A geometric method in nonlinear programming

A differential geometric approach to the constrained function maximization problem is presented. The continuous analogue of the Newton-Raphson method due to Branin for solving a system of nonlinear equations is extended to the case where the system is under-determined. The method is combined with the continuous analogue of the gradient-projection method to obtain a constrained maximization method with enforced constraint restoration. Detailed analysis of the global behavior of both methods is provided. It is shown that the conjugate-gradient algorithm can take advantage of the sparse structure of the problem in the computation of a vector field, which constitutes the main computational task in the methods.This is part of a paper issued as Stanford University, Computer Science Department Report No. STAN-CS-77-643 (Ref. 45), which was presented at the Gatlinburg VII Conference, Asilomar, California, 1977. This work was supported in part by NSF Grant No. NAT BUR OF ECON RES/PO No. 4369 and by Department of Energy Contract No. EY-76-C-02-0016.The main part of this work was presented at the Japan-France Seminar on Functional Analysis and Numerical Analysis, Tokyo, Japan, 1976. The paper was prepared in part while the author was a visitor at the Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 1976–77, and was completed while he was a visitor at the Computer Science Department, Stanford University, Stanford, California, 1977. He acknowledges the hospitality and stimulating environment provided by Professor G. H. Golub, Stanford University, and Professors N. J. Rose and C. D. Meyer, North Carolina State University.     

Preconcentration of samples in sandwich tanks for continuous thin-layer chromatography

It is demonstrated that horizontai sandwich tank with two cover plates can be used to preconcentrate sample solutions. Narrow sharply-defined starting zones are obtained just outside the smaller cover plate when the series of spots of the sample solution are applied on the thin layer and predeveloped with a volatile solvent. After evaporation of the solvent, the whole area of the TLC-plate is covered with two plates and developed with a suitable solvent.     

Limit distributions and one-parameter groups of linear operators on Banach spaces

Let = {U_t: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {U_tn(μ₁ * μ₂*…*μ_n)*δ_xn}, where {μ_n}Q, {x_n}X and measures U_tnμ_j (j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes L_m( ) as follows: L₀( )= ( (X); ), L_m( )= (L_m−1( ); ) for m=1, 2,… and L_∞( )=_m=0^∞L_m( ). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from L_m( ), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.     

Photodynamic Therapy for Cancer Cells Using a Flash Wave Light Xenon Lamp

We determined photodynamic therapy (PDT) efficacy using a flash wave (FW) and a continuous wave (CW) light, of which the fluence rate was 70 W/cm², for murine thymic lymphoma cells (EL-4) cultivated in vitro. The irradiation frequency and the pulse width of the FW light were in the range of 1–32 Hz and less than one millisecond, respectively. 5-Aminolevulinic acid-induced protoporphyrin IX (ALA-PpIX) was used as a photosensitizer. When EL-4 with ALA administration was irradiated by the light for 4 h (irradiation fluence: 1.0J/cm²), the survival rate of EL-4 by the FW light was lower than that by the CW light, except for the FW light with irradiation frequency of 32 Hz, and decreased gradually with decreasing irradiation frequency. Moreover, the FW light, especially at lower irradiation frequency, was superior to the CW light for the generation of singlet oxygen in an aqueous PpIX solution. Therefore, thehigher PDT efficacy for EL-4 of the FW light would be caused by the greater generation of singlet oxygen in the cells.     

An Example of Embedded Singular Continuous Spectrum for One-Dimensional Schrödinger Operators

We present a new example of a potential such that the corresponding Schrödinger operator in the halfaxis has singular continuous spectrum embedded in the absolutely continuous spectrum. The singular part is supported in an essentialy dense set. This generalizes a result of C. Remling [3].Mathematics Subject Classification: 34L40, 81Q10.     

On lineability of sets of continuous functions

We study the existence of vector spaces of dimension at least two of continuous functions on (subsets of) , every non-zero element of which admits one and only one absolute maximum.     

Uniform Continuity of a Family of Weakly Regular Set Functions on Topological Spaces

Conditions for the uniform continuity of a family of weakly regular set functions defined on an algebra of subsets of a -topological space (T,) and taking values in an arbitrary topological space are found.     

Dispersive flow of disks through a two-dimensional Galton board

We report here an experimental and numerical study of the flow properties of disks driven by gravity through a hexagonal lattice of obstacles, i.e. a Galton board. During the fall, particles experience dissipative collisions that scatter them in random directions. A driven-diffusion regime can be achieved under certain conditions. A characteristic length of the motion and its dependence on geometrical parameters of the system is analyzed in the steady regime. The influence of collective effects on the dispersion process is investigated by comparison between single- and many-particle flows. The characterization of the dynamics and the diffusive properties of the flow in a system like a Galton board can be expanded to other granular systems, particularly static solid particle mixers and will give some insight in understanding granular mixing.     

Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation D^su(x)=f(x,u(x)), 0<s<1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804-812), where D^s denotes Riemann-Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential equation:

L(D)u=f(x,u),u(0)=0,0<x<1,     

Extending hysteresis operators to spaces of piecewise continuous functions

We consider continuous-time hysteresis operators, defined to be causal and rate independent operators mapping input signals to output signals . We show how a hysteresis operator defined on the set of continuous piecewise monotone functions can be naturally extended to piecewise continuous piecewise monotone functions. We prove that the extension is also a hysteresis operator and that a number of important properties of the original operator are inherited by the extension. Moreover, we define the concept of a discrete-time hysteresis operator and we show that discretizing continuous-time hysteresis operators using standard sampling and hold operations leads to discrete-time hysteresis operators. We apply the concepts and results described above in the context of sampled-data feedback control of linear systems with input hysteresis.