Showing papers on "Equivalence class published in 2011"

Topological phases of fermions in one dimension

Abstract: In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of one-dimensional systems. We focus on the time-reversal-invariant Majorana chain (BDI symmetry class).While the band classification yields an integer topological index k, it is known that phases characterized by values of k in the same equivalence class modulo 8 can be adiabatically transformed one to another by adding suitable interaction terms. Here we show that the eight equivalence classes are distinct and exhaustive, and provide a physical interpretation for the interacting invariant modulo 8. The different phases realize different Altland-Zirnbauer classes of the reduced density matrix for an entanglement bipartition into two half chains. We generalize these results to the classification of all one-dimensional gapped phases of fermionic systems with possible antiunitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.

Stimulus equivalence and the symbolic control of behavior

Abstract: Six subjects formed two three-member equivalence classes following conditional dis­crimination training in a matching-to-sample procedure using nonsense syllables as stimulii. For three subjects, a stimulus from each equivalence class was then given a distinct dis­criminative function. For the remaining three subjects, a stimulus from each equivalence class was given a conditioned reinforcing function. Six additional subjects served as con­trols. Three received the discriminative control training and three the conditioned reinfor­cement training, but none received the conditional discrimination training required to form the equivalence class. Following training, testing was done using an element of the equivalence classes related symmetrically and transitively to the element used in the dis­criminative or reinforcement training. For the equivalence subjects, but not the control subjects, discriminative control and conditioned reinforcement effects transferred to other members of the equivalence classes. This transfer of discriminative and reinforcing func­tions across members of equivalence classes may provide a more complete account of some types of generalization and maintenance. In addition, stimulus equivalence and related phenomena may help explain the control exerted by symbolic stimuli such as in rule governed behavior

Variations on the Sensitivity Conjecture

Abstract: The sensitivity of a Boolean function f of n Boolean variables is the maximum over all inputs x of the number of positions i such that flipping the i-th bit of x changes the value of f(x). Permitting to flip disjoint blocks of bits leads to the notion of block sensitivity, known to be polynomially related to a number of other complexity measures of f , including the decision-tree complexity, the polynomial degree, and the certificate complexity. A long-standing open question is whether sensitivity also belongs to this equivalence class. A positive answer to this question is known as the Sensitivity Conjecture. We present a selection of known as well as new variants of the Sensitivity Conjecture and point out some weaker versions that are also open. Among other things, we relate the problem to Communication Complexity via recent results by Sherstov (QIC 2010). We also indicate new connections to Fourier analysis.

Signal Estimation Under Random Time-Warpings and Nonlinear Signal Alignment

Abstract: While signal estimation under random amplitudes, phase shifts, and additive noise is studied frequently, the problem of estimating a deterministic signal under random time-warpings has been relatively unexplored. We present a novel framework for estimating the unknown signal that utilizes the action of the warping group to form an equivalence relation between signals. First, we derive an estimator for the equivalence class of the unknown signal using the notion of Karcher mean on the quotient space of equivalence classes. This step requires the use of Fisher-Rao Riemannian metric and a square-root representation of signals to enable computations of distances and means under this metric. Then, we define a notion of the center of a class and show that the center of the estimated class is a consistent estimator of the underlying unknown signal. This estimation algorithm has many applications: (1) registration/alignment of functional data, (2) separation of phase/amplitude components of functional data, (3) joint demodulation and carrier estimation, and (4) sparse modeling of functional data. Here we demonstrate only (1) and (2): Given signals are temporally aligned using nonlinear warpings and, thus, separated into their phase and amplitude components. The proposed method for signal alignment is shown to have state of the art performance using Berkeley growth, handwritten signatures, and neuroscience spike train data.

