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A class of left bialgebroids whose underlying algebra A ♯ H is a smash product of a bialgebra H with a braided commutative Yetter–Drinfeld H -algebra A has recently been studied in relation to models of field theories on noncommutative spaces. In Borowiec and Pachoł (2017 J. Phys. A: Math. Theor. 50 055205) a proof has been presented that the bialgebroid A F ♯ H F where HF and AF are the twists of H and A by a Drinfeld 2-cocycle F = ∑ F 1 ⊗ F 2 is isomorphic to the twist of bialgebroid A ♯ H by the bialgebroid 2-cocycle ∑ 1 ♯ F 1 ⊗ 1 ♯ F 2 induced by F. They assume H is quasitriangular, which is reasonable for many physical applications. However the proof and the entire paper take for granted that the coaction and the prebraiding are both given by special formulas involving the R-matrix. There are counterexamples of Yetter–Drinfeld modules over quasitriangular Hopf algebras which are not of this special form. Nevertheless, the main result essentially survives. We present a proof with a general coaction and the correct prebraiding, and even without the assumption of quasitriangularity. [ABSTRACT FROM AUTHOR]

The authors replies to the comment made by Yang and Zhou (2024 J. Phys. A: Math. Theor.) on his 2015 paper entitled 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras' (Jacobsen 2015 J. Phys. A: Math. Theor. 48 454003). [ABSTRACT FROM AUTHOR]

In 1989 Mason and Newman (Commun. Math. Phys. 121 659–68) proved that there is a 1-1-correspondence between self-dual metrics satisfying Einstein vacuum equation (in complex case or in neutral signature) and pairs of commuting parameter depending vector fields X 1 (λ) , X 2 (λ) which are divergence free with respect to some volume form. Earlier Plebański (1975 J. Math. Phys. 16 2395–402) showed instances of such vector fields depending of one function of four variables satisfying the so-called I or II Plebański heavenly PDEs. Other PDEs leading to Mason–Newman vector fields are also known in the literature: Park (1992 Int. J. Mod. Phys. A 7 1415), Husain (1994 Phys. Rev. Lett. 72 800), Grant (1993 Phys. Rev. D 48 2606–12), Schief (Phys. Lett. A 223 55–62). In this paper we discuss these matters in the context of the web theory, i.e. theory of collections of foliations on a manifold, understood from the point of view of Nijenhuis operators. In particular we show how to apply this theory for constructing new 'heavenly' PDEs based on different normal forms of Nijenhuis operators in 4D, which are integrable similarly to their predecessors. Relation with the Hirota dispersionless systems of PDEs and the corresponding Veronese webs, which was recently observed by Konopelchenko–Schief–Szereszewski, is established in all the cases. We also discuss some higher dimensional generalizations of the 'heavenly' PDEs and the existence of related vacuum Einstein metrics in 4D-case. [ABSTRACT FROM AUTHOR]

In the paper, we are concerned with the nonlinear Cauchy problem on the Vlasov–Poisson–Landau/Boltzmann system around global Maxwellians in a torus or a finite channel. The main goal is to establish the global existence and large time behaviour of small amplitude solutions for a class of low regularity initial data. The molecular interaction type is restricted to the case of hard potentials for two classical collision operators because of the effect of the self-consistent forces. The result extends the one by Duan–Liu–Sakamoto-Strain (Duan et al 2021 Commun. Pure Appl. Math. 74 932–1020) for the pure Landau/Boltzmann equation to the case of the VPL and VPB systems. [ABSTRACT FROM AUTHOR]

In this paper, we present the construction of all nonstandard integrable systems in magnetic fields whose integrals have leading order structure corresponding to the case (i) of theorem 1 in Marchesiello and Šnobl (2022 J. Phys. A: Math. Theor. 55 145203). We find that the resulting systems can be written as one family with several parameters. For certain limits of these parameters the system belongs to intersections with already known standard systems separating in Cartesian and / or cylindrical coordinates and the number of independent integrals of motion increases, thus the system becomes minimally superintegrable. These results generalize the particular example presented in section 3 of Marchesiello and Šnobl (2022 J. Phys. A: Math. Theor. 55 145203). [ABSTRACT FROM AUTHOR]

