Spatially homogeneous solutions of the Vlasov–Nordström–Fokker–Planck system

The Vlasov–Nordström–Fokker–Planck system describes the evolution of self-gravitating matter experiencing collisions with a fixed background of particles in the framework of a relativistic scalar theory of gravitation. We study the spatially-homogeneous system and prove global existence and uniqueness of solutions for the corresponding initial value problem in three momentum dimensions. Additionally, we study the long time asymptotic behavior of the system and prove that even in the absence of friction, solutions possess a non-trivial asymptotic profile. An exact formula for the long time limit of the particle density is derived in the ultra-relativistic case.     

Mean field dynamics of fermions and the time-dependent Hartree–Fock equation

The time-dependent Hartree–Fock equations are derived from the N-body linear Schrödinger equation with the mean-field scaling in the limit N→+∞ and for initial data that are close to Slater determinants. Only the case of bounded, symmetric binary interaction potentials is treated in this work. We prove that, as N→+∞, the first partial trace of the N-body density operator approaches the solution of the time-dependent Hartree–Fock equations (in operator form) in the sense of the trace norm.     

Continued fraction proofs of <Emphasis Type="Italic">m</Emphasis>-versions of some identities of Rogers–Ramanujan–Slater type

We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two q-continued fractions previously investigated by the authors.     

Bishop–Phelps–Bollobás property for certain spaces of operators

We characterize the Banach spaces Y   for which certain subspaces of operators from _L1(μ)$L_1(\mu )$ into Y have the Bishop–Phelps–Bollobás property in terms of a geometric property of Y, namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain that certain ideals of Asplund operators satisfy the Bishop–Phelps–Bollobás property.     

Comments on “Embedded pair of extended Runge–Kutta–Nyström type methods for perturbed oscillators”

In a recent paper Fang [Embedded pair of extended Runge–Kutta–Nyström type methods for perturbed oscillators, Appl. Math. Modell. (2009), doi:10.1016/j.apm.2009.12.004] considered the embedded pair of extended Runge–Kutta–Nyström type methods for perturbed oscillators and analyzed numerical stability and phase properties of the methods. The authors claimed that their methods are based on the order conditions of extended Runge–Kutta–Nyström type methods presented by Yang et al. [Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Commun. 180 (2009) 1777–1794]. However, some careless mistakes have been made in that paper. For this reason we will make some comments on that paper.     

A sharp version of Bauer–Fike’s theorem

In this paper, we present a sharp version of Bauer–Fike’s theorem. We replace the matrix norm with its spectral radius or sign-complex spectral radius for diagonalizable matrices; 1-norm and ∞$\infty$-norm for non-diagonalizable matrices. We also give the applications to the pole placement problem and the singular system.     

WKB analysis of the Schrödinger–KdV system

We consider the behavior of solutions to the water wave interaction equations in the limit ε→0⁺$\epsilon \to 0^{+}$. To justify the semiclassical approximation, we reduce the water wave interaction equation into some hyperbolic-dispersive system by using a modified Madelung transform. The reduced system causes loss of derivatives which prevents us to apply the classical energy method to prove the existence of solution. To overcome this difficulty we introduce a modified energy method and construct the solution to the reduced system.     

Linear invariant relations of the Poincaré–Zhukovskii equations

All the cases of the existence of a linear invariant relation of the Poincaré–Zhukovskii equations with symmetric Hamiltonian matrices are found. A similar geometrical meaning of the existence conditions is indicated for well-known special cases. Precessional motions with a linear invariant relation are identified.     

Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series

We develop a doubly spectral representation of a stationary functional time series, and study the properties of its empirical version. The representation decomposes the time series into an integral of uncorrelated frequency components (Cramér representation), each of which is in turn expanded in a Karhunen–Loève series. The construction is based on the spectral density operator, the functional analogue of the spectral density matrix, whose eigenvalues and eigenfunctions at different frequencies provide the building blocks of the representation. By truncating the representation at a finite level, we obtain a harmonic principal component analysis of the time series, an optimal finite dimensional reduction of the time series that captures both the temporal dynamics of the process, as well as the within-curve dynamics. Empirical versions of the decompositions are introduced, and a rigorous analysis of their large-sample behaviour is provided, that does not require any prior structural assumptions such as linearity or Gaussianity of the functional time series, but rather hinges on Brillinger-type mixing conditions involving cumulants.     

On the convergence of type I Hermite–Padé approximants

Padé approximation has two natural extensions to vector rational approximation through the so-called type I and type II Hermite–Padé approximants. The convergence properties of type II Hermite–Padé approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite–Padé approximants for Nikishin systems of functions.     

A density version of the Halpern–Läuchli theorem

We prove a density version of the Halpern–Läuchli Theorem. This settles in the affirmative a conjecture of R. Laver.     

On Birman's sequence of Hardy–Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_n\}$, for $n\in \mathbb{N}$, valid for functions in $C_0^n((0,\infty ))?L^2((0,\infty ))$. In particular, $I_1$ is the classical (integral) Hardy inequality and $I_2$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_n([0,\infty ))$ of functions defined on $[0,\infty )$. Moreover, $f\in H_n([0,\infty ))$ implies $f^{\prime }\in H_{n?1}([0,\infty ))$; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0$, these inequalities hold on the standard Sobolev space $H_0^n((0,b))$. Furthermore, in all cases, the Birman constants ${[(2n?1)!!]}^2/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^2((0,\infty ))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.