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I have a graph with 2 vertices. They each have a directed weighed edge towards the other vertex. One of the weights is higher than the other. Would this graph count as a graph that has a different shortest path tree than minimum spanning tree? Since you could have the shortest path tree start at the vertex that has a h... |
Consider two positive semidefinite and symmetric matrices, namely $\bf{A}$ and $\bf{B}$. Denote $\Vert\cdot\Vert$ as the operator norm. If we have another positive semidefinite matrix $\bf{B}^*$ that satisfies $\Vert\bf{B}^*\Vert\geqslant\Vert\bf{B}\Vert$, can we get
$$\Vert\bf{A}+\bf{B}\Vert\leqslant\Vert\bf{A}+\bf{B... |
I want to to implement the Blahut-Arimoto algorithm to evaluate the clustering membership probability, $p(\hat{x}|\mathbf{x})$, with fixed number of clusters, $N_c$, and compression-distortion tradeoff parameter, $\beta$. However, I need help on how to calculate the distortion measure $d(x,\hat{x})$.
I have a data ... |
Let $f$ and $g$ be functions in $L^1(\mathbb{R})$ such that
$$
\int_E f \, dm = \int_E g \, dm
$$
for every measurable subset $E$ of $\mathbb{R}$. Prove that $f = g$ almost everywhere on $\mathbb{R}$.
Proof
note that
$$
\int_E f \, dm = \int_E g \, dm
$$
then
$
\int_E f \, dm -\int_E g \, dm = 0... |
I have a graph with 2 vertices. They each have a directed weighed edge towards the other vertex. One of the weights is higher than the other. Would this graph count as a graph that has a different shortest path tree than minimum spanning tree? Since you could have the shortest path tree start at the vertex that has a h... |
> Let $f: D(0,1)\to\mathbb{C}$ be holomorphic s.t $\Re
(f'(z))>0\quad\forall z\in D(0,1)$. Prove that $f$ is injective.
My attempt:\
Since $f'(z)=\dfrac{\partial u}{\partial x}+i\dfrac{\partial v}{\partial x}=\dfrac{\partial v}{\partial y}-i\dfrac{\partial u}{\partial y}$ (Cauchy Riemann), we have $\dfrac{\partia... |
We show that if $\sigma(n)$ is exponential then the series tends towards $1/2$ but retains an oscillation about this value and does not completely converge.
Let $\sigma(n)=3^n$ and sum from $n=0$ to $n=\infty$, thus
$f(x)=\lim_{x\to1^-}\sum\limits_{n=0}^\infty (-1)^n x^{3^n}.$
The following graph shows a plot ... |
Given:
$$
\mathbf{a} = [a_1, a_2] \quad \text{and} \quad \mathbf{b} = [b_1, b_2]
$$
The expression for $\mathbf{c}$:
$$
\mathbf{c} = 3\mathbf{a}^3 - \mathbf{b}^2
$$
We need to find the partial derivative of $\mathbf{c}$ with respect to $\mathbf{a}$. Let's denote $\mathbf{c} = [c_1, c_2]$. Then:
$$
\... |
Let $X$ be a Hausdorff space and $\hat X$ a [compactification](https://en.wikipedia.org/wiki/Compactification_(mathematics)) of $X$ (a compact space that extends $X$). Is it possible for the space $\{f\vert_X:f\in\mathcal{C}(\hat X,\mathbb{R})\}$ to be $\aleph_0$-dimensional as a $\mathbb{R}$-vector subspace of $\mathc... |
Can we make this space $\aleph_0$-dimensional? |
What conditions are necessary and sufficient for a smooth vector field $v: \mathbb R^2 \to \mathbb R^2$ to induce an ODE $y' = f(x,y)$?
My conjecture is below. Is it correct? Can we prove (or fix) it?
