scholarly journals Asymptotically Optimal Box Packing Theorems

10.37236/802 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Michael Reid
Keyword(s):  
Algebraic Geometry ◽  
Ad Hoc ◽  
Necessary Conditions ◽  
Combinatorial Proof ◽  
Original Proof ◽  
Asymptotically Optimal ◽  
Straightforward Calculation ◽  
Complete Answer ◽  
Special Case

Given a protoset of $d$-dimensional polyominoes, we ask which boxes can be packed by the protoset. In some cases, it may be too difficult to give a complete answer to this question, so we ask the easier question about determining all sufficiently large boxes that can be packed. (We say that a box is "sufficiently large" if all edge lengths are ${} \ge C$ for some large $C$.) We give numerous examples (mostly $2$-dimensional) where we can answer this easier question. The various techniques involved are: checkerboard-type colorings/numberings (tile homology), the boundary word method of Conway and Lagarias (tile homotopy), ad hoc geometric arguments, and a very nice theorem of Barnes. Barnes' Theorem asserts that all necessary conditions for a box to be packable can be given in a certain form, and these conditions are also sufficient for large boxes. Barnes' Theorem has not received the appreciation it deserves. We give a new, purely combinatorial proof of this important result. (Barnes' original proof uses techniques of algebraic geometry.) In the special case that all the prototiles are boxes themselves, we show how to determine all sufficiently large boxes that they pack. We prove a theorem based on Barnes' result that reduces this to a straightforward calculation.

Trans‐European Party Groupings: Emergence of New and Alignment of Old Parties in the Light of the Direct Elections to the European Parliament

1979 ◽  
Vol 14 (4) ◽  
pp. 455-478 ◽  
Author(s):  
Paul‐H. Claeys ◽  
Nicole Loeb‐Mayer
Keyword(s):  
Political Parties ◽  
Ad Hoc ◽  
European Parliament ◽  
The Other ◽  
European Level ◽  
Or Groups ◽  
Direct Elections ◽  
Special Case ◽  
Domestic Level

TWO QUESTIONS ARISE WHEN CONSIDERING THE CHANGES that might be brought about by direct elections and by developments in the new European Parliament. One concerns institutionalized cooperation between political parties. To what extent can the three existing European party federations – Socialist, Christian Democrat, Liberal – be considered as a step towards the formation of genuine European political parties? Are they anything more than alignments of traditional parties coordinating their action at European level? The other question is related to parties or groups which have not until now created close-knit ad hoc structures. A special case is that of the Communist parties, which have not organized specific links at Communit level. Another problem is raised by non-traditional parties and groups that have in most cases little or no parlia mentary representation at either national or European level. Will some of them take advantage of the European sphere of action to make more impression than they have been able to do at domestic level, in cooperation with similarly oriented partners in other member countries?

Fragmentation and Coarse-Grained Content

2021 ◽  
pp. 54-77
Author(s):  
Daniel Greco
Keyword(s):  
Possible Worlds ◽  
Ad Hoc ◽  
Model Building ◽  
Coarse Grained ◽  
Coarse Grain ◽  
The Relationship ◽  
Special Case

This chapter defends the possible worlds framework for modeling the contents of belief. Both the threats against which the chapter defends it—the problems of coarse grain—and the ‘fragmentationist’ response it offers are familiar. At least as a sociological matter, the fragmentationist response has been unpersuasive, likely because it can look like an ad hoc patch—an unmotivated epicycle aimed at saving a flailing theory from decisive refutation. The chapter offers two responses to this charge. First, the problems of coarse grain aren’t unique to the possible worlds framework and indeed arise for anyone who accepts certain very attractive views about the relationship between beliefs, desires, and action. Second, the fragmentationist response to these problems is in fact a special case of an independently motivated, ‘modest’ approach to model-building in philosophy.

The Induced Removal Lemma in Sparse Graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 153-162
Author(s):  
Shachar Sapir ◽  
Asaf Shapira
Keyword(s):  
General Result ◽  
Original Proof ◽  
Sparse Graphs ◽  
Vertex Graph ◽  
Removal Lemma ◽  
Special Case

AbstractThe induced removal lemma of Alon, Fischer, Krivelevich and Szegedy states that if an n-vertex graph G is ε-far from being induced H-free then G contains δH(ε) · nh induced copies of H. Improving upon the original proof, Conlon and Fox proved that 1/δH(ε)is at most a tower of height poly(1/ε), and asked if this bound can be further improved to a tower of height log(1/ε). In this paper we obtain such a bound for graphs G of density O(ε). We actually prove a more general result, which, as a special case, also gives a new proof of Fox’s bound for the (non-induced) removal lemma.

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.

