RSS 1.0Mathematics
We are an active university Mathematics Department with a strong teaching and research reputation. We offer students the chance to study at undergraduate or postgraduate level on degree programmes leading to: BSc in Mathematics, BSc/BA joint courses in Mathematics or Applied Statistics and a wide range of other subjects. We have an active research group focusing on Computational Applied Mathematics, with research students studying for the degrees of MPhil and PhD, postdoctoral workers and associated collaborators from across the world.
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Recent Submissions
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A corrected Crank–Nicolson scheme for the time fractional parabolic integro-differential equation with nonsmooth dataThis paper proposes a corrected Crank–Nicolson (CN) scheme for solving time fractional parabolic integro-differential equations which involve Caputo time fractional derivative and fractional Riemann–Liouville (R-L) integral. The weighted and shifted Grünwald–Letnikov (WSGL) formulae is adopted to approximate the time fractional Riemann–Liouville integral. The Crank–Nicolson scheme is applied to approximate the Caputo time fractional derivative. After appropriating corrections, the proposed scheme attains the optimal convergence order of O(\tau^2) with respect to the time step size \tau for both smooth and nonsmooth data at any fixed time $t$. When combined with the Galerkin finite element method for spatial discretization, it forms a fully discrete scheme. The second-order error estimate for this scheme is rigorously established using the Laplace transform technique and verified by some numerical examples.
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Numerical algorithms for nonlinear fractional stochastic Volterra-type equationIn this work, we investigate a class of nonlinear stochastic Volterra-type evolution equations, which can be regarded as an extension of the results reported in Qiao et al. (Fract Calc Appl Anal 27:1136–1161, 2024). For such equations, we propose an Euler scheme and rigorously establish the existence, uniqueness, and regularity of the solution. Moreover, we present the detailed numerical implementation of the scheme and derive the corresponding error estimates.
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Numerical approximation for a stochastic time-fractional cable equationAn efficient numerical method is proposed to address a stochastic time-fractional cable equation driven by fractionally integrated additive noise. Under the reasonable assumptions, we rigorously establish for the first time, the existence, uniqueness, and regularity of the mild solution for this equation. For spatial discretization, a semi-discrete scheme is constructed employing the Galerkin FEM, and the optimal spatial error estimate is derived based on the semigroup approach. In temporal discretization, a piecewise constant function is introduced to approximate the noise, leading to the formulation of a regularized stochastic time-fractional cable equation. A detailed proof of the temporal error estimates is provided via the semigroup approach. Numerical experiments demonstrate that the temporal convergence order attains O ( τ 1 / 2 ) for initial data of either smooth or non-smooth type. The order is independent of the parameters α 1 ∈ ( 0 , 1 ) , α 2 ∈ ( 0 , 1 ) , and β ∈ ( 0 , 1 ) in the equation. These results perfectly align with the theoretical predictions.
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Correction: Existence, uniqueness and regularity for a semilinear stochastic subdiffusion with integrated multiplicative noiseThe original online version of this article was revised: The co-author’s name was misspelled. The co-author's name was spelled Ziqiang Li but should have been Zhiqiang Li. The original version is corrected.
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Numerical approximation for a stochastic Caputo fractional differential equation with multiplicative noiseWe 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.
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Partially-elementary end extensions of countable models of set theoryLet KP denote Kripke–Platek Set Theory and let M be the weak set theory obtained from ZF by removing the collection scheme, restricting separation to $\Delta _0$ -formulae and adding an axiom asserting that every set is contained in a transitive set ( $\mathsf {TCo}$). A result due to Kaufmann [9] shows that every countable model, M of KP+\Pi _n\textsf {-Collection}$ has a proper $\Sigma _{n+1}$ -elementary end extension. We show that for all $n \geq 1$, there exists an $L_\alpha $ (where $L_\alpha $ is the $\alpha ^{\textrm {th}}$ approximation of the constructible universe L ) that satisfies Separation, Powerset and $\Pi _n\textsf {-Collection}$, but that has no $\Sigma _{n+1}$ -elementary end extension satisfying either $\Pi _n\textsf {-Collection}$ or $\Pi _{n+3}\textsf {-Foundation}$. Thus showing that there are limits to the amount of the theory of $\mathcal {M}$ that can be transferred to the end extensions that are guaranteed by Kaufmann’s theorem. Using admissible covers and the Barwise Compactness theorem, we show that if $\mathcal {M}$ is a countable model KP+\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Foundation}$ and T is a recursive theory that holds in $\mathcal {M}$, then there exists a proper $\Sigma _n$ -elementary end extension of $\mathcal {M}$ that satisfies T . We use this result to show that the theory $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\textsf {-Foundation}$ proves $\Sigma _{n+1}\textsf {-Separation}$.
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Numerical study for a stochastic semilinear subdiffusion equation driven by fractional Brownian motionsIn 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.
