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How to prove that {$X \in M : det (AX)=0$} is not a subspace of M? |
To assist OP & at the risk of downvoting , I will add a very simple answer which the Particular numbers involved here allow (The Duplicates do not seem to allow this method) . . . .
Let $X=\sqrt{18} - \sqrt{12}-\sqrt{45} + \sqrt{6}=3\sqrt{2} - 2\sqrt{3}-3\sqrt{5} + \sqrt{2}\sqrt{3}$
Hence $X - 3\sqrt{2} + 2... |
* $f(x) = 2[x]-x$,
* $[x]$ is the floor function,
* the domain of the function is $[-1,2]$,
* its codomain is $\Bbb R$
$$
f(x) = \begin{cases}
-2-x & \text{if } -1 \leq x < 0 \\
-x & \text{if } 0 \leq x < 1 \\
2-x & \text{if } 1 \leq x < 2 \\
4-x & \text{if } x = 2 \\
\end{cases}
$$
a diagonal line is forme... |
I am going to evaluate the integral
$$\int_{\mathbb{R}} \int^T_0 f(x,t) \dfrac{1}{\sqrt{t}}e^{-x^2/2t}dt$$
for some nice enough function $f$. The integrand has a very strong singularity at $t=0$. The function integral2 gives me warnings so often. Do we have some good function for an accurate answer? |
Assume $1\leq p <\infty$. It is known that for all $q>p$
$$
S^{p}
\subset S^{p,\infty}
\subset S^{q}.
$$
Out by curioisity, it is true that $\cap_{q>p} S^{q}=S^{p,\infty}$ ?
Recall that
$$
S^{p,\infty}
=\Big\{T \in S^\infty : s_n(T)=O_{n \to \infty}\Big(n^{-\frac{1}{p}}\Big) \Big\}.
$$ |
Some intersection of Schatten spaces gives a weak Schatten space? |
I have a conceptual problem when we extend the correlation to time series. I understand probability and statistics as a two way route. Either I begin from a random variable (r.v.) $X$ and sample from its probability distribution, or I observe a sample of values $x_1,...,x_n$ drawn from that distribution and I infer som... |
Let's contemplate the two possible definitions of triangulated subcategory that OP is hinting at. For simplicity, the translation functor of a triangulated category (that I shall denote as $[1]$) is always assumed to be an automorphism. The class of distinguished triangles of a (pre)triangulated category $\mathcal{D}$ ... |
**Background**
In the <a href="https://math.stackexchange.com/questions/170747/sum-equals-integral-identities-similar-to-int-01-fracdxxx-sum-n%20">following question</a> and references therein, a number of "Sum equals integral" identities are described.
For instance, we have $$ \sum_{n = -\infty}^{+\infty} {\rm ... |
I am reading [Understanding Machine Learning: From Theory to Algorithms][1] and I don't know how to prove the second part of their **Lemma B.10** (page 427).
Specifically, let $S=\{(x_i,y_i)\}_{i=1}^m$ be a sample that is drawn i.i.d from some distribution $D$ over $\mathcal{X}\times\mathcal{Y}.$ Let $l:\mathcal{Y}\... |
At first, it should be noted that Linear Programming (LP) **has benefited from** many mathematical tools, for a review, see e.g., [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
Moreover, due to its widespread usage, LP has **become a strong driver** for th... |
I must draw a diagram with all "relevant paths" to
- WIN - last four rolls are in order $1234$
- LOSS / GAME OVER - last four rolls are in order among $1235, 1236, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, 3456$.
The modelling probability space is the space of all infinite tuples with entries among $1,2,3,4... |
There is an idea that has been on my mind for a while, and I would like to share it so that it turns into a snowball. Perhaps it will be useful and attractive to engineering enthusiasts. I ask you to help me develop this model that I am thinking of...
Most geometric theorems are proven by putting several theorems to... |
I am reading [Understanding Machine Learning: From Theory to Algorithms][1] and I don't know how to prove the second part of their **Lemma B.10** (page 427).
