Slice regular functions on real alternative algebras

In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C¹.     

The fundamental theorem of algebra for Hamilton and Cayley numbers

In this paper we prove the fundamental theorem of algebra for polynomials with coefficients in the skew field of Hamilton numbers (quaternions) and in the division algebra of Cayley numbers (octonions). The proof, inspired by recent definitions and results on regular functions of a quaternionic and of a octonionic variable, follows the guidelines of the classical topological argument due to Gauss. G. Gentili and F. Vlacci are partially supported by G.N.S.A.G.A. of the I.N.D.A.M. and by M.I.U.R.     

On the Notion of the Bochner–Martinelli Integral for Domains with Rectifiable Boundary

In this paper we discuss the notion of the Bochner–Martinelli kernel for domains with rectifiable boundary in , by expressing the kernel in terms of the exterior normal due to Federer (see [17,18]). We shall use the above mentioned kernel in order to prove both Sokhotski–Plemelj and Plemelj–Privalov theorems for the corresponding Bochner–Martinelli integral, as well as a criterion of the holomorphic extendibility in terms of the representation with Bochner–Martinelli kernel of a continuous function of two complex variables. Explicit formula for the square of the Bochner–Martinelli integral is rediscovered for more general surfaces of integration extending the formula established first by Vasilevski and Shapiro in 1989. The proofs of all these facts are based on an intimate relation between holomorphic function theory of two complex variables and some version of quaternionic analysis. Submitted: September 6, 2006. Accepted: November 1, 2006.     

Functions of a quaternion variable which are gradients of real-valued functions

We consider the collection of functions of one quaternion variable which can be expressed asG(Y) whereY is a real-valued quaternion function andG is a differential operator which corresponds to the gradient of real variable theory. Integral theorems for such functions are given, together with necessary and sufficient conditions for a function to be a gradient function, in terms of its Frechet derivative. The extended complex analytic functions, the Fueter functions, and the momentum-energy density functions are seen to be gradient functions which correspond to biharmonic, harmonic, and wave functions respectively.     

A Greedy Partition Lemma for directed domination

A directed dominating set in a directed graph D is a set S of vertices of V such that every vertex u∈V(D)?S has an adjacent vertex v in S with v directed to u. The directed domination number of D, denoted by γ(D), is the minimum cardinality of a directed dominating set in D. The directed domination number of a graph G, denoted Γ_d(G), is the maximum directed domination number γ(D) over all orientations D of G. The directed domination number of a complete graph was first studied by Erd?s [P. Erd?s On a problem in graph theory, Math. Gaz. 47 (1963) 220–222], albeit in a disguised form. In this paper we prove a Greedy Partition Lemma for directed domination in oriented graphs. Applying this lemma, we obtain bounds on the directed domination number. In particular, if α denotes the independence number of a graph G, we show that α≤Γ_d(G)≤α(1+2ln(n/α)).     

The graph and range singularity spectra of b-adic independent cascade functions

With the “iso-Hölder” sets of a function we naturally associate subsets of the graph and the range of the function. We compute the Hausdorff dimension of these subsets for a class of statistically self-similar multifractal functions, namely the b-adic independent cascade functions.     

Remark on the global existence and large time asymptotics of solutions for the quadratic NLS

We study the global in time existence of small solutions to the nonlinear Schrödinger equation with quadratic interactions (0.1) We prove that if the initial data u₀ satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore, we prove the existence of the usual scattering states and find the large time asymptotics of the solutions.     

Invariant metrics and the boundary behavior of holomorphic functions on domains in\mathbb{C}^n

Invariant metrics are used to provide a unified approach to the study of holomorphic functions in Hardy classes on domains in one and several complex variables. Both approach regions and boundary measures are constructed from the metric. Examples are provided to show how diverse theories can be unified with this approach. The Hartogs extension phenomenon and Fatou’s theorem are seen to be two aspects of the same circle of ideas. Author supported in part by a grant from the National Science Foundation     

Local solutions for a coupled system of Kirchhoff type

We investigate the existence of local solutions of the following coupled system of Kirchhoff equations subject to nonlinear dissipation on the boundary: (∗) Here {Γ₀,Γ₁} is an appropriate partition of the boundary Γ of Ω and ν(x), the outer unit normal vector at x∈Γ₁.By applying the Galerkin method with a special basis for the space where lie the approximations of the initial data, we obtain local solutions of the initial-boundary value problem for (∗).     

Subordination and superordination results for analytic functions with respect to symmetrical points

The theory of symmetric functions has many applications in the investigation of fixed points, estimation of absolute values of some integrals and obtaining the results of the type of Cartan’s uniqueness theorem. In this paper, we solve some differential subordinations and superordinations involving analytic functions with respect to the symmetric points and also derive some sandwich results under certain assumptions on the parameters involved. The various results presented in this paper are shown to apply to yield the corresponding (new or known) results for many simpler function classes.     

Deformations of associative submanifolds with boundary

Let M be a topological G₂-manifold. We prove that the space of infinitesimal associative deformations of a compact associative submanifold Y with boundary in a coassociative submanifold X is the solution space of an elliptic problem. For a connected boundary ∂Y of genus g, the index is given by _∫∂Yc₁(νX)+1−g, where νX denotes the orthogonal complement of T∂Y in TX|∂Y and c₁(νX) the first Chern class of νX with respect to its natural complex structure. Further, we exhibit explicit examples of non-trivial index.     

