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This paper is concerned with optimal control problems governed by constrained hereditary second- order differential inclusions (DFIs). We also focus on delay-neutral and assigned Hale type DFIs, which arise in many fields of applied science and play a vital role in the mathematical modeling of real-life phenomena. To the best of our knowledge, such problems for second-order hereditary DFIs have not been considered in the literature from the viewpoint of optimality conditions. In this way we formulate sufficient conditions of optimality for second-order convex and nonconvex neutral functional-DFIs in Euler-Lagrange and Hamiltonian forms and transversality conditions. The results obtained are applied to some linear problems. [ABSTRACT FROM AUTHOR]

We investigate the tunable multicolor optomechanically induced transparency and the effect of slow/fast light in a hybrid electro-optomechanical system, in which a degenerate optical parametric amplifier (OPA) and a Λ − type three-level atomic ensemble are placed in a optical cavity with two oscillator modes. The system is driven by a strong pump field and a weak probe field, respectively. Multicolor transparency windows appear in the output field under the atom-photon-OPA-phonon-phonon coherent interaction. And the position and width of the transparent port of the output field can be manipulated by properly modifying the different parameters of the subsystem. We also research the slow/fast light effect related to phase and group delay in the probe field. Our proposal provides a great flexibility for phonon storage and has some potential applications in quantum information processing. [ABSTRACT FROM AUTHOR]

We propose a simple procedure to produce energy eigenstates of a Hamiltonian with discrete eigenvalues. We use ancilla qubits and quantum entanglement to separate an energy eigenstate from the other energy eigenstates. We exhibit a few examples derived from the (1 + 1)‐dimensional massless Schwinger model. Our procedure in principle will be applicable for a Hamiltonian with a finite‐dimensional Hilbert space. Choosing an initial state properly, we can in principle produce any energy eigenstate of the Hamiltonian. [ABSTRACT FROM AUTHOR]

Quantum optimization is a significant area of quantum computing research with anticipated near-term quantum advantages. Current quantum optimization algorithms, most of which are hybrid variational-Hamiltonian-based algorithms, struggle to present quantum devices due to noise and decoherence. Existing techniques attempt to mitigate these issues through employing different Hamiltonian encodings or Hamiltonian clause pruning, but they often rely on optimistic assumptions rather than a deep analysis of the problem structure. We demonstrate how to formulate the problem Hamiltonian for a quantum approximate optimization algorithm that satisfies all the requirements to correctly describe the considered tactical aircraft deconfliction problem, achieving higher probabilities for finding solutions compared to previous works. Our results indicate that constructing Hamiltonians from an unconventional, quantum-specific perspective with a high degree of entanglement results in a linear instead of exponential number of entanglement gates instead and superior performance compared to standard formulations. Specifically, we achieve a higher probability of finding feasible solutions: finding solutions in nine out of nine instances compared to standard Hamiltonian formulations and quadratic programming formulations known from quantum annealers, which only found solutions in seven out of nine instances. These findings suggest that there is substantial potential for further research in quantum Hamiltonian design and that gate-based approaches may offer superior optimization performance over quantum annealers in the future. [ABSTRACT FROM AUTHOR]

Within the framework of quantum information theory, the performance of conventional 2-D techniques is often unsatisfactory for the study of Bell states that hold a unique status as maximally entangled states. Therefore, 3-D approaches are increasingly employed to achieve more accurate and detailed analysis, offering improved performance and insights in complex scenarios. Among the diverse background of mixed two-qubit states, certain configurations exhibit unique quantum correlations that harness advanced quantum information processing tasks such as Bell diagonal states. Although, these states may appear superficially simple and exhibit a rich spectrum of correlations. The present research employs a methodology that involves a convex combination of distinct Bell states to generate the entire class of Bell diagonal states. This work explores the time evolution of Bell diagonal states, when exposed to various quantum channels and investigates the dynamics of quantum correlations such as entanglement, discord, and state of separability. Finally, the behaviour of Bell diagonal states is analysed and results are compared between theory and practice. A threedimensional visual approach is used to illustrate a deeper understanding of various quantum features and dynamic behaviour of the Bell diagonal states. [ABSTRACT FROM AUTHOR]

