Restricted Algebras Research Articles

VISUAL LIMITS OF MAXIMAL FLATS IN SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS

Let X be a Riemannian symmetric space of non-compact type or a locally finite, strongly transitive Euclidean building, and let ∂∞X denote the geodesic boundary of X. We reduce the study of visual limits of maximal flats in X to the study of limits of apartments in the spherical building ∂∞X: this defines a natural, geometric compactification of the space of maximal flats of X. We then completely determine the possible degenerations of apartments when X is of rank 1 or associated to a classical group of rank 2 or to PGL(4). In particular, we exhibit remarkable behaviours of visual limits of maximal flats in various symmetric spaces of small rank and the surprising algebraic restrictions that occur.

Symplectic T7, T8 singularities and Lagrangian tangency orders

AbstractWe study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic T7, T8 singularities. We define discrete symplectic invariants (the Lagrangian tangency orders) and compare them with the index of isotropy. We use these invariants to distinguish symplectic singularities of classical T7 singularity. We also give the geometric description of symplectic classes of the singularity.

Singular Boolean networks: Semi-tensor product approach

Singular Boolean networks are introduced in this paper. Via semi-tensor product of matrices and the matrix expression of logical functions, two kinds of the condensed algebraic expressions of singular Boolean networks are obtained. The normalization problem of singular Boolean networks is addressed; that is, under what condition singular Boolean networks can be converted into normal Boolean networks with algebraic restrictions. Then one sufficient condition and one necessary and sufficient condition are derived for the normalization problem. Furthermore, the solvability of singular Boolean networks is discussed and the concept of admissible initial values of singular Boolean networks is presented. Finally, fixed points and cycles of singular Boolean networks are also investigated.

Zero-dimensional symplectic isolated complete intersection singularities

We study the local symplectic algebra of the 0-dimensional isolated complete intersection singularities. We use the method of algebraic restrictions to classify these symplectic singularities. We show that there are non-trivial symplectic invariants in this classification.

On the Covariance Completion Problem Under a Circulant Structure

Covariance matrices with a circulant structure arise in the context of discrete-time periodic processes and their significance stems also partly from the fact that they can be diagonalized via a Fourier transformation. This note deals with the problem of completion of partially specified circulant covariance matrices. The particular completion that has maximal determinant (i.e., the so-called maximum entropy completion) was considered in Carli where it was shown that if a single band is unspecified and to be completed, the algebraic restriction that enforces the circulant structure is automatically satisfied and that the inverse of the maximizer has a band of zero values that corresponds to the unspecified band in the data, i.e., it has the Dempster property. The purpose of the present note is to develop an independent proof of this result which in fact extends naturally to any number of missing bands as well as arbitrary missing elements. More specifically, we show that this general fact is a direct consequence of the invariance of the determinant under the group of transformations that leave circulant matrices invariant. A description of the complete set of all positive extensions of partially specified circulant matrices is also given and certain connections between such sets and the factorization of certain polynomials in many variables, facilitated by the circulant structure, is highlighted.

Global Versus Local Casimir Effect

This paper continues the investigation of the Casimir effect with the use of the algebraic formulation of quantum field theory in the initial value setting. Basing on earlier papers by one of us (AH) we approximate the Dirichlet and Neumann boundary conditions by simple interaction models whose nonlocality in physical space is under strict control, but which at the same time are admissible from the point of view of algebraic restrictions imposed on models in the context of Casimir backreaction. The geometrical setting is that of the original parallel plates. By scaling our models and taking appropriate limit we approach the sharp boundary conditions in the limit. The global force is analyzed in that limit. One finds in Neumann case that although the sharp boundary interaction is recovered in the norm resolvent sense for each model considered, the total force per area depends substantially on its choice and diverges in the sharp boundary conditions limit. On the other hand the local energy density outside the interaction region, which in the limit includes any compact set outside the strict position of the plates, has a universal limit corresponding to sharp conditions. This is what one should expect in general, and the lack of this discrepancy in Dirichlet case is rather accidental. Our discussion pins down its precise origin: the difference in the order in which scaling limit and integration over the whole space is carried out.