It is well-known that any Lennard–Jones type potential energy must have a periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method that rigorously shows such global minimality result among 2-dimensional lattices once the exponents of the potential have been fixed. The method is applied to the widely used classical (12 , 6) Lennard–Jones potential, which is the main result of this work. Furthermore, a new bound on the inverse density (i.e. the co-volume) for which the triangular lattice is minimal is derived, improving those found in (Bétermin and Zhang 2015 Commun. Contemp. Math. 17 1450049) and (Bétermin 2016 SIAM J. Math. Anal. 48 3236–3269). The same results are also shown to hold for other exponents as additional examples and a new conjecture implying the global optimality of a triangular lattice for any parameters is stated. [ABSTRACT FROM AUTHOR]

In this paper, the initial-boundary value problem of the one-dimensional full compressible Navier–Stokes equations with positive constant viscosity but with zero heat conductivity is considered. Global well-posedness is established for any H1 initial data. The initial density is assumed only to be nonnegative, and, thus, is not necessary to be uniformly away from vacuum. Comparing with the well-known result of Kazhikhov and Shelukhin (1977 J. Appl. Math. Mech. 41 273–282), the heat conductive coefficient is zero in this paper, and the initial vacuum is allowed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Levenbergâ€"Marquardt method with general regularization terms that are uniformly convex on bounded sets to solve the ill-posed inverse problems in Banach spaces, where the forward mapping might not Gâteaux differentiable and the image space is unnecessarily reflexive. The method therefore extends the one proposed by Jin and Yang in (2016 Numer. Math. 133 655â€"684) for smooth inverse problem setting with globally uniformly convex regularization terms. We prove a novel convergence analysis of the proposed method under some standing assumptions, in particular, the generalized tangential cone condition and a compactness assumption. All these assumptions are fulfilled when investigating the identification of the heat source for semilinear elliptic boundary-value problems with a Robin boundary condition, a heat source acting on the boundary, and a possibly non-smooth nonlinearity. Therein, the Clarke subdifferential of the non-smooth nonlinearity is employed to construct the family of bounded operators that is a replacement for the non-existing Gâteaux derivative of the forward mapping. The efficiency of the proposed method is illustrated with a numerical example. [ABSTRACT FROM AUTHOR]

Let us say that a three-qubit state u 000|000âź© + u 001|001âź© + â‹Ż + u 111|111âź© is real if all its amplitudes u rst are real numbers. We will prove that for every real three-qubit state | Ď• âź©, there exist three angles θ 0, θ 1 and θ 2 such that R y (θ 2) ⊗ R y (θ 1) ⊗ R y (θ 0)| Ď• âź© is a three-qubit of the form λ 1|000âź© + λ 2|011âź© + λ 3|101âź© + λ 4|110âź© + λ 5|111âź© with the λ i real numbers. In contrast with the general case, the case of three-qubits with complex amplitudes, we proved that for three qubit states, the dimension of the real entanglement space (the space obtained by identifying real qubit states with local orthogonal gates, instead of local unitary gates) is 4 and in this paper we find four linearly independent polynomial invariants of degree 4 which are not possible to find for the different Schmidt representations of three qubit states. See (AcĂ-n et al 2000 Phys. Rev. Lett. 85 1560; AcĂ-n et al 2001 J. Phys. A: Math. Gen. 34 6725; Carteret et al 2000 J. Math. Phys. 41 7932; Sudbery 2001 J. Phys. A: Math. Gen. 34 643). [ABSTRACT FROM AUTHOR]