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A corrected L1 scheme for solving a tempered subdiffusion equation with nonsmooth dataIn this paper, we consider a time semi-discrete scheme for a tempered subdiffusion equation with nonsmooth data. Due to the low regularity of the solution, the optimal convergence rate cannot be achieved when the L1 time-stepping scheme is directly applied to discretize the tempered fractional derivative. By introducing a correction term at the initial time step, we propose a corrected L1 scheme which recover to the optimal convergence rate. Theoretical error estimates and numerical experiments validate the improvement.
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Milstein scheme for a stochastic semilinear subdiffusion equation driven by fractionally integrated multiplicative noiseThis 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.
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Automorphisms of models of set theory and extensions of NFUIn this paper we exploit the structural properties of standard and non-standard models of set theory to produce models of set theory admitting automorphisms that are well-behaved along an initial segment of their ordinals. NFU is Ronald Jensen's modification of Quine's ‘New Foundations’ Set Theory that allows non-sets (urelements) into the domain of discourse. The axioms AxCount, AxCount≤ and AxCount≥ each extend NFU by placing restrictions on the cardinality of a finite set of singletons relative to the cardinality of its union. Using the results about automorphisms of models of subsystems of set theory we separate the consistency strengths of these three extensions of NFU. More specifically, we show that NFU+AxCount proves the consistency of NFU+AxCount≤, and NFU+AxCount≤ proves the consistency of NFU+AxCount≥.
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Decidable fragments of the simple theory of types with infinity and NFWe 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.
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On the relative strengths of fragments of collectionLet M be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ0-separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to M. We focus on two common parameterisations of the collection: -collection, which is the usual collection scheme restricted to -formulae, and strong -collection, which is equivalent to -collection plus -separation. The main result of this paper shows that for all , 1. M + proves that there exists a transitive model of Zermelo Set Theory plus -collection, 2. the theory M + is -conservative over the theory M + strong . It is also shown that (2) holds for when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke-Platek Set Theory with Infinity plus V=L) that does not include the powerset axiom.
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Initial self-embeddings of models of set theoryBy 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 .
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End extending models of set theory via power admissible coversMotivated by problems involving end extensions of models of set theory, we develop the rudiments of the power admissible cover construction (over ill-founded models of set theory), an extension of the machinery of admissible covers invented by Barwise as a versatile tool for generalising model-theoretic results about countable well-founded models of set theory to countable ill-founded ones. Our development of the power admissible machinery allows us to obtain new results concerning powerset-preserving end extensions and rank extensions of countable models of subsystems of ZFC . The canonical extension KP P of Kripke-Platek set theory KP plays a key role in our work; one of our results refines a theorem of Rathjen by showing that Σ 1 P -Foundation is provable in KP P (without invoking the axiom of choice).
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On the strength of a weak variant of the axiom of countingIn 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.
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Largest initial segments pointwise fixed by automorphisms of models of set theoryGiven a model M of set theory, and a nontrivial automorphism j of M, let Ifix(j) be the submodel of M whose universe consists of elements m of M such that j(x)=x for every x in the transitive closure of m (where the transitive closure of m is computed within M). Here we study the class C of structures of the form Ifix(j), where the ambient model M satisfies a frugal yet robust fragment of ZFC known as MOST, and j(m)=m whenever m is a finite ordinal in the sense of M. Our main achievement is the calculation of the theory of C as precisely MOST+Δ0P Collection. The following theorems encapsulate our principal results: Theorem A. Every structure inC satisfies MOST+Δ0P Collection. Theorem B. Each of the following three conditions is sufficient for a countable structure (a) N is a transitive model of MOST+Δ0P Collection. (b) N is a recursively saturated model of MOST+Δ0P Collection. (c) N is a model of ZFC. Theorem C. Suppose M is a countable recursively saturated model of ZFC and I is a proper initial segment of OrdM that is closed under exponentiation and contains ωM. There is a group embedding j⟼j from Aut(Q) into Aut(M) such that I is the longest initial segment of OrdM that is pointwise fixed by jˇ for every nontrivial j∈Aut(Q). In Theorem C, Aut(X) is the group of automorphisms of the structure X, and Q is the ordered set of rationals.
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Feferman’s forays into the foundations of category theoryThis 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.
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Iterated ultrapowers for the massesWe present a novel, perspicuous framework for building iterated ultrapowers. Furthermore, our framework naturally lends itself to the construction of a certain type of order indiscernibles, here dubbed tight indiscernibles, which are shown to provide smooth proofs of several results in general model theory.
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L1 scheme for semilinear stochastic subdiffusion with integrated fractional Gaussian noiseThis 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>0. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method.
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A fractional Adams method for Caputo fractional differential equations with modified graded meshesIn this paper, we introduce an Adams-type predictor–corrector method based on a modified graded mesh for solving Caputo fractional differential equations. This method not only effectively handles the weak singularity near the initial point but also reduces errors associated with large intervals in traditional graded meshes. We prove the error estimates in detail for both 0<α<1 and 1<α<2 cases, where α is the order of the Caputo fractional derivative. Numerical experiments confirm the convergence of the proposed method and compare its performance with the traditional graded mesh approach.
