Transcending classical boundaries: Choice and its equivalences

This article aims to develop a uniform method for solving plane elastic boundary value problems, PEBVPs. Stress or displacement vectors are assumed to be given on the boundary in the classical formulation of PEBVP, while non– classical formulations include boundary conditions in terms of orientations of stresses, forces or displacements. It is shown that the approximation of the complex potentials by the linear combinations of holomorphic functions can be used to obtain solutions in these cases and that the other well known numerical methods can be represented as particular cases of this approach. Two examples are presented.

We shall derive existence, uniqueness and comparison results for the functional differential equationx′(t)=f(t, x), a. e.t∈I, with classical Nicoletti boundary conditionsxi(ti)=yi∈X, i∈A, whereI is a real interval,A is a nonempty set andX is a Banach space.

We report high-resolution measurements of the properties of the velocity boundary layer in turbulent thermal convection using the particle image velocimetry (PIV) technique and measurements of the temperature profiles and the thermal boundary layer. Both velocity and temperature measurements were made near the lower conducting plate of a rectangular convection cell using water as the convecting fluid, with the Rayleigh numberRavarying from 109to 1010and the Prandtl numberPrfixed at 4.3. From the measured profiles of the horizontal velocity we obtain the viscous boundary layer thickness δυ. It is found that δυfollows the classical Blasius-like laminar boundary layer in the present range ofRa, and it scales with the Reynolds numberReas δυ/H= 0.64Re−0.50±0.03(whereHis the cell height). While the measured viscous shear stress and Reynolds shear stress show that the boundary layer is laminar forRa< 2.0 × 1010, two independent extrapolations, one based on velocity measurements and the other on velocity and temperature measurements, both indicate that the boundary layer will become turbulent atRa~ 1013. Just above the thermal boundary layer but within the mixing zone, the measured temperature r.m.s. profiles σT(z) are found to follow either a power law or a logarithmic behaviour. The power-law fitting may be slightly favoured and its exponent is found to depend onRaand varies from −0.6 to −0.77, which is much larger than the classical value of −1/3. In the same region, the measured profiles of the r.m.s. vertical velocity σw(z) exhibit a much smaller scaling range and are also consistent with either a power-law or a logarithmic behaviour. The Reynolds number dependence of several wall quantities is also measured directly. These are the wall shear stress τw~Re1.55, the viscous sublayer δw~Re−0.91, the friction velocityuτ~Re0.80, and the skin-friction coefficientcf~Re−0.34. All of these scaling properties are very close to those predicted for a classical Blasius-type laminar boundary layer, except that ofcf. Similar to classical shear flows, a viscous sublayer is also found to exist in the present system despite the presence of a nested thermal boundary layer. However, velocity profiles normalized by wall units exhibit no obvious logarithmic region, which is likely to be a result of the very limited distance between the edge of the viscous sublayer and the position of the maximum velocity. Compared to traditional shear flows, the peak position of the wall-unit-normalized r.m.s. profiles is found to be closer to the plate (atz+=z/δw≃ 5). Our overall conclusion is that a Blasius-type laminar boundary condition is a good approximation for the velocity boundary layer in turbulent thermal convection for the present range of Rayleigh number and Prandtl number.

Recently, a new viewpoint on the classical c-boundary in Mathematical Relativity has been developed, the relations of this boundary with the conformal one and other classical boundaries have been analyzed and its computation in some classes of spacetimes, as the standard stationary ones, has been carried out. In this paper, we consider the notion of isocausality given by García-Parrado and Senovilla, and introduce a framework to carry out isocausal comparisons with standard stationary spacetimes. As a consequence, the qualitative behavior of the c-boundary (at the three levels: point-set, chronology and topology) of a wide class of spacetimes is obtained.

