Superlocalization of waves and electrons and hopping conductivity on classical and superconductive fractals

We have found an unexpected paradoxical situation in the percolation transition: the superconductive behavior below and above the threshold. We have found also the two different density of states d_s=4/3 and d_v=1.05 and the inverse localization lengths for fractons with the scalar and vector interactions, respectively. In this concept the wave functions of electrons or waves on an incipient percolation cluster and fractal dilute structure exhibit superlocalization behavior of the form ψ(r)∝exp[−r^d_φ] with values of d_φ1=1.73 and d_φ2=2.4 for the former and the latter. Applications of these results for thermally activated hopping conductivity σ(t)∝exp[−(T₀/T)^β] between impurities on a random fractal structure give the values of β=2/5 for the scalar and β=1/2 (Mott's law) for the vector interactions, respectively. Band states are localized in classical and superlocalized in superconductive percolations.     

Gradual and tapered overflow and underflow: A functional differential equation and its approximation

A modified structure for floating-point representation, which can essentially eliminate overflow from floating-point calculations, is analyzed. The main part of the paper deals with a computational examination of a model for floating-point exponents and the probabilities of overflow and underflow. Earlier results in the absence of gradual underflow are presented for comparison but the primary focus is on the effect of gradual underflow, a version of gradual overflow, and of a proposal for an extended treatment of these exceptions which we call tapered overflow and underflow. The latter virtually eliminates the exceptions; its overflow threshold is ${10}^{600000000}$. This paper reviews several models for the distribution of exponents of floating-point numbers and their evolution in the presence of repeated multiplicative operations. A continuous model, which closely approximates the discrete case, is seen to satisfy a functional differential equation with both delay and advance terms.     

Lift and project relaxations for the matching and related polytopes

We compare lift and project methods given by Lovász and Schrijver (the N₊ and N procedures) and by Balas, Ceria and Cornuéjols (the disjunctive procedure) when working on the matching, perfect matching and covering polytopes. When the underlying graph is the complete graph of n=2s+1 nodes we obtain that the disjunctive index for all problems is s², the N₊-index for the matching and perfect matching problems is s (extending a result by Stephen and Tunçel), the N-index for the perfect matching problem is s, and the N₊ and N indices for the covering problem and the N-index for the matching problem are strictly greater than s.     

Codes from lattice and related graphs, and permutation decoding

Codes of length n² and dimension 2n−1 or 2n−2 over the field F_p, for any prime p, that can be obtained from designs associated with the complete bipartite graph K_n,n and its line graph, the lattice graph, are examined. The parameters of the codes for all primes are obtained and PD-sets are found for full permutation decoding for all integers n≥3.     

Tangent and secant numbers and representations of symmetric groups

Expressions for the tangent and secant numbers are obtained, for n=1, 2,…, 10, in terms of the numbers f(λ) and g_θμη occurring in the representation theory of the symmetric group δ_n. Some explicit results are obtained for g_θμη in certain general cases. Characters of δ_n are obtained, for general n, corresponding to the tangent and secant numbers.     

Null spaces and ranges for the classical and spherical Radon transforms

Let R be the classical Radon transform that integrates a function over hyperplanes in Rⁿ and let SM be the transform that integrates a function over spheres containing the origin in Rⁿ. We prove continuity results for both transforms and explicitly give the null space of R for a class of square integrable functions on the exterior of a ball in Rⁿ as well as the null space of SM for square integrable functions on a ball. We show SM: L²(Rⁿ) → L²(Rⁿ) is one-one, and we characterize the range of SM on classes of smooth functions and square integrable functions by certain moment conditions. If g(x) is a Schwartz function on Rⁿ that is zero to infinite order at x = 0, we prove moment conditions sufficient for g to be in the range of SM(C^∞(Rⁿ)). We apply our results on SM to existence and uniqueness theorems for solutions to a characteristic initial value problem for the Darboux partial differential equation.     

Line-digraphs,arborescences, and theorems of tutte and knuth

The line-digraph of a digraph D with vertices V₁, …, V_n is the digraph $\text{D}{}^1$ obtained from D by associating with each edge of D a vertex of $\text{D}{}^1$, and then directing an edge from vertex (V_i, V_j) of $\text{D}{}^1$ to vertex (V_k, V_m) if and only if j = k. This paper extends a characterization given by Harary and Norman for linedigraphs. It is also possible to repeatedly contract vertices of the line-digraph (with a new contraction procedure) so as to obtain the digraph derived from D by deleting all vertices with no incoming edges. Several new identities for arborescences are presented, leading to a combinatorial proof of Knuth's formula for the number of arborescences of a line-digraph. A new proof is given for the fact that in a digraph with every vertex having indegree equal to outdegree, the number of arborescences with root V_j is independent of j. Finally a new proof is presented for Tutte's Matrix Tree Theorem which shows the theorem to be a special case of the principle of inclusion-exclusion.     