Specifically, let $S=\{(x_i,y_i)\}_{i=1}^m$ be a sample that is drawn i.i.d from some distribution $D$ over $\mathcal{X}\times\mathcal{Y}.$ Let $l:\mathcal{Y}\... |
As a [well-known][1] fact, a matrix is circulant if and only if it has a representation of the form $F^{-1}DF$ where $D$ is a diagonal matrix and $F$ is a discrete Fourier transformation.
In other words, the class of matrices diagonalizable by FFT are circulant matrices.
Recall that the FFT algorithm has $\log_2(N)... |
What is the class of matrices diagonalizable by "generalized FFT"? |
Let $(X,d)$ compact metric space, $D=\{x_n: n \in \mathbb{N} \} $ a countable dense subset of $X$. For all n, let $f_n:X \rightarrow [0,1]$ defined by $$f_n(x)=\frac{1}{1+d(x_n,x)}$$
Prove for all $n \in \mathbb{N}$, $A, B$ closed subsets of $X$ such that $A \subsetneq B$ then $f_n(A) \subsetneq f_n(B)$
If $f_n(A... |
I have found two different definitions of the limit of a function ***f*** at a point ***a*** in very popular calculus/analysis texts.
In one, the function ***f*** is asked to be defined on an open interval containing the point ***a***, possibly excluding said point (*Stewart, Wade*).
The other definition, mor... |
Are there two contradictory definitions of the limit of a function at a point? |
I *think* the answer is "no" in general.
Suppose that $M$ is the interior of the Alexander horned sphere. How would you "complete" $M$ to a closed manifold $K$? This completion would need to contain the limit point of the 'arms', but that point would have no disk-like (or half-disk-like) neighborhood in $N$. |
I'm interested in tackling the following integral:
$$\int_{-\ln (2+\sqrt 5)}^{\ln (2+\sqrt 5)} \sqrt{4+\sinh^2(x)} dx$$
While I've attempted various techniques, it appears challenging to find a closed-form solution for this integral. I'm beginning to suspect that it might not have one.
Do you have any insights... |
All values that satisfy $x^2=2^x$ would satisfy $\ln(2)x^3 = x\ln(2)e^{x\ln(2)}$, and would therefore satisfy the relationship $W(\ln(2)x^3) = x\ln(2)$. The problem is that when I graph these functions, there are four real solutions to this relation: $x \approx -0.77$, and $x=4, 2, 0$. The first three are solutions to ... |
So I need to verify
$$\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$
Using Parsevals identity on $f(x)=x$ on $[0,2\pi)$. Does this mean if $a_n$ are the Fourier coefficients then
$$\sum_{n=-\infty}^\infty \vert a_n \vert^2 = \frac{1}{2\pi} \int_0^{2\pi} x^2 dx.$$
Where
$$a_n=\frac{1}{2\pi}\int_0^{2\p... |
I need to understand what follows:
Given
- $X \in \mathbb{R}^{T \times n_0}$
- $W \in \mathbb{R}^{n_0 \times n} : W_{ij}\sim \mathbb{N(0, \frac{1}{n_0})} \space i.i.d.$
- $\sigma : \mathbb{R} \rightarrow \mathbb{R} $ so that $\sigma(x) = max(0,x)$
- $S = \sigma(XW)$ where $\sigma(\cdot)$ is applied elemen... |
Compute $$\int \frac{\sin(x)\cos(x)}{x^2+1}dx$$
**Attempt**
$u=\sin(x)\Rightarrow du=\cos(x)dx$
$$\int\frac{u}{x^2+1}du$$
But, now I have both $u$ and $x$ in the function. How do I resolve?
|
- My understanding about fractional order system is that it can have poles on the right hand side of imaginary axis in s-plane and yet being stable. (statement 1)
There are other theorems about stability that cannot be true at the same time with statement 1.
- The second understanding is that Region Of Convergenc... |
Is the 6 dimensional $ (2,0) $ irrep of $ SU(3) $ maximal in $ SU(6) $?