This paper studies the conductance on the universal homology covering space Z of 2D orbifolds in a strong magnetic field, thereby removing the rationality constraint on the magnetic field in earlier works (Avron et al 1994 Phys. Rev. Lett. 73 3255–3257; Mathai and Wilkin 2019 Lett. Math. Phys. 109 2473–2484; Prieto 2006 Commun. Math. Phys. 265 373–396) in the literature. We consider a natural Landau Hamiltonian on Z and study its spectrum which we prove consists of a finite number of low-lying isolated points and calculate the von Neumann degree of the associated holomorphic spectral orbibundles when the magnetic field B is large, and obtain fractional quantum numbers as the conductance. [ABSTRACT FROM AUTHOR]

In this paper we prove that, after an appropriate rescaling, the sum of moments of an N × N Hermitian matrix H sampled according to the generalized Cauchy (also known as Hua–Pickrell) ensemble with parameter s > 0 is a continuous-Hahn polynomial in the variable k. This completes the picture of the investigation that began in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) where analogous results were obtained for the other three classical ensembles of random matrices, the Gaussian, the Laguerre and Jacobi. Our strategy of proof is somewhat different from the one in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) due to the fact that the generalized Cauchy is the only classical ensemble which has a finite number of integer moments. Our arguments also apply, with straightforward modifications, to the other three cases studied in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) as well. We finally obtain a differential equation for the one-point density function of the eigenvalue distribution of this ensemble and establish the large N asymptotics of the moments. [ABSTRACT FROM AUTHOR]

Godrčche, in Comment on 'Fluctuation dominated phase ordering at a mixed order transition' [2021 J. Phys. A: Math. Theor. 54 038001], has commented on our recent paper Fluctuation dominated phase ordering at a mixed order transition (2019 J. Phys. A: Math. Theor. 52 254001). This comment concerns the prefactor of the cusp-like small-argument singularity of the scaled spin-spin correlation function at criticality. We remark that the approach used in our paper is adequate for computing the cusp exponent, which is what is really needed to establish fluctuation-dominated phase ordering. Computing the precise value of the prefactor of the cusp singularity is irrelevant for this purpose, or for the physics behind the relation between the mixed order phase transition and the fluctuation-dominated phase ordering—the understanding of which was the main purpose of our paper. [ABSTRACT FROM AUTHOR]

We consider the Sasa–Satsuma (SS) and nonlinear Schrödinger (NLS) equations posed along the line, in 1 + 1 dimensions. Both equations are canonical integrable U(1) models, with solitons, multi-solitons and breather solutions Yang (2010 SIAM Mathematical Modeling and Computation). For these two equations, we recognize four distinct localized breather modes: the Sasa–Satsuma for SS, and for NLS the Satsuma–Yajima, Kuznetsov–Ma and Peregrine breathers. Very little is known about the stability of these solutions, mainly because of their complex structure, which does not fit into the classical soliton behavior Grillakis et al (1987 J. Funct. Anal. 74 160–97). In this paper we find the natural H2 variational characterization for each of them. This seems to be the first known variational characterization for these solutions; in particular, the first one obtained for the famous Peregrine breather. We also prove that Sasa–Satsuma breathers are H2 nonlinearly stable, improving the linear stability property previously proved by Pelinovsky and Yang (2005 Chaos 15 037115). Moreover, in the SS case, we provide an alternative understanding of the SS solution as a breather, and not only as an embedded soliton. The method of proof is based in the use of a H2 based Lyapunov functional, in the spirit of Alejo and Muńoz (2013 Commun. Math. Phys. 324 233–62), extended this time to the vector-valued case. We also provide another rigorous justification of the instability of the remaining three nonlinear modes (Satsuma–Yajima, Peregrine and Kuznetsov–Ma), based in the study of their corresponding linear variational structure (as critical points of a suitable Lyapunov functional), and complementing the instability results recently proved e.g. in Muńoz (2017 Proyecciones (Antofagasta) 36 653–83). [ABSTRACT FROM AUTHOR]

