Analytic Equivalence Research Articles

Incomparable, non-isomorphic and minimal Banach spaces

A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to isomorphism of th

Brackets and slashes, stars and dots: understanding the notation of linguistic types

The standard by which token expressions are classified as linguistically type equivalent has persistently attracted both philosophical speculation and linguistic inquiry. Yet while psychophysical, perceptual, structural and semantic accounts have all competed for preeminence, one problem affects every theoretical position: how to actually refer to a linguistic type. The notational technique of phonetic brackets and phonemic slash bars, while developed and deployed within linguistics, was also independently conceived of by the philosopher Wilfrid Sellars [Philosoph. Stud. 1 (1950) 24] in the form of his “star” and better known “dot” quotes. This convergence between phonological theory and philosophical semantics, and the significance of a concretized notation for referring to linguistic types, is highlighted by relating the Descriptivist failure to “reduce the phoneme” within American Structural linguistics to the philosophical challenge (Quine, W.V. 1953. From a Logical Point of View. Harvard University Press, Cambridge, MA) to provide a non-semantic reduction of analytic equivalence.

Analytic equivalence and similarity of operators

The analytic equivalence of two operators is a generalization of similarity. We prove that under some conditions the analytic equivalence between two Hilbert space operatorsT andR implies the similarity of their restrictions on generalized ranges. We also prove that, in certain cases, the similarity ofT to a contraction implies that ofR. An improvement of a well-known criterion of similarity to an isometry due to Sz.-Nagy is given and an extension of a result of Apostol is obtained.

Propagating mode in the photonic gap of 1D resonant Bragg reflector

Concerning “1D resonant photonic crystals”, i.e., 1D periodic arrays of resonant layers with a spacing satisfying the Bragg condition for resonant light, recent topics are summarized together with additional new findings. The central points are about (1) the recent theoretical finding of a new branch in the middle of photonic gap, (2) an approach-dependent controversy about its existence, (3) its consequence as sharp dips in reflectance spectrum and the corresponding internal field patters, indicating its propagating mode character, (4) the comparison between macroscopic and microscopic treatment of resonance effect, and (5) corrected transfer matrix method and its analytical equivalence with nonlocal response theory.

Greg Hjorth and Alexander S. Kechris. Analytic equivalence relations and Ulm-type classifications. The journal of symbolic logic, vol. 60 (1995), pp. 1273–1300. - Greg Hjorth, Alexander S. Kechris, and Alain Louveau. Borel equivalence relations induced by actions of the symmetric group. Annals of pure and applied logic, vol. 92 (1998), pp. 63–112.

Greg Hjorth and Alexander S. Kechris. Analytic equivalence relations and Ulm-type classifications. The journal of symbolic logic, vol. 60 (1995), pp. 1273–1300. - Greg Hjorth, Alexander S. Kechris, and Alain Louveau. Borel equivalence relations induced by actions of the symmetric group. Annals of pure and applied logic, vol. 92 (1998), pp. 63–112. - Volume 7 Issue 4

Relations d'équivalence analytiques complètes

We prove that various concrete analytic equivalence relations arising in model theory or analysis are complete, i.e., maximum in the Borel reducibility ordering. The proofs use some general results concerning the wider class of analytic quasi-orders.

Direct competitive immunoassays for the coplanar polychlorinated biphenyls

The development of competitive enzyme-linked immunosorbent assays (ELISAs) based on polyclonal antibodies for coplanar congeners is described. As a first step to developing the functional ELISAs, hapten mimics with different lengths of carboxylic acid spacers at para-position were derived from coplanar 3,3′,4,4′-tetrachlorobiphenyl and 3,3′,4,4′,5-pentachlorobiphenyl (IUPAC No. 77 (PCB 77) and 126 (PCB 126), respectively). It appeared from the results that the length of the spacer did not affect significantly the sensitivity of the assays established. The direct ELISAs using the antibodies in combination with various hapten–peroxidase tracers showed IC 50-values for polychlorinated biphenyl (PCB) 77 and 126 of 1.3–19.2 μg l −1. The ELISAs were tested for their cross-reactivity profiles using 20 selected congeners. The assays were highly specific for non- ortho-substituted (coplanar) congeners and did not recognize the more abundant but less toxic non-coplanar PCB congeners or organochlorine compounds. The superior assay was found for the antibody against PCB 126(CH 2) 3hapten mimic–bovine serum albumin (BSA) in combination with PCB 77CH 2hapten mimic–peroxidase tracer. The IC 50-values for the assay to PCB 77 and 126 were 2.0 and 5.2 μg l −1, respectively; the highest cross-reactivity values (PCB 126=100%) achieved for non-coplanar structures and metabolites were only 1.3 and 1.0% for 2,3,3′,4,4,′5,5′-heptachlorobiphenyl (PCB 189) and PCB 77 ether, respectively. In order to interpret measured data in terms of analytical or toxicological equivalents, a reliable relationship between assay responses and the coplanar congener content of Aroclor-contaminated samples should be verified and validated.

