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This paper provides characterizations of the value function as the unique lower semicontinuous solution, appropriately defined, to the Hamilton-Jacobi equation, for optimal control problems of Bolza type with state constraints. Notable features of our analysis are that it covers problems in which the running cost is possibly non Lipschitz continuous w.r.t. the state variable and the dynamic constraint takes the form of a differential inclusion that is possibly discontinuous w.r.t. time. Alternative characterizations are given, in terms of lower Dini derivative type solutions and of proximal solutions to the Hamilton-Jacobi equation. Distance estimates, which permit us to approximate an arbitrary state trajectory by a state trajectory that strictly satisfies the state constraint, have a key role in the analysis. Earlier research, treating problems in which the running cost is absent or at least Lipschitz continuous w.r.t. the state, makes use of known distance estimates in which 'approximate' is interpreted in terms of the L ∞ norm. L ∞ distance estimates are inadequate, however, for problems in which the running cost may fail to be Lipschitz continuous. We show that, for this broader class of problems, the analysis can still be carried out, if it is based on stronger distance estimates involving the W 1 , 1 norm. The new W 1 , 1 distance estimates, developed here for this purpose, are of independent interest. Motivation for the study of problems with running costs that are not Lipschitz continuous is provided by a 'growth versus consumption' problem in neo-classical macro-economics. Our theory provides fresh insights into this much studied problem, by extending the class of initial data points for which a solution can be achieved. [ABSTRACT FROM AUTHOR]

A well-balanced second-order finite volume scheme is proposed and analyzed for a 2 × 2 system of non-linear partial differential equations which describes the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. To derive a second-order scheme, we combine a MUSCL type spatial reconstruction with strong stability preserving Runge-Kutta time stepping method. The resulting scheme is ensured to be well-balanced through a modified limiting approach that allows the scheme to revert to a well-balanced first-order scheme if it fails to capture or preserve the discrete steady state solution during its time evolution. Further, the scheme maintains the second-order accuracy away from the steady state. The well-balance property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. Additionally, numerical experiments reveal that the second-order scheme reduces finite time oscillations and provides better resolutions of the physical properties of the state variables, as compared to the existing first-order schemes of the literature. • Analytically proved that the standard second-order scheme for the Hadeler and Kuttler (HK) model equations is not well-balanced in general. • An adaptive second-order scheme for the HK model is developed and analytically proved that the proposed scheme is well-balanced. • Significance of the proposed scheme is demonstrated through its ability to tackle the challenges faced by conventional second-order scheme. [ABSTRACT FROM AUTHOR]

In constrained Hamiltonian dynamics there are two views regarding how first-class constraints generate gauge transformations: individually or only in a certain combination, the Rosenfeld–Anderson–Bergmann–Castellani gauge generator. This gauge generator G preserves Hamilton's equations and changes the canonical action at most by a boundary term; Hamiltonian's equations are the Euler–Lagrange equations for the canonical action. Hence the canonical formalism is equivalent to the Lagrangian formalism, and indeed subsumed within it (in important examples) with many canonical momenta serving as auxiliary fields. G generates transformations basically equivalent to the usual 4-dimensional Lagrangian expressions, such as a 4-gradient in electromagnetism or a space–time coordinate transformation (on-shell) in General Relativity. It has been shown recently that separate first-class constraints lead to inequivalent observables between Proca non-gauge and Stueckelberg gauge massive electromagnetism. There is, however, widespread agreement that second-class constraints do not generate gauge transformations. Here it is shown that in such a sense as the first-class primary constraint in Maxwell's theory generates a gauge transformation, the second-class primary constraint in Proca's massive electromagnetism also generates a gauge transformation. Likewise the second-class primary constraints in various massive spin 2 relatives of General Relativity generate as much of a gauge transformation as do the corresponding first-class primary constraints in GR. Hence the view that first-class constraints typically generate gauge transformations individually faces a puzzle not faced by the gauge generator view. [ABSTRACT FROM AUTHOR]

We investigated several global behaviors of the weak KAM solutions u c (x , t) parametrized by c ∈ H 1 (T , R). For the suspended Hamiltonian H (x , p , t) of the exact symplectic twist map, we could find a family of weak KAM solutions u c (x , t) parametrized by c (σ) ∈ H 1 (T , R) with c (σ) continuous and monotonic and ∂ t u c + H (x , ∂ x u c + c , t) = α (c) , a.e. (x , t) ∈ T 2 , such that sequence of weak KAM solutions { u c } c ∈ H 1 (T , R) is 1/2-Hölder continuity of parameter σ ∈ R. Moreover, for each generalized characteristic (no matter regular or singular) solving { x ˙ (s) ∈ co ∂ p H (x (s) , c + D + u c (x (s) , s + t) , s + t) , x (0) = x 0 , (x 0 , t) ∈ T 2 , we evaluate it by a uniquely identified rotational number ω (c) ∈ H 1 (T , R). This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases. [ABSTRACT FROM AUTHOR]

