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On the singularity of random Bernoulli matrices - novel integer partitions and lower bound expansions derives the lower bound (theorem 1) $$\mathbb{P}\{ \text{det}(A_n) = 0\} \geq 2^{2-n} \binom{n …

Note that the random vector $u=(u_1,u_2,\ldots u_n)$, uniformly distributed on the unit sphere, can be replaced by the ratio $u=y/|y|$, with $y=(y_1,y_2,\ldots y_n)\sim N(0,I_n)$ a multivariate normal …

The established reference (see for example arXiv:0912.0651) to Taubes trick is C.H. Taubes, Seiberg Witten and Gromov invariants for symplectic 4-manifolds (2000). This book collects results from fo …

Why not just do what Google did, digitize it yourself? According to WorldCat the book is present in many libraries. You could borrow it and scan it using a book scanner. This may require an investment …

See Asymptotic expansions of oscillatory integrals with complex phase (2010). The usual formulas carry through even if $\phi(x)$ is complex, provided that it has a non-negative imaginary part.

Let me assume that $M$ is symmetric, then we can work in a basis where it is diagonal, $M=\operatorname{diag}(\mu_1,\mu_2,\ldots \mu_n)$. I denote the vectors $v_i=V_{i1}$ and $u_i=V_{i2}$, $i=1,2\ldo …

Back in 2012, professor Ben Chow gave some advice to a similar question; these include the Lectures on Elliptic and Parabolic Equations in Sobolev Spaces by Krylov [recommended here by Giorgio Metafu …

Professor Atsushi Murase from Kyoto Sangyo University was kind enough to reach out to Professor Kato for the 1978 preprint. It's a 41 page typewritten text in English. I am uncertain whether I can pos …

For a given Hermitian $A$, the matrix $A^+$ is the positive semi-definite matrix $X$ that minimizes the Frobenius norm $\| X-A\|_{\rm F}$, as well as the spectral norm $\| X-A\|_2$, see section 8.1.1, …

Edit: An earlier version of this answer missed a factor of two, now corrected, thanks to Daniele Tampieri. We seek the variation of the functional $$L[u]=\int_\Omega\left(|\nabla u|^2+1\right)\chi_{u> …

On the problem of stability for higher-order derivative Lagrangian systems in Letters in Mathematical Physics (1987) may have the desired level of rigor (see Theorem 1). The proof of the theorem is a …

Erin Pearse's Introduction to dimension theory and fractal geometry may well be suited for this purpose. It introduces the various ways to define and measure a fractional dimension (box counting, Mink …

Cauchy's criterion is a special case$^\ast$ of a more general criterion proven by Kummer in 1857 [1], and by Fueter in 1922 [2]. A description of Kummer's derivation and how it implies the Cauchy crit …

The full physics problem is complex, the vibrating membrane displaces the air, which causes a backreaction and signifantly modifies the response. Moreover, the response also depends sensitively on whe …

Since this question was bumped to the front page, I might address Q1: Can someone provide historical references for it? This goes back to André Weil, who writes in [1] that Ernst Hellinger, a student …