A differential operator D commuting with an S-action is said to be rigid if the non-constant Fourier coefficients of kerD and coker D are the same. Somewhat surprisingly, the study of rigid differential operators turns out to be closely related to the problem of defining Chern numbers on singular varieties. This relationship comes into play when we make use of the rigidity properties of the com...

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories local modules are, again, modular. This generalizes previous work Kirillov Ostrik (Adv Math 171(2):183–227, 2002) semisimple setup. Examples non-semisimple via modules, as well connections to authors’ prior on relative monoidal centers, are provided. In particular, we classify ...

A direction-length framework is a pair (G, p), where G = (V ;D,L) is a ‘mixed’ graph whose edges are labeled as ‘direction’ or ‘length’ edges, and p is a map from V to R for some d. The label of an edge uv represents a direction or length constraint between p(u) and p(v). Let G be obtained from G by adding, for each length edge e of G, a direction edge with the same end vertices as e. We show t...

It is possible, using a first-order sigma-delta modulation scheme, to represent an imager's pixel value as an oversampled bit stream. The goal of the project was to demonstrate the feasibility of computer vision algorithms on such sigma-delta encoded bit streams. The computational modules used were constrained to fit within the footprint of a single pixel. This ensures that fully parallel opera...

in this errata, we reconsider and modify two propositions and their corollaries which were written on epi-retractable and co-epi-retractable modules.

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor HomC(T, −), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equiv...

Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary 2-category. Examples include G-graded fusion 1-categories, G-crossed 1-categories. explore the properties 2-categories modules bimodules over algebra, by giving criterion for existence right left adjoints. Then, we co...

Let $V\subseteq A$ be a conformal inclusion of vertex operator algebras and let $\mathcal{C}$ category grading-restricted generalized $V$-modules that admits the algebraic braided tensor structure Huang-Lepowsky-Zhang. We give conditions under which inherits semisimplicity from $A$-modules in $\mathcal{C}$, vice versa. The most important condition is $A$ rigid $V$-module with non-zero categoric...

The configuration management of a sensor network composed of tens or up to thousands of nodes is difficult due to the limit of memory space and energy consumption. This paper presents an approach for dynamic reconfiguration of application modules in sensor networks based on Nano-Qplus. The version information from proposed methods is registered into NVSync (Nano-Qplus Version Synchronization), ...

‎In this paper‎, ‎some properties of pre-monotone operators are proved‎. ‎It is shown that in a reflexive Banach space‎, ‎a full domain multivalued $sigma$-monotone operator with sequentially norm$times$weak$^*$ closed graph is norm$times$weak$^*$ upper semicontinuous‎. ‎The notion of $sigma$-convexity is introduced and the‎ ‎relations between the $sigma$-monotonicity and $sigma$-convexity is i...