Harder-Narasimhan Filtrations and K-Groups of an Elliptic Curve

Let $X$ be an elliptic curve over an algebraically closed field. We prove that some exact sub-categories of the category of vector bundles over $X$, defined using Harder-Narasimhan filtrations, have the same K-groups as the whole category.Comment: 5 pages, typos correcte

On the Vertices of Indecomposable Modules Over Dihedral 2-Groups

Let $k$ be an algebraically closed field of characteristic 2. We compute the vertices of all indecomposable $kD_8$-modules for the dihedral group $D_8$ of order 8. We also give a conjectural formula of the induced module of a string module from $kT_0$ to $kG$ where $G$ is a dihedral group of order $\geq 8$ and where $T_0$ is a dihedral subgroup of index 2 of $G$. Some cases where we verified this formula are given.Comment: 19 page

Kuelshammer ideals and the scalar problem for blocks with dihedral defect groups

In by now classical work, K. Erdmann classified blocks of finite groups with dihedral defect groups (and more generally algebras of dihedral type) up to Morita equivalence. In the explicit description by quivers and relations of such algebras with two simple modules, several subtle problems about scalars occurring in relations remained unresolved. In particular, for the dihedral case it is a longstanding open question whether blocks of finite groups can occur for both possible scalars 0 and 1. In this article, using Kuelshammer ideals (a.k.a. generalized Reynolds ideals), we provide the first examples of blocks where the scalar is 1, thus answering the above question to the affirmative. Our examples are the principal blocks of PGL_2(F_q), the projective general linear group of 2x2-matrices with entries in the finite field F_q, where q=p^n\equiv \pm 1 mod 8, with p an odd prime number.Comment: 23 page

Auslander-Reiten conjecture for symmetric algebras of polynomial growth

This paper studies self-injective algebras of polynomial growth. We prove that the derived equivalence classification of weakly symmetric algebras of domestic type coincides with the classification up to stable equivalences (of Morita type). As for weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between weakly symmetric algebras of polynomial growth

Numerical approximations of the Cahn-Hilliard and Allen-Cahn Equations with general nonlinear potential using the Invariant Energy Quadratization approach

In this paper, we carry out stability and error analyses for two first-order, semi-discrete time stepping schemes, which are based on the newly developed Invariant Energy Quadratization approach, for solving the well-known Cahn-Hilliard and Allen-Cahn equations with general nonlinear bulk potentials. Some reasonable sufficient conditions about boundedness and continuity of the nonlinear functional are given in order to obtain optimal error estimates. These conditions are naturally satisfied by two commonly used nonlinear potentials including the double-well potential and regularized logarithmic Flory-Huggins potential. The well-posedness, unconditional energy stabilities and optimal error estimates of the numerical schemes are proved rigorously

Verdier quotients of homotopy categories

We study Verdier quotients of diverse homotopy categories of a full additive subcategory $\mathcal E$ of an abelian category. In particular, we consider the categories $K^{x,y}({\mathcal E})$ for $x\in\{\infty, +,-,b\}$, and $y\in\{\emptyset,b,+,-,\infty\}$ the homotopy categories of left, right, unbounded complexes with homology being $0$, bounded, left or right bounded, or unbounded. Inclusion of these categories give a partially ordered set, and we study localisation sequences or recollement diagrams between the Verdier quotients, and prove that many quotients lead to equivalent categories

Ergodic Diffusion Control of Multiclass Multi-Pool Networks in the Halfin-Whitt Regime

We consider Markovian multiclass multi-pool networks with heterogeneous server pools, each consisting of many statistically identical parallel servers, where the bipartite graph of customer classes and server pools forms a tree. Customers form their own queue and are served in the first-come first-served discipline, and can abandon while waiting in queue. Service rates are both class and pool dependent. The objective is to study the limiting diffusion control problems under the long run average (ergodic) cost criteria in the Halfin--Whitt regime. Two formulations of ergodic diffusion control problems are considered: (i) both queueing and idleness costs are minimized, and (ii) only the queueing cost is minimized while a constraint is imposed upon the idleness of all server pools. We develop a recursive leaf elimination algorithm that enables us to obtain an explicit representation of the drift for the controlled diffusions. Consequently, we show that for the limiting controlled diffusions, there always exists a stationary Markov control under which the diffusion process is geometrically ergodic. The framework developed in our earlier work is extended to address a broad class of ergodic diffusion control problems with constraints. We show that that the unconstrained and constrained problems are well posed, and we characterize the optimal stationary Markov controls via HJB equations.Comment: 32 page

Existence of steady multiple vortex patches to the vortex-wave system

In this paper we prove the existence of steady multiple vortex patch solutions to the vortex-wave system in a planar bounded domain. The construction is performed by solving a certain variational problem for the vorticity and studying its asymptotic behavior as the vorticity strength goes to infinity.Comment: 22 page

Steady double vortex patches with opposite signs in a planar ideal fluid

In this paper we consider steady vortex flows for the incompressible Euler equations in a planar bounded domain. By solving a variational problem for the vorticity, we construct steady double vortex patches with opposite signs concentrating at a strict local minimum point of the Kirchhoff-Routh function with $k=2$. Moreover, we show that such steady solutions are in fact local maximizers of the kinetic energy among isovortical patches, which correlates stability to uniqueness.Comment: 17 page

Golden-rule capacity allocation for distributed delay management in peer-to-peer networks

We describe a distributed framework for resources management in peer-to-peer networks leading to golden-rule reciprocity, a kind of one-versus-rest tit-for-tat, so that the delays experienced by any given peer's messages in the rest of the network are proportional to those experienced by others' messages at that peer