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Let $G$ be a triangle-free graph. Let $n_{0} = v(G)$. Now $G$ will have a an independent set of size $\lfloor \sqrt{n_{0}} \rfloor$. You can google wikipedia on triangle-free graph and you will get the proof.
What I would like to do is this. Remove the above independent set, call it $S$. Now $G - S$ is also triangle... |
I'm unable to prove if the following series converge uniformly on $[0,+\infty)$ for the values of the parameter $\beta$ between $2$ and $3$.
$$
\sum_{n\geq 1}\frac{n^\beta x}{x^4+n^4}
$$
The maximum of the summed functions is realized (for fixed $n$) in $\frac{n}{\sqrt[4]{3}}$ and it is $\frac{\sqrt[4]{27}}{4}n^{... |
Given two cones $C_1, C_2$ centered at the vertex of a unit sphere and knowing only the angle $x$ between the axis of the cones, would it is be possible to find the two conical angles at the vertex? I presume there isnt sufficient information for this. So if in addition I know that the diameters of the base circles of ... |
A cube with edge length n is divided by planes parallel to its faces into $n^3$ unit cubes. How many pairs of such unit cubes exist which have no more than two vertices in common?
My solution:
Two cubes have more than two vertices in common if they meet with whole walls
1. Each inner cube of big cube has $6$ ... |
A cube $n*n*n$ is divided into $n^3$ unit cubes. How many pairs of such unit cubes exist which have no more than two vertices in common? |
An $n\times n\times n$ cube is divided into $n^3$ unit cubes. How many pairs of such unit cubes exist which have no more than two vertices in common? |
There is [numerical evidence][1] that $$I=\int_{\arccos(1/4)}^{\pi/2}\arccos\left((\cos x)\left(2\sin^2x+\sqrt{1+4\sin^4x}\right)\right)\mathrm dx=\frac{\pi^2}{40}$$
How can this be proved?
I was trying to [answer another question][2], and I got it down to this integral.
Wolfram [does not evaluate][3] the inde... |
If I have a subset $A \subseteq \mathbb{R}$, a function $f: A \to \mathbb{R}$, and a point $a \in A$, then (according to the definitions I came across) for $f$ to be differentiable at $a$, then there must exist some open interval $I$ so that $a \in I$ and $I \subseteq A$.
I tried understanding why this requirement e... |
Can a set of vectors span P2 and P3? If a set of vectors does not include a value for X^3, but spans P2, can it still span P3? |
I'm reading the Hrbaceck and Jech's book and they say that recursion until $\omega$ does not work for constructing the Borel sets
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/VNX75.png
But I haven't been able to find such sequences, do you have any suggestions?
|
Why is transfinite recursion needed for the construction of Borel sets? |
I have two O.N sets
$\{|e_i\rangle\}_{i=1}^r$ and $\{|\tilde e_i\rangle\}_{i=1}^r$
Then there is gotta be a change of basis matrix C, such that
$|\tilde e_i\rangle = \sum_{j=1}^rc_{j,i}|e_j\rangle$
I was hopping to use this equation to reduce the following expresion
$|\Psi_{AB}\rangle=\sum_{i=1}^r\til... |
For $a,b\in [0,1]$ and $\epsilon\geq 0$, does the following equality hold?
$a^{1+\epsilon}+b^{1+\epsilon}\geq (a-b)^{1+\epsilon}$
I want to know if I can apply this result, I know such a result is true for $\epsilon=1$, i.e. $a^2+b^2\geq (a-b)^2$ |
Thank you in advance for your comments!
Let $(M,g)$ be a Riemannian manifold (in general non-compact, connected). Then the associated Dirichlet Laplacian $\Delta_g$ generates the heat semigroup $(e^{s\Delta_g})_{s\geq 0}$ on $L^2(M,d\mu_g)$, where $\mu_g$ is the measure induced by the metric volume form as usual. Mo... |
Let $h_t:X\to Y$ be a homotopy, we assume both spaces are locally compact and hausdorff, and each $h_t$ is proper, when can we lift it to a homotopy of one point compactification $\bar X\to \bar Y$? I think this is true when $h$ is a isotopy, or at least when $h_t$ is a homeomorphism at each $t$?
