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In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if A is a separable algebra which is either simple or stable, then the corona of A has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit. [ABSTRACT FROM AUTHOR]

In this paper, we first define a new family of Yoneda algebras, called F-Auslander-Yoneda algebras, in triangulated categories by introducing the notion of admissible sets F in N, which includes higher cohomologies indexed by F, and then present a general method to construct a family of new derived equivalences for these F-Auslander-Yoneda algebras (not necessarily Artin algebras), where the choices of the parameters F are rather abundant. Among applications of our method are the following results: (1) if A is a self-injective Artin algebra, then, for any A-module X and for any admissible set F in N, the F-Auslander-Yoneda algebras of A ⊕ X and A ⊕ OA(X) are derived equivalent, where O is the Heller loop operator. (2) Suppose that A and B are representation-finite self-injective algebras with additive generators AX and BY, respectively. If A and B are derived equivalent, then so are the F-Auslander-Yoneda algebras of X and Y for any admissible set F. In particular, the Auslander algebras of A and B are derived equivalent. The converse of this statement is open. Further, motivated by these derived equivalences between F-Auslander-Yoneda algebras, we show, among other results, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factoring out socles. [ABSTRACT FROM AUTHOR]

In an earlier paper, we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras. Here, we consider full quivers (as well as pseudo-quivers) as a combinatoric tool in order to describe PI-varieties of algebras. Each full quiver is naturally associated to a polynomial that encapsulates trace-like properties of the underlying algebra. [ABSTRACT FROM AUTHOR]

Let G be a topological group. Then the based loopspace OG of G is an algebra over the cacti operad, while the double loopspace O2BG of the classifying space of G is an algebra over the framed little discs operad. This paper shows that these two algebras are equivalent, in the sense that they are weakly equivalent E-algebras, where E is an operad weakly equivalent to both framed little discs and cacti. We recover the equivalence between cacti and framed little discs, and Menichi's isomorphism between the BV-algebras H*(OG) and H*(O2BG). [ABSTRACT FROM AUTHOR]

In this paper we study the stack τg of smooth triple covers of a conic; when g ≥ 5 this stack is embedded Mg as the locus of trigonal curves. We show that τ is a quotient [Ug/Γg/], where Γg/ is a certain algebraic group and Ug/ is an open subscheme of a Γg/-equivariant vector bundle over an open subscheme of a representation of Γg/. Using this, we compute the integral Picard group of τg when g/ > 1. τhe main tools are a result of Miranda that describes a flat finite triple cover of a scheme S as given by a locally free sheaf E of rank two on S, with a section of Sym3E ⊗detEv, and a new description of the stack of globally generated locally free sheaves of fixed rank and degree on a projective line as a quotient stack. [ABSTRACT FROM AUTHOR]

In this paper, we first give a slight improvement of Yamanoi's truncated second main theorem for holomorphic maps into abelian varieties. We then use the result to study the uniqueness problem for such maps. The results obtained generalize and improve E. M. Schmid's uniqueness theorem for holomorphic maps into elliptic curves. In the last section, we consider algebraic dependence for a finite collection of holomorphic curves into an abelian variety. [ABSTRACT FROM AUTHOR]

This paper studies quadratic and Koszul duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We present a very general definition of quadratic and Koszul duality functors backed up by explicit examples. This generalizes the work of Beilinson, Ginzburg, and Soergel, 1996, in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also define a ``Koszul'' duality {it functor} for not necessarily Koszul categories. As an illustration of the techniques we reprove the Koszul duality (Ryom-Hansen, 2004) of translation and Zuckerman functors for the classical category $mathcal {O}$ in a quite elementary and explicit way. From this we deduce a conjecture of Bernstein, Frenkel, and Khovanov, 1999. As applications we propose a definition of a ``Koszul'' dual category for integral blocks of Harish-Chandra bimodules and for blocks outside the critical hyperplanes for the Kac-Moody category $mathcal {O}$. [ABSTRACT FROM AUTHOR]

In this paper we let $G$ be a simple algebraic group and $r$ be a natural number, and consider the codimension in $G$ of the variety of elements $g\in G$ satisfying $g^r=1$. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if $G$ is of adjoint type. [ABSTRACT FROM AUTHOR]

