Nontrivial Set Research Articles

Ergodic properties of skew products in infinite measure

Let (\Omega,\mu) be a shift of finite type with a Markov probability, and (Y,\nu) a non-atomic standard measure space. For each symbol i of the symbolic space, let \Phi_i be a measure-preserving automorphism of (Y,\nu). We study skew products of the form (\omega,y) --> (\sigma\omega,\Phi_{\omega_0}(y)), where \sigma =shift map on (\Omega,\mu). We prove that, when the skew product is conservative, it is ergodic if and only if the \Phi_i's have no common non-trivial invariant set. In the second part we study the skew product when \Omega={0,1}^Z, \mu =Bernoulli measure, and \Phi_0,\Phi_1 are R-extensions of a same uniquely ergodic probability-preserving automorphism. We prove that, for a large class of roof functions, the skew product is rationally ergodic with return sequence asymptotic to \sqrt{n}, and its trajectories satisfy the central, functional central and local limit theorem.

On spectral decomposition of Smale–Vietoris axiom A diffeomorphisms

ABSTRACTWe introduce Smale–Vietoris diffeomorphisms that include the classical DE mappings with Smale solenoids. We describe the correspondence between basic sets of axiom A Smale–Vietoris diffeomorphisms and basic sets of non-singular axiom A-endomorphisms. For Smale–Vietoris diffeomorphisms of 3-manifolds, we prove the uniqueness of non-trivial solenoidal basic set. We construct a bifurcation between different types of solenoidal basic sets which can be considered as a destruction (or birth) of Smale solenoid.

Obstacles with non-trivial trapping sets in higher dimensions

Using a well-known example of Livshits, for every \({n > 2}\) we construct obstacles K in \({\mathbb{R}^n}\) such that the set of trapped points for the billiard flow in the exterior of K has a non-empty interior, and therefore a positive Lebesgue measure.

Integrability of a deterministic cellular automaton driven by stochastic boundaries

We propose an interacting many-body space–time-discrete Markov chain model, which is composed of an integrable deterministic and reversible cellular automaton (rule 54 of Bobenko et al 1993 Commun. Math. Phys. 158 127) on a finite one-dimensional lattice , and local stochastic Markov chains at the two lattice boundaries which provide chemical baths for absorbing or emitting the solitons. Ergodicity and mixing of this many-body Markov chain is proven for generic values of bath parameters, implying the existence of a unique nonequilibrium steady state. The latter is constructed exactly and explicitly in terms of a particularly simple form of matrix product ansatz which is termed a patch ansatz. This gives rise to an explicit computation of observables and k-point correlations in the steady state as well as the construction of a nontrivial set of local conservation laws. The feasibility of an exact solution for the full spectrum and eigenvectors (decay modes) of the Markov matrix is suggested as well. We conjecture that our ideas can pave the road towards a theory of integrability of boundary driven classical deterministic lattice systems.

Monte Carlo Tree Search for Multi-Robot Task Allocation

Multi-robot teams are useful in a variety of task allocation domains such as warehouse automation and surveillance. Robots in such domains perform tasks at given locations and specific times, and are allocated tasks to optimize given team objectives. We propose an efficient, satisficing and centralized Monte Carlo TreeSearch based algorithm exploiting branch and bound paradigm to solve the multi-robot task allocation problem with spatial, temporal and other side constraints. Unlike previous heuristics proposed for this problem, our approach offers theoretical guarantees and finds optimal solutions for some non-trivial data sets.

Argyres-Douglas theories, the Macdonald index, and an RG inequality

We conjecture closed-form expressions for the Macdonald limits of the superconformal indices of the (A_1, A_{2n-3}) and (A_1, D_{2n}) Argyres-Douglas (AD) theories in terms of certain simple deformations of Macdonald polynomials. As checks of our conjectures, we demonstrate compatibility with two S-dualities, we show symmetry enhancement for special values of n, and we argue that our expressions encode a non-trivial set of renormalization group flows. Moreover, we demonstrate that, for certain values of n, our conjectures imply simple operator relations involving composites built out of the SU(2)_R currents and flavor symmetry moment maps, and we find a consistent picture in which these relations give rise to certain null states in the corresponding chiral algebras. In addition, we show that the Hall-Littlewood limits of our indices are equivalent to the corresponding Higgs branch Hilbert series. We explain this fact by considering the S^1 reductions of our theories and showing that the equivalence follows from an inequality on monopole quantum numbers whose coefficients are fixed by data of the four-dimensional parent theories. Finally, we comment on the implications of our work for more general N=2 superconformal field theories.

