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Bayesian inference for spatial point patterns is often hindered computationally by intractable likelihoods. In the frequentist literature, estimating equations utilizing pseudolikelihoods have long been used for simulation-free parameter…

Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We prove that, if the irrationality exponent of $\xi$ is equal to $2$ or slightly greater than $2$, then the $b$-ary expansion of $\xi$ cannot be `too simple', in a suitable…

Littlewood polynomials are polynomials with each of their coefficients in $\{-1,1\}$. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro…

In a recent paper of Blomer, Bourgain, Radziwi{\l}{\l} and Rudnick, the authors proved the existence of small gaps between eigenvalues of the Laplacian in a rectangular billiard with sides $\pi$ and $\pi/\sqrt\alpha$, i.e. numbers of the…

Let $\xi$ be an irrational algebraic real number and $(p_k / q_k)_{k \ge 1}$ denote the sequence of its convergents. Let $(u_n)_{n \geq 1}$ be a non-degenerate linear recurrence sequence of integers, which is not a polynomial sequence. We…

It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Pade approximants at infinity by considering rational interpolants, (bi-)orthogonal rational functions and linear…

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of…

We show the existence of ``good'' approximations to a real number $\gamma$ using rationals with denominators formed by digits $0$ and $1$ in base $b$. We derive an elementary estimate and enhance this result by managing exponential sums.

Analytical complexity of quantum wavefunction whose argument is extended into the complex plane provides an important information about the potentiality of manifesting complex quantum dynamics such as time-irreversibility, dissipation and…

We study the problem of computing the $p\rightarrow q$ norm of a matrix $A \in R^{m \times n}$, defined as \[ \|A\|_{p\rightarrow q} ~:=~ \max_{x \,\in\, R^n \setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p} \] This problem generalizes the spectral…

The recent technique for estimating lower bounds of the prime counting function $\pi(x)=#\{p \leq x: p\text{ prime}\}$ by means of the irrationality measures $\mu(\zeta(s)) \geq 2$ of special values of the zeta function claims that $\pi(x)…

Interior point methods (IPMs) are a common approach for solving linear programs (LPs) with strong theoretical guarantees and solid empirical performance. The time complexity of these methods is dominated by the cost of solving a linear…

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite's two approximation problems for functions leads to the Schlesinger…

We obtain new parametric quadrature formulas with variable nodes for integrals of complex rational functions over circles, segments of the real axis and the real axis itself. Basing on these formulas we derive $(q,p)$-inequalities of…

We present a proximal augmented Lagrangian based solver for general convex quadratic programs (QPs), relying on semismooth Newton iterations with exact line search to solve the inner subproblems. The exact line search reduces in this case…

We compute the constant of approximation for an arbitrary rational point on an arbitrary smooth cubic hypersurface $X$ over a number field $k$, provided that there is a $k$-rational line somewhere on $X$. In the process, we verify the Coba…

We prove that Pade approximants yield increasingly accurate predictions of higher-order coefficients in QCD perturbation series whose high-order behaviour is governed by a renormalon. We also prove that this convergence is accelerated if…

Laplace's method is used to approximate intractable integrals in a statistical problems. The relative error rate of the approximation is not worse than $O_p(n^{-1})$. We provide the first statistical lower bounds showing that the $n^{-1}$…

In this paper, we simplify and improve the constant, $c$, that appears in effective irrationality measures, $|(a/b)^{m/n}-p/q|>c|q|^{-(\kappa+1)}$, obtained from the hypergeometric method for $a/b$ near $1$. The dependence of $c$ on $|a|$…

This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all feasible solutions have irrational entries, numerical…