In this paper, we consider a compressible–incompressible two-velocity hydrodynamic system studied by Bresch et al (2015 J. Math. Pure Appl. 104 762–800) and Lions (1996 Mathematical Topics in Fluid Mechanics (Oxford: OUP)). When the density ρ is a small perturbation of a constant, we establish a priori estimates by using some delicate structure of the nonlinear terms and Hardy space. By using these a priori estimates, we prove the existence of global strong solutions and weak solutions. Our results do not require any constraint between the viscosity and the conductivity and improve the results of Bresch et al (2015) and Lions (1996) in the two-dimensional case. As an application of our results we also establish global strong solutions and weak solutions for a model of gaseous mixture and the ghost effect system. [ABSTRACT FROM AUTHOR]

Asymmetric lemon billiards was introduced in Chen et al (2013 Chaos 23 043137), where the billiard table Q(r, b, R) is the intersection of two round disks with radii r ⩽ R, respectively, and b measures the distance between the two centres. It is conjectured Bunimovich et al (2016 Commun. Math. Phys. 341 781–803) that the asymmetric lemon billiards is hyperbolic when the arc Γr is a major arc and R is large. In this paper we prove this conjecture for sufficiently large R. [ABSTRACT FROM AUTHOR]

In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field ϕ on a globally hyperbolic spacetime M with C1,1 metric g. This first entails showing that the (classical) Cauchy problem for the wave equation is well-posed for initial data and sources in Sobolev spaces and then constructing low-regularity advanced and retarded Green operators as maps between suitable function spaces. In specifying the relevant function spaces we need to control the norms of both ϕ and □gϕ in order to ensure that □g◦G± and G±◦□g are the identity maps on those spaces. The causal propagator G = G+ − G− is then used to define a symplectic form ω on a normed space V(M) which is shown to be isomorphic to ker(□g). This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local C*-algebras. [ABSTRACT FROM AUTHOR]

This paper aims to provide a robust model for the well-known phenomenon of Kármán vortex street arising in fluid mechanics. The first theoretical attempt to model this pattern was given by von Kármán (1991 Göttinger Nachr., Math. Phys. Kl. 509–17); von Kármán (1912 Göttinger Nachr., Math. Phys. Kl. 547–56). He considered two parallel staggered rows of point vortices, with opposite strength, that translate at the same speed. Following the ideas of Saffman and Schatzman (1981 SIAM J. Sci. Stat. Comput. 2 285–95), we propose to study this phenomenon in the Euler equations by considering two infinite arrows of vortex patches. The key idea is to desingularize the point vortex model proposed by von Kármán. Our construction is flexible and can be extended to more general incompressible models. [ABSTRACT FROM AUTHOR]

This paper examines classical privacy amplification using a universal family of hash functions. In quantum key distribution, the adversary's measurement can wait until the choice of hash functions is announced, and so the adversary's information may depend on the choice. Therefore the existing result on classical privacy amplification, which assumes the independence of the choice from the other random variables, is not applicable to this case. This paper provides a security proof of privacy amplification which is valid even when the adversary's information may depend on the choice of hash functions. The compression rate of the proposed privacy amplification can be taken to be the same as that of the existing one with an exponentially small loss in secrecy of a final key. [ABSTRACT FROM AUTHOR]

In 2008, the conjecture that structural physical approximations (SPAs) to optimal entanglement witnesses are separable states (in general unnormalized) was posed. In an attempt to disprove it, Ha and Kye (2012 arXiv:1210.1088v3) proposed a decomposable entanglement witness, whose SPA is entangled, and argued that it is optimal. In this paper, which is based on a comment to the latter work (Augusiak et al 2013 arXiv:1304.2040v1), we show, both analytically and numerically, that this entanglement witness is not optimal, and as such it is not a counterexample to the conjecture. To this end, we make use of a method for checking the optimality of entanglement witnesses developed already in Lewenstein et al (2000 Phys. Rev. A 62 052310), however, hardly exploited so far in the literature. [ABSTRACT FROM AUTHOR]

