Canonical conservative state/signal shift realizations of passive discrete time behaviors

A passive linear discrete time invariant s/s (state/signal) system Σ=(V;X,W) consists of a Hilbert (state) space X, a Kre?n (signal) space W, a maximal nonnegative (generating) subspace V of the Kre?n space K:=−X[?]X[?]W. The sets of trajectories (x(⋅);w(⋅)) generated by V on the discrete time intervals I⊂Z are defined by

     

Isotone functions,dual cones,and networks

If the collection of all real-valued functions defined on a finite partially ordered set S of n elements is identified in the natural way with Rⁿ, it is obvious that the subset of functions that are isotone or order preserving with respect to the given partial order constitutes a closed, convex, polyhedral cone K in Rⁿ. The dual cone K^* of K is the set of all linear functionals that are nonpositive of K. This article identifies the important geometric properties of K, and characterizes a nonredundant set of defining equations and inequalities for K^* in terms of a special class of partitions of S into upper and lower sets. These defining constraints immediately imply a set of extreme rays spanning K and K^*. One of the characterizations of K^* involves feasibility conditions on flows in a network. These conditions are also used as a tool in analysis.     

Sets on which several measures agree

It is known that, given n non-atomic probability measures on the space I = [0, 1], and a number α between 0 and 1, there exists a subset K of I that has measure α in each measure. It is proved here that K may be chosen to be a union of at most n intervals. If the underlying space is the circle S¹ instead of I, then K may be chosen to be a union of at most n ? 1 intervals. These results are shown to be best possible for all irrational and many rational values of α. However, there remain many rational values of α for which we are unable to determine the minimum number of intervals that will suffice.     

An analogue of group divisible designs for Moore graphs

We study graphs whose adjacency matrix S of order n satisfies the equation S + S² = J ? K + kI, where J is a matrix of order n of all 1's, K is the direct sum on $\text{n}\text{l}$ matrices of order l of all 1's, and I is the identity matrix. Moore graphs are the only solutions to the equation in the case l = 1 for which K = I. In the case k = l we can obtain Moore graphs from a solution S by a bordering process analogous to obtaining (ν, κ, λ)-designs from some group divisible designs. Other parameters are rare. We are able to find one new interesting graph with parameters k = 6, l = 4 on n = 40 vertices. We show that it has a transitive automorphism group isomorphic to C₄ × S₅.     

Abstract Holography

The problem that is presented and investigated is a natural nonlinear extension of the following linear problem. Let H ⊕ H′ and K ⊕ K′ be two orthogonal Hilbert decompositions of a real Hilbert space X. Let P, P′, Q, Q′ and N′ be the operators of orthogonal projection of X onto H, H′, K, K′ and H′ ∩ K′ respectively. Denoting by Z′ the Hilbert space, Z′ = {(a′, b′) ?H′ × K′: N′a′ = N′b′}, let F be the linear mapping of X into Z′, F(x) = (P′x, Q′x). Under the condition ∥PQ∥ < 1, which proves to be equivalent to H ∩ K = {0} and H + K closed, F is bicontinuous. The problem is then to choose a constructive procedure for the calculation of a = (P ° F^?1) · (a′,b′), and to analyse the continuity of P ° F^?1. One may use an iterative technique depending on a real relaxation parameter ω. Let the “separation angle” between H and K be defined by (H, K) = Arc cos ∥PQ∥. The present analysis stresses the fundamental part played by the separation angles α = (H, K), α′ = (H, K′), β = (H, SH) and β′ = (H′, SH) where S (= 2Q ? I) denotes the operator of orthogonal symmetry with respect to K. In the special case where X and H are complex spaces, and K′ = iK, the analysis of the problem is governed by the separation angles β and β′ only. These angles are involved in what may then be called “the conjugate image effect of H with respect to the orthogonal decomposition of X, K ⊕ iK.” Then, $\text{α = α′ =}\text{β}\text{2}$, and the optimal value of ω is known a priori (ω₀ = 2). This particular problem, which proves to be related to the central problem of Holography, defines what we have called “Abstract Holography”. (One of the main objects of our analysis is to show what underlies the principle of “Wavefront Reconstruction,” which is referred to in Classical Holography, and how it is possible to circumvent certain related difficulties by using an optimal iterative procedure).     

