数学无穷与中介逻辑(上)——Leibniz的割线与切线问题在中介逻辑中的逻辑数学解释方法(英文)

From the perspective of potential infinity (poi) and actual infinity, Ref [4] has confirmed that poi and aci are in ’unmediated opposition’ (P,﹁P ) whether in ZFC or not; it has further been proved that the manners in which a variable infinitely approaches its limit also satisfy the law of intermediate exclusion. With these results as theoretical bases, this paper attempts to provide an accurate and strict logical-mathematical interpretation of the incompatibility of Leibniz’s secant and tangent lines in the medium logic system from the perspective of logical mathematics.     

TIME AND NORM OPTIMAL CONTROLS:A SURVEY OF RECENT RESULTS AND OPEN PROBLEMS

We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagin’s maximum principle. In particular, we discuss optimality (or nonoptimality) of singular controls satisfying the maximum principle and smoothness of the costate in function of smoothness of the target.     

Asymptotic Behavior of a Structure Made by a Plate and a Straight Rod

This paper is devoted to describing the asymptotic behavior of a structure made by a thin plate and a thin perpendicular rod in the framework of nonlinear elasticity. The authors scale the applied forces in such a way that the level of the total elastic energy leads to the Von-K′arm′an’s equations (or the linear model for smaller forces) in the plate and to a one-dimensional rod-model at the limit. The junction conditions include in particular the continuity of the bending in the plate and the stretching in the rod at the junction.     

Analysis of FDTD to UPML for Maxwell equations in polar coordinates

An FDTD system associated with uniaxial perfectly matched layer(UPML) for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic(TM) mode to Maxwell’s equations is obtained by Yee’s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis.     

Property (ω′) and Its Perturbation

In this note we define the property (ω′), a variant of Weyl’s theorem, and establish for a bounded linear operator defined on a Hilbert space the necessary and sufficient conditions for which property (ω′) holds by means of the variant of the essential approximate point spectrum σ1(·) and the spectrum defined in view of the property of consistency in Fredholm and index. In addition, the perturbation of property (ω′) is discussed.     

The Extension Degree Conditions for Fractional Factor

In Gao’s previous work, the authors determined several degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if b = f(x) = g(x) = a for all vertices x in G. In this paper, we continue to discuss these degree conditions for admitting fractional factor in the setting that several vertices and edges are removed and there is a difference Δ between g(x) and f(x) for every vertex x in G. These obtained new degree conditions reformulate Gao’s previous conclusions, and show how Δ acts in the results. Furthermore,counterexamples are structured to reveal the sharpness of degree conditions in the setting f(x) =g(x) + Δ.     

On the error term in Weyl’s law for the Heisenberg manifolds (Ⅱ)

In this paper we study the mean square of the error term in the Weyl’s law of an irrational (2l + 1)-dimensional Heisenberg manifold. An asymptotic formula is established.     

SOME ASYMPTOTIC PROPERTIES OF THE CONVOLUTION TRANSFORMS OF FRACTAL MEASURES

We study the asymptotic behavior near the boundary of u(x,y) = Ky μ(x),defined on the half-space R+ ×RN by the convolution of an approximate identity Ky(.)(y > 0) and a measure μ on RN.The Poisson and the heat kernel are unified as special cases in our setting.We are mainly interested in the relationship between the rate of growth at boundary of u and the s-density of a singular measure μ.Then a boundary limit theorem of Fatou’s type for singular measures is proved.Meanwhile,the asymptotic behavior of a quotient of Kμ and Kν is also studied,then the corresponding Fatou-Doob’s boundary relative limit is obtained.In particular,some results about the singular boundary behavior of harmonic and heat functions can be deduced simultaneously from ours.At the end,an application in fractal geometry is given.     

EXACT EVALUATIONS OF FINITE TRIGONOMETRIC SUMS BY SAMPLING THEOREMS

We use the sampling representations associated with Sturm-Liouville difference operators to derive generalized integral-valued trigonometric sums. This extends the known results where zeros of Chebyshev polynomials of the first kind are involved to the use of the eigenvalues of difference operators, which leads to new identities. In these identities Bernoulli’s numbers play a role similar to that of Euler’s in the old ones. Our technique differs from that of Byrne-Smith (1997) and Berndt-Yeap (2002).     

LOCAL ANALYTIC SOLUTIONS OF A MORE GENERALIZED DHOMBRES EQUATION

We study the local analytic solutions f of the functional equation f(ψ(zf(z)))=(f(z)) for z in some neighborhood of the origin.Whether the solution f vanishes at z=0 or not plays a critical role for local analytic solutions of this equation.In this paper,we obtain results of analytic solutions not only in the case f(0)=0 but also for f(0)≠0.When assuming f(0) =0,for technical reasons,we just get the result for f’(0)≠0.Then when assuming f(0)=ω0≠0,ψ’(0)=s≠0,ψ(z) is analytic at z=0 and(z)is analytic at z=ω0,we give the existence of local analytic solutions f in the case of 0<|sω0|<1 and the case of |sω0|=1 with the Brjuno condition.     

Second-order two-scale method for bending behavior analysis of composite plate with 3-D periodic configuration and its approximation

This paper considers the bending behaviors of composite plate with 3-D periodic configuration.A second-order two-scale(SOTS)computational method is designed by means of construction way.First,by 3-D elastic composite plate model,the cell functions which are defined on the reference cell are constructed.Then the effective homogenization parameters of composites are calculated,and the homogenized plate problem on original domain is defined.Based on the Reissner-Mindlin deformation pattern,the homogenization solution is obtained.And then the SOTS’s approximate solution is obtained by the cell functions and the homogenization solution.Second,the approximation of the SOTS’s solution in energy norm is analyzed and the residual of SOTS’s solution for 3-D original in the pointwise sense is investigated.Finally,the procedure of SOTS’s method is given.A set of numerical results are demonstrated for predicting the effective parameters and the displacement and strains of composite plate.It shows that SOTS’s method can capture the 3-D local behaviors caused by3-D micro-structures well.     

