In 1994 Velazquez constructed a smooth \(O(4)\times O(4)\) invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly bounded. Earlier, Velazquez also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity. Here we prove short time existence of Velazquez formal continuation, and we verify that the mean curvature is also uniformly bounded on the continuation. Combined with the earlier results of Velazquez-Stolarski we therefore show that there exists a solution \(\{M_t^7\subset\R^8 \mid -t_0 <t<t_0\}\) that has an isolated singularity at the origin \(0\in\R^8\), and at \(t=0\); moreover, the mean curvature is uniformly bounded on this solution, even though the second fundamental form is unbounded near the singularity.

In this paper, we review hypergeometric functions $\mathscr{F}^{\rm Dw}_{\underline{a}}(t),$ $\mathscr{F}^{(σ)}_{\underline{a}}(t)$ and $\widehat{\mathscr{F}}^{(σ)} _{\mathbf{a}}(t)$ together with their conjectured transformation formulas, and show that one transformation formula implies the other.

In this article, we analyze a two-level finite element method for the two dimensional time-dependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size $H$ and solving a Stokes problem on a fine grid of size $h, h<<H$. This method gives optimal convergence for velocity in $H^1$-norm and for pressure in $L^2$-norm. The analysis takes in to account the loss of regularity of the solution at $t=0$ of the Navier-Stokes equations.

We consider the distribution of $\argζ(σ+it)$ on fixed lines $σ> \frac12$, and in particular the density \[d(σ) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: |\argζ(σ+it)| > π/2\}|\,,\] and the closely related density \[d_{-}(σ) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: \Reζ(σ+it) < 0\}|\,.\] Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function $ψ_σ(x)$ associated with $\argζ(σ+it)$. We give explicit expressions for $d(σ)$ and $d_{-}(σ)$ in terms of $ψ_σ(x)$. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of $d(σ)$ and $d_{-}(σ)$.

Let $p\geq 3$ be a prime number. In this article, we study the canonical expansion of the primitive $p^n$-th root of unity $ζ_{p^n}$ in $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ for $n\geq 1$. More precisely, we give the explicit formula for the first $\aleph_0$ terms of the expansion of $ζ_{p^n}$ and as an application, we use it to construct a uniformizer of $K_{2,m}=\mathbb{Q}_p\left(ζ_{p^2},p^{1/p^m}\right)$ with $m\geq 1$.

In [AGRS] a multiplicity one theorem is proven for general linear groups, orthogonal groups and unitary groups ($GL, O,$ and $U$) over $p$-adic local fields. That is to say that when we have a pair of such groups $G_n\subseteq G_{n+1}$, any restriction of an irreducible smooth representation of $G_{n+1}$ to $G_n$ is multiplicity free. This property is already known for $GL$ over a local field of positive characteristic, and in this paper we also give a proof for $O,U$, and $SO$ over local fields of positive odd characteristic. These theorems are shown in [GGP] to imply the uniqueness of Bessel models, and in [CS] to imply the uniqueness of Rankin-Selberg models. We also prove simultaniously the uniqeuness of Fourier-Jacobi models, following the outlines of the proof in [Sun]. By the Gelfand-Kazhdan criterion, the multiplicity one property for a pair $H\leq G$ follows from the statement that any distribution on $G$ invariant to conjugations by $H$ is also invariant to some anti-involution of $G$ preserving $H$.

In this work we consider Runge-Kutta discontinuous Galerkin methods (RKDG) for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on GPUs. A mesh curvature approach is presented for the proper resolution of the domain boundary. This approach is based on the linear elasticity equations and enables a boundary approximation with arbitrary, high order. In order to demonstrate the performance of the boundary curvature a massively parallel solver on graphics processors is implemented and utilized for the solution of the Euler equations of gas-dynamics.

We discuss certain special cases of algebraic approximants that are given as zeroes of so-called "effective characteristic polynomials" and their generalization to a multiseries setting. These approximants are useful for the convergence acceleration or summation of quantum mechanical perturbation series. Examples will be given and some properties will be discussed.

