ORTHOGONAL MATRIX POLYNOMIALS WITH RESPECT TO LINEAR MATRIX MOMENT FUNCTION ALS:THEORY AND APPLICATIONS

In this paper orthogonal matrix polynomials with respect to a right matrix moment functional an introduced. Basic results, important examples and applications to the approximation of matrix integrals are studied. Error bounds for the proposed matrix quadrature rules are given.

ON CONSERVATIVE APPROXIMATION BY LINEAR POLYNOMIAL OPERATORS AN EXTENSION OF THE BERNSTEIN'S OPERATOR

In this work we slwly linear polynomial operators preserving some consecutive i-convexities and leaving in-verant the polynomtals up to a certain degree. First we study the existence of an incompatibility between the conservation of cenain i-cotivexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DcVore about the Bernstein's operator ts extended. Finally, from these results a genera'ized Bernstein's operator is obtained.

Analysis of Precipitation Trends and Prediction in Selected Cities in the Southeast Louisiana

The impacts of climate change are being felt in Louisiana, in the form of changing weather patterns that have resulted in changes in floods, hurricanes, tornadoes frequencies of occurrence, and magnitudes, among others resulting in, flooding. The variabilities in rainfall in a drainage basin affect water availability and sustainability. This study analyzed the precipitation data of Southeastern Louisiana, United States, for the period 1990 to 2020. Data used in the study was from, Donaldsonville, Galliano, Lafourche, Gonzales, Ascension, Morgan, New Orleans, Audubon, Plaquemine, and Ponchatoula, Tangipahoa, weather stations. These stations were selected because the differences between each of their highest and lowest average annual rainfall data were greater than 20 inches. To investigate climate patterns and trends for the given weather stations in Southeastern Louisiana, precipitation data were analyzed on annual time scales using data collected from the World Bank Group Climate Change Knowledge Portal for Development Practitioners and Policy Makers and the Applied Climate Information System (ACIS) of the National Weather Service Prediction Center. The data were further aggregated using annual average blocks of 4 years, and linear and polynomial regression was performed to establish trends. The highest and lowest average annual rainfall data for Donaldsonville, Galliano, Lafourche, Gonzales, Ascension, Morgan, New Orleans, Audubon, Plaquemine, and Ponchatoula, Tangipahoa, weather stations were, 75 and 48, 71 and 44, 73.5 and 52.7, 75 and 46.4, 72 and 41.3, 94 and 55.3, Ponchatoula, and 78.6 and 44, respectively. Plaquemine recorded the highest average annual average rainfall while New Orleans, Audubon station recorded the lowest. The projection of the precipitation in 2030 has been carried out to inform scientists and stakeholders about the approximate quantity of rainfall expected and enable them to make their expected impacts on agriculture, economy, etc. The precipitation for 2030 was predicted by extrapolating models for the weather stations. The data used for the modeling was selected based on the data entries most representative. Hence, the coefficient of correlation and the number of data entries were both considered. Extrapolating results for 2030 precipitation in Donaldsonville, Galliano, Gonzales, Morgan, New Orleans, Audubon, and Plaquemine were found to be within the ranges, (85.6 - 86.7), (75.55 - 76.60), (89.7 - 90.67), (99.9 - 100.5), (71.68 - 72.66), and (107.7 - 108.8) inches, respectively. Hence, the average annual precipitations in areas covered by these stations except for Plaquemine station are expected to significantly increase. A restively low increase in average precipitation is expected for Plaquemine station. The increase could impact agriculture negatively or positively depending on the crop’s soil moisture tolerance.

Dembowski-Ostrom Polynomials from Reversed Dickson Polynomials

This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2.The authors also classify almost perfect nonlinear functions among all such Dembowski-Ostrom polynomials based on a general result describing when the composition of an arbitrary linearized polynomial and a monomial of the form x^(2+2^α) is almost perfect nonlinear.It turns out that almost perfect nonlinear functions derived from reversed Dickson polynomials are all extended affine equivalent to the well-known Gold functions.

A New Criterion on k-Normal Elements over Finite Fields

The notion of normal elements for finite fields extension was generalized as k-normal elements by Huczynska et al.(2013).Several methods to construct k-normal elements were presented by Alizadah et al.(2016)and Huczynska et al.(2013),and the criteria on k-normal elements were given by Alizadah et al.(2016)and Antonio et al.(2018).In the paper by Huczynska,S.,Mullen,G.,Panario,D.and Thomson,D.(2013),the number of k-normal elements for a fixed finite field extension was calculated and estimated.In this paper the authors present a new criterion on k-normal elements by using idempotents and show some examples.Such criterion was given for usual normal elements before by Zhang et al.(2015).

On Regular Linear Relations

For a closed linear relation in a Banach space the concept of regularity is introduced and studied. It is shown that many of the results of Mbekhta and other authors for operators remain valid in the context of multivalued linear operators. We also extend the punctured neighbourhood theorem for operators to linear relations and as an application we obtain a characterization of semiFredholm linear relations which are regular.

A FULL-NEWTON STEP INFEASIBLE INTERIOR-POINT ALGORITHM FOR P_*(κ) LINEAR COMPLEMENTARITY PROBLEM

This paper proposes a new infeasible interior-point algorithm with full-Newton steps for P_*(κ) linear complementarity problem(LCP),which is an extension of the work by Roos(SIAM J.Optim.,2006,16(4):1110-1136).The main iteration consists of a feasibility step and several centrality steps.The authors introduce a specific kernel function instead of the classic logarithmical barrier function to induce the feasibility step,so the analysis of the feasibility step is different from that of Roos' s.This kernel function has a finite value on the boundary.The result of iteration complexity coincides with the currently known best one for infeasible interior-point methods for P_*(κ) LCP.Some numerical results are reported as well.

Threshold Signature Scheme with Threshold Verification Based on Multivariate Linear Polynomial

Secret sharing schemes are multi-party protocols related to key establishment. They also facilitate distributed trust or shared control for critical activities (e.g., signing corporate cheques and opening bank vaults), by gating the critical action on cooperation from t(t ∈Z+) of n(n ∈Z+) users. A (t, n) threshold scheme (t < n) is a method by which a trusted party computes secret shares Γi(1 i n) from an initial secret Γ0 and securely distributes Γi to user. Any t or more users who pool their shares may easily recover Γ0, but any group knowing only t-1 or fewer shares may not. By the ElGamal public key cryptophytes and the Schnorr's signature scheme, this paper proposes a new (t,n) threshold signature scheme with (k,m) (k,m ∈Z+) threshold verification based on the multivariate linear polynomial.