In this work, the stability of exact smooth multi-solitons for the Dullin-Gottwald-Holm (DGH) equation is investigated for the first time. Firstly, through the inverse scattering approach, we derive the conservation laws in terms of scattering data and multi-solitons related to the discrete spectrum based on the Lax pair of the DGH equation. Notably, a new series Hamiltonian derived from bi-Hamilton structure are equivalent to the conservation laws, and can also be expressed in terms of scattering data. Thus we successfully connect scattering data (multi-solitons) to the recursion operator between Hamiltonian, and to the Lyapunov functional constructed from Hamiltonian. With this Lyapunov functional, it is identified that these smooth multi-solitons serve as non-isolated constrained minimizers, adhering to a suitable variational nonlocal elliptic equation. Moreover, the integrable properties of the recursion operator are presented, which is the crucial for spectral analysis in this work. Consequently, the investigation of dynamical stability transforms into a problem on the spectrum of explicit linearized systems. It is worth noting that, compared to previous works on Camassa-Holm equation and KdV-type equations, recursion operator derived from existing Hamiltonian for the DGH equation do not possess good integrable properties. We construct a new series of Hamiltonians with both analysable recursion operator and simple initial variation derivative (δ H 1 / δ u = m) to overcome this problem. Furthermore, orbital stability of the smooth double solitons is proved at last of the work. [ABSTRACT FROM AUTHOR]

This study presents a hybrid modeling approach that integrates physics and machine learning for modeling multi-dimensional dynamics of a coupled nonlinear dynamical system. This approach leverages principles from classical mechanics, such as the Euler-Lagrange and Hamiltonian formalisms, to facilitate the process of learning from data. The hybrid model incorporates single or multiple artificial neural networks within a customized computational graph designed based on the physics of the problem. The customization minimizes the potential of violating the underlying physics and maximizes the efficiency of information flow within the model. The capabilities of this approach are investigated for various multidimensional modeling scenarios using different configurations of a coupled nonlinear dynamical system. It is demonstrated that, in addition to improving modeling criteria such as accuracy and consistency with physics, this approach provides additional modeling benefits. The hybrid model implements a physics-based architecture, enabling the direct alteration of both conservative and non-conservative components of the dynamics. This allows for an expansion in the model's input dimensionality and optimal allocation of input variable effects on conservative or non-conservative components of dynamics. [ABSTRACT FROM AUTHOR]

We show that for every complex simple Lie algebra g , the equations of Schubert divisors on the flag variety G / B - give a complete integrable system of the minimal nilpotent orbit O min . The approach is motivated by the integrable system on Coulomb branch as reported by Braverman (arXiv preprint arXiv:1604.03625, 2016).We give explicit computations of these Hamiltonian functions, using Chevalley basis and a so-called Heisenberg algebra basis. For classical Lie algebras we rediscover the lower order terms of the celebrated Gelfand-Zeitlin system. For exceptional types we computed the number of Hamiltonian functions associated to each vertex of Dynkin diagram. They should be regarded as analogs of Gelfand-Zeitlin functions on exceptional type Lie algebras. [ABSTRACT FROM AUTHOR]

This study investigates the kinetic properties of both symmetrical and asymmetrical multilayer and nano-sized semiconductor structures. We develop a theoretical framework using various models and mathematical methods to solve the Schrödinger matrix equation for a system of electrons, taking into account the Bastard condition, which considers the difference in the effective masses of current carriers in adjacent layers. We analyze tunnel-coupled electronic states in quantum wells separated by a narrow tunnel-transparent potential barrier. Our findings provide insights into the electronic properties of semiconductor structures, which are crucial for applications in micro- or nanoelectronics and other areas of solid-state physics.

The paper defines a class of Hamiltonian systems whose phase flows are exact solutions of the two-dimensional hydrodynamics of an incompressible fluid. The properties of this class are considered. An example of a Lagrangian one-dimensional system is given, which after the transition to the Hamiltonian formalism leads to an unsteady flow, that is, to an exact solution of two-dimensional hydrodynamics. The connection between these formalisms is discussed and the Lagrangians that give rise to Lagrangian hydrodynamics are introduced. The obtained results make it possible to obtain accurate solutions, such as phase flows of special Hamiltonian systems.

