scholarly journals PRIMES REPRESENTED BY INCOMPLETE NORM FORMS

2020 ◽  
Vol 8 ◽  
Author(s):  
JAMES MAYNARD
Keyword(s):  
Algebraic Geometry ◽  
Irreducible Polynomial ◽  
Type I ◽  
Type Ii ◽  
Geometry Of Numbers ◽  
Asymptotic Number ◽  
Special Case

Let $K=\mathbb{Q}(\unicode[STIX]{x1D714})$ with $\unicode[STIX]{x1D714}$ the root of a degree $n$ monic irreducible polynomial $f\in \mathbb{Z}[X]$ . We show that the degree $n$ polynomial $N(\sum _{i=1}^{n-k}x_{i}\unicode[STIX]{x1D714}^{i-1})$ in $n-k$ variables takes the expected asymptotic number of prime values if $n\geqslant 4k$ . In the special case $K=\mathbb{Q}(\sqrt[n]{\unicode[STIX]{x1D703}})$ , we show that $N(\sum _{i=1}^{n-k}x_{i}\sqrt[n]{\unicode[STIX]{x1D703}^{i-1}})$ takes infinitely many prime values, provided $n\geqslant 22k/7$ . Our proof relies on using suitable ‘Type I’ and ‘Type II’ estimates in Harman’s sieve, which are established in a similar overall manner to the previous work of Friedlander and Iwaniec on prime values of $X^{2}+Y^{4}$ and of Heath-Brown on $X^{3}+2Y^{3}$ . Our proof ultimately relies on employing explicit elementary estimates from the geometry of numbers and algebraic geometry to control the number of highly skewed lattices appearing in our final estimates.

scholarly journals Construction of the mutually orthogonal extraordinary supersquares

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Cristian Ghiu ◽  
Iulia Ghiu
Keyword(s):  
Information Processing ◽  
Quantum Information ◽  
Quantum Information Processing ◽  
Latin Squares ◽  
Mutually Unbiased Bases ◽  
Type I ◽  
Type Ii ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Set ◽  
Special Case

AbstractOur purpose is to determine the complete set of mutually orthogonal squares of order d, which are not necessary Latin. In this article, we introduce the concept of supersquare of order d, which is defined with the help of its generating subgroup in $$\mathbb{F}_d \times \mathbb{F}_d$$. We present a method of construction of the mutually orthogonal supersquares. Further, we investigate the orthogonality of extraordinary supersquares, a special family of squares, whose generating subgroups are extraordinary. The extraordinary subgroups in $$\mathbb{F}_d \times \mathbb{F}_d$$ are of great importance in the field of quantum information processing, especially for the study of mutually unbiased bases. We determine the most general complete sets of mutually orthogonal extraordinary supersquares of order 4, which consist in the so-called Type I and Type II. The well-known case of d − 1 mutually orthogonal Latin squares is only a special case, namely Type I.

Bifurcation and instability in gravity waves

1986 ◽  
Vol 403 (1825) ◽  
pp. 167-187 ◽  
Keyword(s):  
Normal Mode ◽  
Gravity Waves ◽  
Type I ◽  
Type Ii ◽  
The Family ◽  
The Matrix ◽  
Form Type ◽  
Rates Of Growth ◽  
Special Case ◽  
Frequency Curves

Calculations of the two-dimensional normal-mode perturbations of gravity waves on deep water (Longuet-Higgins, Proc. R. Soc. Lond . A 360, 471-488 (1978 a ); Longuet-Higgins, Proc. R. Soc. Lond . A 360, 489-505 (1978 b )) are here extended to values of the wave steepness ak as high as 0.43. This is achieved ( a ) by using a new method to evaluate the coefficients in Stokes’s series to high order, and ( b ) by rearrangement of the matrix equations so as to reduce the order by half. The behaviour of the normal-mode frequencies σ n in the range 0.35 < ak < 0.43 is clarified. Subharmonics of the form n = ( l / m , 2 - l / m ) Where l and m are integers and l < 2 m are shown to combine in pairs to form type II instabilities with relatively high rates of growth. For these modes, the critical values of ak at which σ vanishes correspond precisely to bifurcation points. In the special case l / m = ½ the two modal curves coincide. The family of frequency curves is bounded by the lowest superharmonic ( n = 2). It is verified that this mode becomes unstable when ak = 0.4292, corresponding to the lowest maximum of the energy density E . The boundaries of the parameter regions for instabilities of both type I and type II are determined.

