Further results on almost resolvable cycle systems and the Hamilton–Waterloo problem

In this paper, we almost completely solve the existence of almost resolvable cycle systems with odd cycle length. We also use almost resolvable cycle systems as well as other combinatorial structures to give some new solutions to the Hamilton–Waterloo problem.     

On the existence of cycle frames and almost resolvable cycle systems

Suppose H is a complete m-partite graph K_m(n₁,n₂,…,n_m) with vertex set V and m independent sets G₁,G₂,…,G_m of n₁,n₂,…,n_m vertices respectively. Let G={G₁,G₂,…,G_m}. If the edges of λH can be partitioned into a set C of k-cycles, then (V,G,C) is called a k-cycle group divisible design with index λ, denoted by (k,λ)-CGDD. A (k,λ)-cycle frame is a (k,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V?G_i for some G_i∈G. Stinson et al. have resolved the existence of (3,λ)-cycle frames of type g^u. In this paper, we show that there exists a (k,λ)-cycle frame of type g^u for k∈{4,5,6} if and only if , , u≥3 when k∈{4,6}, u≥4 when k=5, and (k,λ,g,u)≠(6,1,6,3). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). Lindner et al. have considered the general existence problem of k-ARCS(n) from the commutative quasigroup for . In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k-ARCS(n)s when . We also update the known results and prove that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three known exceptions and four additional possible exceptions.     

Completing the spectrum of almost resolvable cycle systems with odd cycle length

In this paper, we construct almost resolvable cycle systems of order $4k+1$ for odd $k\ge 11$. This completes the proof of the existence of almost resolvable cycle systems with odd cycle length. As a by-product, some new solutions to the Hamilton–Waterloo problem are also obtained.     

A characterization of almost resolvable spaces

A space is said to be resolvable if it has two disjoint dense subsets. It is shown thatX is a Baire space with no resolvable open subsets iff every real function defined onX has a dense set of points of continuity. Thus almost resolvable spaces, as defined by Bolstein, are shown to be characterized as the union of a first category set and a closed resolvable set.     

On large sets of resolvable and almost resolvable oriented triple systems

An MTS(v) [or DTS(v)] is said to be resolvable, denoted by RMTS(v) [or RDTS(v)], if its block set can be partitioned into parallel classes. An MTS(v) [or DTS(v)] is said to be almost resolvable, denoted by ARMTS(v) [or ARDTS(v)], if its block set can be partitioned into almost parallel classes. The large set of RMTS(v) [or RDTS(v) or ARMTS(v) or ARDTS(v)] is denoted by LRMTS(v) [or LRDTS(v) or LARMTS(v) or LARDTS(v)]. In this article we do some preliminary study for their existence, and give several recursive theorems using other combinatorial structures. © 1996 John Wiley & Sons, Inc.     

The generalized Estermann ternary problem for noninteger powers with almost equal summands

An asymptotic formula in the generalized Estermann ternary problem for noninteger powers with almost equal summands dealing with the representation of a sufficiently large natural number as the sum of two primes and the integer part of a noninteger power of a natural number is proved.     

An analog of the generalized Hardy-Littlewood problem with almost prime numbers

We obtain an asymptotic formula for the number of solutions of the equationm–(u, v)=a, wherem is an almost prime number,(u, v) is a given binary quadratic form, anda is an arbitrary fixed integer.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 81–94, January, 1996.     

The Cauchy problem for a generalized spatial Cauchy-Riemann system

We consider the problem on the analytic continuation of a solution to a generalized Cauchy-Riemann system from its values on a part of the domain boundary.     

Almost resolvable k‐cycle systems with

In this paper, we show that if and , then there exists an almost resolvable k‐cycle system of order for all except possibly for and . Thus we give a partial solution to an open problem posed by Lindner, Meszka, and Rosa (J. Combin. Des., vol. 17, pp. 404–410, 2009).     

The Yamabe problem for almost Hermitian manifolds

The conformal class of a Hermitian metric g on a compact almost complex manifold (M^2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature s^J. We ask if the conformal class of g carries a metric with constant s^J, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the affirmative. In fact, we show that (2m−1)s^J−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well. We discuss some applications.     

The generalized dock problem

We show existence and uniqueness for a linearized water wave problem in a two dimensional domain G with corner, formed by two semi-axes Γ₁ and Γ₂ which intersect under an angle α?∈?(0,?π]. The existence and uniqueness of the solution is proved by considering an auxiliary mixed problem with Dirichlet and Neumann boundary conditions. The latter guarantees the existence of the Dirichlet to Neumann map. The water wave boundary value problem is then shown to be equivalent to an equation like v_tt ?+?gΛv?=?P_t with initial conditions, where t stands for time, g is the gravitational constant, P means pressure and Λ is the Dirichlet to Neumann map. We then prove that Λ is a positive self-adjoint operator.     

New generalized almost perfect nonlinear functions

APN (almost perfect non-linear) functions over finite fields of even characteristic are widely studied due to their applications to the design of symmetric ciphers resistant to differential attacks. This notion was recently generalized to GAPN (generalized APN) functions by Kuroda and Tsujie to odd characteristic p. They presented some constructions of GAPN functions, and other constructions were given by Zha et al. We present new constructions of GAPN functions both in the case of monomial and multinomial functions. Our monomial GAPN functions can be viewed as a further generalization of the Gold APN functions. We show that a certain technique used by Hou to construct permutations over finite fields also yields monomial GAPN functions. We also present several new constructions of GAPN functions which are sums of monomial GAPN functions, as well as new GAPN functions of degree p which can be written as the product of two powers of linearized polynomials. For this latter construction we describe some interesting differences between even and odd characteristic and also obtain a classification in certain cases.     

The Estermann cubic problem with almost equal summands

We prove an asymptotic formula for the number of representations of a sufficiently large natural number N as the sum of two primes p ₁ and p ₂ and the cube of a natural numbermsatisfying the conditions |p _i ? N/3| ≤ H, |m ³ ? N/3| ≤ H, H ≥ N ^5/6 ? ¹⁰.     

The elastic generalized assignment problem

The generalized assignment problem (GAP) has been studied by numerous researchers over the past 30 years or so. Simply stated, one must find a minimum-cost assignment of tasks to agents such that each task is assigned to exactly one agent and such that each agent's resource capacity is honoured. The problem is known to be NP-hard. In this paper, we study the elastic generalized assignment problem (EGAP). The elastic version of GAP allows agent resource capacity to be violated at additional cost. Another version allows undertime costs to be assessed as well if an agent's resource capacity is not used to its full extent. The EGAP is also NP-hard. We describe a special-purpose branch-and-bound algorithm that utilizes linear programming cuts, feasible solution generators, Lagrangean relaxation and subgradient optimization. We present computational results on a large collection of randomly generated ‘hard’ problems with up to 4000 binary variables.     

The generalized order complementarity problem

Given an ordered Banach Space (E,K) andm functionsf ₁,f ₂,...,f _m:EE, the generalized order complementarity problem associated with {f _i} andK is to findx ₀K such thatf _i(x ₀)K,i=1,...,m, and (x ₀,f ₁(x ₀),...,f _m(x ₀))=0. The problem is shown to be equivalent to several fixed-point problems and equivalent to the order complementarity problem studied by Borwein and Dempster and by Isac. Existence and uniqueness of solutions and least-element theory are shown in the spacesC(, ) andL _p(, ). For general locally convex spaces, least-element theory is derived, existence is proved, and an algorithm for computing a solution is presented. Applications to the mixed lubrication theory of fluid mechanics are described.