Quantization of formal classical dynamical r-matrices: the reductive case

In this paper we prove the existence of a formal dynamical twist quantization for any triangular and non-modified formal classical dynamical r-matrix in the reductive case. The dynamical twist is constructed as the image of the dynamical r-matrix by a L_∞-quasi-isomorphism. This quasi-isomorphism also allows us to classify formal dynamical twist quantizations up to gauge equivalence.     

Generalized M-Matrices

Let ρ?Rⁿ be a proper cone. From the theory of M-matrices (see e.g. [1]) it is known that if there exist α > 0 and a matrix B: ρ→ρ such that A = B?αI, then the following conditions are equivalent: (i) ? A is ρ-monotone,(ii) A is ρ-seminegative, (iii) Re[Spectrum(A)]<0. In this paper we show that while the condition (e) e^tAρ?ρ ?t≥0 is more general than the structural assumption A = B?αI, conditions (i)-(iii) are nevertheless all equivalent to (iv) {x∈ρ: Ax∈ρ}={0} when (e) holds.     

Quadratic programming with M-Matrices

In this paper, we study the problem of quadratic programming with M-matrices. We describe (1) an effective algorithm for the case where the variables are subject to a lower-bound constraint, and (2) an analogous algorithm for the case where the variables are subject to lower-and-upper-bound constraints. We demonstrate the special monotone behavior of the iterate and gradient vectors. The result on the gradient vector is new. It leads us to consider a simple updating procedure which preserves the monotonicity of both vectors. The procedures uses the fact that an M-matrix has a nonnegative inverse. Two new algorithms are then constructed by incorporating this updating procedure into the two given algorithms. We give numerical examples which show that the new methods can be more efficient than the original ones.     

Longest cycles in r-regular r-connected graphs

Let r ≥ 3 be an integer, and ε > 0 a real number. It is shown that there is an integer N(r, ε) such that for all n ≥ N (if r is even) or for all even n ≥ N (if r is odd), there is an r-connected regular graph of valency r on exactly n vertices whose longest cycles have fewer than εn vertices. That is, the number ε > 0, no matter how small, is a “shortness coefficient” for the family of r-valent regular r-connected graphs.     

The r-major index

The r-major index is a new permutation statistic that is suggested by the work of Carlitz and Gansner on Foulkes' skew-hook rule for computing the r-Eulerian numbers. The new statistic (1) generalizes both the major index and the inversion number of a permutation and (2) leads to a q-analog of the r-Eulerian numbers.     

Basic r-symmetric tropical polynomials

We consider tropical polynomials in nr variables, divided into n blocks of r variables, and especially r-symmetric tropical polynomials, which are invariant under the action of the symmetric group $S_n$ on the blocks. We define a set of basic r-symmetric tropical polynomials and show that the basic 2-symmetric tropical polynomials give coordinates on ${\mathbb{R}}^{2n}/S_n$ more efficiently than known polynomials. Moreover, we present special cases for $r\ge 3$ where the basic polynomials separate orbits.     

On r-potent linear mappings

Let L(E) be the set of all linear mappings of a vector space E. Let Z₊ be the set of all positive integers. A nonzero element ? in L(E) is called an r-potent if $\text{?}{}^{\mathrm{r}}\text{=?}\text{and}\text{?}{}^{\mathrm{i}}\text{≠?}\text{for}\text{1<i<r (i,r∈Z}{}_{\mathrm{+}}\text{)}$. We prove that $\text{S(E)= {?∈L(E): ?}\text{is singular}\text{}}$ is a semigroup generated by the set of all r-potents in S(E), where r is a fixed positive integer with 2?r?n=dim(E).     

r-Strong edge colorings of graphs

Let G be a graph and for any natural number r, denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, . Here we generalize this result and show that . Moreover, we show that if for any two vertices u and v with maximum degree d(u,v)?3, then . Also for any tree T with Δ(T)?3 we prove that . Finally, it is shown that for any graph G with no isolated edges, .     

Existence of r-self-orthogonal Latin squares

Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the transpose of the first one, we say that the first square is r-self-orthogonal, denoted by r-SOLS(v). It has been proved that for any integer v?28, there exists an r-SOLS(v) if and only if v?r?v² and r∉{v+1,v²-1}. In this paper, we give an almost complete solution for the existence of r-self-orthogonal Latin squares.     

r-Qsym is free over Sym

Our main result is a proof of the Florent Hivert conjecture [F. Hivert, Local action of the symmetric group and generalizations of quasi-symmetric functions, in preparation] that the algebras of r-Quasi-Symmetric polynomials in x₁,x₂,…,xn are free modules over the ring of Symmetric polynomials. The proof rests on a theorem that reduces a wide variety of freeness results to the establishment of a single dimension bound. We are thus able to derive the Etingof-Ginzburg [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, Mosc. Math. J. 2 (2002) 555-566] Theorem on m-Quasi-Invariants and our r-Quasi-Symmetric result as special cases of a single general principle. Another byproduct of the present treatment is a remarkably simple new proof of the freeness theorem for 1-Quasi-Symmetric polynomials given in [A.M. Garsia, N. Wallach, Qsym over Sym is free, J. Combin. Theory Ser. A 104 (2) (2003) 217-263].     

Wandering r-tuples for unitary systems

The properties of the set Wr(U) of all complete wandering r-tuples for a system U of unitary operators acting on a Hilbert space are investigated by parameterizing Wr(U) in terms of a fixed wandering r-tuple Ψ and the set of all unitary operators which locally commute with U at Ψ. The special case of greatest interest is the system 〈D,T〉 of dilation (by 2) and translation (by 1) unitary operators acting on L²(R), for which the complete wandering r-tuples are precisely the orthogonal multiwavelets with multiplicity r. We also give some examples for its application.     

r-tuple colorings of uniquely colorable graphs

An r-tuple coloring of a graph is one in which r colors are assigned to each point of the graph so that the sets of colors assigned to adjacent points are always disjoint. We investigate the question of whether a uniquely n-colorable graph can receive an r-tuple coloring with fewer than nr colors. We show that this cannot happen for n=3 and r=2 and that for a given n and r to establish the conjecture that no uniquely n-colorable graph can receive an r-tuple coloring from fewer than nr colors it suffices to prove it for on a finite set of uniquely n-colorable graphs.     

The G/M/r machine interference model

This paper solves the machine interference problem in which N automatic machines are maintained by a team of r operatives. Repair times are assumed to follow a negative exponential distribution, and running times for each of the machines are assumed to be independently and identically distributed. It is shown that the solution to this G/M/r model is identical in most important respects to that for the M/M/r model.     

Generalized Hadamard's inequality and r-convex functions in Carnot groups

In this paper, we establish a generalization of Hadamard's inequality to r-convex functions on Carnot groups.     

Sets of orthogonal hypercubes of class r

A (d,n,r,t)-hypercube is an n×n×?×n (d-times) array on nr symbols such that when fixing t coordinates of the hypercube (and running across the remaining d−t coordinates) each symbol is repeated nd−r−t times. We introduce a new parameter, r, representing the class of the hypercube. When r=1, this provides the usual definition of a hypercube and when d=2 and r=t=1 these hypercubes are Latin squares. If d?2r, then the notion of orthogonality is also inherited from the usual definition of hypercubes. This work deals with constructions of class r hypercubes and presents bounds on the number of mutually orthogonal class r hypercubes. We also give constructions of sets of mutually orthogonal hypercubes when n is a prime power.     

Generalized quadratic mappings of r-type in several variables

Let X,Y be vector spaces. It is shown that if an even mapping satisfies f(0)=0, and

(∗)