The order bidual of <Emphasis Type="Italic">f</Emphasis> -algebras revisited

It is shown that the order bidual X ^~~ of an Archimedean semiprime f -algebra X has a unit element for the Arens multiplication if and only if every positive linear functional on X extends to a positive linear functional on the f -algebra Orth (X) of all orthomorphisms on X.     

Asymptotics of heavy atoms in high magnetic fields: I. Lowest landau band regions

The ground state energy of an atom of nuclear charge Ze in a magnetic field B is evaluated exactly to leading order as Z → ∞. In this and a companion work (see [28]) we show that there are five regions as Z → ∞: B < Z^4/3, B ～ Z^4/3, Z^4/3 < B < Z³, B ～ Z³, B > Z³. Regions 1, 2, 3, and 4 (and conceivably 5) are relevant for neutron stars. Different regions have different physics and different asymptotic theories. Regions 1, 2, and 3 are described by a simple density functional theory of the semiclassical Thomas-Fermi form. Here we concentrate mainly on regions 4 and 5 which cannot be so described, although 3, 4, and 5 have the common feature (as shown here) that essentially all electrons are in the lowest Landau band. Region 5 does have, however, a simple non-classical density functional theory (which can be solved exactly). Region 4 does not, but, surprisingly, it can be described by a novel density matrix functional theory. © 1994 John Wiley & Sons, Inc.     

Enhancement of adjoint topology optimization by approximated L1-regularization for promotion of sparse distributed control

An L¹-penalty term is inserted in the cost functional of an optimal control problem in order to promote a sparse distribution of the control variable. The non-differentiable L¹-regularization term is approximated by differentiable Huber functions. The approach is validated by generic fluid-dynamic test cases and applied to a cabin air outlet of an aircraft. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some remarks on the Walras equilibrium problem in Lebesgue spaces

We introduce a parametric variational inequality in order to model the time dependent Walras economic equilibrium and discuss its relation with an integral formulation in the spaces (L ^∞, L ¹). The role of monotonicity is analysed and, as a classical example, we study the Walras problem using the Cobb–Douglas functions in this new functional setting.     

BOUNDARY VALUE PROBLEMS FOR SECOND ORDER MIXED-TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS

We use the topological degree method to deal with the generalized Sturm-Liouville boundary value problem (BVP) for second order mixed-type functional differential equation x(t)=f(t,xt,xt), 0≤t≤T. Existence principle and theorem for solutions of the BVP are obtained.     

Multiplicity results for the two-vortex Chern-Simons Higgs model on the two-sphere

We consider a Ginzburg-Landau type functional on S ² with a 6 ^th order potential and the corresponding selfduality equations. We study the limiting behavior in the two vortex case when a coupling parameter tends to zero. This two vortex case is a limiting case for the Moser inequality, and we correspondingly detect a rich and varied asymptotic behavior depending on the position of the vortices. We exploit analogies with the Nirenberg problem for the prescribed Gauss curvature equation on S ². Received: December 3, 1997     

Supernumary polylogarithmic ladders and related functional equations

Summary The nature of the polylogarithmic ladder is briefly reviewed, and its close relationship to the associated cyclotomic equation explained. Generic results for the base determined by the family of equationsu ^p +u ^q = 1 are developed, and many new supernumary ladders, existing for particular values ofp andq, are discussed in relation to theirad hoc cyclotomic equations. Results for ordersn from 6 through 9, for which no relevant functional equations are known, are reviewed; and new results for the base , where ³ + = 1, are developed through the sixth order.Special results for the exponentp from 4 through 6 are determined whenever a new cyclotomic equation can be constructed. Only the equationu ⁵+u ³ = 1 has so far resisted this process. The need for the constraint (p,q) = 1 is briefly considered if redundant formulas are to be avoided.The equationu ^6m+1 +u ^6r–1 = 1 is discussed and some valid results deduced. This equation is divisible byu ² –u + 1, and the quotient polynomial is useful for constructing cyclotomic equations. The casem = 1,r = 2 is the first example encountered for which no valid ladders have yet been found.New functional equations to give the supernumary -ladders of index 24 are developed, but their construction runs into difficulty at the third order, apparently requiring the introduction of an adjoint set of variables that blocks the extension to the fourth order.A demonstration, based on the indices of existing accessible and supernumary ladders, indicates that functional equations based on arguments ±z ^m (1–z) ^r (1 +z) ^s are not capable of extension to the sixth order.There are some miscellaneous supernumary ladders that seem incapable, at this time, of analytic proof, and these are briefly discussed. In conclusion, applications of ladders are considered, and attention drawn to the existence of ladders with the base on the unit circle giving rise to Clausenfunction formulas which may play an important role inK-theory.     