This paper is dedicated to a description of the poles of the Igusa local zeta function $Z(s,f,v)$ when $f(x,y)$ satisfies a new non-degeneracy condition called \textit{arithmetic non-degeneracy}. More precisely, we attach to each polynomial $f(x,y)$ a collection of convex sets $\Gamma ^{A}(f)=\left{ \Gamma _{f,1},\dots ,\Gamma _{f,l_{0}}\right} $ called the \textit{arithmetic Newton polygon} of $f(x,y)$, and introduce the notion of \textit{arithmetic non-degeneracy with respect to }$\Gamma ^{A}(f)$. If $L_{v}$ is a $p$-adic field, and $f(x,y)\in L_{v}\left[ x,y \right] $ is arithmetically non-degenerate, then the poles of $Z(s,f,v)$ can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets $\Gamma _{f,1},\dots , \Gamma _{f,l_{0}}$. Moreover, the proof of the main result gives an effective procedure for computing $Z(s,f,v)$. [ABSTRACT FROM AUTHOR]

In this paper we show that the $\varrho$-th order variation operator, for both the Fejér and Poisson kernels, are bounded from $L^p$ to $L^p$, $12$. Counterexamples are given if $\varrho =2$. [ABSTRACT FROM AUTHOR]

Let $A=\oplus_{g\in G}A_g$ be a $G$-graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component $A_e$ to that of the whole of $A$, in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where $A$ is finite dimensional and $A_e$ has polynomial growth. [ABSTRACT FROM AUTHOR]

This paper studies coordinates in two variables over a $\mathbb{Q}$-algebra. It gives several ways to characterize such coordinates. Also, various results about coordinates in two variables that were previously only known for fields, are extended to arbitrary $\mathbb{Q}$-algebras. [ABSTRACT FROM AUTHOR]

Let $\mathcal{G}\to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\mathcal{G}^i(Y)$. These groups satisfy the usual properties of the equivariant $K$-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle $\mathcal{G}\to B$. As an application, we define a gauge-equivariant index for a family of elliptic operators $(P_b)_{b \in B}$ invariant with respect to the action of $\mathcal{G}\to B$, which, in this approach, is an element of $K_\mathcal{G}^0(B)$. We then give another definition of the gauge-equivariant index as an element of $K_0(C^*(\mathcal{G}))$, the $K$-theory group of the Banach algebra $C^*(\mathcal{G})$. We prove that $K_0(C^*(\mathcal{G})) \simeq K^0_\mathcal{G}(\mathcal{G})$ and that the two definitions of the gauge-equivariant index are equivalent. The algebra $C^*(\mathcal{G})$ is the algebra of continuous sections of a certain field of $C^*$-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant $K$-theory groups are thus examples of twisted $K$-theory groups, which have recently turned out to be useful in the study of Ramond--Ramond fields. [ABSTRACT FROM AUTHOR]

Jordan and alternative tori are the coordinate algebras of extended affine Lie algebras of types $\textup{A}_1$ and $\textup{A}_2$. In this paper we show that the derivation algebra of a Jordan torus is a semidirect product of the ideal of inner derivations and the subalgebra of central derivations. In the course of proving this result, we investigate derivations of the more general class of division graded Jordan and alternative algebras. We also describe invariant forms of these algebras. [ABSTRACT FROM AUTHOR]

In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics. The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point $p\in\operatorname{int}{{\mathbf R}^n_+}$ and the carrying simplex of the system lies to one side of its tangent hyperplane at $p$, then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems. [ABSTRACT FROM AUTHOR]

Let $\mu$ be a Radon measure on $\mathbb{R}^d$, which may be nondoubling. The only condition that $\mu$ must satisfy is the size condition $\mu(B(x,r))\leq Cr^n$, for some fixed $0

In this paper we prove that the moduli space of rank $2n$ symplectic instanton bundles on ${\mathbb{P}^{2n+1}}$, defined from the well-known monad condition, is affine. This result was not known even in the case $n=1$, where by Atiyah, Drinfeld, Hitchin, and Manin in 1978 the real instanton bundles correspond to self-dual Yang Mills $Sp(1)$-connections over the $4$-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multidimensional matrices representing the instanton bundles. [ABSTRACT FROM AUTHOR]