The dynamical behavior of the Lakshamanan-Porsezian-Daniel (LPD) model with applications in nonlinear optics is uncovered in this paper. The incorporation of spatio-temporal dispersion into the model, addressing the internet bottleneck issue, is emphasized. A planar dynamical system (DS) is accounted for by applying the Galilean transformation to the considered model. All types of phase portraits are plotted based on specific values of parameters. For the generation of quasi-periodic and chaotic patterns, Acos⁡(Ωτ) with strength and frequency components is added. The resulting quasi-periodic and chaotic orbits are also plotted. Various bifurcation and chaos detecting tools, such as time series analysis, Poincare maps, Lyapunov exponent, bifurcation diagrams, and sensitivity, are employed to examine the behavior of the obtained dynamical system under proposed initial conditions. The Hamiltonian function is calculated and plotted for the unperturbed dynamical system. Multi-stability phenomena demonstrate the coexistence of multi-stable orbits in their dynamic features. The dynamics of pulse propagation are found to be significantly impacted by the spatio-temporal dispersion.

Summary: The paper introduces a computational framework using a novel Arbitrary Lagrangian Eulerian (ALE) formalism in the form of a system of first‐order conservation laws. In addition to the usual material and spatial configurations, an additional referential (intrinsic) configuration is introduced in order to disassociate material particles from mesh positions. Using isothermal hyperelasticity as a starting point, mass, linear momentum and total energy conservation equations are written and solved with respect to the reference configuration. In addition, with the purpose of guaranteeing equal order of convergence of strains/stresses and velocities/displacements, the computation of the standard deformation gradient tensor (measured from material to spatial configuration) is obtained via its multiplicative decomposition into two auxiliary deformation gradient tensors, both computed via additional first‐order conservation laws. Crucially, the new ALE conservative formulation will be shown to degenerate elegantly into alternative mixed systems of conservation laws such as Total Lagrangian, Eulerian and Updated Reference Lagrangian. Hyperbolicity of the system of conservation laws will be shown and the accurate wave speed bounds will be presented, the latter critical to ensure stability of explicit time integrators. For spatial discretisation, a vertex‐based Finite Volume method is employed and suitably adapted. To guarantee stability from both the continuum and the semi‐discretisation standpoints, an appropriate numerical interface flux (by means of the Rankine–Hugoniot jump conditions) is carefully designed and presented. Stability is demonstrated via the use of the time variation of the Hamiltonian of the system, seeking to ensure the positive production of numerical entropy. A range of three dimensional benchmark problems will be presented in order to demonstrate the robustness and reliability of the framework. Examples will be restricted to the case of isothermal reversible elasticity to demonstrate the potential of the new formulation. [ABSTRACT FROM AUTHOR]

We investigate some properties of Cayley fuzzy graphs on groups in terms of algebraic structures. And we also discuss the connectedness on it. In this paper, we prove that the Cayley fuzzy graph induced by (Zm × Zn,+, ν) is the disjoint union of Cay(aH, R) provided ν(h) > 0 h H and (h0) = 0 - h0 - Zm ×Zn\H, where H is the cyclic subgroup of order n in Zm × Zn. [ABSTRACT FROM AUTHOR]

The frame work of this study is the bi-elliptic restricted four body problem, where the largest primary is assumed to be a radiating body and the other two massive bodies and are assumed to be oblate spheroids. The problem is restricted in the sense that the fourth body is assumed to be of infinitesimal mass. The goal of the paper is to study the so-called equilibrium points by generalizing R3BP to a non-coherent but highly practical R4BP model. The location of the planar equilibrium points according to this model is numerically studied for Sun–Earth–Moon system. The position of the triangular equilibrium points are also obtained analytically and graphically compared with numerically obtained values. Both the graphical and analytical studies confirms the high dependence of the position of the triangular equilibrium points on radiation pressure, however the collinear points were found to be less affected. The collinear points were found to be more affected by the oblateness of the second primary. The triangular equilibrium points were found to be stable for the third and fourth order resonance cases when the mass ratio is less than equal to a critical mass ratio. This critical mass ratio is also found to be dependent on the radiation pressure and phase angle . The transition curve in the () plane is plotted to find the value of for which the motion near triangular equilibrium points become unstable. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the conditions for mapping the geometric description of the kinematics of particles that probe a given Hamiltonian in phase space to a description in terms of Finsler geometry (and vice-versa). [ABSTRACT FROM AUTHOR]

