scholarly journals Numbers with simply normal β-expansions

2018 ◽  
Vol 167 (01) ◽  
pp. 171-192 ◽  
Author(s):  
SIMON BAKER ◽  
DERONG KONG
Keyword(s):  
Dynamical Systems ◽  
Combinatorics On Words ◽  
Unique Zero ◽  
Made In

AbstractIn [6] the first author proved that for any β ∈ (1, βKL) everyx∈ (0, 1/(β − 1)) has a simply normal β-expansion, where βKL≈ 1.78723 is the Komornik–Loreti constant. This result is complemented by an observation made in [22], where it was shown that whenever β ∈ (βT, 2] there exists anx∈ (0, 1/(β − 1)) with a unique β-expansion, and this expansion is not simply normal. Here βT≈ 1.80194 is the unique zero in (1, 2] of the polynomialx3−x2− 2x+ 1. This leaves a gap in our understanding within the interval [βKL, βT]. In this paper we fill this gap and prove that for any β ∈ (1, βT], everyx∈ (0, 1/(β − 1)) has a simply normal β-expansion. For completion, we provide a proof that for any β ∈ (1, 2), Lebesgue almost everyxhas a simply normal β-expansion. We also give examples ofxwith multiple β-expansions, none of which are simply normal.Our proofs rely on ideas from combinatorics on words and dynamical systems.

scholarly journals Laguerre-Type Exponentials, Laguerre Derivatives and Applications. A Survey

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2054 ◽  
Author(s):  
Paolo Emilio Ricci
Keyword(s):  
Population Dynamics ◽  
Dynamical Systems ◽  
Special Functions ◽  
Linear Dynamical Systems ◽  
Analytical Functions ◽  
Made In

Laguerrian derivatives and related autofunctions are presented that allow building new special functions determined by the action of a differential isomorphism within the space of analytical functions. Such isomorphism can be iterated every time, so that the resulting construction can be re-submitted endlessly in a cyclic way. Some applications of this theory are made in the field of population dynamics and in the solution of Cauchy’s problems for particular linear dynamical systems.

Maximizing Sensitivity Vector Fields: A Parametric Study

2014 ◽  
Vol 9 (2) ◽  
Author(s):  
Andrew R. Sloboda ◽  
Bogdan I. Epureanu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Vector Fields ◽  
Nonlinear Feedback ◽  
Baseline System ◽  
Nominal Parameter ◽  
Construction Play ◽  
Sensitivity Vector ◽  
System Attractor ◽  
Made In

Sensitivity vector fields (SVFs) have proven to be an effective method for identifying parametric variations in dynamical systems. These fields are constructed using information about how a dynamical system's attractor deforms under prescribed parametric variations. Once constructed, they can be used to quantify any additional variations from the nominal parameter set as they occur. Since SVFs are based on attractor deformations, the geometry and other qualities of the baseline system attractor impact how well a set of SVFs will perform. This paper examines the role attractor characteristics and the choices made in SVF construction play in determining the sensitivity of SVFs. The use of nonlinear feedback to change a dynamical system with the intent of improving SVF sensitivity is explored. These ideas are presented in the context of constructing SVFs for several dynamical systems.

Spectrophotometric measurements of objective-prism spectra

1966 ◽  
Vol 24 ◽  
pp. 118-119
Author(s):  
Th. Schmidt-Kaler
Keyword(s):  
Progress Report ◽  
Two Dimensional ◽  
Objective Prism ◽  
Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

Surface Structure in Microtomed Sections Caused by Glass and Diamond Knives

1971 ◽  
Vol 29 ◽  
pp. 456-457
Author(s):  
J. Temple Black ◽  
William G. Boldosser
Keyword(s):  
Plastic Deformation ◽  
Epoxy Resin ◽  
Carbon Coating ◽  
Surface Damage ◽  
Body Angle ◽  
Normal Manner ◽  
Bottom Side ◽  
Cutting Direction ◽  
Made In ◽  
Diamond Knife

