THE SCHUR HARMONIC CONVEXITY FOR A CLASS OF SYMMETRIC FUNCTIONS

In this article, we prove that the symmetric function Fn（x,r）=∑i1＋i2＋……in=r（x1（i1x2^i2……xn^in）1/r is Schur harmonic convex for x ∈ R＋n and r ∈N -=（1, 2, 3,...} As its applications, some analytic inequalities are established.

Hermite interpolation and its numerical differentiation formula involving symmetric functions

By utilizing symmetric functions,this paper presents explicit representations for Hermite interpolation and its numerical differentiation formula.And the corresponding error estimates are also provided.

INITIAL COEFFICIENT ESTIMATES FOR SOME SUBCLASSES OF M-FOLD SYMMETRIC BI-UNIVALENT FUNCTIONS

In the present investigation, we consider two new general subclasses B∑m（T, λ; α）and B^∑m （τ λ;β） of Em consisting of analytic and m-fold symmetric bi-univalent functions in the open unit disk U. For functions belonging to the two classes introduced here, we derive non-sharp estimates on the initial coefficients [a-~＋ll and │a2＋1│. Several connections to some of the earlier known results are also pointed out.

Schur Convexity for Two Classes of Symmetric Functions and Their Applications

respectively, where r = 1, 2, … , n, and il, i2, … , is are positive integers. In this paper, the Schur convexity of Fn（X, r） and Gn（x, r） are discussed. As applications, by a bijective transformation of independent variable for a Schur convex function, the authors obtain Schur convexity for some other symmetric functions, which subsumes the main results in recent literature; and by use of the theory of majorization establish some inequalities. In particular, the authors derive from the results of this paper the Weierstrass inequalities and the Ky Fan＇s inequality, and give a generalization of Safta＇s conjecture in the n-dimensional space and others.

Generalized Numerical Radius of Real Quaternion Matrices with Symmetric Function

In the paper the generalized numerical radiuses of real quaternion matrices with symmetric function are defined and theirs inequalities are given.

Quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters

This paper is concerned with construction of quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters.The boson-fermion correspondence for these symmetric functions have been presented.In virtue of quantum fields,we derive a series of infinite order nonlinear integrable equations,namely,universal character hierarchy,symplectic KP hierarchy and symplectic universal character hierarchy,respectively.In addition,the solutions of these integrable systems have been discussed.

Walsh Spectrum Properties of Rotation Symmetric Boolean Function

Rotation symmetric function was presented by Pieprzyk. The algebraic configuration of rotation symmetric（RotS） function is special. For a Rots n variables function f（x1, x2, …, xn） we have f（ρn^k （x1, x2, …xn））=f（x1, x2, …, xn） for k=0, 1, …, n-1. In this paper, useing probability method we find that when the parameters of RotS function is under circular translation of indices, its walsh spectrum is invariant. And we prove the result is both sufficient and necessary.

Novel Quantum Algorithms to Minimize Switching Functions Based on Graph Parti

After Google reported its realization of quantum supremacy,Solving the classical problems with quantum computing is becoming a valuable research topic.Switching function minimization is an important problem in Electronic Design Automation(EDA)and logic synthesis,most of the solutions are based on heuristic algorithms with a classical computer,it is a good practice to solve this problem with a quantum processer.In this paper,we introduce a new hybrid classic quantum algorithm using Grover’s algorithm and symmetric functions to minimize small Disjoint Sum of Product(DSOP)and Sum of Product(SOP)for Boolean switching functions.Our method is based on graph partitions for arbitrary graphs to regular graphs,which can be solved by a Grover-based quantum searching algorithm we proposed.The Oracle for this quantum algorithm is built from Boolean symmetric functions and implemented with Lattice diagrams.It is shown analytically and verified by simulations on a quantum simulator that our methods can find all solutions to these problems.

On the 2~m-variable symmetric Boolean functions with maximum algebraic immunity

The properties of the 2m-variable symmetric Boolean functions with maximum al- gebraic immunity are studied in this paper. Their value vectors, algebraic normal forms, and algebraic degrees and weights are all obtained. At last, some necessary conditions for a symmetric Boolean function on even number variables to have maximum algebraic immunity are introduced.

A Class of Rotation Symmetric Boolean Functions with Optimum Algebraic Immunity

For an odd integer n ≥ 7, this paper presented a class of n-variable rotation symmetric Boolean functions （RSBFs） with optimum algebraic immunity. The nonlinearity of the constructed functions is determined.

RESULTS ON PERMUTATION SYMMETRIC BOOLEAN FUNCTIONS

This paper provides a systematic method on the enumeration of various permutation symmetric Boolean functions. The results play a crucial role on the search of permutation symmetric Boolean functions with good cryptographic properties. The proposed method is algebraic in nature. As a by-product, the authors correct and generalize the corresponding results of St^nic~ and Maitra （2008）. Further, the authors give a complete classification of block-symmetric bent functions based on the results of Zhao and Li （2006）, and the result is the only one classification of a certain class of permutation symmetric bent functions after the classification of symmetric bent functions proposed by Savicky （1994）.

