Jordan (Super)Coalgebras and Lie (Super)Coalgebras

We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Koecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra (J,) with a dual algebra J ^*, there exists a Lie supercoalgebra (L ^c (J), _L ) whose dual algebra (L ^c (J))^* is the Lie KKT-superalgebra for the Jordan superalgebra J ^*. It is well known that some Jordan coalgebra J ⁰ can be constructed from an arbitrary Jordan algebra J. We find necessary and sufficient conditions for the coalgebra (L ^c (J ⁰),_L) to be isomorphic to the coalgebra (Loc(L _in (J)⁰), _L ⁰), where L _in (J) is the adjoint Lie KKT-algebra for the Jordan algebra J.     

Existence of a somewhere injective pseudo-holomorphic disc

Given a smooth totally real submanifold L {\cal L} in an almost complex manifold (M,J) and a J-holomorphic disc with boundary in L {\cal L} , by restriction of the initial disc and factorization, one gets a smooth simple J-holomorphic curve still with boundary in L {\cal L} . As a consequence one gets a proof of the Arnold-Givental conjecture for a class of Lagrangian submanifolds in a symplectic manifold.     

The centroid of a weak*-closed inner ideal in a JBW*-triple

It is shown that there exists a ^*-homomorphism from the continuous centroid L^b (A){\cal L}^b (A) of a JBW^*-triple A onto the continuous centroid L^b (J){\cal L}^b (J) of an arbitrary weak^*-closed inner ideal J in A.     

Über Homologiegruppen von Faserbündeln

Let L be a distributive lattice with 0 and C (L) be its lattice of congruences. The skeleton, SC (L), of C (L) consists of all those congruences which are the pseudocomplements of members of C (L), and is a complete BOOLEan lattice. An ideal is the kernel of a skeletal congruence if and only if it is an intersection of relative annihilator ideals, i.e. ideals of the form <r, s>j={x∈L: xΔr≤s} for suitable r, s∈L. The set KSC (L) of all such kernels forms an upper continuous distributive lattice and the map a ? (a={x∈L: x≤a} is a lower regular joindense embedding of L into KSC (L). The relationship between SC (L) and KSC (L) leads to numerous characterizations of disjunctive and generalized BOOLEan lattices. In particular, a distributive lattice L is disjunctive (generalized Boolean) if and only if the map Θ ? ker Θ is a lattice-isomorphism of SC (L) onto KSC (L), whose inverse is the map J ? Θ (J)** (the map J ? Θ(J)). In addition, a study of KSC (L) leads to new simple proofs of results on the completions of special classes of lattices.     

Using Root Functions for Eigenvalue Problems of Ordinary Differential Operators

We study eigenvalue problems for an ordinary differential operator L acting on L ₂(?)-spaces (Problem 1) and on L ₂(J)-spaces (Problem 2). Here J is a bounded but large interval. Assuming that in Problem 1 the spectral parameter s lies in the set of normal points of L, we show that the structure of eigenspaces for both problems is similar to the structure of finite complex-valued matrices. In the case of a finite matrix, the geometry of eigenspaces is described by the Jordan form. In the case of ordinary differential operators, the corresponding geometry is described by a sequence of root functions. Therefore, the main tool of our studies is root functions for complex-valued analytical matrix functions.     

Rhombi inscribed in simple closed curves

We show that given any simple closed curveJ in ² and any lineL, the curveJ contains the four vertices of some rhombusR with two sides parallel toL. Furthermore, the cyclic order of the vertices ofR agrees with their cyclic order onJ. We also show that the diameters of the rhombi so produced (one for each lineL) may be bounded away from zero.     

Jordan triple derivations on J-subspace lattice algebras

Let L be a J-subspace lattice on a Banach space X and Alg L the associated J-subspace lattice algebra. Let A be a standard operator subalgebra (i.e., it contains all finite rank operators in AlgL) of AlgL and M■B(X) the Alg L-bimodule. It is shown that every linear Jordan triple derivation from A into M is a derivation, and that every generalized Jordan (triple) derivation from A into M is a generalized derivation.     

