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Asymmetries in the state of a cold-rolling mill cause lateral displacements of the metal strip, which sometimes result in a major breakdown of the mill. Non-linear model equations, previously employed, are reduced to a linear form via a first-order expansion of the plastic reduction equations, appropriate for small displacements. The continuity equations, normally applied at the entry and exit of a stand, are replaced by simpler equations applied on the neutral line. These simplifications provide explicit formulae for equilibrium lateral displacements, which lead to more general conclusions. Moreover, these formulae do not involve the compliance and reaction of the mill, but relate the displacements to strip conditions and roll-gap asymmetries alone. A strong coupling between the stands and a strong upstream influence are demonstrated. The results agree with that of a non-linear model and with data from a laboratory mill. Increasing entry tension to the first stand, up to moderate levels, has the beneficial effect of reducing the size of the lateral displacements. However, the model predicts that abnormally high strip tension or low friction can lead to divergent displacements, which would crash the mill. The results offer the potential for better control and should be of interest to mill operators.

Previous calculations of transport coefficients of particle systems in two or more dimensions have used computer simulations or various approximations, like those implicit in the Boltzmann equation or in the standard probability theory of diffusion. Here an exactly solved model of crystal growth in three dimensions is transformed into a 2D model of steady particle transport. Then the exact, many-body stationary distribution and exact transport coefficients can be found as well as correlation functions. The model is highly asymmetric. In the transverse direction, particles are strongly attracted, so that long transverse rows of particles tend to form at low temperature. Kinks in these rows are rare, so the transverse positions of kinks effectively move continuously on a large distance scale.

We present a new methodology for the mathematical analysis of 3D paper structure and apply it to a model that can be simulated on a computer. We rigorously derive upper and lower bounds for fiber contact areas, and derive an approximation that is very close to values calculated from the simulations. The method involves the stochastic geometry and combinatorics of large numbers of randomly located sets, which quantify the interactions between fibers. The main calculation involves a sum resembling a partition function with many-body interactions of all orders.

A new Markov process is introduced, describing growth or spread in two dimensions, via the aggregation of particles or the filling of cells. States of the process are configurations of part of the boundary of the growing aggregate, and transitions are captures or escapes of single particles. For suitably chosen transition rates, the process is dynamically reversible, leading to an explicit stationary distribution and a statistical description of the boundary. The growth rate is calculated and growth behaviour described. Different asymptotic relations between transition rates lead to different growth patterns or regimes. Besides the regimes familiar in polymer crystal growth, several new ones are described. The aggregate can have a porous structure resembling thin solid films deposited from vapour. Two measures of porosity, one for the boundary and one for the bulk, are calculated. The process is relevant to growing colonies of bacteria or the like, to the spread of epidemics and grass or forest fires, and to voter models.

Crystal growth at the molecular level can be described by statistical models for the surface structure and for the capture and escape of molecules. For some models the steady growth can be analysed exactly, leading to exact expressions for growth rate and surface characteristics. These are expressed in terms of the dimensions and angles of the surface and the rates of molecular events. Two-dimensional models describe the growth of a single layer on the surface of a three-dimensional crystal and approximate the growth of lamellar polymer crystals. The well known growth regimes for polymer crystals can be derived exactly. A two-dimensional model that can generate a rougher edge and a porous structure provides some insights into fast growth and into deposition of thin films from the vapour. A terrace–step–kink growth model in three dimensions can be exactly solved. For some models that have not been solved exactly, one can find conditions on the rates of molecular capture and escape that ensure steady, stable crystal growth.

A classical theorem of Hardy, Littlewood, and Polya on rearrangements of functions is used to prove the equivalence of a class of variational problems. As a consequence, solutions are shortest paths and can be computed numerically via quadratic programming. Applications include solar cars and reservoirs.

A new Markov process describing crystal growth in three dimensions is introduced. States of the process are configurations of the crystal surface, which has a terrace-edge-kink structure. The states are continuous along edges but discrete across edges, in accordance with the very different rates for the two types of captures of particles. Stationary distributions, describing steady crystal growth, are found in general. To our knowledge, these are the first examples of stationary distributions for layered crystal growth in three dimensions. The steady growth rate and other quantities are obtained explicitly for two interacting edges. For many interacting edges, growth behavior is determined (a) in various asymptotic regimes including thermodynamic limits, (b) via simulations, and (c) using series (cluster) expansions in the slope of the surface, the first three coefficients being computed. The theoretical growth rates show a marked dependence on surface dimensions. This may contribute to the size dependence and dispersion in the observed growth rate of small crystals.