Besides, if $X,Y$ ... |
When can a homotopy lift to its comactification? |
Let $L$ be a framed link in an integral homology 3-sphere $M$. I read in [this paper][1] that if $L$ is algebraically split (pairwise linking number is zero) and unit-framed (framing $\pm 1$), then by doing Dehn surgery along $L$ again yields an integral homology sphere $M_L$, the author says it's easy to confirm the r... |
Let $n \in \mathbb{N}$, $0<r<1$ and $\omega = \exp\left(\frac{2 \pi i}{n}\right)$.
For $j = 0, 1, \ldots, n-1$ define $a_j = r \omega^j$, these are the vertices of a regular $n$-gon inside the circle of radious $r$.
Now for which $z \in \mathbb{C}$ on the unit circle, that is $|z|=1$, does the following expression... |
For $a,b\in [0,1]$ and $\epsilon\geq 0$, does the following equality hold?
$a^{1+\epsilon}+b^{1+\epsilon}\geq |a-b|^{1+\epsilon}$
I want to know if I can apply this result, I know such a result is true for $\epsilon=1$, i.e. $a^2+b^2\geq (a-b)^2$ |
Let $h_t:X\to Y$ be a homotopy, we assume both spaces are locally compact and hausdorff, and each $h_t$ is proper, when can we lift it to a homotopy of one point compactification $\bar X\to \bar Y$? I think this is true when $h$ is a isotopy, or at least when $h_t$ is a homeomorphism at each $t$?
Besides, if $X,Y$ ... |
For $a,b\in [0,1]$ and $\epsilon\geq 0$, does the following equality hold?
$a^{1+\epsilon}+b^{1+\epsilon}\geq |a-b|^{1+\epsilon}$
All I can think to do so far is:
\begin{align*}
|a-b|^{1+\epsilon} = ((a-b)^2)^{(\epsilon+1)/2}
\end{align*}
I want to know if I can apply this result, I know such a result... |
In Jech's Chapter 14, p. 218 (2nd edition), the proof for the generic model theorem (i.e. Theorem 14.5) begins with: *"Let $(P, <) $ be a notion of forcing in the ground model $M$, and let $G \subset P$ be generic over $M$"*.
Afterwards, the corresponding boolean algebra $B=B(P)$ is constructed (for which $P$ is de... |
Let $F\leq E \leq K$ fields. If $K/F$ is a finite extension field then $[E:F]\leq [K:F]$ and $[K:E]\leq [K:F]$
My try, how $K/F$ is a extension field then as a $F-$vector space $E$ is a subspace of $K$ and then $\rm{dim}_F E \leq \rm{dim}_F K$ and then $[E:F]\leq [K:F]$ by definition but for the second part im stu... |
I am studying Jacobson rings, using [this file](https://www.math.uchicago.edu/~emerton/pdffiles/jacobson.pdf) by Matthew Emerton and [this](https://www2.math.upenn.edu/~chai/602f06/hw602f06/jacobson_ring.pdf) source.
I am trying to understand the proof of the following:
> **Theorem.** if $A$ is a Jacobson ring (th... |
The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
>Show that $P(ab>c)=\frac12$.
The result is strongly suggested by simulations, and by my attempt shown below.
The simplicity of the result suggests that there may be an intuitive explan... |
Given two cones $C_1, C_2$ centered at the vertex of a unit sphere and knowing only the angle $x$ between the axis of the cones, would it is be possible to find the two conical angles at the vertex? I presume there isnt sufficient information for this. So if in addition I know that the diameters of the base circles of ... |
In Jech's Chapter 14, p. 218 (2nd edition), the proof for the generic model theorem (i.e. Theorem 14.5) begins with: *"Let $(P, <) $ be a notion of forcing in the ground model $M$, and let $G \subset P$ be generic over $M$"*.
