The Generalized Holditch Theorem for the Homothetic Motions on the Planar Kinematics

W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let E/E be a 1-parameter closed planar Euclidean motion with the rotation number and the period T. Under the motion E/E, let two points A = (0, 0), B = (a + b, 0) E trace the curves k _A, k _B E and let F _A, F _B be their orbit areas, respectively. If F _X is the orbit area of the orbit curve k of the point X = (a, 0) which is collinear with points A and B then

Holditch Theorem and Steiner Formula for the Planar Hyperbolic Motions

The Steiner formula and the Holditch Theorem for one-parameter closed planar Euclidean motions [1, 5] were expressed by Müller, under the one-parameter closed planar motions in the complex sense. Also, Müller had given Holditch theorem in the complex sense, [6].     

平面运动学位似运动Steiner公式的推广

对单参数闭平面复意义位似运动，给出了Muller提出的Steiner公式和混合面公式的表达式。利用推广的Steiner公式得到了位似运动的Holditch定理．作为特例，给出了Muller位似比h≡1时的结果。     

A generalization of the Holditch Theorem for the planar homothetic motions

In this paper, under the one-parameter closed planar homothetic motion, a generalization of Holditch Theorem is obtained by using two different line segments (with fixed lengths) whose endpoints move along two different closed curves.This revised version was published online in April 2005 with a corrected missing date string.     

Hyperbolic system of balance laws modeling dissipative ionic crystals

Based on the classical continuum theory of electroelasticity which includes polarization gradients as independent variables, we propose a constitutive model for ionic crystals accounting for both ionic and electronic contributions to polarization. Dissipation is modeled via internal variables which satisfy suitable evolution equations and the consequences of the second law of thermodynamics are exploited to cast the non-linear problem in the form of a symmetric hyperbolic system of balance laws. The stability of perturbations with respect to unstrained, unpolarized states is discussed. A set of linear equations is also derived for the fully electromagnetic problem which generalizes previous results.     

On the Airy Reproducing Kernel, Sampling Series, and Quadrature Formula

We determine the class of entire functions for which the Airy kernel (of random matrix theory) is a reproducing kernel. We deduce an Airy sampling series and quadrature formula. Our results are analogues of well known ones for the Bessel kernel. The need for these arises in investigating universality limits for random matrices at the soft edge of the spectrum. Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353.     

Some Physics Questions in Hyperbolic Complex Space

In hyperbolic complex space, the Clifford algebra is isomorphic to that of a corresponding Minkowski geometry. We define the hyperbolic imaginary unit j (j² = 1, j ≠   ±  1, j*  =   − j) to generate a class of Clifford algebras. We can introduce a class of non-Euclidean spaces and discuss the general form of 4-dimensional Lorentz transformation, and related special relativistic physics.     

Estimates for Littlewood-Paley Functions and Extrapolation

We prove certain L ^p -estimates for Littlewood-Paley functions arising from rough kernels. The estimates are useful for extrapolation to prove L ^p -boundedness of the Littlewood-Paley functions under a sharp kernel condition.      

Distance and fractional isomorphism in Steiner triple systems

In [8], Quattrochi and Rinaldi introduced the idea ofn ⁻¹-isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. The main result is that for any positive integerN, there existsv ₀(N) such that for all admissiblev≥v ₀(N) and for each STS(v) (sayS), there exists an STS(v) (sayS′) such that for somen>N, S is strictlyn ⁻¹-isomorphic toS′. We also prove that for all admissiblev≥13, there exist two STS(v)s which are strictly 2⁻¹-isomorphic. Define the distance between two Steiner triple systemsS andS′ of the same order to be the minimum volume of a tradeT which transformsS into a system isomorphic toS′. We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly 2⁻¹-isomorphic and 3⁻¹-isomorphic pairs of STS(15)s.     

Regular Problems for Semilinear Hyperbolic Type Equations

In this paper we establish the wellposedness and regularity properties of solutions of Cauchy problems for semilinear hyperbolic equations of second order with unbounded principal operators. An example illustrating how our results apply is given.      

