Oscillation theorems for second-order functional differential equations

This paper presents some comparison theorems on the oscillatory behavior of solutions of second-order functional differential equations. Here we state one of the main results in a simplified form: Let q, τ₁, τ₂ be nonnegative continuous functions on (0, ∞) such that τ₁ ? τ₂ is a bounded function on [1, ∞) and t ? τ₁(t) → ∞ if t → ∞. Then $\text{y}\text{?}\text{(t) + q(t) y(t ? τ}{}_1\text{(t)) = 0}$ is oscillatory if and only if $\text{y}\text{?}\text{(t) + q(t) y(t ? τ}{}_2\text{(t)) = 0}$ is oscillatory.     

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.     

A proof of the q,t-Catalan positivity conjecture

We present here a proof that a certain rational function C_n(q,t) which has come to be known as the “q,t-Catalan” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. The precise form of the conjecture is given in Garsia and Haiman (J. Algebraic Combin. 5(3) (1996) 191), where it is further conjectured that C_n(q,t) is the Hilbert series of the diagonal harmonic alternants in the variables (x₁,x₂,…,x_n;y₁,y₂,…,y_n). Since C_n(q,t) evaluates to the Catalan number at t=q=1, it has also been an open problem to find a pair of statistics a(π),b(π) on Dyck paths π in the n×n square yielding C_n(q,t)=∑_πt^a(π)q^b(π). Our proof is based on a recursion for C_n(q,t) suggested by a pair of statistics a(π),b(π) recently proposed by Haglund. Thus, one of the byproducts of our developments is a proof of the validity of Haglund's conjecture. It should also be noted that our arguments rely and expand on the plethystic machinery developed in Bergeron et al. (Methods and Applications of Analysis, Vol. VII(3), 1999, p. 363).     

Asymptotic analysis of non-self-adjoint Hill operators

We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L _t (q) with a potential q ∈ L ₁[0,1] and t-periodic boundary conditions, t ∈ (?π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L ₂(?∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.     

Some big sets of mutually disjoint subgeometries of PG(d, q t )

In PG(d, q ^t ) we construct a set ? of mutually disjoint subgeometries isomorphic to PG(d, q) almost partitioning the point set of PG(d, q ^t ) such that there is a group of collineations of PG(d, q ^t ) operating simultaneously as a Singer cycle on all elements of ?. In PG(t?1,q ^t ) we construct big subsets ? of ? whose elements are far away from each other in the following sense:

u

? If P ₁, P ₂ ∈ ? _k , then no point of P ₁ lies on ak-dimensional subspace of P ₂.

For example, we get a set ofq - 1 subplanes of orderq of PG(2,q ³) such that no point of one subplane lies on a line of another subplane, and such that no three points of three different subplanes are collinear.     

Extremal sizes of subspace partitions

A subspace partition Π of V?= V(n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. The size of Π is the number of its subspaces. Let σ _q (n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρ _q (n, t) denote the maximum size of a subspace partition of V in which the smallest subspace has dimension t. In this article, we determine the values of σ _q (n, t) and ρ _q (n, t) for all positive integers n and t. Furthermore, we prove that if n ≥?2t, then the minimum size of a maximal partial t-spread in V(n +?t ?1, q) is σ _q (n, t).     

On the Rogers--Ramanujan Periodic Continued Fraction

In the paper, the convergence properties of the Rogers--Ramanujan continued fraction $$1 + \frac{qz}{1 + \tfrac{q^2 z}{1 + \cdots}}$$ are studied for q = exp (2 π i τ), where τ is a rational number. It is shown that the function H _q to which the fraction converges is a counterexample to the Stahl conjecture (the hyperelliptic version of the well-known Baker--Gammel--Wills conjecture). It is also shown that, for any rational τ, the number of spurious poles of the diagonal Padé approximants of the hyperelliptic function H _q does not exceed one half of its genus.     