Learning equivalence classes of acyclic models with latent and selection variables from multiple datasets with overlapping variables

Abstract: While there has been considerable research in learning probabilistic graphical models from data for predictive and causal inference, almost all existing algorithms assume a single dataset of i.i.d. observations for all variables. For many applications, it may be impossible or impractical to obtain such datasets, but multiple datasets of i.i.d. observations for different subsets of these variables may be available. Tillman et al. [2009] showed how directed graphical models learned from such datasets can be integrated to construct an equivalence class of structures over all variables. While their procedure is correct, it assumes that the structures integrated do not entail contradictory conditional independences and dependences for variables in their intersections. While this assumption is reasonable asymptotically, it rarely holds in practice with finite samples due to the frequency of statistical errors. We propose a new correct procedure for learning such equivalence classes directly from the multiple datasets which avoids this problem and is thus more practically useful. Empirical results indicate our method is not only more accurate, but also faster and requires less memory.

Quotient sets and diophantine equations

Abstract: Quotient sets 핌/핌 = {u/u′ : u, u′ ∊ 핌} have been considered several times before in the Monthly. We consider more general quotient sets 핌/핍 and we apply our results to certain simultaneous Diophant...

System and method for evaluating marketer re-identification risk

Abstract: Disclosures of databases for secondary purposes is increasing rapidly and any identification of personal data may from a dataset of database can be detrimental. A re-identification risk metric is determined for the scenario where an intruder wishes to re-identify as many records as possible in a disclosed database, known as a marketer risk. The dataset can be analyzed to determine equivalence classes for variables in the dataset and one or more equivalence class sizes. The re-identification risk metric associated with the dataset can be determined using a modified log-linear model by measuring a goodness of fit measure generalized for each of the one or more equivalence class sizes.

Search-Based Planning with Provable Suboptimality Bounds for Continuous State Spaces

Abstract: Search-based planning is widely used for mobile robot motion planning because of its guarantees of optimality and completeness. In continuous state-spaces, however, most existing approaches have significant limitations in terms of optimality and completeness because of the underlying grid used. We propose an approach that eliminates the dependency on grids by using more general equivalence classes to quickly find an initial solution and instead of pruning states that fall within an equivalence class and have higher cost, we use an inflated heuristic to lower the priority of these states in the search. In further iterations, we reduce the inflated heuristic in a principled way, thus providing fast solutions with provable suboptimality bounds that can be improved as time allows. The proposed approach produces smooth paths with the resolution dictated by the action set. Finer action sets produce higher resolution paths that are more computationally intensive to calculate and coarser action sets produce lower resolution paths that are faster to compute. To the best of our knowledge, this is the first algorithm that is able to plan in continuous state-spaces with provable guarantees on completeness and bounds on suboptimality for a given action set. Experimental results on 3D (x,y,theta) path planning show that, on average, this approach is able to find paths in less than two seconds that are within 2% of the optimal path cost in worlds of up to 1000x1000 m with a minimum step size of one meter.

Equivalence class transformation based mining of frequent itemsets from uncertain data

Abstract: Numerous frequent itemset mining algorithms have been proposed over the past two decades. Most of them mine traditional databases of precise data. However, there are many real-life applications for which data are uncertain. This leads to the mining of uncertain data. In this paper, we propose an equivalence class transformation based algorithm---called UV-Eclat---which transforms probabilistic databases of uncertain data from their usual horizontal format into a vertical format, from which frequent itemsets are mined.

Adjacent transformations in permutations

Abstract: We continue a study of the equivalence class induced on $S_n$ when one is permitted to replace a consecutive set of elements in a permutation with the same elements in a different order. For each possible set of allowed replacements, we characterise and/or enumerate the set of permutations reachable from the identity. In some cases we also count the number of equivalence classes.

Posets from Admissible Coxeter Sequences

Abstract: We study the equivalence relation on the set of acyclic orientations of an undirected graph $\Gamma$ generated by source-to-sink conversions. These conversions arise in the contexts of admissible sequences in Coxeter theory, quiver representations, and asynchronous graph dynamical systems. To each equivalence class we associate a poset, characterize combinatorial properties of these posets, and in turn, the admissible sequences. This allows us to construct an explicit bijection from the equivalence classes over $\Gamma$ to those over $\Gamma'$ and $\Gamma"$, the graphs obtained from $\Gamma$ by edge deletion and edge contraction of a fixed cycle-edge, respectively. This bijection yields quick and elegant proofs of two non-trivial results: $(i)$ A complete combinatorial invariant of the equivalence classes, and $(ii)$ a solution to the conjugacy problem of Coxeter elements for simply-laced Coxeter groups. The latter was recently proven by H. Eriksson and K. Eriksson using a much different approach.

Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions

Abstract: We consider a large family of equivalence relations on permutations in Sn that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of pattern-replacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, subject to the constraint that a third element of a suitable type be in a suitable position. For various instances of the problem, we compute the number of equivalence classes, determine how many n-permutations are equivalent to the identity permutation, or characterise this equivalence class. Although our results feature familiar integer sequences (e.g., Catalan, Fibonacci, and Tribonacci numbers) and special classes of permutations (layered, connected, and 123-avoiding), some of the sequences that arise appear to be new.

Prefix-Free regular languages: closure properties, difference, and left quotient

Abstract: We show that the class of prefix-free languages is closed under intersection, difference, concatenation, square, and the k-th power and is not closed under complement, union, symmetric difference, Kleene star, reversal, cyclic shift, shuffle, and left quotient. Then, we study the state complexity of difference and left quotient of prefix-free regular languages. In both cases we get tight bounds. In the case of difference, the tight bound is mn−m−2n+4 and is met by binary languages. In the case of left quotient, the tight bound is 2n−1. The bound is met by languages over (n−1)-letter alphabet and cannot be met using smaller alphabets.

Quotients of Representation Rings

Abstract: We give a proof, using so-called fusion rings and q-deformations of Brauer algebras that the representation ring of an orthogonal or symplectic group can be obtained as a quotient of a ring Gr(O(\infinity)). This is obtained here as a limiting case for analogous quotient maps for fusion categories, with the level going to \infinity. This in turn allows a detailed description of the quotient map in terms of a reflection group. As an application, one obtains a general description of the branching rules for the restriction of representations of Gl(N) to O(N) and Sp(N) as well as detailed information about the structure of the q-Brauer algebras in the nonsemisimple case for certain specializations.

Scaling up the greedy equivalence search algorithm by constraining the search space of equivalence classes

Abstract: Greedy Equivalence Search (GES) is nowadays the state of the art algorithm for learning Bayesian networks (BNs) from complete data. However, from a practical point of view, this algorithm may not be fast enough to work in high dimensionality domains. This paper proposes some variants of GES aimed to increase its efficiency. Under faithfulness assumption, the modified algorithms preserve the same theoretical properties as the original one, that is, they recover a perfect map of the target distribution in the large sample limit. Moreover, experimental results confirm that, although they carry out much less computations, BNs learnt by those algorithms have the same quality as those learnt by GES.

Integrating history-length interpolation and classes in language modeling

Abstract: Building on earlier work that integrates different factors in language modeling, we view (i) backing off to a shorter history and (ii) class-based generalization as two complementary mechanisms of using a larger equivalence class for prediction when the default equivalence class is too small for reliable estimation. This view entails that the classes in a language model should be learned from rare events only and should be preferably applied to rare events. We construct such a model and show that both training on rare events and preferable application to rare events improve perplexity when compared to a simple direct interpolation of class-based with standard language models.

On extremal quasiconformal mappings of non-landslide type

Abstract: Let S = {(x,y): 0 < x < 1,0 < y < 1}, and let f be a quasiconformal mapping on S. It is proved that there is at least one extremal quasiconformal mapping of non-landslide type in the Teichmuller equivalence class [f]. This gives a positive answer to the problem proposed by Z. Li in a recent paper.

On families of weakly admissible filtered phi-modules and the adjoint quotient of GL_d

Abstract: We study the relation of the notion of weak admissibility in families of filtered phi-modules, as considered in a companion paper, with the adjoint quotient. We show that the weakly admissible subset is an open subvariety in the fibers over the adjoint quotient. Further we determine the image of the weakly admissible set in the adjoint quotient generalizing earlier work of Breuil and Schneider.