For those of you who are interested in context, I started wondering this the other day when I tried to write down the maximal subgroups of $ SU(6) $. My guess so far is that the full list of maximal (proper closed) subgroups of $ SU(6) $ is:
... |
I want to solve for the line integral:
$$\tag{1}\oint \alpha\nabla \phi_i\cdot \hat{\textbf{n}} ds$$
on the square boundary: $(0\le x \le 1, 0), (1,0 \le y \le 1), (1 \le x \le 0, 1),(0,1\le y \le 0)$
Where the function $\phi_i = \sin(n\pi x)\sin(n\pi y)$ and $\alpha = 1$ on the boundary. I don't know how $\ha... |
Could I use Green's Theorem here? |
A parallelepiped is formed using three non collinear vectors whose magnitudes are 1, 2, 3. Angle between any of the vector with normal of the plane determined by other two is π/3. Then the ratio of the surface area of the parallelogram to the volume of the parallelogram is?
My method:
Heres a slightly rough figur... |
In my answer to a recent
[MSE question](https://math.stackexchange.com/q/4875434/)
I mention the elliptic curve
[LMFDB 1728.n4](https://www.lmfdb.org/EllipticCurve/Q/1728a1)
with an infinite order generator and integer points $(-1,\pm1)$. The
[elliptic divisibility sequence](
https://en.wikipedia.org/wiki/Ellipti... |
Choose $(v,v) \in \Delta(T_x(X) \times T_x(X))$, with $v \in T_x(X)$. Then there is a smooth curve $\gamma : I \to X$ such that $\gamma(0) = x$, $\gamma'(0) = v$. Now $(\gamma(t) , \gamma(t))$ is a smooth curve contained in the diagonal of $X \times X$. What happens when you differentiate the curve at $t = 0$? Prove th... |
At first, it should be noted that Linear Programming (LP) **has benefited from** many mathematical tools, for a review, see e.g., [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
Moreover, due to its widespread usage, LP has **become a strong driver** for th... |
Is there a simple way to find the $\gcd$ of $x^2y$ and $xy^2+1$?
I tried adding multiples of $x^2y$ to the other and vice-versa but I found no easy way to find the gcd. For another example I found $(xy,x^3y+1)=(xy,x^3y+1-x^3y)=1$ by adding $-x^2$ of the first to the second. |
It came accros my mind when doing an exercise in calculus. Consider the following inequalities:
$1\leq 2\leq 5$\
$1\leq 3\leq 5$
Adding the equaions up is totally fine. But subtracting them, we get:
$0\leq 2-3\leq 0$ ??
I fail to understand what is wrong with this step. If someone could formally explain it... |
What is wrong about this seemingly simple false proof? |
Let $B$ be a standard Brownian motion on $[0, T]$ and let $\tau$ be the (a.s. unique) moment at which it attains its maximum value: $M = \sup_{t \in [0, T]} B(t) = B (\tau)$.
There is a well-known formula for the joint density of $(M, B(T))$, but I cannot find a joint density of $(M, B(u))$ for $u < T$. Is it possib... |
Let $(X,d)$ compact metric space, $D=\{x_n: n \in \mathbb{N} \} $ a countable dense subset of $X$. For all n, let $f_n:X \rightarrow [0,1]$ defined by $$f_n(x)=\frac{1}{1+d(x_n,x)}$$
Prove exista $n \in \mathbb{N}$ such that for all $A, B$ closed subsets of $X$ such that $A \subsetneq B$ then $f_n(A) \subsetneq f_n(... |
A parallelepiped is formed using three non collinear vectors whose magnitudes are 1, 2, 3. Angle between any of the vector with normal of the plane determined by other two is π/3. Then the ratio of the surface area of the parallelogram to the volume of the parallelogram is?
