A central limit theorem for Gibbs measures relative to Brownian motion

We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position to use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.     

The giant vortex state for a Bose–Einstein condensate in a rotating anharmonic trap: Extreme rotation regimes

We study the fast rotation limit for a Bose–Einstein condensate in a quadratic plus quartic confining potential within the framework of the two-dimensional Gross–Pitaevskii energy functional. As the rotation speed tends to infinity with a proper scaling of the other parameters in the model, a linear limit problem appears for which we are able to derive precise energy estimates. We prove that the energy and density asymptotics of the problem can be obtained by minimizing a simplified one-dimensional energy functional. In the case of a fixed coupling constant we also prove that a giant vortex state appears. It is an annulus with pure irrotational flow encircling a central low-density hole around which there is a macroscopic phase circulation.     

Variations on R. Schwartz��s Inequality for the Schwarzian Derivative

R. Schwartz??s inequality provides an upper bound for the Schwarzian derivative of a parameterization of a circle in the complex plane and on the potential of Hill??s equation with coexisting periodic solutions. We prove a discrete version of this inequality and obtain a version of the planar Blaschke?CSantalo inequality for not necessarily convex polygons. In the proof, we use some formulas from the theory of frieze patterns. We consider a centro-affine analog of Lük???s inequality for the average squared length of a chord subtending a fixed arc length of a curve??the role of the squared length played by the area??and prove that the central ellipses are local minima of the respective functionals on the space of star-shaped centrally symmetric curves. We conjecture that the central ellipses are global minima. In an appendix, we relate the Blaschke?CSantalo and Mahler inequalities with the asymptotic dynamics of outer billiards at infinity.     

Good candidates for least area soap films

Soap films are presented as potential global area minimizers subject to a topological constraint. Experimentally, this constraint is the shape of the soapy water in a soap film experiment. In this context, soap films which are probable area minimizers for rectangular n-prisms are described. By allowing area minimizers which arise as deformations of higher genus surfaces, we are able to discover previously unknown soap films spanning rectangular n-prisms with large aspect ratios and n ≥ 5. For n = 3, 4, 5, we show that the central film contracts to a point as the aspect ratio of the prism increases. We also prove that the area of the central hexagon for a soap film spanning a tall 6-prism approaches zero like (height)^?4 as the height approaches infinity, provided we fix the length of the hexagon base. Finally, we prove that, if the aspect ratio is large enough, the soap film produced experimentally spanning a 4-prism has films which look planar but in reality are non-planar.     

Some non-interior path-following methods based on a scaled central path for linear complementarity problems

In this paper we present some non-interior path-following methods for linear complementarity problems. Instead of using the standard central path we use a scaled central path. Based on this new central path, we first give a feasible non-interior path-following method for linear complementarity problems. And then we extend it to an infeasible method. After proving the boundedness of the neighborhood, we prove the convergence of our method. Another point we should present is that we prove the local quadratic convergence of feasible method without the assumption of strict complementarity at the solution.     

Long-step strategies in interior-point primal-dual methods

In this paper we analyze from a unique point of view the behavior of path-following and primal-dual potential reduction methods on nonlinear conic problems. We demonstrate that most interior-point methods with efficiency estimate can be considered as different strategies of minimizing aconvex primal-dual potential function in an extended primal-dual space. Their efficiency estimate is a direct consequence of large local norm of the gradient of the potential function along a central path. It is shown that the neighborhood of this path is a region of the fastest decrease of the potential. Therefore the long-step path-following methods are, in a sense, the best potential-reduction strategies. We present three examples of such long-step strategies. We prove also an efficiency estimate for a pure primal-dual potential reduction method, which can be considered as an implementation of apenalty strategy based on a functional proximity measure. Using the convex primal dual potential, we prove efficiency estimates for Karmarkar-type and Dikin-type methods as applied to a homogeneous reformulation of the initial primal-dual problem.     

On the Central Paths in Symmetric Cone Programming

This paper is devoted to the study of optimal solutions of symmetric cone programs by means of the asymptotic behavior of central paths with respect to a broad class of barrier functions. This class is, for instance, larger than that typically found in the literature for semidefinite positive programming. In this general framework, we prove the existence and the convergence of primal, dual and primal–dual central paths. We are then able to establish concrete characterizations of the limit points of these central paths for specific subclasses. Indeed, for the class of barrier functions defined at the origin, we prove that the limit point of a primal central path minimizes the corresponding barrier function over the solution set of the studied symmetric cone program. In addition, we show that the limit points of the primal and dual central paths lie in the relative interior of the primal and dual solution sets for the case of the logarithm and modified logarithm barriers.     

On the second cohomology group in semi-abelian categories

We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild-Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and only if it admits a universal central extension; we show how the second Barr-Beck cohomology group classifies isomorphism classes of central extensions; we prove a universal coefficient theorem to explain the relationship with homology.     