scholarly journals New Multigranulation Neutrosophic Rough Set with Applications

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 578 ◽  
Author(s):  
Chunxin Bo ◽  
Xiaohong Zhang ◽  
Songtao Shao ◽  
Florentin Smarandache
Keyword(s):  
Decision Making ◽  
Rough Set ◽  
Neutrosophic Set ◽  
Type I ◽  
Type Ii ◽  
Type Iii ◽  
Proposed Model ◽  
Definition Of ◽  
Two Universes ◽  
Special Case

After the neutrosophic set (NS) was proposed, NS was used in many uncertainty problems. The single-valued neutrosophic set (SVNS) is a special case of NS that can be used to solve real-word problems. This paper mainly studies multigranulation neutrosophic rough sets (MNRSs) and their applications in multi-attribute group decision-making. Firstly, the existing definition of neutrosophic rough set (we call it type-I neutrosophic rough set (NRSI) in this paper) is analyzed, and then the definition of type-II neutrosophic rough set (NRSII), which is similar to NRSI, is given and its properties are studied. Secondly, a type-III neutrosophic rough set (NRSIII) is proposed and its differences from NRSI and NRSII are provided. Thirdly, single granulation NRSs are extended to multigranulation NRSs, and the type-I multigranulation neutrosophic rough set (MNRSI) is studied. The type-II multigranulation neutrosophic rough set (MNRSII) and type-III multigranulation neutrosophic rough set (MNRSIII) are proposed and their different properties are outlined. We found that the three kinds of MNRSs generate tcorresponding NRSs when all the NRs are the same. Finally, MNRSIII in two universes is proposed and an algorithm for decision-making based on MNRSIII is provided. A car ranking example is studied to explain the application of the proposed model.

Heat-Etching with a Bullivant Type II Simple Freeze-Cleave Device

1970 ◽  
Vol 28 ◽  
pp. 106-107 ◽  
Author(s):  
Ronald S. Weinstein ◽  
N. Scott McNutt
Keyword(s):  
New Zealand ◽  
General Hospital ◽  
Massachusetts General Hospital ◽  
Type I ◽  
Type Ii ◽  
Collaborative Effort ◽  
Temperature And Pressure ◽  
The University

The Type I simple cold block device was described by Bullivant and Ames in 1966 and represented the product of the first successful effort to simplify the equipment required to do sophisticated freeze-cleave techniques. Bullivant, Weinstein and Someda described the Type II device which is a modification of the Type I device and was developed as a collaborative effort at the Massachusetts General Hospital and the University of Auckland, New Zealand. The modifications reduced specimen contamination and provided controlled specimen warming for heat-etching of fracture faces. We have now tested the Mass. General Hospital version of the Type II device (called the “Type II-MGH device”) on a wide variety of biological specimens and have established temperature and pressure curves for routine heat-etching with the device.

Labelling of non-A, non-B hepatitis infected chimpanzee hepatocytes with nuclease-gold conjugates

1984 ◽  
Vol 42 ◽  
pp. 250-251
Author(s):  
G. D. Gagne ◽  
M. F. Miller ◽  
D. A. Peterson
Keyword(s):  
Endoplasmic Reticulum ◽  
Experimental Infection ◽  
Uranyl Acetate ◽  
Type I ◽  
Cellular Injury ◽  
Type Ii ◽  
Tubular Structures ◽  
Type Iii ◽  
Type Iv ◽  
Infected Chimpanzee

Experimental infection of chimpanzees with non-A, non-B hepatitis (NANB) or with delta agent hepatitis results in the appearance of characteristic cytoplasmic alterations in the hepatocytes. These alterations include spongelike inclusions (Type I), attached convoluted membranes (Type II), tubular structures (Type III), and microtubular aggregates (Type IV) (Fig. 1). Type I, II and III structures are, by association, believed to be derived from endoplasmic reticulum and may be morphogenetically related. Type IV structures are generally observed free in the cytoplasm but sometimes in the vicinity of type III structures. It is not known whether these structures are somehow involved in the replication and/or assembly of the putative NANB virus or whether they are simply nonspecific responses to cellular injury. When treated with uranyl acetate, type I, II and III structures stain intensely as if they might contain nucleic acids. If these structures do correspond to intermediates in the replication of a virus, one might expect them to contain DNA or RNA and the present study was undertaken to explore this possibility.

