The analytic torsion of a disc

In this article, we study the Reidemeister torsion and the analytic torsion of the m dimensional disc, with the Ray and Singer homology basis (Adv Math 7:145–210, 1971). We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger–Müller theorem. We use a formula proved by Brüning and Ma (GAFA 16:767–873, 2006) that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary (Lück, J Diff Geom 37:263–322, 1993). Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang (Asian J Math 4:695–714, 2000), and we compare the results. The results of these work were announced in the study of Hartmann et al. (BUMI 2:529–533, 2009).     

New monotonicity formulas for Ricci curvature and applications. I

We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov?CHausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop?CGromov volume comparison theorem and Perelman??s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these. Moreover, we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the Green function that we prove. This is parallel to the fact that Perelman??s monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li?CYau. In [CM4] one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross-sections of Einstein manifolds. Finally, there are obvious parallelisms between our monotonicity and the positive mass theorem of Schoen?CYau and Witten.     

A Berger-Green type inequality for compact Lorentzian manifolds

We give a Lorentzian metric on the null congruence associated with a timelike conformal vector field. A Liouville type theorem is proved and a boundedness for the volume of the null congruence, analogous to a well-known Berger-Green theorem in the Riemannian case, will be derived by studying conjugate points along null geodesics. As a consequence, several classification results on certain compact Lorentzian manifolds without conjugate points on its null geodesics are obtained. Finally, several properties of null geodesics of a natural Lorentzian metric on each odd-dimensional sphere have been found.

     

Semi-symmetric Lorentzian metrics and three-dimensional curvature homogeneity of order one

We completely classify three-dimensional semi-symmetric Lorentzian manifolds which are curvature homogeneous up to order one. Curvature restrictions for semi-symmetry turn out to be the same ones which ensure the existence on these manifolds of a degenerate parallel null line field (Chaichi et al., J. Phys. A, Math. Gen. 38, 841–850, 2005). Supported by funds of MURST, GNSAGA and University of Salento.     

A strong form of almost differentiability

We present a uniformization of Reeken’s macroscopic differentiability (see [5]), discuss its relations to uniform differentiability (see [6]) and classical continuous differentiability, prove the corresponding chain rule, Taylor’s theorem, mean value theorem, and inverse mapping theorem. An attempt to compare it with the observability (see [1, 4]) is made too.     

On the Existence of Min-Max Minimal Torus

In this paper, we will study the existence problem of min-max minimal torus. We use classical conformal invariant geometric variational methods. We prove a theorem about the existence of min-max minimal torus in Theorem 5.1. First we prove a strong uniformization result (Proposition 3.1) using the method of Ahlfors and Bers (Ann. Math. 72(2):385–404, 1960). Then we use this proposition to choose good parameterization for our min-max sequences. We prove a compactification result (Lemma 4.1) similar to that of Colding and Minicozzi (Width and finite extinction time of Ricci flow, [math.DG], 2007), and then give bubbling convergence results similar to that of Ding et al. (Invent. math. 165:225–242, 2006). In fact, we get an approximating result similar to the classical deformation lemma (Theorem 1.1).     

An Index Theorem for Band-Dominated Operators with Slowly Oscillating Coefficients (after Deundyak and Shteinberg)

We provide a proof of an index theorem for band-dominated operators with slowly oscillating coefficients. The statement is essentially the same as the main result of the announcement of Deundyak and Shteinberg (Funct Anal Appl 19(4):321–323, 1985), but our methods are very different from those hinted at there. The index theorem we prove can also be seen as a partial generalization to higher dimensions of the main result of the article of Rabinovich et al. (Integr Equ Oper Theory 49:221–238, 2004).     

Generalising Conduché’s Theorem

In a previous paper (Kasangian and Labella, J Pure Appl Algebra, 2009) we proved a form of Conduché’s theorem for LSymcat-categories, where L was a meet-semilattice monoid. The original theorem was proved in Conduché (CR Acad Sci Paris 275:A891–A894, 1972) for ordinary categories. We showed also that the “lifting factorisation condition” used to prove the theorem is strictly related to the notion of state for processes whose semantics is modeled by LSymcat-categories. In this note we resume the content of Kasangian and Labella (J Pure Appl Algebra, 2009) in order to generalise the theorem to other situations, mainly arising from computer science. We will consider PSymcat-categories, where P is slightly more general than a meet-semilattice monoid, in which the lifting factorisation condition for a PSymcat-functor still implies the existence of a right adjoint to its corresponding inverse image functor.     

Perturbations of finite networks and asymptotic periodicity of flow semigroups

We consider a transport process on a finite network and introduce a new special transformation of the underlying graph that is then used to yield an alternative proof of the main theorem on asymptotic periodicity of the flow semigroup proven in Kramar and Sikolya (Math. Z. 249:139–162, 2005). The perturbation technique we develop allows us to prove this theorem starting from a special case, thus greatly simplifying the proof. This technique is in Kunszenti-Kovács (Perturbations of networks and asymptotic periodicity of recurrent flows: the infinite case, Preprint, 2008) adapted to infinite networks, where it allows for significant generalisations of previous results.     

