Congruences for finite triple harmonic sums

Zhao (2003a) first established a congruence for any odd prime p>3, S(1,1,1;p)≡-2Bp-3 (mod p), which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S(α,β,γ;p) (mod p) is considered for all positive integers α,β,γ. We refer to w=α β γ as the weight of the sum, and show that if w is even, S(α,β,γ;p)≡0 (mod p) for p≥w 3; if w is odd, S(α,β,γ;p)≡rBp≥w (mod p) for p≥w, here r is an explicit rational number independent of p. A congruence of Catalan number is obtained as a special case.     

L^p-estimates on a ratio involving a Bessel process

INTRODUCTION Throughout this paper, we shall work with a filtered complete probability space (?,F,(Ft),P) sat-isfying the usual conditions. Let B=(Bt)t≥0 be a stan-dard Brownian motion with B0=0. Denote by ú the set of all non-negative real numbers. Recall that a diffusion process X starting at x≥0 is called the square of a Bessel process of dimension δ>0 if d X t = δd t 2 | X t |d Bt , X 0= x, (1) Clearly, this equation has a unique non-negative strong solution X, i.e., …     

A general version of the Morse-Sard theorem

Let k,m,n be positive integers,and k≥2,α∈(0,1],0＜r＜min {m,n} an integer,d=r (m-r)/(k α),and iff∈ Ck,α(Rm,R″),A =Cr(f)= {x ∈ Rm ,rank(Df(x))≤r},thenf(A)is d-null.Thus the statement posed by Arthur Sard in 1965 can be completely solved when k≥2.     

Projectively flat exponential Finsler metric

INTRODUCTION One century ago, Hilbert announced his famous 23 problems. The Hilbert’s Fourth Problem is to characterize the (not-necessarily-reversible) distance functions on an open subset in ún such that straight lines are shortest paths. Distance functions induced by a Finsler metric are regarded as smooth ones. Thus Hilbert’s Fourth Problem in the smooth case is to characterize Finsler metric on an open subset in ún whose geodesics are straight lines. Finsler metric on an ope…     

On some projectively flat polynomial （α,β）-metrics

In this paper, we consider some polynomial (α,β)-metrics, and discuss the sufficient and necessary conditions for a Finsler metric in the form F=α a1β a2β2/α a4β4/α3 to be projectively flat, where ai (i=1,2,4) are constants with a1≠0, α is a Riemannian metric and β is a 1-form. By analyzing the geodesic coefficients and the divisibility of certain polynomials, we obtain that there are only five projectively flat cases for metrics of this type. This gives a classification for such kind of Finsler metrics.     

Regular-singular crossings on nonlinear systems with turning point

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Precise asymptotics in the law of the logarithm for random fields in Hilbert space

Consider the positive d-dimensional lattice d (d≥2) with partial ordering ≤, let {XK; K ∈ d } be i.i.d. random vari- ables taking values in a real separable Hilbert space (H, ||·||) with mean zero and covariance operator ∑, and set partial sums SN =∑K≤NXK, K , N ∈ d . Under some moment conditions, we obtain the precise asymptotics of a kind of weighted infinite series for partial sums SN as ε 0 by using the truncation and approximation methods. The results are related to the convergence rates of the law of the logarithm in Hilbert space, and they also extend the results of (Gut and Spǎtaru, 2003).     

具有转向点的非线性系统的奇异摄动

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Projectively flat Asanov Finsler metric

In this work, we study the Asanov Finsler metric F=α(β2/α2 gβ/α 1)1/2exp{(G/2)arctan[β/(hα) G/2]}, where α=(αijyiyj)1/2 is a Riemannian metric and β=biyj is a 1-form, g∈(-2,2), h=(1-g2/4)1/2, G=g/h. We give the necessary and sufficient condition for Asanov metric to be locally projectively flat, i.e., α is projectively flat and β is parallel with respect to α. Moreover, we proved that the Douglas tensor of Asanov Finsler metric vanishes if and only if β is parallel with respect to α.     