My method:
Heres a slightly rough figur... |
$f$ is Riemann integrable on $[a,b]$, $\int_a^bf(x)^2>0$. Prove that there is a point of continuity $x$ on $[a,b]$ that $f(x) \neq 0$. I provided the following: almost every point of $[a,b]$ is a continuity point. $F(x) = \int_a^bf(x)^2>0$ is differentiable in every continuity point and in such point $F'(x) = f^2(x)$. ... |
In the context of defining discrepancy for a pair $(X,\Delta)$ Kollár-Mori in Birational Geometry of Algebraic Varieties say that for a birational morphism $f\colon Y\to X$ from a normal variety a divisor over $X$ is simply a prime divisor on $Y$. Then we use this terminology to define the discrepancy and total discrep... |
The dodecahedral graph is the Platonic graph corresponding to the connectivity of the vertices of a dodecahedron, with 20 nodes and 30 edges. Assuming all edges have a length (weight) of 1, I want to find the shortest path that
- visits every node at least once,
- traverses every edge at least once, and
- starts ... |
What is the shortest path that visits every node and edge in the dodecahedral graph? |
$\newcommand{\T}{\mathscr{T}}\newcommand{\C}{\mathscr{C}^\infty}$I've been enjoying Iversen's book on sheaf cohomology. He briefly mentions De Rham cohomology and a sheafy perspective on it but he doesn't mention if the canonical isomorphism is functorial; everyone says that it is, but with the sheaf perspective I'm no... |
For fields of characteristic $0$ one can derive from the Jordan normal form that the trace/determinant of an operator are equal to the sum/product of its eigenvalues. Does this fact remain true when the characteristic is $p>0$? I feel like this statement should be derivable from the definition of the characteristic pol... |
Let $X$ be a smooth projective geometrically integral variety of dimension
$m \geq 1$ over $\mathbb{F}_q$ and a finite subset of points $F \subseteq X$. Then there is a way to construct a smooth projective geometrically integral curve $Y \subseteq X$ s.t. $F \subseteq Y$.
My question is: Does this also work over an a... |
Let $f(x)$ be a cubic whose coefficient of the leading term is positive and $g(x) = e^{\sin(\pi x)} - 1$. The composite function $h(x) = g(f(x))$ is defined in the set of all numbers, and has a local maximum at $x=0$. In the open interval $(0,3)$ the function $h(x)$ intersects the line $y= 1$ seven times.
Given tha... |
Yes, these are indeed two different definitions of the limit of a function at a point. In my experience, your observations can be generalized: most introductory textbooks use the first definition (where the function is required to be defined in a punctured neighborhood of the point, and thus in particular one-sided lim... |
Is there a **non-constant** distribution $\mathbb{D}$ such that if $X_1, ..., X_n \sim \mathbb{D}$ are iid random variables with order statistics $Y_{(1)}, ..., Y_{(n)}$, then the distribution of $Y_{(i)} | Y_{(1)}, ..., Y_{(i-1)}$ is identical?
The constant distribution (say $X_i=1$ w.p $1$) satisfies this. My intu... |
Let $(R, m_R, \kappa_R) \subset (B, m_B, \kappa_B)$ be an extension of discrete valuation rings whose corresponding extension of fields of fractions $K :=\text{Frac}(R) \subset \text{Frac}(B) =: L$ is finite of degree $n:=[L:K]$.
It is well known that in general we have inequality $[L:K] \ge e_{B/R} \cdot [\kappa_B: \... |
Is there a closed-form expression for the series $\sum_{n=0}^\infty \frac{n x^n}{1 - \left( \frac{x}{1+x} \right)^n}$? |
I construct a finitely presented discrete group $G$ in GAP, a normal subgroup $H\triangleleft G$ of finite index $N$, and the factor group $G/H$ of order $N$. Assume for simplicity that $G$ has two generators $\gamma_1,\gamma_2$. For each coset $[g_k]\in G/H$, I wish to construct the $N\times N$ matrices of the left-re... |
Let the digits *a*, *b*, *c* be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?
I figured out that they want permutations of aaabbbccc such that it must contain atleast one of the fo... |
I am trying to show that the function $f(x,y) = x^2 + y^2 + |xy|$ is convex. So I can show that both $x^2 + y^2 + xy$ and $x^2 + y^2 - xy$ are convex because their hessian is a diagonally dominant matrix with strictly positive entries on the diagonal. So this shows that my function is convex within all 4 quadrants. But... |
Compute $$\int \frac{\sin(x)\cos(x)}{x^2+1}dx$$
**Attempt**
$u=\sin(x)\Rightarrow du=\cos(x)dx$
$$\int\frac{u}{x^2+1}du$$
But, now I have both $u$ and $x$ in the function. How do I resolve?
