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Curve fitting is a fundamental technique in engineering and scientific research, serving as a critical tool for extracting insights from data. This study explores the application of various statistical equations to estimate outcomes in…
We study integration and $L^2$-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure…
We define two-dimensional Dirichlet spectrum (with respect to Euclidean norm) as D_2=\lambda\in\mathbf{R} | \exists \mathbf{v}=(v_1,v_2)\in \mathbf {R}^2: \limsup\limits_{t\rightarrow\infty} {t\cdot\psi_{v}^2(t)}=\lambda, where…
Flow matching generates data by integrating a learned velocity field, where the number of integration steps (NFE) directly determines inference cost. We analyze which properties of the velocity field govern integration error by decomposing…
We study a family of approximations to Euler's equation depending on two parameters $\varepsilon,\eta \ge 0$. When $\varepsilon=\eta=0$ we have Euler's equation and when both are positive we have instances of the class of…
It is well known that if a function $f$ satisfies $$\|f(x) e^{\pi \alpha |x|^2}\|_p + \| \widehat{f}(\xi) e^{\pi \alpha |\xi|^2} \|_q<\infty \qquad\qquad\qquad(*)$$ with $\alpha=1$ and $1\le p,q<\infty$, then $f\equiv 0.$ We prove that if…
We describe a factor-revealing convex optimization problem for the integrality gap of the maximum-cut semidefinite programming relaxation: for each $n \geq 2$ we present a convex optimization problem whose optimal value is the largest…
We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $\frac{1}{2} + \Omega(1/\sqrt{D})$ fraction of constraints, where…
Through an inversion approach, we suggest a possible estimation for the absolute value of Mertens function $\vert M(x) \vert$ that $ \left\vert M(x) \right\vert \sim \left[\frac{1}{\pi \sqrt{\varepsilon}(x+\varepsilon)}\right]\sqrt{x}$…
This article deals with error estimates for the finite element approximation of variational normal derivatives and, as a consequence, error estimates for the finite element approximation of Dirichlet boundary control problems with energy…
In this note we present a new special function that behaves like the error function and we provide an approximated accurate closed form for its CDF in terms of both Chebyshev polynomials of the first kind and the error function. Also, we…
The Sinc approximation is a function approximation formula that attains exponential convergence for rapidly decaying functions defined on the whole real axis. Even for other functions, the Sinc approximation works accurately when combined…
The paper is concerned with a class of nonlinear free boundary problems, which are usually solved by variational methods based on primal (or primal-dual) variational settings. We deduce and investigate special relations (error identities).…
We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection-diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verf\"{u}rth indicates that a…
We prove that the backward shift operator on $H^4$ has norm equal to $\sqrt[4]{\varphi}$, with $\varphi = \frac{1 + \sqrt{5}}{2}$. Furthermore, we characterize all extremal functions; they are precisely the functions of the form \[ f(z) =…
Take a random variable X with some finite exponential moments. Define an exponentially weighted expectation by E^t(f) = E(e^{tX}f)/E(e^{tX}) for admissible values of the parameter t. Denote the weighted expectation of X itself by r(t) =…
In this paper functions $f:D\to\mathbb{R}$ satisfying the inequality \[ f\Big(\frac{x+y}{2}\Big)\leq\frac12f(x)+\frac12f(y) +\varphi\Big(\frac{x-y}{2}\Big) \qquad(x,y\in D) \] are studied, where $D$ is a nonempty convex subset of a real…
We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and ``interpolates" the usual Taylor formulas with two consecutive integer orders. This enables us to…
We determine the optimal majorant $M^+$ and minorant $M^-$ of exponential type for the truncation of $x\mapsto (x^2+a^2)^{-1}$ with respect to general de Branges measures. We prove that \[ \int_\mathbb{R} (M^+ - M^-) |E(x)|^{-2}dx =…
We solve variationally certain equations of stellar dynamics of the form $-\sum_i\partial_{ii} u(x) =\frac{|u|^{p-2}u(x)}{{\rm dist} (x,{\mathcal A} )^s}$ in a domain $\Omega$ of $\rn$, where ${\mathcal A} $ is a proper linear subspace of…