arXiv Math @arxiv_math@qoto.org

There are two permutation groups that they share the same character table of order 1344. We take up natural representations on 8 and 14 letters respectively. The purpose of this paper is to examine the semi-simple structure of centralizing algebras in the tensor representation.

The article uses an arithmetic-geometric Fibonacci series to find the expected value of trials needed to observe k consecutive successes for the first time in a Bernoulli experiment using a recurrence relation. It is important to note that this is not a new result, but to the best of the authors' knowledge at the time of uploading, this is a novel derivation of a well-established result. The other derivations of this result are cited in the references section.

Two new matrix classes are introduced; inverse cyclic matrices and bi-diagonal south-west matrices. An interesting relation is established between these classes. Applications to two classes of inverse $Z$-matrices are provided.

In this study, an odour-mediated system is developed and studied. In an odor-mediated systems, the sense of smell or odour of species plays a critical role in the interactions between predators and prey. It is widely recognised in scientific literature that these systems are very common and essential across natural ecosystems. These systems are crucial for various behaviors, including foraging, mating, and avoiding predators. In this paper, it is assumed that the presence of prey odour aids the predator in its hunting efforts. It is assumed that both prey and intermediate predators seek refuge against their respective predators upon detecting the odour of their predators. In other words, the odour of predators assists prey species in evaluating the danger and seeking refuge for hiding. This model incorporates the prey species' harvesting as well. We also explore the impact of fading memory on the system by incorporating fractional derivatives into the model. The conditions for both the existence and local stability of the non-negative equilibria are derived. The current model system undergoes both Hopf and transcritical bifurcation when the parameter values are appropriately chosen. The dynamic behaviour of the system is showcased and thoroughly analysed using a range of diagrams, highlighting the the impact of prey refuge and predator odour parameters. This paper extensively examines the long-term impacts of harvesting within the system. The presence of prey odour plays a significant role in promoting a long-term cohabitation dynamic within this specific system. It has been observed that when individuals within the system have a strong memory, it positively affects the stability of the system. Numerical simulations are conducted in order to demonstrate and validate the usefulness of the model being considered, therefore supporting the analytical conclusions.

We study the problem of nonparametric estimation of the linear multiplier function $θ(t)$ for processes satisfying stochastic differential equations of the type $$dX_t= θ(t)X_t dt+ ε\; σ_1(t,X_t)σ_2(t,Y_t)dW_t, X_0=x_0, 0 \leq t \leq T$$ where $\{W_t, t\geq 0\}$ is a standard Brownian motion, $\{Y_t, t\geq 0\}$ is a process adapted to the filtration generated by the Brownian motion. We study the problem of estimation of the unknown function $θ(.)$ as $ε\rightarrow 0$ based on the observation of the process $\{X_t,0\leq t \leq T\}.$

For the numerical solution of shape optimization problems, particularly those constrained by partial differential equations (PDEs), the quality of the underlying mesh is of utmost importance. Particularly when investigating complex geometries, the mesh quality tends to deteriorate over the course of a shape optimization so that either the optimization comes to a halt or an expensive remeshing operation must be performed before the optimization can be continued. In this paper, we present a novel, semi-discrete approach for enforcing a minimum mesh quality in shape optimization. Our approach is based on Rosen's gradient projection method, which incorporates mesh quality constraints into the shape optimization problem. The proposed constraints bound the angles of triangular and solid angles of tetrahedral mesh cells and, thus, also bound the quality of these mesh cells. The method treats these constraints by projecting the search direction to the linear subspace of the currently active constraints. Additionally, only slight modifications to the usual line search procedure are required to ensure the feasibility of the method. We present our method for two- and three-dimensional simplicial meshes. We investigate the proposed approach numerically for the drag minimization of an obstacle in a two-dimensional flow and for the large-scale, three-dimensional optimization of a structured packing used in a distillation column. Our results show that the proposed method is indeed capable of guaranteeing a minimum mesh quality for both academic examples and challenging industrial applications. Particularly, our approach allows the shape optimization of extremely complex structures while ensuring that the mesh quality does not deteriorate.

