Finite Set Research Articles

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

Rings of functions which are discontinuous on a finite set with countable range

Rings of functions which are discontinuous on a finite set with countable range

Co-Axial Metrics on the Sphere and Algebraic Numbers

In this paper, we consider the following curvature equation $$\Delta u+{\rm e}^u=4\pi\biggl((\theta_0-1)\delta_0+(\theta_1-1)\delta_1 +\sum_{j=1}^{n+m}\bigl(\theta_j'-1\bigr)\delta_{t_j}\biggr)\qquad \text{in}\ \mathbb R^2,$$ $$u(x)=-2(1+\theta_\infty)\ln|x|+O(1)\qquad \text{as} \ |x|\to\infty,$$ where $\theta_0$, $\theta_1$, $\theta_\infty$, and $\theta_{j}'$ are positive non-integers for $1\le j\le n$, while $\theta_{j}'\in\mathbb{N}_{\geq 2}$ are integers for $n+1\le j\le n+m$. Geometrically, a solution $u$ gives rise to a conical metric ${\rm d}s^2=\frac12 {\rm e}^u|{\rm d}x|^2$ of curvature $1$ on the sphere, with conical singularities at $0$, $1$, $\infty$, and $t_j$, $1\le j\le n+m$, with angles $2\pi\theta_0$, $2\pi\theta_1$, $2\pi\theta_\infty$, and $2\pi\theta_{j}'$ at $0$, $1$, $\infty$, and $t_j$, respectively. The metric ${\rm d}s^2$ or the solution $u$ is called co-axial, which was introduced by Mondello and Panov, if there is a developing map $h(x)$ of $u$ such that the projective monodromy group is contained in the unit circle. The sufficient and necessary conditions in terms of angles for the existence of such metrics were obtained by Mondello-Panov (2016) and Eremenko (2020). In this paper, we fix the angles and study the locations of the singularities $t_1,\dots,t_{n+m}$. Let $A\subset\mathbb{C}^{n+m}$ be the set of those $(t_1,\dots,t_{n+m})$'s such that a co-axial metric exists, among other things we prove that (i) If $m=1$, i.e., there is only one integer $\theta_{n+1}'$ among $\theta_j'$, then $A$ is a finite set. Moreover, for the case $n=0$, we obtain a sharp bound of the cardinality of the set $A$. We apply a result due to Eremenko, Gabrielov, and Tarasov (2016) and the monodromy of hypergeometric equations to obtain such a bound. (ii) If $m\ge 2$, then $A$ is an algebraic set of dimension $\leq m-1$.

A long-horizon move-blocking based direct power model predictive control for dynamic enhancement of DC microgrids

This research paper presents a novel long-horizon move-blocking based direct power model predictive control (DPMPC) strategy, uniquely designed to enhance the dynamic performance of boost converters in the grid-connected mode within DC microgrids. A precise dynamic model is developed considering a boost converter connected to a DC-link supplying constant power and resistive loads. To predict the system's dynamic behavior over an extended interval and enhance its performance in the presence of constant power loads, the long-horizon based finite control set model predictive control (FCS-MPC) method is introduced. To address the computational complexity associated with the conventional long-horizon DPMPC approach, a move-blocking (MB) strategy is incorporated into FCS-MPC, reducing the computational burden while improving the dynamic performance of the boost converter. Unlike recent studies that utilize DPMPC with short prediction interval, this paper presents evidence that a longer prediction interval is crucial for maintaining system stable and enhancing the dynamic response and also utilizing MB to make the control implementable in the real-time. The simulation results conducted using MATLAB/Simulink demonstrate the effective performance of the proposed strategy across various operating conditions of the boost converter. Experimental results further validate the strategy's capability to enhance the dynamic performance of the boost converter. Importantly, the experimental findings confirm the feasibility of implementing the proposed long-horizon move-blocking DPMPC strategy in real-time applications.