---
I conjecture that a single condition is both necessary and sufficient: **$v(x,y)$ is never orthogonal to ... |
When does a smooth vector field induce an ODE? |
Let $X$ be a compact Hausdorff space and $A\subseteq X$ a subspace of $X$. Is it possible for the space $\{f\vert_A:f\in\mathcal{C}(X,\mathbb{R})\}$ to be $\aleph_0$-dimensional as a $\mathbb{R}$-vector subspace of $\mathcal{C}(X,\mathbb{R})$? I know that in this case $\mathcal{C}(X,\mathbb{R})$ is, at least, $\mathfra... |
Can we make this subspace $\aleph_0$-dimensional? |
Let $k$ be a field of characteristic $0$ and $R$ be a finite type $k$-algebra. Let $X$ be a smooth $k$-scheme and $f: X \to \text{Spec}(R)$ be a proper birational morphism.
Then, is the $R$-module $f_* \mathcal O_X$ supported at every maximal ideal of $R$, i.e., is the localization of $f_* \mathcal O_X$ non-zero at e... |
[![The international club of a school has 105 members, many of whom speak multiple languages. The most commonly spoken languages in the club are English, Korean, and Chinese. Use the Venn diagram below to determine the probability of selecting students who][1]][1]
>The international club of a school has 105 member... |
I want to find
$$\lim_{z \to i} \frac{2z^4-2z^3+8z^2-2z+5}{z-i}$$
I have a feeling that this limit does not exist, because the numerator term for it is not factorizable in the field of complexes, try to test [here][1]
But the limit does not really refer to a case of zero over zero, since let us note that:
... |
We define a map $P:\mathbb{C}_\infty \to S^2$ by declaring $P(\infty) = (0,0,1)$, and $P(z) = \left(\frac{2x}{|z|^2+1}, \frac{2y}{|z|^2+1}, \frac{|z|^2-1}{|z|^2+1}\right)$ for $z \in \mathbb{C}$ which is the stereographic projection.
Now, we define a metric on $\mathbb{C}_\infty$:
$$ d_\infty = \|P(z) - P(w)\| \ \ ... |
Let $n\ge 2$ and $A\in GL(n,\mathbb R)$ be a matrix and define a linear transformation on $M(n,\mathbb R)$ as follows:
$$f_A: M(n,\mathbb R) \to M(n,\mathbb R),
B \to ABA^{-1}.
$$
Suppose $f_A$ is a unipotent linear transformation on $M(n,\mathbb R) \cong \mathbb R^{n^2}$, namely the only eigenvalue of $f_A$ is... |
This is a self-answer question that was inspired by
[this recently posted question](https://math.stackexchange.com/questions/4887808/in-how-many-ways-can-we-distribute-6-white-and-6-black-ball-in-10-different-boxe).
$\underline{\text{The Question}}$ <br>
Assuming that balls of the same color are indistinguishable... |
In how many ways can we distribute $~B~$ white and $~B~$ black balls into $~U~$ distinguishable urns such that each urn has at least 1 ball? |
There may be a more elegant way of attacking this problem than the one that I will use. I considered
inclusion-exclusion, but found it to be messy. So, I had to settle for the direct approach, which yields a computation that (unfortunately) I am only able to express as a summation.
$\underline{\text{Preliminary ... |
I suddenly start to wonder this question. The main text I would refer to is Baby Rudin, and in chapter 6 about Riemann integral, only $\int_{a}^{b}f(x)dx$, where $a, b$ are real numbers, is defined.
