Continuous stochastic processes

A general review of stochastic processes is given in the introduction; definitions, properties and a rough classification are presented together with the position and scope of the author's work as it fits into the general scheme.

The first section presents a brief summary of the pertinent analytical properties of continuous stochastic processes and their probability-theoretic foundations which are used in the sequel.

The remaining two sections (II and III), comprising the body of the work, are the author's contribution to the theory. It turns out that a very inclusive class of continuous stochastic processes are characterized by a fundamental partial differential equation and its adjoint (the Fokker-Planck equations). The coefficients appearing in those equations assimilate, in a most concise way, all the salient properties of the process, freed from boundary value considerations. The writer’s work consists in characterizing the processes through these coefficients without recourse to solving the partial differential equations.

First, a class of coefficients leading to a unique, continuous process is presented, and several facts are proven to show why this class is restricted. Then, in terms of the coefficients, the unconditional statistics are deduced, these being the mean, variance and covariance. The most general class of coefficients leading to the Gaussian distribution is deduced, and a complete characterization of these processes is presented. By specializing the coefficients, all the known stochastic processes may be readily studied, and some examples of these are presented; viz. the Einstein process, Bachelier process, Ornstein-Uhlenbeck process, etc. The calculations are effectively reduced down to ordinary first order differential equations, and in addition to giving a comprehensive characterization, the derivations are materially simplified over the solution to the original partial differential equations.

In the last section the properties of the integral process are presented. After an expository section on the definition, meaning, and importance of the integral process, a particular example is carried through starting from basic definition. This illustrates the fundamental properties, and an inherent paradox. Next the basic coefficients of the integral process are studied in terms of the original coefficients, and the integral process is uniquely characterized. It is shown that the integral process, with a slight modification, is a continuous Markoff process.

The elementary statistics of the integral process are deduced: means, variances, and covariances, in terms of the original coefficients. It is shown that an integral process is never temporally homogeneous in a non-degenerate process.

Finally, in terms of the original class of admissible coefficients, the statistics of the integral process are explicitly presented, and the integral process of all known continuous processes are specified.

Sektore automobilistikoaren denborazko prezio-analisia

Lan honek 2007 eta 2013 urteen bitartean autoen prezioak nola aldatzen diren erakusten digu. Horretarako, autoen hainbat ezaugarri hartu dira kontuan, hala nola, modeloa, mota, zilindrada, zilindroen posizioa, potentzia, luzera, zabalera, altuera, kontsumoa, abiadura maximoa, azelerazioa eta maleteroaren kapazitatea. Gaur egungo egoera dela eta, hasiera batean pentsa genezake kotxeen prezioak jaitsi egin direla, baina dituzten aurrerapenak kontuan izanda, logikoena izango zen zenbat eta aldagai garatuagoak izan orduan eta garestiagoak izatea. Beraz, lanaren helburu izango da, emaitza horiek bete diren edo ez frogatzea, eredu ekonometriko bat zehaztuz. Ereduaren estimazioa egiteko erabilitako metodologia Karratu Txikien Arruntak izan dira, baina heterozedastizitatearen arazoa agertu denez, eredu honek ez du bariantza txikiena izango eta beraz estimatzaile berria erabili beharko da, kasu honetan, Karratu Txikienen Zabalduen metodoa erabili da. Urte bakoitzerako egokiagoa den eredua hautatu ondoren eredu orokorra egin dut, horretarako 2007 eta 2013ko datuak begiraturik, bakoitzari dagokion aldagai azaltzaile bat eratu diot, hau da, ezaugarri bakoitza bere urtearekin elkartu dut. Murriztutako eredura heltzerakoan, lortutako emaitzak ikusirik ondorio batera heldu naiz, ondorio hori hasieran planteatutako hipotesiarekin bat datorrela ikusi da, hau da, autoen prezioak proportzionalki jaitsi egin dira, eta ondorioz, autoen ezaugarriak gutxiago baloratzen dira.

Routing scheme, permutation properties, and optical implementation of a four-shuffle-exchange-based Omega network

The routing scheme and some permutation properties of a four-shuffle-exchange-based Omega network are discussed. The corresponding optical setup, which is composed of 2-D phase spatial light modulators and calcite plates, is proposed and demonstrated through mapping the inputs to a 2-D array. Instead of one shuffle-exchange followed by one switching operation as in ordinary Omega networks, in our presented system, the shuffle interconnection embraced in the switches is accomplished simply by varying the switching structure of each stage. For the proposed polarization-optical modules, the system is compact in structure, efficient in performance, and insensitive to the environment. (C) 1997 Society of Photo-Optical Instrumentation Engineers.

Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design

The general formulation of double refraction or internal double reflection for any directions of incidence and arbitrary orientation of the optic axis in a uniaxial crystal is analysed in terms of Huygens' principle. Then double refraction and double reflection along the sequential interfaces in a crystal are discussed. On this basis, if the parameters of the interface are chosen appropriately, the range of angular separation between the ordinary ray and extraordinary ray can be much greater, It is useful for crystal element design. Finally, as an example, an optimum design of the Output end interface for a 2 x 2 electro-optic switch is given.

Equivalent differential equations for nonlinear dynamical systems

A technique for obtaining approximate periodic solutions to nonlinear ordinary differential equations is investigated. The approach is based on defining an equivalent differential equation whose exact periodic solution is known. Emphasis is placed on the mathematical justification of the approach. The relationship between the differential equation error and the solution error is investigated, and, under certain conditions, bounds are obtained on the latter. The technique employed is to consider the equation governing the exact solution error as a two point boundary value problem. Among other things, the analysis indicates that if an exact periodic solution to the original system exists, it is always possible to bound the error by selecting an appropriate equivalent system.

Three equivalence criteria for minimizing the differential equation error are compared, namely, minimum mean square error, minimum mean absolute value error, and minimum maximum absolute value error. The problem is analyzed by way of example, and it is concluded that, on the average, the minimum mean square error is the most appropriate criterion to use.

A comparison is made between the use of linear and cubic auxiliary systems for obtaining approximate solutions. In the examples considered, the cubic system provides noticeable improvement over the linear system in describing periodic response.

A comparison of the present approach to some of the more classical techniques is included. It is shown that certain of the standard approaches where a solution form is assumed can yield erroneous qualitative results.

I. Electronic states of heavily-doped molecular crystals--naphthalene. I. Theoretical. II. Experimental. II. Lowest singlet and triplet excited states of pyrazine

PART I

The energy spectrum of heavily-doped molecular crystals was treated in the Green’s function formulation. The mixed crystal Green’s function was obtained by averaging over all possible impurity distributions. The resulting Green’s function, which takes the form of an infinite perturbation expansion, was further approximated by a closed form suitable for numerical calculations. The density-of-states functions and optical spectra for binary mixtures of normal naphthalene and deuterated naphthalene were calculated using the pure crystal density-of-state functions. The results showed that when the trap depth is large, two separate energy bands persist, but when the trap depth is small only a single band exists. Furthermore, in the former case it was found that the intensities of the outer Davydov bands are enhanced whereas the inner bands are weakened. Comparisons with previous theoretical calculations and experimental results are also made.

PART II

The energy states and optical spectra of heavily-doped mixed crystals are investigated. Studies are made for the following binary systems: (1) naphthalene-h₈ and d₈, (2) naphthalene--h₈ and αd₄, and (3) naphthalene--h₈ and βd₁, corresponding to strong, medium and weak perturbations. In addition to ordinary absorption spectra at 4˚K, band-to-band transitions at both 4˚K and 77˚K are also analyzed with emphasis on their relations to cooperative excitation and overall density-of-states functions for mixed crystals. It is found that the theoretical calculations presented in a previous paper agree generally with experiments except for cluster states observed in system (1) at lower guest concentrations. These features are discussed semi-quantitatively. As to the intermolecular interaction parameters, it is found that experimental results compare favorably with calculations based on experimental density-of-states functions but not with those based on octopole interactions or charge-transfer interactions. Previous experimental results of Sheka and the theoretical model of Broude and Rashba are also compared with present investigations.

PART III

The phosphorescence, fluorescence and absorption spectra of pyrazine-h₄ and d₄ have been obtained at 4˚K in a benzene matrix. For comparison, those of the isotopically mixed crystal pyrazine-h₄ in d₄ were also taken. All these spectra show extremely sharp and well-resolved lines and reveal detailed vibronic structure.

The analysis of the weak fluorescence spectrum resolves the long-disputed question of whether one or two transitions are involved in the near-ultraviolet absorption of pyrazine. The “mirror-image relationship” between absorption and emission shows that the lowest singlet state is an allowed transition, properly designated as ¹B_3u ← ¹A_1g. The forbidden component ¹B_2g, predicted by both “exciton” and MO theories to be below the allowed component, must lie higher. Its exact location still remains uncertain.