• A system consisting of the Euler-Bernoulli beam subjected to tendon-induced axial loading is studied. • The system is modelled using a set of partial differential equations derived by Hamilton's principle. • The model is thoroughly experimentally validated for a wide range of loadings and configurations. • The effect of the tendon tension and its mass on the modal properties and frequency loci veering are studied. • Further numerical studies show the effect of the tendon on the torsional modes and structural stability. Theoretical and experimental modal analysis of the system consisting of the Euler-Bernoulli beam axially loaded by a tendon is studied in the paper. The beam-tendon system is modelled using a set of partial differential equations derived by Hamilton's principle and the coupling between the beam and the tendon is ensured by the boundary conditions. Theoretical modal analysis is conducted using a boundary value problem solver and the results are thoroughly experimentally validated using a bench-top experiment. In particular, the effect of the tendon tension on the modal properties of the system is studied. It is found that by increasing the tension, the natural frequencies of the beam decrease while the natural frequencies of the tendon increase. It is also shown that these two sets of modes interact with each other through frequency loci veering. The effect of the tendon mass is also experimentally and numerically studied and it is shown that lighter tendon produces fewer vibration modes in the studied frequency region. Two further numerical studies are conducted to demonstrate the effect of the tendon on the torsional modes of the beam, and to study the structural stability. Overall, an excellent agreement between the numerical and experimental results is obtained, giving the confidence in the derived theoretical model. [ABSTRACT FROM AUTHOR]

Highlights • Hamilton–Jacobi equation. • Kernel based scheme. • Unconditionally stable. • High order accuracy. • Weighted essentially non-oscillatory methodology. Abstract In this paper, a class of high order numerical schemes is proposed for solving Hamilton–Jacobi (H–J) equations. This work is regarded as an extension of our previous work for nonlinear degenerate parabolic equations, see Christlieb et al. [14] , which relies on a special kernel-based formulation of the solutions and successive convolution. When applied to the H–J equations, the newly proposed scheme attains genuinely high order accuracy in both space and time, and more importantly, it is unconditionally stable, hence allowing for much larger time step evolution compared with other explicit schemes and saving computational cost. A high order weighted essentially non-oscillatory methodology and a novel nonlinear filter are further incorporated to capture the correct viscosity solution. Furthermore, by coupling the recently proposed inverse Lax–Wendroff boundary treatment technique, this method is very flexible in handing complex geometry as well as general boundary conditions. We perform numerical experiments on a collection of numerical examples, including H–J equations with linear, nonlinear, convex or non-convex Hamiltonians. The efficacy and efficiency of the proposed scheme in approximating the viscosity solution of general H–J equations is verified. [ABSTRACT FROM AUTHOR]

Abstract We introduce a novel heteronuclear dipolar recoupling based on the R 2 1 − 1 symmetry, which uses the tanh/tan (tt) shaped pulse as a basic inversion element and is denoted R 2 1 − 1 (tt). Using first-order average Hamiltonian theory, we show that this sequence is non-γ-encoded and that it reintroduces the | m | = 1 spatial component of the Chemical Shift Anisotropy (CSA) of the irradiated isotope and its heteronuclear dipolar interactions. Using numerical simulations and one-dimensional (1D) 27Al-{31P} through-space D -HMQC (Dipolar Heteronuclear Multiple-Quantum Correlation) experiments on VPI-5, we compare the performances of this recoupling to those of other non-γ-encoded | m | = 1 heteronuclear recoupling schemes: REDOR (Rotational-Echo DOuble Resonance), SFAM (Simultaneous Frequency and Amplitude Modulation) and R 4 2 − 1 (tt). Such comparison indicates that the R 2 1 − 1 (tt) scheme is more robust to CSA, offset and radiofrequency field inhomogeneities than the other schemes. We take advantage of the high robustness of R 2 1 − 1 (tt) to CSA and offset to demonstrate the possibility to correlate the signals of 207Pb isotope with those of neighboring half-integer spin quadrupolar nuclei. Such approach is demonstrated experimentally by acquiring 11B-{207Pb} D -HMQC 2D spectra of Pb 4 O(BO 3) 2 crystalline powder. Graphical abstract (a) D-HMQC pulse sequence with Inverse-Recoupling performed with (b) REDORxy4, (c) SFAM, and (d) R21−1 (tt). Image 1 Highlights • Indirect recoupling decreases the t1-noise of D-HMQC. • SFAM recoupling is very efficient with D-HMQC if 1H is not detected. • R21−1 symmetry with tanh/tan shaped pulse as a basic inversion is more robust to large CSA in the indirect dimension. • 11B-{207 Pb} D-HMQC can be recorded with R21−1 (tanh/tan) recoupling. [ABSTRACT FROM AUTHOR]