Comparative surface and ultrastructural views of methylotrophic bacteria by TEM, STEM, SE, and freeze-etch electron microscopy.

1987 ◽  
Vol 45 ◽  
pp. 792-793
Author(s):  
T.A. Fassel ◽  
M.J. Schaller ◽  
M.E. Lidstrom ◽  
C.C. Remsen
Keyword(s):  
Cytoplasmic Membrane ◽  
Structural Features ◽  
Methylotrophic Bacteria ◽  
Type I ◽  
Type Ii ◽  
Membrane Complex ◽  
Oxidation Of Methane ◽  
Methane Oxidizing Bacteria ◽  
Wall Structures ◽  
Methylomonas Albus

Methylotrophic bacteria play an Important role in the environment in the oxidation of methane and methanol. Extensive intracytoplasmic membranes (ICM) have been associated with the oxidation processes in methylotrophs and chemolithotrophic bacteria. Classification on the basis of ICM arrangement distinguishes 2 types of methylotrophs. Bundles or vesicular stacks of ICM located away from the cytoplasmic membrane and extending into the cytoplasm are present in Type I methylotrophs. In Type II methylotrophs, the ICM form pairs of peripheral membranes located parallel to the cytoplasmic membrane. Complex cell wall structures of tightly packed cup-shaped subunits have been described in strains of marine and freshwater phototrophic sulfur bacteria and several strains of methane oxidizing bacteria. We examined the ultrastructure of the methylotrophs with particular view of the ICM and surface structural features, between representatives of the Type I Methylomonas albus (BG8), and Type II Methylosinus trichosporium (OB-36).

OPTICAL STUDIES OF TYPE I AND TYPE II RECOMBINATION IN GaAs-AlAs QUANTUM WELLS

1987 ◽  
Vol 48 (C5) ◽  
pp. C5-525-C5-528 ◽  
Author(s):  
K. J. MOORE ◽  
P. DAWSON ◽  
C. T. FOXON
Keyword(s):  
Quantum Wells ◽  
Type I ◽  
Type Ii ◽  
Optical Studies

An old friend is better than two new ones? Approaches to market research in the context of digital transformation for the antitrust laws enforcement

2020 ◽  
pp. 37-55 ◽  
Author(s):  
A. E. Shastitko ◽  
O. A. Markova
Keyword(s):  
Business Models ◽  
Rapid Development ◽  
Digital Transformation ◽  
Market Definition ◽  
Type I ◽  
Type Ii ◽  
Digital Platforms ◽  
Advantages And Disadvantages ◽  
Pass Through ◽  
Type Ii Errors

Digital transformation has led to changes in business models of traditional players in the existing markets. What is more, new entrants and new markets appeared, in particular platforms and multisided markets. The emergence and rapid development of platforms are caused primarily by the existence of so called indirect network externalities. Regarding to this, a question arises of whether the existing instruments of competition law enforcement and market analysis are still relevant when analyzing markets with digital platforms? This paper aims at discussing advantages and disadvantages of using various tools to define markets with platforms. In particular, we define the features of the SSNIP test when being applyed to markets with platforms. Furthermore, we analyze adjustment in tests for platform market definition in terms of possible type I and type II errors. All in all, it turns out that to reduce the likelihood of type I and type II errors while applying market definition technique to markets with platforms one should consider the type of platform analyzed: transaction platforms without pass-through and non-transaction matching platforms should be tackled as players in a multisided market, whereas non-transaction platforms should be analyzed as players in several interrelated markets. However, if the platform is allowed to adjust prices, there emerges additional challenge that the regulator and companies may manipulate the results of SSNIP test by applying different models of competition.

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