
\\
\sum_{i=1}^n \log(x_i) \geq 1
\end{align}
I want to use the ellipsoid method, so I need a [separation oracle][1]. This is a function that, given a vector $x$... |
Why is $p$ (or $p + 1$) not the upper bound of the number of solutions mod $p$ (finite field)? |
In the complex space, I understand that we need to conjugate one of the vectors in the dot product to avoid getting that <**v**,**v**> ≤ 0. What I am unable to understand, is how taking the dot product between **w** as shown in *picture 1* and **v** conjugated still gives the correct dot product? How can the dot produc... |
Why does the poisson regression beta coefficient ML estimate have no closed form solution? |
I am looking for methods to solve the following minimization. Let $A\in \mathbb{R}^{n\times k}$, $B\in \mathbb{R}^{n\times m}$, $C\in \mathbb{R}^{m\times m}$ and $E\in \mathbb{R}^{m\times k}$, where $n>m>k$. Additionally, $C$ is positive definite.
$$G(d)=||A-B(C+\text{diag}(d))^{-1}E||^2_F$$
Is there any hopes of fin... |
I'm plotting stock market charts with matplotlib in Python, where the x-axis represents dates with 0-indexed numbers, and the y-axis shows varying price scales for different stocks.
I draw lines between 2 points of varying lengths on each of these charts and calculated their slope.
$$ m = \frac{{y2 - y1}\text{ }(... |
Why does conjugating one of the complex vectors in the inner product, still give the correct dot product? |
Instead of looking at the collection $X_1,...,X_n$ we can look at the vector $Y = (X_1,...,X_n)$ for which $\sigma(Y) = \sigma(X_1,...,X_n)$, so it is enough to consider case of single random variable $Y$ (but taking values in some product space $\mathbb R^n$ or any different).
Thus, let us assume that $(\Omega,\mat... |
Suppose $\omega$ is a strictly semi-finite weight on a von Neumann algebra $M$ and $N$ is the von Neumann subalgebra of $M$.
Is $\omega|_N$ a strictly semifinite weight on $N$? |
>Let $C(0,0,\sqrt{2})$ and $F(0,\sqrt{2},0)$ be two points in $\Bbb R^3$.\
$AF,BF$ are arbitrary perpendicular lines through $F$ on the plane $y=\sqrt{2}$.\
These lines project to the lines $DF,EF$ under the perspective projection centered at $C$ onto $xy$-plane.\
Then the lines $DF,EF$ perpendicular?
This questi... |
Consider the Gaussian white noise model given by,
$$ dX_{t_{1},...,t_{d}}=f(t_{1},...,t_{d})dt_{1}...dt_{d}+\theta dW_{t_{1},...,t_{d}}$$
with $W$ a $d-$parameters Wiener field, $f:[0,1]^{d}\rightarrow \mathbb{R}$ a signal and $\theta\geq0$ the level of noise. Furthermore suppose that $f$ is $(L,\alpha)$-Hölder conti... |
What are the number of possible combinations for rolling N indistinguishable dice obtaining a total punctuation p? |
I am trying to compute some quantity like a partition function for a harmonic oscillator type system and at the end of the computation I am coming across a sum that looks like,
\begin{align}
S=\sum_{n=0}^{\infty}e^{an +bn^2}
\end{align}
with $a,b \in \mathbb{R}$. How do I find the sum? Any help regarding the sa... |
How to find $\sum_{n=0}^{\infty}e^{a n + b n^2}$? |
Consider the Gaussian white noise model given by,
$$ dX_{t_{1},...,t_{d}}=f(t_{1},...,t_{d})dt_{1}...dt_{d}+\theta dW_{t_{1},...,t_{d}}$$
with $W$ a $d-$parameters Wiener field, $f:[0,1]^{d}\rightarrow \mathbb{R}$ a signal and $\theta\geq0$ the level of noise. Furthermore suppose that $f$ is $(L,\alpha)$-Hölder conti... |
My updated question is
$A$ is the point $(0,0,0)\in\Bbb R^3$.\
$C$ is the point $(1,0,0)$.\
$S$ is the point $(0,1,\tan(\alpha))$.\
A line through $S$ intersects the plane $z=0$ at $B$ and intersects the plane $z=\tan(2\alpha)y$ at $D$.\
Then $\angle CAD=\angle BAC$.
[![enter image description here][1]][1]
... |
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