Elimination in Weyl Algebra and q-Identities

For the algorithmic proof of q-proper-hypergeometric identities, H.Wilf and D.Zeiberggave a theoretical frame work. In [1], they proved that q-proper-hypergeometric termssatisfy recurrence relations with polynomial coefficients and could obtain quite explicitbounds for the order of such a recurrence. But how can we find the recurrence relations?We consider single-variable q-proper-hypergeometric identities based on Zeilberg's basicidea. To find the recurrence relations, an elimination in the non-conuntative Weyl algebra     

A Weighted Likelihood Ratio of Two Related Negative Hypergeomeric Distributions

In this paper we consider some related negative hypergeometric distributions arising from the problem of sampling without replacement from an urn containing balls of different colours and in different proportions but stopping only after some specific number of balls of different colours have been obtained. With the aid of some simple recurrence relations and identities we obtain in the case of two colours the moments for the maximum negative hypergeometric distribution, the minimum negative hypergeometric distribution,the likelihood ratio negative hypergeometric distribution and consequently the likelihood proportional negative hypergeometric distributiuon. To the extent that the sampling scheme is applicable to modelling data as illustrated with a biological example and in fact many situations of estimating Bernoulli parameters for binary traits within a finite population, these are important first-step results.     

Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

In this paper,we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates.We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation.We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions.Furthermore,we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation.We obtain two critical values of r,and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value,while it appears as a damped oscillatory wave if |r| is less than some critical value.By means of analysis and the undetermined coefficients method,we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation.Based on the above discussions and according to the evolution relations of orbits in the global phase portraits,we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method.Finally,using the homogenization principle,we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions.Moreover,we also give the error estimates for these approximate solutions.     

Quasi-binomial coefficients stemming from Nakayama algebras

It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic indecomposable basic Nakayama algebras,we describe the cardinal number of the set of all t-length admissible sequences using a new type of integers called quasi-binomial coefficients.Furthermore,we find some intrinsic relations among binomial coefficients and quasi-binomial coefficients.     

Minimum secondary aberration fractional factorial split-plot designs in terms of consulting designs

It is very powerful for constructing nearly saturated factorial designs to characterize fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary wordlength pattern of a regular symmetrical fractional factorial split-plot (FFSP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial design theory and coding theory, we obtain some general and unified combinatorial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to these identities, we further establish some general and unified rules for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.     

A new nonparametric stability test with an application to major Chinese macroeconomic time series

In this paper, we propose a new test for testing the stability in macroeconomic time series, based on the LASSO variable selection approach and nonparametric estimation of a time-varying model. The wild bootstrap is employed to obtain its data-dependent critical values. We apply the new method to test the stability of bivariate relations among 92 major Chinese macroeconomic time series. We find that more than 70% bivariate relations are significantly unstable.     

Algorithms and Identities for(q,h)-Bernstein Polynomials and(q,h)-Bézier Curves–A Non-Blossoming Approach

We establish several fundamental identities,including recurrence relations,degree elevation formulas,partition of unity and Marsden identity,for quantum Bernstein bases and quantum Bézier curves.We also develop two term recurrence relations for quantum Bernstein bases and recursive evaluation algorithms for quantum Bézier curves.Our proofs use standard mathematical induction and other elementary techniques.     

g-Besselian frames in Hilbert spaces

In this paper, we introduce the concept of a g-Besselian frame in a Hilbert space and discuss the relations between a g-Besselian frame and a Besselian frame. We also give some characterizations of g-Besselian frames. In the end of this paper, we discuss the stability of g-Besselian frames. Our results show that the relations and the characterizations between a g-Besselian frame and a Besselian frame are different from the corresponding results of g-frames and frames.     

On Some Properties of $c$-Frames

In this paper we discuss about $c$-frames, namely continuous frames. Since, $c$-frames are generalizations of discrete frames, we generalize some results of discrete frames to continuous version. We explain some results about relations of projections in Hilbert spaces and $c$-frames to characterize these frames. Also, we will specify (precisely) the synthesis and frame operators of Bochner integrable $c$-frames. Finally, we classify Hilbert-Schmidt operators by $c$-frames and express some new identities for Parseval $c$-frames.     

APPLICATIONS OF HYPERGEOMETRIC SUMMATION THEOREMS OF KUMMER AND DIXON INVOLVING DOUBLE SERIES

Using series iteration techniques identities and apply each of these identities in we derive a number of general double series order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.