Multiple pulse sequences are often used to manipulate spin Hamiltonians in solid-state nuclear magnetic resonance spectroscopy. In this paper, we analyze multiple pulse sequences using the well-known average Hamiltonian theory. We first expand the resulting average Hamiltonian into a reachable set of sub-Hamiltonians and then develop a general procedure using both flip-angle and phase of the applied pulses as control variables to select any of those sub-Hamiltonians. We use this method to analyze solid-echo based sequences and to design new proton-proton homonuclear decoupling sequences in static solids. It is found that this newly designed decoupling scheme, in the presence of finite pulse length, effectively suppresses the 1 H– 1 H homonuclear dipolar interactions while establishes variable scaling factors on the heteronuclear dipolar interactions and chemical shift interactions, depending on the flip-angle of the applied pulses. When the pulse flip-angle is close to 54.7°, this sequence possesses a large scaling factor with relatively low average decoupling field. When the pulse flip-angle becomes ∼120°, the scaling factor is almost zero. A static 15 N-acetyl-valine crystal sample has been used as an example to confirm and validate the performance of this new decoupling scheme. [ABSTRACT FROM AUTHOR]

We continue studying an inverse problem in the theory of periodic homogenization of Hamilton–Jacobi equations proposed in [14] . Let V 1 , V 2 ∈ C ( R n ) be two given potentials which are Z n -periodic, and H ‾ 1 , H ‾ 2 be the effective Hamiltonians associated with the Hamiltonians 1 2 | p | 2 + V 1 , 1 2 | p | 2 + V 2 , respectively. A main result in this paper is that, if the dimension n = 2 , and each of V 1 , V 2 contains exactly 3 mutually non-parallel Fourier modes, then H ‾ 1 ≡ H ‾ 2 ⇔ V 1 ( x ) = V 2 ( x c + x 0 ) for all x ∈ T 2 = R 2 / Z 2 , for some c ∈ Q ∖ { 0 } and x 0 ∈ T 2 . When n ≥ 3 , the scenario is slightly more subtle, and a complete description is provided for any dimension. These resolve partially a conjecture stated in [14] . Some other related results and open problems are also discussed. [ABSTRACT FROM AUTHOR]

We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded domains; moreover, near the origin, the gradient graph of such a solution is continuous if and only if the graph is special Lagrangian. [ABSTRACT FROM AUTHOR]

This paper describes the connections between the notions of Hamiltonian system, contact Hamiltonian system and nonholonomic system from the perspective of differential equations and dynamical systems. It shows that action minimizing curves of nonholonomic system satisfy the dissipative Lagrange system, which is equivalent to the contact Hamiltonian system under some generic conditions. As the initial research of contact Hamiltonian dynamics in this direction, we investigate the dynamics of contact Hamiltonian systems in some special cases including invariants, completeness of phase flows and periodic behavior. [ABSTRACT FROM AUTHOR]

In this paper, we prove the long time existence of the combinatorial Calabi flow in hyperbolic background geometry and the convergence of the flow to a smooth hyperbolic surface under Thurston's combinatorial condition in full generality, which improves the result in [14] . This flow provides a natural algorithm to find Thurston's hyperbolic circle patterns. [ABSTRACT FROM AUTHOR]

The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities. [ABSTRACT FROM AUTHOR]

Let G be a fixed graph. Two paths of length n − 1 on n vertices (Hamiltonian paths) are G-different if there is a subgraph isomorphic to G in their union. In this paper we prove that the maximal number of pairwise triangle-different Hamiltonian paths is equal to the number of balanced bipartitions of the ground set, answering a question of Körner, Messuti and Simonyi. [ABSTRACT FROM AUTHOR]

We prove internal controllability in arbitrary time, for small data, for quasi-linear Hamiltonian NLS equations on the circle. We use a procedure of reduction to constant coefficients up to order zero and HUM method to prove the controllability of the linearized problem. Then we apply a Nash–Moser–Hörmander implicit function theorem as a black box. [ABSTRACT FROM AUTHOR]

The stability of a pretwisted cantilever beam spinning about its longitudinal axis and subjected to non-conservative force is investigated. In this study, it is assumed that the cantilever is embedded in viscoelastic medium, which is modeled by the Kelvin-Voigt foundation. Two different types of the non-conservative force are considered. The governing equations of motion and boundary conditions are derived by using Hamilton's principle. The finite element method is utilized to transform the coupled equations of motion to a general eigenvalue problem. The proposed model is justified by an excellent agreement between the present results and those reported in the literature. The effects of several design parameters including the pretwist angle, the cross section ratio, the viscoelastic parameters and load span length on the stability of the spinning pretwisted cantilevers are also examined. Moreover, the critical load and spinning speed and stability regions of the spinning cantilevers are identified. The results show that the design parameters significantly change the stability of the spinning pretwisted cantilever beams. [ABSTRACT FROM AUTHOR]

