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We propose a new simple and explicit numerical scheme for time-homogeneous stochastic differential equations. The scheme is based on sampling increments at each time step from a skew-symmetric probability distribution, with the level of skewness determined by the drift and volatility of the underlying process. We show that as the step-size decreases the scheme converges weakly to the diffusion of interest. We then consider the problem of simulating from the limiting distribution of an ergodic diffusion process using the numerical scheme with a fixed step-size. We establish conditions under which the numerical scheme converges to equilibrium at a geometric rate, and quantify the bias between the equilibrium distributions of the scheme and of the true diffusion process. Notably, our results do not require a global Lipschitz assumption on the drift, in contrast to those required for the Euler--Maruyama scheme for long-time simulation at fixed step-sizes. Our weak convergence result relies on an extension of the theory of Milstein \& Tretyakov to stochastic differential equations with non-Lipschitz drift, which could also be of independent interest. We support our theoretical results with numerical simulations.

A swarm intelligence-based optimization algorithm, named Duck Swarm Algorithm (DSA), is proposed in this study, which is inspired by the searching for food sources and foraging behaviors of the duck swarm. Two rules are modeled from the finding food and foraging of the duck, which corresponds to the exploration and exploitation phases of the proposed DSA, respectively. The performance of the DSA is verified by using multiple CEC benchmark functions, where its statistical (best, mean, standard deviation, and average running-time) results are compared with seven well-known algorithms like Particle swarm optimization (PSO), Firefly algorithm (FA), Chicken swarm optimization (CSO), Grey wolf optimizer (GWO), Sine cosine algorithm (SCA), and Marine-predators algorithm (MPA), and Archimedes optimization algorithm (AOA). Moreover, the Wilcoxon rank-sum test, Friedman test, and convergence curves of the comparison results are utilized to prove the superiority of the DSA against other algorithms. The results demonstrate that DSA is a high-performance optimization method in terms of convergence speed and exploration-exploitation balance for solving the numerical optimization problems. Also, DSA is applied for the optimal design of six engineering constrained optimization problems and the node optimization deployment task of the Wireless Sensor Network (WSN). Overall, the comparison results revealed that the DSA is a promising and very competitive algorithm for solving different optimization problems.

Symplectic integrators are widely implemented numerical integrators for Hamiltonian mechanics, which preserve the Hamiltonian structure (symplecticity) of the system. Although the symplectic integrator does not conserve the energy of the system, it is well known that there exists a conserving modified Hamiltonian, called the shadow Hamiltonian. For the Nambu mechanics, which is a kind of generalized Hamiltonian mechanics, we can also construct structure-preserving integrators by the same procedure used to construct the symplectic integrators. In the structure-preserving integrator, however, the existence of shadow Hamiltonians is nontrivial. This is because the Nambu mechanics is driven by multiple Hamiltonians and it is nontrivial whether the time evolution by the integrator can be cast into the Nambu mechanical time evolution driven by multiple shadow Hamiltonians. In this paper we present a general procedure to calculate the shadow Hamiltonians of structure-preserving integrators for Nambu mechanics, and give an example where the shadow Hamiltonians exist. This is the first attempt to determine the concrete forms of the shadow Hamiltonians for a Nambu mechanical system. We show that the fundamental identity, which corresponds to the Jacobi identity in Hamiltonian mechanics, plays an important role in calculating the shadow Hamiltonians using the Baker-Campbell-Hausdorff formula. It turns out that the resulting shadow Hamiltonians have indefinite forms depending on how the fundamental identities are used. This is not a technical artifact, because the exact shadow Hamiltonians obtained independently have the same indefiniteness.

In this paper, to address the optimization problem on a compact matrix manifold, we introduce a novel algorithmic framework called the Transformed Gradient Projection (TGP) algorithm, using the projection onto this compact matrix manifold. Compared with the existing algorithms, the key innovation in our approach lies in the utilization of a new class of search directions and various stepsizes, including the Armijo, nonmonotone Armijo, and fixed stepsizes, to guide the selection of the next iterate. Our framework offers flexibility by encompassing the classical gradient projection algorithms as special cases, and intersecting the retraction-based line-search algorithms. Notably, our focus is on the Stiefel or Grassmann manifold, revealing that many existing algorithms in the literature can be seen as specific instances within our proposed framework, and this algorithmic framework also induces several new special cases. Then, we conduct a thorough exploration of the convergence properties of these algorithms, considering various search directions and stepsizes. To achieve this, we extensively analyze the geometric properties of the projection onto compact matrix manifolds, allowing us to extend classical inequalities related to retractions from the literature. Building upon these insights, we establish the weak convergence, convergence rate, and global convergence of TGP algorithms under three distinct stepsizes. In cases where the compact matrix manifold is the Stiefel or Grassmann manifold, our convergence results either encompass or surpass those found in the literature. Finally, through a series of numerical experiments, we observe that the TGP algorithms, owing to their increased flexibility in choosing search directions, outperform classical gradient projection and retraction-based line-search algorithms in several scenarios.

