Partition of <Emphasis Type="Italic">C</Emphasis><Subscript>4</Subscript>-Designs into Minimum and Maximum Number of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-Designs

Let be a C₄-design of order n and index , on the vertex set V, |V|=n. If V₁V_m=V is a partition of the vertex set, such that the intersections of the with V_i form a P₃-design of order |V_i| and the same index , for each 1im, then 2m log₃(2n+1). The minimum bound is best possible for every . The maximum bound is best possible for =2, and hence also for every even .Supported by MIUR, Italy and CNR-GNSAGAAlso affiliated with the Department of Computer Science, University of Veszprém, Hungary; supported in part by the Hungarian Scientific Research Fund, grant OTKA T-32969AMS classification: 05B05     

The -conjecture and equivariant <Emphasis Type="Italic">e</Emphasis><Subscript><Emphasis Type="Italic">C</Emphasis></Subscript>-invariants

Let X be a smooth closed oriented non-spin 4-manifold with even intersection form kE₈nH (n1). The -conjecture states that n is greater than or equal to |k|. In this paper we give a proof of the -conjecture. The strategy of this paper is to use the finite dimensional approximation of the map induced from the Seiberg-Witten equations and equivariant e_C-invariants as in the paper of M. Furuta and Y. Kametani.Mathematics Subject Classification (1991): 57R55This work was supported by Korea Research Foundation Grant (KRF–2002–003–C00011).     

Almost optimal estimates for entropy numbers of <Emphasis Type="Italic">B</Emphasis><Subscript>2,2</Subscript><Superscript>1/2</Superscript> and its consequences

Triebels conjecture on the entropy numbers for the limiting embedding case B_p,_p^1/p E, E being suitable Orlicz spaces, is almost proved for p2 – almost in the sense that a log -factor occurs in the upper estimate. Furthermore, it is shown that the breaking point, where the radii of the -balls in the covering of the B_p,_q^1/p-unit ball stop to be constant, essentially depends on the second parameter q. The methods used in the paper also give improvements of the results till now known in the case p<2, explains a result of Kashin and Temlyakov on entropy numbers of continuous functions of low smoothness, and indicates how the E-scale can be refined.Mathematics Subject Classification (2000) : 46E35, 41A46, 46E30Part of this work was done while this author was visiting the Department of Mathematics at Technical University Darmstadt in the framework of a DAAD Scholarship.     

On the finite sum representations of the Lauricella functions <Emphasis Type="Italic">F</Emphasis><Subscript>D</Subscript>

By using divided differences, we derive two different ways of representing the Lauricella function of n variables F_D⁽ⁿ⁾(a,b₁,b₂,...,b_n;c;x₁,x₂,...,x_n) as a finite sum, for b₁,b₂,...,b_n positive integers, and a,c both positive integers or both positive rational numbers with c–a a positive integer. AMS subject classification 33D45, 40B05, 40C99Jieqing Tan: Research supported by the National Natural Science Foundation of China under Grant No. 10171026 and Anhui Provincial Natural Science Foundation under Grant No. 03046102.Ping Zhou: Corresponding author. Research supported by NSERC of Canada.     

The Exceptional Compact Symmetric Spaces <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">SO</Emphasis>(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.The author was partially supported by the Hungarian National Science and Research Foundation OTKA T032478.     

Simply connected symplectic 4-manifolds with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript><Superscript>+</Superscript>=1 and <Emphasis Type="Italic">c</Emphasis><Subscript>1</Subscript><Superscript>2</Superscript>=2

In this article we construct a new simply connected symplectic 4-manifold with b₂⁺=1 and c₁²=2 which is homeomorphic, but not diffeomorphic, to a rational surface by using rational blow-down technique. As a corollary, we conclude that a rational surface admits an exotic smooth structure. Mathematics Subject Classification (2000) 53D05, 14J26, 57R55, 57R57     

Rationality of the vertex operator algebra <Emphasis Type="Italic">V</Emphasis><Subscript><Emphasis Type="Italic">L</Emphasis></Subscript>+ for a positive definite even lattice <Emphasis Type="Italic">L</Emphasis>

The lattice vertex operator algebra V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from –1-isometry of L. The fixed point set V_L+ of V_L for the automorphism is naturally a vertex operator algebra. We prove that any ₀-graded weak V_L⁺-module is completely reducible.Supported by JSPS Research Fellowships for Young Scientists.     

