Minimal submanifolds from the abelian Higgs model

Given a Hermitian line bundle $$L\rightarrow M$$ L → M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as $$\epsilon \rightarrow 0$$ ϵ → 0 , of couples $$(u_\epsilon ,\nabla _\epsilon )$$ ( u ϵ , ∇ ϵ ) critical for the rescalings $$\begin{aligned} E_\epsilon (u,\nabla )=\int _M\Big (|\nabla u|^2+\epsilon ^2|F_\nabla |^2+\frac{1}{4\epsilon ^2}(1-|u|^2)^2\Big ) \end{aligned}$$ E ϵ ( u , ∇ ) = ∫ M ( | ∇ u | 2 + ϵ 2 | F ∇ | 2 + 1 4 ϵ 2 ( 1 - | u | 2 ) 2 ) of the self-dual Yang–Mills–Higgs energy, where u is a section of L and $$\nabla $$ ∇ is a Hermitian connection on L with curvature $$F_{\nabla }$$ F ∇ . Under the natural assumption $$\limsup _{\epsilon \rightarrow 0}E_\epsilon (u_\epsilon ,\nabla _\epsilon )<\infty $$ lim sup ϵ → 0 E ϵ ( u ϵ , ∇ ϵ ) < ∞ , we show that the energy measures converge subsequentially to (the weight measure $$\mu $$ μ of) a stationary integral $$(n-2)$$ ( n - 2 ) -varifold. Also, we show that the $$(n-2)$$ ( n - 2 ) -currents dual to the curvature forms converge subsequentially to $$2\pi \Gamma $$ 2 π Γ , for an integral $$(n-2)$$ ( n - 2 ) -cycle $$\Gamma $$ Γ with $$|\Gamma |\le \mu $$ | Γ | ≤ μ . Finally, we provide a variational construction of nontrivial critical points $$(u_\epsilon ,\nabla _\epsilon )$$ ( u ϵ , ∇ ϵ ) on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral $$(n-2)$$ ( n - 2 ) -varifolds in an arbitrary closed Riemannian manifold.

Rigid body dynamics using equimomental systems of point-masses

The inertia matrix of any rigid body is the same as the inertia matrix of some system of four point-masses. In this work, the possible disposition of these point-masses is investigated. It is found that every system of possible point-masses with the same inertia matrix can be parameterised by the elements of the orthogonal group in four-dimensional modulo-permutation of the points. It is shown that given a fixed inertia matrix, it is possible to find a system of point-masses with the same inertia matrix but where one of the points is located at some arbitrary point. It is also possible to place two point-masses on an arbitrary line or three of the points on an arbitrary plane. The possibility of placing some of the point-masses at infinity is also investigated. Applications of these ideas to rigid body dynamics are considered. The equation of motion for a rigid body is derived in terms of a system of four point-masses. These turn out to be very simple when written in a 6-vector notation.

Russian Punctuation as a Sign System

The Russian punctuation is a system of signs with its own functions and its own formal organization, different from the organization of alphabetic characters, numbers and other written signs of the Russian language. In terms of formal organization, punctuation marks differ from two-dimension characters in geometry and in onedimensionality or even zero-dimensionality. In terms of function, they are used as signs of articulation and organization of the text (as the metatext by A. Wierzbicka). Punctuation marks are located between graphic (linguistic-visual) and paragraphic (nonlinguistic visual) systems of writing. They perform their function in combination with the system of spaces (word space, arbitrary line end, empty part of the page, empty page) and paragraph tools. The punctuation is similar in function to road signs and computer text markup but unlike them punctuation marks are situated on the line alternately with the main objects of the writing.

The History of Aphasiology

This chapter charts the history of aphasiology from antiquity to the recent past. An arbitrary line is drawn to include paragraphs of only those who passed, to avoid the review of large amounts of currently active research. References are provided for original work as well as other historical reviews, the review groups, scientific and conceptual developments in historical time periods, regions of work, disciplines, and schools of thought. Contemporary aphasiology is part of the neurosciences, both basic and applied, advancing at a pace so precipitous as to be almost disorienting. We are benefiting from today’s incredible advances in MRI, PET, functional MRI, voxel-based morphometry, transcortical stimulation and white matter diffusion imaging, genetic and molecular biology laboratory work, and continuing clinical experience. History is being made at almost every minute and is better recorded than ever before. An update is already overdue.

Approximation of Capture Sets in Visibility-Based Target-Tracking Games for Non-Holonomic Players

In this work, we analyze approximations of capture sets [1] for a visibility based pursuit-evasion game. In contrast to the capture problem, the pursuer tries to maintain a line-of-sight with the evader in free space in our problem. We extend the concept of U set initially proposed in [2] for holonomic players to the scenario in which the pursuer is holonomic. The problem of computing the U set is reduced to that of computing time-optimal paths for the non-holonomic vehicles to an arbitrary line. We characterize the primitives for time-optimal paths for the Dubin’s vehicle, Reed-shepps car and a Differential Drive robot. Based on these primitives, we construct the optimal paths and provide an algorithm to compute the U set.

New Scholarship on Central and Eastern Europe

A steady stream of creative new scholarship on Central and Eastern Europe has been arriving at the desks of reviews editors in Contemporary European History. Commissioning and editing essays on this scholarship has persuasively demonstrated its wider importance, not least in challenging the ambiguous and arbitrary line that continues to divide European historiography between East and West. We have therefore taken the decision to present the following five articles collectively, as a means of reflecting on and interrogating assumptions in European historiography. Why do Germany and France still dominate narratives of the Great War? Can we speak of urban continuities and similarities across the socialist and capitalist spheres? How does studying food offer new insights on the global phenomenon of socialism and socialist production?