97,574 research outputs found
Upper bounds of Hilbert coefficients and Hilbert functions
Let be a -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 -primary ideal that
improves all known upper bounds unless for a finite number of cases. We also
provide new upper bounds of the Hilbert functions of 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
Imperial Users onl
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
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, 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
Gravity and axions from a random UV QFT
It is postulated that the UV QFT is enormous and random. The coupling of the
Standard Model to such QFT is analyzed. It is argued that massless 4d gravity
and axions are general avatars of the postulate. The equivalence principle
emerges naturally as well as a concrete set of sources for its breaking. The
axion scale is related to the 4d Planck scale as , where is the
"number of colors" of the (almost) hidden UV CFT.Comment: Latex, 39 page
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
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