Integral formulas for differential forms of type (p,q) on complex Finsler manifolds

Using the invariant integral kernel introduced by Demailly and Laurent-Thiebaut, complex Finsler metric and nonlinear connection associating with Chern-Finsler connection, we research the integral representation theory on complex Finsler manifolds. The Koppelman and Koppelman-Leray formulas are obtained, and the -equations are solved.     

Hodge decomposition theorem on strongly K?hler Finsler manifolds

Faran posed an open problem about analysis on complex Finsler spaces: Is there an analogue of the -Laplacian? Is there an analogue of Hodge theory? Under the assumption that (M, F) is a compact strongly K?hler Finsler manifold, we define a -Laplacian on the base manifold. Our result shows that the well-known Hodge decomposition theorem in K?hler manifolds is still true in the more general compact strongly K?hler Finsler manifolds. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

On microfunctions at the boundary along CR manifolds

Let X be a complex analytic manifold, a C ² submanifold, an openset with C ² boundary .Denote by (resp. ) the microlocalization along M (resp. ) of the sheaf of holomorphic functions.In the literature (cf. [A-G], [K-S 1,2])one encounters two classical results concerning the vanishing of the cohomology groups .The most general gives the vanishing outside a range of indices j whose length is equal to (with being the number of respectively positive, negative and null eigenvalues for themicrolocal Levi form ).The sharpest result gives the concentration in a single degree, provided that the difference is locally constant for near p (with for z the base point of p).The first result was restated for the complex in [D'A-Z 2], in the case codim We extend it here to any codimension and moreover we also restate for the second vanishing theorem.We also point out that the principle of our proof, related to a criterion for constancy of sheaves due to [K-S 1], is a quite new one.     

Eden-Staudacher and Beisert-Eden-Staudacher equations in the $$\mathcal{N} = 4$$ supersymmetric gauge theory

We investigate the Eden-Staudacher and Beisert-Eden-Staudacher equations for the anomalous dimension of twist-2 operators at a large spin s in the supersymmetric gauge theory. We reduce these equations to a set of linear algebraic equations and calculate their kernels analytically. We demonstrate that in the perturbation theory, the anomalous dimension is a sum of products of the Euler functions ζ(k) having the maximum transcendentality property. We also show that at a large coupling, the “singular” solution of the Beisert-Eden-Staudacher equation reproduces the anomalous dimension constants predicted from the string side of the AdS/CFT correspondence. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 117–129, April, 2008.     

Quasiconformal Mappings and Solutions of the Dispersionless KP Hierarchy

A formalism for studying dispersionless integrable hierarchies is applied to the dispersionless KP (dKP) hierarchy. We relate this formalism to the theory of quasiconformal mappings on the plane and present some classes of explicit solutions of the dKP hierarchy.     

Strong laws of large numbers for weighted sums of $$\tilde \rho $$ -mixing random variables

Strong laws are established for linear statistics that are weighted sums of a -mixing random sample. The results obtained generalize the results of Baxter et al. [SLLN for weighted independent indentically distributed random variables, J. Theoret. Probab. 17 (2004), 165–181] to -mixing random variables. This paper is supported by Key discipline of Zhejiang Province (Key discipline of Statistics of Zhejiang Gongshang University) and National Natural Science Foundation of China.     

A Modal Logic Based on Linearly Ordered f-Spaces

A modal logic associated with the -spaces introduced by Ershov is examined. We construct a modal calculus that is complete w.r.t. the class of all strictly linearly ordered -frames, and the class of all strictly linearly ordered -frames.     

Anomalous dimensions of Wilson operators in the N = 4\mathcal{N} = 4 supersymmetric Yang-Mills theory

We present results for the universal anomalous dimension γ_un(j) of Wilson twist-2 operators in the supersymmetric Yang-Mills theory in the first three orders of the perturbation theory. We obtain these expressions by extracting the most complicated terms from the corresponding anomalous dimensions in QCD. The result obtained agrees with the hypothesis of the integrability of the supersymmetric Yang-Mills theory in the context of the AdS/CFT correspondence. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 249–262, February, 2007.     

