No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Methods of mathematical physics - Vol.1; Vol.2
Abstract
Since the first volume of this work came out in Germany in 1924, this book, together with its second volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's second and final revision of 1953. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. All rights reserved.
... where J α stands for semilinear terms in ϕ and M ab αβ is the principal part of the system. As it is well known, the highest-order terms in derivatives almost completely controls the qualitative behaviour of solutions of a partial differential equation [30]. We obtain ...
... It is well known that when m ab is Lorentzian, the vectors q a such that P # 2 (x, q) = 0 (in a region of spacetime) satisfy the equation of null geodesics w.r.t. the effective metric, see [37], i.e. q a ||b q b = 0 (30) where || denotes covariant differentiation w.r.t. m ab (see, however, [38] for a case where this property fails). ...
... Interestingly, if the map (3) is independent of time, the problem of finding null geodesics in the effective spacetime reduces to that of finding geodesics in an effective Riemannian manifold of lower dimension. Indeed, choosing a coordinate system x a = (t, x i ), i = 1, 2, 3, such that g 0i = 0 and ∂ t ϕ α = 0 one reduces (30) and (31) to the equationsẍ ...
Topological solitons are relevant in several areas of physics [1]. Recently, these configurations have been investigated in contexts as diverse as hydrodynamics [2], Bose-Einstein condensates [3], ferromagnetism [4], knotted light [5] and non-abelian gauge theories [6]. In this paper we address the issue of wave propagation about a static Hopf soliton in the context of the Nicole model. Working within the geometrical optics limit we show that several nontrivial lensing effects emerge due to nonlinear interactions as long as the theory remains hyperbolic. We conclude that similar effects are very likely to occur in effective field theories characterized by a topological invariant such as the Skyrme model of pions.
... As classical in literature (see e.g. [13]), we call nodal domains of v λ α the maximally connected subsets of R 2 for which v λ α does not change sign. From Proposition 1.3 and Lemma 2.1, we obtain the following statement. ...
... Let Ω be a domain such that the Dirichlet Laplacian has an eigenfunction v that can be extended into a functionṽ satisfying ∆ṽ + k 2ṽ = 0 in a neighborhood of Ω. (13) First, for Ω as in (13), due to the implicit functions theorem, ∂Ω must be piecewise analytic. Additionally, we have the following statement (see also [13, Chapter V, §16]). ...
We consider the scattering of waves by a penetrable inclusion embedded in some reference medium. We exhibit examples of materials and geometries for which non-scattering frequencies exist, i.e., for which at some frequencies there are incident fields which produce null scattered fields outside of the inhomogeneity. We show in particular that certain domains with corners or even cusps can support non-scattering frequencies. We relate the latter, for some inclusions, to resonance frequencies for Dirichlet or Neumann cavities. We also find situations where incident non-scattering fields solve the Helmholtz equation in a neighborhood of the inhomogeneity and not in the whole space. Finally, in relation with invisibility, we give examples of inclusions of anisotropic materials which are non-scattering for all real frequencies. We prove that corresponding material indices must have a special structure on the boundary.
... Notably in [3], Kolmogorov related equations, transformed to the backward heat model were under investigation. The solution of equations such as (1) can be found via fundamental solutions [12,7] (a particular solution with specific singularities), which in fact is an invariant solution [13]. ...
... The initial roundness error of the anode workpiece is δ max0 and the radius of the cathode tool is R c = R a = (R max + R min ) /2. The initial coordinates (x 0 K , y 0 K ) of any contour point K (shown in Fig. 2) on the surface of the anode satisfy the following [49]: ...
Roundness error plays an important role of rotating parts in engineering fields and it has a significant influence on the machining quality and accuracy during electrochemical machining (ECM) process. The precision ECM could be processed only if the initial roundness error is decreased or eliminated and the inter-electrode gap (IEG) becomes steady after reaching an equilibrium state. However, a constant voltage is generally used in ECM process. And a long time and a large allowance are required to level the profile error of workblank if the initial profile error is large. In this study, the focus herein is on the acceleration of the leveling process of the rotary workpiece. A counter-rotating electrochemical machining (CRECM) process method with a variable voltage is proposed to improve the leveling ability. For a rotary workpiece with elliptical contours, the machining voltage can be dynamically adjusted based on the IEG through the approximate regulation of sine waves according to modeling-based analysis. The method aims to improve the leveling ability by expanding the difference in the magnitude of electric current between the high and low points on the profile of the anode workpiece under different voltages. The results of experiments confirmed that the proposed method significantly reduced the leveling time from 36 to 7 min (by 81%), and the depth of dissolution of the highest point on the profile from 1.68 to 0.45 mm while reducing the roundness error from 0.5 to 0.05 mm. The leveling ratio increased from 0.26 to 0.99.
