Set Of Isomorphisms Research Articles

BRST cohomology is Lie algebroid cohomology

In this paper we demonstrate that the exterior algebra of an Atiyah Lie algebroid generalizes the familiar notions of the physicist's BRST complex. To reach this conclusion, we develop a general picture of Lie algebroid isomorphisms as commutative diagrams between algebroids preserving the geometric structure encoded in their brackets. We illustrate that a necessary and sufficient condition for such a diagram to define a morphism of Lie algebroid brackets is that the two algebroids possess gauge-equivalent connections. This observation indicates that the aforementioned set of Lie algebroid isomorphisms should be regarded as equivalent to the set of local diffeomorphisms and gauge transformations. Moreover, a Lie algebroid isomorphism being a chain map in the exterior algebra sense ensures that isomorphic algebroids are cohomologically equivalent. The Atiyah Lie algebroids derived from principal bundles with common base manifolds and structure groups may therefore be divided into equivalence classes of isomorphic algebroids. Each equivalence class possesses a local representative which we refer to as the trivialized Lie algebroid, and we show that the exterior algebra of the trivialized algebroid gives rise to the BRST complex. We conclude by illustrating the usefulness of Lie algebroid cohomology in computing quantum anomalies, including applications to the chiral and Lorentz-Weyl (LW) anomalies. In particular, we pay close attention to the fact that the geometric intuition afforded by the Lie algebroid (which was absent in the naive BRST complex) provides hints of a deeper picture that simultaneously geometrizes the consistent and covariant forms of the anomaly. In the algebroid construction, the difference between the consistent and covariant anomalies is simply a different choice of basis.

Fractals and Finite Distributions of Power Sets

In this paper, we are committed to investigating the fractal decomposition of power sets. Our main result is that every power set can be decomposed into a sum of a power set and an isomorphic set that does not intersect with it. For the finite power set, this property can be drawn on the ordinal line by constructing the fractal number axis of the ordinal line, and the fractal distribution and fractal graph of the finite power set can be obtained by using the parallel translation drawing method. Moreover, the distributions do not overlap or cross. The results in this paper provide a new perspective for further investigation of the fractal distribution of power sets.

Single-support serial isomorphous replacement phasing.

The use of single isomorphous replacement (SIR) has become less widespread due to difficulties in sample preparation and the identification of isomorphous native and derivative data sets. Non-isomorphism becomes even more problematic in serial experiments, because it adds natural inter-crystal non-isomorphism to heavy-atom-soaking-induced non-isomorphism. Here, a method that can successfully address these issues (and indeed can benefit from differences in heavy-atom occupancy) and additionally significantly simplifies the SIR experiment is presented. A single heavy-atom soak into a microcrystalline slurry is performed, followed by automated serial data collection of partial data sets. This produces a set of data collections with a gradient of heavy-atom occupancies, which are reflected in differential merging statistics. These differences can be exploited by an optimized genetic algorithm to segregate the pool of data sets into `native' and `derivative' groups, which can then be used to successfully determine phases experimentally by SIR.

The Concept of Optimal Compaction of the Charge in the Gravitation System Using the Grains Triangle for Cokemaking Process

Two isomorphic sets of grains, small and large, were analysed—without specifying their dimensions—under the acronym CMC (Curve of Maximum Compression) and taking into account the effects of segregation CMCS. The proposal is particularly valuable for optimal blend preparation in the gravity system in cokemaking. The main advantage of this work is the proposal of using the grains triangle, which limits the values calculated by the relations: bulk density-share of coarse/fine grains, for different levels of moisture content. Each system of changing shares of coarse grains is characterised by a constant C, but there is no need to determine it. Compliance of the calculated value with the experimentally determined value means that the given arbitrary grain set has reached its maximum density called the “locus”. The grains triangle practically covers the vast majority of laboratory and industrial test results, and geometrically or computationally indicates the ability of a given particle size distribution to reach maximum bulk density. This paper presents analysis of the results of tests on crushing, coal briquettes, and grinding coal blend in selected mechanical systems. Results of tests on coke quality (CRI, CSR) in connection with the grain size triangle are discussed.

The Steenrod algebra from the group theoretical viewpoint

In the paper “The Steenrod algebra and its dual” [2], J. Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme Gp represented by the dual Hopf algebra of the mod p Steenrod algebra. Then, Gp assigns a graded commutative algebra A⁎ over a prime field of finite characteristic p to a set of isomorphisms of the additive formal group law over A⁎, whose group structure is given by the composition of formal power series ([4] Appendix). The aim of this paper is to show some group theoretic properties of Gp by making use of this presentation of Gp(A⁎). We give a decreasing filtration of subgroup schemes of Gp which we use for estimating the length of the lower central series of finite subgroup schemes of Gp. We also give a successive quotient maps Gp→ρ0Gp〈1〉→ρ1Gp〈2〉→ρ2⋯→ρk−1Gp〈k〉→ρkGp〈k+1〉→ρk+1⋯ of affine group schemes over a prime field Fp such that the kernel of ρk is a maximal abelian subgroup.