SOME t-CLOSED PAIRS

Let A ? B be integral domains. (A, B) is called a t-closed pair if each subring of B containing A is t-closed. Let R be a t-closed domain containing a field K and let I be a nonzero proper ideal of R. Let D be a subring of K and let S = D + I. If D is a field then it is shown that (S, R) is a t-closed pair if and only if R is integral over S and I is a maximal ideal of R. If D is not a field then we prove in this note that (S, R) is a t-closed pair if and only if (D, K) is a t-closed pair and R = K + I.     

Waldhausen's theory of k-fold end structures: a survey

Let R⁺ be the space of nonnegative real numbers. F. Waldhausen defines a k-fold end structure on a space X as an ordered k-tuple of continuous maps x_f:X → R⁺, 1 ? j ? k, yielding a proper map x:X → (R⁺)^k. The pairs (X,x) are made into the category E^k of spaces with k-fold end structure. Attachments and expansions in E^k are defined by induction on k, where elementary attachments and expansions in E⁰ have their usual meaning. The category E^k/Z consists of objects (X, i) where i: Z → X is an inclusion in E^k with an attachment of i(Z) to X, and the category E^k6Z consists of pairs (X,i) of E^k/Z that admit retractions X → Z. An infinite complex over Z is a sequence X = {X₁ ? X₂ ? … ? X_n …} of inclusions in E^k6Z. The abelian grou p S₀(Z) is then defined as the set of equivalence classes of infinite complexes dominated by finite ones, where the equivalence relation is generated by homotopy equivalence and finite attachment; and the abelian group S₁(Z) is defined as the set of equivalence classes of X₁, where X ∈ E^k/Z deformation retracts to Z. The group operations are gluing over Z. This paper presents the Waldhausen theory with some additions and in particular the proof of Waldhausen's proposition that there exists a natural exact sequence 0 → S₁(Z × R)→^πS₀(Z) by utilizing methods of L.C. Siebenmann. Waldhausen developed this theory while seeking to prove the topological invariance of Whitehead torsion; however, the end structures also have application in studying the splitting of a noncompact manifold as a product with R[1].     

How to project onto an isotone projection cone

The solution of the complementarity problem defined by a mapping f:Rⁿ→Rⁿ and a cone K⊂Rⁿ consists of finding the fixed points of the operator P_K°(I-f), where P_K is the projection onto the cone K and I stands for the identity mapping. For the class of isotone projection cones (cones admitting projections isotone with respect to the order relation they generate) and f satisfying certain monotonicity properties, the solution can be obtained by iterative processes (see G. Isac, A.B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153(1) (1990) 258-275 and S.Z. Németh, Iterative methods for nonlinear complementarity problems on isotone projection cones, J. Math. Anal. Appl. 350(1) (2009) 340-347). These algorithms require computing at each step the projection onto the cone K. In general, computing the projection mapping onto a cone K is a difficult and computationally expensive problem. In this note it is shown that the projection of an arbitrary point onto an isotone projection cone in Rⁿ can be obtained by projecting recursively at most n-1 times into subspaces of decreasing dimension. This emphasizes the efficiency of the algorithms mentioned above and furnishes a handy tool for some problems involving special isotone projection cones, as for example the non-negative monotone cones occurring in reconstruction problems (see e.g. Section 5.13 in J. Dattorro, Convex Optimization and Euclidean Distance Geometry, Meboo, 2005, v2009.04.11).     

