scholarly journals Bounds for sets with no polynomial progressions

2020 ◽  
Vol 8 ◽  
Author(s):  
Sarah Peluse
Keyword(s):  
General Result ◽  
Constant Term ◽  
Gowers Norms ◽  
The Way

Abstract Let $P_1,\dots ,P_m\in \mathbb{Z} [y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset A of $\{1,\dots ,N\}$ with no nontrivial progressions of the form $x,x+P_1(y),\dots ,x+P_m(y)$ has size $|A|\ll N/(\log \log {N})^{c_{P_1,\dots ,P_m}}$ . Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.

scholarly journals Challenging the majority rule in matters of truth

2014 ◽  
Vol 7 (2) ◽  
pp. 54 ◽  
Author(s):  
Bernd Lahno
Keyword(s):  
General Result ◽  
Majority Rule ◽  
Bayesian Model ◽  
Basic Problem ◽  
Expert Judgment ◽  
Good Evidence ◽  
Recent Debate ◽  
The Way

The majority rule has caught much attention in recent debate about the aggregation of judgments. But its role in finding the truth is limited. A majority of expert judgments is not necessarily authoritative, even if all experts are equally competent, if they make their judgments independently of each other, and if all the judgments are based on the same source of (good) evidence. In this paper I demonstrate this limitation by presenting a simple counterexample and a related general result. I pave the way for this argument by introducing a Bayesian model of evidence and expert judgment in order to give a precise account of the basic problem.

COMPLETE SYSTEMS OF SUBALGEBRAS

2003 ◽  
Vol 13 (01) ◽  
pp. 69-79 ◽  
Author(s):  
ANDRZEJ KISIELEWICZ
Keyword(s):  
General Result ◽  
Extension Property ◽  
Finite Sequence ◽  
Minimal Extension ◽  
Open Problems ◽  
The 1960S ◽  
Free Spectra ◽  
The Way

In the 1960s, G. Grätzer introduced the notion of the minimal extension property (MEP) of a finite sequence in order to investigate pn-sequences and free spectra of algebras. While there are many particular results on the MEP, stating that some sequences or families of sequences have the MEP, no general result has been obtained so far and the main general problems remain open. In this paper we prove a fact about existence of finite complete systems of subalgebras, which generalizes a method used occasionally to prove the minimal extension property of a sequence. Using this we obtain some general results about minimal extensions and, along the way, a rather unexpected solution of one of the central open problems in the area.

DIOPHANTINE APPROXIMATION WITH GAUSSIAN PRIMES

2019 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Recent Result ◽  
Diophantine Approximation ◽  
General Result ◽  
London Math ◽  
Infinitely Many Solutions ◽  
Arbitrary Complex ◽  
Gaussian Primes ◽  
The Way

Abstract In this paper we prove that the exact analogue of the author’s work with real irrationals and rational primes (G. Harman, On the distribution of $\alpha p$ modulo one II, Proc. London Math. Soc. (3) 72, 1996, 241–260) holds for approximating $\alpha \in \mathbb{C}\setminus \mathbb{Q}[i]$ with Gaussian primes. To be precise, we show that for such $\alpha $ and arbitrary complex $\beta $ there are infinitely many solutions in Gaussian primes $p$ to $$\begin{equation*} ||\alpha p + \beta|| <| p|^{-7/22}, \end{equation*}$$where $||\cdot ||$ denotes distance to a nearest member of $\mathbb{Z}[i]$. We shall, in fact, prove a slightly more general result with the Gaussian primes in sectors, and along the way improve a recent result due to Baier (S. Baier, Diophantine approximation on lines in $\mathbb{C}^2$ with Gaussian prime constraints, Eur. J. Math. 3, 2017, 614–649).

Self-sacrifice for in-group's history: A diachronic perspective

2018 ◽  
Vol 41 ◽  
Author(s):  
Maria Babińska ◽  
Michal Bilewicz
Keyword(s):  
Clinical Psychology ◽  
Historical Events ◽  
Emotional Events ◽  
Reciprocal Process ◽  
The Way

AbstractThe problem of extended fusion and identification can be approached from a diachronic perspective. Based on our own research, as well as findings from the fields of social, political, and clinical psychology, we argue that the way contemporary emotional events shape local fusion is similar to the way in which historical experiences shape extended fusion. We propose a reciprocal process in which historical events shape contemporary identities, whereas contemporary identities shape interpretations of past traumas.

What is the purpose of cognition?

2020 ◽  
Vol 43 ◽  
Author(s):  
Aba Szollosi ◽  
Ben R. Newell
Keyword(s):  
Infinite Number ◽  
Human Cognition ◽  
The Way

Abstract The purpose of human cognition depends on the problem people try to solve. Defining the purpose is difficult, because people seem capable of representing problems in an infinite number of ways. The way in which the function of cognition develops needs to be central to our theories.

