Showing papers on "Section (fiber bundle) published in 1983"

Diamond anvil cell and high-pressure physical investigations

Abstract: The present status of high-pressure research with the diamond anvil cell (DAC) is reviewed in this article, mainly from an experimental aspect. After a brief description of the different types of DAC's that are currently in vogue, the techniques used in conjunction with the DAC in modern high-pressure research are presented. These include techniques for low- and high-temperature studies, x-ray diffractometry, spectroscopy with the DAC, and other measurements. Results on selected materials, with a view to illustrating the physics behind high-pressure phenomena, are presented and discussed. These include metal-semiconductor transitions, electronic transitions, phonons and high-pressure lattice dynamics, and phase transitions. A whole section is devoted to the behavior of condensed gases, principally ${\mathrm{H}}_{2}$, ${\mathrm{D}}_{2}$, ${\mathrm{O}}_{2}$, ${\mathrm{N}}_{2}$, and rare-gas solids. The concluding section briefly deals with speculations on ultra-high-pressure research with the DAC in the future.

The Second Homology Group of the Mapping Class Group of an Orientable Surface

Abstract: In I-7] Mumford shows that the Picard group P ic (~ ' ) is isomorphic to H2(F; 2~) and conjectures the latter is rank one, g>3 . We prove this below for g>5 . Another interpretation of this theorem may be obtained by identifying H2(F) as bordism classes of fiber bundles F---*W4---* T where T is a closed oriented surface (Sect. 0). When F is closed every such bundle is bordant to F ~ W' --* T', a bundle admitting a section s: T ' ~ W'. The theorem then says that