Harmonic functions on [IN] and central hypergroups

In this paper we introduce the notions of [I N] and [S I N]-hypergroups and prove a Choquet-Deny type theorem for [I N] and central hypergroups. More precisely, we prove a Liouville theorem for bounded harmonic functions on a class of [I N]-hypergroups. Further, we show that positive harmonic functions on [I N]-hypergroups are integrals of exponential functions. Similar results are proved for [S I N] and central hypergroups.     

关于4-体问题中心构型的一点研究

N-体问题的中心构型是应用数学领域广泛研究的问题．关于N-体问题的中心构型已有许多研究结果．但是对于n≥4,其中心构型解的计算是比较困难的．作者运用Wu-Ritt零点分解方法和子结式序列研究了一般的平面4体中心构型问题,给出了这类4体中心构型问题的解析解,从而证明了一类平面牛顿4-体问题的中心构型个数是有限的．     

Annihilation Theorem and Separation Theorem for basic classical Lie superalgebras

In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

     

Congruence extension,Hamiltonian and Abelian properties in locally finite varieties

We prove that in a locally finite variety with the congruence extension property, locally solvable congruences are central and locally solvable algebras are Hamiltonian. Also, we prove that a maximal subuniverse of a finite algebra in an Abelian variety is identical with an equivalence class of some congruence.Presented by H. P. Gumm.     

Why do so many cubature formulae have so many positive weights?

We consider cubature formulae which are invariant with respect to a transformation group and prove sufficient conditions for such formulae to have positive weights. This is worked out for different symmetries: we consider central symmetric, symmetric and fully symmetric cubature formulae. The theoretical results are illustrated with examples.     

Minami’s Estimate: Beyond Rank One Perturbation and Monotonicity

In this note, we prove Minami’s estimate for a class of discrete alloy-type models with a sign-changing single-site potential of finite support. We apply Minami’s estimate to prove Poisson statistics for the energy level spacing. Our result is valid for random potentials which are in a certain sense sufficiently close to the standard Anderson potential (rank one perturbations coupled with i.i.d. random variables).     

On the Central Limit Theorem and Its Weak Invariance Principle for Strongly Mixing Sequences with Values in a Hilbert Space via Martingale Approximation

In this paper we not only prove an extension to Hilbert spaces of a sharp central limit theorem for strongly real-valued mixing sequences, but also slightly improve it. The proof is mainly based on the Bernstein blocking technique and approximations by martingale differences. Moreover, we derive also the corresponding functional central limit theorem.     

Central semicommutative rings

A ring R is central semicommutative if ab = 0 implies that aRb ? Z(R) for any a, b ∈ R. Since every semicommutative ring is central semicommutative, we study sufficient condition for central semicommutative rings to be semicommutative. We prove that some results of semicommutative rings can be extended to central semicommutative rings for this general settings, in particular, it is shown that every central semicommutative ring is nil-semicommutative. We show that the class of central semicommutative rings lies strictly between classes of semicommutative rings and abelian rings. For an Armendariz ring R, we prove that R is central semicommutative if and only if the polynomial ring R[x] is central semicommutative. Moreover, for a central semicommutative ring R, it is proven that (1) R is strongly regular if and only if R is a left GP-V′-ring whose maximal essential left ideals are GW-ideals if and only if R is a left GP-V′-ring whose maximal essential right ideals are GW-ideals. (2) If R is a left SF and central semicommutative ring, then R is a strongly regular ring.     

Asymptotic behavior of a tagged particle in simple exclusion processes

We review in this article central limit theorems for a tagged particle in the simple exclusion process. In the first two sections we present a general method to prove central limit theorems for additive functional of Markov processes. These results are then applied to the case of a tagged particle in the exclusion process. Related questions, such as smoothness of the diffusion coefficient and finite dimensional approximations, are considered in the last section.     

Relationships between the canonical ascending and descending central series of ideals of an associative algebra

We consider the canonical descending and ascending central series of ideals of an associative algebra. In particular, we prove that some ideal in the descending central series is finite-dimensional if and only if some ideal in the ascending central series is finite-codimensional. This result is the associative algebra analogue of results due to Reinhold Baer and Philip Hall in group theory and Ian Stewart in Lie algebra. We also prove various related results.     

On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials

The present paper is devoted to the study of spectral properties of random Schrödinger operators. Using a finite section method for Toeplitz matrices, we prove a Wegner estimate for some alloy type models where the single site potential is allowed to change sign. The results apply to the corresponding discrete model, too. In certain disorder regimes we are able to prove the Lipschitz continuity of the integrated density of states and/or localization near spectral edges.     

A Fractional Muckenhoupt–Wheeden Theorem and its Consequences

In the 1970s Muckenhoupt and Wheeden made several conjectures relating two weight norm inequalities for the Hardy-Littlewood maximal operator to such inequalities for singular integrals. Using techniques developed for the recent proof of the A ₂ conjecture we prove a related pair of conjectures linking the Riesz potential and the fractional maximal operator. As a consequence we are able to prove a number of sharp one and two weight norm inequalities for the Riesz potential.