Sums of Squares of Integers

We find a formula for the sum of the first n squares:Sn = 12 22 32 42 … n2.We haveS1 = 1, S2 = 12 22 = 5, S3 = l2 22 32 = 14,S4 = 30, S5 = 55, S6 = 91.and so on. To make things a little simpler, we will also use the sum of the squares of the first zero terms , S0 = 02 = 0. Arranging these values in a table,we find thatSince the third differenes △3Sn are constant,these data values must follow a cubic pattern;that is, the formula for the sum of the squares of the first n integers is a cubic function,     

Vertices of π-Irreducible Characters of Groups of Odd Order

If G is a group of odd order and χ ∈ Irr(G) lifts an irreducible Brauer character ?, then we associate to χ a canonical pair (Q, δ) up to G-conjugacy, where Q is a vertex of ? and δ ∈ Irr(Q) is a linear character of Q. We show that (Q, δ) is a Navarro vertex for χ. We also discuss examples.     

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.     

On the Growth of Components of Meromorphic Solutions of Systems of Complex Differential Equations

This paper investigates the problem of the growth of the components of meromorphic solutions of a class of a system of complex algebraic differential equations, and generalized some of N. Toda＇s results concerning the growth of differential equations to the case of systems of differential equations. The paper considers the existence of admissible solutions of the system of differential equations.     

Correlation of zeros of automorphic L-functions

We compute the n-level correlation of normalized nontrivial zeros of a product of L-functions:L(s,π1)···L(s,πk), where πj, j=1,...,k, are automorphic cuspidal representations of GLmj(QA). Here the sizes of the groups GLmj(QA) are not necessarily the same. When these L(s,πj) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s,πj) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m1,...,mk 4,but are under Hypothesis H in other cases.     

Extension of a Theorem of Carleson-Duren

1. Introduction A theorem of Carleson,as generalized by Duren characterizes those positive measure μ on the unit disc U={z∈C:|z|<1} for which the H~p norm domiates the L~q(μ) norm of elements of H~p. Later on, Hasting proved an analogous results with H~p replaced by A~p, the Bergman space of fuctions f     

The Degree of Unsolvability of Models

The purpose of this paper is to describe the complexity of models by theirdegrees of unsolvability,J.Richter defined the degree of a structure to be deg ()=sup{deg(),deg(R_i),i=1,…,n},Where is a model for afinite language L={R_i,i=1,…,n}and the universe of is a subset of ω. Shepointed out that, according to her definition, there can be models which areisomorphic but their degrees are different.Also,her discussions are restricted tofinite languages and models whose universes are subsets of ω.     

Lie Bialgebras of a Family of Lie Algebras of Block Type

Lie bialgebra structures on a family of Lie algebras of Block type are shown to be triangular coboundary.     

Dynamics of Polynomial Automorphisms of C~N

ＤｙｎａｍｉｃｓｏｆＰｏｌｙｎｏｍｉａｌＡｕｔｏｍｏｒｐｈｉｓｍｓｏｆＣ~Ｎ￥ＺｈａｎｇＷｅｎｊｕｎ（ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ，Ｋａｉｆｅｎｇ，Ｐ．Ｒ．Ｃｈｉａｎ，４７５００１）Ａｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｉｓａｓｓｉｇｎｅｄｔｏｄｉｓ?..     

Group of Self-Equivalences of Sum Object

In general category, the group of self-equivalences of sum object can decompose as the product of its two subgroups under certain conditions and also several split short exact sequences are obtained. Subsequently, we apply the above results to the category of space under a space and get some results which are dual to those in fibrewise category.     

