Upper bounds of Hilbert coefficients and Hilbert functions

Let $(R, m)$ be a $d$-dimensional Cohen-Macaulay local ring. In this note we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a $m$-primary ideal $I\subset R$ that improves all known upper bounds unless for a finite number of cases. We also provide new upper bounds of the Hilbert functions of $I$ extending the known bounds for the maximal ideal

Network analysis of complex biological systems : boundedness of weakly reversible chemical reaction networks and conditions for synchronisation of coupled oscillators

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Thick Soergel calculus in type A

Let R be the polynomial ring in n variables, acted on by the symmetric group S_n. Soergel constructed a full monoidal subcategory of R-bimodules which categorifies the Hecke algebra, whose objects are now known as Soergel bimodules. Soergel bimodules can be described as summands of Bott-Samelson bimodules (attached to sequences of simple reflections), or as summands of generalized Bott-Samelson bimodules (attached to sequences of parabolic subgroups). A diagrammatic presentation of the category of Bott-Samelson bimodules was given by the author and Khovanov in previous work. In this paper, we extend it to a presentation of the category of generalized Bott-Samelson bimodules. We also diagrammatically categorify the representations of the Hecke algebra which are induced from trivial representations of parabolic subgroups. The main tool is an explicit description of the idempotent which picks out a generalized Bott-Samelson bimodule as a summand inside a Bott-Samelson bimodule. This description uses a detailed analysis of the reduced expression graph of the longest element of S_n, and the semi-orientation on this graph given by the higher Bruhat order of Manin and Schechtman.Comment: Changed title. Expanded the exposition of the main proof. This paper relies extensively on color figure

The anatomist Hans Elias: A Jewish German in exile

Hans Elias (1907 to 1985) was an anatomist, an educator, a mathematician, a cinematographer, a painter, and a sculptor. Above all, he was a German of Jewish descent, who had to leave his home country because of the policies of the National Socialist (NS) regime. He spent his life in exile, first in Italy and then in the United States. His biography is exemplary for a generation of younger expatriates from National Socialist Germany who had to find a new professional career under difficult circumstances. Elias was a greatly productive morphologist whose artistic talent led to the foundation of the new science of stereology and made him an expert in scientific cinematography. He struggled hard to fulfill his own high expectations of himself in terms of his effectiveness as a scientist, educator, and politically acting man in this world. Throughout his life this strong‐willed and outspoken man never lost his great fondness for Germany and many of its people, while reserving some of his sharpest criticism for fellow anatomists who were active in National Socialist Germany, among them his friend Hermann Stieve, Max Clara, and Heinrich von Hayek. Hans Elias' life is well documented in his unpublished diaries and memoirs, and thus allows fresh insights into a time period when some anatomists were among the first victims of NS policies and other anatomists became involved in the execution of such policies. Clin. Anat. 25:284–294, 2012. © 2011 Wiley Periodicals, Inc.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90396/1/21293_ftp.pd

Photon orbital angular momentum and torque metrics for single telescopes and interferometers

Context. Photon orbital angular momentum (POAM) is normally invoked in a quantum mechanical context. It can, however, also be adapted to the classical regime, which includes observational astronomy. Aims. I explain why POAM quantities are excellent metrics for describing the end-to-end behavior of astronomical systems. To demonstrate their utility, I calculate POAM probabilities and torques from holography measurements of EVLA antenna surfaces. Methods. With previously defined concepts and calculi, I present generic expressions for POAM spectra, total POAM, torque spectra, and total torque in the image plane. I extend these functional forms to describe the specific POAM behavior of single telescopes and interferometers. Results. POAM probabilities of spatially uncorrelated astronomical sources are symmetric in quantum number. Such objects have zero intrinsic total POAM on the celestial sphere, which means that the total POAM in the image plane is identical to the total torque induced by aberrations within propagation media & instrumentation. The total torque can be divided into source- independent and dependent components, and the latter can be written in terms of three illustrative forms. For interferometers, complications arise from discrete sampling of synthesized apertures, but they can be overcome. POAM also manifests itself in the apodization of each telescope in an array. Holography of EVLA antennas observing a point source indicate that ~ 10% of photons in the n = 0 state are torqued to n != 0 states. Conclusions. POAM quantities represent excellent metrics for characterizing instruments because they are used to simultaneously describe amplitude and phase aberrations. In contrast, Zernike polynomials are just solutions of a differential equation that happen to ~ correspond to specific types of aberrations and are typically employed to fit only phases

Existence of long‐time solutions to dynamic problems of viscoelasticity with rate‐and‐state friction

We establish existence of global solutions to a dynamic problem of bilateral contact between a rigid surface and a viscoelastic body, subject to rate‐and‐state friction. The term rate‐and‐state friction describes friction laws where the friction is rate‐dependent and depends on an additional internal state variable defined on the contact surface. Our mathematical conditions rule out certain slip laws, but do cover the ageing law, and thus at least one of the rate‐and‐state friction laws commonly used in the geoscience

On the last Hilbert-Samuel coefficient of isolated singularities

In 1978 Lipman presented a proof of the existence of a desingularization for any excellent surface. The strategy of Lipman's proof is based on the finiteness of the number H(R) defined as the supreme of the second Hilbert-Samuel coefficient I, where I range the set of normal m-primary ideals of a Noetherian complete local ring (R,m). The problem studied in the paper is the extension of the result of Lipman on H(R) to m-primary ideals I of a d-dimensional Cohen-Macaulay ring R such that the associated graded ring of R with respect to I^n is Cohen-Macaulay for n>> 0

Dissecting the string theory dual of QCD

Input from QCD and string theory is used in order to elucidate basic features of the string theory dual of QCD, It is argued that the relevant string theory is a five-dimensional version of the type-0 superstring. The vacuum solution is asymptotically AdS$_5$, and the geometry near the boundary is stringy. The structure of YM perturbation theory however emerges near the boundary. In the IR, the theory is argued to be well-approximated by a two-derivative truncation that takes into account strong coupling effects. This explains the success of previously proposed five-dimensional Eistein-dilaton gravity with an appropriate potential to describe salient features of the strong YM dynamics.Comment: LateX 33 pages, no figures. Based on presentations at various meertings. To appear in the proceedings of the 4th RTN-EU conference, Varna, Bulgaria (v2) Various misprints corrected. Added discussion on the definition of the 't Hooft coupling and issues of scheme dependenc