A generalization of the Oberwolfach problem

Let n > 1 be an integer and let a₂,a₃,…,a_n be nonnegative integers such that . Then can be factored into ‐factors, ‐factors,…, ‐factors, plus a 1‐factor. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 151–161, 2002     

Tauberian theorem for m‐spherical transforms on the Heisenberg group

In this paper we prove a Tauberian type theorem for the space L ( H _n ). This theorem gives sufficient conditions for a L ( H _n ) submodule J ? L ( H _n ) to make up all of L ( H _n ). As a consequence of this theorem, we are able to improve previous results on the Pompeiu problem with moments on the Heisenberg group for the space L^∞( H _n ). In connection with the Pompeiu problem, given the vanishing of integrals ∫ z ^m L _g f ( z , 0) dσ ( z ) = 0 for all g ∈ H _n and i = 1, 2 for appropriate radii r₁ and r₂, we now have the (improved) conclusion f ≡ 0, where = · · · and form the standard basis for T^(0,1)( H _n ). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strengths and Weaknesses of LH Arithmetic

In this paper we provide a new arithmetic characterization of the levels of the og‐time hierarchy (LH). We define arithmetic classes and that correspond to ‐LOGTIME and ‐LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose ‐definable functions are precisely B( ‐LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)‐definable functions are LH, proves LH is not equal to ‐TIME(s) for any m> 0, where 2^s ∈L, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.     

Infinite games in the Cantor space and subsystems of second order arithmetic

In this paper we study the determinacy strength of infinite games in the Cantor space and compare them with their counterparts in the Baire space. We show the following theorems: 1. RCA₀ ? ‐Det* ? ‐Det* ? WKL₀. 2. RCA₀ ? ( )2‐Det* ? ACA₀. 3. RCA₀ ? ‐Det* ? ‐Det* ? ‐Det ? ‐Det ? ATR₀. 4. For 1 < k < ω, RCA₀ ? ( )_k ‐Det* ? ( )_{k –1}‐Det. 5. RCA₀ ? ‐Det* ? ‐Det. Here, Det* (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and ( )_k is the collection of formulas built from formulas by applying the difference operator k – 1 times. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence Laws for Very Sparse Random Structures with Generalized Quantifiers

We prove convergence laws for logics of the form , where is a properly chosen collection of generalized quantifiers, on very sparse finite random structures. We also study probabilistic collapsing of the logics , where is a collection of generalized quantifiers and k ∈ ℕ₊, under arbitrary probability measures of finite structures.     

Quasilinear degenerate parabolic equations of Kirchhoff type

We investigate the evolution problem where H is a Hilbert space, A is a self‐adjoint linear non‐negative operator on H with domain D(A), and is a continuous function. We prove that if , and , then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if for all , then this problem is well posed in H. On the contrary, if for some it happens that for all , then this problem has no solution if with β small enough. We apply these results to degenerate parabolic PDEs with non‐local non‐linearities. Copyright © 1999 John Wiley & Sons, Ltd.     

On the Points on the Unit Circle with Finite b–Adic Expansions

Let be an arbitrary integer base and let be the number of different prime factors of with , . Further let be the set of points on the unit circle with finite –adic expansions of their coordinates and let be the set of angles of the points . Then is an additive group which is the direct sum of infinite cyclic groups and of the finite cyclic group . If in case of the points of are arranged according to the number of digits of their coordinates, then the arising sequence is uniformly distributed on the unit circle. On the other hand, in case of the only points in are the exceptional points (1, 0), (0, 1), (–1, 0), (0, –1). The proofs are based on a canonical form for all integer solutions of .     

On (ε,k)‐min‐wise independent permutations

A family of permutations of [n] = {1,2,…,n} is (ε,k)‐min‐wise independent if for every nonempty subset X of at most k elements of [n], and for any x ∈ X, the probability that in a random element π of , π(x) is the minimum element of π(X), deviates from 1/∣X∣ by at most ε/∣X∣. This notion can be defined for the uniform case, when the elements of are picked according to a uniform distribution, or for the more general, biased case, in which the elements of are chosen according to a given distribution D. It is known that this notion is a useful tool for indexing replicated documents on the web. We show that even in the more general, biased case, for all admissible k and ε and all large n, the size of must satisfy as well as This improves the best known previous estimates even for the uniform case. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

On the Range of Two Convolution Operators

Let _(ω)(ℝ) denote the non–quasianalytic class of Beurling type on ℝ. For μ, ν ∈ ′_(ω)(ℝ) we give necessary conditions for the inclusion T_ν( _(ω)(ℝ)) ⊂ T_μ( _(ω)(ℝ)), thus extending previous work of Malgrange and Ehrenpreis .     