Estimation method for mixed-effect coefficient semiparametric regression model

Consider the mixed-effect coefficient semiparametric regression modelZ=X'α+Y'β+g(T)+e, whereX, Y andT are random vectors onR ^p×R ^q×[0, 1], α is a p-dimensional fixed-effect parameter, β is a q-dimensional random-effect parameter (Eβ=b, Cov(β)=Σ),g(.) is an unknown function on [0, 1], e is a random error with mean zero and variance σ², and (X, Y, T) and (β,e), β ande are mutually independent. We estimate α,b andg(.) by the nearest neighbor and the least square method. In this paper, we prove that estimations of α,b have asymptotic normality and obtain the best convergence raten ^−1/3 for the estimation ofg(.). Project supported by the NSFC and the Natural Science Foundation of Zhejiang Province.     

On Turán type inequality with doubling weights andA * weights

LetH _n be the set of real algebraic polynomials of degreen, whose zeros all lie in the interval [−1,1]. The well known Turán type inequalities tell us that forf(x)∈H _n , it holds ‖f′‖≥C√n‖f‖. This note deals with the weighted Turán type inequalities with the weights having inner singularities underL ^p norm for 0<p≤∞. Our results essentially extend the result of Wang and Zhou (2002), and the method used in this paper is simpler and more direct than that of Wang and Zhou (2002). The results and methods have their own values in approximation theory and computation.     

An empirical study of students’ understanding of a logical structure in the definition of limit via the <Emphasis Type="Italic">ε</Emphasis>-strip activity

This study explored students’ understanding of a logical structure in defining the limit of a sequence, focusing on the relationship between ε and N. The subjects of this study were college students who had already encountered the concept of limit but did not have any experience with rigorous proofs using the ε–N definition. This study suggested two statements, each of which is written by using a relationship between ε and N, similar to the ε–N definition. By analyzing the students’ responses to the validity of the statements as definitions of the limit of a sequence, students’ understanding of such a relationship was classified into five major categories. This paper discusses some essential components that students must conceptualize in order properly to understand the relationship between ε and N in defining the limit of a sequence.     

Ted Hughes: The Development of a Children’s Poet

This article looks at how Ted Hughes’ poetry for children developed over more than 30 years of publication. It traces the movement from his earlier, more conventional rhyming poems, such as Meet My Folks! (1961) and Nessie the Mannerless Monster (1964), to the mature, free verse “animal poems” for older readers of Season Songs (1976c), Under the North Star (1981) and the “farmyard fable” What is the Truth? (1984). The article argues that the later lyrical poems for younger readers where Hughes returned to rhyme, The Cat and the Cuckoo (1987) and The Mermaid’s Purse (1993), represent an undervalued final phase of Hughes’ work for children which is rarely discussed by critics. The discussion considers Hughes’ changing attitude to the concept of the “children’s poet” at different periods of his career. Reference is made throughout to Hughes’ own writing about children and poetry, such as Poetry in the Making (1967), and to parallel developments in his poetry for adults.     

Axioms, Essences, and Mostly Clean Hands: Preparing to Teach Chemistry with Libavius and Aristotle

Andreas Libavius’ (c. 1555–1616) three part collection of letters, the Rerum chymicarum epistolica forma ... liber (1595–1599) is a particularly important text in fashioning the subject of chemistry as a demonstrative science and as a didactic discipline. Where Libavius’ Alchemia, which some have claimed to be the first textbook of chemistry, had mostly a humanist agenda, the Rerum chymicarum ... liber more directly sought to wrest the subject of “chemistry” away from Paracelsian adepts, and established the methodological basis for a specific form of knowledge suitable to the university. Making use of Aristotle’s Posterior analytics Libavius created a “floor-plan” for chemistry that integrated practical experience with natural philosophy, and could thus, he claimed, penetrate more deeply into the structure of nature than other academic disciplines.     

在n维赋范空间中单位球体积及其渐进性质

Let Bnp={x ∈ Rn‖|x ‖p≤l} be the unit ball ofp norm in the n-dimensional normed space p. The formula for the volume of Bnp was obtained and its asymptotic properties were found out as n→∞and p→∞.     

