Bénard Polygons

We prove that if a neighborhood of a polygonal region admits a two-dimensional eigenfunction of the Laplacian, having a nonzero eigenvalue and such that its normal derivative vanishes on all the bounding edges, then the polygonal region must be a union of complete pieces of a tiling of the plane by congruent rectangles, or by congruent (45°, 45°, 90°) or (30°, 60°, 90°) triangles. Hydrodynamically, this means that during critical convection in a horizontal fluid layer uniformally heated from below, the mere occurrence of one arbitrary closed vertical polygonal fluid surface across which there is no transportation of fluid automatically guarantees the presence of one of the usual special convection patterns. In addition it shows that linear convection theory seldom predicts a regular fluid pattern: e.g., for the case of a triangular container having angles substantially different from (45°, 45°, 90°), (30°, 60°, 90°), (60°, 60°, 60°) or (30°, 30°, 120°), it predicts that the convection cells not touching the boundary, if any, should be noticeably nonpolygonal. We also consider a nonlinear generalization and the noneuclidean analogues of such polygons.     

Fundamental regular semigroups with inverse transversals

Let C be a regular semigroup with an inverse transversal C° and let C be generated by its idempotents. Following W. D. Munn and T. E. Hall’s idea, in this paper, a fundamental regular semigroup T _C,C° with an inverse transversal T _C,C° ^° is constructed such that the following holds. For any regular semigroup S with an inverse transversal S° and 〈E(S)〉 = C, C° = C ∩ S°, there is a homomorphism φ from S to T _C,C° such that the kernel of φ is the maximum idempotent-separating congruence on S, and φ satisfies: (1) φ| _C is a homomorphism from C onto 〈E(T _C,C°)〉 ; (2) φ| _S° is a homomorphism from S° to T _C,C° °. In particular, S is fundamental if and only if S is isomorphic to a full subsemigroup of T _C,C°. Our fundamental regular semigroup T _C,C° is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation—although not the usual composition—is defined by means of composition.     

Operator valued weights in von Neumann algebras,I

An operator valued weight is a kind of generalized conditional expectation from a von Neumann algebra M to a sub von Neumann algebra N. If T is a n.f.s. (normal, faithful, semifinite) operator valued weight from M to N, and φ is a n.f.s. weight on N, then φ ° T defines a n.f.s. weight on M. Our main result is that σ_t^φ°T extends σ_t^φ, and that the map φ → φ ° T preserves cocycle Radon Nikodym derivatives.     

Periodicities in the series of cyclonic disturbances over India

The time series of yearly occurrence frequency of cyclonic disturbances over the Indian region for 1891–1970 is subjected to power spectrum analysis by the maximum entropy method (MEM) of Burg. Earlier analyses by the conventional Blackman-Tukey method had indicated periodicities atT= 2·3-2·5, 5 and 30–45 years at about 90–95% significance level. The present analysis shows peaks atT = 1·2, 2·4, 2·9, 3·7, 4·7, 6·4, 7·4, 9·2, 10·5, 20 and 40–60 years but the significance level may not be better than the earlier method. When data are analysed separately for first half and latter half, periodicities observed are quite different, the earlier half revealingT= 1· 2, 2·3, 2·9, 4·7, 9·0 and 23 years and the latter half revealingT= 1·2, 2·1, 2·4, 3·8, 6·7 and 21 years. Thus, a change of pattern before and after about 1920 is indicated. However, the power in the various peaks is small and a random origin cannot be ruled out. An analysis of several artificially produced samples of pseudo-Gaussian random white noise of the same mean, standard deviation and length (80 values) as the above physical samples indicated that similar peaks could occur by pure chance also. Physically, T=l·2 years corresponds to the Chandler’s wobble and other peaks could be harmonics of the sunspot cycleT= 11 years or double sunspot cycleT = 22 years. However, no peak is noticed at or nearT= 11 years whereasT near 22 years is noticed.     

Characterization of joint ergodicity for non-commuting transformations

The study of jointly ergodic transformations, begun in [2] and [1], is continued. The main result is that, ifT ₁,T ₂, …,T _s are arbitrary measure preserving transformations of a probability space (X, ?,μ), then , if and only if the following conditions are satisfied:

1. T ₁×T ₂×…×T _s is ergodic.
2. .

     

A Unifying Approach to Weyl Type Theorems for Banach Space Operators

Weyl type theorems have been proved for a considerably large number of classes of operators. In this paper, by introducing the class of quasi totally hereditarily normaloid operators, we obtain a theoretical and general framework from which Weyl type theorems may be promptly established for many of these classes of operators. This framework also entails Weyl type theorems for perturbations f(T + K), where K is algebraic and commutes with T, and f is an analytic function, defined on an open neighborhood of the spectrum of T + K, such that f is non constant on each of the components of its domain.     

