Computing generalized Langevin equations and generalized Fokker–Planck equations

The Mori–Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker–Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori–Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.     

On the London equations

The classical London equations for superconductors are written in differential form notation and recast in relativistic form, where they yield the Proca equation. In particular, the field itself acts as its own charge carrier.     

Venous hemodynamics. Basic equations

The hydrodynamics in tubes with inert walls collapsed by a negative transmural pressure (inside pressure minus outside pressure) set up a fundamental and original approach in venous hemodynamics. Mechanisms of flow regulation and flow limitation can occur especially when the upstream pressure is held constant or when the average flow speed in a cross-section equals the local speed of the pressure waves. Three baseline equations on mechanics are required to demonstrate these properties: the tube law, the equation of motion, the equation of continuity. It is shown that the behaviour of the tube varies with the nature of the flow (subcritical, critical or super-critical, subsonic, sonic or supersonic, according to the aerodynamics). Though this study does not take all the physiological and biological data into account, it includes indeed the three fundamental aspects on mechanics of the veins. The rheologic equation on the state of the venous wall, normal or pathologic, plays a major and determinant role.     

Uniqueness of solutions of semilinear Poisson equations

Let Rⁿ denote n-dimensional Euclidean space, with n > 1. We study the uniqueness of positive solutions u(x), x Rⁿ, of the semilinear Poisson equation Δu + f(u) = 0 under the assumption that u(x) → 0 as |x| → ∞. This type of problem arises in phase transition theory, in population genetics, and in the theory of nucleon cores, with various different forms of the driving term f(u). For the important model case f(u) = −u + u^p, where p is a constant greater than 1, our results show (i) that when the dimension n of the underlying space is 2, there is at most one solution (up to translation) for any given p and (ii) that when the dimension n is 3, there is at most one solution when 1 < p ≤ 3. In both cases, the solution is radially symmetric and monotonically decreasing as one moves outward from the center. For dimensions other than 2 or 3, and indeed for the analogous cases of a real dimensional parameter n > 1, we obtain corresponding results. We note finally, again for the model case, that existence holds for 1 < p < (n + 2)/(n − 2); thus, there remains an interesting difference between the parameter ranges for which existence and uniqueness are established.     

Venous hemodynamics. Base equations

The hydrodynamics in tubes with inert walls collapsed by a negative transmural pressure (inside pressure minus outside pressure) set up a fundamental and original approach in venous hemodynamics. Mechanisms of flow regulation and flow limitation can occur especially when the upstream pressure is held constant or when the average flow speed in a cross-section equals the local pressure waves speed. The three baseline equations on mechanics are required to demonstrate these properties: --the tube law, --the equation of motion, --the equation of continuity. It is shown that the behaviour of the tube varies with the nature of the flow (subcritical, critical or supercritical, subsonic, sonic or supersonic, according to the aerodynamics). Though this study does not take all the physiological and biological data into account, it includes indeed the three fundamental aspects on mechanics of the veins. The rheologic equation on the state of the venous wall, normal or pathologic, plays a major and determinant role.     

Benjamin-Ono-related equations and their solutions

It is shown that the Benjamin—Ono equation is one of a hierarchy of nonlinear partial differential singular integral equations. Explicit multisoliton solutions are constructed for all equations of the hierarchy.     

Equipartition of energy for higher-order hyperbolic equations

Let A₀, A₁,...,A_2^N-1 be commuting skew-adjoint operators on a Hilbert space [unk]. Then the equation Π_j=0^2N-1 (d/dt - A_j)v(t) = 0 (t real) admits equipartition of energy [in the sense that the jth partial energy E_j(t) of any solution at time t satisfies lim_t→±∞E_j(t) = 2^-N·(total energy) for each of the 2^N values of j] if and only if the closure B_jk of A_j - A_k satisfies weak-operator-limit exp(tB_jk) = 0 as t → ±∞ whenever j ≠ k.     

Strongly nonlinear integral equations of hammerstein type

This paper studies the solution of the nonlinear Hammerstein equation u(x) + k(x,y)f[y,u(y)]mu(dy) = h(x) in the singular case, i.e., where the linear operator K with kernel k(x,y) is not defined for all the range of the nonlinear mapping F given by Fu(y) = f[y,u(y)] over the whole class X of functions u which are potential solutions of the equation. An existence theorem is derived under relatively minimal assumptions upon k and f, namely that (Ku,u) >/= 0, that K maps L(1) into L(1) (loc) and is compact from L(1) [unk] L(infinity) into L(1) (loc), that f(y,s) has the same sign as s for s >/= R, and that for each constant r > 0, f(y,s) 相似文献   