Afterwards, the corresponding boolean algebra $B=B(P)$ is constructed (for which $P$ is de... |
Let $m$ and $n$ be two integers and $m \le n$. There are a matrix $A$ of $m$-by-$m$ with $A(i,j) = 1/(2n+2j-2i)!$ and a vector $r$ of $m$ entries with $r(i) = 2/(2n+2i)!$.
Is there a formula for the inner product of $r$ and the first column of the inverse of $A$?
Or, can it be shown that the inner product afore... |
When can a homotopy lift to its compactification? |
I'm afraid this is a stupid question — I'm not a mathematician, so please correct me when I'll be saying something wrong — but I've been stuck at this point for so long that I thought it would be wise to ask for help.
Hereafter is an excerpt from *Introduction to Smooth Manifold* by John Lee. The last line is where ... |
Let be $K$ the finite splitting field of $f(x) (\in \Bbb Q[x])$ over the field, $\Bbb Q$(rational number set)
And say $E_H$ is a fixed field of $H\subset \operatorname{Gal}(K/\mathbb{Q}) $.
Main Question) Find the fixed field $E_H$
(1) $f(x) = x^4 -2$, $H= \langle \sigma \rangle$ with $\sigma(\alpha) = -\a... |
Consider two integer numbers $a$ and $b$, and let $n$ be the maximal natural number (or $0$) such that $a^n$ still divides $b$. Is there some standard term and/or notation for this $n$ (as a function of $a$ and $b$)?
Of course, this notion applies not only to integers: $a$ and $b$ could be elements of any commutativ... |
They can in general, but in your specific case they can't because they cannot generate $x^3$.
Perhaps, to expand on this, you may also be interested in an answer to the question: what is the meaning of $x$? Here is one possible explanantion (also given [here][1]): If $V$ is the vector space of all infinite real sequ... |
I'm afraid this is a stupid question — I'm not a mathematician, so please correct me when I'll be saying something wrong — but I've been stuck at this point for so long that I thought it would be wise to ask for help.
Hereafter is an excerpt from *Introduction to Smooth Manifold* by John Lee. The last line is where ... |
I'm afraid this is a stupid question — I'm not a mathematician, so please correct me when I'll be saying something wrong — but I've been stuck at this point for so long that I thought it would be wise to ask for help.
Hereafter is an excerpt from *Introduction to Smooth Manifold* by John Lee. The last line is where ... |
I have been interested in the gamma function and its derivatives recently and was surprised to see these constants appear in the values of the derivatives of the gamma function:
$$\Gamma'(1) = -\gamma$$
$$\Gamma''(1) = \gamma^2 + \zeta(2)$$
$$\Gamma'''(1) = -2\zeta(3)-\gamma^3-\frac{\gamma\pi^2}{2}$$
I have read... |
Why do the constants $\gamma$, $\zeta(2)$ and $\zeta(3)$ appear in the values of $\Gamma''(1)$ and $\Gamma'''(1)$? |
They can in general, but in your specific case they can't because they cannot generate $x^3$.
Perhaps, to expand on this, you may also be interested in an answer to the question: what is the meaning of $x$? Here is one possible explanation in the case of real polynomials. (See also [here][1], and a more advanced dis... |
I'm afraid this is a stupid question — I'm not a mathematician, so please correct me when I'll be saying something wrong — but I've been stuck at this point for so long that I thought it would be wise to ask for help.
Hereafter is an excerpt from *Introduction to Smooth Manifold* by John Lee. The last line is where ... |
Let's take it that under the assumption of A, ~C is true. And let's also take it that if ~C is true and if B is derivable from A, then C is true.
I want to show that B is not derivable from A. Is this possible under these assumptions? It seems like we can assume that B is derivable from A for the sake of obtaining ... |
How do I simplify $\sqrt{1+\sqrt5}/\sqrt2$? |
How do I simplify $\sqrt{1-\sqrt5}/\sqrt2$? |
I'm afraid this is a stupid question — I'm not a mathematician, so please correct me when I'll be saying something wrong — but I've been stuck at this point for so long that I thought it would be wise to ask for help.