On the cycle structure of certain classes of nonlinear shift registers

When m = qt, g(x_t+1, x_2t+1,…, x_(q?1)t+1) is a linear combination of only odd (or only even) elementary symmetric functions, then every cycle of the nonlinear shift register with feedback function f(x₁, x₂,…, x_m) = x₁ + g(x_t+1, x_2t+1,…, x_(q?1)t+1) has a minimal period dividing m(q+1). It is also shown that when g is derived from a cyclic code with minimum distance ?3, every cycle of this shift register has a minimal period dividing m(q + 1).     

Dirichlet operators with convex potentials on the half-line

We consider the Dirichlet operator H _t =?d ²/dx ²+q(x) on L ²([t,??)), where q is a convex potential with q(x)???? as x????. We show that the eigenvalue gap ??(t) of H _t is monotone increasing as t increases from ??? to ??. We also show that ??(t) is strictly increasing if q is not linear at infinity. An asymptotic estimate of ??(t) for quadratic potentials is obtained.     

The minimum size of a finite subspace partition

A subspace partition of P=PG(n,q) is a collection of subspaces of P whose pairwise intersection is empty. Let σ_q(n,t) denote the minimum size (i.e., minimum number of subspaces) in a subspace partition of P in which the largest subspace has dimension t. In this paper, we determine the value of σ_q(n,t) for . Moreover, we use the value of σ_q(2t+2,t) to find the minimum size of a maximal partial t-spread in PG(3t+2,q).     

Lévy functional and jump process martingales

An arbitrary jump process is considered without any assumption about the jump times and allowing the jump times to have accumulation points of arbitrary order. Certain basic martingales q(t, A) and the related Lévy system describing the jumps are introduced, and a notion of quadratic integration with respect to the predictable quadratic variation <q, q> of the basic martingales is defined. If M_t is a (locally) square integrable martingale with respect to the family of σ-fields generated by the jump process it is shown that M_t can be represented as a stochastic integral with respect to the basic martingales and with an integrand (locally) square integrable with respect to <q, q>.     

On a comparison theorem

This paper gives a generalization of the Sturm comparison theorem for differential equations (p): y″ = p(t)y, (q): y″ = q(t)y under the assumption that the function p ? q changes its sign exactly once on [a, b] or ∝_t^bp ? q, ∝_a^tp ? q maintain the sign on [a, b]. The results are used for investigating the distributions of zeros of solutions and the derivative of solutions of (p), (q).     

The complexity of the fixed point property

It is NP-complete to determine whether a given ordered set has a fixed point free order-preserving self-map. On the way to this result, we establish the NP-completeness of a related problem: Given ordered sets P and Q with t-tuples (p ₁, ... , p _t) and (q ₁, ... , q _t) from P and Q respectively, is there an order-preserving map f: P→Q satisfying f(p _i)≥q _i for each i=1, ... , t?     

A second-order nonlinear difference equation: Oscillation and asymptotic behavior

Discrete analogues are investigated for well-known results on oscillation, growth, and asymptotic behavior of solutions of y″ + q(t) y^γ = 0, for q(t) ? 0 and for q(t) ? 0. The analogue of Atkinson's oscillation criterion is shown to be true for Δ²y_{n ? 1} + q_ny_n^γ = 0, but the analogue for Atkinson's nonoscillation criterion is shown to be false.     

Subordination,rank, and determinism of multivariate stationary sequences

Necessary and sufficient conditions for an arbitrary q-variate stationary sequence x_t, t ∈ $\text{Z}$, to be deterministic are presented. A characterization of the rank r(x) of x_t, t ∈ $\text{Z}$, and a method to construct the Wold-Cramér decomposition for x_t, t ∈ $\text{Z}$, are given. Subordination of q-variate bounded orthogonally scattered vector measures is considered.     

Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D₄,E₆,E₇,E₈). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z??Z², where ? is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation H(t,q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H(t,q) for D₄ is the Cherednik algebra of type C^∨C₁, which was studied by Noumi, Sahi, and Stokman, and controls Askey-Wilson polynomials. We prove that H(t,q) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C[G]. We also show that if q is a root of unity, then for generic t the algebra H(t,q) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra eH(t,q)e provides a quantization of such surfaces. We also discuss connections of H(t,q) with preprojective algebras and Painlevé VI.     