Tournois infinis et critiques

Given a tournament T=(V,A), a subset X of V is an interval of T provided that for every a,b∈X and x∈V?X, (a,x)∈A if and only if (b,x)∈A. For example, ?, {x} (x∈V) and V are intervals of T, called trivial intervals. A tournament all the intervals of which are trivial is called indecomposable; otherwise, it is decomposable. An indecomposable tournament T=(V,A) is then said to be critical if for each x∈V, T(V?{x}) is decomposable and if there are x≠y∈V such that T(V?{x,y}) is indecomposable. We introduce the operation of expansion which allows us to describe a process of construction of critical and infinite tournaments. It follows that, for every critical and infinite tournament T=(V,A), there are x≠y∈V such that T and T(V?{x,y}) are isomorphic. To cite this article: I. Boudabbous, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Elastoplastic stress analysis of thick laminated metal-matrix composite plates by the finite-element method

The elastoplastic stress state of a laminated stainless-steel-fiber-reinforced aluminum-matrix plates, with or without a hole, subjected to a pressure on their top is examined by using the finite-element method. The analysis is carried out for three layouts: (0/90/0/90)_s, (45/-45/45/-45)_s, and (30/60/30/60)_s. The Newton-Raphson method is used to solve the nonlinear problem. The distributions of equivalent stresses and the plastic zones of the plates without a hole and with a hole of various diameters are determined. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 4, pp. 531–544, July–August, 2006.     

On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

To each once-punctured-torus bundle, Tφ, over the circle with pseudo-Anosov monodromy φ, there are associated two tessellations of the complex plane: one, Δ(φ), is (the projection from ∞ of) the triangulation of a horosphere at ∞ induced by the canonical decomposition into ideal tetrahedra, and the other, CW(φ), is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through ∞. In this paper, we fully describe the relation between Δ(φ) and CW(φ).     

Super-simple, Pan-orientable, and Pan-decomposable BIBDs with Block Size 4 and Related Structures

Let v, k, and μ be positive integers. A tournament T of order k, briefly k-tournament, is a directed graph on k vertices in which there is exactly one directed edge between any two vertices. A (v, k, λ = 2μ)-BIBD is called T-orientable if for each of its blocks B, it is possible to replace B by a copy of T on the set B so that every ordered pair of distinct points appears in exactly μ k-tournaments. A (v, k, λ = 2μ)-BIBD is called pan-orientable if it is T-orientable for every possible k-tournament T. In this paper, we continue the earlier investigations and complete the spectrum for (v, 4, λ = 2μ)-BIBDs which possess both the pan-orientable property and the pan-decomposable property first introduced by Granville et al. (Graphs Comb 5:57–61, 1989). For all μ, we are able to show that the necessary existence conditions are sufficient. When λ = 2 and v > 4, our designs are super-simple, that is they have no two blocks with more than two common points. One new corollary to this result is that there exists a (v, 4, 2)-BIBD which is both super-simple and directable for all v ≡ 1, 4 (mod 6), v > 4. Finally, we investigate the existence of pan-orientable, pan-decomposable (v, 4, λ = 2μ)-BIBDs with a pan-orientable, pan-decomposable (w, 4, λ = 2μ)-BIBD as a subdesign; here we obtain complete results for λ = 2, 4, but there remain several open cases for λ = 6 (mostly for v < 4w), and the case λ = 12 still has to be investigated.     

COMPLETE CONVERGENCE THEORY OF THE CONTACT PROCESS ON Td × Z1

The author considers the contact process on a branching plane T_d × Z, which is the product of a regular tree T_d and the line Z. It is shown that above the second critical point, the complete convergence theory holds.     

Real class sizes

Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle [?T,T] × [a, b] is asymptotically between cT and CT², with positive constants c and C. We also supply with conditions (in terms of the spectral measure) under which the variance grows asymptotically linearly with T, as a quadratic function of T, or has intermediate growth. The results are compared with known results for real stationary Gaussian processes and other models.     

The cycle space of an embedded graph

Let G be a connected graph with edge set E embedded in the surface ∑. Let G° denote the geometric dual of G. For a subset d of E, let τd denote the edges of G° that are dual to those edges of G in d. We prove the following generalizations of well-known facts about graphs embedded in the plane. (1) b is a boundary cycle in G if and only if τb is a cocycle in G°. (2) If T is a spanning tree of G, then τ(E/T) contains a spanning tree of G°. (3) Let T be any spanning tree of G and, for e ? E/T, let T(e) denote the fundamental cycle of e. Let U ∪ E/T. Then τU is a spanning tree of G° if and only if the set of face boundaries, less any one, together with the set {T(e); e ? E/(T ∪ U)} is a basis for the cycle space of G.     

