Tensor product of operads and iterated loop spaces

A algebraic characterization of an n-fold loop space in terms of its n different 1-fold loop structures is established. This amounts to describing the higher homotopy commutativity for such a space as a strict partial commutativity of the 1-fold loop structures. The tensor product of operads (a special case of the construction for algebraic theories) is ideally suited for this. In particular we show that the operad of little n-cubes C_n is homotopy equivalent to the n-fold tensor product Cⁿ₁, i.e., ‘tensoring these A_∞-structures yields an iterated loop structure’. This is not true for arbitrary A_∞-operads.     

Intersections of nest algebras in finite dimensions

If are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dim H. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n(n+1)/2, where n=dim H. For any two maximal nests there exists a basis {f₁,f₂,…,f_n} of H and a permutation π such that and where M_i=  span{f₁,f₂,…,f_i} and N_i= span{f_π(1),f_π(2),…,f_π(i)}. The intersection of the corresponding nest algebras has minimum dimension, namely dim H, precisely when π(j)=n−j+1,1jn. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras.     

The smallest degree sum that yields potentially kCℓ-graphic sequences

Gould et al. (Combinatorics, Graph Theory and Algorithms, Vol. 1, 1999, pp. 387–400) considered a variation of the classical Turán-type extremal problems as follows: For a given graph H, determine the smallest even integer σ(H,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(H,n) has a realization G containing H as a subgraph. In this paper, for given integers k and ℓ, ℓ7 and 3kℓ, we completely determine the smallest even integer σ(_kC_ℓ,n) such that each n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(_kC_ℓ,n) has a realization G containing a cycle of length r for each r, krℓ.     

Concerning two conjectures on the set of fixed points of a complete rotation of a Cayley digraph

In 1996, J.C. Bermond, T. Kodate, S. Perennes and N. Marlin conjectured that the set F_σ of fixed points of some complete rotation σ of the toroidal mesh TM(p)^k is not separating (that is F_σ does not disconnect TM(p)^k). They also conjectured that the set F_ω of fixed points of any complete rotation ω of any Cayley digraph is not separating. In this paper, we prove the first conjecture and disprove the second one.     

A nonterminating 8φ7 summation for the root system Cr

Using multiple q-integrals and a determinant evaluation, we establish a nonterminating ₈φ₇ summation for the root system C_r. We also give some important specializations explicitly.     

On the vital role played by the electron-volt units system in high energy physics and Mach’s principle of “Denkökonomie”

The work highlights within the framework of E. Mach’s “Denkökonomie” the crucial role played by the electron-volt unit system in uncovering terce relationships in the various fundamental interactions. In particular it is shown that a great deal of insight and analytical simplicity is gained from the number coincidence between the E^(∞) dimensionless inverse fine structure constant and the mass of a postulated expectation π-meson m_π=(m_π^±+m_π⁰)/2 when measured in Mega-electron-volt. This line of thought could be continued to the k-meson and relate it to the exceptional Lie group of super strings and unification namely E₈  E₈. We show further that m_π could be viewed as a two quark building block relating the old Yukawa theory to an extended quark model based on E-infinity Cantorian spacetime.     

Summation theorems for multidimensional basic hypergeometric series by determinant evaluations

We derive summation formulas for a specific kind of multidimensional basic hypergeometric series associated to root systems of classical type. We proceed by combining the classical (one-dimensional) summation formulas with certain determinant evaluations. Our theorems include A_r extensions of Ramanujan's bilateral ₁ψ₁ sum, C_r extensions of Bailey's very-well-poised ₆ψ₆ summation, and a C_r extension of Jackson's very-well-poised ₈φ₇ summation formula. We also derive multidimensional extensions, associated to the classical root systems of type A_r, B_r, C_r, and D_r, respectively, of Chu's bilateral transformation formula for basic hypergeometric series of Gasper–Karlsson–Minton type. Limiting cases of our various series identities include multidimensional generalizations of many of the most important summation theorems of the classical theory of basic hypergeometric series.     

An extension theorem for t-designs

In 1994, van Trung (Discrete Math. 128 (1994) 337–348) [9] proved that if, for some positive integers d and h, there exists an S_λ(t,k,v) such that

then there exists an S_λ(v−t+1)(t,k,v+1) having v+1 pairwise disjoint subdesigns S_λ(t,k,v). Moreover, if B_i and B_j are any two blocks belonging to two distinct such subdesigns, then d|B_i∩B_j|<k−h. In 1999, Baudelet and Sebille (J. Combin. Des. 7 (1999) 107–112) proved that if, for some positive integers, there exists an S_λ(t,k,v) such that

where m=min{s,v−k} and n=min{i,t}, then there exists an

having pairwise disjoint subdesigns S_λ(t,k,v). The purpose of this paper is to generalize these two constructions in order to produce a new recursive construction of t-designs and a new extension theorem of t-designs.     

Sometimes slow growing is fast growing

The slow growing hierarchy is commonly defined as follows: G₀(x) = 0, G_x−1(x) := G_x(x) + 1 and G_λ(x) := G_λ[x](x) where λ<₀ is a limit and ·[·]:₀∩ Lim × ω → ₀ is a given assignment of fundamental sequences for the limits below ₀. The first obvious question which is encountered when one looks at this definition is: How does this hierarchy depend on the choice of the underlying system of fundamental sequences? Of course, it is well known and easy to prove that for the standard assignment of fundamental sequence the hierarchy (G_x)_x<0 is slow growing, i.e. each G_x is majorized by a Kalmar elementary recursive function.