On uniformly continuous Nemytskij operators generated by set-valued functions

Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all Hölder functions ${\varphi : I \to C}Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all H?lder functions j: I ? C{\varphi : I \to C} into the set H _β (I, clb(Z)) of all H?lder set-valued functions f: I ? clb(Z){\phi : I \to clb(Z)} and is uniformly continuous, then

F(x,y)=A(x,y) \text+* B(x),       x ? I, y ? CF(x,y)=A(x,y) \stackrel{*}{\text{+}} B(x),\qquad x \in I, y \in C      10. A harmonic analysis approach to essential normality of principal submodules      Ronald G. Douglas 《Journal of Functional Analysis》2011,261(11):3155-3180 Guo and the second author have shown that the closure [I] in the Drury-Arveson space of a homogeneous principal ideal I in C[z1,…,zn] is essentially normal. In this note, the authors extend this result to the closure of any principal polynomial ideal in the Bergman space. In particular, the commutators and cross-commutators of the restrictions of the multiplication operators are shown to be in the Schatten p-class for p>n. The same is true for modules generated by polynomials with vector-valued coefficients. Further, the maximal ideal space XI of the resulting C?-algebra for the quotient module is shown to be contained in Z(I)∩∂Bn, where Z(I) is the zero variety for I, and to contain all points in ∂Bn that are limit points of Z(I)∩Bn. Finally, the techniques introduced enable one to study a certain class of weight Bergman spaces on the ball.      11. Subdifferentials of a minimum time function in Banach spaces      Yiran He  Kung Fu Ng 《Journal of Mathematical Analysis and Applications》2006,321(2):896-910 In general Banach space setting, we study the minimum time function determined by a closed convex set K and a closed set S (this function is simply the usual Minkowski function of K if S is the singleton consisting of the origin). In particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function.      12. Sequence entropy and mixing      Alan Saleski 《Journal of Mathematical Analysis and Applications》1977,60(1):58-66 The relationship between sequence entropy and mixing is examined. Let T be an automorphism of a Lebesgue space X, $\text{L}$0 denote the set of all partitions of X possessing finite entropy, and $\text{S}$ denote the set of all increasing sequences of positive integers. It is shown that: (1) T is mixing /a2 supA ? BhA(T, α) = H(α) for all B∈I and α∈Z0. (2) T is weakly mixing /a2 supAhA(T, α) = H(α) for all α∈Z0. (3) If T is partially mixing with constant $\text{c (1 ?}\text{1}\text{e}\text{< c < 1)}$, then supA ? BhA(T, α) > cH(α) for all B∈I and nontrivial α∈Z0. (4) If supA ? BhA(T, α) > 0 for all B∈I and nontrivial α∈Z0, then T is weakly mixing.      13. Affine hypersurfaces admitting a pointwise symmetry      Y. Lu  C. Scharlach 《Results in Mathematics》2005,48(3-4):275-300 An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T p M) for all p ∈ M, which preserves (pointwise) the affine metric h, the difference tensor K (resp. the cubic form) and the affine shape operator S. In this paper, we deal with locally strongly convex affine hypersurfaces of dimension three. First we solve an algebraic problem. We determine the non-trivial stabilizers G of the pair (K, S) under the action of SO(3) on a Euclidean vector space (V, h) and find a representative (canonical form of K and S) of each SO(3)/G-orbit. Then, we classify hypersurfaces admitting a pointwise G-symmetry for all non-trivial stabilizers G (apart of Z 2). Besides well-known hypersurfaces (for Z 2 × Z 2 we get the locally homogeneous hypersurface (x 1 ?) 1/2x 32 (x 2 ?) 