Set-valued contractions and fixed points

A common fixed point theorem is proved for a family of set-valued contraction mappings in gauge spaces. This result is related to a recent result of Frigon for ‘generalized contractions’ and it includes a method for approximating the fixed point. The remainder of the paper is devoted to results for families of set-valued contraction mappings in hyperconvex spaces. It is proved, for example, that if M is a hyperconvex metric space and f_α is a family of set-valued contractions indexed over a directed set Λ and taking values in the space of all nonempty admissible subsets of M endowed with the Hausdorff metric, then the condition f_β(x)⊆f_α(x) for all x∈M and β⩾α implies that the set of points x∈M for which x∈⋂_α∈Λf_β(x) is nonempty and hyperconvex.     

Flocks and ovals

An infinite family of q-clans, called the Subiaco q-clans, is constructed for q=2^e. Associated with these q-clans are flocks of quadratic cones, elation generalized quadrangles of order (q ², q), ovals of PG(2, q) and translation planes of order q ² with kernel GF(q). It is also shown that a q-clan, for q=2^e, is equivalent to a certain configuration of q+1 ovals of PG(2, q), called a herd.W. Cherowitzo gratefully acknowledges the support of the Australian Research Council and has the deepest gratitude and warmest regards for the Combinatorial Computing Research Group at the University of Western Australia for their congenial hospitality and moral support. I. Pinneri gratefully acknowledges the support of a University of Western Australia Research Scholarship.     

Coderivative calculus and metric regularity for constraint and variational systems

This paper provides new developments in generalized differentiation theory of variational analysis with their applications to metric regularity of parameterized constraint and variational systems in finite-dimensional and infinite-dimensional spaces. Our approach to the study of metric regularity for these two major classes of parametric systems is based on appropriate coderivative constructions for set-valued mappings and on extended calculus rules supporting their computation and estimation. The main attention is paid in this paper to the so-called reversed mixed coderivative, which is of crucial importance for efficient pointwise characterizations of metric regularity in the general framework of set-valued mappings between infinite-dimensional spaces. We develop new calculus results for the latter coderivative that allow us to compute it for large classes of parametric constraint and variational systems. On this basis we derive verifiable sufficient conditions, necessary conditions as well as complete characterizations for metric regularity of such systems with computing the corresponding exact bounds of metric regularity constants/moduli. This approach allows us to reveal general settings in which metric regularity fails for major classes of parametric variational systems. Furthermore, the developed coderivative calculus leads us also to establishing new formulas for computing the radius of metric regularity for constraint and variational systems, which characterize the maximal region of preserving metric regularity under linear (and other types of) perturbations and are closely related to conditioning aspects of optimization.     

Multiobjective optimization problems with modified objective functions and cone constraints and applications

In this paper, we consider a differentiable multiobjective optimization problem with generalized cone constraints (for short, MOP). We investigate the relationship between weakly efficient solutions for (MOP) and for the multiobjective optimization problem with the modified objective function and cone constraints [for short, (MOP) _η (x)] and saddle points for the Lagrange function of (MOP) _η (x) involving cone invex functions under some suitable assumptions. We also prove the existence of weakly efficient solutions for (MOP) and saddle points for Lagrange function of (MOP) _η (x) by using the Karush-Kuhn-Tucker type optimality conditions under generalized convexity functions. As an application, we investigate a multiobjective fractional programming problem by using the modified objective function method.     

Identifying codes and locating-dominating sets on paths and cycles

Let G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define N_r[x]={y∈V:d(x,y)≤r} and D_r(x)=N_r[x]∩D, where d(x,y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x∈V (x∈V?D, respectively), D_r(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72-82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969-987].     

Bounded point evaluations and capacity

Let E be a compact set in the complex plane with positive Lebesgue measure, and denote by R^p(E), p ? 1, the closure in the L^p(E) norm of the rational functions with poles off E. A point z?E is said to be a bounded point evaluation for R^p(E) if the map z   ?(z), defined for the rational functions, can be extended to a bounded linear functional on R^p(E). For p < 2 there are no other bounded point evaluations for R^p(E) than the interior points of E, but for p ? 2 there may be bounded point evaluations on the boundary, ∂E. We give a condition, in terms of capacity, which is necessary and sufficient for a point on ∂E to be a bounded point evaluation for R^p(E), 2 < p < ∞, and close to necessary and sufficient when p = 2. We also treat bounded point derivations, and the corresponding problems for L_p-spaces of analytic functions on open sets.     

