Infinite Set Research Articles

Signature-based standard basis algorithm under the framework of GVW algorithm

The GVW algorithm, one of the most important so-called signature-based algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the GVW algorithm, and up to now, it applies only to a certain class of monomial orders, namely global orders and a special class of local orders. In this paper we extend this theorem to any semigroup order, which can be either global, local or even mixed. Building upon the pioneering idea of the Mora normal form algorithm, we propose a more comprehensive and general proof for the cover theorem while bypassing the need to choose a minimal element from an infinite set of monomials in all the existing proofs. Therefore, the algorithm for signature-based standard bases is presented for any semigroup order under the framework of the GVW algorithm, and an example is given to provide an illustration of the algorithm.

Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin - [formula omitted] chain

It is known that for the Heisenberg XXZ spin - 12 chain in the critical regime, the scaling limit of the vacuum Bethe roots yields an infinite set of numbers that coincide with the energy spectrum of the quantum mechanical 3D anharmonic oscillator. The discovery of this curious relation, among others, gave rise to the approach referred to as the ODE/IQFT (or ODE/IM) correspondence. Here we consider a multiparametric generalization of the Heisenberg spin chain, which is associated with the inhomogeneous six-vertex model. When quasi-periodic boundary conditions are imposed the lattice system may be explored within the Bethe Ansatz technique. We argue that for the critical spin chain, with a properly formulated scaling limit, the scaled Bethe roots for the ground state are described by second order differential equations, which are multi-parametric generalizations of the Schrödinger equation for the anharmonic oscillator.

Implications of Ramsey Choice principles in ZF$\mathsf {ZF}$

AbstractThe Ramsey Choice principle for families of ‐element sets, denoted , states that every infinite set has an infinite subset with a choice function on . We investigate for which positive integers and the implication is provable in . It will turn out that beside the trivial implications , under the assumption that every odd integer is the sum of three primes (known as ternary Goldbach conjecture), the only non‐trivial implication which is provable in is .

Quantum Annular Homology and Bigger Burnside Categories

As part of their construction of the Khovanov spectrum, Lawson, Lipshitz and Sarkar assigned to each cube in the Burnside category of finite sets and finite correspondences, a finite cellular spectrum. In this paper, we extend this assignment to cubes in Burnside categories of infinite sets. This is later applied to the work of Akhmechet, Krushkal and Willis on the quantum annular Khovanov spectrum with an action of a finite cyclic group: we obtain a quantum annular Khovanov spectrum with an action of the infinite cyclic group.

The effect of obstacle length and height in subcritical free-surface flow

Two-dimensional free-surface flow past a submerged rectangular disturbance in an open channel is considered. The forced Korteweg–de Vries model of Binder et al. (Theor Comput Fluid Dyn 20:125–144, 2006) is modified to examine the effect of varying obstacle length and height on the response of the free-surface. For a given obstacle height and flow rate in the subcritical flow regime an analysis of the steady solutions in the phase plane of the problem determines a countably infinite set of discrete obstacle lengths for which there are no waves downstream of the obstacle. A rich structure of nonlinear behaviour is also found as the height of the obstacle approaches critical values in the steady problem. The stability of the steady solutions is investigated numerically in the time-dependent problem with a pseudospectral method.

The uncertainty principle and classical amplitudes

We study the variance in the measurement of observables during scattering events, as computed using amplitudes. The classical regime, characterised by negligible uncertainty, emerges as a consequence of an infinite set of relationships among multileg, multiloop amplitudes in a momentum-transfer expansion. We discuss two non-trivial examples in detail: the six-point tree and the five-point one-loop amplitudes in scalar QED. We interpret these relationships in terms of a coherent exponentiation of radiative effects in the classical limit which generalises the eikonal formula, and show how to recover the impulse, including radiation reaction, from this generalised eikonal. Finally, we incorporate the physics of spin into our framework.