This paper studies the equivariant cobordism classification of all involutions fixing a disjoint union of an odd-dimensional real projective space $\mathbb{RP}^j$ with its normal bundle nonbounding and a Dold manifold $P(h,i)$ with $h>0$ and $i>0$. For odd $h$, the complete analysis of the equivariant cobordism classes of such involutions is given except that the upper and lower bounds on codimension of $P(h,i)$ may not be best possible; for even $h$, the problem may be reduced to the problem for even projective spaces. [ABSTRACT FROM AUTHOR]

Recently Johnson and Kollár determined the complete list of anticanonically embedded quasi-smooth log del Pezzo surfaces in weighted projective $3$-spaces. They also proved that many of those surfaces admit a Kähler-Einstein metric, and that some of them do not have tigers. The aim of this paper is to settle the question of the existence of Kähler-Einstein metrics and tigers for those surfaces for which the question was left open. In order to do so, we will use techniques developed earlier by Nadel, Demailly and Kollár. [ABSTRACT FROM AUTHOR]

Let $\mathcal{S}$ be a finite set of powers of $p$ containing 1. It is known that for some choices of $\mathcal{S}$, if $P$ is a finite $p$-group whose set of character degrees is $\mathcal{S}$, then the nilpotence class of $P$ is bounded by some integer that depends on $\mathcal{S}$, while for some other choices of $\mathcal{S}$ such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set $\mathcal{S}$ is class bounding if and only if $p\notin\mathcal{S}$. In this article we provide a new approach to this problem. Our main result shows the relevance of certain $p$-adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets $\mathcal{S}$ such that $p\notin\mathcal{S}$. [ABSTRACT FROM AUTHOR]

This paper concerns the rational cohomology ring of the complement $M$ of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for $M$. Inside it we find a much smaller subalgebra $D$ quasi-isomorphic to the whole algebra. $D$ is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice $L$ whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of $H^*(M)$. The algebra $D$ has a natural integral version that is a good candidate for an integral model of $M$. If the rational local homology of $L$ can be computed explicitly we obtain an explicit presentation of the ring $H^*(M,{\mathbf Q})$. For example, this is done for the cases where $L$ is a geometric lattice and where $M$ is a $k$-equal manifold. [ABSTRACT FROM AUTHOR]

In this paper we introduce recurrent dimensions of discrete dynamical systems and we give upper and lower bounds of the recurrent dimensions of the quasi-periodic orbits. We show that these bounds have different values according to the algebraic properties of the frequency and we investigate these dimensions of quasi-periodic trajectories given by solutions of a nonlinear PDE. [ABSTRACT FROM AUTHOR]

In this paper we give solutions to the Hamburger and Stieltjes moment problems in several variables, in algebraic terms, via extended sequences. Some characterizations of the uniqueness of the solutions are also presented. [ABSTRACT FROM AUTHOR]

We give a new method for construction of unconditional bases for general classes of Triebel-Lizorkin and Besov spaces. These include the $L_p$, $H_p$, potential, and Sobolev spaces. The main feature of our method is that the character of the basis functions can be prescribed in a very general way. In particular, if $\Phi$ is any sufficiently smooth and rapidly decaying function, then our method constructs a basis whose elements are linear combinations of a fixed (small) number of shifts and dilates of the single function $\Phi$. Typical examples of such $\Phi$'s are the rational function $ \Phi (\cdot) = (1 + |\cdot|^2)^{-N} $ and the Gaussian function $ \Phi (\cdot) = e^{-|\cdot|^2}. $ This paper also shows how the new bases can be utilized in nonlinear approximation. [ABSTRACT FROM AUTHOR]

Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities. [ABSTRACT FROM AUTHOR]

The goal of this paper is to show that for almost all choices of $n$ masses, $m_i$, there are only finitely many central configurations of the Newtonian $n$-body problem for which the bodies span a space of dimension $n-2$ (such a central configuration is called a Dziobek configuration). The result applies in particular to two-dimensional configurations of four bodies and three-dimensional configurations of five bodies. [ABSTRACT FROM AUTHOR]