The phosphorescence spectrum when compared with the excitation phosphorescence spectra, indicates that the lowest triplet state is also symmetry allowed, showing a strong 0-0 band and a “mirror-image relationship” between absorption and emission. In accordance with previous work, the triplet state is designated as ³B_3u.

The vibronic structure of the phosphorescence spectrum is very complicated. Previous work on the analysis of this spectrum all concluded that a long progression of v_6a exists. Under the high resolution attainable in our work, the supposed v_6a progression proves to have a composite triplet structure, starting from the second member of the progression. Not only is the v_9a hydrogen-bending mode present as shown by the appearance of the C-D bending mode in the d₄ spectrum, but a band of 1207 cm^-1 in the pyrazine in benzene system and 1231 cm^-1 in the mixed crystal system is also observed. This band is assigned as 2v_6b and of a_1g symmetry. Its anonymously strong intensity in the phosphorescence spectrum is interpreted as due to the Fermi resonance with the 2v_6a and v_9a band.

To help resolve the present controversy over the crystal phosphorescence spectrum of pyrazine, detailed vibrational analyses of the emission spectra were made. The fluorescence spectrum has essentially the same vibronic structure as the phosphorescence spectrum.

Range of validity of the method of averaging

Sufficient conditions are derived for the validity of approximate periodic solutions of a class of second order ordinary nonlinear differential equations. An approximate solution is defined to be valid if an exact solution exists in a neighborhood of the approximation.

Two classes of validity criteria are developed. Existence is obtained using the contraction mapping principle in one case, and the Schauder-Leray fixed point theorem in the other. Both classes of validity criteria make use of symmetry properties of periodic functions, and both classes yield an upper bound on a norm of the difference between the approximate and exact solution. This bound is used in a procedure which establishes sufficient stability conditions for the approximated solution.

Application to a system with piecewise linear restoring force (bilinear system) reveals that the approximate solution obtained by the method of averaging is valid away from regions where the response exhibits vertical tangents. A narrow instability region is obtained near one-half the natural frequency of the equivalent linear system. Sufficient conditions for the validity of resonant solutions are also derived, and two term harmonic balance approximate solutions which exhibit ultraharmonic and subharmonic resonances are studied.

Experimental culture of algal balls (Cladophora sauteri). Retention of original shape [Translation from: Bull.Jap.Soc.Phycol. Vol. 10, (1), p. 23-27, 1962]

It was on July 1960 when 10 algal balls were acquired for exhibition at Suma Aquarium, Kobe. Permission to remove the specimens from the Lake Akan Reserve was given by the National Nature Reserve Committee. Algal balls, as a rule, lose their natural beauty when they are kept in an ordinary tank for a certain length of time. In an effort to retain the natural beauty it was decided to exhibit them in culture. This paper summarises the findings of this experiments with Cladophora sauteri. The author concludes that serious consideration has to be given as to the intensity of light, the sunlight, the water temperature and the nutrition for algal balls in culture in order to retain the natural beauty and shape.

Studies on photoblastic germination in lettuce seed

It was shown, with the aid of osmotic inhibition of germination, that the action of the far-red-absorbing form of phytochrome (Pf) in promoting germination can be completed even if the seed is held under conditions where germination is not possible. An effect of the continuing action of Pf beyond the point of complete germination promotion was demonstrated by enhancement of germination rate after removal of the osmotically active solute.

Previous reports that the rate of growth in water of seeds freed from the expansion-restricting endosperm is independent of the state of phytochrome were confirmed. However, a marked, phytochrome-mediated enhancement of the growth potential of such seeds was demonstrated through restricting water uptake by incubation in an osmoticum.

An experimental system, utilizing the appearance of a geotropic curvature in the radicle of the excised axial portion of the seed, was developed for more detailed studies of the phytochrome-enhanced growth potential. It was possible to demonstrate the light effect in water as well as in osmotica; this apparently is not possible with de-endospermed entire seeds. As in intact seeds, the effect of the continuing action of Pf is to enhance the rate of the response. Secretion of a chemical inhibitor of growth by the endosperm as a possible mechanism of induction of light sensitivity has been ruled out.

The phytochrome-dependent rate of appearance of geotropic curvature in osmotica is paralleled in time by a similar dependence of the rate of early extension growth of the embryonic axis. Only the first small increment of growth is a differentially responsive to red (R) and far-red (F); the rate of later increase in length is independent of the light regime.

