New light on Hensel''s lemma

The historical development of Hensel's lemma is briefly discussed (Section 1). Using Newton polygons, a simple proof of a general Hensel's lemma for separable polynomials over Henselian fields is given (Section 3). For polynomials over algebraically closed, valued fields, best possible results on continuity of roots (Section 4) and continuity of factors (Section 6) are demonstrated. Using this and a general Krasner's lemma (Section 7), we give a short proof of a general Hensel's lemma and show that it is, in a certain sense, best possible (Section 8). All valuations here are non-Archimedean and of arbitrary rank. The article is practically self-contained.     

Memory estimation of inverse operators

We use methods of harmonic analysis and group representation theory to estimate the memory decay of the inverse operators in Banach spaces. The memory of the operators is defined using the notion of the Beurling spectrum. We obtain a general continuous non-commutative version of the celebrated Wiener's Tauberian Lemma with estimates of the “Fourier coefficients” of inverse operators. In particular, we generalize various estimates of the elements of the inverse matrices. The results are illustrated with a variety of examples including integral and integro-differential operators.     

Topological differential fields

We consider first-order theories of topological fields admitting a model-completion and their expansion to differential fields (requiring no interaction between the derivation and the other primitives of the language). We give a criterion under which the expansion still admits a model-completion which we axiomatize. It generalizes previous results due to M. Singer for ordered differential fields and of C. Michaux for valued differential fields. As a corollary, we show a transfer result for the NIP property. We also give a geometrical axiomatization of that model-completion. Then, for certain differential valued fields, we extend the positive answer of Hilbert’s seventeenth problem and we prove an Ax-Kochen-Ershov theorem. Similarly, we consider first-order theories of topological fields admitting a model-companion and their expansion to differential fields, and under a similar criterion as before, we show that the expansion still admits a model-companion. This last result can be compared with those of M. Tressl: on one hand we are only dealing with a single derivation whereas he is dealing with several, on the other hand we are not restricting ourselves to definable expansions of the ring language, taking advantage of our topological context. We apply our results to fields endowed with several valuations (respectively several orders).     

Another proof of the kuhn-tucker theorem ∗

A proof of the Kuhn-Tucker theorem is given using Zorn's Lemma     

Adequate predimension inequalities in differential fields

In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be adequate. Adequacy was informally introduced by Zilber, and here we give a precise definition in a quite general context. We also discuss the connection of this problem to definability of derivations in the reducts of differentially closed fields. The Ax-Schanuel inequality for the exponential differential equation (proved by Ax) and its analogue for the differential equation of the j-function (established by Pila and Tsimerman) are our main examples of predimensions. We carry out a Hrushovski construction with the latter predimension and obtain a natural candidate for the first-order theory of the differential equation of the j-function. It is analogous to Kirby's axiomatisation of the theory of the exponential differential equation (which in turn is based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences. In joint work with Sebastian Eterovi? and Jonathan Kirby we have recently proven that the axiomatisation obtained in this paper is indeed an axiomatisation of the theory of the differential equation of the j-function, that is, the Ax-Schanuel inequality for the j-function is adequate.     

李超代数的Schur引理

对经典文献中李超代数的Schur引理的各种表述进行比对研究.给出李超代数的Schur引理的明晰表达及证明;作为应用,刻画了一类Heisenberg李超代数的不可约模的自同态.进而利用奇偶函子和符号映射刻画了李超代数模同态的各种定义之间的关系.     

A new approach for the derivation of bounds for the Jensen difference

There are many useful applications of Jensen's inequality in several fields of science, and due to this reason, a lot of results are devoted to this inequality in the literature. The main theme of this article is to present a new method of finding estimates of the Jensen difference for differentiable functions. By applying definition of convex function, and integral Jensen's inequality for concave function in the identity pertaining the Jensen difference, we derive bounds for the Jensen difference. We present integral version of the bounds in Riemann sense as well. The sharpness of the proposed bounds through examples are discussed, and we conclude that the proposed bounds are better than some existing bounds even with weaker conditions. Also, we present some new variants of the Hermite–Hadamard and Hölder inequalities and some new inequalities for geometric, quasi-arithmetic, and power means. Finally, we give some applications in information theory.     

An extension of Lewis's Lemma,renormalization of harmonic and analytic functions,and normal families

An extension of a lemma due to J. Lewis is established and is used to give rapid proofs of some classical theorems in complex function theory such as Montel's theorem and Miranda's theorem. Another application of Lewis's Lemma yields a general normality criterion for families of harmonic and holomorphic functions. This criterion permits a quick proof of Bloch's classical covering theorem for holomorphic functions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A combinatorial proof of the Removal Lemma for Groups

Green [B. Green, A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005) 340-376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.     