Algebroid plane curves whose Milnor and Tjurina numbers differ by one or two

In this paper we describe all irreducible plane algebroid curves, defined over an algebraically closed field of characteristic zero, modulo analytic equivalence, having the property that the difference between their Milnor and Tjurina numbers is 1 or 2. Our work extends a previous result of O. Zariski who described such curves when this difference is zero.

Rigidity Theorem for Degenerated Singular Points of Germs of Holomorphic Vector Fields in the Complex Plane

We consider the class νn of germs of holomorphic vector fields in ({\Bbb C}^2, 0) with vanishing (n − 1)-jet at zero. We prove that the formal equivalence of two generic germs in νn implies their analytic equivalence. This result is analogous to the one obtained in l16r for the case of orbital analytic equivalence.

Optimization Criteria and Geometric Algorithms for Motion and Structure Estimation

Prevailing efforts to study the standard formulation of motion and structure recovery have recently been focused on issues of sensitivity and robustness of existing techniques. While many cogent observations have been made and verified experimentally, many statements do not hold in general settings and make a comparison of existing techniques difficult. With an ultimate goal of clarifying these issues, we study the main aspects of motion and structure recovery: the choice of objective function, optimization techniques and sensitivity and robustness issues in the presence of noise. We clearly reveal the relationship among different objective functions, such as “(normalized) epipolar constraints,” “reprojection error” or “triangulation,” all of which can be unified in a new “optimal triangulation” procedure. Regardless of various choices of the objective function, the optimization problems all inherit the same unknown parameter space, the so-called “essential manifold.” Based on recent developments of optimization techniques on Riemannian manifolds, in particular on Stiefel or Grassmann manifolds, we propose a Riemannian Newton algorithm to solve the motion and structure recovery problem, making use of the natural differential geometric structure of the essential manifold. We provide a clear account of sensitivity and robustness of the proposed linear and nonlinear optimization techniques and study the analytical and practical equivalence of different objective functions. The geometric characterization of critical points and the simulation results clarify the difference between the effect of bas-relief ambiguity, rotation and translation confounding and other types of local minima. This leads to consistent interpretations of simulation results over a large range of signal-to-noise ratio and variety of configurations.

Linearization of definable order relations

We prove that if ≼ is an analytic partial order then either ≼ can be extended to a Δ 2 1 linear order similar to an antichain in 2 < ω 1 , ordered lexicographically, or a certain Borel partial order ⩽ 0 embeds in ≼. Similar linearization results are presented, for κ-bi-Souslin partial orders and real-ordinal definable orders in the Solovay model. A corollary for analytic equivalence relations says that any (lightface) Σ 1 1 equivalence relation E, such that E 0 does not embed in E, is fully determined by intersections with E -invariant Borel sets coded in L.

On non-wellfounded iterations of the perfect set forcing

AbstractWe prove that ifIis a partially ordered set in a countable transitive modelof ZFC thencan be extended by a generic sequence of realsai,i∈I, such thatis preserved and everyaiis Sacks generic over[〈aj:j&lt;i〉]. The structure of the degrees of-constructibility of reals in the extension is investigated.As applications of the methods involved, we define a cardinal invariant to distinguish product and iterated Sacks extensions, and give a short proof of a theorem (by Budinas) that in ω2-iterated Sacks extension of L the Burgess selection principle for analytic equivalence relations holds.

Explicit Variational Solution of the Cubic-Quintic Nonlinear Schrödinger Equation for Pulse Propagation in Optical Fibers

With a variational method, we study the cubic-quintic nonlinear Schrödinger equation for pulse propagation in optical fibers. Anomalous dispersion that gives rise to fundamental bright solitary waves is considered. It is shown that within the specified limits, the present variational model demonstrates (for 1 picosecond pulse) good agreements with its analytical equivalent. It is proved that for negative coefficient of the fifth-order nonlinearity existence of the two-state solitary wave solution is a peculiarity of the variational formulation.