In this paper the zero reaction limit of the hyperbolic conservation law with stiff source term of monostable type \[\partial _{t} u + \partial _{x} f(u) = \frac {1}{\epsilon } u(1-u)\] is studied. Solutions of Cauchy problems of the above equation with initial value $0\le u_{0}(x)\le 1$ are proved to converge, as $\epsilon \to 0$, to piecewise constant functions. The constants are separated by either shocks determined by the Rankine-Hugoniot jump condition, or a non-shock jump discontinuity that moves with speed $f'(0)$. The analytic tool used is the method of generalized characteristics. Sufficient conditions for the existence and non-existence of traveling waves of the above system with viscosity regularization are given. The reason for the failure to capture the correct shock speed by first order shock capturing schemes when underresolving $\epsilon >0$ is found to originate from the behavior of traveling waves of the above system with viscosity regularization. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the $\operatorname{Ext}$-algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the ``no loop'' conjecture. [ABSTRACT FROM AUTHOR]

Let $\mathfrak n$ be a nilpotent Lie algebra, over a field of characteristic zero, and $\mathcal U$ its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of $\mathcal U$ related to finitely generated $\mathcal U$-modules $V$, and in particular the set $\operatorname{Ass}V$ of associated primes for such $V$ (note that now $\operatorname{Ass}V$ is equal to the set $\operatorname{Annspec}V$ of annihilator primes for $V$); (2) the problem of nontriviality for the modules $V/\mathcal PV$ when $\mathcal P$ is a (maximal) prime of $\mathcal U$, and in particular when $\mathcal P$ is the augmentation ideal $\mathcal U\mathfrak n$ of $\mathcal U$. We define the support of $V$, as a natural generalization of the same notion from commutative theory, and show that it is the object of primary interest when dealing with (2). We also introduce and study the reduced localization and the reduced support, which enables to better understand the set $\operatorname{Ass}V$. We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when $\mathfrak N$, $\mathfrak N$ as in the theorem, are abelian. We also present some of its interesting consequences. \noindent\par\textbf{Theorem.} \emph{Let $\mathfrak Q$ be a finite-dimensional Lie algebra over a field of characteristic zero, and $\mathfrak N$ an ideal of $\mathfrak Q$; denote by $U(\mathfrak N)$ the universal enveloping algebra of $\mathfrak N$. Let $V$ be a $\mathfrak Q$-module which is finitely generated as an $\mathfrak N$-module. Then every annihilator prime of $V$, when $V$ is regarded as a $U(\mathfrak N)$-module, is $\mathfrak Q$-stable for the adjoint action of $\mathfrak Q$ on $U(\mathfrak N)$.} [ABSTRACT FROM AUTHOR]

A construction of Bernstein associates to each cocharacter of a split $p$-adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calculate the Bernstein functions explicitly in as many cases as possible, with a view towards testing Kottwitz' conjecture. In this paper we prove a characterization of the Bernstein function associated to a \emph{minuscule} cocharacter (the case of interest for Shimura varieties). This is used to write down the Bernstein functions explicitly for some minuscule cocharacters of $Gl_n$; one example can be used to verify Kottwitz' conjecture for a special class of Shimura varieties (the ``Drinfeld case''). In addition, we prove some general facts concerning the support of Bernstein functions, and concerning an important set called the ``$\mu$-admissible'' set. These facts are compatible with a conjecture of Kottwitz and Rapoport on the shape of the special fiber of a Shimura variety with parahoric type bad reduction. [ABSTRACT FROM AUTHOR]

Let $X$ be a curve defined over an algebraically closed field $k$ with $\operatorname{char}(k)=p>0$. Assume that $X/k$ is reduced. In this paper we study the unipotent part $U$ of the Jacobian $J_{X/k}$. In particular, we prove that if $p$ is large in terms of the dimension of $U$, then $U$ is isomorphic to a product of additive groups $\mathbb{G}_a$. [ABSTRACT FROM AUTHOR]

In this paper we construct a finitely based variety, whose equational theory is undecidable, yet whose word problems are recursively solvable, which solves a problem stated by G. McNulty (1992). The construction produces a discriminator variety with the aforementioned properties starting from a class of structures in some multisorted language (which may include relations), axiomatized by a finite set of universal sentences in the given multisorted signature. This result also presents a common generalization of the earlier results obtained by B. Wells (1982) and A. Mekler, E. Nelson, and S. Shelah (1993). [ABSTRACT FROM AUTHOR]