... It is assumed that the steady state is within the set of reachable states, which allows the elimination of the final state weighting term. Using the calculus of variations, the minimization of (5) is attained when the associated Euler-Lagrange equations are satisfied [Courant and Hilbert (1953) ...
This paper develops a new method for computing the state feedback gain of a Linear Quadratic Regulator (LQR) with input derivative weighting that circumvents solving the Riccati equation. The additional penalty on the derivatives of the input introduces intuitively tunable weights and enables smoother control characteristics without the need of model extension. This is motivated by position controlled mechanical systems. The physical limitations of these systems are usually their velocity and acceleration rather than the position itself. The presented algorithm is based on a discretization approach to the calculus of variations and translating the original problem into a least-squares with equality constraints problem. The control performance is analyzed using a laboratory setup of an underactuated crane-like system.
... The elements of potential theory and the boundary integral methods can be found in many textbooks on fluid dynamics. Rigorous mathematical treatments of the potential theory are also available in the publications like Courant & Hilbert (1962). Since there are so many different ways to use these integral equations, the choice of the integral equations for our present cavity flow problem is the main discussion in the section. ...
This thesis deals with the theory and the numerical simulation of sheet cavity
flows on arbitrary lifting bodies like hydrofoils and propeller blades. The
objective of this research is the accurate prediction of the cavity volume and
the volume variations in time when these lifting bodies travel in a gust or are
subject to an ambient pressure change. This volume change plays an important
role in the pressure excitation on neighboring structures like hull of a ship and
in the radiation of underwater noise.
The physical phenomenon of sheet cavitation on lifting bodies is described
according to experimental observations at the beginning of this thesis. The
specific features of sheet cavitation, different from bubble and cloud cavitation,
are addressed. The basic cavity flow theory within the frame of the potential
theory is described and all the boundary conditions are discussed in order to
obtain the solution of the problem. For predicting the dynamics of the cavity,
numerical methods to find the solution in time are studied.
All the numerical algorithms for solving the problem are discussed in detail
and checked extensively by numerical tests. A higher order panel method is
described and evaluated. Emphasis is given to the problems of the analytical
calculation of the influence coefficients. The system of equations of the
fully wetted and cavitated flows are established under different Kutta conditions.
The detachment condition and its influence on the cavity flow are
studied. Cavity planform searching, grid updating and cavity-body intersection
are described. Other highly-related numerical methods for panel methods
and cavity flows, like the Kutta condition and wake alignment, are investigated
and checked by numerical tests. Specific attention is paid to the influence of
these numerical algorithms on the calculated results.
The present method for predicting steady sheet cavity flows on two-dimensional
and three-dimensional hydrofoils and on propeller blades is used extensively
and validated by experimental results in this thesis. Good agreement
between the calculations and the experiments is achieved. The dynamics of
sheet cavitation is predicted by the present method for a hydrofoil moving
into a sinusoidal gust and for a propeller rotating in a sharp wake peak. The
present method demonstrates the ability of capturing the dynamic movement
of the sheet cavitation.
At the end of the thesis, the conclusion is drawn that the present method
has the potential to predict the cavity topology and the cavity dynamics. After
improvement of some numerical algorithms, the efficiency of the method can
be enhanced to such a level that it can be applied in the early stage of ship
propeller design in order to prevent excessive cavitation and vibrations.
... This bound was originally demonstrated for oriented hypergraphs in [15], and we simply extend it to weighted hypergraphs. For our purposes its primary significance is only as a stepping stone to Theorem 2. We begin by defining the Rayleigh-Ritz quotient and stating the Min-Max theorem, proven in [7]. ...
A Bayesian Network (BN) is a probabilistic model that represents a set of variables using a directed acyclic graph (DAG). Current algorithms for learning BN structures from data focus on estimating the edges of a specific DAG, and often lead to many `likely' network structures. In this paper, we lay the groundwork for an approach that focuses on learning global properties of the DAG rather than exact edges. This is done by defining the structural hypergraph of a BN, which is shown to be related to the inverse-covariance matrix of the network. Spectral bounds are derived for the normalized inverse-covariance matrix, which are shown to be closely related to the maximum indegree of the associated BN.
