Growth of number of periodic orbits of one family of skew product maps

In this article we introduce a one-parameter family of skew product (G_t)_{t ∈ [?ε, ε]} maps exhibiting a heterodimensional cycle such that the number of isolated periodic orbits inside it has not super-exponential growth. The dynamics in the central direction of the maps G_t is described by a one-parameter family of system of iterated functions.     

C 1-stably ℒ p shadowable homoclinic classes

Let p be a hyperbolic saddle of diffeomorphism f on closed manifold M and H(p, f) be the homoclinic class associated with it. In this article, we introduce the notion of C ¹-stably ? _p shadowing and prove that if f is C ¹-stably ? _p shadowable on a homoclinic class H(p, f) then, H(p, f) has a dominated splitting. Moreover, we prove that if f is C ¹-stably Lipschitz ? _p shadowable on H(p, f) and H(p, f)-germ of f is expansive then the homoclinic class is hyperbolic.     

Fracture mechanics and notch sensitivity

A diagram valid for the analysis of the fatigue limit of cracks and notches centred in an infinite plate was recently proposed by the authors of the present work with the aim to make explicit the bridging at the fatigue limit between defect sensitivity (correlated to the length parameter a₀, according to El Haddad–Topper–Smith's definition) and notch sensitivity (correlated to a*, where a* is a particular notch depth corresponding to the intersection between the ΔK_th and Δσ₀/K_t curves). The expression being valid, defect sensitivity and notch sensitivity were seen as two sides of the same medal. Such a diagram is now extended to finite size components by simply introducing the shape factor α commonly used in fracture mechanics. The obtained critical defect size is termed a_D, which is a material and geometry dependent parameter, in order to distinguish it from a₀, which is a material parameter. As a consequence the critical notch depth a_N is introduced, such that . This results in the proposal of a ‘universal’ diagram able to summarize experimental data related to different materials, geometry and loading conditions. The diagram, the validity of which is checked by means of several results available in the literature, is applied both to the interpretation of the scale effect and to the surface finishing effect.     

On Methods of Constructing Sets of Mutually Orthogonal Latin Squares Using a Computer. I

This paper deals with the problem of finding sets of mutually orthogonal Latin squares of order 4t (where 4t – 1 is a prime power) based on orthogonal mappings of a group. For the group G we take the module G(2, 2t) whose elements are vectors (a ₁, a ₂,) where a ₁ is a residue class (mod 2) and a ₂ is a residue class (mod 2t), the addition being defined by (a ₁, a ₂) + (b ₁, b ₂) = (c ₁, c ₂) where a ₁ + b ₁ = C ₁ (mod 2) and a ₂ + b ₂ = c ₂ (mod 2t). Then the search for orthogonal mappings is materially simplified by using a configuration based on the balanced incomplete block design with parameters v = b = 4t – 1, r = k = 2t – 1, λ = t – 1. Using this method, two sets of five mutually orthogonal Latin squares of order 12 were obtained.     

First hyperbolic times for intermittent maps with unbounded derivative

We establish some statistical properties of the hyperbolic times for a class of non-uniformly expanding dynamical systems. The maps arise as factors of area preserving maps of the unit square via a geometric baker’s map-type construction, exhibit intermittent dynamics, and have unbounded derivatives. The geometric approach captures various examples from the literature over the last 30 years. The statistics of these maps are controlled by the order of tangency (linked to a single parameter α, where 0 < α < ∞) that a certain ‘cut function’ makes with the boundary of the square. Previously, a direct Young tower construction has been used to obtain optimal correlation decay rates of O(n^?1/α) for Hölder observables and all values of the parameter α. A central limit theorem (CLT) is obtained when 0 < α < 1.

The asymptotics of a natural hyperbolic time for this family of maps are analysed via the same Young tower. By using a large deviations result of Melbourne and Nicol, we prove that the first hyperbolic time is integrable if and only if the parameter satisfies 0 < α < 1. Furthermore, within this restricted range of parameters, concentration inequalities recently established by Chazottes and Gouëzel imply sharp O(n^?1/α) bounds on the tail distribution of first hyperbolic times. As shown by Alves, Viana, and others, knowledge of the tail distribution of the hyperbolic times leads to upper bounds on the rate of decay of correlations and derivation of a CLT. Comparing to the results obtained directly for this family of maps, the latter estimates via hyperbolic times are suboptimal, even over the restricted range of parameters 0 < α < 1.     

