A lower closure theorem for autonomous orientor fields

1988 ◽  
Vol 110 (3-4) ◽  
pp. 249-254 ◽  
Author(s):  
Luigi Ambrosio
Keyword(s):  
Differential Inclusion ◽  
Closure Property ◽  
Closure Theorem ◽  
Set Valued Mapping ◽  
Orientor Fields ◽  
Lower Closure ◽  
Lower Closure Theorem ◽  
Image Position

SynopsisGiven a set valued mapping ∑: ℝ → ∑n, we prove a closure property with respect to -convergence for the differential inclusionunder very mild assumptions on ∑.

scholarly journals Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

2021 ◽  
Author(s):  
Rafał Kamocki
Keyword(s):  
Optimal Control ◽  
Fractional Laplacian ◽  
Nonlinear Pde ◽  
Optimal Solutions ◽  
Dirichlet Laplacian ◽  
Existence Of Optimal Solutions ◽  
Closure Theorem ◽  
Orientor Fields ◽  
Lower Closure ◽  
Lower Closure Theorem

AbstractWe consider an optimal control problem containing a control system described by a partial nonlinear differential equation with the fractional Dirichlet–Laplacian, associated to an integral cost. We investigate the existence of optimal solutions for such a problem. In our study we use Filippov’s approach combined with a lower closure theorem for orientor fields.

scholarly journals On weak solutions of random differential inclusions

1995 ◽  
Vol 8 (4) ◽  
pp. 393-396
Author(s):  
Mariusz Michta
Keyword(s):  
Probability Measure ◽  
Differential Inclusion ◽  
Weak Solutions ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Initial Condition ◽  
Set Valued Mapping ◽  
Nonempty Compact ◽  
Convex Subsets

In the paper we study the existence of solutions of the random differential inclusion x˙t∈G(t,xt)     P.1,t∈[0,T]-a.e.x0=dμ, where G is a given set-valued mapping value in the space Kn of all nonempty, compact and convex subsets of the space ℝn, and μ is some probability measure on the Borel σ-algebra in ℝn. Under certain restrictions imposed on F and μ, we obtain weak solutions of problem (I), where the initial condition requires that the solution of (I) has a given distribution at time t=0.

scholarly journals Existence of solutions of functional differential inclusions

1992 ◽  
Vol 5 (4) ◽  
pp. 315-323
Author(s):  
A. Anguraj ◽  
K. Balachandran
Keyword(s):  
Fixed Point ◽  
Differential Inclusion ◽  
Existence Of Solutions ◽  
Integral Inclusion ◽  
Functional Differential Inclusion ◽  
Variation Of Parameters ◽  
Set Valued Mapping ◽  
Functional Differential ◽  
Variation Of Parameters Formula ◽  
Functional Differential Inclusions

We prove the existence of solutions of a functional differential inclusion. By using the variation of parameters formula we convert the functional differential inclusion into an integral inclusion and prove the existence of a fixed point of the set-valued mapping with the help of the Kakutani-Bohnenblust-Karlin fixed point theorem.

scholarly journals A DIRECT PROOF OF SCHWICHTENBERG’S BAR RECURSION CLOSURE THEOREM

2018 ◽  
Vol 83 (1) ◽  
pp. 70-83 ◽  
Author(s):  
PAULO OLIVA ◽  
SILVIA STEILA
Keyword(s):  
Direct Proof ◽  
Explicit Construction ◽  
Logical Relation ◽  
Original Proof ◽  
Concrete System ◽  
Closure Theorem ◽  
Primitive Recursion ◽  
Image Position ◽  
Recursive Definitions

AbstractIn [12], Schwichtenberg showed that the System T definable functionals are closed under a rule-like version Spector’s bar recursion of lowest type levels 0 and 1. More precisely, if the functional Y which controls the stopping condition of Spector’s bar recursor is T-definable, then the corresponding bar recursion of type levels 0 and 1 is already T-definable. Schwichtenberg’s original proof, however, relies on a detour through Tait’s infinitary terms and the correspondence between ordinal recursion for $\alpha < {\varepsilon _0}$ and primitive recursion over finite types. This detour makes it hard to calculate on given concrete system T input, what the corresponding system T output would look like. In this paper we present an alternative (more direct) proof based on an explicit construction which we prove correct via a suitably defined logical relation. We show through an example how this gives a straightforward mechanism for converting bar recursive definitions into T-definitions under the conditions of Schwichtenberg’s theorem. Finally, with the explicit construction we can also easily state a sharper result: if Y is in the fragment Ti then terms built from $BR^{\mathbb{N},\sigma } $ for this particular Y are definable in the fragment ${T_{i + {\rm{max}}\left\{ {1,{\rm{level}}\left( \sigma \right)} \right\} + 2}}$.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

Optimizing conditions for x-ray microchemical analysis in analytical electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 548-549 ◽  
Author(s):  
N. J. Zaluzec
Keyword(s):  
Solid Angle ◽  
Analytical Electron Microscopy ◽  
Incident Beam ◽  
Thin Foil ◽  
Experimental Conditions ◽  
X Ray ◽  
Microchemical Analysis ◽  
Analytical Electron ◽  
Image Position ◽  
Incident Beam Energy

The ultimate sensitivity of microchemical analysis using x-ray emission rests in selecting those experimental conditions which will maximize the measured peak-to-background (P/B) ratio. This paper presents the results of calculations aimed at determining the influence of incident beam energy, detector/specimen geometry and specimen composition on the P/B ratio for ideally thin samples (i.e., the effects of scattering and absorption are considered negligible). As such it is assumed that the complications resulting from system peaks, bremsstrahlung fluorescence, electron tails and specimen contamination have been eliminated and that one needs only to consider the physics of the generation/emission process.The number of characteristic x-ray photons (Ip) emitted from a thin foil of thickness dt into the solid angle dΩ is given by the well-known equation

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