arXiv Math @arxiv_math@qoto.org

In this paper, we shall compare two metrics in terms of orderly dependence, a notion developed in exponential vector space in the article 'Basis and Dimension of Exponential Vector Space' by Jayeeta Saha and Sandip Jana in Transactions of A. Razmadze Mathematical Institute Vol. 175 (2021), issue 1, 101-115. Exponential vector space, in short 'evs', is a partially ordered space associated with a commutative semigroup structure and a compatible scalar multiplication. In the present paper we shall show that the collection $\mathcal{D(\mathbf X)}$ of all metrics on a non-empty set $\mathbf X$, together with the constant function zero $O$, forms a topological exponential vector space. We shall discuss the orderly dependence of two metrics through our findings of a basis of $\mathcal{D(\mathbf X)}\smallsetminus\{O\}$ in different scenario. We shall characterise orderly independence of two elements of a topological evs in terms of the comparing function, another mechanism developed in topological exponential vector space, which can measure the degree of comparability of two elements of an evs. Finally, we shall discuss existence of orderly independent norms on a linear space. For an infinite dimensional linear space we shall construct a large number of orderly independent norms depending on the dimension of the linear space. Orderly independent norms are precisely those which are totally non-equivalent, in the sense that they produce incomparable topologies.

This paper aims to make a mark in the future of sustainable robotics, where efficient algorithms are required to carry out tasks like environmental monitoring and precision agriculture efficiently. We proposed a hybrid algorithm that combines Artificial Bee Colony (ABC) with Levy flight to optimize adaptive sensor placement alongside an important notion of hotspots from domain knowledge experts. By enhancing exploration and exploitation, our approach significantly improves the identification of critical hotspots. This algorithm also finds its usecases for broader search and rescue operations applications, demonstrating its potential in optimization problems across various domains.

We establish interior regularity results for first-order, stationary, local mean-field game (MFG) systems. Specifically, we study solutions of the coupled system consisting of a Hamilton-Jacobi-Bellman equation $H(x, Du, m) = 0$ and a transport equation $-\operatorname{div}(m D_pH(x, Du, m)) = 0$ in a domain $Ω\subset \mathbb{R}^d$. Under suitable structural assumptions on the Hamiltonian $H$, without requiring monotonicity of the system, convexity of the Hamiltonian, separability in variables, or smoothness beyond basic continuity in $(p,m)$, we introduce a notion of weak solutions that allows the application of techniques from elliptic regularity theory. Our main contribution is to prove that the value function $u$ is locally Hölder continuous in $Ω$. The proof leverages the connection between first-order MFG systems and quasilinear equations in divergence form, adapting classical techniques to handle the specific structure of MFG systems.

We construct a series of classic vorticity solutions for incompressible Euler equation on $\mathbb S^2$, which constitute the $C^1$ type regularization for a general traveling point vortex system. The construction is accomplished by applying tangent mapping on $\mathbb S^2$ and Lyapunov--Schmidt reduction argument. Using the fixed-point theorem and a finite dimensional equation on vortex dynamics, we prove that the vortices are located near a nondegenerate critical point of Kirchhoff--Routh function. Moreover, in the tangent space at each vortex center, the scaled stream function is verified as a perturbation of the ground state for generalized plasma problem. Some other qualitative and quantitative estimates for the regularization series are also obtained in this paper.

We consider a generalized derivative nonlinear Schr\''odinger equation. We prove existence of wave operator under an explicit smallness of the given asymptotic states. Our method bases on studying the associated system used in \cite{Tinpaper4}. Moreover, we show that if the initial data is small enough in $H^2(\mathbb{R})$ then the associated solution scatters up to a Gauge transformation.

The random population control decision problem asks for the existence of a controller capable of gathering almost-surely a whole population of identical finite-state agents simultaneously in a final state. The controller must be able to satisfy this requirement however large the population, provided that it is finite. The problem was previously known to be decidable and EXPTIME-hard. This paper tackles the exact complexity: the problem is EXPTIME-complete.

We implement the quantum inverse scattering method for the 4-vertex model. In comparison to previous works of the author which examined the 6-vertex, and 20-vertex, models, the 4-vertex model exhibits different characteristics, ranging from L-operators expressed in terms of projectors and Pauli matrices to algebraic and combinatorial properties, including Poisson structure and boxed plane partitions. With far fewer computations with an L-operator provided for the 4-vertex model by Bogoliubov in 2007, in comparison to those for L-operators of the 6, and 20, vertex models, from lower order expansions of the transfer matrix we derive a system of relations from the structure of operators that can be leveraged for studying characteristics of the higher-spin XXX chain in the weak finite volume limit. In comparison to quantum inverse scattering methods for the 6, and 20, vertex models which can be used to further study integrability, and exact solvability, an adaptation of such an approach for the 4-vertex model can be used to approximate, asymptotically in the weak finite volume limit, sixteen brackets which generate the Poisson structure. From explicit relations for operators of the 4-vertex transfer matrix, we conclude by discussing corresponding aspects of the Yang-Baxter algebra, which is closely related to the operators obtained from products of L-operators for approximating the transfer, and quantum monodromy, matrices. The structure of computations from L-operators of the 4-vertex model directly transfers to L-operators of the higher-spin XXX chain, revealing a similar structure of another Yang-Baxter algebra of interest.