We prove that the probability of “A or B”, denoted as p(A or B), where A and B are events or hypotheses that may be recursively dependent, is given by a “Hyperbolic Sum Rule” (HSR), which is relationally isomorphic to the hyperbolic tangent double-angle formula. We also prove that this HSR is Maximum Entropy (MaxEnt). Since this recursive dependency is commutative, it maintains the symmetry between the two events, while the recursiveness also represents temporal symmetry within the logical structure of the HSR. The possibility of recursive probabilities is excluded by the “Conventional Sum Rule” (CSR), which we have also proved to be MaxEnt (with lower entropy than the HSR due to its narrower domain of applicability). The concatenation property of the HSR is exploited to enable analytical, consistent, and scalable calculations for multiple hypotheses. Although they are intrinsic to current artificial intelligence and machine learning applications, such calculations are not conveniently available for the CSR, moreover they are presently considered intractable for analytical study and methodological validation. Where, for two hypotheses, we have p(A|B) > 0 and p(B|A) > 0 together (where “A|B” means “A given B”), we show that either {A,B} is independent or {A,B} is recursively dependent. In general, recursive relations cannot be ruled out: the HSR should be used by default. Because the HSR is isomorphic to other physical quantities, including those of certain components that are important for digital signal processing, we also show that it is as reasonable to state that “probability is physical” as it is to state that “information is physical” (which is now recognised as a truism of communications network engineering); probability is not merely a mathematical construct. We relate this treatment to the physics of Quantitative Geometrical Thermodynamics, which is defined in complex hyperbolic (Minkowski) spacetime.
For a one-dimensional semilinear wave equation with a free term that is a solution value at one given point (a Dirac potential), we consider the Cauchy problem in the upper half-plane. We construct the solution using the method of characteristics in implicit analytical form as a solution of some integral equations. The solvability of these equations, as well the smoothness of their solutions, is studied. For the problem in question, we prove the uniqueness of the solution, and establish the conditions under which its classical solution exists.
Simple closed-form analytical expressions for tilted astigmatic wave beams and wavepackets that are exact solutions of the wave equation are constructed. They are obtained through two different but equivalent derivations, one is based on a complex shift in the Bateman-type solutions, the other employs their Lorentz transformation. Analytic expressions for the propagation invariants: energy, momentum and orbital angular momentum of tilted waveobjects with general astigmatism are presented.
In the presented research work, we have solved a new kind of problem of forced shock waves in a compressible inviscid perfect gas having dirty (dust) particles of small size in a one‐dimensional unsteady adiabatic flow. The approach that we have used is referred to as the generalized geometry approach. Here, we investigated how the density of the zone, which is undisturbed, changes as a function of the position from the point of the source of explosion. In addition, we have obtained analytically a novel solution to the problem in the form of a new rule of power of time and distance. Further, we have investigated the energy behavior of forced shock waves and interaction within the environment containing dust particles. Also, the behavior of the entire energy of a forced shock wave is expounded at different Mach numbers, respectively, for planar geometry, cylindrically symmetric geometry, and spherically symmetric geometry under a dusty gas medium. Furthermore, the findings show that dust particles in a gas produce a more sophisticated representation rather than the standard gas dynamics.
In recent years, our research on bone mechanics has revealed that the fundamental laws in physical fractal space can be characterized by fractal operators. Based on the invariant properties of bone fractal operators, we used the error function as the core and derived the fractional-order correlation between different special functions. This paper is a continuation of the previous work. Inspired by bone fractal operators, we aim to logically construct a Golden Meta-Spring to illustrate the interconnections between various disciplines. Specifically, the following contents are included: (1) originating from the Golden Ratio, we present the construction process of Golden Meta-Spring; (2) based on the continued fraction theory, we discuss the properties, characteristics, and interdisciplinary insights provided by various types of Meta-Springs; (3) using the bone fractal operators as the link, we demonstrate the correlations between different disciplines.
One‐dimensional distributed‐parameter systems consisting of several convection‐reaction equations, which are distributedly coupled with a reaction equation, are considered. To solve the open‐loop control problem for these kind of systems, an inversion‐based approach is proposed. Along the common characteristic projection, the system can be reduced to a single partial differential equation (PDE) whose boundary conditions (BCs) result from the particular choice of the output to be tracked. The solution of this latter equation constitutes the basis of the control design. The particular properties of the scalar boundary value problem (BVP) to be solved depend on the coupling structure of the originally given convection‐reaction system. While in most cases the output has a relative degree of zero, for practical applications a relative degree of one is of particular interest. In this case, the obtained scalar BVP which describes the internal dynamics, possesses the structure of an infinite‐dimensional differential‐algebraic equation (DAE). Within the paper, the properties of the solution of both problems are discussed from a system analytical point of view.