Ultramicrotomy produces plastic deformation in the surfaces of microtomed TEM specimens which can not generally be observed unless special preparations are made. In this study, a typical biological composite of tissue (infundibular thoracic attachment) infiltrated in the normal manner with an embedding epoxy resin (Epon 812 in a 60/40 mixture) was microtomed with glass and diamond knives, both with 45 degree body angle. Sectioning was done in Portor Blum Mt-2 and Mt-1 microtomes. Sections were collected on formvar coated grids so that both the top side and the bottom side of the sections could be examined. Sections were then placed in a vacuum evaporator and self-shadowed with carbon. Some were chromium shadowed at a 30 degree angle. The sections were then examined in a Phillips 300 TEM at 60kv.Carbon coating (C) or carbon coating with chrom shadowing (C-Ch) makes in effect, single stage replicas of the surfaces of the sections and thus allows the damage in the surfaces to be observable in the TEM. Figure 1 (see key to figures) shows the bottom side of a diamond knife section, carbon self-shadowed and chrom shadowed perpendicular to the cutting direction. Very fine knife marks and surface damage can be observed.

Iron storage sites in the myocardium as revealed by cytohistochemistry

1988 ◽  
Vol 46 ◽  
pp. 394-395
Author(s):  
M. Ashraf ◽  
F. Thompson ◽  
S. Miki ◽  
P. Srivastava
Keyword(s):  
Cardiac Tissue ◽  
Coronary Perfusion ◽  
Intracellular Iron ◽  
Ether Anesthesia ◽  
Intense Staining ◽  
Iron Storage ◽  
Thin Sections ◽  
Rat Hearts ◽  
Cacodylate Buffer ◽  
Made In

Iron is believed to play an important role in the pathogenesis of ischemic injury. However, the sources of intracellular iron in myocytes are not yet defined. In this study we have attempted to localize iron at various cellular sites of the cardiac tissue with the ferrocyanide technique.Rat hearts were excised under ether anesthesia. They were fixed with coronary perfusion with 3% buffered glutaraldehyde made in 0.1 M cacodylate buffer pH 7.3. Sections, 60 μm in thickness, were cut on a vibratome and were incubated in the medium containing 500 mg of potassium ferrocyanide in 49.5 ml H2O and 0.5 ml concentrated HC1 for 30 minutes at room temperature. Following rinses in the buffer, tissues were dehydrated in ethanol and embedded in Spurr medium.The examination of thin sections revealed intense staining or reaction product in peroxisomes (Fig. 1).

Microanalysis of precipitates in a ferritic steel

1978 ◽  
Vol 36 (1) ◽  
pp. 618-619
Author(s):  
J.M. Titchmarsh
Keyword(s):  
Ferritic Steel ◽  
Particle Identification ◽  
Energy Dispersive ◽  
Objective Lens ◽  
Ferritic Steels ◽  
Replica Technique ◽  
X Ray ◽  
Large Numbers ◽  
Scanning Transmission ◽  
Made In

The advances in recent years in the microanalytical capabilities of conventional TEM's fitted with probe forming lenses allow much more detailed investigations to be made of the microstructures of complex alloys, such as ferritic steels, than have been possible previously. In particular, the identification of individual precipitate particles with dimensions of a few tens of nanometers in alloys containing high densities of several chemically and crystallographically different precipitate types is feasible. The aim of the investigation described in this paper was to establish a method which allowed individual particle identification to be made in a few seconds so that large numbers of particles could be examined in a few hours.A Philips EM400 microscope, fitted with the scanning transmission (STEM) objective lens pole-pieces and an EDAX energy dispersive X-ray analyser, was used at 120 kV with a thermal W hairpin filament. The precipitates examined were extracted using a standard C replica technique from specimens of a 2¼Cr-lMo ferritic steel in a quenched and tempered condition.