It was shown that the high concentrations of gibberellic acid required for germination promotion in the intact seed are due at least in part to a diffusion barrier in the endosperm, and that the occasional reports in the literature of the ineffectiveness of kinetin are probably due to the same phenomenon. It was shown that gibberellin, like red light, enhances the growth potential of the axis, but kinetin does not. The difference in rates of response obtained after R-irradiation or gibberellin treatment, together with other results reported in the literature, strongly suggests that gibberellic acid and red light promote germination by different means. The idea that kinetin promotes germination by yet another mechanism, probably operating in the cotyledons, was supported through two different experimental approaches.

The phenomenon of temperature-dependent dark germination was examined in detail, using a wide range of both temperatures and incubation times. With the aid of the half-seed system, it was demonstrated that the promotive effect of low temperature on germination could not be due to a low optimum temperature for early growth of the radicle, since the rate of that process increased with increasing temperature, up to the highest temperature used.

It was shown that phytochrome does not function at high temperatures. This fact is of considerable importance in interpreting the phenomenon of thermodormancy, since in the literature only a small part of the effect of high temperature has been ascribed to an effect on phytochrome, and at that, only to an acceleration of dark reversion of Pf to the red-absorbing form of phytochrome (Pr). Partial denaturation of phytochrome may also make some contribution.

It was shown that the germination-promoting effect of low temperature depends on the presence of Pf, and concluded that low temperatures act by delaying or preventing transformation of Pf. Support for the assumption that Pf, not Pr, is the active form of phytochrome in lettuce seeds was drawn from the same evidence.

Attempts to stimulate germination by repeated irradiation with F over relatively prolonged incubation times resulted in failure, as have similar attempts reported in the literature. However, an enhancement of growth potential in the half-seed system by the maintenance of a small amount of Pf over long periods at ordinary temperatures by repeated irradiation with F was demonstrated.

It was observed that cold storage of the dry seed prevents or delays loss of dark dormancy during post-harvest storage. No change in the response of the half-seed in osmoticum to R and F was observed in seeds that has lost dark dormancy; that is, no internal change took place to measurably increase the growth potential of the embryonic axis. This suggests that the endosperm is the seat of changes responsible for after-ripening of photoblastic lettuce seed.

I. Nuclear spin-internal-rotation-coupling. II. Fluorine spin-rotation interaction and magnetic shielding in fluorobenzene

Part I.

The interaction of a nuclear magnetic moment situated on an internal top with the magnetic fields produced by the internal as well as overall molecular rotation has been derived following the method of Van Vleck for the spin-rotation interaction in rigid molecules. It is shown that the Hamiltonian for this problem may be written

H_SR = Ῑ · M · Ĵ + Ῑ · M” · Ĵ”

Where the first term is the ordinary spin-rotation interaction and the second term arises from the spin-internal-rotation coupling.

The F¹⁹ nuclear spin-lattice relaxation time (T₁) of benzotrifluoride and several chemically substituted benzotrifluorides, have been measured both neat and in solution, at room temperature by pulsed nuclear magnetic resonance. From these experimental results it is concluded that in benzotrifluoride the internal rotation is crucial to the spin relaxation of the fluorines and that the dominant relaxation mechanism is the fluctuating spin-internal-rotation interaction.

Part II.

The radiofrequency spectrum corresponding to the reorientation of the F¹⁹ nuclear moment in flurobenzene has been studied by the molecular beam magnetic resonance method. A molecular beam apparatus with an electron bombardment detector was used in the experiments. The F¹⁹ resonance is a composite spectrum with contributions from many rotational states and is not resolved. A detailed analysis of the resonance line shape and width by the method of moments led to the following diagonal components of the fluorine spin-rotational tensor in the principal inertial axis system of the molecule:

F/Caa = -1.0 ± 0.5 kHz

F/Cbb = -2.7 ± 0.2 kHz

F/Ccc = -1.9 ± 0.1 kHz

From these interaction constants, the paramagnetic contribution to the F¹⁹ nuclear shielding in C₆H₅F was determined to be -284 ± ppm. It was further concluded that the F¹⁹ nucleus in this molecule is more shielded when the applied magnetic field is directed along the C-F bond axis. The anisotropy of the magnetic shielding tensor, σ_” - σ_⊥, is +160 ± 30 ppm.

Particle fluid mechanics in shear flows, acoustic waves, and shock waves

Three different categories of flow problems of a fluid containing small particles are being considered here. They are: (i) a fluid containing small, non-reacting particles (Parts I and II); (ii) a fluid containing reacting particles (Parts III and IV); and (iii) a fluid containing particles of two distinct sizes with collisions between two groups of particles (Part V).

