scholarly journals Continuous CM-regularity of semihomogeneous vector bundles

2020 ◽  
Vol 20 (3) ◽  
pp. 401-412
Author(s):  
Alex Küronya ◽  
Yusuf Mustopa
Keyword(s):  
Vector Bundle ◽  
Abelian Variety ◽  
Upper Bound ◽  
Vector Bundles ◽  
Projective Variety ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Piecewise Constant ◽  
Generic Vanishing

AbstractWe ask when the CM (Castelnuovo–Mumford) regularity of a vector bundle on a projective variety X is numerical, and address the case when X is an abelian variety. We show that the continuous CM-regularity of a semihomogeneous vector bundle on an abelian variety X is a piecewise-constant function of Chern data, and we also use generic vanishing theory to obtain a sharp upper bound for the continuous CM-regularity of any vector bundle on X. From these results we conclude that the continuous CM-regularity of many semihomogeneous bundles — including many Verlinde bundles when X is a Jacobian — is both numerical and extremal.

scholarly journals Positivity of vector bundles on homogeneous varieties

2020 ◽  
Vol 31 (12) ◽  
pp. 2050097
Author(s):  
Indranil Biswas ◽  
Krishna Hanumanthu ◽  
D. S. Nagaraj
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Algebraic Group ◽  
Abelian Variety ◽  
Maximal Torus ◽  
Vector Bundles ◽  
Projective Variety ◽  
Closed Curve ◽  
Seshadri Constant ◽  
Higher Rank

We study the following question: Given a vector bundle on a projective variety [Formula: see text] such that the restriction of [Formula: see text] to every closed curve [Formula: see text] is ample, under what conditions [Formula: see text] is ample? We first consider the case of an abelian variety [Formula: see text]. If [Formula: see text] is a line bundle on [Formula: see text], then we answer the question in the affirmative. When [Formula: see text] is of higher rank, we show that the answer is affirmative under some conditions on [Formula: see text]. We then study the case of [Formula: see text], where [Formula: see text] is a reductive complex affine algebraic group, and [Formula: see text] is a parabolic subgroup of [Formula: see text]. In this case, we show that the answer to our question is affirmative if [Formula: see text] is [Formula: see text]-equivariant, where [Formula: see text] is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on [Formula: see text].

scholarly journals Current-phase relation in layered superconducting structures of SIS’IS type

2021 ◽  
Vol 24 (2) ◽  
pp. 23701
Author(s):  
A. M. Shutovskyi ◽  
V. E. Sakhnyuk
Keyword(s):  
Phase Difference ◽  
Current Density ◽  
Phase Relation ◽  
Dielectric Layer ◽  
Intermediate Layer ◽  
Order Parameter ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Current Phase ◽  
Piecewise Constant

The dependence of the current density on the phase difference is investigated considering the layered superconducting structures of a SIS’IS type. To simplify the calculations, the quasiclassical equations for the Green’s functions in a t-representation are derived. An order parameter is considered as a piecewise constant function. To consider the general case, no restrictions on the dielectric layer transparency and the thickness of the intermediate layer are imposed. It was found that a new analytical expression for the current-phase relation can be used with the aim to obtain a number of previously known results arising in particular cases.

scholarly journals A theorem of Matsushima

1974 ◽  
Vol 54 ◽  
pp. 123-134 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Vector Bundle ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Complex Torus ◽  
Frobenius Endomorphism

In [7], Matsushima studied the vector bundles over a complex torus. One of his main theorems is: A vector bundle over a complex torus has a connection if and only if it is homogeneous (Theorem (2.3)). The aim of this paper is to prove the characteristic p > 0 version of this theorem. However in the characteristic p > 0 case, for any vector bundle E over a scheme defined over a field k with char, k = p, the pull back F*E of E by the Frobenius endomorphism F has a connection. Hence we have to replace the connection by the stratification (cf. (2.1.1)). Our theorem states: Let A be an abelian variety whose p-rank is equal to the dimension of A. Then a vector bundle over A has a stratification if and only if it is homogeneous (Theorem (2.5)).

