How old are the Platonic Solids?

Recently a belief has spread that the set of five Platonic Solids has been known since prehistoric times, in the form of carved stone balls from Scotland, dating from the Neolithic period. A photograph of a group of these objects has even been claimed to show mathematical understanding of the regular solids, a millennium or so before Plato. I argue that this is not so. The archaeological and statistical evidence do not support this idea, and it has been shown that there are problems with the photograph. The high symmetry of many of these objects can readily be explained without supposing any particular mathematical understanding on the part of the creators, and there seems to be no reason to doubt that the discovery of the set of five regular solids is contemporary with Plato.     

What Are Equivalent Equations?

Theequationsx =6 38,4 32and 6 0 6 ,90 0 -x =- 31,532areequivalentequationsbecausetheyhavethesamesolu tions .Whenyouwritedownthestepstosolveanequation ,eachstepshouldbeanequationequivalenttotheequationsintheprevioussteps .ExampleMultiplechoice .Whichequationisnotequivalenttotheothers?(a)x +3=8　 (b) 2x =10　 (c) - 1=4 -x　 (d)x =5　 (e)x - 0 .4 =4 .96SolutionEquations (a) ,(b) ,(c)and (d)allhavethesinglesolution 5.Thesolutiontoequation (e)is5.36 .Equatin (e)isnotequivalenttotheothers .Ofalleq…     

Are Bare Particulars Constituents?

In this article I examine an as yet unexplored aspect of J.P. Moreland’s defense of so-called bare particularism — the ontological theory according to which ordinary concrete particulars (e.g., Socrates) contain bare particulars as individuating constituents and property ‘hubs.’ I begin with the observation that if there is a constituency relation obtaining between Socrates and his bare particular, it must be an internal relation, in which case the natures of the relata will necessitate the relation. I then distinguish various ways in which a bare particular might be thought to have a nature and show that on none of these is it possible for a bare particular to be a constituent of a complex particular. Thus, Moreland’s attempt to resurrect bare particulars as ontologically indispensable entities is not wholly without difficulties.     

Are Chain-Complete Posets Co-wellpowered?

Order - We show that the category CPO of chain-complete posets is not co-wellpowered but that it is weakly co-wellpowered. This implies that CPO is nearly locally presentable.     

Are Operads Algebraic Theories?

I exhibit a pair of non-symmetric operads that, while not themselvesisomorphic, induce isomorphic monads. The existence of sucha pair implies that if ‘algebraic theory’ is understoodas meaning ‘monad’ then operads cannot be regardedas algebraic theories of a special kind. 2000 Mathematics SubjectClassification 18D50 (primary), 18C10, 18C15 (secondary).     

Are Biseparable Extensions Frobenius?

In Secion 1 we describe what is known of the extent to which a separable extension of unital associative rings is a Frobenius extension. A problem of this kind is suggested by asking if three algebraic axioms for finite Jones index subfactors are dependent. In Section 2 the problem in the title is formulated in terms of separable bimodules. In Section 3 we specialize the problem to ring extensions, noting that a biseparable extension is a two-sided finitely generated projective, split, separable extension. Some reductions of the problem are discussed and solutions in special cases are provided. In Section 4 various examples are provided of projective separable extensions that are neither finitely generated nor Frobenius and which give obstructions to weakening the hypotheses of the question in the title. In Section 5 we show that characterizations of the separable extensions among Frobenius extensions are special cases of a result for adjoint functors. Mathematics Subject Classifications (2000): 16L60, 16H05.     

Are All Uniform Algebras Amnm?

A Banach algebra a is AMNM if whenever a linear functional on a and a positive number satisfy |(ab)–(a)(b)|||a||·||b||for all a, b a, there is a multiplicative linear functional on a such that ||–||=o(1) as 0. K. Jarosz [1] asked whetherevery Banach algebra, or every uniform algebra, is AMNM. B.E. Johnson [3] studied the AMNM property and constructed a commutativesemisimple Banach algebra that is not AMNM. In this note weconstruct uniform algebras that are not AMNM. 1991 MathematicsSubject Classification 46J10.     

Are Hamiltonian flows geodesic flows?

When a Hamiltonian system has a ``Kinetic + Potential' structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure.

We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the -body problem. We show that the flow of the reduced planar -body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.

     

How Many Communities Are There?

Stochastic blockmodels and variants thereof are among the most widely used approaches to community detection for social networks and relational data. A stochastic blockmodel partitions the nodes of a network into disjoint sets, called communities. The approach is inherently related to clustering with mixture models; and raises a similar model selection problem for the number of communities. The Bayesian information criterion (BIC) is a popular solution, however, for stochastic blockmodels, the conditional independence assumption given the communities of the endpoints among different edges is usually violated in practice. In this regard, we propose composite likelihood BIC (CL-BIC) to select the number of communities, and we show it is robust against possible misspecifications in the underlying stochastic blockmodel assumptions. We derive the requisite methodology and illustrate the approach using both simulated and real data. Supplementary materials containing the relevant computer code are available online.     

Are complex systems hard to evolve?

Evolutionary complexity is measured here by the number of trials/evaluations needed for evolving a logical gate in a nonlinear medium. Behavioral complexity of the gates evolved is characterized in terms of cellular automata behavior. We speculate that hierarchies of behavioral and evolutionary complexities are isomorphic up to some degree, subject to substrate specificity of evolution, and the spectrum of evolution parameters. © 2009 Wiley Periodicals, Inc. Complexity, 2009     

Are regularly estimable functionals always differentiable?

Summary. According to van der Vaart (1991), regularly estimable functionals k \kappa are necessarily differentiable, if lim_{t? 0}t^-1(k(P_t)-k(P₀)) \lim\limits_{t\to 0}t^{-1}(\kappa(P_t)-\kappa(P_0)) exists for every differentiable path. In the present paper, a comparable result is obtained under slightly weaker conditions. A counterexample shows that these conditions are minimal.