An equivalent formulation of chromatic quasi-polynomials

Given a central integral arrangement, the reduction of the arrangement modulo a positive integer q gives rise to a subgroup arrangement in Zqℓ. Kamiya et al. (2008) introduced the notion of characteristic quasi-polynomial, which enumerates the cardinality of the complement of this subgroup arrangement. Chen and Wang (2012) found a similar but more general setting that replacing the integral arrangement by its restriction to a subspace of Rℓ, and evaluating the cardinality of the q-reduced complement will also lead to a quasi-polynomial in q. On an independent study, Brändén and Moci (2014) defined the so-called chromatic quasi-polynomial, and initiated the study of q-colorings on a finite list of elements in a finitely generated abelian group. The main purpose of this paper is to verify that the Chen–Wang quasi-polynomial and the Brändén–Moci chromatic quasi-polynomial are equivalent in the sense that the quasi-polynomials enumerate the cardinalities of isomorphic sets. Some applications including periodicity of the intersection posets of Zq-arrangements, an answer to a problem of Chen–Wang, and computation on the characteristic polynomials of R-arrangements will also be discussed.

Stochastic Graphlet Embedding.

Graph-based methods are known to be successful in many machine learning and pattern classification tasks. These methods consider semistructured data as graphs where nodes correspond to primitives (parts, interest points, and segments) and edges characterize the relationships between these primitives. However, these nonvectorial graph data cannot be straightforwardly plugged into off-the-shelf machine learning algorithms without a preliminary step of-explicit/implicit-graph vectorization and embedding. This embedding process should be resilient to intraclass graph variations while being highly discriminant. In this paper, we propose a novel high-order stochastic graphlet embedding that maps graphs into vector spaces. Our main contribution includes a new stochastic search procedure that efficiently parses a given graph and extracts/samples unlimitedly high-order graphlets. We consider these graphlets, with increasing orders, to model local primitives as well as their increasingly complex interactions. In order to build our graph representation, we measure the distribution of these graphlets into a given graph, using particular hash functions that efficiently assign sampled graphlets into isomorphic sets with a very low probability of collision. When combined with maximum margin classifiers, these graphlet-based representations have a positive impact on the performance of pattern comparison and recognition as corroborated through extensive experiments using standard benchmark databases.

First-Best-World Economic Theory and the Second-Best-World of Public Sector Outsourcing: The Reinvention of the Soviet Kombinat by Other Means

This paper examines how public sector outsourcing has performed in the UK, one of its leading exponents. It sets out the theoretical economic logic behind it, the unanticipated risks in its conception, and the deepening problems with its intensification. It shows how, when we put the market rhetoric of New Public Management to one side, outsourcing necessitates the central planning of private actors, and how the success of this venture hinges on the viability of the outsourcing contract as a fully effective junction of instruction and control. As contract theory tells us, however, the more complex and dynamic the good, the less a contract can guarantee effective control over its production. Moreover, as the critical economics of Soviet central planning teaches us, the resulting asymmetries in information and leverage are just the start of bargaining games that the state (and taxpayer) cannot win. As the paper shows, a state that outsources its complex tasks puts itself at a chronic informational disadvantage, renders itself dependent on poorly controlled private monopoly service providers for essential services that form part of a matrix of interdependent services, and cannot exit failing contracts under acceptable terms. In the USSR a remarkably isomorphic set of hazards had driven Nikita Khrushchev back to the drawing board by 1965.

WEAK CAYLEY TABLE GROUPS OF SOME CRYSTALLOGRAPHIC GROUPS

AbstractFor a group G, a weak Cayley table isomorphism is a bijection f : G → G such that f(g1g2) is conjugate to f(g1)f(g2) for all g1, g2 ∈ G. The set of all weak Cayley table isomorphisms forms a group (G) that is the group of symmetries of the weak Cayley table of G. We determine (G) for each of the 17 wallpaper groups G, and for some other crystallographic groups.

