Nontrivial Set Research Articles

Homeomorphisms of the real line with singularities.

Given a real number a ≠ 0, we consider the set of homeomorphisms f: R\{0}→ R \{a} where{(x, y):x=0}is a vertical asymtote, {(x, y):y=a} is a horizontal asymtote and f is strictly increasing in each connected component (−∞,0) and (0,+∞). In this context, similar to circle homeomorphisms, all possible dynamics are shown. It is established the relationship between existence of periodic orbits and the limit sets. Also, whenever f−n(0) ≠a for all n ∈ N, then the non-existence of periodic orbits leads to a non-trivial limit set, which is either the whole line R or perfect and nowhere dense. It is shown a notion of separation of points that leads to transitivity

A Bayesian theory of market impact

Abstract The available liquidity at any time in financial markets falls largely short of the typical size of the orders that institutional investors would trade. In order to reduce the impact on prices due to the execution of large orders, traders in financial markets split large orders into a series of smaller ones, which are executed sequentially. The resulting sequence of trades is called a meta-order. Empirical studies have revealed a non-trivial set of statistical laws on how meta-orders affect prices, which include (i) the square-root behaviour of the expected price variation with the total volume traded, (ii) its crossover to a linear regime for small volumes and (iii) a reversion of average prices towards its initial value, after the sequence of trades is over. Here we recover this phenomenology within a minimal theoretical framework where the market sets prices by incorporating all information on the direction and speed of trade of the meta-order in a Bayesian manner. The simplicity of this derivation lends further support to the robustness and universality of market impact laws. In particular, it suggests that the square-root impact law originates from over-estimation of order flows originating from meta-orders.

Nonexistence of energy function for surface diffeomorphisms with horseshoe basic sets

The Lyapunov function is a powerful tool for studying dynamical systems. In particular, the existence of such a function for any dynamical system given on compact manifold, established by C. Conley, is the basis for proving the Ω-stability criterion of dynamical systems. Moreover, the Lyapunov function, whose set of critical points of aligns with the chain-recurrent set of the system, which called an energy function, qualitatively determines the dynamics of the system. For continuous-time systems, which models dissipative processes, such a function naturally arises as an energy of the system. The existence of an energy function for arbitrary flows on compact manifolds is known. Examples of Morse–Smale diffeomorphism without energy function exist on n-dimensional manifolds starting with n>2. For systems with chaotic dynamics on surfaces, the existence of energy functions has also been established in cases where the dimension of non-trivial basic sets is greater than or equal to 1. When the chain-recurrent set of a diffeomorphism contains at least one zero-dimensional non-trivial basic set, the question of the existence of the energy function is still open. To date, it is known that the answer is negative if the zero-dimensional non-trivial basic set does not contain pairs of conjugated points. The present paper is devoted to the question of existence of an energy function for systems having a basic set with pairs of conjugated points. The primary outcome of this study is that there is no energy function for a dynamical system containing a Smale horseshoe as a basic set.

On non-trivial hyperbolic sets and their bifurcations in families of diffeomorphisms of a two-dimensional torus.

We propose a simple model-two-parameter family of diffeomorphisms of a two-dimensional torus. Combining analytical and numerical methods, we find regions in the parameter plane such that each diffeomorphism of the family is hyperbolic and describe the structure of the corresponding hyperbolic sets. We also study bifurcations on the boundaries of these regions, which lead to the change of hyperbolicity type (from Anosov diffeomorphisms to DA-diffeomorphisms).

The Wiener Index of Prime Graph $PG(\mathbb{Z}_n)$

Let G = (V, E) be a simple graph and (R, +, ·) be a ring with zero element 0R. The Wiener index of G, denoted by W(G), is defined as the sum of distances of every vertex u and v, or half of the sum of all entries of its distance matrix. The prime graph of R, denoted by P G(R), is defined as a graph with V (P G(R)) = R such that uv ∈ E(P G(R)) if and only if uRv = {0R} or vRu = {0R}. In this article, we determine the Wiener index of P G(Zn) in some cases n by constructing its distance matrix. We partition the set Zn into three types of sets, namely zero sets, nontrivial zero divisor sets, and unit sets. There are two objectives to be achieved. Firstly,we revise the Wiener index formula of P G(Zn) for n = p2 and n = p3 for prime number p and we compare this results with the results carried out by previous researchers. Secondly, we determine the Wiener index formula of P G(Zn) for n = pq, n = p2q, n =p 2q2, and n = pqr for distinct prime numbers p, q, and r.

