Mixing “in the sense of ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. II

In the first part of the present paper, we established estimates for the rate of approach of the integrals of a family of “physical” white noises to a family of Wiener processes. We use this result to establish the estimate for the rate of approach of a family of solutions of ordinary differential equations perturbed by some “physical” white noises to a family of solutions of the corresponding It? equations. We consider both the case where the coefficient of random perturbation is separated from zero and the case where it is not separated from zero.     

Spectrum of the Kerzman–Stein operator for a family of smooth regions in the plane

The Kerzman–Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on the boundary. For bounded regions with smooth boundary, the Kerzman–Stein operator is compact on the Hilbert space of square integrable functions. Here we give an explicit computation of its Hilbert–Schmidt norm for a family of simply connected regions. We also give an explicit computation of the Cauchy operator acting on an orthonormal basis, and we give estimates for the norms of the Kerzman–Stein and Cauchy operators on these regions. The regions are the first regions that display no apparent Möbius symmetry for which there now is explicit spectral information.     

Bogoliubov’s metric as a global characteristic of the family of metrics in the Hilbert algebra of observables

We comparatively analyze a one-parameter family of bilinear complex functionals with the sense of “deformed” Wigner-Yanase-Dyson scalar products on the Hilbert algebra of operators of physical observables. We establish that these functionals and the corresponding metrics depend on the deformation parameter and the extremal properties of the Kubo-Martin-Schwinger and Wigner-Yanase metrics in quantum statistical mechanics. We show that the Bogoliubov-Kubo-Mori metric is a global (integral) characteristic of this family. It occupies an intermediate position between the extremal metrics and has the clear physical sense of the generalized isothermal susceptibility. We consider the example for the SU(2) algebra of observables in the simplest model of an ideal quantum spin paramagnet.     

An approximation of the Mumford–Shah energy by a family of discrete edge-preserving functionals

We show the Γ$\Gamma$-convergence of a family of discrete functionals to the Mumford and Shah image segmentation functional. The functionals of the family are constructed by modifying the elliptic approximating functionals proposed by Ambrosio and Tortorelli. The quadratic term of the energy related to the edges of the segmentation is replaced by a nonconvex functional.     

Asymptotic behavior on (−∞, 0) of the spectral measures of a family of singular Sturm-Liouville operators

We find the asymptotics as λ/? → ?∞ of the density of the spectral measure of the Sturm-Liouville operator in L ²(0,+∞) generated by the expression ?y″ + ?q(x)y, ? > 0, with the boundary condition y(0) cos α+y′(0) sinα = 0. The potential q(x) tends to ?∞ as x → +∞ and is assumed to satisfy the Sears condition and some additional regularity conditions.     

Threefolds in ℙ5 with a 3-dimensional family of plane curveswith a 3-dimensional family of plane curves

A classification theorem is given of smooth threefolds of ?⁵ covered by a family of dimension at least three of plane integral curves of degreed≧2. It is shown that for such a threefoldX there are two possibilities:

1. X is any threefold contained in a hyperquadric;
2. d≦3 andX is either the Bordiga or the Palatini scroll.

     

One-parameter family of integrable solutions of a system of nonlinear integral equations of the Hammerstein–Volterra type in the supercritical case

We study a system of nonlinear integral equations of the Hammerstein–Volterra type on a half-line in the supercritical case. We show that this system has a one-parameter family of positive integrable bounded solutions. We describe the structure of each solution in this family. The monotone dependence of the solutions on the parameter is proved.     

Stability of the sufficient statistic “the value of the observation in the last moment” for a family of random processes with independent increments

One investigates the problem of the stability of the sufficiency property of the statistic the value in the last moment for processes with independent increments relative to small perturbations of the Poisson spectra of these processes.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 174–176, 1982.     

Some properties of the family Γ of modular Lie superalgebras

In this paper, we continue to investigate some properties of the family Γ of finite-dimensional simple modular Lie superalgebras which were constructed by X. N. Xu, Y. Z. Zhang, L. Y. Chen (2010). For each algebra in the family, a filtration is defined and proved to be invariant under the automorphism group. Then an intrinsic property is proved by the invariance of the filtration; that is, the integer parameters in the definition of Lie superalgebras Γ are intrinsic. Thereby, we classify these Lie superalgebras in the sense of isomorphism. Finally, we study the associative forms and Killing forms of these Lie superalgebras and determine which superalgebras in the family are restrictable.     

