Local Isomorphism Research Articles

A family of ternary decagonal tilings

A new family of decagonal quasiperiodic tilings are constructed by the use of generalized point substitution processes, which is a new substitution formalism developed by the author [N. Fujita, Acta Cryst. A 65, 342 (2009)]. These tilings are composed of three prototiles: an acute rhombus, a regular pentagon and a barrel shaped hexagon. In the perpendicular space, these tilings have windows with fractal boundaries, and the windows are analytically derived as the fixed sets of the conjugate maps associated with the relevant substitution rules. It is shown that the family contains an infinite number of local isomorphism classes which can be grouped into several symmetry classes (e.g., C10, D5, etc.). The member tilings are transformed into one another through collective simpleton flips, which are associated with the reorganization in the window boundaries.

ON THE ISOMORPHIC DESCRIPTION OF CHIRAL SYMMETRY BREAKING BY NONUNITARY LIE GROUPS

It is well known that chiral symmetry breaking (χSB) in QCD with Nf = 2 light quark flavors can be described by orthogonal groups as O (4) → O (3), due to local isomorphisms. Here we discuss the question how specific this property is. We consider generalized forms of χSB involving an arbitrary number of light flavors of continuum or lattice fermions, in various representations. We search systematically for isomorphic descriptions by nonunitary, compact Lie groups. It turns out that there are a few alternative options in terms of orthogonal groups, while we did not find any description entirely based on symplectic or exceptional Lie groups. If we adapt such an alternative as the symmetry breaking pattern for a generalized Higgs mechanism, we may consider a Higgs particle composed of bound fermions and trace back the mass generation to χSB. In fact, some of the patterns that we encounter appear in technicolor models. In particular if one observes a Higgs mechanism that can be expressed in terms of orthogonal groups, we specify in which cases it could also represent some kind of χSB of techniquarks.

Electric fields on quasiperiodic potentials

The effects of an electric field on the electronic spectrum and localization properties ofquasiperiodic chains are studied. As quasiperiodic systems, we use the Harper and theFibonacci potentials since we prove that both are closely interrelated. In the limit of astrong field, a ladder spectrum with localized states is observed. The ladder structure canbe understood by using perturbation theory. Then each local isomorphism class of thequasiperiodic potential reproduces its structure in the ladder. In the case of a weak field,we observed that the singular spectrum of the quasiperiodic potential tends to besmoothed, and the gaps decrease linearly with the field. Such an effect can be understoodusing a variational approach, perturbation theory and a series of approximants.When the electric field and the quasiperiodic potential have the same order ofmagnitude, it is possible to observe a delocalization effect due to local resonances.

Arguments for the compactness and multiple connectivity of our cosmic spacetime

Some global topological as well as quantative arguments are given, indicating that our universe is most probably compact, multiply connected and without boundaries. The analysis leading to this tentative conclusion is based on a combination of Nash Euclidean embedding theorems, the local isomorphism theorem, cosmic crystallography and the theory of fractal-Cantorian spacetime. It is shown that the correct topological dimension D = 4 of space is derived from the Euclidean embedding of spacetime quanta when the corresponding manifold is assumed to be compact. This and other conclusions regarding multi-connectivity seems to reinforce the findings of relatively recent research results on topological cosmology by Luminet et al. (see Nature 425;9 Oct. 2003:593–95).

Arithmetical D-degrees

Description is given of the isomorphism types of the principal ideals of the join semilattice of m-degrees which are generated by arithmetical sets. A result by Lachlan of 1972 on computably enumerable m-degrees is extended to the arbitrary levels of the arithmetical hierarchy. As a corollary, a characterization is given of the local isomorphism types of the Rogers semilattices of numberings of finite families, and the nontrivial Rogers semilattices of numberings which can be computed at the different levels of the arithmetical hierarchy are proved to be nonisomorphic provided that the difference between levels is more than 1.

Identifiability of discrete-time nonlinear systems: The local state isomorphism approach

A new theorem is provided to test the identifiability of discrete-time systems with polynomial nonlinearities. That extends to discrete-time systems the local state isomorphism approach for continuous-time systems. Two examples are provided to illustrate the approach.