Idempotent cellular automata and their natural order

Motivated by the search for idempotent cellular automata (CA), we study CA that act almost as the identity unless they read a fixed pattern p. We show that constant and symmetrical patterns always generate idempotent CA, and we characterize the quasi-constant patterns that generate idempotent CA. Our results are valid for CA over an arbitrary group G. Moreover, we study the semigroup theoretic natural partial order defined on idempotent CA. If G is infinite, we prove that there is an infinite independent set of idempotent CA, and if G has an element of infinite order, we prove that there is an infinite increasing chain of idempotent CA.

On the diameter of semigroups of transformations and partitions

AbstractFor a semigroup whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right‐), the right diameter of is a parameter that expresses how ‘far apart’ elements of can be from each other, in a certain sense. To be more precise, for each finite generating set for the universal right congruence on , we have a metric space where is the minimum length of derivations for as a consequence of pairs in ; the right diameter of with respect to is the diameter of this metric space. The right diameter of is then the minimum of the set of all right diameters with respect to finite generating sets. We develop a theoretical framework for establishing whether a semigroup of transformations or partitions on an arbitrary infinite set has a finitely generated universal right/left congruence, and, if it does, determining its right/left diameter. We apply this to prove results such as the following. Each of the monoids of all binary relations on , of all partial transformations on , and of all full transformations on , as well as the partition and partial Brauer monoids on , have right diameter 1 and left diameter 1. The symmetric inverse monoid on has right diameter 2 and left diameter 2. The monoid of all injective mappings on has right diameter 4, and its minimal ideal (called the Baer–Levi semigroup on ) has right diameter 3, but neither of these two semigroups has a finitely generated universal left congruence. On the other hand, the semigroup of all surjective mappings on has left diameter 4, and its minimal ideal has left diameter 2, but neither of these semigroups has a finitely generated universal right congruence.

Learning the Optimal Control for Evolving Systems with Converging Dynamics

We consider a principle or controller that can pick actions from a fixed action set to control an evolving system with converging dynamics. The converging dynamics means that, if the principle holds the same action, the system will asymptotically converge to a unique stable state determined by this action. In our model, the dynamics of the system are unknown to the principle, and the principle can only receive bandit feedback (maybe noisy) on the impacts of his actions. The principle aims to learn which stable state yields the highest reward while adhering to specific constraints and to immerse the system into this state as quickly as possible. We measure the principle's performance in terms of regret and constraint violation. In cases where the action set is finite, we propose an algorithm Optimistic-Pessimistic Convergence and Confidence Bounds (OP-C2B) that ensures sublinear regret and constraint violation simultaneously. Particularly, OP-C2B achieves logarithmic regret and constraint violation when the system convergence rate is linear or superlinear. Furthermore, we generalize our algorithm OP-C2B to the case of an infinite action set and demonstrate its ability to maintain sublinear regret and constraint violation.

Quantum mechanical bootstrap on the interval: Obtaining the exact spectrum

We show that for a particular model, the quantum mechanical bootstrap is capable of finding exact results. We consider a solvable system with Hamiltonian H=SZ(1−Z)S, where Z and S satisfy canonical commutation relations. While this model may appear unusual, using an appropriate coordinate transformation, the Schrödinger equation can be cast into a standard form with a Pöschl-Teller-type potential. Since the system is defined on an interval, it is well-known that S is not self-adjoint. Nevertheless, the bootstrap method can still be implemented, producing an infinite set of positivity constraints. Using a certain operator ordering, the energy eigenvalues are only constrained into bands. With an alternative ordering, however, we find that a finite number of constraints is sufficient to fix the low-lying energy levels exactly. Published by the American Physical Society 2024

Modeling NP-problems with families of extended graph-based reaction systems

In this paper, we continue the investigation of graph-based reaction systems. We extend the notion by input and output states as well as admitted context sequences to model explicitly input–output relations and decision problems on the inputs. Moreover, we combine extended graph-based reaction systems into families to cover infinite input–output relations and decision problems on infinite sets of graphs. This is used to model NP-problems on graphs and reductions between them as well as to prove their correctness.

