Numerical study of the transition to chaos of a buoyant plume from a two-dimensional open cavity heated from below

The transition to a chaotic plume from a two-dimensional (2D) open cavity heated from below has been investigated using numerical simulation. A large range of Rayleigh numbers (Ra) pertaining to an aspect ratio of A = 1, and Prandtl number (Pr) of Pr = 0.71 (air) is numerically investigated. It is shown that there exists a complex transition of the plume from a steady reflection symmetry to a chaotic flow with a sequence of bifurcations. As the Rayleigh number increases, the plume from the open cavity undergoes a supercritical pitchfork bifurcation from a steady reflection symmetry to a steady reflection asymmetry flow. Once the Rayleigh number exceeds 7 × 10³, the plume appears as a distinct flapping namely, a Hopf bifurcation, and then as a distinct puffing. The chaotic plume has the possibility to exhibit an alternate appearance of flapping and puffing in the event the Rayleigh number exceeds 8 × 10⁴. Moreover, the dynamics of the plume from the open cavity is discussed, and the dependence on the Rayleigh number of heat and mass transfer of the plume from the open cavity is quantified.     

Numerical study of formation of a series of bubbles at a submerged orifice

Bubble formation from a submerged orifice is widely applied in bio-process and chemical reaction systems. In this study, the effects of different orifice diameters and contact angles in Period-I and Period-II regimes are studied systematically on a 2D axisymmetric domain. Simulation results are presented from the formation of the first bubble and explained by means of the surrounding fluid field, bubble interaction, and bubble aspect ratio.The orifice diameter is varied from 0.6 mm to 3 mm. The numerical results show that the detachment time of all bubbles remains constant (in time) for smaller orifice diameters (d_a ≤ 1.5 mm), while the detachment time of the first bubble is different from the rest of the bubbles for larger orifice diameters (d_a ≥ 2 mm), which is due to the different surrounding flow field. Contact angles from 60° to 165° are considered for the gas flow rates in the regime of bubble pairing, and it is observed that the bubble detachment time decreases when the contact angle increases, and it converges to a constant value when the contact angle is larger than 135°. In addition, the transition from period doubling to deterministic chaos (in which there is a variable number of bubbles within each period) is observed.A new scenario of inserting a submerged tube upward into the liquid is considered and compared to the previous cases. It is observed that when the tube is vertically inserted into the liquid, the bubble detachment time is even smaller because of higher influence from the surrounding liquid field, leading to a different phenomenon from the non-inserted tube cases.     

Matrix transformations of summability domains of generalized matrix methods in Banach spaces

The necessary and sufficient conditions for a matrix M to be a transform from the summability domain of generalized matrix method A into the summability domain of another generalized matrix method B are proved. The elements of Mare continuous linear operators from a Banach space X into another Banach space Y, and the elements of A and B are continuous linear operators from X into X and from Y into Y, respectively. As a special case there are considered the case when A is the generalized Riesz method.     

Embeddings and hyperplanes of the Lie incidence geometry of type E7,1

This paper continues a program to show that for most of the standard Lie incidence geometries, all geometric hyperplanes arise from a necessarily absolutely universal embedding, by addingE _7,1 to the list. It follows from [5, 12] that any projective embedding of this point line geometry is a homomorphic image of the one afforded by the 56-dimensional module for the groupE ₇(K).This work was supported by a grant from the National Science Foundation.     

On divisors of Lucas and Lehmer numbers

Let u _n be the nth term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of u _n which is of the form n exp(log n/104 log log n). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erd?s from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.     

Estimating the value of a linear functional of a distribution density

One considers the problem of the estimation of a continuous linear functional L(f) in L_p at an unknown point f from a sample X₁,..., X_n of size n from a distribution with density f.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 45–59, 1986.     