A semi-recursion for the number of involutions in special orthogonal groups over finite fields

Let I(n) be the number of involutions in a special orthogonal group SO(n,Fq) defined over a finite field with q elements, where q is the power of an odd prime. Then the numbers I(n) form a semi-recursion, in that for m>1 we haveI(2m+3)=(q2m+2+1)I(2m+1)+q2m(q2m−1)I(2m−2). We give a purely combinatorial proof of this result, and we apply it to give a universal bound for the character degree sum for finite classical groups defined over Fq.     

Global differential equations for slice regular functions

In the present paper, we give a system of global differential equations which are satisfied by slice regular functions on a real alternative algebra. By means of the concepts of stem function and slice function, we are able to improve some results obtained recently in the quaternionic and slice monogenic case and to extend them to this general setting. In particular, we describe the precise relation existing between the global differential equations and the condition of slice regularity.     

Stability results for approximation by positive definite functions on SO(3)

We consider interpolation methods defined by positive definite functions on a locally compact group G. Estimates for the smallest and largest eigenvalue of the interpolation matrix in terms of the localization of the positive definite function on G are presented, and we provide a method to get positive definite functions explicitly on compact semisimple Lie groups. Finally, we apply our results to construct well-localized positive definite basis functions having nice stability properties on the rotation group SO(3).     

Superharmonic functions are locally renormalized solutions

We show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent. As an application we characterize singular solutions of multidimensional Riccati type partial differential equations.     

On functions convex in the direction of the real axis with a fixed second coefficient

In the paper we consider the class Γ of analytic and univalent functions f in the unit disk Δ, normalized by f(0) = f′(0) − 1 = 0, having real coefficients and such that f(Δ) is convex in the direction of the real axis. We are especially interested in some subclasses of Γ. The most important of them is Γ(c) consisting of those functions which have the second coefficients of the Taylor expansion fixed and equal to c. We obtain the Koebe set for this class as well as for the classes Γ⁺(c) and Γ⁻(c) of functions which are in some sense convex in the direction of positive and negative axes respectively.     

On fine fractal properties of generalized infinite Bernoulli convolutions

The paper is devoted to the investigation of generalized infinite Bernoulli convolutions, i.e., the distributions μξ of the following random variables: where ak are terms of a given positive convergent series; ξk are independent random variables taking values 0 and 1 with probabilities p0k and p1k correspondingly.We give (without any restriction on {an}) necessary and sufficient conditions for the topological support of ξ to be a nowhere dense set. Fractal properties of the topological support of ξ and fine fractal properties of the corresponding probability measure μξ itself are studied in details for the case where ak?rk:=ak+1+ak+2+? (i.e., rk−1?2rk) for all sufficiently large k. The family of minimal dimensional (in the sense of the Hausdorff–Besicovitch dimension) supports of μξ for the above mentioned case is also studied in details. We describe a series of sets (with additional structural properties) which play the role of minimal dimensional supports of generalized Bernoulli convolutions. We also show how a generalization of M. Cooper's dimensional results on symmetric Bernoulli convolutions can easily be derived from our results.     

Positive curvature property for some hypoelliptic heat kernels

In this note, we look at some hypoelliptic operators arising from nilpotent rank 2 Lie algebras. In particular, we concentrate on the diffusion generated by three Brownian motions and their three Lévy areas, which is the simplest extension of the Laplacian on the Heisenberg group H. In order to study contraction properties of the heat kernel, we show that, as in the case of the Heisenberg group, the restriction of the sub-Laplace operator acting on radial functions (which are defined in some precise way in the core of the paper) satisfies a non-negative Ricci curvature condition (more precisely a CD(0,∞) inequality), whereas the operator itself does not satisfy any CD(r,∞) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel.     

On the singular Bochner-Martinelli integral

The Bochner-Martinelli (B.-M.) kernel inherits, forn2, only some of properties of the Cauchy kernel in . For instance it is known that the singular B.-M. operatorM _n is not an involution forn2. M. Shapiro and N. Vasilevski found a formula forM ₂ ² using methods of quaternionic analysis which are essentially complex-twodimensional. The aim of this article is to present a formula forM _n ² for anyn2. We use now Clifford Analysis but forn=2 our formula coincides, of course, with the above-mentioned one.     

Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic–parabolic systems

Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic–parabolic systems including the Navier–Stokes equations of compressible gas, and magnetohydrodynamics with inflow or outflow boundary conditions, establishing that linear and nonlinear stability are both equivalent to an Evans function, or generalized spectral stability, condition. The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in particular, it has been shown by Costanzino, Humpherys, Nguyen, and Zumbrun to hold for sufficiently large-amplitude layers for isentropic ideal gas dynamics, with general adiabiatic index γ?1. Together with these previous results, our results thus give nonlinear stability of large-amplitude isentropic boundary layers, the first such result for compressive (“shock-type”) layers in other than the nearly-constant case. The analysis, as in the strictly parabolic case, proceeds by derivation of detailed pointwise Green function bounds, with substantial new technical difficulties associated with the more singular, hyperbolic behavior in the high-frequency/short time regime.