Occasionalism and the Cartesian Metaphysic of Motion*

1975 ◽  
Vol 1 (1) ◽  
pp. 29-40 ◽  
Author(s):  
T. M. Lennon
Keyword(s):  
Body Problem ◽  
Ad Hoc ◽  
Body Interaction ◽  
General Account ◽  
Causal Connections ◽  
Mind Body Problem ◽  
The Mind ◽  
Special Case

Occasionalism is often taken by historians of philosophy to have been an ad hoc hypothesis to establish the mind-body causal connections which on Cartesian principles are thought otherwise impossible. My aim in this paper is to show that this view is utterly without historical foundation, that, on the contrary, the view that only God can be a real cause of mind-body interaction was but a special case of a claim argued on grounds transcending the mind-body problem, and, what will be part of this, that the logical character of occasionalism anyhow precluded it from the role into which it was later miscast. More specifically, I shall show that occasionalism was but a consequence of the metaphysics adopted by the Cartesians in their general account of change. Though the same case could be made for the views of Clauberg and Geulincx, my concern will be with the occasionalism of Malebranche. My case here will be that his view is the historical and logical dénouement of principles more or less explicit both in Descartes and in two of his lesser known disciples, LaForge and Cordemoy.

Asymptotic stability of heteroclinic cycles in systems with symmetry

1995 ◽  
Vol 15 (1) ◽  
pp. 121-147 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Ad Hoc ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Higher Dimensions ◽  
Systematic Investigation ◽  
Heteroclinic Cycles ◽  
General Sufficient Condition ◽  
Higher Dimensional ◽  
The Stability

AbstractSystems possessing symmetries often admit heteroclinic cycles that persist under perturbations that respect the symmetry. The asymptotic stability of such cycles has previously been studied on an ad hoc basis by many authors. Sufficient conditions, but usually not necessary conditions, for the stability of these cycles have been obtained via a variety of different techniques.We begin a systematic investigation into the asymptotic stability of such cycles. A general sufficient condition for asymptotic stability is obtained, together with algebraic criteria for deciding when this condition is also necessary. These criteria are always satisfied in ℝ3 and often satisfied in higher dimensions. We end by applying our results to several higher-dimensional examples that occur in mode interactions with O(2) symmetry.

Research and Simulation of Routing Protocol in Different VANET Scenarios

2011 ◽  
Vol 217-218 ◽  
pp. 451-456
Author(s):  
Meng Jun Tong ◽  
Li Yu ◽  
Chang Heng Shu ◽  
Qi Fen Dong ◽  
Feng Gao
Keyword(s):  
Routing Protocol ◽  
Traffic Management ◽  
Ad Hoc ◽  
Road Traffic ◽  
Simulation Software ◽  
Ad Hoc Routing ◽  
Scenarios Simulation ◽  
Stable Performance ◽  
Set Up ◽  
Special Case

VANET is a special case of MANET , and will play an increasingly important role in road traffic management. VANET networks have different motion characters in different scenarios. Through the extension of the current network simulation software NS2, the different simulation scenarios were set up. Several typical Ad hoc routing protocols were simulated and analyzed in the streets and highways scenarios. Simulation results show that AODV and DSR have a relatively stable performance. The results have value in the research and application of VANET to traffic management.

scholarly journals Cubical Convex Ear Decompositions

10.37236/83 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Russ Woodroofe
Keyword(s):  
Finite Group ◽  
Necessary Conditions ◽  
Partition Lattice ◽  
Ear Decomposition ◽  
Interesting Special Case ◽  
Order Complexes ◽  
Special Case ◽  
The Way

We consider the problem of constructing a convex ear decomposition for a poset. The usual technique, introduced by Nyman and Swartz, starts with a $CL$-labeling and uses this to shell the 'ears' of the decomposition. We axiomatize the necessary conditions for this technique as a "$CL$-ced" or "$EL$-ced". We find an $EL$-ced of the $d$-divisible partition lattice, and a closely related convex ear decomposition of the coset lattice of a relatively complemented finite group. Along the way, we construct new $EL$-labelings of both lattices. The convex ear decompositions so constructed are formed by face lattices of hypercubes. We then proceed to show that if two posets $P_{1}$ and $P_{2}$ have convex ear decompositions ($CL$-ceds), then their products $P_{1}\times P_{2}$, $P_{1}\check{\times} P_{2}$, and $P_{1}\hat{\times} P_{2}$ also have convex ear decompositions ($CL$-ceds). An interesting special case is: if $P_{1}$ and $P_{2}$ have polytopal order complexes, then so do their products.

scholarly journals Tensor fields associated with integrable systems of chiral

2019 ◽  
Vol 21 (4) ◽  
pp. 405-412
Author(s):  
Alexander V. Balandin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Necessary Conditions ◽  
Differential Forms ◽  
Type Systems ◽  
Lax Representation ◽  
Killing Tensor ◽  
Tensor Fields ◽  
Linear Differential ◽  
Special Case

This article describes necessary conditions for chiral-type systems to admit Lax representation with values in simple compact Lie algebras. These conditions state that there exists a covariant constant tensor field with an additional property. It is proposed to construct in an invariant way some covariant tensor fields using the Lax representation of the system under consideration. These fields are constructed by taking linear differential forms with values in the Lie algebra that are constructed using the Lax representation of the system and substituting them into an arbitrary Ad-invariant form on the Lie algebra. The paper proves that such tensor fields are Killing tensor fields or covariant constant fields. The discovered necessary conditions for the existence of the Lax representation are obtained using a special case of such tensor fields associated with the Killing metric of the Lie algebra.

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