The Bar-Natan skein module of the solid torus and the homology of (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">n</Emphasis>) Springer varieties

This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n, n) Springer variety. The results build on Khovanov’s work with crossingless matchings and the cohomology of the (n, n) Springer variety. We also give a formula for comultiplication in the Bar-Natan skein module for this specific three-manifold and boundary curve system.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

(<Emphasis Type="Italic">q</Emphasis>,<Emphasis Type="Italic">t</Emphasis>)-Deformations of multivariate hook product formulae

We generalize multivariate hook product formulae for P-partitions. We use Macdonald symmetric functions to prove a (q,t)-deformation of Gansner’s hook product formula for the generating functions of reverse (shifted) plane partitions. (The unshifted case has also been proved by Adachi.) For a d-complete poset, we present a conjectural (q,t)-deformation of Peterson–Proctor’s hook product formula.     

<Emphasis Type="Italic">P</Emphasis>-adic continued fractions

We study Schneider’s p-adic continued fraction algorithms. For p=2, we give a combinatorial characterization of rational numbers that have terminating expansions. For arbitrary p, we give data showing that rationals with terminating expansions are relatively rare. Finally, we prove an analogue of Khinchin’s theorem.     

Generalized fuzzy <Emphasis Type="Italic">n</Emphasis>-ary subhypergroups of a commutative <Emphasis Type="Italic">n</Emphasis>-ary hypergroup

In this paper, by means of a new idea, the concept of (invertible) (∈,∈∨q)-fuzzy n-ary subhypergroups of a commutative n-ary hypergroup is introduced and some related properties are investigated. A kind of n-ary quotient hypergroup by an (∈, ∈∨q)-fuzzy n-ary subhypergroup is provided and the relationships among (∈, ∈∨q)-fuzzy n-ary subhypergroups, n-ary quotient hypergroups and homomorphism are investigated. Several isomorphism theories of n-ary hypergroups are established.     

A general method for construction of (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-threshold visual secret sharing schemes for color images

This paper is concerned with the construction of basis matrices of visual secret sharing schemes for color images under the (t, n)-threshold access structure, where n ≥ t ≥ 2 are arbitrary integers. We treat colors as elements of a bounded semilattice and regard stacking two colors as the join of the two corresponding elements. We generate n shares from a secret image with K colors by using K matrices called basis matrices. The basis matrices considered in this paper belong to a class of matrices each element of which is represented by a homogeneous polynomial of degree n. We first clarify a condition such that the K matrices corresponding to K homogeneous polynomials become basis matrices. Next, we give an algebraic scheme for the construction of basis matrices. It is shown that under the (t, n)-threshold access structure we can obtain K basis matrices from appropriately chosen K − 1 homogeneous polynomials of degree n by using simple algebraic operations. In particular, we give basis matrices that are unknown so far for the cases of t = 2, 3 and n − 1.     

The <Emphasis Type="Italic">s</Emphasis>-energy of spherical designs on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S ². A spherical n-design is a point set on S ² that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy E _s (X) of a point set of m distinct points is the sum of the potential for all pairs of distinct points . A sequence Ξ = {X _m } of point sets X _m ⊂S ², where X _m has the cardinality card(X _m )=m, is well separated if for each pair of distinct points , where the constant λ is independent of m and X _m . For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy E _s (X _m(n)) for well separated sequences Ξ = {X _m(n)} of spherical n-designs X _m(n) with card(X _m(n))=m(n).      

Classification of regular embeddings of <Emphasis Type="Italic">n</Emphasis>-dimensional cubes

An orientably-regular map is a 2-cell embedding of a connected graph or multigraph into an orientable surface, such that the group of all orientation-preserving automorphisms of the embedding has a single orbit on the set of all arcs (incident vertex-edge pairs). Such embeddings of the n-dimensional cubes Q _n were classified for all odd n by Du, Kwak and Nedela in 2005, and in 2007, Jing Xu proved that for n=2m where m is odd, they are precisely the embeddings constructed by Kwon in 2004. Here, we give a classification of orientably-regular embeddings of Q _n for all n. In particular, we show that for all even n (=2m), these embeddings are in one-to-one correspondence with elements σ of order 1 or 2 in the symmetric group S _n such that σ fixes n, preserves the set of all pairs B _i ={i,i+m} for 1≤i≤m, and induces the same permutation on this set as the permutation B _i ↦ B _f(i) for some additive bijection f:ℤ _m →ℤ _m . We also give formulae for the numbers of embeddings that are reflexible and chiral, respectively, showing that the ratio of reflexible to chiral embeddings tends to zero for large even n.     

On proper secrets, (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">k</Emphasis>)-bases and linear codes

This paper contains three parts where each part triggered and motivated the subsequent one. In the first part (Proper Secrets) we study the Shamir’s “k-out-of-n” threshold secret sharing scheme. In that scheme, the dealer generates a random polynomial of degree k−1 whose free coefficient is the secret and the private shares are point values of that polynomial. We show that the secret may, equivalently, be chosen as any other point value of the polynomial (including the point at infinity), but, on the other hand, setting the secret to be any other linear combination of the polynomial coefficients may result in an imperfect scheme. In the second part ((t, k)-bases) we define, for every pair of integers t and k such that 1 ≤ t ≤ k−1, the concepts of (t, k)-spanning sets, (t, k)-independent sets and (t, k)-bases as generalizations of the usual concepts of spanning sets, independent sets and bases in a finite-dimensional vector space. We study the relations between those notions and derive upper and lower bounds for the size of such sets. In the third part (Linear Codes) we show the relations between those notions and linear codes. Our main notion of a (t, k)-base bridges between two well-known structures: (1, k)-bases are just projective geometries, while (k−1, k)-bases correspond to maximal MDS-codes. We show how the properties of (t, k)-independence and (t, k)-spanning relate to the notions of minimum distance and covering radius of linear codes and how our results regarding the size of such sets relate to known bounds in coding theory. We conclude by comparing between the notions that we introduce here and some well known objects from projective geometry.      