Compartmental Modeling and Tracer Kinetics

Abstract: Section 1. Compartmental Systems.- 1A. Introduction.- 1B. Preliminary definitions.- 1C. Tracer experiments.- 1D. History of compartmental analysis.- Section 2. Elementary Compartmental Models.- 2A. Drug kinetics.- 2B. Leaky fluid tanks.- 2C. Diffusion.- 2D. Solute mixture.- Section 3. First-Order Chemical Reactions.- Section 4. Environmental Studies.- 4A. Kinetics of lead in the body.- 4B. The Aleut ecosystem.- Section 5. Nonlinear Compartmental Models.- 5A. Continuous flow chemical reactor.- 5B. Reaction order.- 5C. Other nonlinear compartmental models.- Section 6. The General Compartmental Model.- Section 7. Tracer Kinetics in Steady-State Systems.- 7A. The tracer equations.- 7B. Linear compartmental models..- Section 8. Uptake of Potassium by Red Blood Cells.- Section 9. Standard Types of Tracer Experiments.- 9A. Tracer concentration equations.- 9B. Tracer specific activity equations.- Section 10. Analytical Solution of the Tracer Model.- 10A. The general solution of the model.- 10B. Nonnegativity of the solution.- Section 11. System Structure and Connectivity.- 11A. The connectivity diagram.- 11B. Common compartmental systems.- 11C. Strongly connected systems.- Section 12. System Eigenvalues and Stability..- 12A. Nonpositive eigenvalues ..- 12B. The smallest magnitude eigenvalue.- 12C. Symmetrizable compartmental matrices and real eigenvalues.- 12D. Distinct eigenvalues.- 12E. Compartmental model stability.- 12F. Bounds on the extreme eigenvalues.- Section 13. The Inverse of a Compartmental Matrix.- 13A. Invertibility conditions.- 13B. A Neumann series for the inverse matrix.- 13C. Matrix inequalities.- Section 14. Mean Times and the Inverse Matrix.- 14A. Mean residence times.- 14B. The compartmental matrix exponential.- 14C. Further properties of mean residence time.- 14D. System Mean residence time.- Section 15. Solution of the Steady-State Problem for SEC Systems.- 15A. The tracer steady-state problem.- 15B. Ill-conditioned SEC systems.- 15C. An iterative procedure for SEC systems.- 15D. Updating the algorithm.- Section 16. Structural Identification of the Model.- 16A. The system (A, B, C).- 16B. The structural identification problem.- 16C. A simple identification example.- 16D. Realizations of impulse response functions.- 16E. Impulse response function structure.- 16F. Nonlinear identification equations.- 16G. A three compartment model.- 16H. A four compartment model.- Section 17. Necessary and Sufficient Conditions for Identifiability.- 17A. Model identifiability.- 17B. Necessary conditions.- 17C. Sufficient conditions.- Section 18. A Simple Test for Nonidentifiability.- 18A. Counting nonzero transfer function coefficients.- 18B. Coefficient structure.- 18C. Further refinements of formula (18.4).- 18D. The nonidentifiability test.- 18E. Tighter bounds on the number of independent equations.- Section 19. Computation of the Model Parameters.- 19A. Local identifiability.- 19B. Newton'Section 1. Compartmental Systems.- 1A. Introduction.- 1B. Preliminary definitions.- 1C. Tracer experiments.- 1D. History of compartmental analysis.- Section 2. Elementary Compartmental Models.- 2A. Drug kinetics.- 2B. Leaky fluid tanks.- 2C. Diffusion.- 2D. Solute mixture.- Section 3. First-Order Chemical Reactions.- Section 4. Environmental Studies.- 4A. Kinetics of lead in the body.- 4B. The Aleut ecosystem.- Section 5. Nonlinear Compartmental Models.- 5A. Continuous flow chemical reactor.- 5B. Reaction order.- 5C. Other nonlinear compartmental models.- Section 6. The General Compartmental Model.- Section 7. Tracer Kinetics in Steady-State Systems.- 7A. The tracer equations.- 7B. Linear compartmental models..- Section 8. Uptake of Potassium by Red Blood Cells.- Section 9. Standard Types of Tracer Experiments.- 9A. Tracer concentration equations.- 9B. Tracer specific activity equations.- Section 10. Analytical Solution of the Tracer Model.- 10A. The general solution of the model.- 10B. Nonnegativity of the solution.- Section 11. System Structure and Connectivity.- 11A. The connectivity diagram.- 11B. Common compartmental systems.- 11C. Strongly connected systems.- Section 12. System Eigenvalues and Stability..- 12A. Nonpositive eigenvalues ..- 12B. The smallest magnitude eigenvalue.- 12C. Symmetrizable compartmental matrices and real eigenvalues.- 12D. Distinct eigenvalues.- 12E. Compartmental model stability.- 12F. Bounds on the extreme eigenvalues.- Section 13. The Inverse of a Compartmental Matrix.- 13A. Invertibility conditions.- 13B. A Neumann series for the inverse matrix.- 13C. Matrix inequalities.- Section 14. Mean Times and the Inverse Matrix.- 14A. Mean residence times.- 14B. The compartmental matrix exponential.- 14C. Further properties of mean residence time.- 14D. System Mean residence time.- Section 15. Solution of the Steady-State Problem for SEC Systems.- 15A. The tracer steady-state problem.- 15B. Ill-conditioned SEC systems.- 15C. An iterative procedure for SEC systems.- 15D. Updating the algorithm.- Section 16. Structural Identification of the Model.- 16A. The system (A, B, C).- 16B. The structural identification problem.- 16C. A simple identification example.- 16D. Realizations of impulse response functions.- 16E. Impulse response function structure.- 16F. Nonlinear identification equations.- 16G. A three compartment model.- 16H. A four compartment model.- Section 17. Necessary and Sufficient Conditions for Identifiability.- 17A. Model identifiability.- 17B. Necessary conditions.- 17C. Sufficient conditions.- Section 18. A Simple Test for Nonidentifiability.- 18A. Counting nonzero transfer function coefficients.- 18B. Coefficient structure.- 18C. Further refinements of formula (18.4).- 18D. The nonidentifiability test.- 18E. Tighter bounds on the number of independent equations.- Section 19. Computation of the Model Parameters.- 19A. Local identifiability.- 19B. Newton's method and modifications.- 19C. The Kantorovich conditions.- 19D. An example.- Section 20. An Alternative Approach to Identification.- 20A. A new identification method.- 20B. The component matrices of A.- 20C. The identification technique using component matrices.- 20D. Identification of a lipoprotein model.- 20E. The identification technique using modal matrices.- 20F. Identification of a pharmacokinetic system.- 20G. Spectral sensitivity of a linear model.- Section 21. Controllability, Observability, and Parameter Identifiability.- 21A. The control problem.- 21B. Completely controllable systems.- 21C. Completely observable systems.- 21D. Realizations and identifiability.- 21E. A third method of identifiability.- Section 22. Model Identification from the Transfer Function Equations.- 22A. Form of the nonlinear equations.- 22B. Coefficients of the transfer fuction.- 22C. The nonlinear algebraic equations for the identification problem.- 22D. Necessary conditions for positive solutions of the nonlinear system.- 22E. Refined necessary conditions.- 22F. Additional properties of the nonlinear algebraic system.- 22G. An iterative scheme for solving F(?) = 0.- 22H. Triangularization of F(?) = 0.- 22I. Uniqueness of solution F(?) = 0.- Section 23. The Parameter Estimation Problem.- 23A. The basic estimation problem.- 23B. A lipoprotein metabolism model.- 23C. Nonlinear least-squares.- 23D. Initial parameter estimates.- 23E. Method of moments.- 23F. Other methods of parameter estimation.- 23G. Positive amplitudes.- 23H. Curve-fitting sums of exponentials is ill-posed.- 23I. Fitting the differential equation model directly to data.- 23J. Modulating function method.- 23K. An antigen - antibody reaction example.- 23L. Additional literature on fitting of differential equations to data.- Section 24. Numerical Simulation of the Model.- 24A. Compartmental model simulation.- 24B. A three compartment thyroxine model.- 24C. Numerical integration methods and some inadequacies.- 24D. Implicit methods ..- 24E. Determining model stiffness.- Section 25. Identification of Compartment Volumes.- 25A. The basic single exit compartmental model.- 25B. Readily identifiable parameters.- 25C. The catenary single exit system.- 25D. Estimation of compartmental volumes.- 25E. Creatinine clearance model.- 25F. Shock therapy.- 25G. Bounds and approximations on compartmental volumes.- Section 26. A Discrete Time Stochastic Model of a Compartmental System.- 26A. The Markov chain model.- 26B. The liver disease model.- 26C. Simulation of the hepatic system.- 26D. Mathematical analysis of the model.- 26E. Parameter estimation.- 26F. Discussion.- Section 27. Closing Remarks.