1/2x 42) = 1) we obtain e.g. warped products of two-dimensional affine spheres (resp. quadrics) and curves.      14. Ergodic Actions of Semisimple Lie Groups on Compact Principal Bundles      Dave Witte Morris  Robert J. Zimmer 《Geometriae Dedicata》2004,106(1):11-27 Let G = SL(n, ?) (or, more generally, let G be a connected, noncompact, simple Lie group). For any compact Lie group K, it is easy to find a compact manifold M, such that there is a volume-preserving, connection-preserving, ergodic action of G on some smooth, principal K-bundle P over M. Can M can be chosen independent of K? We show that if M = H/Λ is a homogeneous space, and the action of G on M is by translations, then P must also be a homogeneous space H′Λ′. Consequently, there is a strong restriction on the groups K that can arise over this particular M.      15. The inflation class operator      A. Christie  Q. Wang  S. L. Wismath 《Algebra Universalis》2007,56(1):107-118 We consider the inflation class operator, denoted by F, where for any class K of algebras, F(K) is the class of all inflations of algebras in K. We study the interaction of this operator with the usual algebraic operators H, S andP, and describe the partially-ordered monoid generated by H, S, P andF (with the isomorphism operator I as an identity). Received February 3, 2004; accepted in final form January 3, 2006.      16. Derivations on ideals in commutative AW*-algebras      V. I. Chilin  G. B. Levitina 《Siberian Advances in Mathematics》2014,24(1):26-42 LetA be a commutativeAW*-algebra.We denote by S(A) the *-algebra of measurable operators that are affiliated with A. For an ideal I in A, let s(I) denote the support of I. Let Y be a solid linear subspace in S(A). We find necessary and sufficient conditions for existence of nonzero band preserving derivations from I to Y. We prove that no nonzero band preserving derivation from I to Y exists if either Y ? Aor Y is a quasi-normed solid space. We also show that a nonzero band preserving derivation from I to S(A) exists if and only if the boolean algebra of projections in the AW*-algebra s(I)A is not σ-distributive.      17. The Topology of Limits of Direct Systems Induced by Maps      Ivan Ivanšić  Leonard R. Rubin 《Mediterranean Journal of Mathematics》2014,11(4):1261-1273 Let Z, H be spaces. In previous work, we introduced the direct system X induced by the set of maps between the spaces Z and H. Now we will consider the case that X is induced by possibly a proper subset of the maps of Z to H. Our objective is to explore conditions under which X = dirlim X will be T1, Hausdorff, regular, completely regular, pseudo-compact, normal, an absolute co-extensor for some space K, or will enjoy some combination of these properties.      18. Coverings, the graded center and Hochschild cohomology      Edward L. Green 《Journal of Pure and Applied Algebra》2008,212(12):2691-2706       19. Note on bounds for multiplicities      Tim Römer 《Journal of Pure and Applied Algebra》2005,195(1):113-123 Let S=K[x1,…,xn] be a polynomial ring and R=S/I be a graded K-algebra where I⊂S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.      20. Operator spaces with prescribed sets of completely bounded maps      Timur Oikhberg 《Journal of Functional Analysis》2005,224(2):296-315 Suppose A is a dual Banach algebra, and a representation π:A→B(?2) is unital, weak* continuous, and contractive. We use a “Hilbert-Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ?2⊗?2, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A)⊗I?2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K1(CB(X)) contains Z2 as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.