• A coupling dynamic model of the non-symmetric spatial dumbbell-type system is developed. • A structure-preserving numerical iteration method is proposed to solve the coupling problems concerned. • The effects of the masses of additional particles on orbit-attitude-vibration coupling behaviors of the model are revealed. The structural symmetric breaking of the spatial structure will enhance the coupling dynamic behaviors and bring new challenges for the dynamic analysis on the spatial structure inevitably. The main contribution of this paper is proposing a structure-preserving iteration method to investigate the effects of the dynamic symmetry-breaking factors on the dynamic behaviors of the rigid-flexible coupling systems. Firstly, the spatial flexible damping beam assembled with two particles on both ends is simplified as an asymmetric spatial dumbbell-type model. For this model, the coupling dynamic equations are presented based on the Hamiltonian variational principle. Then, connecting the symplectic precise integration method for the ordinary differential equations mainly controlling the plane motion of the model and the generalized multi-symplectic scheme for the partial differential equation mainly controlling the transverse vibration of the beam, a structure-preserving numerical iteration method is proposed, which provides a new way to analysis the dynamic behaviors of the coupling problems with the symmetric breaking. Finally, the effects of the absolute mass and the mass ratio of the additional particles on the orbit radius, the orbit true anomaly velocity, the attitude angle velocity of the model and the transverse vibration of the flexible beam are reproduced by using the structure-preserving numerical iteration method in the numerical simulations. Particularly, the effects of the additional particles on the attitude stability of the model and the vibration dissipation of the flexible damping beam are investigated, which gives some guidance for the attitude adjustment strategy and the vibration control method for the spatial flexible structure with the structural symmetric breaking. [ABSTRACT FROM AUTHOR]

Held in Dublin, Ireland, IS2024 invites us to refresh understanding of contributions to systems engineering by Ireland's greatest mathematician–Sir William Rowan Hamilton (1805 ‐ 1865), Professor of Astronomy at Trinity College Dublin and Royal Astronomer of Ireland. His profound contributions to STEM deserve greater systems community attention. Supporting theory and practice, they intersect Foundations and Applications streams of INCOSE's Future of Systems Engineering (FuSE) program. Strikingly, key aspects apply to systems of all types, including socio‐technical and information systems. Hamilton abstracted the energy‐like generator of dynamics for all systems, while also generalizing momentum. Applied to the INCOSE Innovation Ecosystem Pattern as dynamics of learning, development, and life cycle management, this suggests an architecture for integration of the digital thread and machine learning in innovation enterprises, along with foundations of systems engineering as a dynamical system. [ABSTRACT FROM AUTHOR]

This paper develops a theoretical basis and a systematic process for resolving all inertia forces along generalized coordinates from the overall energy equation of a dynamical system. The theory is developed for natural systems with scleronomic constraints, where the potential energy is independent of generalized velocities. The process involves expansion of the energy equation, and specifically a special expansion of the kinetic energy term, from which the inertia forces emerge. The expansion uses fundamental kinematic identities of the phase space. It is also guided by insights drawn from the structure of the Hamiltonian function. The resulting equation has the structure of the D'Alembert's equation but involving generalized coordinates, from which the Lagrange's equations of motion are obtained. The expansion process elucidates how certain inertia forces manifest in the energy equation as composite terms that must be accurately resolved along different generalized coordinates. The process uses only the system energy equation, and neither the Hamiltonian nor the Lagrangian function are required. Extension of this theory to non-autonomous and non-holonomic systems is an area of future research. [ABSTRACT FROM AUTHOR]

