On tame and wild bocses

We first give an alternative proof of the well-known Drozd's wild Theorem, which lowers down the dimension 43 to 20. Then we list more minimally wild bocses and discuss the possible differentials of the first arrow of a tame bocs, which is useful for reductions of bocses.     

Representation type of local and two-vertex bocses

Let κ be an algebraically closed field. It has been proved by Zhang and Xu that if a bocs is of tame representation type, then the degree of the diffierential of the first solid arrow must be less than or equal to 3. We will prove in the present paper that: The bocs is still wild when the degree of the diffierential of the first arrow is equal to 3. Especially, the bocs with only one solid arrow is of tame type if and only if the degree of the diffierential of the arrow is less than or equal to 2. Moreover,...     

Derived dynkin extensions

The finite dimensional tame hereditary algebras are associated with the extended Dynkin diagrams. An indecomposable module over such an algebra is either preprojective or preinjective or lies in a family of tubes whose tubular type is the corresponding Dynkin diagram. The study of one-point extensions by simple regular modules in such tubes was initiated in [Ri].

We generalise this approach by starting out with algebras which are derived equivalent to a tame hereditary algebra and considering one-point extensions by modules which are simple regular in tubes in the derived category. If the obtained tubular type is again a Dynkin diagram these algebras are called derived Dynkin extensions.

Our main theorem says that a representation infinite algebra is derived equivalent to a tame hereditary algebra iff it is an iterated derived Dynkin extension of a tame concealed algebra. As application we get a new proof of a theorem in [AS] about domestic tubular branch enlargements which uses the derived category instead of combinatorial arguments.     

Minimal algebras of infinite representation type with preprojective component

Let A be a finite dimensional, basic and connected algebra (associative, with 1) over an algebraically closed field k. Denote by e₁,...,e_n a complete set of primitive orthogonal idempotents in A and by A_i= A/Ae_iA. A is called a minimal algebra of infinite representation type provided A is itself of infinite representation type,whereas all A_i, 1≤i≤n,are of finite representation type. The main result gives the classification of the minimal algebras having a preprojective component in their Auslander-Reiten quiver. The classification is obtained by realizing that these algebras are essentially given by preprojective tilting modules over tame hereditary algebras.     

△-tame quasi-hereditary algebras

Let (K, M, H) be an upper triangular bimodule problem. Brüstle and Hille showed that the opposite algebra A of the endomorphism algebra of a projective generator P of the matrices category of (K, M, H) is quasi-hereditary, and there is an equivalence between the category of △-good modules of A and Mat(K, M). In this note, based on the tame theorem for bimodule problems, we show that if the algebra A associated with an upper triangular bimodule problem is of △-tame representation type, then the category F(△) has the homogeneous property, i.e. almost all modules in F(△) are isomorphic to their Auslander-Reiten translations. Moreover, if (K, M, H) is an upper triangular bipartite bimodule problem, then A is of △-tame representation type if and only if F(△) is homogeneous.     

Symplectic Geometry of Manifolds with Almost Nowhere Vanishing Closed 2-Form

We study local geometric properties of manifolds equipped with a closed 2-form nondegenerate at all points of a dense proper subset. We introduce the natural notion of tame singular point, at which the matrix of the 2-form degenerates in a regular way. We find a condition for Hamiltonian dynamical systems to be extended smoothly to tame singular points, generalize the Darboux theorem about the local reduction of the matrix of the 2-form to canonical form, and study the singular behavior of directional gradients.     

Chebyshev polynomials on symmetric matrices

In this paper we evaluate Chebyshev polynomials of the second kind on a class of symmetric integer matrices, namely on adjacency matrices of simply laced Dynkin and extended Dynkin diagrams. As an application of these results we explicitly calculate minimal projective resolutions of simple modules of symmetric algebras with radical cube zero that are of finite and tame representation type.     

TAME TWO-POINT ALGEBRAS WITHOUT LOOPS

We determine the representation type of the algebras whose quiver has precisely two vertices and admits no loops by listing all minimal wild algebras of this form. It turns out that such an algebra A is tame if and only if A/rad³ A is tame, and in this case A degenerates to a special biserial algebra. Moreover, A is wild if and only if it is controlled wild.     

Slice theorem and orbit type stratification in infinite dimensions

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.     

Geometric criteria for tame ramification

We prove an A’Campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field K. As a first application, we relate the orders of the tame monodromy eigenvalues on the ?-adic cohomology of a K-curve to the geometry of a relatively minimal sncd-model, and we show that the semi-stable reduction theorem and Saito’s criterion for cohomological tameness are immediate consequences of this result. As a second application, we compute the error term in the trace formula for smooth and proper K-varieties. We see that the validity of the trace formula would imply a partial generalization of Saito’s criterion to arbitrary dimension.     