Domain and Range Operations in Semigroups and Rings

A D-semigroup S is a semigroup equipped with an operation D satisfying laws asserting that for a ∈ S, D(a) is the smallest e in some set of idempotents U ? S for which ea = a. D-semigroups correspond to left-reduced U-semiabundant semigroups. The basic properties and many examples of D-semigroups are given. Also considered are D-rings, whose multiplicative semigroup is a D-semigroup. Rickart *-rings provide important examples, and the most general D-rings for which the elements of the form D(a) constitute a lattice under the same meet and join operations as for Rickart *-rings are described.     

On a lemma due to Ladyzhenskaya and Solonnikov and some applications

We present some applications of a lemma by Ladyzhenskaya and Solonnikov [Determination of solutions of boundary value problems for stationary Stokes and Navier–Stokes equations having an unbounded Dirichlet integral, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 (1980) 117–160 (English Transl.: J. Soviet Math. 21 (1983) 728–761)]. Some other results in that paper referring to stationary Navier–Stokes equations are extended to a non-Newtonian fluid, the so-called micropolar fluid. This model depends on the microrotational viscosity _νr${\nu }_{\mathrm{r}}$ which vanishes for a Navier–Stokes fluid. We use the lemma in full to show that, as _νr${\nu }_{\mathrm{r}}$ tends to zero, the solutions of the Ladyzhenskaya–Solonnikov problem converge to the solutions of the corresponding problem for Navier–Stokes equations. In addition, we obtain a similar convergence regarding the Leray problem for micropolar fluids.     

Evasion and prediction

 Say that a function π:n ^<ω →n (henceforth called a predictor) k-constantly predicts a real xn ^ω if for almost all intervals I of length k, there is iI such that x(i)=π(x↾i). We study the k-constant prediction number v _n ^const (k), that is, the size of the least family of predictors needed to k-constantly predict all reals, for different values of n and k, and investigate their relationship. Received: 27 June 2001 / Revised version: 10 September 2001 / Published online: 10 October 2002 RID="*" ID="*" Supported by Grant–in–Aid for Scientific Research (C)(2)12640124, Japan Society for the Promotion of Science RID="†" ID="†" Supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Publication 762     

Finite-Dimensional Motives and the Conjectures of Beilinson and Murre

We relate the notion of finite dimensionality of the Chow motive M(X) of a smooth projective variety X (as defined by S. Kimura) with the conjectures of Beilinson, Bloch and Murre on the existence of a filtration on the Chow ring CH*(X). We show (Theorem 3) that finite dimensionality of M(X) implies uniqueness, up to isomorphism, of Murre's decomposition of M(X). Conversely (Theorem 4), Murre's conjecture for X ^m ×X ^m (for a suitable m) implies finite-dimensionality of M(X). We also show (Theorem 7) that, for a surface X with p _g = 0, the motive M(X) is finite-dimensional if and only if the Chow group of 0-cycles of X is finite-dimensional in the sense of Mumford, i.e. iff the Bloch conjecture holds for X.The second named author is a member of GNSAGA of CNR.     

Uniqueness and nonuniqueness of the Stokes and Oseen flows

The uniqueness class for the solutions to the Cauchy problem for flows modeled by the time-dependent Stokes and Oseen systems of equations is determined as in the growth class C exp(α|x|²). An example of a type considered by Tychonoff [Mat. Sb. 42 (1935), 199–216] is given that establishes the lack of uniqueness for such Stokes and Oseen systems. Even for the incompressible Navier–Stokes system, an example shows that rapidly growing nonphysical mathematical solutions exist.     

Special domains and nonmanipulability

We investigate domains on which a nonmanipulable, nondictatorial social choice function exists, having at least three distinct values. We do not make the assumptions of Kalai and Muller (1977). We classify all such 2-person functions on the domain which is the cyclic group Z_m. We show that for any domain containing Z_m, existence for 2 voters and existence for some n > 2 voters are equivalent. We show that for an n-person, onto, nonmanipulable social choice function F on Z_m, F(P₁, P₂,…, P_n) {x₁, x₂,…, x_n} always, x_i being the most preferred alternative under preference P_i. We show that no domain containing the dihedral group admits such a social choice function. We show that there exists a domain on which all k-tuples are free for arbitrarily large k, for which such a social choice function does exist.