Deoxyribonucleic acid of cells undergoing apoptosis is cleaved by a calcium-dependent endonuclease into oligonucleosomal-sized fragments. These fragments can be labeled using the enzyme terminal deoxynucleotidyl transferase so that the cells can be visualized immunohistochemically. Few investigators have evaluated this method in disease processes of the human central nervous system. The Tdt-mediated dUTP-biotin nick end labeling (TUNEL) technique has been investigated in preliminary studies of a variety of pathologic conditions of the human brain (e.g., gliomas, traumatic brain injury, Parkinson's disease, Parkinson's-Alzheimer's complex, multisystem atrophy, striatonigral degeneration). We focus, however, on Huntington's disease (HD) because of the availability of well-characterized pathological stages for study, and also because of the neurodegenerative diseases studied to date, only Huntington's disease revealed significant and consistent labeling with this method. This implies a possibly unique nature to the mechanism of cell death in Huntington's disease compared to the other neurodegenerative diseases studied. TUNEL+ neurons were found in Grade 1-4 HD neostriatum, while labeled astrocytes were found predominantly in the Grade 1 and 2 cases studied to date. TUNEL+ cells were also found in glioblastoma multiforme and traumatic brain injury. We conclude that while there appear to be several limitations associated with this technique, it may be useful for identifying both apoptosis and necrosis in certain neuropathological conditions.

Using a Markov rate-process model, exact expressions are found for the steady growth rate of an edge of a two-dimensional crystal in terms of the numberM of particles along the edge, the height difference (or number of permanent steps)K along the edge, the nucleation rate α, and the speed α+ν of movement of steps. The familiar growth regimes can be identified with asymptotic regimes for the parametersK, (v/α)1/2, andM. From a mathematical viewpoint, there are seven basic regimes, of which the known physical regimes are special cases.

We investigate properties of solid-on-solid models for crystal growth, involving general microscopic rates of capture of atoms by the crystal surface and of escape of atoms. The rates in this Markov process influence the stability of the growing surface. We prove, for various different ranges of the rate parameters, stability (i.e., ergodicity) and instability (i.e., nullity) of the growth process. Symmetry properties of the process, such as reversibility, dynamic reversibility, and reflection invariance, are proved or disproved under various conditions. We give a measure of surface smoothness that distinguishes between stable and unstable growth.

A sudden loss of lateral stability (strip track-off) is a serious operational problem in cold rolling of steel that often leads to catastrophic consequences such as mill crashes and roll damage. This paper studies the causes of instability in lateral motion of a metal strip in a single span between the uncoiler and the first stand of a cold rolling mill. A mathematical model of strip lateral dynamics that includes a rolling mill model and a simplified physically based model of strip buckling is formulated. The numerical analysis of the model reveals that in the presence of strip buckling, the lateral motion becomes unstable once a critical level of asymmetry in rolling conditions is exceeded. The critical level of asymmetry increases with the increase in the tension applied to the strip. Below this critical level of asymmetry, a sudden transition to unstable motion occurs if a critical lateral deviation is reached. This paper demonstrates that the buckling of the strip can be responsible for lateral instability. The results of this paper explain the sudden onset of instability observed in metal rolling and the stabilizing effect of the tension applied to the strip, known in rolling practice.

We consider solid-on-solid models of two- and three-dimensional crystal growth described by Markov rate processes whose states are representations of the entire crystal surface and whose transitions are captures of single atoms by this surface or escapes of atoms. Under natural conditions on the transition probability rates, we prove ergodicity of the Markov process. This implies the existence of states of steady growth and statistically stationary surface structure. Microscopic surface instabilities do not therefore occur under such conditions. For two dimensions, our conditions are very general; for three dimensions, they are less general.

A plantation is modelled in terms of discs of random size centred on the points of a two-dimensional lattice, together with rules for partitioning the union of these discs. The resulting sets are functions of random sequences, and some properties of these functions are deduced. Results are given which relate various properties of measurements on a plantation to measurements of individual plants and of plants in isolation. These include limit theorems for complete plantations and for samples. They also lead to a rigorous demonstration of certain well-known empirical relations between typical measurements and survival frequencies and to some new relations which are amenable to test. The model is a formulation of computer simulations which have had success in describing competition in plantations, but are too costly for routine forest management.

