Conjectured statistics for the q,t-Catalan numbers

We introduce the distribution function F_n(q,t) of a pair of statistics on Catalan words and conjecture F_n(q,t) equals Garsia and Haiman's q,t-Catalan sequence C_n(q,t), which they defined as a sum of rational functions. We show that F_n,s(q,t), defined as the sum of these statistics restricted to Catalan words ending in s ones, satisfies a recurrence relation. As a corollary we are able to verify that F_n(q,t)=C_n(q,t) when t=1/q. We also show the partial symmetry relation F_n(q,1)=F_n(1,q). By modifying a proof of Haiman of a q-Lagrange inversion formula based on results of Garsia and Gessel, we obtain a q-analogue of the general Lagrange inversion formula which involves Catalan words grouped according to the number of ones at the end of the word.     

Algebraic approach to q,t-characters

Frenkel and Reshetikhin (in: Recent Developments in Quantum Affine Algebras and Related Topics, Contemporary Mathematics, Vol. 248, 1999, pp. 163-205) introduced q-characters to study finite dimensional representations of the quantum affine algebra . In the simply laced case Nakajima (in: Physics and Combinatorics, Proceedings of the Nagoya 2000 International Workshop, World Scientific, Singapore, 2001, pp. 181-212; Preprint arXiv:math.QA/0105173) defined deformations of q-characters called q,t-characters. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case. In this article we propose an algebraic general (non-necessarily simply laced) new approach to q,t-characters motivated by the deformed screening operators (Internat. Math. Res. Not. 2003 (8) (2003) 451). The t-deformations are naturally deduced from the structure of : the parameter t is analog to the central charge . The q,t-characters lead to the construction of a quantization of the Grothendieck ring and to general analogues of Kazhdan-Lusztig polynomials in the same spirit as Nakajima did for the simply laced case.     

On caps with weighted points in PG(t, q)

In this note the study of caps with weighted points inPG(t,q), t?2, is investigated. Some relations between parameters of a(k>,nf)-cap and its characters are found. Monoidal caps are also considered and some examples are given. Finally, the(k,n;f)-caps of type (0,n>) are characterized.     

On the solutions to x′(t) = A(t) x (t) over all A(t), where P ⩽ A(t) ⩽ Q

Let P_ij and q_ij be positive numbers for i ≠ j, i, j = 1, …, n, and consider the set of matrix differential equations x′(t) = A(t) x(t) over all A(t), where a_ij(t) is piecewise continuous, a_ij(t) = ?∑_{i ≠ j}a_ij(t), and p_ij ? a_ij(t) ? q_ij all t. A solution x is also to satisfy ∑_{i = 1}ⁿx_i(0) = 1. Let C_t denote the set of all solutions, evaluated at t to equations described above. It is shown that $\text{C}{}_{\mathrm{t}}$, the topological closure of C_t, is a compact convex set for each t. Further, the set valued function $\text{C}{}_{\mathrm{t}}$, of t is continuous and $\text{limit}{}_{\mathrm{t\; \to \; \infty }}\text{C}\text{?}{}_{\mathrm{t}}\text{= ∩}\text{C}\text{?}{}_{\mathrm{t}}$.     

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.     

A proof of the Hoggatt-Bergum conjecture

It is proved that if and are positive integers such that the product of any two distinct elements of the set

increased by is a perfect square, then has to be . This is a generalization of the theorem of Baker and Davenport for .

     

A proof of the integral identity conjecture,II

In this note, using Cluckers–Loeser's theory of motivic integration, we prove the integral identity conjecture with framework a localized Grothendieck ring of varieties over an arbitrary base field of characteristic zero.     

A proof of MIKOTA's conjecture

Coming from MIKOTA's conjecture [1] a proof is given by examinating the properties of the occuring matrices M , K and M ^–1 K and deducing the natural frequencies of the chain structured mass-spring vibration system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A proof of Nogura's conjecture

Answering a question of T. Nogura (1985), we show using the Open Coloring Axiom that the weak diagonal sequence property is preserved by taking products whenever the products themselves are Fréchet. As an application we show, using the same assumption, that the product of two Fréchet groups is Fréchet provided it is sequential. Recall that the product of two Fréchet groups may not be sequential.

     

A proof of Coleman's conjecture

Almost thirty years ago Coleman made a conjecture that for any convex lattice polygon with v vertices, g (g?1) interior lattice points and b boundary lattice points we have b?2g-v+10. In this note we give a proof of the conjecture. We also aim to describe all convex lattice polygons for which the bound b=2g-v+10 is attained.     

A new representation of Student's t as a function of independent t's,with a generalization to the matrix t

A new synthetic representation of Student's t statistic is given which exhibits the random variable as a nonlinear function of two independent t variables. A generalization is given to the matricvariate t with an interpretation in terms of “natural conjugate” inference for multivariate normal sampling.