This paper, in the first step, develops the system of bipolar fuzzy relational equations (FRE) to the most general case where the bipolar FREs are defined by an arbitrary continuous t-norm. Also, since fuzzy relational equations are special cases of the bipolar FREs, the proposed system can be also interpreted as a generalization of traditional FREs where the fuzzy compositions are generally defined by any continuous t-norm. The consistency of the continuous bipolar FREs is initially investigated and some necessary and sufficient conditions are derived for determining the feasibility of the proposed system. Subsequently, the feasible solutions set of the problem is completely characterized. It is shown that unlike FREs and those bipolar FREs defined by continuous Archimedean t-norms, the feasible solutions set of the generalized bipolar FREs is formed as the union of a finite number of compact sets that are not necessarily connected. Moreover, five techniques have also been introduced with the aim of simplifying the current problem, and then an algorithm is accordingly presented to find the feasible region of the problem. In the second step, a new class of optimization models is studied where the constraints are defined by the continuous bipolar FREs and the objective function covers a wide range of (non)linear functions such as maximum function, geometric mean function, log-sum-exp function, maximum eigenvalue of a symmetric matrix, support function of a set, etc. It is proved that the problem has a finite number of local optimal solutions and a global optimal solution can always be obtained by choosing a point having the minimum objective value compared to all the local optimal solutions. Finally, to illustrate the discussed study, a step-by-step example is described in several sections, whose constraints are a system of the bipolar FRES defined by Dubois-Prade t-norm.

We consider the uniqueness of the following positive solutions of anisotropic elliptic equation: \begin{equation}\nonumber \left\{ \begin{aligned} -Δ^F _p u&=u^q \quad \text{in} \quad Ω, u&=0 \quad \text{on} \quad \partial Ω, \end{aligned} \right. \end{equation} where $p>\frac{3}{2}$ is a constant. We utilize the linearized method to derive the uniqueness results, which extends the conclusion obtained by L. Brasco and E. Lindgren.

We classify real or complex finite-dimensional $C^*$-algebras and their underlying fields from the properties of Birkhoff-James orthogonality. Application to strong Birkhoff-James orthogonality preservers is also given.

If $f: \mathbb{N} \rightarrow \{\pm1\}$ is a sample of the random completely multiplicative function, we show that almost surely $\sum_{n \le x} \frac{f(n)}{\sqrt{n}}$ changes signs infinitely many times, answering a question of Aymone.

The Herglotz representation theorem for holomorphic functions with non-negative real part is a fundamental result in the theory of holomorphic functions. In this paper, we reinterpret the Herglotz representation in the context of modern techniques, specifically realization formula. This reinterpretation is then extended to operator-valued functions on arbitrary sets, in association with a collection of test functions.

In this paper, we provide the following simple equivalent condition for a nonsymmetric Algebraic Riccati Equation to admit a stabilizing cone-preserving solution: an associated coefficient matrix must be stable. The result holds under the assumption that said matrix be cross-positive on a proper cone, and it both extends and completes a corresponding sufficient condition for nonnegative matrices in the literature. Further, key to showing the above is the following result which we also provide: in order for a monotonically increasing sequence of cone-preserving matrices to converge, it is sufficient to be bounded above in a single vectorial direction.

Scenario tree reduction techniques are essential for achieving a balance between an accurate representation of uncertainties and computational complexity when solving multistage stochastic programming problems. In the realm of available techniques, the Kovacevic and Pichler algorithm (Ann. Oper. Res., 2015 [1]) stands out for employing the nested distance, a metric for comparing multistage scenario trees. However, dealing with large-scale scenario trees can lead to a prohibitive computational burden due to the algorithm's requirement of solving several large-scale linear problems per iteration. This study concentrates on efficient approaches to solving such linear problems, recognizing that their solutions are Wasserstein barycenters of the tree nodes' probabilities on a given stage. We leverage advanced optimal transport techniques to compute Wasserstein barycenters and significantly improve the computational performance of the Kovacevic and Pichler algorithm. Our boosted variants of this algorithm are benchmarked on several multistage scenario trees. Our experiments show that compared to the original scenario tree reduction algorithm, our variants can be eight times faster for reducing scenario trees with 8 stages, 78 125 scenarios, and 97 656 nodes.