Probing non-covalent interactions with a second generation energy decomposition analysis using absolutely localized molecular orbitals.

An energy decomposition analysis (EDA) separates a calculated interaction energy into as many interpretable contributions as possible; for instance, permanent and induced electrostatics, Pauli repulsions, dispersion and charge transfer. The challenge is to construct satisfactory definitions of all terms in the chemically relevant regime where fragment densities overlap, rendering unique definitions impossible. Towards this goal, we present an improved EDA for Kohn-Sham density functional theory (DFT) with properties that have previously not been simultaneously attained. Building on the absolutely localized molecular orbital (ALMO)-EDA, this second generation ALMO-EDA is variational and employs valid antisymmetric electronic wavefunctions to produce all five contributions listed above. These contributions moreover all have non-trivial complete basis set limits. We apply the EDA to the water dimer, the T-shaped and parallel-displaced benzene dimer, the p-biphthalate dimer "anti-electrostatic" hydrogen bonding complex, the biologically relevant binding of adenine and thymine in stacked and hydrogen-bonded configurations, the triply hydrogen-bonded guanine-cytosine complex, the interaction of Cl(-) with s-triazine and with the 1,3-dimethyl imidazolium cation, which is relevant to the study of ionic liquids, and the water-formaldehyde-vinyl alcohol ter-molecular radical cationic complex formed in the dissociative photoionization of glycerol.

Periodicity forcing words

The Dual Post Correspondence Problem asks, for a given word α, if there exists a non-periodic morphism g and an arbitrary morphism h such that g(α)=h(α). Thus α satisfies the Dual PCP if and only if it belongs to a non-trivial equality set. Words which do not satisfy the Dual PCP are called periodicity forcing, and are important to the study of word equations, equality sets and ambiguity of morphisms. In this paper, a ‘prime’ subset of periodicity forcing words is presented. It is shown that when combined with a particular type of morphism it generates exactly the full set of periodicity forcing words. Furthermore, it is shown that there exist examples of periodicity forcing words which contain any given factor/prefix/suffix. Finally, an alternative class of mechanisms for generating periodicity forcing words is developed, resulting in a class of examples which contrast those known already.

Two global several-parameter bifurcation theorems and their applications

In this paper, we investigate the structure of the nontrivial solution set for the following nonlinear operator equationu=L(λ)u+H(λ,u),(λ,u)∈Rm×X, where m is a positive integer, X is a Banach space, L(⋅):X→X is a (positively) homogeneous compact operator and H:Rm×X→X is compact with H=o(‖u‖) near u=0 uniformly on bounded λ sets. We obtain some results involving (unilateral) global bifurcation. Two examples of p-Laplacian problem with jumping nonlinearity and nonlocal boundary value problem are given to demonstrate how the theory can be applied.

ON MAXIMAL, MINIMAL OPEN AND CLOSED SETS

We obtain some conditions for disconnectedness of a topolog- ical space in terms of maximal and minimal open sets, and some similar results in terms of maximal and minimal closed sets along with interre- lations between them. In particular, we show that if a space has a set which is both maximal and minimal open, then either this set is the only nontrivial open set in the space or the space is disconnected. We also obtain a result concerning a minimal open set on a subspace.

Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

In this paper we prove an almost sure local well-posedness result for the periodic 3D quintic nonlinear Schrödinger equation in the supercritical regime, that is below the critical space H^1(\mathbb T^3) . We also prove a long time existence result; more precisely we prove that for fixed T>0 there exists a set \Sigma_T , \mathbb P(\Sigma_T) > 0 such that any data \phi^{\omega}(x) \in H^{\gamma}(\mathbb T^3), \gamma<1, \omega \in \Sigma_T , evolves up to time T into a solution u(t) with u(t) - e^{it\Delta} \phi^{\omega} \in C([0,T]; H^s(\mathbb T^3)) , s=s(\gamma)>1 . In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space H^1(\mathbb T^3) , that is in the supercritical scaling regime.