Abundance of elliptic dynamics on conservative three-flows

We consider a compact three-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C ¹-residual (dense G _δ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flow-setting counterpart of Newhouse's Theorem 1.3 (S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 (1977), pp. 1061–1087). Our result follows from two theorems, the first one says that if Λ is a hyperbolic invariant set for some class C ¹ zero divergence vector field X on M, then either X is Anosov, or else Λ has empty interior. The second one says that, if X is not Anosov, then for any open set U ? M there exists Y arbitrarily close to X such that Y ^t has an elliptical closed orbit through U.     

Cooperative Atomic Behaviour and Oscillator Formation in a Squeezed Vacuum

Abstract

Time-dependent (numerical) results are presented for super-radiant behaviour in the Dicke model of N _a = 2, 3 atoms in a broad band squeezed vacuum. This concerns the fluctuations and the intensity of the fluorescent radiation as well as the atomic population inversion of the system with atoms initially in an atomic coherent state. In the steady state, and in the N _a → ∞, we show that the ‘atomic’ Dicke model behaves like a ‘giant quantum oscillator’, in which the number of excited atoms asymptotically approaches the average number of photons in the resonant mode of the squeezed vacuum, just as in the thermally driven case.     

The indentation size effect of sintered Fe/3.3 wt-%Cu + CnHm measured by Vickers scale

ABSTRACT

The aim of the submitted work is to study the influence of applied loads (test forces) between 0.09807 and 2.9421?N on the measured value of micro-hardness of sintered Fe/3.3?wt-%Cu?+?C_nH_m. The Indentation Size Effect (ISE), i.e. the influence of the load on the micro-hardness is expected. The results were evaluated by Meyer’s index n, t-test, and non-parametric tests. The applied load has a statistically significant influence on the type and size of the ISE. Tested sintered material shows ‘normal’ ISE with Meyer’s index n?=?1.7588.     

Crystal growth nucleation and Fermi energy equalization of intrinsic spherical nuclei in glass-forming melts

Abstract

The energy saving resulting from the equalization of Fermi energies of a crystal and its melt is added to the Gibbs free-energy change ΔG_2ls associated with a crystal formation in glass-forming melts. This negative contribution being a fraction ε _ls(T) of the fusion heat is created by the electrostatic potential energy ?U₀ resulting from the electron transfer from the crystal to the melt and is maximum at the melting temperature T_m in agreement with a thermodynamics constraint. The homogeneous nucleation critical temperature T₂, the nucleation critical barrier ΔG_2ls?/k_BT and the critical radius R?_2ls are determined as functions of ε_ls(T). In bulk metallic glass forming melts, ε_ls(T) and T₂ only depend on the free-volume disappearance temperature T_0l, and ε_ls(T_m) is larger than 1 (T_0l>T_m/3); in conventional undercooled melts ε_ls(T_m) is smaller than 1 (T_0l>T_m/3). Unmelted intrinsic crystals act as growth nuclei reducing ΔG_2ls?/k_BT and the nucleation time. The temperature-time transformation diagrams of Mg₆₅Y₁₀Cu₂₅, Zr_41.2Ti_13.8Cu_12.5Ni₁₀Be_22.5, Pd₄₃Cu₂₇ Ni₁₀P₂₀, Fe₈₃B₁₇ and Ni melts are predicted using classic nucleation models including time lags in transient nucleation, by varying the intrinsic nucleus contribution to the reduction of ΔG_2ls?/k_BT. The energy-saving coefficient ε _nm(T) of an unmelted crystal of radius R_nm is reduced when R_nm ?R?_2ls; ε_nm is quantified and corresponds to the first energy level of one s-electron moving in vacuum in the same spherical attractive potential ?U₀ despite the fact that the charge screening is built by many-body effects.     