A harmonic analysis approach to essential normality of principal submodules

Guo and the second author have shown that the closure [I] in the Drury-Arveson space of a homogeneous principal ideal I in C[z₁,…,zn] is essentially normal. In this note, the authors extend this result to the closure of any principal polynomial ideal in the Bergman space. In particular, the commutators and cross-commutators of the restrictions of the multiplication operators are shown to be in the Schatten p-class for p>n. The same is true for modules generated by polynomials with vector-valued coefficients. Further, the maximal ideal space XI of the resulting C^?-algebra for the quotient module is shown to be contained in Z(I)∩∂Bn, where Z(I) is the zero variety for I, and to contain all points in ∂Bn that are limit points of Z(I)∩Bn. Finally, the techniques introduced enable one to study a certain class of weight Bergman spaces on the ball.     

Subdifferentials of a minimum time function in Banach spaces

In general Banach space setting, we study the minimum time function determined by a closed convex set K and a closed set S (this function is simply the usual Minkowski function of K if S is the singleton consisting of the origin). In particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function.     

Sequence entropy and mixing

The relationship between sequence entropy and mixing is examined. Let T be an automorphism of a Lebesgue space X, $\text{L}$₀ denote the set of all partitions of X possessing finite entropy, and $\text{S}$ denote the set of all increasing sequences of positive integers. It is shown that: (1) T is mixing /a2 sup_{A ? B}h_A(T, α) = H(α) for all B∈I and α∈Z₀. (2) T is weakly mixing /a2 sup_Ah_A(T, α) = H(α) for all α∈Z₀. (3) If T is partially mixing with constant $\text{c (1 ?}\text{1}\text{e}\text{< c < 1)}$, then sup_{A ? B}h_A(T, α) > cH(α) for all B∈I and nontrivial α∈Z₀. (4) If sup_{A ? B}h_A(T, α) > 0 for all B∈I and nontrivial α∈Z₀, then T is weakly mixing.     

Affine hypersurfaces admitting a pointwise symmetry

An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T _p M) for all p ∈ M, which preserves (pointwise) the affine metric h, the difference tensor K (resp. the cubic form) and the affine shape operator S. In this paper, we deal with locally strongly convex affine hypersurfaces of dimension three. First we solve an algebraic problem. We determine the non-trivial stabilizers G of the pair (K, S) under the action of SO(3) on a Euclidean vector space (V, h) and find a representative (canonical form of K and S) of each SO(3)/G-orbit. Then, we classify hypersurfaces admitting a pointwise G-symmetry for all non-trivial stabilizers G (apart of Z ₂). Besides well-known hypersurfaces (for Z ₂ × Z ₂ we get the locally homogeneous hypersurface (x ₁ ?) 1/2x ₃2 (x ₂ ?) 1/2x ₄2) = 1) we obtain e.g. warped products of two-dimensional affine spheres (resp. quadrics) and curves.     

Ergodic Actions of Semisimple Lie Groups on Compact Principal Bundles

Let G = SL(n, ?) (or, more generally, let G be a connected, noncompact, simple Lie group). For any compact Lie group K, it is easy to find a compact manifold M, such that there is a volume-preserving, connection-preserving, ergodic action of G on some smooth, principal K-bundle P over M. Can M can be chosen independent of K? We show that if M = H/Λ is a homogeneous space, and the action of G on M is by translations, then P must also be a homogeneous space H′Λ′. Consequently, there is a strong restriction on the groups K that can arise over this particular M.     

The inflation class operator

We consider the inflation class operator, denoted by F, where for any class K of algebras, F(K) is the class of all inflations of algebras in K. We study the interaction of this operator with the usual algebraic operators H, S andP, and describe the partially-ordered monoid generated by H, S, P andF (with the isomorphism operator I as an identity). Received February 3, 2004; accepted in final form January 3, 2006.     

Derivations on ideals in commutative AW*-algebras

LetA be a commutativeAW*-algebra.We denote by S(A) the *-algebra of measurable operators that are affiliated with A. For an ideal I in A, let s(I) denote the support of I. Let Y be a solid linear subspace in S(A). We find necessary and sufficient conditions for existence of nonzero band preserving derivations from I to Y. We prove that no nonzero band preserving derivation from I to Y exists if either Y ? Aor Y is a quasi-normed solid space. We also show that a nonzero band preserving derivation from I to S(A) exists if and only if the boolean algebra of projections in the AW*-algebra s(I)A is not σ-distributive.     

The Topology of Limits of Direct Systems Induced by Maps

Let Z, H be spaces. In previous work, we introduced the direct system X induced by the set of maps between the spaces Z and H. Now we will consider the case that X is induced by possibly a proper subset of the maps of Z to H. Our objective is to explore conditions under which X = dirlim X will be T₁, Hausdorff, regular, completely regular, pseudo-compact, normal, an absolute co-extensor for some space K, or will enjoy some combination of these properties.     

Note on bounds for multiplicities

Let S=K[x₁,…,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I⊂S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.     

Operator spaces with prescribed sets of completely bounded maps

Suppose A is a dual Banach algebra, and a representation π:A→B(?₂) is unital, weak* continuous, and contractive. We use a “Hilbert-Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ?₂⊗?₂, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A)⊗I_?2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K₁(CB(X)) contains Z₂ as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.