Exceptional Modules for Tubular Canonical Algebras

Let Λ be a tubular canonical algebra of quiver type over a field. We show that each exceptional Λ-module can be exhibited by matrices involving as coefficients 0, 1 and –1 if Λ is of type (3,3,3), (2,4,4) or (2,3,6) and by matrices involving as coefficients 0, 1, –1, λ, –λ and λ–1 if Λ is of type (2,2,2,2) and defined by a parameter λ. Presented by Claus M. Ringel.     

Amemiya Norm Equals Orlicz Norm in Musielak-Orlicz Spaces

Let （Ω,μ） be a a-finite measure space and Φ ： Ω × [0,∞） → [0, ∞] be a Musielak-Orlicz function. Denote by L^Φ（Ω） the Musielak-Orlicz space generated by Φ. We prove that the Amemiya norm equals the Orlicz norm in L^Φ（Ω）.     

Positive versus free boundary solutions to a singular elliptic equation

The equation-δu = χ uo(-1/u^Β + λf (x, u)) in Ω with Dirichlet boundary condition on ∂Ω has a maximal solution uλ ≥0 for every λ 0. For λ less than a constant λ*, the solution vanishes inside the domain; and for λ λ*, the solution is positive. We obtain optimal regularity ofuλ even in the presence of the free boundary. Supported in part by H. J. Sussmann’s NSF Grant DMS01-03901. Supported by FAPESP. He also thanks Rutgers University.     

Locally Closed Semirings

 We call a semiring S locally closed if for all a ∈ S there is some integer k such that 1 + a + ⋯ + a ^k  =1 + a + ⋯ + a ^k + 1 . In any locally closed semiring we may define a star operation a ↦ a ^*, where a ^* is the above finite sum. We prove that when S is locally closed and commutative, then S is an iteration semiring. Partially supported by grant no. T30511 from the National Foundation of Hungary for Scientific Research and the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Partially supported by the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Received March 16, 2001     

Decompositions of semigroups induced by identities

In this paper we consider decompositions of semigroups induced by identities. Here we give some new characterizations of a semilattice of Archimedean semigroups and, using this, we describe all identities which induce decompositions into a semilattice of Archimedean semigroups. Also, we give a solution for one problem ofШеврин andСуханов [27]. Supported by Grant 0401A of RFNS through Math. Inst. SANU     

Quaternion Polar Representation with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form

We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex ‘modulus’ and a complex ‘argument’. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of −1), but the complex phase is multiplied by a different complex root of −1 in the exponential function. We show how to calculate the ‘modulus’ and ‘argument’ from an arbitrary quaternion in Cartesian form.     

A consistency result on thin-very tall Boolean algebras

It was proved by Baumgartner and Shelah that Con (ZFC)→Con (ZFC + “there is a superatomic Boolean algebra of width ω and height ω₂”). In this paper we improve Baumgartner-Shelah’s theorem, showing that Con (ZFC)→Con (ZFC+“for every α<ω₃ there is a superatomic Boolean algebra of width ω and height α”). The preparation of this paper was supported by DGICYT Grant PB98-1231.     

Comparison of two kinds of Hankel type operators

We compare two kinds of Hankel type operators, the “BJP-operators” introduced by Boman—Janson—Peetre and the “Z-operators” considered by Zhang. We show that every BJP-operator comes as an orthogonal sum ofZ-operators.     

A note on radial graphs with constant mean curvature

 Let Ω be a smooth domain on the unit sphere 𝕊ⁿ whose closure is contained in an open hemisphere and denote by ℋ the mean curvature of ∂Ω as a submanifold of Ω with respect to the inward unit normal. It is proved that for each real number H that satisfies inf ℋ > − H ≥ 0, there exists a unique radial graph on Ω bounded by ∂Ω with constant mean curvature H. The orientation on the graph is based on the normal that points on the opposite side as the radius vector. Received: 5 June 2001 / Revised version: 9 April 2002 Research partially supported by a DGICYT Grant No. BFM2001-2967. Mathematics Subject Classification (2000): 53A10, 53C42, 49Q05, 49Q10     

On shape preserving extension and approximation of functions

We establish the concept of shapes of functions by using partial differential inequalites. Our definition about shapes includes some usual shapes such as convex, subharmonic, etc., and gives many new shapes of functions. The main results show that the shape preserving approximation has close relation to the shape preserving extension. One of our main results shows that if f∈C(Ω) has some shape defined by our definition, then f can be uniformly approximated by polynomials P_n ∈ ℙ_n (n∈ℕ) which have the same shape in Ω, and the degree of the approximation is Cω(f,n^−β) with constants C,β>0.     

