scholarly journals Flat connections and resonance varieties: From rank one to higher ranks

2016 ◽  
Vol 369 (2) ◽  
pp. 1309-1343 ◽  
Author(s):  
Daniela Anca Măcinic ◽  
Ştefan Papadima ◽  
Clement Radu Popescu ◽  
Alexander I. Suciu
Keyword(s):  
Flat Connections ◽  
Rank One ◽  
Resonance Varieties

scholarly journals Non-abelian cohomology jump loci from an analytic viewpoint

2014 ◽  
Vol 16 (04) ◽  
pp. 1350025 ◽  
Author(s):  
Alexandru Dimca ◽  
Ştefan Papadima
Keyword(s):  
Minimal Model ◽  
Nilpotent Groups ◽  
Exponential Map ◽  
Flat Connections ◽  
Differential Graded Algebra ◽  
Irreducible Components ◽  
Rank One ◽  
Positive Weights ◽  
Projective Manifolds ◽  
Representation Varieties

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber–Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart.

scholarly journals Characteristic varieties and logarithmic differential 1-forms

2010 ◽  
Vol 146 (1) ◽  
pp. 129-144 ◽  
Author(s):  
Alexandru Dimca
Keyword(s):  
Elliptic Curve ◽  
Recent Work ◽  
Configuration Space ◽  
Zero Set ◽  
Characteristic Variety ◽  
Local Systems ◽  
Complex Variety ◽  
Rank One ◽  
Characteristic Varieties ◽  
Resonance Varieties

AbstractWe introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety M. A logarithmic resonance variety is also considered and, as an application, we determine the first characteristic variety of the configuration space of n distinct labeled points on an elliptic curve. Finally, for a logarithmic 1-form α on M we investigate the relation between the resonance degree of α and the codimension of the zero set of α on a good compactification of M. This question was inspired by the recent work by Cohen, Denham, Falk and Varchenko.

Derivation of the Wave and Scattering Operators for an Interaction of Rank One

1970 ◽  
Vol 11 (8) ◽  
pp. 2415-2424 ◽  
Author(s):  
M. Anthea Grubb ◽  
D. B. Pearson
Keyword(s):  
Rank One ◽  
Scattering Operators

scholarly journals Ramification Theory for Extensions of Degree p. II

1972 ◽  
Vol 46 ◽  
pp. 97-109
Author(s):  
Susan Williamson
Keyword(s):  
Valuation Ring ◽  
Quotient Field ◽  
Root Of Unity ◽  
Ramification Index ◽  
Ramification Theory ◽  
Rank One ◽  
The Absolute

Let k denote the quotient field of a complete discrete rank one valuation ring R of unequal characteristic and let p denote the characteristic of R̅; assume that R contains a primitive pth root of unity, so that the absolute ramification index e of R is a multiple of p — 1, and each Gallois extension K ⊃ k of degree p may be obtained by the adjunction of a pth root.

scholarly journals Algorithms and Applications to Weighted Rank-one Binary Matrix Factorization

2020 ◽  
Vol 11 (2) ◽  
pp. 1-33
Author(s):  
Haibing Lu ◽  
Xi Chen ◽  
Junmin Shi ◽  
Jaideep Vaidya ◽  
Vijayalakshmi Atluri ◽  
...  
Keyword(s):  
Matrix Factorization ◽  
Binary Matrix ◽  
Rank One ◽  
Binary Matrix Factorization ◽  
Weighted Rank

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

2021 ◽  
Vol 143 (2) ◽  
pp. 301-335
Author(s):  
Jendrik Voss ◽  
Ionel-Dumitrel Ghiba ◽  
Robert J. Martin ◽  
Patrizio Neff
Keyword(s):  
Differential Inequalities ◽  
Two Dimensional ◽  
Ellipticity Condition ◽  
One Dimensional ◽  
Suitable Sense ◽  
Precise Analysis ◽  
Rank One ◽  
Isotropic Elasticity ◽  
Convexity Conditions

AbstractWe consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\frac{\mu }{2} \hspace{0.07em} \frac{\lVert F \rVert ^{2}}{\det F}+f(\det F)$ W ( F ) = μ 2 ∥ F ∥ 2 det F + f ( det F ) ; such an energy is rank-one convex if and only if the function $f$ f is convex.

scholarly journals Optimal Multi-Antenna Transmission for the Cooperative Non-Orthogonal Multiple-Access System

2021 ◽  
Vol 11 (5) ◽  
pp. 2203
Author(s):  
Duckdong Hwang ◽  
Janghoon Yang ◽  
Sung Sik Nam ◽  
Hyoung-Kyu Song
Keyword(s):  
Interference Cancellation ◽  
Multiple Access ◽  
Average Power ◽  
Access Point ◽  
Power Performance ◽  
Multiple User ◽  
Structure Simulation ◽  
Rank One ◽  
Multiple Access System ◽  
Optimal Beamforming

We investigate the beamforming for the multi antenna cooperative non-orthogonal multiple access (NOMA) system, where an access point (AP) delivers messages for multiple user terminals (UT) with successive interference cancellation (SIC) reception method. Some UTs with multiple antennas cooperate with the AP transmission to improve the diversity and the average power performance. We formally present two optimal beamforming schemes at the AP and at the cooperative UTs. One scheme has no power limitation for the cooperative UTs, while the other one does have such limitation. We guarantee that the rank one beamformer is sufficient to achieve the optimal points so that the proposed schemes have rank one semi-definite programming (SDP) structure. Simulation results show the performance gain of the multi-antenna cooperative NOMA schemes in the sense of diversity and the average power.

scholarly journals BPS Quivers of Five-Dimensional SCFTs, Topological Strings and q-Painlevé Equations

2021 ◽  
Author(s):  
Giulio Bonelli ◽  
Fabrizio Del Monte ◽  
Alessandro Tanzini
Keyword(s):  
Topological String ◽  
Cluster Algebra ◽  
Painlevé Equations ◽  
Partition Functions ◽  
Quantum Field Theories ◽  
Particle Spectrum ◽  
Yang Mills ◽  
Painleve Equations ◽  
Rank One ◽  
Bilinear Equations

AbstractWe study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $$\tau $$ τ -functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU(2) pure super Yang–Mills and $$N_f=2$$ N f = 2 on a circle.

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