In this paper we try to extract stable components in the extended-range forecast for the coming 10–30 days by using empirical orthogonal function (EOF) analysis, similarity coefficient, and some other methods based on the National Center for Environmental Prediction (NCEP)/National Center for Atmospheric Research (NCAR) reanalysis daily data. The comparisons of the coefficient of variance of climatological background field and truth data in winter between 2010 and 2011 are made. The method of extracting stable components and climatological background field can be helpful to increase forecasting skill. The forecasting skill improvement of air temperature is better than geopotential height at 500 hPa. Moreover, this method improves the predictability better in the Pacific Ocean. In China, the forecast in winter in Northeast China is more uncertain than in the other parts. [ABSTRACT FROM AUTHOR]

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

n-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. In this paper, we construct (n+1)-Lie algebras by two-dimensional extensions of metric n-Lie algebras, and construct (n + 1)-Lie algebras in terms of n-Lie algebras and certain linear functions. Along this approach, we obtain (n+1)-Lie algebras by n-Lie algebras with the generating index n, and 4-Lie algebras by non-nilpotent 3-Lie algebras with the generating index 4. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the 'singular statistics' of pseudointegrable Šeba billiards, i.e. billiards perturbed by zero-range perturbations. We have shown that the computation of a spectrum is reduced to the calculation of the uniquely defined renormalized Green's function. We relate a spectrum of the billiard to the scattering length, which is the only parameter describing the perturbation. We show that taking into account the growing number of resonances, one observes a transition from 'semi-Poissonian'-like statistics to Poissonian. This observation is in agreement with the argument that a classical particle does not feel a point perturbation. [ABSTRACT FROM AUTHOR]

This paper proves that it is impossible to identify orthogonally time-separated Bell states. If two qubits of A Bell state interact with the measurement apparatus at different time, any attempt to identify this state will disturb it. [ABSTRACT FROM AUTHOR]

Based on the weak Noether symmetry proposed by Mei F X, this paper discusses the weak Noether symmetry for nonholonomic system of non-Chetaev type, and presents expressions of three kinds of conserved quantities by weak Noether symmetry. Finally, the application of this new results is showed by a practical example. [ABSTRACT FROM AUTHOR]

One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought. [ABSTRACT FROM AUTHOR]

In this paper we investigate the exponential growth of products of two matrices A,B \in {{\rm SL}(2,\mathbb{R})} . We prove, assuming A is a fixed hyperbolic matrix, that for Lebesgue almost every B, products of length n involving less than na, 0 [?] a < 1/2 matrices B are uniformly bounded from below by gn for some g > 1. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the topological structure of orbits of the Rabinovich system \dot{x}=hy-v_1x+yz, \dot{y}=hx-v_2y-xz and \dot{z}=v_3z+xy having an invariant algebraic surface. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the asymptotic speed of spread and travelling waves for a spatially discrete SIS epidemic model. By appealing to the theory of spreading speeds and travelling waves for monotonic semiflows, we establish the existence of asymptotic speed of spread and show that it coincides with the minimal wave speed for monotonic travelling waves. This also gives an affirmative answer to an open problem presented by Rass and Radcliffe (2003 Spatial Deterministic Epidemics (Mathematical Surveys and Monographs vol 102) (Providence, RI: American Mathematical Society)) in the case of discrete spatial habitat. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we generalize the results of the author (Liu 2006 Nonlinearity 19 277-91, 2007 Nonlinearity 20 1017-30) from the random flow case to the random semiflow case, i.e. we obtain the Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose the invariant random compact set (e.g. global random attractor) into random Morse sets and connecting orbits between them, which generalizes the Morse decomposition of invariant sets originated from Conley (Conley 1978 Isolated Invariant sets and the Morse Index (Conf. Board Math. Sci. vol 38) (Providence, RI: American Mathematical Society)) to the random semiflow setting and gives a positive answer to an open problem put forward by Caraballo and Langa (Caraballo and Langa 2002 Stochastic Partial Differential Equations and Applications (Trento, 2002) (Lecture Notes in Pure and Applied Mathematics) vol 227 (New York: Dekker) pp 89-104). [ABSTRACT FROM AUTHOR]

