Related papers: The higher order Carath\'eodory--Julia theorem and…
The development of high-degree interpolation polynomials which use the values of the function and its subsequent derivatives is reformulated. Also, we present a variant of new formula in barycentric form.
Using the new conformable fractional derivative, which differs from the Riemann-Liouville and Caputo fractional derivatives, we reformulate the second-order conjugate boundary value problem in this new setting. Utilizing the corresponding…
In this paper we study the existence of solutions for a new class of nonlinear differential equations with three-point boundary conditions. Existence of solutions are obtained by using the Leray-Schauder degree.
We establish a C^1,alpha Schauder estimate of a non-standard degenerate elliptic equation and use it to give another proof of the higher order boundary Harnack inequality. As an application, we obtain the analyticity of the free boundary in…
In this note we introduce higher order polar loci as natural generalizations of the classical polar loci, replacing the role of tangent spaces by that of higher order osculating spaces. The close connection between polar loci and dual…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
We develop the machinery of boundary triplets for one-dimensional operators generated by formally self-adjoint quasi-differential expression of arbitrary order on a finite interval. The technique are then used to describe all maximal…
We consider free higher derivative theories of scalars and Dirac fermions in the presence of a boundary in general dimension. We establish a method for finding consistent conformal boundary conditions in these theories by removing certain…
Let $w(\zeta)$ be a function analytic on $\mathbb D$, $|w(\zeta)|\le 1$. Let $|t_0|=1$. Assume that $w$ and $w'$ have nontangential boundary values $w_0$ and $w'_0$, respectively, at $t_0$, $|w_0|=1$. Then (Carath\'eodory - Julia)…
We investigate the existence and multiplicity of solutions for higher order discrete boundary value problems via critical point theory.
Sufficient boundary asymptotic conditions are established for a generalized Schur function $f$ to be identically equal to a given rational function $g$ unimodular on the unit circle. Similar rigidity statements are presented for generalized…
Surface physics dominated by bulk properties has been one of the central interests in modern condensed matter physics, from electric polarization to bulk-boundary correspondence of topological insulators and superconductors. Here, we extend…
The Carath\'{e}odory problem in the $N$-variable non-commutative Herglotz--Agler class and the Carath\'{e}odory--Fej\'{e}r problem in the $N$-variable non-commutative Schur--Agler class are posed. It is shown that the Carath\'{e}odory…
We give new improvements to the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves. Our approach features three…
We introduce the Schur class of functions, discrete analytic on the integer lattice in the complex plane. As a special case, we derive the explicit form of discrete analytic Blaschke factors and solve the related basic interpolation…
In this paper we are concerned with singular points of solutions to the {\it unstable} free boundary problem $$ \Delta u = - \chi_{\{u>0\}} \qquad \hbox{in} B_1. $$ The problem arises in applications such as solid combustion, composite…
Using class field theory, we prove a restriction on the intersection of the maximal abelian extensions associated with different number fields. This restriction is then used to improve a result of Rosen and Silverman about the linear…
We consider the boundary value problem associated to the curl operator, with vanishing Dirichlet boundary conditions. We prove, under mild regularity of the data of the problem, existence of classical solutions.
We describe a generalization of the notion of a Hilbert space model of a function in the Schur class of the bidisc. This generalization is well adapted to the investigation of boundary behavior at a mild singularity of the function on the…
Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S-class numbers of intermediate fields, arising from Brauer-Kuroda relations, as a unit…