肾小球滤过率评估公式的比较

肾小球滤过率(GFR)是评价肾功能的重要指标,也是慢性肾脏病(CKD)分期的重要依据。使用公式评估GFR方法简便,已被广泛应用,其中Cockcroft-Gault(CG)公式和MDRD公式最为常用,CKD-EPI、改良MDRD公式、瑞金公式及基于胱抑素C的公式等次之,通过比较其偏差、精确度、准确性等指标发现,CG公式更适用于健康人群,但在肾功能不全患者适用性较差,且受体重影响明显;MDRD公式适用于肾功能不全患者,但在GFR水平较高时可靠性较差;CKD-EPI克服了MDRD公式在GFR较高时会低估真实值的缺点;改良MDRD公式和瑞金公式则更适用于中国人;而基于胱抑素C的公式在疾病早期有较好适用性。因此,尚需进一步改进或者发掘更好地评估GFR的公式。     

Conjugate points of second-order linear differential equations

The paper of which the present article is itself an abstract is concerned with locating zeros of nonnull solutions of differential systems of the formcplx=fwhere r(x) and p(x) are continous and r(x) is positive for x ≥ a. When r(x) = 1 and p(x) = αx + β, necessary conditions and sufficient conditions are pro-vided that the interval (a,b] contain n zeros. The results are, of course, of importanceconsiderably beyond the case when p(x) is linear.     

Some nonlinear elliptic equations from geometry

We describe some recent work on certain nonlinear elliptic equations from geometry. These include the problem of prescribing scalar curvature on (n), the Yamabe problem on manifolds with boundary, and the best Sobolev inequality on Riemannian manifolds.     

Spirometric reference equations for older adults

The objective of this study was to develop spirometric reference equations for healthy, never-smoking, older adults. It was designed as a cross-sectional observational study consisting of 1510 Seventh Day Adventists, ages 43-79 years enrolled in a study of health effects of air pollutants. Individuals were excluded from the reference group (n = 565) for a history of current respiratory illness, smoking, or chronic respiratory disease, and for a number of 'non-respiratory' conditions which were observed in these data to be related to lower values of FEV1. Gender-specific reference equations were developed for the entire reference group and for a subset above 65 years of age (n = 312). Controlling for height and age, lung function was found to be positively related to the difference between armspan and height, and in males was found to be quadratically related to age. The predicted values for this population generally fell within the range of those of other population groups containing large numbers of adults over the age of 65 years. Individuals with lung function below the 5th percentile in this sample, however, could not be reliably identified by using the lower limits of normal predictions commonly used in North America and Europe.     

Two functional equations preserving functional forms

Two functional equations are considered that are motivated by three considerations: work in utility theory and psychophysics, questions concerning when pairs of degree 1 homogeneous functions can be homomorphic and calculating their homomorphisms, and the link of the latter questions to quasilinear mean values. The first equation is h(σ(y)x + [1 - σ(y)]y) = τ(y)h(x) + [1 - τ(y)]h(y) (x ≥ y ≥ 0), where h maps [0, ∞[into a subset of [0, ∞[and is strictly increasing and continuously differentiable; the functions σ and τ map [0, ∞[continuously into [0,1], σ(y) > 0 for y > 0 but σ is not 1 on]0, ∞[. The solutions are fully determined. (Recently Zsolt Páles has eliminated the differentiability assumption.) The second equation is h[y + f(x - y)] = h(y) + g[h(x) - h(y)] (x ≥ y ≥ 0), where h maps [0, ∞[onto a subinterval of positive length of [0, ∞[and is strictly increasing and twice continuously differentiable, f and g map [0, ∞[onto[0, ∞[and are twice differentiable, and either f"(0) ≠ 0 or g"(0) ≠ 0. The solutions are fully determined under these conditions. When f"(0) = g"(0) = 0 and h" is not identically zero, we determine the solutions under the added assumption of analyticity. It remains an open problem to find the solutions in the latter case under the assumption of only second order differentiability. A more general open problem is to eliminate all differentiability conditions for the second equation.     

Comparison of Indian reference equations for spirometry interpretation

BACKGROUND AND OBJECTIVES: It would be desirable in a large country such as India that a single set of reference equations be used to interpret lung function tests performed across the entire country. This study compared north, west and south reference equations in interpreting spirometry results in north Indian patients. METHODS: Spirometric records of 27,383 patients aged 16-65 years were assessed. Spirometric values for FVC, FEV(1) and FEV(1)%FVC values derived from north, west and south Indian reference equations were compared. Differences in the lower limit of normal (LLN) were studied across the age and height range of the study group to determine if there was any clinically significant difference in the three derived values. RESULTS: The north and west Indian equations was discordant in 22.1% instances, and the north and south Indian equations in 12.9% instances, with kappa estimates of agreement being 0.626 and 0.781, respectively. Most of the patients with abnormal spirometry using north Indian equations were erroneously interpreted to have normal spirometry using west or south Indian equations. The south Indian equations underpredicted LLN for FVC and FEV(1) for most men and women. The west Indian equations underpredicted LLN for FVC and FEV(1) in all men, and in younger and short statured women. CONCLUSIONS: North, west and south Indian reference equations do not yield equivalent results for spirometry interpretation in north Indian patients.