Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+h^l+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ²+h^l+1).     

Waiting times and stopping probabilities for patterns in Markov chains

Suppose that C is a finite collection of patterns. Observe a Markov chain until one of the patterns in C occurs as a run. This time is denoted by τ. In this paper, we aim to give an easy way to calculate the mean waiting time E(τ) and the stopping probabilities P(τ = τA)with A ∈ C, where τA is the waiting time until the pattern A appears as a run.     

C 0 elements for generalized indefinite Maxwell equations

In this paper we develop the C ⁰ finite element method for a generalized curlcurl-grad div indefinite Maxwell problem in a Lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the H ^r regularity for some r?<?1. The ingredients of our method are that two ??mass-lumping?? L ² projectors are applied to curl and div operators in the problem and that C ⁰ linear element or isoparametric bilinear element enriched with one element-bubble on every triangle element or with two-element-bubbles on every quadrilateral element, respectively, is employed for each component of the nonsmooth solution. Due to the fact that the element-bubbles can be statically eliminated at element levels, our method is essentially three-nodes or four-nodes C ⁰ Lagrange element method. With two Fortin-type interpolations established, a very technical duality argument is elaborated to estimate the error for the indefinite problem. For the nonsmooth solution having the H ^r regularity where r may vary in the interval [0, 1), we obtain the error bound ${{\mathcal O}(h^r)}$ in an energy norm. Some numerical experiments are performed to confirm the theoretical error bounds.     

Direct sums of finitary permutation groups

The interpolation problem under consideration is connected with the finite element method in ?³. In most cases, when finite elements are constructed by means of the partition of a given domain in ?² into triangles and interpolation of the Hermite or Birkhoff type, the sine of the smallest angle of the triangle appears in the denominators of the error estimates for the derivatives. In the case of ? ^m (m ≥ 3), the ratio of the radius of the inscribed sphere to the diameter of the simplex is used as an analog of this characteristic. This makes it necessary to impose constraints on the triangulation of the domain. The recent investigations by a number of authors reveal that, in the case of triangles, the smallest angle in the error estimates for some interpolation processes can be replaced by the middle or the greatest one, which makes it possible to weaken the triangulation requirements. There are fewer works of this kind for m ≥ 3, and the error estimates are given there in terms of other characteristics of the simplex. In this paper, methods are suggested for constructing an interpolation third-degree polynomial on a simplex in ?³. These methods allow one to obtain estimates in terms of a new characteristic of a rather simple form and weaken the triangulation requirements.     

On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

For a finitely triangulated closed surface M ², let α_x be the sum of angles at a vertex x. By the well-known combinatorial version of the 2- dimensional Gauss-Bonnet Theorem, it holds Σ_x(2π - α_x) = 2πχ(M ²), where χ denotes the Euler characteristic of M ², α_x denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ(τ) as the normed dihedral angle defect around a simplex τ. Our main result is Σ_τ (-1)^dim(τ)δ(τ) = χ(K), where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature κ(x) at a vertex and we prove the vertex-version Σ _x∈K0 κ(x) = χ(K) of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial (n + 1)-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B:

$$\chi \left( W \right) - \frac{1}{2}\chi \left( B \right) = \sum {_{\tau \in W - B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}} + \sum {_{\tau \in B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}}\rho \left( \tau \right)$$

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On 2D Newest Vertex Bisection: Optimality of Mesh-Closure and H 1-Stability of L 2-Projection

Newest vertex bisection (NVB) is a popular local mesh-refinement strategy for regular triangulations that consist of simplices. For the 2D case, we prove that the mesh-closure step of NVB, which preserves regularity of the triangulation, is quasi-optimal and that the corresponding L ₂-projection onto lowest-order Courant finite elements (P1-FEM) is always H ¹-stable. Throughout, no additional assumptions on the initial triangulation are imposed. Our analysis thus improves results of Binev et al. (Numer. Math. 97(2):219–268, 2004), Carstensen (Constr. Approx. 20(4):549–564, 2004), and Stevenson (Math. Comput. 77(261):227–241, 2008) in the sense that all assumptions of their theorems are removed. Consequently, our results relax the requirements under which adaptive finite element schemes can be mathematically guaranteed to convergence with quasi-optimal rates.     