The General Case

Abstract: My object in this section is to present Blackwell’s (1958) example of a standard stochastic semigroup, all of whose states are instantaneous. For other examples of this phenomenon, see Sections 3.3 of ACM and Section 2.12 of B & D. To begin with, consider the matrix $$ Q = \left( {\begin{array}{*{20}{c}} { - \lambda } & \lambda \\ \mu & { - \mu } \\ \end{array} } \right) $$ on {0, 1}, with λ and μ nonnegative, λ + μ positive. There is exactly one standard stochastic semigroup P on {0, 1} with P′(0) = Q, namely: $$ P\left( {t,0,0} \right) = \frac{\mu }{{\mu + \lambda }} + \frac{\lambda }{{\mu + \lambda }}{e^{{ - \left( {\mu + \lambda } \right)t}}}P\left( {t,0,1} \right) = 1 - P\left( {t,0,0} \right)P\left( {t,1,1} \right) = \frac{\lambda }{{\mu + \lambda }} + \frac{\mu }{{\mu + \lambda }}{e^{{ - 1\left( {\mu + \lambda } \right)t}}}P\left( {t,1,0} \right) = 1 - P\left( {t,1,1} \right) $$ (1) One way to see this is to use (5.29): define P by (1); check P is continuous, P(0) is the identity, P′(0) = Q, and P(t + s) = P(t) · P(s). Dull computations in the last step can be avoided by thinking: it is enough to do μ + λ = 1 by rescaling time; since P(u) is 2 × 2 and stochastic when u is t or s or t + s, it is enough to check that P(t + s) = P(t) · P(s) on the diagonal; by interchanging μ and λ, so 0 and 1, it is enough to check the (0, 0) position. This is easy.