Empirical likelihood inference for semi-parametric estimating equations

Qin and Lawless (1994) established the statistical inference theory for the empirical likelihood of the general estimating equations. However, in many practical problems, some unknown functional parts h(t) appear in the corresponding estimating equations EFG(X, h(T), β) = 0. In this paper, the empirical likelihood inference of combining information about unknown parameters and distribution function through the semiparametric estimating equations are developed, and the corresponding Wilk’s theorem is established. The simulations of several useful models are conducted to compare the finite-sample performance of the proposed method and that of the normal approximation based method. An illustrated real example is also presented.     

Modeling and targeting an essential metabolic pathway of Plasmodium falciparum in apicoplast using Petri nets

Petri net(PN) is one of the promising computational and mathematical formalisms used to represent and study the behavior of complex metabolic networks. The various available analysis techniques of PN could be used to validate and analyze the network in different scenarios. Plasmodium falciparum is one of the threatening parasites which causes malaria, a deadly disease affecting a large number of today’s world population. The development of antimalarial drug resistance is an emerging global threat, highlighting the need to discover novel antimalarial targets. The fatty acid biosynthesis of malarial parasite is one of the essential metabolic pathways required for its growth and is present in apicoplast, a non-photosynthetic plastid. The malarial parasite obtains fatty acids by using type two fatty acid synthase(FAS II) enzyme,which is different from type one enzyme used by human host, making it an ideal drug target.This article proposes and studies the PN model of the parasite’s FAS II pathway to analyze the mechanism of potential drug targets in this pathway. The proposed PN model can serve as a base for further findings in the field of antimalarial drug targets to decrease the malaria mortality rate.     

Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell’s equations

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain(ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell’s equations.Precisely,for the case with a perfectly electric conducting(PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete H 1-norm for the ADI-FDTD scheme,and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero,then the discrete L 2-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time.The key ingredient is two new discrete modified energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell’s equations introduced in this paper.Furthermore,we prove that,in addition to two known discrete modified energy identities which are second-order in time perturbations of two known energy conservation laws,the ADI-FDTD scheme also satisfies two new discrete modified energy identities which are second-order in time perturbations of the two new energy conservation laws.This means that the ADI-FDTD scheme is unconditionally stable under the four discrete modified energy norms.Experimental results which confirm the theoretical results are presented.     

The Gaussian approximation for generalized Friedman’s urn model with heterogeneous and unbalanced updating

The Friedman’s urn model is a popular urn model which is widely used in many disciplines.In particular,it is extensively used in treatment allocation schemes in clinical trials.Its asymptotic properties have been studied by many researchers.In literature,it is usually assumed that the expected number of balls added at each stage is a constant in despite of what type of balls are selected,that is,the updating of the urn is assumed to be balanced.When it is not,the asymptotic property of the Friedman’s urn model is stated in the book of Hu and Rosenberger(2006) as one of open problems in the area of adaptive designs.In this paper,we show that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a general multi-color Friedman type urn model with heterogeneous and unbalanced updating.The Gaussian process is a solution of a stochastic differential equation.As an application,we obtain the asymptotic properties including the asymptotic normality and the exact law of the iterated logarithm.     

Testing for the parametric parts in a single-index varying-coefficient model

Single-index varying-coefficient models (SIVCMs) are very useful in multivariate nonparametric regression.However,there has less attention focused on inferences of the SIVCMs.Using the local linear method,we propose estimates of the unknowns in the SIVCMs.In this article,our main purpose is to examine whether the generalized likelihood ratio (GLR) tests are applicable to the testing problem for the index parameter in the SIVCMs.Under the null hypothesis our proposed GLR statistic follows the chi-squared distribution asymptotically with scale constant and degree of freedom independent of the nuisance parameters or functions,which is called as Wilks’ phenomenon (see Fan et al.,2001).A simulation study is conducted to illustrate the proposed methodology.     

Boundedness of Solutions of Quasi-periodic p-Laplacian Equations with Jumping Nonlinearity

In this article, we prove the existence of quasi-periodic solutions and the boundedness of all solutions of the p-Laplacian equation(φ_p(x’))’ + aφp(x⁺)-bφp(x^-) = g(x, t) + f(t), where g(x, t)and f(t) are quasi-periodic in t with Diophantine frequency. A new method is presented to obtain the generating function to construct canonical transformation by solving a quasi-periodic homological equation.     

The Equivalence between Property (ω) and Weyl’s Theorem

We call T ∈ B(H) consistent in Fredholm and index (briefly a CFI operator) if for each B ∈ B(H),T B and BT are Fredholm together and the same index of B,or not Fredholm together.Using a new spectrum defined in view of the CFI operator,we give the equivalence of Weyl’s theorem and property (ω) for T and its conjugate operator T* .In addition,the property (ω) for operator matrices is considered.     

A second order nonconforming rectangular finite element method for approximating Maxwell’s equations

The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell’s equations.Then the corresponding optimal error estimates are derived.The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O(h 3 ) ,properly one order higher than that of its interpolation error O(h 2 ) in the broken energy norm,where h is the subdivision parameter tending to zero.