Hierarchies of higher order kernels

Summary Recent literature on functional estimation has shown the importance of kernels with vanishing moments although no general framework was given to build kernels of increasing order apart from some specific methods based on moment relationships. The purpose of the present paper is to develop such a framework and to show how to build higher order kernels with nice properties and to solve optimization problems about kernels. The proofs given here, unlike standard variational arguments, explain why some hierarchies of kernels do have optimality properties. Applications are given to functional estimation in a general context. In the last section special attention is paid to density estimates based on kernels of order (m, r), i.e., kernels of orderr for estimation of derivatives of orderm. Convergence theorems are easily derived from interpretation by means of projections inL ² spaces.     

Onsager-Machlup functional for the fractional Brownian motion

 Let X be the solution of the stochastic differential equation where B ^H is a fractional Brownian motion with Hurst parameter H. In this paper we compute the Onsager-Machlup functional of X for the supremum norm and H?lder norms of order β with in the case and for H?lder norms of order β with when . Received: 16 July 2001 / Revised version: 12 March 2002 / Published online: 10 September 2002     

Minimization of asymptotic energy of Müller's functional endowed with local nonlinear lower-order term

We solve a minimization problem associated to a generalization of the Müller functional studied in the paper G. Alberti, S. Müller: A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 , 761–825 (2001), whereby the lower order term ∫¹₀a(s)v²(s)ds (involving a primitive of the mass density function, v = v(s) , and the weight function a = a(s) ) is replaced by ∫¹₀a(s, v(s), v′(s))v²(s)ds (where a belongs to a suitable Carathéodory class). We calculate the rescaled asymptotic energy of the functional as small parameter epsilon tends to zero. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two Positive Solutions of a Quasilinear Elliptic Dirichlet Problem

For a class of second order quasilinear elliptic equations we establish the existence of two non–negative weak solutions of the Dirichlet problem on a bounded domain, Ω. Solutions of the boundary value problem are critical points of C ¹–functional on H₀¹(W){H_0^1(\Omega)}. One solution is a local minimum and the other is of mountain pass type.     

Quadratic invariant curves for a planar mapping

A planar mapping was derived from a second order delay differential equation with a piecewise constant argument. Invariant curves for the planar mapping reflects on the dynamics of the differential equation. Results were reported on a planar mapping admitting quadratic invariant curves y=x ²+C, except for the case -3/4≥C≤0. This remaining case is now resolved, and we describe the solutions of the functional equation K(x ²+C)+k(x)=x by iterations of y.     

The effect of numerical integration on the finite element approximation of linear functionals

In this paper, we have studied the effect of numerical integration on the finite element method based on piecewise polynomials of degree k, in the context of approximating linear functionals, which are also known as “quantities of interest”. We have obtained the optimal order of convergence, O(h^2k){\mathcal{O}(h^{2k})}, of the error in the computed functional, when the integrals in the stiffness matrix and the load vector are computed with a quadrature rule of algebraic precision 2k − 1. However, this result was obtained under an increased regularity assumption on the data, which is more than required to obtain the optimal order of convergence of the energy norm of the error in the finite element solution with quadrature. We have obtained a lower bound of the error in the computed functional for a particular problem, which indicates the necessity of the increased regularity requirement of the data. Numerical experiments have been presented indicating that over-integration may be necessary to accurately approximate the functional, when the data lack the increased regularity.     

Nucleation of bulk superconductivity close to critical magnetic field

We consider the two-dimensional Ginzburg–Landau functional with constant applied magnetic field. For applied magnetic fields close to the second critical field HC₂ and large Ginzburg–Landau parameter, we provide leading order estimates on the energy of minimizing configurations. We obtain a fine threshold value of the applied magnetic field for which bulk superconductivity contributes to the leading order of the energy. Furthermore, the energy of the bulk is related to that of the Abrikosov problem in a periodic lattice. A key ingredient of the proof is a novel L^∞-bound which is of independent interest.     

