Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

We consider Holder continuous circulant (2 × 2) matrix functions \({{\bf G}^1_2}\) defined on the fractal boundary Γ of a Jordan domain Ω in \({\mathbb{R}^{2n}}\). The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitian Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitian Dirac operators. In Brackx et al. (Bull Braz Math Soc 40(3): 395–416, 2009) a Hermitian Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from which a Hilbert transform was derived in Brackx et al. (J Math Anal Appl 344: 1068–1078, 2008) for the case of domains with smooth boundary. However, crucial parts of the method are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitian Hilbert transform in that case. As a consequence, we give necessary and sufficient conditions for the Hermitian monogenicity of a circulant matrix function \({{\bf G}^1_2}\) in the interior and exterior of Ω, in terms of its boundary value \({{\bf g}^1_2={\bf G}^1_2|_\Gamma}\), extending in this way also results of Abreu Blaya et al. (Bound. Value Probl. 2008: 2008) (article ID 425256), (article ID 385874), where Γ is required to be Ahlfors–David regular.

Clifford analysis is a higher-dimensional function theory offering a refinement of classical harmonic analysis. It is centred around monogenic functions, i.e. null solutions of the rotation invariant vector-valued Dirac operator. In this context, generalizations of classical orthogonal polynomials on the real line have been introduced in order to use them as building blocks for wavelets, as has been the case, e.g. for the generalized Clifford–Hermite polynomials. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the standard Euclidean case; it focuses on simultaneous null solutions of two Hermitean Dirac operators which are invariant under the action of the unitary group. In this Hermitean setting, generalized Hermitean Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Hermitean Dirac operators. In this paper, we establish their mutual orthogonality relations w.r.t. a well-chosen weight function.

In the d-dimensional Euclidean space, any set of n + 1 independent random points, uniformly distributed in the interior of a unit ball of center O, determines almost surely a circumsphere of center C and radius Ω (1 ≤ n ≤ d) and an n-flat (1 ≤ n ≤ d − 1). The orthogonal projection of O onto this flat is called O′ while Δ designates the distance O′C. The classical problem of the distance between two random points in a unit ball corresponds to n = 1. The focus is set on the family of circumspheres which are contained in this unit ball. For any d≥2 and 1≤n≤d−1, the joint probability density function of the distance Δ≡O′C and circumradius Ω has a simple closed-form expression. The marginal probability density functions of Δ and Ω are both products of powers and a Gauss hypergeometric function. Stochastic representations of the latter random variables are described in terms of geometric means of two independent beta random variables. For n=d≥1, Δ and Ω have a joint Dirichlet distribution with parameters (d, d2, 1) while Δ and Ω are beta distributed. Results of Monte Carlo simulations are in very good agreement with their calculated counterparts. The tail behavior of the circumradius probability density function has been studied by Monte Carlo simulations for 2≤n=d≤9, where all circumspheres are this time considered, regardless of whether or not they are entirely contained in the unit ball.

Options have been developed for constructing circulant matrices of any M-sequence (MS) based on automorphic multiplicative groups of the extended Galois field, constructed using an irreducible primitive polynomial, on the basis of which the original MS is formed. The result of this approach is the identification of new methods for transforming MS circulant matrices to a matrix of Walsh functions, ordered by the powers of the antiderivative element of the field. It is shown for the first time that, depending on the initial conditions of the transformation, a set of any number of any cyclic shifts of the MP, shifted relative to each other by one symbol, can be transformed to any rows of the ordered matrix of Walsh functions, following one another. This circumstance makes it possible to simplify the MS synchronization algorithm for a known range of its cyclic shifts, especially in the case of large periods of its repetition, and also to reduce the computational complexity of the processing algorithm when working in a truncated basis of Walsh–Hadamard functions.

Let ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> H$ be a closed half-space of $n$-dimensional Euclidean space. Suppose $S$ is a unit sphere in $H$ that touches the supporting hyperplane of $H$. The one-sided kissing number $B(n)$ is the maximal number of unit nonoverlapping spheres in $H$ that can touch $S$. Clearly, $B(2)=4$. It was proved that $B(3)=9$. Recently, K. Bezdek proved that $B(4)=18$ or 19, and conjectured that $B(4)=18$. We present a proof of this conjecture.

We show that an equiangular tight frame of m unit vectors in the n–dimensional Euclidean space is a 4th order semidesign if and only if m = n(n+1)/2. For an important example we present an equiangular tight frame of n + 1 vertices of a regular simplex inscribed into the unit sphere (the Mercedes–Benz frame). We construct a system of the projections of the midpoints of edges of a simplex onto the sphere and show that this system is a tight frame for any $$ n \geqslant 2 $$ and is equiangular only if n = 2 and n = 7. In the last two cases, we find an equiangular 4th order semidesign. Bibliography: 6 titles.