Applying the Sine-Cosine Optimization Algorithm to the Parametric Estimation Problem in Three-Phase Induction Motors

The steady-state analysis of electrical machines requires a detailed characterization of their equivalent electrical circuit, which adequately represents the transformation and interaction between electrical and mechanical energy. This research aims to characterize the equivalent circuit of three-phase induction motors by minimizing the mean square error between the measured and calculated torque variables. These torques are obtained from data provided by the manufacturer, including starting, peak, and full-load torques. A metaheuristic optimization technique is applied to solve the resulting nonlinear programming model based on the interactions between the sine and cosine functions. The numerical results obtained with this algorithm demonstrate its efficiency in terms of response quality, reaching objective function values of less than \(1\times10^{-8}\) with regard to the measured and calculated variables. Simulation results in two test systems allow concluding that the parametric estimation problem in three-phase induction motors is a multimodal optimization problem. This implies a potentially infinite set of solutions that minimize the root mean square error and adequately represent the behavior of the motor's output torque under various probable operating conditions.

Geochemical Pathways Defined by Predictive Regolith-Landform Models Using TanDEM-X Data in the Tanami Region, Australia

The Granites-Tanami Orogen (GTO) is a significant auriferous province located in the poorly exposed southwestern part of the North Australian Craton. This paper looks at the data sources that are suited to studying regolith-landform in areas such as the Tanami Region. One of the most useful datasets for regional regolith-landform mapping is a digital elevation model (DEM). The DEM data that were reviewed demonstrated the utility of TanDEM over other sources, and quantitative results focused on these data for identifying regolith-landform patterns that would aid in geochemical sampling and other land use studies. This paper will demonstrate that neither the classical approach of mapping boundaries based on visual estimates of breaks in slope, nor the alternative approach of producing maps and models derived from algorithms, can be used in isolation. Visual estimates of many boundaries are complex, unreliable and time consuming, and algorithmic mapping is driven by parameters than can produce an infinite set of models. This paper shows that simple landform visualisation is often the most powerful tool for map and model creation, supported by geomorphometric analysis and remotely sensed spectral imagery. The use of TanDEM data is shown to produce the best regolith-landform maps and models of the Tanami Region, which can then improve mineral exploration success.

Learning the Optimal Control for Evolving Systems with Converging Dynamics

We consider a principle or controller that can pick actions from a fixed action set to control an evolving system with converging dynamics. The actions are interpreted as different configurations or policies. We consider systems with converging dynamics, i.e., if the principle holds the same action, the system will asymptotically converge (possibly requiring a significant amount of time) to a unique stable state determined by this action. This phenomenon can be observed in diverse domains such as epidemic control, computing systems, and markets. In our model, the dynamics of the system are unknown to the principle, and the principle can only receive bandit feedback (maybe noisy) on the impacts of his actions. The principle aims to learn which stable state yields the highest reward while adhering to specific constraints (i.e., optimal stable state) and to immerse the system into this state as quickly as possible. A unique challenge in our model is that the principle has no prior knowledge about the stable state of each action, but waits for the system to converge to the suboptimal stable states costs valuable time. We measure the principle's performance in terms of regret and constraint violation. In cases where the action set is finite, we propose a novel algorithm, termed Optimistic-Pessimistic Convergence and Confidence Bounds (OP-C2B), that knows to switch an action quickly if it is not worth waiting until the stable state is reached. This is enabled by employing "convergence bounds" to determine how far the system is from the stable states, and choosing actions through maintaining a pessimistic assessment of the set of feasible actions while acting optimistically within this set. We establish that OP-C2B can ensure sublinear regret and constraint violation simultaneously. Particularly, OP-C2B achieves logarithmic regret and constraint violation when the system convergence rate is linear or superlinear. Furthermore, we generalize our algorithm OP-C2B to the case of an infinite action set and demonstrate its ability to maintain sublinear regret and constraint violation. We finally show two game control problems including mobile crowdsensing and resource allocation that our model can address.