In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be characterized by fractal operators, and each type of operator possesses rich invariant properties. Based on the invariant properties of fractal operators, we discovered that the symmetry evolution laws of functional fractal trees in the physical fractal space can reveal the intrinsic correlations between special functions. This article explores the fractional-order correlation between special functions inspired by bone fractal operators. Specifically, the following contents are included: (1) showing the intrinsic expression in the convolutional kernel function of bone fractal operators and its correlation with special functions; (2) proving the following proposition: the convolutional kernel function of bone fractal operators is still related to the special functions under different input signals (external load, external stimulus); (3) using the bone fractal operators as the background and error function as the core, deriving the fractional-order correlation between different special functions.
In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we abstracted a family of fractal operators and investigate the kernel function and properties thereof. Based on fractal operators, the intrinsic relation between Bessel function and Struve function was revealed, and some new special functions were found. The results provide mathematical tools for biomechanics and automatic control.
The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this branch of mathematical analysis. This paper presents two non-classical, distinct methods for solving such problems. The first method is based on the differential transform method (DTM), which seeks an analytical solution in the form of a certain functional series. The second method, on the other hand, is based on the physics-informed neural network (PINN), where artificial intelligence in the form of a neural network is used to solve the differential equation. In addition to describing both methods, this paper also presents numerical examples along with a comparison of the obtained results.Comparingthe two methods, DTM produced marginally more accurate results than PINNs. While PINNs exhibited slightly higher errors, their performance remained commendable. The key strengths of neural networks are their adaptability and ease of implementation. Both approaches discussed in the article are effective for addressing the examined problems.
More often, in the mathematical literature, the injectivity of the spherical Radon transform (SRT) for compactly supported functions is considered. In this article, an additional condition, for the reconstruction of an unknown function (the support can be noncompact) using the circular Radon transform (CRT) over circles centered on a smooth simple curve is found. It is proved that this problem is equivalent to the injectivity of a so‐called two‐data CRT over circles centered on a smooth curve (can be a segment). Also, we present an inversion formula of the transform that uses the local data of the circular integrals to reconstruct the unknown function. Such inversions are the mathematical base of modern modalities of imaging, such as thermo‐ and photoacoustic tomography and radar imaging, and have theoretical significance.
This paper employs Nevanlinna value distribution theory in several complex variables to explore the characteristics and form of finite as well as infinite order solutions of Circular-type nonlinear partial differential-difference equations in Cn. The results obtained in this paper contribute to the improvement and generalization of some recent results. In addition, illustrative examples are provided to validate the conclusions drawn from the main results. Moreover, we investigate the infinite-order solutions of Circular-type functional equations in Cn.
This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: p+α2-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators.
Because of the benefits of solar energy, solar photovoltaic (PV) technology is being deployed at an unprecedented rate and the number of photovoltaic panels is sharply increasing. Agrophotovoltaic systems (solar farms) seem to be the most sustainable tools to create renewable energy without compromising agricultural production. However, utility-scale solar energy development is land intensive and its large-scale installation can have negative impacts on the environment. Moreover, its impacts on soil and on relative hydrological processes have been poorly studied. This paper aims to evaluate the impact of solar panels on the runoff generation process, which is directly linked to the soil erosion process. Using a rainfall simulator, runoff measurements for a rainfall intensity equal to 56 mm h-1 were carried out by assuming different panel arrangements with respect to the maximum slope direction of the field (cross slope and aligned slope). Results were compared to a control reference of the same plot, with no panels (bare soil). Physical models found in the literature were then applied and calibrated, to upscale the models to a much higher hillslope length. Results showed that solar panels increase the peak discharge by about 11 times compared to the reference hillslope. A moderate effect of PV panel arrangement was observed on the peak discharges (11.7 and 11.5 times higher, for cross slope and aligned slope panels, respectively), whereas the time to runoff was the lowest for aligned slope panels (0.3 h), higher for cross slope panels (0.62 h), and the highest (1.2 h), for the bare soil hillslope. As it would be expected, upscaling the models to longer hillslopes resulted in increases in outlet discharges, and in the time to runoff, with an exception for aligned slope panels.