Structural and Organizational Strategies of Locomotor Modules during Landing in Patients with Chronic Ankle Instability

Background: Human locomotion involves the coordinated activation of a finite set of modules, known as muscle synergy, which represent the motor control strategy of the central nervous system. However, most prior studies have focused on isolated muscle activation, overlooking the modular organization of motor behavior. Therefore, to enhance comprehension of muscle coordination dynamics during multi-joint movements in chronic ankle instability (CAI), exploring muscle synergies during landing in CAI patients is imperative. Methods: A total of 22 patients with unilateral CAI and 22 healthy participants were recruited for this research. We employed a recursive model for second-order differential equations to process electromyographic (EMG) data after filtering preprocessing, generating the muscle activation matrix, which was subsequently inputted into the non-negative matrix factorization model for extraction of the muscle synergy. Muscle synergies were classified utilizing the K-means clustering algorithm and Pearson correlation coefficients. Statistical parameter mapping (SPM) was employed for temporal modular parameter analyses. Results: Four muscle synergies were identified in both the CAI and healthy groups. In Synergy 1, only the gluteus maximus showed significantly higher relative weight in CAI compared to healthy controls (p = 0.0035). Synergy 2 showed significantly higher relative weights for the vastus lateralis in the healthy group compared to CAI (p = 0.018), while in Synergy 4, CAI demonstrated significantly higher relative weights of the vastus lateralis compared to healthy controls (p = 0.030). Furthermore, in Synergy 2, the CAI group exhibited higher weights of the tibialis anterior compared to the healthy group (p = 0.042). Conclusions: The study suggested that patients with CAI exhibit a comparable modular organizational framework to the healthy group. Investigation of amplitude adjustments within the synergy spatial module shed light on the adaptive strategies employed by the tibialis anterior and gluteus maximus muscles to optimize control strategies during landing in patients with CAI. Variances in the muscle-specific weights of the vastus lateralis across movement modules reveal novel biomechanical adaptations in CAI, offering valuable insights for refining rehabilitation protocols.

The Bhargava Greedoid as a Gaussian Elimination Greedoid

Inspired by Manjul Bhargava's theory of generalized factorials, Grinberg and Petrov have defined the Bhargava greedoid - a greedoid (a matroid-like set system on a finite set) assigned to any "ultra triple" (a somewhat extended variant of a finite ultrametric space). Here we show that the Bhargava greedoid of a finite ultra triple is always a Gaussian elimination greedoid over any sufficiently large (e.g., infinite) field; this is a greedoid analogue of a representable matroid. We find necessary and sufficient conditions on the size of the field to ensure this.

There are no post-quantum weakly pseudo-free families in any nontrivial variety of expanded groups

Let [Formula: see text] be a finite set of finitary operation symbols and let [Formula: see text] be a nontrivial variety of [Formula: see text]-algebras. Assume that for some set [Formula: see text] of group operation symbols, all [Formula: see text]-algebras in [Formula: see text] are groups under the operations associated with the symbols in [Formula: see text]. In other words, [Formula: see text] is assumed to be a nontrivial variety of expanded groups. In particular, [Formula: see text] can be a nontrivial variety of groups or rings. Our main result is that there are no post-quantum weakly pseudo-free families in [Formula: see text], even in the worst-case setting and/or the black-box model. In this paper, we restrict ourselves to families [Formula: see text] of computational and black-box [Formula: see text]-algebras (where [Formula: see text]) such that for every [Formula: see text], each element of [Formula: see text] is represented by a unique bit string of length polynomial in the length of [Formula: see text]. In our main result, we use straight-line programs to represent nontrivial relations between elements of [Formula: see text]-algebras. Note that under certain conditions, this result depends on the classification of finite simple groups. Also, we define and study some types of post-quantum weak pseudo-freeness for families of computational and black-box [Formula: see text]-algebras.

Distributionally Robust Observable Strategic Queues

This paper presents an extension of Naor’s analysis on the join-or-balk problem in observable M/M/1 queues. Although all other Markovian assumptions still hold, we explore this problem assuming uncertain arrival rates under the distributionally robust settings. We first study the problem with the classical moment ambiguity set, where the support, mean, and mean-absolute deviation of the underlying distribution are known. Next, we extend the model to the data-driven setting, where decision makers only have access to a finite set of samples. We develop three optimal joining threshold strategies from the perspectives of an individual customer, a social optimizer, and a revenue maximizer such that their respective worst-case expected benefit rates are maximized. Finally, we compare our findings with Naor’s original results and the traditional sample average approximation scheme. Funding: This research was supported by the National Science Foundation [Grants 2342505 and 2343869].