Ample Vector Bundles of Curve Genus One

1999 ◽  
Vol 42 (2) ◽  
pp. 209-213 ◽  
Author(s):  
Antonio Lanteri ◽  
Hidetoshi Maeda
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Ample Vector Bundle ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Curve Genus ◽  
Genus One ◽  
Complex Projective ◽  
Zero Locus

AbstractWe investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.

scholarly journals PERIODIC ORBITS OF SINGLE NEURON MODELS WITH INTERNAL DECAY RATE 0 < Β ≤ 1

2013 ◽  
Vol 18 (3) ◽  
pp. 325-345 ◽  
Author(s):  
Aija Anisimova ◽  
Maruta Avotina ◽  
Inese Bula
Keyword(s):  
Dynamical System ◽  
Decay Rate ◽  
Single Neuron ◽  
Discrete Dynamical System ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Sigmoid Function ◽  
Signal Function ◽  
Neuron Models ◽  
Piecewise Constant

In this paper we consider a discrete dynamical system x n+1=βx n – g(x n ), n=0,1,..., arising as a discrete-time network of a single neuron, where 0 &lt; β ≤ 1 is an internal decay rate, g is a signal function. A great deal of work has been done when the signal function is a sigmoid function. However, a signal function of McCulloch-Pitts nonlinearity described with a piecewise constant function is also useful in the modelling of neural networks. We investigate a more complicated step signal function (function that is similar to the sigmoid function) and we will prove some results about the periodicity of solutions of the considered difference equation. These results show the complexity of neurons behaviour.

scholarly journals The Kirchhoff Transformation and the Fick’s Second Law with Concentration-dependent Diffusion Coefficient

2021 ◽  
Vol 16 ◽  
pp. 59-67
Author(s):  
R. M. S. Gama ◽  
R. Pazetto S. Gama
Keyword(s):  
Diffusion Coefficient ◽  
Mathematical Structure ◽  
Constant Function ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Piecewise Constant Function ◽  
Second Law ◽  
Piecewise Constant ◽  
Numerical Examples ◽  
Kirchhoff Transformation

In this work it is considered the Fick’s second law in a context in which the diffusion coefficient depends on the concentration. It is employed the Kirchhoff transformation in order to simplify the mathematical structure of the Fick’s second law, giving rise to a more convenient description. In order to provide a general protocol, the diffusion coefficient will be assumed a piecewise constant function of the concentration. Exact formulas are presented for both the Kirchhoff transformation and its inverse, in such a way that there is no limit of accuracy. Some numerical examples are presented with the aid of a semi-implicit procedure associated with a finite difference approximation.

scholarly journals Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

2021 ◽  
Vol 20 (1) ◽  
pp. 95-119
Author(s):  
Nathan Grieve
Keyword(s):  
New York ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Index Theorem ◽  
Polynomial Functions ◽  
Algebraic Varieties ◽  
Wedderburn Decomposition ◽  
Generic Vanishing ◽  
Roch Theorem

Abstract We study the property of continuous Castelnuovo-Mumford regularity, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in A. Küronya and Y. Mustopa [Adv. Geom. 20 (2020), no. 3, 401-412]. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety’s endo-morphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that was established in N. Grieve [New York J. Math. 23 (2017), 1087-1110]. In a complementary direction, we explain how these topics pertain to the Index and Generic Vanishing Theory conditions for simple semihomogeneous vector bundles. In doing so, we refine results from M. Gulbrandsen [Matematiche (Catania) 63 (2008), no. 1, 123–137], N. Grieve [Internat. J. Math. 25 (2014), no. 4, 1450036, 31] and D. Mumford [Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100].

scholarly journals A Production Inventory Model for deteriorating items with backlog-dependent demand

2019 ◽  
Author(s):  
Chayanika Rout ◽  
Debjani Chakraborty ◽  
Prof Adrijit Goswami
Keyword(s):  
Inventory Model ◽  
Constant Function ◽  
Deteriorating Items ◽  
Piecewise Constant Function ◽  
High Quality ◽  
Piecewise Constant ◽  
Production Inventory ◽  
Demand Rate ◽  
Theoretical Results ◽  
Dependent Demand

This paper investigates a production inventory model under classical EPQ framework with the assumption that the customer demand during the stock out period is affected by the accumulated back-orders. The backlog rate is not fixed; instead, the demand rate during stock-out is assumed to decrease proportionally to the existing backlog which is thereby approximated by a piecewise constant function. Deteriorating items are taken into consideration in this proposed work. For better illustration of the theoretical results and to highlight managerial insights, numerical examples arepresentedwhicharethencomparedtotheresultsobtainedbyconsideringanexact (non-approximated) backlogging rate (from literature). The comparisons indicate high quality results for the approximated model.

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