Balanced ternary designs with block size three,any Λ and anyR

A balanced ternary design onV elements is a collection ofB blocks (which are multisets) of sizeK, such that each element occurs 0, 1 or 2 times per block andR times altogether, and such that each unordered pair of distinct elements occurs times. (For example, in the blockxxyyz, the pairxy is said to occur four times and the pairsxz, yz twice each.) It is straightforward to show that each element has to occur singly in a constant number of blocks, say ₁, and so each element also occurs twice in a constant number of blocks, say ₂, whereR= ₁+2 ₂. If ₂=0 the design is a balanced incomplete block design (binary design), so we assume ₂>0, andK<2V (corresponding to incompleteness in the binary case). Necessarily >1 if ₂>0 (andK>2).In 1980 and 1982 the author gave necessary and sufficient conditions for the existence of balanced ternary designs withK=3, =2 and ₂=1, 2 or 3. In this paper work on the existence of balanced ternary designs with block size three is concluded, in that necessary and sufficient conditions for the existence of a balanced ternary design withK=3, any >1 and any ₂ are given.     

Apollonian circle packings: Number theory II. Spherical and hyperbolic packings

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space it is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. There are infinitely many different integral packings; these were studied in Part I (J. Number Theory 100, 1–45, 2003). Integral circle packings also exist in spherical and hyperbolic space, provided a suitable definition of curvature is used and again there are an infinite number of different integral packings. This paper studies number-theoretic properties of such packings. This amounts to studying the orbits of a particular subgroup of the group of integral automorphs of the indefinite quaternary quadratic form . This subgroup, called the Apollonian group, acts on integer solutions . This paper gives a reduction theory for orbits of acting on integer solutions to valid for all integer k. It also classifies orbits for all k≡0 (mod 4) in terms of an extra parameter n and an auxiliary class group (depending on n and k), and studies congruence conditions on integers in a given orbit. Much of this work was done while the authors were at AT&T Labs-Research, whom the authors thank for support. N. Eriksson was also supported by an NDSEG fellowship and J.C. Lagarias by NSF grant DMS-0500555.     

On the geometry of supermanifolds with even and odd Kählerian structures

Even and odd Kählerian structures are constructed on supermanifolds associated with the tangent bundles of Kählerian manifolds. Mechanics that are bi-Hamiltonian with respect to the corresponding Poisson brackets are found; they determine Killing vectors of the Kählerian structures. An analog of the operator in the Batalin—Vilkovisky quantization method is constructed; it corresponds to the divergence operator of the base manifold.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 140–149, July, 1993.     

Orthogonal geometries over finite fields with characteristic ≠2 and block designs

By taking as blocks certain subspace-pairs of an orthogonal geometry over a finite field with characteristic 2 we construct some new types of BIB designs and PBIB designs whose parameters are also given.     

Small embeddings of partial directed triple systems and partial triple systems with even λ

In 1976, Lindner and Rosa (Ars Combin. 1 (1976), 159–166) showed that a partial triple system with λ > 1 can be embedded in a finite triple system with the same λ. This result is improved in the case when λ is even by embedding a partial triple system on υ symbols in a triple system on t symbols, t ≡ 0,1 (mod 3), for all t >/ 3(λυ² + υ(2 ? λ) + 1). In the process, it is shown that for any λ >/ 1, a partial directed triple system on υ symbols can be embedded in a directed triple system on t symbols, t ≡ 0, 1 (mod 3), for all t ? 6λv² + 6v(1 ? λ) + 3, thus generalizing a result of Hamm (Proceedings, 14th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Boca Raton, Florida, 1983).     

Schwarz’s lemma for the circle packings with the general combinatorics

Rodin (1987) proved the Schwarz’s lemma analog for the circle packings based on the hexagonal combinatorics. In this paper, we prove the Schwarz’s lemma for the circle packings with the general combinatorics and our proof is more simpler than Rodin’s proof. At the same time, we obtain a rigidity property for those packings with the general combinatorics.     

Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras

We develop a theory of multigraded (i.e., ℕ ^l -graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar et al. (Compos. Math. 142:1–30, 2006). In particular we introduce the notion of canonical k-odd and k-even subalgebras associated with any multigraded combinatorial Hopf algebra, extending simultaneously the work of Aguiar et al. and Ehrenborg. Among our results are specific categorical results for higher level quasisymmetric functions, several basis change formulas, and a generalization of the descents-to-peaks map.     

Full-rank block LDL ? decomposition and the inverses of n×n block matrices

Full-rank block LDL ^? decomposition of a Hermitian n×n block matrix A is examined, where the iterative procedure evaluating the sub-matrices appearing in L and D is provided. This factorization is used to evaluate the inverse and Moore-Penrose inverse of a Hermitian n×n block matrix. The method for the calculation of the Moore-Penrose inverse of an arbitrary 2×2 block matrix is also provided. Therefore, matrix products A ^? A and AA ^? and the corresponding full-rank block LDL ^? factorizations are observed. Also, a simple explicit formulae calculating the solution vector components of the normal system of equations is stated, where the LDL ^? decomposition of the system matrix is done.     