Hereafter is an excerpt from *Introduction to Smooth Manifold* by John Lee. The last line is where ... |
A puzzle I’ve wondered about but never got around to solving/verifying:
In the game Pokemon Let’s Go, the Pokemon Abra instantly teleports away if the user is detected within its line of sight.
Let’s say you’re in a square of length $s$ and one Abra randomly spawns in the square, with a random orientation, and h... |
Where is the optimal point to “hide” in a shape? |
How do I simplify $\sqrt{1-\sqrt5\5}/\sqrt2$? |
How do I simplify $\sqrt{\frac{1-\frac{\sqrt5 }5}2}$? |
Given a multivariate Gaussian over the domain, $\Bbb{R}^n$ with mean $\mu$ and covariance matrix, $\Sigma$. For simplicity sake, we can consider the case where $\Sigma$ is a diagonal matrix. I'd like to find the probability that this distribution lies in the region: $\sum c_i x_i \leq b$. We can assume the $c_i$'s and ... |
Given a multivariate Gaussian over the domain, $\Bbb{R}^n$ with mean $\mu$ and covariance matrix, $\Sigma$. For simplicity sake, we can consider the case where $\Sigma$ is a diagonal matrix. I'd like to find the probability that this distribution lies in the region: $\sum c_i x_i \leq b$. We can assume the $c_i$'s and ... |
F is a vector field with unknown potential A and there's a surface with the boundary of a unit circle at the origin. Also, the circulation A around the unit circle oriented clockwise is a positive constant. Given this information, I know that the flux through the unit circle will give me the flux through the surface, o... |
I am finding the Mordell-Weil group of the elliptic curve of the form $y^2=x^3+ax+b$ over the rational field. Using Magma, I found that many of them have the form $Z/2\times Z/2\times Z$ or $Z/2\times Z/2\times Z\times Z$. Can we determine full candidates of the group form? From <https://math.stackexchange.com/question... |
Let $p$ be a prime and let $f(x) \in F[x]$ have degree $p$. Let $K$ be the splitting field for $f(x)$ over $F$. Suppose $[K:F] =tp$ for some $t \in \mathbb{N}$. Prove that $f(x)$ is irreducible over $F$.
Here's what I have so far.
Suppose not. Suppose instead that $f(x)$ is *reducible*. Then $f(x) = g(x)h(x)$ f... |
Let's take it that under the assumption of A, ~C is true. And let's also take it that if ~C is true and if B is derivable from A, then C is true.
I want to show that B is not derivable from A. Is this possible under these assumptions? It seems like we can assume that B is derivable from A for the sake of obtaining ... |
Let $p$ be a prime and let $f(x) \in F[x]$ have degree $p$. Let $K$ be the splitting field for $f(x)$ over $F$. Suppose $[K:F] =tp$ for some $t \in \mathbb{N}$. Prove that $f(x)$ is irreducible over $F$.
Here's what I have so far.
Suppose not. Suppose instead that $f(x)$ is *reducible*. Then $f(x) = g(x)h(x)$ f... |
How can one prove/disprove this argument for an arbitrary number $n$ of subspaces?
> Prove or disprove that for subspaces $U_1, U_2, ... U_n$ of a vector space $V$ over $F$,
$$U_1 \cup U_2 \cup... U_{n-1} \cup U_n$$ is a subspace of $V$ if and only if $U_1 \cup U_2 \cup... U_{n-1} \cup U_n$ is $U_j$ for some $j =... |
Becaus $$det(tI-A)=t^n+...$$
So the characteristic polynomial is positive for big enough values of $t$.
If the characteristic polynomial is negative for some number greater than the spectral radius then it woul be 0 for some point by the intermediate value theorem.
Is this correct? |
Is the characteristic polynomial positive for values greater than the spectral radius of the matrix? |
I've never taken a projective geometry course, and I'm trying to understand the real projective plane $\mathbb{P}^2$ and its description using homogeneous coordinates, and how these relate to rational Bézier curves.
