We investigate a numerical method for approximating stochastic Caputo fractional differential equations driven by multiplicative noise. The nonlinear functions f and g are assumed to satisfy the global Lipschitz conditions as well as the linear growth conditions. The noise is approximated by a piecewise constant function, yielding a regularized stochastic fractional differential equation. We prove that the error between the exact solution and the solution of the regularized equation converges in the L2((0,T)×Ω) norm with an order of O(Δtα−1/2), where α∈(1/2,1] is the order of the Caputo fractional derivative, and Δt is the time step size. Numerical experiments are provided to confirm that the simulation results are consistent with the theoretical convergence order.

In this work, we consider the Galerkin finite element method for solving the stochastic semilinear subdiffusion equation driven by additive fractional Brownian motion with Hurst parameter H ∈ ( 0 , 1 ) , where the fractional Brownian motion has a Wiener integration representation. The existence and uniqueness of a mild solution are proved using the Banach fixed point theorem. The temporal and spatial regularity of the solution are studied via the semigroup approach. The finite element method is used to approximate the spatial variable. The Caputo fractional time derivative and the Riemann–Liouville integral are approximated using the Grünwald–Letnikov schemes, respectively, and the noise is discretized using the Euler method. The optimal error estimates of the fully discrete scheme are established using the discrete Laplace transform method. Under the assumption that the noise is in the trace class, we prove that the time convergence order is O ( τ min { α , H + α + γ − 1 , 1 / 2 } ) when H ∈ ( 0 , 1 / 2 ) , and O ( τ min { α , H + α + γ − 1 } ) when H ∈ ( 1 / 2 , 1 ) . Here, τ denotes the time step size. Numerical experiments are conducted to validate the theoretical results.

This paper investigates the strong convergence of a Milstein scheme for a stochastic semilinear subdiffusion equation driven by fractionally integrated multiplicative noise. The existence and uniqueness of the mild solution are established via the Banach fixed point theorem. Temporal and spatial regularity properties of the mild solution are derived using the semigroup approach. For spatial discretization, the standard Galerkin finite element method is employed, while the Grünwald–Letnikov method is used for time discretization. The Milstein scheme is utilized to approximate the multiplicative noise. For sufficiently smooth noise, the proposed scheme achieves the temporal strong convergence order of O(τα), α∈(0,1). Numerical experiments are presented to verify that the computational results are consistent with the theoretical predictions.

We identify complete fragments of the simple theory of types with infinity (TSTI) and Quine's new foundations (NF) set theory. We show that TSTI decides every sentence φ in the language of type theory that is in one of the following forms: (A) φ = ∀x r11 ⋯ ∀xrkk ∃ys11 ⋯ ∃ys11 θ where the superscripts denote the types of the variables, s1 > ⋯ > s1, and θ is quantifier-free, (B) φ = ∀x r11 ⋯ ∀xrkk ∃ys11 ⋯ ∃ys11 θ where the superscripts denote the types of the variables and θ is quantifier-free. This shows that NF decides every stratified sentence φ in the language of set theory that is in one of the following forms: (A′) φ = ∀x1 ⋯ ∀xk ∃y1 ⋯ ∃yl θ where θ is quantifier-free and φ admits a stratification that assigns distinct values to all of the variables y1,⋯, yl, (B′) φ = ∀x1 ⋯ ∀xk ∃y1 ⋯ ∃yl θ where θ is quantifier-free and <p admits a stratification that assigns the same value to all of the variables y1,⋯, yl.

By a classical theorem of Harvey Friedman (1973), every countable nonstandard model MM of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding j, i.e., j is a self-embedding of MM such that j[M]⊊Mj[M]⊊M , and the ordinal rank of each member of j[M]j[M] is less than the ordinal rank of each element of M∖j[M]M∖j[M] . Here, we investigate the larger family of proper initial-embeddings j of models MM of fragments of set theory, where the image of j is a transitive submodel of MM . Our results include the following three theorems. In what follows, ZF−ZF− is ZFZF without the power set axiom; WOWO is the axiom stating that every set can be well-ordered; WF(M)WF(M) is the well-founded part of MM ; and Π1∞-DCαΠ∞1-DCα is the full scheme of dependent choice of length αα . Theorem A. There is an ωω -standard countable nonstandard model MM of ZF−+WOZF−+WO that carries no initial self-embedding j:M⟶Mj:M⟶M other than the identity embedding. Theorem B. Every countable ωω -nonstandard model MM of ZF ZF is isomorphic to a transitive submodel of the hereditarily countable sets of its own constructible universe LMLM . Theorem C. The following three conditions are equivalent for a countable nonstandard model MM of ZF−+WO+∀α Π1∞-DCαZF−+WO+∀α Π∞1-DCα . 1. (I) There is a cardinal in MM that is a strict upper bound for the cardinality of each member of WF(M)WF(M) . 2. (II) WF(M)WF(M) satisfies the powerset axiom. 3. (III) For all n∈ωn∈ω and for all b∈Mb∈M , there exists a proper initial self-embedding j:M⟶Mj:M⟶M such that b∈rng(j)b∈rng(j) and j[M]≺nMj[M]≺nM .