We outline a new tool that can promote coherence within and across higher education mathematics courses by focusing on problem-solving: the Mathematical Problem-Solving Pipeline or MPSP. The MPSP can be used for teaching mathematics and mathematical reasoning authentically. It can be used across courses to develop problem solving skills that can be generalized beyond the course and topic. It has a specific focus on documentation and communication that lets students leverage skills they grow, and use, in other courses or fields. The MPSP can be used in singleton (service) courses, throughout a curriculum, and/or within a capstone experience.

Let $x$ and $n$ be positive integers. We prove a non-trivial lower bound for $x$, dependant only on $ω_n$, the number of distinct prime factors of $x^n-1$. By considering the divisibility of $φ\mid x^n-1$ for $φ\mid n$, we obtain a further refinement. This bound has applications for existence problems relating to primitive elements in finite fields.

We present a new Clifford-valued linear canonical Stockwell transform aimed at providing efficient and focused representation of Clifford-valued functions in high-dimensional time-frequency analysis. This transform improves upon the windowed Fourier and wavelet transforms by incorporating angular, scalable, and localized windows, allowing for greater directional flexibility in multi-scale signal analysis within the Clifford domain. Using operator theory, we explore the core properties of the proposed transform, such as the inner product relation, reconstruction formula, and interval theorem. Practical examples are included to confirm the validity of the derived results.

Carleman linearization is a technique that embeds systems of ordinary differential equations with polynomial nonlinearities into infinite dimensional linear systems in a procedural way. In this paper we generalize the method for systems of partial differential equations with quadratic nonlinearities, while maintaining the original structure of Carleman linearization. Furthermore, we apply our approach to Burger's equation and to the Vlasov equation as examples.

We present a fast, robust algorithm for applying a matrix transfer function of a linear time invariant system (LTI) in time domain. Computing $L$ states of a multiple-input multiple-output (MIMO) LTI appears to require $L$ matrix-vector multiplications. We demonstrate that, for any finite user-selected accuracy, the number of matrix-vector multiplications can be reduced to $\mathcal{O}\left(\log_{2}L\right)$ (within an $\mathcal{O}\left(L\right)$ algorithm). The algorithm uses an approximation of the rational transfer function in the z-domain by a matrix polynomial of degree $2^{N+1}-1$, where $N$ is chosen to achieve any user-selected accuracy. Importantly, using a cascade implementation in time domain, applying the transfer function requires only $N+1$ matrix-vector multiplications. We note that LTI systems are used in state space models (SSMs) for modeling long range dependencies where $L$ is large. In applications where the state matrix of LTI system is approximated by a structured matrix, the computational cost is further reduced. We briefly describe several structured approximations of matrices that can be used for such purpose.

In this paper, we study the almost sure well-posedness theory and orbital stability for the nonlinear Schrödinger equation with potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+Δu-V(x)u+|u|^{2}u=0,\ (x, t) \in \mathbb{R}^4 \times \mathbb{R}, \\ \left.u\right|_{t=0}=f \in H ^s(\mathbb{R}^4), \end{array}\right. \end{equation*} where $\frac{1}{3}<s<1$ and $V(x):\mathbb{R}^4\rightarrow \mathbb{R}$ satisfies appropriate conditions. The main idea in the proofs is based on Strichartz spaces as well as variants of local smoothing, inhomogeneous local smoothing and maximal function spaces. To our best knowledge, this is the first almost sure orbital stability result.

In this paper, we give an explicit formula for the rank of the $Q$-walk matrix of the Dynkin graph $A_n$. Moreover, we prove that its Smith normal form is $$ \mathrm{diag}\left( \underset{r=\lceil \frac{n}{2} \rceil}{\underbrace{1,2,2,...,2}},0,...,0 \right), $$ where $r$ is the rank of the $Q$-walk matrix $W_Q\left( A_n \right) $ of the Dynkin graph $A_n$.