We investigate the correlation functions and the critical exponentν for Ising models and spherical models ond-dimensional hypercubic lattices in the limitd → ∞ Our results include a generalization of the Ornstein-Zernike theory, and an alternative explanation of the crossover phenomenon described by Baker.

The problem of partitioning the union of disks, which arises in theories of competition between plants, is analysed from a general point of view. Some simple and natural assumptions such as connectedness, invariance under translation, rotation and change of scale, and monotonicity and continuity properties of the partition sets are imposed. From these it is proved that the partition is specified by giving a boundary curve between two disks with the very explicit form: R α x –∣ r – x ∣ α = R α y –∣ r – y ∣ α , α ≥ 1, where R x , R y are the radii of the disks, x , y are their centres and r is a point on the boundary. This includes the Johnson–Mehl construction( α =1), the common chord ( α = 2) and the perimeter of the larger disc ( α →∞). Larger disks dominate smaller ones if and only if α > 1, so that ƞ = 1 — 1/ α , 0 ≤ ƞ < 1, is a natural index measuring the strength of domination. It is proved that a partition implies the existence of three-dimensional regions which are such that the projection on the plane of their exposed upper surface yields the partition (the reverse implication is trivial). Thus the partition implies, in a sense, the existence of a related three-dimen­sional crown shape. The partition rules make severe restrictions on these shapes. Making the further assumption that crowns of different sizes are ‘similar’ results in the specific crown surface ( z /( βR ) α + (∣ r – x ∣ R ) α = 1, where z > 0, α ≤ 1 and β > 0. The obvious special cases are ƞ = 0 (cones, for conifers), ƞ = 1/2 (ellipsoids of revolution, used by other authors for broadleaf trees) and ƞ = 1 (cylinders or umbrellas, perhaps for flat topped plants). The results apply equally to N- dimensional spheres, while some apply to more general shapes. The disk partitions also generate partitions of (R 2 , giving a natural prescription for ‘growing space’ which is simpler than those of some other authors.

AbsrtactIn this, the first of three papers, we examine conditions, derived previously, which specify the equilibrium solutions of an adjustment process for N players engaged in a game with continuous (in fact, continuously differentiable) payoff functions, where each player's strategy is to choose a single real number. It is equivalent to the basic form of quantity-variation competition between N firms. The conditions are related to a new optimum which takes account of the ability of firms, or coalitions of firms, to discipline another firm that tries to increase its own profit. Closely related optima are also introduced and analysed. The new optima occupy N-dimensional regions in the strategy space, and contain the optima of Cournot, Pareto, von-Neumann and Morgenstern, and Nash as special cases.

The new optima and equilibria discussed in the preceding two papers are compared with the results of bargaining experiments between two and three players performed by Fouraker and Siegel. Experiments where players have complete or incomplete information are considered. There is clear evidence that the new optima are operating, and that traditional optima–Cournot, Pareto and competitive (threat)–are less satisfactory in explaining the course of the whole bargaining process.

A new perturbation expansion for the hydrostatic density distribution of a fluid in a steep field gradient is presented. It is a power series in the scale γ of an external potential ψ (γx), and is asymptotically valid as γ→0, although convergence is not proved. Significant corrections to the conventional hydrostatic equations are found near the critical point of a fluid, even in the gravitational field.

We derive asymptotic forms for the distributions of k-point scan statistics as the interval L under study becomes infinite, while k and the window length are held fixed. In the Poisson case the intensity is also held fixed. In the uniform case the number of points N becomes infinite and N/L tends to a limit, representing a limiting intensity. These results are made explicit for k = 3, and in the Poisson case provide approximations which are typically accurate to six or seven figures, even for small L.

A Markov rate process whose transitions are captures and escapes of single atoms from the edge of a two-dimensional crystal is introduced. The stochastic equilibrium states of this process describe steady crystal growth, crystal-fluid equilibrium, and steady crystal decrescence. Exact and asymptotic growth rates are found. This extends recent results which dealt only with capture events. One application is to the growth of lamellar crystals from polymers.