The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary space–time dimensions n + 1 ≥ 3. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.     

Multidimensional BSDEs with weak monotonicity and general growth generators

This paper aims at solving a multidimensional backward stochastic differential equation （BSDE） whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonieity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonieity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results.     

Veraverbeke’s theorem at large: on the maximum of some processes with negative drift and heavy tail innovations

Veraverbeke’s (Stoch Proc Appl 5:27–37, 1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke’s theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process, conditionally on a large maximum. Those results are obtained by using a general scheme of proof which we present in some detail and should be of value in other related problems.     

Modified Proximal-Point Algorithm for Maximal Monotone Operators in Banach Spaces

We introduce an iterative sequence for finding the solution to 0∈T(v), where T : E⇉E ^* is a maximal monotone operator in a smooth and uniformly convex Banach space E. This iterative procedure is a combination of iterative algorithms proposed by Kohsaka and Takahashi (Abstr. Appl. Anal. 3:239–249, 2004) and Kamamura, Kohsaka and Takahashi (Set-Valued Anal. 12:417–429, 2004). We prove a strong convergence theorem and a weak convergence theorem under different conditions respectively and give an estimate of the convergence rate of the algorithm. An application to minimization problems is given. This work was partially supported by the National Natural Sciences Grant 10671050 and the Heilongjiang Province Natural Sciences Grant A200607. The authors thank the referees for useful comments improving the presentation and Professor K. Kohsaka for pointing out Ref. 7.     

A Report on Canonical Null Curves and Screen Distributions for Lightlike Geometry

The general theory of lightlike submanifolds makes use of a non-degenerate screen distribution which is not unique and, therefore, the induced objects (starting from null curves) depend on the choice of a screen, which creates a problem. The purpose of this paper is to report on the existence of a canonical representation of null curves of Lorentzian manifolds and the choice of a canonical or a good screen for large classes of lightlike hypersurfaces of semi-Riemannian manifolds. We also prove a new theorem on the existence of an integrable canonical screen, subject to a geometric condition, and supported by a physical application.      

Eigenvalue problems and fixed point theorems for a class of positive nonlinear operators

We study the eigenvalue problems for a class of positive nonlinear operators defined on a cone in a Banach space. Using projective metric techniques and Schauder’s fixed-point theorem, we establish existence, uniqueness, monotonicity and continuity results for the eigensolutions. Moreover, the method leads to a result on the existence of a unique fixed point of the operator. Applications to nonlinear boundary-value problems, to differential delay equations and to matrix equations are considered.      

Lord Kelvin’s method of images in semigroup theory

We show that Lord Kelvin’s method of images is a way to prove generation theorems for semigroups of operators. To this end we exhibit three examples: a more direct semigroup-theoretic treatment of abstract delay differential equations, a new derivation of the form of the McKendrick semigroup, and a generation theorem for a semigroup describing kinase activity in the recent model of Kaźmierczak and Lipniacki (J. Theor. Biol. 259:291–296, 2009).     

Nonvanishing of Hecke L-Functions for CM Fields and Ranks of Abelian Varieties

In this paper we prove a nonvanishing theorem for central values of L-functions associated to a large class of algebraic Hecke characters of CM number fields. A key ingredient in the proof is an asymptotic formula for the average of these central values. We combine the nonvanishing theorem with work of Tian and Zhang [TiZ] to deduce that infinitely many of the CM abelian varieties associated to these Hecke characters have Mordell–Weil rank zero. Included among these abelian varieties are higher-dimensional analogues of the elliptic \mathbb Q{{\mathbb Q}} -curves A(D) of B. Gross [Gr].     

Perelman’s Entropy and Doubling Property on Riemannian Manifolds

The purpose of this work is to study some monotone functionals of the heat kernel on a complete Riemannian manifold with nonnegative Ricci curvature. In particular, we show that on these manifolds, the gradient estimate of Li and Yau (Acta Math. 156, 153–201, 1986), the gradient estimate of Ni (J. Geom. Anal. 14(1), 87–100, 2004), the monotonicity of the Perelman’s entropy and the volume doubling property are all consequences of an entropy inequality recently discovered by Baudoin and Garofalo, , 2009. The latter is a linearized version of a logarithmic Sobolev inequality that is due to D. Bakry and M. Ledoux (Rev. Mat. Iberoam. 22, 683–702, 2006).     

Differential calculus and integration of generalized functions over membranes

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144:13–29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144:13–29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green’s theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.     

The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes, we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective \mathbb Q{\mathbb Q}-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev’s structure theorem for the cone of nef curves of projective terminal threefolds.