The applications of spherical Radon transforms include synthetic aperture radar, sonar tomography, and medical imaging modalities. A spherical Radon transform maps a function to its integrals over a family of spheres. Recently, several types of incomplete spherical Radon transforms have received attention in research. This study examines two types of quarter-spherical Radon transforms that assign a function to its integral over a quarter of a sphere: 1) center of a quarter sphere of integration on a plane, and 2) center on a line and the rotation of the quarter sphere. Furthermore, we present inversion formulas for these two quarter-spherical Radon transforms.
In a bounded cylinder with a rough interface we study the asymptotic behaviour of the spectrum and its associated eigenspaces for a stationary heat propagation problem. The main novelty concerns the proof of the uniform a priori estimates for the eigenvalues. In fact, due to the peculiar geometry of the domain, standard techniques do not apply and a suitable new approach is developed.
This article mainly deals with the blow-up properties of nonnegative solutions for a reaction–diffusion system coupled with norm-type sources under positive boundary value conditions. The local existence of a nonnegative solution and the comparison principle are given. The criteria for the global existence or finite time blow-up of the solutions are obtained by constructing new functions and utilizing the super- and -sub-solution method. The results reveal a correlation between the blow-up profiles of the solutions and the size of the domain, as well as the positive boundary value.
In this paper, we propose an empirical model for the VIX index. Our findings indicate that the VIX has a long-term empirical distribution. To model its dynamics, we utilize a continuous-time Markov process with a uniform distribution as its invariant distribution and a suitable function h. We determined that h is the inverse function of the VIX data's empirical distribution. Additionally, we use the method of variables of separation to get the exact solution to the pricing problem for VIX futures and call options.
This article studies the convolutional kernel function of fractal operators in bone fibers. On the basis of the micro-nano composite structure of compact bone, we abstracted the physical fractal space of bone fibers and derived the fractal operators. The article aims to construct the convolutional analytical expression of bone fractal operators and proves that the error function is the core component of the convolution kernel function in the fractal operators. In other words, bone mechanics is the fractional mechanics controlled by error function.
In this paper, we study the stability analysis of thermodynamic configurations in the space of matrices, arising from an embedding of real thermodynamic spaces under fluctuations of the model parameters. The model parameters under the present consideration are the thermal expansion parameter, and the isothermal and/or adiabatic compressibilities. Under parametric fluctuations, we examine the matrix valued Fourier series and Fourier transforms by invoking the limiting local and global stability structures. The resulting embeddings are taken as the specific heat capacity at a constant pressure and that of at a constant volume, their ratio, and Mayer’s relation. In conclusion, by examining the Fourier series and Fourier transforms in the space of regular matrices, we have explicated the stability properties of such configurations under fluctuations of experimentally measurable quantities. To be specific, our method determines positions of the underling critical phenomena such as the phase transition points/ curves, bifurcations, cross-overs and others. Indeed, we provide highlights for possible geometric extensions by discussing the standing notions, such as the Onsager region, local stability structures and global parameter space invariants, and others. Moreover, an extension of our approach for the thermodynamic systems away from attractor fixed points may be examined for moduli space configurations through certain geometric and algebraic techniques. Finally, we have discussed prospective research directions toward the existence of the series representations of matrix valued functions, series expansions with discontinuities, integral kernels, and their multi-matrix extensions in the light of thermodynamic (in)stabilities under fluctuations of the system parameters.
This paper considers a class of hyperbolic-parabolic Partial Differential Equation (PDE) system withsome interior mixed-coupling terms, a rather unexplored family of systems. The family of systems we explore contains several interior-coupling terms, which makes controller design more challenging. Our goal is to design a boundary controller to exponentially stabilize the coupled system. For that, we propose a controller whose design is based on the backstepping method. Under this controller, we analyse the stability of the closed loop in the
sense. A set of (highly coupled) backstepping kernel equations are derived, and their well-posedness is shown in the appropriate spaces by an infinite induction energy series, which has not been used before in this setting. Moreover, we show the invertibility of transformations by displaying the inverse transformations, as required for closed-loop well-posedness and stability. Finally, a numerical simulation is implemented, and the result illustrates that the control law designed by the backstepping transformation can stabilize the mixed PDE system exponentially.
