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37 votes
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How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?

Integrate by parts: \begin{align} \int_x^{x+1}\sin(e^t)dt & =\int_x^{x+1}e^{-t}d(-\cos(e^t)) \\ & =e^{-x}\cos e^x-e^{-x-1}\cos e^{x+1}-\int_x^{x+1}e^{-t}\cos e^{t}dt\\ & =e^{-x}\cos e^x-e^{...
Fedor Petrov's user avatar
35 votes
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Can a real quartic polynomial in two variables have more than 4 isolated local minima?

Recent addition: Inspired by DimaPasechnik's and Matt F.'s comments about sum of squares decompositions, I tried the following very natural idea: Try to find $f$ of the form $f(x,y)=A(x,y)^2+B(x,y)^2$,...
Peter Mueller's user avatar
31 votes

Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

This is an instance of Holte's Amazing matrix. Consider addition of binary digits. Start with a carry of $c \in \{0,1,\ldots,2(m-1)\}$. Choose $2m-1$ bits uniformly at random, and add their sum to $c$....
Mark Wildon's user avatar
26 votes

Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

The Lindstrom-Gessel-Viennot lemma says that the number of families of nonintersecting lattice paths can be counted by a determinant. Let $a_i = (2m-i,i)$. Let $b_j = (2m-2j,-2m+2j)$. Then the number ...
Douglas Zare's user avatar
  • 27.9k
21 votes

Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

Here is a very low-brow answer to the original question. Consider the lower-triangular matrix \begin{equation*} V = [V_{ij}] = \left[\binom{i-1}{j-1}\right]\quad \text{for}\quad i \ge j. \end{...
Suvrit's user avatar
  • 28.5k
21 votes
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Prove that this expression is greater than 1/2

Let $$f(x,y):=4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Bigl( \sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}(\pi y-\pi x) \Bigr).$$ I will show that $$\min_{0\leq x\leq y\leq 1}f(x,y)=\min_{0\leq x\leq 1/...
GH from MO's user avatar
  • 102k
15 votes
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Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$

The question is about the signature of a quadratic form $$ \sum_{i=1}^n x_i^2 + \frac12\sum_{1 \le \mathrm{dist}(i,j) \le p} x_ix_j $$ (or about the spectrum of the corresponding linear operator). The ...
Ivan Izmestiev's user avatar
15 votes

Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

Let $A_n(x,\lambda)$ be the $n\times n$ matrix $$\left[\binom{x}{2j-i+\lambda}\right]_{i,j=1}^n.$$ Let's "generalize to trivialize". Sometimes, generalizations offer more elbow room to maneuver, such ...
T. Amdeberhan's user avatar
14 votes

Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

The matrix $A_n(x,\lambda)$ is obtained from the dual Jacobi matrix for the partition $\mu=(n+\lambda,n-1+\lambda,...,1+\lambda)$ by setting $x$ variables equal to 1 and the remaining variables equal ...
Richard Stanley's user avatar
14 votes
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Finding the closest matrix to $\text{SO}_n$ with a given determinant

(Basically) Full answer For $s \geq 1$ we always take the matrix with diagonals $s^{1/n}$. For $s < 1$ and we have two possibilities. For $n$ small enough we take (still) the matrix with ...
Tim Carson's user avatar
14 votes
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Can a cubic polynomial in two real variables have three saddle points?

The cubic $x^3 - xy^2 - 2x^2 + x$ has critical points in $(1,0)$, $(0,-1)$, $(0,1)$ and $(1/3, 0)$. The determinant of the Hessian matrix is $-4(3x^2 + y^2 - 2x)$. It assumes the values $-4$, $-4$, $-...
Peter Mueller's user avatar
14 votes
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How many saddle points can a quartic polynomial in two real variables have? All 9?

By (3.1) of Counting Critical Points of Real Polynomials in Two Variables by Alan Durfee, Nathan Kronefeld, Heidi Munson, Jeff Roy, Ina Westby a degree $d$ polynomial with only nondegenerate critical ...
Will Sawin's user avatar
  • 141k
13 votes

How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?

Here is a method that will allow one to find the exact upper and lower bounds on $g(z)$ over $z>0$ with any degree of accuracy. Take any real $z>0$. Since \begin{equation*} \frac1y=\int_0^\...
Iosif Pinelis's user avatar
13 votes
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An elementary inequality for three complex numbers

I will prove the original inequality. First, performing the change of variables $x=1/a$, etc., and inverting the harmonic mean, we need $$ \sum \left|\frac{yz}{x(y+z-x)}\right|\geq \frac32. $$ Next, ...
Ilya Bogdanov's user avatar
11 votes
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Min problem on integers