Unexpected Properties of the Klein Configuration of 60 Points in P3

Felix Klein in course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of $60$ points in ${\mathbb P}^3$. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the $60$ reflection planes in the group $G_{31}$ in the Shephard-Todd list. In the present note we show that the $60$ points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree $6$. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of $60$ points is a cone with a single singularity of multiplicity $6$ and the other has three singular points of multiplicities $4,2$ and $2$. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in ${\mathbb P}^3$ with the surprising property that their general projection to ${\mathbb P}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of $24$ points in ${\mathbb P}^3$ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property. \

Wormholes and surface defects in rational ensemble holography

We study wormhole contributions to the bulk path integral in holographic models which are dual to ensembles of rational free boson conformal field theories. We focus on the path integral on a geometry connecting two toroidal boundaries, which should capture the variance of the ensemble distribution. We show that this requirement leads to a nontrivial set of constraints which generically picks out the uniform, maximum entropy, ensemble distribution. Furthermore, we show that the two-boundary path integral should receive contributions from ‘exotic’ wormholes, which arise from the inclusion of topological surface defects.

Uncertainty-Aware Deep Neural Representations for Visual Analysis of Vector Field Data.

The widespread use of Deep Neural Networks (DNNs) has recently resulted in their application to challenging scientific visualization tasks. While advanced DNNs demonstrate impressive generalization abilities, understanding factors like prediction quality, confidence, robustness, and uncertainty is crucial. These insights aid application scientists in making informed decisions. However, DNNs lack inherent mechanisms to measure prediction uncertainty, prompting the creation of distinct frameworks for constructing robust uncertainty-aware models tailored to various visualization tasks. In this work, we develop uncertainty-aware implicit neural representations to model steady-state vector fields effectively. We comprehensively evaluate the efficacy of two principled deep uncertainty estimation techniques: (1) Deep Ensemble and (2) Monte Carlo Dropout, aimed at enabling uncertainty-informed visual analysis of features within steady vector field data. Our detailed exploration using several vector data sets indicate that uncertaintyaware models generate informative visualization results of vector field features. Furthermore, incorporating prediction uncertainty improves the resilience and interpretability of our DNN model, rendering it applicable for the analysis of non-trivial vector field data sets.

A Decision Procedure for a Theory of Finite Sets with Finite Integer Intervals

In this paper we extend a decision procedure for the Boolean algebra of finite sets with cardinality constraints (ℒ |⋅| ) to a decision procedure for ℒ |⋅| extended with set terms denoting finite integer intervals (ℒ [] ). In ℒ [] interval limits can be integer linear terms including unbounded variables . These intervals are a useful extension because they allow to express non-trivial set operators such as the minimum and maximum of a set, still in a quantifier-free logic. Hence, by providing a decision procedure for ℒ [] it is possible to automatically reason about a new class of quantifier-free formulas. The decision procedure is implemented as part of the { log } (‘setlog’) tool. The paper includes a case study based on the elevator algorithm showing that { log } can automatically discharge all its invariance lemmas, some of which involve intervals.

Non-linear oscillators with Kuramoto-like local coupling: Complexity analysis and spatiotemporal pattern generation

Can a simple oscillator system, as in cellular automata, sustain complex nature upon discretization in time and space? The answer is by no means trivial as even the most simple, two-state, nearest neighbours cellular automata can lead to Universal Turing Machine (UTM) computing power. This study analyses a recently proposed model consisting of a ring of identical excitable Adler-type oscillators with local Kuramoto-like coupling in terms of its complexity. Regions with non-trivial, complex behaviour have been identified, where spatiotemporal maps closely resemble those found in elementary cellular automata, not only from the visual perspective but also from entropic measures characterization. Also, the possibility of enhanced computation at the edge of chaos is explored by monitoring the effective complexity measure, entropy density and informational distance, following previous approaches. Distance matrix, and the corresponding dendrograms, show that in the complex regions, a non-trivial set of hierarchies emerges where local communities can appear and sustain themself in time. The fact that coupled oscillators can be realized through dedicated electronic circuitry or the result of natural processes can make finding such complex dynamics, with potential computational power, an important one in a broad scope of applications and implications.

Energy Function for Direct Products of Discrete Dynamical Systems

This paper is devoted to the construction of an energy function, i.e. a smooth Lyapunov function, whose set of critical points coincides with the chain-recurrent set of a dynamical system — for a cascade that is a direct product of two systems. One of the multipliers is a structurally stable diffeomorphism given on a two-dimensional torus, whose non-wandering set consists of a zero-dimensional non-trivial basic set without pairs of conjugated points and without fixed source and sink, and the second one is an identical mapping on a real axis. It was previously proved that if a non-wandering set of a dynamical system contains a zero-dimensional basic set, as the diffeomorphism under consideration has, then such a system does not have an energy function, namely, any Lyapunov function will have critical points outside the chain-recurrent set. For an identical mapping, the energy function is a constant on the entire real line. In this paper, it is shown that the absence of an energy function for one of the multipliers is not a sufficient condition for the absence of such a function for the direct product of dynamical systems, that is, in some cases it is possible to select the second cascade in such a way that the direct product will have an energy function.