This paper is concerned with the Sobolev type weak solutions of one class of second order quasilinear parabolic partial differential equations (PDEs, for short). First of all, similar to Feng, Wang and Zhao [9] and Wu and Yu [29], we use a family of coupled forward-backward stochastic differential equations (FBSDEs, for short) which satisfy the monotonous assumption to represent the classical solutions of the quasilinear PDEs. Then, based on the classical solutions of a family of PDEs approximating the weak solutions of the quasilinear PDEs, we prove the existence of the weak solutions. Moreover, the principle of norm equivalence is employed to link FBSDEs and PDEs to obtain the uniqueness of the weak solutions. In summary, we provide a probabilistic interpretation for the weak solutions of quasilinear PDEs, which enriches the theory of nonlinear PDEs.
Source extension is a reformulation of inverse problems in wave propagation, that at least in some cases leads to computationally tractable iterative solution methods. The core subproblem in all source extension methods is the solution of a linear inverse problem for a source (right hand side in a system of wave equations) through minimization of data error in the least squares sense with soft imposition of physical constraints on the source via an additive quadratic penalty. For an acoustic formulation for sources supported on a surface, with a soft contraint enforcing concentration at a point, a variant of the time reversal method from photoacoustic tomography provides an approximate solution. This approximate inverse can be used to precondition Krylov space iteration for rapid convergence to the solution of the core subproblem in this setting. Numerical examples illustrate the effectiveness of this preconditioner.
The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhuis and Haantjes, offers new tools for the study of normal forms of operator fields. We prove a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level m vanishes, there exists a local chart where all operators can be simultaneously block-diagonalized. We also introduce the notion of generalized Haantjes algebra, consisting of operators with a vanishing higher-level torsion, as a new algebraic structure naturally generalizing standard Haantjes algebras.
The paper presents some new results of investigation developing the approach aimed at applying the group theory methods to radiation transfer problems. It consists of two separate parts. In the first part we derive new properties of supersymmetry of fundamental supermatrices - representations of composition and translation groups. It is shown that these supermatrices, which determine the layers adding to the two opposite boundaries of inhomogeneous medium, are connected with each other by the procedure of parity transposition. It is also demonstrated that, by analogy with the common second order matrices, the considered supermatrices can be factorized yielding the product of triangular supermatrices. The second part generalizes and applies the concept of composition groups to the case of media with spherical symmetry.
The transmutation operator method is extended to the case of functions of two variables. The transmutation operator flattens the function, i.e. the transmutation operator replaces a function with discontinuous partial derivatives on the coordinate axes by a continuously differentiable function. The work reveal the properties of the transmutation operator, and prove the commutativity of the transmutation operator and the Laplace operator. It was found that the Cauchy problem for the Laplace equation with internal conjugations in an unbounded domain can be replaced with the model Cauchy problem for the twodimensional Laplace equation. As a result, a new analytical method for solving initial-boundary value problems for a two-dimensional heat equation has been developed. The factorization of the transmutation operator is established as a product of two one-dimensional transmutation operators. The form of the transmutation operator establishing the isomorphism of two mathematical models of heat conduction in unbounded media with different physical characteristics was found and descrfibed.
A bstract
We study dynamic processes through which the scalar hair of black holes is generated or detached in a theory with a scalar field non-minimally coupled to Gauss-Bonnet and Ricci scalar invariants. We concentrate on the nonlinear temporal evolution of a far-from-equilibrium gravitational system. In our simulations, we choose the initial spacetime to be either a bald Schwarzschild or a scalarized spherically symmetric black hole. Succeeding continuous accretion of the scalar field onto the original black hole, the final fate of the system displays intriguing features, which depend on the initial configurations, strengths of the perturbation, and specific metric parameters. In addition to the scalarization process through which the bald black hole addresses scalar hair, we observe the dynamical descalarization, which removes scalar hair from an original hairy hole after continuous scalar field accretion. We examine the temporal evolution of the scalar field, the metrics, and the Misner-Sharp mass of the spacetime and exhibit rich phase structures through nonlinear dynamical processes.
In previous work a variational integration scheme for mechanical systems containing linear dissipation was developed [1]. The key idea is to use a transmission line as the solution to a power-balancing control problem-in the physics literature controllers of this type are known as ‘heat baths’. The power-balanced closed-loop system is thereby made amenable to analysis with Hamilton's principle. As a consequence, variational principles can be used to find symplectic integration schemes for dissipative systems. The outcome of this work is a coherent physical explanation as to why variational/symplectic integration algorithms for open dissipative system can perform as well as their conservative counterparts. An important application of these integration algorithms is the solution of long-time-horizon optimal control problems. In this paper these ideas are extended to systems with smooth memoryless nonlinear dissipation. This extension employs a nonlinear lossless transmission line, which is again the solution to a power-balancing control problem. As in the case of linear dissipation, the resulting variational integration schemes are demonstrated to have excellent numerical properties, which are significantly less demanding computationally than their implicit non-variational counterparts.