Let us denote $$\sigma_\ell:=\sum_{i=1}^\ell q_i\qquad\text{and}\qquad\tau_\ell:=\sum_{i=\ell+1}^s\frac{1}{q_i}.$$ Then $$\prod_{\ell=1}^{s-1}\left(\frac{q_\ell}{q_{\ell+1}}\cdot\frac{\sigma_{\ell+1}}{...
GH from MO's user avatar
  • 102k
10 votes
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Surprisingly simple minimum of a rational function on $\mathbb R_+^n$

I'm not sure about stationary points, but the global minimum is certainly there. Let's do it for $n=5$. Write $$ f(x)=1+\frac{x_2}{2x_1}+\frac{x_2}{2x_1}+\frac{x_2x_4}{2x_1x_3}+\frac{x_2x_4}{2x_1x_3}\,...
fedja's user avatar
  • 60.9k
10 votes

Prove that this expression is greater than 1/2

Let $F(x,y)$ denote the left-hand side of your inequality. It is easy to see that $|\nabla F(x,y)|\sqrt2/n<0.002$ if $0<x<y<1$, where $n:=6600$. A direct calculation shows that $F(i/n,j/n)&...
Iosif Pinelis's user avatar
10 votes

Elementary inhomogeneous inequality for three non-negative reals

Write $x = 1-X$, $y=1-Y$, $z=1-Z$. Then the inequality reduces to $$2XYZ \leq X^2 + Y^2 + Z^2$$ for $X, Y, Z \leq 1$. If $X, Y, Z < 0$ then the inequality is trivial, since LHS < 0. Otherwise ...
Sean Eberhard's user avatar
8 votes
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Elementary inhomogeneous inequality for three non-negative reals

Denote $x^2=a^3,y^2=b^3,z^2=c^3$. By AM-GM we have $1+2xyz=1+(abc)^{3/2}+(abc)^{3/2}\geqslant 3\sqrt[3]{1\cdot (abc)^{3/2}\cdot (abc)^{3/2}}=3abc$, so LHS is not less then $$a^3+b^3+c^3+3abc\geqslant ...
Fedor Petrov's user avatar
7 votes
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How to solve optimization problems on manifolds?

To complement Christian Clason's comment: there is usually no need to compute geodesics to optimize over manifolds directly. The usual replacement used for optimization purposes is called a retraction....
Nicolas Boumal's user avatar
7 votes

An intuition for three different types of subgradients (proximal, regular, limiting)

This might not be a satisfying answer, but this is how I personally deal with this issue (at least in the topic of optimization). I think these concepts are not made for actually computing them but ...
xel's user avatar
  • 181
7 votes
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Maximize the Euclidean norm of a matrix times a vector on unit sub-spheres

You cannot hope for anything like a closed form solution, or even an exact efficient algorithm for this problem, because it is NP-hard. The reduction is from the max-cut problem. Let's look at the ...
Sasho Nikolov's user avatar
7 votes
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What's the best orthonormal matrix to align two matrices in the operator norm sense?

The operator norm version of this problem is considered in: The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms, by G. A. Watson, Advances in Computational ...
Suvrit's user avatar
  • 28.5k
7 votes
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Is there a point in 6-dimensional space satisfying these polynomial inequalities?

All the conditions hold for $$(a, b, p, q, x, y)=\left(\frac{211}{500},\frac{531}{1000},\frac{96106069}{341750000},\frac{281961}{1000000},\frac{23996819}{170875000},\frac{149721291}{1000000000}\right)....
Iosif Pinelis's user avatar
7 votes
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On a certain norm of the identity operator on $\mathbb R^2$

Simply observe that $$\|x\|_{a,b}=\|x_1a+x_2b\|_1\,.$$ Thus, by orthogonality of $a,b$ and the easily-derived inequality $\|y\|_2\le\|y\|_1\le\sqrt{n}\|y\|_2$ for any $y\in\mathbb{R}^n$, we have \...
Jack L.'s user avatar
  • 1,443
7 votes
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Reference request: importance of Lipschitz continuity

In Mathematical/High Dimensional Statistics: One fairly amazing result is for $X=(X_1,\dots,X_n)$ where the coordinates are i.i.d. standard Gaussians, and $f:R^n \to R$ a $L$-Lipschitz function (w.r.t....
WeakLearner's user avatar
7 votes
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Does this maximisation problem admit a finite upper bound?

The answer is "yes" and it is quite a nice linear algebra problem, but let me restate it first in a less intimidating way. We'll deal with $\mathbb R^n$ for any finite $n$. The first thing I ...
fedja's user avatar
  • 60.9k
6 votes
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Levenberg's original article "A method for the solution of certain problems in least squares"

This link might be better. The paper is now open access and the pdf is searchable.
Niclas Börlin's user avatar
6 votes
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Is the solution of this optimization problem always positive semidefinite?

No, it is not always attained at a positive semidefinite matrix. The simplest example I have been able to find to demonstrate this is as follows: \begin{align*} U = \{ (1,0), (0,1), \tfrac{1}{\...
Nathaniel Johnston's user avatar

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