Quasiclassical dynamics of trapped ions is characterized by applying the time dependent variational principle (TDVP) on coherent state orbits, in case of quadrupole and octupole combined (Paul and Penning) or radiofrequency (RF) traps. A dequantization algorithm is proposed, by which the classical Hamilton (energy) function associated to the system results as the expectation value of the quantum Hamiltonian on squeezed coherent states. We develop such method and particularize the quantum Hamiltonian for both combined and RF nonlinear traps, that exhibit axial symmetry. We also build the classical Hamiltonian functions for the particular traps we considered, and find the classical equations of motion. [ABSTRACT FROM AUTHOR]

A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this form the non-Hermitian oscillators can be studied in much the same way as in the Hermitian approaches. Two different nonlinear algebras generated by properly constructed ladder operators are found and the corresponding generalized coherent states are obtained. The non-Hermitian oscillators can be steered to the conventional one by the appropriate selection of parameters. In such limit, the generators of the nonlinear algebras converge to generalized ladder operators that would represent either intensity-dependent interactions or multi-photon processes if the oscillator is associated with single mode photon fields in nonlinear media. [ABSTRACT FROM AUTHOR]

A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occurs in a pairwise fashion. It is also shown that with the choice of a suitable co-ordinate, the Hamiltonian can always be reformulated in the background of a pseudo-Euclidean metric. If the equations of motion of some of the well-known many-body systems like Calogero models are generalized to include balanced loss and gain, it appears that the same may not be amenable to a Hamiltonian formulation. A few exactly solvable systems with balanced loss and gain, along with a set of integrals of motion are constructed. The examples include a coupled chain of nonlinear oscillators and a many-particle Calogero-type model with four-body inverse square plus two-body pair-wise harmonic interactions. For the case of nonlinear oscillators, stable solution exists even if the loss and gain parameter has unbounded upper range. Further, the range of the parameter for which the stable solutions are obtained is independent of the total number of the oscillators. The set of coupled nonlinear equations are solved exactly for the case when the values of all the constants of motions except the Hamiltonian are equal to zero. Exact, analytical classical solutions are presented for all the examples considered. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that three-dimensional stochastic Maxwell equations with multiplicative noise are stochastic Hamiltonian partial differential equations possessing a geometric structure (i.e. stochastic multi-symplectic conservation law), and the energy of system is a conservative quantity almost surely. We propose a stochastic multi-symplectic energy-conserving method for the equations by using the wavelet collocation method in space and stochastic symplectic method in time. Numerical experiments are performed to verify the excellent abilities of the proposed method in providing accurate solution and preserving energy. The mean square convergence result of the method in temporal direction is tested numerically, and numerical comparisons with finite difference method are also investigated. [ABSTRACT FROM AUTHOR]

We provide a criterion in order to decide the stability of non-degenerate equilibrium states of completely integrable systems. More precisely, given a Hamilton–Poisson realization of a completely integrable system generated by a smooth n -dimensional vector field, X , and a non-degenerate regular (in the Poisson sense) equilibrium state, x ‾ e , we define a scalar quantity, I X ( x ‾ e ) , whose sign determines the stability of the equilibrium. Moreover, if I X ( x ‾ e ) > 0 , then around x ‾ e , there exist one-parameter families of periodic orbits shrinking to { x ‾ e } , whose periods approach 2 π / I X ( x ‾ e ) as the parameter goes to zero. The theoretical results are illustrated in the case of the Rikitake dynamical system. [ABSTRACT FROM AUTHOR]

This paper deals with the limit cycles of a class of cubic Hamiltonian systems under polynomial perturbations. We suppose that the corresponding Hamiltonian system which has at least one center has finite singular points and is symmetrical with respect to both x-axis and y-axis, and also the origin is a nilpotent singular point. Hence, the Hamiltonian H ( x , y ) can be written as H ( x , y ) = − x 2 + ( a x 4 + b x 2 y 2 + c y 4 ) , ( a , b , c ) ∈ R 3 , c ≠ 0 . For the above H ( x , y ) , the corresponding vector field is x ˙ = 2 y ( b x 2 + 2 c y 2 ) , y ˙ = 2 x ( 1 − 2 a x 2 − b y 2 ) . We first obtain that the above vector fields with at least one center can be divided into 8 classes by its topological phase portraits. For the following perturbed system x ˙ = 2 y ( b x 2 + 2 c y 2 ) + ε f ( x , y ) , y ˙ = 2 x ( 1 − 2 a x 2 − b y 2 ) + ε g ( x , y ) , where 0 < | ε | ≪ 1 , f ( x , y ) and g ( x , y ) are polynomials in ( x , y ) with degree n , we give an estimation of the number of isolated zeros of the corresponding Abelian integral. The number of limit cycles follows from Poincaré–Pontryagin Theorem. [ABSTRACT FROM AUTHOR]

