f-Class two graphs whose f-cores have maximum degree two

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v∈V(G) at most f(v) times.The f-core of G is the subgraph of G induced by the vertices v of degree d(v)=f(v) maxv∈V(G){ d(v)/f(v) }.In this paper,we find some necessary conditions for a simple graph,whose f-core has maximum degree two,to be of class 2 for f-colorings.     

Mixed-integer sets from two rows of two adjacent simplex bases

In Andersen et al. (Lecture Notes in Computer Science, vol. 4513, Springer, Berlin, pp. 1–15, 2007) we studied a mixed-integer set arising from two rows of a simplex tableau. We showed that facets of such a set can be obtained from lattice point free triangles and quadrilaterals associated with either three or four variables. In this paper we generalize our findings and show that, when upper bounds on the non-basic variables are also considered, further classes of facets arise that cannot be obtained from triangles and quadrilaterals. Specifically, when exactly one upper bound on a non-basic variable is introduced, stronger inequalities that can be derived from pentagons involving up to six variables also appear.     

Slim exceptional set for sums of two squares,two cubes,and two biquadrates of primes

We prove that, with at most O\left(N^{\frac{\mathrm{17}}{\mathrm{192}}+\epsilon }\right) exceptions, all even positive integers up to Nare expressible in the form p_{\mathrm{1}}^{\mathrm{2}}+p_{\mathrm{2}}^{\mathrm{2}}+p_{\mathrm{3}}^{\mathrm{3}}+p_{\mathrm{4}}^{\mathrm{3}}+p_{\mathrm{5}}^{\mathrm{4}}+p_{\mathrm{6}}^{\mathrm{4}},where p_{\mathrm{1}},\mathrm{}{p}_{\mathrm{2}},.\mathrm{}.\mathrm{}.\mathrm{},\mathrm{}{p}_{\mathrm{6}} are prime numbers. This gives large improvement of a recent result O\left(N^{\frac{\mathrm{13}}{\mathrm{16}}+\epsilon }\right) due to M. Zhang and J. J. Li.     

On the product of two skew-Hamiltonian or two skew-symmetric matrices

We show that the product C of two skew-Hamiltonian matrices obeys the Stenzel conditions. If at least one of the factors is nonsingular, then the Stenzel conditions amount to the requirement that every elementary divisor corresponding to a nonzero eigenvalue of C occurs an even number of times. The same properties are valid for the product of two skew-pseudosymmetric matrices. We observe that the method proposed by Van Loan for computing the eigenvalues of real Hamiltonian and skew-Hamiltonian matrices can be extended to complex skew-Hamiltonian matrices. Finally, we show that the computation of the eigenvalues of a product of two skew-symmetric matrices reduces to the computation of the eigenvalues of a similar skew-Hamiltonian matrix. Bibliography: 8 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 45–51.     

Between two moments

In this short note, we draw attention to a relation between two Horn polytopes which is proved in [3] as a result, on the one hand, of a deep combinatorial result in [5] and, on the other hand, of a simple computation involving complex structures. This suggests an inequality between Littlewood-Richardson coefficients, which we prove using the symmetric characterization of these coefficients given in [1].     

The two Pappus-tori

Two polyhedral embeddings of the regular maps {3, 6}_3,0 and {6, 3}_3,0 in E ³ are given. They are the smallest combinatorially regular tori of type {3, 6} and {6, 3} embedded in E ³, and their automorphism group is that of the famous Pappus-configuration.     

On two inequalities similar to Opial's inequality in two independent variables

In the present paper we establish two new integral inequalities similar to Opial's inequality in two independent variables. The inequalities established in this paper are similar to the analogues of Calvert's generalizations of Opial's inequality, in two independent variables and contains in the special case the analogue of Opial's inequality given by G. S. Yang in two independent variables.     

A retrial system with two input streams and two orbit queues

Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type- $i$ job ( $i=1,2$ ) finds the server busy, it is blocked and routed to a separate type- $i$ retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a Riemann–Hilbert boundary value problem. Various performance measures are calculated and numerical results are presented. In particular, numerical results demonstrate that the proposed multiple access system with two types of jobs and constant retrial rates provides incentives for the users to respect their contracts.     

