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

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

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

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

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

Dose verification of dynamic MLC-tracked radiotherapy using small PRESAGE (R) 3D dosimeters and a motion phantom

With the increasing complexity of radiotherapy treatments typical 1D and 2D quality assurance (QA) detectors may fail to detect out-of-plane dose discrepancies, in particular in the presence of motion. In this work, small samples of the PRESAGE® 3D radiochromic dosimeter were used in combination with a motion phantom to measure real-time multileaf collimator (MLC)-tracked radiotherapy treatments. A different sample of PRESAGE® was irradiated for each of three different irradiation scenarios: (1) static: static sample, without tracking (2) motion: moving sample, without tracking and (3) tracking: moving sample, with tracking. Our in-house software DynaTrack dynamically moves the linac's MLC leafs based on the target position. The doses delivered to the samples were reconstructed based on the recorded positions of the MLC and phantom during the beam delivery. PRESAGE® samples were imaged with an in-house optical-CT scanner. Comparison between simulated and measured 3D dose showed good agreement for all three irradiation scenarios (static: 99.2%; motion: 99.7%; tracking: 99.3% with a 3%, 2 mm and a 10% threshold local gamma criterion), failing only at the edges of the PRESAGE® samples (~ 6 mm). Given that the dose distributions deposited using the DynaTrack system have been independently verified, this experiment demonstrates the ability of PRESAGE to measure 3D doses correctly in a tracking context. We conclude that this methodology could be used in the future to validate the delivery of dynamic MLC-tracked radiotherapy

Consistent and invertible deformation vector fields for a breathing anthropomorphic phantom: a post-processing framework for the XCAT phantom.

Breathing motion is challenging for radiotherapy planning and delivery. This requires advanced four-dimensional (4D) imaging and motion mitigation strategies and associated validation tools with known deformations. Numerical phantoms such as the XCAT provide reproducible and realistic data for simulation-based validation. However, the XCAT generates partially inconsistent and non-invertible deformations where tumours remain rigid and structures can move through each other. We address these limitations by post-processing the XCAT deformation vector fields (DVF) to generate a breathing phantom with realistic motion and quantifiable deformation. An open-source post-processing framework was developed that corrects and inverts the XCAT-DVFs while preserving sliding motion between organs. Those post-processed DVFs are used to warp the first XCAT-generated image to consecutive time points providing a 4D phantom with a tumour that moves consistently with the anatomy, the ability to scale lung density as well as consistent and invertible DVFs. For a regularly breathing case, the inverse consistency of the DVFs was verified and the tumour motion was compared to the original XCAT. The generated phantom and DVFs were used to validate a motion-including dose reconstruction (MIDR) method using isocenter shifts to emulate rigid motion. Differences between the reconstructed doses with and without lung density scaling were evaluated. The post-processing framework produced DVFs with a maximum [Formula: see text]-percentile inverse-consistency error of 0.02 mm. The generated phantom preserved the dominant sliding motion between the chest wall and inner organs. The tumour of the original XCAT phantom preserved its trajectory while deforming consistently with the underlying tissue. The MIDR was compared to the ground truth dose reconstruction illustrating its limitations. MIDR with and without lung density scaling resulted in small dose differences up to 1 Gy (prescription 54 Gy). The proposed open-source post-processing framework overcomes important limitations of the original XCAT phantom and makes it applicable to a wider range of validation applications within radiotherapy

Edge effects in 3D dosimetry: characterisation and correction of the non-uniform dose response of PRESAGE®.

Previous work has shown that PRESAGE® can be used successfully to perform 3D dosimetric measurements of complex radiotherapy treatments. However, measurements near the sample edges are known to be difficult to achieve. This is an issue when the doses at air-material interfaces are of interest, for example when investigating the electron return effect (ERE) present in treatments delivered by magnetic resonance (MR)-linac systems. To study this effect, a set of 3.5 cm-diameter cylindrical PRESAGE® samples was uniformly irradiated with multiple dose fractions, using either a conventional linac or an MR-linac. The samples were imaged between fractions using an optical-CT, to read out the corresponding accumulated doses. A calibration between TPS-predicted dose and optical-CT pixel value was determined for individual dosimeters as a function of radial distance from the axis of rotation. This data was used to develop a correction that was applied to four additional samples of PRESAGE® of the same formulation, irradiated with 3D-CRT and IMRT treatment plans, to recover significantly improved 3D measurements of dose. An alternative strategy was also tested, in which the outer surface of the sample was physically removed prior to irradiation. Results show that for the formulation studied here, PRESAGE® samples have a central region that responds uniformly and an edge region of 6-7 mm where there is gradual increase in dosimeter response, rising to an over-response of 24%-36% at the outer boundary. This non-uniform dose response increases in both extent and magnitude over time. Both mitigation strategies investigated were successful. In our four exemplar studies, we show how discrepancies at edges are reduced from 13%-37% of the maximum dose to between 2 and 8%. Quantitative analysis shows that the 3D gamma passing rates rise from 90.4, 69.3, 63.7 and 43.6% to 97.3, 99.9, 96.7 and 98.9% respectively