Remarks on the Zeros of the Associated Legendre Functions with Integral Degree

We present some formulas for the computation of the zeros of the integral-degree associated Legendre functions with respect to the order. Work supported in part by the ‘Anillo Ecuaciones Asociadas a Reticulados’, financed by the World Bank through the ‘Programa Bicentenario de Ciencia y Tecnología’, and by the ‘Programa Reticulados y Ecuaciones’ of the Universidad de Talca.     

On strongly and weakly defined boolean terms

A problem of Gaifman about strongly and weakly defined Boolean terms is solved by finding a Boolean algebra ℱ with a complete subalgebra ℰ such that some element of ℱ not in ℰ can be obtained from elements of ℰ by meets and joins in the normal completion of ℱ.     

Nonexistence of Proper Holomorphic Maps Between Certain Classical Bounded Symmetric Domains

The author,motivated by his results on Hermitian metric rigidity,conjectured in [4] that a proper holomorphic mapping f：Ω→Ω′from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ′is necessarily totally geodesic provided that r′：=rank（Ω′）≤rank（Ω）：=r.The Conjecture was resolved in the affirmative by I.-H.Tsai [8].When the hypothesis r′≤r is removed,the structure of proper holomorphic maps f：Ω→Ω′is far from being understood,and the complexity in studying such maps depends very much on the difference r′-r,which is called the rank defect.The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H.Tu [10],in which a rigidity theorem was proven for certain pairs of classical domains of type I,which implies nonexistence theorems for other pairs of such domains.For both results the rank defect is equal to 1,and a generaliza- tion of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of （Ω→Ω′） of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and L-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case.     

Realization of a sum of sequences by a sum graph

It is shown that the realizability of the sequences ϕ=(a ₁,…, _a ), ψ=(b ₁,…,b _n ) and ϕ+ψ is a sufficient condition for the realizability of ϕ+ψ by a graph with a ϕ-factor ifb _i ≦1 fori=1,…,n. The condition is not sufficient in general. A necessary and sufficient condition for the realizability of ϕ+ψ by a graph with a ϕ-factor is given for the case that ϕ is realizable by a star and isolated vertices.     

THE ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION IN MULTIPLE LINEAR REGRESSION MODEL

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Compactness for a class of hit-and-miss hyperspaces

In the study of some kind of generalized Vietoris-type topologies for the hyperspace of all nonempty closed subsets of a topological space (X, τ), namely the so called Δ-hit-and-miss-topologies with Δ⊇Cl(X) (or Δ-topologies), which was initiated by the second author in 1965, it is obvious, that the non-compactness of such a hyperspace often depends on the non-compactness even in the lower-semifinite topology (induced by the “hit-sets”), which is contained in all hypertopologies of this type. Otherwise, compactness for these topologies is easily obtained from the compactness of (X, τ) by well-known theorems, if the “miss-sets” are induced either by compact or closed subsets. To obtain a similar result for topologies with “miss-sets” generated by subsets with a property which generalizes both, closedness and compactness especially in the non-Hausdorff case, we use consequently a quite set-theoretical lemma, stated at the beginning.     

Constrained energy problems with applications to orthogonal polynomials of a discrete variable

Given a positive measure Σ with gs > 1, we write Με ℳ^Σ if Μ is a probability measure and Σ—Μ is a positive measure. Under some general assumptions on the constraining measure Σ and a weight functionw, we prove existence and uniqueness of a measure λ^Σ _w that minimizes the weighted logarithmic energy over the class ℳ^Σ. We also obtain a characterization theorem, a saturation result and a balayage representation for the measure λ^Σ _w As applications of our results, we determine the (normalized) limiting zero distribution for ray sequences of a class of orthogonal polynomials of a discrete variable. Explicit results are given for the class of Krawtchouk polynomials. The research done by this author is in partial fulfillment of the Ph.D. requirements at the University of South Florida. The research done by this author was supported, in part, by U.S. National Science Foundation under grant DMS-9501130.     

Homoclinic and Periodic Orbits Arising Near the Heteroclinic Cycle Connecting Saddle-focus and Saddle Under Reversible Condition

In this paper, we study the dynamical behavior for a 4-dimensional reversible system near its heteroclinic loop connecting a saddle-focus and a saddle. The existence of infinitely many reversible 1-homoclinic orbits to the saddle and 2-homoclinic orbits to the saddle-focus is shown. And it is also proved that, corresponding to each 1-homoclinic （resp. 2-homoclinic） orbit F, there is a spiral segment such that the associated orbits starting from the segment are all reversible 1-periodic （resp. 2-periodic） and accumulate onto F. Moreover, each 2-homoclinic orbit may be also accumulated by a sequence of reversible 4-homoclinic orbits.     

The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.