Second attempt:
using Taylor series expansion for sin(x) and cos(x),
sin(x)=1-x^3/3!+x^5/5!...
cos(x)=1-x^2/2!+x^4/4!.... |
For simplicity, assume we are working with simple regression where the predictor $x\in\mathbb{R}$.
First write $y=E[y \mid x]+u$, where the variance of $u$ is a constant, and $E[u|x]=0$. I understand $E(y \mid x)$ minimizes the mean square prediction error (MSPE). We call $E[y \mid x]$ the true regression.
I und... |
Compute $$\int \frac{\sin(x)\cos(x)}{x^2+1}dx$$
**Attempt**
$u=\sin(x)\Rightarrow du=\cos(x)dx$
$$\int\frac{u}{x^2+1}du$$
But, now I have both $u$ and $x$ in the function. How do I resolve?
Second attempt:
using Maclaurin series for sin(x) and cos(x),
sin(x)=1-x^3/3!+x^5/5!...
cos(x)=1-x^2/2!+x^4/4!...
si... |
Let $(R, m_R, \kappa_R) \subset (B, m_B, \kappa_B)$ be an extension of discrete valuation rings whose corresponding extension of fields of fractions $K :=\text{Frac}(R) \subset \text{Frac}(B) =: L$ is finite of degree $n:=[L:K]$.
It is well known that in general we have only inequality $[L:K] \ge e_{B/R} \cdot [\kappa... |
In my mechanics class we had a differential equation arise from a spring oscillator via its energy ($x^2_{max}$ is some constant maximum length) :
$$\begin{align}\frac{1}{2}kx^2_{max}&=\frac{1}{2}mv^2+\frac{1}{2}kx^2\\\frac{k}{m}x^2_{max}&=\dot{x}^2+\frac{k}{m}x^2\\\end{align}$$
So we have $\dot{x}^2+\frac{k}{m}x... |
While trying to evaluate $\int_0^1 k^2K(k)dk$ related to elliptic integral of the [first kind][1], by integral switching method, I reached the trigonometric integral
$$\int_0^{\frac\pi 2}\frac{\theta-\cos\theta\sin\theta}{2\sin^3\theta}d\theta$$
which is evaluated to $\frac{2C+1}4$ by Wolfram Alpha. Here, $C$ is the ... |
so I had to convert a general form equation**(3x+4y-12=0)** into its perpendicular form **(xcost+using=p)**. for this we compared the two equations and as they are both equalivent equations, the coefficients of **x, y and xy^0** were equal. but then we also equated the ratios of these coefficients, and considered them ... |
comparing equations and equating the ratios of their coefficients? |
Dick Dastardly is taking part in an infinite wacky race. What is infinite about it, you ask? Well, just everything! There are infinitely many racers, every one of which can run infinitely fast and the finish line is infinitely far away.
To be more precise, the race will take place on the natural numbers, starting at... |
Compute $$\int \frac{\sin(x)\cos(x)}{x^2+1}dx$$
**Attempt**
$u=\sin(x)\Rightarrow du=\cos(x)dx$
$$\int\frac{u}{x^2+1}du$$
But, now I have both $u$ and $x$ in the function. How do I resolve?
Second attempt:
using Maclaurin series for sin(x) and cos(x),
sin(x)=1-x^3/3!+x^5/5!...
cos(x)=1-x^2/2!+x^4/4!...
si... |
**Polar form relative to center**
In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate $\theta$ measured from the major axis, the ellipse's equation is
$r(\theta)=\dfrac{b}{\sqrt{1-(e\cos\theta)^2}}$
[![an ellipse with the angle and distance marked from the cente... |
***Interviewer:***
Assume $a\sim U[0,1]\,,$ and conditional on $a\,,$ $$b\sim U[a,1]\,.$$
What is the unconditional distribution of $b\,?$
***Candidate*** (sweating):
\begin{align}
&\mathbb P\big\{b\le y\big\}=\int_0^1\mathbb P\big\{b\le y\,\big|\,a=x\big\}\,dx
=\int_0^1\frac{y-x}{1-x}1_{\{x\le y\}}\,dx=\i... |
Let $X$ be a topological space. Let $(S_\lambda)_{\lambda\in\Lambda}$ be a net of closed subspaces of $X$, and suppose that for all $\lambda\le\mu$, $S_\lambda\subseteq S_\mu$.