Lagrangian dynamics and possible isochronous behavior in several classes of non-linear second order oscillators via the use of Jacobi last multiplier

In this paper, we employ the technique of Jacobi Last Multiplier (JLM) to derive Lagrangians for several important and topical classes of non-linear second-order oscillators, including systems with variable and parametric dissipation, a generalized anharmonic oscillator, and a generalized Lane–Emden equation. For several of these systems, it is very difficult to obtain the Lagrangians directly, i.e., by solving the inverse problem of matching the Euler–Lagrange equations to the actual oscillator equation. In order to facilitate the derivation of exact solutions, and also investigate possible isochronous behavior in the analyzed systems, we next invoke some recent theoretical results and attempt to map the potential term to either the simple harmonic oscillator or the isotonic potential for specific values of the coefficient parameters of each non-linear oscillator. We find non-trivial parameter sets corresponding to isochronous dynamics in some of the considered systems, but none in others. Finally, the Lagrangians obtained here are coupled to Noether׳s theorem, leading to non-trivial conservation laws for several of the oscillators.

EH-suprema of tournaments with no nontrivial homogeneous sets

A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph H there exists ϵ(H)>0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or stable set of size at least nϵ(H).The conjecture has directed equivalent version stating that for every tournament H there exists ϵ(H)>0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least nϵ(H). For a fixed tournament H, define ξ(H) to be the supremum of all ϵ for which the following holds: for some n0 and every n>n0 every tournament with n≥n0 vertices not containing H as a subtournament has a transitive subtournament of size at least nϵ. We call ξ(H) the EH-supremum of H. The Erdös–Hajnal conjecture is true if and only if ξ(H)>0 for every H. If the conjecture is false then the smallest counterexample has no nontrivial so-called homogeneous sets (to be defined below). Therefore of interest are EH-suprema of those tournaments. In [5] it was proven that there exists a constant η>0 such that ξ(H)≤4h(1+ηlog⁡(h)h) for almost every h-vertex tournament H. However this result does not say anything about ξ(H) for an arbitrarily chosen tournament with no nontrivial homogeneous sets. We address that problem in this paper, proving that there exists C>0 such that every h-vertex tournament H with no nontrivial homogeneous sets satisfies ξ(H)≤Clog⁡(h)h. We will also give upper bounds on sizes of families of h-vertex tournaments with big EH-suprema. In [1] Alon, Pach and Solymosi proposed a procedure that produces bigger graphs satisfying the conjecture from smaller ones. All graphs obtained in such a way have nontrivial homogeneous sets. For a long time that was the only method to obtain infinite families of graphs satisfying the conjecture. Recently Berger, the author and Chudnovsky (see [2]) constructed a new infinite family of tournaments (so-called galaxies, to be defined below) that satisfies the conjecture and with no nontrivial homogeneous sets. Therefore it cannot be obtained by the procedure described in [1]. In this paper we construct a new infinite family of tournaments satisfying the conjecture – the family of so-called constellations (to be defined below). These results extend the results of [2] since every galaxy is a constellation.

Formal linguistics as a cue to demographic history.

Beyond its theoretical success, the development of molecular genetics has brought about the possibility of extraordinary progress in the study of classification and in the inference of the evolutionary history of many species and populations. A major step forward was represented by the availability of extremely large sets of molecular data suited to quantitative and computational treatments. In this paper, we argue that even in cognitive sciences, purely theoretical progress in a discipline such as linguistics may have analogous impact. Thus, exactly on the model of molecular biology, we propose to unify two traditionally unrelated lines of linguistic investigation: 1) the formal study of syntactic variation (parameter theory) in the biolinguistic program; 2) the reconstruction of relatedness among languages (phylogenetic taxonomy). The results of our linguistic analysis have thus been plotted against data from population genetics and the correlations have turned out to be largely significant: given a non-trivial set of languages/populations, the description of their variation provided by the comparison of systematic parametric analysis and molecular anthropology informatively recapitulates their history and relationships. As a result, we can claim that the reality of some parametric model of the language faculty and language acquisition/transmission (more broadly of generative grammar) receives strong and original support from its historical heuristic power. Then, on these grounds, we can begin testing Darwin's prediction that, when properly generated, the trees of human populations and of their languages should eventually turn out to be significantly parallel.