Infinite multisets: Basic properties and cardinality

This research work presents the topic of infinite multisets, their basic properties and cardinality from a somewhat different perspective. In this work, a new property of multisets, ‘m-cardinality’, is defined using multiset functions. M-cardinality unifies and generalizes the definitions of cardinality, injection, bijection, and surjection to apply to multisets. M-cardinality takes into account both the number of distinct elements in a multiset and the number of copies of each element (i.e., the multiplicity of the elements). Based on m-cardinality, ‘m-cardinal numbers’ are defined as a generalization of cardinal numbers in the context of multisets. Some properties of m-cardinal numbers associated with finite and infinite msets have been researched. Concrete examples of transfinite m-cardinal numbers are given, corresponding to infinite msets which are less than ℵ0 (the cardinality of the countably infinite set). It has been established that between finite numbers and ℵ0 there exist hierarchies of transfinite m-cardinals, corresponding to infinite msets. Furthermore, there are examples of infinite msets with negative multiplicity that have a cardinality less than zero. We prove that there is a decreasing sequence of transfinite m-cardinal numbers, corresponding to infinite msets with negative multiplicity, and in this sequence, there is not a smallest transfinite m-cardinal number.

Closed-loop Koopman operator approximation

This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop (CL) system, to simultaneously identify the CL and plant systems. The advantages of the proposed CL Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper.

Analytic bootstrap for magnetic impurities

We study the O(3) critical model and the free theory of a scalar triplet in the presence of a magnetic impurity. We use analytic bootstrap techniques to extract results in the ε-expansion. First, we extend by one order in perturbation theory the computation of the beta function for the defect coupling in the free theory. Then, we analyze in detail the low-lying spectrum of defect operators, focusing on their perturbative realization when the defect is constructed as a path-ordered exponential. After this, we consider two different bulk two-point functions and we compute them using the defect dispersion relation. For a free bulk theory, we are able to fix the form of the correlator at all orders in ε. In particular, taking ε → 1, we can show that in d = 3 one does not have a consistent and non-trivial defect CFT. For an interacting bulk, we compute the correlator up to second order in ε. Expanding these results in the bulk and defect block expansions, we are able to extract an infinite set of defect CFT data. We discuss low-spin ambiguities that affect every result computed through the dispersion relation and we use a combination of consistency conditions and explicit diagrammatic calculations to fix this ambiguity.

Markoff–Lagrange spectrum of one‐sided shifts

AbstractFor the Lagrange spectrum and other applications, we determine the smallest accumulation point of binary sequences that are maximal in their shift orbits. This problem is trivial for the lexicographic order, and its solution is the fixed point of a substitution for the alternating lexicographic order. For orders defined by cylinders, we show that the solutions are ‐adic sequences, where is a certain infinite set of substitutions that contains Sturmian morphisms. We also consider a similar problem for symmetric ternary shifts, which is applicable to the multiplicative version of the Markoff–Lagrange spectrum.

Katětov order between Hindman, Ramsey and summable ideals

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Synesth: Comprehensive Syntenic Reconciliation with Unsampled Lineages

We present Synesth, the most comprehensive and flexible tool for tree reconciliation that allows for events on syntenies (i.e., on sets of multiple genes), including duplications, transfers, fissions, and transient events going through unsampled species. This model allows for building histories that explicate the inconsistencies between a synteny tree and its associated species tree. We examine the combinatorial properties of this extended reconciliation model and study various associated parsimony problems. First, the infinite set of explicatory histories is reduced to a finite but exponential set of Pareto-optimal histories (in terms of counts of each event type), then to a polynomial set of Pareto-optimal event count vectors, and this eventually ends with minimum event cost histories given an event cost function. An inductive characterization of the solution space using different algebras for each granularity leads to efficient dynamic programming algorithms, ultimately ending with an O(mn) time complexity algorithm for computing the cost of a minimum-cost history (m and n: number of nodes in the input synteny and species trees). This time complexity matches that of the fastest known algorithms for classical gene reconciliation with transfers. We show how Synesth can be applied to infer Pareto-optimal evolutionary scenarios for CRISPR-Cas systems in a set of bacterial genomes.