In this paper Lou's direct perturbation method is applied to the perturbed coupled nonlinear Schrödinger equations to obtain their asymptotical solutions, which include not only the zero-order solutions but also the first-order modifications. Based on the asymptotical solutions, the effects of perturbations on soliton parameters and the collision between two solitons are then discussed in brief. Furthermore, we directly simulate the perturbed coupled nonlinear Schrödinger equations by split-step Fourier method to check the validity of the direct perturbation method. It turns out that our analytical results are well supported by the numerical calculations. [ABSTRACT FROM AUTHOR]

Lie symmetry algebra of linear nonconservative dynamical systems is studied in this paper. By using 1-1 mapping, the Lie point and Lie contact symmetry algebras are obtained from two independent solutions of the one-dimensional linear equations of motion. [ABSTRACT FROM AUTHOR]

This paper discusses the extent to which one can determine the spacetime metric from a knowledge of a certain subset of the (unparametrized) geodesics of its Levi-Civita connection, that is, from the experimental evidence of the equivalence principle. It is shown that, if the spacetime concerned is known to be vacuum, then the Levi-Civita connection is uniquely determined and its associated metric is, generically, uniquely determined up to a choice of units of measurement, by the specification of these geodesics. A special case arises here concerning pp-waves and is dealt with. It is further demonstrated that if two spacetimes share a similar subset of unparametrized geodesics they share all geodesics (that is they are projectively related). Furthermore, if one of these spacetimes is assumed vacuum then their Levi-Civita connections are again equal (and so the other metric is also a vacuum metric) and the first result above is recovered. [ABSTRACT FROM AUTHOR]

In this paper, we consider the transmission of classical information through a class of quantum channels with long-term memory, which are convex combinations of memoryless channels. Hence, the memory of such channels can be considered to be given by a Markov chain which is aperiodic but not irreducible. We prove the coding theorem and weak converse for this class of channels. The main techniques that we employ are a quantum version of Feinstein's fundamental lemma (Feinstein A 1954 IRE Trans. PGIT 4 2-22, Khinchin A I 1957 Mathematical Foundations of Information Theory: II. On the Fundamental Theorems of Information Theory (New York: Dover) chapter IV) and a generalization of Helstrom's theorem (Helstrom CW1976 Quantum detection and estimation theory Mathematics in Science and Engineering vol 123 (London: Academic)). [ABSTRACT FROM AUTHOR]

The focus is on two parties with Hilbert spaces of dimension d, i.e. 'qudits'. In the state space of these two possibly entangled qudits an analogue to the wellknown tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them are included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications. Its rich symmetry—related to that of a classical phase space—helps to study entanglement, to construct witnesses and perform partial transpositions. This simplex has been explored in detail for d = 3. In this paper, the mathematical background and extensions to arbitrary dimensions are analysed. [ABSTRACT FROM AUTHOR]

In this paper, we consider feedback control algorithms for the rapid purification of a bipartite state consisting of two qubits, when the observer has access to only one of the qubits. We show(1) the algorithm maximizing the average purification rate is different from that for a projective measurement and (2) that the weak measurement protocol for bipartite systems contrasts with known single qubit protocols. We also reveal that the rate of increase in purity may be made deterministic for bipartite states. [ABSTRACT FROM AUTHOR]

In this paper, a new mechanism for the emergence of scale-free distribution is proposed. It is more realistic than the existing mechanism. Based on our mechanism, a model responsible for the scale-free distribution with an exponent in a range of 3-to-5 is given. Moreover, this model could also reproduce the exponential distribution that is discovered in some real networks. Finally, the analytical result of the model is given and the simulation shows the validity of our result. [ABSTRACT FROM AUTHOR]

In this paper, a Birkhoff-Noether method of solving ordinary differential equations is presented. The differential equations can be expressed in terms of Birkhoff's equations. The first integrals for differential equations can be found by using the Noether theory for Birkhoffian systems. Two examples are given to illustrate the application of the method. [ABSTRACT FROM AUTHOR]