Let now
$$
x\in \mathrm{closure} \left( \bigcup_{\lambda\in\Lambda} S_\lambda \right) .
$$
Can we express $x$ as a limit of a net $(x_... |
What is the inclusion property between the Besov space $B^n_{\infty,1}(\mathbb{T})$ and $C^n(\mathbb T)$? |
Let $E$ be a vector bundle over a smooth manifold $M$ and $\nabla$ a connection on $E$. If $A$ is an $\text{End}(E)$-valued $1$-form do we have that $$F_{\nabla+A}=F_\nabla+\nabla A+ A \wedge A?$$
I'm a bit confused about the term $\nabla A$ in this expression as $A$ is not really a section of $E$ so I do not know w... |
Suppose $f(x,y,z)$ is a real-valued function of three real variables that satisfies the Laplace equation, $\nabla^2f=0$. Consider the Hessian matrix
$$
\Delta(x,y,z)=
\begin{bmatrix}
f_{xx} & f_{xy} & f_{xz}\\
f_{yx} & f_{yy} & f_{yz}\\
f_{zx} & f_{zy} & f_{zz}\\
\end{bmatrix},
$$
which is real symmetric. Ther... |
Let $(R, m_R, \kappa_R) \subset (B, m_B, \kappa_B)$ be an extension of discrete valuation rings whose corresponding extension of fields of fractions $K :=\text{Frac}(R) \subset \text{Frac}(B) =: L$ is finite of degree $n:=[L:K]$.
It is well known that in general we have only inequality $[L:K] \ge e_{B/R} \cdot [\kappa... |
What about this? Using matrix definition.
To find |(XG)^2|_max, the largest element of (XG)^2, we can express (XG)^2 as:
(XG)^2 = XG * XG
= X * G * X * G
Let's consider the (i, j)th element of (XG)^2, denoted as ((XG)^2)_ij.
((XG)^2)_ij = Σ_k Σ_l (X_ik * G_kk * X_lj * G_ll)
Since G is a diagonal matrix, G_kk = g_... |
Suppose $f(x,y,z)$ is a real-valued function of three real variables that satisfies the Laplace equation, $\nabla^2f=0$. Consider the Hessian matrix
$$
\Delta(x,y,z)=
\begin{bmatrix}
f_{xx} & f_{xy} & f_{xz}\\
f_{yx} & f_{yy} & f_{yz}\\
f_{zx} & f_{zy} & f_{zz}\\
\end{bmatrix},
$$
which is real symmetric. Ther... |
What about this? Using matrix definition.
To find $|(XG)^2|_{max}$, the largest element of $(XG)^2$, we can express $(XG)^2$ as:
$(XG)^2$ = XG * XG
= X * G * X * G
Let's consider the $(i, j)$ th element of $(XG)^2$, denoted as $((XG)^2)_{ij}$.
$((XG)^2)_ij$ = $Σ_k Σ_l (X_ik * G_kk * X_lj * G_ll)$
Since $G$ ... |
What about this? Using matrix definition.
To find $|(XG)^2|_{max}$, the largest element of $(XG)^2$, we can express $(XG)^2$ as:
$(XG)^2$ = XG * XG
= X * G * X * G
Let's consider the $(i, j)$ th element of $(XG)^2$, denoted as $((XG)^2)_{ij}$.
$((XG)^2)_ij$ = $Σ_k Σ_l (X_{ik} * G_{kk} * X_{lj} * G_{ll})$
Si... |
What about this? Using matrix definition.
To find $|(XG)^2|_{max}$, the largest element of $(XG)^2$, we can express $(XG)^2$ as:
$(XG)^2$ = XG * XG
= X * G * X * G
Let's consider the $(i, j)$ th element of $(XG)^2$, denoted as $((XG)^2)_{ij}$.
