XXIV.—Sets of Integers Containing not more than a Given Number of Terms in Arithmetical Progression

1961 ◽  
Vol 65 (4) ◽  
pp. 332-344 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Analogous Problem ◽  
Positive Density ◽  
Arithmetical Progression ◽  
Zero Density

SYNOPSISSets of integers are constructed having the property that n members are in arithmetical progression only if they are all equal; here n is any integer greater than or equal to 3. Previous results have been obtained only for n=3. The problem is generalized in various ways. The analysis can also be applied to construct sets for the analogous problem of geometrical progressions. These sets are of positive density, unlike those of the first kind, which have zero density.

Generalized Polynomials and Mild Mixing

2009 ◽  
Vol 61 (3) ◽  
pp. 656-673 ◽  
Author(s):  
Randall McCutcheon ◽  
Anthony Quas
Keyword(s):  
Positive Density ◽  
Weak Mixing ◽  
Generalized Polynomials ◽  
Zero Density ◽  
Floor Function

Abstract.An unsettled conjecture of V. Bergelson and I. Håland proposes that if (X ,A, μ, T ) is an invertible weak mixing measure preserving system, where μ(X ) < ∞, and if p1, p2, … , pk are generalized polynomials (functions built out of regular polynomials via iterated use of the greatest integer or floor function) having the property that no pi, nor any pi−pj, i ≠ j, is constant on a set of positive density, then for any measurable sets A0, A1, … , Ak, there exists a zero-density set E ⊂ Zsuch thatWe formulate and prove a faithful version of this conjecture for mildly mixing systems and partially characterize, in the degree two case, the set of families ﹛p1, p2, … , pk﹜ satisfying the hypotheses of this theorem.

scholarly journals MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

2021 ◽  
Author(s):  
LUCAS FRESSE ◽  
IVAN PENKOV
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Interesting Case ◽  
Analogous Problem ◽  
Restrictive Condition ◽  
Dimensional Case ◽  
Essential Difference ◽  
Parabolic Subgroups ◽  
Finite Dimensional ◽  
Finite Dimensional Case

AbstractLet G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

Minimal invariant spaces in formal topology

1997 ◽  
Vol 62 (3) ◽  
pp. 689-698 ◽  
Author(s):  
Thierry Coquand
Keyword(s):  
Direct Consequence ◽  
Arithmetical Progression ◽  
Standard Result ◽  
Continuous Map ◽  
Compact Topological Space ◽  
Combinatorial Theorem ◽  
Minimal Property ◽  
Special Instance ◽  
Invariant Spaces ◽  
Formal Topology

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.

The preserved plume conduits of the Caribbean Large Igneous Plateau and their relation with the Galápagos hotspot back to 90 Ma

2021 ◽  
Author(s):  
Angela Maria Gomez Garcia ◽  
Eline Le Breton ◽  
Magdalena Scheck-Wenderoth ◽  
Gaspar Monsalve ◽  
Denis Anikiev
Keyword(s):  
Reference Frames ◽  
Thermal Anomaly ◽  
Caribbean Region ◽  
Positive Density ◽  
South American ◽  
Kinematic Models ◽  
Tomographic Data ◽  
The Caribbean ◽  
Moho Depths ◽  
South American Continent

&lt;p&gt;Remnants of the Caribbean Large Igneous Plateau (C-LIP) are found as thickened zones of oceanic crust in the Caribbean Sea, that formed during strong pulses of magmatic activity around 90 Ma. Previous studies have proposed the Gal&amp;#225;pagos hotspot as the origin of the thermal anomaly responsible for the development of this igneous province. Particularly, geochemical signature relates accreted C-LIP fragments along northern South America with the well-known hotspot material.&lt;/p&gt;&lt;p&gt;In this research, we use 3D lithospheric-scale structural and density models of the Caribbean region, in which up-to-date geophysical datasets (i.e.: tomographic data, Moho depths, sedimentary thickness, and bathymetry) have been integrated. Based on the gravity residuals (modelled minus observed EIGEN6C-4 dataset), we reconstruct density heterogeneities both in the crust and the uppermost oceanic mantle (&lt; 50km).&lt;/p&gt;&lt;p&gt;Our results suggest the presence of two positive mantle density anomalies in the Colombian and the Venezuelan basins, interpreted as the preserved plume material which migrated together with the Proto-Caribbean plate from the east Pacific. Such bodies have never been identified before, but a positive density trend is also observed in the mantle tomography, at least down to 75 km depth.&lt;/p&gt;&lt;p&gt;Using recently published regional plate kinematic models and absolute reference frames, we test the hypothesis of the C-LIP origin in the Gal&amp;#225;pagos hotspot. However, misfits of up to ~3000 km between the present hotspot location and the mantle anomalies, reconstructed back to 90 Ma, is observed, as other authors reported in the past.&lt;/p&gt;&lt;p&gt;Therefore, we discuss possible sources of error responsible for this offset and pose two possible interpretations: 1. The Gal&amp;#225;pagos hotspot migrated (~1200-3000 km) westward while the Proto-Caribbean moved to the northeast, or 2. The C-LIP was formed by a different plume, which &amp;#8211; if considered fixed - would be nowadays located below the South American continent.&lt;/p&gt;

scholarly journals A cryptic Allee effect: spatial contexts mask an existing fitness–density relationship

2015 ◽  
Vol 2 (6) ◽  
pp. 150034 ◽  
Author(s):  
Akira Terui ◽  
Yusuke Miyazaki ◽  
Akira Yoshioka ◽  
Shin-ichiro S. Matsuzaki
Keyword(s):  
Allee Effect ◽  
Local Level ◽  
Freshwater Mussel ◽  
Fertilization Rate ◽  
Positive Density ◽  
Allee Effects ◽  
Flow Conditions ◽  
Conspecific Density ◽  
Sperm Density ◽  
Density Relationship

Current theories predict that Allee effects should be widespread in nature, but there is little consistency in empirical findings. We hypothesized that this gap can arise from ignoring spatial contexts (i.e. spatial scale and heterogeneity) that potentially mask an existing fitness–density relationship: a ‘cryptic’ Allee effect. To test this hypothesis, we analysed how spatial contexts interacted with conspecific density to influence the fertilization rate of the freshwater mussel Margaritifera laevis . This sessile organism has a simple fertilization process whereby females filter sperm from the water column; this system enabled us to readily assess the interaction between conspecific density and spatial heterogeneity (e.g. flow conditions) at multiple spatial levels. Our findings were twofold. First, positive density-dependence in fertilization was undetectable at a population scale (approx. less than 50.5 m 2 ), probably reflecting the exponential decay of sperm density with distance from the sperm source. Second, the Allee effect was confirmed at a local level (0.25 m 2 ), but only when certain flow conditions were met (slow current velocity and shallow water depth). These results suggest that spatial contexts can mask existing Allee effects.

scholarly journals Three primes in arithmetical progression

1961 ◽  
Vol 37 (7) ◽  
pp. 329-330
Author(s):  
Saburô Uchiyama
Keyword(s):  
Arithmetical Progression
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