# Tag Info

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^{...
• 104k
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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 $f(x,y)=A(x,y)^2+B(x,y)^2$,...
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### 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 $c$....
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### 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 ...
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### 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{...
• 28.5k
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• 102k
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• 104k
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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....

### 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
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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 ...
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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 ...
• 28.5k
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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....
• 206
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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 ...
• 60.9k
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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.