In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motion of high degree, or symmetry operators of high order, have been found and studied. Most of these Hamiltonians, in the classical case, can be included in the family of extended Hamiltonians, geometrically characterized by the structure of warped manifold of their configuration manifold. For the extended Hamiltonians, the characteristic constants of motion of high degree are polynomial in the momenta of determined form. We consider here a different form of the constants of motion, based on the factorization procedure developed for systems of two degrees of freedom by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians, with arbitrary dimension, admits factorized constants of motion and we determine their expression. The classical constants can be polynomial or non-polynomial in the momenta, but the factorization procedure allows, in a type of extended Hamiltonians, their quantization via shift and ladder operators, for systems of any finite dimension. [ABSTRACT FROM AUTHOR]

In this study, the effects of next-nearest neighbor interaction on a one-dimensional mixed spin 3 - spin 3/2 Ising model was investigated using Cellular Automaton (CA). The Ising model hamiltonian contains the antiferromagnetic nearest neighbor interaction (J 1 ), the ferromagnetic next- nearest neighbor interaction (J 2 ) and the external magnetic field (h = H/J 1 ). Magnetization (M) of the mixed spin system was obtained in the interval 0 ≤ R ≤ 1 of the interaction ratio (R = J 2 /J 1 ) using field cooling (FC) and zero-field cooling (ZFC) processes. Hysteresis curves were drawn for several R values in the interval −0.1 ≤ h ≤ 0.1 using FC results. The functional behavior for coercive field (H C ) was determined depending on R. Thus, the mixed spin system became a hard magnetic material and the lattice geometry also changed from one dimensional linear chain to triangular chain with increasing R value. [ABSTRACT FROM AUTHOR]

Between 1947 and 1950, Laurent Schwartz (1915–2002) went from being almost unknown outside of France to being an international mathematical celebrity. This paper accounts for Schwartz's rapid ascent by focusing on the social, institutional, and mathematical contexts of his crucial trajectory from Nancy, via Copenhagen, to the world stage, culminating in his 1950 Fields Medal awarded in Cambridge, Massachusetts. We identify, based on new archival findings, the pivotal role of Danish mathematician Harald Bohr along this trajectory. Our analysis reveals the emerging dynamics of early postwar international mathematics, and explains how certain individuals and theories could rise to prominence in this period. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of quasi-periodic response solutions for a class of reversible forced harmonic oscillators with two basic frequencies ω = ( 1 , α ) , α is an irrational number. Since we do not impose usual Diophantine or Brjuno conditions on ω , it can also be Liouvillean. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for reversible systems. [ABSTRACT FROM AUTHOR]

We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary N -body interactions. We identify four types of symmetries such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non -ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold classes, but also obtains the canonical representations of these symmetries in each of the ten classes. We also provide a group cohomological perspective of our results in terms of projective representations. We then use the canonical representations of the symmetries to obtain the structure of Hamiltonians with arbitrary N -body interactions in each of the ten classes. We show that the space of N -body Hamiltonians has an affine subspace (of a vector space) structure in classes which have either or both charge conjugation and sublattice symmetries. Our results can help address many open questions including the topological classification of interacting fermionic systems. [ABSTRACT FROM AUTHOR]

In this work, we present the study of monolayer MoS 2 band structure using k.p method based on infinitesimal basis transformations perturbation theory. We also study the effect of spin-orbit coupling on the band structure. We have calculated the low energy k.p Hamiltonian and derived the dispersion relations for the last valance band and the first conduction band at K and –K points of the first Brillouin zone. The results clearly show the electron-hole asymmetry and trigonal warping effects. The results also indicate the spin-splitting of the first conduction band. [ABSTRACT FROM AUTHOR]

A detailed exposition of highly efficient and accurate method for the propagation of the time-dependent Schrödinger equation [50] is presented. The method is readily generalized to solve an arbitrary set of ODE's. The propagation is based on a global approach, in which large time-intervals are treated as a whole, replacing the local considerations of the common propagators. The new method is suitable for various classes of problems, including problems with a time-dependent Hamiltonian, nonlinear problems, non-Hermitian problems and problems with an inhomogeneous source term. In this paper, a thorough presentation of the basic principles of the propagator is given. We give also a special emphasis on the details of the numerical implementation of the method. For the first time, we present the application for a non-Hermitian problem by a numerical example of a one-dimensional atom under the influence of an intense laser field. The efficiency of the method is demonstrated by a comparison with the common Runge–Kutta approach. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the semiclassical analysis of linear Schrödinger equations from a Gabor frame perspective. We consider (time-dependent) smooth Hamiltonians with at most quadratic growth. Then we construct higher order parametrices for the corresponding Schrödinger equations by means of ħ -Gabor frames, as recently defined by M. de Gosson, and we provide precise L 2 -estimates of their accuracy, in terms of the Planck constant ħ . Nonlinear parametrices, in the spirit of the nonlinear approximation, are also presented. Numerical experiments are exhibited to compare our results with the early literature. [ABSTRACT FROM AUTHOR]