On a linear transcendence measure for the solutions of a universal differential equation at algebraic points

In this paper the author continues his work on arithmetic properties of the solutions of a universal differential equation at algebraic points. Every real continuous function on the real line can be uniformly approximated by C^∞-solutions of a universal differential equation. An algebraic universal differential equation of order five and degree 11 is explicitly given, such that every finite set of nonvanishing derivatives y^(k₁)(τ),…,y^(k_r)(τ) (1?k₁<?<k_r) at an algebraic point τ is linearly independent over the field of algebraic numbers. A linear transcendence measure for these values is effectively computed.     

Number of divisors of the central binomial coefficient

Asymptotic formulas are derived for the following expressions: log τ(C _2n ⁿ ) and log τ([1, ... , n]).     

Finite element method for space-time fractional diffusion equation

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order β (1 < β ≤ 2), and the first order time derivative by a Caputo fractional derivative of order γ (0 < γ ≤ 1). For the 0 < γ < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(τ^2?γ + h²) and O(τ² + h²), respectively, in which τ is the time step size and h is the space step size. And for the case γ = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(τ² + h²) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.     

Structure spaces of approximately finite-dimensional C1-algebras

A necessary and sufficient condition for a topological space Y to be the primitive ideal space of an approximately finite dimensional C¹-algebra is proved. The condition is that Y can be obtained from a second countable, totally disconnected, compact Hausdorff space X by equipping X with a topology τ coarser than the original topology and setting Y = (X, τ)/~. ~ is the equivalence relation on X defined by x ~ y iff all sets $\text{o}$ ? τ that contain x also contain y, and conversely. The topology τ must satisfy some additional requirements.     

Homogenization of hyperbolic equations

We consider a self-adjoint matrix elliptic operator A _{ε, ε} > 0, on L ₂(R ^d ;C ⁿ ) given by the differential expression b(D)*g(x/ε)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τA _ε ^1/2 ), where τ ∈ R. It is shown that, as ε → 0, the operator cos(τA _ε ^1/2 ) converges to cos(τ(A ⁰)^1/2) in the norm of operators acting from the Sobolev space H ^s (R ^d ;C ⁿ ) (with a suitable s) to L ₂(R ^d ;C ⁿ ). Here A ₀ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ? _τ ² u _ε (x, τ) = ?A _ε u _ε (x, τ).     

Geométrie des suites de Kronecker

Let Θ be a element of the d-dimensional torus $\mathbb{T}$ ^d andτ the translationτ(x)=x + Θ. When d=1 there existe some partitions of $\mathbb{T}$ ¹ which are associated withτ. We prove the existence of partitions of $\mathbb{T}$ ^d which enjoyed the same kind of properties and whose elements (A _i ) _i≤n are convex polytopes. We also give a lower bound for the isotropic discrepancy of the sequence (nΘ) _nε?.     

High accuracy analysis of the lowest order <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-Galerkin mixed finite element method for nonlinear sine-Gordon equations

The lowest order H¹-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart-Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h²)/O(h² + τ²) in H¹-norm and H(div;Ω)-norm are deduced for the semi-discrete and the fully-discrete schemes, where h, τ denote the mesh size and the time step, respectively, which improve the results in the previous literature.     

Pointwise estimates of polynomials orthogonal on a circle with respect to a weight not belonging to the spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript> (<Emphasis Type="Italic">r</Emphasis> &gt; 1)

Two-sided pointwise estimates are established for polynomials that are orthogonal on the circle |z| = 1 with respect to the weight ?(τ): = h(τ)|sin(τ/2)|^?1 g(|sin(τ/2)|) (τ ∈ ?), where g(t) is a concave modulus of continuity slowly changing at zero such that t ^?1 g(t) ∈ L ¹[0, 1] and h(τ) is a positive function from the class C _2π with a modulus of continuity satisfying the integral Dini condition. The obtained estimates are applied to find the order of the distance from the point t = 1 to the greatest zero of a polynomial orthogonal on the segment [?1, 1].     