Some Nonlinear Evolution Equations

Abstract: In this section we will study the following semilinear initial value problem: $$\left\{ {\begin{array}{*{20}{c}} {\frac{{du(t)}}{{dt}} + Au(t) = f(t,u(t)), t > {{t}_{0}}} \hfill \\ {u({{t}_{0}}) = {{u}_{0}}} \hfill \\ \end{array} } \right.$$ (1.1) where -A is the infinitesimal generator of a C0semigroup T(t), t ≥ 0, on a Banach space X and f: [t 0 ,T] X X→ X is continuous in t and satisfies a Lipschitz condition in u.

Design considerations for the structural optimization of a single-mode fiber

Abstract: Design considerations are made for the structural optimization of single-mode fibers used in high-bit-rate and long-haul transmission systems in the long-wavelength region. As the basic fiber parameters, a combination of the spot size W 0 and the effective cutoff wavelength λ ce is newly chosen, because the combination is found to suitably describe various actual index profiles which deviate from an ideal step-index profile. A procedure to specify the usable range of W 0 and λ ce is established, whereby the overall transmission-line loss in one repeater section is calculated using simple expressions for fiber intrinsic loss, excess loss in the cabling process, and splice loss, etc. The optimum values for a 400 Mbit/s transmission system operating at 1.3 μm with a repeater spacing of 20 km are obtained as W_{0} = 5.0 \pm 0.5\mu m and 1.1 μm \leq\lambda_{ce}\leq 1.28 \mu m taking into consideration the additional requirement for the possible use at \lambda=1.55 \mu m

Applications to Partial Differential Equations—Linear Equations

Abstract: In this section we consider a simple application of the results of Section 6.1 to the initial value problem for the following nonlinear Schrodinger equation in ∝2 $$\left\{ {_{u(x,0) = {u_0}(x)in{\mathbb{R}^2}}^{\frac{1}{i}\frac{{\partial u}}{{\partial t}} - \Delta u + k{{\left| u \right|}^2}u = 0in]0,\infty [x{\mathbb{R}^2}}} \right.$$ (1.1) where u is a complex valued function and k a real constant. The space in which this problem will be considered is L2(R2). Defining the linear operator A 0 by D(A 0 ) = H 2 (R2)and A 0 u = — i & u for u ϵ D(A0) the initial value problem (1.1) can be rewritten as $$\left\{{_{u(0) = {u_0}}^{\frac{{du}}{{dt}} + {A_0}u + F(u) = 0fort>0}}\right.$$ (1.2)

Mean-value of the Riemann zeta-function and other remarks III

Abstract: The results given in these papers continue the theme developed in part I of this series. In Part III we prove $M(\frac{1}{2})>\!\!\!>_k (\log H_0/q_n)^{k^2}$, where $p_m/q_m$ is the $m$th convergent of the continued fraction expansion of $k$, and $n$ is the unique integer such that $q_nq_{n+1}\geq \log\log H_0 > q_nq_{n-1}$. Section 4 of part III discusses lower bounds of mean values of Titchmarsh series.

Forming method of curved surface

Abstract: PURPOSE: To obtain a curved surface even with a three-dimensional reference curve by producing the surface of a three-dimensional curved surface body from plural intermediate section curved surfaces. CONSTITUTION: At the outset, a three-dimensional curve 31a is specified and fed, and at the same time the data on sections 11 and 12 as well as section curves 11a and 12a are fed. Then a dividing point Si is obtained to divide the length of the curve 31a into m:n, and at the same time a tangent line TLN of the curve 31a is obtained at the point Si. An intermediate section 41 is formed so that it is vertical to the line TLN. This process to form the section 41 is repeated while changing the dividing ratio m:n successively from 0:1 through 1:0. As a result, many intermediate section curves 41a are obtained. These curves are connected to each other to obtain a smooth curved surface. COPYRIGHT: (C)1985,JPO&Japio

The Stable Case

Abstract: In this chapter, unless I say otherwise, let P be a standard stochastic semigroup on the countable set I, with all states stable: $$ q(i) = - P'\left( {0,i,i} \right) < \infty $$ for all i The first problem is to create a P-chain X all of whose sample functions are regular: continuous from the right, with limits from the left at all times, when discrete I has been compactified by adjoining the point at infinity φ. This is done in Section 2. To see why compactification is a good idea, look at (6.142).