Invariant subspaces for operators in a general II<Subscript>1</Subscript>-factor

Let ℳ be a von Neumann factor of type II₁ with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ _T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace affiliated with ℳ, such that the Brown measure of is concentrated on B and the Brown measure of is concentrated on ℂ∖B. Moreover, is T-hyperinvariant and the trace of is equal to μ _T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit exists in the strong operator topology, and the projection onto is equal to 1_[0,r](A), for every r>0. Supported by The Danish National Research Foundation.     

Point sets with low <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-discrepancy

In this paper we study the L _p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L _p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L _p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L ₂-discrepancy of digitally shifted Hammersley point sets which show that the value of the L ₂-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.     

Homology of GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>: injectivity conjecture for GL<Subscript>4</Subscript>

The homology of GL _n (R) and SL _n (R) is studied, where R is a commutative ‘ring with many units’. Our main theorem states that the natural map H ₄(GL₃(R), k) → H ₄(GL₄(R), k) is injective, where k is a field with char(k) ≠ 2, 3. For an algebraically closed field F, we prove a better result, namely, is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the K-group K ₄(R).     

<Emphasis Type="Italic">p</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-sequences of algebras with one fundamental operation

We show that if (p₀, p₁, ...) is the p_n-sequence of a nontrivial algebra with one fundamental operation, then p₁ ≥ p₀. Moreover, if , then p₁ > 2p₀. Received April 21, 2003; accepted in final form November 28, 2005.     

Topological hypercovers and <Superscript>1</Superscript>-realizations

We show that if U_* is a hypercover of a topological space X then the natural map hocolim U_* X is a weak equivalence. This fact is used to construct topological realization functors for the ¹-homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.Mathematics Subject Classification (2000):55U35, 14F20, 14F42The second author was supported by an NSF Postdoctoral Research Fellowship     

Geometric dual formulation for first-derivative-based univariate cubic <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> splines

With the objective of generating “shape-preserving” smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based -smooth univariate cubic L ₁ splines. An L ₁ spline minimizes the L ₁ norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L ₁ spline is a nonsmooth non-linear convex program. Via Fenchel’s conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.     

Maximal regularity for evolution equations in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem enjoys maximal regularity in weighted L _p -spaces with weights , where , if and only if it has the property of maximal L _p -regularity. Moreover, it is also shown that the derivation operator admits an -calculus in weighted L _p -spaces. Received: 26 February 2003     

Recognition of the groups <Emphasis Type="Italic">E</Emphasis><Subscript>7</Subscript>(2) and <Emphasis Type="Italic">E</Emphasis><Subscript>7</Subscript>(3) by prime graph

It is proved that the groups E ₇(2) and E ₇(3) are recognizable by their prime graphs. As a corollary, this completes the proof of V.D. Mazurov’s conjecture that every finite simple group whose prime graph has at least three connected components is either recognizable by spectrum or isomorphic to A ₆.     

Entropy conditions for <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">r</Emphasis></Subscript>-convergence of empirical processes

The law of large numbers (LLN) over classes of functions is a classical topic of empirical processes theory. The properties characterizing classes of functions on which the LLN holds uniformly (i.e. Glivenko–Cantelli classes) have been widely studied in the literature. An elegant sufficient condition for such a property is finiteness of the Koltchinskii–Pollard entropy integral, and other conditions have been formulated in terms of suitable combinatorial complexities (e.g. the Vapnik–Chervonenkis dimension). In this paper, we endow the class of functions with a probability measure and consider the LLN relative to the associated L _r metric. This framework extends the case of uniform convergence over , which is recovered when r goes to infinity. The main result is a L _r -LLN in terms of a suitable uniform entropy integral which generalizes the Koltchinskii–Pollard entropy integral.      

Paired-Domination in <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by , is the minimum cardinality of a paired-dominating set of G. In [1], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, the critical cases are not claws but subdivided stars. We here give a bound for graphs containing no induced P ₅, which seems to be the critical case.     

Complete Minors In <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">s,s</Emphasis></Subscript>-Free Graphs

We prove that for a fixed integer s2 every K _s,s -free graph of average degree at least r contains a K _p minor where . A well-known conjecture on the existence of dense K _s,s -free graphs would imply that the value of the exponent is best possible. Our result implies Hadwigers conjecture for K _s,s -free graphs whose chromatic number is sufficiently large compared with s.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Boundedness of general index transforms

We establish the boundedness properties in L _p for a class of integral transformations with respect to an index of hypergeometric functions. In particular, by using the Riesz-Thorin interpolation theorem, we get the corresponding results in L _p (R ₊), 1 p 2, for the Kontorovich-Lebedev, Mehler-Fock, and Olevskii index transforms. An inversion theorem is proved for a general index transformation. The case p=2 is known as the Plancherel-type theory for this class of transformations.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 127–147, January–March, 2005.