The set of numerical semigroups of a given genus

Abstract: In this paper we present a new approach to construct the set of numerical semigroups with a fixed genus. Our methodology is based on the construction of the set of numerical semigroups with fixed Frobenius number and genus. An equivalence relation is given over this set and a tree structure is defined for each equivalence class. We also provide a more efficient algorithm based in the translation of a numerical semigroup to its so-called Kunz-coordinates vector.

Almost overlap-free words and the word problem for the free Burnside semigroup satisfying x 2 =x 3

Abstract: We study the word problem for the free Burnside semigroup satisfying x2 = x3 and having two generators. The elements of this semigroup are classes of equivalent words. A natural way to solve the word problem is to select a unique "canonical" representative for each equivalence class. We prove that overlap-free words and "almost" overlap-free words can serve as canonical representatives of their equivalence classes. We show that such a word in a given class, if any, can be efficiently found. As a result, we construct a linear-time algorithm that partially solves the word problem for the semigroup under consideration.

Growing words in the free group on two generators

Abstract: This paper is concerned with minimal-length representatives of equivalence classes of words in $F_2$ under Aut $F_2$. We give a simple inequality characterizing words of minimal length in their equivalence class. We consider an operation that “grows” words from other words, increasing the length, and we study root words—minimal words that cannot be grown from other minimal words. Root words are “as minimal as possible” in the sense that their characterization is the boundary case of the minimality inequality. The property of being a root word is respected by equivalence classes, and the length of each root word is divisible by 4.

Classification of Ternary Logic Functions by Self-Dual Equivalence Classes

Abstract: Self-dual equivalence class of ternary logic functions is investigated using ternary parametron logic. Based on multiplex or (MUX) realization of a function, a self-dual function is constructed by introducing Goto's self-dualizing variable which feeds constant values to MUX circuit. Self-dual operation is an operation to transform from one function to another among functions realized by different values of self-dualizing variable. It is defined as a combination of permutation group on values and positions of variables and that on values of function. It can be formulated as an exponentiation group by these groups and the equivalence class count is obtained from the cycle index of these groups.

Algorithm based on equivalence class to resolve short-circuit problem in IILVS

Abstract: The present invention discloses an algorithm based on equivalence class for solving HLVS short circuit problem, belonging to the field of integrated circuit computer aided design, especially relating to the field of layout verification. In HLVS, short circuit test is always an improtant problem for consistency test of a layout and a schematic diagram. Since a port can pass connection relation information in hierarchical processing method, short circuit problem is not liminted in a unit, but needs to be considered in inter-hierarchy mode. According to the present invention, short circuit problem are divided into two cases, i.e. Inside Short and Outside Short. In HLVS, the essential is an equivalence class problem no matter Inside Short or Outside Short , i.e. the problem boils down to dividing a certain set into equivalence classes according to given equivalence relation, thereby correctly representing circuit connection relation. Therefore, core problem of solving HLVS short circuit is converted into problem of resolving equivalence class. The invention is mainly used in solving HLVS short circuit problem and preparations for circuit comparison.

The module of pseudo-rational relations of a quotient divisible group

Abstract: It is proved that the module of pseudo-rational relations determines a quotient divisible group up to quasi-isomorphism.

Equivalence classes of multiplicative central (pn, pn, pn, 1)-relative difference sets

Abstract: We show by explicit construction that the equivalence classes of multiplicative central (p n , p n , p n , 1)-RDSs relative to ${\mathbb Z}_p^n$ in groups E with $E/{\mathbb Z}_p^n \cong {\mathbb Z}_p^n$ are in one-to-one correspondence with the strong isotopism classes of presemifields of order p n . We also show there are 1,446 equivalence classes of central (16, 16, 16, 1)-RDS relative to ${\mathbb Z}_2^4$ , in groups E for which $E/{\mathbb Z}_2^4 \cong {\mathbb Z}_2^4$ . Only one is abelian.