Ultrametric and Tree Potential

In this article we study which infinite matrices are potential matrices. We tackle this problem in the ultrametric framework by studying infinite tree matrices and ultrametric matrices. For each tree matrix, we show the existence of an associated symmetric random walk and study its Green potential. We provide a representation theorem for harmonic functions that includes simple expressions for any increasing harmonic function and the Martin kernel. For ultrametric matrices, we supply probabilistic conditions to study its potential properties when immersed in its minimal tree matrix extension. C. Dellacherie thanks support from Nucleus Millennium P04-069-F for his visit to CMM-DIM at Santiago. The research of S. Martinez is supported by Nucleus Millennium Information and Randomness P04-069-F and by the BASAL CONICYT Program. The research of J. San Martin is supported by FONDAP and by the BASAL CONICYT Program.     

Maximal (Minimal) Conditional Expectation and European Option Pricing with Ambiguous Return Rate and Volatility

In this paper, we consider the problem of option pricing when return rate and volatility are ambiguous. Firstly we illustrate how to describe this ambiguous option pricing model by using set-valued differential inclusion and how to change the discussion of pricing bound problems of options into that of maximal and minimal conditional expectations. Secondly we discuss the properties of maximal and minimal conditional expectations, especially the representation theorem of maximal and minimal expectations. Finally we give the bounds of the European option pricing by using above theorems.     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.     

Cartan determinants for gentle algebras

The determinant of the Cartan matrix of a finite dimensional algebra is an invariant of the derived category and can be very helpful for derived equivalence classifications. In this paper we determine the determinants of the Cartan matrices for all gentle algebras. This is a class of algebras of tame representation type which occurs naturally in various places in representation theory. The definition of these algebras is of a purely combinatorial nature, and so are our formulae for the Cartan determinants.Received: 29 October 2004     

Δ-tame quasi-hereditary algebras

Let (K, M, H) be an upper triangular biomodule problem. Brüstle and Hille showed that the opposite algebra A of the endomorphism algebra of a projective generator P of the matrices category of (K, M, H) is quasi-hereditary, and there is an equivalence between the category of Δ-good modules of A and Mat(K, M). In this note, based on the tame theorem for bimodule problems, we show that if the algebra A associated with an upper triangular bimodule problem is of Δ-tame representation type, then the category F(Δ) has the homogeneous property, i.e. almost all modules in F(Δ) are isomorphic to their Auslander-Reiten translations. Moreover, if (K, M, H) is an upper triangular bipartite bimodule problem, then A is of Δ-tame representation type if and only if F(Δ) is homogeneous.     

Effective Flow Potentials for Anisotropic Polycrystals

In this paper a micromechanically based flow potential for anisotropic fcc polycrystals is derived that takes into account the crystallite orientation distribution function (codf) in terms of tensorial texture coefficients. The effective flow potential is based on a representation theorem for anisotropic scalar functions depending on a 2nd-order tensor. A priori unknown functions in the representation are determined by defining and solving explicitly a minimization problem over SO(3). Important analytical properties of the coefficient matrix of the minimization problem are discussed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

局部对称的黎曼流形中的极小子流形

主要研究了局部对称的黎曼流形中的定向紧致无边极小子流形的内蕴刚性问题,利用一个矩阵不等式,得到了这类子流形的一个刚性定理．所得结果部分改进了已有的一个结论．     

Uniqueness cases in odd-type groups of finite Morley rank

There is a longstanding conjecture, due to Gregory Cherlin andBoris Zilber, that all simple groups of finite Morley rank aresimple algebraic groups. One of the major theorems in the areais Borovik's trichotomy theorem. The ‘trichotomy’here is a case division of the generic minimal counterexampleswithin odd type, that is, groups with a large and divisibleSylow° 2-subgroup. The so-called ‘uniqueness case’in the trichotomy theorem is the existence of a proper 2-generatedcore. It is our aim to drive the presence of a proper 2-generatedcore to a contradiction, and hence bind the complexity of theSylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilberconjecture. This paper shows that the group in question is aminimal connected simple group and has a strongly embedded subgroup,a far stronger uniqueness case. As a corollary, a tame counterexampleto the Cherlin–Zilber conjecture has Prüfer rankat most two.     

A Berry-Esseen Bound for <Emphasis Type="Italic">U</Emphasis>-Statistics

The rate of convergence in the central limit theorem for nondegenerate U-statistics of n independent random variables is investigated under minimal sufficient moment conditions on canonical functions of the Hoeffding representation. Bibliography: 12 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 298, 2003, pp. 54–79.