We study the mixing processes inside a forced fountain using data from direct numerical simulation. The outer boundary of the fountain with the ambient is a turbulent/non-turbulent interface. Inside the fountain, two internal boundaries, both turbulent/turbulent interfaces, are identified: (i) the classical boundary between upflow and downflow which is composed of the loci of points of zero mean vertical velocity; and (ii) the streamline that separates the mean flow emitted by the source from the entrained fluid from the ambient (the separatrix). We show that entrainment due to turbulent fluxes across the internal boundary is at least as important as that by the mean flow. However, entrainment by the turbulence behaves substantively differently from that by the mean flow and cannot be modelled using the same assumptions. This presents a challenge for existing models of turbulent fountains and other environmental flows that evolve inside turbulent environments.

This paper explores quantum and classical chaos in the stadium billiard using Matlab simulations to investigate the behavior of wave functions in the stadium and the corresponding classical orbits believed to underlie wave function scarring. The simulations use three complementary methods. The quantum wave functions are modeled using a cellular automaton simulating a Hamiltonian wave function with discrete (square pixel) boundary conditions approaching the stadium in the classical limit. The classical orbits are computed by solving the reflection equations at the classical boundary thus giving direct insights into the wave functions and eigenstates of the quantum stadium. Finally, a simplified semi-classical algorithm is developed to show the comparison between this and the quantum wave function method. Quanta 2014; 3: 16–31.

In topological terms, we compute the spectral flow of an arbitrary family of self-adjoint Dirac-type operators with classical (local) boundary conditions on a compact Riemannian manifold with boundary under the assumption that the initial and terminal operators of the family are conjugate by an automorphism of the bundle in which the operators act. We use this result to study conditions for the existence of a nonzero spectral flow of a family of self-adjoint Dirac-type operators with local boundary conditions in a two-dimensional domain with a nontrivial topology and discuss possible physical realizations of a nonzero spectral flow.

This paper proposes an improved differential quadrature finite element method (DQFEM) by combining the virtual boundary spring technique with the standard DQFEM, in order to deal with free vibration of Mindlin plates with arbitrary elastic constraints. The incorporation of the virtual boundary spring technique makes it easy to impose general elastic constraints including some classical boundary conditions and avoids the deficiencies of the classical elimination method, which is widely used to process boundary conditions. The improved DQFEM formulation for rectangular and curvilinear quadrilateral elements is established. Convergence characteristics of the present approach are discussed, and the minimum number of nodes to derive convergent results and the appropriate value of the boundary spring stiffness to simulate classical boundary are obtained. Numerical examples are carried out for Mindlin plates with various boundary conditions and thickness ratios, covering both rectangular and irregular geometries. By numerical comparisons, the high accuracy and the remarkable efficiency of the present method are demonstrated. Additionally, adopting the virtual spring boundary can improve the efficiency to some extent compared with the standard DQFEM.

It has long been known that certain integral transforms and Fourier-type series can be used to solve many classical boundary and initial value problems in separable coordinate systems. More recently, it has been shown that these classical transforms and series are spectral representations associated with an ordinary differential system which results on applying separation of variables to the given boundary value problem. This has been the basis for recent work concerned with systematically generating the proper spectral representation needed to solve a given problem. See [1]-[4] for a list of references. 
\n
\nWe shall consider the problem of finding the associated spectral representation when the resulting ordinary differential system has the eigenvalue parameter occurring in both the equation and one boundary condition. Moreover, the differential equation is to be satisfied on a semi-infinite interval, thus leading to a singular problem which does not seem to have been studied before. In §2 by using a transformation due to B. Friedman (which we modify appropriately for our singular case), we first give a formal derivation of the spectral representation, and then we rigorously prove the result. 
\n
\nIn §3 our representation is applied to solve an initial-boundary value problem arising in the theory of diffusion and heat flow in one dimension. We should bear in mind that even in cases where solutions are already known, our method systematically yields alternative representations which are often more rapidly convergent and from which asymptotic expansions of solutions with respect to parameters can often be found. In the problem to be considered a representation of the solution can also be found easily by a straight-forward application of the Laplace transform in time t, while the new transform derived in §2 yields the solution when applied in space x. The new transform, however, can be applied when the coefficients in the boundary value problem are time dependent, a situation which, in general, precludes the use of the Laplace transform.. As a general rule [1] there should be a spectral representation associated with each ordinary differential system resulting from applying separation of variables to the original boundary value problem, and each spectral representation should lead to a different representation of the solution of the original problem.