An Infinite Presentation of the Torelli Group

In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all “separating twists”, all “bounding pair maps”, and all “commutators of simply intersecting pairs” and whose relations all come from a short list of topological configurations of these generators on the surface. Aside from a few obvious ones, all of these relations come from a set of embeddings of groups derived from surface groups into the Torelli group. In the process of analyzing these embeddings, we derive a novel presentation for the fundamental group of a closed surface whose generating set is the set of all simple closed curves.     

Construction of Euclidian Monopoles

This paper describes a procedure for the construction of monopoleson three-dimensional Euclidean space, starting from their rationalmaps. A companion paper, ‘Euclidean monopoles and rationalmaps’, to appear in the same journal, describes the assignmentto a monopole of a rational map, from CP¹ to a suitable flagmanifold. In describing the reverse direction, this paper completesthe proof of the main theorem therein. A construction of monopoles from solutions to Nahm's equations(a system of ordinary differential equations) has been well-knownfor certain gauge groups for some time. These solutions arehard to construct however, and the equations themselves becomeincreasingly unwieldy when the gauge group is not SU(2). Here, in contrast, a rational map is the only initial data.But whereas one can be reasonably explicit in moving from Nahmdata to a monopole, here the monopole is only obtained fromthe rational map after solving a partial differential equation. A non-linear flow equation, essentially just the path of steepestdescent down the Yang-Mills-Higgs functional, is set up. Itis shown that, starting from an ‘approximate monopole’- constructed explicitly from the rational map - a solutionto the flow must exist, and converge to an exact monopole havingthe desired rational map. 1991 Mathematics Subject Classification:53C07, 53C80, 58D27, 58E15, 58G11.     

Version of the Grothendieck theorem for subspaces of analytic functions in lattices

A version of Grothendieck’s inequality says that any bounded linear operator acting from a Banach lattice X to a Banach lattice Y acts from X(ℓ²) to Y (ℓ²) as well. A similar statement is proved for Hardy-type subspaces in lattices of measurable functions. Namely, let X be a Banach lattice of measurable functions on the circle, and let an operator T act from the corresponding subspace of analytic functions X_A to a Banach lattice Y or, if Y is also a lattice of measurable functions on the circle, to the quotient space Y/Y_A. Under certain mild conditions on the lattices involved, it is proved that T induces an operator acting from X_A(ℓ²) to Y (ℓ²) or to Y/Y_A(ℓ2), respectively. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 5–16.     

Topological realizations of ortho-projection graphs

We discuss the production of ortho-projection graphs from alternating knot diagrams, and introduce a more general construction of such graphs from “splittings” of closed, non-orientable surfaces. As our main result, we prove that this new topological construction generates all ortho-projection graphs. We present a minimal example of an ortho-projection graph that does not arise from a knot diagram, and provide a surface-splitting that realizes this graph.     

Fusions of Character Tables II: p-Groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. In a previous article, we gave examples of Camina pairs that fuse from abelian groups. In this article, we give more general examples of Camina triples that fuse from abelian groups. We use this result to give an example of a group which fuses from an abelian group, but which has a subgroup that does not. We also give an example of a powerful 2-group which does not fuse from an abelian group and of a regular 3-group which does not fuse from an abelian group.     

Branching of symmetric solution of molecular field equation

Branching of a completely symmetric solution of the molecular field equation is investigated under the assumption that the configuration space of a molecule is a certain groupQ. The scheme can be used to describe orientationally ordered and space-modulated structures. Phase transitions from an isotropic liquid to the nematic, cholesteric, and conic phases are considered. The relationship between the phase-transition characteristics (order, Curie point, symmetry of the newly formed phase) and the potential of the intermolecular interaction is established. Investigation of the isolated isotropic-liquid—two-axis-nematic phase transition point has shown that this is not always a second-order transition. The possible existence of a one-dimensional modulated structure having smectic-A symmetry is predicted. The corresponding phase transition follows a scenario different from that of the isotropic-liquid—smectic-A phase transition.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 2, pp. 246–258, February, 1992.     