An intrinsic characterization of 2 + 2 warped spacetimes

We give several equivalent conditions that characterize the 2 + 2 warped spacetimes: imposing the existence of a Killing–Yano tensor A subject to complementary algebraic restrictions, in terms of the projector v (or of the canonical 2-form U) associated with the 2-planes of the warped product. These planes are the principal planes of the Weyl and/or Ricci tensors and can be explicitly obtained from them. Therefore, we obtain the necessary and sufficient (local) conditions for a metric tensor to be a 2 + 2 warped product. These conditions exclusively involve explicit concomitants of the Riemann tensor. We present a similar analysis for the conformally 2 + 2 product spacetimes and give an invariant classification of them. The warped products correspond to two of these invariant classes. The more degenerate class is the set of product metrics which are also studied from an invariant point of view. The interest of these results in labeling specific solutions of the Einstein equations is outlined.

Discussion on: “A Decentralized Sliding Control Approach for Distributed Simulation of Differential-Algebraic Equation Systems”

The article by Azizi et al. deals with an application of the decentralized control methodology to a class of dynamical processes governed by differential-algebraic equations (DAEs) with an arbitrary finite index. The proposed approach is developed in a constructive framework of the distributed simulation techniques and is motivated by the high complexity of the classical analytical and computational methods related to the differential-algebraic systems. The main mathematical idea of this contribution consists of an equivalent representation of the algebraic constraints in the form of some appropriate sliding manifolds. This interpretation of the given algebraic restrictions makes it possible to apply the well-established sliding mode control (SMC) analysis/design. And, this development is exactly the topic under investigation in the above article. Let us recall that the SMC usually implies strong stability and robustness properties of the resulting systems. Moreover, the SMC approach also provides a basis for some effective computational methods associated with the feedback-type design of controllers. The authors use this fact and apply a decentralized SMC algorithm to simulations of the deformable surfaces. Moreover, they also propose a numerically stable solution procedure for the associated Jacobian inversion problem. The published article addresses an interesting and important subject of the modern control engineering and also studies some related aspects of the computational control science. The above-mentioned main point of the article, namely, the

At infinity of finite-dimensional CAT(0) spaces

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space $X$ has a non-empty intersection in the visual bordification $ \bar{X} = X \cup \partial X$. Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.

Cremmer‐Gervais quantum Lie algebra

AbstractWe describe a quantum Lie algebra based on the Cremmer‐Gervais R‐matrix. The algebra arises upon a restriction of an infinite‐dimensional quantum Lie algebra.

Local symplectic algebra of quasi-homogeneous curves

We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic sin- gularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of a K-analytic curve is a nite-dimensional vector space. We also show that the action of local dieomorphisms preserving the quasi-homogeneous curve on this vector space is determined by the innitesimal action of liftable vector elds. We apply these results to obtain a complete symplectic classication of curves with semigroups (3; 4; 5), (3; 5; 7), (3; 7; 8).

Fuzzy toric geometries

We describe a construction of fuzzy spaces which approximate projective toric varieties. The construction uses the canonical embedding of such varieties into a complex projective space: The algebra of fuzzy functions on a toric variety is obtained by a restriction of the fuzzy algebra of functions on the complex projective space appearing in the embedding. We give several explicit examples for this construction; in particular, we present fuzzy weighted projective spaces as well as fuzzy Hirzebruch and del Pezzo surfaces. As our construction is actually suited for arbitrary subvarieties of complex projective spaces, one can easily obtain large classes of fuzzy Calabi-Yau manifolds and we comment on fuzzy K3 surfaces and fuzzy quintic three-folds. Besides enlarging the number of available fuzzy spaces significantly, we show that the fuzzification of a projective toric variety amounts to a quantization of its toric base.

Symplectic singularities of varieties: The method of algebraic restrictions

We study germs of singular varieties in a symplectic space. In (A1), V. Arnol'd discovered so called ''ghost'' symplectic invariants which are induced purely by singularity. We introduce algebraic restrictions of dierential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the sym- plectic form. This follows from our generalization of Darboux-Givental' theorem from non-singular submanifolds to arbitrary quasi-homogeneous varieties in a symplectic space. Using algebraic restrictions we introduce new symplectic invariants and explain their geo- metric meaning. We prove that a quasi-homogeneous variety N is contained in a non- singular Lagrangian submanifold if and only if the algebraic restriction of the symplectic form to N vanishes. The method of algebraic restriction is a powerful tool for various classification problems in a symplectic space. We illustrate this by complete solutions of symplectic classification problem for the classical A, D, E singularities of curves, the S5 sin- gularity, and for regular union singularities.