In this paper, analytic solutions and dynamic properties of a variety of Boussinesq-type equations are established via the complete discrimination system for polynomial method. All the existing single traveling wave solutions to these equations as well as some new solutions are shown, and the Hamiltonian and topological properties to these equations are also presented. Considering the significance of the Boussinesq-type equations, our results would have wide applications in ocean engineering and fluid mechanics, like describing and predicting the solitary and periodic waves in various shallow water models. [ABSTRACT FROM AUTHOR]

A nondispersive, conservative regularisation of the inviscid Burgers equation is proposed and studied. Inspired by a related regularisation of the shallow water system recently introduced by Clamond and Dutykh, the new regularisation provides a family of Galilean-invariant interpolants between the inviscid Burgers equation and the Hunter–Saxton equation. It admits weakly singular regularised shocks and cusped traveling-wave weak solutions. The breakdown of local smooth solutions is demonstrated, and the existence of two types of global weak solutions, conserving or dissipating an H 1 energy, is established. Dissipative solutions satisfy an Oleinik inequality like entropy solutions of the inviscid Burgers equation. As the regularisation scale parameter ℓ tends to 0 or ∞ , limits of dissipative solutions are shown to satisfy the inviscid Burgers or Hunter–Saxton equation respectively, forced by an unknown remaining term. [ABSTRACT FROM AUTHOR]

The dynamical behavior of the Lakshamanan-Porsezian-Daniel (LPD) model with applications in nonlinear optics is uncovered in this paper. The incorporation of spatio-temporal dispersion into the model, addressing the internet bottleneck issue, is emphasized. A planar dynamical system (DS) is accounted for by applying the Galilean transformation to the considered model. All types of phase portraits are plotted based on specific values of parameters. For the generation of quasi-periodic and chaotic patterns, A cos ⁡ (Ω τ) with strength and frequency components is added. The resulting quasi-periodic and chaotic orbits are also plotted. Various bifurcation and chaos detecting tools, such as time series analysis, Poincare maps, Lyapunov exponent, bifurcation diagrams, and sensitivity, are employed to examine the behavior of the obtained dynamical system under proposed initial conditions. The Hamiltonian function is calculated and plotted for the unperturbed dynamical system. Multi-stability phenomena demonstrate the coexistence of multi-stable orbits in their dynamic features. The dynamics of pulse propagation are found to be significantly impacted by the spatio-temporal dispersion. [ABSTRACT FROM AUTHOR]

The generalized Sierpiński gasket graphs $S[G,t]$ are introduced as the graphs obtained from the Sierpiński graphs $S(G,t)$ by contracting single edges between copies of previous phases. The family $S[G,t]$ is a generalization of a previously studied class of generalized Sierpiński gasket graphs $S[n,t]$. In this paper, several properties of $S[G,t]$ are studied. In particular, adjacency of vertices, degree sequence, general first Zagreb index, hamiltonicity, and Eulerian.

Variational Integrator (VI) is a numerical technique, in which the Lagrangian of the system is used as the action integral. It is a special type of numerical solution that preserves the energy and momentum of the system. In this paper, we retrieve numerical solutions for heat and wave equation with the help of all possible combinations of finite difference scheme like forward–forward, forward–backward, forward–centered, backward–forward, backward–backward, backward–centered, centered–forward, centered–backward, centered–centered. We also use Lagrangian approach along with the projection technique to obtain approximate solutions of these linear models. This approach provides the best approximate solutions as well as preserves the energy of the system while the finite difference scheme gives only the numerical solutions. We also draw a comparison of existing exact solution with all approximate solutions for both models and provide graphical representation of these solutions. [ABSTRACT FROM AUTHOR]

The paper defines a class of Hamiltonian systems whose phase flows are exact solutions of the two-dimensional hydrodynamics of an incompressible fluid. The properties of this class are considered. An example of a Lagrangian one-dimensional system is given, which after the transition to the Hamiltonian formalism leads to an unsteady flow, that is, to an exact solution of two-dimensional hydrodynamics. The connection between these formalisms is discussed and the Lagrangians that give rise to Lagrangian hydrodynamics are introduced. The obtained results make it possible to obtain accurate solutions, such as phase flows of special Hamiltonian systems. [ABSTRACT FROM AUTHOR]