Simplification d’idéaux fractionnaires

Riassunto SianoI, J, N eL degli ideali frazionari di un dominioA. Mostriamo che se dueA-moduli moltiplicativi (I ⊕J, 〈,〉) e (L ⊕N, 〈,〉) sono isometrici suA alloraI è isomorfo aL (rispettivamenteJ≈N) o aN (rispettivamenteJ≈L); se ne deduce un criterio che permette di sapere se dueA-moduli quadratici, isotropici di rango 2, sono isometrici o no su un anello di Prüfer.

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Summary LetI, J, L andN be fractional ideals of a domainA. We prove that if two ?Multiplicative modules? (I⊕J, 〈,〉) and (L⊕N, 〈,〉) are isometric, thenI is isomorphic toL (respectivelyJ is isomorphic toN) or toN (respectively toL). As a consequence, we can know if two isotropic quadratic spaces of rank 2 are isometric on a Prüfer domain.

     

Embeddings into Orthomodular Lattices with Given Centers,State Spaces and Automorphism Groups

We prove that, given a nontrivial Boolean algebra B, a compact convex set S and a group G, there is an orthomodular lattice L with the center isomorphic to B, the automorphism group isomorphic to G, and the state space affinely homeomorphic to S. Moreover, given an orthomodular lattice J admitting at least one state, L can be chosen such that J is its subalgebra.     

Semimodularity in lattices of finite length

For a lattice L of finite length we denote by J(L) the set of all join-irreducible elements (≠0) of L. By u′ we mean the uniquely determined lower cover of an element u?J(L). Our main result is the following theorem: A lattice L of finite length is (upper) semimodular if and only if it satisfies the exchange property (EP): c?b∨u and $\text{c?b∨u′}$ imply u?b∨c∨u′ (b, c?L;u?J(L)).     

Discrete-Time Dichotomous Well-Posed Linear Systems and Generalized Schur–Nevanlinna–Pick Interpolation

We introduce a class of matrix-valued functions W called “L²- regular”. In case W is J-inner, this class coincides with the class of “strongly regular J-inner” matrix functions in the sense of Arov–Dym. We show that the class of L²-regular matrix functions is exactly the class of transfer functions for a discrete-time dichotomous (possibly infinite-dimensional) input-state-output linear system having some additional stability properties. When applied to J-inner matrix functions, we obtain a state-space realization formula for the resolvent matrix associated with a generalized Schur–Nevanlinna–Pick interpolation problem. Communicated by Daniel Alpay Submitted: August 20, 2006; Accepted: September 13, 2006     

Determinantal representations for the J transformation

Summary. The iterative J transformation [Homeier, H. H. H. (1993): Some applications of nonlinear convergence accelerators. Int. J. Quantum Chem. 45, 545-562] is of similar generality as the well-known E algorithm [Brezinski, C. (1980): A general extrapolation algorithm. Numer. Math. 35, 175-180. Havie, T. (1979): Generalized Neville type extrapolation schemes. BIT 19, 204-213]. The properties of the J transformation were studied recently in two companion papers [Homeier, H. H. H. (1994a): A hierarchically consistent, iterative sequence transformation. Numer. Algo. 8, 47-81. Homeier, H. H. H. (1994b): Analytical and numerical studies of the convergence behavior of the J transformation. J. Comput. Appl. Math., to appear]. In the present contribution, explicit determinantal representations for this sequence transformation are derived. The relation to the Brezinski-Walz theory [Brezinski, C., Walz, G. (1991): Sequences of transformations and triangular recursion schemes, with applications in numerical analysis. J. Comput. Appl. Math. 34, 361-383] is discussed. Overholt's process [Overholt, K. J. (1965): Extended Aitken acceleration. BIT 5, 122-132] is shown to be a special case of the J transformation. Consequently, explicit determinantal representations of Overholt's process are derived which do not depend on lower order transforms. Also, families of sequences are given for which Overholt's process is exact. As a numerical example, the Euler series is summed using the J transformation. The results indicate that the J transformation is a very powerful numerical tool. Received May 24, 1994 / Revised version received November 11, 1994     

An exact Daugavet type inequality for small into isomorphisms in L 1

The well known Daugavet property for the space L ₁ means that || I  +  K || = 1+ || K || for any weakly compact operator K : L ₁ → L ₁, where I is the identity operator in L ₁. We generalize this theorem to the case when we consider an into isomorphism J : L ₁ → L ₁ instead of I and a narrow operator T. Our main result states that , where d  =  || J|| || J ⁻¹||. We also give an example which shows that this estimate is exact. Received: 21 August 2007     