For the class of continuous games where σi and fi {σi, φ(σ1, …, σN)} are the strategy of and payoff to player i for i = 1, …, N, it is proved that the set of weak type I optima defined in Paper I conicide with the set of solution of a matrix condition. The latter condition restricts the equilibrium solutions of an adjustment process. Numerical results for N = 2 and N = 3 indicate that the set of all equilibrium solutions coincides with the above sets. The optima of types I to IV from Paper I are described fairly completely for the given class of games.

Diffusion laws and diffusion coefficients are given for random walks on lattices whose scattering probabilities differ from site to site. Models are considered in which the direction of the particle after scattering depends on its direction before scattering.

We introduce and analyse a new and special case of the Lorentz gas or the Wind-Tree model of Ehrenfest. This model (which has a number-theoretic character) is shown to exhibit normal diffusion, the diffusion coefficient C(ϱ) being obtained in closed form as a function of the density ϱ. The function ϱ C(ϱ) turns out to be an entire analytic function of ϱ, in spite of the model's non-classical high density behaviour. A “collision expansion” which is appropriate for high densities is also given.

For a large electricity grid comprising many units (plants) of various types, such as coal, oil, nuclear, hydro, etc., with known unreliabilities (outage rates) we study the optimal (i.e. the cheapest) total capacity, or numbers, of each type of unit. Existing treatments of the problem involve numerical methods and approximations of unknown accuracy. For a range of cases, we find explicit solutions. This extends the known explicit solutions, which are confined to completely reliable units. The cases we analyse are (I) a demand (load) which has a shifted Rayleigh distribution—a good approximation to the real load-duration curve—with some restriction on reliability (big units are more reliable) and (II) an exponential load distribution—which is unrealistic—with no restrictions on reliability. In both cases, the solutions reduce to transformed versions of the exact solutions for totally reliable units and, like the latter, can be exhibited by means of a cost polygon.

A special case of the adjustment process for competing firms, introduced elsewhere, is analysed. It is proved that all equilibrium states are characterised by the vanishing of the Jacobian of the payoff functions of the market game, and that under suitable initial conditions the process is a contraction mapping, implying convergence to equilibrium.

Some lattice versions of the wind‐tree model of Ehrenfest are introduced and studied. They include two versions in which overlapping of tree particles is forbidden, a third in which it is allowed, and a fourth in which it is allowed but results in a finite repulsive force. In every case it is found that the mean‐square displacement Δ(t) of the wind particles at time t is bounded independently of t at sufficiently high density of the trees. This is in sharp contrast with the Einstein relation Δ(t) = O(t) as t → ∞, which might be expected to hold at low densities. Randomization of the initial velocity of a wind particle is also shown to occur in a certain sense, and upper bounds on a recurrence time are obtained at high density. On the other hand, it is shown that thermalization does not occur even at low densities.

A nu -dimensional system of particles with two-body potential q(r)+ gamma nu K( gamma r) is considered. Various correlation functions are defined and evaluated in the limit gamma to 0. Some of the results describe two-phase states, and others are closely related to the Ornstein- Zernike theory.

We examine the limiting free energy density $$a(\varrho ,0 + ) \equiv \mathop {\lim }\limits_{\gamma \to 0} a(\varrho ,\gamma )$$ of a classical system of particles with the two-body potentialq(r)+γ v K (γr), at density ϱ inv dimensions. Starting from a variational formula fora(ϱ, 0 + ), obtained in Part I of these papers, we obtain a new upper bound ona(ϱ, 0 + ) given by $$a(\varrho ,0 + ) \mathbin{\lower.3ex\hbox{$\buildrel

A theoretical discussion of helicon wave propagation in non-resistive plasmas is presented. The boundary conditions are derived and applied in cylindrical geometry to give results consistent with experiment. The problem is more complicated in spherical geometry because of the non-separability of the wave equation. However, by separating in cylindrical coordinates and rearranging in a series of spherical waves, solutions are obtained in the form of an infinite set of integral equations. Approximate solutions are obtained and the first few natural frequencies are identified. Wave fields for the principal mode are calculated and displayed graphically.

The free energy and crystal structure of some particular cases of a cell model of melting are obtained. These include (i) a class of one-dimensional cases with interactions extending to many neighbouring cells, (ii) the plane hexagonal, bcc and triangular lattices with nearest-neighbour interactions, (iii) a case of the plane, square lattice with first- and second-neighbour (diagonal) interactions, and others. The solutions for the simple cubic lattices in all dimensions were obtained in a previous paper (abstr. A46569 of 1971). It is shown that some of the 2, 3, etc dimensional cases are closely related ('isomorphic') to one-dimensional cases.