$((XG)^2)_ij$ = $Σ_k Σ_l (X_{ik} * G_{kk} * X_{lj} * G_{ll})$
Si... |
What about this? Using matrix definition.
To find $|(XG)^2|_{max}$, the largest element of $(XG)^2$, we can express $(XG)^2$ as:
$(XG)^2$ = XG * XG
= X * G * X * G
Let's consider the $(i, j)$ th element of $(XG)^2$, denoted as $((XG)^2)_{ij}$.
$((XG)^2)_ij$ = $Σ_k Σ_l (X_{ik} * G_{kk} * X_{lj} * G_{ll})$
Si... |
Let X be a discrete random variable drawn according to probability
mass function $p(x)$ over alphabet $X$ , and let random variables $Y_1$ and $Y_2$ take value in alphabet $Y$ with probability $p_1(y)$ and $p_2(y)$, respectively. The divergence and conditional divergence in this notation are: [![enter image descriptio... |
The exponential generating function for the number of permutations of $n$ elements is (see [Wikipedia][1])
\begin{eqnarray*}
\frac1{1-z}
&=&
\exp\left(\log\left(\frac1{1-z}\right)\right)
\\
&=&
\exp\left(\sum_{k\ge1}\frac{z^k}k\right)\;,
\end{eqnarray*}
where the term $\frac{z^k}k=\frac{(k-1)!z^k}{k!}$ rep... |
I am looking for conditions on a function $f(x)$ for its Fourier transform $f(k)$ to not be always $\geq 0$ for all $k$; i.e. a condition on $f(x)$ such that $f(k)<0$ for some k, where $f(k)$ is the Fourier transform of $f(x)$.
In my case the function should be a kernel, so conditions compatible with that ($f(x)$ ne... |
I have the following excercise:
"Let S = {$v_1, v_2, ... v_n$} is a basis for a vectorial space V of dimension equal to n.
prove { $[v_1]_S$, $[v_s]_S$, ... $[v_n]_S$} is a basis for $R^n$."
I know that a coordenate vector $[v]_s$ is the secuence of coordinates {$c_1, c_2,...c_n$} from **w**=$c_1 v_1 + c_2v_... |
I do not know the actual history of determinant, but I think it is very well motivated. From the way I look at it, it's actually those properties of determinant that make sense. Then you derive the formula from them.
Let me start by trying to define the "signed volume" of a hyper-parallelepiped whose sides are $(u_1... |
Is it possible to generate all possible rankings in a concatenated lists created via linear combination of two sorted lists (with overlapping elements) when ignoring the raw values of the elements and using instead ranks (or log-ranks) in each list? |
Is it possible to generate all possible rankings in a concatenated list created via merging two sorted lists (with overlapping elements) using linear combinations of elements scores, if we ignore the raw values of the elements scores and rely instead on their ranks (or log-ranks) per list? |
Suppose that there is an $n$ such that the following set: $\{\varphi_i\}_{i\in I}$ where for all L-formulas $\varphi(x_1, x_2, \ldots, x_n)$ there is a $i \in I$ such that $T \models \forall x_1, x_2, \ldots, x_n(\varphi(x_1, x_2, \ldots, x_n) \leftrightarrow \varphi_i(x_1, x_2, \ldots, x_n)) $ is an infinite set. Then... |
I have the following exercise:
"Let S = {$v_1, v_2, ... v_n$} is a basis for a vectorial space V of dimension equal to n.
prove { $[v_1]_S$, $[v_s]_S$, ... $[v_n]_S$} is a basis for $R^n$."
I know that a coordinate vector $[v]_s$ is the sequence of coordinates {$c_1, c_2,...c_n$} from **w**=$c_1 v_1 + c_2v_2... |
Q)**What is the formula for finding the summation of the sequence $1,2,5,12,26,51,...$ upto $n$ terms ?**
I know how to find the summation of sequences like $1,2,3,...,$ upto $n$ terms ; $1,2,4,8,...,$ upto $n$ terms, $1,2,4,7,11,...$, upto $n$ terms.