The game problem for an input-output system governed by a Volterra integral equation with respect to a quadratic performance functional is an untouched open problem. In this paper, it is studied by a new approach called projection causality. The main result is the causal synthesis which provides a causal feedback implementation of the optimal strategies in the saddle point sense. The linear feedback operator is determined by the solution of a Fredholm integral operator equation, which is independent of data functions and control functions. Two application examples are included. The first one is quadratic differential games of a linear system with arbitrary finite delays in the state variable and control variables. The second is the standard linear-quadratic differential games, for which it is proved that the causal synthesis can be reduced to a known result where the feedback operator is determined by the solution of a differential Riccati operator equation. [ABSTRACT FROM AUTHOR]

In this paper we study finite group extensions represented by special cohomology classes. As an application, we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if $(Z/p)^r$ acts freely on $(S^1)^k$, then $r \leq k$. [ABSTRACT FROM AUTHOR]

In this paper we prove the following result: Let $X$ be a complex torus and $M$ a normally generated line bundle on $X$; then, for every $p \geq 0$, the line bundle $M^{p+1}$ satisfies Property $ N_{p}$ of Green-Lazarsfeld. [ABSTRACT FROM AUTHOR]

In 1980, Lusztig posed the problem of showing the existence of a unipotent support for the irreducible characters of a finite group of Lie type. This problem was solved by Lusztig in the case where the characteristic of the field over which the group is defined is large enough. The first named author extended this to the case where the characteristic is good. It is the purpose of this paper to remove this condition as well, so that the existence of unipotent supports is established in complete generality. [ABSTRACT FROM AUTHOR]

In this paper, we investigate $*$-polynomial identities of minimal degree for the algebra of $n\times n$ matrices over a field, where $n<5$ and $*$ is the transpose involution. We first present some basic generators, and then proceed to show that all other minimal degree identities can be derived from those. [ABSTRACT FROM AUTHOR]

Let $I$ be an interval. $I$ need not be compact or bounded. Let $f:I\rightarrow I$ be a continuous map, and $(x_0, x_1, \cdots, x_n)$ be a trajectory of $f$ with $x_n\leq x_0

The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie. We generalize part of this result by proving that all locally finite varieties generated by nilpotent algebras that have a trivial locally strongly solvable subvariety are congruence permutable. [ABSTRACT FROM AUTHOR]

In this paper we study the kernel of a non-zero locally nilpotent $R$-derivation of the polynomial ring $R[X,Y]$ over a noetherian integral domain $R$ containing a field of characteristic zero. We show that if $R$ is normal then the kernel has a graded $R$-algebra structure isomorphic to the symbolic Rees algebra of an unmixed ideal of height one in $R$, and, conversely, the symbolic Rees algebra of any unmixed height one ideal in $R$ can be embedded in $R[X,Y]$ as the kernel of a locally nilpotent $R$-derivation of $R[X,Y]$. We also give a necessary and sufficient criterion for the kernel to be a polynomial ring in general. [ABSTRACT FROM AUTHOR]

Let $\mathcal{I}$ be the ideal in the enveloping algebra of $\mathfrak{sp}(n,\mathbb C)$ generated by the maximal compact subalgebra of $\mathfrak{sp}(n-1,1)$. In this paper we construct an analog of $\mathcal I$ in the quantized enveloping algebra $\U$ corresponding to a type $C_{n}$ diagram at generic $q$. We find generators for $\mathcal{I}$ and explicit bases for $\U/\mathcal{I}$. [ABSTRACT FROM AUTHOR]

In this paper we study symbolic dynamics over alphabets which are modules over a principal ideal domain, considering topological shifts which are also submodules. Our main result is the classification, up to algebraic and topological conjugacy, of the torsion-free, transitive, finite memory shifts. [ABSTRACT FROM AUTHOR]

In this paper we study the space $S_{H}(A)$ of unital completely positive linear maps from a $C^{*}$-algebra $A$ to the algebra $B(H)$ of continuous linear operators on a complex Hilbert space $H$. The state space of $A$, in this notation, is $S_{\mathbb{C}}(A)$. The main focus of our study concerns noncommutative convexity. Specifically, we examine the $C^{*}$-extreme points of the $C^{*}$-convex space $S_{H}(A)$. General properties of $C^{*}$-extreme points are discussed and a complete description of the set of $C^{*}$-extreme points is given in each of the following cases: (i) the cases $S_{{\mathbb{C}}^{2}}(A)$, where $A$ is arbitrary ; (ii) the cases $S_{{\mathbb{C}}^{r}}(A)$, where $A$ is commutative; (iii) the cases $S_{{\mathbb{C}}^{r}}(M_{n})$, where $M_{n}$ is the $C^{*}$-algebra of $n\times n$ complex matrices. An analogue of the Krein-Milman theorem will also be established. [ABSTRACT FROM AUTHOR]