This study investigates the kinetic properties of both symmetrical and asymmetrical multilayer and nano-sized semiconductor structures. We develop a theoretical framework using various models and mathematical methods to solve the Schrödinger matrix equation for a system of electrons, taking into account the Bastard condition, which considers the difference in the effective masses of current carriers in adjacent layers. We analyze tunnel-coupled electronic states in quantum wells separated by a narrow tunneltransparent potential barrier. Our findings provide insights into the electronic properties of semiconductor structures, which are crucial for applications in micro- or nanoelectronics and other areas of solid-state physics. [ABSTRACT FROM AUTHOR]

A hyperlink is a finite set of non-intersecting simple closed curves in R 4 ≡ R × R 3 , and each curve is either a matter or geometric loop. We consider an equivalence class of such hyperlinks, up to time-like isotopy, preserving time-ordering. Using an equivalence class and after coloring each matter component loop with an irreducible representation of s u (2) × su (2) , we can define its Wilson loop observable using an Einstein–Hilbert action, which is now thought of as a functional acting on the set containing equivalence classes of hyperlink. Construct a vector space using these functionals, which we now term as quantum states. To make it into a Hilbert space, we need to define a counting probability measure on the space containing equivalence classes of hyperlinks. In our previous work, we defined area, volume and curvature operators, corresponding to given geometric objects like surface and a compact solid spatial region. These operators act on the quantum states and, by deliberate construction of the Hilbert space, are self-adjoint and possibly unbounded operators. Using these operators and Einstein's field equations, we can proceed to construct a quantized stress operator and also a Hamiltonian constraint operator for the quantum system. We will also use the area operator to derive the Bekenstein entropy of a black hole. In the concluding section, we will explain how loop quantum gravity predicts the existence of gravitons, implies causality and locality in quantum gravity and formulates the principle of equivalence mathematically in its framework. [ABSTRACT FROM AUTHOR]

Dragonfly networks have significant advantages in data exchange due to the small network diameter, low cost and modularization. A graph |$G$| is |$vertex$| - |$pancyclic$| if for any vertex |$u\in V(G)$|⁠ , there exist cycles through |$u$| of every length |$\ell $| with |$3\leq \ell \leq |V(G)|$|⁠. A graph |$G$| is |$Hamiltonian$| - |$connected$| if there exists a Hamiltonian path between any two distinct vertices |$u,v\in V(G)$|⁠. In this paper, we mainly research the pancyclic and Hamiltonian properties of the dragonfly network |$D(n,h)$|⁠ , and find that it is Hamiltonian with |$n\geq 1,\,\,h\geq 2$|⁠ , pancyclic, vertex-pancyclic and Hamiltonian-connected with |$n\geq 4,\,\,h\geq 2$|⁠. [ABSTRACT FROM AUTHOR]

The main purpose of the present paper is the characterization, in the finite dimensional setting, of weak and strong invariance of closed sets with respect to a differential inclusion governed by time-dependent maximal monotone operators and multi-valued perturbation, by the use of the corresponding Hamiltonians. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Hamiltonian regularization of scalar conservation laws, which is parametrized by $ \ell>0 $ and conserves an $ H^1 $ energy. We prove the existence of global weak solutions for this regularization. Furthermore, we demonstrate that as $ \ell $ approaches zero, the unique entropy solution of the original scalar conservation law is recovered, providing justification for the regularization.This regularization belongs to a family of non-diffusive, non-dispersive regularizations that were initially developed for the shallow-water system and extended later to the Euler system. This paper represents a validation of this family of regularizations in the scalar case. [ABSTRACT FROM AUTHOR]