Multilevel Monte Carlo for stochastic differential equations with additive fractional noise

We adopt the multilevel Monte Carlo method introduced by M. Giles (Multilevel Monte Carlo path simulation, Oper. Res. 56(3):607–617, 2008) to SDEs with additive fractional noise of Hurst parameter H>1/2. For the approximation of a Lipschitz functional of the terminal state of the SDE we construct a multilevel estimator based on the Euler scheme. This estimator achieves a prescribed root mean square error of order ε with a computational effort of order ε ⁻².     

Error analysis of the L2 least‐squares finite element method for incompressible inviscid rotational flows

In this article we analyze the L² least‐squares finite element approximations to the incompressible inviscid rotational flow problem, which is recast into the velocity‐vorticity‐pressure formulation. The least‐squares functional is defined in terms of the sum of the squared L² norms of the residual equations over a suitable product function space. We first derive a coercivity type a priori estimate for the first‐order system problem that will play the crucial role in the error analysis. We then show that the method exhibits an optimal rate of convergence in the H¹ norm for velocity and pressure and a suboptimal rate of convergence in the L² norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Higher order Willmore hypersurfaces in Euclidean space

Let x ： Mn^n→ R^n＋1 be an n（≥2）-dimensional hypersurface immersed in Euclidean space Rn＋1. Let σi（0≤ i≤ n） be the ith mean curvature and Qn = ∑i=0^n（-1）^i＋1 （n^i）σ1^n-iσi. Recently, the author showed that Wn（x） = ∫M QndM is a conformal invariant under conformal group of R^n＋1 and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional Wn is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in Ra which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.     

Holomorphic functional calculus of Hodge-Dirac operators in L p

We study the boundedness of the H ^∞ functional calculus for differential operators acting in L ^p (R ⁿ ; C ^N ). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients, we extend our recent results for the L ^p theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients Π _B as treated in L ²(R ⁿ ; C ^N ) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that Π _B has a bounded H ^∞ functional calculus, in terms of randomized boundedness conditions of its resolvent. This allows us to deduce stability under small perturbations of this functional calculus.     

Polylogarithmic functional equations: A new category of results developed with the help of computer algebra (MACSYMA)

The interrelation of polylogarithmic functional equations and certain numerical results, known as ladders, is discussed, and leads to a consideration of three new, single-variable functional equations at the second order. Two of these families each contain six leading terms whose interrelationship constitutes a constraint on the integration process, but the third has only a single leading term with no such constraints. It is shown how this functional equation can be integrated to the third order, and the process reduced to an algorithm — actually a sequence of instructions — for incorporation into a computer program for symbolic manipulation. The procedure utilizes results from Kummer's equations to cancel out, in sequence, terms which do not vanish, or do vanish, with the variablez. Arguments are all of the form ±z ^p (1–z) ^q (1+z) ^r , and the process is algebraicized by using a (p,q,r,s) notation (withs=±1) to represent such terms. Application of the procedure leads to an integration to the fourth and fifth orders, the latter exhibiting 55 transcendental terms. The first step for the transition to the sixth order can also be achieved but the subsequent steps are frustrated by the restricted forms that the Kummer equations take at the fifth order — it is not possible to create the needed equations in a form which vanishes withz; this corresponding to the elimination of the (5) constant in the extension of the numerically determined ladders to the sixth and higher orders. The existence of the higher-order ladders strongly suggests functional equations af these orders, but the present process has not yet been successful in finding them. The new equations have, however, produced ladders that were inaccessible from Kummer's equations, and had heretofore been only obtainable numerically, up to the fifth order. The method which was developed should be capable of generalization to other systems of equations characterized by the appearance of arguments with recurrent factors. Some new feature, however, will need to be determined before the barrier to the sixth order can be breached.     

Finite p-Groups in Which the Number of Subgroups of Possible Order is Less Than or Equal to $p^3 $

In this paper, groups of order pn in which the number of subgroups of possible order is less than or equal to p3 are classified. It turns out that if p 2, n ≥ 5, then the classification of groups of order pn in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order pn with a cyclic subgroup of index p2 are the same.