4,043 research outputs found
Density Functional Theory for Hard Particles in N Dimensions
Recently it has been shown that the heuristic Rosenfeld functional derives
from the virial expansion for particles which overlap in one center. Here, we
generalize this approach to any number of intersections. Starting from the
virial expansion in Ree-Hoover diagrams, it is shown in the first part that
each intersection pattern defines exactly one infinite class of diagrams.
Determining their automorphism groups, we sum over all its elements and derive
a generic functional. The second part proves that this functional factorizes
into a convolute of integral kernels for each intersection center. We derive
this kernel for N dimensional particles in the N dimensional, flat Euclidean
space. The third part focuses on three dimensions and determines the
functionals for up to four intersection centers, comparing the leading order to
Rosenfeld's result. We close by proving a generalized form of the Blaschke,
Santalo, Chern equation of integral geometry.Comment: 2 figure
Relay: A New IR for Machine Learning Frameworks
Machine learning powers diverse services in industry including search,
translation, recommendation systems, and security. The scale and importance of
these models require that they be efficient, expressive, and portable across an
array of heterogeneous hardware devices. These constraints are often at odds;
in order to better accommodate them we propose a new high-level intermediate
representation (IR) called Relay. Relay is being designed as a
purely-functional, statically-typed language with the goal of balancing
efficient compilation, expressiveness, and portability. We discuss the goals of
Relay and highlight its important design constraints. Our prototype is part of
the open source NNVM compiler framework, which powers Amazon's deep learning
framework MxNet
Centers and homotopy centers in enriched monoidal categories
We consider a theory of centers and homotopy centers of monoids in monoidal
categories which themselves are enriched in duoidal categories. Duoidal
categories (introduced by Aguillar and Mahajan under the name 2-monoidal
categories) are categories with two monoidal structures which are related by
some, not necessary invertible, coherence morphisms. Centers of monoids in this
sense include many examples which are not `classical.' In particular, the
2-category of categories is an example of a center in our sense. Examples of
homotopy center (analogue of the classical Hochschild complex) include the
Gray-category Gray of 2-categories, 2-functors and pseudonatural
transformations and Tamarkin's homotopy 2-category of dg-categories,
dg-functors and coherent dg-transformations.Comment: 52 page
Commutativity
We describe a general framework for notions of commutativity based on
enriched category theory. We extend Eilenberg and Kelly's tensor product for
categories enriched over a symmetric monoidal base to a tensor product for
categories enriched over a normal duoidal category; using this, we re-find
notions such as the commutativity of a finitary algebraic theory or a strong
monad, the commuting tensor product of two theories, and the Boardman-Vogt
tensor product of symmetric operads.Comment: 48 pages; final journal versio
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A modular hybrid simulation framework for complex manufacturing system design
For complex manufacturing systems, the current hybrid Agent-Based Modelling and Discrete Event Simulation (ABM–DES) frameworks are limited to component and system levels of representation and present a degree of static complexity to study optimal resource planning. To address these limitations, a modular hybrid simulation framework for complex manufacturing system design is presented. A manufacturing system with highly regulated and manual handling processes, composed of multiple repeating modules, is considered. In this framework, the concept of modular hybrid ABM–DES technique is introduced to demonstrate a novel simulation method using a dynamic system of parallel multi-agent discrete events. In this context, to create a modular model, the stochastic finite dynamical system is extended to allow the description of discrete event states inside the agent for manufacturing repeating modules (meso level). Moreover, dynamic complexity regarding uncertain processing time and resources is considered. This framework guides the user step-by-step through the system design and modular hybrid model. A real case study in the cell and gene therapy industry is conducted to test the validity of the framework. The simulation results are compared against the data from the studied case; excellent agreement with 1.038% error margin is found in terms of the company performance. The optimal resource planning and the uncertainty of the processing time for manufacturing phases (exo level), in the presence of dynamic complexity is calculated
Berezin-Toeplitz Quantization and Star Products for Compact Kaehler Manifolds
For compact quantizable K\"ahler manifolds certain naturally defined star
products and their constructions are reviewed. The presentation centers around
the Berezin-Toeplitz quantization scheme which is explained. As star products
the Berezin-Toeplitz, Berezin, and star product of geometric quantization are
treated in detail. It is shown that all three are equivalent. A prominent role
is played by the Berezin transform and its asymptotic expansion. A few ideas on
two general constructions of star products of separation of variables type by
Karabegov and by Bordemann--Waldmann respectively are given. Some of the
results presented is work of the author partly joint with Martin Bordemann,
Eckhard Meinrenken and Alexander Karabegov. At the end some works which make
use of graphs in the construction and calculation of these star productsComment: 39 pages, Based on a talk presented in the frame of the Thematic
Program on Quantization, Spring 2011, at the University of Notre Dame, USA.
In the revised version some additional references are given in relation to
the role of the metaplectic correction and quotients. Also now there is an
additional section about applications and related reference
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