What is the Substitution Rule？

When I learned the following basic formula of indefinite integrals in the calculus, at that time I was a teenager, the strange ideas had jump out my head.     

The best quadrature formula based on Hermite information for the class KW~2[a,b]

The best quadrature formula has been found in the following sense:for afunction whose norm of the second derivative is bounded by a given constant and thebest quadrature formula for the approximate evaluation of integration of that function canminimize the worst possible error if the values of the function and its derivative at certainnodes are known.The best interpolation formula used to get the quadrature formula aboveis also found.Moreover,we compare the best quadrature formula with the open compoundcorrected trapezoidal formula by theoretical analysis and stochastic experiments.     

Volume of Domains in Symmetric Spaces

The authors derive a formula for the volume of a compact domain in a symmetric space from normal sections through a special submanifold in the symmetric space.This formula generalizes the volume of classical domains as tubes or domains given as motions along the submanifold.Finally,some stereological considerations regarding this formula are provided.     

A kinematic formula for integral invariant of degree 4 in real space form

In this paper,kinematic formulae in a real space form are investigated.A kinematic formula for homogeneous polynomial of degree 4 on the second fundamental forms in a real space form is obtained.The formula obtained is a concrete form of the result of Howard and the analogue of the known formula of Chen and Zhou.     

Inverse radon transform with one-dimensional wavelet transform

1. IntroductionSince Radon obtained the inverse formula of Radon transform in 1917, different inversemethods such as Fourier inversion, convolution back-projection inversion etc. have beeninvestigated['l']. Wavelet as a useful tool is interested in the inversion of Radon transformin recent years['--']. The application of wavelet analysis to Radon transform was proposedin I4] and [5]. An inversion formula based on continuous wavelet transform was derivedin [6] and [7]. This formula was based…     

WEIGHTED KOPPELMAN－LERAY－NORGUET FORMULAS ON A LOCAL q－CONCAVE WEDGE IN A COMPLEX MANIFOLD

A weighted Koppelman-Leray-Norguet formula of (r, s) differential forms on a local q-concave wedge in a complex manifold is obtained. By constructing the new weighted kernels, the authors give a new weighted Koppelman-Leray-Norguet formula with-out boundary integral of (r, s) differential forms, which is different from the classical one. The new weighted formula is especially suitable for the case of the local q-concave wedge with a non-smooth boundary, so one can avoid complex estimates of boundary integrals and the density of integral may be not defined on the boundary but only in the domain. Moreover, the weighted integral formulas have much freedom in applications such as in the intervolation of functions.     

TWO REMARKS ON SCHWARZ FORMULA

This paper discusses two problems. Firstly the authors give the Schwarz formula for a holomorphic function in unit disc when the boundary value of its real part is in the class H of generalized functions in the sense of Hua. Secondly the authors use the classical Schwarz formula to give a new proof of the zero free region of the Riemann zeta-function.     

Toeplitz operators on connected domains"

The proof of the index formula of the Toeplitz operator with a continuous symbol on the Hardy space for the unit circle in the complex plane depends on the Hopf theorem. However, the analogue result of the Hopf theorem does not hold on a general connected domain. Hence, the extension of the index formula of the Toeplitz operator on a general domain needs a method which is different from that for the case of the unit circle. In the present paper, the index formula of the Toeplitz operator with a continuous symbol on the finite complex connected domain in the complex plane is obtained, and the cohomology groups of Toeplitz algebras on general domains are discussed. In addition, the Toeplitz operators with symbols in QC are also discussed.     

$L^{\Phi}(I,X)$中的最佳同时逼近

Let X be a Banach space and Ф be an Orlicz function. Denote by L^Ф（I,X） the space of X-valued （I）-integrable functions on the unit interval I equipped with the Luxemburg norm. For f1,f2,... ,fm ∈ L^Ф（I,X）, a distance formula distv（f1,f2,... ,fm,L^Ф（I,G）） is presented, where G is a close subspace of X. Moreover, some existence and characterization results concerning the best simultaneous approximation of L^Ф （I, G） in L^Ф （I, X） axe given.     

Distribution of Primitive λ-Roots of Composite Moduli II

We improve estimates for the distribution of primitiveλ-roots of a composite modulus q yielding an asymptotic formula for the number of primitiveλ-roots in any interval I of length |I| >> q1/2 c. Similar results are obtained for the distribution of ordered pairs (x,x-1) with x a primitive A-root, and for the number of primitiveλ-roots satisfying inequalities such as |x - x-1| < B.     