We study the spectral and ergodic properties of Schrödinger operators on a compact connected Riemannian manifold M without boundary in case that the examined system possesses certain symmetries. More precisely, if M carries an isometric and effective action of a compact connected Lie group G , we prove a generalized equivariant version of the semiclassical Weyl law with an estimate for the remainder, using a semiclassical functional calculus for h -dependent functions and relying on recent results on singular equivariant asymptotics. We then deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry-reduced Hamiltonian flow on the principal stratum of the singular symplectic reduction of M is ergodic. In particular, we obtain an equivariant version of the Shnirelman–Zelditch–Colin-de-Verdière theorem, as well as a representation theoretic equidistribution theorem. If M / G is an orbifold, similar results were recently obtained by Kordyukov. When G is trivial, one recovers the classical results. [ABSTRACT FROM AUTHOR]

Time dependent quantum problems defined by quadratic Hamiltonians are solved using canonical transformations. The Green’s function is obtained and a comparison with the classical Hamilton–Jacobi method leads to important geometrical insights like exterior differential systems, Monge cones and time dependent Gaussian metrics. The Wei–Norman approach is applied using unitary transformations defined in terms of generators of the associated Lie groups, here the semi-direct product of the Heisenberg group and the symplectic group. A new explicit relation for the unitary transformations is given in terms of a finite product of elementary transformations. The sequential application of adequate sets of unitary transformations leads naturally to a new separation of variables method for time dependent Hamiltonians, which is shown to be related to the Inönü–Wigner contraction of Lie groups. The new method allows also a better understanding of interacting particles or coupled modes and opens an alternative way to analyze topological phases in driven systems. [ABSTRACT FROM AUTHOR]

We extend a result on renormalized oscillation theory, originally derived for Sturm–Liouville and Dirac-type operators on arbitrary intervals in the context of scalar coefficients, to the case of general Hamiltonian systems with block matrix coefficients. In particular, this contains the cases of general Sturm–Liouville and Dirac-type operators with block matrix-valued coefficients as special cases. The principal feature of these renormalized oscillation theory results consists in the fact that by replacing solutions by appropriate Wronskians of solutions, oscillation theory now applies to intervals in essential spectral gaps where traditional oscillation theory typically fails. [ABSTRACT FROM AUTHOR]

We propose a scheme of controlling entanglement dynamics of a quantum system by applying the external classical driving field for two atoms separately located in a single-mode photon cavity. It is shown that, with a judicious choice of the classical-driving strength and the atom–photon detuning, the effective atom–photon interaction Hamiltonian can be switched from Jaynes–Cummings model to anti-Jaynes–Cummings model. By tuning the controllable atom–photon interaction induced by the classical field, we illustrate that the evolution trajectory of the Bell-like entanglement states can be manipulated from entanglement-sudden-death to no-entanglement-sudden-death, from no-entanglement-invariant to entanglement-invariant. Furthermore, the robustness of the initial Bell-like entanglement can be improved by the classical driving field in the leaky cavities. This classical-driving-assisted architecture can be easily extensible to multi-atom quantum system for scalability. [ABSTRACT FROM AUTHOR]

The experimental techniques have evolved to a stage where various examples of nanostructures with non-trivial shapes have been synthesized, turning the dynamics of a constrained particle and the link with geometry into a realistic and important topic of research. Some decades ago, a formalism to deduce a meaningful Hamiltonian for the confinement was devised, showing that a geometry-induced potential (GIP) acts upon the dynamics. In this work we study the problem of prescribed GIP for curves and surfaces in Euclidean space R 3 , i.e., how to find a curved region with a potential given a priori . The problem for curves is easily solved by integrating Frenet equations, while the problem for surfaces involves a non-linear 2nd order partial differential equation (PDE). Here, we explore the GIP for surfaces invariant by a 1-parameter group of isometries of R 3 , which turns the PDE into an ordinary differential equation (ODE) and leads to cylindrical, revolution, and helicoidal surfaces. Helicoidal surfaces are particularly important, since they are natural candidates to establish a link between chirality and the GIP. Finally, for the family of helicoidal minimal surfaces, we prove the existence of geometry-induced bound and localized states and the possibility of controlling the change in the distribution of the probability density when the surface is subjected to an extra charge. [ABSTRACT FROM AUTHOR]