Hierarchical activation of quantum nonlocality: Stronger than local indistinguishability

The quantum state discrimination primitive becomes highly nontrivial in the limited measurement setting and leads to different classes of impossibility, viz., indistinguishability, unmarkability, irreducibility, etc. These phenomena, often referred to as another nonlocal aspect of quantum theory, have utmost importance in the domain of data hiding, secret sharing, etc. Motivated by this, recently a significant effort has been devoted to activate local indistinguishability from locally distinguishable nontrivial sets of quantum states. In the present work, we introduce other stronger notions of quantum nonlocality and depict a series of hierarchical nonlocality activation from nontrivial sets of locally distinguishable entangled states. Moreover, we appropriately moderate the strict condition of nontriviality, introduced in earlier literature, and come up with interesting examples in support of our claim.

On a Classification of Chaotic Laminations which are Nontrivial Basic Sets of Axiom A Flows

We prove that, given any $n\geqslant 3$ and $2\leqslant q\leqslant n-1$, there is a closed $n$-manifold $M^{n}$ admitting a chaotic lamination of codimension $q$ whose support has the topological dimension ${n-q+1}$. For $n=3$ and $q=2$, such chaotic laminations can be represented as nontrivial 2-dimensional basic sets of axiom A flows on 3-manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2-dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repeller-attractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with one-dimensional basic sets obtained previously by V. Grines, R. Plykin and Yu. Zhirov [16, 28, 31].

Dependence-Robust Confidence Intervals for Capture-Recapture Surveys.

Capture-recapture (CRC) surveys are used to estimate the size of a population whose members cannot be enumerated directly. CRC surveys have been used to estimate the number of Coronavirus Disease 2019 (COVID-19) infections, people who use drugs, sex workers, conflict casualties, and trafficking victims. When k-capture samples are obtained, counts of unit captures in subsets of samples are represented naturally by a contingency table in which one element-the number of individuals appearing in none of the samples-remains unobserved. In the absence of additional assumptions, the population size is not identifiable (i.e., point identified). Stringent assumptions about the dependence between samples are often used to achieve point identification. However, real-world CRC surveys often use convenience samples in which the assumed dependence cannot be guaranteed, and population size estimates under these assumptions may lack empirical credibility. In this work, we apply the theory of partial identification to show that weak assumptions or qualitative knowledge about the nature of dependence between samples can be used to characterize a nontrivial confidence set for the true population size. We construct confidence sets under bounds on pairwise capture probabilities using two methods: test inversion bootstrap confidence intervals and profile likelihood confidence intervals. Simulation results demonstrate well-calibrated confidence sets for each method. In an extensive real-world study, we apply the new methodology to the problem of using heterogeneous survey data to estimate the number of people who inject drugs in Brussels, Belgium.

Affine semigroups of maximal projective dimension

A submonoid of \( {\mathbb {N}}^d \) is of maximal projective dimension (\({\text {MPD}}\)) if the associated affine semigroup ring has the maximum possible projective dimension. Such submonoids have a nontrivial set of pseudo-Frobenius elements. We generalize the notion of symmetric semigroups, pseudo-symmetric semigroups, and row-factorization matrices for pseudo-Frobenius elements of numerical semigroups to the case of \({\text {MPD}}\)-semigroups in \({\mathbb {N}}^d\). Under suitable conditions, we prove that these semigroups satisfy the generalized Wilf’s conjecture. We prove that the generic nature of the defining ideal of the associated semigroup ring of an \({\text {MPD}}\)-semigroup implies uniqueness of row-factorization matrix for each pseudo-Frobenius element. Further, we give a description of pseudo-Frobenius elements and row-factorization matrices of gluing of \({\text {MPD}}\)-semigroups. We prove that the defining ideal of gluing of \({\text {MPD}}\)-semigroups is never generic.

Higgledy-Piggledy Sets in Projective Spaces of Small Dimension

This work focuses on higgledy-piggledy sets of $k$-subspaces in $\text{PG}(N,q)$, i.e. sets of projective subspaces that are 'well-spread-out'. More precisely, the set of intersection points of these $k$-subspaces with any $(N-k)$ subspace $\kappa$ of $\text{PG}(N,q)$ spans $\kappa$ itself.
 We highlight three methods to construct small higgledy-piggledy sets of $k$-subspaces and discuss, for $k\in\{1,N-2\}$, 'optimal' sets that cover the smallest possible number of points.
 Furthermore, we investigate small non-trivial higgledy-piggledy sets in $\text{PG}(N,q)$, $N\leqslant5$. Our main result is the existence of six lines of $\text{PG}(4,q)$ in higgledy-piggledy arrangement, two of which intersect. Exploiting the construction methods mentioned above, we also show the existence of six planes of $\text{PG}(4,q)$ in higgledy-piggledy arrangement, two of which maximally intersect, as well as the existence of two higgledy-piggledy sets in $\text{PG}(5,q)$ consisting of eight planes and seven solids, respectively. Finally, we translate these geometrical results to a coding- and graph-theoretical context.