Strongly dependent theories

We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e.g., the first order theory of the p-adics is strongly dependent. The most notable result is: if |A| + |T| ≤ µ, I ? ? and |I|≥?_|T|+(µ), then some J ? I of cardinality µ⁺ is an indiscernible sequence over A.     

Convex n-ominoes

Unit squares having their vertices at integer points in the Carresian plane are called cells. A connected union of n distinct cells having no finite cut set is an n-omino. Two n-ominoes are the same if one is mapped onto the other by a translation of the plane. An n-omino is convex if the cells in each row and each column to an a connected strip. When viewed from a distance, most convex n-ominoes resemble rods tilted 45° from the vertical with horizontal (and vertical) thickness roughly equal to 2.37597. If c(n) denotes the number of convex n-ominoes, then c(n) ～ fyⁿ, where y = 2.30914 and f = 2.67564. (It is understood that all constants are accurate to within -$\text{1}\text{2}$ in the last place.)     

Features of electron density and temperature in the 500–3500 km region of the plasmasphere—(1) Dawn and dusk sectors

Dawn-dusk features of the plasmasphere are examined for intervals in February and September 1969, using electrostatic probe data ofN _e andT _e from the ISIS-I satellite. Clear plasmatrough formation is seen in the vicinity of 70° geomagnetic latitude in both dawn and dusk sectors in the 1500–3500 km region, but the plasmatrough is absent in the altitude range 500–1500 km. The plasmatrough minimum near 70°φ exhibits no asymmetry between dawn and dusk sectors in its latitudinal position. TheT _e peak associated with the plasmatrough is more pronounced in the dawn sector. DawnN _e is less than duskN _e, but dawnT _e exceeds duskT _e. The influence of processes in the magnetosphere in causing these features is examined.     

Spectral radius of Hadamard product versus conventional product for non-negative matrices

We prove an inequality for the spectral radius of products of non-negative matrices conjectured by X. Zhan. We show that for all n×n non-negative matrices A and B, ρ(A°B)?ρ((A°A)(B°B))^1/2?ρ(AB), in which ° represents the Hadamard product.     

Distributive equations of implications based on nilpotent triangular norms

In this paper, we explore the distributive equations of implications, both independently and along with other equations. In detail, we consider three classes of equations. (1) By means of the section of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T(y, z)) = T(I(x, y), (x, z)) based on a nilpotent triangular norm T and an unknown function I, which indicates that there are no continuous solutions satisfying the boundary conditions of implications. Under the assumptions that I is continuous except the vertical section I(0, y), y ∈ [0, 1), we get its complete characterizations. (2) We prove that there are no solutions for the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, I(y, z)) = I(T(x, y), z). (3) We obtain the sufficient and necessary conditions on T and I to be solutions of the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, y) = I(N(y), N(x)).     

Supergroups for six series of hyperbolic simplex groups

There are investigated supergroups of some hyperbolic space groups with simplicial fundamental domain. Six simplices considered here from [9] are collected in families F9 (T ₂₃, T ₆₄), F10 (T ₂₁, T ₄₉, T ₆₁), F29 (T ₃₄). All of them have the same symmetry by half-turn h, with axis through the midpoints of edges A ₀ A ₁ and A ₂ A ₃. Since that isometry identifies pairs of points, if a supergroup with such smaller fundamental domain exists, it is of index 2. At the side pairings of T ₃₄ this half-turn implies additional reflections, equal parameters 2a = 6b, and leads to Family 2, considered in [9]. Other possibility to find supergroups is when the simplices have vertices out of the absolute. In that case we can truncate them by polar planes of the vertices and the new polyhedra are fundamental domains of richer groups.     

Cyclicity of Vectors with Orbital Limit Points for Backward Shifts

On a separable, infinite dimensional Banach space X, a bounded linear operator T : X → X is said to be hypercyclic, if there exists a vector x in X such that its orbit Orb(T, x) = {x, Tx, T ² x, …} is dense in X. In a recent paper (Chan and Seceleanu in J Oper Theory 67:257–277, 2012), it was shown that if a unilateral weighted backward shift has an orbit with a single non-zero limit point, then it possesses a dense orbit, and hence the shift is hypercyclic. However, the orbit with the non-zero limit point may not be dense, and so the vector x inducing the orbit need not be hypercyclic. Motivated by this result, we provide conditions for x to be a cyclic vector.