This paper reviews and addresses a variety of issues relating to inclusive fitness. The main question is: are there limits to the generality of inclusive fitness, and if so, what are the perimeters of the domain within which inclusive fitness works? This question is addressed using two well-known tools from evolutionary theory: the replicator dynamics, and adaptive dynamics. Both are combined with population structure. How generally Hamilton's rule applies depends on how costs and benefits are defined. We therefore consider costs and benefits following from Karlin and Matessi's (1983) “counterfactual method”, and costs and benefits as defined by the “regression method” ( Gardner et al., 2011 ). With the latter definition of costs and benefits, Hamilton's rule always indicates the direction of selection correctly, and with the former it does not. How these two definitions can meaningfully be interpreted is also discussed. We also consider cases where the qualitative claim that relatedness fosters cooperation holds, even if Hamilton's rule as a quantitative prediction does not. We furthermore find out what the relation is between Hamilton's rule and Fisher's Fundamental Theorem of Natural Selection. We also consider cancellation effects – which is the most important deepening of our understanding of when altruism is selected for. Finally we also explore the remarkable (im)possibilities for empirical testing with either definition of costs and benefits in Hamilton's rule. [ABSTRACT FROM AUTHOR]

We present commuting projector Hamiltonian realizations of a large class of ( 3 + 1 )D topological models based on mathematical objects called unitary G-crossed braided fusion categories. This construction comes with a wealth of examples from the literature of symmetry-enriched topological phases. The spacetime counterparts to our Hamiltonians are unitary state sum topological quantum fields theories (TQFTs) that appear to capture all known constructions in the literature, including the Crane–Yetter–Walker–Wang and 2-Group gauge theory models. We also present Hamiltonian realizations of a state sum TQFT recently constructed by Kashaev whose relation to existing models was previously unknown. We argue that this TQFT is captured as a special case of the Crane–Yetter–Walker–Wang model, with a premodular input category in some instances. [ABSTRACT FROM AUTHOR]

We study the Dirac–Maxwell model quantized in the Lorenz gauge. In this gauge, the space of quantum mechanical state vectors inevitably adopt an indefinite metric so that the canonical commutation relation (CCR) is realized in a Lorentz covariant manner. In order to obtain a physical subspace, in which no negative norm state exists, the method first proposed by Gupta and Bleuler is applied with mathematical rigor. It is proved that a suitably defined physical subspace has a positive semi-definite metric, and naturally induces a physical Hilbert space with a positive definite metric. Then, the original Dirac–Maxwell Hamiltonian defines an induced Hamiltonian on the physical Hilbert space which is essentially self-adjoint. [ABSTRACT FROM AUTHOR]

In this article, we present a unified framework of discontinuous Galerkin (DG) discretizations for Hamiltonian ODEs and PDEs. We show that with appropriate numerical fluxes the numerical algorithms deduced from DG discretizations can be combined with the symplectic methods in time to derive the multi-symplectic PRK schemes. The resulting numerical discretizations are applied to the linear and nonlinear Schrödinger equations. Some conservative properties of the numerical schemes are investigated and confirmed in the numerical experiments. [ABSTRACT FROM AUTHOR]

Motivated by recent developments in the fields of large deviations for interacting particle systems and mean field control, we establish a comparison principle for the Hamilton–Jacobi equation corresponding to linearly controlled gradient flows of an energy function E defined on a metric space (E , d). Our analysis is based on a systematic use of the regularizing properties of gradient flows in evolutional variational inequality (EVI) formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian at hand and, in combination with the use of the Tataru's distance, for establishing the key estimates needed to bound the difference of the Hamiltonians in the proof of the comparison principle. Our abstract results apply to a large class of examples only partially covered by the existing theory, including gradient flows on Hilbert spaces and the Wasserstein space equipped with a displacement convex energy functional E satisfying McCann's condition. [ABSTRACT FROM AUTHOR]

Using the Keeton and Petters approach, we determine the deflection angle. We also investigate the motion of photons around a parameterized black hole in the presence of non-magnetized cold plasma by using a new ray-tracing algorithm. In spherically symmetric spacetime, we examine the influence of the plasma by applying the Hamiltonian equation on the deflection angle as well as shadow. It is examine to derive the rays analytically from Hamilton's equation by separating the metric and the plasma frequency. We study that the presence of plasma affects the deflection angle as well as shadow for the parameterized black hole and they depend on plasma frequency. If the plasma frequency is significantly lower than the photon frequency, the photon sphere and shadow radius expressions can be linearized around the values found for space light rays. Furthermore, we have graphically analyze the behavior of shadow for distinct positions under the effects of plasma frequency as well as low density plasma medium. • Introductory review of parameterized black hole. • From Keeton and Petters approach to obtain deflection angle. • Detailed description of ray-tracing algorithm, to examine the influence of the plasma by applying the Hamiltonian equation on the deflection angle as well as shadow. [ABSTRACT FROM AUTHOR]

This article considers non-relativistic charged particle dynamics in both static and non-static electromagnetic fields, which are governed by nonseparable, possibly time-dependent Hamiltonians. For the first time, explicit symplectic integrators of arbitrary high-orders are constructed for accurate and efficient simulations of such mechanical systems. Performances superior to the standard non-symplectic method of Runge–Kutta are demonstrated on two examples: the first is on the confined motion of a particle in a static toroidal magnetic field used in tokamak; the second is on how time-periodic perturbations to a magnetic field inject energy into a particle via parametric resonance at a specific frequency. [ABSTRACT FROM AUTHOR]