Inverse pseudo orbit tracing property for robust diffeomorphisms

Let f : M → M be a diffeomorphism of a compact smooth manifold M. Herein, we demonstrate that (i) if f has the C1 robustly inverse pseudo orbit tracing property on the chain recurrent set CR(f), then CRf is hyperbolic of f and (ii) if f has the C1 robustly inverse pseudo orbit tracing property on a nontrivial transitive set Λ ⊂ M, then Λ is hyperbolic for f.111991 Mathematics Subject Classification. 37C50; 37D20.

Generalized signed graphs of large girth and large chromatic number

Assume G is a graph and S is a set of permutations of positive integers. An S-labelling of G is a pair (D,σ), where D is an orientation of G and σ:E(D)→S is a mapping which assigns to each arc e=(u,v) of D a permutation σe∈S. A proper k-colouring of (D,σ) is a mapping f:V(G)→[k]={1,2,...,k} such that σe(f(x))≠f(y) for each arc e=(x,y). We say S is non-trivial if for every positive integer i, there is a permutation π∈S such that π(i)=i, and S is transitive if for every pair (i,j) of positive integers, there is a permutation π∈S such that π(i)=j or π(j)=i. The S-chromatic number of a graph G is the minimum integer k such that any S-labelling (D,σ) of G has a proper k-colouring. This paper constructs, for any non-trivial set S of permutations of integers, for any integers k,g, a graph of girth at least g and S-chromatic number greater than k. We further prove that if S is a non-trivial set of permutations of positive integers, and 2≤k′≤k and g are positive integers, then the following two statements are equivalent: (1) there exists a graph of girth at least g, of chromatic number k′ and S-chromatic number greater than k, (2) for any k′-subset I of [k], there exist a,b∈I and π∈S such that a≠b and either π(a)=b or π(b)=a. As a consequence, there is a bipartite graph of arbitrary large girth and arbitrary large S-chromatic number if and only if S is transitive. In particular, there is a bipartite graph G of arbitrary large girth and arbitrary large group chromatic number.

Characterizing bipartite consensus on signed matrix-weighted networks via balancing set

In contrast with the scalar-weighted networks, where bipartite consensus can be achieved if and only if the underlying signed network is structurally balanced, the structural balance property is no longer a graph-theoretic equivalence to the bipartite consensus in the case of signed matrix-weighted networks. To re-establish the relationship between the network structure and the bipartite consensus solution, the non-trivial balancing set is introduced which is a set of edges whose sign negation can transform a structurally imbalanced network into a structurally balanced one and the weight matrices associated with edges in this set have a non-trivial intersection of null spaces. We show that necessary and/or sufficient conditions for bipartite consensus on matrix-weighted networks can be characterized by the uniqueness of the non-trivial balancing set, while the contribution of the associated non-trivial intersection of null spaces to the steady-state of the matrix-weighted network is examined. Moreover, for matrix-weighted networks with a positive–negative spanning tree, necessary and sufficient condition for bipartite consensus using the non-trivial balancing set is obtained. Simulation examples are provided to demonstrate the theoretical results.

Interval Privacy: A Framework for Privacy-Preserving Data Collection

The emerging public awareness and government regulations of data privacy motivate new paradigms of collecting and analyzing data that are transparent and acceptable to data owners. We present a new concept of privacy and corresponding data formats, mechanisms, and theories for privatizing data during data collection. The privacy, named Interval Privacy, enforces the raw data conditional distribution on the privatized data to be the same as its unconditional distribution over a nontrivial support set. Correspondingly, the proposed privacy mechanism will record each data value as a random interval (or, more generally, a range) containing it. The proposed interval privacy mechanisms can be easily deployed through survey-based data collection interfaces, e.g., by asking a respondent whether its data value is within a randomly generated range. Another unique feature of interval mechanisms is that they obfuscate the truth but do not perturb it. Using narrowed range to convey information is complementary to the popular paradigm of perturbing data. Also, the interval mechanisms can generate progressively refined information at the discretion of individuals, naturally leading to privacy-adaptive data collection. We develop different aspects of theory such as composition, robustness, distribution estimation, and regression learning from interval-valued data. Interval privacy provides a new perspective of human-centric data privacy where individuals have a perceptible, transparent, and simple way of sharing sensitive data.