In the exercise 8 of that chapter, it says if $\int_{a}^{b}f(x)dx$ exists for every $b>a$ where $a$ is fixed, then $\i... |
How is the Riemann integral over the real line defined? |
[![The international club of a school has 105 members, many of whom speak multiple languages. The most commonly spoken languages in the club are English, Korean, and Chinese. Use the Venn diagram below to determine the probability of selecting students who][1]][1]
>The international club of a school has 105 member... |
Let $f=\sum_{i=0}^\infty f_i T^i\in \mathbb{F}[[T]]$ be a formal power series over a field $\mathbb{F}$ of characteristic $p$. For every $k\geq 0$, we define the $k$-th Hasse derivative on $\mathbb{F}[[T]]$:
$$\partial_T^{[k]}\colon f\longmapsto \sum_{i=0}^\infty f_i\binom{i}{k}T^{i-k}.$$
Then an easy observation is ... |
I have just started learning about Brownian motion $(B_t)_{t \geq 0}$. The book I'm following defines a filtration $(\mathcal F_t)\_{t \geq 0}$ by $\mathcal F_t := \sigma(B_r, 0 \leq r \leq t)$. I think it's fairly clear that this is in fact a filtration.
What I'm confused about is that the author defines $\mathcal ... |
I am sorry if this isn’t obvious to me. I was doing questions on limits of multivariable functions and encountered such functions. Is this because both numerator and denominator decay at same rate so they are bounded?
Thanks for your time. |
Why is that quotient of homogeneous polynomials each of same degree in multivariable case is bounded? |
In orthogonal cartesian coordinates $c\cdot c=|c|^2$ is the square of length of the vector $c$.
In oblique coordinates the square of the length of $c = c * g * c$ where g is the metric tensor and $g*c$ is a covector.
If $e_x$ and $e_y$ are the basis vectors of vector $c$ then how does one calculate the new basis ... |
I am sorry if this isn’t obvious to me. I was doing questions on limits of multivariable functions and encountered such functions.
Is this because both numerator and denominator decay at same rate as they approach to (0,0), so they are bounded?
And what can you say about the limit about the point (0,0) in this c... |
Why the polynomial $x^{p+1}=y$ has exactly $p+1$ solutions in $F_{p^2}^*$ for $y \in F_p^*$? |
I have a system of 6 ODE representing infected subpopulations of a model of epidemic\
$$
\begin{cases}
\dfrac{de}{dt} =\dfrac{dy_2}{dt} = \phi y_1(y_4+y_5)-p_2 y_2\\
\dfrac{dq^*}{dt} =\dfrac{dy_3}{dt} = \mu y_2 -P_3 y_3\\
\dfrac{ di_s}{dt} =\dfrac{dy_4}{dt} = \psi y_2 +\alpha y_3 -p_4 y_4\\
\dfrac{di_a}{dt} =\d... |
For an epidemic model, I got a negative 'basic reproduction number' obtained by next-generation matrix method. How this can be corrected? |
[![The international club of a school has 105 members, many of whom speak multiple languages. The most commonly spoken languages in the club are English, Korean, and Chinese. Use the Venn diagram below to determine the probability of selecting students who][1]][1]
>The international club of a school has 105 member... |
> Let $( X_1 , X_2 )$ be a bivariate normal random variable. Assume that the correlation coefficient $|\rho[ X_1 , X_2 ] | \ne 1$. Show that $X_1$ and $X_2$ are Normal random variables by calculating their marginal densities.
Can someone help me with this question? I can show $X_1$ and $X_2$ are normal as $( X_1 , X... |
this is the solution provided from my class lecture notes. I have a question related to the graph drawing. Based on the calculation, we have two vectors, vector r(2) = <2,4> and vector r'(2) = <1,4>. According to the calculation of vectors, shouldn't those two vector have the same Y axis on the graph? Many thanks.
[!... |
>where is $Q$ in that case?
When you move $P$ to $T$, then $R$ moves to $R'$ where $\vec{RR'}=\vec{PT}$ implies $R'_x=R_x+T_x-P_x=-9.9$ and $R'_y=R_y+T_y-P_y=-233.5$.
The equation of the circle whose center is $R'$ with radius $250$ is given by
$$(x+9.9)^2+(y+233.5)^2=250^2$$
So, the slope of the tangent line... |
I am sorry if this isn’t obvious to me. I was doing questions on limits of multivariable functions and encountered such functions.
