The Nakayama Property of a Module and Related Concepts

Three related properties of a module are investigated in this article, namely the Nakayama property, the Maximal property, and the S-property. A module M has the Nakayama property if 𝔞M = M for an ideal 𝔞 implies that sM = 0 for some s ∈ 𝔞 + 1. A module M has the Maximal property if there is in M a maximal proper submodule, and finally, M is said to have the S-property if S ^?1 M = 0 for a multiplicatively closed set S implies that sM = 0 for some s ∈ S.     

A Remark on Perfect Gaussian Elimination of Symmetric Matrices

Let M be a symmetric matrix with non-zero diagonal entries. A result of Golumbic [3, 5] states that if M has a perfect elimination scheme then it also has a perfect elimination scheme with the additional property that all pivots are chosen along the main diagonal. However, the proof given by Golumbic seems to be incomplete. In the present note, we refine Golumbic's proof, thus obtaining a complete version of it.     

Wallman compactification and matrices of zeroes and ones

Let M be an m by n matrix (where m and n are any finite or infinite cardinals) such that the entries of M are 0's or 1's and M contains the zero row $\text{0}$ and the rows of M are closed under coordinatewise multiplication. We prove that M can be extended to an m by n′ ? n matrix M′ such that the entries of M′ are 0's or 1's and M′ contains the zero row 0?′ and the extension preserves the zero products. Moreover, the newly introduced columns (if any) are pairwise distinct. Furthermore, if E′ is any set of rows of M′ with the property that for every finite subset of rows r′_i of E′ there exists j < n′ such that r′_ij = 1, then every subset of rows of E′ has the same property. We rephrase this by saying that if E′ has the finite intersection property then E′ has a nonempty intersection. We also show (this time by Zorn's lemma) that there exists an extension of M with all the abovementioned properties such that the extension preserves products sums, complements and the newly introduced columns (if any) are pairwise distinct in a stricter sense. In effect, our result shows that the classical Wallman compactification theorem can be formulated purely combinatorially requiring no introduction of any topology on n.     

On the spectral synthesis property and its application to partial differential equations

LetM be a (n?1)-dimensional manifold inR ⁿ with non-vanishing Gaussian curvature. Using an estimate established in the early work of the author [4], we will improve the known result of Y. Domar on the weak spectral synthesis property by reducing the smoothness assumption upon the manifoldM. Also as an application of the method, a uniqueness property for partial differential equations with constant coefficients will be proved, which for some specific cases recovers or improves Hörmander's general result.     

Géométrie des espaces de Banach ayant des approximations de l'identité contractantes

Let a, c ≥ 0 and let B be a compact set of scalars. We introduce property M^* (a, B, c) of Banach spaces X which is a geometric property of Banach spaces generalizing property (M^*) due to Kalton. Using M^*(a, B, c) with max ¦B¦ + c > 1, we characterize intrinsically a large class of shrinking approximations of the identity, including those related to M-, u-, and h-ideals of compact operators. We also show that the existence of these approximations of the identity is separably determined. As an application, we study ideals of compact and approximable operators. In particular, this provides an alternative unified and easier approach to the theories of M-, u-, and h-ideals of compact operators.     

On the property of kelley in the hyperspace and Whitney continua

In this paper, we introduce the notion of property [K]¹ which implies property [K], and we show the following: Let X be a continuum and let ω be any Whitney map for C(X). Then the following are equivalent. (1) X has property [K]¹. (2) C(X) has property [K]¹. (3) The Whitney continuum ω⁻¹(t) (0⩽t<ω(X)) has property [K]¹.As a corollary, we obtain that if a continuum X has property [K]¹, then C(X) has property [K] and each Whitney continuum in C(X) has property [K]. These are partial answers to Nadler's question and Wardle's question ([10, (16.37)] and [11, p. 295]).Also, we show that if each continuum X_n (n=1,2,3,…) has property [K]¹, then the product ∏X_n has property [K]¹, hence C(∏X_n) and each Whitney continuum have property [K]¹. It is known that there exists a curve X such that X has property [K], but X×X does not have property [K] (see [11]).     

Diffeomorphisms with C 1-stably average shadowing

Let M be a closed smooth manifold M, and let f : M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f . We show that if f has the C1-stably average shadowing property on Λ, then Λ admits a dominated splitting.     

