Parametric Polyhedra with at least kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup \mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{ Sg}(A) such that $P_A(b) \cap {\mathbb Z}^n$ has at least $k$ solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ are fixed natural numbers, one can compute in polynomial time an encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives

An investigation of the Mach Phenomenon and its relation to visual image evaluation

The Mach Phenomenon*(as perceived on photographic film) is herein investigated as a function of edge acutance, contrast and illumination level for the purpose of determining its probable effect in the area of visual image evaluation. The results indicate that dark Mach band widths increase as acutance decreases and dark band width is independent of contrast and illumination level while light band widths remain constant, over the ranges investigated. Using these findings as a basis, a dark band width predictor equation was found, taking the form; dark band width (in microns) =K/acutance, when average edge gradient is the measure of acutance. The probable effects (which increase in importance as acutance decreases) of the Mach bands on visual image evaluation is then discussed. * a visual sensation of density bands near edge

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result

AQFT from n-functorial QFT

There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to "extended" functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra "of observables", the latter uses n-functors which assign to each patch a "propagator of states". In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2-dimensional extended Minkowskian QFT 2-functor ("parallel surface transport") naturally yields a local net. This is obtained by postcomposing the propagation 2-functor with an operation that mimics the passage from the Schroedinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general pseudo-Riemannian structure and higher dimensions.Comment: 39 pages; further examples added: Hopf spin chains and asymptotic inclusion of subfactors; references adde

Kitaev's quantum double model from a local quantum physics point of view

A prominent example of a topologically ordered system is Kitaev's quantum double model $\mathcal{D}(G)$ for finite groups $G$ (which in particular includes $G = \mathbb{Z}_2$, the toric code). We will look at these models from the point of view of local quantum physics. In particular, we will review how in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the different superselection sectors of the model. In this way one finds that the charges are in one-to-one correspondence with the representations of $\mathcal{D}(G)$, and that they are in fact anyons. Interchanging two of such anyons gives a non-trivial phase, not just a possible sign change. The case of non-abelian groups $G$ is more complicated. We outline how one could use amplimorphisms, that is, morphisms $A \to M_n(A)$ to study the superselection structure in that case. Finally, we give a brief overview of applications of topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J. Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015). Mainly revie

From conformal embeddings to quantum symmetries: an exceptional SU(4) example

We briefly discuss several algebraic tools that are used to describe the quantum symmetries of Boundary Conformal Field Theories on a torus. The starting point is a fusion category, together with an action on another category described by a quantum graph. For known examples, the corresponding modular invariant partition function, which is sometimes associated with a conformal embedding, provides enough information to recover the whole structure. We illustrate these notions with the example of the conformal embedding of SU(4) at level 4 into Spin(15) at level 1, leading to the exceptional quantum graph E4(SU(4)).Comment: 22 pages, 3 color figures. Version 2: We changed the color of figures (ps files) in such a way that they are still understood when converted to gray levels. Version 3: Several references have been adde

M2-Branes and Fano 3-folds

A class of supersymmetric gauge theories arising from M2-branes probing Calabi-Yau 4-folds which are cones over smooth toric Fano 3-folds is investigated. For each model, the toric data of the mesonic moduli space is derived using the forward algorithm. The generators of the mesonic moduli space are determined using Hilbert series. The spectrum of scaling dimensions for chiral operators is computed.Comment: 128 pages, 39 figures, 42 table

Lattice width directions and Minkowski's 3^d-theorem

We show that the number of lattice directions in which a d-dimensional convex body in R^d has minimum width is at most 3^d-1, with equality only for the regular cross-polytope. This is deduced from a sharpened version of the 3^d-theorem due to Hermann Minkowski (22 June 1864--12 January 1909), for which we provide two independent proofs.Comment: 1 figure, 10 page

Comparing nuclear power trajectories in Germany and the UK: from ‘regimes' to ‘democracies’ in sociotechnical transitions and Discontinuities

This paper focuses on arguably the single most striking contrast in contemporary major energy politics in Europe (and even the developed world as a whole): the starkly differing civil nuclear policies of Germany and the UK. Germany is seeking entirely to phase out nuclear power by 2022. Yet the UK advocates a ‘nuclear renaissance’, promoting the most ambitious new nuclear construction programme in Western Europe.Here,this paper poses a simple yet quite fundamental question: what are the particular divergent conditions most strongly implicated in the contrasting developments in these two countries. With nuclear playing such an iconic role in historical discussions over technological continuity and transformation, answering this may assist in wider understandings of sociotechnical incumbency and discontinuity in the burgeoning field of‘sustainability transitions’. To this end, an ‘abductive’ approach is taken: deploying nine potentially relevant criteria for understanding the different directions pursued in Germany and the UK. Together constituted by 30 parameters spanning literatures related to socio-technical regimes in general as well as nuclear technology in particular, the criteria are divided into those that are ‘internal’ and ‘external’ to the ‘focal regime configuration’ of nuclear power and associated ‘challenger technologies’ like renewables. It is ‘internal’ criteria that are emphasised in conventional sociotechnical regime theory, with ‘external’ criteria relatively less well explored. Asking under each criterion whether attempted discontinuation of nuclear power would be more likely in Germany or the UK, a clear picture emerges. ‘Internal’ criteria suggest attempted nuclear discontinuation should be more likely in the UK than in Germany– the reverse of what is occurring. ‘External’ criteria are more aligned with observed dynamics –especially those relating to military nuclear commitments and broader ‘qualities of democracy’. Despite many differences of framing concerning exactly what constitutes ‘democracy’, a rich political science literature on this point is unanimous in characterising Germany more positively than the UK. Although based only on a single case,a potentially important question is nonetheless raised as to whether sociotechnical regime theory might usefully give greater attention to the general importance of various aspects of democracy in constituting conditions for significant technological discontinuities and transformations. If so, the policy implications are significant. A number of important areas are identified for future research, including the roles of diverse understandings and specific aspects of democracy and the particular relevance of military nuclear commitments– whose under-discussion in civil nuclear policy literatures raises its own questions of democratic accountability

The Role of Individual Variables, Organizational Variables and Moral Intensity Dimensions in Libyan Management Accountants’ Ethical Decision Making

This study investigates the association of a broad set of variables with the ethical decision making of management accountants in Libya. Adopting a cross-sectional methodology, a questionnaire including four different ethical scenarios was used to gather data from 229 participants. For each scenario, ethical decision making was examined in terms of the recognition, judgment and intention stages of Rest’s model. A significant relationship was found between ethical recognition and ethical judgment and also between ethical judgment and ethical intention, but ethical recognition did not significantly predict ethical intention—thus providing support for Rest’s model. Organizational variables, age and educational level yielded few significant results. The lack of significance for codes of ethics might reflect their relative lack of development in Libya, in which case Libyan companies should pay attention to their content and how they are supported, especially in the light of the under-development of the accounting profession in Libya. Few significant results were also found for gender, but where they were found, males showed more ethical characteristics than females. This unusual result reinforces the dangers of gender stereotyping in business. Personal moral philosophy and moral intensity dimensions were generally found to be significant predictors of the three stages of ethical decision making studied. One implication of this is to give more attention to ethics in accounting education, making the connections between accounting practice and (in Libya) Islam. Overall, this study not only adds to the available empirical evidence on factors affecting ethical decision making, notably examining three stages of Rest’s model, but also offers rare insights into the ethical views of practising management accountants and provides a benchmark for future studies of ethical decision making in Muslim majority countries and other parts of the developing world