To improve the convergence speed and optimization accuracy of the Dung Beetle Optimizer (DBO), this paper proposes an improved algorithm based on circle mapping and longitudinal-horizontal crossover strategy (CICRDBO). First, the Circle method is used to map the initial population to increase diversity. Second, the longitudinal-horizontal crossover strategy is applied to enhance the global search ability by ensuring the position updates of the dung beetle. Simulations were conducted on 10 benchmark test functions, and the results demonstrate that the improved algorithm performs well in both convergence speed and optimization accuracy. The improved algorithm is further applied to the hyperparameter selection of the Random Forest classification algorithm for binary classification prediction in the retail industry. Various combination comparisons prove the practicality of the improved algorithm, followed by SHapley Additive exPlanations (SHAP) analysis.

For a poset $(P,\leqslant)$ we consider the first-order theory, that is defined by set $P$ and relation $\leqslant$. The problem of undecidability of combinatorial theories attracts significant attention. Recently A. Wires proved the undecidability of the elementary theory of Young lattice and also established the maximal definability property of this theory. The purpose of this article is to obtain the same results for another graded lattice, which has much in common with Young lattice: Young--Fibonacci lattice. As Wires does for Young lattice, for the proof of undecidability we define Arithmetic into this theory.

This paper develops a time-split linearized explicit/implicit approach for solving a two-dimensional hydrodynamic flow model with appropriate initial and boundary conditions. The time-split technique is employed to upwind the convection term and to treat the friction slope so that the numerical oscillations and stability are well controlled. A suitable time step restriction for stability and convergence accurate of the new algorithm is established using the $L^{\infty}(0,T; L^{2})$-norm. Under a time step requirement, some numerical examples confirm the theoretical studies and suggest that the proposed computational technique is spatial fourth-order accurate and temporal second-order convergent. An application to floods observed in the far north region of Cameroon is considered and discussed.

Theorems relating permutations with objects in other fields of mathematics are often stated in terms of avoided patterns. Examples include various classes of Schubert varieties from algebraic geometry (Billey and Abe 2013), commuting functions in analysis (Baxter 1964), beta-shifts in dynamical systems (Elizalde 2011) and homology of representations (Sundaram 1994). We present a new algorithm, BiSC, that, given any set of permutations, outputs a conjecture for describing the set in terms of avoided patterns. The algorithm automatically conjectures the statements of known theorems such as the descriptions of smooth (Lakshmibai and Sandhya 1990) and forest-like permutations (Bousquet-M{é}lou and Butler 2007), Baxter permutations (Chung et al. 1978), stack-sortable (Knuth 1975) and West-2-stack-sortable permutations (West 1990). The algorithm has also been used to discover new theorems and conjectures related to the dihedral and alternating subgroups of the symmetric group, Young tableaux, Wilf-equivalences, and sorting devices.

It was shown by Kutnar, Maru\v si\v c and Zhang in 2012 that every connected vertex-transitive graph of order $10p$, where $p$ is a prime and $p\ne 7$, contains a Hamilton path, except for graphs $X$ arising from the action of PSL$(2, s^m)$ on cosets of $\mathbb{Z}_s^m\rtimes \mathbb{Z}_{\frac{s^m-1}{10}}$, where $s$ is a prime. In this paper, Hamilton cycles of these exceptions $X$ will be found.

We continue our investigation of cardinal sequences associated with locally Lindelof, scattered, Hausdorff P-spaces (abbreviated as LLSP spaces). We outline a method for constructing LLSP spaces from cone systems and partial orders with specific properties. Additionally, we establish limitations on the cardinal sequences of LLSP spaces. Finally, we present both a necessary condition and a distinct sufficient condition for a sequence $\langle κ_α: α< ω_2 \rangle$ to be the cardinal sequence of an LLSP space.

We study a new class of polyominoes, called $p$-Fibonacci polyominoes, defined using $p$-Fibonacci words. We enumerate these polyominoes by applying generating functions to capture geometric parameters such as area, semi-perimeter, and the number of inner points. Additionally, we establish bijections between Fibonacci polyominoes, binary Fibonacci words, and integer compositions with certain restrictions.