Mixed finite element methods for generalized Forchheimer flow in porous media

Mixed finite element methods are analyzed for the approximation of the solution of the system of equations that describes the flow of a single‐phase fluid in a porous medium in ?^d, d ≤ 3, subject to Forchhheimer's law—a nonlinear form of Darcy's law. Existence and uniqueness of the approximation are proved, and optimal order error estimates in L^∞(J; L²(Ω)) and in L^∞(J; H(div; Ω)) are demonstrated for the pressure and momentum, respectively. Error estimates are also derived in L^∞(J; L^∞(Ω)) for the pressure. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Probability measures on [SIN] groups and some related ideals in group algebras

Given a locally compact group G, let J(G){\cal J}(G) denote the set of closed left ideals in L ¹(G), of the form J _μ = [L¹(G) * (δ _e − μ)]⁻, where μ is a probability measure on G. Let J_d(G)={\cal J}_d(G)= {J_m;m is discrete}\{J_{\mu};\mu\ {\rm is discrete}\} , J_a(G)={J_m;m is absolutely continuous}{\cal J}_a(G)=\{J_{\mu};\mu\ {\rm is absolutely continuous}\} . When G is a second countable [SIN] group, we prove that J(G)=J_d(G){\cal J}(G)={\cal J}_d(G) and that J_a(G){\cal J}_a(G) , being a proper subset of J(G){\cal J}(G) when G is nondiscrete, contains every maximal element of J(G){\cal J}(G) . Some results concerning the ideals J _μ in general locally compact second countable groups are also obtained.     

Optimal location of the support of the control for the 1-D wave equation: numerical investigations

We consider in this paper the homogeneous 1-D wave equation defined on Ω⊂ℝ. Using the Hilbert Uniqueness Method, one may define, for each subset ω⊂Ω, the exact control v _ω of minimal L ²(ω×(0,T))-norm which drives to rest the system at a time T>0 large enough. We address the question of the optimal position of ω which minimizes the functional . We express the shape derivative of J as an integral on ∂ ω×(0,T) independently of any adjoint solution. This expression leads to a descent direction for J and permits to define a gradient algorithm efficiently initialized by the topological derivative associated with J. The numerical approximation of the problem is discussed and numerical experiments are presented in the framework of the level set approach. We also investigate the well-posedness of the problem by considering a relaxed formulation.     

Existence and uniqueness of solutions of stationary transport equations

We consider the abstract kinetic equation dψ /dx = –JLψ, x ∈ [0, τ ], in a Hilbert space H. It is supposed that J = J * = J^–1, L = L * ≥ 0, ker L = 0. The following theorem is proved: if JL is similar to a self-adjoint operator, then an associated boundary problem has a unique solution. We apply this theorem to the stationary equation of Brownian motion (sgn μ)|μ |α (∂ψ /∂x) (x,μ) = (∂²ψ /∂μ²) (x,μ), 0 < x < τ, μ ∈ ℝ. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite distributive lattices are congruence lattices of almost-geometric lattices

A semimodular lattice L of finite length will be called an almost-geometric lattice if the order J(L) of its nonzero join-irreducible elements is a cardinal sum of at most two-element chains. We prove that each finite distributive lattice is isomorphic to the lattice of congruences of a finite almost-geometric lattice.     

Sharp metastability threshold for two-dimensional bootstrap percolation

 In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p→0 and L→∞ simultaneously of the probability I(L,p) that the entire square is eventually active. We prove that I(L,p)→1 if , and I(L,p)→0 if , where λ=π²/18. We prove the same behaviour, with the same threshold λ, for the probability J(L,p) that a site is active by time L in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold λ^′=π²/6. The existence of the thresholds λ,λ^′ settles a conjecture of Aizenman and Lebowitz [3], while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony [2]. Received: 12 May 2002 / Revised version: 12 August 2002 / Published online: 14 November 2002 Research funded in part by NSF Grant DMS-0072398 Mathematics Subject Classification (2000): Primary 60K35; Secondary 82B43 Key words or phrases: Bootstrap percolation – Cellular automaton – Metastability – Finite-size scaling     

New a posteriori L∞(L2) and L2(L2)-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L^∞(J; L²Ω)-norm and L²(J; L²Ω)-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.