The purpose of this article is to prove the following theorem:Let n be a positive integer larger than or equal to 2, and let S be the unit sphere in the 2n + 1 dimensional Euclidean space. Given a compact Riemann surface, we can always find a conformal and minimal immersion of the surface into S whose image is not lying in any 2n — 1 dimensional hyperplane.This is a partial generalization of the result by R. L. Bryant. In this papers, he demonstrates the existence of a conformal and minimal immersion of a compact Riemann surface into S2n, which is generically 1:1, when n = 2 ([2]) and n = 3 ([1]).

Christoffel’s paper [1865] „Ueber die Bestimmung der Gestalt einer krummen Oberflache...“, discussed in W. Klingenberg’s lecture, treats the following questions. Given a function o over the unit sphere Ω of directions u, is there a convex surface E whose sum of principal radii of curvature equals o(u) at that point where the outer normal direction to E is u? To what extent is E unique? Here convex surface means the whole boundary of a convex compactum: the setting is Euclidean space of dimension n ≥ 3. Christoffel answered these questions by solving appropriate Poisson equations by Green’s function methods. For this he required the facts that the integral of uo(u) over Ω to be the zero vector and the integral of o(u) over each hemisphere to be positive. His work was incomplete because it neglected the convexity assumption, which occurs quite indirectly in the paper. Christoffel also showed that E is unique up to a translation.

The set of primitive vectors on large spheres in the euclidean space of dimension d>2 equidistribute when projected on the unit sphere. We consider here a refinement of this problem concerning the direction of the vector together with the shape of the lattice in its orthogonal complement. Using unipotent dynamics we obtained the desired equidistribution result in dimension d>5 and in dimension d=4,5 under a mild congruence condition on the square of the radius. The case of d=3 is considered in a separate paper.

<p style='text-indent:20px;'>Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball <inline-formula><tex-math id="M1">$ B $</tex-math></inline-formula> are studied. The weights can be singular at <inline-formula><tex-math id="M2">$ x = 0 \in B $</tex-math></inline-formula>. Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point <inline-formula><tex-math id="M3">$ x = 0 $</tex-math></inline-formula>.</p>

Generalized Taylor series in hermitian Clifford analysis

In metric Diophantine approximation, one frequently encounters the problem of showing that a limsup set has positive or full measure. Often it is a set of points in m m -dimensional Euclidean space, or a set of n n -by- m m systems of linear forms, satisfying some approximation condition infinitely often. The main results of this paper are bootstraps: if one can establish positive measure for such a limsup set in m m -dimensional Euclidean space, then one can establish positive or full measure for an associated limsup set in the setting of n n -by- m m systems of linear forms. Consequently, a class of m m -dimensional results in Diophantine approximation can be bootstrapped to corresponding n n -by- m m -dimensional results. This leads to short proofs of existing, new, and hypothetical theorems for limsup sets that arise in the theory of systems of linear forms. We present several of these.

We introduce two new linear differential operators which are invariant with respect to the unitary group SU(n). They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator \( \partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} } \) and \( \partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} }^\dag \) and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.

In a recent series of short papers [2, 3, 41 I have been discussing the relationships of uniform convexity with certain other properties of normed vector spaces. In this paper I propose to discuss relationships between uniform convexity, factor spaces, and conjugate spaces. Because of the large number of special results which are needed for two dimensional spaces-and which are false in general-the paper will be divided into two parts: In Part I (??2-4) B is two dimensional; in Part II (??5-7) this restriction is removed and B may be any normed vector space. To recall the definition, B is said to be uniformly convex if there exists a function 6 such that 0 e, is a modulus of convexity for B. Note that if 51 is nowhere greater than 5, a modulus of convexity for B, then 5l is a modulus of convexity for B. Because of the pointwise nature of the definition of uniform convexity, it is clear that B is uniformly convex with modulus of convexity a if and only if all the two dimensional subspaces of B have the common modulus of convexity 6. One of the most useful results of this investigation (Theorem 5.5) is the complementary fact that B is uniformly convex if and only if all the two dimensional factor spaces of B have a common modulus of convexity. In the study of the effect of uniform convexity of B on the nature of the conjugate space B*, this result makes it possible to reduce the problem to the study of two dimensional spaces. In such a space B uniform convexity is equivalent to strict convexity: that is, a two (or finite) dimensional space is uniformly convex if and only if there does not exist a line segment of positive length all of whose points are of norm one. It has been observed [1, Footnote 13] that such a line segment on the unit sphere is equivalent to the existence of a sharp edge on the unit sphere in B*. The attempt to describe a sharp edge of the unit sphere in a finite dimensional space B in terms of the norm in that space leads to the condition that there exists a k > 0 such that for any e > 0 a pair of points bi and b2 exists such that II b -b2 11 k 11 bl-be 11 . Contradicting this suggests the following condition, a sort of dual concept to uniform convexity. A space B is said to be uniformly flattened if there exists a function v7 positive for 0 < e ? 2 such that lim.b.o 7(e) = 0 while (211 b + b2 11)/i b2 11 < 7(e) if J]b11 = JJ b1 = 1 and 11 bb2 ? || ; 375

Spin actions in Euclidean and Hermitian Clifford analysis in superspace