In a previous work we provided the explicit form of the nonlinear PDEs, subjected to the appropriate boundary conditions, which have to be satisfied by transport coefficients for systems out of Onsager’s region. Since the proposed PDEs are obtained without neglecting any term present in the balance equations (i.e., the mass, momentum, and energy balance equations), we propose them as a good candidate for describing also transport in thermodynamic systems in turbulent regime. As a special case, we derive the nonlinear PDEs for transport coefficients when the thermodynamic system is subjected to two thermodynamic forces. In this case, the obtained PDE is, in thermodynamical field theory (TFT), analogous to Liouville’s equation in Riemannian (or pseudo-Riemannian) geometry. The validity of our model is tested by analysing a concrete example where Onsager’s relations manifestly disagree with experience: transport in Tokamak-plasmas. More specifically, we compute the electron mass and energy losses in turbulent FTU (Frascati Tokamak Upgrade)-plasmas. We show the agreement between the theoretical predictions and experimental observations. This approach allows to predict the values of the Bohm and the gyro-Bohm coefficients. To the best of our knowledge, it is the first time that such coefficients have been evaluated analytically. The aim of this series of works is to apply our approach to the Divertor Tokamak Test facility (DTT), to be built in Italy, and to ITER.
All causal Lie products of solutions of the Klein-Gordon equation and the wave equation in Minkowski space are determined. The results shed light on the origin of the algebraic structures underlying quantum field theory.
We introduce a metasurface-based system for the microwave hyperthermia treatment of brain tumors, where the metasurface is designed by the time-reversal technique for optimal focusing at the location of the tumor. Compared to previous time-reversal microwave hyperthermia systems, which are based on antenna arrays, the proposed system offers two advantages: 1) it provides superior focal resolution, due to the finer sampling capability of its deeply-subwavelength metasurface particles compared to that of typical array antenna elements, 2) it features much lower complexity and cost, due to its direct illumination principle compared to the heavy feeding network of the array system. The system leverages reflecting metallic walls and a rich scattering structure in combination with the metasurface to achieve high hyperthermia performance with maximal simplicity. The metasurface system is demonstrated by full-wave simulation results, which show that it represents a viable novel noninvasive hyperthermia technology.
We compute the scattering of unsteady acoustic waves about complex three-dimensional bodies with high order accuracy. The geometry of a scattering body is defined with the help of CAD. Its surface is represented as a collection of non-overlapping patches, each parameterized independently by means of high order splines (NURBS). As a specific example, we consider a submarine-like scatterer constructed using five different patches.
The acoustic wave equation on the region exterior to the scatterer is solved by first reducing it to a system of Calderon's boundary operator equations. The latter are obtained using the method of difference potentials coupled with a compact fourth order accurate finite difference scheme. When solving the boundary operator equations, we employ Huygens' principle. It allows us to work on a sliding time window of non-increasing duration rather than keep the entire temporal history of the solution at the boundary.
The proposed methodology demonstrates grid-independent computational complexity at the boundary and sub-linear complexity with respect to the grid dimension. It efficiently handles complex non-conforming geometries on Cartesian grids with no penalty for either accuracy or stability due to the cut cells. Its performance does not deteriorate over arbitrarily long simulation times. The exact treatment of artificial outer boundaries is inherently built in. Finally, multiple similar problems can be solved efficiently at a low individual cost per problem. This is important when, for example, the boundary condition on the surface changes but the scattering body stays the same.
This work is to investigate terminal value problem for a stochastic time fractional wave equation, driven by a cylindrical Wiener process on a Hilbert space. A representation of the solution is obtained by basing on the terminal value data u(T,x)=φ(x) and the spectrum of the fractional Laplacian operator (−Δ)s/2 (in a bounded domain 𝕏⊂ℝd, 0<s<2). First, we show the existence and uniqueness of a mild solution in Lp(0,T;L2(Ω,V))∩C((0,T];L2(Ω,L2(𝕏))), for a suitable sub‐space V V of L2(𝕏). A limitation of this result is the lack of time continuity at t=0 t=0 . Second, we study the inverse problem (IP) of recovering u(0,x) when the terminal value data φ and the source f f are given. We give an explanation why the time continuity of the solution at t=0 t=0 could not derived. The main reason comes from unboundedness of a solution operator, so the problem (IP) is then ill‐posed, that is, recovery u(0,x) cannot be obtained in general. Hence, we propose a truncation regularization method with a suitable choice of the regularization parameter. Finally, we present a numerical example to demonstrate our proposed method.
