Difference sets with <Emphasis Type="Italic">n</Emphasis> = 5<Emphasis Type="Italic"> p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p ^r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.      

On Analytic Campanato and <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">s</Emphasis>) Spaces

We investigate the relation between analytic Campanato spaces \(\mathcal {AL}_{p,s}\) and the spaces F(p, q, s), characterize the bounded and compact Riemann–Stieltjes operators from \(\mathcal {AL}_{p,s}\) to \(F(p,p-s-1,s)\). We also describe the corona theorem and the interpolating sequences for the class \(F(p,p-2,s)\), which is the Möbius invariant subspace of the analytic Besov type spaces \(B_p(s)\).     

Derivatives for smooth representations of <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℝ) and <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℂ)

The notion of derivatives for smooth representations of GL(n, ? _p ) was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the “adduced” representation. In this paper we define derivatives of all orders for smooth admissible Fréchet representations of moderate growth. The real case is more problematic than the p-adic case; for example, arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation.In the companion paper [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations.We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS13a].     

Equitable total coloring of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> ⋁ <Emphasis Type="Italic">W</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The minimum number of total independent partition sets of V ∪ E of graph G(V,E) is called the total chromatic number of G denoted by χ _t (G). If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χ _et (G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order. Supported by the National Natural Science Foundation of China (No. 10771091).     

On common values of <Emphasis Type="Italic">φ</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) and <Emphasis Type="Italic">σ</Emphasis>(<Emphasis Type="Italic">m</Emphasis>). I

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x/(log x)^1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sum-of-divisors function.     

Regular<Emphasis Type="Italic">n</Emphasis>-simplices in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with vertices in ℤ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝ ⁿ are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝ ⁿ isn = 2m ² - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝ ⁿ and combinational design is given.     

<Emphasis Type="Italic">p</Emphasis>-Adic Brownian motion over ℚ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we generalize the result of Bikulov and Volovich (1997) and construct a p-adic Brownian motion over ℚ _p . First, we construct directly a p-adic white noise over ℚ _p by using a specific complete orthonormal system of (ℚ _p ). A p-adic Brownian motion over ℚ _p is then constructed by the Paley-Wiener method. Finally, we introduce a p-adic random walk and prove a theorem on the approximation of a p-adic Brownian motion by a p-adic random walk.     

Free involutions on <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Topological free involutions on S ¹ × S ⁿ are classified up to conjugation. We prove that this is the same as classifying quotient manifolds up to homeomorphism. There are exactly four possible homotopy types of such quotients, and surgery theory is used to classify all manifolds within each homotopy type.     

Factorization properties of (<Emphasis Type="Italic">n</Emphasis> × <Emphasis Type="Italic">n</Emphasis>) Boolean matrices

It is proved that every (n × n) Boolean matrix can be expressed as a product of primes and elementary matrices in the semigroup of Boolean matrices.     

On the Spectrum of the Sizes of Maximal Partial Line Spreads in <Emphasis Type="Italic">PG</Emphasis>(2<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">n</Emphasis> ≥ 3

A lot of research has been done on the spectrum of the sizes of maximal partial spreads in PG(3,q) [P. Govaerts and L. Storme, Designs Codes and Cryptography, Vol. 28 (2003) pp. 51–63; O. Heden, Discrete Mathematics, Vol. 120 (1993) pp. 75–91; O. Heden, Discrete Mathematics, Vol. 142 (1995) pp. 97–106; O. Heden, Discrete Mathematics, Vol. 243 (2002) pp. 135–150]. In [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129], results on the spectrum of the sizes of maximal partial line spreads in PG(N,q), N 5, are given. In PG(2n,q), n 3, the largest possible size for a partial line spread is q^2n-1+q^2n-3+...+q³+1. The largest size for the maximal partial line spreads constructed in [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129] is (q²ⁿ⁺¹–q)/(q²–1)–q³+q²–2q+2. This shows that there is a non-empty interval of values of k for which it is still not known whether there exists a maximal partial line spread of size k in PG(2n,q). We now show that there indeed exists a maximal partial line spread of size k for every value of k in that interval when q 9.J. Eisfeld: Supported by the FWO Research Network WO.011.96NP. Sziklai: The research of this author was partially supported by OTKA D32817, F030737, F043772, FKFP 0063/2001 and Magyary Zoltan grants. The third author is grateful for the hospitality of Ghent University.     

Weak-Polynomial Convergence on Spaces ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is concerned with the study of the set P ^-1(0), when P varies over all orthogonally additive polynomials on _p and L _p spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials.     