In this paper we study the inverse boundary value problem for the Schrödinger equation with a potential and the conductivity equation using partial Cauchy data. We derive stability estimates for these inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, the problem of uniqueness concerning the inverse scattering problem in two-dimensional acoustics for one incident plane wave and one wavenumber is considered. Using the fact that the optimal lower estimate for the eigenvalues of the Laplacian for a domain is given by the Faber–Krahn inequality, which relates the area of the domain to the first eigenvalue of a disc of equal area, it is proved that the uniqueness holds under the restriction that the possible scatterers do not deviate ‘too much’ in area. Also an improvement of the results due to Colton and Sleeman (1983 IMA J. Appl. Math. 31 253–9) is presented, based on the a priori information that the unknown scatterers lie inside a given ball and that the far field is known for a finite number of incident plane waves. The main advantage of this work is that it provides uniqueness for the half number of the needed incoming waves in Colton and Sleeman (1983 IMA J. Appl. Math. 31 253–9). For the case of one incoming plane wave uniqueness is satisfied if the scatterers are contained in a ball of radius R such that kR < t10 = 4.4939, where t10 is the first root of the spherical Bessel function of first order j1(x). The result of local uniqueness is applied to a class of star-shaped scatterers which are smooth perturbations of discs with common centre in [inline image] for one incident plane-wave direction. Numerical implementations are presented for smooth perturbations of discs. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the interaction of boundary shear stress, velocity distribution and secondary currents in open channel flows. A method for computing boundary shear stress and velocity distribution in steady, uniform and fully developed turbulent flows is developed by applying an order-of-magnitude analysis to the Reynolds equations. A simplified relationship between the wall-tangential and wall-normal terms in the Reynolds equation is hypothesized, then the Reynolds equations become solvable. This analysis suggests that the energy from the main flow is transported towards the nearest boundary to be dissipated through a minimum relative distance or normal distance of the boundary. The equations governing the boundary shear stress and Reynolds shear stress distributions are obtained, and the influence of wall-normal velocity on the streamwise velocity is assessed. It is found that the classical log-law is valid only when the wall-normal velocity is zero, the non-zero wall-normal velocity results in the derivation of measured streamwise velocity from the classical log-law. The derived equations are in good agreement with existing experimental data available in the literature. [Copyright &y& Elsevier]

This article focuses on views of the author, who is the winner of 2003 Olin Palladium Award, on conducting research in the digital era. At the University of Pennsylvania, Philadelphia the author met Srinivasan Subramanian, called Srini for short, who has been his good friend ever since. When there was an equation to solve Srini would look at it, think a minute, and write down the solution. He was the mathematical wizard for students. They spend time for nearly three years, learned a lot from each other, wrote a few papers together, and consoled each other when the going got rough.

We discuss a comment recently made by C F Lo concerning validity of one of the examples used in our paper (2013 J. Phys. A: Math. Theor.46 235301). While we agree with Lo’s statement regarding inapplicability of the example to the proposed schema, we disagree with the claim concerning mathematical correctness of that example. We are not convinced by Lo’s arguments leading to the conclusion that the model is mathematically ill-defined. [ABSTRACT FROM AUTHOR]

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter is analyzed, when the external force is periodic in space and given by , both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force , and the soliton width ls, which is a function of the nonlinearity as well as the initial frequency of the solitary wave, there is a transition (at fixed ) from trapped to unbound behavior of the soliton, which depends on the parameters r and K of the external force and the nonlinearity parameter . We previously studied this phenomena when (Quintero et al 2019 J. Phys. A: Math. Theor. 52 285201) where we showed that for the soliton oscillates in an effective potential, while for it moves uniformly as a free particle. In this paper we focus on the dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of r and K when at fixed value of the frequency . Since the solitary wave gets narrower for fixed as a function of , we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase . [ABSTRACT FROM AUTHOR]