Mainly what I am trying to say is that I know how to find the ... |
What is the formula for finding the summation of the sequence : $1,2,5,12,26,51,...$ upto $n$ terms? |
Every non-empty set of cardinals has a least element, but it is not true that every non-empty set of cardinals has a greatest element. Thus it is preferable to ask "what is the least cardinal $\kappa$ such that there exists a set of racers of cardinality $\kappa$ which Dick cannot win against?" rather than "what is the... |
I've been delving into the properties of tensor ($\otimes$) and external tensor products ($\boxtimes$) within the context of coalgebra, particularly examining how the coproduct $\Delta$ applies to tensor products of elements from a vector space $V$ and its extension. The operation $\otimes$ is understood as the tenso... |
Let $\{e_1, e_2, ..., e_n\}$ be the set of elements. Let $\{s_1(e_1), s_1(e_2), ..., s_1(e_n)\}$, $\{s_2(e_1), s_2(e_2), ..., s_2(e_n)\}$ be the scores of elements after applying two ranking functions $s_1$, $s_2$. Let $\{w\cdot s_1(e_1) + (1-w)\cdot s_2(e_1), w\cdot s_1(e_2) + (1-w)\cdot s_2(e_2), ..., w\cdot s_1(e_n)... |
Every non-empty set of cardinals has a least element, but it is not true that every non-empty set of cardinals has a greatest element. Thus it is preferable to ask "what is the least cardinal $\kappa$ such that there exists a set of racers of cardinality $\kappa$ which Dick cannot win against?" rather than "what is the... |
Let $\{e_1, e_2, ..., e_n\}$ be the set of elements. Let $\{s_1(e_1), s_1(e_2), ..., s_1(e_n)\}$, $\{s_2(e_1), s_2(e_2), ..., s_2(e_n)\}$ be the scores of elements after applying two ranking functions $s_1$, $s_2$. Let $\{w\cdot s_1(e_1) + (1-w)\cdot s_2(e_1), w\cdot s_1(e_2) + (1-w)\cdot s_2(e_2), ..., w\cdot s_1(e_n)... |
Dobner in his paper defines (https://arxiv.org/abs/2005.05142) some complicated function $\Phi_F$ (Eq. 8) and then a page after defines $H_t(z) = \int_{-\infty}^{\infty} e^{tu^2} \Phi_F(u) e^{izu} du$. Then he defines the set $Z$ to be the set of those real $t$ such that all the roots of $H_t$ are real. Then claims th... |
What is Hurwitz Theorem and how is it applied? |
I was reading [this][1] short book about Modular arithmetic, towards the end (Chapter 6.3) it provided a fast way to solve a linear system of congruences, instead of solving two of them at a time. Here's an example of how it works:
$x ≡ 2 (mod 3)≡ 2 (mod 4)≡ 1 (mod 5)$
then it write x as a "sum of three modular ... |
$ \frac{y}{x}+ (y^3-\log(x))\frac{dy}{dx}=0 \\ y(1)=\sqrt{2}$
I have the IVP above and I'm trying to solve it, I know that $y=k\log(|x|); k \in \mathbb{R}$ is a solution of the similar linear-homogeneous equation $\frac{y}{x} -\log(x)\frac{dy}{dx}=0$ but this doesn't seem to help me find the solution of this prob... |
Let's take the following random variables $X,Y,Z_1,...,Z_K$, on the same probability space (or associated with the same experiment), where we know that
$$ Y = \sum_{k=1}^KZ_k$$
Let's say I am interested in computing the conditional variance of $X$ knowing $Y$, similarly as in this [question][1]. Applying the law o... |
Suppose that there is an $n$ such that the set of $L$-formulas in variables $x_1,\dots,x_n$ up to $T$-equivalence is finite. That is, there is so finite set of formulas $\{\varphi_i\}_{i\in I}$ (with $I$ finite) such that for all $L$-formulas $\psi(x_1, x_2, \ldots, x_n)$ there is a $i \in I$ such that $$T \models \for... |
If $F=\{0,1,a,b\}$ is a field (where the four elements are distinct), then:
1. What is the characteristic of $F$?
2. Write $b$ in terms of the other elements.
3. What are the multiplication and addition tableau of these operations? |
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