Supersymmetry-inspired low-energy <Emphasis Type="Italic">α</Emphasis>–<Emphasis Type="Italic">p</Emphasis> elastic scattering phases

We consider an effective potential model consisting of an electromagnetic part plus a nuclear part as the ground state interaction for an α–p system. The next few higher partial wave interactions are generated using the formalism of supersymmetric quantum mechanics. We adapt the phase function method to compute α–p elastic scattering phases up to 12 MeV, including the effect of the electromagnetic interaction quite rigorously in our phase shift calculation. With the further incorporation of some energy-dependent correction factors to our interactions, we obtain a good agreement with the experimental data.     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

On the Crossing Number of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> □ <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Crossing numbers of graphs are in general very difficult to compute. There are several known exact results on the crossing number of the Cartesian products of paths, cycles or stars with small graphs. In this paper we study cr(K_m □ P_n), the crossing number of the Cartesian product K_m □ P_n. We prove that for m ≥ 3,n ≥ 1 and cr(K_m □ P_n)≥ (n − 1)cr(K_m+2 − e) + 2cr(K_m+1). For m≤ 5, according to Klešč, Jendrol and Ščerbová, the equality holds. In this paper, we also prove that the equality holds for m = 6, i.e., cr(K₆ □ P_n) = 15n + 3. Research supported by NFSC (60373096, 60573022).     

The critical group of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> × <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

In this paper, the structure of the critical group of the graph K _m × C _n is determined, where m, n ≥ 3.     

On the equation Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)<Emphasis Type="Italic">u</Emphasis> = 0

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δu + q(x)u = 0, where either q(x) ≠ const or q(x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.     

Hochschild Cohomology of Deformation Quantizations over ℤ/<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>ℤ

Let A ₁ be an Azumaya algebra over a smooth affine symplectic variety X over Spec F _p , where p is an odd prime. Let A be a deformation quantization of A ₁ over the p-adic integers. In this note we show that for all n ≥ 1, the Hochschild cohomology of A/p ⁿ A is isomorphic to the de Rham-Witt complex \(W_{n}{\Omega }^{\ast }_{X}\) of X over \(\mathbb {Z}/p^{n}\mathbb {Z}\). We also compute the center of deformations of certain affine Poisson varieties over F _p .     

On coherent systems of type (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">d</Emphasis>, <Emphasis Type="Italic">n</Emphasis> + 1) on Petri curves

We study coherent systems of type (n, d, n + 1) on a Petri curve X of genus g ≥ 2. We describe the geometry of the moduli space of such coherent systems for large values of the parameter α. We determine the top critical value of α and show that the corresponding “flip” has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of α, proving in many cases that the condition for non-emptiness is the same as for large α. We give some detailed results for g ≤ 5 and applications to higher rank Brill–Noether theory and the stability of kernels of evaluation maps, thus proving Butler’s conjecture in some cases in which it was not previously known. The authors are members of the research group VBAC (Vector Bundles on Algebraic Curves). The first two authors were supported by EPSRC grant GR/T22988/01 for a visit to the University of Liverpool. The second author acknowledges the support of CONACYT grant 48263-F. The third author thanks CIMAT, Guanajuato, México and California State University Channel Islands, where a part of this paper was completed, and acknowledges support from the Academia Mexicana de Ciencias, under its exchange agreement with the Royal Society of London.     

The crossing number of <Emphasis Type="Italic">G</Emphasis> ▭ <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> for the graph <Emphasis Type="Italic">G</Emphasis> on six vertices

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. The crossing numbers of G□C _n for some graphs G on five and six vertices and the cycle C _n are also given. In this paper, we extend these results by determining the crossing number of the Cartesian product G □ C _n , where G is a specific graph on six vertices.     

Restriction of Representations of GL (<Emphasis Type="Italic">n</Emphasis> + 1, <Emphasis Type="Italic">ℂ</Emphasis>) to GL (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">ℂ</Emphasis>) and Action of the Lie Overalgebra

Consider a restriction of an irreducible finite dimensional holomorphic representation of \(\text {GL}(n + 1,\mathbb {C})\) to the subgroup \(\text {GL}(n,\mathbb {C})\). We write explicitly formulas for generators of the Lie algebra \(\mathfrak {g}\mathfrak {l}(n + 1)\) in the direct sum of representations of \(\text {GL}(n,\mathbb {C})\). Nontrivial generators act as differential-difference operators, the differential part has order n ??1, the difference part acts on the space of parameters (highest weights) of representations. We also formulate a conjecture about unitary principal series of \(\text {GL}(n,\mathbb {C})\).