Part I

A numerical solution is obtained for a fluid containing small particles flowing over an infinite disc rotating at a constant angular velocity. It is a boundary layer type flow, and the boundary layer thickness for the mixture is estimated. For large Reynolds number, the solution suggests the boundary layer approximation of a fluid-particle mixture by assuming W = W_p. The error introduced is consistent with the Prandtl’s boundary layer approximation. Outside the boundary layer, the flow field has to satisfy the “inviscid equation” in which the viscous stress terms are absent while the drag force between the particle cloud and the fluid is still important. Increase of particle concentration reduces the boundary layer thickness and the amount of mixture being transported outwardly is reduced. A new parameter, β = 1/Ω τ_v, is introduced which is also proportional to μ. The secondary flow of the particle cloud depends very much on β. For small values of β, the particle cloud velocity attains its maximum value on the surface of the disc, and for infinitely large values of β, both the radial and axial particle velocity components vanish on the surface of the disc.

Part II

The “inviscid” equation for a gas-particle mixture is linearized to describe the flow over a wavy wall. Corresponding to the Prandtl-Glauert equation for pure gas, a fourth order partial differential equation in terms of the velocity potential ϕ is obtained for the mixture. The solution is obtained for the flow over a periodic wavy wall. For equilibrium flows where λ_v and λ_T approach zero and frozen flows in which λ_v and λ_T become infinitely large, the flow problem is basically similar to that obtained by Ackeret for a pure gas. For finite values of λ_v and λ_T, all quantities except v are not in phase with the wavy wall. Thus the drag coefficient C_D is present even in the subsonic case, and similarly, all quantities decay exponentially for supersonic flows. The phase shift and the attenuation factor increase for increasing particle concentration.

Part III

Using the boundary layer approximation, the initial development of the combustion zone between the laminar mixing of two parallel streams of oxidizing agent and small, solid, combustible particles suspended in an inert gas is investigated. For the special case when the two streams are moving at the same speed, a Green’s function exists for the differential equations describing first order gas temperature and oxidizer concentration. Solutions in terms of error functions and exponential integrals are obtained. Reactions occur within a relatively thin region of the order of λ_D. Thus, it seems advantageous in the general study of two-dimensional laminar flame problems to introduce a chemical boundary layer of thickness λ_D within which reactions take place. Outside this chemical boundary layer, the flow field corresponds to the ordinary fluid dynamics without chemical reaction.

Part IV

The shock wave structure in a condensing medium of small liquid droplets suspended in a homogeneous gas-vapor mixture consists of the conventional compressive wave followed by a relaxation region in which the particle cloud and gas mixture attain momentum and thermal equilibrium. Immediately following the compressive wave, the partial pressure corresponding to the vapor concentration in the gas mixture is higher than the vapor pressure of the liquid droplets and condensation sets in. Farther downstream of the shock, evaporation appears when the particle temperature is raised by the hot surrounding gas mixture. The thickness of the condensation region depends very much on the latent heat. For relatively high latent heat, the condensation zone is small compared with Ʌ_D.

For solid particles suspended initially in an inert gas, the relaxation zone immediately following the compression wave consists of a region where the particle temperature is first being raised to its melting point. When the particles are totally melted as the particle temperature is further increased, evaporation of the particles also plays a role.

The equilibrium condition downstream of the shock can be calculated and is independent of the model of the particle-gas mixture interaction.

Part V

For a gas containing particles of two distinct sizes and satisfying certain conditions, momentum transfer due to collisions between the two groups of particles can be taken into consideration using the classical elastic spherical ball model. Both in the relatively simple problem of normal shock wave and the perturbation solutions for the nozzle flow, the transfer of momentum due to collisions which decreases the velocity difference between the two groups of particles is clearly demonstrated. The difference in temperature as compared with the collisionless case is quite negligible.