Rainich theory for type D aligned Einstein–Maxwell solutions

The original Rainich theory for the non-null Einstein–Maxwell solutions consists of a set of algebraic conditions and the Rainich (differential) equation. We show here that the subclass of type D aligned solutions can be characterized just by algebraic restrictions.

Local energy decay for some hyperbolic equations with initial data decaying slowly near infinity

A uniform local energy decay property is discussed to a linear hyperbolic equation with spatial variable coefficients. We shall deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we assume algebraic order weight restrictions as $\vert x\vert \to +\infty$ on the initial data in order to derive the uniform local energy decay, and its proof is quite simple.

Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra

AbstractIn 1999 V. Arnol’d introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a 1-form to a singular submanifold. We prove that a germ of any subset N of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to N of the contact structure. This is a generalization of the Darboux-Givental’ theoremfor smooth submanifolds of a contactmanifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V. Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold N in terms of the algebraic restriction of the contact structure to N. In particular, the algebraic restriction is zero if and only if N is contained in a smooth Legendrian submanifold of M.

Softness of MALL proof-structures and a correctness criterion with Mix

We show that every MALL proof-structure [9] satisfies the property of softness, originally a categorical notion introduced by Joyal. Furthermore, we show that the notion of hereditary softness precisely captures Girard’s algebraic restriction of the technical condition on proof-structures. Relying on this characterization, we prove a MALL+Mix sequentialization theorem by a proof-theoretical method, using Girard’s notion of jump. Our MALL+Mix correctness criterion subsumes the Danos/Fleury-Retore criterion [6] for MLL+Mix.

Gropes and the rational lift of the Kontsevich integral

In this note, we calculate the leading term of the rational lift of the Kontsevich integral, introduced by Garoufalidis and Kricker, on the boundary of an embedded grope of class 2n. We observe that it lies in the subspace spanned by connected diagrams of Euler degree 2n-2 which have a bead t-1 on a single edge. This places severe algebraic restrictions on the sort of knots that can bound gropes, and in particular implies the two main results of the author's thesis, at least over the rationals.

The architecture of the GroEL-GroES-(ADP)(7) chaperonin complex. II. Heptagrammal characterization of the folding.

The heptagrammal forms derived in part I [Janner (2003a). Acta Cryst. D59, 783-794] enclose chain segments of symmetry-related monomers in the GroEL-GroES-(ADP)(7) chaperonin complex. A chain reaching the boundary of a given form either ends, proceeds to a neighbouring form or has to fold. C(alpha) atoms corresponding to these folding points are identified in each of the nine forms of the chaperonin and are approximated by ideal positions having integral coordinates (the indices) with respect to a symmetry-adapted basis. Mutual structural relations between the indexed positions are derived in terms of integral scale-rotations (similar to those that leave the form invariant). The magnesium ions at the binding sites of the nucleotides ADP and ATP are shown to be symmetry-related to these folding points. The change in folding (polymorphism) observed in the cis ring of GroEL arising from binding to GroES is discussed. In particular, the form segmentation is conserved in the polymorphic transition. The geometric and algebraic restrictions imposed on the indexed positions and on their structural relations by the integrality condition are presented in an appendix.

Modeling of Transport Processes in the Presence of Substance-Locking Effects

Mass-transport processes under the action of an external field in a medium with piecewise-constant properties and special equations of state are investigated both analytically and numerically. An analytical solution of hyperbolic conservation law on which some algebraic restrictions are imposed is presented. Special emphasis is given to the construction of a correct mathematical model for transport processes. The technique of solving the generalized Riemann problem with two initial discontinuities is developed and rigorously revised. The influence of diffusion effects on the solution profile, especially in the vicinity of a stationary contact discontinuity, is studied by numerical methods. An example of a real mass-transport process under the impact of an external force in which a part of the substance is entrapped between stationary discontinuities is presented. This physical effect was observed experimentally for the first time by the authors in electrophoresis---a separation method for chemically reactive multicomponent mixtures by means of an applied electric field.