In this paper we describe a one--variable Mal\textprime cev--like condition satisfied by any locally finite minimal variety. We prove that a locally finite variety is minimal if and only if it satisfies this Mal\textprime cev--like condition and it is generated by a strictly simple algebra which is nonabelian or has a trivial subalgebra. Our arguments show that the strictly simple generator of a minimal locally finite variety is unique, it is projective and it embeds into every member of the variety. We give a new proof of the structure theorem for strictly simple abelian algebras that generate minimal varieties. [ABSTRACT FROM AUTHOR]

The paper is devoted to computation of the cohomology of the complex of logarithmic differential forms with coefficients in rational functions whose poles are located on the union of several hyperplanes of a linear space over a field of characteristic zero. The main result asserts that for a vast class of hyperplane arrangements, including all free and generic arrangements, the cohomology algebra coincides with the Orlik-Solomon algebra. Over the field of complex numbers, this means that the cohomologies coincide with the cohomologies of the complement of the union of the hyperplanes. We also prove that the cohomologies do not change if poles of arbitrary multiplicity are allowed on some of the hyperplanes. In particular, this gives an analogue of the algebraic de Rham theorem for an arbitrary arrangement over an arbitrary field of zero characteristic. [ABSTRACT FROM AUTHOR]

A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically closed field of arbitrary characteristic. Applications, to be treated in separate papers, include questions involving: points in good position, birational models of rational surfaces in projective space, and resolutions for 0-dimensional subschemes of $\mathbf{P}^{2}$ defined by complete ideals. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and nonexistence of global positive solutions to $u_t=\Delta\ln(1+u)$ in $\Omega \times (0, +\infty)$, \[\frac{\partial\ln(1+u)}{\partial n}=\sqrt{1+u}(\ln(1+u))^{\alpha} \quad\text{on} \partial \Omega \times (0, +\infty),] and $u(x, 0)=u_0(x)$ in $\Omega$. Here $\alpha\geq 0$ is a parameter, $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain. After pointing out the mistakes in {\em Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary}, SIAM J. Math. Anal. {\bf 24} (1993), 317--326, by N. Wolanski, which claims that, for $\Omega=B_R$ the ball of $\mathbb{R}^N$, the positive solution exists globally if and only if $\alpha\leq 1$, we reconsider the same problem in general bounded domain $\Omega$ and obtain that every positive solution exists globally if and only if $\alpha\leq {1/2}$. [ABSTRACT FROM AUTHOR]

Let $f(x)$ be a given positive function in $R^{n+1}$. In this paper we consider the existence of convex, closed hypersurfaces $X$ so that its Gauss-Kronecker curvature at $x\in X$ is equal to $f(x)$. This problem has variational structure and the existence of stable solutions has been discussed by Tso (J. Diff. Geom. 34 (1991), 389--410). Using the Mountain Pass Lemma and the Gauss curvature flow we prove the existence of unstable solutions to the problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to a study of the subfactors arising from vertex models constructed out of `biunitary' matrices which happen to be permutation matrices. After a discussion on the computation of the higher relative commutants of the associated subfactor (in the members of the tower of Jones' basic construction), we discuss the principal graphs of these subfactors for small sizes ($ N=k \leq 3 $) of the vertex model. Of the 18 possibly inequivalent such biunitary matrices when $ N = 3$, we compute the principal graphs completely in 15 cases, all of which turn out to be finite. In the last section, we prove that two of the three remaining cases lead to subfactors of infinite depth and discuss their principal graphs. [ABSTRACT FROM AUTHOR]

In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of the Iwahori-Hecke algebras of types $A_{n-1}, B_n$, and $D_n$. Our method is a generalization of a derivation of the Murnaghan-Nakayama formula for the irreducible characters of the symmetric group given by Curtis Greene. Greene's approach is to sum up the diagonal entries of the matrices of certain cycle permutations in Young's seminormal representations. The analogues of the Young seminormal representations for the Iwahori-Hecke algebras of types $A_{n-1},B_n$, and $D_n$ were given by Hoefsmit. [ABSTRACT FROM AUTHOR]