Ulm Classification of Analytic Equivalence Relations in Generic Universes

AbstractWe prove that if every real belongs to a set generic extension of L, then every Σ equivalence relation E on reals either admits a Δ1 reduction to the equality on the set 2&lt; ω1 of all countable binary sequences, or the Vitali equivalence E0 continuously embeds in E. The proofs are based on a topology generated by OD sets.

On measures ergodic with respect to an analytic equivalence relation

In this paper, we prove that the set of probability measures which are ergodic with respect to an analytic equivalence relation is an analytic set. This is obtained by approximating analytic equivalence relations by measures, and is used to give an elementary proof of an ergodic decomposition theorem of Kechris.

Nonstandard models and analytic equivalence relations

We improve a result of Hjorth concerning the nature of thin analytic equivalence relations. The key lemma uses a weakly compact cardinal to construct certain nonstandard models, which Hjorth obtained using # \# ’s.

Do Household Scanner Data Provide Representative Inferences from Brand Choices: A Comparison with Store Data

The authors investigate whether household scanner data provide representative inferences about brand choice behavior. They show the (analytical) equivalence of the homogeneous brand choice logit and the multinomial store sales models. Then, they determine how results for the homogeneous logit model estimated with actual household data deviate from results for the multinomial model estimated with actual store data from the same community. They also perform this comparison using model specifications that provide average estimated parameters while accommodating unobserved household heterogeneity. The authors find statistical support for the hypothesis that panelist households are not representative, whether household heterogeneity is or is not accommodated. Substantively, however, the average estimated price elasticities are close for the household and store data analyzed, if the household data are selected on the basis of purchase selection. An alternative selection procedure, called household selection, which is used for the analysis of complete household purchase records, provides results that strongly differ from the purchase selection results.

Some applications of illfoundedness

It is possible to completely characterize which countable models generated by 0# exist inL. This in turn has applications in the study of analytic equivalence relations; for instance, ifE is∑ 1 1 and every invariant∑ 1 1 (0#) set isΔ 1 1 , thenE has at most ℵ0 many equivalence classes.

Analytic equivalence relations and Ulm-type classifications

Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.The original Glimm-Effros dichotomy, established by Effros [Ef], [Ef1], who generalized work of Glimm [G1], asserts that if an Fσ equivalence relation on a Polish space X is induced by the continuous action of a Polish group G on X, then exactly one of the following alternatives holds:(I) Elements of X can be classified up to E-equivalence by “concrete invariants” computable in a reasonably definable way, i.e., there is a Borel function f: X → Y, Y a Polish space, such that xEy ⇔ f(x) = f(y), or else(II) E contains a copy of a canonical equivalence relation which fails to have such a classification, namely the relation xE0y ⇔ ∃n∀m ≥ n(x(n) = y(n)) on the Cantor space 2ω (ω = {0,1,2, …}), i.e., there is a continuous embedding g: 2ω → X such that xE0y ⇔ g(x)Eg(y).Moreover, alternative (II) is equivalent to:(II)′ There exists an E-ergodic, nonatomic probability Borel measure on X, where E-ergodic means that every E-invariant Borel set has measure 0 or 1 and E-nonatomic means that every E-equivalence class has measure 0.

Real Analytic Fibre Bundles: Some Results on Classification and Homotopies

AbstractThe classification problem for holomorphic fibre bundles over Stein spaces was solved by H. GRAUERT. Along the same lines, the real coherent analytic case was considered by A. TOGNOLI and V. ANCONA.In this paper we propose a different approach, based on classifying spaces, in order to study the previous problem for real analytic fibre bundles over C‐analytic subspaces of Rm.So, let X be a C‐analytic subspace of Rm and G a compact Lie group. The main result is a characterization of the real analytic G‐principal fibre bundles over X for which the analytic and topological equivalence coincide.Moreover, we prove that these bundles can be classified also by means of homotopy classes of analytic maps of X into classifying spaces.Among the others results, are worth recording: a relative approximation theorem of continuous cross sections by analytic ones, a theorem about the equivalence between analytical and topological homotopy between cross sections and a covering homotopy theorem.