Eigenvalue location using certain matrix functions and geometric curves

The main inertia theorem gives necessary and sufficient conditions that an n×n complex matrix A have no eigenvalues on the imaginary axis of the complex plane. In this paper corresponding necessary and sufficient conditions are given that A have no eigenvalues on any arbitrary circle, line, and certain other curves in the complex plane. Generalizations of the second part of the main inertia theorem give inclusion regions for the eigenvalues of A.     

The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers

In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli-Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form.     

On the geometric form of solutions of a free boundary problem involving galvanization

In [6], T. I. Vogel studied a free boundary problem originating in the galvanization process. He showed that if the given boundary Γ* is starlike or convex, then so is the free boundary solution Γ. Our purpose is to generalize Vogel's second result by showing (under certain assumptions) that Γ cannot have more (local) maxima or minima (relative to a given direction) than Γ*; also that Γ cannot have more inflection points or greater total curvature than Γ*. The author has already proven analogous results for the Bernoulli free boundary problem in [1], [2] and [3].     

Flat connections,geometric invariants and energy of harmonic functions on compact Riemann surfaces

A geometric invariant is associated to the space of flat connections on a G-bundle over a compact Riemann surface and is related to the energy of harmonic functions.     

The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions

In a previous Note the author gave a generalisation of Witten's proof of the Morse inequalities to the model of a singular complex algebraic curve X and a stratified Morse function f. In this Note a geometric interpretation of the complex of eigenforms of the Witten Laplacian corresponding to small eigenvalues is provided in terms of an appropriate subcomplex of the complex of unstable cells of critical points of f. To cite this article: U. Ludwig, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

An estimate of the curvature of level curves in a class of univalent functions

In this note a class of normalized functions which are univalent in the unit disk is considered. An exact solution of the problem of an upper estimate of curvature of level curves for functions of this class is obtained. In particular, an exact solution of this problem for univalent functions with real coefficients at the points of the interval (0, 1) is obtained.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 381–388, March, 1976.The author is thankful to the referee for taking interest in this note.     

On the approximation of harmonic functions by harmonic polynomials

Siberian Mathematical Journal -     

Gaussian curvature estimates for the convex level sets of p‐harmonic functions

We give a positive lower bound for the Gaussian curvature of the convex level sets of p‐harmonic functions with the Gaussian curvature of the boundary and the norm of the gradient on the boundary. Combining the deformation process, this estimate gives a new approach to studying the convexity of the level sets of the p‐harmonic function. © 2010 Wiley Periodicals, Inc.     

Harnack inequality for harmonic functions relative to a nonlinear p-homogeneous Riemannian Dirichlet form

We state a Wiener criterion for the regularity of points with respect to a relaxed Dirichlet problem for a p-homogeneous Riemannian Dirichlet form.     

On the geometric form of Bernoulli configurations

In References 4 and 5, the author studied the geometric form of the free boundary Γ of an annular domain Ω, characterized by the Bernoulli condition that |?U|=1 on Γ, where U is the capacity potential in Ω. It was shown, for example, that Γ has at most as many ν-extrema (for a given direction ν) or inflection points as the other boundary component Γ* of Ω, which is assumed given. Our purpose here is to extend the results in References 4 and 5 (wherever possible) to multiply connected regions for which some boundary components are given and others are free boundaries characterized by the Bernoulli condition. We present both positive results and counterexamples.     

The harmonic and geometric mean of vectors

New vector operations are defined that generalize the classical harmonic and geometric means of scalars. The definition of the harmonic mean involves an explicit vector formula, whereas the geometric mean of two vectors is defined by an iterative procedure. Two special cases are described in some detail, the case of 2-vectors which are equivalent to the complex numbers, and the case of 4-vectors which are a representation of the quaternions.     

On the minimum of harmonic functions

Letu be a function harmonic in the unit disc or in the plane, and letu(z)≤M(|z|) for a majorantM. We formulate conditions onM that guarantee thatu(z)≥−(1+o(1))M(|z|) for |z|→1 in the disc and for |z|→∞ in the plane.     

Deriving the beta-constraints for geometric continuity of parametric curves

Parametric splines curves are typically constructed so that the firstn parametric derivatives agree where the curve segments abut. This type of continuity condition has become known asC ⁿ orn ^th orderparametric continuity. It has previously been shown that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves. We definen ^th ordergeometric continuity (Gⁿ), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity.G ⁿ continuity provides for the introduction ofn quantities known asshape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices. Several applications of the theory are discussed, along with topics of future research.     

A geometric problem involving the nearest neighbour algorithm

It is shown that for arbitrary large n there exist sets L in the unit square of R² with |L| ≤ n, such that starting at any chosen point P₁ ϵ L and touring L according to the Nearest Neighbour Algorithm, the sum of the squares of the distances exceeds c·log₂ n, with c > 0.     

Certain geometric properties of the Mittag-Leffler functions

In the present investigation, the Mittag-Leffler functions with their normalization are considered. Several sufficient conditions are obtained so that the Mittag-Leffler functions have certain geometric properties including univalency, starlikeness, convexity and close-to-convexity in the open unit disk. Partial sums of Mittag-Leffler functions are also studied. The results obtained are new and their usefulness is depicted by deducing several interesting corollaries and examples.