This study presents a novel approach to controlling cocoa black pod disease, caused by Phytophthora megakarya. Unlike previous methods, we employ a mathematical modeling framework based on Pontryagin’s Maximum Principle to optimize control strategies that minimize disease impact. The model incorporates the dynamics of healthy and infected pods while considering the combined effects of three key interventions: Infected pod removal, Targeted fungicide application and Promoting a healthy growing environment. Through numerical simulations, we identify the optimal timing and intensity for each intervention to minimize infected pods over time. This study highlights the following novelties: Synergistic effect of combined control which demonstrates that combining all three strategies is significantly more effective than relying on individual methods, Optimal strategies which involves dynamically adjusting control measures throughout the growing season to adapt to disease progression, Prompt action after disease detection proves crucial for successful control. These findings offer valuable data-driven recommendations for cocoa farmers and disease management professionals. By strategically implementing a combination of infected pod removal, targeted fungicide use, and environmental management, farmers can significantly reduce disease severity, enhance cocoa production, and promote a more sustainable cocoa industry.

This work studies a mathematical model of a quantum cluster consisting of three qubits and being in thermal equilibrium with the environment. The effective Hamiltonian is invariant under permutations of qubits and consists of two parts. The first part is similar to the Heisenberg XYZ-model with internal two-qubit interaction, while the second includes three-qubit interaction with the thermostat. Such a quantum system admits a fully analytical investigation and is considered in the context of mathematical modeling of quantum metamaterials, in which nanoclusters are elementary structural units with the strong internal interaction of qubits and the relatively weak coupling with the environment. For the Hamiltonian, we construct an orthonormal basis of eigenvectors, which includes the maximally entangled W-state. We also obtain the density operator of the cluster state in explicit form, and study the temperature dependences of the thermodynamic characteristics of the cluster: the partition function, entropy, and free energy. It is shown that the conditions of thermal equilibrium in this quantum system are satisfied at temperatures from 0,2 K to microkelvins, which correspond to the operating range of modern quantum logic elements and quantum simulators.

Quantum optimization is a significant area of quantum computing research with anticipated near-term quantum advantages. Current quantum optimization algorithms, most of which are hybrid variational-Hamiltonian-based algorithms, struggle to present quantum devices due to noise and decoherence. Existing techniques attempt to mitigate these issues through employing different Hamiltonian encodings or Hamiltonian clause pruning, but they often rely on optimistic assumptions rather than a deep analysis of the problem structure. We demonstrate how to formulate the problem Hamiltonian for a quantum approximate optimization algorithm that satisfies all the requirements to correctly describe the considered tactical aircraft deconfliction problem, achieving higher probabilities for finding solutions compared to previous works. Our results indicate that constructing Hamiltonians from an unconventional, quantum-specific perspective with a high degree of entanglement results in a linear instead of exponential number of entanglement gates instead and superior performance compared to standard formulations. Specifically, we achieve a higher probability of finding feasible solutions: finding solutions in nine out of nine instances compared to standard Hamiltonian formulations and quadratic programming formulations known from quantum annealers, which only found solutions in seven out of nine instances. These findings suggest that there is substantial potential for further research in quantum Hamiltonian design and that gate-based approaches may offer superior optimization performance over quantum annealers in the future.

Fault detection and localization are vital for ensuring the stability of data center networks (DCNs). Specifically, adaptive fault diagnosis is deemed a fundamental technology in achieving the fault tolerance of systems. The highly scalable data center network (HSDC) is a promising structure of server-centric DCNs, as it exhibits the capacity for incremental scalability, coupled with the assurance of low cost and energy consumption, low diameter, and high bisection width. In this paper, we first determine that both the connectivity and diagnosability of the m-dimensional complete HSDC, denoted by H S D C m (m) , are m. Further, we propose an efficient adaptive fault diagnosis algorithm to diagnose an H S D C m (m) within three test rounds, and at most N + 4 m (m − 2) tests with m ≥ 3 (resp. at most nine tests with m = 2 ), where N = m · 2 m is the total number of nodes in H S D C m (m) . Our experimental outcomes demonstrate that this diagnosis scheme of HSDC can achieve complete diagnosis and significantly reduce the number of required tests. [ABSTRACT FROM AUTHOR]