I understand that the elements of $\mathbb{P}^2$ can be identified with lines through the origin in $... |
Because $$\det(tI-A)=t^n+...$$
So the characteristic polynomial is positive for big enough values of $t$.
If the characteristic polynomial is negative for some number greater than the spectral radius then it would be 0 for some point by the intermediate value theorem.
Is this correct? |
Let $S$ be the set of points whose coordinates $x,$ $y$ are integers that satisfy $0\le x\le3,$ $0\le y\le4,$. Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is?
My approach:
I tried finding the probability of the points ***no... |
Assume A $\supset$ B. (for contradiction)
Assume ($\lnot$ C $\land$ (A $\supset$ B)) $\supset$ C. (gathered from your question)
($\lnot$ C $\land$ True) $\supset$ C. (replacing A implies B with its truth value)
$\lnot$ C $\supset$ C. (reducing "and True")
$\lnot$ C only implies C if $\lnot$ C is false. ... |
Considering $y=a^x$, $a>1$, showing that $\frac{2(a-1)}{(a+1)} < ln(a) < -1 + \sqrt{2a-1}$
My idea is to write $y=a^x = \exp{(x*lna)}$, then find the derivative of y at $x=0$ is $ln a$.
But I dunno how to continue from there .
Any idea on using geometry or something to prove it? |
I'm playing a game right now, and one of the formulas for damage they have is:
Your character has 11 equipment slots. Let's call it head, neck, body, back, hand, hand, ring, ring, arm, waist, leg. Your goal is to find out how to maximize your damage. The "formula" for damage is as follows:
You have 1100 "total" p... |
Consider function $y=a^x$, $a>1$, and we need to show that
$$\frac{2(a-1)}{(a+1)} < \ln(a) < -1 + \sqrt{2a-1}$$
My idea is to use $y=a^x = \exp{(x\ln a)}$, then find the derivative of $y$ at $x=0$, which is $ln a$.
But I dunno how to continue from there .
Any idea on using geometry or something to prove it... |
Consider function $y=a^x$, $a>1$, and we need to show that
$$\frac{2(a-1)}{(a+1)} < \ln(a) < -1 + \sqrt{2a-1}$$
My idea is to use $y=a^x = \exp{(x\ln a)}$, then find the derivative of $y$ at $x=0$, which is $\ln a$.
But I dunno how to continue from there .
Any idea on using geometry or something to prove i... |
$F$ is a vector field with unknown potential $A$ and there's a surface with the boundary of a unit circle at the origin. Also, the circulation $A$ around the unit circle oriented clockwise is a positive constant. Given this information, I know that the flux through the unit circle will give me the flux through the surf... |
Consider function $y=a^x$, $a>1$, and we need to show that
$$\frac{2(a-1)}{(a+1)} < \ln(a) < -1 + \sqrt{2a-1}$$
My idea is to use $y=a^x = \exp{(x\ln a)}$, then find the derivative of $y$ at $x=0$, which is $\ln a$.
But I don’t know how to continue beyond this.
Any idea on using geometry or something to p... |
I do not know much about the Gamma function, $\Gamma(n)$, except it's series expansion (I am from the physics stackexchange), but I was wondering if there is a way to reduce $\Gamma(1-d/2)$ for $d\in\mathbb{C}$ (namely for $d\geq 0$) to something that looks likes $(1-d/2)\Gamma(-d/2)$? I know there are properties such ... |
Properties of the Gamma function? |
If $X$ is a random variable with cdf $F_X$ and pdf $f_X$, and $Y=ln(f_X(X))$, then the differential entropy of $X$ is $E(-ln(f_X(X)))=-\int_{range(X)}f_X(x)ln(f_X(x))dx$. I'm making a mistake in this calculation to get the right hand side.