Is this because both numerator and denominator decay at same rate as they approach the origin, so they are bounded?
And what can you say about the limit as it tends to origin in this... |
Working on Divisors Sum Efficient calulcation topic. Accidentaly discovered one interesting relation which is accurate up to $10^{17}$ order.
$$\sum_{i=1}^{\infty}{\frac{\sigma(i)}{e^{i}}}\approx\frac{\pi^2}{6}-\frac{1}{2}+\frac{1}{24}$$
To get things more clear look at the below numbers:
$$\sum_{i=1}^{\infty}{\... |
I am sorry if this isn’t obvious to me. I was doing questions on limits of multivariable functions and encountered such functions.
Is this because both numerator and denominator **decay** at same rate as they approach the origin, so they are bounded?
And what can you say about the limit as it tends to origin in ... |
Why is that quotient of homogeneous polynomials (each of same degree) in multivariable case bounded? |
By using
$$\int_0^\infty\frac{\ln^{2n}(x)}{1+x^2}dx=|E_{2n}|\left(\frac{\pi}{2}\right)^{2n+1}\tag{a}$$
and
$$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(2k)}{(a-2k)!}\ln^{a-2k}(z)\tag{b}$$
I managed to find:
$$\int_0^\infty\frac{\ln^{2n}(x)}{1+yx^2}dx=\frac{... |
I am attempting to solve the following integral
$\mathrm{Im}\int_{0}^{\infty} \frac{f(x)}{(x-x_0+ia)(x-x_0\pm ib)} dx = ??? $
where $a,b>0$ and then take the $a,b \rightarrow 0$ limit. Also note that $f(x)$ is a real function for every $x \in \mathbb{R}$ and $x_0>0$. I realize that because we are purely intereste... |
Let $\phi: G \to F(X)$ be a group homomorphism suppose that $\phi$ maps a subset $S$ of $G$ bijectively onto $X$. Then $F(X) $ is isomorphic to $<S>,$ where $F(X)$ free group with basis $X$.
Idea of the proof : $\phi:S\to X$ is a bijection . $\phi^{-1}:X\to S\subset G$ is injective . Then by definition of free gro... |
Let $\phi: G \to F(X)$ be a group homomorphism suppose that $\phi$ maps a subset $S$ of $G$ bijectively onto $X$. Then $F(X) $ is isomorphic to $\langle S\rangle$, where $F(X)$ free group with basis $X$.
Idea of the proof : $\phi:S\to X$ is a bijection . $\phi^{-1}:X\to S\subset G$ is injective . Then by definitio... |
> Let $F:\mathbb{R}^2\to\mathbb{R}, F(x,y)=\sin |x^2+y|$.
>
> a) Prove that $F(x,y)$ is not continuously differentiable at $(0,0)$.
>
> b) Is $F$ differentiable at $(0,0)$?
I can do part b):\
If $F$ is differentiable at $(0,0)$ then $\dfrac{\partial F}{\partial y}(0,0)$ must exist. However, $\dfrac{\partial... |
Let there be an antisymmetric tensor field $\Omega_{ab}(q)$ where $q^i$ are coordinates on a 2N dimensional manifold. For context, this is a general symplectic form on phase space. I want to find a coordinate transformation that takes this matrix to its canonical form $$\Omega^0=\begin{bmatrix}0&-\mathbb{I}_{N\times N}... |
Do coordinate changes only affect antisymmetric matrices linearly? |
A sequence of reals $x_1,...,x_n$ is said to have a **turning point** at index-value $i$ ($1\lt i\lt n$) iff $x_{i-1}\lt x_{i}\gt x_{i+1}$ or $x_{i-1}\gt x_{i}\lt x_{i+1}$. The number of turning points in the sequence is denoted $T(x_1,...,x_n)$, and we define the **turning-point fraction** as $$R(x_1,...,x_n)={\text{... |
Say you have a list of $n$ random integers in the range $[1, m]$. The binomial PMF can find the probability of exactly $k$ occurrences of a number with $p = \frac{1}{m}$
$$ {n \choose k} p^k (1-p)^{n-k} $$
Now say the list is constructed in the following way: the $i$th element in the list is a random number in $[1,... |
The four Latustectum of two hyperbolas forms four distinct sides of a rectangle. If $e,E$ ate their eccentricities then which of the following are true?