There are three different forms of adjoint inclusions that appear in the most advanced necessary optimality conditions for optimal control problems involving differential inclusions: Euler--Lagrange inclusion (with partial convexification) [A. D. Ioffe, J. Optim. Theory Appl., 182 (2019), pp. 285--309], fully convexified Hamiltonian inclusion [F. H. Clarke, Mem. Amer. Math. Soc., 173 (2005), 816], and partially convexified Hamiltonian inclusion [P. D. Loewen and R. T. Rockafellar, SIAM J. Control Optim., 34 (1996), pp. 1496--1511], [A. D. Ioffe, Trans. Amer. Math. Soc., 349 (1997), pp. 2871--2900], [R. B. Vinter, SIAM J. Control Optim., 52 (2014), pp. 1237--1250] (for convex-valued differential inclusions in the first two references). This paper addresses all three types of necessary conditions for problems with (in general) nonconvex-valued differential inclusions. The first of the two main theorems, with the Euler--Lagrange inclusion, is equivalent to the main result of [A. D. Ioffe, J. Optim. Theory Appl., 182 (2019), pp. 285--309] but proved in a substantially different and much more direct way. The second theorem contains conditions that guarantee necessity of both types of Hamiltonian conditions. It seems to be the first result of such a sort that covers differential inclusions with possibly unbounded values and contains the most recent results of [F. H. Clarke, Mem. Amer. Math. Soc., 173 (2005), 816] and [R. B. Vinter, SIAM J. Control Optim., 52 (2014), pp. 1237--1250] as particular cases. And again, the proof of the theorem is based on a substantially different approach. [ABSTRACT FROM AUTHOR]

In control theory, there is growing interest in the evolution of sets, especially integral funnels and reachable sets at a certain time. In this paper, we establish sufficient conditions for the convexity of reachable sets for an object whose behavior is described by the second-order differential inclusions. This fact is proved using the concavity of the Hamilton function in the first argument. Further, in connection with the usefulness of the Hamilton function, some of its properties, such as continuity and Lipschitz property, are investigated. At the end of the article, the results obtained are demonstrated with some examples. [ABSTRACT FROM AUTHOR]

Ferroelectric crystals show spontaneous electric polarization even in the absence of an electric field. Lead hydrogen arsenate (LHA) is a member of the group of Lead hydrogen phosphate (LHP) type ferroelectric crystals. We have modified earlier two sub-lattice pseudospin model (PLCM) Hamiltonian for LHA crystal. phonon anharmonic terms, four body interaction terms, and extra spin-lattice terms have been added to the model. For the LHA crystal, formulas for Cochran's frequency, dielectric constant, and loss tangent have been determined using Green's function approach, and a strong agreement is displayed when comparing our theoretical results to other researchers' experimental findings. [ABSTRACT FROM AUTHOR]

We consider the three-particle Schrödinger operator , , associated with a system of three particles (of which two are bosons with mass and one is arbitrary with mass ) coupled by pairwise contact potentials and on the three-dimensional lattice . We prove that there exist critical mass ratio values and such that for sufficiently large and fixed the operator , , has at least one eigenvalue lying to the left of the essential spectrum for , at least two such eigenvalues for , and at least four such eigenvalues for . [ABSTRACT FROM AUTHOR]

This paper studies the dynamics and conservation of energy. It evaluates the validity of Noether's theorem as a formal argument supporting the law of conservation of energy in physical systems. The analysis examines the role of nonconvexity in energy dynamics. The paper argues that nonconvexity can arise in the presence of catalytic effects or in situations of transitions between multiple regimes. With the introduction of nonconvexity, the analysis relies on a generalized Lagrangian and generalized Hamiltonian. The investigation applies under general conditions, allowing for multiple types of energy with dynamics driven by multiple state variables. Our key result is to show that the conservation of energy (Noether's theorem) holds under convexity but not under nonconvexity. This identifies situations where energy in isolated systems is not necessarily constant over time. By relaxing the law of conservation of energy, our analysis provides new insights into energy dynamics. It offers new directions for scientific inquiries, including improved understanding about the origin of life, the evolution of the early universe and the nature of space and time.