Enumeration of (p,q)-parking functions

Parking functions are central in many aspects of combinatorics. We define in this communication a generalization of parking functions which we call (p₁,…,p_k)-parking functions. We give a characterization of them in terms of parking functions and we show that they can be interpreted as recurrent configurations in the sandpile model for some graphs. We also establish a correspondence with a Lukasiewicz language, which enables to enumerate (p₁,…,p_k)-parking functions as well as increasing ones.     

An integral transform in L2-cohomology for the ladder representations of U(p, q)

Starting from the realization of the Fock space as L²-cohomology of $\text{C}$^{p + q}, $\text{H}$^0,p($\text{C}$^{p + q}) = ⊕_m?$\text{Z}$$\text{H}$_m^0,p($\text{C}$^{p + q}), an integral transform is constructed which is a direct-image mapping from $\text{H}$_m^{0,p($\text{C}$p + q) into the space of holomorphic sections of some vector bundle E_m over M ≈ U(p, q)/(U(q) × U(p)), m ? 0}. The transform intertwines the natural actions of U(p, q) and is injective if m ? 0, so it provides a geometric realization of the ladder representations of U(p, q). The sections in the image of the transform satisfy certain linear differential equations, which are explicitly described. For example, Maxwell's equations are of this form if p = q = 2 and m = 2. Thus, this transform is analogous to the Penrose correspondence.     

Linear operators preserving the (p,q)-numerical range

Let $\text{C}{}_{\mathrm{n\times n}}$ and $\text{H}{}_{\mathrm{n}}$ denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p?q?n. For $\text{A?}\text{C}{}_{\mathrm{n\times n}}$, the (p,q)-numerical range of A is the set

$\text{W}{}_{\mathrm{p,q}}\text{(A)={trC}{}_{\mathrm{p}}\text{(J}{}_{\mathrm{q}}\text{UAU}{}^1\text{):U}\text{unitary}\text{}}$

, where C_p(X) is the pth compound matrix of X, and Jq is the matrix I_q?O_n-q. Let $\text{L}$ denote $\text{H}$_n or $\text{C}{}_{\mathrm{n\times n}}$. The problem of determining all linear operators T: $\text{L}$→$\text{L}$ such that

$\text{W}{}_{\mathrm{p,q}}\text{(T(A))=W}{}_{\mathrm{p,q}}\text{(A)}\text{for all}\text{A?}\text{L}$

is treated in this paper.     

Sharp regularity for functionals with (p,q) growth

We prove regularity theorems for minimizers of integral functionals of the Calculus of Variations

     

The proximate type and λ-proximate type of an entire Dirichlet series with index-pair (p, q)

The notion of proximate type for an entire Dirichlet series with index-pair (p, q) is introduced and its existence is proved. In order to study the lower proximate type, the idea has been extended to the case of entire functions of irregular (p, q)-growth. The proximate type for a class of entire Dirichlet series under certain conditions is constructed.     

On perfect codes in the hamming schemes H(n, q) with q arbitrary

For each e ? 3, there are at most finitely many nontrivial perfect e-codes in the Hamming schemes H(n, q) where n and q are arbitrary.     

Characterization of H(n, q) by the parameters

Distance-regular graphs which have the same parameters as the Hamming scheme H(n, q) are classified. If q ≠ 4, H(n, q) is the only such graph. If q = 4, there are precisely $\text{[}\text{n}\text{2}\text{]}$ (isomorphism classes of) such graphs other than H(n, q).     

Generalized composition operators from F(p, q, s) spaces to Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we investigate the boundedness and compactness of the generalized composition operator

     

On properties of discrete (r, q) and (s, T) inventory systems

We consider single-item (r, q) and (s, T) inventory systems with integer-valued demand processes. While most of the inventory literature studies continuous approximations of these models and establishes joint convexity properties of the policy parameters in the continuous space, we show that these properties no longer hold in the discrete space, in the sense of linear interpolation extension and L^?-convexity. This nonconvexity can lead to failure of optimization techniques based on local optimality to obtain the optimal inventory policies. It can also make certain comparative properties established previously using continuous variables invalid. We revise these properties in the discrete space.     

Orthogonal resolutions of lines in AG(n,q)

A resolution of the lines of AG(n,q) is a partition of the lines classes (called resolution classes) such that every point of the geometry is on exactly one line of each resolution class. Two resolutions R,R' of AG(n,q) are orthogonal if any resolution class from R has at most one line in common with any class from R'. In this paper, we construct orthogonal resolutions on AG(n,q) for all n=2ⁱ⁺¹, i=1,2,…, and all q>2 a prime power. The method involves constructing AG(n,q) from a finite projective plane of order q^n-1 and using the structure of the plane to display the orthogonal resolutions.     

L-Poisson integral representations of solutions of the Hua system on the bounded symmetric domain SU(n,n)/S(U(n)×U(n))

Let positive definite} be the matrix ball of rank n and let H_D be the associated Hua operator. For a complex number λ, such that Reiλ>n−1 we give a necessary and sufficient condition on solutions F of the following Hua system of differential equations on D:

     

Packing the maximum number of m × n tiles in a large p × q rectangle

For relatively prime m and n we determine precisely the wasted area when a large p × q rectangle is packed with m × n tiles in the most efficient manner. The case m = 1 is considered first, and we derive a formula for the wasted area depending only on the residue classes of p and q (mod n). This result has also been obtained by various other authors. Then by regarding an m × n tile as a union of 1 × n or 1 × m tiles a lower bound for the wasted area is obtained. Finally, by a series of explicit constructions, we show that for sufficiently large p, q, this lower bound is actually the correct value.     

Composition operators from F(p, q, s) spaces to the nth weighted-type spaces on the unit disc

In this note we characterize the boundedness and compactness of the composition operator from the general function space F(p, q, s) to the nth weighted-type space on the unit disk, where the nth weighted-type space has been recently introduced by Stevo Stevi?.     

Eigenvalues and the One-Dimensional p-Laplacian

We consider the boundary value problem (?_p(u′))′ + λF(t, u) = 0, with p > 1, t ∈ (0, 1), u(0) = u(1) = 0, and with λ > 0. The value of λ is chosen so that the boundary value problem has a positive solution. In addition, we derive an explicit interval for λ such that, for any λ in this interval, the existence of a positive solution to the boundary value problem is guaranteed. In addition, the existence of two positive solutions for λ in an appropriate interval is also discussed.