Effect of cycle asymmetry on the cyclic strength of polymers

Cyclic strength diagrams have been obtained for polymethyl methacrylate and kaprolon at different degrees of loading asymmetry (from a pure compression to a pure tension cycle) on the interval from –100 to +80° C at 2·10³ and 3·10⁵ cycles.M. I. Kalinin Leningrad Polytechnic Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 948–951, September–October, 1968.     

Method of averaging in the problem of stability of motion of a body suspended from a string

On the basis of the Bogolyubov-Mitropol’skii method of averaging, we study the problem of stability of the vertical rotation of a body suspended from a string. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 467–475, April, 2007.     

On a system of partial differential equations of Monge-Kantorovich type

We consider a system of PDEs of Monge-Kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors. The existence of a solution of such system (in a regular open domain Ω⊂Rn), whose construction is based on an asymmetric Minkowski distance from the boundary of Ω, was already established in [G. Crasta, A. Malusa, The distance function from the boundary in a Minkowski space, Trans. Amer. Math. Soc., submitted for publication]. In this paper we prove that this solution is essentially unique. A fundamental tool in our analysis is a new regularity result for an elliptic nonlinear equation in divergence form, which is of some interest by itself.     

Minimax estimation of a variance

The nonparametric problem of estimating a variance based on a sample of sizen from a univariate distribution which has a known bounded range but is otherwise arbitrary is treated. For squared error loss, a certain linear function of the sample variance is seen to be minimax for eachn from 2 through 13, exceptn=4. For squared error loss weighted by the reciprocal of the variance, a constant multiple of the sample variance is minimax for eachn from 2 through 11. The least favorable distribution for these cases gives probability one to the Bernoulli distributions.     

Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces

On a complete metric space X, we solve the problem of constructing an algorithm (in general, nonunique) of successive approximations from any point in space to a given closed subsetA. We give an estimate of the distance from an arbitrary initial point to the corresponding limit points. We consider three versions of the subset A: (1) A is the complete preimage of a closed subspace H under a mapping from X into the metric space Y; (2) A is the set of coincidence points of n (n > 1) mappings from X into Y; (3) A is the set of common fixed points of n mappings of X into itself (n = 1, 2, …). The problems under consideration are stated conveniently in terms of a multicascade, i.e., of a generalized discrete dynamical system with phase space X, translation semigroup equal to the additive semigroup of nonnegative integers, and the limit set A. In particular, in case (2) for n = 2, we obtain a generalization of Arutyunov’s theorem on the coincidences of two mappings. In case (3) for n = 1, we obtain a generalization of the contraction mapping principle.     

On the expressive power of CNF formulas of bounded tree- and clique-width

We study representations of polynomials over a field K from the point of view of their expressive power. Three important examples for the paper are polynomials arising as permanents of bounded tree-width matrices, polynomials given via arithmetic formulas, and families of so called CNF polynomials. The latter arise in a canonical way from families of Boolean formulas in conjunctive normal form. To each such CNF formula there is a canonically attached incidence graph. Of particular interest to us are CNF polynomials arising from formulas with an incidence graph of bounded tree- or clique-width.We show that the class of polynomials arising from families of polynomial size CNF formulas of bounded tree-width is the same as those represented by polynomial size arithmetic formulas, or permanents of bounded tree-width matrices of polynomial size. Then, applying arguments from communication complexity we show that general permanent polynomials cannot be expressed by CNF polynomials of bounded tree-width. We give a similar result in the case where the clique-width of the incidence graph is bounded, but for this we need to rely on the widely believed complexity theoretic assumption #P?FP/poly.     

Periodic behaviour of generalized threshold functions

It is shown that, for a function Δ from {0, 1}ⁿ to {0, 1}ⁿ whose components from a symmetric set of threshold functions the repeated application of Δ, leads either to a fixed point or to a cycle of length two.     

Construction of polynomial superalgebras

In the algebraA, defined equationally, we construct a basis (standard) with with a suitable law of multiplication and we give the transition formula from this basis to the original one so thatA has the structure of a Z₂-graded space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 257–261, February, 1990.