Hardy-Littlewood system and representations of integers by an invariant polynomial

Let f be an integral homogeneous polynomial of degree d, and let be the level set for each . For a compact subset in ), set

Combinatorial and Geometric Methods in Topology

Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central r?le played by hyperbolic geometry within three-dimensional topology. We will also point out the striking difference with the two dimensional case, and we will review some of the results of the combinatorial and computational approach to three-manifolds developed by different mathematicians over the last several years. Lecture held by Carlo Petronio in the Seminario Matematico e Fisico di Milano on April 23, 2007     

Compact Hankel Forms on Planar Domains

A Hankel form on a Hilbert function space is a bounded, symmetric, bilinear form [., .] satisfying [fx, y] = [x, fy] for a class of multipliers f. We prove analogs of Weyl–Horn and Ky Fan inequalities for compact Hankel forms, and apply them to estimate the related eigenvalues, both for Hardy–Smirnov and Bergman spaces norms associated to multiply connected planar domains. In the case of the unit disk, we investigate the asymptotic of some measures constructed by eigenfunctions of Hankel operators with certain Markov functions as symbols. Submitted: May 2, 2008. Accepted: June 28, 2008.     

Reflections, Rotations, and Pythagorean Numbers

In this article, simple reflections, rotations and the Cartan theorem are handled using Clifford algebras. With this tool we provide a constructive proof of the Cartan theorem and the relationship with Pythagorean numbers is discussed.      

Distance sets corresponding to convex bodies

Suppose that is a 0-symmetric convex body which denes the usual norm

A Clifford Algebraic Framework for $$\mathfrak{sp}(m)$$-Invariant Differential Operators

We introduce a framework for studying differential operators which are invariant with respect to the real (complex) symplectic Lie algebra ( ), associated to a quaternionic structure on a vector space . To do so, these algebras are realized within the orthogonal Lie algebra . This leads in a natural way to a refinement of the recently introduced notion of complex Hermitean Clifford analysis, in which four variations of the classical Dirac operator play a dominant role. David Eelbode: Postdoctoral fellow supported by the F.W.O. Vlaanderen (Belgium).     

Non-Commutative Harmonic and Subharmonic Polynomials

The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x = (x ₁,…, x _g ). The Laplacian Lap[p, h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[p, h] = h ²Δ _x p where Δ _x p is the usual Laplacian. A symmetric polynomial in symmetric variables will be called harmonic if Lap[p, h] = 0 and subharmonic if the polynomial q(x, h) :=  Lap[p, h] takes positive semidefinite matrix values whenever matrices X ₁,…, X _g , H are substituted for the variables x ₁,…, x _g , h. In this paper we classify all homogeneous symmetric harmonic and subharmonic polynomials in two symmetric variables. We find there are not many of them: for example, the span of all such subharmonics of any degree higher than 4 has dimension 2 (if odd degree) and 3 (if even degree). Hopefully, the approach here will suggest ways of defining and analyzing other partial differential equations and inequalities. Dedicated to Israel Gohberg on the occasion of his 80^th birthday. All authors were partially supported by J.W. Helton’s grants from the NSF and the Ford Motor Co. and J. A. Hernandez was supported by a McNair Fellowship.     

On a conjecture of Schmutz

In this work we discuss Schmutz’s conjecture that in dimension 2 to 8 the distinct norms that occur in the lattices with the best known sphere packings are strictly greater than those in any other lattice of the same covolume. We see that the ternary conjecture is not true. However, it seems that there is but one exception: one lattice, where for one length the conjecture fails. Received: 11 February 2008, Revised: 20 May 2008     

Maximal subnear-rings of functions

For a group G, let M(G) denote the near-ring of functions on G. We characterize all maximal subnear-rings of M(G) and show that for many classes of groups, E(G), the near-ring generated by the semigroup, End(G) of G, is never maximal as a subnear-ring of M ₀ (G). Received: 25 April 2008