Peculiarity Parameters

1976 ◽  
Vol 32 ◽  
pp. 233-254
Author(s):  
H. M. Maitzen
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Theoretical Work ◽  
Observational Evidence ◽  
Rotator Model ◽  
Peculiar Behaviour ◽  
Field Spectrum ◽  
Oblique Rotator ◽  
Ap Stars ◽  
The Way

Ap stars are peculiar in many aspects. During this century astronomers have been trying to collect data about these and have found a confusing variety of peculiar behaviour even from star to star that Struve stated in 1942 that at least we know that these phenomena are not supernatural. A real push to start deeper theoretical work on Ap stars was given by an additional observational evidence, namely the discovery of magnetic fields on these stars by Babcock (1947). This originated the concept that magnetic fields are the cause for spectroscopic and photometric peculiarities. Great leaps for the astronomical mankind were the Oblique Rotator model by Stibbs (1950) and Deutsch (1954), which by the way provided mathematical tools for the later handling pulsar geometries, anti the discovery of phase coincidence of the extrema of magnetic field, spectrum and photometric variations (e.g. Jarzebowski, 1960).

The mensuration of interfaces

1992 ◽  
Vol 50 (1) ◽  
pp. 216-217
Author(s):  
W.M. Stobbs
Keyword(s):  
Electron Microscopy ◽  
50Th Anniversary ◽  
Numerical Data ◽  
Industrial Applications ◽  
Dynamical Theory ◽  
Large Strains ◽  
Reasonable Approach ◽  
Lattice Resolution ◽  
Analogue To Digital ◽  
The Way

I do not have access to the abstracts of the first meeting of EMSA but at this, the 50th Anniversary meeting of the Electron Microscopy Society of America, I have an excuse to consider the historical origins of the approaches we take to the use of electron microscopy for the characterisation of materials. I have myself been actively involved in the use of TEM for the characterisation of heterogeneities for little more than half of that period. My own view is that it was between the 3rd International Meeting at London, and the 1956 Stockholm meeting, the first of the European series , that the foundations of the approaches we now take to the characterisation of a material using the TEM were laid down. (This was 10 years before I took dynamical theory to be etched in stone.) It was at the 1956 meeting that Menter showed lattice resolution images of sodium faujasite and Hirsch, Home and Whelan showed images of dislocations in the XlVth session on “metallography and other industrial applications”. I have always incidentally been delighted by the way the latter authors misinterpreted astonishingly clear thickness fringes in a beaten (”) foil of Al as being contrast due to “large strains”, an error which they corrected with admirable rapidity as the theory developed. At the London meeting the research described covered a broad range of approaches, including many that are only now being rediscovered as worth further effort: however such is the power of “the image” to persuade that the above two papers set trends which influence, perhaps too strongly, the approaches we take now. Menter was clear that the way the planes in his image tended to be curved was associated with the imaging conditions rather than with lattice strains, and yet it now seems to be common practice to assume that the dots in an “atomic resolution image” can faithfully represent the variations in atomic spacing at a localised defect. Even when the more reasonable approach is taken of matching the image details with a computed simulation for an assumed model, the non-uniqueness of the interpreted fit seems to be rather rarely appreciated. Hirsch et al., on the other hand, made a point of using their images to get numerical data on characteristics of the specimen they examined, such as its dislocation density, which would not be expected to be influenced by uncertainties in the contrast. Nonetheless the trends were set with microscope manufacturers producing higher and higher resolution microscopes, while the blind faith of the users in the image produced as being a near directly interpretable representation of reality seems to have increased rather than been generally questioned. But if we want to test structural models we need numbers and it is the analogue to digital conversion of the information in the image which is required.

Process \'prǎ|,ses\n: The Action of Moving Forward Progressively from One Point to Another on the Way to Completion

1979 ◽  
Vol 44 (1) ◽  
pp. 3-30 ◽  
Author(s):  
Carol A. Pruning
Keyword(s):  
Process Model ◽  
Communicative Behaviors ◽  
Adult Level ◽  
Remedial Programs ◽  
Stage Process ◽  
The Way

A rationale for the application of a stage process model for the language-disordered child is presented. The major behaviors of the communicative system (pragmatic-semantic-syntactic-phonological) are summarized and organized in stages from pre-linguistic to the adult level. The article provides clinicians with guidelines, based on complexity, for the content and sequencing of communicative behaviors to be used in planning remedial programs.

From the President: Working Early-Intervention Magic in Community Settings

ASHA Leader ◽  
2013 ◽  
Vol 18 (2) ◽  
pp. 5-7 ◽  
Author(s):  
Patty Prelock
Keyword(s):  
Early Intervention ◽  
Children With Disabilities ◽  
Community Settings ◽  
The Way

Children with disabilities benefit most when professionals let families lead the way.

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