Extensions and Refinements of Adamovic's Inequality

In this article, by means of the theory of majorization, Adamovic's inequality is extended to the cases of the general elementary symmetric functions and its duals, and the refined and reversed forms are also given. As applications, some new inequalities for simplex are established.

Exponential Generalization of Newman's Inequality and Klamkin's Inequality

Exponential generalizations of Newman＇s inequality and Klamkin＇s inequality are established by the Wang Wan-lan＇s inequality, and they are extended to the cases involving general elementary symmetric functions. As an application, some new inequalities for a simplex are established. In addition, an open problem is posed.

Tau functions of the charged free bosons

We study bosonic tau functions in relation with the charged free bosonic fields.It is proved that up to a constant the only tau function in the Fock space M is the vacuum vector,and some tau functions are given in the completion M by using Schur functions.We also give a new proof of Borchardt’s identity and obtain several q-series identities by using the boson-boson correspondence.

Some results on the algebraic immunity of Boolean functions

From the motivation of algebraic attacks on stream and block ciphers,the concept of algebraic immunity（AI） of a Boolean function was introduced and studied extensively.High algebraic immunity is a necessary condition for resisting algebraic attacks.In this paper,we give some lower bounds on the algebraic immunity of Boolean functions.The results are applied to give lower bounds on the AI of symmetric Boolean functions and rotation symmetric Boolean functions.Some balanced rotation symmetric Boolean functions with their AI near the maximum possible value「n/2」are constructed.

Kernel Function-Based Primal-Dual Interior-Point Methods for Symmetric Cones Optimization

In this paper, we present a large-update primal-dual interior-point method for symmetric cone optimization（SCO） based on a new kernel function, which determines both search directions and the proximity measure between the iterate and the center path. The kernel function is neither a self-regular function nor the usual logarithmic kernel function. Besides, by using Euclidean Jordan algebraic techniques, we achieve the favorable iteration complexity O（ √r（1/2）（log r）^2 log（r/ ε））, which is as good as the convex quadratic semi-definite optimization analogue.

The research and progress of the enumeration of lattice paths

The enumeration of lattice paths is an important counting model in enumerative combinatorics.Because it can provide powerful methods and technical support in the study of discrete structural objects in different disciplines,it has attracted much attention and is a hot research field.In this paper,we summarize two kinds of the lattice path counting models that are single lattice paths and family of nonintersecting lattice paths and their applications in terms of the change of dimensions,steps,constrained conditions,the positions of starting and end points,and so on.(1)The progress of classical lattice path such as Dyck lattice is introduced.(2)A method to study the enumeration of lattice paths problem by generating function is introduced.(3)Some methods of studying the enumeration of lattice paths problem by matrix are introduced.(4)The family of lattice paths problem and some counting methods are introduced.(5)Some applications of family of lattice paths in symmetric function theory are introduced,and a related open problem is proposed.

Recursions in Calogero-Sutherland Model Based on Virasoro Singular Vectors

The present work is much motivated by finding an explicit way in the construction of the Jack symmetric function,which is the spectrum generating function for the Calogero-Sutherland (CS) model.To accomplish this work,the hidden Virasoro structure in the CS model is much explored.In particular,we found that the Virasoro singular vectors form a skew hierarchy in the CS model.Literally,skew is analogous to coset,but here specifically refer to the operation on the Young tableaux.In fact,based on the construction of the Virasoro singular vectors,this hierarchical structure can be used to give a complete construction of the CS states,i.e.the Jack symmetric functions,recursively.The construction is given both in operator formalism as well as in integral representation.This new integral representation for the Jack symmetric functions may shed some insights on the spectrum constructions for the other integrable systems.

Bonding of the chain structure for novel boranes B_(3k+)H_(5k+p+3)^-

The molecular orbitals obtained from conventional quantum chemistry calculations, are expressed in terms of symmetrized valence bond functions of fragment, and a direct picture of chemical bonding can be drawn easily. This method is utilized, together with extended Huckel calculations, to interpret the bonding properties of a centipede-like chain structure for novel laser-producing boranes B3k+pH5k+p+3- which is constructed from the repeated unit B3H5 linked to each other by three B-H-B bonds.

Distribution Estimation with Smoothed Auxiliary Information

Distribution estimation is very important in order to make statistical inference for parameters or its functions based on this distribution. In this work we propose an estimator of the distribution of some variable with non-smooth auxiliary information, for example, a symmetric distribution of this variable, A smoothing technique is employed to handle the non-differentiable function. Hence, a distribution can be estimated based on smoothed auxiliary information. Asymptotic properties of the distribution estimator are derived and analyzed. The distribution estimators based on our method are found to be significantly efficient than the corresponding estimators without these auxiliary information. Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.