We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel algorithms and changing dimension algorithms. More precisely, the spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms induced by an underlying anchored function space decomposition. Here the weights model the relative importance of different groups of variables. The error criterion used is the deterministic worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points, and two classes of weights, namely product and order-dependent (POD) weights and the newly introduced weights with finite active dimension. We show for these classes of weights that multilevel algorithms achieve the optimal rate of convergence in the first cost model while changing dimension algorithms achieve the optimal convergence rate in the second model. As an illustrative example, we discuss the anchored Sobolev space with smoothness parameter $α$ and provide new optimal quasi-Monte Carlo multilevel algorithms and quasi-Monte Carlo changing dimension algorithms based on higher-order polynomial lattice rules.

We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on weighted Hilbert spaces with an underlying anchored decomposition and arbitrary weights. We focus on randomized algorithms and the randomized worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points. Multilevel algorithms behave very well with respect to the first cost model, while changing dimension algorithms and also dimension-wise quadrature methods, which are based on a similar idea, can take advantage of the more generous second cost model. We prove the first non-trivial lower error bounds for randomized algorithms in these cost models and demonstrate their quality in the case of product weights. In particular, we show that the randomized changing dimension algorithms provided in [L. Plaskota, G. W. Wasilkowski, J. Complexity 27 (2011), 505--518] achieve convergence rates arbitrarily close to the optimal convergence rate.

The Operator axioms have produced new real numbers with new operators. New operators naturally produce new equations and thus extend the traditional mathematical models which are selected to describe various scientific rules. So new operators help to describe complex scientific rules which are difficult described by traditional equations and have an enormous application potential. As to the equations including new operators, engineering computation often need the approximate solutions reflecting an intuitive order relation and equivalence relation. However, the order relation and equivalence relation of real numbers are not as intuitive as those of base-b expansions. Thus, this paper introduces numerical computations to approximate all real numbers with base-b expansions.

Nearest-site distances arise in many applications involving spherical or directional domains, including global geospatial analysis, wireless communications, spherical clustering, and cosine-similarity-based data analysis. In this paper, we study the distributional and computational properties of $L_2$, the minimal angular great-circle distance from a uniformly distributed random point on a sphere to a set of prespecified sites on the same sphere. We first derive the cumulative distribution function (CDF) and probability density function (PDF) of $L_0$, the angular great-circle distance from a fixed vertex of a spherical triangle to a random point uniformly distributed within that triangle. We then extend these triangle-level results to convex spherical polygons and use spherical Voronoi diagrams, triangulations of Voronoi cells, and numerical integration to obtain computable distributional and moment formulas for $L_2$. In addition, we derive explicit formulas for selected moments of $\cos(L_2)$, which are relevant to cosine similarity and spherical data analysis. Extensive Monte Carlo simulations validate the proposed CDF, PDF, and moment formulas and demonstrate computational efficiency of our method relative to generic numerical integration and simulation-based alternatives.

We introduce new kind of $p$-adic hypergeometric functions. We show these functions satisfy congruence relations, so they are convergent functions. And we show that there is a transformation formula between our new $p$-adic hypergeometric functions and $p$-adic hypergeometric functions of logarithmic type defined by M. Asakura in a particular case.

We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions. We review the various non-degeneracy concepts in the literature. For finite characteristic we conjecture that there are nowild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $x^p+x^q$ in characteristic $p$, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

It is well known that a fixed point iteration for solving a linear equation system converges if and only if the spectral radius of the iteration matrix is less than one. A method is presented which guarantees the Fixed Point, even if this condition is not ("spectral radius <1") fulfilled and demonstrated through calculation examples.

Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, we establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, we illustrate our results for several SDEs from finance, physics, biology and chemistry.