1) $eE\geq 2$
2) $(e-\frac{1}{e})(E-\frac{1}{E})=1$
3) $eE\geq 3$
4) both hyperbolas can never have same eccentricities
I have tried to make figures but ... |
Let $g$ be a quadratic form of a bilinear function that is given by $g(x,y) = -x^2+y^2$. Suppose I have another quadratic form of another bilinear function given by $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ for which $f \geq 0$ whenever $|x|=|y|$.
* how do I demonstrate the bound $f \geq c |g|$?
* how do I gen... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satisfying... |
>Let $f : A \rightarrow B$ and $g : A \rightarrow C$ be homomorphisms of Noetherian rings.Suppose \
1) $B \otimes_A C$ is Noetherian,\
2) $f$ is flat and\
3) $g$ is non-degenerate.\
Then $1_B \otimes g : B \rightarrow B \otimes C$ is also non-degenerate.
This is a proposition in Mat80, but is left as... |
Determine the general term of the sequence $(a_n)_{n>=1}$, strictly decreasing, of strictly positive numbers, which satisfies the properties:
(a) $n \cdot a_n \in \mathbb{N} \setminus \{0\}$ for every $n \in \mathbb{N} \setminus \{0\}$.
(b) $\sum_{k=1}^{n} (-1)^{k-1} \cdot a_k \cdot \binom{n}{k} = \sum_{k=1}^{... |
Determine the general term of the sequence $(a_n)_{n\ge1}$, strictly decreasing, of strictly positive numbers, which satisfies the properties:
(a) $na_n \in \mathbb{N} \setminus \{0\}$ for every $n \in \mathbb{N} \setminus \{0\}$.
(b) $\sum_{k=1}^{n} (-1)^{k-1} a_k \binom{n}{k} = \sum_{k=1}^{n} a_k$.
I trie... |
Just for curiosity I want to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$.
----
Since $xP_n= P_{n+1} +nP_n$ this proves that it is possible for any $x^n$ to be represented as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$ where $1\le k \le n$, $ \ n \in \mathbb{N}$. We have the form $x^n = \sum\limits_{... |
How to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$? |
Recently found an exercise for high school students as follows:
Let $A,B$ be $2\times 2$ matrices with real entries and suppose $A^2=B^2$, $\operatorname{tr}(A)\neq0\neq\operatorname{tr}(B)$. Show that $AB=BA$.
Since it only ask the case for $2\times 2$ matrix and hence we can simply bash out all the equations.
... |
Let $p$ be a prime, and consider the finite field $\mathbb{F}_p$. Fix any $n\ge1$, and consider the field extension $\mathbb{F}_{p^n}/\mathbb{F}_p$. If $\alpha\in\mathbb{F}_{p^n}$ is a multiplicative generator, then $\alpha$ is also a primitive element, that is, $\mathbb{F}_{p^n}=\mathbb{F}_p(\alpha)$, and further, t... |
Just for curiosity I want to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$.
----
Since $xP_n= P_{n+1} +nP_n$, this proves that it is possible for any $x^n$ to be represented as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$ where $1\le k \le n$, $ \ n \in \mathbb{N}$. We have the form $x^n = \sum\limits_... |
A few days ago, I suddenly came up with a question about polynomials: Let $x_k=-1+\dfrac{2k}{n}$, where $k=0,1,\cdots,n$, be $n+1$ equidistant nodes on $[-1,1]$. Consider polynomial $p_n$ with order less or equal than $n$ satisfying
$$
|p(x_k)|\leq M,
$$
where $M>0$ is a given real number. I want to know something... |
Let $C_1$ and $C_2$ be uniformly integrable collections of functions
on a measure space $(\Omega,\mathscr{F},\mu)$,
and let $C$ be the collection defined as
\begin{equation*}
C = \{f_1 + f_2:f_1\in C_1, f_2\in C_2\}.