The article discusses the possibility of detecting the grounds of the aesthetization process in modern culture in the structure of the Kantian philosophy. The connection of Kant's aesthetics with science and morality is traced, and also Kant's influence on the romantic philosophy of art is revealed. The loss of its classical boundaries by modern aesthetics is shown as originated in the Kantian philosophy, though it was this very philosophy in which these boundaries were once defined.

The fluorescence of a resonantly driven superconducting qubit is measured in the time domain, providing a weak probe of the qubit dynamics. Prior preparation and final, single-shot measurement of the qubit allows us to average fluorescence records conditionally on past and future knowledge. The resulting interferences reveal purely quantum features characteristic of weak values. We demonstrate conditional averages that go beyond classical boundaries and probe directly the jump operator associated with relaxation. The experimental results are remarkably captured by a recent theory, which generalizes quantum mechanics to open quantum systems whose past and future are known.

In this chapter, we discuss the results obtained in [38] with the use of the relative index superposition principle for the spectral flow of families of self-adjoint Dirac type operators with classical elliptic boundary conditions on a compact Riemannian manifold with boundary. In the two-dimensional case, such operators arise, for example, when describing electron states in graphene [39] or in topological insulators [33,68], and their spectral flow has an important physical meaning, being related to the creation of electron–hole (or, more generally, particle–antiparticle) pairs.

A geometry-based mechanism for generating steady-state internal circulation of local thermal currents is demonstrated by harmonically coupled quantum oscillators formulated by the Redfield quantum master equation(RQME) and the Bose-Hubbard model (BHM) of phonons formulated by the Lindblad quantum master equation(LQME) using the simple multipath geometry of a triangle. Driven by two reservoirs at different temperatures, both systems can exhibit an atypical local thermal current flowing against the total current. However, the total thermal current behaves normally. While the RQME of harmonically coupled quantum oscillators allows an analytical solution, the LQME of the interacting BHM can be solved numerically. The emergence of the geometry-based circulation in both systems demonstrates the ubiquity and robustness of the mechanism. In the high-temperature limit, the results agree with the classical results, confirming the generality of the geometric-based circulation across the quantum and classical boundary. The geometry-based circulation also emerges from a quantum Langevin equationcalculation. Possible experimental implications and applications are briefly discussed.

We propose a definition of wavefunction "branchings": quantum superpositions which can't be feasibly distinguished from the corresponding mixed state, even under time evolution. Our definition is largely independent of interpretations, requiring only that it takes many more local gates to swap branches than to distinguish them. We give several examples of states admitting such branch decompositions. Under our definition, we argue that attempts to get relative-phase information between branches will fail without frequent active error correction, that branches are effectively the opposite of good error-correcting codes, that branches effectively only grow further apart in time under natural evolution, that branches tend to absorb spatial entanglement, that branching is stronger in the presence of conserved quantities, and that branching implies effective irreversibility. Identifying these branch decompositions in many-body quantum states could shed light on the emergence of classicality, provide a metric for experimental tests at the quantum/ classical boundary, and allow for longer numerical time evolution simulations. We see this work as a generalization of the basic ideas of environmentally-induced decoherence to situations with no clear system/ environment split.

We investigate the transition from integrable to chaotic dynamics in the quantum mechanical wave functions from the point of view of the nodal structure by employing a two-dimensional quartic oscillator. We find that the number of nodal domains is drastically reduced as the dynamics of the system changes from integrable to nonintegrable, and then gradually increases as the system becomes chaotic. The number of nodal intersections with the classical boundary as a function of the level number shows a characteristic dependence on the dynamics of the system, too. We also calculate the area distribution of nodal domains and study the emergence of the power law behavior with the Fisher exponent in the chaotic limit.