Privacy-Preserving Quantum Two-Party Geometric Intersection

Privacy-preserving computational geometry is the research area on the intersection of the domains of secure multi-party computation (SMC) and computational geometry. As an important field, the privacy-preserving geometric intersection (PGI) problem is when each of the multiple parties has a private geometric graph and seeks to determine whether their graphs intersect or not without revealing their private information. In this study, through representing Alice’s (Bob’s) private geometric graph G_A (G_B) as the set of numbered grids S_A (S_B), an efficient privacy-preserving quantum two-party geometric intersection (PQGI) protocol is proposed. In the protocol, the oracle operation O_A (O_B) is firstly utilized to encode the private elements of S_A =(a₀,a₁,…,a_M-1) (S_B =(b₀,b₁,…,b_N-1)) into the quantum states, and then the oracle operation O_f is applied to obtain a new quantum state which includes the XOR results between each element of S_A and S_B. Finally, the quantum counting is introduced to get the amount (t) of the states |a_i⊕b_j| equaling to |0|, and the intersection result can be obtained by judging t >0 or not. Compared with classical PGI protocols, our proposed protocol not only has higher security, but also holds lower communication complexity.     

Accelerated life-test experiments on low-power rectifier diodes

Low-power rectifier diodes were stressed by different forward currents I_F, reverse voltages U_R, and ambient temperatures _a for a time interval (0, t*) in order to analyse the resulting distributions. Owing to the limitations in the test time, t*, the gathering of comprehensive information about their performance requires high stress levels without exceeding either physical limits or the limits for initiating thermal runaway. First, the results of lifetime experiments related to the middle stress region were described by a Weibull distribution with constant shape parameter and stress-dependent scale parameter. This model, however, did not fit sufficiently the typical features of the real-life distribution. Therefore a non-parametric measure called the average initial reliability in (O,t*) was defined using the experimental results. Its stress dependence was estimated. The results are guiding figures of stress conditions both for accelerated tests and for circuits with high demands on reliability.     

Accelerated Life Testing for a Class of Linear Hazard Rate Type Distributions

Accelerated life testing for distributions with hazard rate functions of the form r(t) = Ag(t) + Bh(t) are considered. Let V ₁, …, V _k be stress levels larger than V ₀—the stress level under normal conditions [V ₀ > 0]—and let a(v) be a nondecreasing function on (0, ∞). We discuss a generalization of the common accelerated models (the power rule model and the Arrhenius model) by assuming that the hazard rate under the stress level V, is of the form (a(V _t )) ^P (Ag(t) + Bh(t)). The maximum likelihood estimators of A, B and P for complete and censored samples are studied. The estimation procedure reduces to a solution of one equation with one unknown parameter. The estimation procedure under the assumption of aging is also described. The asymptotic variance-covariance matrix is given.     

A strange attractor in the unfolding of an orbit-flip homoclinic orbit

An orbit-flip homoclinic orbit o of a vector field defined on R 3 is a homoclinic orbit to an equilibrium point for which the one-dimensional unstable manifold of the equilibrium point is connected to the one-dimensional strong stable manifold. In this paper, we show that in a generic unfolding of such a homoclinic orbit, there exists a positive Lebesgue measure set in the parameter space for which the corresponding vector field possesses a suspended strange attractor. To prove the result, we propose a rescaling in the phase space and a blowing up in the parameter space, and in the new system, we show that the Poincaré return map is close to the map (x,y) M (1 - āx2,bx) when b is close to 0. With a similar rescaling/blowing up, we also obtain a similar result in the case where o is an inclination-flip homoclinic orbit.     

Microscopic Scenario for x = 1/8 Anomaly in La2 − x Sr x CuO4

We adopt a t ₁-t ₂-t ₃-J-G model for explanation of x = 1/8 anomaly in La_{2 ? x} Sr _x CuO₄ family compound. The calculated charge susceptibility shows a maximum near Q = (π, π) at intermediate temperatures and near (π, π/2) as temperature approaches zero, in agreement with neutron scattering experiments. Coulomb repulsion G between the first neighbors turns out to be the source of Charge Density Waves (CDW) in narrow band t ^eff ₁, t ^eff ₂, t ^eff ₃ < G. For physically realistic hopping values we obtain the CDW amplitude e _Q = x. The in-phase domain structure as a candidate for “stripe” picture is proposed.     