Fiducial inference in the pivotal family of distributions

In this paper a family, called the pivotal family, of distributions is considered. A pivotal family is determined by a generalized pivotal model. Analytical results show that a great many parametric families of distributions are pivotal. In a pivotal family of distributions a general method of deriving fiducial distributions of parameters is proposed. In the method a fiducial model plays an important role. A fiducial model is a function of a random variable with a known distribution, called the pivotal random element, when the observation of a statistic is given. The method of this paper includes some other methods of deriving fiducial distributions. Specially the first fiducial distribution given by Fisher can be derived by the method. For the monotone likelihood ratio family of distributions, which is a pivotal family, the fiducial distributions have a frequentist property in the Neyman-Pearson view. Fiducial distributions of regular parametric functions also have the above frequentist property. Some advantages of the fiducial inference are exhibited in four applications of the fiducial distribution. Many examples are given, in which the fiducial distributions cannot be derived by the existing methods.     

On a family of results concerning direction of vorticity and regularity for the Navier–Stokes equations

These notes concern existence, and suitable formulation, of meaningful conditions on the direction of the vorticity which guarantee the regularity of the solutions to the evolution Navier–Stokes equations. A main concern here is to compare the different situations which appear in considering slip and no-slip boundary conditions. The paper reviews mainly results obtained in some of the references cited.     

Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction–diffusion equations

We consider front propagation in a family of scalar reaction–diffusion equations in the asymptotic limit where the polynomial degree of the potential function tends to infinity. We investigate the Gevrey properties of the corresponding critical propagation speed, proving that the formal series expansion for that speed is Gevrey-1 with respect to the inverse of the degree. Moreover, we discuss the question of optimal truncation. Finally, we present a reliable numerical algorithm for evaluating the coefficients in the expansion with arbitrary precision and to any desired order, and we illustrate that algorithm by calculating explicitly the first ten coefficients. Our analysis builds on results obtained previously in [F. Dumortier, N. Popovi?, T.J. Kaper, The asymptotic critical wave speed in a family of scalar reaction–diffusion equations, J. Math. Anal. Appl. 326 (2) (2007) 1007–1023], and makes use of the blow-up technique in combination with geometric singular perturbation theory and complex analysis, while the numerical evaluation of the coefficients in the expansion for the critical speed is based on rigorous interval arithmetic.     

Local connectivity of Julia sets for a family of biquadratic polynomials

The local connectivity of Julia sets for the family of biquadratic polynomials f_c(z)= (z~2-2c~2)z~2 with a parameter c is discussed.It is proved that for any parameter c,the boundary of the immediately attracting domain of f_c is a Jordan curve.     

Schrödinger equation. The theorem concerning the ansatz representation of a solution concentrated in a neighborhood of a minimum of the potential

The one-dimensional Schrödinger equation is considered on the segment [–l,l] It is assumed that the potential v(x) of this equation has one minimum v(0)=v(0)=0, v(0)>0 v(x)>0 for x0; v(x)h>0 outside some neighborhood of zero. It is proved that there exists a solution of the form where is a parabolic cylinder function, and is a smooth function which is bounded on [–l,l] together with derivatives through third order by a constant not depending on . The function and the real number E admit a known asymptotic expansion as 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 137–150, 1984.     

Multiplicative cones — a family of three eigenvalue graphs

The possibilities for a graph onn vertices with spectrum >>– and some vertex of valencen–1 are presented. Some of these are constructed.     

The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant

In this article, we study small perturbations of the family of Friedmann–Lemaître–Robertson–Walker cosmological background solutions to the 1 + 3 dimensional Euler–Einstein system with a positive cosmological constant. These background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. Our nonlinear analysis shows that under the equation of state ${p = c^2_s \rho}$ , ${0 < c^2_s < 1/3}$ , the background solutions are globally future-stable. In particular, we prove that the perturbed spacetime solutions, which have the topological structure ${[0,\infty) \times \mathbb{T}^3}$ , are future-causally geodesically complete. These results are extensions of previous results derived by the author in a collaboration with I. Rodnianski, in which the fluid was assumed to be irrotational. Our novel analysis of a fluid with non-zero vorticity is based on the use of suitably defined energy currents.     

On a family involving R.L. Cohen's ζ-element(II)

In 1981, Cohen constructed an infinite family of homotopy elements ζk∈π*(S) represented by h0bk∈ Ext3,2(p-1)(pk+1+1)A(Z/p, Z/p) in the Adams spectral sequence, where p 2 and k≥1. In this paper,we make use of the Adams spectral sequence and the May spectral sequence to prove that the composite map ζn-1β2γs+3is nontrivial in the stable homotopy groups of spheres πt(s,n)-s-8(S), where p≥7, n 3,0≤s p- 5 and t(s, n) = 2(p- 1)[pn+(s + 3)p2 +(s + 4)p +(s + 3)] + s.