双中心全息记录的矢量分析与记录方向优化

对描述双掺杂晶体非挥发性全息记录动力学过程的Kukhtarev方程进行了矢量分析, 分析中考虑了体光生伏特效应和外加电场的作用。在小信号近似的基础上给出了双中心全息记录中记录与固定阶段空间电荷场的矢量解析解。在综合考虑空间电荷场的各向异性以及晶体有效电光系数的各向异性后, 给出了双中心全息记录的优化记录方向。结果表明, 对(Fe, Mn):LiNbO3晶体633 nm寻常光记录, 优化记录方向主要由有效电光系数决定, 光栅波矢与光轴夹角为22°, 方位角为30°;对(Fe, Mn):LiNbO3晶体633

A nonlinear systems model for the control mechanism of free fatty acid-glucose metabolosm in normal humans

A mathematical model is proposed in this thesis for the control mechanism of free fatty acid-glucose metabolism in healthy individuals under resting conditions. The objective is to explain in a consistent manner some clinical laboratory observations such as glucose, insulin and free fatty acid responses to intravenous injection of glucose, insulin, etc. Responses up to only about two hours from the beginning of infusion are considered. The model is an extension of the one for glucose homeostasis proposed by Charette, Kadish and Sridhar (Modeling and Control Aspects of Glucose Homeostasis. Mathematical Biosciences, 1969). It is based upon a systems approach and agrees with the current theories of glucose and free fatty acid metabolism. The description is in terms of ordinary differential equations. Validation of the model is based on clinical laboratory data available at the present time. Finally procedures are suggested for systematically identifying the parameters associated with the free fatty acid portion of the model.

On the Cesari fixed point method in a Banach space

In a paper published in 1961, L. Cesari [1] introduces a method which extends certain earlier existence theorems of Cesari and Hale ([2] to [6]) for perturbation problems to strictly nonlinear problems. Various authors ([1], [7] to [15]) have now applied this method to nonlinear ordinary and partial differential equations. The basic idea of the method is to use the contraction principle to reduce an infinite-dimensional fixed point problem to a finite-dimensional problem which may be attacked using the methods of fixed point indexes.

The following is my formulation of the Cesari fixed point method:

Let B be a Banach space and let S be a finite-dimensional linear subspace of B. Let P be a projection of B onto S and suppose Г≤B such that pГ is compact and such that for every x in PГ, P^-1x∩Г is closed. Let W be a continuous mapping from Г into B. The Cesari method gives sufficient conditions for the existence of a fixed point of W in Г.

Let I denote the identity mapping in B. Clearly y = Wy for some y in Г if and only if both of the following conditions hold:

(i) Py = PWy.

(ii) y = (P + (I - P)W)y.

Definition. The Cesari fixed paint method applies to (Г, W, P) if and only if the following three conditions are satisfied:

(1) For each x in PГ, P + (I - P)W is a contraction from P^-1x∩Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P^-1x∩Г which satisfies the equation y(x) = Py(x) + (I-P)Wy(x).

(2) The function y just defined is continuous from PГ into B.

(3) There are no fixed points of PWy on the boundary of PГ, so that the (finite- dimensional) fixed point index i(PWy, int PГ) is defined.

Definition. If the Cesari fixed point method applies to (Г, W, P) then define i(Г, W, P) to be the index i(PWy, int PГ).

The three theorems of this thesis can now be easily stated.

Theorem 1 (Cesari). If i(Г, W, P) is defined and i(Г, W, P) ≠0, then there is a fixed point of W in Г.

Theorem 2. Let the Cesari fixed point method apply to both (Г, W, P₁) and (Г, W, P₂). Assume that P₂P₁=P₁P₂=P₁ and assume that either of the following two conditions holds:

(1) For every b in B and every z in the range of P₂, we have that ‖b=P₂b‖ ≤ ‖b-z‖

(2)P₂Г is convex.

Then i(Г, W, P₁) = i(Г, W, P₂).

Theorem 3. If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinite-dimensional) Leray-Schauder index i_LS(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = i_LS(W, Ω).

Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finite-dimensional and the Leray-Schauder indexes. These and other properties of indexes will be listed before the theorem in which they are used.

Coma measurement by use of an alternating phase-shifting mask mark with a specific phase width

The correlation between the coma sensitivity of the alternating phase-shifting mask (Alt-PSM) mark and the mark's structure is studied based on the Hopkins theory of partially coherent imaging and positive resist optical lithography (PROLITH) simulation. It is found that an optimized Alt-PSM mark with its phase width being two-thirds its pitch has a higher sensitivity to coma than Alt-PSM marks with the same pitch and the different phase widths. The pitch of the Alt-PSM mark is also optimized by PROLITH simulation, and the structure of p - 1.92 lambda/NA and pw = 2p/3 proves to be with the highest sensitivity. The optimized Alt-PSM mark is used as a measurement mark to retrieve coma aberration from the projection optics in lithographic tools. In comparison with an ordinary Alt-PSM mark with its phase width being a half its pitch, the measurement accuracies of Z(7) and Z(14) apparently increase. (C) 2009 Optical Society of America