Apparatus for manufacturing fasciated yarn

Abstract: Apparatus for manufacturing fasciated spun yarn by false-twisting and detwisting a bundle of fibers is provided. The apparatus has a fiber-diffusing section which utilizes differential fluid flow to separate and transfer free fibers in a stable manner for subsequent wrapping about the fiber bundle as the bundle is detwisted.

Continuous incidence relations of topological planes

Abstract: 0 In topological planes the link between the geometrical and the topological structure is established by the requirement that the partial external operations "joining of points" and "intersecting of lines" be (op-) continuous (for the terminology, see Section 1) It seems that nowhere in the literature one has discussed the question, if and under which conditions the incidence relation (or other relations derived from it) underlying to the notion of a topological plane is (resp are) "continuous" in some natural sense, or whether, in a more general setting (see Remark 6), the incidence relation should be required to be "continuous" In this paper, it turns out that the incidence relation of each topological plane and its inverse are lowersemicontinuous and they are continuous (in the sense introduced by the author in [3]) in certain classes of topological planes (Theorem 4, Propositions 5 through 8) As a byproduct in this paper, we characterize (in Theorem 1) the continuity of certain relations between a topological space and a product of topological spaces (which has a simple consequence in Proposition 9) and describe (in Theorem 3) the topology of the set of lines (of a topological plane) in terms of limits (via the notion of the power of a topology)

Routing past unions of disjoint linear barriers

Abstract: Given a disjoint planar set, {Li, = PiQi, i = 1,…, n}, of line segments called barriers, we consider the question of finding a path Γ of minimal length which connects two given points A and B and which does not “cut” any of the barriers. In Section III we show that such a minimal path exists and that it is polygonal with its bend points lying in W = {Pi, Qi: i = 1,…, n}. The problem is thus reduced to finding the shortest path between two points A and B in an approximate network with vertex set V = W ∪ {A, B}. The latter can be solved by a network routing algorithm such as Dantzig's. Section IV presents Algorithms for reducing the size of the network.

The weil conjectures in finite geometry

Abstract: In the first section the Weil conjectures for non-singular primals are stated and several examples are given. Particularities for curves are described in section two. The remaining sections are devoted to elliptic cubic curves. In particular, the number of points that a cubic can have is precisely given, as well as the number of inequivalent curves with a fixed number of points.

Structural Identification of the Model

Abstract: A central problem in linear compartmental studies is discussed in this section. The question is whether or not the system parameters aij in the model \( {\dot x = Ax(t) + b,\;A \equiv [a_{ij} ]} \) can be uniquely determined from experimental observation or measurement of certain components of the solution vector x. The closely related problem of getting actual numerical estimates of the aijs will be considered later in Section 19.

Reflected Brownian Motions

Abstract: The process \( |B| \) is called the reflection of the Brownian motion B at zero. By (8.1), the pair \( (|B|,L) \) is almost surely the solution of a certain problem of reflection for \( \hat{B} \), discussed in Section 8.2. This yields an alternative representation, directly in terms of \( \hat{B} \), for L and hence for \( |B| \) The Ito formula will be used in Section 8.3, to make the connection with the analytical theory of this alternative representation.