In this paper NFU-AC is used to denote Jensen's modification of Quine's ‘new foundations’ set theory (NF) fortified with a type-level pairing function but without the axiom of choice. The axiom AxCount>_ is the variant of the axiom of counting which asserts that no finite set is smaller than its own set of singletons. This paper shows that NFU-AC + AxCount>_ proves the consistency of the simple theory of types with infinity (TSTI). This result implies that NF + AxCount>_ proves that consistency of TSTI, and that NFU-AC + AxCount>_ proves the consistency of NFU-AC.

This paper is primarily concerned with assessing a set-theoretical system, S*, for the foundations of category theory suggested by Solomon Feferman. S* is an extension of NFU, and may be seen as an attempt to accommodate unrestricted categories such as the category of all groups (without any small/large restrictions), while still obtaining the benefits of ZFC on part of the domain. A substantial part of the paper is devoted to establishing an improved upper bound on the consistency strength of S*. The assessment of S* as a foundation of category theory is framed by the following general desiderata (R) and (S). (R) asks for the unrestricted existence of the category of all groups, the category of all categories, the category of all functors between two categories, etc., along with natural implementability of ordinary mathematics and category theory. (S) asks for a certain relative distinction between large and small sets, and the requirement that they both enjoy the full benefits of the ZFC axioms. S* satisfies (R) simply because it is an extension of NFU. By means of a recursive construction utilizing the notion of strongly cantorian sets, we argue that it also satisfies (S). Moreover, this construction yields a lower bound on the consistency strength of S*. We also exhibit a basic positive result for category theory internal to NFU that provides motivation for studying NFU-based foundations of category theory.

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter H∈(1/2,1). The finite element method is employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula are used to approximate the Caputo time-fractional derivative of order α∈(0,1) and the Riemann–Liouville time-fractional integral of order γ∈(0,1), respectively. Using the semigroup approach, we establish the temporal and spatial regularity of the mild solution to the problem. The fully discrete solution is expressed as a convolution of a piecewise constant function with the inverse Laplace transform of a resolvent-related function. Based on the Laplace transform method and resolvent estimates, we prove that the proposed numerical scheme has the optimal convergence order O(τmin{H+α+γ−1−ε,α}),ε&gt;0. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method.

This paper develops a discrete competitive Lotka–Volterra system with single diffusion under Neumann boundary conditions. It establishes the conditions for Turing instability and identifies the precise Turing bifurcation when the diffusion coefficient is used as a bifurcation parameter. Within Turing unstable regions, a variety of Turing patterns are explored via numerical simulations, encompassing lattice, nematode, auspicious cloud, spiral wave, polygon, and stripe patterns, as well as their combinations. The periodicity and complexity of these patterns are verified through bifurcation simulations, Lyapunov exponent analysis, trajectory or phase diagrams. These methods are also applicable to other single diffusion systems, including partial dissipation systems.

A high-order time discretization scheme to approximate the time-fractional wave equation with the Caputo fractional derivative of order $\alpha \in (1, 2)$ is studied. We establish a high-order formula for approximating the Caputo fractional derivative of order $\alpha \in (1, 2)$. Based on this approximation, we propose a novel numerical method to solve the time-fractional wave equation. Remarkably, this method corrects only one starting step and demonstrates second-order convergence in both homogeneous and inhomogeneous cases, regardless of whether the data is smooth or nonsmooth. We also analyze the stability region associated with the proposed numerical method. Some numerical examples are given to elucidate the convergence analysis.