On asymmetric coverings and covering numbers

An asymmetric covering is a collection of special subsets S of an n‐set such that every subset T of the n‐set is contained in at least one special S with . In this paper we compute the smallest size of any for We also investigate “continuous” and “banded” versions of the problem. The latter involves the classical covering numbers , and we determine the following new values: , , , , and . We also find the number of non‐isomorphic minimal covering designs in several cases. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 218–228, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10022     

The generalized Randić index of trees

The generalized Randi?; index of a tree T is the sum over the edges of T of where is the degree of the vertex x in T. For all , we find the minimal constant such that for all trees on at least 3 vertices, , where is the number of vertices of T. For example, when . This bound is sharp up to the additive constant—for infinitely many n we give examples of trees T on n vertices with . More generally, fix and define , where is the number of leaves of T. We determine the best constant such that for all trees on at least 3 vertices, . Using these results one can determine (up to terms) the maximal Randi?; index of a tree with a specified number of vertices and leaves. Our methods also yield bounds when the maximum degree of the tree is restricted. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 270–286, 2007     

On the Spectrum of the Product of Closed Operators

In this note we study the connection between the spectra of the products AB and BA of unbounded closed operators A and B acting in Banach spaces. Under the condition that the resolvent sets of these products are not empty we show that the spectra of AB and BA coincide away from zero and prove the commutation relation 𝕀 . Further, we prove statements concerning the relationship between the spectra of the operator AB and the block operator matrix .     

Comparison of Inertial Manifolds and Application to Modulated Systems

We consider two dissipative systems having inertial manifolds and give estimates which allow us to compare the flows on the two inertial manifolds. As an example of a modulated system we treat the Swift–Hohenberg equation , ∈ ℝ, with periodic boundary conditions on the interval . Recent results in the theory of modulation equation show that the solutions of this equation can be described over long time scales by those of the associated Ginzburg–Landau equation ∈ ℂ, with suitably generalized periodic boundary conditions on . We prove that both systems have an inertial manifold of the same dimension and that the flows on these finite dimensional manifolds converge against each other for .     

Ryser's embedding problem for Hadamard matrices

What is the minimum order of a Hadamard matrix that contains an a by b submatrix of all 1's? Newman showed that where c^? denotes the smallest order greater than or equal to c for which a Hadamard matrix exists. It follows that if 4 divides both a and b, and if the Hadamard conjecture is true, then . We establish the improved bounds for min {a,b} ≥ 2. The Hadamard conjecture therefore implies that if 4 divides both 2ab and ?a/2? ?b/2?, then (a, b) = 2 · max {?a/2?b, ?b/2?a}. Our lower bound comes from a counting argument, while our upper bound follows from a sub‐multiplicative property of : Improvements in our upper bound occur when suitable conference matrices or Bush‐type Hadamard matrices exist. We conjecture that any (1,?1)‐matrix of size a by b occurs as a submatrix of some Hadamard matrix of order at most . © 2005 Wiley Periodicals, Inc. J Combin Designs     

Compactness Conditions for the Green Operator in Lp(ℝ) Corresponding to a General Sturm–Liouville Operator

We consider the equation ℝ, where , for ℝ, (ℝ), (ℝ), (ℝ), (ℝ) := C(ℝ)). We give necessary and sufficient conditions under which, regardless of , the following statements hold simultaneously: I) For any (ℝ) Equation (0.1) has a unique solution (ℝ) where $\int ^{\infty}_{-\infty}$ ℝ. II) The operator (ℝ) → (ℝ) is compact. Here is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm–Liouville operator.     

Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Law of large numbers for increasing subsequences of random permutations

Let the random variable Z_n,k denote the number of increasing subsequences of length k in a random permutation from S_n, the symmetric group of permutations of {1,…,n}. We show that Var(Z) = o((EZ)²) as n → ∞ if and only if . In particular then, the weak law of large numbers holds for Z if ; that is, We also show the following approximation result for the uniform measure U_n on S_n. Define the probability measure μ on S_n by where U denotes the uniform measure on the subset of permutations that contain the increasing subsequence {x₁,x₂,…,x}. Then the weak law of large numbers holds for Z if and only if where ∣∣˙∣∣ denotes the total variation norm. In particular then, (*) holds if . In order to evaluate the asymptotic behavior of the second moment, we need to analyze occupation times of certain conditioned two‐dimensional random walks. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣Γ = 0 for Integrable Tensor-Coefficients G and Applications to Elasticity

Let be bounded Lipschitz and relatively open. We show that the solution to the linear first order system 1 : (1) vanishes if and , (e.g. ). We prove to be a norm if with , for some p, q > 1 with 1/p + 1/q = 1 and . We give a new proof for the so called ‘in-finitesimal rigid displacement lemma’ in curvilinear coordinates: Let , satisfy for some with . Then there are and a constant skew-symmetric matrix , such that . (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New Aspects For Bifurcation Problems in Continuous Mechanics

Consider a semilinear eigenvalue problem where λ ∈ R , the linear operator is defined in a real Hilbert space H and : H → H is generaly a nonlinear perturbation. We can define a coincidence degree of the pair ( ) under some conditions weaker than the ones when the classical coincidence degree was defined. Our final purpose is to extend the results to the case of the operators from the Banach space X into its dual X*, using the representation theorem due to Browder and Ton. We use these results to study resonance problems in mechanics of continua, such as the buckling in finite elastostatics and the steady state flow of incompressible fluids. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)