Dependence over subgroups of free groups

Given a finitely generated subgroup [Formula: see text] of a free group [Formula: see text], we present an algorithm which computes [Formula: see text], such that the set of elements [Formula: see text], for which there exists a non-trivial [Formula: see text]-equation having [Formula: see text] as a solution is precisely the disjoint union of the double cosets [Formula: see text]. Moreover, we present an algorithm which, given a finitely generated subgroup [Formula: see text] and an element [Formula: see text], computes a finite set of elements from [Formula: see text] (of the minimum possible cardinality) generating, as a normal subgroup, the “ideal” [Formula: see text] of all “polynomials” [Formula: see text], such that [Formula: see text]. The algorithms, as well as the proofs, are based on the graph-theoretic techniques introduced by Stallings and on the more classical combinatorial techniques of Nielsen transformations. The key notion here is that of dependence of an element [Formula: see text] on a subgroup [Formula: see text]. We also study the corresponding notions of dependence sequence and dependence closure of a subgroup.

On the Structure of Finite Sets with Identical Representation Functions

On the Structure of Finite Sets with Identical Representation Functions

How real is incomputability in physics?

A physical system is determined by a finite set of initial conditions and “laws” represented by equations. The system is computable if we can solve the equations in all instances using a “finite body of mathematical knowledge”. In this case, if the laws of the system can be coded into a computer program, then given the initial conditions of the system, one can compute the system's evolution.Are there incomputable physical systems? This question has been theoretically studied in the last 30–40 years.In this paper, we experimentally show for the first time the strong incomputability of a quantum experiment, namely the outputs of a quantum random number generator. Moreover, the experimental results are robust and statistically significant.

Aggregating inconclusive data sets

An administrator is provided with data collected by several practitioners. These data may include inconclusive observations. The administrator is required to form a frequency distribution on the states of nature that would be approved by external auditors as long as it is compatible with the available information. We state a novel result on the compatibility of a probability with a finite set of capacities. We use this result to provide necessary and sufficient conditions for the compatibility of the administrator's frequency distribution with the data collected by the practitioners, according to two auditing criteria.

Distribution-free Inferential Models: Achieving finite-sample valid probabilistic inference, with emphasis on quantile regression

This paper presents a novel distribution-free Inferential Model (IM) construction that provides valid probabilistic inference across a broad spectrum of distribution-free problems, even in finite sample settings. More specifically, the proposed IM has the capability to assign (imprecise) probabilities to assertions of interest about any feature of the unknown quantities under examination, and these probabilities are well-calibrated in a frequentist sense. It is also shown that finite-sample confidence regions can be derived from the IM for any such features. Particular emphasis is placed on quantile regression, a domain where uncertainty quantification often takes the form of set estimates for the regression coefficients in applications. Within this context, the IM facilitates the acquisition of these set estimates, ensuring they are finite-sample confidence regions. It also enables the provision of finite-sample valid probabilistic assignments for any assertions of interest about the regression coefficients. As a result, regardless of the type of uncertainty quantification desired, the proposed framework offers an appealing solution to quantile regression.

Holographic renormalization of Horndeski gravity

We study the renormalization of a particular sector of Horndeski theory. In particular, we focus on the nonminimal coupling of a scalar field to the Gauss-Bonnet term and its kinetic coupling to the Einstein tensor. Adopting a power expansion on the scalar function that couples the Gauss-Bonnet term, we find specific conditions on their coefficients such that the action and charges are finite. To accomplish the latter, we add a finite set of intrinsic boundary terms. The contribution of the nonminimal coupling generates an effective scalar mass, allowing us to recover a modified Breitenlohner-Freedman bound. Furthermore, we compute the holographic 1-point functions and Ward identities associated with the scalar field and the metric. We constrain the parameter space of the theory by taking into account the preservation of scaling symmetry at the boundary.