Dimensions of students’ views of themselves as learners of mathematics

Students’ views of themselves as learners of mathematics are a decisive parameter for their engagement and success in school. We are interested in students’ experiences with mathematics encompassing cognitive, emotional and motivational aspects. In particular, we focus on capturing the structural properties of affect related to mathematics. Participants in our study were 1,436 randomized chosen students of secondary schools from overall Finland. In the Finnish upper secondary school, there are two different syllabi for mathematics: the general and the advanced one. Schools were invited to organize the survey by one of their year 2 general syllabus courses and one of their year 2 advanced syllabus courses in grade 11. By means of factor analysis, we obtained seven dimensions in which students’ hold beliefs and emotions about mathematics partly intertwined with their motivational orientations. These dimensions are described by reliable scales, which allow outlining an average image of Finnish students’ views of themselves as learners of mathematics. Moreover, we analyzed relations between the seven dimensions and what kind of structure they generate. Thereby, a core of three high correlating dimensions could be identified, yielding different accentuations with regard to course choice.     

The radius of convergence of Poincaré series of loop spaces

LetR _S (resp.R _A) be the radius of convergence of the Poincaré series of a loop space (S) (resp. of the Betti-Poincaré series of a noetherian connected graded commutative algebraA over a field of characteristic zero).IfS is a finite 1-connected CW-complex, the rational homotopy Lie algebra ofS is finite dimensional if and only ifR _S-1. OtherwiseR _S<1.There is an easily computable upper bound (usually less than 1) forR _S ifS is formal or coformal.On the other handR _A=+ if and only ifA is a polynomial algebra andR _A=1 if and only ifA is a complete intersection (Golod and Gulliksen conjecture). OtherwiseR _A<1 and the sequence dim Tor _p ^H grows exponentially withp.     

Method of Variations of Potential of Quasi-Periodic Schrödinger Equations

We study the one-dimensional discrete quasi-periodic Schr?dinger equation

Multiplicity of fixed points and growth of ε-neighborhoods of orbits

We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on ε of the length of ε-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered in Elezovi?, ?ubrini? and ?upanovi? (2007) [5] in the differentiable case, and related to the box dimension of the orbit.Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity of fixed points and the dependence on ε of the length of ε-neighborhoods of orbits in non-differentiable cases.Applications include in particular Poincaré maps near homoclinic loops and hyperbolic 2-cycles, and Abelian integrals. This is a new approach to estimate the cyclicity, by computing the length of the ε-neighborhood of one orbit of the Poincaré map (for example numerically), and by comparing it to the appropriate scale.     

Estimates of the approximation characteristics of the classes B p,θ Ω of periodic functions of several variables with given majorant of mixed moduli of continuity

We obtain order-sharp estimates of the orthogonal projection widths of the classes B _p,θ ^Ω of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers.     

Characterizations of connected orthogonality graphs of projections of Rickart ∗-rings

In this paper, we study the orthogonality graphs (see Definition 1.2) of ortholattices. We provide a graph theoretic condition for an ortholattice to be orthomodular. We prove that, the orthogonality graphs of two orthomodular lattices are isomorphic if and only if the lattices are isomorphic. As an application, it is proved that the zero-divisor graph of a Rickart ?-ring is obtained by successively duplicating the vertices of the orthogonality graph of the lattice of projections in the ring. We characterize the finite Rickart ?-rings for which the orthogonality graph of projections is connected.     

Efficient simulation of tail probabilities of sums of correlated lognormals

We consider the problem of efficient estimation of tail probabilities of sums of correlated lognormals via simulation. This problem is motivated by the tail analysis of portfolios of assets driven by correlated Black-Scholes models. We propose two estimators that can be rigorously shown to be efficient as the tail probability of interest decreases to zero. The first estimator, based on importance sampling, involves a scaling of the whole covariance matrix and can be shown to be asymptotically optimal. A further study, based on the Cross-Entropy algorithm, is also performed in order to adaptively optimize the scaling parameter of the covariance. The second estimator decomposes the probability of interest in two contributions and takes advantage of the fact that large deviations for a sum of correlated lognormals are (asymptotically) caused by the largest increment. Importance sampling is then applied to each of these contributions to obtain a combined estimator with asymptotically vanishing relative error.