The Lorentz group and its finite field analogs: Local isomorphism and approximation

Finite Lorentz groups acting on four-dimensional vector spaces coordinatized by finite fields with a prime number of elements are represented as homomorphic images of countable, rational subgroups of the Lorentz group acting on real four-dimensional space-time. Bounded subsets of the real Lorentz group are retractable with arbitrary precision to finite subsets of such rational subgroups. These finite retracts correspond, via local isomorphisms, to well-behaved subsets of Lorentz groups over finite fields. This establishes a relationship of approximation between the real Lorentz group and Lorentz groups over very large finite fields.

Isomorphism and local isomorphism of clones of spaces

We construct metric spaces X and Y such that their clones clo X and clo Y (i.e. the categories of all continuous maps on their respective finite powers) are not isomorphic while any initial segment of clo X is isomorphic to the corresponding segment of clo Y.

Strings in five-dimensional anti-de Sitter space with a symmetry

The equation of motion of an extended object in spacetime reduces to an ordinary differential equation in the presence of symmetry. By properly defining the symmetry with the notion of cohomogeneity, we discuss the method for classifying all these extended objects. We carry out the classification for the strings in the five-dimensional anti-de Sitter space by the effective use of the local isomorphism between $SO(4,2)$ and $SU(2,2)$. In the case where the string is described by the Nambu-Goto action, we present a general method for solving the trajectory. We then apply the method to one of the classification cases, where the spacetime naturally obtains a Hopf-like bundle structure, and find a solution. The geometry of the solution is analyzed and found to be a timelike, helicoidlike surface.

Flow prolongation of some tangent valued forms

We study the prolongation of semibasic projectable tangent valued k-forms on fibered manifolds with respect to a bundle functor F on local isomorphisms that is based on the flow prolongation of vector fields and uses an auxiliary linear r-th order connection on the base manifold, where r is the base order of F. We find a general condition under which the Frolicher-Nijenhuis bracket is preserved. Special attention is paid to the curvature of connections. The first order jet functor and the tangent functor are discussed in detail. Next we clarify how this prolongation procedure can be extended to arbitrary projectable tangent valued k-forms in the case F is a fiber product preserving bundle functor on the category of fibered manifolds with m-dimensional bases and local diffeomorphisms as base maps.

The Isomorphism of Local Development Policy: A Case Study of the Formation and Transformation of National Development Zones in Post-Mao Jiangsu, China

It is often the case, although it has not been well-documented, that local governments aiming for economic growth tend to formulate and implement very similar, or even identical, policies. This phenomenon is referred to as the `isomorphism of local development policy', or `local policy isomorphism'. Five mechanisms are theoretically elaborated in order to explain the phenomenon of local policy isomorphism. The different regions formulate similar development strategies for the following reasons: they face similar pressure from the central government (or international organisations); they are competing for investments that are highly mobile; they engage in mimetic learning attributable to the uncertainties of development; experienced personnel move from advanced regions to underdeveloped areas; and, professional organisations are involved on a consultancy basis. The case of the formation and transformation of national development zones in the Jiangsu province in post-Mao China is used to illustrate the empirical applications of these five mechanisms.

The effect of corporate-level organizational factors on the transfer of human resource management practices: European and US MNCs and their Greek subsidiaries

One of the central questions in the literature on MNCs is the extent to which their subsidiaries act and behave as local firms (local isomorphism) versus the extent to which their practices resemble those of the parent company or some other global standard (internal consistency). Drawing on the resource-based view and resource-dependency theory, this paper aims to provide an insight into the interplay of several corporate-level organizational factors that affect the transfer of HRM practices across borders. Data collected from 80 European and US multinationals with subsidiaries in Greece are used to test specific hypotheses. Our results indicate that the level of importance attached to HRM by the MNC's top management and international experience have the highest explanatory power for the transfer of HRM practices, while international competitive strategy, informal control and the presence of expatriates also have a marginally significant influence.

Transient response, model structure and systematic errors in hybrid respirometers: structural identifiability analysis based on OUR and DO measurements

Experimental respirometry shows oxygen uptake rate (OUR) transients which are not predicted by standard ASM models. These transients were analysed from a detailed modelling of hybrid respirometers. Structural identifiability analysis of several models based on local isomorphism transformation approach is carried out. It can be concluded that the knowledge of the OUR does not help positively on an accurate determination of biokinetic parameters. The measure of dissolved oxygen (DO) is just enough to calculate the group of identifiable parameters from the models in a calibration process. The analysis suggests to use only the DO data to avoid the systematic errors associated to OUR calculations. Furthermore, the existence of OUR transients after a precise modelling of the hybrid respirometer is deduced. The duration of these transients are the result of the coupling between the kinetics of the biological process and the residence time in the respiration chamber of the instrument. An expression for the duration of these transients is proposed.

