Two cardinal models for singular μ

We deal here with colorings of the pair (μ⁺, μ), when μ is a strong limit and singular cardinal. We show that there exists a coloring c with no refinement. It follows that the properties of colorings of (μ⁺, μ) when μ is singular differ in an essential way from the case of regular μ (although the identities may be the same). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New steps in the amazing world of sequences and schedules

In this paper we consider classical shop problems:n jobs have to be processed onm machines. The processing timep _i,j of jobi on machinej is given for all operations (i, j). Each machine can process at most one job at a time and each job can be processed at most on one machine at a given time. The machine orders are fixed (job-shop) or arbitrary (open-shop). We have to determine a feasible combination of machine and job orders, a so-called sequence, which minimizes the makespan. We introduce a partial order on the set of sequences with the property that there exists at least one optimal sequence in the set of minimal elements of this partial order independent of the given processing times. The set of minimal elements (set of irreducible sequences) can be in detail described in the case of the two machine open-shop problem. The cardinality is calculated. We will show which sequences are generated by the well-known polynomial algorithms for the construction of optimal schedules. Furthermore, we investigate the problemO∥C _max on an operation set with spanning tree structure. Supported by Deutsche Forschungsgemeinschaft, Project ScheMA     

A Technique for the Search for Vertices of the Components of a k G-Permutation Module

In this paper we present a necessary condition for a p-group V ≤ G to be a vertex of some indecomposable direct summand of the permutation module k _H  ↑ ^G , where H ≤ G, and G is a finite group. We call this condition H-suitability and present a method how to check for it. In an example, we determine all H-suitable groups. In fact, in this example every H-suitable group is the vertex of some indecomposable direct summand of k _H  ↑ ^G .     

Space complexity of Abelian groups

We develop a theory of LOGSPACE structures and apply it to construct a number of examples of Abelian Groups which have LOGSPACE presentations. We show that all computable torsion Abelian groups have LOGSPACE presentations and we show that the groups , and the additive group of the rationals have LOGSPACE presentations over a standard universe such as the tally representation and the binary representation of the natural numbers. We also study the effective categoricity of such groups. For example, we give conditions are given under which two isomorphic LOGSPACE structures will have a linear space isomorphism.      

Induced Decompositions of Graphs

We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.     

Ramsey theory on Steiner triples

We call a partial Steiner triple system C (configuration) t‐Ramsey if for large enough n (in terms of $C,t$), in every t‐coloring of the blocks of any Steiner triple system STS(n) there is a monochromatic copy of C. We prove that configuration C is t‐Ramsey for every t in three cases:

• C is acyclic
• every block of C has a point of degree one
• C has a triangle with blocks 123, 345, 561 with some further blocks attached at points 1 and 4

This implies that we can decide for all but one configurations with at most four blocks whether they are t‐Ramsey. The one in doubt is the sail with blocks 123, 345, 561, 147.     

Reconstructing trees from two cards

Let ?? be the class of unlabeled trees. An unlabeled vertex‐deleted subgraph of a tree T is called a card. A collection of cards is called a deck. We say that the tree T has a deck D if each card in D can be obtained by deleting distinct vertices of T. If T is the only unlabeled tree that has the deck D, we say that T is ??‐reconstructible from D. We want to know how large of a deck D is necessary for T to be ??‐reconstructible. We define ??rn(T) as the minimum number of cards in a deck D such that T is ??‐reconstructible from D. It is known that ??rn(T)≤3, but it is conjectured that ??rn(T)≤2 for all trees T. We prove that the conjecture holds for all homeomorphically irreducible trees. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 243–257, 2010     

Zeros of Asai-Eisenstein series

We generalize a result of Max Deuring on the zeros of Zeta-function of quadratic forms to Asai's non-holomophic Eisenstein series E(z,s) of the Hilbert modular group. We prove that inside the rectangular and −1≤Re(s) ≤ 2 the function E(z,s) has only simple zeros on the line Re(s)=1/2 and two simple real zeros, if |N(y)| is large. The research was supported by a fellowship within the Post-doc-Program of the DAAD (German Academic Exchange Service)     

Lieb–Thirring inequalities for higher order differential operators

We derive Lieb–Thirring inequalities for the Riesz means of eigenvalues of order γ ≥ 3/4 for a fourth order operator in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of such operators, in dimensions greater than one. For the critical case γ = 1 – 1/(2l) in dimension d = 1 with l ≥ 2 we prove the inequality L⁰_l,γ,d < L_l,γ,d , which holds in contrast to current conjectures. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The polynomial representation of the type An−1 rational Cherednik algebra in characteristic p | n

We study the polynomial representation of the rational Cherednik algebra of type A_n?1 with generic parameter in characteristic p for p | n. We give explicit formulas for generators for the maximal proper graded submodule, show that they cut out a complete intersection, and thus compute the Hilbert series of the irreducible quotient. Our methods are motivated by taking characteristic p analogues of existing characteristic 0 results.     