A corollary of the b-function lemma

Let X be an algebraic variety, f a regular function, ${j:U\hookrightarrow X}$ the complement to the locus of vanishing of f, and M a holonomic D-module on U. Consider the D _U [s]-module ${M\otimes ``{f^{s}}''}$ . The goal of this note is to describe all D <sub>X</sub> [s] submodules ${N\hookrightarrow j_*(M\otimes ``{f^{s}}'')}$ such that ${j^*(N)\simeq M\otimes ``{f^{s}}''}$ .     

A Characterization of Maximal Monotone Operators

It is shown that a set-valued map $M:\mathbb{R}^{q} \rightrightarrows \mathbb{R}^{q}$ is maximal monotone if and only if the following five conditions are satisfied: (i) M is monotone; (ii) M has a nearly convex domain; (iii) M is convex-valued; (iv) the recession cone of the values M(x) equals the normal cone to the closure of the domain of M at x; (v) M has a closed graph. We also show that the conditions (iii) and (v) can be replaced by Cesari’s property (Q).     

On the maximal order of cyclicity of antisymmetric directed graphs

A directed graph G without loops or multiple edges is said to be antisymmetric if for each pair of distinct vertices of G (say u and v), G contains at most one of the two possible directed edges with end-vertices u and v. In this paper we study edge-sets M of an antisymmetric graph G with the following extremal property: By deleting all edges of M from G we obtain an acyclic graph, but by deleting from G all edges of M except one arbitrary edge, we always obtain a graph containing a cycle. It is proved (in Theorem 1) that if M has the above mentioned property, then the replacing of each edge of M in G by an edge with the opposite direction has the same effect as deletion: the graph obtained is acyclic. Further we study the order of cyclicity of G (= theminimalnumberofedgesinsuchasetM) and the maximal order of cyclicity in an antisymmetric graph with given number n of vertices. It is shown that for n < 10 this number is equal to the maximal number of edge-disjoint circuits in the complete (undirected) graph with n vertices and for n = 10 (and for an infinite set of n's) the first number is greater than the latter.     

Fixed point properties of semigroups of nonexpansive mappings

We prove that if H is a Hilbert space then the Schatten (trace) class operators on H has the weak^∗ fixed point property for left reversible semigroups. This answered positively a problem raised by A.T.-M. Lau. We also prove that if M is a finite von Neumann algebra then any nonempty bounded convex subset of the non-commutative L¹-space associated to M that is compact for the measure topology has the fixed point property for left reversible semigroups.     

L 1-Approximation and Finding Solutions with Small Support

In this paper, we study an interesting property of L ¹-approximation. For many subspaces M, there exist ?? ^?(M)>0 with the following property: if f vanishes off a set of measure at most ?? ^?(M), then the zero function is a best L ¹-approximant to f from M. We explain this phenomenon, provide estimates for ?? ^?(M) in many cases, and present some open questions.     

On the identities of the Grassmann algebras in characteristicp>0

In this note we exhibit bases of the polynomial identities satisfied by the Grassmann algebras over a field of positive characteristic. This allows us to answer the following question of Kemer: Does the infinite dimensional Grassmann algebra with 1, over an infinite fieldK of characteristic 3, satisfy all identities of the algebraM ₂(K) of all 2×2 matrices overK? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemer's question. All this allows us to obtain some information about the identities satisfied by the algebraM ₂(K) over an infinite fieldK of positive odd characteristic, and to conjecture bases of theidentities ofM ₂(K).     

Régularité Besov-Orlicz du temps local brownien

Let (B_t, t ε [0, 1]) be a linear Brownian motion starting from 0 and denote (L_t(x), t ≥0, x ∈ ℝ) its local time. We prove that, for all t > 0, a.s. the function xL_t (x) belongs to the Besov-Orlicz space B^½M_{2, ∞} with M₂(x)=e^|x|2 -1 and doesn't belong a.s. to B^½,0M_{2, ∞}.     