A geostatistical approach for dynamic life tables: The effect of mortality on remaining lifetime and annuities

Dynamic life tables arise as an alternative to the standard (static) life table, with the aim of incorporating the evolution of mortality over time. The parametric model introduced by Lee and Carter in 1992 for projected mortality rates in the US is one of the most outstanding and has been used a great deal since then. Different versions of the model have been developed but all of them, together with other parametric models, consider the observed mortality rates as independent observations. This is a difficult hypothesis to justify when looking at the graph of the residuals obtained with any of these methods.Methods of adjustment and prediction based on geostatistical techniques which exploit the dependence structure existing among the residuals are an alternative to classical methods. Dynamic life tables can be considered as two-way tables on a grid equally spaced in either the vertical (age) or horizontal (year) direction, and the data can be decomposed into a deterministic large-scale variation (trend) plus a stochastic small-scale variation (residuals).Our contribution consists of applying geostatistical techniques for estimating the dependence structure of the mortality data and for prediction purposes, also including the influence of the year of birth (cohort). We compare the performance of this new approach with different versions of the Lee-Carter model. Additionally, we obtain bootstrap confidence intervals for predicted q_xt resulting from applying both methodologies, and we study their influence on the predictions of e_65t and a_65t.     

On limit distributions of first crossing points of Gaussian sequences

Let {X_k, k?Z} be a stationary Gaussian sequence with EX₁ – 0, EX²_k = 1 and EX₀X_k = r_k. Define τ_x = inf{k: X_k >– βk} the first crossing point of the Gaussian sequence with the function – βt (β > 0). We consider limit distributions of τ_x as β→0, depending on the correlation function r_k. We generalize the results for crossing points τ_x = inf{k: X_k >β?(k)} with ?(– t)?t^γL(t) for t→∞, where γ > 0 and L(t) varies slowly.     

Embedding of a pseudo-residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let v, t and λ₁, 0 ? i ? t, be nonnegative integers, and K be a subset of nonnegative integers such that every member of K is at most v. A pair (X, U) is called a (λ₀, λ₁,…, λ_t; K, υ)t-design if (1) |X| = υ, (2) every i-subset of X is contained in exactly λ_t blocks, 0 ? i ? t, and (3) for every block A in $\text{U}$, |A| ?K. It is well-known that if K consists of a singleton k, then λ₀,…, λ_{t ? 1} can be determined from υ, t, k and λ_t. Hence, we shall denote a (λ₀,…, λ_t; {k}, υ)t-design by S_λ(t, k, υ), where λ = λ_t. A Möbius plane M is an S₁(3, q + 1, q² + 1), where q is a positive integer. Let A be a fixed block in M. If A is deleted from M together with the points contained in A, then we obtain a residual design M′ with parameters λ₀ = q³ + q ? 1, λ₁ = q² + q, λ₂ = q + 1, λ₃ = 1, K = {q + 1, q, q ? 1}, and υ = q² ? 1. We define a design to be a pseudo-block-residual design of order q (abbreviated by PBRD(q)) if it has these parameters. We consider the reconstruction problem of a Möbius plane from a given PBRD(q). Let B and B′ be two blocks in a residual design M′. If B and B′ are tangent to each other at a point x, and there exists a block C of size q + 1 such that C is tangent to B at x and is secant to B′, then we say B is r-tangent to B′ at x. A PBRD(q) is said to satisfy the r-tangency condition if for every block B of size q, and any two points x and y not in B, there exists at most one block which is r-tangent to B and contains x and y. We show that any PBRD(q)D can be uniquely embedded into a Möbius plane if and only if D satisfies the r-tangency condition.     

On the pointwise matrix product and the mean value theorem

Bounds for the pointwise or Schur product of two matrices are derived with respect to the spectral norm 6·6. For real symmetric and positive semidefinite matrices Q=(q_ij) one of them gives a bound of 6|Q6, |Q|=(|q_ij|). Two of these bounds are applied to obtain a mean value theorem for g: t→f(A(t)) where A(t) is a matrix depending on a parameter t and f is a function on the spectrum of A(t).