Processing apparatus for a paper sheet bundle with a tape wound therearound

Abstract: Provided is a processing apparatus for a paper sheet bundle (P) with a tape wound therearound. The bundle is carried one by one into a bundle receiving section (21) provided in the apparatus. In the bundle receiving section (21) the bundle is retained by a vertical stop member (50) and the tape around the bundle is cut by a knife member (86) of a tape cutting mechanism. During this period of time, an urging roller (54) is pressed against the upper surface of the bundle so as to keep the bundle in its retained position. The paper sheets of the untied bundle, from a lowermost one thereof, are sequentially delivered from the receiving section and, at the section of shredder rolls (56, 57), are cut into fine pieces to destroy their paper value. The paper pieces thus produced are accumulated in a receiving chamber (15) located in the lower part of the apparatus.

Ultra-Distributions and Weighted Spaces of Entire Analytic Functions

Abstract: In Chapter 6 we continue the studies on entire analytic functions which we began in the first chapter. A description of the problems treated has been given in Section 1.1. The goal is the proof of estimates of type (1.1/5) and related maximal inequalities. Chapter 1 dealt essentially with the unweighted case. This was not only convenient but completely sufficient for the main purpose of this book: the study of unweighted spaces of type \(B_{p,q}^S \) and \(F_{p,q}^S \) on R n and on domains, and their applications to elliptic boundary value problems in Part I. On the other hand, weighted spaces of several types have attracted much attention in recent times.

The Inverse of a Compartmental Matrix

Abstract: In the last section we saw that under the condition of a constant input vector b, the compartmental model $$ {\dot x(t) = Ax(t) + b} $$ (13.1) comes to a natural resting spot or equilibrium provided A-1 exists. A constant vector xe is an equilibrium point of model (13.1) provided it has the property that once the state vector is equal to x it remains equal to that vector for all future time [153]. For the system (13.1), an equilibrium point xe satisfies the equation $$ {\text{0 = Ax}}_{\text{e}} {\text{ + b}}.{\text{ }} $$ (13.2)

Some Aspects of Theoretical Physics Relating to Harmonic Maps

Abstract: Let (M,g) be a compact Riemannian manifold without boundary, and consider an action which, for simplicity, we suppose has the form $$I\left( \phi \right) = \int\limits_M {L\left( {{j^1}\left( \phi \right)} \right)} \;dx\;, $$ (1.1) , where Φ is a section of a Riemannian fiber bundle π: E → M, and L: J1 (E) → ℝ possibly depends on the metric of E.

On Quotients of Lp-Means

Abstract: In this paper we propose a method for establishing upper bounds for functions of the form $$\Phi (f) = \frac{{{{\left( {\int {{f^p}d\mu } } \right)}^{1/p}}}}{{{{\left( {\int {{f^q}d\lambda } } \right)}^{1/q}}}}$$ , where μ and λ are probability measures and q is less than p. Specifically, it is shown that, subject to certain conditions, ϕ is a quasiconvex function and therefore satisfies a boundary-maximum principle. This yields a unified explanation of the “vertex phenomenon” in the theory of complementary inequalities (cf. [6], Section 2). The lower bound for ϕ is also determined.

Aircraft and method of constructing it.

Abstract: The aircraft (6) is a section from a basic body (4) which is formed by rotating an aerofoil section (1) with its exit edge about a centre of formation (B). The diameter (d) of the aircraft (6) is 15 - 40% larger than the radius (R) of the basic body (4) and touches the circumferential line (5) thereof at a point (7). The aerofoil sections (1) cut from the circumference of the aircraft (6) form section fragments whose underside is curved with respect to the topside of the section. This results in an aircraft which is inherently stable by contrast with other aircraft, and whose large area is particularly suitable for flying using energy from solar cells.

Linear Ordinary Differential Equations with Quasiperiodic Coefficients

Abstract: In this section we wish to study the general form of the solution matrix Q(t) of the differential equation $$ \dot Q\left( t \right) = M\left( t \right)Q\left( t \right) $$ (3.1.1) where M is a complex-valued m × m matrix which can be expressed as a Fourier series of the form $$ M\left( t \right) = \sum\limits_{n1,n2, \ldots ,nN} {M_{n1,n2, \ldots ,nN} \exp } \left( {{\text{i}}\omega _1 n_1 t + {\text{i}}\omega _N n_N t} \right). $$ (3.1.2) .