\end{equation*}
How to show that $C$ is uniformly integrable?
Rogers & Williams state this (as... |
If $C_1$ and $C_2$ are uniformly integrable then $\{f_1+f_2:f_1\in C_1, f_2\in C_2\}$ is? |
Well assuming that the [Prime Number Theorem][1] is true, when substituting $\delta(x)$, the density of primes near $x$—which I am being vague of what it means 'cause I don't have enough foundation about the topic—is asymptotic to $\frac{1}{\log(x)}$. Then, when I graphed the difference between $\log(x!)$ & $x\cdot\sum... |
How to correct the error between $\log(x!)\approx x\sum_{n\leq x}\delta(n)\frac{\log(n)}{n}$, where $\delta(n)$ is the density of primes near x? |
Continuing from your relaxed equation,
$m^2 - mn - n^2 = \pm 1$
$m^2 - mn - n^2 \pm 1 = 0 $
m = $ \frac{n \pm {\sqrt {5n^2 \pm 4}}}{2} $
As a property, x is a fibonacci number if atleast one of $5x^2 + 4$ and $5x^2 - 4$ is a perfect square.
So, for a valid m to exist, $5n^2 \pm 4$ must be a perfect sq... |
I need some help on the existence of solutions of an equation system modulo some prime $p$. The equation system has three parametes $n$, $N$ and $p$ and the equations are of the form
$$
\sum_{i=1}^n x_i^d = 1 \bmod p \qquad \text{for $d = 1, \dots, N$.}
$$
The $x_i$ are elements of $\mathbb{Z}_p$. A trivial solutio... |
Proof of the first inequality
$$S=\sin(x_1+x_2) + \sin(x_2+x_3) + \sin(x_3+x_4) + \sin(x_4+x_5)+\sin(x_5+x_1) \ge 0 \tag{1}$$
in the following case :
$$\text{(H) for any } k, \ 0<x_k<\pi$$
Let us give a geometric interpretation allowing to establish (1).
Under hypothese (H), angles $x_k$, having thei... |
There is an old question [here][1] on the Elchanan Mossel's Dice Paradox:
> You throw a fair six-sided die until you get 6. What is the expected number of throws (including the throw giving 6) conditioned on the event that all throws gave even numbers?
There are plenty of good answers to that question. I have a... |
In one dimension, we can represent a point as the solution set of a single algebraic equation.
In two dimensions, we can represent a line through a single equation, and points as the common solution set of certain lines.
In three dimensions, we can represent plane as a single equaiton, lines as common solutions o... |
Why is that in n dimensions, we can't represent lower dimensional entities with a single equation? |
An $8 \times 8$ matrix $M$ is directly computed by computing its determinant $|M|$, and by dividing it into chunked matrices $ M = \left [ \begin{matrix}
A & B
\\ C & D
\end{matrix} \right ] _{4 \times 4} $,$A, B, C, D$ are the $2 \times 2$ matrix, compute determinant value of $ \left | \begin{matrix}
A & B
... |
Is the value of the determinant of a large matrix the same as the value of the determinant of its chunked matrix? |
>**Question.**
Let $R$ be a Noetherian ring. Let $N\subset M$ be finitely generated $R$-modules such that $\dim(M/N)>0$. Take a prime filtration $0=K_0\subsetneq K_1\subsetneq ...\subsetneq K_t = M/N$, and let $P_i\in\text{Spec}(R)$ such that $K_i/K_{i-1}\cong R/P_i$ for $1\leq i\leq t$.