On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages

The existence and construction of symplectic 2s-stage variable coefficients Runge-Kutta (RK) methods that integrate exactly IVPs whose solution is a trigonometrical polynomial of order s with a given frequency ω is considered. The resulting methods, that can be considered as trigonometrical collocation methods, are fully implicit, symmetric and symplectic RK methods with variable nodes and coefficients that are even functions of ν=ω h (h is the step size), and for ω→0 they tend to the conventional RK Gauss methods. The present analysis extends previous results on two-stage symplectic exponentially fitted integrators of Van de Vyver (Comput. Phys. Commun. 174: 255–262, 2006) and Calvo et al. (J. Comput. Appl. Math. 218: 421–434, 2008) to symmetric and symplectic trigonometrically fitted methods of high order. The algebraic order of the trigonometrically fitted symmetric and symplectic 2s-stage methods is shown to be 4s like in conventional RK Gauss methods. Finally, some numerical experiments with oscillatory Hamiltonian systems are presented.     

Directed graphs and combinatorial properties of completely regular semigroups

In this paper, we investigate the divisibility graphs and power graphs of completely regular semigroups. We give the structures of these two kinds of graphs and describe a combinatorial property of completely regular semigroups defined in terms of divisibility graphs and power graphs, respectively.     

A flexible block classical Gram–Schmidt skeleton with reorthogonalization

We investigate a variant of the reorthogonalized block classical Gram–Schmidt method for computing the QR factorization of a full column rank matrix. Our aim is to bound the loss of orthogonality even when the first local QR algorithm is only conditionally stable. In particular, this allows the use of modified Gram–Schmidt instead of Householder transformations as the first local QR algorithm. Numerical experiments confirm the stable behavior of the new variant. We also examine the use of non-QR local factorization and show by example that the resulting variants, although less stable, may also be applied to ill-conditioned problems.     

Embeddings of Pure Mendelsohn Triple Systems and Pure Directed Triple Systems

A λ-fold triple system of order v,denoted TS(v,λ),is a pair(V,A)where V is a v-set and A is a collection of 3-subsets(called triples)of V such that each 2-subset of V is contained in exactly λ triples.A triple system is called simple if itcontains no repeated triples. There are two related classes of triple systems,namely,Mendelsohn triple sys-tems and directed triple systems.     

Example of the matrix of size 6 × 6 with different tropical rank and Kapranov rank

We present an example of a 6 × 6 matrix A such that rk _t (A) = 4, rk _K (A) = 5. This disproves the conjecture formulated by M. Chan, A. Jensen, and E. Rubei.     

Tubes of even order and flat π.C 2 geometries

Atube of even orderq=2 ^d is a setT={L, } ofq+3 pairwise skew lines in PG(3,q) such that every plane onL meets the lines of in a hyperoval. Thequadric tube is obtained as follows. Take a hyperbolic quadricQ=Q ₃ ⁺ (q) in PG(3,q); letL be an exterior line, and let consist of the polar line ofL together with a regulus onQ.In this paper we show the existence of tubes of even order other than the quadric one, and we prove that the subgroup of PL(4,q) fixing a tube {L, } cannot act transitively on . As pointed out by a construction due to Pasini, this implies new results for the existence of flat .C ₂ geometries whoseC ₂-residues are nonclassical generalized quadrangles different from nets. We also give the results of some computations on the existence and uniqueness of tubes in PG(3,q) for smallq. Further, we define tubes for oddq (replacing hyperoval by conic in the definition), and consider briefly a related extremal problem.Dedicated to luigi antonio rosati on the occasion of his 70th birthday     

Minimal coverings and maximal packings of (k−1)-subsets by k-subsets

This paper studies the asymptotic behavior of functions M(n, k, k–1, ) and m(n, k, k–1, ), equal to the respective cardinalities of the minimal -covering and maximal -packing of all (k–1)-subsets of the n-element set of its k-subsets. It is shown that, if sequence k=k(n) is such that k(n)/n 0 as then m(n, k, k–1, ).( _k–1 ⁿ ).k¹, and if as n , thenM(n,k,k–1,).( _k–1 ⁿ ).k^–1. A consequence of these results is the validity of the Erdös-Hanani conjecture concerning the asymptotic behavior of functions M(n, k, k–1, 1) and m(n, k, k–1, 1).Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 565–571, April, 1977.In conclusion, the author wishes to thank A. A. Sapozhenko under whose direction the present work was achieved.     

The nonlinear Schrödinger harmonic oscillator problem with small odd or even disturbances

In this paper, we present research for the nonlinear Schrödinger harmonic oscillator problem with small odd or even disturbances. The analytical solutions obtained by the homotopy analysis method and the Adomian decomposition technique are in close agreement. The effects of involved parameters on the wave function are analyzed and discussed.     

Strong and Weak Disorder for Lévy Directed Polymers in Random Environment

Abstract

We consider Lévy directed polymers in the Poisson random environment. We give conditions for strong or weak disorder in terms of the Lévy exponent of symmetric Lévy process.