The paper focuses on a problem arising in nonlinear guided wave optics. The problem for Maxwell's equations with nonhomogeneous nonlinear constitutive law is reduced to an eigenvalue problem for a system of nonlinear ordinary differential equations with local boundary conditions. We suggest a novel approach to study eigenvalue problems for nonlinear nonautonomous equations based on studying an integral characteristic equation (ICE) of the problem. Using asymptotical analysis of the ICE, we prove existence of infinitely many eigenvalues and find their distribution. It is important that the corresponding linear problem has only a finite number of eigenvalues.
In many engineering applications it is often necessary to determine the flow of shear stresses in the cross-sections of beamlike bodies. Taking a cue from Jourawski's well-known formula, several scholars have proposed expressions for evaluating the shear stresses in non-prismatic linear elastic beams, where longitudinal variations in the size and shape of the cross-sections produces complex stress fields. In the present paper, a new shear formula, derived using a mechanical model developed in a previous work, is presented for tapered beams subject to even large displacements and small strains. Numerical examples and comparisons with results obtained using other formulas in the literature and non-linear 3D-FEM simulations show how the new formula constitutes an important generalization of the previous ones and is able to provide particularly accurate results.
In this paper, we study the boundedness of multilinear and multiparameter spectral multipliers on a product space of Lie groups of polynomial growth G=G1×⋯×GM. We firstly prove the boundedness from the product of homogeneous Besov spaces B˙p1,q1s1(G)×B˙p2,q2s2(G)×⋯×B˙pN,qNsN(G) to Lebesgue spaces Lp(G) with p1,…,pN,q1,…,qN,p⩾1 and s1,…,sN∈RM. Then we prove the boundedness from the product of homogeneous Triebel–Lizorkin spaces T˙p1,q1s1(G)×T˙p2,q2s2(G)×⋯×T˙pN,qNsN(G) to Lebesgue spaces Lp(G) with p1,…,pN,q1,…,qN>1, p⩾1, s1,…,sN∈RM.
In the previous chapter, we discuss general soft matter using the theory and method developed for solving soft-matter quasicrystals, where we emphasized one must consider the structure of concrete soft matter. In this chapter, we study a concrete soft matter, i.e., the smectic A liquid crystal and its dislocation and crack problem. These are interesting topics in soft matter. Apart from this, we hope to explore a longstanding puzzle, perhaps a paradox. The solution to the paradox may yield some beneficial results and lessons.
The objective of this paper is, for the first time, to extend the fractional Laplacian (−△) s u ( x ) over the space C k ( R ⁿ ) (which contains S ( R ⁿ ) as a proper subspace) for all s > 0 and s ≠ 1, 2, …, based on the normalization in distribution theory, Pizzetti’s formula and surface integrals in R ⁿ . We further present two theorems showing that our extended fractional Laplacian is continuous at the end points 1, 2, … . Two illustrative examples are provided to demonstrate computational techniques for obtaining the fractional Laplacian using special functions, Cauchy’s residue theorem and integral identities. An application to defining the Riesz derivative in the classical sense at odd numbers is also considered at the end.
Let be the semigroup of linear operators generated by a Schrödinger operator on Heisenberg group , where is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class and Q is the homogeneous dimension of . Let \int _0^{\infty } \uplambda dE_{{\mathcal {L}}}(\uplambda ) be the spectral resolution of , we prove that if a function F satisfies a Mihlin condition with exponent then the operator F({\mathcal {L}})=\int _0^\infty F(\uplambda )dE_{{\mathcal {L}}}(\uplambda ) is bounded on Hardy space .
We propose a new, infinite class of brackets generalizing the Frölicher–Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We also prove that the vanishing of a higher-level Nijenhuis torsion of an operator field is a sufficient condition for the integrability of its eigen-distributions. This result (which does not require any knowledge of the spectral properties of the operator) generalizes the celebrated Haantjes theorem. The same vanishing condition also guarantees that the operator can be written, in a local chart, in a block-diagonal form.