Three algorithms of Gram-Schmidt type are given that produce an orthogonal decomposition of finite $d$-dimensional symmetric, alternating, or Hermitian forms over division rings. The first uses $d^3/3+O(d^2)$ ring operations with very simple implementation. Next, that algorithm is adapted in two new directions. One is an optimal sequential algorithm whose complexity matches the complexity of matrix multiplication. The other is a parallel NC algorithm with similar complexity.

Consider a customer who needs to fulfill a shopping list, and also a personal shopper who is willing to buy and resell to customers the goods in their shopping lists. It is in the personal shopper's best interest to find (shopping) routes that (i) minimize the time serving a customer, in order to be able to serve more customers, and (ii) minimize the price paid for the goods, in order to maximize his/her potential profit when reselling them. Those are typically competing criteria leading to what we refer to as the Personal Shopper's Dilemma query, i.e., to determine where to buy each of the required goods while attempting to optimize both criteria at the same time. Given the query's NP-hardness we propose a heuristic approach to determine a subset of the sub-optimal routes under any linear combination of the aforementioned criteria, i.e., the query's approximate linear skyline set. In order to measure the effectiveness of our approach we also introduce two new metrics, optimality and coverage gaps w.r.t. an optimal, but computationally expensive, baseline solution. Our experiments, using realistic city-scale datasets, show that our proposed approach is two orders of magnitude faster than the baseline and yields low values for the optimality and coverage gaps.

The paper is concerned with the adaptive finite element solution of linear elliptic differential equations using equidistributing meshes. A strategy is developed for defining this type of mesh based on residual-based a posteriori error estimates and rigorously analyzing the convergence of a linear finite element approximation using them. The existence and computation of equidistributing meshes and the continuous dependence of the finite element approximation on mesh are also studied. Numerical results are given to verify the theoretical findings.

We prove explicit, i.e. non-asymptotic, error bounds for Markov chain Monte Carlo methods. The problem is to compute the expectation of a function f with respect to a measure π. Different convergence properties of Markov chains imply different error bounds. For uniformly ergodic and reversible Markov chains we prove a lower and an upper error bound with respect to the L2 -norm of f . If there exists an L2 -spectral gap, which is a weaker convergence property than uniform ergodicity, then we show an upper error bound with respect to the Lp -norm of f for p > 2. Usually a burn-in period is an efficient way to tune the algorithm. We provide and justify a recipe how to choose the burn-in period. The error bounds are applied to the problem of the integration with respect to a possibly unnormalized density. More precise, we consider the integration with respect to log-concave densities and the integration over convex bodies. By the use of the Metropolis algorithm based on a ball walk and the hit-and-run algorithm it is shown that both problems are polynomial tractable.

The paper describes several efficient parallel implementations of the one-sided hyperbolic Jacobi-type algorithm for computing eigenvalues and eigenvectors of Hermitian matrices. By appropriate blocking of the algorithms an almost ideal load balancing between all available processors/cores is obtained. A similar blocking technique can be used to exploit local cache memory of each processor to further speed up the process. Due to diversity of modern computer architectures, each of the algorithms described here may be the method of choice for a particular hardware and a given matrix size. All proposed block algorithms compute the eigenvalues with relative accuracy similar to the original non-blocked Jacobi algorithm.

We describe two main classes of one-sided trigonometric and hyperbolic Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian matrices. These types of algorithms exhibit significant advantages over many other eigenvalue algorithms. If the matrices permit, both types of algorithms compute the eigenvalues and eigenvectors with high relative accuracy. We present novel parallelization techniques for both trigonometric and hyperbolic classes of algorithms, as well as some new ideas on how pivoting in each cycle of the algorithm can improve the speed of the parallel one-sided algorithms. These parallelization approaches are applicable to both distributed-memory and shared-memory machines. The numerical testing performed indicates that the hyperbolic algorithms may be superior to the trigonometric ones, although, in theory, the latter seem more natural.