Graph fibrations, graph isomorphism, and PageRank

PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the values that PageRank can assume. Using our results, we show that a recently defined class of graphs that admit a polynomial-time isomorphism algorithm based on the computation of PageRank is really a subclass of fibration-prime graphs, which possess simple, entirely discrete polynomial-time isomorphism algorithms based on classical techniques for graph isomorphism. We discuss efficiency issues in the implementation of such algorithms for the particular case of web graphs, in which O(n) space occupancy (where n is the number of nodes) may be acceptable, but O(m) is not (where m is the number of arcs).

On the classification of all self-dual additive codes over GF(4) of length up to 12

We consider additive codes over GF ( 4 ) that are self-dual with respect to the Hermitian trace inner product. Such codes have a well-known interpretation as quantum codes and correspond to isotropic systems. It has also been shown that these codes can be represented as graphs, and that two codes are equivalent if and only if the corresponding graphs are equivalent with respect to local complementation and graph isomorphism. We use these facts to classify all codes of length up to 12, where previously only all codes of length up to 9 were known. We also classify all extremal Type II codes of length 14. Finally, we find that the smallest Type I and Type II codes with trivial automorphism group have length 9 and 12, respectively.

Inverse Problem for the Discrete 1D Schrödinger Operator with Small Periodic Potentials

Consider the discrete 1D Schr\"odinger operator on $\Z$ with an odd $2k$ periodic potential $q$. For small potentials we show that the mapping: $q\to $ heights of vertical slits on the quasi-momentum domain (similar to the Marchenko-Ostrovski maping for the Hill operator) is a local isomorphism and the isospectral set consists of $2^k$ distinct potentials. Finally, the asymptotics of the spectrum are determined as $q\to 0$.

Non-Periodic Rhomb Substitution Tilings that Admit Order n Rotational Symmetry

For each n ∈ ℕ we construct a substitution rule using the set of rhombs with angles (πk)/n. These substitution rules generate a local isomorphism class of tilings closed under rotation of order 2n, and also admit singular tilings fixed under a rotation of order n. The scaling factors for this set of substitution rules includes algebraic numbers of every rank.

PROJECTIVE PRESENTATIONS OF FINITELY GENERATED MODULES WITH LARGE ANNIHILATORS

ABSTRACT Let R be an integral domain, and let A be an “almost” finitely presented R -module such that R / Ann ( A ) is a finite direct sum of quasilocal rings. We characterize the relationship between the modules that can occur as kernels of finite rank projective presentations of A, and we give a criterion for when local isomorphism of torsionless modules implies an equivalence of the two modules up to summands of projective R-modules.

Identifiability of homogeneous systems using the state isomorphism approach

New results are presented concerning the state isomorphism approach to global identifiability analysis of parameterized classes of nonlinear state space systems with specified initial states. In particular we study the class of homogeneous systems, for which, under certain conditions, the local state isomorphism for a pair of indistinguishable parameter vectors is shown to be homogeneous of degree one. For homogeneous polynomial systems, conditions are given under which the local state isomorphism becomes linear. Here, the issue of whether or not the observability rank condition holds at the origin is shown to be of key importance. The scope of the results, which extend to the multivariable case, is discussed and illustrated by a number of worked examples. This demonstrates how the developed theory can be put to use to investigate the global identifiability properties of parameterized model classes.

Multiparty entanglement in graph states

Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They are many-body spin states of distributed quantum systems that play a significant role in quantum error correction, multiparty quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which we provide upper and lower bounds in graph theoretical terms. Several examples and classes of graphs will be discussed, where these bounds coincide. These examples include trees, cluster states of different dimensions, graphs that occur in quantum error correction, such as the concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier transform in the one-way computer. We also present general transformation rules for graphs when local Pauli measurements are applied, and give criteria for the equivalence of two graphs up to local unitary transformations, employing the stabilizer formalism. For graphs of up to seven vertices we provide complete characterization modulo local unitary transformations and graph isomorphisms.