New Monotonicity Formulae for Semi-linear Elliptic and Parabolic Systems

The authors establish a general monotonicity formula for the following elliptic system
△ui＋fi（x,ui,…,um）=0 in Ω,
where Ω belong to belong to R^n is a regular domain, （fi（x, u1,... ,um）） = △↓F（x,→↑u）, F（x,→↑u ） is a given smooth function of x ∈ R^n and →↑u = （u1,…,um） ∈ R^m. The system comes from understanding the stationary case of Ginzburg-Landau model. A new monotonicity formula is also set up for the following parabolic system
δtui-△ui-fi（x,ui,…,um）=0 in（ti,t2）×R^n,
where t1 〈 t2 are two constants, （fi（x,→↑u ） is given as above. The new monotonicity formulae are focused more attention on the monotonicity of nonlinear terms. The new point of the results is that an index β is introduced to measure the monotonicity of the nonlinear terms in the problems. The index β in the study of monotonieity formulae is useful in understanding the behavior of blow-up sequences of solutions. Another new feature is that the previous monotonicity formulae are extended to nonhomogeneous nonlinearities. As applications, the Ginzburg-Landau model and some different generalizations to the free boundary problems are studied.     

GENERALIZED LERAY FORMULA ON POSITIVE COMPLEX LAGRANGE-GRASSMANN MANIFOLDS

A full proof of a matrix lemma stated in[1]is given,and the notions concerningcannonical argument and signature of a triple of the Lagrange planes in a complex phasespace is formulated.Then a formula is established,which generalizes that one of J.Leray'sin real phase space case.Finally,some applications of the formula are given.     

THE POINCAR-BERTRAND FORMULA ON THE BUILDING DOMAIN OF COMPLEX BIBALLS

The Poincaré-Bertrand formula takes an important position in the study of complex singular integral.The Poincaré-Bertrand formula on the complex sphere in the multidimensional complex Euclidian spaces was given by Sheng Gong.Using the method of solid angular coefficient,the authors extend the Poincaré-Bertrand formula on the complex sphere to the building domain of the complex biballs,and obtain a more general Poincaré-Bertrand formula with the solid angular coefficients.     

Best Simultaneous Approximation in L~Φ(I,X)

Let X be a Banach space and Φ be an Orlicz function.Denote by LΦ(I,X) the space of X-valued Φ-integrable functions on the unit interval I equipped with the Luxemburg norm.For f1,f2,...,fm ∈ LΦ(I,X),a distance formula distΦ(f1,f2,...,fm,LΦ(I,G)) is presented,where G is a close subspace of X.Moreover,some existence and characterization results concerning the best simultaneous approximation of LΦ(I,G) in LΦ(I,X) are given.     

The Greneralized Weierstrass Formula for Surfaces in R~3

The basic formula for minimal surfaces in R~3 is the Weierstrass formula whichhas a lot of deep application([1]).In 1979 K.Kenmotsu gave a representationformula for a surface with prescribed mean curvature in terms of its Gauss map([2]).We will generalize the Weierstraes formula for minimal surfaces to thecase of general surfaces,and our results cover that of Kenmotsu's.     

Positive energy theorem for (4+1)-dimensional asymptotically anti-de Sitter spacetimes

We define the total energy-momenta for(4+1)-dimensional asymptotically anti-de Sitter spacetimes,which comes from the boundary terms at infinity in the integral form of the Weitzenbck formula.Then we prove the positive energy theorem for such spacetimes,following Witten’s original argumentsfor the positive energy theorem in asymptotically flat spacetimes.     

The Gelfand-Kirillov dimension of a unitary highest weight module

During the last decade, a great deal of activity has been devoted to the calculation of the HilbertPoincar′e series of unitary highest weight representations and related modules in algebraic geometry. However,uniform formulas remain elusive—even for more basic invariants such as the Gelfand-Kirillov dimension or the Bernstein degree, and are usually limited to families of representations in a dual pair setting. We use earlier work by Joseph to provide an elementary and intrinsic proof of a uniform formula for the Gelfand-Kirillov dimension of an arbitrary unitary highest weight module in terms of its highest weight. The formula generalizes a result of Enright and Willenbring(in the dual pair setting) and is inspired by Wang's formula for the dimension of a minimal nilpotent orbit.     

The Existence of p-blocks of a Finite Group With a Given Defect Group

Brauer's problem 19[B] -- in terms of group theoretical properties count the number of p-blocks with a given defect group -- has been studied by many authors. Among them G. R.Robinson gave a pricise formula for the number. of course it is by no means easy to calculateby the formula. So it is still interesting to find other formula counting the number or conditionsfor the existence of p-blocks. In [St], for an element g of a finite group G, the author defined a(g,p)-vector of G (in this pape…     

TWO-GATE FORMULA

The two gate formula for a diffusion X_t in R~n is considered.The formula gives an expressionof the density function of X_t in terms of the path integral of the Brownian bridge starting from xand ending on y at time t.     

On Ramanujan Sums over a Dedekind Domain with Finite Norm Property

In this paper, we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property. We define the Ramanujan’s sums η(a, A) and η(B, A), where a is an arbitrary element in a Dedekind domain, B is an ideal and A is a non-zero ideal. In particular, we discuss the Kluyver formula and Hèolder formula for η(a, A) and η(B, A). We also prove the reciprocity formula enjoyed byη(B, A) and the orthogonality relations for η(a, A) in the last two parts.