Property T＊＊ for C＊-algebras

Extending the notion of property T of finite von Neumann algebras to general von Neumann algebras, we define and study in this paper property T** for (possibly non-unital) C* -algebras. We obtain several results of property T** parallel to those of property T for unital C* -algebras. Moreover, we show that a discrete group Γ has property T if and only if the group C* -algebra Cr* (Γ) (or equivalently, the reduced group C* -algebra Cr* (Γ)) has property T**. We also show that the compact operators K(l2 ) has property T** but c0 does not have property T**.     

On the existence and exactness of the associated sheaf functor

Let $\text{C}$ be a category with inverse limits. A category $\text{x}$is called an $\text{A}$-topos if there is a site (?, τ), i.e. a small category ? together with a Grothendieck topology τ such that $\text{x}$is equivalent to the category Sh_τ[$\text{C}$⁰. $\text{A}$] of τ-sheaves on $\text{C}$ with values in $\text{C}$. If $\text{x}$is an $\text{C}$-topos, then so is Sh_τ'[?⁰, $\text{x}$] for any site (?', τ'). It is shown that if for every site (?,τ) the associated sheaf functor from presheaves to τ-sheaves with values in $\text{A}$ exists (and preserves finite inverse limits), then the same holds if $\text{A}$is replaced by any $\text{A}$-topos $\text{x}$. Roughly speaking, the main result is that for a site (?,τ) the associated sheaf functor [?⁰, $\text{A}$] → Sh_τ [?⁰, $\text{A}$] exists and preserves finite inverse limits, provided $\text{A}$ has filtered direct limits which commute with finite inverse limits, e.g. if $\text{A}$ is a Grothendieck category or a category of sheaves with values in a locally finitely presentable category [8. 7.1]. Analogous results hold in the additive case.     

On recognition by spectrum of the simple groups B 3(q), C 3(q), and D 4(q)

The spectrum of a finite group is the set of its element orders. Two groups are isospectral whenever they have the same spectra. We consider the classes of finite groups isospectral to the simple symplectic and orthogonal groups B ₃(q), C ₃(q), and D ₄(q). We prove that in the case of even characteristic and q > 2 these groups can be reconstructed from their spectra up to isomorphisms. In the case of odd characteristic we obtain a restriction on the composition structure of groups of this class.     

Stabilized finite element method for the viscoelastic Oldroyd fluid flows

In this paper, a new stabilized finite element method based on two local Gauss integrations is considered for the two-dimensional viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newtonian fluid flows. This new stabilized method presents attractive features such as being parameter-free, or being defined for non-edge-based data structures. It confirms that the lowest equal-order P ₁???P ₁ triangle element and Q ₁???Q ₁ quadrilateral element are compatible. Moreover, the long time stabilities and error estimates for the velocity in H ¹-norm and for the pressure in L ²-norm are obtained. Finally, some numerical experiments are performed, which show that the new method is applied to this model successfully and can save lots of computational cost compared with the standard ones.     

Unique representation in convex sets by extraction of marked components

After introducing the basic concepts of extraction and marking for convex sets, the following marked representation theorem is established: Let C be a lineally closed convex set without lines, the face lattice of which satisfies some descending chain condition, and let μ be some marking on C. Then every point of C can be represented in unique way as a convex (nonnegative) linear combination of points (directions) of C which are μ-independent, and this representation can be determined by an algorithm of successive extractions. In particular, if C is a finite dimensional closed convex set without lines and μ marks extreme points (directions) only, then the marked representation theorem contains some well-known results of convex analysis as special cases, and it yields in the case where C is a polyhedral triangulation which extends available results on polytopes to the unbounded case. The triangulation of unbounded polyhedra then is applied to a certain class of parametric linear programs.