We consider the thermodynamic pressurep(μ, γ) of a classical system of particles with the two-body interaction potentialq(r)+γvK(γr), where υ is the number of space dimensions, γ is a positive parameter, and μ is the chemical potential. The temperature is not shown in the notation. We prove rigorously, for hard-core potentialsq(r) and for a very general class of functionsK(s), that the limit γ→0 of the pressurep(μ, γ) exists and is given by where the limit and the supremum can be interchanged. Here ℛ is a certain class of nonnegative, Riemann integrable functions,D is a cube of volume |D|, anda0(ϱ) is the free energy density of a system withK=0 and density ϱ. A similar result is proved for the free energy.

A new cell model for classical particle systems is presented and analyzed. In this model the particles are confined to congruent, interconnected, cubic cells of volume ω centered on the points of a cubic lattice with lattice spacing 1/γ. The particles interact via a 2‐body potential of the form q(r) + ω−1K(γr). The paper deals with the limiting form of this model in which the cells are very large but their separation is much larger. The free energy density is defined by a(ρ,T)≡lim lim ω→∞lim lim γ→0a(ρ,T,γ,ω), where a(ρ, T, γ, ω) is the free energy density at density ρ, temperature T, and arbitrary γ and ω. For a very general class of functions q and K, it is proved that a(ρ, T) is given by a variational principle. For a certain class of functions K (including K ≤ 0), a(ρ, T) is given by the Lebowitz‐Penrose generalization of the van der Waals‐Maxwell theory. For a different class of functions K the system has crystalline states. When K is chosen so that only particles in nearest‐neighbor cells interact a...

Asymptotic series are derived for two integrals using a Gaussian identity and Laplace’s method, demonstrating an improvement over earlier methods.

Three lattice versions of the wind‐tree model of Ehrenfest are studied. It is shown that various moments, including the recurrence time and the Cesaro limits limT→∞ (1/T)∑t=1TΔ(t) of the mean‐square displacement Δ(t) and of the one‐particle distribution ρ(t, x) at time t, are analytic functions of the reciprocal of the fugacity of the trees, or equivalently of the deviation 1‐ρ of the density ρ of the trees from their close packing density 1, in certain disks in the complex plane. Two of the models were considered in Paper I, but the third is new.

Wound healing normally follows an uninterrupted course but, for the purposes of description, it can be considered in four phases -- hemostasis, acute inflammation, repair and consolidation. Arrest of healing may occur at any phase and lead to a number of clinical problems. Repair cannot begin until the phase of acute inflammation has begun to subside; this fact is of special importance in clinical surgery. Wound healing in gastrointestinal tract and abdominal incisions is emphasized because healing problems are common in these areas and failure to heal results in morbidity and a regrettable mortality.

Bacteriophage T4 is neutralized by univalent antibody fragment Fab′ with apparent two-target neutralization kinetics. By the use of conditional lethal (amber) T4 mutants, it was shown that partially neutralized preparations of phages adsorb to bacteria with kinetics suggesting a mixture of fast and slow adsorbers; the ratio of fast to slow phages fits well that predicted by Poisson distribution of phages with, respectively, no (zero hit) and one (one hit) Fab′ complexed. A portion of the infectivity of one-hit phages seems due to phage–Fab′ dissociation, but the larger portion (80%) of the infectivity is due to weak adsorption of phages complexed still with one Fab′. It was shown that antibody to tail fiber is responsible for this effect. Phage adsorption heterogeneity was a possible explanation of kinetics obtained using phages which had not been treated with Fab′.

A deterministic mathematical model is developed for competition between two plant species (or genotypes) in the presence of a host-specific pathogen capable of attacking one of the species only. The complexity of the plant-plant-pathogen interaction precluded precise mathematical description of all possible long-term outcomes. However, extensive computer simulation shows that stability is not an inevitable consequence of the differential interaction, rather, the outcome depends greatly upon the growth rates and relative competitive abilities of the plant species and on the transmission efficiency of the pathogen and its effect on individual host plants. The model confirmed the general validity of previous intuitive arguments but clearly indicated that the occurrence of inflected yield curves is not a sufficient indication in itself of long-term stability.