Characterization of the Jaccard dissimilarity metric and a generalization

The Jaccard dissimilarity metric identifies the distance between two finite sets by the number of their unique elements as a proportion of their joint cardinality. This note gives an elementary characterization of the Jaccard metric by means of three simple axioms. Relaxing the most substantial one allows for a general family of new dissimilarity quasi-metrics to emerge that encompasses Jaccard’s metric.

Online Fault Detection of Open-Circuit Faults in a DTP-PMSM Using Double DQ Current Prediction

This research proposes a strategy to diagnose open-phase faults (OPF) and open-switching faults (OSF) in dual three-phase permanent magnet synchronous motor (DTP-PMSM) inverters. The method is based on the dual d–q predictive current model and involves establishing a mathematical model and utilizing the finite control set model predictive current extraction technique to predict the motor current. It then analyzes the characteristics of the switching-tube current under both normal and fault conditions. Finally, a fault predictive current model is introduced and the residual is calculated based on the predicted fault current value and the actual measured current value to diagnose the inverter fault. The proposed method effectively overcomes misjudgment issues encountered in traditional open-circuit fault diagnosis of inverters. It enhances the system’s response speed during dynamic processes and strengthens the robustness of diagnosis algorithm parameters. The experimental results demonstrate that the proposed method can rapidly, effectively, and accurately diagnose open-circuit faults presented in this paper fastest within one-fifth of a current cycle. It achieves a diagnostic accuracy rate of 97% in the dual three-phase permanent magnet synchronous motor drive system.

Generic families of finite metric spaces with identical or trivial 1-dimensional persistence

AbstractPersistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of persistent homology provides an upper bound for the change of persistence in the bottleneck distance under perturbations of points, but without giving a lower bound. This paper clarifies the possible limitations persistent homology may have in distinguishing finite metric spaces, which is evident for non-isometric point sets with identical persistence. We describe generic families of point sets in metric spaces that have identical or even trivial one-dimensional persistence. The results motivate stronger invariants to distinguish finite point sets up to isometry.

On sets of rational functions which locally represent all of [formula omitted]

We investigate finite sets of rational functions {f1,f2,…,fr} defined over some number field K satisfying that any t0∈K is a Kp-value of one of the functions fi for almost all primes p of K. We give strong necessary conditions on the shape of functions appearing in a minimal set with this property, as well as numerous concrete examples showing that these necessary conditions are in a way also close to sufficient. We connect the problem to well-studied concepts such as intersective polynomials and arithmetically exceptional functions.

On Conditional Axioms and Associated Inference Rules

In the present paper, we address the following general question in the framework of classical first-order logic. Assume that a certain mathematical principle can be formalized in a first-order language by a set E of conditional formulas of the form α(v)→β(v). Given a base theory T, we can use the set of conditional formulas E to extend the base theory in two natural ways. Either we add to T each formula in E as a new axiom (thus obtaining a theory denoted by T+E) or we extend T by using the formulas in E as instances of an inference rule (thus obtaining a theory denoted by T+E–Rule). The theory T+E will be stronger than T+E–Rule, but how much stronger can T+E be? More specifically, is T+E conservative over T+E–Rule for theorems of some fixed syntactical complexity Γ? Under very general assumptions on the set of conditional formulas E, we obtain two main conservation results in this regard. Firstly, if the formulas in E have low syntactical complexity with respect to some prescribed class of formulas Π and in the applications of E–Rule side formulas from the class Π and can be eliminated (in a certain precise sense), then T+E is ∀B(Π)-conservative over T+E–Rule. Secondly, if, in addition, E is a finite set with m conditional sentences, then nested applications of E–Rule of a depth at most of m suffice to obtain ∀B(Π) conservativity. These conservation results between axioms and inference rules extend well-known conservation theorems for fragments of first-order arithmetics to a general, purely logical framework.

Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains

AbstractThe Delaunay triangulation of a set of points P on a hyperbolic surface is the projection of the Delaunay triangulation of the set $$\widetilde{P}$$ P ~ of lifted points in the hyperbolic plane. Since $$\widetilde{P}$$ P ~ is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than $$12g-6$$ 12 g - 6 with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space.