Microscopic Scenario for x = 1/8 Anomaly in La2 ? xSrxCuO4

We adopt a t ₁-t ₂-t ₃-J-G model for explanation of x = 1/8 anomaly in La_{2 – x} Sr _x CuO₄ family compound. The calculated charge susceptibility shows a maximum near Q = (, ) at intermediate temperatures and near (, /2) as temperature approaches zero, in agreement with neutron scattering experiments. Coulomb repulsion G between the first neighbors turns out to be the source of Charge Density Waves (CDW) in narrow band t ^eff ₁, t ^eff ₂, t ^eff ₃ < G. For physically realistic hopping values we obtain the CDW amplitude e _Q = x. The in-phase domain structure as a candidate for stripe picture is proposed.     

Nonlinear dynamics of structures assembled by bolted joints

Summary The nonlinear transfer behaviour of an assembled structure such as a large lightweight space structure is caused by the nonlinear influence of structural connections. Bolted or riveted joints are the primary source of damping compared to material damping, if no special damping treatment is added to the structure. Simulation of this damping amount is very important in the design phase of a structure. Several well known lumped parameter joint models used in the past to describe the dynamic transfer behaviour of isolated joints by Coulomb friction elements are capable of describing global states of slip and stick only.The present paper investigates the influence of joints by a mixed experimental and numerical strategy. A detailed Finite Element model is established to provide understanding of different slip-stick mechanisms in the contact area. An advanced lumped parameter model is developed and identified by experimental investigations for an isolated bolted joint. This model is implemented in a Finite Element program for calculating the dynamic response of assembled structures incorporating the influence of micro- and macroslip of several bolted joints.List of symbols a acceleration - E ₀, E_t material moduli - F ₀ mass weighted excitation force - F _t tangential joint force - F generalized force - F _exc ^* excitation force - F _exc amplitude of excitation force - F _C0 spring element force - F _R0 friction element force - K _A, K_B normal stiffness - K _t tangential stiffness - L length of contact area - M _t transmitted joint torque - m _red reduced mass - p normal contact pressure - r effective radius - q generalized coordinate - z internal variable - x coordinate in the contact area - u relative displacement - u relative velocity - relative angle - friction coefficient - damping ratio - material parameter - ₀ equivalent slip limit - microslip parameter - excitation frequency Dedicated to Prof. Dr. Dr. h. c. Franz Ziegler on the occasion of this 60th birthday     

Computation of the relaxation and creep functions of elastomers from harmonic shear modulus

The purpose of this paper is to compute the relaxation and creep functions from the data of shear complex modulus, G ^∗(iν). The experimental data are available in the frequency window ν∈[ν_min,ν_max] in terms of the storage G′(ν) and loss G″(ν) moduli. The loss factor h( n) = \fracG"( n)G￠(n)\eta( \nu) = \frac{G'( \nu )}{G'(\nu )} is asymmetrical function. Therefore, a five-parameter fractional derivative model is used to predict the complex shear modulus, G ^∗(iν). The corresponding relaxation spectrum is evaluated numerically because the analytical solution does not exist. Thereby, the fractional model is approximated by a generalized Maxwell model and its rheological parameters (G _k ,τ _k ,N) are determined leading to the discrete relaxation spectrum G(t) valid in time interval corresponding to the frequency window of the input experimental data. Based on the deterministic approach, the creep compliance J(t) is computed on inversing the relaxation function G(t).     

A three-dimensional multilayer composite finite element for stress analysis of composite laminates

A three-dimensional multilayer composite finite element method has been developed based on a composite variational functional which takes three in-plane strains ε_x, ε_x, ε_xy and three transverse stresses σ_z, σ_yz, σ_xz as the basic variables. The continuity of the transverse stresses σ_z, σ_yz, σ_xz across the laminate thickness is assured a priori by introducing a partial stress field parameter α which is associated with the lower and upper surfaces of a lamina in a laminate. A method has been developed to form the partial stress field based on the assumed displacement field. With this method, a three dimensional (3-D) multilayer composite finite element is formulated for stress analysis of composite laminates. A numerical example is given, which shows some advantages of this composite element.