Yang-Mills action from minimally coupled bosons on R^4 and on the 4D Moyal plane

We consider bosons on Euclidean R^4 that are minimally coupled to an external Yang-Mills field. We compute the logarithmically divergent part of the cut-off regularized quantum effective action of this system. We confirm the known result that this term is proportional to the Yang-Mills action. We use pseudodifferential operator methods throughout to prepare the ground for a generalization of our calculation to the noncommutative four-dimensional Moyal plane (also known as noncommutative flat space). We also include a detailed comparison of our cut-off regularization to heat kernel techniques. In the case of the noncommutative space, we complement the usual technique of asymptotic expansion in the momentum variable with operator theoretic arguments in order to keep separated quantum from noncommutativity effects. We show that the result from the commutative space R^4 still holds if one replaces all pointwise products by the noncommutative Moyal product.Comment: 37 pages, v2 contains an improved treatment of the theta function in Appendix A.

Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic pseudo-differential operators

We obtain a complete asymptotic expansion of the integrated density of states of operators of the form H =(-\Delta)^w +B in R^d. Here w >0, and B belongs to a wide class of almost-periodic self-adjoint pseudo-differential operators of order less than 2w. In particular, we obtain such an expansion for magnetic Schr\"odinger operators with either smooth periodic or generic almost-periodic coefficients.Comment: 47 pages. arXiv admin note: text overlap with arXiv:1004.293

Uniformization and an Index Theorem for Elliptic Operators Associated with Diffeomorphisms of a Manifold

We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely differential operators with shifts induced by the action of an isometric diffeomorphism. The key to the solution is the method of uniformization: We assign to the nonlocal problem a pseudodifferential operator with the same index, acting in sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah-Singer index theorem.Comment: 16 pages, no figure

Innovative solutions to protect scientific and technological equipment from vibrations

Необходимость получения предельных разрешений, манипуляции с нано и микрообъектами, а также создание наноструктур, требуют технических решений как для максимального подавления шумов от источников вибраций, так и максимальной защиты (изоляции) исполнительных элементов технологического оборудования и чувствительных элементов измерительного оборудования от воздействия шумового спектра в местах расположения. Компания Остек предлагает комплекс инновационных технических решений для борьбы с источниками вибраций и их воздействием для защиты атомно-силовых микроскопов, электронных просвечивающих и сканирующих микроскопов, оптических и интерференционных микроскопов высокого разрешения, зондовых станций, установок проекционной литографии, сверхточных систем измерения веса и др. приборов и устройств, требующих максимальной виброзащищенности для успешной эксплуатации.The need to obtain maximum resolution, manipulations with nano and microobjects, as well as the creation of nanostructures, require technical solutions both for maximum suppression of noise from vibration sources and for maximum protection (isolation) of executive elements of process equipment and sensing elements of measuring equipment against noise spectrum in the places of location. Ostec company offers a range of innovative technical solutions for controlling vibration sources and their impact for the protection of atomic force microscopes, electronic transmission and scanning microscopes, high-resolution optical and interference microscopes, probe stations, projection lithography sets, ultra-precise weighing systems and other devices that require maximum vibration protection for successful operation

Conormal distributions in the Shubin calculus of pseudodifferential operators

We characterize the Schwartz kernels of pseudodifferential operators of Shubin type by means of an FBI transform. Based on this we introduce as a generalization a new class of tempered distributions called Shubin conormal distributions. We study their transformation behavior, normal forms and microlocal properties.Comment: 23 page

Fin ray patterns at the fin-to-limb transition

The fin-to-limb transition was marked by the origin of digits and the loss of dermal fin rays. Paleontological research into this transformation has focused on the evolution of the endoskeleton, with little attention paid to fin ray structure and function. To address this knowledge gap, we study the dermal rays of the pectoral fins of 3 key tetrapodomorph taxa—Sauripterus taylori (Rhizodontida), Eusthenopteron foordi (Tristichopteridae), and Tiktaalik roseae (Elpistostegalia)—using computed tomography. These data show several trends in the lineage leading to digited forms, including the consolidation of fin rays (e.g., reduced segmentation and branching), reduction of the fin web, and unexpectedly, the evolution of asymmetry between dorsal and ventral hemitrichia. In Eusthenopteron, dorsal rays cover the preaxial endoskeleton slightly more than ventral rays. In Tiktaalik, dorsal rays fully cover the third and fourth mesomeres, while ventral rays are restricted distal to these elements, suggesting the presence of ventralized musculature at the fin tip analogous to a fleshy “palm.” Asymmetry is also observed in cross-sectional areas of dorsal and ventral rays. Eusthenopteron dorsal rays are slightly larger than ventral rays; by contrast, Tiktaalik dorsal rays can be several times larger than ventral rays, and degree of asymmetry appears to be greater at larger sizes. Analysis of extant osteichthyans suggests that cross-sectional asymmetry in the dermal rays of paired fins is plesiomorphic to crown group osteichthyans. The evolution of dermal rays in crownward stem tetrapods reflects adaptation for a fin-supported elevated posture and resistance to substrate-based loading prior to the origin of digits

The Gabor wave front set of compactly supported distributions

We show that the Gabor wave front set of a compactly supported distribution equals zero times the projection on the second variable of the classical wave front set

Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-M\"uller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L2 analytic and Reidemeister torsions are equal.Comment: 78 pages, AMSTe