Interaction-induced connectivity of disordered two-particle states

We study the interaction-induced connectivity in the Fock space of two particles in a disordered one-dimensional potential. Recent computational studies showed that the largest localization length $\xi_2$ of two interacting particles in a weakly random tight binding chain is increasing unexpectedly slow relative to the single particle localization length $\xi_1$, questioning previous scaling estimates. We show this to be a consequence of the approximate restoring of momentum conservation of weakly localized single particle eigenstates, and disorder-induced phase shifts for partially overlapping states. The leading resonant links appear among states which share the same energy and momentum. We substantiate our analytical approach by computational studies for up to $\xi_1 = 1000$. A potential nontrivial scaling regime sets in for $ \xi_1 \approx 400$, way beyond all previous numerical attacks.

Hyperbolicity of generic high-degree hypersurfaces in complex projective space

We use two ingredients to prove the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements in the complex projective space. One is the pullbacks of appropriate low pole order meromorphic jet differentials on the complex projective space to a hypersurface. The other is slanted vector fields of low vertical pole order on the vertical jet space of the universal hypersurface. We also present a number of related results, obtained by the same methods, such as: (i) a Big-Picard-Theorem type statement concerning extendibility, across the puncture, of holomorphic maps from a punctured disk to a generic hypersurface of high degree, (ii) nonexistence of nontrivial sets of entire functions satisfying certain polynomial equations with slowly varying coefficients, and (iii) Second Main Theorems for jet differentials and slowly moving targets.

Grounded Fixpoints

Algebraical fixpoint theory is an invaluable instrument for studying semantics of logics. For example, all major semantics of logic programming, autoepistemic logic, default logic and more recently, abstract argumentation have been shown to be induced by the different types of fixpoints defined in approximation fixpoint theory (AFT). In this paper, we add a new type of fixpoint to AFT: a grounded fixpoint of lattice operator O : L → L is defined as a lattice element x ∈ L such that O(x) = x and for all v ∈ L such that O(v ∧ x) ≤ v, it holds that x ≤ v. On the algebraical level, we show that all grounded fixpoints are minimal fixpoints approximated by the well-founded fixpoint and that all stable fixpoints are grounded. On the logical level, grounded fixpoints provide a new mathematically simple and compact type of semantics for any logic with a (possibly non-monotone) semantic operator. We explain the intuition underlying this semantics in the context of logic programming by pointing out that grounded fixpoints of the immediate consequence operator are interpretations that have no non-trivial unfounded sets. We also analyse the complexity of the induced semantics. Summarised, grounded fixpoint semantics is a new, probably the simplest and most compact, element in the family of semantics that capture basic intuitions and principles of various non-monotonic logics.

Experimental construction of generic three-qubit states and their reconstruction from two-party reduced states on an NMR quantum information processor

We experimentally explore the state space of three qubits on a nuclear magnetic resonance (NMR) quantum-information processor. We construct a scheme to experimentally realize a canonical form for general three-qubit states up to single-qubit unitaries. This form involves a nontrivial combination of Greenberger-Horne-Zeilinger (GHZ) and $W$-type maximally entangled states of three qubits. The general circuit that we have constructed for the generic state reduces to those for GHZ and $W$ states as special cases. The experimental construction of a generic state is carried out for a nontrivial set of parameters and the good fidelity of preparation is confirmed by complete state tomography. The GHZ and $W$ states are constructed as special cases of the general experimental scheme. Further, we experimentally demonstrate a curious fact about three-qubit states, where for almost all pure states, the two-qubit reduced states can be used to reconstruct the full three-qubit state. For the case of a generic state and for the $W$ state, we demonstrate this method of reconstruction by comparing it to the directly tomographed three-qubit state.