$F_Y(y)=P(Y\leq y)=P(ln(f_X(X))\leq y))=P(X\leq f_X^{-1}(e^y))=f_X(f_X^{-1}(e... |
I do not know much about the Gamma function, $\Gamma(n)$, except it's series expansion (I am from the physics stackexchange), but I was wondering if there is a way to reduce $\Gamma(1-d/2)$ for $d\in\mathbb{C}$ (namely for $d\geq 0$) to something that looks likes $(1-d/2)\Gamma(-d/2)$? I know there are properties such ... |
I do not know much about the Gamma function, $\Gamma(n)$, except its series expansion (I am from the physics stackexchange), but I was wondering if there is a way to reduce $\Gamma(1-d/2)$ for $d\in\mathbb{C}$ (namely for $d\geq 0$) to something that looks likes $(1-d/2)\Gamma(-d/2)$? I know there are properties such a... |
I was doing this question as a homework from a lecture. I have proven that for all prime numbers, the square root is irrational using Euler’s lemma. However, I’m stuck on how to prove it for all non perfect squares. How should I approach this problem? Any help or solutions would be appreciated! Thank you so much! |
How to prove that the square root of all non perfect squares is irrational? |
Let $f:(0, 1] \to \Bbb R$ given by $f(x)=\sqrt{x}$, i want to show that $f$ is locally Liptschitz, this means (for $\lambda>0$, and $x, y \in B(p;r)$)
$$|\sqrt x-\sqrt y| \leq \lambda |x-y|$$
If we take $x, y \in B(0, r)$ for any $r>0$, then $|x|<r$ and $|y|<r$. As $x,y \in (0, 1]$ then $|\sqrt{x}|<|x|$ and $|\sqrt... |
I'm trying to model the process scheduling mechanism in Linux using queueing theory models.
Assuming that both the arrival and processing times of processes in the system follow a Poisson distribution of, and the system contains `c` independent logical cores, this scenario approximates an `M/M/c` model.
The chal... |
How to calculate the response time with M/M/c/PS(Processor Sharing) models in queueing theory? |
I do know that every normed vector space is also a topological vector space.
Is every metric (vector) space also a topological vector space.
I.e. is there a metric d, such that the induced topology is not continuous with the vector space operations? |
Does metric vector space imply topological vector space? |
I am trying to prove the following statements.
First, some definitions - here is my attempt to define a montonic function:
A function $f(x)$ is monotonic on an interval $I$ if for any two points $x_1, x_2 \in I$, the following conditions hold:
- Monotonically Increasing: If $x_1 < x_2$ then $f(x_1) \leq f(x_2)... |
When does the sum of parts equal to the whole? |
I am trying to understand the proof of this statement.
> Any Hausdorff topology on a finite-dimensional vector space with
> respect to which vector space operations are continuous is equivalent
> to the usual one.
Source: https://personal.math.ubc.ca/~cass/research/pdf/TVS.pdf
In the proof it is stated that:... |
Proposition 4.4.12 Pederson Analysis Now, Why does this operator commute? |
Let $f:(0, 1] \to \Bbb R$ given by $f(x)=\sqrt{x}$, I want to show that $f$ is locally Liptschitz, this means (for $\lambda>0$, and $x, y \in B(p;r)$)
$$|\sqrt x-\sqrt y| \leq \lambda |x-y|$$
If we take $x, y \in B(0, r)$ for any $r>0$, then $|x|<r$ and $|y|<r$. As $x,y \in (0, 1]$ then $|\sqrt{x}|<|x|$ and $|\sqrt... |
Let $R, S$ be commutative Noetherian rings such that $R$ is a subring of $S$. If $S$ is a normal domain, then it is standard that $R$ is also a normal domain. This leads me to ask the following question:
Let $X, Y$ be Noetherian schemes such that $X$ is an integral and normal, and there exists a proper birational mo... |
I just learned how finite $p$-th moment control the distribution of random variables. That is, given any random variable $X$ if $E[|X^p|]<\infty$ (or we say $X\in L^p$) for non-negative integer $p$ then $$E[|X|^p]=\int_0^{\infty}px^{p-1}\mathbb{P}(X>x)dx\tag{*}$$. The finite $p$-th moment tells us distribution of $X$ e... |
Here, I write $\mu$ as the Borel measure on $\mathbb{R^d}$ satisfying $\mu([a_1, b_1] \times \cdots \times [a_d, b_d]) = (b_1 - a_1) \cdots (b_d - a_d)$ ($a_i, b_i \in \mathbb{R}$) and $\lambda$ as the Lebesgue measure on $\mathbb{R}^d$.