I am trying to show that ... |
I came across this summation while solving harmonic chain.
\begin{align}
\sum_{l=1}^N \sin^2{(\frac{m\pi}{N+1}\cdot l)}
\end{align}
for an integer m that can range from 1 to N.
So the question is if the value of the summation changes with m? I checked for until N=5. And it remains the same for all integer... |
Write, assuming $a\neq0$,
$$ax^2+bx+c=a\left(x^2+\frac bax+\frac ca\right)=a\left(x+\frac{b}{2a}\right)^2+\frac{4ac-b^2}{4a}=aP(x)-\frac{D}{4a}$$
where $P(x)$ is a function that has range $[0,\infty)$.
When $a>0$, the range of $aP(x)$ is also $[0,\infty)$ (why?) and then the range of $aP(x)-D/4a$ is $\left[-\frac{... |
In one dimension, we can represent a point as the solution set of a single algebraic equation.
In two dimensions, we can represent a line through a single equation, and points as the common solution set of certain lines.
In three dimensions, we can represent plane as a single equaiton, lines as common solutions o... |
Why is that in n dimensions, we can't represent lower dimensional entities with a single equation in cartesian coordinates? |
In one dimension, we can represent a point as the solution set of a single algebraic equation.
In two dimensions, we can represent a line through a single equation, and points as the common solution set of certain lines.
In three dimensions, we can represent plane as a single equaiton, lines as common solutions o... |
Given a function
$g(x)=log(\sum_{i=1}^{m}e^{\frac{f_{i}(\boldsymbol{x})}{\lambda_{i}}})$,
where $f_{i}$ is a $L_{i}$ smooth function. $\lambda \in \mathbb{R}_{+}^{m}$ and $\sum_{i=1}^{m}\lambda_{i}=1$.
How to prove the smoothness and covexity of $g(x)$? |
## Question: Find the number of solutions of equation $f(x)=2^x-x^2+x+\cos x=0$ ##
----------
Given answer is $1$
*My Attempt*
I have checked the derivative of it. $$f'(x) = 2^x\ln2-2x+1-\sin x\\f''(x)=2^x(\ln2)^2-2-\cos x$$ but $f'(x)$ is both +ve and -ve so I could not make any conclusion. I had projecte... |
Just for curiosity I want to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$.
----
Since $x=P_1,\ xP_n= P_{n+1} +nP_n$, this proves that it is possible for any $x^n$ to be represented as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$ where $1\le k \le n$, $ \ n \in \mathbb{N}$. We have $$x^n = \sum\limits_... |
An $8 \times 8$ matrix $M$ is directly computed by computing its determinant $|M|$, and by dividing it into blocks $ M = \left [ \begin{matrix}
A & B
\\ C & D
\end{matrix} \right ] _{4 \times 4} $,$A, B, C, D$ are the $2 \times 2$ matrix, compute determinant value of $ \left | \begin{matrix}
A & B
\\ C & D... |
Why is that in n dimensions, we can't represent lower dimensional entities with a single linear equation in cartesian coordinates? |
> Does this hold for $n\times n$ matrices?
Of course not. The problem statement seems to insinuate that $A$ and $B$ may possibly not commute when one of them has a zero trace. So, suppose you can find an example in which both $A$ and $B$ have zero trace and $AB\ne BA$. Then $\pmatrix{1\\ &A}$ and $\pmatrix{1\\ &B}$ ... |
Given a function $g(x)$,
$g(x)=log(\sum_{i=1}^{m}e^{\frac{f_{i}(\boldsymbol{x})}{\lambda_{i}}})$,
where $f_{i}$ is a $L_{i}$ smooth function. $\lambda \in \mathbb{R}_{+}^{m}$ and $\sum_{i=1}^{m}\lambda_{i}=1$.