On the sum of two primes and k powers of two

Let X be a large integer. We prove that, for any fixed positiveinteger k, a suitable asymptotic formula for the number of representationsof an even integer N [1,X] as the sum of two primes and k powersof 2 holds with at most exceptions.     

Cohen macaulay rings of global dimension two and krull dimension two

Abstract

We construct an example of an IA-automorphism of the free metabelian group of rank n ≥ 3 without nontrivial fixed points. That gives a negative answer to the question raised by Shpilrain in [Shpilrain, V. (1998). Fixed points of endomorphisms of a free metabelian group. Math. Proc. Cambridge Philos. Soc. 123(1): 75–83. MR 98k:20056]. By a result of Bachmuth [Bachmuth, S. (1965). Automorphisms of free metabelian groups. Trans. Am. Math. Soc. 118:93–1104. MR 31 #4831], such an automorphism does not exist if the rank is equal to 2.     

On the existence of graphs of diameter two and defect two

In the context of the degree/diameter problem, the ‘defect’ of a graph represents the difference between the corresponding Moore bound and its order. Thus, a graph with maximum degree d and diameter two has defect two if its order is n=d²−1. Only four extremal graphs of this type, referred to as (d,2,2)-graphs, are known at present: two of degree d=3 and one of degree d=4 and 5, respectively. In this paper we prove, by using algebraic and spectral techniques, that for all values of the degree d within a certain range, (d,2,2)-graphs do not exist.The enumeration of (d,2,2)-graphs is equivalent to the search of binary symmetric matrices A fulfilling that AJ_n=dJ_n and A²+A+(1−d)I_n=J_n+B, where J_n denotes the all-one matrix and B is the adjacency matrix of a union of graph cycles. In order to get the factorization of the characteristic polynomial of A in Q[x], we consider the polynomials F_i,d(x)=f_i(x²+x+1−d), where f_i(x) denotes the minimal polynomial of the Gauss period , being ζ_i a primitive ith root of unity. We formulate a conjecture on the irreducibility of F_i,d(x) in Q[x] and we show that its proof would imply the nonexistence of (d,2,2)-graphs for any degree d>5.     

Flow of a fluid of grade two between two eccentric rotating spheres

The isothermal, stationary and isochoric flow of a fluid of grade two between a pair of rotating eccentric spheres is investigated. The equations of motion of first and second order are formulated and solved for the first order only. However, the equation of second order indicates the presence of secondary flow. The stress distributions are computed and used to determine the resultant forces and torques acting on the stationary outer sphere. An important result for rheometry is that the resultant torques can be used to determine the coefficient of viscosity, while the resultant force in the direction of the axis of symmetry may be employed to determine the second normal stress difference.     

Rounding errors in numerical solutions of two linear equations in two unknowns

New condition numbers and stability constants for the numerical behaviour of Cramer's rule and Gaussian elimination for solving two linear equations in two unknowns under data perturbations and rounding errors of floating-point arithmetic are established. By these means fundamental error estimates and stability theorems are proved. The error estimates are illustrated by a series of numerical examples.     

Groups with two normalizers

To B. Huppert on his sixtieth birthday     

Discriminants in characteristic two

The Kronecker-Weierstrass theory of pencils is extended to give a necessary and sufficient condition that two 2×m×n tensors are equivalent. The connection between equivalence class representatives and the triple transitivity of PGL(2,F) is discussed. One consequence of the discussion is that the number of inequivalent 2×3×n tensors is finite. An efficient algorithm is given for testing the condition which ultimately depends on a fast pattern matching algorithm.     

Competition between two species

Although the classical work on competition between species goes back to Lotka [3] and Volterra [7], very few results have been obtained for the stochastic model. We obtain state probabilities, and hence moments, for the competition process in which growth of new individuals does not occur.     

Cutting two graphs simultaneously

Consider two graphs, and , on the same vertex set V, with and having edges for . We give a simple algorithm that partitions V into sets A and B such that and . We also show, using a probabilistic method, that if and belong to certain classes of graphs, (for instance, if and both have a density of at least 2/, or if and are both regular of degree at most with n sufficiently large) then we can find a partition of V into sets A and B such that for . © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 19–32, 2008