37
votes
Accepted
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^{...
- 99k
35
votes
Accepted
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 $f(x,y)=A(x,y)^2+B(x,y)^2$,...
- 20.3k
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 $c$....
- 10.7k
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 ...
- 27.8k
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{...
- 28.3k
21
votes
Accepted
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
$$\min_{0\leq x\leq y\leq 1}f(x,y)=\min_{0\leq x\leq 1/...
- 95.7k
15
votes
Accepted
Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$
The question is about the signature of a quadratic form
$$
\sum_{i=1}^n x_i^2 + \frac12\sum_{1 \le \mathrm{dist}(i,j) \le p} x_ix_j
$$
(or about the spectrum of the corresponding linear operator). The ...
- 6,217
15
votes
Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Let $A_n(x,\lambda)$ be the $n\times n$ matrix
$$\left[\binom{x}{2j-i+\lambda}\right]_{i,j=1}^n.$$
Let's "generalize to trivialize". Sometimes, generalizations offer more elbow room to maneuver, such ...
- 41.2k
14
votes
Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
The matrix $A_n(x,\lambda)$ is obtained from the dual Jacobi matrix for the partition $\mu=(n+\lambda,n-1+\lambda,...,1+\lambda)$ by setting $x$ variables equal to 1 and the remaining variables equal ...
- 48.7k
14
votes
Accepted
Finding the closest matrix to $\text{SO}_n$ with a given determinant
(Basically) Full answer
For $s \geq 1$ we always take the matrix with diagonals $s^{1/n}$.
For $s < 1$ and we have two possibilities. For $n$ small enough we take (still) the matrix with ...
- 846
14
votes
Accepted
Can a cubic polynomial in two real variables have three saddle points?
The cubic $x^3 - xy^2 - 2x^2 + x$ has critical points in $(1,0)$, $(0,-1)$, $(0,1)$ and $(1/3, 0)$. The determinant of the Hessian matrix is $-4(3x^2 + y^2 - 2x)$. It assumes the values $-4$, $-4$, $-...
- 20.3k
14
votes
Accepted
How many saddle points can a quartic polynomial in two real variables have? All 9?
By (3.1) of Counting Critical Points of Real Polynomials in Two Variables by Alan Durfee, Nathan Kronefeld, Heidi Munson, Jeff Roy, Ina Westby a degree $d$ polynomial with only nondegenerate critical ...
- 131k
13
votes
How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?
Here is a method that will allow one to find the exact upper and lower bounds on $g(z)$ over $z>0$ with any degree of accuracy.
Take any real $z>0$. Since
\begin{equation*}
\frac1y=\int_0^\...
- 111k
13
votes
Accepted
An elementary inequality for three complex numbers
I will prove the original inequality.
First, performing the change of variables $x=1/a$, etc., and inverting the harmonic mean, we need
$$
\sum \left|\frac{yz}{x(y+z-x)}\right|\geq \frac32.
$$
Next, ...
- 22.1k
11
votes
Accepted
Min problem on integers
Let us denote
$$\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}}{...
- 95.7k
10
votes
Accepted
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
$$
f(x)=1+\frac{x_2}{2x_1}+\frac{x_2}{2x_1}+\frac{x_2x_4}{2x_1x_3}+\frac{x_2x_4}{2x_1x_3}\,...
- 58.2k
10
votes
Prove that this expression is greater than 1/2
Let $F(x,y)$ denote the left-hand side of your inequality. It is easy to see that $|\nabla F(x,y)|\sqrt2/n<0.002$ if $0<x<y<1$, where $n:=6600$. A direct calculation shows that $F(i/n,j/n)&...
- 111k
10
votes
Elementary inhomogeneous inequality for three non-negative reals
Write $x = 1-X$, $y=1-Y$, $z=1-Z$. Then the inequality reduces to
$$2XYZ \leq X^2 + Y^2 + Z^2$$
for $X, Y, Z \leq 1$. If $X, Y, Z < 0$ then the inequality is trivial, since LHS < 0. Otherwise ...
- 8,903
8
votes
Accepted
Elementary inhomogeneous inequality for three non-negative reals
Denote $x^2=a^3,y^2=b^3,z^2=c^3$. By AM-GM we have $1+2xyz=1+(abc)^{3/2}+(abc)^{3/2}\geqslant 3\sqrt[3]{1\cdot (abc)^{3/2}\cdot (abc)^{3/2}}=3abc$, so LHS is not less then $$a^3+b^3+c^3+3abc\geqslant ...
- 99k
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 ...
- 181
7
votes
Accepted
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 ...
- 757
7
votes
Accepted
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 ...
- 28.3k
7
votes
Accepted
Is there a point in 6-dimensional space satisfying these polynomial inequalities?
All the conditions hold for
$$(a, b, p, q, x, y)=\left(\frac{211}{500},\frac{531}{1000},\frac{96106069}{341750000},\frac{281961}{1000000},\frac{23996819}{170875000},\frac{149721291}{1000000000}\right)....
- 111k
7
votes
Accepted
On a certain norm of the identity operator on $\mathbb R^2$
Simply observe that
$$\|x\|_{a,b}=\|x_1a+x_2b\|_1\,.$$
Thus, by orthogonality of $a,b$ and the easily-derived inequality $\|y\|_2\le\|y\|_1\le\sqrt{n}\|y\|_2$ for any $y\in\mathbb{R}^n$, we have
\...
- 1,425
7
votes
Accepted
Reference request: importance of Lipschitz continuity
In Mathematical/High Dimensional Statistics:
One fairly amazing result is for $X=(X_1,\dots,X_n)$ where the coordinates are i.i.d. standard Gaussians, and $f:R^n \to R$ a $L$-Lipschitz function (w.r.t....
- 186
6
votes
Accepted
Levenberg's original article "A method for the solution of certain problems in least squares"
This link might be better. The paper is now open access and the pdf is searchable.
- 208
6
votes
Accepted
How to solve optimization problems on manifolds?
To complement Christian Clason's comment: there is usually no need to compute geodesics to optimize over manifolds directly. The usual replacement used for optimization purposes is called a retraction....
- 390
6
votes
Accepted
Is the solution of this optimization problem always positive semidefinite?
No, it is not always attained at a positive semidefinite matrix. The simplest example I have been able to find to demonstrate this is as follows:
\begin{align*}
U = \{ (1,0), (0,1), \tfrac{1}{\...
- 5,558
6
votes
Nearest matrix orthogonally similar to a given matrix
Not a full answer, but some pointers on how to get a numerical method: If $\|\cdot\|_2$ denotes the spectral norm, then, by unitary invariance, your problem is equal to
$$
\min_{T\in O(n)}\|AT-TB\|_2.
...
- 12.2k
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