How to prove the smoothness and covexity of $g(x)$? |
Given a function $g(\boldsymbol{x})$,
$g(\boldsymbol{x})=log(\sum_{i=1}^{m}e^{\frac{f_{i}(\boldsymbol{x})}{\lambda_{i}}})$,
where $\boldsymbol{x} \in \mathbb{R}^{n}$, and $f_{i}$ is a $L_{i}$-smooth function. $\lambda \in \mathbb{R}_{+}^{m}$ and $\sum_{i=1}^{m}\lambda_{i}=1$.
How to prove the smoothness and cove... |
Let $X_n$ be a sequence of positive random variables that converge almost surely to a constant $a$, and have bounded expected value $\mathbb{E}X_n=\mathbb{E}|X_n|<C$ for some constant $C$ and all $n$. Does this imply that $\mathbb{E}X_n\rightarrow a$? |
almost sure convergence of positive random variables with bounded expectation implies uniform integranility? |
> Prove that for every positive integer $n$, $\sum\limits_{d|n}d.\phi
(d).\sigma \left(\dfrac{n}{d}\right)=\sum\limits_{d|n}d^2$.
I can prove in the case of $n=p^a$, where $p$ is a prime. I'm stuck at the general case when $n=\prod\limits_{i=\overline{1,k}}p^{a_i}_i$.\
Can someone show me how to deal with this ca... |
> Does this hold for $n\times n$ matrices?
Of course not. The problem statement seems to insinuate that $A$ and $B$ may possibly not commute when one of them has a zero trace. So, suppose you can find an example in which both $A$ and $B$ have zero trace and $AB\ne BA$. Then $\pmatrix{1\\ &A}$ and $\pmatrix{1\\ &B}$ ... |
>$b,c\in\{0,1,2,3,4,5,6,7,8,9\}$ so $b,c$ has $10$ choices each
This is where you went wrong. Consider a divisibility test for an integer by $9$, which is that the sum of its digits must itself be a multiple of $9$. Clearly it doesn't matter what $a$ is; there are $9$ of them in our template, and something added to ... |
I need your help with the following exercise:
The sphere $x^{2}+y^{2}+z^{2}=4$ is made of certain material whose density is given by $\rho(x,y,z)=y^{2}+xy+2$. I need to find the extreme values of $\rho(x,y,z)$ using the Lagrange multiplier method.
If $g(x,y,z) = x^{2}+y^{2}+z^{2}-4=0$, $\rho_{x}(x,y,z) = y$, $\rh... |
It's probably a stupid question but if you have a function with variables x,y,z and a single output.
Does the order of the partial derivatives in the gradient matter? As in, is the same (∂x,∂y,∂z) and(∂z,∂x,∂y)? |
Does the order of the partial derivatives in a gradient vector matter? |
I am currently exploring an intriguing topic related to quaternion-based wave functions and have encountered a mathematical challenge that I hope to get some insights on. The concept is detailed in an article on arXiv, which presents a two-component wave function represented as a quaternion:
$$\psi(\boldsymbol{x},t)=a... |
Suppose that $u$ is the solution to
$\begin{cases}
u_t -u_{xx} =2 , &0<x<1, t>0 \\
u(x,0)=0, & 0\le x \le 1 \\
u(0,t) = u(1,t)=0, &t\ge0
\end{cases}$
Show that $u(x,t)\le x(1-x), 0\le x \le 1$
While I am studying heat equation, I think it looks trivial because, say $k(x,t)=x(1-x)$,
then
$\begin{cas... |
>where is $Q$ in that case?
When you move $P$ to $T$, then $R$ moves to $R'$ where $\vec{RR'}=\vec{PT}$ implies $R'_x=R_x+T_x-P_x=-9.9$ and $R'_y=R_y+T_y-P_y=-233.5$.
The equation of the circle whose center is $R'$ with radius $250$ is given by
$$(x+9.9)^2+(y+233.5)^2=250^2$$
So, the slope of the tangent line... |
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