流动儿童城市适应状况及过程:一项质性研究的结果

This article employs qualitative research methods to explore the urban adaptation and adaptation processes of Chinese migrant children. Through twenty-one in-depth interviews with migrant children, the researchers discovered: The participant migrant children showed a fairly high level of adaptation to the city; their process of urban adaptation emerged as three categories: the U-Type, the J-type, and the “flat line” type. Also explored were long-term acquisition of explicit behaviors, implicit notions, adaptation to the socio-cultural environment, and success in education as difficult aspects of adaptation. After comparing these three processes, the research team developed a theory of migrant children urban adaptation. It hypothesizes that during the urban adaptation process, migrant children will encounter four stages of development: excitement and curiosity, shock and resistance, exploration and adaptation, and finally integration and assimilation. __________ Translated by Amanda Weiss from Beijing Shifan Daxue Xuebao (Shehui Kexue Ban) 北京师范大学学报 (社会科学版) (Journal of Beijing Normal University (Social Sciences)), 2008, (3): 9–20, revised by Deane Neubauer, Emeritus Professor, University of Hawaii     

A study of the relationship between expression level of TRF1 protein and telomerase activity in human acute leukemia

INTRODUCTION Telomeres are distinctive DNA-protein struc-tures that cap the ends of linear chromosomes.It is very important to keep the chromosomes stabilization.Telomerase activity is closely linked to attainment of cellular immortality,a step in carcinogenesis,while lack of such activity contributes to cellular senes-cence.Telomerase is activated in more than85%ofmalignant tumors(Hiayma et al.,1997).Human te-lomeric repeat binding factor1(TRF1)is a telomere associated with proteins a…     

Generalized Derivations in Prime Rings

Let R be a ring, a ,b ∈ R, ( D , α ) and (G , β ) be two generalized derivations of R . It is proved that if aD ( x ) = G ( x )b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b is contained in C , then α = β= 0 and there exist p , q ∈ Qr ( RC) such that D ( x )= px and G ( x )= qx for all x ∈ R;(ii) If both a and b are contained in C , then either a = b= 0 or D and G are C-linearly dependent;(iii) If neither a nor b is contained in C , then there exist p , q ∈ Qr ( RC) and w ∈ Qr ( R) such that α ( x ) = [ q ,x] and β ( x ) = [ x ,p] for all x ∈ R, whence D ( x )= wx-xq and G ( x )= xp + avx with v ∈ C and aw-pb= 0.     

Projectively flat arctangent Finsler metric

INTRODUCTION In recent years, significant progress has been made in the study of Finsler metris which have straight lines in local coordinates (Hamel, 1903). Finsler metrics with this property are called locally projectively flat metrics. An important fact about these metrics is that locally projectively flat Finsler metrics have scalar flag curvature, a natural extension of the sectional curvature in Riemannian geometry. In general, it also depends on the direction (flag pole) in the se…     

Over ten years after the implementation of the Teacher’s Law: watching and expecting

It has been over ten years since the Teacher’s Law took effect on January 1, 1994, and its promulgation and implementation have somewhat helped in protecting teachers’ legal rights and interests. Undeniably, however, the Teacher’s Law is defective in many aspects, such as the absence of teachers’ legal identity, its failure to safeguard a teacher’s essential interests and rights, serious laggardness of the teacher’s appointment system and obstacles to dealing with teachers’ complaints and litigations. All such defectiveness make the legislative purpose “to safeguard the teacher’s legal rights and interests” in Article 1 of the Teacher’s Law less effective in practice. Therefore, in order to judicially assist teachers in safeguarding their own legal rights and interests, it is necessary to modify the Teacher’s Law and especially give priorities to: (1) defining the identity of the state staff member for teachers; (2) treating teachers owed to in arrears as peasant-workers, and paying and remunerating them punctually (3) formulating detailed measures on the implementation of the teacher’s appointment system as soon as possible, solving problems of unemployment of teachers arising from the inequality between the teacher’s rights, the school’s power and the absence of the teacher’s identity; (4) establishing a lawsuit system connected with teachers’ complaints and personnel arbitration. __________ Translated from Theory and Practice of Education, 2005:2