Exact Recovery with Symmetries for the Doubly Stochastic Relaxation

Graph matching, or quadratic assignment, is the problem of labeling the vertices of two graphs so that they are as similar as possible. A common method for approximately solving the NP-hard graph matching problem is relaxing it to a convex optimization problem over the set of doubly stochastic (DS) matrices. Recent analysis has shown that for almost all pairs of isomorphic and asymmetric graphs, the DS relaxation succeeds in correctly retrieving the isomorphism between the graphs. Our goal in this paper is to analyze the case of symmetric isomorphic graphs. This goal is motivated by shape matching applications where the graphs of interest usually have reflective symmetry. For symmetric problems the graph matching problem has multiple isomorphisms, and so convex relaxations admit all convex combinations of these isomorphisms as viable solutions. If the convex relaxation does not admit any additional superfluous solution, we say that it is convex exact. We show that convex exactness depends strongly on the symmetry group of the graphs; for a fixed symmetry group $G$, the DS relaxation either will be convex exact for almost all pairs of isomorphic graphs with symmetry group $G$, or will fail for all such pairs. We show that for reflective groups with at least one full orbit, convex exactness holds almost everywhere. The importance of this result stems from the fact that such groups are the most common symmetry groups in shape matching applications. Finally, we provide some simple examples of nonreflective symmetry groups for which convex exactness always fails. These examples also show that the property of having generic convex exactness is not invariant under group isomorphisms. When convex exactness holds, the isomorphisms of the graphs are the extreme points of the convex solution set and the task of finding an isomorphism in polynomial time is tractable. We discuss several common algorithms capable of retrieving an isomorphism in this case. We also show that interior point algorithms will not return an isomorphism in this case but will return the centroid of the set of isomorphisms, which gives a “fuzzy encoding” of the symmetries of the shape. In fact, for any symmetry group $G$, the centroid solution can be recovered efficiently for almost all pairs of isomorphic graphs with symmetry group $G$.

Finite Closed Sets of Functions in Multi-valued Logic

The article is devoted to classification of close sets of functions in k-valued logic. We build the classification of finite closed sets. The sets contain only constants and unary functions since sets containing even a two-ary function are infinite. Formulas of the number of finite sets and minimal sets exist for all natural k. We find the numbers of closed sets containing the identity function f (x)=x or a constant for all k. We give the number of sets on levels of inclusions at k up to 10. The inclusion diagrams are present at k up to 5, at k=6 we give inclusion diagrams of sets containing only function f (x)=x and constant 0. We find isomorphic sets and use only one of the isomorphic sets to build the diagrams.

Reconsidering Pairs and Functions as Sets

We give representations for ordered pairs and functions in set theory with the property that ordered pairs are functions from the finite ordinal 2. We conjecture that these representations are useful for formalized mathematics since certain isomorphic sets are identified. The definitions, theorems and proofs have been formalized in the proof assistant Coq using only the simply typed features of Coq. We describe the development within the context of an intuitionistic simply typed (higher-order) version of (well-founded) Zermelo-Fraenkel set theory without the axiom of infinity.

Design tools for interdisciplinary translation of material experiences

Designers increasingly have the opportunity to influence the development of materials as they emerge from the laboratory. In order for this to be successful, designers need to be able to communicate effectively with materials scientists so that materials can be developed with desired functionalities and properties. This paper reviews evidence in favour of using isomorphic sets of material stimuli as tools to bridge the disciplinary gap between designers and materials scientists. We show how these isomorphic sets and their accompanying experiments can be used to translate between the two communities, and to systematically explore the relationship between the technical attributes of materials and subjective experiences of their sound, taste and feel. This paper also explores the limitations of psychophysical approaches and other quantitative techniques for elucidating material experience, and suggests new possibilities for interdisciplinary collaborations that draw on ethnographic approaches.

Do the sizes of isomorphic sets affect the discriminability of presented pairs of figures?

Do the sizes of isomorphic sets affect the discriminability of presented pairs of figures?

Bisdithiazolyl Radical Spin Ladders

A series of four bisdithiazolyl radicals 1a-d (R(1) = Pr, Bu, Pn, Hx; R(2) = F) has been prepared and characterized by X-ray crystallography. The crystal structure of 1a (R(1) = Pr) belongs to the tetragonal space group P42(1)m and consists of slipped π-stack arrays of undimerized radicals packed about 4 centers running along the z-direction, an arrangement identical to that found for 1 (R(1) = Et; R(2) = F). With increasing chain length of the R(1) substituent, an isomorphous set 1b-d is generated. All three compounds crystallize in the P2(1)/c space group and consist of pairs of radical π-stacks locked together by strong intermolecular F···S' bridges to create spin ladder arrays. The slipped π-stack alignment of radicals produces close S···S' interactions which serve as the "rungs" of a spin ladder, and the long chain alkyl substituents (R(1)) serve as buffers which separate the ladders from each other laterally. Variable temperature magnetic susceptibility measurements indicate that 1a behaves as an antiferromagnetically coupled Curie-Weiss paramagnet, the behavior of which may be modeled as a weakly coupled AFM chain. Stronger antiferromagnetic coupling is observed in 1b-d, such that the Curie-Weiss fit is no longer applicable. Analysis of the full data range (T = 2-300 K) is consistent with the Johnston strong-leg spin ladder model. The origin of the magnetic behavior across the series has been explored with broken-symmetry Density Functional Theory (DFT) calculations of individual pairwise exchange energies. These confirm that strong antiferromagnetic interactions are present within the ladder "legs" and "rungs", with only very weak magnetic exchange between the ladders.