On relations between weak approximation properties and their inheritances to subspaces

It is shown that for the separable dual X^∗ of a Banach space X, if X^∗ has the weak approximation property, then X^∗ has the metric weak approximation property. We introduce the properties W^∗D and MW^∗D for Banach spaces. Suppose that M is a closed subspace of a Banach space X such that M^⊥ is complemented in the dual space X^∗, where for all m∈M}. Then it is shown that if a Banach space X has the weak approximation property and W^∗D (respectively, metric weak approximation property and MW^∗D), then M has the weak approximation property (respectively, bounded weak approximation property).     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

Codes with the identifiable parent property for multimedia fingerprinting

Let \({\mathcal {C}}\) be a q-ary code of length n and size M, and \({\mathcal {C}}(i) = \{\mathbf{c}(i) \ | \ \mathbf{c}=(\mathbf{c}(1), \mathbf{c}(2), \ldots , \mathbf{c}(n))^{T} \in {\mathcal {C}}\}\) be the set of ith coordinates of \({\mathcal {C}}\). The descendant code of a sub-code \({\mathcal {C}}^{'} \subseteq {\mathcal {C}}\) is defined to be \({\mathcal {C}}^{'}(1) \times {\mathcal {C}}^{'}(2) \times \cdots \times {\mathcal {C}}^{'}(n)\). In this paper, we introduce a multimedia analogue of codes with the identifiable parent property (IPP), called multimedia IPP codes or t-MIPPC(n, M, q), so that given the descendant code of any sub-code \({\mathcal {C}}^{'}\) of a multimedia t-IPP code \({\mathcal {C}}\), one can always identify, as IPP codes do in the generic digital scenario, at least one codeword in \({\mathcal {C}}^{'}\). We first derive a general upper bound on the size M of a multimedia t-IPP code, and then investigate multimedia 3-IPP codes in more detail. We characterize a multimedia 3-IPP code of length 2 in terms of a bipartite graph and a generalized packing, respectively. By means of these combinatorial characterizations, we further derive a tight upper bound on the size of a multimedia 3-IPP code of length 2, and construct several infinite families of (asymptotically) optimal multimedia 3-IPP codes of length 2.     

Geometry of D’Atri spaces of type k

A Riemannian n-dimensional manifold M is a D’Atri space of type k (or k-D’Atri space), 1 ≤ k ≤ n ? 1, if the geodesic symmetries preserve the k-th elementary symmetric functions of the eigenvalues of the shape operators of all small geodesic spheres in M. Symmetric spaces are k-D’Atri spaces for all possible k ≥ 1 and the property 1-D’Atri is the D’Atri condition in the usual sense. In this article we study some aspects of the geometry of k-D’Atri spaces, in particular those related to properties of Jacobi operators along geodesics. We show that k-D’Atri spaces for all k = 1, . . ., l satisfy that ${{\rm{tr}}(R_{v}^{k})}$ , v a unit vector in TM, is invariant under the geodesic flow for all k = 1, . . ., l. Further, if M is k-D’Atri for all k = 1, . . ., n ? 1, then the eigenvalues of Jacobi operators are constant functions along geodesics. In the case of spaces of Iwasawa type, we show that k-D’Atri spaces for all k = 1, . . ., n ? 1 are exactly the symmetric spaces of noncompact type. Moreover, in the class of Damek-Ricci spaces, the symmetric spaces of rank one are characterized as those that are 3-D’Atri.     

On the lower limits of maxima and minima of wiener process and partial sums

Let M ⁺ (t) and ?M ^? (t) be the maximum and minimum of a Wiener process on the interval (O,t). This paper gives an integral test for P(M ⁺(t)?(t)t i.o.)=0 or 1. The case of i.i.d. random variables is also treated here. If a(t)=b(t), then our result gives Chung's law of the iterated logarithm [5], while b(t)=∞ corresponds to Hirsch's theorem [9]. Finally, a converse to Chung's LIL is given.     

Small tight sets of hyperbolic quadrics

An x-tight set of a hyperbolic quadric Q ⁺(2n + 1, q) can be described as a set M of points with the property that the number of points of M in the tangent hyperplanes of points of M is as big as possible. We show that such a set is necessarily the union of x mutually disjoint generators provided that x ≤ q and n ≤ 3, or that x < q, n ≥ 4 and q ≥ 71. This unifies and generalizes many results on x-tight sets that are presently known, see (J Comb Theory Ser A 114(7):1293–1314 [1], J Comb Des 16(4):342–349 [5], Des Codes Cryptogr 50:187–201 [4], Adv Geom 4(3):279–286 [8], Bull Lond Math Soc 42(6):991–996 [11]).