We detail for the first time a complete explicit description of the quasi-cyclic structure of all classical finite generalized quadrangles. Using these descriptions we construct families of quasi-cyclic LDPC codes derived from the point-line incidence matrix of the quadrangles by explicitly calculating quasi-cyclic generator and parity check matrices for these codes. This allows us to construct parity check and generator matrices of all such codes of length up to 400000. These codes cover a wide range of transmission rates, are easy and fast to implement and perform close to Shannon's limit with no visible error floors. We also include some performance data for these codes. Furthermore, we include a complete explicit description of the quasi-cyclic structure of the point-line and point-hyperplane incidences of the finite projective and affine spaces.

This paper is devoted to the study of a novel mixed Finite Element Method for approximating the solutions of fourth order variational problems subjected to a constraint. The first problem we consider consists in establishing the convergence of the error of the numerical approximation of the solution of a biharmonic obstacle problem. The contents of this section are meant to generalise the approach originally proposed by Ciarlet \& Raviart, and then complemented by Ciarlet \& Glowinski. The second problem we consider amounts to studying a two-dimensional variational problem for linearly elastic shallow shells subjected to remaining confined in a prescribed half-space. We first study the case where the parametrisation of the middle surface for the linearly elastic shallow shell under consideration has non-zero curvature, and we observe that the numerical approximation of this model via a mixed Finite Element Method based on conforming elements requires the implementation of the additional constraint according to which the gradient matrix of the dual variable has to be symmetric. However, differently from the biharmonic obstacle problem previously studied, we show that the numerical implementation of this result cannot be implemented by solely resorting to Courant triangles. Finally, we show that if the middle surface of the linearly elastic shallow shell under consideration is flat, the symmetry constraint required for formulating the constrained mixed variational problem announced in the second part of the paper is not required, and the solution can thus be approximated by solely resorting to Courant triangles. The theoretical results we derived are complemented by a series of numerical experiments.

This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial values, the regularity of the mild solution is investigated, and an error estimate is derived with the spatial $ L^2 $-norm. For smooth initial values, two error estimates with the general spatial $ L^q $-norms are established.

Given a finite set of matrices with integer entries, the matrix mortality problem asks if there exists a product of these matrices equal to the zero matrix. We consider a special case of this problem where all entries of the matrices are nonnegative. This case is equivalent to the NFA mortality problem, which, given an NFA, asks for a word $w$ such that the image of every state under $w$ is the empty set. The size of the alphabet of the NFA is then equal to the number of matrices in the set. We study the length of shortest such words depending on the size of the alphabet. We show that this length for an NFA with $n$ states can be at least $2^n - 1$, $2^{(n - 4)/2}$ and $2^{(n - 2)/3}$ if the size of the alphabet is, respectively, equal to $n$, three and two.

We present a class of high-order Eulerian-Lagrangian Runge-Kutta finite volume methods that can numerically solve Burgers' equation with shock formations, which could be extended to general scalar conservation laws. Eulerian-Lagrangian (EL) and semi-Lagrangian (SL) methods have recently seen increased development and have become a staple for allowing large time-stepping sizes. Yet, maintaining relatively large time-stepping sizes post shock formation remains quite challenging. Our proposed scheme integrates the partial differential equation on a space-time region partitioned by linear approximations to the characteristics determined by the Rankine-Hugoniot jump condition. We trace the characteristics forward in time and present a merging procedure for the mesh cells to handle intersecting characteristics due to shocks. Following this partitioning, we write the equation in a time-differential form and evolve with Runge-Kutta methods in a method-of-lines fashion. High-resolution methods such as ENO and WENO-AO schemes are used for spatial reconstruction. Extension to higher dimensions is done via dimensional splitting. Numerical experiments demonstrate our scheme's high-order accuracy and ability to sharply capture post-shock solutions with large time-stepping sizes.

The present paper is devoted to study the effect of connected and disconnected rotations of G\"odel algebras with operators grounded on directly indecomposable structures. The structures resulting from this construction we will present are nilpotent minimum (with or without negation fixpoint, depending on whether the rotation is connected or disconnected) with special modal operators defined on a directly indecomposable algebra. In this paper we will present a (quasi-)equational definition of these latter structures. Our main results show that directly indecomposable nilpotent minimum algebras (with or without negation fixpoint) with modal operators are fully characterized as connected and disconnected rotations of directly indecomposable G\"odel algebras endowed with modal operators.