It is shown in this paper that the slow growing hierarchy (G_x)_x<0 — when it is defined with respect to the norm-based assignment of fundamental sequences which is defined in the article by Cichon (1992, pp. 173–193) — is actually fast growing, i.e. each PA-provably recursive function is eventually dominated by G_x for some <₀. The exact classification of this hierarchy, i.e. the problem whether it is slow or fast growing, has been unsolved since 1992. The somewhat unexpected result of this paper shows that the slow growing hierarchy is extremely sensitive with respect to the choice of the underlying system of fundamental sequences.

The paper is essentially self-contained. Only little knowledge about ordinals less than ₀ — like the existence of Cantor normal forms, etc. and the beginnings of subrecursive hierarchy theory as presented, for example, in the 1984 textbook of Rose — is assumed.     

Enumeration of irreducible binary words

Overlap free words over two letters are called irreducible binary words. Let d(n) denote the number of irreducible binary words of length n. In this paper we show that there are positive constants C₁ and C₂ such that C₁n^1.155<d(n)<C₂n^1.587 holds for all n>0.     

On the complex zeros of Hμ(z), J(z), J(z) for real or complex order

Propositions about the nonexistence of complex zeros of the functions H_μ(z)=J_μ(z)+zJ^′_μ(z),J^′_μ(z),J^″_μ(z), where J^′_μ(z) and J^″_μ(z) are the first two derivatives of the Bessel functions J_μ(z), for μ in general complex are proved. Bounds for the purely imaginary zeros of the above functions assuming their existence are given. Thus for the range of values for which these bounds are violated there are no purely imaginary zeros of the above functions. Finally, some known results from previous work are generalized in the present paper.     

On the surface viscosity at the boundary between phases

The equations of motion of the interphase boundary are considered. It is shown that the conditions at the surface separating the phases obtained in /1, 2/ by different methods, are identical. The study of the dynamics of the fluid-fluid interface was initiated by Bussinesq /3/ who postulated a linear relationship between the surface stress tensor T_β and the strain rate tensor S_β, assigning two viscosity coefficients to the surface, the dilatation coefficient k (the analog of volume viscosity) and the two-dimensional shear viscosity . In the three-dimensional coordinate system two of whose axes u¹ and usu2 coincide with the axes of any coordinate system at the surface and whose third axis u³ is perpendicular to the surface, his results have the form Tβ = [γ + (k - )θ]a_β + S_β , θ = a^βS_β, V_{, β} = r_{. β} — v_sb_β,      

Analysis and numerics for a parabolic equation with impulsive forcing

The paper considers a one-dimensional particle-continuum model, with impulsive interaction between the fluid and a number of pointwise particles. A simplification results in a system of ODEs coupled with a parabolic PDE forced by a nonlinear term involving a sum of Dirac delta functions. The existence of a mild solution is proved using a combination of energy estimates and semigroup theory. However, the regularity of these solutions is shown to be limited to C^0,1 by the impulsive terms. The convergence of a Galerkin method is established simultaneously with a proof of continuous dependence, and thus uniqueness, of solutions for the underlying system. The peculiarities of the system imply this analysis must be performed in L_∞. The C^0,1 regularity of the solution determines a suboptimal rate of convergence for the Galerkin method. The theoretical results are verified by MATLAB computations.     

On an explicit formula for the distribution of the supremum

Let M_∞ be the supremum of a random walk drifting to -∞ which is generated by the partial sums of a sequence of independent identically distributed random variables with a common distribution F. We prove that the moment generating function E exp(sM_∞) is a rational function if and only if the function ∫₀^∞ exp(sx)F(dx) is rational.     

A theorem on planar continua and an application to automorphisms of the field of complex numbers

Let K be a continuum in the plane which does not lie on a line. Then the set of differences, K - K, contains an open set. Let ψ be an automorphism of the field of complex numbers which is bounded on an F_σ set of positive inductive dimension. Then ψ is continuous.     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

How many maxima can there be?

Let C be a planar region. Choose n points p₁,,p_nI.I.D. from the uniform distribution over C. Let M^C_n be the number of these points that are maximal. If C is convex it is known that either E(M^C_n)=Θ(√n)> or E(M^C_n)=O(log n). In this paper we will show that, for general C, there is very little that can be said, a-priori, about E(M^C_n). More specifically we will show that if g is a member of a large class of functions then there is always a region C such that E(M^C_n)=Θ(g(n)). This class contains, for example, all monotically increasing functions of the form g(n)= nln^βn, where 0<<1 and β0. This class also contains nondecreasing functions like g(n)=ln^*n. The results in this paper remain valid in higher dimensions.     

Decompositions of positive self-dual boolean functions

A coterie, which is used to realize mutual exclusion in distributed systems, is a family C of subsets such that any pair of subsets in C has at least one element in common, and such that no subset in C contains any other subset in C. Associate with a family of subsets C a positive Boolean function f_c such that f_c(x) = 1 if the Boolean vector x is equal to or greater than the characteristic vector of some subset in C, and 0 otherwise. It is known that C is a coterie if and only if f_c is dual-minor, and is a non-dominated (ND) coterie if and only if f_c is self-dual. We study in this paper the decomposition of a positive self-dual function into smaller positive self-dual functions, as it explains how to represent and how to construct the corresponding ND coterie. A key step is how to decompose a positive dual-minor function f into a conjunction of positive self-dual functions f₁,f₂,…, f_k. In addition to the general condition for this decomposition, we clarify the condition for the decomposition into two functions f₁, and f₂, and introduce the concept of canonical decomposition. Then we present an algorithm that determines a minimal canonical decomposition, and a very simple algorithm that usually gives a decomposition close to minimal. The decomposition of a general self-dual function is also discussed.