<Emphasis Type="Italic">p</Emphasis>-groups having few almost-rational irreducible characters

We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ _p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p ² and p ² + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p ² + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p ², and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p ³ are exceptional.     

The Dirichlet series for the exterior square <Emphasis Type="Italic">L</Emphasis>-function on GL(<Emphasis Type="Italic">n</Emphasis>)

We present an elementary derivation of the Jacquet–Shalika construction for the exterior square L-function on GL(n), as a classical Dirichlet series in the Fourier coefficients A(m ₁,…,m _n−1).     

Weierstrass Type Representation of Willmore Surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we reformulate the Euler-Lagrange equations of Willmore surfaces in S^n as the flatness of a family of certain loop algebra-valued 1-forms. Therefore we can give the Weierstrass type representation of conformal Willmore surfaces. We also discuss the relations between conformal Willmore surfaces in S^n and minimal surfaces in constant curvature spaces S^n, R^n, H^n, and prove that some special Willmore surfaces can be derived from minimal surfaces in S^n, R^n, H^n.     

Picard values of <Emphasis Type="Italic">p</Emphasis>-adic meromorphic functions

We investigate Picard-Hayman behavior of derivatives of meromorphic functions on an algebraically closed field K, complete with respect to a non-trivial ultrametric absolute value. We present an analogue of the well-known Hayman’s alternative theorem both in K and in any open disk. Here the main hypothesis is based on the behaviour of |f|(r) when r tends to +∞ on properties of special values and quasi-exceptional values.We apply this study to give some sufficient conditions on meromorphic functions so that they satisfy Hayman’s conjectures for n = 1and for n = 2. Given a meromorphic transcendental function f, at least one of the two functions f′f and f′f ² assumes all non-zero values infinitely often. Further, we establish that if the sequence of residues at simple poles of a meromorphic transcendental function on K admits no infinite stationary subsequence, then either f′ + af ² has infinitely many zeros that are not zeros of f for every a ∈ K* or both f′ + bf ³ and f′ + bf ⁴ have infinitely many zeros that are not zeros of f for all b ∈ K*. Most of results have a similar version for unbounded meromorphic functions inside an open disk.     

Getting on Top of Oneself: Comments on <Emphasis Type="Italic">Self-Expression</Emphasis>

This paper is a critical review of Mitchell Green’s Self-Expression. The principal focus is on Green’s contention that all expression is at route, a form of signalling by an agent or by some mechanism of the organism which has been evolutionary selected for signalling. Starting from the idea that in some but not all expression an agent seeks to express his or her self, I question the centrality of communication to the idea of expression.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

Optimal (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">n</Emphasis>) visual cryptographic schemes for general <Emphasis Type="Italic">k</Emphasis>

In (k, n) visual cryptographic schemes (VCS), a secret image is encrypted into n pages of cipher text, each printed on a transparency sheet, which are distributed among n participants. The image can be visually decoded if any k(≥2) of these sheets are stacked on top of one another, while this is not possible by stacking any k − 1 or fewer sheets. We employ a Kronecker algebra to obtain necessary and sufficient conditions for the existence of a (k, n) VCS with a prior specification of relative contrasts that quantify the clarity of the recovered image. The connection of these conditions with an L ₁-norm formulation as well as a convenient linear programming formulation is explored. These are employed to settle certain conjectures on contrast optimal VCS for the cases k = 4 and 5. Furthermore, for k = 3, we show how block designs can be used to construct VCS which achieve optimality with respect to the average and minimum relative contrasts but require much smaller pixel expansions than the existing ones.     

Semiclassical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Estimates of Quasimodes on Curved Hypersurfaces

Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L ² normalized family of functions such that P(h)u(h) is O(h) in L ²(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L ^p norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch ^−δ(n,k,p) where k=dim H (except for a logarithmic divergence in the case k=n−2, p=2). When H is a hypersurface, i.e., k=n−1, we have δ(n,n−1, 2)=1/4, which is sharp when M is the round n-sphere and H is an equator.     

On the Riesz means of coefficients of <Emphasis Type="Italic">m</Emphasis>th symmetric power <Emphasis Type="Italic">L</Emphasis>-functions

Let c _n be the Fourier coefficients of L(sym ^m   f, s), and Δρ(x; sym ^m   f) be the error term in the asymptotic formula for ∑ _n≪x c _n . In this paper, we study the Riesz means of Δρ(x; sym ^m   f) and obtain a truncated Voronoi-type formula under the hypothesis Nice(m, f).     

Cocyclic <Emphasis Type="Italic">n</Emphasis>-groups

We describe all cocyclic n-groups and the structure of (n, 2)-rings of endomorphisms of cocyclic n-groups. We prove that a cocyclic n-group is defined uniquely by its (n, 2)-ring of endomorphisms.