On the Generality of the Topological Theory of Visual Shape Perception

This study used a series of six closely related experiments to examine whether individuals use topological structures to discriminate figures. Strict control was exerted over the selection of stimuli, which were a specific type of randomly generated lined figures that can be classified using isomorphic sets defined by graph theory. Any two figures within an isomorphic set possessed the same topological structure. The experiments described here used a same/different discrimination task with simultaneously presented pairs of figures: (a) identical pairs (Id pairs), in which each pair of figures had the same topological and superficial properties; (b) nonidentical and isomorphic pairs (Iso pairs), in which each pair had the same topological but different superficial properties; and (c) nonidentical and nonisomorphic pairs (Noniso pairs), in which each pair had different topological properties. Within these experiments I varied the conditions related to the intersecting line segments, presentation of points defining each figure, figure complexity, stimulus aspect ratios, and the parity of the total line-segment lengths between the figures in each pair. These variations showed that the latencies for making accurate discriminations were shorter for Noniso pairs than for Iso pairs, suggesting that individuals are sensitive to topology when distinguishing figures.

Algebraic properties of complete residuated lattice valued tree automata

This paper investigates tree automata based on complete residuated lattice valued (referred to as L-valued) logic. First, we define the notions of L-valued set of pure subsystems and L-valued set of strong pure subsystems, as well as, their relation is considered. Also, L-valued n-tuple operator consist of n successors is defined, some of its properties are examined and its relation with pure subsystem is analyzed. Furthermore, we investigate some concepts such as L-valued set of (strong) homomorphisms, L-valued set of (strong) isomorphisms, and L-valued set of admissible relations. Moreover, we discuss bifuzzy topological characterization of L-valued tree automata. Finally, the relations of homomorphisms between the L-valued tree automata to continuous mappings and open mappings is examined.

Order isomorphisms in windows

We characterize order preserving transforms on the class of lower-semi-continuous convex functions that are defined on a convex subset of $\mathbb{R}^n$ (a "window") and some of its variants. To this end, we investigate convexity preserving maps on subsets of $\mathbb{R}^n$. We prove that, in general, an order isomorphism is induced by a special convexity preserving point map on the epi-graph of the function. In the case of non-negative convex functions on $K$, where $0\in K$ and $f(0) = 0$, one may naturally partition the set of order isomorphisms into two classes; we explain the main ideas behind these results.

Symmetrized Induced Ramsey Theory

We prove induced Ramsey theorems in which the monochromatic induced subgraph satisfies that all members of a prescribed set of its partial isomorphisms extend to automorphisms of the colored graph (without requirement of preservation of colors). We consider vertex and edge colorings, and extensions of partial isomorphisms in the set of all partial isomorphisms between singletons as considered by Babai and Sos (European J Combin 6(2):101–114, 1985), the set of all finite partial isomorphisms as considered by Hrushovski (Combinatorica 12(4):411–416, 1992), Herwig (Combinatorica 15:365–371, 1995) and Herwig-Lascar (Trans Amer Math Soc 5:1985–2021, 2000), and the set of all total isomorphisms. We observe that every finite graph embeds into a finite vertex transitive graph by a so called bi-embedding, an embedding that is compatible with a monomorphism between the corresponding automorphism groups. We also show that every countable graph bi-embeds into Rado’s universal countable graph Γ.

Extending conceptualisation modes for generalised Formal Concept Analysis

Formal Concept Analysis (FCA) is an exploratory data analysis technique for boolean relations based on lattice theory. Its main result is the existence of a dual order isomorphism between two set lattices induced by a binary relation between a set of objects and a set of attributes. Pairs of dually isomorphic sets of objects and attributes, called formal concepts, form a concept lattice, but actually model only a conjunctive mode of conceptualisation. In this paper we augment this formalism in two ways: first we extend FCA to consider different modes of conceptualisation by changing the basic dual isomorphism in a modal-logic motivated way. This creates the three new types of concepts and lattices of extended FCA, viz., the lattice of neighbourhood of objects, that of attributes and the lattice of unrelatedness. Second, we consider incidences with values in idempotent semirings—concretely the completed max-plus or schedule algebra R ¯ max, + —and focus on generalising FCA to try and replicate the modes of conceptualisation mentioned above. To provide a concrete example of the use of these techniques, we analyse the performance of multi-class classifiers by conceptually analysing their confusion matrices.