Commutativity of the Arens product in lattice ordered algebras

Let be an Abelian Archimedean lattice ordered algebra. The order bidual furnished with the Arens product is again a lattice ordered algebra. We show that the order continuous order bidual is Abelian. This solves an open problem and improves a result of Scheffold, who proved it for the case of normed lattice ordered algebras. The proof is based on the up-down-up approximation of positive elements in the order continuous order bidual by elements in the canonical image of in Components of positive elements in are characterized and the result is applied to the Arens product of -and almost -algebras.     

On indecomposable modules over the Virasoro algebra

It is proved that an indecomposable Harish- Chandra module over the Virasoro algebra must be (i) a uniformly bounded module, or (ii) a module in Category , or (iii) a module in Category , or ( iv) a module which contains the trivial module as one of its composition factors.     

Projective modules and prime submodules

In this paper, we use Zorn’s Lemma, multiplicatively closed subsets and saturated closed subsets for the following two topics (i) The existence of prime submodules in some cases (ii) The proof that submodules with a certain property satisfy the radical formula. We also give a partial characterization of a submodule of a projective module which satisfies the prime property.     

The Abelian category of quotients of $$\mathcal{L}\mathfrak{F}$$ -spaces

We construct the category of quotients of -spaces and we show that it is Abelian. This answers a question of L. Waelbroeck from 1990.     

Variational measures related to local systems and the Ward property of \mathcal{P}-adic path bases

Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a -adic path system that defines a differentiation basis which does not possess Ward property.     

On the parameter dependence of the real cubic surfaces in arnold’s form

We consider the topological classification of cubic surfaces which are obtained as intersection of the sphere with the algebraic variety defined by the zeroes of a homogeneous cubic polynomial in Arnold’s normal form. This classification is based on the parameters appearing in this normal form, obtaining a correspondence between the parameters of the surface and its topological type. General classifications of cubic surfaces are made in the projective space ℙ³(ℝ), but our method, based on a very simple combinatorial procedure is easier to implement in . We split the cubic surfaces parameter space into ten equivalence classes.     

Polynomials with All Zeros Real and in a Prescribed Interval

We provide a characterization of the real-valued univariate polynomials that have only real zeros, all in a prescribed interval [a,b]. The conditions are stated in terms of positive semidefiniteness of related Hankel matrices.     

Optimal Ternary Linear Rate 1/2 Codes

In this paper, we classify all optimal linear[n, n/2] codes up to length 12. We show that thereis a unique optimal [10, 5, 5] code up to equivalence.     

How to realize a Lie algebra by vector fields

We describe an algorithm for embedding a finite-dimensional Lie algebra (superalgebra) into a Lie algebra (superalgebra) of vector fields that is suitable for a ground field of any characteristic and also a way to select the Cartan, complete, and partial prolongations of the Lie algebra of vector fields using differential equations. We illustrate the algorithm with the example of Cartan’s interpretation of the exceptional simple Lie algebra (2) as the Lie algebra preserving a certain nonintegrable distribution and also several other examples. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 450–469, June, 2006.     

Canonical connections on paracontact manifolds

The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A -homothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is proved that their scalar curvature is a constant, and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with -homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.      

The Local Stark Conjecture at a Real Place

A refinement of the rank 1 Abelian Stark conjecture has been formulated by B.Gross. This conjecture predicts some -adic analytic nature of a modification of the Stark unit. The conjecture makes perfect sense even when is an Archimedean place. Here we consider the conjecture when is a real place, and interpret it in terms of 2-adic properties of special values of L-functions. We prove the conjecture for CM extensions; here the original Stark conjecture is uninteresting, but the refined conjecture is nontrivial. In more generality, we show that, under mild hypotheses, if the subgroup of the Galois group generated by complex conjugations has less than full rank, then the refined conjecture implies that the Stark unit should be a square. This phenomenon has been discovered by Dummit and Hayes in a particular type of situation. We show that it should hold in much greater generality.     

A Korovkin type approximation theorems via $$\mathcal{I}$$ -convergence

Using the concept of -convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class.