AEM investigation of ZrO2 solid solution formation in ZrO2/mullite composites

1984 ◽  
Vol 42 ◽  
pp. 408-409
Author(s):  
T. R. Dinger
Keyword(s):  
Solid Solution ◽  
Ion Beam ◽  
Analytical Electron Microscopy ◽  
Transformation Toughening ◽  
Solid Solution Formation ◽  
Solution Formation ◽  
Propagating Crack ◽  
Analytical Electron ◽  
Microcrack Toughening ◽  
Made In

Zirconia (ZrO2) is often added to ceramic compacts to increase their toughness. The mechanisms by which this toughness increase occurs are generally accepted to be those of transformation toughening and microcracking. The mechanism of transformation toughening is based on the presence of metastable tetragonal ZrO2 which transforms to the monoclinic allotrope when stressed by a propagating crack. The decrease in volume which accompanies this transformation effectively relieves the applied stress at the crack tip and toughens the material; microcrack toughening arises from the deflection of a propagating crack around sharply angular inclusions.These mechanisms, however, do not explain the toughness increases associated with the class of composites investigated here. Analytical electron microscopy (AEM) has been used to determine whether solid solution effects could be the cause of this increased toughness. Specimens of a mullite (3Al2O3·2SiO2) + 15 vol. % ZrO2 were prepared by the usual technique of mechanical thinning followed by ion beam milling. All observations were made in a Philips EM400 TEM/STEM microscope fitted with EDXS and EELS spectrometers.

A study of interface sliding at F.C.C.-B.C.C. boundaries in a Cu-Zn alloy

1978 ◽  
Vol 36 (1) ◽  
pp. 422-423
Author(s):  
F. Monchoux ◽  
A. Rocher ◽  
J.L. Martin
Keyword(s):  
Face Centered Cubic ◽  
Creep Tests ◽  
High Voltage Electron Microscopy ◽  
Diffusion Treatment ◽  
Voltage Electron ◽  
The Face ◽  
Face Centered ◽  
Interface Sliding ◽  
Zn Alloy ◽  
Made In

Interphase sliding is an important phenomenon of high temperature plasticity. In order to study the microstructural changes associated with it, as well as its influence on the strain rate dependence on stress and temperature, plane boundaries were obtained by welding together two polycrystals of Cu-Zn alloys having the face centered cubic and body centered cubic structures respectively following the procedure described in (1). These specimens were then deformed in shear along the interface on a creep machine (2) at the same temperature as that of the diffusion treatment so as to avoid any precipitation. The present paper reports observations by conventional and high voltage electron microscopy of the microstructure of both phases, in the vicinity of the phase boundary, after different creep tests corresponding to various deformation conditions.Foils were cut by spark machining out of the bulk samples, 0.2 mm thick. They were then electropolished down to 0.1 mm, after which a hole with thin edges was made in an area including the boundary

Searching for the causes why some of the paintings by Mihaly Munkacsy are darkening

1990 ◽  
Vol 48 (2) ◽  
pp. 204-205
Author(s):  
Imre Pozsgai ◽  
Klara Erdöhalmi-Torok
Keyword(s):  
Electron Beam ◽  
Cross Section ◽  
Polyester Resin ◽  
National Gallery ◽  
X Ray ◽  
Art Historians ◽  
Made In ◽  
Grinding And Polishing

The paintings by the great Hungarian master Mihaly Munkacsy (1844-1900) made in an 8-9 years period of his activity are deteriorating. The most conspicuous sign of the deterioration is an intensive darkening. We have made an attempt by electron beam microanalysis to clarify the causes of the darkening. The importance of a study like this is increased by the fact that a similar darkening can be observed on the paintings by Munkacsy’s contemporaries e.g Courbet and Makart. A thick brown mass the so called bitumen used by Munkacsy for grounding and also as a paint is believed by the art historians to cause the darkening.For this study, paint specimens were taken from the following paintings: “Studio”, “Farewell” and the “Portrait of the Master’s Wife”, all of them are the property of the Hungarian National Gallery. The paint samples were embedded in a polyester resin “Poly-Pol PS-230” and after grinding and polishing their cross section was used for x-ray mapping.

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