Obstruction Theory and Coincidences in Positive Codimension

Let f, g ： X→Y be two maps between closed manifolds with dim X ≥ dim Y = n ≥ 3. We study the primary obstruction on（f, g） to deforming f and g to be coincidence free on the n-th skeleton of X. We give examples for which obstructions to deforming f and g to be coincidence free are detected by on （f, g）.     

A Hecke Algebra Quotient and Some Combinatorial Applications

Let (W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient \-H of H and show that it has a basis parametrized by a certain subset W _cof the Coxeter group W. Specifically, W _cconsists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommuting generators in S. We determine which Coxeter groups have finite W _cand compute the cardinality of W _cwhen W is a Weyl group. Finally, we give a combinatorial application (which is related to the number of reduced expressions for w W _cof an exponential formula of Lusztig which utilizes a specialization of a subalgebra of \-H.     

Embeddings with multiple regularity

We introduce (k,l)-regular maps, which generalize two previously studied classes of maps: affinely k-regular maps and totally skew embeddings. We exhibit some explicit examples and obtain bounds on the least dimension of a Euclidean space into which a manifold can be embedded by a (k,l)-regular map. The problem can be regarded as an extension of embedding theory to embeddings with certain non-degeneracy conditions imposed, and is related to approximation theory.     

Galois Theory of Difference Equations with Periodic Parameters

We develop a Galois theory for systems of linear difference equations with periodic parameters, for which we also introduce linear difference algebraic groups. We apply this to constructively test if solutions of linear q-difference equations, with q ∈ ?* and q not a root of unity, satisfy any polynomial ζ-difference equations with ζ ^t  = 1, t ≥ 1.     

Multiplicity results for the two-vortex Chern-Simons Higgs model on the two-sphere

We consider a Ginzburg-Landau type functional on S ² with a 6 ^th order potential and the corresponding selfduality equations. We study the limiting behavior in the two vortex case when a coupling parameter tends to zero. This two vortex case is a limiting case for the Moser inequality, and we correspondingly detect a rich and varied asymptotic behavior depending on the position of the vortices. We exploit analogies with the Nirenberg problem for the prescribed Gauss curvature equation on S ². Received: December 3, 1997     

Sum-Avoiding Subsets

We estimate the maximal number of elements that can be selected from a set of n numbers so that the sum of two selected elements are never in the original set. We improve Choi’s upper bound of n^2/5 to e^{c√log n n}.To Professor Nicolas, on the occasion of his 60th birthday.2000 Mathematics Subject Classification: Primary—11B75Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 38396, T43623 and T42750.     

Rings of SL2( $\Bbb{C}$ )-characters and the Kauffman bracket skein module

Let M be a compact orientable 3-manifold. The set of characters of SL ₂()-representations of forms a closed affine algebraic set. We show that its coordinate ring is isomorphic to a specialization of the Kauffman bracket skein module, modulo its nilradical. This is accomplished by realizing the module as a combinatorial analog of the ring in which tools of skein theory are exploited to illuminate relations among characters. We conclude with an application, proving that a small manifold's specialized module is necessarily finite dimensional. Received: April 18, 1996     

On the hyperdeterminant for 2×2×3 arrays

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest non-constant invariant polynomial in the entries of a general 2?×?2?×?3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients ±1, ±2 in the 12 indeterminates x _ijk where i, j?=?1,?2 and k?=?1,?2,?3. This invariant can be regarded as a natural generalization of Cayley's hyperdeterminant for 2?×?2?×?2 arrays.     

Quantized Heisenberg Space

We investigate the algebra F _q (N) introduced by Faddeev, Reshetikhin and Takhadjian. In the case where q is a primitive root of unity, the degree, the center, and the set of irreducible representations are found. The Poisson structure is determined and the De Concini–Kac–Procesi Conjecture is proved for this case.     

On pseudolinearity and generic pairs

We continue the study of the connection between the “geometric” properties of SU ‐rank 1 structures and the properties of “generic” pairs of such structures, started in [8]. In particular, we show that the SU‐rank of the (complete) theory of generic pairs of models of an SU ‐rank 1 theory T can only take values 1 (if and only if T is trivial), 2 (if and only if T is linear) or ω, generalizing the corresponding results for a strongly minimal T in [3]. We also use pairs to derive the implication from pseudolinearity to linearity for ω ‐categorical SU ‐rank 1 structures, established in [7], from the conjecture that an ω ‐categorical supersimple theory has finite SU ‐rank, and find a condition on generic pairs, equivalent to pseudolinearity in the general case (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)