F-Polynomials in Quantum Cluster Algebras

F-polynomials and g-vectors were defined by Fomin and Zelevinsky to give a formula which expresses cluster variables in a cluster algebra in terms of the initial cluster data. A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra. In this paper, we define and prove the existence of analogous quantum F-polynomials for quantum cluster algebras. We prove some properties of quantum F-polynomials. In particular, we give a recurrence relation which can be used to compute them. Finally, we compute quantum F-polynomials and g-vectors for a certain class of cluster variables, which includes all cluster variables in type A_n\mbox{A}_{n} quantum cluster algebras.     

Combinatorial invariance of Kazhdan-Lusztig polynomials on intervals starting from the identity

We show that for Bruhat intervals starting from the identity in Coxeter groups the conjecture of Lusztig and Dyer holds, that is, the R-polynomials and the Kazhdan-Lusztig polynomials defined on [e,u] only depend on the isomorphism type of [e,u]. To achieve this we use the purely poset-theoretic notion of special matching. Our approach is essentially a synthesis of the explicit formula for special matchings discovered by Brenti and the general special matching machinery developed by Du Cloux.     

A Unified Approach to Shifts Induced by Right Invertible Operators

In this paper, shifts for a right invertible operator D induced by an analytic function acting in a linear complete metric space are considered. The case when these shifts coincide with the operator - valued function on a set which contains the set of all D-polynomials and the set of all exponentials is studied. It is shown that in this case these shifts are R-shifts and D-shifts (cf. [1], [10]).     

The stability function for multistep collocation methods

Summary C-polynomials for rational approximation to the exponential function was introduced by Nørsett [7] to study stability properties of one-step methods. For one-step collocation methods theC-polynomial has a very simple form. In this paper we studyC-polynomials for multistep collocation methods and obtain results that generalize those in the one-step case, and provide a way to analyze linear stability of such methods.     

Solutions of 2nd-order linear differential equations subject to Dirichlet boundary conditions in a Bernstein polynomial basis

An algorithm for approximating solutions to 2nd-order linear differential equations with polynomial coefficients in B-polynomials (Bernstein polynomial basis) subject to Dirichlet conditions is introduced. The algorithm expands the desired solution in terms of B-polynomials over a closed interval [0, 1] and then makes use of the orthonormal relation of B-polynomials with its dual basis to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of B-polynomials, and the procedure is much simpler compared to orthogonal polynomials for solving differential equations. The current procedure is implemented to solve five linear equations and one first-order nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations.     

Hall–Littlewood polynomials, alcove walks, and fillings of Young diagrams

A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. The inversion statistic, which is the more intricate one, suffices for specializing a closely related formula to one for the type A Hall–Littlewood Q-polynomials (spherical functions on p-adic groups). An apparently unrelated development, at the level of arbitrary finite root systems, led to Schwer’s formula (rephrased and rederived by Ram) for the Hall–Littlewood P-polynomials of arbitrary type. The latter formula is in terms of so-called alcove walks, which originate in the work of Gaussent–Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, we relate the above developments, by deriving a Haglund–Haiman–Loehr type formula for the Hall–Littlewood P-polynomials of type A from Ram’s version of Schwer’s formula via a “compression” procedure.     

A q-Analogue of the Euler Gamma Integral

We discuss q-analogues of the Euler reflection formula and the Euler gamma integral. The central role here is played by the Ramanujan q-extension of the Euler integral representation for the gamma function, which allows deriving the Mellin integral transformations for the q-polynomials of Stieltjes–Wigert, Rogers–Szegö, Laguerre, and Wall, for the alternative q-polynomials of Charlier, and for the little q-polynomials of Jacobi.     

On the combinatorial invariance of Kazhdan-Lusztig polynomials

In this paper, we solve the conjecture about the combinatorial invariance of Kazhdan-Lusztig polynomials for the first open cases, showing that it is true for intervals of length 5 and 6 in the symmetric group. We also obtain explicit formulas for the R-polynomials and for the Kazhdan-Lusztig polynomials associated with any interval of length 5 in any Coxeter group, showing in particular what they look like in the symmetric group.     

A relation of Kauffman's f-polynomials of virtual links

For an oriented virtual link diagram, Kauffman defined the f-polynomial. In this paper we give a relation of the f-polynomials of virtual link diagrams of a virtual skein triple. Then the f-polynomials for some one-component knots are computed by use of resolution trees of one-component knots.     

Bivariate fibonacci like p-polynomials

In this article, we study the bivariate Fibonacci and Lucas p-polynomials (p ? 0 is integer) from which, specifying x, y and p, bivariate Fibonacci and Lucas polynomials, bivariate Pell and Pell-Lucas polynomials, Jacobsthal and Jacobsthal-Lucas polynomials, Fibonacci and Lucas p-polynomials, Fibonacci and Lucas p-numbers, Pell and Pell-Lucas p-numbers and Chebyshev polynomials of the first and second kind, are obtained. Afterwards, we obtain some properties of the bivariate Fibonacci and Lucas p-polynomials.     

