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Based on a quantitative version of the inverse function theorem and an appropriate saddle-point formulation we derive a quasi-optimal error estimate for the finite element approximation of harmonic maps into spheres with a nodal…

We establish boundedness results for bilinear singular integral operators with rough homogeneous kernels whose restriction to the unit sphere belongs to the Orlicz space $L(\log L)^\alpha$. This improves the previously best known condition…

We present convergence theory for corrected quadrature rules on uniform Cartesian grids for functions with a point singularity. We begin by deriving an error estimate for the punctured trapezoidal rule, and then derive error expansions. We…

We give a direct proof of the local $Tb$ Theorem, in the Euclidean setting, and under the assumption of dual exponents. This Theorem provides a flexible framework for proving the boundedness of a Calder\'on-Zygmund operator, supposing the…

Solutions to the $n$-dimensional Laplace equation which are constant on a central quadric are found. The associated twistor description of the case $n=3$ is used to characterise Gibbons-Hawking metrics with tri-holomorphic $SL(2, \C)$…

We prove a complexity dichotomy theorem for Holant Problems on 3-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted…

This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic…

We offer a conjecture on sharp estimation of a definite improper integral depend on a parameter $\lambda \in (0,+\infty)$ by means of given estimate of other definite integral depend on parameters $t\in [0,+\infty)$ and $\lambda$. Such…

Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the…

Singular and oscillatory functions feature in numerous applications. The high-accuracy approximation of such functions shall greatly help us develop high-order methods for solving applied mathematics problems. This paper demonstrates that…

We study the Schwarz lemma for harmonic functions and prove sharp versions for the cases of real harmonic functions and the norm of harmonic mappings.

In this work the implicit function theorem is used for searching local symbolic resolution of differential equations. General results of existence for first order equations are proven and some examples, one relative to cavitation in a…

We prove a duality theorem the computation of certain Bellman functions is usually based on. As a byproduct, we obtain sharp results about the norms of monotonic rearrangements. The main novelty of our approach is a special class of…

In this paper, we prove common fixed point results for a self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results modify, unify, extend and generalize…

Relativistic secular perturbation theory has ignited significant interest in uncovering intricate cross-term effects, especially the interplay between 1PN and quadrupole terms. While most existing studies rely on the Lagrangian planetary…

A local Tb Theorem provides a flexible framework for proving the boundedness of a Calder\'on-Zygmund operator T. One needs only boundedness of the operator T on systems of locally pseudo-accretive functions \{b_Q\}, indexed by cubes. We…

We introduce averaging operators on lattices $\mathbb{Z}^d$ and study the Liouville property for functions satisfying mean value properties associated to such operators. This framework encloses discrete harmonic, $p$-harmonic,…

An improvement of the Liouville theorem for discrete harmonic functions on $\mathbb{Z}^2$ is obtained. More precisely, we prove that there exists a positive constant $\varepsilon$ such that if $u$ is discrete harmonic on $\mathbb{Z}^2$ and…

In this paper we prove an analogue in the discrete setting of \Bbb Z^d, of the spherical maximal theorem for \Bbb R^d. The methods used are two-fold: the application of certain "sampling" techniques, and ideas arising in the study of the…

We prove a sharp integral gradient estimate for harmonic functions on noncompact K\"ahler manifolds. As application, we obtain a sharp estimate for the bottom of spectrum of the p-Laplacian and prove a splitting theorem for manifolds…