A Geometric Proof of the Gap Theorem

We present an extended analytic technique for calculating the permeability and permittivity spectra of granular materials. It addresses the relationship between grain properties, their size relative to the wavelength, and the permeability and permittivity spectra of polycrystalline material. The scattered multipolar fields about a single sphere are related to the polarizability of an ordered congregation of such spheres. The product of the external wave vector k and sphere radius a is small and the product of the internal wave vector ${\mathit{k}}_{\mathit{i}}$ and a unrestricted. The Clausius-Mossotti relation is used in combination with the scattering results to yield an equation that permits us to calculate the effective permeability and permittivity spectra of a cubic array. The result is a variety of possible spectral types, including complex permittivity and permeability spectra that have been measured and explained either by including multiple atomic-level sources or by statistical weighting over an ensemble of grains. This theory predicts both ``classical'' and ``anomalous'' spectra from a single source, as well as certain conglomerate permeability spectra that have been measured and for which no direct explanation is available. Our results show that an understanding of the permeability and permittivity spectra of composite materials cannot be complete without inclusion of a dependence on the ratio of the grain size to the wavelength.

The dynamics of a sphere fluidized in a nearly levitating upflow of air were previously found to be identical to those of a Brownian particle in a two-dimensional harmonic trap, consistent with a Langevin equation [Ojha et al., Nature (London) 427, 521 (2004)]. The random forcing, the drag, and the trapping potential represent different aspects of the interaction of the sphere with the air flow. In this paper we vary the experimental conditions for a single sphere, and report on how the force terms in the Langevin equation scale with air flow speed, sphere radius, sphere density, and system size. We also report on the effective interaction potential between two spheres in an upflow of air.

Hydrodynamic interaction of a spherical particle with a planar boundary I. Free surface

SummaryIn 1693, Gottfried Wilhelm Leibniz published in the Acta Eruditorum a geometrical proof of the fundamental theorem of the calculus. It is shown that this proof closely resembles Isaac Barrow's proof in Proposition 11, Lecture 10, of his Lectiones Geometricae, published in 1670. This comparison provides evidence that Leibniz gained substantial help from Barrow's book in formulating and presenting his geometrical formulation of this theorem. The analysis herein also supports the work of J. M. Child, who in 1920 studied the early manuscripts of Leibniz and concluded that he had frequently copied his diagrams from Barrow's book, but without acknowledgement. It is also shown that the diagram of Leibniz associated with his 1693 proof has often been reproduced with errors that make some aspects of his text difficult to comprehend.

Geometric proofs of some theorems of Schur-Horn type

Recently, Baker and Norine (Advances in Mathematics, 215(2): 766-788, 2007) found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph $G$. In this paper, we develop a general Riemann-Roch theory for sub-lattices of the root lattice $A_n$ analogue to the work of Baker and Norine, and establish connections between the Riemann-Roch theory and the Voronoi diagrams of lattices under certain simplicial distance functions. In this way, we obtain a geometric proof of the Riemann-Roch theorem for graphs and generalise the result to other sub-lattices of $A_n$. In particular, we provide a new geometric approach for the study of the Laplacian of graphs. We also discuss some problems on classification of lattices with a Riemann-Roch formula as well as some related algorithmic issues.

We give a short geometric proof of the Kochen-Specker no-go theorem for non-contextual hidden variables models. Note added to this version: I understand from Jan-Aake Larsson that the construction we give here actually contains the original Kochen-Specker construction as well as many others (Bell, Conway and Kochen, Schuette, perhaps also Peres).

A recent theorem of Hyde proves that the factorizations statistics of a random polynomial over a finite field are governed by the action of the symmetric group on the configuration space of $n$ distinct ordered points in $\mathbb R^3$. Hyde asked whether this result could be explained geometrically. We give a geometric proof of Hyde's theorem as an instance of the Grothendieck--Lefschetz trace formula applied to an interesting, highly nonseparated algebraic space. An advantage of our method is that it generalizes uniformly to an arbitrary Weyl group. In the process we study certain non-Hausdorff models for complements of hyperplane arrangements, first introduced by Proudfoot.

We present new sampling theorems for surfaces and higher dimensional manifolds. The core of the proofs resides in triangulation results for manifolds with boundary, not necessarily bounded. The method is based upon geometric considerations that are further augmented for 2-dimensional manifolds (i.e surfaces). In addition, we show how to apply the main results to obtain a new, geometric proof of the classical Shannon sampling theorem, and also to image analysis.

A geometric proof of the flyping theorem

This paper offers a new reading of the method of hypothesis in the Divided Line of Plato's Republic that preserves a clear distinction between the objects of that method and dialectic. Properties, or Forms, are hypothesized when they are predicated of particulars whose character is to be investigated, as in the ekthesis of a construction based geometric proof. Hypothesis is the basis for rigorous proof of general theorems, expressed entirely in terms of particulars, such as geometric figures. Only in dialectic, centered on the ‘What is F?’ question, do we come to investigate universal properties directly.

We give elementary geometric proofs of the structure theorems for the (small) quantum cohomology of partial flag varieties SL ⁡ ( n ) / P \operatorname {SL}(n)/P , including the quantum Pieri and quantum Giambelli formulas and the presentation.

We give geometric proofs for the Grobman-Hartman theorem for diffeomorphisms and ODEs. Proofs use covering relations and cone conditions for maps and isolating segments and cone conditions for ODEs. We establish topological versions of the Grobman-Hartman theorem as the existence of some semiconjugaces.

Let (M;F) be a smooth foliated manifold. We prove that there exists a triangulation of M such that each simplex is a distinguished chart for the foliation. This result enables us to give a complete geometric proof of the stability theorem. We also show that the relation between C(M;F) and the Calgebra of a regular covering, is a stability result.

Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck-Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We show (quasi-)polynomiality and reciprocity results for the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar-Ardila (2017), Billera-Jia-Reiner (2009), and Karaboghossian (2022). Applying this reciprocity theorem to hypergraphic polytopes allows to give a geometric proof of a combinatorial reciprocity theorem for hypergraph colorings by Aval-Karaboghossian-Tanasa (2020). This proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.