The zero set of a twistor spinor in any metric signature

Using tractor methods, we exhibit the local structure of the zero set of a twistor spinor in any metric signature. It is given as the image under the exponential map of a distinguished totally lightlike subspace. Based on these results we are able to construct starting from a conformal structure admitting a twistor spinor with zero a projective structure on its zero set in a natural way. The construction is presented both explicitly, using a fixed metric in the conformal class, and more abstractly on the level of associated parabolic Cartan geometries. Finally, we use known results about pseudo-Riemannian extensions of 2-dimensional projective structures to \((2,2)\)-ASD conformal structures to find first examples of non-conformally flat, non Riemannian manifolds admitting a twistor spinor with nontrivial zero set.

A combinatorial approach for mapping the interactions of multiple inputs to a biological system in a tractable number of experiments

Event Abstract Back to Event A combinatorial approach for mapping the interactions of multiple inputs to a biological system in a tractable number of experiments Vladimir Brezina1* and Brian Fulton-Howard1 1 Icahn School of Medicine at Mount Sinai, Department of Neuroscience, United States Multiple simultaneous inputs (e.g., neurochemical agents, drugs, therapeutic interventions) act together to affect the output of complex biological systems such as neurons and neural circuits or organs such as the heart. Describing how all these inputs act together on a system is difficult because the inputs often interact in ways that cannot be predicted from their individual actions. This happens because biological systems are nonlinearly coupled networks of many components through which the presence of one input can change the effect of another on the output. Full experimental mapping of all interactions between inputs is currently impractical for nontrivial input sets because of the combinatorial explosion: there are simply too many combinations of inputs to test each combination individually. To solve the problem, we have developed a block-design approach in which we construct a small number of test sets, each containing a number of pairs (or higher tuples) of the inputs, such that together all the test sets contain all possible pairs multiple times. Similar algorithms are used in communications and computer systems testing, but our algorithm must satisfy the additional need for repeated testing to deal with inter-preparation variability seen in biological systems. Our algorithm furthermore adaptively reduces the number of experiments required depending on the results obtained so far. Finally, it incorporates statistical tests to evaluate how nonlinearly “unexpected” each pairwise interaction is, relative to the prediction from the linear combination of effects of the two (or more) inputs alone. The statistic ranks all of the pairwise interactions from the most to the least unexpected. The algorithm thus functions as a global screen for the most unexpected interactions in the input set. Even large numbers of inputs require only small, experimentally tractable, numbers of test sets. Altogether, our approach promises to be able to guide the experimental discovery and global mapping of interactions between inputs to a system without any knowledge of the internal structure of the system. We are testing and further refining the approach using the Luo-Rudy computational model of mammalian ventricular myocytes, and then applying it experimentally to map the interactions between the many neuromodulators of a crustacean cardiac system. Acknowledgements Funded by NSF IOS 1146019 Keywords: neuromodulators, drugs, Multiple inputs, Combinatorial explosion, Block design, Computational Biology, biological networks Conference: Neuroinformatics 2015, Cairns, Australia, 20 Aug - 22 Aug, 2015. Presentation Type: Poster, to be considered for oral presentation Topic: Computational neuroscience Citation: Brezina V and Fulton-Howard B (2015). A combinatorial approach for mapping the interactions of multiple inputs to a biological system in a tractable number of experiments. Front. Neurosci. Conference Abstract: Neuroinformatics 2015. doi: 10.3389/conf.fnins.2015.91.00051 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 07 Apr 2015; Published Online: 05 Aug 2015. * Correspondence: Dr. Vladimir Brezina, Icahn School of Medicine at Mount Sinai, Department of Neuroscience, New York, NY, United States, vladimir.brezina@gmail.com Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Vladimir Brezina Brian Fulton-Howard Google Vladimir Brezina Brian Fulton-Howard Google Scholar Vladimir Brezina Brian Fulton-Howard PubMed Vladimir Brezina Brian Fulton-Howard Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.