In this paper, a new control scheme, called additive state decomposition based tracking control, is proposed to solve the tracking (rejection) problem for rotational position of the TORA (a nonlinear nonminimum phase system). By the additive state decomposition, the tracking (rejection) task for the considered nonlinear system is decomposed into two independent subtasks: a tracking (rejection) subtask for a linear time invariant (LTI) system, leaving a stabilization subtask for a derived nonlinear system. By the decomposition, the proposed tracking control scheme avoids solving regulation equations and can tackle the tracking (rejection) problem in the presence of any external signal (except for the frequencies at +1 or -1) generated by a marginally stable autonomous LTI system. To demonstrate the effectiveness, numerical simulation is given.

This paper presents a new control, namely additive-state-decomposition dynamic inversion stabilized control, that is used to stabilize a class of multi-input multi-output (MIMO) systems subject to nonparametric time-varying uncertainties with respect to both state and input. By additive state decomposition and a new definition of output, the considered uncertain system is transformed into a minimum-phase uncertainty-free system with relative degree one, in which all uncertainties are lumped into a new disturbance at the output. Subsequently, dynamic inversion control is applied to reject the lumped disturbance. Performance analysis of the resulting closed-loop dynamics shows that the stability can be ensured. Finally, to demonstrate its effectiveness, the proposed control is applied to two existing problems by numerical simulation. Furthermore, in order to show its practicability, the proposed control is also performed on a real quadrotor to stabilize its attitude when its inertia moment matrix is subject to a large uncertainty.

Besides parametric uncertainties and disturbances, the unmodeled dynamics and time delay at the input are often present in practical systems, which cannot be ignored in some cases. This paper aims to solve output feedback tracking control problem for a class of nonlinear uncertain systems subject to unmodeled high-frequency gains and time delay at the input. By the additive decomposition, the uncertain system is transformed to an uncertainty-free system, where the uncertainties, disturbance and effect of unmodeled dynamics plus time delay are lumped into a new disturbance at the output. Sequently, additive decomposition is used to decompose the transformed system, which simplifies the tracking controller design. To demonstrate the effectiveness, the proposed control scheme is applied to three benchmark examples.

Recently, Dyachenko & Zakharov (2011) have derived a compact form of the well known Zakharov integro-differential equation for the third order Hamiltonian dynamics of a potential flow of an incompressible, infinitely deep fluid with a free surface. Special traveling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. Further, unstable traveling waves with wedge-type singularities, viz. peakons, are numerically discovered. To gain insights into the properties of singular traveling waves, we consider the academic case of a perturbed version of the compact equation, for which analytical peakons with exponential shape are derived. Finally, by means of an accurate Fourier-type spectral scheme it is found that smooth solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation.

In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints. Our derivation procedure does not explicitly involve any small parameter and is straightforward. The novel system is a non-dispersive non-hydrostatic extension of the classical Saint-Venant equations. A key feature of the new model is that, like the classical NSWE, it is hyperbolic and thus similar numerical methods can be used. We also propose a finite volume discretisation of the obtained hyperbolic system. Several test-cases are presented to highlight the added value of the new model. Some implications to tsunami wave modelling are also discussed.

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of them is the Galilean symmetry, which is not present in important models such as the BBM equation and the Peregrine (Classical Boussinesq) system. In this paper we propose a mechanism to modify the above mentioned classical models and derive new, Galilean invariant models. We present some properties of the new equations, with special emphasis on the computation and interaction of their solitary-wave solutions. The comparison with full Euler solutions shows the relevance of the preservation of Galilean invariance for the description of water waves.

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical, experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.

Historically the finite volume methods have been developed for the numerical integration of conservation laws. In this study we present some recent results on the application of such schemes to dispersive PDEs. Namely, we solve numerically a representative of Boussinesq type equations in view of important applications to the coastal hydrodynamics. Numerical results of the runup of a moderate wave onto a non-uniform beach are presented along with great lines of the employed numerical method (see D. Dutykh et al. (2011) for more details).