My question is: If $f \colon \mathbb{R} \to \mathbb{R}$ is a Borel measurable f... |
An integral of a Borel measurable function w.r.t. Borel measure and that w.r.t. Lebesgue measure coincide with each other? |
Question: Four persons independently solve a certain problem correctly with probabilities 1/2, 3/4, 1/4 and 1/8.Then the probability that the problem is solved correctly by at least one
of them is ?
Solution:
say P(A)=1/2, P(B)=3/4, P(C)=1/4, P(D)=1/8
and P(A')=1 -1/2 = 1/2, P(B')= 1 - 3/4 = 1/4, P(C') = 1 - ... |
A puzzle I’ve wondered about but never got around to solving/verifying:
In the game Pokemon Let’s Go, the Pokemon Abra instantly teleports away if the player is detected within its line of sight.
Let’s say you’re in a square of length $s$ and one Abra randomly spawns in the square, with a random orientation, and... |
Consider a real positive definite matrix $A\in R^{d\times d}$.
Let $S_d^+$ be the set of symmetric positive matrices.
Is it always true that
$$(x,B) \to x^TB^{-1/2} A B^{-1/2} x$$
is jointly convex in $(x,B)\in R^d \times S_d^+$?
It is true in the case $A=I$, see https://math.stackexchange.com/questions/1070011... |
For a fixed positive definite $A$, is $(x,B) \to x^TB^{-1/2} A B^{-1/2} x$ always jointly convex? |
I am reading the Gortz's Algebraic Geometry, proof of Lemma 15.19 ( or Lemma 14.18 in the second edition ) and stuck at some statement. I've searched for math stack exchnage, but there doesn't seem to be any related questions, so I'm asking this.
**Q.1** Let $X$ be a reduced scheme with finitely many irreducible ... |
Dominant separated morphism between reduced schemes induces injective stalk homomorphism ? ( Gortz's Algebraic Geometry, Lemma 15.19 ) |
Let $X_1, \ldots, X_n$ be metric spaces and $\|\cdot\|$ be a norm on $\mathbb R^n$. Then it's easily seen that if $\|\cdot\|$ is monotonic in each coordinate while (others fixed) in the orthant $[0, +\infty)^n$, then $d(x, y) := \bigl\| \bigl( d_1(x_1, y_1), \ldots, d_n(x_n, y_n) \bigr) \bigr\|$ defines a metric on $X_... |
Any norm on $\mathbb R^n$ induces product metric? |
I know and can show that
\begin{equation*}
{}_{1}F_{1}\left(1,s,z\right)=1+\frac{z}s\,{}_{1}F_{1}\left(1,s+1,z\right)
\end{equation*}
From here, I think I get:
\begin{equation}
\begin{split}
{}_{1}F_{1}\left(1,s+1,z\right)&= \frac sz\left\{{}_{1}F_{1}\left(1,s,z\right)-1\right\} \\
& = \frac sz\left[\fr... |
**Question**: What is the probability that the coefficients of a quadratic equation with real roots form the sides of triangle? Assume that the coefficients are uniformly distributed and positive.
Note that it is sufficient to assume that the coefficients are uniformly distributed in $(0,1)$ since we can always divi... |
What is the probability that the coefficients of a quadratic equation with real roots form a triangle? |
I am trying to reduce this sum of these hypergeometric functions
$${{}_2{F_1}\left( {\frac{1}{2}, - \frac{1}{3};\frac{1}{6};x} \right) + {}_2{F_1}\left( { - \frac{1}{2},\frac{1}{3};\frac{1}{6};x} \right) + {}_2{F_1}\left( { - \frac{1}{2}, - \frac{1}{3};\frac{1}{6};x} \right)}$$
I tried :
$$\eqalign{
& 1.{}_2{F... |
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