Henselian valued fields: a constructive point of view

This article is a logical continuation of the Henri Lombardi and Franz‐Viktor Kuhlmann article [9]. We address some classical points of the theory of valued fields with an elementary and constructive point of view. We deal with Krull valuations, and not simply discrete valuations. First of all, we show how (in the spirit of [9]) to construct the Henselization of a valued field; we restrict to fields in which one has at one's disposal algorithmic tools to test the nullity or the valuation ring membership. It is therefore a work that differs as much in spirit as in field of application from that of Mines, Richman and Bridges (cf. [10]), who address the framework of Heyting fields and discrete valuation. We show then in a constructive way a batch of classical results in Henselian fields, notably factorization criteria and Krasner's Lemma. We conclude by a construction of the inertia field of a valued field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on kernels and Sperner’s Lemma

The kernel-solvability of perfect graphs was first proved by Boros and Gurvich, and later Aharoni and Holzman gave a shorter proof. Both proofs were based on Scarf’s Lemma. In this note we show that a very simple proof can be given using a polyhedral version of Sperner’s Lemma. In addition, we extend the Boros–Gurvich theorem to h-perfect graphs and to a more general setting.     

Lie symmetry analysis to Fisher's equation with time fractional order

The aim of this letter is to apply the Lie group analysis method to the Fisher''s equation with time fractional order. We considered the symmetry analysis, explicit solutions to the time fractional Fisher''s(TFF) equations with Riemann-Liouville (R-L) derivative. The time fractional Fisher''s is reduced to respective nonlinear ordinary differential equation(ODE) of fractional order. We solve the reduced fractional ODE using an explicit power series method.     

黑洞附近修正Stefan-Boltzmann定律

该文在一般球对称静态黑洞背景下,用WKB近似法得到黑洞附近修正的Stefan-Boltzmann定律. 发现在黑洞事件视界附近,由于场自旋的存在,结果中除了类似于平直时空的主导项外, 多了一个和局域温度二次方成正比的附加项.该项的出现暗示黑洞的辐射可能不是精确热的.     

An Axiomatisation of the Conditionals of Post's Many Valued Logics

The paper provides a method for a uniform complete Hilbert-style axiomatisation of Post's (m, u)-conditionals and Post's negation, where m is the number of truth values and u is the number of designated truth values (cf. [5]). The main feature of the technique which we employ in this proof generalises the well-known Kalmár Lemma which was used by its author in his completeness argument for the ordinary, two-valued logic (cf. [2]).     

关于Morse引理的推广

Morse Lemma是奇点理论中一个极为重要的结论。[1]的作者称其文中的定理1和定理2是Morse Lemma的推广。为此我们愿就[1]中的几个问题与[1]的作者商榷。     

Reformulation of Hensel’s Lemma and extension of a theorem of Ore

In this paper, we extend the theorem of Ore regarding factorization of polynomials over p-adic numbers to henselian valued fields of arbitrary rank thereby generalizing the main results of Khanduja and Kumar (J Pure Appl Algebra 216:2648–2656, 2012) and Cohen et al. (Mathematika 47:173–196, 2000). As an application, we derive the analogue of Dedekind’s Theorem regarding splitting of rational primes in algebraic number fields as well as of its converse for general valued fields extending similar results proved for discrete valued fields in Khanduja and Kumar (Int J Number Theory 4:1019–1025, 2008). The generalized version of Ore’s Theorem leads to an extension of a result of Weintraub dealing with a generalization of Eisenstein Irreducibility Criterion (cf. Weintraub in Proc Am Math Soc 141:1159–1160, 2013). We also give a reformulation of Hensel’s Lemma for polynomials with coefficients in henselian valued fields which is used in the proof of the extended Ore’s Theorem and was proved in Khanduja and Kumar (J Algebra Appl 12:1250125, 2013) in the particular case of complete rank one valued fields.     

A study on the convergence conditions of generalized differential transform method

This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non‐linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd.     

Bishop's Lemma

Bishop's Lemma is a centrepiece in the development of constructive analysis. We show that

• 1. its proof requires some form of the axiom of choice; and that
• 2. the completeness requirement in Bishop's Lemma can be weakened and that there is a vast class of non‐complete spaces that Bishop's Lemma applies to.

     

THE HENSTOCK INTEGRAL FOR FUNCTIONS WITH VALUES IN NUCLEAR SPACES AND THE HENSTOCK LEMMA

This paper shows that Henstock‘s Lemma holds for functions with values in a countably Hilbert space, where the Henstock integral is defined as a natural extension of the resl valued case.     

Non-integrability of the Karabut system

In order to characterize the solitary wave in a fluid of finite depth, Witting introduced a specific power series (the Witting series). Karabut demonstrated that the problem of summation of the Witting series is brought to the integration of a particular system of ordinary differential equations and solved this system in the cases when the number of the equations is three or four. We give a simple proof that the Karabut system of five equations is already non-integrable in non-Hamiltonian sense using the Differential Galois approach.