We complete the derivation of the sausage model NLIE by giving a proof of the crucial relation (3.24) of the original paper based on the analytic properties of Q and . [ABSTRACT FROM AUTHOR]

In this paper we consider the slightly L2-supercritical gKdV equations , with the nonlinearity and . In the previous work of the author, we know that there exists a stable self-similar blow-up dynamics for slightly L2-supercritical gKdV equations. Such solutions can be viewed as solutions with a single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time. [ABSTRACT FROM AUTHOR]

In this paper, we study a large time behavior of the global spherically or cylindrically symmetric solutions in H1 for the compressible viscous radiative and reactive gas in multi-dimension with large initial data. Precisely, if the initial data are spherically symmetric or cylindrically symmetric, the smallness of initial data is not needed. The main concern of the present paper is to investigate the exponential stability of a solution toward the stationary solution as time goes to infinity. We obtain the uniform positive lower and upper bounds of the density by using different methods. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give a characterization of the structure of non-autonomous attractors of the problem when the parameter varies. Also, we answer a question proposed in Carvalho et al (2012 Proc. Am. Math. Soc. 140 2357–73), concerning the complete description of the structure of the pullback attractor of the problem when and, more generally, for , . We construct global bounded solutions, ‘non-autonomous equilibria’, connections between the trivial solution and these ‘non-autonomous equilibria’ and characterize the -limit and -limit set of global bounded solutions. As a consequence, we show that the global attractor of the associated skew-product flow has a gradient structure. The structure of the related pullback an uniform attractors are derived from that. [ABSTRACT FROM AUTHOR]

The topological entropy of upper recurrent, lower recurrent points and Birkhoff regularity were considered in Tian (2016 Adv. Math. 288 464–526) but Banach upper recurrence points are not considered. In this paper we mainly consider whether the points that are Banach upper recurrent but not upper recurrent can carry full topological entropy. Moreover, we use statistical -limit sets to obtain entropy results on the refined orbit distribution of Banach recurrence. In this process, we strengthen Pfister and Sullivan’s result of Pfister and Sullivan (2007 Ergod. Theor. Dynam. Syst. 27 929–56) from saturated property to transitively-saturated property (and from single-saturated property to transitively-convex-saturated property). We also establish an abstract framework on the combination of Banach recurrence and multifractal analysis of general observable functions and in particular, how these results can be applied in the combination of Banach recurrence Lyapunov regularity to obtain a generalized and refined theory of Barreira and Doutor (2009 J. Math. Pures Appl. 92 1–17); Barreira and Gelfert (2006 Commun. Math. Phys. 267 393–418); Barreira and Saussol (2001 Trans. Am. Math. Soc. 353 3919–44); Feng and Huang (2010 Commun. Math. Phys. 297 1–43); Feng (2003 J. Math. 138 353–76); Feng (2009 Israel J. Math. 170 355–94); Feng and Lau (2002 Math. Res. Lett. 9 363–78) etc, on mixed Lyapunov multifractal analysis such as irregular sets and level sets of Lyapunov exponents. [ABSTRACT FROM AUTHOR]

This paper is the first part of a series which provides a systematic treatment of the space-filling curves of self-similar sets. In the present paper, we introduce a notion of linear graph-directed IFS (linear GIFS in short). We show that to construct a space-filling curve of a self-similar set, it amounts to exploring its linear GIFS structures. Compared to the previous methods, such as the L-system or recurrent set method, the linear GIFS approach is simpler, more rigorous and leads to further studies on this topic. We also propose a new algorithm for the beautiful visualization of space-filling curves. In a series of papers Dai et al (2015 arXiv:1511.05411 [math.GN]), Rao and Zhang (2015) and Rao and Zhang (2015), we investigate for a given self-similar set how to get ‘substitution rules’ for constructing space-filling curves, which was obscure in the literature. We solve the problem for self-similar sets of finite type, which covers most of the known results on constructions of space-filling curves. [ABSTRACT FROM AUTHOR]