Deciding bank interest rates has been a long-standing challenge in finance. It is crucial to ensure that the selected rates balance market share and profitability. However, traditional approaches typically focus on the interest rate changes of individual banks, often neglecting the interactions with other banks in the market. This work proposes a novel framework that models the interest rate problem as a major-minor mean field game within the context of an interbank game. To incorporate the complex interactions between banks, we utilize mean-field theory and employ impulsive control to model the overhead in rate adjustments. Ultimately, we solve this optimal control problem using a new deep Q-network method, which iterates the parameterized action value functions for major and minor players and updates the networks in a Fictitious Play way. Our proposed algorithm converges, offering a solution that enables the analysis of strategies for major and minor players in the market under the Nash Equilibrium.

In an amalgamation Nim, players are allowed to use a move from the traditional form of Nim or to amalgamate two piles when they are not empty. No formula that describes the set of P-positions of Amalgamation Nim is known. The author gives a condition on the amalgamation of two piles. Players can amalgamate two piles only when the number of stones in these two piles is equal to or more than 2. Then, we get a formula that describes the set of P-positions for this variant of amalgamation Nim.

Given a graph G=(V,E), a connected Grundy coloring is a proper vertex coloring that can be obtained by a first-fit heuristic on a connected vertex sequence. A first-fit coloring heuristic is one that attributes to each vertex in a sequence the lowest-index color not used for its preceding neighbors. A connected vertex sequence is one in which each element, except for the first one, is connected to at least one element preceding it. The connected Grundy coloring problem consists of obtaining a connected Grundy coloring maximizing the number of colors. In this paper, we propose two integer programming (IP) formulations and a local-search enhanced biased random-key genetic algorithm (BRKGA) for the connected Grundy coloring problem. The first formulation follows the standard way of partitioning the vertices into color classes while the second one relies on the idea of representatives in an attempt to break symmetries. The BRKGA encompasses a local search procedure using a newly proposed neighborhood. A theoretical neighborhood analysis is also presented. Extensive computational experiments indicate that the problem is computationally demanding for the proposed IP formulations. Nonetheless, the formulation by representatives outperforms the standard one for the considered benchmark instances. Additionally, our BRKGA can find high-quality solutions in low computational times for considerably large instances, showing improved performance when enhanced with local search and a reset mechanism. Moreover we show that our BRKGA can be easily extended to successfully tackle the Grundy coloring problem, i.e., the one without the connectivity requirements.

Inspired by recent work of Ferone and Volzone arXiv:2007.13195, we derive sufficient conditions for the validity and non-validity of a boundary version of Talenti's comparison principle in the context of Dirichlet-Poisson problems for the fractional Laplacian $(-Δ)^s$ in the unit ball $Ω= B_1(0) \subset \mathbb{R}^N$. In particular, our results imply a universial failure of the classical pointwise Talenti inequality in the fractional radial context which sheds new light on the one-dimensional counterexamples given in arXiv:2007.13195.

We consider a branched transport problem with weakly imposed boundary conditions. This problem arises as a reduced model for pattern formation in type-I superconductors. For this model, it is conjectured that the dimension of the boundary measure is non-integer. We prove this conjecture in a simplified 2D setting, under the (strong) assumption of Ahlfors regularity of the irrigated measure. This work is the first rigorous proof of a singular behaviour for irrigated measures resulting from minimality.

Burn injuries present a significant global health challenge. Among the most severe long-term consequences are contractures, which can lead to functional impairments and disfigurement. Understanding and predicting the evolution of post-burn wounds is essential for developing effective treatment strategies. Traditional mathematical models, while accurate, are often computationally expensive and time-consuming, limiting their practical application. Recent advancements in machine learning, particularly in deep learning, offer promising alternatives for accelerating these predictions. This study explores the use of a deep operator network (DeepONet), a type of neural operator, as a surrogate model for finite element simulations, aimed at predicting post-burn contraction across multiple wound shapes. A DeepONet was trained on three distinct initial wound shapes, with enhancement made to the architecture by incorporating initial wound shape information and applying sine augmentation to enforce boundary conditions. The performance of the trained DeepONet was evaluated on a test set including finite element simulations based on convex combinations of the three basic wound shapes. The model achieved an $R^2$ score of $0.99$, indicating strong predictive accuracy and generalization. Moreover, the model provided reliable predictions over an extended period of up to one year, with speedups of up to 128-fold on CPU and 235-fold on GPU, compared to the numerical model. These findings suggest that DeepONets can effectively serve as a surrogate for traditional finite element methods in simulating post-burn wound evolution, with potential applications in medical treatment planning.