A unified presentation of the quasi-exact solutions of the Bohr Hamiltonian with a sextic oscillator potential is offered by evidencing the similarities and the differences between them. An extended study of the Bohr Hamiltonian with sextic potential for γ -rigid triaxial nuclei is conducted, focusing on the evolution of the energy spectra and B ( E 2 ) transition probabilities when the order of the quasi-exactly solvable system is increased. The effect is analyzed throughout the shape phase transition from a γ -rigid triaxial vibrator to an anharmonic one and especially in its critical point. The new results are tested by several applications for experimental data, improving the previous calculations and providing new possible triaxial nuclei. [ABSTRACT FROM AUTHOR]

Identity operation in the form of π pulses is widely used in NMR spectroscopy. For an isolated single spin system, a sequence of even number of π pulses performs an identity operation, leaving the spin state essentially unaltered. For multi-spin systems, trains of π pulses with appropriate phases and time delays modulate the spin Hamiltonian to perform operations such as decoupling and recoupling. However, experimental imperfections often jeopardize the outcome, leading to severe losses in sensitivity. Here, we demonstrate that a newly designed Genetic Algorithm (GA) is able to optimize a train of π pulses, resulting in a robust identity operation. As proof-of-concept, we optimized the recoupling sequence in the transferred-echo double-resonance (TEDOR) pulse sequence, a key experiment in biological magic angle spinning (MAS) solid-state NMR for measuring multiple carbon-nitrogen distances. The GA modified TEDOR (GMO-TEDOR) experiment with improved recoupling efficiency results in a net gain of sensitivity up to 28% as tested on a uniformly 13 C, 15 N labeled microcrystalline ubiquitin sample. The robust identity operation achieved via GA paves the way for the optimization of several other pulse sequences used for both solid- and liquid-state NMR used for decoupling, recoupling, and relaxation experiments. [ABSTRACT FROM AUTHOR]

The renormalized cyclotron mass of a strong coupling polaron in a three-dimensional (3D) anisotropic quantum dot is investigated using the Landau-Pekar variational approach in which a 3D anisotropic harmonic potential and electron wave function are included in the Hamiltonian to obtain ground state (GS) and excited-state (ES) energies. The expressions of the GS and ES energies under investigation depict a rich variety of dependent relationship with the variational parameters in three different limiting case based on them. It is demonstrated that the Landau-Pekar variational approach provides a reasonable description of the observed properties, in particular, detailed relations between the cyclotron masses and the magnetic field strength, the confinement lengths in the xy -plane and the z direction are discussed. [ABSTRACT FROM AUTHOR]

We prove that, for large n , every 3-connected D -regular graph on n vertices with D ≥ n / 4 is Hamiltonian. This is best possible and verifies the only remaining case of a conjecture posed independently by Bollobás and Häggkvist in the 1970s. The proof builds on a structural decomposition result proved recently by the same authors. [ABSTRACT FROM AUTHOR]

We deal with a singularly perturbed optimal control problem with slow and fast variable depending on a parameter ε . We study the asymptotics, as ε goes to 0, of the corresponding value functions, and show convergence, in the sense of weak semilimits, to sub and supersolution of a suitable limit equation containing the effective Hamiltonian. The novelty of our contribution is that no compactness condition is assumed on the fast variable. This generalization requires, in order to perform the asymptotic procedure, an accurate qualitative analysis of some auxiliary equations posed on the space of fast variable. The task is accomplished using some tools of Weak KAM theory, and in particular the notion of Aubry set. [ABSTRACT FROM AUTHOR]

A novel 6 degree of freedom (6-DOF) passive vibration isolator is studied theoretically and validated with experiments. Based on the Stewart platform configuration, the 6-DOF isolator is constructed by 6 X-shape structures as legs, which can realize very good and tunable vibration isolation performance in all 6 directions with a passive manner. The mechanic model is established for static analysis of the working range, static stiffness and loading capacity. Thereafter, the equation of motion of the isolator is derived with the Hamilton principle. The equivalent stiffness and the displacement transmissibility in the six decoupled DOFs direction are then discussed with experimental results for validation. The results reveal that (a) by designing the structure parameters, the system can possess flexible stiffness such as negative, quasi-zero and positive stiffness, (b) due to the combination of the Stewart platform and the X-shape structure, the system can have very good vibration isolation performance in all the 6 directions and in a passive manner, and (c) compared with the simplified linear-stiffness legs, the nonlinearity of the X-shape structures enhance the passive isolator to have much better vibration isolation performance. [ABSTRACT FROM AUTHOR]

We consider the completely resonant non-linear Schrödinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix s > 1 . We show the existence of solutions of this equation which achieve arbitrarily large growth of H s Sobolev norms. We also give estimates for the time required to attain this growth. [ABSTRACT FROM AUTHOR]