Cutting Polytopes and Flag f-Vectors

We show how the flag f -vector of a polytope changes when cutting off any face, generalizing work of Lee for simple polytopes. The result is in terms of explicit linear operators on cd-polynomials. Also, we obtain the change in the flag f -vector when contracting any face of the polytope. Received July 13, 1998, and in revised form April 14, 1999.     

On the q-polynomials: a distributional study

In this paper we present a unified distributional study of the classical discrete q-polynomials (in the Hahn's sense). From the distributional q-Pearson equation we will deduce many of their properties such as the three-term recurrence relations, structure relations, etc. Also several characterizations of such q-polynomials are presented.     

Polynomial time algorithms for some classes of constrained nonconvex quadratic problems

In this paper we consider different classes of noneonvex quadratic problems that can be solved in polynomial time. We present an algorithm for the problem of minimizing the product of two linear functions over a polyhedron P in R ⁿ The complexity of the algorithm depends on the number of vertices of the projection of P onto the R ² space. In the worst-case this algorithm requires an exponential number of steps but its expected computational time complexity is polynomial. In addition, we give a characterization for the number of isolated local minimum areas for problems on this form.

Furthermore, we consider indefinite quadratic problems with variables restricted to be nonnegative. These problems can be solved in polynomial time if the number of negative eigenvalues of the associated symmetric matrix is fixed.     

A New Algorithm for Simulating a Correlation Matrix Based on Parameter Expansion and Reparameterization

The correlation matrix (denoted by R) plays an important role in many statistical models. Unfortunately, sampling the correlation matrix in Markov chain Monte Carlo (MCMC) algorithms can be problematic. In addition to the positive definite constraint of covariance matrices, correlation matrices have diagonal elements fixed at one. In this article, we propose an efficient two-stage parameter expanded reparameterization and Metropolis-Hastings (PX-RPMH) algorithm for simulating R. Using this algorithm, we draw all elements of R simultaneously by first drawing a covariance matrix from an inverse Wishart distribution, and then translating it back to a correlation matrix through a reduction function and accepting it based on a Metropolis-Hastings acceptance probability. This algorithm is illustrated using multivariate probit (MVP) models and multivariate regression (MVR) models with a common correlation matrix across groups. Via both a simulation study and a real data example, the performance of the PX-RPMH algorithm is compared with those of other common algorithms. The results show that the PX-RPMH algorithm is more efficient than other methods for sampling a correlation matrix.     

Computing elliptic integrals by duplication

Summary Logarithms, arctangents, and elliptic integrals of all three kinds (including complete integrals) are evaluated numerically by successive applications of the duplication theorem. When the convergence is improved by including a fixed number of terms of Taylor's series, the error ultimately decreases by a factor of 4096 in each cycle of iteration. Except for Cauchy principal values there is no separation of cases according to the values of the variables, and no serious cancellations occur if the variables are real and nonnegative. Only rational operations and square roots are required. An appendix contains a recurrence relation and two new representations (in terms of elementary symmetric functions and power sums) forR-polynomials, as well as an upper bound for the error made in truncating the Taylor series of anR-function.     

Combinatorial expressions for F-polynomials in classical types

We give combinatorial formulas for F-polynomials in cluster algebras of classical types in terms of the weighted paths in certain directed graphs. As a consequence we prove the positivity of F-polynomials in cluster algebras of classical types.     

On the linearization problem involving Pochhammer symbols and their q-analogues

In this paper we present a simple recurrent algorithm for solving the linearization problem involving some families of q-polynomials in the exponential lattice x(s)=c₁q^s+c₃. Some simple examples are worked out in detail.     

Simplicial algorithm to solve the nonlinear complementarity problem onS n×R + m

In this paper, a simplicial algorithm is developed to solve the nonlinear complementarity problem onS ⁿ×R ₊ ^m . Furthermore, a condition for convergence is formulated. The triangulation which underlies the algorithm is a combination of the V-triangulation ofS ⁿ and the K-triangulation ofR ₊ ^m . Therefore, we will call it the VK-triangulation.The author wishes to thank Professor G. van der Laan for his valuable comments.     

Stability analysis of a general toeplitz systems solver

We show that a fast algorithm for theQR factorization of a Toeplitz or Hankel matrixA is weakly stable in the sense thatR ^T R is close toA ^T A. Thus, when the algorithm is used to solve the semi-normal equationsR ^TRx=A^T b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear systemAx=b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem min ||Ax-b||₂.